Well-Posed Nonlinear Problems: A Study of Mathematical Models of Contact (Advances in Mechanics and Mathematics, 50) 3031414152, 9783031414152

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Advances in Mechanics and Mathematics 50

Mircea Sofonea

Well-Posed Nonlinear Problems A Study of Mathematical Models of Contact

Advances in Mechanics and Mathematics Volume 50

Series Editors David Gao, Federation University Australia, Mt. Helen, VIC, Australia Tudor Ratiu, Shanghai Jiao Tong University, Minhang District, Shanghai, China Editorial Board Members Anthony Bloch, University of Michigan, Ann Arbor, MI, USA John Gough, Aberystwyth University, Aberystwyth, UK Darryl D. Holm, Imperial College London, London, UK Peter Olver, University of Minnesota, Minneapolis, MN, USA Juan-Pablo Ortega, University of St. Gallen, St. Gallen, Switzerland Genevieve Raugel, CNRS and University-Paris Sud, Orsay Cedex, France Jan Philip Solovej, University of Copenhagen, Copenhagen, Denmark Michael Z. Zgurovsky, Igor Sikorsky Kyiv Polytechnic Institute, Kyiv, Ukraine Jun Zhang, University of Michigan, Ann Arbor, MI, USA Enrique Zuazua, Universidad Autónoma de Madrid and DeustoTech, Madrid, Spain

Mechanics and mathematics have been complementary partners since Newton’s time, and the history of science shows much evidence of the beneficial influence of these disciplines on each other. Driven by increasingly elaborate modern technological applications, the symbolic relationship between mathematics and mechanics is continually growing. Mechanics is understood here in the most general sense of the word, including - besides its standard interpretation - relevant physical, biological, and information-theoretical phenomena, such as electromagnetic, thermal, and quantum effects; bio- and nano-mechanics; information geometry; multiscale modelling; neural networks; and computation, optimization, and control of complex systems. Topics with multi-disciplinary range, such as duality, complementarity, and symmetry in sciences and mathematics, are of particular interest. Advances in Mechanics and Mathematics (AMMA) is a series intending to promote multidisciplinary study by providing a platform for the publication of interdisciplinary content with rapid dissemination of monographs, graduate texts, handbooks, and edited volumes on the state-of-the-art research in the broad area of modern mechanics and applied mathematics. All contributions are peer reviewed to guarantee the highest scientific standards. Monographs place an emphasis on creativity, novelty, and innovativeness in the field; handbooks and edited volumes provide comprehensive surveys of the state-of-the-art in particular subjects; graduate texts may feature a combination of exposition and electronic supplementary material. The series is addressed to applied mathematicians, engineers, and scientists, including advanced students at universities and in industry.

Mircea Sofonea

Well-Posed Nonlinear Problems A Study of Mathematical Models of Contact

Mircea Sofonea LAMPS University of Perpignan via Domitia Perpignan, France

ISSN 1571-8689 ISSN 1876-9896 (electronic) Advances in Mechanics and Mathematics ISBN 978-3-031-41415-2 ISBN 978-3-031-41416-9 (eBook) https://doi.org/10.1007/978-3-031-41416-9 Mathematics Subject Classification: 49J40, 47H10, 35Q74, 74M10, 49J20 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

To my family and to the memory of my uncle, Liviu

Preface

The contents of this book are structured around two main topics: the Mathematical Theory of the Well-posed Problems (MTWP) and the Mathematical Theory of Contact Mechanics (MTCM). A short description of these research topics is the following. The concept of well-posedness originates in the papers of Hadamard [76, 77] at the beginning of the last century. There, a partial differential equation was said to be well-posed if it has a unique solution which depends continuously on the problem’s data. Later, in early 1960s, Tykhonov [243] introduced a concept of wellposedness for minimization problems, based on two main ingredients: the existence of a unique minimizer and the convergence to it of any approximating sequence. A version of this concept, known as the Levitin-Polyak well-posedness concept, was introduced in [129] in the analysis of constrained minimization problems. Various extensions of the Tykhonov and Levitin-Polyak well-posedness concepts have been considered in [62, 104, 136, 137] in the study of variational inequalities and [71, 244, 248], in the study of hemivariational inequalities and inclusions problems. Wellposedness results for fixed point problems have been obtained in [126, 127, 188]. Comprehensive references in the field are the books [56, 135]. There, basic aspects of the study of well-posed problems in scalar optimization are presented in a unified way. Motivated by both theoretical and numerical approximation, the MTWP has undergone an important development. It deals with different well-posedness results and concepts in the study of various nonlinear problems in abstract spaces. It also provides convergence results for such problems which lay the background of their numerical approximation. Contact phenomena between deformable bodies abound in industry and everyday life. A few simple examples are brake pads in contact with wheels, tires on roads, and pistons with skirts. Because of the importance of contact processes in structural and mechanical systems, considerable effort has been put into their modelling, analysis, and numerical simulations and, consequently, the MTCM has undergone a great development, especially in the last decades. It deals with the study of mathematical models of contact and provides the analysis of such models, including results of existence, uniqueness, and continuous dependence of the solutions with ix

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respect to the data and parameters. It also deals with the optimal control of the corresponding models as well as with their numerical approximation. The first recognized publication on contact between deformable bodies was that of Hertz [92]. The next one was by Signorini [203] where an elastic contact problem was posed, in what is now termed a weak or variational formulation, and was subsequently solved by Fichiera [63, 64]. However, the MTCM began with the monograph by Duvaut and Lions [59], who presented variational formulations of contact problems and proved some basic existence and uniqueness results. References in the field include the books [34, 60, 84, 114, 125, 157, 184, 191, 200, 218, 222] and the special issues [139, 161, 193, 199, 245], for instance. The link between the MTCM and the MTWP is based on the following ingredients. First, stated as strongly nonlinear boundary value problems which usually do not have classical solutions, the mathematical models of contact lead to a large variety of weak formulations, expressed in terms of inequality problems, fixed point problems, inclusions, or optimization problems. Providing existence and uniqueness result for such formulations establishes a first link between these theories. Second, recall that convergence results represent an important topic which arises both in Contact Mechanics and in the study of nonlinear abstract problems. Some examples are the following: the convergence of the solution of a viscoelastic problem to the solution of an elastic problem as the viscosity coefficient converges to zero, and the convergence of a solution of a frictional contact problem to the solution of a frictionless contact problem as the coefficient of friction converges to zero, the convergence of a solution of a contact problem with a deformable foundation to the solution of a contact problem with a rigid foundation as the stiffness coefficient of the foundation converges to infinity, one one hand, and the convergence of the solution of a penalty problem to the solution of the original problem when the penalty parameter converges to zero, the convergence of the solution of a regularized problem to the solution of a nonsmooth problem when the regularization parameter converges, the convergence of the solution of a discrete problem to the solution of the continuous problem when the time-step or the spatial discretization parameter converges to zero, on the other hand. It follows from here that the study of convergence results represents a second link betwen the MTCM and the MTWP. A careful analysis of the well-posedness concepts in the literature of MTWP shows that all of them imply convergence results (the convergence of approximating sequences, for instance). Nevertheless, it is easy to see that the convergence of some sequences to the solution of a nonlinear problem cannot be recovered by the Hadamard, Tykhonov, and Levitin-Polyak well-posedness results or their extensions. In this book, we try to fill this gap, since we introduce a new concept of well-posedness which is able to describe any convergence result to the solution of a nonlinear problem. More precisely, we construct a well-posedness theory in which every convergence result can be interpreted as a well-posedness result. The theory we consider provides necessary and sufficient conditions which guarantee the convergence to the solution of a nonlinear problem, unifies different convergence

Preface

xi

results, and provides a framework in which the link between different problems can been established. Its main ingredient consists in a new mathematical concept, the so-called Tykhonov triple, denoted in this book by .T . The corresponding wellposedness concept will be called well-posedness with .T or .T -well-posedness. It can be used in the study of a large class of nonlinear problems for which well-posedness results obtained by using classical concepts are available. The book is divided into three parts which correspond to three different purposes. The first one is to introduce the new concept of .T -well-posedness (Part I). This concept is formulated in an abstract framework of metric spaces and extends some classical well-posedness concepts, already used in the literature. The second purpose is to present new results of well-posedness for a number of nonlinear problems, such as fixed point problems, variational inequalities, hemivariational inequalities, inclusions, minimization, and optimal control problems, in the abstract framework of reflexive Banach and Hilbert spaces (Part II). This allows us to obtain various convergence results which express the continuous dependence of the solution with respect to the data and parameters, on the one hand, and establish a link between different problems, on the other hand. Finally, our third purpose is to illustrate the use of these abstract results in the MTCM (Part III). There, we prove the well-posedness of some static and quasistatic contact problems together with various convergence results and describe the link between different mathematical models of contact. In this way, we initiate a bridge between the MTWP and the MTCM, providing an example of cross-fertilization between the Nonlinear Functional Analysis, on the one hand, and the mathematical modelling in Contact Mechanics, on the other hand. A brief description of the three parts of the book follows. Part I is devoted to general results of the well-posedness theory. We start with some necessary preliminary results of Nonlinear and Nonsmooth Analysis, including properties of nonlinear operators and the subdifferential of convex and locally Lipschitz functions. We then revisit a number of nonlinear problems for which we recall existence, uniqueness, and classical well-posedness results. Then, we introduce the concepts of Tykhonov triple and well-posedness with a Tykhonov triple, state and prove some general results, and provide elementary examples. We also show that these concepts extend and unify some classical concepts of wellposedness already used in the literature. Part II is devoted to the study of well-posedness of representative nonlinear problems. Although the theory introduced in Part I can be used in the analysis of various problems in a general framework, we restrict ourselves to study only fixed point problems, variational and hemivariational inequalities, inclusions, optimization, and optimal control problems, in Banach and Hilbert spaces. This choice was inspired by the examples of mathematical models of contact we consider in Part III of this book. For each problem, we choose appropriate Tykhonov triples and prove wellposedness results which, in turn, allows us to deduce various convergence results. We present examples, counter-examples, and some direct applications. Part III deals with the study of static and quasistatic frictional and frictionless contact problems. We model the material’s behavior with an elastic or viscoelastic

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constitutive law. The contact is either bilateral or with normal compliance and sometimes is associated with unilateral constraints and versions of Coulomb’s law of dry friction. For each problem we provide classical and variational formulations. Then we use the abstract results in Parts I and II in order to establish existence, uniqueness, and convergence results, for which we provide the corresponding mechanical interpretations. The current book is intended as a unified and readily accessible source for advanced graduate students, mathematicians, applied mathematicians, engineers, and scientists. Its reading requires only basic knowledge of General Topology, Functional Analysis, and Solid Mechanics. Most of the results presented can be extended to more general cases. However, since our aim is to provide an accessible presentation, we avoid giving the results in the most general abstract form, so that it is easier for the reader to understand more clearly the essential ideas involved. For instance, in Parts I and II, we voluntary avoid some results on generalized wellposedness of various problems, which require arguments of pseudomonotonicity, compactness, and Haussdorff distance of sets; in the study of the variational inequalities presented in Part II of the book, we restrict ourselves to the Hilbertian case and strongly monotone Lipschitz continuous operators; in the study of historydependent inequalities and inclusions, we avoid using the Bochner-Sobolev spaces k,p (0, T ; X) and we restrict ourselves to using spaces of continuous functions .W defined on a time interval with values in X; in Part III, we discuss only static and quasistatic models of contact and avoid considering dynamic models of contact whose study requires additional functional arguments. Nevertheless, we refer the reader to the section of bibliographical notes at the end of the book, where we provide references in which more information on the topics related to the body of the text can be found. Part of the material presented in this book originates or is related to my joint work with several collaborators and friends to whom I express my thanks: Prof. Weimin Han (Iowa City, IA, USA), Prof. Zhenhai Liu (Nanning, P.R. China), Prof. Andaluzia Matei (Craiova, Romania), Prof. Stanislaw Migórski (Krakow, Poland), Prof. Anna Ochal (Krakow, Poland), Prof. Meir Shillor (Rochester, MI, USA), Prof. Domingo A. Tarzia (Rosario, Argentina), Prof. Yibin Xiao (Chengdu, P.R. China). I also thank Joëlle Sulian (Perpignan, France), who prepared the figures for this book, as well as Rong Hu (Chengdu, P.R. China) and Anna Ochal (Krakow, Poland), who checked the whole manuscript and provided pertinent corrections. I extend my gratitude to the unknown Referees for their useful remarks and corrections. Finally, I thank Chris Eder, Dorothy Mazlum, and their Springer staff for their help in bringing this book in your hand. Perpignan, France May 2023

Mircea Sofonea

Acknowledgements

This work has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 823731 CONMECH.

xiii

Contents

Part I An Introduction to Well-Posedness of Nonlinear Problems 1

Nonlinear Problems and Their Classical Well-Posedness . . . . . . . . . . . . . . 1.1 Elements of Nonsmooth Analysis in Banach Spaces . . . . . . . . . . . . . . . 1.1.1 Normed Spaces and Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 History-Dependent Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Convex Lower Semicontinuous Functions . . . . . . . . . . . . . . . . 1.1.4 Locally Lipschitz Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Variational–Hemivariational Inequalities . . . . . . . . . . . . . . . . . . 1.1.6 Miscellaneous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Elements of Nonlinear Analysis in Hilbert Spaces . . . . . . . . . . . . . . . . . 1.2.1 Nonlinear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Convex Sets and Convex Functions. . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Classical Well-Posedness Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Minimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Fixed Point Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 7 9 13 15 18 19 19 22 25 28 28 29 35 38 40

2

Tykhonov Triples and Associated Well-Posedness Concept . . . . . . . . . . . 2.1 A New Well-Posedness Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Basic Definitions and General Properties . . . . . . . . . . . . . . . . . 2.1.3 Classical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Metric Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 An Elementary Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Well-Posedness of Split and Dual Problems . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Split Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Dual Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 45 46 53 58 63 70 70 76

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2.3

2.4

Extended Classical Well-Posedness Concepts . . . . . . . . . . . . . . . . . . . . . 2.3.1 Extended Levitin–Polyak Well-Posedness Concept . . . . . . 2.3.2 Extended Hadamard Well-Posedness Concept . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 89 92

Part II Relevant Examples of Well-Posed Problems 3

Fixed Point Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Case of Contractive Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Well-Posedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Applications to Elliptic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 An Elliptic Variational Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 A Stationary Inclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Case of History-Dependent Operators . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Well-Posedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Applications to History-Dependent Problems . . . . . . . . . . . . . . . . . . . . . . 3.4.1 A Volterra-Type Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 A History-Dependent Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Case of Almost History-Dependent Operators . . . . . . . . . . . . . . . . 3.5.1 Well-Posedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 99 99 103 107 107 112 116 116 119 122 123 124 126 126 131

4

Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Elliptic Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 A Well-Posedness Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 History-Dependent Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Well-Posedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Two Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Split and Dual Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 A Split History-Dependent Variational Inequality . . . . . . . . 4.3.2 An Example of Dual Variational Inequalities . . . . . . . . . . . . . 4.4 Mixed Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 An Existence and Uniqueness Result . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Well-Posedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137 137 137 144 147 147 152 159 162 162 167 172 172 173

5

Hemivariational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 A First Elliptic Hemivariational Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Well-Posedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 A Second Elliptic Hemivariational Inequality . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Existence, Uniqueness, and Compactness Results . . . . . . . . 5.2.2 Generalized Well-Posedness Results . . . . . . . . . . . . . . . . . . . . . .

177 177 177 183 186 186 189

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5.3

A Variational–Hemivariational Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 193 5.3.1 Well-Posedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 5.3.2 Some Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

6

Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Stationary Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Problem Statement and Preliminaries . . . . . . . . . . . . . . . . . . . . . 6.1.2 Some Well-Posedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 An Additional Well-Posedness Result . . . . . . . . . . . . . . . . . . . . . 6.1.4 A Convergence Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 History-Dependent Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 An Existence and Uniqueness Result . . . . . . . . . . . . . . . . . . . . . . 6.2.2 A Well-Posedness Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 A Dual History-Dependent Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Well-Posedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 205 208 213 220 223 223 228 230 230 234

7

Minimization and Optimal Control Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Minimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Well-Posedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Optimal Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Well-Posedness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241 241 241 244 248 250 250 251 261

Part III Well-Posed Contact Problems 8

Mathematical Modeling in Contact Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Physical Setting and Mathematical Models . . . . . . . . . . . . . . . 8.1.2 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Modeling of Static Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Basic Equations and Boundary Conditions. . . . . . . . . . . . . . . . 8.2.2 Interface Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 A One-Dimensional Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Modeling of Quasistatic Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Basic Equations and Boundary Conditions. . . . . . . . . . . . . . . . 8.3.2 Interface Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Two One-Dimensional Examples . . . . . . . . . . . . . . . . . . . . . . . . . .

267 267 268 269 273 273 276 285 289 289 294 295

9

Static Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 A Contact Problem with Unilateral Constraints . . . . . . . . . . . . . . . . . . . . 9.1.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 A Well-Posedness Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

301 301 301 305 308

xviii

Contents

9.2

A Contact Problem with Bilateral Constraints . . . . . . . . . . . . . . . . . . . . . . 9.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Dual Variational Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two Frictionless Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Existence and Uniqueness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Nonsmooth Contact Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 A Convergence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311 311 313 319 323 323 327 329 332 332 335

Quasistatic Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Two Frictionless Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 A Fixed Point Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.4 A Volterra-Type Variational Formulation. . . . . . . . . . . . . . . . . . 10.1.5 An Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 A Frictional Contact Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 A Well-Posedness Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 A Contact Problem for Rate-Type Materials . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 A Fixed Point Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.4 A Well-Posedness Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.5 Additional Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . .

339 339 340 341 344 346 349 352 352 353 358 363 363 364 367 370 376

9.3

9.4

10

Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

Part I

An Introduction to Well-Posedness of Nonlinear Problems

Chapter 1

Nonlinear Problems and Their Classical Well-Posedness

We start this chapter with some preliminary material from functional analysis which will be used subsequently. This concerns the Banach and Hilbert spaces, various classes of nonlinear and history-dependent operators, convex functions, and locally Lipschitz functions, including the properties of convex and Clarke subdifferential. Most of the results are stated without proofs since their proofs are standard and can be found in various references. We also recall some existence and uniqueness results in the study of variational and hemivariational inequalities. We indicate recent references where the proof of these results can be found. We then proceed with some examples of classical well-posedness concepts in the literature: the Hadamard, the Tykhonov, and the Levitin–Polyak well-posedness concepts. We illustrate these concepts in the study of minimization problems, fixed point problems, inclusions, and variational inequalities.

1.1 Elements of Nonsmooth Analysis in Banach Spaces 1.1.1 Normed Spaces and Banach Spaces The linear spaces used in this book are assumed to be real. For a normed space X, we denote by .·X its norm, by .X∗ its topological dual, and by .·, ·X∗ ×X the duality pairing of .X∗ and X. Sometimes, when no confusion could arise, we simply write .·, · instead of .·, ·X ∗ ×X . We use .0X and .0X ∗ to represent the zero element of X and .X∗ , respectively, .IX for the identity operator on X, and .2X to denote the set of parts of X. Moreover, we use .int M for the interior of the set .M ⊂ X in the strong topology of X and, when no confusion arises, we sometimes write “.∞” instead of “.+∞.” The symbols “.→” and “.” will represent the strong and the weak convergence in the spaces X and .X∗ , while .Xw will denote the space X equipped with the weak © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Sofonea, Well-Posed Nonlinear Problems, Advances in Mechanics and Mathematics 50, https://doi.org/10.1007/978-3-031-41416-9_1

3

4

1 Nonlinear Problems and Their Classical Well-Posedness

∗ for the space .X ∗ equipped with the weak.∗ topology. We also use the notation .Xw ∗ topology. The limits, lower limits, and upper limits are considered as .n → ∞, even if we do not mention it explicitly. For a Hilbert space, we denote by .(·, ·)X its inner product and by . · X the associated norm. Notation .L(X, Y ) will represent the space of linear continuous operators from X to a normed space Y . In the particular case when .X = Y , we simply write .L(X) instead of .L(X, Y ) and we use the notation .·L(X) for the norm on the space .L(X). For a linear operator L, we usually write .L(v) as Lv, but sometimes we also write Lv even when L is not linear. For the subsets .K ⊂ X, we recall the following definition.

Definition 1.1 Let X be a normed space. A subset .K ⊂ X is called: (a) (Strongly) closed if the limit of each convergent sequence of elements of K belongs to K, that is, {un } ⊂ K,

.

un → u in X ⇒ u ∈ K.

(b) Weakly closed if the limit of each weakly convergent sequence of elements of K belongs to K, that is, {un } ⊂ K,

.

un  u

in X ⇒ u ∈ K.

(c) Compact if every sequence of elements of K contains a subsequence which converges to an element of K, that is, {un } ⊂ K

.

⇒

∃ {unk } ⊂ {un }, u ∈ K

s.t. unk → u in X.

(d) Weakly compact if every sequence of elements of K contains a subsequence which converges weakly to an element of K, that is, {un } ⊂ K

.

⇒

∃ {unk } ⊂ {un }, u ∈ K

s.t. unk  u in X.

(e) Convex if it has the property u, v ∈ K ⇒ (1 − t) u + t v ∈ K

.

∀ t ∈ [0, 1].

Evidently, every compact subset is closed and every weakly compact subset is weakly closed. Moreover, every weakly closed subset of X is (strongly) closed, but the converse is not true, in general. An exception is provided by the class of convex subsets of a Banach space, as shown in the following result. Theorem 1.1 (The Mazur Theorem) A convex subset of a Banach space is (strongly) closed if and only if it is weakly closed.

1.1 Elements of Nonsmooth Analysis in Banach Spaces

5

We also need the following definition for the convergence of sequences of subsets of X, introduced in [169]. Definition 1.2 Let X be a normed space and .{Kn } a sequence of nonempty subsets of X, and let K be a nonempty subset of X. We say that the sequence .{Kn } converges M

to K in the sense of Mosco and we write .Kn −→ K if the following conditions hold: (a) For each .u ∈ K, there exists a sequence .{un } such that .un ∈ Kn for each .n ∈ N and .un → u in X. (b) For each sequence .{un } such that .un ∈ Kn for each .n ∈ N and .un  u in X, we have .u ∈ K. The following classical result will be used repeatedly in this book. Theorem 1.2 (The Smulyan Theorem) Let X be a reflexive Banach space. Then every bounded sequence in X contains a weakly convergent subsequence. It follows that if X is a reflexive Banach space and the sequence .{un } ⊂ X is bounded, that is, .supn un X < ∞, then there exist a subsequence .{unk } ⊂ {un } and an element .u ∈ X such that .unk  u in X. Furthermore, if the limit u is independent of the subsequence, then the whole sequence .{un } converges weakly to u, as stated in the following result. Theorem 1.3 Let X be a reflexive Banach space and let .{un } be a bounded sequence of elements in X such that each weakly convergent subsequence of .{un } converges weakly to the same limit .u ∈ X. Then .un  u in X. We now introduce various classes of nonlinear operators in normed spaces. Definition 1.3 Let X be a normed space. An operator .A : X → X∗ is called: (a) Bounded if A maps bounded sets of X into bounded sets of .X∗ . (b) Monotone if .Au − Av, u − v ≥ 0 for all u, .v ∈ X. (c) Strongly monotone if there exists .m > 0 such that Au − Av, u − v ≥ mu − v2X

∀ u, v ∈ X.

.

(d) Pseudomonotone if it is bounded and .un .lim sup Aun , un − u ≤ 0 implies that .



lim inf Aun , un − v ≥ Au, u − v

u in X together with ∀ v ∈ X.

(e) Demicontinuous if .un → u in X implies .Aun  Au in .X∗ . (f) Hemicontinuous if for all u, v, .w ∈ X, the function .λ → A(u + λv), w is continuous on .[0, 1]. (g) Of type .(S+ ) if .un  u in X together with .lim sup Aun , un − u ≤ 0 implies that .un → u in X.

6

1 Nonlinear Problems and Their Classical Well-Posedness

Examples and various properties of the nonlinear operators which satisfy the definitions above can be found in [21, 29, 50, 52, 53, 157, 254], for instance. In particular, examples of nonlinear operators of type .(S+ ) can be found in [50, page 430] and [53, page 97]. For the following result, we refer the reader to [53, Section 1.9] and [254, Proposition 27.6]. Proposition 1.1 Let A, .B : X → X∗ be operators on a reflexive Banach space X. Then, the following statements hold: (a) If A is monotone, then A is demicontinuous if and only if A is hemicontinuous. (b) If A is bounded, hemicontinuous, and monotone, then it is pseudomonotone. (c) If A and B are pseudomonotone, then .A + B is pseudomonotone. The following result will be used in Sect. 5.2.1. Proposition 1.2 Let .A : X → X∗ be a strongly monotone operator in the normed space X. Then A is an operator of type .(S+ ). Proof Let .{un } ⊂ X be a sequence which converges weakly in X to .u ∈ X such that .

lim sup Aun , un − u ≤ 0.

(1.1)

We use the strong monotonicity of A with constant .m > 0 to see that mun − u2X ≤ Aun − Au, un − u = Aun , un − u + Au, u − un 

.

for each .n ∈ N. We now pass to the upper limit in this inequality and use (1.1) together with the convergence .un  u in X to deduce that .un → u in X, which   concludes the proof. We end this subsection with a result on linear compact operators. Definition 1.4 Let X and Y be normed spaces. An operator .A : X → Y is called: (a) Compact if for every bounded sequence .{un } ⊂ X there exists a subsequence .{unk } ⊂ {un } such that the sequence .{Aunk } converges in Y . (b) Completely continuous if .un  u in X implies .Aun → Au in Y . Proposition 1.3 Let X and Y be normed spaces and let .A : X → Y be a linear compact operator. Then A is completely continuous. Proposition 1.3 is a direct consequence of Definition 1.4. It will be used in Sect. 5.2 as well in various places in Part III of the book.

1.1 Elements of Nonsmooth Analysis in Banach Spaces

7

1.1.2 History-Dependent Operators We now introduce two special classes of nonlinear operators, the so-called almost history-dependent operators and history-dependent operators, respectively. To this end, we need some preliminaries. Let .(X,  · X ) be a normed space and consider an interval of interest .I0 . Here and below we assume that either .I0 = [0, T ] with .T > 0 or .I0 = R+ , where .R+ = [0, +∞). We start with the case when .I0 = [0, T ] with .T > 0. We denote by .C([0, T ]; X) the space of continuous functions defined on .[0, T ] with values in X, that is, C([0, T ]; X) =

.



 v : [0, T ] → X : v is continuous .

The space .C([0, T ]; X) will be equipped with the norm vC([0,T ];X) = max v(t)X .

.

t∈[0,T ]

(1.2)

It is well known that if X is a Banach space, then .C([0, T ]; X) is also a Banach space. Moreover, it follows from above that the convergence of a sequence .{vn } to an element v, in the space .C([0, T ]; X), can be described as follows:  .

vn → v in C([0, T ]; X) as n → ∞ if and only if max vn (t) − v(t)X → 0 as n → ∞.

(1.3)

t∈[0,T ]

For a subset .K ⊂ X, we still use the symbol .C([0, T ]; K) for the set of continuous functions defined on .[0, T ] with values in K. Definition 1.5 Let X and Y be normed spaces. An operator .S : C([0, T ]; X) → C([0, T ]; Y ) is said to be an almost history-dependent operator if there exist .l ∈ [0, 1) and .L > 0 such that .

Su(t) − Sv(t)Y ≤ l u(t) − v(t)X (1.4)  t +L u(s) − v(s)X ds ∀ u, v ∈ C([0, T ]; X), t ∈ [0, T ]. 0

If inequality (1.4) holds with .l = 0, then the operator .S is said to be a historydependent operator. Note that here and below, when no confusion arises, we use the shorthand notation .Su(t) to represent the value of the function .Su at the point t, i.e., .Su(t) = (Su)(t). We now consider the case when .I0 = R+ . We use the notation .C(R+ ; X) for the space of continuous functions defined on .R+ with values in X, that is,

8

1 Nonlinear Problems and Their Classical Well-Posedness

C(R+ ; X) =

.



 u : R+ → X : u is continuous .

For a subset .K ⊂ X, we use the symbol .C(R+ ; K) for the set of continuous functions defined on .R+ with values on K. It is well known that if X is a Banach space, then .C(R+ ; X) can be organized in a canonical way as a Fréchet space, i.e., a complete metric space in which the corresponding topology is induced by a countable family of seminorms. The details could be found in [49, 142]. Here we restrict ourselves to recall that the convergence of a sequence .{vn } to an element v, in the space .C(R+ ; X), can be described as follows:  .

vn → v in C(R+ ; X) as n → ∞ if and only if max vn (t) − v(t)X → 0 as n → ∞, ∀ m ∈ N.

(1.5)

t∈[0,m]

In other words, using (1.3), it is easy to see that the sequence .{vn } converges to the element v in the space .C(R+ ; X) if and only if it converges to v in the space .C([0, m]; X) for all .m ∈ N. The equivalence (1.5) will be used repeatedly in the next chapters, in order to prove convergence results for various problems, stated in the context of an unbounded interval of time. We now extend Definition 1.5 to operators defined on the space .C(R+ ; X). Definition 1.6 Let X and Y be normed spaces. An operator .S : C(R+ ; X) → C(R+ ; Y ) is said to be an almost history-dependent operator if for any .m ∈ N there exist .lm ∈ [0, 1) and .Lm > 0 such that .

Su1 (t) − Su2 (t)Y ≤ lm u1 (t) − u2 (t)X (1.6)  t +Lm u1 (s) − u2 (s)X ds ∀ u1 , u2 ∈ C(R+ ; X), t ∈ [0, m]. 0

If inequality (1.6) holds with .lm = 0, then the operator .S is said to be a historydependent operator. History-dependent and almost history-dependent operators arise in Functional Analysis, Solid Mechanics, and Contact Mechanics, as well. General properties, examples, and mechanical interpretations can be found in [222]. The terms “historydependent operator” and “almost history-dependent operator” have been introduced in [217] and [222], respectively. Since then, they have been used in many papers (see [159, 215, 218, 227], for instance). In what follows, as already mentioned, we use notation .I0 to denote either a bounded interval of the form .[0, T ] with .T > 0 or the unbounded interval .R+ = [0, +∞). Moreover, .C(I0 ; X) will represent either the space .C([0, T ]; X) or the space .C(R+ ; X), respectively, and, in addition, .C(I0 ; K) will represent either the set .C([0, T ]; K) or the set .C(R+ ; K). The following fixed point property represents one of the main properties of the almost history-dependent operators.

1.1 Elements of Nonsmooth Analysis in Banach Spaces

9

Theorem 1.4 Let X be a Banach space and let .S : C(I0 ; X) → C(I0 ; X) be an almost history-dependent operator. Then .S has a unique fixed point, i.e., there exists a unique element .η∗ ∈ C(I0 ; X) such that .Sη∗ = η∗ . A proof of Theorem 1.4 can be found in [212] as well as in [222, p. 41], based on the Banach fixed point principle. Note that since history-dependent operators represent a particular case of almost history-dependent operators, Theorem 1.4 represents the first main property of history-dependent operators, too. We now turn to a second property of such operators and, to this end, we assume that X is a Hilbert spaces. Moreover, we assume that ⎧ ⎨ A : X → X is a linear operator and . there exist LA , mA > 0 such that ⎩ AuX ≤ LA uX , (Au, u)X ≥ mA u2X ∀ u ∈ X.

(1.7)

Note that this assumption implies that A is invertible, its inverse .A−1 : X → X is continuous, and, in addition, it satisfies the inequality A−1 uX ≤

.

1 uX mA

∀ u ∈ X.

Consider now the sum .A + S, where .S : C(I0 ; X) → C(I0 ; X) is a historydependent operator. It turns out that this operator is invertible and it has the same structure. More precisely, we have the following result. Theorem 1.5 Let .(X, (·, ·)X ,  · X ) be a Hilbert space. Assume that A is an operator which satisfies condition (1.7) and consider a history-dependent operator .S : C(I0 ; X) → C(I0 ; X). Then the operator .A + S : C(I0 ; X) → C(I0 ; X) is invertible and its inverse is of the form .A−1 + R : C(I0 ; X) → C(I0 ; X), where .R : C(I0 ; X) → C(I0 ; X) is a history-dependent operator. Theorem 1.5 represents a particular case of Theorem 36 in [222, p. 55]. Its proof is based on a result on nonlinear implicit equations in Banach spaces.

1.1.3 Convex Lower Semicontinuous Functions We introduce here the basic properties of convex lower semicontinuous functions, including the notion of convex subdifferential. Let X be a normed space and .ϕ : X → R ∪ {+∞}. We denote by .dom ϕ the (effective) domain of .ϕ, i.e.,   dom ϕ = x ∈ X : ϕ(x) < +∞ .

.

10

1 Nonlinear Problems and Their Classical Well-Posedness

Definition 1.7 The function .ϕ is said to be: (a) Proper if .dom ϕ = ∅. (b) Convex if ϕ((1 − t)u + tv) ≤ (1 − t)ϕ(u) + tϕ(v)

.

(1.8)

for all u, .v ∈ dom ϕ and .t ∈ [0, 1]. (c) Strictly convex if the inequality in (1.8) is strict for all u, .v ∈ dom ϕ, .u = v, and .t ∈ (0, 1). (d) Strongly convex if there exists .m > 0 such that (1 − t)ϕ(u) + tϕ(v) ≥ ϕ((1 − t)u + tv) + mt (1 − t)u − v2X

.

(1.9)

for all u, .v ∈ dom ϕ and .t ∈ [0, 1]. (e) Coercive if for any sequence .{un } ⊂ X such that .un X → +∞ we have .ϕ(un ) → +∞. (f) Lower semicontinuous (l.s.c.) at .u ∈ X, if .un → u in X implies .

lim inf ϕ(un ) ≥ ϕ(u).

(1.10)

(g) Lower semicontinuous (l.s.c.) if it is l.s.c. at every .u ∈ X. (h) Weakly lower semicontinuous (weakly l.s.c.) at .u ∈ X if inequality (1.10) holds whenever .un  u in X. (i) Weakly lower semicontinuous (weakly l.s.c.) if it is weakly l.s.c. at every .u ∈ X. We note that if .ϕ : X → R is a continuous function, then it is also lower semicontinuous. The converse is not true and a lower semicontinuous function can be discontinuous. Since the strong convergence in X implies the weak convergence, it follows that a weakly lower semicontinuous function .ϕ : X → R ∪ {+∞} is lower semicontinuous. Moreover, the following result holds. Proposition 1.4 Let .(X,  · X ) be a Banach space and let .ϕ : X → R ∪ {+∞} be a convex function. Then .ϕ is lower semicontinuous if and only if it is weakly lower semicontinuous. Convex lower semicontinuous functions can always be minorized by affine functions. The proof of the following classical result is based on a separation theorem and can be found in [52, Proposition 5.2.25]. Proposition 1.5 If X is a normed space and .ϕ : X → R∪{+∞} is a proper, convex, and l.s.c. function, then .ϕ is bounded from below by an affine function, i.e., there are ∗ .l ∈ X and .b ∈ R such that .ϕ(v) ≥ l, v + b for all .v ∈ X. The continuity of a convex lower semicontinuous function is guaranteed by the following result which represents a direct consequence of [61, Corollary 2.5]. Proposition 1.6 Let .(X,  · X ) be a Banach space and let .ϕ : X → R be a convex lower semicontinuous function. Then .ϕ is continuous.

1.1 Elements of Nonsmooth Analysis in Banach Spaces

11

Next, we recall the notion of the subdifferential of a convex function. Definition 1.8 Let .ϕ : X → R∪{+∞} be a proper convex function. The (generally ∗ multivalued) mapping .∂c ϕ : X → 2X defined by  ∂c ϕ(u) = u∗ ∈ X∗ : u∗ , v − u ≤ ϕ(v) − ϕ(u)

.

∀v ∈ X



is called the convex subdifferential of .ϕ. An element .u∗ ∈ ∂c ϕ(u) (if any) is called a subgradient of .ϕ at u. We denote in what follows by .D(∂c ϕ) the domain of the subdifferential of the function .ϕ, i.e.,   D(∂c ϕ) = u ∈ X : ∂ϕc (u) = ∅ .

.

It can be shown (see Corollary 2.38 of [21] and Section 5.2 of [52], for instance) that .int(dom ϕ) ⊂ D(∂c ϕ) ⊂ dom ϕ where, recall, .int M denotes the interior of the set M in the strong topology of X. The following is an important example of a convex function. Example 1.1 Given a nonempty subset K of a Banach space X, the function .ψK on X, defined by  ψK (u) =

.

0

if u ∈ K,

+∞

if u ∈ / K,

is called the indicator function of K. The following statements hold: (i) The subset K of X is convex if and only if .ψK is convex. (ii) The subset K of X is closed if and only if .ψK is l.s.c. (iii) If K is a closed convex subset of X, then .D(∂c ψK ) = K, ∂c ψK (u) = { u∗ ∈ X∗ : u∗ , v − u ≤ 0

.

∀ v ∈ K } ∀ u ∈ K,

and .∂c ψK (u) = {0X } for each .u ∈ int K. Below in this chapter, we need the following result on strongly convex lower semicontinuous functions. Proposition 1.7 Let X be a normed space and let .J : X → R be a strongly convex lower semicontinuous function. Then J is coercive. Proof Let .{un } ⊂ X be a sequence such that un X → ∞

.

(1.11)

12

1 Nonlinear Problems and Their Classical Well-Posedness

and let .u0 ∈ X be fixed. We fix .n ∈ N and use (1.9) with .ϕ = J and .t = that .

u + u m 1 1 n 0 + un − u0 2X . J (un ) + J (u0 ) ≥ J 2 2 4 2

1 2

to see

(1.12)

Now, since J is proper, convex, and lower semicontinuous, Proposition 1.5 shows that there exist .l ∈ X∗ and .b ∈ R such that .J (v) ≥ l, v + b for all .v ∈ X and, therefore, J

.

u + u

1 un + u0 n 0  + b ≥ − lX∗ un + u0 X + b. ≥ l, 2 2 2

Then, using the inequality .un + u0 X ≤ un X + u0 X , we find that J

.

u + u

 1 n 0 ≥ − lX∗ un X + u0 X + b. 2 2

(1.13)

Next, the inequality



un X − u0 X ≤ un − u0 X

.

implies that un − u0 2X ≥ un 2X − 2un X u0 X + u0 2X .

.

(1.14)

We now combine inequalities (1.12)–(1.14) to deduce that there exist some constants .a, a  , and .a  which do not depend on n such that .a > 0 and J (un ) ≥ aun 2X + a  un X + a  .

.

Finally, we use assumption (1.11) to see that .J (un ) → +∞ as .n → ∞, which concludes the proof.   We end this subsection by recalling the following version of the Weierstrass theorem, together with some consequences. Theorem 1.6 Let X be a reflexive Banach space, K a nonempty weakly closed subset of X, and .J : X → R a weakly lower semicontinuous function. In addition, assume that either K is bounded or J is coercive, i.e., .J (v) → +∞ as .vX → +∞. Then, there exists at least an element u such that u ∈ K,

.

J (u) ≤ J (v)

∀ v ∈ K.

(1.15)

1.1 Elements of Nonsmooth Analysis in Banach Spaces

13

Remark 1.1 A proof of Theorem 1.6 can be found in [123], for instance. Moreover, it is easy to see that if, in addition, K is a convex set and J is a strictly convex function, then the solution of the minimization problem (1.15) is unique. Corollary 1.1 Let X be a reflexive Banach space, K a nonempty, closed, convex subset of X, and .J : X → R a strongly convex lower semicontinuous function. Then, there exists a unique element u such that (1.15) holds. Proof We use Theorem 1.1 to see that the set K is weakly closed. On the other hand, Propositions 1.4 and 1.7 show that the function J is weakly lower semicontinuous and coercive. Therefore, we are in a position to use Theorem 1.6 in order to deduce the existence of a solution to the minimization problem (1.15). The uniqueness is a consequence of the strict convexity of J , guaranteed by its strong convexity. A direct proof, which uses the strong convexity of J , is as follows. Assume that u and  .u are two solutions of inequality (1.15) and let ω = min J (v).

.

v∈K

We now use (1.9) with .ϕ = J , .v = u and .t = .

1 2

to see that

u + u m 1 1 + u − u 2X . J (u) + J (u ) ≥ J 2 2 4 2 

Next, using equalities .J (u) = J (u ) = ω and inequality .J ( u+u 2 ) ≥ ω, we deduce   that .u − u 2X ≤ 0, which implies that .u = u .

1.1.4 Locally Lipschitz Functions In this subsection, we recall the basic definitions and properties of the generalized subdifferential in the sense of Clarke [43]. Definition 1.9 Let X be a Banach space. A function .j : X → R is said to be locally Lipschitz if for every .x ∈ X, there exist .Ux a neighborhood of x and a constant .Lx > 0 such that |j (y) − j (z)| ≤ Lx y − zX

.

for all y, .z ∈ Ux . The constant .Lx in the previous inequality is called the Lipschitz constant of j near x. We note that a convex continuous function .j : X → R is locally Lipschitz. More generally, a convex function .j : X → R which is bounded from above on a neighborhood of some point is locally Lipschitz (see [44, p. 34], for instance). Also,

14

1 Nonlinear Problems and Their Classical Well-Posedness

if a function .j : X → R is Lipschitz continuous on bounded sets of X, then it is locally Lipschitz, while the converse does not hold, in general. The following result can be proved as in [52, Proposition 5.2.10]. Proposition 1.8 Let X be a Banach space and let .ϕ : X → R ∪ {+∞} be a proper convex lower semicontinuous function. Then .ϕ is locally Lipschitz on the interior of .dom ϕ. We now proceed with the following definition. Definition 1.10 Let .j : X → R be a locally Lipschitz function. The generalized (Clarke) directional derivative of j at .x ∈ X in the direction .v ∈ X, denoted by 0 .j (x; v), is defined by j (y + λv) − j (y) . λ y→x, λ↓0

j 0 (x; v) = lim sup

.

The subdifferential in the sense of Clarke (or, equivalently, the generalized gradient) of j at x, denoted by .∂j (x), is a subset of the dual space .X∗ given by   ∂j (x) = ζ ∈ X∗ : j 0 (x; v) ≥ ζ, v ∀ v ∈ X .

.

A locally Lipschitz function j is said to be regular (in the sense of Clarke) at .x ∈ X if for all .v ∈ X the one-sided directional derivative j  (x; v) = lim

.

λ↓0

j (x + λv) − j (x) λ

exists and .j 0 (x; v) = j  (x; v). The following result collects basic properties of the generalized directional derivative and the generalized gradient. Proposition 1.9 Assume that .j : X → R is a locally Lipschitz function on a Banach space X. Then, the following hold: (i) For every .x ∈ X, the function .X  v → j 0 (x; v) ∈ R is positively homogeneous, i.e., .j 0 (x; λv) = λj 0 (x; v) for all .λ ≥ 0, subadditive, i.e., 0 0 0 .j (x; v1 + v2 ) ≤ j (x; v1 ) + j (x; v2 ) for all .v1 , .v2 ∈ X, and satisfies the 0 inequality .|j (x; v)| ≤ Lx vX with .Lx > 0 being the Lipschitz constant of j near x. Moreover, it is Lipschitz continuous and .j 0 (x; −v) = (−j )0 (x; v) for all .v ∈ X. (ii) The function .X × X  (x, v) → j 0 (x; v) ∈ R is upper semicontinuous, i.e., for all x, .v ∈ X, .{xn }, .{vn } ⊂ X such that .xn → x and .vn → v in X, we have 0 0 .lim sup j (xn ; vn ) ≤ j (x; v).   (iii) For every .x, v ∈ X, we have .j 0 (x; v) = max ζ, v : ζ ∈ ∂j (x) .

1.1 Elements of Nonsmooth Analysis in Banach Spaces

15

(iv) For every .x ∈ X, the generalized gradient .∂j (x) is a nonempty, convex, and weakly . ∗ compact subset of .X∗ which is bounded by the Lipschitz constant .Lx > 0 of j near x. ∗ topology, (v) The graph of the generalized gradient .∂j is closed in the .X × Xw ∗ ∗ i.e., if .{xn } ⊂ X and .{ζn } ⊂ X are sequences such that .ζn ∈ ∂j (xn ) and ∗ ∗ .xn → x in X, .ζn → ζ weakly. in .X , then .ζ ∈ ∂j (x). (vi) If .j : X → R is convex, then the subdifferential in the sense of Clarke .∂j (x) at any .x ∈ X coincides with the convex subdifferential .∂c j (x). We recall the following result which corresponds to Propositions 2.3.1 and 2.3.3 in [44]. Proposition 1.10 Let j , .j1 , .j2 : X → R be locally Lipschitz functions on X. Then, (i) (Scalar multiples) The equality .∂(λj )(x) = λ∂j (x) holds, for all .λ ∈ R and all .x ∈ X. (ii) (Sum rule) The inclusion .∂(j1 + j2 )(x) ⊆ ∂j1 (x) + ∂j2 (x) holds for all .x ∈ X or, equivalently, .(j1 + j2 )0 (x; v) ≤ j10 (x; v) + j20 (x; v) for all x, .v ∈ X. (iii) If .j1 and .j2 are regular at .x ∈ X, then .j1 + j2 is regular at .x ∈ X and the inclusion and the inequality in (ii) hold with equalities. We refer to [44, 52, 157] for additional results on the generalized gradient, its relation to classical notions of differentiability, and other calculus rules.

1.1.5 Variational–Hemivariational Inequalities In this subsection, we assume that X is a reflexive Banach space. Let .K ⊂ X, A : X → X∗ ϕ : X × X → R, .j : X → R, and .f ∈ X∗ . We also assume that j is locally Lipshitz and we use notation .j 0 (u; v) for the generalized directional derivative of j at .u ∈ X, in the direction .v ∈ X. Under these assumptions, we formulate the following inequality problem: find an element u such that

.

.

u ∈ K,

Au, v − u + ϕ(u, v) − ϕ(u, u) + j 0 (u; v − u)

(1.16)

≥ f, v − u ∀ v ∈ K. Note that, below, the function .ϕ is supposed to be convex with respect to the second argument, and, for this reason, following the terminology in [184], we refer to inequality (1.16) as a variational–hemivariational inequality. In the case when j vanishes, (1.16) represents a (pure) variational inequality and, in the case when .ϕ vanishes, it is a (pure) hemivariational inequality. In the study of (1.16), we consider the following assumptions:

16

1 Nonlinear Problems and Their Classical Well-Posedness

K is a nonempty, closed, . convex subset of X. ⎧ ⎪ A : X → X∗ is pseudomonotone and strongly monotone, i.e., ⎪ ⎪ ⎪ ⎪ (a) A is bounded and un  u in X ⎪ ⎪ ⎨ with lim sup Aun , un − u ≤ 0 . ⎪ implies lim inf Aun , un − v ≥ Au, u − v ∀ v ∈ X. ⎪ ⎪ ⎪ ⎪ (b) There exists mA > 0 such that ⎪ ⎪ ⎩ Au − Av, u − v ≥ mA u − v2X ∀ u, v ∈ X. ⎧ ⎪ ϕ : X × X → R is such that: ⎪ ⎪ ⎪ ⎪ (a) ϕ(η, ·) : X → R is convex and lower semicontinuous ⎪ ⎪ ⎨ for all η ∈ X. . ⎪ (b) There exists αϕ ≥ 0 such that ⎪ ⎪ ⎪ ⎪ ϕ(η1 , v2 ) − ϕ(η1 , v1 ) + ϕ(η2 , v1 ) − ϕ(η2 , v2 ) ⎪ ⎪ ⎩ ≤ αϕ η1 − η2 X v1 − v2 X ∀ η1 , η2 , v1 , v2 ∈ X. ⎧ j : X → R is such that: ⎪ ⎪ ⎪ ⎪ ⎪ (a) j is locally Lipschitz. ⎪ ⎪ ⎪ ⎪ ⎨ (b) ξ X∗ ≤ c0 + c1 vX for all v ∈ X, ξ ∈ ∂j (v) . with c0 , c1 ≥ 0. ⎪ ⎪ ⎪ ≥ 0 such that (c) There exists α j ⎪ ⎪ ⎪ 0 (v ; v − v ) + j 0 (v ; v − v ) ≤ α v − v 2 ⎪ j ⎪ 1 2 1 2 1 2 j 1 2 X ⎪ ⎩ ∀ v1 , v2 ∈ X.

(1.17)

.

(1.18)

(1.19)

(1.20)

f ∈ X∗ ..

(1.21)

αϕ + αj < mA .

(1.22)

The unique solvability of the inequality (1.16) is given by the following result, proved in [159, 222] by using a surjectivity argument for pseudomonotone multivalued operators. Theorem 1.7 Assume (1.17)–(1.22). Then, inequality (1.16) has a unique solution u ∈ K.

.

We now move to a first particular case of inequality (1.16), obtained when the function .ϕ does not depend on the solution. The statement of the corresponding problem is as follows: find an element u such that .

u ∈ K,

Au, v − u + ϕ(v) − ϕ(u) + j 0 (u; v − u) ≥ f, v − u

(1.23)

∀ v ∈ K.

In the study of this problem, we consider the following specific assumption: .

ϕ : X → R is a convex l.s.c. function..

(1.24)

αj < mA .

(1.25)

1.1 Elements of Nonsmooth Analysis in Banach Spaces

17

We have the following existence and uniqueness result which, obviously, represents a consequence of Theorem 1.7. Corollary 1.2 Assume (1.17), (1.18), (1.20), (1.21), (1.24), and (1.25). Then, inequality (1.23) has a unique solution .u ∈ K. A second particular case is obtained when the function .ϕ vanishes, and therefore, inequality (1.16) is stated as follows: find an element u such that u ∈ K,

.

Au, v − u + j 0 (u; v − u) ≥ f, v − u

∀ v ∈ K.

(1.26)

For this pure hemivariational inequality, we have the following direct consequence of Theorem 1.7. Corollary 1.3 Assume (1.17), (1.18), (1.20), (1.21), and (1.25). Then, inequality (1.26) has a unique solution .u ∈ K. Note that various applications lead to hemivariational inequalities in which the nonsmooth function j depends on the solution via a compact operator. This is the case with several contact models, in which the compact operator is the Sobolev trace operator. To present such kind of inequality problems, besides the reflexive space X, we consider a reflexive space Y . Let K be a nonempty subset of X, .A : X → X∗ , ∗ .j : Y → R, .γ : X → Y , and .f ∈ X . Then, the hemivariational inequality we consider is stated as follows: find an element u such that u ∈ . K,

Au, v − u + j 0 (γ u; γ v − γ u) ≥ f, v − u ∀ v ∈ K.

(1.27)

In the study of (1.27), we consider the following assumptions on the data: .

K is a closed convex subset of X such that 0X ∈ K.. ⎧ A : X → X∗ is such that: ⎪ ⎪ ⎨ (a) A is pseudomonotone. ⎪ (b) There exist cA > 0, dA ≥ 0, and eA ≥ 0 such that ⎪ ⎩ Au, u ≥ cA u2X − dA uX − eA ∀ u ∈ X.

(1.28)

.

(1.29)

γ : X → Y is a linear compact operator.. ⎧ ⎪ ⎪ j : Y → R is such that: ⎪ ⎪ ⎪ (a) j is locally Lipschitz. ⎪ ⎪ ∗ ⎨ (b) ∂j : Y → 2Y is bounded. ⎪ (c) There exist cj , dj , ej ∈ R such that ⎪ ⎪ ⎪ ⎪ (i) j 0 (w; −w) ≤ cj w2Y + dj wY + ej ∀ w ∈ Y. ⎪ ⎪ ⎩ (ii) cj γ 2 < cA .

(1.30)

f ∈ X∗ .

(1.32)

.

(1.31)

18

1 Nonlinear Problems and Their Classical Well-Posedness

We note that, in assumption (1.31)(c)(ii), .γ  represents the norm of the operator γ . Moreover, we note that the assumption .0X ∈ K is made only for simplicity and can be easily replaced by assumption .K = ∅. We have the following result.

.

Theorem 1.8 Assume (1.28)–(1.32). Then there exists at least one solution to the hemivariational inequality (1.27). A proof of Theorem 1.8 can be found in [134] by using the well-known Knaster– Kuratowski–Marzurkiewicz (KKM) principle [120]. A second proof can be found in [36] based on a surjectivity argument for multivalued pseudomonotone operators.

1.1.6 Miscellaneous Results In this short subsection, we present miscellaneous results we need in the rest of the book. We start with the following version of the well-known Banach fixed point theorem. Theorem 1.9 Let X be a Banach space and K a nonempty, closed, subset of X and let .Λ : K → K be a contraction, i.e., there exists .α ∈ [0, 1) such that Λu − ΛvX ≤ αu − vX

.

∀ u, v ∈ K.

Then, there exists a unique element .u ∈ K such that .Λu = u. We now proceed with some inequalities which shall be repeatedly used in this book. The first one is the following elementary inequality, valid for all .x, a, b ≥ 0: x 2 ≤ ax + b

⇒

.

x≤a+

√ b.

(1.33)

The second inequality is the well-known Gronwall inequality. To introduce it, we use the notation .C(I0 ) for the space of real-valued continuous functions defined on the interval .I0 ⊂ R, that is, .C(I0 ) = C(I0 ; R). Lemma 1.1 (The Gronwall Inequality) Let f , .g ∈ C(I0 ) and assume that there exists .c > 0 such that  t .f (t) ≤ g(t) + c f (s) ds ∀ t ∈ I0 . 0

Then 

t

f (t) ≤ g(t) + c

.

0

g(s) ec (t−s) ds

∀ t ∈ I0 .

1.2 Elements of Nonlinear Analysis in Hilbert Spaces

19

Moreover, if g is nondecreasing, then f (t) ≤ g(t) ec t

.

∀ t ∈ I0 .

Lemma 1.1 represents a useful tool in proving various estimates and uniqueness results in the study of evolutionary and time-dependent equations or variational inequalities. Its proof could be found in [218, p. 60], for instance.

1.2 Elements of Nonlinear Analysis in Hilbert Spaces Even if they could be stated in more general settings, part of the results in this book will be formulated in the framework of Hilbert spaces. We made this choice for two reasons. The first one is that considering the Hibertian case allows us to present proofs based on elementary arguments which avoids to introduce more advanced concepts and results. These proofs are more accessible to graduate students and engineers. The second reason is that the mathematical models we present in Part III of the book are formulated in the Hilbertian framework. Therefore, everywhere in this section, X represents a real Hilbert space endowed with the inner product .(·, ·)X and the associated norm . · X . Moreover, as usual, we denote by .IX the identity map of X.

1.2.1 Nonlinear Operators The well-known Riesz representation theorem allows us to identify a Hilbert space with its dual and, therefore, with its bidual which, roughly speaking, shows that each Hilbert space is reflexive. Moreover, it allows to reformulate Definition 1.3 by considering operators .A : X → X instead of operators .A : X → X∗ . Therefore, below in this section and everywhere in the rest of the book, for the Hilbertian framework, we shall use the following definition. Definition 1.11 Let X be a Hilbert space. An operator .A : X → X is called: (a) Monotone if .(Au − Av, u − v)X ≥ 0 .∀ u, v ∈ X. (b) Strongly monotone if there exists .m > 0 such that (Au − Av, u − v)X ≥ mu − v2X

.

(c) Pseudomonotone if it is bounded and .un .lim sup (Aun , un − u)X ≤ 0 implies that .

∀ u, v ∈ X. 

lim inf (Aun , un − v)X ≥ (Au, u − v)X

(1.34)

u in X together with ∀ v ∈ X.

20

1 Nonlinear Problems and Their Classical Well-Posedness

(d) Hemicontinuous if for all u, v, .w ∈ X, the function .λ → (A(u + λv), w)X is continuous on .[0, 1]. (e) Symmetric if .(Au, v)X = (u, Av)X for all u, .v ∈ X. The following result involving monotone operators will be repeatedly used in the analysis of various inequality problems. Proposition 1.11 Let X be a Hilbert space and let .A : X → X be a bounded, hemicontinuous, and monotone operator. Then A is pseudomonotone. Proposition 1.11 represents a direct consequence of Proposition 1.1 (b). A direct proof in the Hilbertian framework can be found in [218, p. 21], for instance. We now recall the following definition. Definition 1.12 An operator .A : X → X is called Lipschitz continuous if there exists .M > 0 such that Au − AvX ≤ Mu − vX

.

∀ u, v ∈ X.

(1.35)

If, in particular, inequality (1.35) holds with .M = 1, then the operator A is said to be nonexpansive. Part of the results we shall present in this book concern problems with strongly monotone Lipschitz continuous operators in Hilbert spaces. For such operators, we sometimes denote by .mA and .MA or .mA and .LA the constants m and M in (1.34) and (1.35), respectively. The main properties of these operators we need in the next chapters are the following. Proposition 1.12 Let .B : X → X be a strongly monotone Lipschitz continuous operator with constants .mB and .MB , respectively. Then B is invertible and its inverse .B −1 : X → X is also strongly monotone with constant . mB2 and Lipschitz MB

continuous with constant . m1B .

Proposition 1.13 Let .B : X → X be a strongly monotone Lipschitz continuous operator with constants .mB and .MB . Then, for any .ρ > 0, the operator .Θρ : X → X defined by .Θρ u = u − ρBu for all .u ∈ X satisfies the inequality Θρ u − Θρ vX ≤ k(ρ)u − vX

.

∀ u, v ∈ X

 1 with .k(ρ) = (1 − 2ρmB + ρ 2 MB2 ) 2 . Moreover, if .ρ ∈ 0, 2m2B , then .0 ≤ k(ρ) < 1, and therefore, .Θρ is a contraction on X.

MB

Proposition 1.14 Let .B : X → X be a strongly monotone Lipschitz continuous operator. Then B is pseudomonotone. A proof of Propositions 1.12 and 1.13 can be found in [218], pages 22–24. Moreover, Proposition 1.14 is a direct consequence of Proposition 1.11. We now proceed with some results concerning multivalued operators defined on the space X. To this end, we recall that given a multivalued operator .T : X → 2X , its domain .D(T ), range .R(T ), and graph .Gr(T ) are the sets defined by

1.2 Elements of Nonlinear Analysis in Hilbert Spaces

21

  D(T ) = v ∈ X : T v =  ∅ ,   R(T ) = f ∈ X : ∃ v ∈ D(T ) s.t. f ∈ T v ,   Gr(T ) = (v, v ∗ ) ∈ X × X : v ∗ ∈ T v ,

.

respectively. Moreover, we recall the following definition. Definition 1.13 The operator .T : X → 2X is said to be: (a) Monotone if (u∗1 − u∗2 , u1 − u2 )X ≥ 0

.

∀ (u1 , u∗1 ), (u2 , u∗2 ) ∈ Gr(T ).

(b) Strongly monotone if there exists a constant .m > 0 such that (u∗1 − u∗2 , u1 − u2 )X ≥ mu1 − u2 2X

.

∀ (u1 , u∗1 ), (u2 , u∗2 ) ∈ Gr(T ).

(c) Maximal monotone if it is monotone, and for any .v ∈ X, .v ∗ ∈ X∗ , the implication below holds: (u∗ − v ∗ , u − v)X ≥ 0 ∀ u ∈ D(T ), u∗ ∈ T u ⇒ v ∈ D(T ) and v ∗ ∈ T v.

.

There is a close connection between the property of maximal monotonicity of T and the surjectivity property of the operator .IX + λT with .λ > 0. The fundamental result in this direction is the celebrated theorem of Minty [162] that we recall below. Theorem 1.10 (The Minty Thorem) Let .T : X → 2X be a maximal monotone operator and let .λ > 0. Then, for any .f ∈ X, there exists a unique element .u ∈ D(T ) such that .u + λT u  f . Theorem 1.10 allows us to consider the resolvent operator .Tλ : X → D(T ) defined by Tλ f = u ⇐⇒ u ∈ D(T ) and u + λT u  f

.

(1.36)

for all .f ∈ X. In other words, .Tλ is the inverse of the operator .IX + λT , i.e., Tλ = (IX + λT )−1 . Note that the resolvent operator exists for each .λ > 0 and is a single-valued operator. Below in this subsection, given a single-valued     operator .A : X → X, we identify A with the multivalued operator .u → Au : X → 2X , where notation . Au represents the set having the single element .Au ∈ X. This allows us to write .A : X → 2X , too. Then, we have the following result.

.

Proposition 1.15 Let .A : X → X be a monotone hemicontinuous operator. Then A : X → 2X is a maximal monotone operator.

.

22

1 Nonlinear Problems and Their Classical Well-Posedness

Proof Assume that .v, v ∗ ∈ X are such that (Au − v ∗ , u − v)X ≥ 0

.

∀ u ∈ X.

Then, taking .u = v + tw with .t > 0 and w arbitrary in X, we find that (A(v + tw) − v ∗ , w)X ≥ 0

∀ w ∈ X.

.

We now pass to the limit as .t → 0 and use the hemicontinuity of A to deduce that (Av − v ∗ , w)X ≥ 0

.

∀ w ∈ X.

This inequality implies that .Av = v ∗ , and therefore, Definition 1.13 (c) shows that A is maximal monotone.   Remark 1.2 Proposition 1.15 guarantees that if .A : X → X is a monotone Lipschitz continuous operator, then .A : X → 2X is a maximal monotone operator. We shall use this property in Sect. 2.2.2, for instance.

1.2.2 Convex Sets and Convex Functions The projection operators represent an important class of nonlinear operators defined in a Hilbert space; to introduce them, we recall the following existence and uniqueness result. Proposition 1.16 (The Projection Lemma) Let K be a nonempty closed convex subset of a Hilbert space X. Then, for each .f ∈ X, there exists a unique element .u ∈ K such that u − f X = min v − f X .

.

v∈K

(1.37)

Proposition 1.16 allows us to introduce the following definition. Definition 1.14 Let K be a nonempty, closed, convex subset of a Hilbert space X. Then, for each .f ∈ X, the element u which satisfies (1.37) is called the projection of f on K and is usually denoted by .PK f . Moreover, the operator .PK : X → K is called the projection operator on K. It is well known that the following equivalence holds, for all .u, f ∈ X: u = PK f

.

⇐⇒ u ∈ K,

(f − u, v − u)X ≤ 0 ∀ v ∈ K.

(1.38)

Moreover, using inequality (1.38), it is easy to prove that the projection operator is monotone and nonexpansive, i.e., for any .f1 , f2 ∈ X the inequalities below hold:

1.2 Elements of Nonlinear Analysis in Hilbert Spaces .

23

(PK f1 − PK f2 , f1 − f2 )X ≥ 0, .

(1.39)

PK f1 − PK f2 X ≤ f1 − f2 X .

(1.40)

The projection operator can be used to characterize the Mosco convergence in Hilbert spaces. Proposition 1.17 Let .{Kn } be a sequence of nonempty closed convex subsets of a Hilbert space X and let K be a nonempty closed convex subset of X. Then, the following statements are equivalent: {Kn } converges to K in the sense of Mosco. PKn u → PK u in X ∀ u ∈ X.

(a) (b)

. .

A proof of Proposition 1.17, together with additional results on the Mosco convergence in Hilbert spaces, can be found in [242]. A generalization of the projection operators in Hilbert spaces is provided by the proximality operators, introduced in [164]. Their definition follows from the following result. Proposition 1.18 Let K be a nonempty closed convex subset of a Hilbert space X and let .ϕ : X → R be a convex lower semicontinuous function. Then, for each .f ∈ X, there exists a unique element u such that u ∈ K,

.

(u, v − u)X + ϕ(v) − ϕ(u) ≥ (f, v − u)X

∀ v ∈ K.

(1.41)

A proof of Proposition 1.18 can be found in [218, p.40], based on the Weierstrass theorem. Using this proposition, we introduce the following definition. Definition 1.15 Let X be a Hilbert space, .K ⊂ X a nonempty, closed, convex subset, and .ϕ : X → R a convex lower semicontinuous function. Then, for each .f ∈ X, the solution u of the variational inequality (1.41), denoted by .proxK,ϕ f , is called the proximal element of f with respect to the set K and the function .ϕ. The operator .proxK,ϕ : X → K defined by .f → u = proxK,ϕ f is called the proximality operator associated with the set K and the function .ϕ. We recall that, by definition, for all .u, f ∈ X, we have u = proxK,ϕ f ⇐⇒

.

(1.41) holds.

(1.42)

Using this equivalence, it is easy to prove that the proximality operator is monotone and nonexpansive, i.e., for any .f1 , f2 ∈ X, the inequalities below hold: .

(proxK,ϕ f1 − proxK,ϕ f2 , f1 − f2 )X ≥ 0, proxK,ϕ f1 − proxK,ϕ f2 X ≤ f1 − f2 X .

Finally, using (1.42) and (1.38), it is easy to see that if .ϕ vanishes, then the corresponding proximality operator becomes the projection operator on K. In other

24

1 Nonlinear Problems and Their Classical Well-Posedness

words, .proxK,0 = PK , which shows that the projection operators represent a particular type of proximality operators. Remark 1.3 Below in this book, we shall use the shorthand notation .proxϕ for the proximality operator .proxX,ϕ . Therefore, if .ϕ : X → R is a convex lower semicontinuous function and .f ∈ X, we have .u = proxϕ f iff u ∈ X,

.

(u, v − u)X + ϕ(v) − ϕ(u) ≥ (f, v − u)X

∀ v ∈ X.

(1.43)

We now turn to the properties of the subdifferential of convex functions in Hilbert spaces. To this end, we start by reformulating Definition 1.8 in the case of Hilbert spaces. Definition 1.16 The subdifferential of a proper convex function .ϕ : X → R ∪ {+∞} is the multivalued operator .∂c ϕ : X → 2X defined by   ∂c ϕ(u) = η ∈ X : ϕ(v) − ϕ(u) ≥ (η, v − u)X ∀ v ∈ X

.

(1.44)

for each .u ∈ X. The following result (see [21, p.94], for instance) represents an important property of the subdifferential of a convex function. Proposition 1.19 Assume that .ϕ : X → R is a convex lower semicontinuous function. Then .ϕ is subdifferentiable on X, that is, .D(∂c ϕ) = X. Moreover, the subdifferential operator .∂c ϕ : X → 2X is a maximal monotone operator. A relevant example of subdifferential operator is provided by the indicator function .ψK defined in Example 1.1. The subdifferential of this function is denoted by .NK and represents the so-called outward normal cone of K. Therefore, .NK = ∂c ψK : X → 2X . Moreover, using (1.44), we deduce that for all .u, f ∈ X, the following equivalence holds: .

f ∈ NK (u) ⇐⇒ u ∈ K,

(f, v − u)X ≤ 0

∀ v ∈ K.

(1.45)

We now end this subsection with the following result which completes Proposition 1.19. Proposition 1.20 Let X be a real Hilbert space ant let .ϕ : X → R be a strongly convex lower semicontinuous function. Then, the subgradient operator .∂c ϕ : X → 2X is a strongly monotone operator. Proof Let .u, v ∈ X and let .f ∈ ∂c ϕ(u), .g ∈ ∂c ϕ(v). We have .

ϕ(w) − ϕ(u) ≥ (f, w − u)X

∀ w ∈ X,

ϕ(w) − ϕ(v) ≥ (g, w − v)X

∀ w ∈ X.

1.2 Elements of Nonlinear Analysis in Hilbert Spaces

25

Let .w ∈ X and .t ∈ [0, 1]. We multiply the first inequality by .1 − t and the second by t and add the resulting inequalities to find that .

(1 − t)ϕ(w) + tϕ(w) ≥ (1 − t)ϕ(u) + tϕ(v) +(1 − t)(f, w − u)X + t (g, w − v)X .

Then, using (1.9), we deduce that .

(1 − t)ϕ(w) + tϕ(w) ≥ ϕ((1 − t)u + tv) + mt (1 − t)u − v2X +(1 − t)(f, w − u)X + t (g, w − v)X .

We now take .w =

u+v 2

and .t =

1 2

in the previous inequality and find that

(f − g, u − v)X ≥ mu − v2X ,

.

 

which concludes the proof. We shall use Proposition 1.20 in the proof of Theorem 1.17 below.

1.2.3 Variational Inequalities We start with an existence and uniqueness result for a class of variational inequalities which is needed in the rest of the book. Everywhere in this subsection, X will represent a Hilbert space, .K ⊂ X, .A : X → X, .ϕ : X × X → R, and .f ∈ X. Then, we formulate the following inequality problem: find an element u such that u ∈ K,

.

(Au, v − u)X + ϕ(u, v) − ϕ(u, u) ≥ (f, v − u)X

∀ v ∈ K. (1.46)

Note that the function .ϕ is supposed to depend on the solution and, for this reason, we refer to inequality (1.46) as a quasivariational inequality. In the study of (1.46), we consider the following assumptions:

.

K is a nonempty, closed, . convex subset of X. (1.47) ⎧ ⎪ A : X → X is strongly monotone and Lipschitz continuous, i.e., ⎪ ⎪ ⎪ ⎪ ⎨ (a) There exists mA > 0 such that . (1.48) (Au − Av, u − v)X ≥ mA u − v2X ∀ u, v ∈ X. ⎪ ⎪ ⎪ (b) There exists LA > 0 such that ⎪ ⎪ ⎩ Au − AvX ≤ LA u − vX ∀ u, v ∈ X.

26

1 Nonlinear Problems and Their Classical Well-Posedness

⎧ ⎪ ϕ : X × X → R is such that: ⎪ ⎪ ⎪ ⎪ (a) ϕ(η, ·) : X → R is convex and lower semicontinuous ⎪ ⎪ ⎨ for all η ∈ X. ⎪ (b) There exists αϕ ≥ 0 such that ⎪ ⎪ ⎪ ⎪ , ϕ(η 1 v2 ) − ϕ(η1 , v1 ) + ϕ(η2 , v1 ) − ϕ(η2 , v2 ) ⎪ ⎪ ⎩ ≤ αϕ η1 − η2 X v1 − v2 X ∀ η1 , η2 , v1 , v2 ∈ X.

.

(1.49)

αϕ < mA ..

(1.50)

f ∈ X.

(1.51)

The unique solvability of inequality (1.46) is provided by the following result. Theorem 1.11 Assume (1.47)–(1.51). Then, inequality (1.46) has a unique solution u ∈ K.

.

A proof of Theorem 1.11 can be found in [218, p. 49], based on the Banach fixed point theorem. A different proof will be provided on page 107, based on the properties of the proximality operators. Next, in the particular case when the function .ϕ does not depend on the solution, the inequality problem (1.46) is stated as follows: find an element u such that u ∈ . K,

(Au, v − u)X + ϕ(v) − ϕ(u) ≥ (f, v − u)X

∀ v ∈ K.

(1.52)

Then, a direct consequence of Theorem 1.11 is the following. Corollary 1.4 Let K be a nonempty, closed, convex subset of X, .A : X → X a strongly monotone Lipschitz continuous operator, .ϕ : X → R a convex lower semicontinuous function, and .f ∈ X. Then, inequality (1.52) has a unique solution. Another particular case of inequality (1.46) arises when the function .ϕ vanishes. In this case, the problem can be stated as follows: find an element u such that u ∈ .K,

(Au, v − u)X ≥ (f, v − u)X

∀ v ∈ K.

(1.53)

Then, a direct consequence of Theorem 1.11 is the following. Corollary 1.5 Let K be a nonempty, closed, convex subset of X, .A : X → X a strongly monotone Lipschitz continuous operator, and .f ∈ X. Then, inequality (1.53) has a unique solution. As usual in the literature, we refer to inequality (1.52) as an elliptic variational inequality of the second kind, while inequality (1.53) is an elliptic variational inequality of the first kind. We now move to the history-dependent case. To this end, besides the Hilbert space X, the set .K ⊂ X, and the operator .A : X → X, we consider a normed space

1.2 Elements of Nonlinear Analysis in Hilbert Spaces

27

Y and a time interval .I0 of the form .I0 = [0, T ] with .T > 0 or .I0 = R+ . We also consider an operator .S : C(I0 ; X) → C(I0 ; Y ) and two functions .ϕ : Y × X × X → R, .f : I0 → X. We deal with the problem of finding a function .u : I0 → X such that .

u(t) ∈ K,

(Au(t), v − u(t))X + ϕ(Su(t), u(t), v)

−ϕ(Su(t), u(t), u(t)) ≥ (f (t), v − u(t))X

(1.54)

∀ v ∈ K, t ∈ I0 .

We refer to inequality (1.54) as a history-dependent variational inequality. In the study of this problem, besides (1.47) and (1.48), we consider the following assumptions:

.

⎧ ϕ : Y × X × X → R is such that: ⎪ ⎪ ⎪ ⎪ ⎪ (a) ϕ(y, u, ·) : X → R is convex and lower semicontinuous, ⎪ ⎪ ⎪ ⎪ for all y ∈ Y and u ∈ X. ⎨ .(1.55) (b) There exist αϕ > 0 and βϕ > 0 such that ⎪ ⎪ ⎪ , u , v ) − ϕ(y , u , v ) + ϕ(y , u , v ) − ϕ(y , u , v ) ϕ(y 1 1 2 1 1 1 2 2 1 2 2 2 ⎪ ⎪ ⎪ ⎪ y − y  v − v  + β u − u  v ≤ α ⎪ ϕ 1 2 Y 1 2 X ϕ 1 2 X 1 − v2 X ⎪ ⎩ ∀ y1 , y2 ∈ Y, u1 , u2 ∈ X, v1 , v2 ∈ X. S : C(I0 ; X) → C(I0 ; Y ) is a history-dependent operator..

(1.56)

f ∈ C(I0 ; X).

(1.57)

Moreover, we assume that (1.58)

βϕ < mA ,

.

where .mA and .βϕ are the constants in (1.48) and (1.55), respectively. The unique solvability of the history-dependent variational inequality (1.54) is given by the following existence and uniqueness result. Theorem 1.12 Assume (1.47), (1.48), and (1.55)–(1.58). Then, inequality (1.54) has a unique solution .u ∈ C(I0 ; K). Theorem 1.12 was proved in [218] in the case when .I0 = [0, T ] and in [227] in the case when .I0 = R+ . A particular version of this theorem is obtained when .Y = X and .ϕ(y, u, v) = (y, v)X for each .y, v ∈ X. In this case, problem (1.54) is stated as follows: find a function .u : I0 → K such that, for all .t ∈ I0 , the following inequality holds: (Au(t), v − u(t))X + (Su(t), v − u(t))X ≥ (f (t), v − u(t))X

.

Then, a direct consequence of Theorem 1.12 is the following.

∀ v ∈ K.

(1.59)

28

1 Nonlinear Problems and Their Classical Well-Posedness

Corollary 1.6 Assume (1.47), (1.48), and (1.57), and moreover, assume that .S : C(I0 ; X) → C(I0 ; X) is a history-dependent operator. Then, inequality (1.59) has a unique solution .u ∈ C(I0 ; K). We shall use Corollary 1.6 in Sect. 4.2 in the case when .I0 = R+ .

1.3 Classical Well-Posedness Concepts 1.3.1 Preliminaries As already mentioned in Preface, the first fundamental concept of well-posedness comes from the classical ideas of Hadamard [76, 77] at the beginning of the last century. It requires existence and uniqueness of the solution of a system of partial differential equations together with its continuous dependence on problem’s data. In early 60s, Tykhonov [243] introduced a concept of well-posedness for minimization problems, based on two ingredients: the existence of a unique minimizer and the convergence to it of every minimizing sequence. A version of this concept in the study of constrained minimization problems has been considered by Levitin and Polyak [129]. In this concept, the solution of the problem is unique, the constraints are satisfied approximatively, and the so-called generalized minimizing sequences converge to the solution. In the last decades, these classical concepts have been extended and new wellposedness concepts have been introduced. They vary from author to author and from paper to paper and have been applied to various problems like fixed point problems, inclusions, variational and hemivariational inequalities, mixed problems, optimal control problems, equilibrium problems, and saddle point problems, for instance. Since the literature in the field is extensive, in this section we restrict ourselves to present only results in the study of the following representative problems: minimization problems (with or without constraints), fixed point problems, inclusions, and variational inequalities. This choice is also motivated by the examples we present in Parts II and III of the book. Moreover, we deal only with the Hadamard, Tykhonov, and Levitin–Polyak well-posedness concepts. Our study in this section is carried out in a metric space .(X, d). We use notation .d(x, A) for the distance function from the element .x ∈ X to the set .A ⊂ X, that is, d(x, A) = inf d(x, a).

.

a∈A

Moreover, we use notation “.→” for the convergence in the space X, i.e., un → u in X

.

⇐⇒

d(un , u) → 0.

Sometimes in this section, X will be a real normed space endowed with the norm  · X and, in particular, a real Hilbert space endowed with the inner product .(·, ·)X

.

1.3 Classical Well-Posedness Concepts

29

and the associated norm . · X . In this case, we use “.→” for the strong convergence in the space X, i.e., un → u in X

.

⇐⇒

un − uX → 0.

Here and below the limits are taken when .n → ∞, even if we do not mention it explicitly.

1.3.2 Minimization Problems Let .(X, d) be a metric space and .K ⊂ X a nonempty set and consider a function J : X → R. We associate with the pair (.K, J ) the following minimization problem.

.

Problem 1.1 Find .u ∈ K such that J (u) ≤ J (v)

.

∀ v ∈ K.

Tykhonov well-posedness Let ω = inf J (v).

.

v∈K

(1.60)

Then, the well-posedness concept introduced by Tykhonov [243] in the study of Problem 1.1 is given by the following definition. Definition 1.17 (a) A sequence .{un } is called a minimizing sequence for Problem 1.1 if .{un } ⊂ K and .J (un ) → ω. (b) Problem 1.1 is well-posed in the sense of Tykhonov (or, equivalently, is Tykhonov well-posed) if there exists a unique element .u ∈ K such that .J (u) = ω and every minimizing sequence .{un } ⊂ K converges in X to u, i.e., J (un ) → ω

.

⇒

un → u in X.

This definition is motivated from the numerical approximation of the solution to Problem 1.1. A minimization problem that does not satisfy the above definition will be called ill-posed. Remark 1.4 Note that if any minimizing sequence converges, then Problem 1.1 has a unique solution. Indeed, if u and .u are two elements of K such that .J (u) = J (u ) = ω, then the sequence .{un } given by  un =

.

u u

if n = 2k if n = 2k + 1

30

1 Nonlinear Problems and Their Classical Well-Posedness

(.k ∈ N) is a minimizing sequence. Thus, assuming that any minimizing sequence converges in X, it follows that the sequence .{un } above converges and, therefore,  .u = u . We now provide two elementary examples. Example 1.2 Let .X = K = R and let .J (u) = |u| for all .u ∈ R. Then Problem 1.1 is Tykhonov well-posed. Example 1.3 Let .X = K = R and .a > 0 and let  J (u) =

.

(u + a)2 if u ≤ 0, u if u > 0.

Then Problem 1.1 is ill-posed. Indeed, the only minimum point of J is .u = −a, but the sequence .{ n1 } is a minimizing sequence which does not converge to u. We now state and prove the following result. Theorem 1.13 Let X be a reflexive Banach space and K a nonempty, closed, convex subset of X and let .J : X → R be a strongly convex lower semicontinuous function. Then, Problem 1.1 is Tykhonov well-posed. Proof The unique solvability of Problem 1.1 follows from Corollary 1.1 on page 13. Denote by u the solution of this problem, and let m be the constant of strong convexity of J , see Definition (1.7) (d). Consider now a minimizing sequence, i.e., a sequence .{un } ⊂ K such that .J (un ) → J (u). Let .n ∈ N and .t ∈ [0, 1]. We use the strong convexity of J in points .un and u to see that (1 − t)J (un ) + tJ (u) ≥ J ((1 − t)un + tu) + mt (1 − t)un − u2X ,

.

and therefore, inequality .J ((1 − t)un + tu) ≥ J (u) yields (1 − t)J (un ) + tJ (u) ≥ J (u) + mt (1 − t)un − u2X .

.

Next, we take .t =

1 2

in the previous inequality and deduce that .

m un − u2X ≤ J (un ) − J (u). 2

We now pass to the limit as .n → ∞ and use the convergence .J (un ) → J (u) to deduce that .un → u in X, which concludes the proof.   Levitin–Polyak well-posedness Some numerical optimization methods for constrained problems produce a sequence .{un } ⊂ X which does not satisfy the constraint .un ∈ K but tend asymptotically to fulfil this constraint, while the corresponding values of .J (un ) approximate the optimal one. A well-known example

1.3 Classical Well-Posedness Concepts

31

is provided by the penalty methods. To take care of such cases, one can strengthen the Tykhonov well-posedness concept, as follows. Definition 1.18 (a) The sequence .{un } is a generalized (or LP -) minimizing sequence for Problem 1.1 if un ∈ X,

J (un ) → ω,

.

d(un , K) → 0.

(b) Problem 1.1 is well-posed in the sense of Levitin–Polyak (or, equivalently, is Levitin–Polyak well-posed) if it has a unique solution .u ∈ K and for any LP minimizing sequence .{un } one has .un → u in X. It is easy to see that every minimizing sequence for Problem 1.1 is an LP minimizing sequence. Therefore, the Levitin–Polyak well-posedness of Problem 1.1 implies its Tykhonov well-posedness. The converse is true provided that J is uniformly continuous, as easily checked. It fails to be true if J is only continuous as it follows from the following example. Example 1.4 Let .X = R2 , .K = R × {0}, and .J (x, y) = x 2 − (x 4 + x)y 2 , and consider the generalized minimizing sequence .{un }, where .un = (n, n1 ), for each .n ∈ N. Then, it is easy to see that the corresponding Problem 1.1 is Tykhonov well-posed but is not Levitin–Polyak well-posed. The previous example suggests us to complete Theorem 1.13 with the following result. Theorem 1.14 Let X be a reflexive Banach space and .K ⊂ X a closed nonempty convex set and let .J : X → R be a strongly convex lower semicontinuous function. Then, Problem 1.1 is Levitin–Polyak well-posed. Proof The unique solvability of the problem is a direct consequence of Corollary 1.1. Let .u ∈ K be the solution of this problem, i.e., u ∈ K,

.

J (u) ≤ J (v)

∀ v ∈ K.

(1.61)

Assume now that .{un } is an LP -minimizing sequence. Then .d(un , K) → 0, and therefore, there exist two sequences .{vn } ⊂ X and .{wn } ⊂ X such that vn ∈ K,

.

un = vn + wn

∀ n ∈ N,

wn X → 0

as n → ∞.

(1.62)

Moreover, J (un ) → J (u).

.

(1.63)

We claim that the sequence .{vn } is bounded in X. Arguing by contradiction, if .{vn } is not bounded, then, passing to a subsequence still denoted by .{vn }, we

32

1 Nonlinear Problems and Their Classical Well-Posedness

can assume that .vn X → +∞. Using now (1.62), we write .vn X = un − wn X ≤ un X + wn X , which implies that .un X → +∞. Therefore, using the coercivity of J guaranteed by Proposition 1.7, we deduce that .J (un ) → +∞, which contradicts (1.63). Now, since the sequence .{vn } is bounded in X, using Theorem 1.2 we deduce that there exist an element . u ∈ X and a subsequence of the sequence .{vn }, again denoted by .{vn }, such that vn   u in X.

.

(1.64)

Recall that K is a closed convex subset of X. Then K is weakly closed and, since {vn } ⊂ K, the convergence (1.64) implies that

.

 u ∈ K.

.

(1.65)

Moreover, (1.62) and (1.64) show that un   u in X.

.

(1.66)

We now use inequality (1.61), regularity (1.65), the weak lower semicontinuity of J , and the convergences (1.66) and (1.63) to see that J (u) ≤ J ( u) ≤ lim inf J (un ) = J (u).

.

We conclude from there that . u is a solution to Problem 1.1 and, by the uniqueness of the solution, we have . u = u. A careful analysis based on the arguments above reveals the fact that any weakly convergent subsequence of the sequence .{vn } converges to the same limit u. On the other hand, the sequence .{vn } is bounded in X. Therefore, using Theorem 1.3, we find that the whole sequence .{vn } converges weakly in X to u. This implies that the whole sequence .{un } converges weakly in X to u. Therefore, for each .n ∈ N, we have u + u

u + u

n n ≤ lim sup J . .J (u) ≤ lim inf J (1.67) 2 2 On the other hand, the convexity of J guarantees that J

.

u + u 1 1 n ≤ J (un ) + J (u), 2 2 2

and using (1.63), we deduce that .

lim sup J

u + u

n ≤ J (u). 2

(1.68)

1.3 Classical Well-Posedness Concepts

33

We now combine the inequalities (1.67) and (1.68) to find that J

.

u + u

n → J (u). 2

(1.69)

Finally, by the strong convexity of J we deduce that .

u + u

u + u

1 m 1 n n un − u2X ≤ J (un ) − J + J (u) − J 4 2 2 2 2

with some .m > 0. We pass to the limit in this inequality and use the convergences (1.63) and (1.69) to deduce that .un → u in X, which ends the proof.   Hadamard well-posedness Even if it was introduced initially in the study of systems of partial differential equations, the concept of well-posedness in the sense of Hadamard can be extended to minimization problems. Consider a minimization problem of the form .(K, J ), see Problem 1.1. We say that this problem is well-posed in the sense of Hadamard if it has a unique solution that depends continuously on the data. Formulated in this way, the Hadamard concept of well-posedness is quite fuzzy since we do not precise which are the problem data and in which sense the continuity of the solution is understood. For this reason, in this section we restrict ourselves to a simple example and we shall provide more details in Sect. 2.3 of Chap. 2. Let X be a Hilbert space, .K ⊂ X, .A : X → X, and .ϕ : X → R, and for any f ∈ X, let .Jf : X → R be the functional given by

.

Jf (v) =

.

1 (Av, v)X + ϕ(v) − (f, v)X 2

∀ v ∈ X.

(1.70)

We now focus on the following minimization problem. Problem 1.2 Find .u ∈ K such that Jf (u) ≤ Jf (v)

.

∀ v ∈ K.

(1.71)

In the study of this problem, we consider the following assumptions: .

K is a nonempty, closed, convex subset of X..  A is a symmetric continuous operator and there exists mA > 0 such that (Au, u)X ≥ mA u2X ∀ u ∈ X. ϕ is a convex lower semicontinuous function.

We have the following result.

(1.72) .

(1.73) (1.74)

34

1 Nonlinear Problems and Their Classical Well-Posedness

Theorem 1.15 Let X be a Hilbert space and assume (1.72)–(1.74). Then, for any f ∈ X, Problem 1.2 has a unique solution .u = u(f ) ∈ K. Moreover, the mapping .f → u(f ) : X → X is Lipschitz continuous. .

Proof Let .f ∈ X and .u, v ∈ K and let .t ∈ [0, 1]. An elementary computation based on definition (1.70) and equality (Au, v)X = (u, Av)X

.

∀ u, v ∈ X

shows that .

(1 − t)Jf (u) + tJf (v) − Jf (u + t (v − u)) =

t (1 − t) (A(u − v), u − v)X 2 +(1 − t)ϕ(u) + tϕ(v) − ϕ(u + t (v − u)).

We now use assumption (1.73) and the convexity of the function .ϕ to deduce that (1 − t)Jf (u) + tJf (v) − Jf (u + t (v − u)) ≥

.

mA t (1 − t) u − v2X , 2

(1.75)

which shows that the function .Jf is strongly convex. Moreover, assumptions (1.73) and (1.74) imply that .Jf is lower semicontinuous. The existence of a unique solution to Problem 1.2 is a direct consequence of Corollary 1.1. Consider now two elements .f1 , f2 ∈ X and denote .u(f1 ) = u1 and .u(f2 ) = u2 . We use inequality (1.75) with .t = 12 , .f = f1 , .u = u1 , and .v = u2 to see that .

mA u1 − u2 2X 8 u + u  1  u + u  1 1 2 1 2 + Jf1 (u2 ) − Jf1 ≤ Jf1 (u1 ) − Jf1 2 2 2 2

and, since .Jf1 (u1 ) ≤ Jf1 (v) for each .v ∈ K, we find that .

u + u  mA 1 1 2 . u1 − u2 2X ≤ Jf1 (u2 ) − Jf1 2 2 8

Next, the identity .

u + u

1 2 2 u + u 1 1 2 = Jf2 (u2 ) − Jf2 + (f1 − f2 , u1 − u2 )X 2 2

Jf1 (u2 ) − Jf1

combined with inequality .Jf2 (u2 ) ≤ Jf2 (v), valid for each .v ∈ K, yields

(1.76)

1.3 Classical Well-Posedness Concepts

35

u + u 1 1 2 ≤ (f1 − f2 , u1 − u2 )X . 2 2

Jf1 (u2 ) − Jf1

.

(1.77)

We now combine inequalities (1.76) and (1.77) to see that mA u1 − u2 2X ≤ 2(f1 − f2 , u1 − u2 )X .

.

This implies that u1 − u2 X ≤

.

2 f1 − f2 X , mA

which concludes the proof.

 

Note that Theorem 1.15 provides the existence of the unique solution to problem (1.71) as well as its continuous dependence with respect to the element f . Therefore, it represents a well-posedness result in the sense of Hadamard. Additional Hadamard-type well-posedness results for Problem 1.2 can be obtained by considering the dependence of the solution with respect to K or with respect to both f and K. We shall consider such kind of dependence in Sect. 7.1, in a more general framework.

1.3.3 Fixed Point Problems As explained in the previous subsection, the concept of Tykhonov well-posedness introduced for minimization problems is based on the following ingredients: the existence of a unique solution u to Problem 1.1 and the convergence to u of any minimizing sequence. Based on this remark, we can easily extend this concept to other problems, provided that we replace the minimizing sequences with an appropriate class of sequences, the so-called approximating sequences. In this subsection, we consider the case of the fixed point problems. The functional framework is the following: .(X, d) is a metric space, .K ⊂ X, and .Λ : K → K is a given operator. We associate with the pair (.K, Λ) the following fixed point problem. Problem 1.3 Find .u ∈ K such that .Λu = u. We now introduce the following definition. Definition 1.19 (a) A sequence .{un } ⊂ K is called an approximating sequence for Problem 1.3 if .d(Λun , un ) → 0. (b) Problem 1.3 is well-posed in the sense of Tykhonov (or, equivalently, is Tykhonov well-posed) if there exists a unique element .u ∈ K such that .Λu = u and every approximating sequence converges to u in X.

36

1 Nonlinear Problems and Their Classical Well-Posedness

Note that the notion of approximating sequence in Definition 1.19(a) plays the role of minimizing sequence in Definition 1.17(a). We now provide two elementary examples. Example 1.5 Let .X = K = R and let .Λu = u3 + u − 1 for any .u ∈ R. Then problem .(X, Λ) is Tykhonov well-posed. Example 1.6 Let .X = K = R and let  Λu =

.

u2 1−u

if u < 0, if u ≥ 0.

Then problem .(X, Λ) is ill-posed. Indeed, the only fixed point of .Λ is .u = 12 , but the sequence .{− n1 } is an approximating sequence which does not converge to u. We now proceed with the following result. Theorem 1.16 Let X be a real Hilbert space and .ϕ : X → R a strongly convex lower semicontinuous function and let .λ > 0. Denote by .Λ the operator given by .Λ = proxλϕ : X → X, see page 24. Then, Problem 1.3 is Tykhonov well-posed. Proof First, we use inequality .λ > 0, definition (1.43), and equality .Λ = proxλϕ to see that the following equivalences hold: .

ϕ(v) ≥ ϕ(u)

∀ v ∈ X ⇐⇒ λϕ(v) ≥ λϕ(u) ∀ v ∈ X

⇐⇒ (u, v − u)X + λϕ(v) − λϕ(u) ≥ (u, v − u)X

∀v ∈ X

⇐⇒ proxλϕ u = u ⇐⇒ Λu = u. On the other hand, we note that Theorem 1.13 implies that there exists a unique element .u ∈ X such that .ϕ(v) ≥ ϕ(u) for all .v ∈ X. We combine these two ingredients to deduce that Problem 1.3 has a unique solution. Consider now an approximating sequence, i.e., a sequence .{un } ⊂ X such that .un − Λun → 0X in X. For each .n ∈ N, let .vn = Λun . Then un − vn → 0X

.

in

(1.78)

X,

and moreover, (1.43) implies that (vn , w − vn )X + λϕ(w) − λϕ(vn ) ≥ (un , w − vn )X

.

∀ w ∈ X, n ∈ N.

(1.79)

Recall also that the solution u of Problem 1.3 satisfies λϕ(w) − λϕ(u) ≥ 0

.

∀ w ∈ X.

(1.80)

Let .w ∈ X, .t ∈ [0, 1], and .n ∈ N. We multiply inequality (1.79) with .(1 − t) and inequality (1.80) with t, and then we add the resulting inequalities to deduce that

1.3 Classical Well-Posedness Concepts .

37

(1 − t)(vn , w − vn )X + (1 − t)λϕ(w) + tλϕ(w)

(1.81)

≥ (1 − t)λϕ(vn ) + tλϕ(u) + (1 − t)(un , w − vn )X . Next, we use the strong convexity of function .λϕ to find that .

(1 − t)λϕ(vn ) + tλϕ(u)

(1.82)

≥ λϕ((1 − t)vn + tu) + mt (1 − t)vn − u2X with some .m > 0. Therefore, combining (1.81) and (1.82), we deduce that .

(1 − t)(vn , w − vn )X + λϕ(w) ≥

(1.83)

λϕ((1 − t)vn + tu) + mt (1 − t)vn − u2X +(1 − t)(un , w − vn )X .

We now take .w =

vn +u 2

and .t =

1 2

in (1.83) to obtain that

(vn , u − vn )X ≥ mvn − u2X + (un , u − vn )X .

.

Thus, vn − u2X ≤

.

1 (vn − un , u − vn )X , m

and therefore, vn − uX ≤

.

1 vn − un X . m

We now use the convergence (1.78) to see that vn − u → 0X

.

in

X.

Finally, we combine (1.78) and (1.84) to find that .un → u in X.

(1.84)  

We conclude this section with the remark that, in the case when .K ⊂ X, .K = X, and .Λ : X → X, we can define the concept of Levitin–Polyak well-posedness for the problem of finding .u ∈ K such that .Λu = u. As in the case of minimization problems, this concept is based on the notion of LP -approximating sequences. Moreover, if the operator .Λ depends on a parameter, we can define the concept of Hadamard well-posedness for this fixed point problem. Nevertheless, to avoid repetitions, we skip to introduce a detailed presentation of these classical wellposedness concepts.

38

1 Nonlinear Problems and Their Classical Well-Posedness

1.3.4 Inclusions We now extend the concept of Tykhonov well-posedness to inclusions. To this end, besides the metric space .(X, d), we consider a normed space .(Y,  · Y ). Below we use the notation .0Y for the zero element of Y , .2Y for the set of parts of Y , and .dY (y, A) for the distance function from the element .y ∈ Y to the set .A ⊂ Y , that is, dY (y, A) = inf y − aY .

.

a∈A

Let .T : X → 2Y be a multivalued mapping. In this functional framework, we consider the following inclusion problem. Problem 1.4 Find .u ∈ X such that .0Y ∈ T u. On occasion, for simplicity, we use the short-hand notation .(X, Y, T ) for Problem 1.4. We now introduce the following definition. Definition 1.20 (a) A sequence .{un } ⊂ X is called an approximating sequence for Problem 1.4 if .dY (0Y , T un ) → 0 or, equivalently,  .

there exists a sequence {ξn } such that ξn ∈ T un ∀ n ∈ N and ξn Y → 0.

(1.85)

(b) Problem 1.4 is well-posed in the sense of Tykhonov (or, equivalently, is Tykhonov well-posed) if there exists a unique element .u ∈ X such that .0Y ∈ T u and any approximating sequence converges in X to u. We now provide some elementary examples of Tykhonov well-posed inclusion problems. Example 1.7 Let .X = Y = R, and let  Tu =

.

[0, 1] 1

if u = 0, if u = 0.

Then problem .(X, Y, T ) is well-posed in the sense of Tykhonov. Indeed, its unique solution is .u = 0 and any approximating sequence converges to 0 since it contains only a finite number of terms that do not vanish. Example 1.8 Let .X = Y = R, and let ⎧ ⎨ [0, 1] .T u = 2 ⎩ u

if u = 1, if u = 0, otherwise.

1.3 Classical Well-Posedness Concepts

39

Then problem .(X, Y, T ) is ill-posed. Indeed, its unique solution is .u = 1, but the sequence .{un } with .un = n1 for any .n ∈ N is an approximating sequence which does not converge to u. Example 1.9 Let X be a normed space and .Λ : X → X, and take .Y = X and T = Λ − IX , where .IX denotes the identity mapping on X. Then it is easy to see that Problem 1.4 is well-posed in the sense of Definition 1.20 if and only if the resulting Problem 1.3 is well-posed in the sense of Definition 1.19.

.

We now proceed with the following result. Theorem 1.17 Let X be a real Hilbert space and .Y = X and let .ϕ : X → R be a strongly convex lower semicontinuous function. Denote by T the operator .∂c ϕ : X → 2X . Then, Problem 1.4 is Tykhonov well-posed. Proof First, we use the definition of the subdifferential (1.44) to see that the following equivalence holds: ϕ(v) ≥ ϕ(u)

.

∀v ∈ X

⇐⇒

0X ∈ ∂c ϕ(u)

⇐⇒

0X ∈ T u.

On the other hand, we note that Theorem 1.13 implies that there exists a unique element .u ∈ X such that .ϕ(v) ≥ ϕ(u) for all .v ∈ X. We combine these two ingredients to deduce that Problem 1.4 has a unique solution. Consider now an approximating sequence .{un } ⊂ X. Then (1.85) shows that there exists a sequence .{ξn } such that .ξn ∈ ∂c ϕ(un ) for all .n ∈ N and .ξn X → 0. Let .n ∈ N. We use the strong monotonicity of the subdifferential operator in points .un and u, guaranteed by Proposition 1.20, to see that (ξn − f, un − u)X ≥ m un − u2X

.

∀f ∈ ∂c ϕ(u),

where, recall, .m > 0. Next, since .0X ∈ ∂c ϕ(u), we are allowed to take .f = 0X in the previous inequality. So, we deduce that (ξn , un − u)X ≥ mun − u2X

.

and, therefore, un − uX ≤

.

1 ξn X , m

which implies that .un → u in X. This proves that Problem 1.4 is Tykhonov wellposed.   We end this section with the remark that given .K ⊂ X and .T : X → 2Y , it is possible to consider the problem of finding .u ∈ K such that .0Y ∈ T u. For such an inclusion problem, we can define the concept of Levitin–Polyak well-posedness, using the notion of LP -approximating sequences. Moreover, if the operator T

40

1 Nonlinear Problems and Their Classical Well-Posedness

depends on a parameter, we can define the concept of Hadamard well-posedness for this inclusion problem. Nevertheless, to avoid repetitions, we skip introducing these concepts.

1.3.5 Variational Inequalities The concept of Tykhonov well-posedness can be easily extended to variational inequalities. To fix the ideas, in this subsection we restrict ourselves to consider a simple example. Thus, we assume that X is a real Hilbert space, .K ⊂ X, .A : X → X, .ϕ : X → R, and .f ∈ X and consider the following problem. Problem 1.5 Find .u ∈ X such that u ∈ K,

(Au, v − u)X + ϕ(v) − ϕ(u) ≥ (f, v − u)X

.

∀ v ∈ K.

(1.86)

We now introduce the following definition. Definition 1.21 (a) A sequence .{un } ⊂ X is called an approximating sequence for Problem 1.5 if there exists a sequence .{θn } ⊂ R+ such that .θn → 0 .

un ∈ K,

(Aun , v − un )X + ϕ(v) − ϕ(un ) + θn v − un X ≥ (f, v − un )X

(1.87)

∀ v ∈ K, n ∈ N.

(b) Problem 1.5 is well-posed in the sense of Tykhonov (or, equivalently, is Tykhonov well-posed) if there exists a unique element .u ∈ X such that (1.86) holds and any approximating sequence converges in X to u. We now proceed with the following result. Theorem 1.18 Let X be a real Hilbert space, K a nonempty, closed, convex subset of X, .A : X → X a strongly monotone Lipschitz continuous operator, .ϕ : X → R a convex lower semicontinuous function, and .f ∈ X. Then, Problem 1.5 is Tykhonov well-posed. Proof The existence and uniqueness of the solution is a direct consequence of Corollary 1.4 on page 26. Consider now an approximating sequence .{un } ⊂ X. Then there exists a sequence .{θn } ⊂ R+ such that .θn → 0 and (1.87) holds. Let .n ∈ N. We take .v = un in (1.86) and then .v = u in (1.87) and add the resulting inequalities to find that (Aun − Au, u − un )X + θn u − un X ≥ 0.

.

This implies that (Aun − Au, un − u)X ≤ θn un − uX ,

.

1.3 Classical Well-Posedness Concepts

41

and using the strong monotonicity of the operator A with constant .mA > 0, we deduce that θn .un − uX ≤ . mA Now, since .θn → 0, we see that .un → u in X, which concludes the proof.   Remark 1.5 A careful examination of the proof of Theorem 1.18 shows that if X is a Hilbert space, .K ⊂ X, .ϕ : X → R, and .A : X → X a strongly monotone operator, then for any .f ∈ X the following statements are equivalent: (1) Problem 1.5 is Tykhonov well-posed. (2) Problem 1.5 has a unique solution. We now establish the link between the well-posedness of an inequality problem of the form (1.86) and an inclusion problem. To this end, we assume in what follows that .K = X, and we denote by .T : X → 2X the multivalued operator given by .T u = Au+∂c ϕ(u)−f . Then, using the definition of the subdifferential, Problem 1.5 can be formulated, equivalently, as follows. Problem 1.6 Find .u ∈ X such that T u = Au + ∂c ϕ(u) − f  0X .

(1.88)

.

We have the following result. Theorem 1.19 If Problem 1.5 is Tykhonov well-posed in the sense of Definition 1.21, then Problem 1.6 is Tykhonov well-posed in the sense of Definition 1.20. Proof Using (1.44), we find that the inclusion .Au + ∂c ϕ(u) − f  0X holds if and only if u ∈ X,

.

(Au, v − u)X + ϕ(v) − ϕ(u) ≥ (f, v − u)X

∀ v ∈ X.

(1.89)

Assume that Problem 1.5 is Tykhonov well-posed in the sense of Definition 1.21. Then, there exists a unique element u such that (1.89) holds. Therefore, the equivalence above implies that u is the unique solution to Problem 1.6. Consider now an approximating sequence .{un } ⊂ X for Problem 1.6 in the sense of Definition 1.20. There exists a sequence .{ξn } ⊂ X such that ξn ∈ T un = Aun + ∂c ϕ(un ) − f

.

∀n ∈ N

(1.90)

and, moreover, ξn X → 0

.

as n → ∞.

(1.91)

42

1 Nonlinear Problems and Their Classical Well-Posedness

Let .n ∈ N and .v ∈ X. Then (1.90) shows that .ξn − Aun + f ∈ ∂ϕc (un ) and, therefore, ϕ(v) − ϕ(un ) ≥ (ξn − Aun + f, v − un )X

.

or, equivalently, (Aun , v − un )X + ϕ(v) − ϕ(un ) + (ξn , un − v)X ≥ (f, v − un )X .

.

This implies that (Aun , v − un )X + ϕ(v) − ϕ(un ) + ξn X v − un X ≥ (f, v − un )X .

.

(1.92)

We now denote .θn = ξn X and use (1.91) and (1.92) to deduce that .{un } ⊂ X is an approximating sequence for Problem 1.5 in the sense of Definition 1.21. Next, the Tykhonov well-posedness of Problem 1.5 in the sense of Definition 1.21 implies that .un → u in X. We conclude from here that Problem 1.6 is Tykhonov well-posed   in the sense of Definition 1.20. Various numerical procedures used to approach the solutions of variational inequalities lead to approximating sequences which usually fail to lie in the constraint set K, but their general term gets closer and closer to the constraint set. To take care of such cases, one can strengthen the Tykhonov well-posedness concept for Problem 1.5, as follows. Definition 1.22 (a) A sequence .{un } ⊂ X is called a generalized (or LP -) approximating sequence for Problem 1.5 if there exist two sequences .{wn } ⊂ X and .{θn } ⊂ R+ such that .wn → 0X in X, .θn → 0, and moreover, un + w. n ∈ K,

(Aun , v − un )X + ϕ(v) − ϕ(un ) + θn v − un X ≥ (f, v − u)X

∀ v ∈ K, n ∈ N.

(b) Problem 1.5 is well-posed in the sense of Levitin–Polyak (or, equivalently, is Levitin-Polyak well-posed) if there exists a unique element .u ∈ X such that (1.86) holds and any LP -approximating sequence converges in X to u. A Levitin–Polyak well-posedness result for a class of variational–hemivariational inequalities which contain as a particular case inequality (1.86) will be presented in Sect. 5.3.2. Moreover, it is easy to see that every approximating sequence for inequality (1.86) is a generalized approximating sequence. Therefore, the Levitin– Polyak well-posedness of Problem 1.5 implies its Tykhonov well-posedness. We end this section with the following remark.

1.3 Classical Well-Posedness Concepts

43

Remark 1.6 Keep the assumption in Theorem 1.18 and denote by .u1 and .u2 the solution to Problem 1.5 for the data .f = f1 and .f = f2 ∈ X, respectively. Then, it is easy to prove that u1 − u2 X ≤

.

1 f1 − f2 X . mA

(1.93)

Inequality (1.93) shows that the solution of Problem 1.5 depends Lipschitz continuously on .f ∈ X. This result could be interpreted as a well-posed result for Problem 1.5, in the sense of Hadamard.

Chapter 2

Tykhonov Triples and Associated Well-Posedness Concept

Inspired by the examples presented in Sect. 1.3, in this chapter we introduce the notion of Tykhonov triple with which we associate a new well-posedness concept for abstract problems in the framework of metric spaces. We present the basic properties of Tykhonov triples, and then we state and prove necessary and sufficient conditions of well-posedness. We illustrate this concept in the study of elementary examples, and we show that it extends the classical well-posedness concepts introduced in Sect. 1.3. Then, we pay a particular attention to the study of two special types of problems, the so-called split and dual problems, respectively. Finally, we present an extension of the Levitin–Polyak and Hadamard well-posedness concepts together with several examples. We end this chapter with some concluding remarks. This chapter plays a crucial role since it introduces definitions and results in a general framework, laying in this way the background for the study of various nonlinear problems considered in the next chapters of this book.

2.1 A New Well-Posedness Concept 2.1.1 Preliminaries A brief analysis of the Tykhonov, Levitin–Polyak, and Hadamard well-posedness concepts presented in Sect. 1.3 shows that, besides the unique solvability of the corresponding problems, they require some convergence results: the convergence of approximating sequences in the case of the Tykhonov well-posedness concept, the convergence of the LP -approximating sequences in the case of the Levitin–Polyak well-posedness concept, and the convergence of the solution .u(fn ) to the solution .u(f ) when .fn converges to f , in the case of the Hadamard well-posedness concept. On the other hand, over the time, a considerable effort was done to obtain convergence results in the study of various mathematical problems including © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Sofonea, Well-Posed Nonlinear Problems, Advances in Mechanics and Mathematics 50, https://doi.org/10.1007/978-3-031-41416-9_2

45

46

2 Tykhonov Triples and Associated Well-Posedness Concept

nonlinear equations, inequality problems, inclusions, fixed point problems, and optimization problems, among others. The literature in the field is extensive and the corresponding results have been obtained by using different methods and functional arguments. Nevertheless, most of these results are stated in the following functional framework: given a space X, a problem .P that has a unique solution .u ∈ X, and a sequence .{uθ } ⊂ X or a family of problems .{Pθ } such that .uθ is a solution of Problem .Pθ , the aim consists in proving that .uθ converges to u in X, as .θ converges. A careful analysis of this description reveals that, in practice, the functional framework above has to be completed by providing details on the following three items: (i) The set I to which the parameter .θ belongs. (ii) The problem .Pθ or its sets of solutions, denoted by .Ω(θ ), for each .θ ∈ I . (iii) The meaning we give to the convergence of the parameter .θ , this convergence being, in general, governed by a set of sequences .C. Then, collecting the three ingredients above, we arrive in a natural way to consider triples of the form .T = (I, Ω, C) that we call Tykhonov triples. Note that this concept was introduced in [250], in the functional framework of metric spaces. Tykhonov triples represent a useful mathematical tool in the analysis of various problems. Indeed, as we shall see below in this section, if a Problem .P is well-posed with respect to a Tykhonov triple .T , then all the approximating sequences generated by .T , the so-called .T -approximating sequences, converge to the unique solution of .P. Among these sequences, we may identify some remarkable ones, and in this way we implicitly deduce their convergence. To conclude, the interest in using the mathematical tool provided by the Tykhonov triples arises from the fact that it allows us to obtain general convergence results, on the one hand, and unifies convergence results previously obtained by different functional arguments, on the other hand. The difficulty in using this tool consists in the choice of the appropriate Tykhonov triple, which needs to be large enough in order to be used to recover a specific convergence result, but small enough in order to guarantee the well-posedness of the considered problem.

2.1.2 Basic Definitions and General Properties We consider an abstract mathematical object .P, called generic “problem,” associated with a metric space .(X, d). Problem .P could be an equation, a minimization problem, a fixed point problem, an inclusion, or an inequality problem, for instance. Note that in this book the statement of .P will vary from subsection to subsection and from example to example. Nevertheless, within a given subsection (or example), we shall consider only one problem (usually denoted by .P), and the results presented in the corresponding subsection (or example) will exclusively refer to that problem. In this way, no confusion arises, even if below in this book, we shall use the same notation for various different problems.

2.1 A New Well-Posedness Concept

47

We associate with Problem .P the concept of “solution” which follows from the 0 the set of solutions to Problem .P. Thus, Problem context. We also denote by .SP 0 0 .P has a unique solution iff .S P has a unique element, i.e., .SP is a singleton. The concept of well-posedness for Problem .P is related to a so-called Tykhonov triple and is defined as follows. Definition 2.1 (a) A Tykhonov triple is a mathematical object of the form .T = (I, Ω, C), where I is a given nonempty set, .Ω : I → 2X is a set-valued mapping such that .Ω(θ ) = ∅ for each .θ ∈ I , and .C is a nonempty subset of sequences with elements in I . (b) Given a Tykhonov triple .T = (I, Ω, C), a sequence .{un } ⊂ X is called a .T approximating sequence if there exists a sequence .{θn } ∈ C, such that .un ∈ Ω(θn ) for each .n ∈ N. (c) Given a Tykhonov triple .T = (I, Ω, C), Problem .P is said to be .T -well-posed (or, equivalently, well-posed with .T ) if it has a unique solution and every .T approximating sequence converges in X to this solution. Remark 2.1 Let .T = (I, Ω, C) be a Tykhonov triple. Then, we refer to I as the set of parameters. A typical element of I will be denoted by .θ . We refer to the family of sets .{Ω(θ )}θ∈I as the family of approximating sets. Note that .T approximating sequences always exist since, by assumption, .C =  ∅, and, in addition, for any sequence .{θn } ∈ C and any .n ∈ N, the set .Ω(θn ) is not empty. We also remark that the concept of approximating sequence above depends on the Tykhonov triple .T , and for this reason, everywhere in this book we use the terminology “.T approximating sequence.” As a consequence, the concept of well-posedness for Problem .P is not an intrinsic one since it depends on the Tykhonov triple .T . For this reason, we refer to it as “well-posedness with .T ” or “.T -well-posedness,” as mentioned in Definition 2.1(c). Remark 2.2 Definition 2.1(a) shows that any approximating set of a Tykhonov triple is nonempty. In many cases this condition is obviously satisfied. In other cases, it can be verified easily and this is what we shall do from the beginning, every time it will be possible. Nevertheless, in several cases, checking this condition requires the proof of an existence result, and for this reason, it will be verified a posteriori. However, we shall refer to the corresponding triple .T = (I, Ω, C) as a Tykhonov triple, even if the proof of the validity of the condition .Ω(θ ) =  ∅ for each .θ ∈ I is postponed. The following example illustrates the previous definitions and some of the related comments above. Example 2.1 Let .X = R endowed with the usual distance, let .f ∈ R, and consider the problem .P of finding .u ∈ R such that .2u = f . We consider the Tykhonov triples .T1 = (I, Ω1 , C) and .T2 = (I, Ω2 , C) defined by

48

.

2 Tykhonov Triples and Associated Well-Posedness Concept

I = [0, +∞), Ω1 (θ ) :=

f 2

  C = {θn } ⊂ I : θn → 0 ,  f +θ , 2

− θ,

Ω2 (θ ) :=

f 2

Ω1 , Ω2 : I → 2X , − θ − 1,

 f +θ +1 2

∀ θ ≥ 0.

Then, using Definition 2.1, it is easy to see that Problem .P is well-posed with .T1 but fails to be well-posed with .T2 . Next, we assume that Problem .P has a unique solution .u ∈ X, and we denote by SP the set of all sequences of X which converge to u, that is,

.

.

  SP = {un } ⊂ X : un → u in X .

(2.1)

Moreover, given a Tykhonov triple .T = (I, Ω, C), we use the notation .ST for a set of .T -approximating sequences, that is, .

  ST = {un } ⊂ X : {un } is a T -approximating sequence .

(2.2)

To avoid any confusion, we underline that in Example 2.1, in (2.1), (2.2), and everywhere in this book we use the notation .{ωn } for the sequence of general  term  .ωn , and we use big parentheses for sets, i.e., for instance, we write .A = a, b, c for the set A of elements .a, b, and .c. The examples below show that no particular inclusion holds between the sets of sequences .SP and .ST . Example 2.2 Let .(X, · X ) be a normed space, .a, b ∈ X with .a = b, and let J (v) = v − a X for all .v ∈ X. In this framework, we consider the following minimization problem:

.

Problem .P. Find .u ∈ X such that .J (u) ≤ J (v) for all .v ∈ X. Moreover, we consider two Tykhonov triples .T1 = (I1 , Ω1 , C1 ) and .T2 = (I2 , Ω2 , C2 ) defined by .

I1 = R+ = [0, +∞),

.

I2 = R+ = [0, +∞),

  Ω1 : I1 → 2X , Ω1 (θ ) =  u ∈ X :  u − a X ≥ θ ∀ θ ≥ 0,   C1 = {θn } ⊂ I1 : θn → 1 ,

  Ω2 : I2 → 2X , Ω2 (θ ) =  u ∈ X :  u − b X ≤ θ   C2 = {θn } ⊂ I2 : θn → 0 .

∀ θ ≥ 0,

2.1 A New Well-Posedness Concept

49

Let .{u1n } ⊂ X be the sequence defined by .u1n = a + n1 b for all .n ∈ N. Since 1 1 .un → a in X and a is the unique solution to Problem .P, it follows that .{un } ∈ SP . 1 Nevertheless, .{un } ∈ ST1 , and therefore, .SP ⊂ ST1 . Let .{u2n } ⊂ X be the sequence defined by .u2n = b for all .n ∈ N. It follows that 2 .{un } is a .T2 -approximating sequence which does not converge to the solution of Problem .P. Therefore, .ST2 ⊂ SP . Next, we turn back to the study of an abstract problem .P, assumed to have a unique solution .u ∈ X. We use Definition 2.1(c) and equalities (2.1) and (2.2) to see that Problem P is T -well-posed if and only if ST ⊂ SP .

.

(2.3)

Moreover, the set .ST of .T -approximating sequences suggests us to introduce the following definition. Definition 2.2 Given two Tykhonov triples .T = (I, Ω, C) and .T = (I , Ω , C ), we say that: (a) .T and .T are equivalent if their sets of approximating sequences are the same, i.e., .ST = ST . In this case, we write .T ∼ T . (b) .T is smaller than .T if .ST ⊂ ST . In this case, we write .T ≤ T . (c) .T is strictly smaller than .T If .ST ⊂ ST and .ST ⊂ ST . In this case, we write .T < T . Using (2.3), we deduce that the following statements hold: If T ∼. T , then P is T -well-posed if and only if P is T -well-posed..



If T ≤ T and P is T -well-posed, then P is T -well-posed, too.

(2.4) (2.5)

It is easy to see that “.∼” represents a relation of equivalence on the set of Tykhonov triples, denoted by .T. Let .Tˆ be the set of equivalence classes of Tykhonov triples, with respect to the relation “.∼,” that is,   Tˆ = Tˆ : T ∈ T ,

.

where   Tˆ = T ∈ T : T ∼ T

.

∀ T ∈ T.

On the set .Tˆ , we introduce the following relation: ˆ Tˆ ⇐⇒ T ≤ T in the sense of Definition 2.2(b), i.e., ST ⊂ ST . Tˆ ≤

.

(2.6)

50

2 Tykhonov Triples and Associated Well-Posedness Concept

ˆ ˆ is well defined and is a relation of order on the set .T. Then, it is easy to see that “.≤” Examples that illustrate Definition 2.2 will be provided in Sects. 3.5.1 and 6.1.2, for instance. We now provide two special examples of Tykhonov triples associated with Problem .P. Example 2.3 Assume that .{un } is a given sequence of elements in X. Moreover, consider the Tykhonov triple .T0 = (I0 , Ω0 , C0 ) defined as follows: .

  I0 = N = 1, 2, , . . . , n, . . . ,  Ω0 : I0 → 2X , Ω0 (n) = un } ∀ n ∈ N,   C0 = {kn } ⊂ I0 : k1 < k2 < . . . < kn . . . .

Then, using Definition 2.1(a), it is easy to see that a sequence .{ un } ⊂ X is a .T0 approximating sequence if and only if .{ un } is a subsequence of the sequence .{un }. Therefore, by Definition 2.1(c), we deduce that .un → u in X if and only if Problem .P is well-posed with the Tykhonov triple .T0 . Example 2.4 Let .TP = (IP , ΩP , CP ) where IP = R. + = [0, +∞), .



ΩP : IP → 2X , ΩP (θ ) =  u ∈ X : d( u, u) ≤ θ   CP = {θn } ⊂ IP : θn → 0 .



(2.7) ∀ θ ≥ 0, .

(2.8) (2.9)

Then, Problem .P is .TP -well-posed. To prove this statement, we shall prove that S TP = S P .

.

(2.10)

Let .{un } be a .TP -approximating sequence, i.e., .{un } ∈ STP . Then, using (2.7)– (2.9), we deduce that there exists .{θn } ⊂ R+ such that .θn → 0 and .d(un , u) ≤ θn for each .n ∈ N. This implies that .un → u in X, and using definition (2.1), we deduce that .{un } ∈ SP . Conversely, assume that .{un } ∈ SP . Then, (2.1) implies that .un → u in X. Denote .θn = d(un , u), for each .n ∈ N. It follows from here that .{θn } ∈ CP and, moreover, .{un } ∈ ΩP (θn ), for each .n ∈ N. This shows that .{un } is a .TP -approximating sequence, and using definition (2.2), we deduce that .{un } ∈ ST . It results from above that equality (2.10) holds, as claimed. We now P use the equivalence (2.3) to deduce that Problem .P is .TP -well-posed. We proceed with several comments related to Examples 2.3 and 2.4. Remark 2.3 It follows from Example 2.3 that a specific convergence result to the solution of Problem .P (say .un → u in X) is equivalent to the well-posedness

2.1 A New Well-Posedness Concept

51

of .P with a specific Tykhonov triple (say .T0 = (I0 , Ω0 , C0 )) in which .Ω0 (·) is a singleton. Nevertheless, in practice, we construct Tykhonov triples .T = (I, Ω, C) for which the approximating sets .Ω(θ ) with .θ ∈ I are not singletons. Usually, the sets .Ω(θ ) are large enough, and therefore, the set of .T -approximating sequences is quite large. Then, the well-posedness of Problem .P with such a Tykhonov triple implicitly provides more than one convergence result since, by definition, all the .T -approximating sequences converge to the solution u of .P. We conclude that the well-posedness concept introduced in Definition 2.1 provides a framework which allows us to obtain various convergence results. ˆ the set of classes .Tˆ ∈ Tˆ of Remark 2.4 We denote in what follows by .(AP , ≤) Tykhonov triples .T with which Problem .P is well-posed, endowed with the relation ˆ TˆP of order (2.6). Then, it follows from (2.3) and (2.10) that .TˆP ∈ AP and .Tˆ ≤ for all .Tˆ ∈ AP . This shows that the equivalence class of the Tykhonov triple .TP ˆ Moreover, equivalence in Example 2.4 is the greatest element of the set .(AP , ≤). (2.3) and equality (2.10) show that among all the elements in .AP , the equivalence class of the Tykhonov triple (2.7)–(2.9), denoted by .TˆP , represents the class of equivalent triples which generates the largest set of approximating sequences since all the sequences that converge to the solution u of problem .P are .TP -approximating sequences. Even if these properties could seem to be interesting, the choice of the Tykhonov triple .TP to represent its equivalence class is not convenient to study the well-posedness of Problem .P. Indeed, some difficulties could arise from the fact that the definition (2.8) uses the solution u of Problem .P which is, in principle, unknown. A reasonable definition of the approximating sets would use Problem .P itself, or some of its perturbation, not its solution. For this reason, it is important to introduce Tykhonov triples which that equivalent to .TP (in the sense of Definition 2.2(a)) but are defined without any explicit mention to the solution u of Problem .P. And this is what we shall do in several places in the rest of the book as, for instance, in Sects. 3.1.1 and 6.1.4. We now turn to the construction of a relevant example of Tykhonov triple which will be used in the next sections. Let .p ∈ N and .Ti = (Ii , Ωi , Ci ) be Tykhonov triples with .i = 1, . . . , p, and consider the following assumptions:  .

For each i = 1, . . . , p, there exists ci ∈ Ii such that the sequence θ i = {θni } defined by θni = ci (for each n ∈ N) belongs to Ci .

⎧ There exists a multifunction Ω : I1 × I2 × . . . × Ip → 2X ⎪ ⎪ ⎪ ⎪ ⎪ such that Ω(θ1 , θ2 , . . . , θn ) = ∅ for each θi ∈ Ii , i = 1, . . . , p ⎪ ⎪ ⎪ ⎪ ⎨ and, moreover, . Ω1 (θ1 ) ⊂ Ω(θ1 , c2 , c3 , . . . , cp−1 , cp ) for all θ1 ∈ I1 , ⎪ ⎪ ⎪ Ω2 (θ2 ) ⊂ Ω(c1 , θ2 , c3 , . . . , cp−1 , cp ) for all θ2 ∈ I2 , ⎪ ⎪ ⎪ ⎪ ...... ⎪ ⎪ ⎩ Ωp (θp ) ⊂ Ω(c1 , c2 , c3 , . . . , cp−1 , θp ) for all θp ∈ Ip .

(2.11)

(2.12)

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2 Tykhonov Triples and Associated Well-Posedness Concept

Let .I = I1 × I2 × · · · × Ip and .C = C1 × C2 × · · · × Cp , and consider the Tykhonov triple .T = (I, Ω, C), where .Ω : I → 2X is the multifunction provided by assumption (2.12). We have the following result which will be used on page 179 and which has some interest of its own. Theorem 2.1 Assume (2.11) and (2.12). Then .Ti ≤ T , for all .i = 1, . . . , p. Proof Let .{u1n } be a .T1 -approximating sequence for Problem .P. Then, it follows from Definition 2.1(b) that there exists a sequence .{θn1 } ∈ C1 such that .u1n ∈ Ω1 (θn1 ) for each .n ∈ N. We now use the first inclusion in assumption (2.12) to see that u1n ∈ Ω(θn1 , c2 , c3 , . . . , cp−1 , cp )

.

∀ n ∈ N.

(2.13)

On the other hand, assumption (2.11) guarantees that the sequence .{θ n } with θ n = (θn1 , c2 , c3 , . . . , cp−1 , cp ) belongs to .C. Combining this result with (2.13), we deduce that

.

{θ n } ∈ C

.

and

u1n ∈ Ω(θ n )

∀ n ∈ N.

(2.14)

We now use (2.14) and Definition 2.1(b) to see that the sequence .{u1n } is a .T approximating sequence for Problem .P. It follows from here that .ST1 ⊂ ST , and using Definition 2.2(b), we find that .T1 ≤ T . A similar argument leads to the inequalities .T2 ≤ T , . . . , Tp ≤ T , which concludes the proof.   We end this section with an extension of Definition 2.1(c), in the case of normed spaces. Definition 2.3 Let X be a normed space and .T = (I, Ω, C) a Tykhonov triple and let .P be an abstract problem associated with X. Then, (a) Problem .P is said to be weakly .T -well-posed (or, equivalently, weakly wellposed with .T ) if it has a unique solution and every .T -approximating sequence converges weakly in X to this solution. (b) Problem .P is said to be strongly generalized .T - well-posed (or, equivalently, 0 is not empty strongly generalized well-posed with .T ) if its set of solutions .SP and every .T -approximating sequence has a subsequence which converges 0. strongly in X to some point of .SP (c) Problem .P is said to be weakly generalized .T -well-posed (or, equivalently, 0 is not empty and weakly generalized well-posed with .T ) if its set of solutions .SP every .T -approximating sequence has a subsequence which converges weakly in 0. X to some point of .SP Note that, to avoid any confusion, we sometimes refer to the .T -well-posedness of Problem .P as strongly .T -well-posedness. Therefore, using the Definitions 2.1(c) and 2.3, it is easy to see that if Problem .P is .T -well-posed, then it is weakly .T -wellposed and strongly generalized .T - well-posed. Moreover if Problem .P is strongly generalized .T -well-posed, then it is weakly generalized .T -well-posed, and if it is

2.1 A New Well-Posedness Concept

53

weakly .T -well-posed, then it is weakly generalized .T -well-posed. In general, the reverse of these statements is not true, as it follows from the examples below.   Example 2.5 Let X be a Hilbert space and .K = v ∈ X : v X ≤ k with .k > 0 and let .PK : X → K be the projection map on the set K. We define the functional .J : X → R by the equality J (v) = v − PK v X

.

∀v ∈ X

and consider the following minimization problem. Problem .P Find .u ∈ X such that .J (u) ≤ J (v) for all .v ∈ X. We now introduce the Tykhonov triple .T = (I, Ω, C) defined as follows: .

I = R+ = [0, +∞),   Ω(θ ) = v ∈ X : v X ≤ k + θ ∀ θ > 0,   C = {θn } ⊂ I : θn → 0 .

With this notation, we claim that Problem .P is weakly generalized .T -well-posed. Indeed, if .{un } is a .T -approximating sequence, then Definition 2.1(b) implies that there exists a sequence .{θn } ⊂ R+ such that .θn → 0 and . un X ≤ k + θn for each .n ∈ N. This implies that the sequence .{un } is bounded, and therefore, there exist a subsequence, still denoted by .{un }, and an element .u ∈ X such that .un  u in X. We now use the weak lower semicontinuity of the norm to see that

u X ≤ lim inf un X ≤ lim inf(k + θn ) = k,

.

and therefore, we deduce that .u ∈ K. On the other hand, it is easy to see that the set 0 . We now of solutions of Problem .P coincides with the set K, and therefore .u ∈ SP use Definition 2.3(c) to deduce that Problem .P is weakly generalized .T -well-posed. This problem is neither weakly .T -well-posed nor strongly .T -well-posed since its set of solutions is not a singleton.

2.1.3 Classical Examples In this subsection we return to the classical well-posed problems introduced in Sect. 1.3 and prove that the corresponding well-posedness concepts can be recovered by the .T -well-posedness concept introduced above, with the choice of a convenient Tykhonov triple .T .

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2 Tykhonov Triples and Associated Well-Posedness Concept

Example 2.6 (Tykhonov well-posedness of minimization problems) Definition 1.17 shows that a sequence .{un } is a minimization sequence for Problem 1.1 if and only if .{un } ⊂ K and there exists a sequence .{θn } ⊂ R+ such that .θn → 0 and .|J (un ) − ω| ≤ θn for each .n ∈ N. Recall that, here and below, .ω is defined by (1.60). Therefore, with the notation .T = (I, Ω, C) where .

I = R+ = [0, +∞),

  Ω : I → 2X , Ω(θ ) =  u ∈ K : |J ( u) − ω| ≤ θ ∀ θ ≥ 0,   C = {θn } ⊂ I : θn → 0 ,

we are in a position to reformulate Definition 1.17 as follows: Problem 1.1 is wellposed in the sense of Tykhonov if the statements (a) and (b) below hold: (a) There exists a unique element .u ∈ K such that .J (u) = ω. (b) If .{un } ⊂ K is a sequence with the property that there exists .{θn } ∈ C such that .un ∈ Ω(θn ) for each .n ∈ N, then .un → u in X. This shows that Problem 1.1 is well-posed in the sense of Definition 1.17 if and only if it is well-posed in the sense of Definition 2.1, with the Tykhonov triple .T introduced above. Example 2.7 (Levitin–Polyak well-posedness of minimization problems) Note that a sequence .{un } ⊂ X is an LP -minimization sequence for Problem 1.1 (in the sense of Definition 1.18) iff there exists a sequence .{θn } ⊂ R+ such that .θn → 0 and .d(un , K) ≤ θn , .|J (un ) − ω| ≤ θn for each .n ∈ N. Therefore, with the notation .T = (I, Ω, C) where .

I = R+ = [0, +∞),

  Ω : I → 2X , Ω(θ ) =  u ∈ X : d( u, K) ≤ θ, |J ( u) − ω| ≤ θ ∀ θ ≥ 0,   C = {θn } ⊂ I : θn → 0 ,

we are in a position to reformulate Definition 1.18 as follows: Problem 1.1 is wellposed in the sense of Levitin–Polyak if the statements (a) and (b) below hold: (a) There exists a unique element .u ∈ K such that .J (u) = ω. (b) If .{un } ⊂ X is a sequence with the property that there exists .{θn } ∈ C such that .un ∈ Ω(θn ) for each .n ∈ N, then .un → u in X. This shows that Problem 1.1 is well-posed in the sense of Definition 1.18 if and only if it is well-posed in the sense of Definition 2.1, with the Tykhonov triple .T introduced above.

2.1 A New Well-Posedness Concept

55

Example 2.8 (Hadamard well-posedness of Problem 1.2) We recall that Problem 1.2 is well-posed in the sense of Hadamard if, by definition, the following hold: (a) For all .f ∈ X, Problem 1.2 has a unique solution .u = u(f ). (b) For any .f ∈ X and any sequence .{fn } ⊂ X such that .fn → f in X, one has .u(fn ) → u(f ) in X. Therefore, with the notation .Tf = (I, Ω, Cf ) where .

I = X,

  Ω : I → 2X , Ω(f) = u(f) ∀ f ∈ X,   Cf = {fn } ⊂ I : fn → f in X ,

we are in a position to reformulate the above definition as follows: Problem 1.2 is well-posed in the sense of Hadamard if the statements (a) and (b) below hold: (a) For any .f ∈ X, Problem 1.2 has a unique solution .u = u(f ). (b) If .{un } ⊂ K is a sequence with the property that there exists .{fn } ∈ Cf such that .un ∈ Ω(fn ) for each .n ∈ N, then .un → u in X. This shows that Problem 1.2 is well-posed in the sense of Hadamard if and only if for any .f ∈ X it is well-posed in the sense of Definition 2.1, with the Tykhonov triple .Tf introduced above. Example 2.9 (Tykhonov well-posedness of fixed point problems) Note that a sequence .{un } ⊂ K is an approximating sequence for Problem 1.3 in the sense of Definition 1.19 if there exists a sequence .{θn } ⊂ R+ such that .θn → 0 and .d(Λun , un ) ≤ θn for each .n ∈ N. Therefore, with the notation .T = (I, Ω, C) where .

I = R+ = [0, +∞),

  Ω : I → 2X , Ω(θ ) =  u ∈ K : d(Λ u,  u) ≤ θ ∀ θ ≥ 0,   C = {θn } ⊂ I : θn → 0 ,

we are in a position to reformulate Definition 1.19 as follows: Problem 1.3 is wellposed in the sense of Tykhonov if the statements (a) and (b) below hold: (a) There exists a unique element .u ∈ K such that .Λu = u. (b) If .{un } ⊂ K is a sequence with the property that there exists .{θn } ∈ C such that .un ∈ Ω(θn ) for each .n ∈ N, then .un → u in X.

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2 Tykhonov Triples and Associated Well-Posedness Concept

This shows that Problem 1.3 is well-posed in the sense of Definition 1.19 if and only if it is well-posed in the sense of Definition 2.1, with the Tykhonov triple .T introduced above. Example 2.10 (Tykhonov well-posedness of inclusions) Note that a sequence {un } ⊂ X is an approximating sequence for Problem 1.4 in the sense of Definition 1.20 if there exists a sequence .{θn } ⊂ R+ such that .θn → 0 and .dY (0Y , T un ) ≤ θn , for each .n ∈ N. Therefore, with the notation .T = (I, Ω, C) where .

.

I = R+ = [0, +∞),

  Ω : I → 2X , Ω(θ ) =  u ∈ X : dY (0Y , T  u) ≤ θ ∀ θ ≥ 0,   C = {θn } ⊂ I : θn → 0 ,

we are in a position to reformulate Definition 1.20 as follows: Problem 1.4 is wellposed in the sense of Tykhonov if the statements (a) and (b) below hold: (a) There exists a unique element .u ∈ X such that .0Y ∈ T u. (b) If .{un } ⊂ X is a sequence with the property that there exists .{θn } ∈ C such that .un ∈ Ω(θn ) for each .n ∈ N, then .un → u in X. This shows that Problem 1.4 is well-posed in the sense of Definition 1.20 if and only if it is well-posed in the sense of Definition 2.1, with the Tykhonov triple .T introduced above. Example 2.11 (Tykhonov well-posedness of variational inequalities) Consider the notation .T = (I, Ω, C) where .

I = R+ = [0, +∞), Ω : I → 2X ,

 u ∈ Ω(θ ) iff  u ∈ K and

(A u, v −  u)X + ϕ(v) − ϕ( u) + θ v −  u X ≥ (f, v −  u)X

∀ v ∈ K,

for all θ ∈ I,   C = {θn } ⊂ I : θn → 0 . Then, we are in a position to reformulate Definition 1.21 as follows: Problem 1.5 is well-posed in the sense of Tykhonov if the statements (a) and (b) below hold: (a) There exists a unique element .u ∈ X such that (1.86) holds. (b) If .{un } ⊂ K is a sequence with the property that there exists .{θn } ∈ C such that .un ∈ Ω(θn ) for each .n ∈ N, then .un → u in X.

2.1 A New Well-Posedness Concept

57

This shows that Problem 1.5 is well-posed in the sense of Definition 1.22 if and only if it is well-posed in the sense of Definition 2.1, with the Tykhonov triple .T introduced above. Example 2.12 (Levitin–Polyak well-posedness of variational inequalities) Consider the notation .T = (I, Ω, C) where .

I = R+ = [0, +∞), Ω : I → 2X ,

 u ∈ Ω(θ ) iff  u ∈ X, d( u, K) ≤ θ and

(A u, v −  u)X + ϕ(v) − ϕ( u) + θ v −  u X ≥ (f, v −  u)X

∀ v ∈ K,

for all θ ∈ I,   C = {θn } ⊂ I : θn → 0 . Then, we are in a position to reformulate Definition 1.22 as follows: Problem 1.5 is well-posed in the sense of Levitin–Polyak if the statements (a) and (b) below hold: (a) There exists a unique element .u ∈ K such that (1.86) holds. (b) If .{un } ⊂ X is a sequence with the property that there exists .{θn } ∈ C such that .un ∈ Ω(θn ) for each .n ∈ N, then .un → u in X. This shows that Problem 1.5 is well-posed in the sense of Definition 1.22 if and only if it is well-posed in the sense of Definition 2.1, with the Tykhonov triple .T introduced above. Example 2.13 (Hadamard well-posedness of Problem 1.5) We recall that Problem 1.5 is well-posed in the sense of Hadamard if the statements (a) and (b) below hold: (a) For all .f ∈ X, inequality (1.86) has a unique solution .u = u(f ). (b) For any .f ∈ X and any sequence .{fn } ⊂ X such that .fn → f in X, one has .u(fn ) → u(f ) in X. Therefore, with the notation .Tf = (I, Ω, Cf ) where .

I = X,

  Ω : I → 2X , Ω(f) = u(f) ∀ f ∈ X,   Cf = {fn } ⊂ I : fn → f in X ,

we are in a position to reformulate the above definition as follows: Problem 1.5 is well-posed in the sense of Hadamard if the following statements hold: (a) For any .f ∈ X, inequality (1.86) has a unique solution .u = u(f ). (b) If .{un } ⊂ K is a sequence with the property that there exists .{fn } ∈ Cf such that .un ∈ Ω(fn ) for each .n ∈ N, then .un → u in X.

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2 Tykhonov Triples and Associated Well-Posedness Concept

This shows that Problem 1.5 is well-posed in the sense of Hadamard if and only if for any .f ∈ X it is well-posed in the sense of Definition 2.1, with the Tykhonov triple .Tf introduced above.

2.1.4 Metric Characterization We now return back to the abstract setting in Sect. 2.1.2. Therefore, we assume in what follows that .(X, d) is a metric space and .P is an abstract problem on X. Our aim in this subsection is to provide necessary and sufficient conditions which guarantee the well-posedness of .P with a given Tykhonov triple .T = (I, Ω, C). To this end, we assume in what follows that 0 SP ⊂ Ω(θ )

.

∀ θ ∈ I.

(2.15)

Note that, in most of the examples presented above in this book, this inclusion is strict. The metric properties of the sets .Ω(θn ) ⊂ X as .{θn } ∈ C will play a crucial role in the characterization of the .T -well-posedness of Problem .P, and as shown in Theorems 2.2 and 2.3, we state and prove below in this subsection. To present these theorems, we recall that the diameter of a nonempty set .A ⊂ X, denoted by .diam(A), is defined by equality diam(A) = sup d(a, b).

.

a, b∈A

Our first result in this section is the following. Theorem 2.2 Let .T = (I, Ω, C) be a Tykhonov triple which satisfies condition 0 is (2.15). Then, Problem .P is .T -well-posed if and only if its set of solutions .SP nonempty and .diam(Ω(θn )) → 0 as .n → ∞, for any sequence .{θn } ∈ C. 0 is a Proof Assume that Problem .P is .T -well-posed. Then, by definition, .SP 0 singleton and, therefore, .SP = ∅. Arguing by contradiction, assume in what follows that there exists .{θn } ∈ C such that diam.(Ω(θn )) → 0 as .n → ∞. Then, there exist .δ0 > 0 and two sequences .{un }, .{vn } ⊂ X such that

un , vn ∈ Ω(θn ),

.

d(un , vn ) ≥ δ0

∀ n ∈ N.

(2.16)

Now, since both .{un } and .{vn } are .T -approximating sequences for Problem .P, the T -well-posedness of .P implies that .un → u and .vn → u in X, where u denotes the 0 . This implies that .d(u , v ) → 0 which is in contradiction unique element of .SP n n with (2.16). We conclude from here that diam.(Ω(θn )) → 0 for any sequence .{θn } ∈ C. 0 is nonempty and diam.(Ω(θ )) → 0 as .n → ∞, for Conversely, assume that .SP n 0 is a singleton. Indeed, let .u, u ∈ S 0 and any sequence .{θn } ∈ C. We claim that .SP P

.

2.1 A New Well-Posedness Concept

59

let .{θn } ∈ C. Then, using (2.15), we deduce that .u, u ∈ Ω(θn ) for any .n ∈ N. Thus, d(u, u ) ≤ diam(Ω(θn )) → 0,

.

which implies that .u = u and proves the claim. We conclude from here that .P has a unique solution, denoted in what follows by u. Let now .{un } ⊂ X be a .T approximating sequence. Then there exists a sequence .{θn } ∈ C such that .un ∈ Ω(θn ) for each .n ∈ N. We use (2.15) to see that .u ∈ Ω(θn ) for each .n ∈ N and, therefore, d(u, un ) ≤ diam(Ω(θn )) → 0.

.

This implies that .un → u in X, which shows that Problem .P is .T -well-posed and concludes the proof.   A consequence of Theorem 2.2 is the following. Corollary 2.1 Assume now that .T = (I, Ω, C) and .T = (I, Ω , C) are two Tykhonov triples such that 0 SP ⊂ Ω(θ ),

.

0 SP ⊂ Ω (θ )

∀ θ ∈ I,

and, moreover, for any sequence .{θn } ∈ C, the implication below holds: diam(Ω(θn )) → 0 as n → ∞ ⇒ diam(Ω (θn )) → 0 as n → ∞.

.

(2.17)

Then, the .T -well-posedness of Problem .P implies its .T -well-posedness. Corollary 2.1 is a direct consequence of Theorem 2.2 combined with assumption (2.17). Note that Theorem 2.2 provides a necessary and sufficient condition which guarantees the .T -well-posedness of Problem .P. Nevertheless, checking this condition 0 = ∅, i.e., to prove an existence result. For this reason, requires to prove that .SP in what follows we present a second characterization of the .T -well-posedness 0 = ∅ is removed, but, in turn, additional of Problem .P in which condition .SP conditions to the multivalued function .Ω are imposed. To this end, we introduce the following definition. Definition 2.4 A Tykhonov triple .T = (I, Ω, C) is called regular (with respect to Problem .P) if it satisfies condition (2.15) and, in addition, the following hold: (a) For all .θ1 , θ2 ∈ I , either the inclusion .Ω(θ1 ) ⊂ Ω(θ2 ) or the inclusion .Ω(θ2 ) ⊂ Ω(θ1 ) is satisfied. 0 , i.e., (b) Any convergent .T -approximating sequence converges to an element of .SP if .{un } is a .T -approximating sequence and there exists .u ∈ X such that .un → u 0. in X as .n → ∞, then .u ∈ SP

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2 Tykhonov Triples and Associated Well-Posedness Concept

We underline that the notion of regularity depends on the Problem .P since a given Tykhonov triple can be regular with respect to Problem .P and could fail to be  regular with respect to another problem .P. Example 2.14 Let .X = R endowed with the usual distance, let .f > 0, and  of finding .u ∈ R such that .|2u − f | ≤ 2. Then it is easy consider the Problem .P to see that the Tykhonov triple .T2 defined in Example 2.1 is regular with respect to  Nevertheless, the same Tykhonov triple fails to be regular with respect Problem .P. to Problem .P in Example 2.1. The previous example suggests us to underline that everywhere in this book we assume that Problem .P is given, even if its statement varies from place to place. Therefore, for simplicity, when no confusion arises, we refer to a Tykhonov triple which is “regular with respect to Problem .P” as a “regular” Tykhonov triple. Our second result in this section is the following. Theorem 2.3 Assume that .(X, d) is a complete metric space, and .T = (I, Ω, C) is a regular Tykhonov triple with respect to Problem .P. Then, .P is .T -well-posed if and only if diam.(Ω(θn )) → 0 as .n → ∞, for any sequence .{θn } ∈ C. Proof Assume that Problem .P is .T -well-posed, and note that, since .T is regular with .P, condition (2.15) holds. Then, we use Theorem 2.2 to see that diam.(Ω(θn )) → 0 as .n → ∞, for any sequence .{θn } ∈ C. Conversely, assume that diam.(Ω(θn )) → 0 as .n → ∞, for any sequence .{θn } ∈ C. Let .{un } be a .T -approximating sequence. Then, there exists a sequence .{θn } ∈ C such that .un ∈ Ω(θn ) for all .n ∈ N. Let .δ > 0 be arbitrary fixed. Since diam.(Ω(θn )) → 0, there exists a positive integer .Nδ such that diam(Ω(θn )) < δ

.

∀ n ≥ Nδ .

(2.18)

Let .n, m ∈ N be such that .n, m ≥ Nδ . Using condition (a) in Definition 2.4, we may assume that .Ω(θn ) ⊂ Ω(θm ). We have .un , um ∈ Ω(θm ), and therefore, (2.18) implies that d(un , um ) < δ.

.

This inequality holds if .Ω(θm ) ⊂ Ω(θn ), too, since in this case .un , um ∈ Ω(θn ). We conclude from here that .{un } is a Cauchy sequence in X, and since .(X, d) is assumed to be complete, there exists .u ∈ X such that un → u

.

in X.

0 This convergence combined with condition (b) in Definition 2.4 shows that .u ∈ SP 0 and, therefore, .SP = ∅. We now use Theorem 2.2 to deduce that Problem .P is .T well-posed, which concludes the proof.  

2.1 A New Well-Posedness Concept

61

It follows from Theorem 2.2 that the .T -well-posedness of a Problem .P is related to the metric properties of the approximating sets .Ω(θ ). Note that, in general, it is not easy to describe explicitly these sets. The two elementary examples we present in what follows have the merit that in each case we can clearly determine these sets. This allows us to use Theorem 2.3 in order to see that, for some data, the corresponding problems are well-posed and, for other data, they fail to be. Example 2.15 Let .X = R and let .p : R → R be the function defined by ⎧ ⎨u .p(u) = 2−u ⎩ u−2

if u < 1, if 1 ≤ u ≤ 2, if u > 2.

(2.19)

The problem .P we consider consists in finding .u ∈ R such that u + p(u) = f.

(2.20)

.

We also consider the Tykhonov triple .T = (I, Ω, C) where .

I = R+ = [0, +∞),   Ω(θ ) =  u ∈ R : | u + p( u) − f | ≤ θ   C = {θn } ⊂ I : θn → 0

∀ θ ∈ I,

and note that .T is regular with respect to Problem .P. Now, using (2.19), we find that the set of solutions of the nonlinear equation (2.20) is given by ⎧   f ⎪ ⎪ ⎨ 2 0 .S P = ⎪ [1,  2]  ⎪ ⎩ f +2 2

if f < 2, if

f = 2,

(2.21)

if f > 2.

This shows that equation (2.20) has a unique solution if .f < 2 or .f > 2 and for f = 2 it has an infinity of solutions, since in this case any element .u ∈ [1, 2] satisfies equation (2.20). We now specify the sets .Ω(θ ). Let .θ ≥ 0 and .u ∈ R. We have

.

u ∈ Ω(θ )

.

⇐⇒

f − θ ≤ u + p(u) ≤ f + θ,

(2.22)

and since .θ is destinated to converge to zero, we have to consider only the cases f < 2 − θ , .f = 2 and .f > 2 + θ . Thus, we use equivalence (2.22) and a graphic method to see that

.

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2 Tykhonov Triples and Associated Well-Posedness Concept

⎧ f −θ f +θ ⎪ ⎪ , if f < 2 − θ, ⎪ ⎪ 2 ⎪ ⎨ 2 2 − θ θ + 4 .Ω(θ ) = , if f = 2, ⎪ 2 2 ⎪ ⎪   ⎪ ⎪ ⎩ f +2−θ,f +2+θ if f > 2 + θ. 2 2 It follows from here that ⎧ ⎨0 . lim diam(Ω(θ )) = 1 ⎩ θ→0 0

if f < 2, if f = 2, if f > 2.

We now apply Theorem 2.3 to see that equation (2.20) is .T -well-posed if and only if .f = 2. This result is in agreement with our previous computations since, recall, the solution of (2.20) is given by (2.21). Example 2.16 Our second example concerns problem (2.20), again, where now p : R → R is the function defined by

.

 p(u) =

.

−2u + 3 1

if u < 1, if u ≥ 1.

(2.23)

In this case, the set of solutions to this problem is given by ⎧ ∅ ⎪ ⎪ ⎨  0 1 .S P = ⎪  ⎪ ⎩ 3 − f, f − 1

if f < 2, if f = 2,

(2.24)

if f > 2.

This shows that the inequality has a unique solution if .f = 2, two solutions if f > 2, and no solution if .f < 2. In the study of equation (2.20) with p given by (2.23), we assume that .f ≥ 2 and we use the Tykhonov triple .T = (I, Ω, C) in Example 2.15. Then, the set .Ω(θ ) can be determined, for any .θ ≥ 0, by using arguments similar to those used in the previous example. We have

.

 Ω(θ ) =

.

[1 − θ, 1 + θ ] if f = 2, [3 − f − θ, 3 − f + θ ] ∪ [f − 1 − θ, f − 1 + θ ] if f > 2 + θ.

It follows from here that .

lim diam(Ω(θ )) = 0 if

θ→0

f =2

2.1 A New Well-Posedness Concept

63

and .

lim diam(Ω(θ )) = 2f − 4 > 0 if

θ→0

f > 2.

We now apply Theorem 2.2 to see that equation (2.20) with p given by (2.23) is well-posed if and only if .f = 2. This result is in agreement with our previous computations since, recall, the solution of this equation is given by (2.24). Moreover, it is in contrast with the situation in our previous example since, there, equation (2.20) is .T -well-posed if and only if .f = 2. We end this section with the following result. Proposition 2.1 Let .{Pn } be a sequence of problems on X and, for each .n ∈ N, 0 the set of solutions of Problem .P . Assume that: denote by .SP n n (a) Problem .P is well-posed with the Tykhonov triple .T = (I, Ω, C). 0 ⊂ Ω(θ ) for each .n ∈ N. (b) There exists .{θn } ∈ C such that .SP n n 0 for all .n ∈ N converges to the Then, any sequence .{un } ⊂ X such that .un ∈ SP n unique solution of Problem .P. 0 for all .n ∈ N. Then, assumption (b) Proof Let .{un } ⊂ X be such that .un ∈ SP n implies that .{un } ⊂ X is a .T -approximating sequence. Therefore, assumption (a) and Definition 2.1(c) imply that .un → u in X, where u denotes the unique solution of Problem .P, which concludes the proof.  

Proposition 2.1 can be used to obtain the convergence of the solution of Problem P with respect to the data. An example will be presented in Remark 2.6 below.

.

2.1.5 An Elementary Example We now present an elementary example which illustrates the abstract theory above. To this end, everywhere in this subsection X represents a Hilbert space endowed with the inner product .(·, ·)X and the associated norm . · X . The Problem .P we study in this subsection can be formulated as follows. Problem .P. Find .u ∈ X such that .Au = f . Here .f ∈ X and .A : X → X is a strongly monotone Lipschitz continuous operator with constants .mA and .LA , respectively, that is, it satisfies the conditions in (1.48). The unique solvability of Problem .P follows from Proposition 1.12. Next, we consider the Tykhonov triples .T1 = (I1 , Ω1 , C1 ) and .T2 = (I2 , Ω2 , C2 ) defined as follows:

64

2 Tykhonov Triples and Associated Well-Posedness Concept .

I1 = R+ = [0, +∞),   Ω1 (θ ) =  u ∈ X : A u − f X ≤ θ ∀ θ ∈ I1 ,   C1 = {θn } ⊂ I1 : θn → 0 ,   I2 = θ = (fθ , εθ ) : fθ ∈ X, εθ ≥ 0 ,   Ω2 (θ ) =  ∀ θ = (fθ , εθ ) ∈ I2 , u ∈ X : A u − fθ X ≤ εθ   C2 = {θ n } : θ n = (fn , εn ) ∈ I2 ∀ n ∈ N, fn → f in X, εn → 0 .

Denote by u the solution of equation .Au = f . Then .u ∈ Ω1 (θ ) for each .θ ∈ I1 , which proves that .Ω1 (θ ) = ∅. Similar arguments show that .Ω2 (θ ) = ∅ for each .θ ∈ I2 . Our first result concerning the well-posedness of Problem .P is the following. Theorem 2.4 Assume (1.48) and let .f ∈ X. Then, the following statements hold: (a) (b) (c) (d)

The Tykhonov triples .T1 and .T2 are equivalent, i.e., .T1 ∼ T2 . Problem .P is well-posed with both Tykhonov triples .T1 and .T2 . .diam(Ω1 (θn )) → 0 for any sequence {θn } ∈ C1 . .diam(Ω2 (θ n )) → 0 for any sequence {θ n } ∈ C2 .

Proof (a) Let .{un } be a .T1 -approximating sequence for Problem .P. Then there exists a sequence .{θn } such that .0 ≤ θn → 0 and . Aun − f X ≤ θn for each .n ∈ N. This shows that .un ∈ Ω2 (θ n ) with .θ n = (f, θn ) ∈ I2 , for each .n ∈ N. It follows from here that .{un } is a .T2 -approximating sequence, too. Conversely, assume now that .{un } is a .T2 -approximating sequence. Then there exists a sequence .{θ n } such that .θ n = (fn , εn ) ∈ I2 , . Aun − fn X ≤ εn for all .n ∈ N, and, moreover, .fn → f in X, .εn → 0. Let .n ∈ N be fixed. We have

Aun − f X ≤ Aun − fn X + fn − f X ≤ εn + fn − f X .

.

This shows that .un ∈ Ω1 (θn ) with .0 ≤ θn = εn + fn − f X → 0. It follows from here that .{un } is a .T1 -approximating sequence. To conclude, we proved that .ST1 = ST2 , and using Definition 2.2(a), we deduce that .T1 ∼ T2 . (b) Let .{un } be a .T1 -approximating sequence for Problem .P. Then there exists a sequence .{θn } such that .0 ≤ θn → 0 and . Aun − f X ≤ θn for each .n ∈ N. Using assumption (1.48)(a) and equality .Au = f , we deduce that .

mA un − u 2X ≤ (Aun − Au, un − u)X = (Aun − f, un − u)X ≤ Aun − f X un − u X ≤ θn un − u X ,

2.1 A New Well-Posedness Concept

65

and therefore,

un − u X ≤

.

θn . mA

(2.25)

The well-posedness of Problem .P with the Tykhonov triple .T1 is a direct consequence of inequality (2.25) and the convergence .θn → 0. The well-posedness of Problem .P with the Tykhonov triple .T2 is now a direct consequence of the equivalence (2.4). (c) The well-posedness of Problem .P with the Tykhonov triple .T1 , guaranteed by part (b) of the current theorem, allows us to use Theorem 2.2. In this way we deduce that (c) holds. (d) Note that condition (2.15) is not satisfied for the Tykhonov triple .T2 since, for a given .θ = (fθ , εθ ) ∈ I2 , inequality . f − fθ X ≤ εθ is not guaranteed. For this reason, in this case we cannot invoke the use of Theorem 2.2. The proof of (d) is as follows. Let .{θ n } ∈ C2 with .θ n = (fn , εn ) for all .n ∈ N. We fix .n ∈ N, and we consider two elements .u, v ∈ Ω(θ n ). We have .

mA u − v 2X ≤ (Au − Av, u − v)X ≤ Au − Av X u − v X

≤ Au − fn X + Av − fn X u − v X ,

and using inequalities . Au − fn X ≤ εn , . Av − fn X ≤ εn , we find that

u − v X ≤

.

2εn . mA

This implies that diam(Ω2 (θ n )) =

.

sup u,v∈Ω(θ n )

u − v X ≤

2εn , mA

and since .εn → 0, we deduce that .diam(Ω2 (θ n )) → 0, which concludes the proof.   Remark 2.5 Using the properties of the operator A, it is easy to see that .un → u in X if and only if .Aun → f in X. This property combined with Theorem 2.4(a) implies that the Tykhonov triples .T1 and .T2 are equivalent to the Tykhonov triple .TP introduced in Example 2.4. We now proceed with the following elementary result which can be interpreted as a consequence of Theorem 2.4. Corollary 2.2 Assume (1.48). Then, the solution u of Problem .P depends continuously on .f ∈ X, i.e., if .fn ∈ X, .Aun = fn for all .n ∈ N, and .fn → f in X, then .un → u in X.

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2 Tykhonov Triples and Associated Well-Posedness Concept

Proof Let .n ∈ N. We use assumption (1.48)(a) and equalities .Aun = fn and .Au = f to find that

un − u X ≤

.

1

fn − f X . mA

(2.26)

It is obvious to see that the convergence .un → u in X follows from (2.26) since fn → f in X. Nevertheless, it can be recovered by Theorem 2.4 in three steps, as follows:

.

(a) Inequality (2.26) shows that .un ∈ Ω1 (θn ) with .θn = m1A fn − f X . (b) .{un } is a .T1 -approximating sequence for Problem .P since .fn → f in X, and therefore, .θn → 0. (c) Theorem 2.4(b) and Definition 2.1(c) imply that .un → u in X.   Remark 2.6 For each .n ∈ N, denote by .Pn the problem of finding .un ∈ X such that .Aun = fn . Then it is easy to see that Corollary 2.2 is a consequence of Proposition 2.1, used with the Tykhonov triple .T = T2 . Corollary 2.2 provides a sequence of elements .{un } which converges in X to the solution of Problem .P. Nevertheless, additional convergence results to this solution are known in the literature, related to the internal approximation of the space X, for instance. Details can be found in [68]. Such convergence results can be recovered in a unified way by considering the well-posedness of Problem .P with respect to a Tykhonov triple which generates appropriate approximating sequences. Our aim in what follows is to construct such a triple. To this end, we denote by .X the family of closed subspaces of X, and for a sequence .{Xn } ⊂ X , we use the following convergence: ⎧ M ⎪ Xn −→ ⎨ X as n → ∞ if for each v ∈ X . there exists a sequence {vn } ⊂ X such that ⎪ ⎩ vn ∈ Xn for each n ∈ N and vn → v in X.

(2.27)

Remark 2.7 Note that the convergence (2.27) represents the convergence of the sequence .{Xn } ⊂ X to X, in the sense of Mosco, as it follows from Definition 1.2. In addition, this convergence shows that the sequence .{Xn } represents an internal approximation of the space X. In practice the spaces .Xn are finite-dimensional spaces constructed by using the finite element method, for instance. Consider now the Tykhonov triple .T = (I, Ω, C) defined as follows:   I= . θ = (Xθ , fθ , εθ ) : Xθ ∈ X , fθ ∈ X, εθ ≥ 0 , .   Ω(θ ) = u ∈ Xθ : (Au, v)X + εθ v X ≥ (fθ , v)X ∀ v ∈ Xθ , ∀ θ = (Xθ , fθ , εθ ) ∈ I,

(2.28) (2.29)

2.1 A New Well-Posedness Concept

67

 C = {θ n } : θ n = (Xn , fn , εn ) ∈ I ∀ n ∈ N,

.

M

Xn −→ X,

fn → f

in X,

(2.30)

 εn → 0 .

Note that the solvability of the variational equation uθ ∈ Xθ ,

.

(Auθ , v)X = (fθ , v)X

∀ v ∈ Xθ

for each .Xθ ∈ X and .fθ ∈ X, guaranteed by assumption (1.48), shows that .Ω(θ ) = ∅, for each .θ ∈ I . We have the following result. Theorem 2.5 Assume (1.48) and let .f ∈ X. Then, Problem .P is well-posed with the Tykhonov triple (2.28)–(2.30). Proof Let .{un } be a .T -approximating sequence for Problem .P. Then by definition, there exists a sequence .{θ n } such that .θ n = (Xn , fn , εn ), un ∈ Xn ,

.

(Aun , v)X + εn v X ≥ (fn , v)X

∀ v ∈ Xn ,

(2.31)

for all .n ∈ N, and moreover, the convergences in (2.30) hold. We take .v = −un in (2.31) and use the strong monotonicity of the operator A, (1.48)(a), to deduce that

un X ≤

.

1

fn X + A0X X + εn mA

∀ n ∈ N.

(2.32)

We combine estimate (2.32) and the convergences in (2.30) to see that the sequence {un } is bounded in X. This implies that there exist an element . u ∈ X and a subsequence of .{un }, again denoted by .{un }, such that

.

un   u in X.

.

(2.33)

Let .v ∈ X. Using (2.27), consider a sequence .{vn } ⊂ X such that .vn ∈ Xn for each .n ∈ N, and moreover, vn → v

.

in X.

(2.34)

We use (2.31) to see that (Aun , un − vn )X ≤ (fn , un − vn )X + εn vn − un X ,

.

and keeping in mind (2.30), (2.33), and (2.34), we deduce that .

lim sup (Aun , un − vn )X ≤ (f,  u − v)X .

(2.35)

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2 Tykhonov Triples and Associated Well-Posedness Concept

On the other hand, we write (Aun , un − v)X = (Aun , un − vn )X + (Aun , vn − v)X ,

.

and since the operator A is bounded, the convergences (2.33) and (2.34) imply that .

lim sup (Aun , un − v)X = lim sup (Aun , un − vn )X .

(2.36)

We now use relations (2.35) and (2.36) to deduce that .

lim sup (Aun , un − v)X ≤ (f,  u − v)X

∀ v ∈ X.

(2.37)

Next, we take .v =  u in (2.37) to find that .lim sup (Aun , un −  u)X ≤ 0, and combining this inequality with the convergence (2.33), Proposition 1.14, and Definition 1.11(c), we find that .

lim inf (Aun , un − v)X ≥ (A u,  u − v)X

∀ v ∈ X.

(2.38)

We now use inequalities (2.37) and (2.38) to see that (A u,  u − v)X ≤ (f,  u − v)X

.

∀ v ∈ X,

which shows that .A u = f . Therefore, the unique solvability of Problem .P implies that . u = u. Now, a careful examination of the proof above reveals that any weakly convergent subsequence of the sequence .{un } converges weakly in X to u, as .n → ∞. Moreover, recall that the sequence .{un } is bounded. Therefore, Theorem 1.3 shows that the whole sequence .{un } converges weakly in X to u. Next, we use (2.38) and (2.37) with .v = u, and since . u = u, we deduce that 0 ≤ lim inf (Aun , un − u)X ≤ lim sup (Aun , un − u)X ≤ 0,

.

which implies that (Aun , un − u)X → 0.

.

(2.39)

Finally, we use condition (1.48)(a) to see that mA un − u 2X ≤ (Aun , un − u)X − (Au, un − u)X .

.

Therefore, the convergences (2.33) and (2.39) and equality . u = u show that .un → u in X, which concludes the proof.  

2.2 Well-Posedness of Split and Dual Problems

69

Some direct consequences of Theorem 2.5 are provided by the following result. Corollary 2.3 Assume (1.48) and denote by u the solution to Problem .P for .f ∈ X. The following statements hold: (a) If .Xn ∈ X and .un represents the solution of the variational equation un ∈ Xn ,

(Aun , v)X = (f, v)X

.

∀ v ∈ Xn ,

(2.40)

M

for all .n ∈ N, then .Xn −→ X implies that .un → u in X. (b) If .fn ∈ X and .un represents the solution of the equation un ∈ X,

.

Aun = fn ,

(2.41)

for all .n ∈ N, then .fn → f in X implies that .un → u in X. (c) If .εn ≥ 0 and .un is a solution of the variational inequality un ∈ X,

.

(Aun , v)X + εn v X ≥ (f, v)X

∀ v ∈ X,

(2.42)

for all .n ∈ N, then .εn → 0 implies that .un → u in X. Proof The convergences in Corollary 2.3 follow by using the well-posedness of Problem .P with the Tykhonov triple .T defined by (2.28)–(2.30), with an appropriate choice of .T -approximating sequences. The details are presented below. M

(a) Assume that .Xn −→ X. Then, it follows that the sequence .{θ n } defined by .θ n = (Xn , f, 0) belongs to .C. Moreover, using (2.29) and (2.40), we find that .un ∈ Ω(θ n ) for all .n ∈ N. This shows that .{un } is a .T -approximating sequence, and therefore, Theorem 2.5 implies that .un → u in X. (b), (c) We use the same arguments as above by choosing the sequence .{θ n } defined by .θ n = (X, fn , 0) and .θ n = (X, f, εn ) for all .n ∈ N, respectively.   We end this subsection with the following comments. First, Corollary 2.3(a) shows the convergence of the solution of the (possible discrete) scheme (2.40) to the solution of Problem .P, provided that .{Xn } represents an internal approximation of the space X. Such kind of convergence results are important in the numerical analysis of Problem .P. Next, we recall that the convergence in Corollary 2.3(b) was already obtained in Corollary 2.2, in a different way. Finally, the convergence in Corollary 2.3(c) is related to a well-posedness result obtained in [229] where approximating sequences .{un } defined by using inequalities of the form (2.42) have been considered.

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2 Tykhonov Triples and Associated Well-Posedness Concept

2.2 Well-Posedness of Split and Dual Problems In this section we deal with the well-posedness of a special type of problems, the so-called split and dual problems, respectively. Here and below, by a split problem, we understand a mathematical object .M, composed of two problems .P and .Q with solutions u and .σ , respectively, associated with an implicit equation of the form .G(u, σ ) = f . For such a problem, we use the notation .M = M(P, Q, G, f ). We use the framework developed in the previous section to provide necessary and sufficient conditions which guarantee the well-posedness of the split problem .M, expressed either in terms of metric characterization of a family of approximating sets or in terms of the well-posedness of the problems .P and .Q. Then we introduce the concept of dual problems which represent a particular case of split problems and for which we present additional results. Even if the theory we present in this section can be constructed in the case of metric spaces, for simplicity, everywhere in this section we shall work in the framework of normed spaces.

2.2.1 Split Problems Everywhere in this subsection, X, Y , and Z are assumed to be normed spaces. The elements of X will be denoted by .u, v, . . . and the elements of Y by .σ, τ, . . . . The norms on these spaces will be denoted by . · X , . · Y , and . · Z , respectively. We also use the notation .X for the product of the spaces X and Y , i.e., .X = X × Y . The elements of .X will be denoted by .x = (u, σ ), .y = (v, τ ), .. . . We endow .X with the norm . · X given by

x X = u X + σ Y

.

∀ x = (u, σ ) ∈ X .

(2.43)

Moreover, for a sequence .{θn } ⊂ R such that .θn ≥ 0 for all .n ∈ N and .θn → 0 as n → ∞, we use the short-hand notation .0 ≤ θn → 0. Consider now two abstract problems .P and .Q, defined on the spaces X and Y , 0 the set of solutions to Problem .P, and respectively. As usual, we denote by .SP 0 we use .SQ for the set of solutions to Problem .Q. We associate with Problem .P a Tykhonov triple .TP = (IP , ΩP , CP ) such that

.

0 SP ⊂ ΩP (θ )

.

∀ θ ∈ IP .

(2.44)

Similarly, we associate with Problem .Q a Tykhonov triple .TQ = (IQ , ΩQ , CQ ) such that 0 SQ ⊂ ΩQ (ω)

.

∀ ω ∈ IQ .

(2.45)

2.2 Well-Posedness of Split and Dual Problems

71

Note that below in this subsection we assume that (2.44) and (2.45) hold, even if we do not mention it explicitly. Finally, we assume that .G : X × Y → Z is a given operator and f is a given element of Z. With these data, we construct a new problem which can be stated as follows. 0 , .σ ∈ S 0 , and Problem .M Find an element .x = (u, σ ) ∈ X such that .u ∈ SP Q .G(u, σ ) = f .

We say in what follows that .M represents a split problem. Note that this new mathematical object is constructed by using the problems .P and .Q, the operator G and the element f . For this reason, we write M = M(P, Q, G, f ).

.

(2.46)

Nevertheless, for simplicity, when no confusion arises, for a split problem .M = M(P, Q, G, f ), we shall use the short-hand notation .M. 0 the set of solutions of problem .M, i.e., We denote in what follows by .SM   0 0 0 SM = x = (u, σ ) : u ∈ SP , σ ∈ SQ and G(u, σ ) = f .

.

(2.47)

Note that the concept of split problem introduced above is quite general and includes as particular cases a number of split problems already studied in the literature, as explained on page 383. Following the references mentioned there, if both .P and .Q represent variational inequalities, we say that .M represents a split variational inequality, and if .P and .Q are hemivariational inequalities, we say that .M represents a split hemivariational inequality. Nevertheless, we stress that the functional framework described above is flexible and allows us to study a large number of split problems, in which .P and .Q could have different features. For instance, .P could be a variational inequality and .Q an inclusion, .P could be a fixed point problem and .Q an optimization problem, and so on. Also, we underline that, in contrast with the references in the literature (where an explicit equation of the form .σ = T u was assumed), the concept of split problem we consider here is more general since it is based on an implicit equation of the form .G(u, σ ) = f . Finally, note that, in general, a split problem .M = M(P, Q, G, f ) is overdetermined and has no solution. Indeed, assume that Problem .P has .u0 as the unique solution and Problem .Q has .σ0 as the unique solution. Then, if the element .f ∈ Z is chosen such that .f = G(u0 , σ0 ), it follows that the couple .x0 = (u0 , σ0 ) is the unique solution to the split Problem .M. Nevertheless, if .f = G(u0 , σ0 ), then Problem .M has no solution. For this reason, it is important to identify classes of split problems .M = M(P, Q, G, f ) which have solutions, without having additional information on these solutions. And this is what we shall do in Sect. 4.3.1 where we present an example of well-posed split variational inequality.

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2 Tykhonov Triples and Associated Well-Posedness Concept

Next, we consider the triple .TM = (IM , ΩM , CM ) defined as follows: .

IM = IP × IQ × R+ ,   ΩM (θ ) = x = (u, σ ) : u ∈ ΩP (θ ), σ ∈ ΩQ (ω), G(u, σ ) − f Z ≤ ε 

∀ θ = (θ, ω, ε) ∈ IM ,



(2.48)

CM = {θ n } : θ n = (θn , ωn , εn ), θn ∈ CP , ωn ∈ CQ , 0 ≤ εn → 0 . Note that .ΩM (θ ) ⊂ ΩP (θ ) × ΩQ (ω). Moreover, (2.47), (2.44), (2.45), and (2.48) imply that 0 SM ⊂ ΩM (θ )

.

∀ θ ∈ IM .

(2.49)

Assume that ΩM (θ ) = ∅

.

∀ θ ∈ IM ,

(2.50)

which implies that .TM is a Tykhonov triple. Then, using inclusion (2.49), Theorem 2.2, and notation diam.(ΩM (θ n )) for the diameter of the set .ΩM (θ n ) in the space .X = X × Y , we deduce the following result. Theorem 2.6 Assume (2.50). Then, the split problem .M = M(P, Q, G, f ) is .TM 0 is nonempty and well-posed if and only if its solution set .SM .diam(ΩM (θ n )) → 0 as n → ∞, for each sequence .{θ n } ∈ CM . Theorem 2.6 provides a necessary and sufficient condition for the .TM -wellposedness of Problem .M. Nevertheless, checking this condition requires to prove 0 = ∅, i.e., to prove the solvability of Problem .M. In what follows, we that .SM introduce a second characterization for the well-posedness of Problem .M in which 0 = ∅ is removed. To this end, as in the statement of Theorem 2.3, we condition .SM need to work with the so-called regular Tykhonov triples that we introduce in the following definition. Definition 2.5 Assume (2.50). The Tykhonov triple .TM is said to be regular (with respect to the split problem .M) if the following hold: (a) For all .θ 1 = (θ1 , ω1 , ε1 ), .θ 2 = (θ2 , ω2 , ε2 ) ∈ IM , we have either .ΩM (θ 1 ) ⊂ ΩM (θ 2 ) or .ΩM (θ 2 ) ⊂ ΩM (θ 1 ). (b) If .{un } is a .TP -approximating sequence and there exists .u ∈ X such that .un → u 0. in X as .n → ∞, then .u ∈ SP (c) If .{σn } is a .TQ -approximating sequence and there exists .σ ∈ Y such that .σn → 0. σ in Y as .n → ∞, then .σ ∈ SQ Our second result in this section is the following.

2.2 Well-Posedness of Split and Dual Problems

73

Theorem 2.7 Assume (2.50). Moreover, assume that X and Y are Banach spaces, the Tykhonov triple .TM is regular (with respect to the split problem .M), and .G : X × Y → Z is a continuous operator. Then, the split problem .M is .TM -well-posed if and only if diam.(ΩM (θ n )) → 0 as .n → ∞, for each sequence .{θ n } ∈ CM . Proof Assume that Problem .M is .TM -well-posed. We use Theorem 2.6 to see that diam.(ΩM (θ n )) → 0 as .n → ∞, for each sequence .{θ n } ∈ CM . Conversely, assume that diam.(ΩM (θ n )) → 0 as .n → ∞, for each sequence .{θ n } ∈ CM . Let .{xn } ⊂ X be a .TM -approximating sequence for .M, with .xn = (un , σn ). Then there exists a sequence .{θ n } ⊂ CM with .θ n = (θn , ωn , εn ) such that .xn ∈ Ω(θ n ), i.e., un ∈ ΩP (θn ),

.

σn ∈ ΩQ (ωn ),

G(un , σn ) − f Z ≤ εn

∀ n ∈ N.

(2.51)

Let .δ > 0 be arbitrary fixed. Since diam.(ΩM (θ n )) → 0, there exists a positive integer .Nδ such that diam(ΩM (θ n )) < δ

.

∀ n ≥ Nδ .

(2.52)

Let .n, m ∈ N with .n, m ≥ Nδ . Then, using condition (a) in Definition 2.5 and (2.51), we have that either .ΩM (θ m ) ⊂ ΩM (θ n ) or .ΩM (θ n ) ⊂ ΩM (θ m ). In the first case, we have .xm ∈ ΩM (θ n ). Thus, since .xn , xm ∈ ΩM (θ n ), inequality (2.52) yields

xm − xn X < δ.

.

This inequality holds even if .ΩM (θ n ) ⊂ ΩM (θ m ) since, in this case, .xn , xm ∈ ΩM (θ m ). We conclude from here that .{xn } is a Cauchy sequence in .X which is a Banach space since, recall, X and Y are assumed to be Banach spaces. It follows from here that there exists .x = (u, σ ) ∈ X such that .xn → x in .X or, equivalently, un → u in X,

.

σn → σ

in Y.

(2.53)

Note also that (2.51) shows that .{un } is a .TP -approximating sequence and {σn } is a .TQ -approximating sequence. Then, the convergences (2.53) combined 0 and .σ ∈ with conditions (b) and (c) in Definition 2.5 show that .u ∈ SP 0 . Moreover, (2.51), the convergences (2.53), .ε → 0, and the continuity of SQ n the operator G imply that .G(u, σ ) = f . It follows now from (2.47) that the element 0 0 .x = (u, σ ) belongs to .S M , and therefore, .SM = ∅. We are now in a position to use Theorem 2.6 to deduce that Problem .M is well-posed, which concludes the proof.   .

We consider in what follows the following conditions:

74

2 Tykhonov Triples and Associated Well-Posedness Concept

⎧ ⎪ For any sequence {θn } ∈ CP , there exist two sequences ⎪ ⎪ ⎪ ⎪ ⎨ {ωn }, {εn } such that, denoting θ n = (θn , ωn , εn ), we have: . (a) {θ n } ∈ CM . ⎪ ⎪ ⎪ (b) For any sequence {un } with un ∈ ΩP (θn ), there exists a ⎪ ⎪ ⎩ sequence {σn } such that (un , σn ) ∈ ΩM (θ n ), for all n ∈ N. ⎧ ⎪ ⎪ For any sequence {ωn } ∈ CQ there exist two sequences ⎪ ⎪ ⎪ ⎨ {θn }, {εn } such that, denoting θ n = (θn , ωn , εn ), we have: . (a) {θ n } ∈ CM . ⎪ ⎪ ⎪ (b) For any sequence {σn } with σn ∈ ΩQ (ωn ), there exists a ⎪ ⎪ ⎩ sequence {un } such that (un , σn ) ∈ ΩM (θ n ), for all n ∈ N.

(2.54)

(2.55)

The following result shows the link between the .TM -well-posedness of the split problem .M, on the one hand, and the .TP -, .TQ -well-posedness of its components .P and .Q, respectively, on the other hand. Theorem 2.8 Let X and Y be Banach spaces and let .P and .Q be two problems defined on X and Y , respectively. (1) Assume that Problems .P and .Q are .TP - and .TQ -well-posed, respectively, denote by u and .σ their solutions, and assume that .G(u, σ ) = f . Then (2.50) holds, and moreover, the split problem .M is .TM -well-posed. (2) Assume that (2.50) holds, the split problem .M is .TM -well-posed, and moreover, (2.54) and (2.55) hold. Then, Problems .P and .Q are .TP - and .TQ -well-posed, respectively. In addition, their solutions, denoted by u and .σ , satisfy the equation .G(u, σ ) = f . Proof 0 , .σ ∈ S 0 , and .G(u, σ ) = f , it follows (1) Let .x = (u, σ ). Then, since .u ∈ SP Q 0 , and therefore, from (2.47) that .x ∈ SM 0 SM = ∅.

(2.56)

.

We now use (2.49) and (2.56) to see that (2.50) holds. Let .{θ n } ∈ CM with θ n = (θn , ωn , εn ) for any .n ∈ N. Then, the definition of the set .CM implies that .{θn } ∈ CP , .{ωn } ∈ CQ , and since Problems .P and .Q are assumed to be well-posed with the Tykhonov triples .TP and .QQ , respectively, Theorem 2.2 shows that .

diam(ΩP (θn )) → 0

.

and

diam(ΩQ (ωn )) → 0

as

n → ∞.

(2.57)

Next, using the inclusion .ΩM (θ n ) ⊂ ΩP (θn ) × ΩQ (ωn ) and the definition (2.43) of the norm in the space .X , it is easy to see that for each .n ∈ N we have diam(ΩM (θ n )) ≤ diam(ΩP (θn )) + diam(ΩQ (ωn )).

.

2.2 Well-Posedness of Split and Dual Problems

75

Therefore, (2.57) implies that diam(ΩM (θ n )) → 0 as

.

n → ∞.

(2.58)

Relations (2.56) and (2.58) allow us to use Theorem 2.2 in order to deduce that Problem .M is .TM -well-posed. (2) Assume now that (2.50) holds, the split problem .M is .TM -well-posed, and conditions (2.54) and (2.55) are satisfied. Then, it follows from Theorem 2.6 that 0 .S M = ∅ and (2.58) holds, for any sequence .{θ n } ∈ CM . Therefore, there exists an element .x = (u, σ ) such that 0 u ∈ SP ,

.

0 σ ∈ SQ ,

G(u, σ ) = f.

(2.59)

This shows that .

0 SP = ∅

(2.60)

0 SQ = ∅.

(2.61)

and .

Let .{θn } ∈ CP , and assume that diam.(ΩP (θn )) → 0 as .n → ∞. Then, there exists a subsequence of the sequence .{θn }, still denoted by .{θn }, with the following property: there exist .δ0 > 0 and two sequences .{un }, .{vn } ⊂ X such that .un , vn ∈ ΩP (θn ) for any .n ∈ N and, moreover,

un − vn X ≥ δ0

.

∀ n ∈ N.

(2.62)

We now use assumption (2.54) to deduce that there exist two sequences .{ωn } and {εn } such that, denoting .θ n = (θn , ωn , εn ), we have that .{θ n } ∈ CM . Moreover, there exist two sequences .{σn } and .{τn } ⊂ Y such that, denoting .xn = (un , σn ) and .yn = (vn , τn ), we have the inclusions .xn , yn ∈ ΩM (θ n ), for each .n ∈ N. Then, (2.43) and (2.62) imply that

.

xn − yn X ≥ δ0

.

∀n ∈ N

and, therefore, diam(ΩM (θ n )) ≥ δ0

.

∀ n ∈ N.

This is in contradiction with the convergence (2.58) which holds from the .TM -wellposedness of the split problem .M. We conclude from here that diam(ΩP (θn )) → 0 as

.

n → ∞.

(2.63)

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2 Tykhonov Triples and Associated Well-Posedness Concept

Using similar arguments and assumption (2.55), we deduce that for any sequence {ωn } ∈ CQ we have the convergence

.

diam(ΩQ (ωn )) → 0 as

.

n → ∞.

(2.64)

We now use (2.60), (2.63), and Theorem 2.2 to see that Problem .P is .TP -wellposed. Next, we use (2.61) and (2.64) to see that Problem .Q is .TQ -well-posed. We now recall equality .G(u, σ ) = f in (2.59) to conclude the proof.  

2.2.2 Dual Problems We now introduce a concept of dual problems in normed spaces. Then we consider the case of Banach spaces for which we state and prove an equivalence result concerning the well-posedness of dual problems with appropriate Tykhonov triples. We check the applicability of this result in the study of two relevant examples in Hilbert spaces: a variational inequality (for which the dual problem is a minimization problem) and a nonlinear equation (for which the dual problem is a fixed point problem). Consider two abstract problems .P and .Q formulated in the normed spaces .(X, ·

X ) and .(Y, · Y ), respectively. We start with the following definition. Definition 2.6 Problems .P and .Q are said to be dual of each other if there exists an operator .D : X → Y such that: (a) D is bijective. (b) Both .D : X → Y and its inverse .D −1 : Y → X are continuous. (c) .u ∈ X is the solution of Problem .P if and only if .σ := Du ∈ Y is the solution of Problem .Q. If conditions (a)–(c) hold, we say that problems .P and .Q are dual problems with operator D or, equivalently, problems .Q and .P are dual problems with operator −1 . Moreover, Problem .Q is a dual problem of Problem .P (with operator D) and, .D conversely, Problem .P is a dual problem of Problem .Q (with operator .D −1 ). We assume in what follows that .P and .Q are dual problems (with operator D) and consider the function .G : X × Y → Y defined by G(u, σ ) = σ − Du

.

∀ u ∈ X, σ ∈ Y.

Moreover, let .TP = (I, ΩP , C) and .TQ = (I, ΩQ , C) be two Tykhonov triples associated with these problems such that 0 SP ⊂ ΩP (θ ),

.

0 SQ ⊂ ΩQ (θ )

∀ θ ∈ I.

(2.65)

2.2 Well-Posedness of Split and Dual Problems

77

Then, following the ingredients on page 72, we construct in a canonical way a split problem .M = M(P, Q, G, 0Y ) and an associated triple .TM = (IM , ΩM , CM ) as follows:   . IM = θ = (θ, θ, ε) : θ ∈ I, ε ∈ R+ , . (2.66) ΩM (θ ) = . (2.67)   σ ∈ ΩQ (θ ),  σ − D u Y ≤ ε ,  x = ( u,  σ) ∈ X :  u ∈ ΩP (θ ),  ∀ θ = (θ, θ, ε) ∈ IM ,   CM = {θ n } : θ n = (θn , θn , εn ), θn ∈ C, 0 ≤ εn → 0 .

(2.68)

We conclude that any couple of dual problems generates a canonic split problem, constructed above. Moreover, Theorem 2.8(1) guarantees that if X and Y are Banach spaces and problems .P and .Q are well-posed with .TP and .TQ , respectively, then .TM is a Tykhonov triple (since condition (2.50) holds) and the split problem .M is .TM well-posed. To obtain additional properties we introduce the following definition. Definition 2.7 Assume that .P and .Q are dual problems (with operator D) and TP = (I, ΩP , C) and .TQ = (I, ΩQ , C) are Tykhonov triples associated with these problems. Then .TP and .TQ are said to be dual Tykhonov triples (with operator D) if the multivalued functions .ΩP : I → 2X and .ΩQ : I → 2Y are such that for any .θ ∈ I the equivalence below holds: .

 u ∈ ΩP (θ )

.

⇐⇒

 σ := D u ∈ ΩQ (θ ).

(2.69)

We complete this definition with the following remark. Remark 2.8 Assume that the Tykhonov triples .TP and .TQ are dual triples. Then the set (2.67) satisfies condition (2.50) and, therefore, the triple .TM = (IM , ΩM , CM ) given by (2.66)–(2.68) is a Tykhonov triple. The interest in Definition 2.7 is given by the following equivalence result. Theorem 2.9 Let X and Y be Banach spaces, .P and .Q be dual problems (with operator D), and .TP = (I, ΩP , C) and .TQ = (I, ΩQ , C) be dual Tykhonov triples (with operator D). Then, the following statements are equivalent: (1) Problem .P is .TP -well-posed. (2) Problem .Q is .TQ -well-posed. (3) Problem .M is .TM -well-posed. Proof (1) .⇐⇒ (2). Assume that Problem .P is .TP -well-posed. This implies that Problem .P has a unique solution .u ∈ X. Moreover, it follows from properties (a) and (c) in Definition 2.6 that .σ = Du is the unique solution of Problem .Q.

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2 Tykhonov Triples and Associated Well-Posedness Concept

Let .{σn } ⊂ Y be a .TQ -approximating sequence for Problem .Q. Then there exists a sequence .{θn } ∈ C such that, for each .n ∈ N, .σn ∈ ΩQ (θn ). We now use (2.69) to see that .D −1 σn ∈ ΩP (θn ), which means that .{D −1 σn } ⊂ X is a .TP -approximating sequence for Problem .P. Therefore, using the .TP -well-posedness of Problem .P, we deduce that .D −1 σn → u in X, and by the continuity of D, we find that .σn → Du = σ in Y . We conclude from above that Problem .Q is .TQ -well-posed. Similar arguments show that if Problem .Q is .TQ -well-posed, then Problem .P is .TP -well-posed, which concludes the proof of this equivalence. (1), (2) .⇐⇒ (3) We use Theorem 2.8 with .Z = Y , .f = 0Y , and .G(u, σ ) = σ − Du. Assume that the statements (1) and (2) hold and denote by u and .σ the solutions of problems .P and .Q, respectively. Then, since problems .P and .Q are dual problems (with operator D), we deduce that .σ = Du. Hence, Theorem 2.8(1) implies that the split problem .M is .TM -well-posed. Conversely, assume that (3) holds, i.e., the split problem .M is .TM -well-posed. Note that by the construction of the Tykhonov triple .TM and the duality of the Tykhonov triples .TP and .TQ , it follows that conditions (2.54) and (2.55) are satisfied. For instance, if .{θn } ∈ CP , we can take .ωn = θn , .εn = 0, and denoting .θ n = (θn , θn , εn ), we deduce that .{θ n } ∈ CM , which shows that condition (2.54)(a) holds. Moreover, if .{un } is a sequence such that .un ∈ ΩP (θn ), taking .σn = Dun and using (2.69) and (2.67), it follows that .(un , σn ) ∈ ΩM (θ n ) for each .n ∈ N. This shows that condition (2.54) (b) holds, too. Using now Remark 2.8, it follows that condition (2.50) is satisfied. Thus, we are in a position to use Theorem 2.8 (2) to see that Problems .P and .Q are .TP - and .TQ -well-posed, respectively. This implies that   the statements (1) and (2) hold, which concludes the proof. Remark 2.9 Theorem 2.9 shows that, in the framework above, if either .P or .Q is well-posed (with the Tykhonov triples .TP and .TQ , respectively), then the duality of .TP and .TQ represents a sufficient condition which implies the well-posedness of the split problem .M (with the Tykhonov triple .TM ). This condition is not necessary, as we shall see in Remark 4.4 on page 171. In what follows, we use notations .TP and .TQ for the Tykhonov triples defined in Example 2.4 on page 50, associated with Problems .P and .Q, respectively. We also use the equivalence relation “.∼” in Definition 2.2(a) and notation (2.2) to complete our study of dual problems with the following result. Proposition 2.2 Let .P and .Q be dual problems (with operator D) and let .TP = (I, ΩP , C) and .TQ = (I, ΩQ , C) be dual Tykhonov triples (with operator D). Moreover, assume that one of the statements (1) or (2) in Theorem 2.9 holds. Then .TP ∼ TP if and only if .TQ ∼ TQ . Proof Recall that the statements (1) and (2) in Theorem 2.9 are equivalent. Therefore, we assume in what follows that both (1) and (2) hold and we denote by u and .σ the solutions of Problems .P and .Q, respectively. Moreover, we assume that .TP ∼ TP or, equivalently, S TP = S TP .

.

(2.70)

2.2 Well-Posedness of Split and Dual Problems

79

Let .{σn } ∈ STQ . Then .σn → σ in Y , which implies that .un := D −1 σn → D −1 σ in X. This shows that .{un } ∈ STP , and using equality (2.70), we deduce that .{un } ∈ STP , i.e., .{un } is a .TP -approximating sequence. Now, using the duality of the Tykhonov triples .TP and .TQ , we find that .{σn } ∈ STQ , i.e., .{σn } is a .TQ approximating sequence. Therefore, we proved that .STQ ⊂ STQ . On the other hand, the .TQ -well-posedness of Problem .Q, guaranteed by assumption 2) in Theorem 2.9, shows that .STQ ⊂ STQ . It follows from above that S TQ = S TQ ,

.

which means that .TQ ∼ TQ . To conclude, we proved that if .TP ∼ TP , then .TQ ∼ TQ . The converse   implication follows from similar arguments. In Sect. 6.3, we shall use Theorem 2.9 in the study of a history-dependent variational inequality with time-dependent constraints, for which the dual problem is in a form of a history-dependent inclusion. Here, to end this section, we restrict ourselves to present two representative elementary examples of problems which are dual of each other. Everywhere in the rest of this section, X will be a real Hilbert space. We use .(·, ·)X and . · X for the inner product and the associated norm of space X. A variational inequality Let .K ⊂ X, .A : X → X, and .ϕ : X → R, and assume that conditions (1.72)–(1.74) hold. Let .f ∈ X and .J : X → R be the function defined by J (v) =

.

1 (Av, v)X + ϕ(v) − (f, v)X 2

∀ v ∈ X.

(2.71)

In this framework, we consider the following two problems. Problem .P Find .u ∈ K such that (Au, v − u)X + ϕ(v) − ϕ(u) ≥ (f, v − u)X

.

∀ v ∈ K.

Problem .Q Find .σ ∈ K such that J (σ ) ≤ J (v)

.

∀ v ∈ K.

In the study of these problems, we consider the Tykhonov triples .TP = (I, ΩP , C) and .TQ = (I, ΩQ , C) defined as follows: .

I = R+ = [0, +∞),  ΩP (θ ) =  u ∈ K : (A u, v −  u)X + ϕ(v) − ϕ( u) + θ v −  u X  ≥ (f, v −  u)X ∀ v ∈ K ∀ θ ∈ I,

80

2 Tykhonov Triples and Associated Well-Posedness Concept

  ΩQ (θ ) =  σ ∈ K : J ( σ ) ≤ J (v) + θ v −  σ X ∀ v ∈ K ∀ θ ∈ I,   C = {θn } ⊂ I : θn → 0 . We have the following result. Theorem 2.10 Under the previous assumptions, Problems .P and .Q are dual problems (with the identity operator .IX ) and the Tykhonov triples .TP and .TQ are dual triples (with the identity operator .IX ). Moreover, Problem .P is .TP -well-posed and Problem .Q is .TQ -well-posed. Proof We start by proving that ΩP (θ ) = ΩQ (θ )

.

∀ θ ∈ I,

(2.72)

and to this end, in what follows, we fix an element .θ ∈ I . Assume that .u ∈ ΩP (θ ), and let v be an arbitrary element on K. Then, (Au, v − u)X + ϕ(v) − ϕ(u) + θ v − u X ≥ (f, v − u)X ,

.

which implies that ϕ(v) − ϕ(u) − (f, v − u)X ≥ −(Au, v − u)X − θ v − u X .

.

(2.73)

On the other hand, definition (2.71) shows that J (v) − J (u) =

.

1 1 (Av, v)X − (Au, u)X + ϕ(v) − ϕ(u) − (f, v − u)X , 2 2

and using (2.73), we find that J (v) − J (u) ≥

.

1 1 (Av, v)X − (Au, u)X − (Au, v − u)X − θ v − u X . 2 2

Thus, by the properties of the operator A, we deduce that J (v) + θ v − u X ≥ J (u) +

.

mA 1 (A(v − u), v − u)X ≥ J (u) +

v − u 2X . 2 2

We conclude that J (u) ≤ J (v) + θ v − u X ,

.

which shows that .u ∈ ΩQ (θ ).

2.2 Well-Posedness of Split and Dual Problems

81

Conversely, assume now that .u ∈ ΩQ (θ ). Let .v ∈ K and .t ∈ (0, 1]. Since u + t (v − u) ∈ K, we have that

.

J (u) ≤ J (u + t (v − u)) + tθ v − u X .

.

We now use definition (2.71) to see that .

t2 (A(u − v), u − v)X 2 +ϕ(u + t (v − u)) − t (f, v − u)X + tθ v − u X ,

ϕ(u) ≤ t (Au, v − u)X +

and the use of the convexity of .ϕ together with assumption .t > 0 yields .

t (f, v − u)X ≤ (Au, v − u)X + (A(u − v), u − v)X 2 +ϕ(v) − ϕ(u) + θ v − u X .

We now pass to the limit as .t → 0 to find that (f, v − u)X ≤ (Au, v − u)X + ϕ(v) − ϕ(u) + θ v − u X ,

.

which implies that .u ∈ ΩP (θ ). We now use equality (2.72) with .θ = 0 to see that u is a solution of Problem .P if and only if u is a solution of Problem .Q. Therefore, according to Definition 2.6, we deduce that problems .P and .Q are duals problems with the identity operator .D = IX . Moreover, equality (2.72) and Definition 2.7 show that the Tykhonov triples .TP and .TQ are dual triples, too. In addition, Theorem 1.18 on page 40 guarantees that Problem .P is .TP -wellposed. Therefore, it follows from Theorem 2.9 that Problem .Q is .TQ -well-posed, too.   A nonlinear equation Let .A : X → X be a nonlinear operator and .f ∈ X. Then, the problem we consider below, already considered on page 63, is stated as follows. Problem .P Find .u ∈ X such that .Au = f . In the study of this problem, we assume that A is a strongly monotone Lipschitz continuous operator, i.e., there exist .mA , .LA > 0 such that .

(Av1 − Av2 , v1 − v2 )X ≥ mA v1 − v2 2X

Av1 − Av2 X ≤ LA v1 − v2 X

∀ v1 , v2 ∈ X.

∀ v1 , v2 ∈ X.

 : X → 2X given Then, using Remark 1.2 on page 22, we find that the operator .A by

82

2 Tykhonov Triples and Associated Well-Posedness Concept

 = Au − f Au

.

∀u ∈ X

(2.74)

is maximal monotone. Therefore, using (1.36), for every .λ > 0, we are in a position to define the resolvent operator .Jλ : X → X by equality  = σ. Jλ σ = u ⇐⇒ u + λAu

.

(2.75)

Note that this equivalence holds for all .σ, u ∈ X. We now combine relations (2.74) and (2.75) to see that Jλ σ = u ⇐⇒ u + λ(Au − f ) = σ

.

(2.76)

for all .σ, u ∈ X. Next, we consider the following fixed point problem. Problem .Q Given .λ > 0, find .σ ∈ X such that .Jλ σ = σ . Let .D : X → X be the identity operator, i.e., .D = IX . Then, using the equivalence (2.76), it is easy to see that D satisfies conditions (a)–(c) in Definition 2.6 with .Y = X. It follows from here that problems .P and .Q are dual problems (with operator .D = IX ). In the study of these problems, we consider the Tykhonov triples .TP = (I, ΩP , C) TQ = (I, ΩQ , C) defined as follows: .

I = R+ = [0, +∞),   ΩP (θ ) =  u ∈ X : A u − f X ≤ θ ∀ θ ∈ I,   ΩQ (θ ) =  σ ∈ X : Jλ σ − σ X ≤ λθ ∀ θ ∈ I,   C = {θn } ⊂ I : θn → 0 .

Note that, in this case, condition (2.65) is obviously satisfied. Moreover, it follows from Theorem 2.4 that Problem .P is .TP -well-posed. Nevertheless, the .TQ well-posedness of Problem .Q cannot be obtained by using the .TP -well-posedness of Problem .P combined with Theorem 2.9 since the Tykhonov triples .TP and .TQ are not dual triples (with operator .D = IX ). A counter-example that justifies this statement follows. Example 2.17 Take .A = IX , .f ∈ X, .u ∈ X, and .λ, θ > 0. Then, using (2.76), it is easy to see that Jλ u =

.

which implies that

λf + u , 1+λ

2.3 Extended Classical Well-Posedness Concepts

Jλ u − u X =

.

λ

u − f X . 1+λ

83

(2.77)

Assume now that .u = f +g, where g is an element of X such that . g X = θ (λ+1). Then (2.77) shows that . Jλ u − u X ≤ λθ , and therefore .u ∈ ΩQ (θ ). Nevertheless, . Au − f X = u − f X = θ (λ + 1) > θ , which implies that .u ∈ ΩP (θ ). This shows that condition (2.69) is not satisfied and, therefore, the corresponding Tykhonov triples .TP and .TQ are not dual triples (with operator .D = IX ). Note that in Sects. 4.3.2 and 6.3.1 we shall give additional examples of problems which are dual of each other. There, in contrast with the examples in this subsection, the operator D will not be the identity operator.

2.3 Extended Classical Well-Posedness Concepts The concept of .T -well-posedness introduced in Sect. 2.1 represents an extension of the Tykhonov concept for minimization problems, fixed point problems, inclusions, and variational inequalities, as explained in Sect. 2.1.3. It can be used to recover the Levitin–Polyak and Hadamard well-posedness concepts, as we proved on pages 54, 55, and 57 in the study of some particular problems. In this section, we study the well-posedness of two abstract problems with specific Tykhonov triples and show that the corresponding well-posedness concepts represent extensions of Levitin– Polyak and Hadamard concepts, respectively. The .TKLP —well-posedness concept introduced below in this section concerns an abstract problem .PK , associated with a set of constraints K. The Hadamard concept of well-posedness that we consider concerns a family of abstract problems .(Pθ ), depending on a parameter .θ .

2.3.1 Extended Levitin–Polyak Well-Posedness Concept Below in this subsection, we assume that X is a normed space, and we use the M

symbol “.−→” for the Mosco convergence of sets, see Definition 1.2. We consider a given nonempty set .K ⊂ X and denote by .d(u, K) the distance of the element .u ∈ X to K. Let I and .C be the sets given by I = R+ = [0, +∞),

.

  C = {θn } ⊂ I : θn → 0 .

(2.78)

0 : I → 2X is a multivalued mapping, and we In addition, we assume that .ΩK LP : I → 2X , consider two additional multivalued mappings .ΩK : I → 2X and .ΩK defined by equalities

84

2 Tykhonov Triples and Associated Well-Posedness Concept

.

  0 0 ΩK (θ ) = u ∈ K : u ∈ ΩK (θ ) = K ∩ ΩK (θ ), .   LP 0 ΩK (θ ) = u ∈ X : d(u, K) ≤ θ, u ∈ ΩK (θ ) ,

(2.79) (2.80)

 ∅, for any .θ ∈ I , which, obviously, implies for all .θ ∈ I . Assume that .ΩK (θ ) = LP (θ ) = that .ΩK  ∅, for any .θ ∈ I . Denote TK = (I, ΩK , C),

.

LP TKLP = (I, ΩK , C).

Then it is easy to see that .TK and .TKLP are Tykhonov triples in the sense of Definition 2.1(a). We refer to .TKLP as a Tykhonov–Levitin–Polyak triple. Moreover, we have the following result. Proposition 2.3 The Tykhonov triple .TK is smaller than the Tykhonov triple .TKLP , that is, TK ≤ TKLP .

.

(2.81)

Proof Let .{un } ⊂ X be a .TK -approximating sequence. Then, there exists a 0 (θ ) for each .n ∈ N. This sequence .{θn } ∈ C such that .un ∈ K and .un ∈ ΩK n implies that .d(un , K) = 0, and therefore, .{un } is a .TKLP -approximating sequence, too. We now use Definition 2.2(b) to conclude the proof.   Remark 2.10 Note that, in general, inequality (2.81) is strict. Indeed, let .X = R2 and .K = R × {0} and let   0 ΩK (θ ) = u ∈ X : |J (u)| ≤ θ ,

.

where .J (u) = x 2 − (x 4 + x)y 2 for all .u = (x, y) ∈ R2 . Let .{un } ⊂ X be the sequence given by .un = (n, n1 ) for each .n ∈ N. Then it is easy to see that 1 1 1 LP .d(un , K) = n , .J (un ) = − n , and therefore, .un ∈ ΩK (θn ) with .θn = n . It follows from here that .{un } is a .TKLP -approximating sequence. Nevertheless, .{un } is not a .TK -approximating sequence since .un ∈ / K, for each .n ∈ N. K , CK ) defined as follows: We now consider the Tykhonov triple .TK = (I, Ω .

  0  ∩ ΩK  θ ) ∈ 2X × I : K (θ ) = ∅ , I = θ = (K,   0 0  : u ∈ ΩK  ∩ ΩK K : I → 2X , Ω K (θ) = u ∈ K Ω (θ ) = K (θ ) 

 θ ) ∈ I, ∀ θ = (K,

 M CK = {θ n } ⊂ I : θ n = (Kn , θn ) ∀ n ∈ N, Kn −→ K, {θn } ∈ C .

(2.82)

2.3 Extended Classical Well-Posedness Concepts

85

Note that (2.79) and assumption .ΩK (θ ) = ∅ imply that .(K, θ ) ∈ I for any .θ ∈ I and, therefore, .I =  ∅. Moreover, by the definition of the set .I, it follows that K (θ) = .Ω  ∅, for any .θ ∈ I. We have the following results. Proposition 2.4 The Tykhonov triple .TK is smaller than the Tykhonov triple .TK , that is, .TK ≤ TK . Proof Let .{un } ⊂ X be a .TK -approximating sequence. Then there exists a sequence 0 (θ ) for all .n ∈ N. This shows that the {θn } ∈ C such that .un ∈ K and .un ∈ ΩK n sequence .{θ n } with .θ n = (K, θn ) belongs to .CK , and moreover, for any .n ∈ N, we K (θ n ). Therefore .{un } ⊂ X is a .TK -approximating sequence, which have .un ∈ Ω shows that .TK ≤ TK .  

.

Proposition 2.5 If K is a weakly closed subset of X, then the Tykhonov triple .TKLP is smaller than the Tykhonov triple .TK , that is, .TKLP ≤ TK . Proof We start with a preliminary result. To this end, for each .ω ≥ 0, define the set Kω by equality .Kω = K + B(0X , ω), where .B(0X , ω) represents the ball of radius .ω centered in .0X , i.e.,   .B(0X , ω) = v ∈ X : v X ≤ ω . (2.83) .

We claim that the following property holds: M

ωn → 0 ⇒ Kωn − → K.

.

(2.84)

Indeed, assume that .ωn → 0. Since .K ⊂ Kωn for each .n ∈ N, it follows that condition (a) in Definition 1.2 is satisfied. Next, let .v ∈ K and let .{vn } ⊂ X be a sequence such that .vn ∈ Kωn for each .n ∈ N and .vn  v in X. Then, for each .n ∈ N, there exist some elements  .vn and .wn such that vn =  vn + wn ,

.

 vn ∈ K,

wn ∈ B(0X , ωn ).

This means that . wn X ≤ ωn and, therefore .wn → 0X in X. We combine this convergence with the convergence .vn  v in X and equality .vn =  vn + wn to see .vn  v in X. Thus, since K is weakly closed, we deduce that .v ∈ K. This that  proves that condition (b) in Definition 1.2 is satisfied, which concludes the proof of the implication (2.84). Next, let .{un } ⊂ X be a .TKLP -approximating sequence. Then, there exists a 0 (θ ) for each .n ∈ N. sequence .{θn } ∈ C such that .d(un , K) ≤ θn and .un ∈ ΩK n This implies that for each .n ∈ N there exist some elements .vn and .wn such that un = vn + wn ,

.

vn ∈ K,

wn ∈ B(0X , 2θn ).

86

2 Tykhonov Triples and Associated Well-Posedness Concept

This means that .un ∈ K + B(0X , 2θn ) = K2θn . Next, for each .n ∈ N, we denote by θ n the element of .I given by .θ n = (K2θn , θn ). Then, using (2.84), it is easy to see that .{θ n } ∈ CK . 0 (θ ), and .θ = Summarizing, for each .n ∈ N, we have .un ∈ K2θn , .un ∈ ΩK n n (K2θn , θn ) ∈ CK . This shows that .{un } is a .TK - approximating sequence, which   concludes the proof.

.

Note that, in general, the inequality in the statement of Proposition 2.5 is strict, as it follows from the following example. Example 2.18 Let .X = lR2 , that is, ∞    X = u = {un } ⊂ R : u2n < ∞ ,

.

n=0

equipped with the canonical Hilbert structure defined by the inner product (u, v)X =

∞ 

.

un vn

∀ u = {un }, v = {vn } ∈ X.

n=0

Moreover, let .K = {0X }. We use notation (2.78), and, in addition, we denote by 0 : I → 2X the multivalued mapping given by ΩK

.

0 ΩK (θ ) = X

.

∀ θ ∈ I.

Then, equalities (2.79) and (2.80) imply that .

0 ΩK (θ ) = K ∩ ΩK (θ ) = K, .   LP (θ ) = u ∈ X : d(u, K) ≤ θ , ΩK

(2.85) (2.86)

for all .θ ∈ I . On the other hand, (2.82) shows that  K (θ) = K Ω

 θ ) with K  = ∅, θ ∈ I. ∀ θ = (K,

.

  Consider now the sequence .{en } ⊂ X, where . e1 , e2 , . . . , en , . . . represents the canonical base of the space .X = lR2 . Then, .d(en , K) = 1 for each .n ∈ N, which implies that .{en } is not a .TKLP -approximating sequence. Indeed, arguing by contradiction, if .{en } is a .TKLP -approximating sequence, then (2.86) implies that .d(e n , K) → 0, which is a contradiction. Nevertheless, we claim that .{en } is a .TK -approximating sequence. To prove this statement, for each .n ∈ N we define the set .Kn ⊂ X by equality   Kn = λen : λ ∈ [0, 1] .

.

2.3 Extended Classical Well-Posedness Concepts

87 M

Our aim in what follows is to prove that .Kn − → K. First, it is easy to see that the sequence .{un } with .un = n1 en is such that .un ∈ Kn for each .n ∈ N and, moreover, .un → 0X in X. Therefore, condition (a) in Definition 1.2 is satisfied. On the other hand, assume that .{un } ⊂ X is a sequence such that .un ∈ Kn for each .n ∈ N and .un  u ∈ X. Then, .un = λn e n with .λn ∈ [0, 1] for each .n ∈ N and .(un , v)X = λn vn → (u, v)X for each .v = {vn } ∈ X. Let .v = {vn } ∈ X. Then, .vn → 0 and, therefore, .λn vn → 0. We deduce from here that .(un , v)X → 0, which implies that .(u, v)X = 0. This equality shows that .u = 0X , i.e., .u ∈ K. It follows M

from here that condition (b) in Definition 1.2 is satisfied and, therefore, .Kn − → K. Take .θn = 0, for each .n ∈ N. Then, to resume, we proved that for each .n ∈ N there M K (θ n ) = Kn , which exists .θ n = (Kn , θn ) such that .Kn − → K, .θn → 0, and .en ∈ Ω  shows that .{en } is a .TK -approximating sequence, as claimed. It follows from above that .{en } ∈ STK and .{en } ∈ / ST LP , which implies that K .S  ⊂ S LP . On the other hand, Proposition 2.5 shows that .S LP ⊂ S  . We now TK TK TK TK use Definition 2.2(c) to see that .T LP < TK . K

Consider now an abstract problem, denoted by .PK , associated with the set K. Then, Propositions 2.3–2.5 combined with Definition 2.1(c) lead to the following result. Theorem 2.11 (a) Assume that Problem .PK is .TKLP -well-posed. Then it is .TK -well-posed, too. (b) Assume that Problem .PK is .TK -well-posed. Then it is .TK -well-posed, too. (c) Assume that Problem .PK is .TK -well-posed, and moreover, K is a weakly closed set. Then Problem .PK is .TKLP -well-posed, too. We complete the statement of Theorem 2.11 with the following remark. Remark 2.11 First, note that the implication “Problem PK is TKLP -well-posed ⇒ Problem PK is TK -well-posed”

.

was already mentioned on page 31, in the particular context of Problem 1.1. Second, several elementary examples can be constructed to prove that the reverse of the statements (a)–(c) in Theorem 2.11 is not true. For instance, Example 1.4 on page 31 provides an example of problem .PK which is .TK -well-posed but is not .TKLP well-posed. Finally, an example of Problem .PK which is .TKLP -well-posed but is not K -well-posed is provided below. .T Example 2.19 Keep the notation from Example 2.18, and consider the operator T : X → X given by

.

T u = (0, u1 , u2 , . . . , un , . . .)

.

∀ u = (u1 , u2 , . . . , un , . . .) ∈ X.

88

2 Tykhonov Triples and Associated Well-Posedness Concept

We also consider the problem .PK of finding .u ∈ K such that T u = u.

(2.87)

.

It is easy to see that .0X is the unique solution of this fixed point problem. Moreover, if .{un } ⊂ X is a .TKLP -approximating sequence, then (2.86) implies that .d(un , K) → 0 and, therefore, .un → 0X in X. We conclude from here that LP problem On the other hand, the sequence .{en } ⊂ X,  (2.87) is .TK -well-posed. 

where . e1 , e2 , . . . , en , . . . represents the canonical base of the space .X = lR2 is a K -approximating sequence (as shown in Example 2.18), which does not converge .T to .0X . We conclude from here that problem (2.87) is not .TK -well-posed.

We now provide two examples of problems .PK for which Theorem 2.11 can be applied. Example 2.20 (A Levitin–Polyak well-posed minimization problem) Let K be a nonempty subset of a normed space X and let Problem .PK be Problem 1.1 on page 29. Also, with the notation used there, let   0 ΩK (θ ) = u ∈ X : |J (u) − ω)| ≤ θ ∀ θ ≥ 0.

.

Then, using (2.79) and (2.80), it follows that   ΩK (θ ) = u ∈ K : |J (u) − ω)| ≤ θ

.

∀θ ≥ 0

and   LP ΩK (θ ) = u ∈ X : d(u, K) ≤ θ, |J (u) − ω)| ≤ θ ∀ θ ≥ 0.

.

We conclude that in this particular case the Tykhonov triple .TK is the Tykhonov triple .T in Example 2.6 and the Tykhonov triple .TKLP is the Tykhonov triple .T in Example 2.7. Therefore, the well-posedness of Problem 1.1 with respect to the Tykhonov triple .TK reduces to its classical Tykhonov well-posedness studied on page 29. Moreover, the well-posedness of Problem 1.1 with respect to the Tykhonov triple .TKLP reduces to its Levitin–Polyak well-posednes studied on page 31. Therefore, Theorem 2.11 can be used in this case, under assumptions of Theorem 1.14. Then, Theorem 2.11(a) shows that the Levitin–Polyak wellposedness of Problem 1.1 implies its Tykhonov well-posedness, as already noted on page 31. The converse does not hold, as it follows from Example 1.4. Example 2.21 (A Levitin–Polyak well-posed variational inequality) Let K be a nonempty subset of a Hilbert space X, and let Problem .PK be Problem 1.5 on page 40. Also, with the notation used there, let

2.3 Extended Classical Well-Posedness Concepts

.

89

 0 ΩK (θ ) = u ∈ X : (Au, v − u)X + ϕ(v) − ϕ(u) + θ v − u X  ≥ (f, v − u)X ∀ v ∈ K ∀ θ ∈ I.

Then, using (2.79) and (2.80), it follows that .

 ΩK (θ ) = u ∈ K : (Au, v − u)X + ϕ(v) − ϕ(u) + θ v − u X  ≥ (f, v − u)X ∀ v ∈ K ∀θ ∈ I

and .

 LP ΩK (θ ) = u ∈ X : d(u, K) ≤ θ, (Au, v − u)X + ϕ(v) − ϕ(u)  +θ v − u X ≥ (f, v − u)X ∀ v ∈ K ∀ θ ∈ I.

We conclude that in this particular case the Tykhonov triple .TK is the Tykhonov triple .T in Example 2.11 and the Tykhonov triple .TKLP is the Tykhonov triple .T in Example 2.12. Therefore, the well-posedness of Problem 1.5 with respect to the Tykhonov triple .TK reduces to its classical Tykhonov well-posednes studied on page 40. Moreover, the well-posedness of Problem 1.5 with respect to the Tykhonov triple .TKLP reduces to its Levitin–Polyak well-posedness in the sense of Definition 1.22 in page 42. Therefore, Theorem 2.11 can be used in this case, provided that Problem 1.5 is Levitin–Polyak well-posed. Then, Theorem 2.11(a) shows that the Levitin–Polyak well-posedness of Problem 1.5 implies its Tykhonov well-posedness, as already mentioned on page 42. Remark 2.12 Inspired by the previous examples, given an abstract problem .PK associated with a set of constraints K, we refer to a Tykhonov triple .TKLP = LP , C) defined by (2.78) and (2.80) as a Tykhonov–Levitin–Polyak triple. The (I, ΩK well-posedness of Problem .PK with respect to such a triple .TKLP represents an extension of the classical Levitin–Polyak well-posedness concept. Recall that this 0 , assumed to be given. Besides concept is constructed based on the multifunction .ΩK Examples 2.20 and 2.21 presented above, an additional example of Levitin–Polyak well-posed problem will be provided on page 202.

2.3.2 Extended Hadamard Well-Posedness Concept Everywhere in this section, unless stated otherwise, we assume that X is a metric space. We consider a family of abstract problems on X, denoted by .(Pθ ). Each problem in the family depends on a parameter .θ ∈ I , where I is a nonempty set endowed with a convergence, denoted by .θn → θ in I . For instance, I could be

90

2 Tykhonov Triples and Associated Well-Posedness Concept

a metric space endowed with its usual convergence or a normed space endowed with the strong or weak convergence. Moreover, it could be a set of nonempty parts of a normed space endowed with the Mosco convergence, as well. Based on the examples considered in Sect. 2.1, we introduce the following definition. Definition 2.8 The family .(Pθ ) is well-posed in the sense of Hadamard if the following conditions are satisfied: (a) For any .θ ∈ I , Problem .Pθ has a unique solution .u(θ ) ∈ X. (b) For any .θ ∈ I and any sequence .{θn } ⊂ I such that .θn → θ in I , we have .u(θn ) → u(θ ) in X. In other words, less precise but more intuitive, we say that the family of problems (Pθ ) is well-posed in the sense of Hadamard iff for each .θ ∈ I Problem .Pθ has a unique solution that depends continuously on .θ . Note that this definition has been applied in the particular setting on minimization problems and variational inequalities, see pages 55 and 57, respectively. For this reason, we can consider the well-posedness concept in Definition 2.8 as an extension of the classical Hadamard well-posedness concept. Our aim in what follows is to show that the well-posedness of the family .(Pθ ) in the sense of Hadamard is equivalent to the .Tθ -well-posedness of Problem .Pθ , for each .θ ∈ I , where .Tθ represents an appropriate Tykhonov triple. To this end, we assume that for each .θ ∈ I Problem .Pθ has at least one solution and, as usual, we 0 the set of solutions of this problem. Then, for each denote in what follows by .SP θ .θ ∈ I , we are in a position to consider the Tykhonov triple .Tθ = (I, Ω, Cθ ) defined as follows: .

.

0 Ω : I → 2X , Ω( θ ) = SP ∀ θ ∈ I, .  θ   Cθ = {θn } ⊂ I, θn → θ in I .

(2.88) (2.89)

Remark 2.13 We refer to a Tykhonov triple of the form .Tθ = (I, Ω, Cθ ) in which I is a nonempty set, and .Ω and .Cθ are given by (2.88) and (2.89), respectively, as a Tykhonov-Hadamard triple. We shall use this concept in Sects. 7.1 and 7.2 below. We have the following result. Theorem 2.12 The family .(Pθ ) is well-posed in the sense of Hadamard if and only if Problem .Pθ is .Tθ -well-posed, for each .θ ∈ I . Proof Assume that the family .(Pθ ) is well-posed in the sense of Hadamard and 0 is a singleton, that is, .S 0 = let .θ ∈ I . Then, Definition 2.8(a) guarantees that .SP Pθ θ   u(θ ) . Let .{un } ⊂ X be a .Tθ -approximating sequence for Problem .Pθ . Then, there exists a sequence .{θn } ∈ Cθ such that .un ∈ Ω(θn ) for each .n ∈ N. Therefore, (2.88) implies that .un = u(θn ), for each .n ∈ N. We now use (2.89) and Definition 2.8(b) to see that .un = u(θn ) → u(θ ) in X, which implies that Problem .Pθ is .Tθ -well-posed.

2.3 Extended Classical Well-Posedness Concepts

91

Conversely, assume that Problem .Pθ is .Tθ -well-posed, for any .θ ∈ I . Then, for any .θ ∈ I , Problem .Pθ has a unique solution denoted by .u(θ ), and therefore, condition (a) in Definition 2.8 is satisfied. Fix .θ ∈ I and let .{θn } ⊂ I be a sequence such that .θn → θ in I . Then, by (2.88), it follows that .u(θn ) ∈ Ω(θn ). Moreover, the definition (2.89) of the set .Cθ shows that .{θn } ∈ Cθ . These ingredients imply that the sequence .{u(θn )} ⊂ X is a .Tθ -approximating sequence, and therefore, the .Tθ -wellposedness of Problem .Pθ implies that .u(θn ) → u(θ ) in X. Therefore, condition (b) in Definition 2.8 is satisfied. It follows from here that the family .(Pθ ) is well-posed in the sense of Hadamard, which concludes the proof.   We end this subsection with three extensions of Definition 2.8 in the case of normed spaces. To this end, we assume in what follows that X is a normed space. Definition 2.9 The family .(Pθ ) is weakly well-posed in the sense of Hadamard if the following conditions are satisfied: (a) For any .θ ∈ I , Problem .Pθ has a unique solution .u(θ ) ∈ X. (b) For any .θ ∈ I and any sequence .{θn } ⊂ I such that .θn → θ in I , we have .u(θn )  u(θ ) in X. Definition 2.10 The family .(Pθ ) is (strongly) generalized well-posed in the sense of Hadamard if the following conditions are satisfied: 0 , is not empty. (a) For any .θ ∈ I , the set of solutions of Problem .Pθ , denoted by .SP θ (b) For any .θ ∈ I and any sequences .{θn } ⊂ I , .{un } ⊂ X such that .θn → θ in I and .un ∈ SPθn for each .n ∈ N, there exists a subsequence of the sequence .{un } 0 . which converges strongly (in X) to some point of .SP θ

Definition 2.11 The family .(Pθ ) is weakly generalized well-posed in the sense of Hadamard if the following conditions are satisfied: 0 , is not empty. (a) For any .θ ∈ I , the set of solutions of Problem .Pθ , denoted by .SP θ (b) For any .θ ∈ I and any sequences .{θn } ⊂ I , .{un } ⊂ X such that .θn → θ in I and .un ∈ SPθn for each .n ∈ N, there exists a subsequence of the sequence .{un } 0 . which converges weakly (in X) to some point of .SP θ

Remark 2.14 We note that the concepts of well-posedness in the sense of Hadamard in Definitions 2.9–2.11 are not an intrinsic since they depends on the choice of the convergence .θn → θ in I . Recall that a similar comment was made in Remark 2.1 on page 47, concerning the concept of .T -well-posedness. We are now in a position to provide the following extensions of Theorem 2.12. Theorem 2.13 Assume that X is a normed space. Then, the following statements hold: (a) The family .(Pθ ) is weakly well-posed in the sense of Hadamard if and only if Problem .Pθ is weakly .Tθ -well-posed, for each .θ ∈ I . (b) The family .(Pθ ) is strongly generalized well-posed in the sense of Hadamard if and only if Problem .Pθ is strongly generalized .Tθ -well-posed, for each .θ ∈ I .

92

2 Tykhonov Triples and Associated Well-Posedness Concept

(c) The family .(Pθ ) is weakly generalized well-posed in the sense of Hadamard if and only if Problem .Pθ is weakly generalized .Tθ -well-posed, for each .θ ∈ I . The proof of Theorem 2.13 is based on arguments similar to those used in the proof of Theorem 2.12 combined with Definitions 2.9–2.11 and 2.3. We end this section with the following example. Example 2.22 Let X be a Hilbert space, .K ⊂ X, .A : X → X, and .ϕ : X → R. Let .I = X, and, for each .θ ∈ I , consider the functional .Jθ : K → R given by Jθ (v) =

.

1 (Av, v)X + ϕ(v) − (θ, v)X 2

∀ v ∈ K,

together with the following minimization problem. Problem .Pθ . Find .u ∈ K such that .Jθ (u) ≤ Jθ (v)

∀ v ∈ K.

Assume (1.72)–(1.74). Then, Theorem 1.15 guarantees that the family .(Pθ ) is wellposed in the sense of Hadamard given in Definition 2.8. Additional examples of families of problems .(Pθ ) which are well-posed in the sense of Hadamard will be presented in Sects. 7.1 and 7.2, in the study of minimization and optimal control problems, respectively.

2.4 Concluding Remarks In this chapter we introduced the concept of Tykhonov triple, denoted by .T , together with the concept of .T -approximating sequence. Then, given an abstract problem .P in a metric space X, we introduced the concept of .T -well-posedness, as follows: Problem .P is .T -well-posed if and only if it has a unique solution .u ∈ X and any .T -approximating sequence converges in X to u. We also proved that this concept represents an extension of the classical well-posedness concepts in the sense of Tykhonov, Levitin–Polyak, and Hadamard, in the study of minimization problems, fixed point problems, inclusions, and variational inequalities. We proved that the Tykhonov triples are closely related to the convergence of sequences to the solution u since: (a) For each sequence .{un }, we can find a Tykhonov triple .T0 such that .un → u in X if and only if Problem .P is well-posed with .T0 (Example 2.3 on page 50). (b) There exists a Tykhonov triple .TP such that .P is .TP -well-posed and any sequence which converges to u is a .TP -approximating sequence (Example 2.4 on page 50). Next, we introduced an equivalence relation on the set of Tykhonov triples as well as an order relation on the set of equivalence classes of Tykhonov triples. We proved that the equivalence class of the Tykhonov triple .TP is the greatest element

2.4 Concluding Remarks

93

of the set of equivalence classes of Tykhonov triples with which Problem .P is well-posed (Remark 2.4). We also identified a situation in which an upper bound of a finite family of Tykhonov triples can be constructed (Theorem 2.1). Then, we provided two metric characterizations of .T -well-posedness, expressed in terms of the diameter of the .T -approximating sets (Theorems 2.2 and 2.3). We used this characterization in the study of the .T -well-posedness of a system of problems, the so-called split problems (Theorems 2.7 and 2.8). Then, we introduced the concepts of dual problems and dual Tykhonov triples and studied the relationship which exists between their .T -well-posedness (Theorem 2.9). For the problems associated with a set of constraints K, we introduced and studied the well-posedness with respect to a special Tykhonov triple, .TKLP , providing in this way an extension of the classical Levitin–Polyak well-posedness concept (Remark 2.12). In addition, we studied the well-posedness of a family of problems .(Pθ ) with respect to a special Tykhonov triple .Tθ , which depends on a parameter .θ and introduced a rigorous definition of the concept of well-posedness in the sense of Hadamard (Definition 2.8). Finally, in the case of normed spaces, we extended the concepts of .T -well-posedness to the concepts of weakly, strongly generalized and weakly generalized well-posedness (Definitions 2.3, 2.9, 2.10, and 2.11). We illustrated the abstract concepts above within various examples and counterexamples. In particular, in Sect. 2.1.5, we studied the .T -well-posedness of a nonlinear equation in Hilbert spaces for which we obtained various convergence results. In Part II of the book, we shall illustrate the mathematical tools in this chapter in the study of five relevant classes of problems: fixed point problems, variational inequalities, hemivariational inequalities, inclusions, and minimization problems, respectively, defined on a Banach or Hilbert space X. A problem of this type is usually denoted by .P, and it is associated with appropriate assumptions which guarantee that it has a unique solution, denoted by u. Then, our study will go in one of the directions (a) and (b) described below. (a) Given a sequence .{un }, we are interested to prove that the convergence .un → u in X holds. To this end, we use the following two steps strategy: • First, we identify an appropriate Tykhonov triple .T and prove that Problem .P is .T -well-posed. • Second, we prove that the given sequence .{un } is a .T -approximating sequence. Then the .T -well-posedness of Problem .P implies the validity of the convergence .un → u in X. This strategy is illustrated in Fig. 2.1a. (b) Given the solution u of Problem .P, we are interested to identify some relevant sequences which converge to u. To this end, our strategy will be the following: • First, we consider a Tykhonov triple .T and prove that Problem .P is .T -wellposed. • Second, we identify some relevant .T -approximating sequence.

94

2 Tykhonov Triples and Associated Well-Posedness Concept

SP

SP {un}

{un}

u

{un’’}

{un’}

ST

u

ST

a)

b)

Fig. 2.1 The strategies (a) and (b): (a) .{un } is given; we find .T such that .{un } ∈ ST ⊂ SP . Then → u in X. (b) .T is constructed such that .ST ⊂ SP ; we identify relevant sequences .{un }, .{u n }, .{un } such that .{un }, {un }, {un } ∈ ST ⊂ SP . Then .un → u, .un → u, .un → u in X .un

Then using the .T -well-posedness of Problem .P, again, we deduce that the corresponding sequences converge to the solution u of Problem .P. This strategy is illustrated in Fig. 2.1b. The general method we presented above allows us to obtain four kinds of results, as follows: (i) The dependence of the solution with respect to the data. This kind of results are convergence results of the form .un → u in X, where .un is the solution of a problem .Pn which has the same structure as .P and is obtained by considering a perturbation of the data. For instance, if .P is an equation of the form .Au = f , .Pn could be an equation of the form .An un = fn , obtained from .P by perturbing the data A and f . The general framework for this type of results was given in Proposition 2.1 on page 63. (ii) The link between the solutions of problems which have a different structure. This kind of results are convergence results of the form .un → u in X, where .un is the solution of a problem .Pn with a structure which is different from the structure of the original problem .P. For instance, if .P is a constrained variational inequality, .Pn could be an unconstrainted penalty inequality, if .P is a hemivariational inequality, .Pn could be a variational–hemivariational inequality, and so on. (iii) The statement of criteria of convergence. In some cases, we are in a position to identify Tykhonov triples .T which are equivalent to the Tykhonov triple .TP defined in Example 2.4. Then, a sequence .{un } ⊂ X converges to the solution u of Problem .P if and only if .{un } is a .T -approximating sequence. In this way, we provide a necessary and sufficient condition which guarantees the convergence of a sequence to the solution u.

2.4 Concluding Remarks

95

(iv) Recovering classical well-posedness results. Finally, on occasion, we recover some classical well-posedness results, and we compare them with .T -wellresults results, obtained using a convenient Tykhonov triple .T . The tools developed in this chapter could be applied in Physics, Mechanics, and Engineering Sciences. In Part III of this book, we illustrate their applications in the study of Mathematical Models of Contact. In this way we deduce continuous dependence results of the solution with respect to the data and establish the link between mathematical models used to describe various contact processes.

Part II

Relevant Examples of Well-Posed Problems

Chapter 3

Fixed Point Problems

In this chapter, we deal with the well-posedness of fixed point problems of the form Λu = u. We consider three particular cases: the case when .Λ is a contraction defined on a closed nonempty subset of a Banach space, the case when .Λ is a history-dependent operator, and the case when .Λ is an almost history-dependent operator. In both cases, we study the stability of the solution with respect to a perturbation of the operator .Λ, under various assumptions. We also apply these results in the analysis of four relevant problems that have a fixed point structure: a stationary inclusion, an elliptic variational inequality, a Volterra-type equation, and a history-dependent inclusion. For these problems, we prove existence, uniqueness, and continuous dependence results of the solution with respect to the data. An application of the results in this chapter will be presented in Sect. 10.1 in the study of a frictionless viscoelastic contact problem.

.

3.1 The Case of Contractive Mappings Everywhere in this section we assume that .(X,  · X ) is a real Banach space. We denote by .0X and .IX the zero element of X and the identity operator of X, respectively.

3.1.1 Well-Posedness Results Let .K ⊂ X and .Λ : K → K. We consider the following fixed point problem. Problem .P Find an element .u ∈ K such that .Λu = u.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Sofonea, Well-Posed Nonlinear Problems, Advances in Mechanics and Mathematics 50, https://doi.org/10.1007/978-3-031-41416-9_3

99

100

3 Fixed Point Problems

We study the well-posedness of Problem .P in the case when .Λ is a contraction. To this end, we consider the following assumptions: K is a nonempty closed subset of X..  Λ : K → K is a contraction, i.e., there exists α ∈ [0, 1) such that Λu − ΛvX ≤ αu − vX for all u, v ∈ K.

.

(3.1) (3.2)

We now provide two examples of relevant Tykhonov triples in the study of the fixed point problem .P. Example 3.1 Assume (3.1), (3.2) and take .T = (I, Ω, C) where .

I = R+ = [0, +∞), .   Ω(θ ) =  u ∈ K : Λ u − uX ≤ θ ∀ θ ∈ I, .   C = {θn } : θn ∈ I ∀ n ∈ N, θn → 0 as n → ∞ .

(3.3) (3.4) (3.5)

Note that the Banach principle (Theorem 1.9) guarantees that the operator .Λ has a unique fixed point .u ∈ K and, therefore, .u ∈ Ω(θ ) for each .θ ∈ I . This shows that .Ω(θ ) = ∅ for each .θ ∈ I and, according to Definition 2.1(a), .T is a Tykhonov triple. Example 3.2 Assume (3.1), (3.2) and take .T  = (I, Ω  , C), where I and .C are defined by (3.3) and (3.5), respectively, and .

  Ω  (θ ) =  u ∈ K : Λ u − uX ≤ θ ( uX + 1)

∀ θ ∈ I.

Note that, again, using Theorem 1.9, it follows that .Ω  (θ ) = ∅, for each .θ ∈ I . The following result is useful in order to compare the Tykhonov triples .T and T .

.

Lemma 3.1 Assume (3.1), (3.2) and let .{un } be a sequence of elements of the set K. Then, .{un } is a .T -approximating sequence if and only if .{un } is a .T  -approximating sequence. Proof Let .{un } be a .T  -approximating sequence. Then, there exists a sequence .{θn } ⊂ R+ such that θn → 0

.

as

n→∞

(3.6)

and, for any .n ∈ N, the following inequality holds:   Λun − un X ≤ θn un X + 1 .

.

(3.7)

3.1 The Case of Contractive Mappings

101

Let .n ∈ N and let .u0 ∈ K. We write un X ≤ un − Λun X + Λun − Λu0 X + Λu0 X ,

.

and then we use inequalities (3.7) and (3.2) to find that un X ≤ θn + (α + θn )un X + αu0 X + Λu0 X

.

or, equivalently, (1 − α − θn )un X ≤ θn + αu0 X + Λu0 X .

(3.8)

.

Next, (3.6) implies that for n large enough we can assume that .θn ≤ 12 (1 − α). Then, 1 .1 − α − θn ≥ (1 − α) and, therefore, inequality (3.8) yields 2 un X ≤

.

2(θn + αu0 X + Λu0 X ) . 1−α

Using now the convergence (3.6), we deduce that there exists .M > 0 such that un X ≤ M

.

∀ n ∈ N.

(3.9)

We now combine inequalities (3.7) and (3.9) to deduce that Λun − un X ≤ ωn

.

(3.10)

with .ωn = θn (M + 1). The convergence (3.6) shows that .ωn → 0 as .n → ∞ and, therefore, (3.10), (3.4), and Definition 2.1(b) imply that .{un } is a .T -approximating sequence. On the other hand, it is easy to see that if .{un } is a .T -approximating sequence, then .{un } is a .T  -approximating sequence and this completes the proof. 

Lemma 3.1 shows that the Tykhonov triples .T and .T  are equivalent, in the sense of Definition 2.2(a). Therefore, using (2.4), we deduce that Problem .P is .T -wellposed if and only if it is .T  -well-posed. We are now in a position to introduce our main result in this section. Theorem 3.1 Assume (3.1) and (3.2). Then Problem .P is .T —and .T  —well-posed. Proof First, we recall that the existence of a unique solution to problem .P follows from Theorem 1.9. Let .{un } be a .T -approximating sequence. Then, using Definition 2.1(b), we deduce that there exists a sequence .{θn } ⊂ R+ such that (3.6) holds and, moreover, Λun − un X ≤ θn .

.

(3.11)

102

3 Fixed Point Problems

Let .n ∈ N. We use equality .u = Λu and write un − uX ≤ un − Λun X + Λun − ΛuX ,

.

and then we use inequalities (3.11) and (3.2) to find that un − uX ≤ θn + αun − uX

.

or, equivalently, un − uX ≤

.

θn . 1−α

(3.12)

We now combine inequality (3.12) and convergence (3.6) to see that .un → u in X, as .n → ∞. This convergence together with Definition 2.1(c) implies that Problem .P is well-posed with the Tykhonov triple .T in Example 3.1. The well-posedness of Problem .P with the Tykhonov triple .T  in Example 3.2 is now a consequence of Lemma 3.1. 

Remark 3.1 We assume in what follows that .K = X, denote by u the solution of Problem .P, and consider the Tykhonov triple .TP = (IP , ΩP , CP ) given by (2.7)– (2.9), where .d( u, u) =  u − uX . Then Theorem 3.1 shows that .T ≤ TP . The converse inequality is also true. Indeed, assume that .{un } ∈ SP . Then, .un → u in X and using (3.2) together with equality .Λu = u, we find that Λun − un X ≤ Λun − ΛuX + u − un X ≤ (α + 1)u − un X .

.

We now take .θn = (α +1)u−un X to see that .{θn } ∈ C and, moreover, .un ∈ Ω(θn ) for each .n ∈ N. This shows that .{un } ∈ ST and, therefore, .TP ≤ T . We conclude from above that the triples .T and .TP are equivalent. Moreover, Lemma 3.1 shows that these triples are equivalent with the Tykhonov triple .T  , too. The importance of this result arises from the properties of the equivalence class of the Tykhonov triple .TP , see Remark 2.4. Indeed, it follows from above that a sequence .{un } ⊂ X converges to the solution u of Problem .P if and only if it is a .T -approximating sequence or, equivalently, a .T  -approximating sequence. And, this represents a convergence criterion that can be formulated as follows: un → u in X ⇐⇒

.

Λun − un → 0X

We shall use this equivalence in Sect. 6.1.4 below.

in X.

3.1 The Case of Contractive Mappings

103

3.1.2 Convergence Results We now use Theorem 3.1 to deduce some well-known convergence results of the form un → u in X,

.

as

n → ∞,

(3.13)

where u is the solution of Problem .P and .{un } represents a sequence of elements in X, constructed by using an iterative method. Our results in this subsection are presented in the form of corollaries. The first one is the following. Corollary 3.1 Assume (3.1), (3.2) and let .u0 ∈ K. Then the convergence (3.13) holds in the following cases: (a) The sequence .{un } is the sequence of successive approximations (Picard iterations [189]) defined by un+1 = Λun

.

∀ n ∈ N.

(3.14)

(b) K is a convex set, .a ∈ (0, 1], and the sequence .{un } is the sequence of Krasnoselskii iterations [121] defined by un+1 = (1 − a)un + aΛun

.

∀ n ∈ N.

(3.15)

(c) K is a convex set, .{an } ⊂ [α0 , 1] with .α0 ∈ (0, 1], and the sequence .{un } is the sequence of Mann iterations [138] defined by un+1 = (1 − an )un + an Λun

.

∀ n ∈ N.

(3.16)

Proof (a) Let .n ∈ N. Then, using (3.14) and (3.2), we deduce that un+1 − un X ≤ αun − un−1 X ≤ α 2 un−1 − un−2 X

.

≤ · · · ≤ α n u1 − u0 X = α n Λu0 − u0 X , which implies that .Λun − un X ≤ θn with .θn = α n Λu0 − u0 X → 0. This shows that .{un } is a .T -approximating sequence and, therefore, Theorem 3.1 and Definition 2.1(c) imply the convergence (3.13). (b) We first note that, since K is convex and .Λ : K → K, the sequence (3.15) is well defined, since .un ∈ K for each .n ∈ N. We start with a direct proof of the convergence result (3.13). Let .n ∈ N. We write .

Λun − un X ≤ Λun − Λun−1 X + Λun−1 − un X ≤ αun − un−1 X + Λun−1 − un X ,

104

3 Fixed Point Problems

and then we use (3.15) to substitute .un , twice, in order to see that   Λun − un X ≤ 1 + (α − 1)a Λun−1 − un−1 X .

.

Next, a recurrence argument shows that  n Λun − un X ≤ 1 + (α − 1)a Λu0 − u0 X .

.

(3.17)

Note that .0 ≤ 1 + (α − 1)a < 1 and, therefore, inequality (3.17) shows that .{un } is a .T -approximating sequence. We now use Theorem 3.1 and Definition 2.1(c) to deduce that the convergence (3.13) holds. Next, we provide a second proof for this convergence result. To this end, we consider the operator .Λa : K → K defined by Λa u = (1 − a)IX + aΛu

.

∀ u ∈ K.

Then, it is easy to see that the operator .Λa is a contraction on K with constant αa = 1 + (α − 1)a ∈ [0, 1). Moreover, .u ∈ K is a fixed point of .Λ if and only if .u ∈ K is a fixed point of .Λa and, in addition, the sequence of Krasnoselskii iterations (3.15) is the Picard iteration corresponding to the operator .Λa . We now use the point (a) of the current corollary to conclude the second proof of (b). (c) The proof is similar to that we presented in (b): we use (3.16) and (3.2) to see that .

  Λun − un X ≤ 1 + (α − 1)an−1 Λun−1 − un−1 X

.

for any .n ∈ N, which implies that     Λun − un X ≤ 1 + (α − 1)an−1 · · · 1 + (α − 1)a0 Λu0 − u0 X

.

for any .n ∈ N. Using now the inequalities .0 ≤ 1 + (α − 1)ak ≤ 1 + (α − 1)α0 with .k = 0, 1, 2, . . . , n − 1, we find that n  Λun − un X ≤ 1 + (α − 1)α0 Λu0 − u0 X

.

for any .n ∈ N. Note that .0 ≤ 1 + (α − 1)α0 < 1 and, therefore, the sequence {un } is a .T -approximating sequence, which concludes the proof. 

.

Remark 3.2 Note that for .a = 1 the Krasnoselskii iteration (3.15) reduces to the Picard iteration (3.14). Moreover, for .an = a, the Mann iteration (3.16) reduces to the Krasnoselskii iteration (3.15). We now study the stability of the solution of Problem .P with respect to perturbations of the operator .Λ. To this end, for each .n ∈ N, we consider the following fixed point problem.

3.1 The Case of Contractive Mappings

105

Problem .Pn Given an operator .Λn : K → K, find .un ∈ K such that .Λn un = un . Besides (3.1), we assume that the operator .Λn is a contraction, i.e., 

There exists αn ∈ [0, 1) such that Λn u − Λn vX ≤ αn u − vX ∀ u, v ∈ K.

.

(3.18)

Then, it follows that Problem .Pn has a unique solution .un ∈ K, for each .n ∈ N. Moreover, we consider the following assumptions:  .

(a) Λn v → Λv in X for all v ∈ K. (b) There exists α0 < 1 such that αn ≤ α0 ∀ n ∈ N. ⎧ ⎨ For each n ∈ N there exists θn > 0 such that (a) Λn v − ΛvX ≤ θn (vX + 1) ∀ v ∈ K. ⎩ (b) θn → 0 as n → ∞.

.

(3.19)

(3.20)

We have the following convergence result. Corollary 3.2 Assume (3.1), (3.2), (3.18), and either (3.19) or (3.20). Then the solution of Problem .Pn converges to the solution of Problem .P, i.e., the convergence (3.13) holds. Proof Let .n ∈ N and assume that (3.19) holds. We use equalities .Λn un = un and Λu = u to write

.

un − uX = Λn un − ΛuX ≤ Λn un − Λn uX + Λn u − ΛuX .

.

This inequality combined with assumptions (3.18) and (3.19)(b) implies that un − uX ≤

.

1 Λn u − ΛuX 1 − α0

and, using assumption (3.19)(a), we deduce that the convergence (3.13) holds. Assume now that (3.20) holds. We use equality .Λn un = un and assumption (3.20)(a) to write Λun − un X = Λun − Λn un X ≤ θn (un X + 1).

.

This inequality combined with assumption (3.20)(b) shows that the sequence .{un } is a .T  -approximating sequence. The convergence result (3.13) is now a consequence 

of the .T  -well-posedness of Problem .P, guaranteed by Theorem 3.1. We now indicate a particular situation in which Corollary 3.2 can be applied. To  ⊂ Y , and this end, we consider a normed space .(Y,  · Y ), a nonempty subset .K two operators R and S that satisfy the following conditions:

106

3 Fixed Point Problems

 .



 → K and there exists LR > 0 such that R:K  Rη − Rξ X ≤ LR η − ξ Y ∀ η, ξ ∈ K.  and there exists LS > 0 such that S:K→K Su − SvY ≤ LS u − vX ∀ u, v ∈ K.

(3.21)

.

(3.22)

.

LR LS < 1.

(3.23)

 → K and .Sn : K → K  Next, for each .n ∈ N, we consider the operators .Rn : K that satisfy conditions (3.21)–(3.23) with constants .LRn and .LSn , respectively. We refer to these conditions as conditions (3.21).n , (3.22).n , and (3.23).n , respectively. Moreover, assume that

.

⎧ ⎨ For each n ∈ N there exists ωn > 0 such that  (a) Rn ξ − Rξ X ≤ ωn (ξ Y + 1) ∀ ξ ∈ K. ⎩ (b) ωn → 0 as n → ∞. ⎧ ⎨ For each n ∈ N there exists εn > 0 such that (a) Sn v − SvY ≤ εn (vX + 1) ∀ v ∈ K. ⎩ (b) εn → 0 as n → ∞.

.

(3.24)

(3.25)

We have the following result. Corollary 3.3 Assume (3.1), (3.21)–(3.23), (3.21).n –(3.23).n and, for each .n ∈ N, denote .Λ = RS, .Λn = Rn Sn . Then: (a) Problem .P has a unique solution u and, for each .n ∈ N, Problem .Pn has a unique solution .un . (b) If, in addition, (3.24) and (3.25) hold, then the convergence (3.13) holds, too. Proof (a) Let .n ∈ N. We note that assumptions (3.23) and (3.23).n guarantee that the operators .Λ and .Λn are contractions defined on K with values in K. Therefore, the unique solvability of Problems .P and .Pn is a direct consequence of the Banach fixed point principle. (b) Assume now that (3.24) and (3.25) hold and let .u0 , v ∈ K, .n ∈ N. Then, a simple calculation based on the conditions (3.21), (3.22), (3.24), and (3.25) shows that .

Λn v − ΛvX = Rn (Sn v) − R(Sv)X ≤ Rn (Sn v) − R(Sn v)X + R(Sn v) − R(Sv)X ≤ ωn (Sn vY + 1) + LR Sn v − SvY

3.2 Applications to Elliptic Problems

107

≤ ωn (Sn v − SvY + Sv − Su0 Y + Su0 Y + 1) + εn LR (vX + 1) ≤ (ωn εn + ωn LS + εn LR )vX + ωn (Su0 Y + LS u0 X ) +ωn εn + ωn + εn LR . Now, since .ωn → 0 and .εn → 0, it follows from here that condition (3.20) is satisfied with a convenient choice of .θn . The convergence (3.13) is now a direct consequence of Corollary 3.2. 

3.2 Applications to Elliptic Problems Everywhere in this section X represents a Hilbert space endowed with the inner product .(·, ·)X and its associated norm . · X . In this framework, we provide some applications of the results in Sect. 3.1 in the study of two elliptic problems: an elliptic variational inequality and a stationary inclusion.

3.2.1 An Elliptic Variational Inequality We keep the functional framework and the notation introduced in Sect. 1.2.3. Therefore, .K ⊂ X, .A : X → X, .ϕ : X × X → R, and .f ∈ X. In this framework, we consider the following inequality problem: Problem .VI. Find an element u such that u ∈ K,

.

(Au, v − u)X + ϕ(u, v) − ϕ(u, u) ≥ (f, v − u)X

∀ v ∈ K.

(3.26)

We show that Problem .VI has a fixed structure and, therefore, its unique solvability as well as the continuous dependence of its solution with respect to the data A, f , and K can be studied by using our abstract results in Sect. 3.1. The unique solvability of Problem .VI is provided by Theorem 1.11. Nevertheless, we repeat below this theorem since, to the best of our knowledge, the proof we present here is different from the proofs that can be found in the literature. Theorem 3.2 Assume (1.47)–(1.51). Then, there exists a unique element u such that (3.26) holds. Proof First, we underline that, for simplicity, the constants .mA and .LA in condition (1.48) will be denoted by m and M, respectively. We introduce the product space .X × X endowed with the canonical Hilbertian structure given by the inner product

108

3 Fixed Point Problems

(η, ξ )X×X = (η1 , ξ1 )X + (η2 , ξ2 )X

.

∀ η = (η1 , η2 ), ξ = (ξ1 , ξ2 ) ∈ X × X.

The associated norm will be denoted by . · X×X . Let .ρ > 0 to be determined later, and note that, for a given .ξ = (ξ1 , ξ2 ) ∈ X ×X, the function .ρϕ(ξ2 , ·) : X → R is convex and lower semicontinuous. This allows us to consider the operator R defined as follows: R : X × X → K,

.

Rξ = proxK,ρϕ(ξ2 ,·) ξ1

∀ ξ = (ξ1 , ξ2 ) ∈ X × X.

(3.27)

Here, we recall that .proxK,ψ represents the proximality operator defined on page 23. Consider also the operators S and .Λ given by .

S : K → X × X, Λ : K → K,

Su = (ρf − ρAu + u, u)

Λu = RSu

∀ u ∈ K..

∀ u ∈ K.

(3.28) (3.29)

Note that the operators R, S, and .Λ above depend on .ρ, but, for simplicity, we do not mention explicitly this dependence. The rest of the proof is structured in three steps, as follows: (i) We prove that the following estimate holds. .

Rη − Rξ X ≤ η1 − ξ1 X + ραϕ η2 − ξ2 X

(3.30)

∀ η = (η1 , η2 ), ξ = (ξ1 , ξ2 ) ∈ X × X. Indeed, let .η = (η1 , η2 ), ξ = (ξ1 , ξ2 ) ∈ X × X. Then, using the definition (3.27) of the operator R and equivalence (1.42), we obtain that .

Rη ∈ K,

(Rη, v − Rη)X + ρϕ(η2 , v) − ρϕ(η2 , Rη) ≥ (η1 , v − Rη)X ,

Rξ ∈ K,

(Rξ, v − Rξ )X + ρϕ(ξ2 , v) − ρϕ(ξ2 , Rξ ) ≥ (ξ1 , v − Rξ )X ,

for all .v ∈ K. We now take .v = Rξ in the first inequality, then .v = Rη in the second one, and add the resulting inequalities to find that .

Rη − Rξ 2X ≤ ρϕ(η2 , Rξ ) − ρϕ(η2 , Rη) + ρϕ(ξ2 , Rη) − ρϕ(ξ2 , Rξ ) +(η1 − ξ1 , Rη − Rξ )X .

Next, we use assumption (1.49)(b) on the function .ϕ to deduce that (3.30) holds. (ii) We prove that, with a convenient choice of .ρ, the operator .Λ : K → K is a contraction. Indeed, assume that .u, v ∈ K and denote by .S1 , .S2 the components of the operator S. We have .

Su = (S1 u, S2 u)

with S1 u = ρf − ρAu + u, S2 u = u, . (3.31)

Sv = (S1 v, S2 v)

with S1 v = ρf − ρAv + v, S2 v = v.

(3.32)

3.2 Applications to Elliptic Problems

109

Using now (3.29) and (3.30), we find that Λu − ΛvX = R(Su) − R(Sv)X ≤ S1 u − S1 vX + ραϕ S2 u − S2 vX

.

and, therefore, (3.31), (3.32) yield Λu − ΛvX ≤ (u − ρAu) − (v − ρAv)X + ραϕ u − vX .

.

We now use assumption (1.48) and Proposition 1.13 on page 20 to deduce that Λu − ΛvX ≤ (k(ρ) + ραϕ )u − vX ,

.

(3.33)

1

where .k(ρ) = (1 − 2ρm + ρ 2 M 2 ) 2 . Consider now the real-valued function given by 1

F (ρ) = k(ρ) + ραϕ = (1 − 2ρm + ρ 2 M 2 ) 2 + ραϕ

.

 2m ∀ ρ ∈ 0, 2 . M

Then, using the smallness assumption (1.50), we deduce that .F  (0) = αϕ −m < 0. This implies that F is strictly decreasing in a neighborhood of .ρ = 0 and, since .F (0) = 1, we deduce that for .ρ > 0 small enough, we can assume that .F (ρ) < 1. We now use inequality (3.33) to see that for such .ρ the operator .Λ is a contraction, as claimed. (iii) We now prove the existence of a unique solution to Problem .VI. First, using definitions (3.27)–(3.29), we deduce that Λu = R(Su) = R(ρf − ρAu + u, u) = proxK,ρϕ(u,·) (ρf − ρAu + u)

.

for all .u ∈ K, and, using the equivalence (1.42), it follows that .

u ∈ K,

u = Λu ⇐⇒

u ∈ K,

(u, v − u)X + ρϕ(u, v) − ρϕ(u, u) ≥ (ρf − ρAu + u, v − u)X

∀v ∈ K

or, equivalently, u ∈ K,

.

u = Λu ⇐⇒ u is a solution to Problem VI.

(3.34)

On the other hand, the previous step combined with the Banach fixed point theorem shows that, with a convenient choice of .ρ, there exists a unique element .u ∈ K such that .u = Λu. The unique solvability of Problem .VI follows now from the equivalence (3.34). 

110

3 Fixed Point Problems

Remark 3.3 Inequality (3.30) implies that the operator R given by (3.27) is a Lipschitz continuous operator. Its Lipschitz constant depends on .ρ. Nevertheless, since no confusion arises, we do not mention explicitly this dependence and, in what follows, we denote this constant by .LR . The solution of variational inequality (3.26) depends on the data K, A, .ϕ, and f . Even if a continuous dependence result with respect to all these data can be obtained, below we assume that K and .ϕ are fixed and we restrict ourselves to present a result that shows the continuous dependence of the solution with respect to only A and f . To this end, we assume in what follows that (1.47), (1.49) hold and we consider two sequences .{An } and .{fn } such that, for each .n ∈ N,  .

An : X → X satisfies condition (1.48) with constants mn and Mn .

(3.35)

.

fn ∈ X.

(3.36)

Moreover, we assume that αϕ < mn

.

for each n ∈ N.

(3.37)

Then, using Theorem 3.2, it follows that for each .n ∈ N there exists a unique solution to the following inequality problem: Problem .VIn . Find an element .un such that .

un ∈ K,

(An un , v − un )X + ϕ(un , v) − ϕ(un , un ) ≥ (fn , v − un )

(3.38)

∀ v ∈ K.

In addition, consider the following assumptions:

.

⎧ ⎨ For each n ∈ N there exists δn > 0 such that (a) An v − AvX ≤ δn (vX + 1) ∀ v ∈ X. ⎩ (b) δn → 0 as n → ∞.  There exist c0 > 0 and d0 > 0 such that . c0 ≤ mn ≤ Mn ≤ d0 ∀ n ∈ N.

(3.39)

(3.40) (3.41)

αϕ < c0 .. fn → f

.

in X.

(3.42)

The following example provides a sequence of operators .{An } that satisfy conditions (3.35) and (3.39).

3.2 Applications to Elliptic Problems

111

Example 3.3 Assume (1.48), let .{ωn } be a sequence of positive real numbers such that .ωn → 0, and let .T : X → X be a monotone Lipschitz continuous operator, i.e., an operator that satisfies the inequalities (T u − T v, u − v)X ≥ 0,

.

T u − T vX ≤ LT u − vX

∀ u, v ∈ X

with .LT > 0. For all .n ∈ N, consider the operator .An : X → X defined by An v = Av + ωn T v for all .v ∈ X. Then, it is easy to see that the sequence of operators .{An } satisfies condition (3.35) with .mn = m and .Mn = M + ωn LT , for each .n ∈ N. Moreover, conditions (3.39) and (3.40) hold, too.

.

Theorem 3.3 Assume (1.47)–(1.51), (3.35)–(3.37), and (3.39)–(3.42). Then, the solution of the variational inequality (3.38) converges to the solution of the variational inequality (3.26), i.e., (3.13) holds. Proof Let .n ∈ N be fixed and define the operators .Sn , .Λn by equalities .

Sn : K → X × X, Λn : K → K,

Sn u = (ρfn − ρAn u + u, u)

Λn u = RSn u

∀ u ∈ K, .

∀ u ∈ K,

(3.43) (3.44)

where, recall, R is the operator defined by (3.27) and .ρ is a positive parameter. Note that assumptions (3.40), (3.41) imply that we can find .m , M  > 0 that do not depend on n such that the operators .An and A are strongly monotone and Lipschitz continuous with constants .m and .M  , respectively. Therefore, the proof of Theorem 3.2 shows that we can choose a convenient value of .ρ, independent of n, for which the operators .Λ and .Λn are contractions on the space X. Moreover, recall that the solutions .un and u of the variational inequalities (3.38) and (3.26), respectively, satisfy the equalities un = Λn un ,

.

u = Λu.

(3.45)

Let .v ∈ K. Then, it is easy to see that .

Sn v − SvX×X = (ρfn − ρAn v + v, v) − (ρf − ρAv + v, v)X×X = (ρ(fn − f ) − ρ(An v − Av)X ≤ ρfn − f X + ρAn v − AvX

and, using (3.39)(a), we deduce that .

Sn v − SvX×X ≤ ρfn − f X + ρδn (vX + 1).

We now use assumptions (3.42) and (3.39)(b) to see that condition (3.25) is satisfied with .εn = ρfn − f X + ρδn . Then, using (3.29), (3.44), we have Λn v − ΛvX×X ≤ LR εn (vX + 1),

.

112

3 Fixed Point Problems

where .LR represents the Lipschitz constant of the operator (3.27) (see Remark 3.3). We conclude from here that (3.20) holds with .θn = LR εn . Theorem 3.3 is now a direct consequence of Corollary 3.3. 

3.2.2 A Stationary Inclusion Let .K ⊂ X, .A : X → X, .f ∈ X and denote by .NK the outward normal cone of K, defined on page 24. Our aim in this subsection is to study the following inclusion problem, referred below as a stationary inclusion. Problem .I. Find an element .u ∈ X such that .

 − u ∈ NK Au + f ).

(3.46)

More precisely, we show that Problem .I has a fixed structure and, therefore, its unique solvability, as well as the continuous dependence of the solution with respect to the data A, f , and K can be studied by using Corollary 3.2. To this end, we consider the following assumptions: K is a nonempty . closed convex subset of X.. ⎧ A : X → X is a strongly monotone and Lipschitz continuous ⎪ ⎪ ⎨ operator, i.e., there exist m > 0 and M > 0 such that: ⎪ (a) (Au − Av, u − v)X ≥ mu − v2X ∀ u, v ∈ X. ⎪ ⎩ (b) Au − AvX ≤ Mu − vX ∀ u, v ∈ X. f ∈ X.

(3.47)

.

(3.48)

(3.49)

The unique solvability of Problem .I is provided by the following existence and uniqueness result. Theorem 3.4 Assume (3.47)–(3.49). Then there exists a unique element .u ∈ X such that (3.46) holds. Proof Note that assumption (3.48) and Proposition 1.12 on page 20 allow us to define the operator .B : X → X by the equality Bξ = A−1 (ξ − f )

.

∀ ξ ∈ X,

(3.50)

where .A−1 denotes the inverse of the operator A. Moreover, B is a strongly monotone Lipschitz continuous operator with constants .mB = Mm2 and .MB = m1 . Let .ρ =

mB MB2

=

m3 M2

and let .Λ : X → K be the operator defined by Λξ = PK (ξ − ρBξ )

.

∀ ξ ∈ X.

(3.51)

3.2 Applications to Elliptic Problems

113

Then, using the nonexpansivity of the projector operator .PK , (1.40), and Proposition 1.13, we deduce that the operator .Λ satisfies the inequality Λη − Λξ X ≤ k(ρ)η − ξ X

.

∀ η, ξ ∈ X

(3.52)

1

with .k(ρ) = (1 − 2ρmB + ρ 2 MB2 ) 2 . Moreover, replacing the values of .ρ, .mB , and .MB , we find that  k(ρ) =

.

1−

m4 . M4

(3.53)

It follows from (3.52) and (3.53) that .Λ : X → K is a contraction on the space X. Therefore, there exists a unique element .η ∈ K such that .Λη = η. Let u = Bη

(3.54)

η = Au + f.

(3.55)

.

and note that (3.50) implies that .

Moreover, using (3.51), (1.38), and (1.45), we deduce that the following equivalences hold: .

Λη = η ⇐⇒ η = PK (η − ρBη) ⇐⇒ η ∈ K,

((η − ρBη) − η, ξ − η)X ≤ 0

⇐⇒ η ∈ K,

(ξ − η, Bη)X ≥ 0

∀ξ ∈ K

∀ ξ ∈ K ⇐⇒ −Bη ∈ NK (η).

We now use equalities (3.54), (3.55) to see that Λη = η ⇐⇒ −u ∈ NK (Au + f ).

.

(3.56)

The unique solvability of Problem .I follows now from the existence of a unique 

fixed point for the operator .Λ, proved above.  Remark 3.4 It is easy to see that for any .ρ ∈ 0, 2m2B , we have .0 < k(ρ) < 1. MB

Therefore, inequality (3.52) shows that the operator .Λ is a contraction on the space  2mB X, for any .ρ ∈ 0, 2 . MB

We proceed our analysis with a result that shows the continuous dependence of the solution with respect to the data K, A, and f . To this end, we consider three sequences .{Kn }, .{An }, and .{fn } such that, for each .n ∈ N, conditions (3.35), (3.36) hold and, moreover,

114

3 Fixed Point Problems .

Kn is a nonempty closed convex subset of X.

(3.57)

Then, using Theorem 3.4, it follows that for each .n ∈ N there exists a unique solution to the following inclusion problem: Problem .In . Find an element .un ∈ X such that .

 − un ∈ NKn An un + fn ).

(3.58)

We now assume that M

Kn −→ K,

(3.59)

.

M

where, recall, the symbol “.−→” denotes the convergence in the sense of Mosco (see Definition 1.2). We have the following convergence result. Theorem 3.5 Assume (3.35), (3.36), (3.39), (3.40), (3.42), (3.47)–(3.49), (3.57), and (3.59). Then, the solution of the inclusion (3.58) converges to the solution of the inclusion (3.46), i.e., (3.13) holds. Proof Let .n ∈ N be fixed, denote by .A−1 n the inverse of the operator .An , and define the operators .Bn , .Λn by the equalities Bn ξ = A−1 n (ξ − fn )

.

Λn ξ = PKn (ξ − ρBn ξ )

∀ ξ ∈ X, . ∀ ξ ∈ X.

(3.60) (3.61)

Next, we successively prove pointwise convergence results for these operators. −1 Let .ξ ∈ X. We denote .A−1 n (ξ − f ) = vn , .A (ξ − f ) = v, which implies that .ξ − f = An vn = Av and, using the strong monotonicity of the operator .An , we write .

mn vn − v2X ≤ (An vn − An v, vn − v)X = (Av − An v, vn − v)X ≤ An v − AvX vn − vX .

Using now assumptions (3.40) and (3.39), it follows that vn − vX ≤

.

1 1 An v − AvX ≤ An v − AvX → 0, mn c0

(3.62)

−1 A−1 n (ξ − f ) − A (ξ − f )X → 0.

(3.63)

which implies that .

3.2 Applications to Elliptic Problems

115

1 Recall also that .A−1 n is a Lipschitz continuous operator with constant . mn ≤ Therefore, writing

1 c0 .

−1 −1 B . n ξ − Bξ X = An (ξ − fn ) − A (ξ − f )X −1 −1 −1 ≤ A−1 n (ξ − fn ) − An (ξ − f )X + An (ξ − f ) − A (ξ − f )X

≤

1 −1 fn − f X + A−1 n (ξ − f ) − A (ξ − f )X c0

and, using assumption (3.42) combined with the convergence (3.63), we deduce that Bn ξ − Bξ X → 0.

.

(3.64)

On the other hand, writing .

Λn ξ − Λξ X = PKn (ξ − ρBn ξ ) − PK (ξ − ρBξ )X ≤ PKn (ξ − ρBn ξ ) − PKn (ξ − ρBξ )X +PKn (ξ − ρBξ ) − PK (ξ − ρBξ )X

and, using the nonexpansivity of the operator .PKn , we find that Λn ξ − Λξ X ≤ ρBn ξ − Bξ X + PKn (ξ − ρBξ ) − PK (ξ − ρBξ )X .

.

We now use the convergence (3.64), assumption (3.59), and Proposition 1.17 on page 23 to deduce that Λn ξ − Λξ X → 0.

.

(3.65)

Finally, note that (3.52) and (3.53)  show that the operator .Λn is a contraction on 1−

the space X, with constant .αn =

m4n . Therefore, assumption (3.40) guarantees Mn4

that  αn ≤

.

1−

c04 d04

< 1.

(3.66)

It follows from (3.65) and (3.66) that we are in a position to use Corollary 3.2. This implies that ηn → η

.

in X,

(3.67)

where .ηn and .η denote the fixed points of the operators .Λn and .Λ, respectively. Moreover, the proof of Theorem 3.4 shows that .un = Bn ηn , .u = Bη (see (3.54)).

116

3 Fixed Point Problems

We now write un − uX = Bn ηn − BηX ≤ Bn ηn − Bn ηX + Bn η − BηX

.

and, since the operator .Bn is Lipschitz continuous with constant deduce that un − uX ≤

.

.

1 mn

≤

1 c0 ,

we

1 ηn − ηX + Bn η − BηX . c0

We now use (3.67) and (3.64) to see that (3.13) holds, which concludes the proof. 

Note that a different proof of Theorem 3.5 will be provided on page 217. There, in contrast with the proof presented here (based on the fixed point structure of Problem .I), we use a consequence of a different .T -well-posedness result (obtained by performing a direct study of the inclusion .I).

3.3 The Case of History-Dependent Operators Everywhere in this section, we assume that .(V ,  · V ) is a real Banach space, .0V denotes the zero element of V , and .T > 0. Moreover, we denote by X the space of continuous functions defined on .[0, T ] with values in V , i.e., .X = C([0, T ]; V ). Recall that X is a Banach space with the norm of the uniform convergence (see (1.2)).

3.3.1 Well-Posedness Results Let .Λ : X → X. We now consider the following fixed point problem. Problem .P h . Find an element .u ∈ X such that .Λu(t) = u(t) for all .t ∈ [0, T ]. In the study of Problem .P h , we assume that .Λ is a history-dependent operator, i.e., ⎧ There exists L > 0 such that ⎪ ⎪ ⎨ t . Λu(t) − Λv(t)V ≤ L u(s) − v(s)V ds ⎪ 0 ⎪ ⎩ ∀ u, v ∈ X, t ∈ [0, T ].

(3.68)

We now provide two examples of relevant Tykhonov triples in the study of Problem .P h .

3.3 The Case of History-Dependent Operators

117

Example 3.4 Assume that (3.68) holds and take .T = (I, Ω, C), where I = R+. = [0, +∞), .   Ω(θ ) =  u ∈ X : Λ u(t) −  u(t)V ≤ θ ∀ t ∈ [0, T ] ∀ θ ∈ I, .   C = {θn }n : θn ∈ I ∀ n ∈ N, θn → 0 as n → ∞ .

(3.69) (3.70) (3.71)

Note that Theorem 1.4 guarantees that the operator .Λ has a unique fixed point u and, therefore, .u ∈ Ω(θ ) for each .θ ∈ I . This shows that .Ω(θ ) = ∅ for each .θ ∈ I and, according to Definition 2.1(a), .T is a Tykhonov triple. Example 3.5 Assume that (3.68) holds and take .T  = (I, Ω  , C), where I and .C are defined by (3.69) and (3.71), respectively, and Ω  (θ ) =   t   u ∈ X : Λ u(t) −  u(t)V ≤ θ  u(s)V ds + 1 ∀ t ∈ [0, T ]

.

0

for all .θ ∈ I . Note that, again, using Theorem 1.4, it follows that .Ω  (θ ) = ∅, for each .θ ∈ I . The following result is useful in order to compare the Tykhonov triples .T and T .

.

Lemma 3.2 Assume (3.68) and let .{un } ⊂ X. Then, .{un } is a .T -approximating sequence if and only if .{un } is a .T  -approximating sequence. Proof Assume that .{un } is a .T  -approximating sequence. Then, there exists a sequence .{θn } ⊂ R+ such that θn → 0

.

as

n→∞

(3.72)

and, for any .n ∈ N, the following inequality holds: Λun (t) − un (t)V ≤ θn



.

t

un (s)V ds + 1

∀ t ∈ [0, T ].

(3.73)

0

Let .n ∈ N and .t ∈ [0, T ]. We write un (t)V ≤ un (t) − Λun (t)V + Λun (t) − Λ0V (t)V + Λ0V (t)V ,

.

and then we use inequalities (3.73) and (3.68) to find that 

t

un (t)V ≤ θn + (L + θn )

.

0

un (s)V ds + Λ0V (t)V .

(3.74)

118

3 Fixed Point Problems

Note that here and below in this book, given .u0 ∈ V , we keep the notation .u0 for the constant function .t → u0 for all .t ∈ [0, T ] that, obviously, defines an element of X. Therefore, notation .Λ0V makes sense and defines an element of X, too. Next, with the notation F0 = max Λ0V (t)V ,

.

t∈[0,T ]

inequality (3.74) yields 

t

un (t)V ≤ (θn + F0 ) + (L + θn )

.

un (s)V ds.

0

We now use the Gronwall argument (Lemma 1.1 on page 18) to find that un (t)V ≤ (θn + F0 )e(L+θn ) t

.

and, using the convergence (3.72) and inequality .t ≤ T , we deduce that there exists M > 0 such that

.

un (t)V ≤ M

.

∀ t ∈ [0, T ], n ∈ N.

(3.75)

We now combine inequalities (3.73) and (3.75) to deduce that Λun (t) − un (t)V ≤ ωn

.

∀ t ∈ [0, T ],

(3.76)

with .ωn = θn (MT + 1). Next, the convergence (3.72) shows that .ωn → 0 as n → ∞ and, therefore, (3.76), (3.70), and Definition 2.1(b) imply that .{un } is a .T -approximating sequence. On the other hand, it is easy to see that if .{un } is a .T -approximating sequence, 

then .{un } is a .T  -approximating sequence, and this completes the proof. .

Lemma 3.2 shows that the Tykhonov triples .T and .T  are equivalent, in the sense that they generate the same set of approximating sequences (see Definition 2.2(a) on page 49). Therefore, using Definition 2.1(c), we deduce that Problem .P is .T -wellposed if and only if it is .T  -well-posed. We are now in a position to introduce our main result in this section. Theorem 3.6 Assume that .Λ : X → X is a history-dependent operator. Then Problem .P h is .T - and .T  -well-posed. Proof First, we recall that the existence of a unique solution to Problem .P, denoted by u, follows from Theorem 1.4. Let .{un } be a .T -approximating sequence. Then, using Definition 2.1(b), we deduce that there exists a sequence .{θn } ⊂ R+ such that (3.72) holds and, moreover, Λun (t) − un (t)V ≤ θn

.

∀ t ∈ [0, T ].

(3.77)

3.3 The Case of History-Dependent Operators

119

Let .n ∈ N and .t ∈ [0, T ]. We use equality .u(t) = Λu(t) and write un (t) − u(t)V ≤ un (t) − Λun (t)V + Λun (t) − Λu(t)V ,

.

and then we use inequalities (3.77) and (3.68) to find that 

t

un (t) − u(t)V ≤ θn + L

.

un (s) − u(s)V ds.

0

Using now the Gronwall inequality yields un (t) − u(t)V ≤ θn eLt .

(3.78)

.

We now combine inequality (3.78) and convergence (3.72) to see that .

max un (t) − u(t)V → 0 as n → ∞

t∈[0,T ]

or, equivalently, .un → u in X = C([0, T ]; V ). This convergence combined with Definition 2.1(c) implies that Problem .P h is well-posed with the Tykhonov triple .T in Example 3.4. The well-posedness of Problem .P with the Tykhonov triple .T  in Example 3.5 is now a direct consequence of Lemma 3.2. 

3.3.2 Convergence Results We now use Theorem 3.6 to deduce a well-known convergence result of the form (3.13) where .{un } represents a sequence of elements in X constructed by using an iterative method and u is the solution of Problem .P h . Corollary 3.4 Assume (3.68). Let .u0 : [0, T ] → V be a continuous function, i.e., u0 ∈ X, and let .{un } be the sequence of Picard iterations defined by (3.14). Then, the convergence (3.13) holds.

.

Proof The proof is similar to that of Corollary 3.1(a). Let .n ∈ N and .t ∈ [0, T ]. Then, using (3.14) and (3.68), we recursively deduce that  .

t

Λun (t) − un (t)V = Λun (t) − Λun−1 (t)V ≤ L ≤L

n

 t  0

0

v

···

 0

s



un (v) − un−1 (v)V dv

0

p

u1 (r) − u0 (r)V dr dp . . . dv.

0

Note that in the previous inequality we have n iterate integrals. We deduce from here that

120

3 Fixed Point Problems .

Λun (t) − un (t)V  t  v  n ≤L ··· 0

0

s



0

p

dr dp . . . dv u1 − u0 X .

0

On the other hand, since  t  .

0

v

···

0



s



0

p

0

Tn tn ≤ , dr dp . . . dv = n! n!

we obtain that Λun (t) . − un (t)V ≤

Ln T n u1 − u0 X . n!

(3.79)

Next, it is easy to see that .

Ln T n =0 n→∞ n! lim

and, therefore, inequality (3.79) shows that .{un } is a .T -approximating sequence for Problem .P h where, recall, .T is the Tykhonov triple in Example 3.4. Finally, we use Theorem 3.6 and Definition 2.1(c) to see that the convergence (3.13) holds. 

We now study the stability of the solution of Problem .P h with respect to perturbations of the operator .Λ. To this end, for each .n ∈ N, we consider the following fixed point problem: Problem .Pnh . Given an operator .Λn : X → X, find a function .un ∈ X such that Λn un (t) = un (t)

.

∀ t ∈ [0, T ].

(3.80)

We assume that the operator .Λn is a history-dependent operator, i.e., it satisfies condition (3.68) with constant .Ln . Below we refer to this condition as condition (3.68).n . Then, using Theorem 1.4, it follows that Problem .Pnh has a unique solution .un ∈ X, for each .n ∈ N. Moreover, we consider the following assumption that links the operators .Λ and .Λn . ⎧ For each n ∈ N there exists θn  > 0 such that: ⎪ ⎪ ⎪  t ⎪ ⎨ (a) Λn v(t) − Λv(t)V ≤ θn v(s)V ds + 1 . 0 ⎪ ⎪ ∀ v ∈ X, t ∈ [0, T ]. ⎪ ⎪ ⎩ (b) θn → 0 as n → ∞.

(3.81)

We have the following convergence result. Corollary 3.5 Assume (3.68), (3.68).n , and (3.81). Then the convergence (3.13) holds.

3.3 The Case of History-Dependent Operators

121

Proof Let .n ∈ N and .t ∈ [0, T ]. We use (3.80) and (3.81)(a) to write Λun (t) − un (t)V = Λun (t) − Λn un (t)V ≤ θn



.

t

un (s)V ds + 1 .

0

This inequality combined with assumption (3.81)(b) shows that the sequence .{un } is a .T  -approximating sequence. The convergence (3.13) is now a consequence of the .T  -well-posedness of Problem .P h , guaranteed by Theorem 3.6. 

We now indicate a particular situation in which Corollary 3.5 can be applied. To this end, besides the spaces V and X already introduced on page 116, we consider a normed space .(W,  · W ) and denote by .(Y,  · Y ) the space .Y = C([0, T ]; W ), equipped with the norm of the uniform convergence. Moreover, we consider two operators R and S that satisfy the following conditions:

.

⎧ ⎨ R : Y → X and there exists LR > 0 such that . Rη(t) − Rξ(t)V ≤ LR η(t) − ξ(t)W ⎩ ∀ η, ξ ∈ Y, t ∈ [0, T ]. ⎧ S : X → Y and there exists ⎪ ⎪  t LS > 0 such that ⎨ Su(t) − Sv(t)W ≤ LS u(s) − v(s)V ds ⎪ 0 ⎪ ⎩ ∀ u, v ∈ X, t ∈ [0, T ].

(3.82)

(3.83)

Next, for each .n ∈ N, we consider the operators .Rn : Y → X and Sn : X → Y that satisfy conditions (3.82) and (3.83) with constants .LRn and .LSn , respectively. We refer to these conditions as conditions (3.82).n and (3.83).n , respectively. Moreover, we consider the following assumptions: .

.

⎧ For each n ∈ N there exists ωn > 0 such that: ⎪ ⎪ ⎨ (a) Rn ξ(t) − Rξ(t)V ≤ ωn (ξ(t)W + 1) . ⎪ ∀ ξ ∈ Y, t ∈ [0, T ]. ⎪ ⎩ (b) ωn → 0 as n → ∞. ⎧ For each n ∈ N there exists εn> 0 such that: ⎪ ⎪ ⎪  t ⎪ ⎨ (a) Sn v(t) − Sv(t)W ≤ εn v(s)V ds + 1 0 ⎪ ⎪ ∀ v ∈ X, t ∈ [0, T ]. ⎪ ⎪ ⎩ (b) εn → 0 as n → ∞.

(3.84)

(3.85)

We have the following result. Corollary 3.6 Assume (3.82), (3.83), (3.82).n , (3.83).n and denote .Λ = RS, .Λn = Rn Sn , for each .n ∈ N. Then:

122

3 Fixed Point Problems

(a) Problem .P h has a unique solution u and, for each .n ∈ N Problem .Pnh has a unique solution .un . (b) If, in addition, (3.84) and (3.85) hold, then the convergence (3.13) holds, too. Proof (a) Let .n ∈ N. We note that assumptions (3.82), (3.83), (3.82).n , and (3.83).n guarantee that the operators .Λ and .Λn are history-dependent operators on the space X. Therefore, the unique solvability of Problems .P h and .Pnh is a direct consequence of Theorem 1.4. (b) Assume now that (3.84) and (3.85) hold, and let .v ∈ X, .t ∈ [0, T ] be given. Then, a simple calculation based on the conditions (3.82), (3.82).n , (3.83), (3.83).n (3.84) and (3.85) shows that .

Λn v(t) − Λv(t)V = Rn (Sn v)(t) − R(Sv)(t)V ≤ Rn (Sn v)(t) − R(Sn v)(t)V + R(Sn v)(t) − R(Sv)(t)V ≤ ωn (Sn v(t)W + 1) + LR Sn v(t) − Sv(t)W ≤ ωn (Sn v(t) − Sv(t)W + Sv(t) − S0X (t)W + S0X (t)W + 1)  t +εn LR v(s)V ds + 1 0  t v(s)V ds ≤ (ωn εn + ωn LS + εn LR ) 0

+ωn εn + ωn S0X (t)W + ωn + εn LR . Now, since .ωn → 0 and .εn → 0, it follows from here that condition (3.81) is satisfied with a convenient choice of .θn . The convergence (3.13) is now a direct consequence of Corollary 3.5. 

3.4 Applications to History-Dependent Problems The results presented in Sect. 3.3 can be used in the study of a large number of history-dependent problems. To provide some elementary examples, we illustrate their applicability in the study of two relevant problems: a Volterra-type integral equation and a history-dependent version of the inclusion (3.46). To this end, below in this section, we keep the functional framework and the notation used in Sect. 3.3, that is, V is a real Banach space, .T > 0, and .X = C([0, T ]; V ). We also denote by Z the space of real-valued continuous functions defined on .[0, T ], i.e., .Z = C([0, T ]; R), endowed with the norm of the uniform convergence.

3.4 Applications to History-Dependent Problems

123

3.4.1 A Volterra-Type Integral Equation We consider the following problem: Problem .V. Find an element .u ∈ X such that  t .Au(t) + b(t − s)u(s) ds = f (t)

∀ t ∈ [0, T ].

0

In addition, for each .n ∈ N, we consider the following perturbation of Problem .V: Problem .Vn . Find an element .un ∈ X such that 

t

Aun (t) +

.

bn (t − s)un (s) ds = fn (t)

∀ t ∈ [0, T ].

0

In the study of these problems, we consider the following assumptions:  .

A : X → Xis a strongly monotone and Lipschitz continuous operator with constants m > 0 and M > 0.

.

(3.86)

b, bn ∈ Z..

(3.87)

f, fn ∈ X..

(3.88)

bn → b

in Z..

(3.89)

fn → f

in X.

(3.90)

We have the following existence, uniqueness, and convergence result. Theorem 3.7 Assume (3.86)–(3.88). Then: (a) Problem .V has a unique solution u and, for each .n ∈ N Problem .Vn has a unique solution .un . (b) If, in addition, (3.89) and (3.90) hold, then the convergence (3.13) holds, too. Proof (a) We consider the operators R, S, and .Λ defined as follows: R : X. → X, S : X → X,

Rη(t) = A−1 (f (t) − η(t))  t Su(t) = b(t − s)u(s) ds

∀ η ∈ X, t ∈ [0, T ], . (3.91) ∀ u ∈ X, t ∈ [0, T ], . (3.92)

0

Λ : X → X,

Λu(t) = RSu(t)

∀ u ∈ X, t ∈ [0, T ].

(3.93)

Moreover, for each .n ∈ N, we define the operators .Rn , .Sn , and .Λn by equalities Rn : . X → X,

Rn η(t) = A−1 (fn (t) − η(t))

∀ η ∈ X, t ∈ [0, T ],

(3.94) .

124

3 Fixed Point Problems

 Sn : X → X,

Sn u(t) =

t

bn (t − s)u(s) ds

∀ u ∈ X, t ∈ [0, T ], . (3.95)

0

Λn : X → X,

Λn u(t) = Rn Sn u(t)

∀ u ∈ X, t ∈ [0, T ].

(3.96)

Then, it is easy to see that conditions (3.82), (3.83), (3.82).n , and (3.83).n are satisfied with .W = V and .Y = X. It follows now from Corollary 3.6(a) that there exists a unique function .u ∈ X such that .Λu = u and, for each .n ∈ N, there exists a unique function .un ∈ X such that .Λn un = un . We conclude the proof of this part of the theorem by using the equivalences .

Λu = u ⇐⇒ u is a solution to Problem V..

(3.97)

Λn un = un ⇐⇒ un is a solution to Problem Vn .

(3.98)

(b) Assume now that, in addition, (3.89) and (3.90) hold. Then, with the notation above, it is easy to see that the conditions (3.84) and (3.85) are satisfied. Indeed, fix .n ∈ N. We use definitions (3.91), (3.94), and the properties of the operator A to see that Rn ξ(t) − Rξ(t)V ≤

.

1 fn (t) − f (t)V m

∀ ξ ∈ Y, t ∈ [0, T ].

(3.99)

Next, definitions (3.92) and (3.95) yield 

t

Sn v(t) − Sv(t)V ≤ bn (t) − b(t)Z

.

v(s)V ds

(3.100)

0

for all .v ∈ Y, t ∈ [0, T ]. We now combine inequalities (3.99), (3.100) with conditions (3.89), (3.90) to see that conditions (3.84) and (3.85) are satisfied. To conclude, it follows from above that we are in a position to use Corollary 3.6(b) to see that the convergence (3.13) holds, .un and u being the solutions of the fixed point problems .Λn un = un and .Λu = u, respectively. We now use equivalences (3.97) and (3.98) to conclude the proof. 

3.4.2 A History-Dependent Inclusion We now move to the study of a history-dependent inclusion. We work in the space X = C([0, T ]; V ) and let .K : [0, T ] → 2V , .A : V → V , and .S : X → X. Our aim is to study the following problem:

.

Problem .I h . Find an element .u ∈ X such that .

 − u(t) ∈ NK(t) Au(t) + Su(t))

∀ t ∈ [0, T ].

(3.101)

3.4 Applications to History-Dependent Problems

125

To this end, we consider the following assumptions: ⎧ ⎨ (a) K : [0, T ] → 2V has nonempty closed convex values. (b). The mapping t → PK(t) u : [0, T ] → V is continuous, ⎩ for all u ∈ V .

.

S : X → X is a history-dependent operator.

(3.102) (3.103)

Note that Proposition 1.17 shows that condition (3.102)(b) is satisfied if and only if for each .t ∈ [0, T ] and each sequence .{tn } ⊂ [0, T ] such that .tn → t we have M

Kn −→ K. The unique solvability of Problem .I h is provided by the following existence and uniqueness result.

.

Theorem 3.8 Assume (3.48), (3.102), and (3.103). Then there exists a unique element .u ∈ X such that (3.101) holds. Proof Fix a function .f ∈ X. Then, assumptions (3.48), (3.102)(a) allow us to use Theorem 3.4 in order to see that for each .t ∈ [0, T ] there exists a unique element .uf (t) ∈ V such that .

  − uf (t) ∈ NK(t) Auf (t) + f (t) .

(3.104)

Moreover, using assumption (3.102)(b) and Theorem 3.5, we deduce that the function .t → uf (t) : [0, T ] → V is continuous, i.e., it belongs to X. This allows us to consider the operators R and .Λ defined by .

R : X → X,

Rf (t) = uf (t)

∀ f ∈ X, t ∈ [0, T ], .

Λ : X → X,

Λu(t) = RSu(t)

∀ u ∈ X, t ∈ [0, T ].

(3.105) (3.106)

Assume now that .η, ξ ∈ X and let .u = Rη, .v = Rξ , .t ∈ [0, T ]. Then .

  − u(t) ∈ NK(t) Au(t) + η(t) ,

  −v(t) ∈ NK(t) Av(t) + ξ(t)

and, using (1.45), we deduce that .

Au(t) + η(t) ∈ K(t),

(u(t), w − (Au(t) + η(t)))V ≥ 0 ∀ w ∈ K(t),

Av(t) + ξ(t) ∈ K(t),

(v(t), w − (Av(t) + ξ(t)))V ≥ 0 ∀ w ∈ K(t).

Next, by a standard argument, we find that (Au(t) − Av(t), u(t) − v(t))V ≤ (ξ(t) − η(t), u(t) − v(t))V

.

126

3 Fixed Point Problems

and, using assumption (3.48)(a), we obtain that u(t) − v(t)V ≤

.

1 η(t) − ξ(t)V . m

We conclude from above that condition (3.82) is satisfied with .W = V and Y = X. Moreover, condition (3.83) holds, too. Then, using notation (3.106) and Corollary 3.6(a), it follows that there exists a unique element .u ∈ X such that .Λu = u or, equivalently, .R(Su) = u. We now use (3.105) and (3.104) to see that u is the unique solution of the history-dependent inclusion (3.101), which concludes the proof. 

.

We end this section with the following remarks. First, using the results in Sect. 3.3.2, it is easy to deduce various continuous dependence results for the solution of Problem .I h . Nevertheless, to avoid repetitions, we skip them. Second, in Sect. 6.3, we shall prove that inclusion (3.101) represents a dual problem of a history-dependent variational inequality with time-dependent constraints. Finally, in Sect. 6.2, we shall study a version of this inclusion in the framework of an unbounded interval of time, i.e., in the case when .t ∈ [0, +∞).

3.5 The Case of Almost History-Dependent Operators The results in this section represent a continuation of the results obtained in Sect. 3.3 where, recall, we studied the fixed point problem .Λu = u in the case when .Λ : C([0, T ]; V ) → C([0, T ]; V ) is a history-dependent operator, with V being a given Banach space. Moreover, in Sect. 3.3, we used the notation .X = C([0, T ]; V ). In contrast, in this section, we assume that X is a given Banach space, and we consider the fixed point problem above in the case when .Λ : C(R+ ; X) → C(R+ ; X). In addition, we denote by .0X the zero element of both spaces X and .C(R+ ; X). The results in this section will be used in Sect. 10.3, in the study of a mathematical model of contact with rate-type materials.

3.5.1 Well-Posedness Results Everywhere in Sect. 3.5 we assume that .Λ : C(R+ ; X) → C(R+ ; X) is an almost history-dependent operator, i.e.,

3.5 The Case of Almost History-Dependent Operators

.

127

⎧ For each m ∈ N there exist l m ∈ [0, 1), Lm > 0 such that ⎪ ⎪ ⎪ ⎪ ⎨ Λu(t) − Λv(t)X ≤ l m u(t) − v(t)X ⎪ ⎪ ⎪ ⎪ ⎩

t

+Lm

u(s) − v(s)X ds

(3.107)

0

∀ u, v ∈ C(R+ ; X), t ∈ [0, m].

Moreover, we consider the following fixed point problem: Problem .P ah . Find a function .u ∈ C(R+ ; X) such that .Λu(t) = u(t) for all .t ∈ R+ . We use Theorem 1.4 to see that Problem .P ah has a unique solution. In order to study its well-posedness, among various choices available, we consider the Tykhonov triples presented in the four examples below. Example 3.6 Take .T1 = (I1 , Ω1 , C1 ) where I1 = .R+ = [0, +∞), .   Ω1 (θ ) =  u ∈ C(R+ ; X) : Λ u(t) −  u(t)X ≤ θ ∀ t ∈ R+ . 

∀ θ ∈ I1 ,

C1 = {θn }n : θn ∈ I1 ∀ n ∈ N,

θn → 0 as

 n→∞ .

(3.108) (3.109)

(3.110)

Note that for each .θ ∈ I1 the fixed point u obtained in Theorem 1.4 belongs to Ω1 (θ ); hence, .Ω1 (θ ) = ∅. Therefore, according to Definition 2.1(a), it follows that .T1 is a Tykhonov triple. .

Example 3.7 Take .T2 = (I2 , Ω2 , C2 ), where I2 = .R+ = [0, +∞), .  Ω2 (θ ) =  u ∈ C(R+ ; X) : Λ u(t) −  u(t)X ≤ θ ( u(t)X + 1).  ∀ t ∈ R+ ∀ θ ∈ I2 ,   C2 = {θn }n : θn ∈ I2 ∀ n ∈ N, θn → 0 as n → ∞ .

(3.111) (3.112)

(3.113)

Note that, again, using Theorem 1.4 it follows that .Ω2 (θ ) = ∅, for each .θ ∈ I2 . Example 3.8 Take .T3 = (I3 , Ω3 , C3 ), where   I3 = . θ = {θ m }m : θ m ∈ R+ ∀ m ∈ N , .  Ω3 (θ ) =  u ∈ C(R+ ; X) : Λ u(t) −  u(t)X ≤ θ m .  ∀ t ∈ [0, m], m ∈ N ∀ θ = {θ m }m ∈ I3 ,

(3.114) (3.115)

128

3 Fixed Point Problems

 C3 = {θ n }n : θ n = {θnm }m ∈ I3 ∀ n ∈ N, θnm → 0 as

(3.116) 

n → ∞,

∀m ∈ N .

Note that .Ω3 (θ) = ∅, for each .θ ∈ I3 . Example 3.9 Take .T4 = (I4 , Ω4 , C4 ), where  I4 = θ = {θ m }m : θ m ∈ R+

 ∀. m ∈ N ,

(3.117)

 Ω4 (θ ) .=  u ∈ C(R+ ; X) : .

(3.118)

Λ u(t) −  u(t)X ≤ θ m ( u(t)X + 1) ∀ t ∈ [0, m], m ∈ N 



∀ θ = {θ m }m ∈ I4 ,

C4 = {θ n }n : θ n = {θnm }m ∈ I4 ∀ n ∈ N, θnm → 0 as

n → ∞,

(3.119) 

∀m ∈ N .

Note that .Ω4 (θ) = ∅, for each .θ ∈ I4 . The following result is useful in order to compare the Tykhonov triples .T3 and T4 .

.

Lemma 3.3 Assume that .Λ : C(R+ ; X) → C(R+ ; X) is an almost history operator and let .{un } be a .T4 -approximating sequence. Then, for each .m ∈ N there exists .Z m > 0 such that un (t)X ≤ Z m

.

∀ t ∈ [0, m], n ∈ N.

Proof Using Definition 2.1(b), (3.118), and (3.119), we deduce that there exists a sequence .{θ n }n with .θ n = {θnm }m ⊂ R+ for any .n ∈ N, such that θnm → 0

.

as

n → ∞,

∀m ∈ N

(3.120)

and, for any .m, n ∈ N, the following inequality holds: Λun (t) − un (t)X ≤ θnm (un (t)X + 1)

.

∀ t ∈ [0, m].

(3.121)

Let .m ∈ N, .t ∈ [0, m], and .n ∈ N. We write un (t)X ≤ un (t) − Λun (t)X + Λun (t) − Λ0X (t)X + Λ0X (t)X ,

.

3.5 The Case of Almost History-Dependent Operators

129

and then we use inequalities (3.121) and (3.107) to find that  un (t)X ≤ θnm (un (t)X + 1) + l m un (t)X + Lm

t

.

un (s)X ds + Λ0X (t)X

0

or, equivalently,  (1 − θnm − l m )un (t)X ≤ θnm + Lm

t

.

un (s)X ds + Λ0X (t)X .

(3.122)

0

Next, (3.120) and the inequality .l m < 1 imply that for n large enough we can m m m assume that . 1−l 2 ≤ 1 − θn − l . Therefore, with the notation F m = max Λ0X (t)X ,

.

t∈[0,m]

(3.123)

inequality (3.122) yields un (t)X ≤

.

2Lm 2(θnm + F m ) + 1 − lm 1 − lm



t

un (s)X ds.

0

We now use the Gronwall argument to find that un (t)X ≤

.

2(θnm + F m ) 2Lmm t e 1−l 1 − lm

and, using the convergence (3.120) together with inequality .t ≤ m, we conclude the proof. 

Next, we recall notation (2.2) and remark that, obviously, .ST1 ⊂ ST2 ⊂ ST4 and, moreover, .ST3 ⊂ ST4 . In addition, using Lemma 3.3, it is easy to see that .ST4 ⊂ ST3 and, therefore, .ST3 = ST4 . We now use Definition 2.2 to see that T1 ≤ T2 ≤ T3

.

and

T3 ∼ T4 .

(3.124)

We are now in a position to introduce our main result in this subsection. Theorem 3.9 Assume that .Λ : C(R+ ; X) → C(R+ ; X) is an almost historydependent operator. Then Problem .P ah is well-posed with the Tykhonov triples .T1 , .T2 , .T3 , and .T4 in Examples 3.6, 3.7, 3.8, and 3.9, respectively. Proof First, we recall that the existence of a unique solution to Problem .P ah , needed for the well-posedness of Problem .P ah with any Tykhonov triple, follows from Theorem 1.4.

130

3 Fixed Point Problems

Consider now the Tykhonov triple .T3 in Example 3.8 and let .{un } be a .T3 approximating sequence. Then, using Definition 2.1(b), we deduce that there exists a sequence .{θ n }n with .θ n = {θnm }m ⊂ R+ for all .n ∈ N, such that (3.120) holds and, moreover, Λun (t) − un (t)X ≤ θnm

.

∀ t ∈ [0, m].

(3.125)

Let .n ∈ N, .m ∈ N and let .t ∈ [0, m]. We use equality .Λu(t) = u(t) and write un (t) − u(t)X ≤ un (t) − Λun (t)X + Λun (t) − Λu(t)X ,

.

then we use inequalities (3.125) and (3.107) to find that  un (t) − u(t)X ≤ θnm + l m un (t) − u(t)X + Lm

t

.

un (s) − u(s)X ds

0

or, equivalently,  (1 − l m )un (t) − u(t)X ≤ θnm + Lm

t

.

un (s) − u(s)X ds.

0

Therefore, un (s) − u(t)X ≤

.

θnm Lm + m 1−l 1 − lm



t

un (s) − u(s)X ds,

0

which implies that un (t) − u(t)X ≤

.

Lm θnm t 1−l m . e 1 − lm

(3.126)

We now combine inequality (3.126) and convergence (3.120) to see that .

max un (t) − u(t)X → 0 as n → ∞

t∈[0,m]

and, since .m ∈ N is arbitrary, using (1.5) we deduce that un → u in C(R+ ; X) as n → ∞.

.

(3.127)

The convergence (3.127) implies that Problem .P ah is well-posed with the Tykhonov triple .T3 in Example 3.8. The well-posedness of Problem .P ah with the Tykhonov triples .T1 , .T2 , and .T4 is now a direct consequence of relations (3.124), (2.4), and (2.5). 

3.5 The Case of Almost History-Dependent Operators

131

3.5.2 Convergence Results In this subsection, we study the stability of the solution of Problem .P ah with respect to perturbations of the operator .Λ. To this end, for each .n ∈ N, we consider the following fixed point problem: Problem .Pnah . Given an operator .Λn : C(R+ ; X) → C(R+ ; X), find a function .un ∈ C(R+ ; X) such that Λn un (t) = un (t)

.

for all t ∈ R+ .

(3.128)

We assume that for each .n ∈ N the operator .Λn is an almost history-dependent operator, i.e.,

.

⎧ For each n, m ∈ N there exist lnm ∈ [0, 1), Lm ⎪ n > 0 such that ⎪ ⎪ ⎪ ⎨ Λn u(t)− Λn v(t)X ≤ lnm u(t) − v(t)X ⎪ ⎪ ⎪ ⎪ ⎩

+Lm n

t

u(s) − v(s)X ds

(3.129)

0

∀ u, v ∈ C(R+ ; X), t ∈ [0, m].

Then, using Theorem 1.4, it follows that Problem .Pnah has a unique solution .un ∈ C(R+ ; X), for each .n ∈ N. Our aim in what follows is to state various conditions that link the operators .Λn and .Λ and guarantee that the solution of Problem .Pnah converges to the solution of Problem .P ah , that is, un → u in C(R+ ; X),

.

as n → ∞.

(3.130)

To this end, we use the Tykhonov triples .T1 , .T2 , .T3 above together with the two steps strategy described on page 93, namely: (i) First, we impose various conditions that guarantee that the sequence .{un } is a .Ti -approximating sequence for some .i ∈ {1, 2, 3}. (ii) Second, we use the .Ti -well-posedness of Problem .P ah , obtained in Theorem 3.9, to deduce that the convergence (3.130) holds. The conditions we consider are the following. ⎧ ⎨ For each n ∈ N there exists θn > 0 such that: (a) . Λn v(t) − Λv(t)X ≤ θn ∀ v ∈ C(R+ ; X), t ∈ R+ . ⎩ (b) θn → 0 as n → ∞. ⎧ For each n ∈ N there exists θn > 0 such that: ⎪ ⎪ ⎨ (a) Λn v(t) − Λv(t)X ≤ θn (v(t)X + 1) . ⎪ ∀ v ∈ C(R+ ; X), t ∈ R+ . ⎪ ⎩ (b) θn → 0 as n → ∞.

.

(3.131)

(3.132)

132

3 Fixed Point Problems

⎧ For each n, m ∈ N there exists θnm > 0 such that: ⎪ ⎪  t ⎪  ⎪ ⎨ (a) Λn v(t) − Λv(t)X ≤ θnm 1 + v(t)X + v(s)X ds 0 ⎪ ⎪ ∀ v ∈ C(R+ ; X), t ∈ [0, m]. ⎪ ⎪ ⎩ (b) θnm → 0 as n → ∞, for each m ∈ N.

(3.133)

We have the following convergence result. Theorem 3.10 Assume that .Λ, Λn : C(R+ ; X) → C(R+ ; X) are almost historydependent operators, for each .n ∈ N. Then: (a) Under assumption (3.131), the sequence .{un } is a .T1 -approximating sequence. Moreover, the convergence (3.130) holds. (b) Under assumption (3.132), the sequence .{un } is a .T2 -approximating sequence. Moreover, the convergence (3.130) holds. (c) Under assumption (3.133), the sequence .{un } is a .T3 -approximating sequence. Moreover, the convergence (3.130) holds. Proof (a) Let .n ∈ N and .t ∈ R+ . We use (3.128) and (3.131)(a) to write Λun (t) − un (t)X ≤ Λun (t) − Λn un (t)X ≤ θn .

.

This inequality combined with assumption (3.131)(b) and definitions (3.108)– (3.110) shows that the sequence .{un } is a .T1 -approximating sequence. The convergence (3.130) is now a consequence of the .T1 -well-posedness of Problem ah , guaranteed by Theorem 3.9. .P (b) Let .n ∈ N and .t ∈ R+ . We use (3.128) and (3.132)(a) to write Λun (t) − un (t)X ≤ Λun (t) − Λn un (t)X ≤ θn (un (t)X + 1).

.

This inequality combined with assumption (3.132)(b) and definitions (3.111)– (3.113) shows that in this case the sequence .{un } is a .T2 -approximating sequence. The convergence (3.130) is now a consequence of the .T2 -wellposedness of Problem .P ah , guaranteed by Theorem 3.9. (c) We first prove that for each .m ∈ N there exists .U m > 0 such that un (t)X ≤ U m

.

∀ t ∈ [0, m], n ∈ N.

(3.134)

Indeed, let .m ∈ N, .n ∈ N and let .t ∈ [0, m]. We use (3.128) and write un (t)X ≤ Λn un (t) − Λun (t)X + Λun (t) − Λ0X (t)X + Λ0X (t)X ,

.

and then we use inequalities (3.133)(a) and (3.107) to find that

3.5 The Case of Almost History-Dependent Operators

.

un (t)X ≤

θnm



133



t

1 + un (t)X +

un (s)X ds



0



t

+l m un (t)X + Lm

un (s)X ds + Λ0X (t)X .

0

Combining this inequality with notation (3.123), we obtain that  (1 − θnm − l m )un (t)X ≤ θnm + (Lm + θnm )

t

.

m .θn

un (s)X ds + F m .

(3.135)

0

Next, (3.133)(b) and inequality .l m < 1 imply that for n large enough we have m m m < 1 and . 1−l 2 ≤ 1 − θn − l and, therefore, (3.135) yields 2(F m + 1) 2(Lm + 1) .un (t)X ≤ + 1 − lm 1 − lm



t

un (s)X ds.

0

We now use the Gronwall argument to find that un (t)X ≤

.

2(F m + 1) 2(Lm m+1) t e 1−l 1 − lm

and, using inequality .t ≤ m, we conclude that (3.134) holds with Um =

.

2(F m + 1) 2(Lm m+1) m . e 1−l 1 − lm

Next, we use (3.128) and (3.133)(a) to write Λun (t) − un (t)X ≤

.

θnm

 t  1 + un (t)X + un (s)X ds 0

and, using the bound (3.134), we find that Λun (t) − un (t)X ≤ θnm (1 + U m + U m m).

.

(3.136)

Consider now the sequence .ωn = {ωnm }m ⊂ R+ defined by ωnm = θnm (1 + U m + U m m)

.

(3.137)

for each .m, n ∈ N. Then inequality (3.136) implies that .un ∈ Ω3 (ωn ) where, recall, .Ω3 (ω) is the set defined by (3.115), for each .ω = {ωm }m ⊂ R+ . On the other hand, assumption (3.133)(b) and definition (3.137) imply that .ωnm → 0 as m .n → ∞, for each .m ∈ N. This shows that .ω n = {ωn }m ∈ C3 , where .C3 is given by (3.116). It follows from above that the sequence .{un } is a .T3 -approximating

134

3 Fixed Point Problems

sequence for Problem .P ah . We now use Theorem 3.9 and Definition 2.1(c) to deduce the convergence (3.130), which concludes the proof. 

We now remark that assumption (3.132) does not guarantee that the sequence {un } is a .T1 -approximating sequence. Moreover, assumption (3.133) does not guarantee that the sequence .{un } is a .T2 -approximating sequence. An evidence of these statements is provided by the two examples below.

.

Example 3.10 Let .X = R and let .Λn , Λ : C(R+ ; X) → C(R+ ; X) be the operators defined by 

t

Λn u(t) =

.

0

1 1 , u(t) + u(s) ds + n+1 n+1



t

Λu(t) =

u(s) ds

(3.138)

0

for all .u ∈ C(R+ ; X), .t ∈ R+ , and .n ∈ N. Then, it is easy to see that .Λn and .Λ are almost history-dependent operators that satisfy condition (3.132). Moreover, with this choice, the solution of Problem .Pnad is given by un (t) =

.

1 n+1 t e n n

∀ n ∈ N, t ∈ R+ .

(3.139)

Arguing by contradiction, assume that the sequence .{un } is a .T1 -approximating sequence. Then, there exists .{θn } ⊂ R+ such that .θn → 0 and |Λun (t) − un (t)| ≤ θn

.

∀ n ∈ N, t ∈ R+ .

(3.140)

Using now (3.138)–(3.140), we deduce that .

n+1 1 1 e n t+ ≤ θn n(n + 1) n+1

∀ n ∈ N, t ∈ R+ .

We now take .t = n2 in the previous inequality, and then we pass to the limit as .n → ∞ and arrive to a contradiction. We conclude from here that .{un } is not a .T1 -approximating sequence. Nevertheless, Theorem 3.10(b) implies that .{un } is a .T2 -approximating sequence. Therefore, with the notation in Definition 2.2(c), we have .T1 < T2 . Example 3.11 Let .X = R, and let .Λn , Λ : C(R+ ; X) → C(R+ ; X) be the operators defined by 1 .Λn u(t) = 1 − n



t

u(s) ds,

Λu(t) = 1

(3.141)

0

for all .u ∈ C(R+ ; X), .t ∈ R+ , and .n ∈ N. Then, it is easy to see that .Λn and .Λ are history-dependent operators that satisfy condition (3.133). Moreover, with this choice, the solutions of Problems .Pnad and .P ah are given by

3.5 The Case of Almost History-Dependent Operators t

un (t) = e− n

∀ n ∈ N, t ∈ R+ ,

.

135

u(t) = 1 ∀ t ∈ R+ ,

respectively. Arguing by contradiction, assume that the sequence .{un } is a .T2 -approximating sequence. Then, there exists .{θn } ⊂ R+ such that .θn → 0 and |Λun (t) − un (t)| ≤ θn (|un (t)| + 1)

.

∀ n ∈ N, t ∈ R+ .

(3.142)

Using now (3.141)–(3.142), we deduce that t

t

1 − e− n ≤ θn (e− n + 1)

.

∀ n ∈ N, t ∈ R+ .

We now take .t = n in the previous inequality, and then we pass to the limit as n → ∞ and arrive to a contradiction. We conclude from here that .{un } is not a .T2 -approximation sequence. Nevertheless, Theorem 3.10(c) implies that .{un } is a .T3 -approximating sequence. We deduce from here that .T2 < T3 . .

It follows from Example 3.10 that, under condition (3.132), the .T1 -wellposedness of Problem .P ad cannot be used to provide the convergence (3.130), in the framework of the two steps strategy described on page 131. Similarly, Example 3.11 shows that, under condition (3.133), the .T2 -well-posedness of Problem .P ad cannot be used to provide the convergence (3.130), in the framework of the same strategy. We conclude from above that the choice of the Tykhonov triple plays a crucial role in proving convergence results.

Chapter 4

Variational Inequalities

In this chapter we present well-posedness results for variational inequalities. We consider both elliptic and history-dependent inequalities, for which we prove their well-posedness with various Tykhonov triples. This allows us to deduce convergence results which, in particular, show the dependence of the solution with respect to the data. We then provide examples of well-posed split and dual variational inequalities. Finally, we discuss the well-posedness of a mixed elliptic variational inequality. Everywhere in this chapter, we assume that X is a Hilbert space, unless stated otherwise.

4.1 Elliptic Variational Inequalities In this section we study the well-posedness of an elliptic variational inequality with a specific Tykhonov triple and provide various convergence results.

4.1.1 A Well-Posedness Result We consider the functional framework defined in Sect. 1.2.3. Therefore, .K ⊂ X, A : X → X, .ϕ : X × X → R, and .f ∈ X. Then, we formulate the following inequality problem.

.

Problem .P Find an element .u ∈ K such that (Au, v − u)X + ϕ(u, v) − ϕ(u, u) ≥ (f, v − u)X

.

∀ v ∈ K.

(4.1)

The unique solvability of the variational inequality (4.1) is given by Theorem 1.11 on page 26, under assumptions (1.47)–(1.51). Our aim in what follows © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Sofonea, Well-Posed Nonlinear Problems, Advances in Mechanics and Mathematics 50, https://doi.org/10.1007/978-3-031-41416-9_4

137

138

4 Variational Inequalities

is to study the well-posedness of this problem in order to deduce some relevant  be a set and G be an operator such that convergence results. To this end, let .K .

 is a nonempty, closed, convex subset of X.. K

(4.2)

G : X → X is a monotone Lipschitz continuous operator.

(4.3)

Moreover, for each .θ = (λ, ε) ∈ (0, +∞) × [0, +∞), consider the following inequality problem.  such that Problem .Pθ Find an element .u ∈ K .

1 (Gu, v − u)X + ϕ(u, v) − ϕ(u, u) λ  +εv − uX ≥ (f, v − u)X ∀ v ∈ K.

(Au, v − u)X +

(4.4)

Our preliminary result in this section is the following. Proposition 4.1 Assume (1.47)–(1.51), (4.2), and (4.3). Then, for each .θ =  to Problem (λ, ε) ∈ (0, +∞) × [0, +∞), there exists at least one solution .u ∈ K .Pθ . Moreover, the solution is unique if .ε = 0. Proof Let .θ = (λ, ε) ∈ (0, +∞) × [0, +∞). Assumptions (1.48) and (4.3) and the inequality .λ > 0 imply that the operator .A + λ1 G : X → X is Lipschitz  satisfies continuous and strongly monotone. On the other hand, recall that the set .K condition (4.2). It follows from above that we are in a position to use Theorem 1.11  and .A + 1 G instead of K and A, respectively. In this way, we deduce the with .K λ  such that existence of a unique element .u ∈ K .

1 (Gu, v − u)X + ϕ(u, v) − ϕ(u, u) λ  ≥ (f, v − u)X ∀ v ∈ K.

(Au, v − u)X +

(4.5)

This proves the unique solvability of Problem .Pθ in the case when .ε = 0. Next, for .ε > 0, it follows that the solution u of (4.5) satisfies inequality (4.4), too. This proves the existence of at least one solution to Problem .Pθ and concludes the proof.

We now consider the triple .T = (I, Ω, C), defined as follows: .

I = (0, +∞) × [0, +∞), .    such that (4.4) holds ∀ θ = (λ, ε) ∈ I, . Ω(θ ) = u ∈ K   C = {θn } ⊂ I : θn = (λn , εn ) ∀ n ∈ N, λn → 0, εn → 0 .

(4.6) (4.7) (4.8)

4.1 Elliptic Variational Inequalities

139

We use Proposition 4.1 to see that .Ω(θ ) = ∅ for each .θ ∈ I , which shows that .T is a Tykhonov triple in the sense of Definition 2.1(a). Next, we consider the following additional assumptions: .

. K ⊂ K.

(4.9)

 v ∈ K.. (Gu, v − u)X ≤ 0 ∀ u ∈ K, ⎧ ⎨ One of the two conditions below holds:  = X and u ∈ X, Gu = 0X ⇒ u ∈ K. (a) K ⎩  (Gu, v − u)X = 0 ∀ v ∈ K ⇒ u ∈ K. (b) u ∈ K,  For each u ∈ K, there exists cϕ (u) ≥ 0 such that ϕ(u, v1 ) − ϕ(u, v2 ) ≤ cϕ (u)v1 − v2 X ∀ v1 , v2 ∈ X.

(4.10) (4.11)

.

(4.12)

Our main result in this section is the following. Theorem 4.1 Assume (1.47)–(1.51), (4.2), (4.3), and (4.9)–(4.12). Then, Problem P is well-posed with the Tykhonov triple .T (4.6)–(4.8).

.

Proof Denote by u the solution of Problem .P, and let .{un } be a .T -approximating sequence. Then, there exists a sequence .{θn } such that .θn = (λn , εn ) with .λn > 0, .εn ≥ 0 for all .n ∈ N, .

λn → 0, .

(4.13)

εn → 0,

(4.14)

and moreover, .

 un ∈ K,

(Aun , v − un )X +

1 (Gun , v − un )X λn

+ϕ(un , v) − ϕ(un , un ) + εn v − un X ≥ (f, v − un )X

(4.15)  ∀ v ∈ K.

Let .n ∈ N. We start by considering the auxiliary problem of finding an element  such that  un ∈ K

.

.

(A un , v −  un )X +

1 (G un , v −  un )X + ϕ(u, v) − ϕ(u,  un ) λn

≥ (f, v −  un )X

(4.16)

 ∀ v ∈ K.

Note that the variational inequality (4.16) is similar to the variational inequality (4.15), with the difference arising on the fact that in (4.16) the first argument of .ϕ is the solution u of Problem .P and, in addition, .εn = 0. The existence of a unique solution to inequality (4.16) follows from Corollary 1.4, by using the same

140

4 Variational Inequalities

arguments as those used in the proof of the unique solvability of inequality (4.5). Next, we divide the rest of the proof into four steps, as follows:  and a subsequence of .{ Step (i) We prove that there exist an element . u∈K un }, still un }, such that . un   u in X, as .n → ∞. denoted by .{ To prove this statement, we show that the sequence .{ un } is bounded in X. Let n ∈ N and let .u0 be a given element in K. We use assumption (4.9) to deduce that

.

.

(A un ,  un − u0 )X ≤

1 (G un , u0 −  un )X + ϕ(u, u0 ) − ϕ(u,  un ) λn

+(f,  un − u0 )X . Then, by the strong monotonicity of the operator A, we obtain that .

mA  un − u0 2X ≤ (Au0 , u0 −  un )X +

1 (G un , u0 −  un )X λn

(4.17)

+ϕ(u, u0 ) − ϕ(u,  un ) + (f,  un − u0 )X . Next, assumption (4.10) implies that 1 (G un , u0 −  un )X ≤ 0 λn

(4.18)

ϕ(u, u0 ) − ϕ(u,  un ) ≤ cϕ (u) un − u0 X .

(4.19)

.

and assumption (4.12) yields .

On the other hand, obviously, (Au0 , u0 −  un )X + (f,  un − u0 )X ≤ f − Au0 X  un − u0 X .

.

(4.20)

We now combine inequalities (4.17)–(4.20) to find that mA  un − u0 X ≤ cϕ (u) + f − Au0 X .

.

The previous inequality implies that .{ un } is a bounded sequence in X. Therefore, by the reflexivity of X, there exist an element . u ∈ X and a subsequence of .{ un }, still  for each .n ∈ N. Then, un }, such that . un   u in X. Recall that . un ∈ K denoted by .{  assumption (4.2) implies that . u ∈ K. Step (ii) We prove that . u is the solution to Problem .P, i.e., . u = u.  We now use To prove this statement, we consider an element .v ∈ K. inequality (4.16) and assumptions (1.48)(a) and (4.12) to see that

4.1 Elliptic Variational Inequalities

.

141

1 (G un ,  un − v)X ≤ (A un − Av, v −  un )X + ϕ(u, v) − ϕ(u,  un ) λn +(f,  un − v)X + (Av, v −  un )X ≤ ϕ(u, v) − ϕ(u,  un ) + (f − Av,  un − v)X ≤ cϕ (u) un − vX + f − AvX  un − vX = (cϕ (u) + f − AvX ) un − vX .

Then, due to the boundedness of sequence .{ un } we deduce that there exists a constant .D > 0 which does not depend on n such that (G un ,  un − v)X ≤ λn D.

.

Passing to the upper limit in the above inequality and using assumption (4.13), we have .

lim sup (G un ,  un − v)X ≤ 0.

(4.21)

 in (4.21), we deduce that Taking now .v =  u∈K .

lim sup (G un ,  un −  u)X ≤ 0.

Then, using the convergence . un   u in X and the pseudomonotonicity of G, guaranteed by assumption (4.3) and Proposition 1.11, we deduce that (G u,  u − v)X ≤ lim inf (G un ,  un − v)X .

.

(4.22)

We now combine inequalities (4.21) and (4.22) to see that (G u,  u − v)X ≤ 0.

.

(4.23)

 Recall that this inequality is valid for any .v ∈ K. Assume that condition (4.11)(a) is satisfied. Then, inequality (4.23) implies that .(G u,  u − v)X ≤ 0 for all .v ∈ X, which yields .G u = 0X , and using (4.11)(a), again, we find that . u ∈ K. Assume now that condition (4.11)(b) is satisfied. Then, using conditions (4.9) and (4.23), we obtain that (G u,  u − v)X ≤ 0

.

∀ v ∈ K.

On the other hand, from the assumption (4.10), we have (G u, v −  u)X ≤ 0

.

∀ v ∈ K.

142

4 Variational Inequalities

The last two inequalities imply that .(G u, v −  u)X = 0 for all .v ∈ K, and using (4.11)(b), we infer that . u ∈ K. We conclude from above that if either (4.11)(a) or (4.11)(b) holds, then  u ∈ K.

(4.24)

.

Let .n ∈ N. Then, using (4.9) and inequality (4.16), we find that .

(A un , v −  un )X +

1 (G un , v −  un )X λn

+ϕ(u, v) − ϕ(u,  un ) ≥ (f, v −  un )X

∀ v ∈ K.

Therefore, assumption (4.10) yields (A un ,  un − v)X ≤ ϕ(u, v) − ϕ(u,  un ) + (f,  un − v)X

.

∀ v ∈ K.

We now pass to the upper limit in this inequality and use the convergence . un   u in X and the lower semicontinuity of .ϕ with respect to its second argument to infer that .

lim sup (A un ,  un − v)X ≤ ϕ(u, v) − ϕ(u,  u) + (f,  u − v)X

∀ v ∈ K.

(4.25)

We take .v =  u in (4.25) to find that .

lim sup (A un ,  un −  u)X ≤ 0.

(4.26)

Therefore, using the pseudomonotonicity of the operator A, we deduce that .

lim inf (A un ,  un − v)X ≥ (A u,  u − v)X

∀ v ∈ X.

(4.27)

We now combine inequalities (4.27) and (4.25) to obtain that (A u,  u − v)X ≤ ϕ(u, v) − ϕ(u,  u) + (f,  u − v)X

.

∀ v ∈ K.

(4.28)

Finally, we use (4.24) and (4.28) to see that . u is a solution to Problem .P, and by the uniqueness of the solution, we have that . u = u, as claimed. Step (iii) We prove the convergence of the whole sequence .{ un } to u. A careful analysis of the proof in step (ii) reveals that every subsequence of .{ un } which converges weakly in X has the same weak limit u. Moreover, we recall that the sequence .{ un } is bounded in X. Therefore, using Theorem 1.3, we deduce that the whole sequence .{ un } converges weakly in X to u, as .n → ∞. This shows that all the statements in step (ii) are valid for the whole sequence .{ un }. Now, assumption (1.48)(a) yields

4.1 Elliptic Variational Inequalities

143

mA  un − u2X ≤ (A un − Au,  un − u)X ,

.

which implies that mA  un − u2X ≤ (A un ,  un − u)X + (Au, u −  un )X .

.

(4.29)

We take .v =  u in (4.27) and use (4.26) together with equality . u = u to see that (A un ,  un − u)X → 0. We now use this convergence, the convergence . un  u in X, and inequality (4.29) to find that

.

 un − uX → 0,

.

(4.30)

which shows that . un → u in X as .n → ∞, as claimed. Step (iv) We prove that .un → u in X, as .n → ∞. Let .n ∈ N. We test with .v =  un in (4.15) and .v = un in (4.16), and then we add the resulting inequalities to see that .

(Aun − A un , un −  un )X ≤

1 (G un − Gun , un −  un )X λn

+ϕ(un ,  un ) − ϕ(un , un ) + ϕ(u, un ) − ϕ(u,  un ) +εn  un − un . Next, using assumptions (4.3) and (1.49)(b), we deduce that .

(Aun − A un , un −  un )X ≤ αϕ un − uX  un − un X + εn  un − un X ,

and therefore, the strong monotonicity of the operator A yields mA  un − un X ≤ αϕ un − uX + εn .

.

We now write αϕ un − uX ≤ αϕ un −  un X + αϕ  un − uX

.

and substitute this inequality in (4.31) to deduce that (mA − αϕ ) un − un X ≤ αϕ  un − uX + εn .

.

Then, using the smallness assumption (1.50), we obtain that .

 un − un X ≤

αϕ εn .  un − uX + mA − αϕ mA − αϕ

(4.31)

144

4 Variational Inequalities

This inequality combined with the convergences (4.30) and (4.14) implies that  un − un X → 0.

.

(4.32)

Finally, writing un − uX ≤ un −  un X +  un − uX

.

and using the convergences (4.30) and (4.32), we deduce that .un → u in X, which concludes the proof.

We end this subsection with the remark that Theorem 4.1 represents an extension of Theorem 1.18 on page 40. Indeed, the well-posedness result in Theorem 1.18 can  = K, the operator G be recovered from Theorem 4.1 in the particular case when .K vanishes, and the function .ϕ does not depend on the solution u.

4.1.2 Convergence Results In this subsection we use Theorem 4.1 to deduce various convergence results to the solution of inequality (4.1). To this end, we assume in what follows that (1.47)– (1.51) hold, even if we do not mention it explicitly, and we denote by u the solution of Problem .P obtained in Theorem 1.11.  = X and .εn = (a) A first penalty method. Our first convergence result is when .K 0 for each .n ∈ N. Thus, for each .n ∈ N, let .λn > 0 and consider the following problem. Problem .Pn Find an element .un ∈ X such that (Au . n , v − un )X +

1 (Gun , v − un )X + ϕ(un , v) − ϕ(un , un ) λn

≥ (f, v − un )X

(4.33)

∀ v ∈ X.

The following result represents a direct consequence of Proposition 4.1 and Theorem 4.1. Corollary 4.1 Assume (1.47)–(1.51), (4.3), and (4.12). Moreover, assume that for any .u ∈ X, the following hold: (Gu, v − u)X ≤ 0

.

∀ v ∈ K,

Gu = 0X ⇒ u ∈ K.

Then Problem .Pn has a unique solution, for each .n ∈ N. In addition, if .λn → 0, then .un → u in X.

4.1 Elliptic Variational Inequalities

145

A brief comparison between inequalities (4.1) and (4.33) shows that (4.34) is obtained from (4.1) by replacing the set K with the set X and the operator A with the operator .A + λ1n G, in which .λn is a penalty parameter. For this reason, we refer to (4.33) as a (classical) penalty problem of (4.1). Corollary 4.1 guarantees the unique solvability of the penalty problem as well as the convergence of its solution to the solution of Problem .P, as the penalty parameter converges to zero.  = X and (b) A second penalty method. Our second convergence result is when .K .εn = 0 for each .n ∈ N. For each .n ∈ N, let .λn > 0 and consider the following problem. n Find an element .un ∈ K  such that Problem .P (Au . n , v − un )X +

1 (Gun , v − un )X + ϕ(un , v) − ϕ(un , un ) λn

(4.34)

 ∀ v ∈ K.

≥ (f, v − un )X

We now use Proposition 4.1 and Theorem 4.1, again, to obtain the following result. Corollary 4.2 Assume (1.47)–(1.51), (4.2), (4.3), (4.9), (4.10), (4.11)(b), and n has a unique solution .un , for each .n ∈ N. In addition, (4.12). Then Problem .P if .λn → 0, then .un → u in X. A brief comparison between inequalities (4.1) and (4.34) shows that (4.34) is  and the operator A with obtained from (4.1) by replacing the set K with the set .K 1 the operator .A + λn G, in which .λn is a penalty parameter. For this reason, we refer to (4.34) as a penalty problem of (4.1). Corollary 4.2 establishes the link between the solutions of these problems. Roughly speaking, it shows that, in the limit when .n → ∞, a partial relaxation of the set of constraints can be compensated by a convenient perturbation of the nonlinear operator which governs Problem .P.  = K, G (c) A continuous dependence result. Our third particular case is when .K vanishes, and f is replaced by .fn . We assume that .

fn ∈ X

∀ n ∈ N, .

(4.35)

fn → f

in X,

(4.36)

and for each .n ∈ N, we consider the following problem. Problem .P n Find an element .un ∈ K such that (Aun , v. − un )X + ϕ(un , v) − ϕ(un , un ) ≥ (fn , v − un )X

∀ v ∈ K.

We have the following existence, uniqueness, and convergence result.

(4.37)

146

4 Variational Inequalities

Corollary 4.3 Assume (1.47)–(1.51), (4.12), (4.35), and (4.36). Then Problem .P n has a unique solution .un , for each .n ∈ N. In addition, .un → u in X. Proof The existence and uniqueness part follows from Proposition 4.1. Let .n ∈ N. Then, using (4.37), it is easy to see that .

(Aun , v − un )X + ϕ(un , v) − ϕ(un , un ) +(f − fn , v − un )X ≥ (f, v − un )X

∀ v ∈ K,

and denoting .εn = f − fn X , it follows that .

(Aun , v − un )X + ϕ(un , v) − ϕ(un , un ) +εn v − un X ≥ (f, v − un )X

∀ v ∈ K.

Moreover, since G vanishes, it follows that conditions (4.3) and (4.10) hold. In  = K, it follows that conditions (4.2) and (4.11) also hold, and addition, since .K finally, assumption (4.36) implies that .εn → 0. We are now in a position to use Theorem 4.1 with .λn = n1 , for instance, to deduce the convergence .un → u in X,

which concludes the proof. Remark 4.1 A direct proof of the convergence result .un → u in X in Corollary 4.3 can be given, based on assumptions (1.48)(a) and (1.49)(b). It follows from here that assumption (4.12) can be skipped in the statement of this corollary. Note that Corollary 4.3 represents a continuous dependence result of the solution to Problem .P with respect to the element f . Various additional convergence results can be obtained by considering perturbation of the operator A and function .ϕ. References in the field include the papers [249, 255] where such kind of results have been obtained in the more general case of variational–hemivariational inequalities. We end this section with an example of operator G which satisfies conditions (4.3), (4.10), and (4.11)  are nonempty closed convex subsets of X such Example 4.1 Assume that K and .K  and denote by .PK : X → K and .PK : X → K  the projection operators that .K ⊂ K  respectively. Then, we claim that the operator .G : X → X on the sets K and .K, given by G = 2IX − PK − PK

.

(4.38)

satisfies conditions (4.3), (4.10), and (4.11). Indeed, using the nonexpansivity property of the projection operator (see (1.40)), we deduce that the operators .IX − PK and .IX − PK are monotone and Lipschitz continuous. Therefore, using (4.38), it follows that G satisfies condition (4.3).  and .v ∈ K, using equalities .P u = u, .PK v = v, and the Next, for any .u ∈ K K monotonicity of the operator .IX − PK , we have

4.2 History-Dependent Variational Inequalities

147

(Gu, v − u)X = ((u − PK u) − (v − PK v), v − u)X ≤ 0,

.

which shows that G satisfies condition (4.10).  = X, and note that in this case, we have .Gu = u − PK u. Assume now that .K Then, the equality .Gu = 0X implies that .u = PK u, and therefore, .u ∈ K. Assume  and .(Gu, v − u)X = 0 for all .v ∈ K. We take .v = PK u in this now that .u ∈ K equality, and since .Gu = u − PK u, we deduce that .(u − PK u, PK u − u)X ≤ 0. We conclude from here that .u − PK u2K ≤ 0, which implies that .u = PK u and, again, .u ∈ K. It follows from here that the operator G satisfies condition (4.11), too.

4.2 History-Dependent Variational Inequalities In this section we study the well-posedness of a history-dependent variational inequality with various Tykhonov triples and provide a number of convergence results.

4.2.1 Well-Posedness Results Let .K ⊂ X, .A : X → X, .S : C(R+ ; X) → C(R+ ; X), and .f : R+ → X. With these data, the history-dependent inequality we study in this section is stated as follows. Problem .P. Find a function .u ∈ C(R+ ; K) such that, for all .t ∈ R+ , the following inequality holds: (Au(t), v − u(t))X + (Su(t), v − u(t))X ≥ (f (t), v − u(t))X

.

∀ v ∈ K.

(4.39)

Recall that here and below, as usual, we use the short-hand notation .Su(t) to represent the value of the function .Su at the point t, i.e., .Su(t) = (Su)(t). In the study of this problem, we assume (1.47) and (1.48) and, moreover

.

⎧ S : C(R+ ; X) → C(R+ ; X) is a history-dependent operator, i.e., ⎪ ⎪ ⎪ m ⎪ ⎨ for each m ∈ N, there exists

L > 0 such that t

(4.40)

f ∈ C(R+ ; X).

(4.41)

⎪ Su(t) − Sv(t)X ≤ Lm u(s) − v(s)X ds ⎪ ⎪ ⎪ 0 ⎩ ∀ u, v ∈ C(R+ ; X), t ∈ [0, m]. .

148

4 Variational Inequalities

Recall that Corollary 1.6 on page 27 guarantees the unique solvability of Problem P. We now provide three relevant Tykhonov triples that are useful in the study of Problem .P.

.

Example 4.2 Take .T1 = (I1 , Ω1 , C1 ) where I1 = R+ = [0, +∞), .  C1 = {θn }n : θn ∈ I1 ∀ n ∈ N,

.

θn → 0 as



n→∞ ,

(4.42) (4.43)

and for each .θ ∈ I1 , the set .Ω1 (θ ) is defined as follows: ⎧ ⎨ u ∈ Ω1 (θ ) if and only if u ∈ C(R+ ; K) and . (Au(t), v − u(t))X + (Su(t), v − u(t))X + θ v − u(t)X ⎩ ≥ (f (t), v − u(t))X ∀ v ∈ K, t ∈ R+ .

(4.44)

Note that for each .θ ∈ I1 the solution u obtained in Corollary 1.6 belongs to .Ω1 (θ ), and therefore, .Ω1 (θ ) = ∅. Hence, according to Definition 2.1(a), .T1 is a Tykhonov triple. Example 4.3 Take .T2 = (I2 , Ω2 , C2 ) where .

I2 = R+ = [0, +∞), .  C2 = {θn }n : θn ∈ I ∀ n ∈ N,



θn → 0

as

n→∞ ,

(4.45) (4.46)

and for each .θ ∈ I2 , the set .Ω2 (θ ) is defined as follows: ⎧ u ∈ Ω2 (θ ) if and only if u ∈ C(R+ ; K) and ⎪ ⎪ ⎨ (Au(t), v − u(t))X + (Su(t), v − u(t))X . ⎪ +θ (u(t)X + 1)v − u(t)X ⎪ ⎩ ≥ (f (t), v − u(t))X ∀ v ∈ K, t ∈ R+ .

(4.47)

Note that, again, using Corollary 1.6, it follows that .Ω2 (θ ) = ∅, for each .θ ∈ I2 . Example 4.4 Take .T3 = (I3 , Ω3 , C3 ) where .

   I3 = θ = θ m }m : θ m ∈ R+ ∀ m ∈ N , .  C3 = {θ n }n : θ n = {θnm }m ∈ I3 ∀ n ∈ N, θnm → 0 as

n → ∞,

(4.48) (4.49) 

∀m ∈ N ,

and for each .θ = {θ m }m ∈ I3 , the set .Ω3 (θ ) is defined as follows:

4.2 History-Dependent Variational Inequalities

149

⎧ ⎨ u ∈ Ω3 (θ ) if and only if u ∈ C(R+ ; K) and . (Au(t), v − u(t))X + (Su(t), v − u(t))X + θ m v − u(t)X ⎩ ≥ (f (t), v − u(t))X ∀ v ∈ K, t ∈ [0, m], m ∈ N.

(4.50)

Note that, again, using Corollary 1.6, it follows that .Ω3 (θ) = ∅, for each .θ ∈ I3 . Our main result in this subsection is the following. Theorem 4.2 Assume (1.47), (1.48), (4.40), and (4.41). Then Problem .P is wellposed with the Tykhonov triples .T1 , .T2 , and .T3 in Examples 4.2, 4.3, and 4.4, respectively. Proof First, we recall that the existence of a unique solution to Problem .P, needed for the well-posedness of Problem .P with any Tykhonov triple, follows from Corollary 1.6. Consider now the Tykhonov triple .T3 in Example 4.4, and let .{un } be a .T3 approximating sequence. Then, using Definition 2.1(b), we deduce that there exists a sequence .{θ n }n such that .θ n = {θnm }m ⊂ R+ for all .n ∈ N, θnm → 0

.

as

n → ∞,

∀ m ∈ N,

(4.51)

and, moreover, .un ∈ Ω3 (θ n ) for each .n ∈ N. Let .n ∈ N and .m ∈ N and let .t ∈ [0, m]. Using the definition (4.50) of the set .Ω3 (θ n ), it follows that .un ∈ C(R+ ; K) and, moreover, .

(Aun (t), v − un (t))X + (Sun (t), v − un (t))X +θnm v

− un (t)X ≥ (f (t), v − un (t))X

(4.52) ∀ v ∈ K.

We now take .v = u(t) in (4.52) and then .v = un (t) in (4.39) and add the resulting inequalities to find that .

(Sun (t) − Su(t), u(t) − un (t))X + θnm un (t) − u(t)X ≥ (Aun (t) − Au(t), un (t) − u(t))X .

Therefore, using assumption (1.48)(a), we deduce that .

mA un (t) − u(t)2X ≤ Sun (t) − Su(t)X un (t) − u(t)X + θnm un (t) − u(t)X .

This inequality combined with condition (4.40) yields Lm θnm .un (t) − u(t)X ≤ + mA mA

0

t

un (s) − u(s)X ds,

150

4 Variational Inequalities

and using Lemma 1.1, it follows that un (t) − u(t)X ≤

.

θnm mLm t e A . mA

(4.53)

Inequality (4.53) and convergence (4.51) show that max un (t) − u(t)X → 0 as n → ∞,

.

t∈[0,m]

and since .m ∈ N is arbitrary, using (1.5), we deduce that un → u in C(R+ ; X) as n → ∞.

.

(4.54)

The convergence (4.54) combined with Definition 2.1(c) implies that Problem .P is well-posed with respect to the Tykhonov triple .T3 in Example 4.4. We now focus on the Tykhonov triple .T2 in Example 4.3, and to this end, we use a claim that we state here and prove at the end of this subsection. Claim 1 Let .{un } be a .T2 -approximating sequence. Then, for each .m ∈ N, there exists .Z m > 0 such that un (t)X ≤ Z m

.

∀ t ∈ [0, m], n ∈ N.

(4.55)

Assume now that .{un } is a .T2 -approximating sequence. Then, using Definition 2.1(b) and (4.47), we deduce that there exists a sequence .{αn } ⊂ R+ such that .αn → 0, and for any .n ∈ N, the following inequality holds: .

(Aun (t), v − un (t))X + (Sun (t), v − un (t))X

(4.56)

+αn (un (t)X + 1)v − un (t)X ≥ (f (t), v − un (t))X

∀ v ∈ K, t ∈ R+ .

Consider the sequence .{θ n } where θ n = {θnm }m ⊂ R+ ,

.

θnm = αn (Z m + 1)

∀ m, n ∈ N.

(4.57)

Then, (4.56) and (4.57) imply that .

(Aun (t), v − u(t))X + (Sun (t), v − un (t))X + θnm v − un (t)X ≥ (f (t), v − un (t))X

∀ v ∈ K, m, n ∈ N, t ∈ [0, m],

and since .θnm → 0 as .n → ∞, for any .m ∈ N, (4.49) and (4.50) imply that .{un } is a .T3 -approximating sequence. We now use the notation (2.2) to conclude that

4.2 History-Dependent Variational Inequalities

151

S T2 ⊂ S T3 .

(4.58)

.

On the other hand, using (4.44) and (4.47), it is easy to see that .Ω1 (θ ) ⊂ Ω2 (θ ) for any .θ ≥ 0, and therefore, (2.2) implies that S T1 ⊂ S T2 .

(4.59)

.

Finally, since Problem .P is .T3 -well-posed, we deduce from (2.3) that S T3 ⊂ S P ,

(4.60)

.

where, recall, .SP is the set of sequences defined by (2.1). The well-posedness of Problem .P with respect to the Tykhonov triples .T1 and .T2 is now a direct consequence of (2.3) and inclusions (4.58)–(4.60).

We end this subsection with the proof of the bound (4.55). Proof of Claim 1. Let .{un } be a .T2 -approximating sequence. Then, using Definition 2.1(b), (4.46), and (4.47), we deduce that there exists a sequence .{θn } ⊂ R+ such that .θn → 0, and for any .n ∈ N, the following inequality holds: .

(Aun (t), v − un (t))X + (Sun (t), v − un (t))X

(4.61)

+θn (un (t)X + 1)v − un (t)X ≥ (f (t), v − un (t))X

∀ v ∈ K, t ∈ R+ .

Let .m ∈ N, .t ∈ [0, m], .n ∈ N, and .u0 ∈ K. Then, (4.61) implies that .

(Aun (t), un (t) − u0 )X ≤ (Sun (t), u0 − un (t))X +θn (un (t)X + 1)u0 − un (t)X + (f (t), un (t) − u0 )X .

Using this inequality and assumption (1.48)(a), we write .

mA un (t) − u0 2X ≤ (Aun (t), un (t) − u0 )X − (Au0 , un (t) − u0 )X ≤ (Sun (t), u0 − un (t))X + θn (un (t)X + 1)u0 − un (t)X +(f (t), un (t) − u0 )X + (Au0 , u0 − un (t))X ,

which implies that .

mA un (t) − u0 X ≤ Sun (t)X + θn (un (t)X + 1) + f (t)X + Au0 X

and, moreover,

152

4 Variational Inequalities .

mA un (t) − u0 X ≤ Sun (t) − Su0 (t)X + Su0 (t)X

(4.62)

+θn (un (t) − u0 X + u0 X + 1) + f (t)X + Au0 X . Recall that here and below in this book we keep the notation .u0 for the constant function .t → u0 for all .t ∈ R+ , and therefore, the notation .Su0 makes sense. Next, since .θn → 0, for n large enough, we can assume that .θn ≤ m2A and, therefore (4.62) yields mA un (t) − u0 X ≤ Sun (t) − Su0 (t)X + Su0 (t)X 2 mA + (u0 X + 1) + f (t)X + Au0 X . 2

.

(4.63)

Denote  mA (u0 X + 1) + f (t)X + Au0 X . Su0 (t)X + t∈[0,m] 2

F m = max

.

(4.64)

Then, using (4.63), (4.64), and assumption (4.40), we find that .

2Lm un (t) − u0 X ≤ mA



t

un (s) − u0 X ds +

0

2F m . mA

We now use the Gronwall argument to find that un (t) − u0 X ≤

.

m 2F m 2L t e mA . mA

This inequality implies that there exists .Y m > 0 which does not depend on n and t such that un (t) − u0 X ≤ Y m .

.

(4.65)

We now use (4.65) to obtain the bound (4.55) with .Z m = Y m + u0 X , which concludes the proof.



4.2.2 Convergence Results In this subsection we use the well-posedness of Problem .P with respect to the Tykhonov triples .T1 , .T2 , and .T3 in order to deduce several convergence results. These results are presented as corollaries of Theorem 4.2. Below in this section we keep the assumptions of this theorem, even if we do not mention it explicitly.

4.2 History-Dependent Variational Inequalities

153

Our first convergence result concerns the dependence of the solution with respect to the function f . To this end, we consider a sequence of functions .{fn } such that, for each .n ∈ N, fn ∈ C(R+ ; X).

.

(4.66)

Moreover, we consider the following variational problem. Problem .Pn1 Find a function .un ∈ C(R+ ; K) such that, for all .t ∈ R+ , the following inequality holds: .

(Aun (t), v − un (t))X + (Sun (t), v − un (t))X ≥ (fn (t), v − un (t))X

(4.67)

∀ v ∈ K.

Then, using Corollary 1.6, it follows that Problem .Pn1 has a unique solution, for each .n ∈ N. Assume now that ⎧ ⎨ (a) For each n ∈ N there exists θn > 0 such that . (4.68) fn (t) − f (t)X ≤ θn ∀ t ∈ R+ . ⎩ (b) θn → 0 as n → ∞. We have the following convergence result. Corollary 4.4 Assume (1.47), (1.48), (4.40), (4.41) (4.66), and (4.68). Then, the solution .un of Problem .Pn1 converges to the solution u of Problem .P, i.e., un → u in C(R+ ; X).

.

(4.69)

Proof Consider the sequence .{θn } provided by assumption (4.68)(a) and let .n ∈ N, t ∈ R+ . Inequality (4.67) implies that

.

.

(Aun (t), v − un (t))X + (Sun (t), v − un (t))X + (f (t) − fn (t), v − un (t))X ≥ (f (t), v − un (t))X

∀ v ∈ K,

and the use of (4.68)(a) yields (Aun (t), . v − un (t))X + (Sun (t), v − un (t))X + θn v − un (t)X ≥ (f (t), v − un (t))X

(4.70)

∀ v ∈ K.

We now combine (4.70) and (4.44) to see that .un ∈ Ω1 (θn ), and therefore, (4.68)(b) implies that .{un } is a .T1 -approximating sequence for Problem .P. We now use Theorem 4.2 and Definition 2.1(c) to deduce the convergence (4.69), which concludes the proof.



154

4 Variational Inequalities

Our second convergence result concerns the dependence of the solution with respect to the operator A. To this end, we consider a sequence of operators .{An } such that, for each .n ∈ N, the following condition holds:

.

⎧ An : X → X is a strongly monotone Lipschitz continuous ⎪ ⎪ ⎨ operator, i.e., there exist mn , Mn > 0 such that: ⎪ (a) (An u − An v, u − v)X ≥ mn u − v2X ∀ u, v ∈ X. ⎪ ⎩ (b) An u − An vX ≤ Mn u − vX ∀ u, v ∈ X.

(4.71)

Moreover, we consider the following variational problem. Problem .Pn2 Find a function .un ∈ C(R+ ; K) such that, for all .t ∈ R+ , the following inequality holds: .

(An un (t), v − un (t))X + (Sun (t), v − un (t))X ≥ (f (t), v − un (t))X

(4.72)

∀ v ∈ K.

Then, using Corollary 1.6, it follows that Problem .Pn2 has a unique solution, for each n ∈ N. Consider now the following condition:

.

⎧ ⎨ (a) For each n ∈ N there exists θn > 0 such that . An u − AuX ≤ θn (uX + 1) ∀ u ∈ X. ⎩ (b) θn → 0 as n → ∞.

(4.73)

We have the following convergence result. Corollary 4.5 Assume (1.47), (1.48), (4.40), (4.41), (4.71), and (4.73). Then, the solution .un of Problem .Pn2 converges to the solution u of Problem .P, i.e., (4.69) holds. Proof Consider the sequence .{θn } provided by assumption (4.73)(a) and let .t ∈ R+ , n ∈ N. We write

.

.

(Aun (t), v − un (t))X + (Sun (t), v − un (t))X = (An un (t), v − un (t))X + (Sun (t), v − un (t))X +(Aun (t) − An un (t), v − un (t))X

∀ v ∈ K,

then we use inequality (4.72) to find that .

(Aun (t), v − un (t))X + (Sun (t), v − un (t))X ≥ (f (t), v − un (t))X + (Aun (t) − An un (t), v − un (t))X ≥ (f (t), v − un (t))X − Aun (t) − An un (t)X v − un (t)X

∀ v ∈ K,

4.2 History-Dependent Variational Inequalities

155

and the use of (4.73)(a) yields (Aun (t), v − un (t))X + (Sun (t), v − un (t))X

.

+θn (un (t)X + 1)v − un (t)X ≥ (f (t), v − un (t))X

(4.74) ∀ v ∈ K.

We now combine (4.74) and (4.47) to see that .un ∈ Ω2 (θn ), and therefore, (4.73)(b) implies that .{un } is a .T2 -approximating sequence for Problem .P. Finally, we use Theorem 4.2 and Definition 2.1(c) to deduce the convergence (4.69), which concludes the proof.

Our third convergence result concerns the dependence of the solution with respect to the operator .S. To this end, we consider a sequence of operators .{Sn } such that, for each .n ∈ N, the following condition holds:

.

⎧ Sn : C(R+ ; X) → C(R+ ; X) is a history-dependent operator, i.e., ⎪ ⎪ ⎪ ⎪ ⎨ for any m ∈ N, there exists Lm n > 0 such that t

⎪ Sn u(t) − Sn v(t)X ≤ Lm u(s) − v(s)X ds ⎪ n ⎪ ⎪ 0 ⎩ ∀ u, v ∈ C(R+ ; X), t ∈ [0, m].

(4.75)

Moreover, we consider the following variational problem. Problem .Pn3 Find a function .un ∈ C(R+ ; K) such that, for all .t ∈ R+ , the following inequality holds: .

(Aun (t), v − un (t))X + (Sn un (t), v − un (t))X ≥ (f (t), v − un (t))X

(4.76)

∀ v ∈ K.

Then, using Corollary 1.6, it follows that Problem .Pn3 has a unique solution, for each .n ∈ N. Assume now that ⎧ m (a) For each m ∈ N and n ∈ N there ⎪ ⎪

t exists αn > 0 such that ⎪ ⎪ ⎨ Sn u(t) − Su(t)X ≤ αnm u(s)X ds . (4.77) 0 ⎪ ⎪ ∀ u ∈ C(R ; X), t ∈ [0, m]. ⎪ + ⎪ ⎩ (b) αnm → 0 as n → ∞, for each m ∈ N. We have the following convergence result. Corollary 4.6 Assume (1.47), (1.48), (4.40), (4.41), (4.75), and (4.77). Then, the solution .un of Problem .Pn3 converges to the solution u of Problem .P, i.e., (4.69) holds.

156

4 Variational Inequalities

Proof Let .n ∈ N, .m ∈ N, and .t ∈ [0, m] and let .v ∈ K. We write (Aun (t), v − un (t))X + (Sun (t), v − un (t))X

.

= (Aun (t), v − un (t))X + (Sn un (t), v − un (t))X +(Sun (t) − Sn un (t), v − un (t))X , and then we use inequality (4.76) to see that .

(Aun (t), v − un (t))X + (Sun (t), v − un (t))X ≥ (f (t), v − un (t))X + (Sun (t) − Sn un (t), v − un (t))X .

Therefore, .

(Aun (t), v − un (t))X + (Sun (t), v − un (t))X +(Sn un (t) − Sun (t), v − un (t))X ≥ (f (t), v − un (t))X ,

and using assumption (4.77)(a), we find that .

(Aun (t), v − un (t))X + (Sun (t), v − un (t))X  t m +αn un (s)X ds v − un (t)X ≥ (f (t), v − un (t))X .

(4.78)

0

Next, we use a claim that we state here and prove below in this subsection. Claim 2 For .n ∈ N, denote by .un the solution of Problem .Pn3 . Then, for each m > 0 such that .m ∈ N, there exists .U un (t)X ≤ U m

.

∀ t ∈ [0, m], n ∈ N.

(4.79)

We now use (4.78) and (4.79) to see that .

(Aun (t), v − un (t))X + (Sun (t), v − un (t))X

(4.80)

+αnm U m mv − un (t)X ≥ (f (t), v − un (t))X . Consider now the sequence .θ n = {θnm }m ⊂ R+ , where θnm = αnm U m m

.

(4.81)

for each .m, n ∈ N. Then (4.80) implies that .un ∈ Ω3 (θ n ), where, recall, .Ω3 (θ ) is the set defined by (4.50), for each .θ = {θ m }m ⊂ R+ . On the other hand, assumption (4.77)(b) and definition (4.81) imply that .θnm → 0 as .n → ∞, for each m .m ∈ N. This implies that .θ n = {θn }m ∈ C3 , where .C3 is given by (4.49). It follows from above that the sequence .{un } is a .T3 -approximating sequence for Problem .P. We now use Theorem 4.2 and Definition 2.1(c) to deduce the convergence (4.69), which concludes the proof.



4.2 History-Dependent Variational Inequalities

157

We now proceed with the proof of the bound (4.79). Proof of Claim 2. Let .m ∈ N, .t ∈ [0, m], .n ∈ N, and .u0 ∈ K. Then, using (4.76), we find that (Aun (t), un (t) − u0 )X ≤ (Sn un (t), u0 − un (t))X + (f (t), un (t) − u0 )X .

.

Using this inequality and assumption (1.48)(a), we write 2 mA u . n (t) − u0 X ≤ (Aun (t), un (t) − u0 )X − (Au0 , un (t) − u0 )X

≤ (Sn un (t), u0 − un (t))X + (f (t), un (t) − u0 )X + (Au0 , u0 − un (t))X , which implies that .

mA un (t) − u0 2X ≤ (Sn un (t) − Sun (t), u0 − un (t))X +(Sun (t) − Su0 (t), u0 − un (t))X + (Su0 (t), u0 − un (t))X +(f (t), un (t) − u0 )X + (Au0 , u0 − un (t))X

and, moreover, .

mA un (t) − u0 X ≤ Sn un (t) − Sun (t)X + Sun (t) − Su0 (t)X +Su0 (t)X + f (t)X + Au0 X .

We now use assumptions (4.77) and (4.40) to find that

t m . mA un (t) − u0 X ≤ αn un (s)X ds 0



t

+Lm

un (s) − u0 X ds + Su0 (t)X + f (t)X + Au0 X ,

0

and therefore,

.

mA un (t) − u0 X ≤ (αnm + Lm )

+αnm

t

t

un (s) − u0 X ds

(4.82)

0

u0 X ds + Su0 (t)X + f (t)X + Au0 X .

0

On the other hand, assumption (4.77)(b) shows that for n large enough we can assume that .αnm ≤ 1, and using this inequality in (4.82), we find that

.

t

mA un (t) − u0 X ≤ (Lm + 1)

un (s) − u0 X ds

0

+m u0 X + Su0 (t)X + f (t)X + Au0 X .

(4.83)

158

4 Variational Inequalities

Denote .

Gm = max

t∈[0,m]

 m u0 X + Su0 (t)X + f (t)X + Au0 X .

(4.84)

Then, (4.83) and (4.84) imply that .

Lm + 1 un (t) − u0 X ≤ mA



t

un (s) − u0 X ds +

0

Gm . mA

We now use the Gronwall argument to find that un (t) − u0 X ≤

.

Gm Lmm +1 t e A . mA

This inequality implies that there exists .V m > 0 which does not depend on n and t such that un (t) − u0 X ≤ V m .

.

(4.85)

We now use (4.85) to obtain the bound (4.79) with .U m = V m + u0 X , which concludes the proof.

We end this section with a convergence result that includes as particular cases the convergence results presented in Corollaries 4.4–4.6. To this end, we consider two sequences of operators .{An } and .{Sn } and a sequence of functions .{fn } such that, for each .n ∈ N, the conditions (4.66), (4.71), and (4.75) hold. Moreover, we consider the following variational problem. Problem .Pn4 Find a function .un ∈ C(R+ ; K) such that, for all .t ∈ R+ , the following inequality holds: .

(An un (t), v − un (t))X + (Sn un (t), v − un (t))X ≥ (fn (t), v − un (t))X

(4.86)

∀ v ∈ K.

Then, using Corollary 1.6, it follows that Problem .Pn4 has a unique solution, for each .n ∈ N. Consider now the convergence fn → f

.

in C(R+ ; X).

(4.87)

We have the following convergence result. Corollary 4.7 Assume (1.47), (1.48), (4.40), (4.41), (4.66), (4.71), (4.73), (4.75), (4.77), and (4.87). Then, the solution .un of Problem .Pn4 converges to the solution u of Problem .P, i.e., (4.69) holds.

4.2 History-Dependent Variational Inequalities

159

Proof The proof is similar to that of Corollary 4.6, and therefore, we skip the details. We restrict ourselves to recall that it is based on the following ingredients: first, we prove that inequality (4.79) still holds where now .un is the solution to history-dependent inequality (4.86), for each .n ∈ N; then, we prove that the sequence .{un } is a .T3 -approximating sequence for Problem .P; and finally, we use Theorem 4.2 and Definition 2.1(c) to deduce the convergence (4.69), which concludes the proof.



4.2.3 Two Examples In this subsection we complete the results in Sect. 4.2.2 with two examples and some comments. First, we state that Corollary 4.5 cannot be proved by using the wellposedness of Problem .P with the Tykhonov triple .T1 in Example 4.2. Moreover, Corollary 4.6 cannot be proved by using the well-posedness of Problem .P with the Tykhonov triple .T2 in Example 4.3. An evidence of these statements is provided by the two examples below. Example 4.5 Consider Problem .P and the Tykhonov triple .T1 in Example 4.2 in the particular case when .K = X and .Au = u for all .u ∈ X, .S ≡ 0, and .f (t) = tf0 for all .t ∈ R+ , where .f0 ∈ X, .f0 = 0X . Note that in this particular case inequality (4.39) becomes (u(t), v − u(t))X ≥ (f (t), v − u(t))X

.

∀ v ∈ X, t ∈ R+ .

(4.88)

Assume now that .An u = Au+ n1 u, for each .n ∈ N. Then, inequality (4.72) becomes (un (t) +

.

1 un (t), v − un (t))X ≥ (f (t), v − un (t))X n

∀ v ∈ X, t ∈ R+ .

(4.89)

Next, it is easy to see that conditions (4.71) and (4.73) hold, and therefore, Corollary 4.5 guarantees the convergence (4.69). This convergence can be proved directly. Indeed, the solutions of inequalities n (4.88) and (4.89) are .u(t) = f (t) and .un (t) = n+1 f (t), respectively, for all .t ∈ R+ and .n ∈ N. Therefore, un (t) − u(t)X =

.

1 f (t)X n+1

∀ t ∈ R, n ∈ N,

which implies (4.69). Nevertheless, we claim that the sequence .{un } is not a .T1 -approximating sequence for Problem .P. Indeed, arguing by contradiction, assume that .{un } is a .T1 -approximating sequence. Then, using (4.43) and (4.44), we deduce that there exists a sequence .{θn } ⊂ R+ such that .θn → 0, and for each .n ∈ N and .t ∈ R+ , the following inequality holds:

160

4 Variational Inequalities

(un (t), v − un (t))X + θn v − un (t)X ≥ (f (t), v − un (t))X

.

∀ v ∈ X.

(4.90)

n f (t) in (4.90), and then we take .v = f (t) in the We now substitute .un (t) = n+1 resulting inequality to deduce that

.

1 f (t)X ≤ θn . n+1

Moreover, since .f (t) = tf0 , we find that .

t f0 X ≤ θn . n+1

Recall that this inequality holds for each .n ∈ N and .t ∈ R+ . Thus, taking .t = n + 1, we deduce that .f0 X ≤ θn for each .n ∈ N, and using the convergence .θn → 0, we find that .f0 = 0X , which is in contradiction with assumption .f0 = 0X . We conclude from above that the sequence .{un } is not a .T1 -approximating sequence for Problem .P, as claimed. This implies that the .T1 -well-posedness of inequality (4.39) combined with Definition 2.1(c) cannot be used to prove Corollary 4.5. Example 4.6 Consider Problem .P and the Tykhonov triple .T2 in Example 4.3 in the particular case when .K = X and .Au = u for all .u ∈ X, .S ≡ 0, and .f (t) = f0 for all .t ∈ R+ , where .f0 = 0X . Note that in this particular case inequality (4.39) becomes (u(t), v − u(t))X ≥ (f0 , v − u(t))X

.

∀ v ∈ X, t ∈ R+ .

(4.91)

Assume now that

t

Sn u(t) = αn

u(s) ds

.

0

∀ u ∈ C(R+ ; X), t ∈ R+

for each .n ∈ N, where .αn > 0 and, moreover, .αn → 0 as .n → ∞. Then, inequality (4.76) becomes

.

t

(un (t) + αn

un (s) ds, v − un (t))X ≥ (f0 , v − un (t))X

(4.92)

0

for all .v ∈ X, .t ∈ R+ , and .n ∈ N. Next, it is easy to see that conditions (4.75) m and (4.77) hold with .Lm n = αn = αn , and therefore, Corollary 4.6 guarantees the convergence (4.69). This convergence can be proved directly. Indeed, inequality (4.92) is equivalent to the integral equation

t

un (t) + αn

.

0

un (s) ds = f0

∀ t ∈ R+ ,

4.2 History-Dependent Variational Inequalities

161

and therefore, its solution is .un (t) = e−αn t f0 for all .t ∈ R+ and .n ∈ N. On the other hand, the solution of inequality (4.91) is .u(t) = f0 , for all .t ∈ R+ . Therefore, 

un (t) − u(t)X = 1 − e−αn t f0 X

.

∀ t ∈ R+ ,

which implies (4.69). Nevertheless, we claim that the sequence .{un } is not a .T2 -approximating sequence for Problem .P. Indeed, arguing by contradiction, assume that .{un } is a .T2 -approximating sequence. Then, using (4.46) and (4.47), we deduce that there exists a sequence .{θn } ⊂ R+ such that .θn → 0, and for each .n ∈ N and .t ∈ R+ , the following inequality holds: .

(un (t), v − un (t))X + θn (un (t)X + 1)v − un (t)X ≥ (f0 , v − un (t))X

(4.93)

∀ v ∈ X.

We now substitute .un (t) = e−αn t f0 on (4.93), and then we take .v = f0 in the resulting inequality to deduce that



 1 − e−αn t f0 X ≤ θn e−αn t f0 X + 1 .

.

Recall that this inequality holds for each .n ∈ N and .t ∈ R+ . Thus, taking .t = α1n , we deduce that .(1 − e−1 )f0 X ≤ θn (e−1 f0 X + 1) for each .n ∈ N. We now use the convergence .θn → 0 to deduce that .f0 = 0X , which contradicts the assumption .f0 = 0X . We conclude from above that the sequence .{un } is not a .T2 -approximating sequence for Problem .P, as claimed. Therefore, the .T2 -wellposedness of the inequality (4.39) cannot be used to prove Corollary 4.6. Examples 4.5 and 4.6 show that, in the framework of the strategy described on page 93, the choice of the Tykhonov triple plays a crucial role in obtaining convergence results. Indeed, it follows from above that the Tykhonov triple .T1 generates enough approximating sequences to guarantee the proof of Corollary 4.4, but, on the other hand, it does not generate enough approximating sequences to be used in the proof of Corollary 4.5. Similarly, the Tykhonov triple .T2 generates enough approximating sequences to guarantee the proof of Corollary 4.5, but it does not generate enough approximating sequences to be used in the proof of Corollary 4.6. In addition, the inclusions (4.58) and (4.59) show that the Tykhonov triples .T2 and .T3 can be used in the proof of Corollary 4.4 and the Tykhonov triple .T3 can be used in the proof of Corollary 4.5. It follows from here that, among the Tykhonov triples .T1 , .T2 , and .T3 , the Tykhonov triple .T3 is the most convenient in the analysis of the history-dependent variational inequality (4.39) since it can be used to obtain the largest variety of convergence results for this inequality.

162

4 Variational Inequalities

4.3 Split and Dual Variational Inequalities In this section we present examples of well-posed split and dual variational inequalities in Hilbert spaces. Such kind of inequalities arise in the study of contact problems with unilateral constraints for elastic, viscoelastic, or rate-type viscoplastic materials. A representative example will be provided in Sect. 9.2 of this book.

4.3.1 A Split History-Dependent Variational Inequality In this subsection we present an example of well-posed split variational inequality with history-dependent operators. Let .T > 0 and denote by .C([0, T ]; X) the space of continuous functions on .[0, T ] with values in X, endowed with its canonical norm (1.2). Following Definition 1.5, we recall that an operator .Λ : C([0, T ]; X) → C([0, T ]; X) is said to be history-dependent if there exists .LΛ > 0 such that ⎧ ⎨ .

⎩

Λu1 (t) − Λu2 (t)X ≤ LΛ

t

u1 (s) − u2 (s)X ds

0

(4.94)

∀ u1 , u2 ∈ C([0, T ]; X), t ∈ [0, T ].

The split problem we consider is governed by a set .K ⊂ X, two operators A : X → X and .B : [0, T ] × X × X → X, and a function .p : [0, T ] → X, which are assumed to satisfy the following conditions:

.

.

K is a. nonempty, convex, and closed set of X.. ⎧ g such that ⎨ There exists an element  . (a)  g ∈ K. ⎩ (b) 2v −  g ∈ K ∀ v ∈ K. ⎧ ⎨ A : V → V is a linear operator and there exist LA , mA > 0 such that ⎩ AuV ≤ LA uV , (Au, u)V ≥ mA u2V ∀ u ∈ V . ⎧ B : [0, T ] × X × X → X is such that: ⎪ ⎪ ⎪ ⎪ ⎪ (a) The mapping t → B(t, u, σ ) : [0, T ] → X is continuous, ⎪ ⎪ ⎪ ⎪ ∀ u ∈ X, σ ∈ X. ⎨ (b) There exists LB > 0 such that ⎪ ⎪ ⎪ B(t, u1 , σ1 ) − B(t, u2 , σ2 )X ⎪ ⎪ ⎪ ⎪ ≤ LB (u1 − u2 X + σ1 − σ2 X ) ⎪ ⎪ ⎩ ∀ u1 , u2 ∈ X, σ1 , σ2 ∈ X, t ∈ [0, T ].

(4.95)

p ∈ C([0, T ]; X).

(4.99)

.

(4.96)

(4.97)

(4.98)

4.3 Split and Dual Variational Inequalities

163

Note that the assumption (4.97) implies that the operator A is invertible and its inverse .A−1 : X → X is linear and positively defined and satisfies the inequality A−1 σ X ≤

.

1 σ X mA

Next, consider the implicit integral equation

t .σ (t) = Au(t) + B(s, u(s), σ (s)) ds

∀ σ ∈ X.

∀ t ∈ [0, T ],

(4.100)

(4.101)

0

for which we recall the following result. Theorem 4.3 Assume that X is a Banach space and (4.97) and (4.98) hold. Then there exists a history-dependent operator .S : C([0, T ]; X) → C([0, T ]; X) such that for all functions .u ∈ C([0, T ]; X) and .σ ∈ C([0, T ]; X), equality (4.101) holds if and only if σ (t) = Au(t) + Su(t)

.

∀ t ∈ [0, T ].

(4.102)

Theorem 4.3 represents a particular case of two more general results obtained in [222]. For instance, its proof can be obtained by taking .Y = X in Theorems 34 and 35 in [222]. The importance of this theorem arises from the fact that it allows us to express one of the unknowns of the implicit equation (4.101) as a function of the other one. Moreover, using Theorem 1.5 on page 9, we deduce that there exists a history-dependent operator .R : C([0, T ]; X) → C([0, T ]; X) such that for all functions .u ∈ C([0, T ]; X) and .σ ∈ C([0, T ]; X), equality (4.102) holds if and only if u(t) = A−1 σ (t) + Rσ (t)

.

∀ t ∈ [0, T ].

(4.103)

Therefore, we deduce the existence (and, obviously, the uniqueness) of two operators .S and .R which will be involved in the statement of the split problem we consider in this subsection. Note that the explicit expressions for the operators .S and .R can be derived in several particular cases, as shown in [222]. Finally, for each .t ∈ [0, T ], we define the set   .Σ(t) = τ ∈ X : (τ, v −  g )X ≥ (p(t), v −  g )X ∀ v ∈ K . (4.104) With these data, we consider the following problem. Problem .M Find .x = (u, σ ) ∈ C([0, T ]; X × X) such that, for all .t ∈ [0, T ], u(t) ∈ . K, (Au(t) + Su(t), v − u(t))X ≥ (p(t), v − u(t))X

∀ v ∈ K, . (4.105)

(τ − σ (t), A−1 σ (t) + Rσ (t) −  g )X ≥ 0 ∀ τ ∈ Σ(t), . (4.106)

t σ (t) = Au(t) + B(s, u(s), σ (s)) ds. (4.107) σ (t) ∈ Σ(t),

0

164

4 Variational Inequalities

It is easy to see that Problem .M is a split variational inequality of the form M = M(P, Q, G, f ) defined on page 71 in which the spaces X, Y , and Z are replaced by the space .C([0, T ]; X). Here, Problem .P is the history-dependent variational inequality (4.105), Problem .Q is given by the history-dependent variational inequality (4.106), the operator .G : C([0, T ]; X) × C([0, T ]; X) → C([0, T ]; X) is defined by

.

G(u, σ )(t) = Au(t) +

.

t

B(s, u(s), σ (s)) ds − σ (t),

0

and finally, f is the zero element of the space .C([0, T ]; X). The sets of solutions of 0 and .S 0 , as usual. these inequalities will be denoted by .SP Q We now associate with Problems .P and .Q the Tykhonov triples .TP = (IP , ΩP , CP ) and .TQ = (IQ , ΩQ , CQ ) defined as follows: .

IP = IQ = R+ = [0, +∞),   CP = CQ = {θn } ⊂ R+ : θn → 0 as n → ∞ ,  ΩP (θ ) = u ∈ C([0, T ]; X) : u(t) ∈ K, (Au(t) + Su(t) − p(t), v − u(t))X ≥ −θ u(t) − vX

.

(4.108)  ∀ v ∈ K, t ∈ [0, T ]

∀ θ ≥ 0,  ΩQ (ω) = σ ∈ C([0, T ]; X) : σ (t) ∈ Σ(t), (τ − σ (t), A−1 σ (t) + Rσ (t) −  g )X ≥ −ωσ (t) − τ X ∀ τ ∈ Σ(t), t ∈ [0, T ]



∀ ω ≥ 0. Note that .ΩP (θ ) = ∅ for all .θ ≥ 0 and .ΩQ (ω) = ∅ for all .θ ≥ 0, as it follows from the solvability of inequalities (4.105) and (4.106), guaranteed by the proof of Theorem 4.4 below. We now follow the construction on page 72 and consider the Tykhonov triple .TM = (IM , IM , CM ) defined as follows: IM =. R+ × R+ × R+ ,  ΩM (θ ) = x = (u, σ ) : u ∈ ΩP (θ ), σ ∈ ΩQ (ω),

t     B(s, u(s), σ (s)) ds − σ (t) ≤ ε Au(t) + 0



∀ θ = (θ, ω, ε) ∈ IM ,

X

∀ t ∈ [0, T ]



 CM = {θ n } ⊂ R3+ : θ n = (θn , ωn , εn ) ∀ n ∈ N, θn , ωn , εn → 0 .

4.3 Split and Dual Variational Inequalities

165

Our result in the study of Problem .M is the following. Theorem 4.4 Assume (4.95)–(4.99). Then, the split history-dependent variational inequality .M is .TM -well-posed. Proof We shall use Theorem 2.8, and to this end, we start by proving the wellposedness of the history-dependent variational inequalities (4.105) and (4.106). First, since X is a Hilbert space and .S is a history dependent operator, Corollary 1.6 guarantees that problem (4.105) has a unique solution .u ∈ C([0, T ]; X). Assume now that .{un } ⊂ C([0, T ]; X) is a .TP -approximating sequence, i.e., .un ∈ ΩP (θn ) with .0 ≤ θn → 0. Let .n ∈ N and .t ∈ [0, T ]. We use (4.105) and (4.108) to see that .

(Au(t) + Su(t) − p(t), un (t) − u(t))X ≥ 0, (Aun (t) + Sun (t) − p(t), u(t) − un (t))X ≥ −θn un (t) − u(t)X .

We now add the above two inequalities to find that .

(Aun (t) − Au(t), un (t) − u(t))X

(4.109)

≤ θn un (t) − u(t)X + (Sun (t) − Su(t), u(t) − un (t))X . On the other hand, since .S : C([0, T ]; X) → C([0, T ]; X) is a history-dependent operator, inequality (4.94) implies that

t

Sun (t) − Su(t)X ≤ LS

.

un (s) − u(s)X ds

(4.110)

0

with some .LS > 0. We now combine (4.109) and (4.110) and use assumption (4.97) to obtain

θn LS t .un (t) − u(t)X ≤ + un (s) − u(s)X ds. mA mA 0 We now use the Gronwall argument to deduce that un (t) − u(t)X ≤

.

θn mLS t e A , mA

which, together with (1.2) and the convergence .θn → 0, implies that .un → u in .C([0, T ]; X). Therefore, using Definition 2.1(c), we deduce that the variational inequality (4.105) is .TP -well-posed. Next, for any .t ∈ [0, T ], it is easy to see that set .Σ(t) defined by (4.104) satisfies the equality Σ(t) = p(t) + Σ0 ,

.

(4.111)

166

4 Variational Inequalities

where .Σ0 is the time-independent set given by .

 Σ0 = τ ∈ X : (τ, v −  g )X ≥ 0

 ∀v ∈ K .

(4.112)

Using this property and the regularity (4.99), it is easy to see that a function .σ ∈ C([0; T ]; X) is the solution of the history-dependent inequality (4.106) if and only if the function .σ0 = σ − p : [0, T ] → X is continuous and satisfies the following auxiliary problem: find .σ0 : [0, T ] → X such that .

σ0 (t) ∈ Σ0 ,

(4.113) −1

(τ − σ0 (t), A

−1

σ0 (t) + A

p(t) + R(σ0 (t) + p(t)) −  g )X ≥ 0

∀ τ ∈ Σ0 , t ∈ [0, T ]. On the other hand, it is easy to see that .Σ0 is a nonempty, closed, convex subset of X. Recall also that .A−1 : X → X is a linear continuous positively defined operator and .R : C([0, T ]; X) → C([0, T ]; X) is a history-dependent operator. Therefore, using again Corollary 1.6 and arguments similar to those used in the study of the history-dependent variational inequality (4.105), we deduce that the auxiliary 0 , C ), where problem (4.113) is well-posed with the Tykhonov triple .TQ0 = (IQ , ΩQ Q .

 0 ΩQ (ω) = σ0 ∈ C([0, T ]; X) : σ0 (t) ∈ Σ0 , (τ − σ0 (t), A−1 σ0 (t) + A−1 p(t) + R(σ0 (t) + p(t)) −  g )X  ≥ −ω σ0 (t) − τ X ∀ τ ∈ Σ0 , t ∈ [0, T ] ∀ ω ≥ 0.

Then, using the particular structure (4.111) of the set .Σ(t), we find that the historydependent variational inequality (4.106) is .TQ -well-posed. We denote in what follows by .σ its unique solution. Let .u ∈ C([0, T ]; K) be the solution of inequality (4.105), .t ∈ [0, T ], and moreover, let  σ (t) = Au(t) + Su(t).

(4.114)

.

Then, using the equivalence between equalities (4.102) and (4.103), we deduce that u(t) = A−1 σ (t) + R σ (t),

.

(4.115)

and in addition, (4.105) yields ( σ (t), v − u(t))X ≥ (p(t), v − u(t))X

.

∀ v ∈ K.

(4.116)

4.3 Split and Dual Variational Inequalities

167

We now use assumptions (4.96) (a) and (b) and test successively on (4.116) with v = 2u(t) −  g and .v =  g . As a result, we find that

.

( σ (t), u(t) −  g )X = (p(t), u(t) −  g )X .

.

(4.117)

We now add inequality (4.116) and equality (4.117) to see that ( σ (t), v −  g )X ≥ (p(t), v −  g )X

∀ v ∈ K,

.

which shows that  σ ∈ Σ(t).

(4.118)

.

Let .τ ∈ X. We use relations (4.115) and (4.117) to see that .

(τ −  σ (t), A−1 σ (t) + R σ (t) −  g )X = (τ −  σ (t), u(t) −  g )X = (τ, u(t) −  g )X − (p(t), u(t) −  g )X ,

and since .u(t) ∈ K, definition (4.104) yields (τ −  σ (t), A−1 σ (t) + R σ (t) −  g )X ≥ 0

.

∀ τ ∈ Σ(t).

(4.119)

We now combine relations (4.118) and (4.119) To deduce that the function . σ , which clearly belongs to .C([0, T ]; X), is a solution of the history-dependent variational inequality (4.106). On the other hand, we know that this inequality has a unique solution, previously denoted by .σ . It follows from here that . σ = σ , and therefore, using (4.114), we find that .σ (t) = Au(t) + Su(t). Finally, using Theorem 4.3, we deduce that (4.107) holds. To conclude, we proved that inequalities (4.105) and (4.106) are .TP -wellposed and .TQ -well-posed, respectively, and their solutions are such that (4.107) holds. We are now in a position to use Theorem 2.8(1) to conclude that the split history-dependent variational inequality .M is .TM -well-posed, which completes the proof.



4.3.2 An Example of Dual Variational Inequalities In this subsection we present an example of well-posed dual variational inequalities in Hilbert spaces. To this end, we shall use the abstract framework in Sect. 2.2.2 in the particular case when X is a Hilbert space and .X = Y . We denote by .(·, ·)X and .0X the inner product and the zero element of the space X, respectively, and we consider a strongly monotone Lipschitz continuous operator A defined on the space X, i.e.,

168

4 Variational Inequalities

⎧ ⎪ A : X → X and: ⎪ ⎪ ⎪ ⎪ ⎨ (a) There exists mA > 0 such that . (Av1 − Av2 , v1 − v2 )X ≥ mA v1 − v2 2X ∀ v1 , v2 ∈ X. ⎪ ⎪ ⎪ (b) There exists LA > 0 such that ⎪ ⎪ ⎩ Av1 − Av2 X ≤ LA v1 − v2 X ∀ v1 , v2 ∈ X.

(4.120)

Recall that, since A is strongly monotone Lipschitz continuous on X, it follows from Proposition 1.12 on page 20 that A is invertible and its inverse .A−1 : X → X is strongly monotone and Lipschitz continuous, too. Moreover, we consider a function j such that  .

j : X → R+ is convex, lower semicontinuous, and positively homogeneous, i.e., j (λv) = λj (v) ∀ v ∈ X, λ ≥ 0.

(4.121)

Note that any continuous seminorm on the space X satisfies condition (4.121). Finally, we assume that f ∈X

(4.122)

.

and denote by .Σ the set defined by  Σ = τ ∈ X : (τ, v)X + j (v) ≥ (f, v)X

.

 ∀v ∈ X .

(4.123)

With these data, we consider the following variational problems. Problem .P Find u such that u ∈ . X,

(Au, v − u)X + j (v) − j (u) ≥ (f, v − u)X

∀ v ∈ X.

(4.124)

Problem .Q Find .σ such that .

σ ∈ Σ,

(A−1 σ, τ − σ )X ≥ 0

∀ τ ∈ Σ.

(4.125)

Our first result in this section is the following. Theorem 4.5 Assume (4.120), (4.121), and (4.122). Then Problems .P and .Q are dual of each other. Proof We use Definition 2.6 on page 20 with .D = A : X → X. It follows from Proposition 1.12 that conditions (a) and (b) in this definition are satisfied. Let u be the solution of inequality (4.124) and let .σ = Au, which implies that −1 σ = u. Then, .A (σ, v − u)X + j (v) − j (u) ≥ (f, v − u)X

.

∀ v ∈ X.

(4.126)

4.3 Split and Dual Variational Inequalities

169

We now take .v = 2u and .v = 0X in (4.126) and use assumption (4.121) to deduce that (σ, u)X + j (u) = (f, u)X .

(4.127)

.

Next, we combine relations (4.126) and (4.127) to see that (σ, v)X + j (v) ≥ (f, v)X

∀ v ∈ X,

.

which shows that σ ∈ Σ.

(4.128)

.

Let .τ ∈ Σ. Then, using equality .A−1 σ = u and (4.127), we find that (A−1 σ, τ − σ )X = (τ − σ, u)X = (τ, u)X + j (u) − (f, u)X ,

.

and using definition (4.123), we deduce that (A−1 σ, τ − σ )X ≥ 0.

(4.129)

.

We now combine relations (4.128) and (4.129) to see that .σ is a solution to the variational inequality (4.125). Conversely, assume that .σ is a solution of the variational inequality (4.125) and let .u = A−1 σ . We have .σ = Au, and therefore, the regularity .σ ∈ Σ implies that (Au, v)X + j (v) ≥ (f, v)X

∀ v ∈ X.

.

(4.130)

On the other hand, using equality .σ = Au and (4.125), we find that (τ, u)X ≥ (Au, u)

.

∀ τ ∈ Σ.

(4.131)

Next, assumption (4.121) and Proposition 1.19 guarantee that j is a subdifferentiable function on X, and therefore, there exists an element .ξ ∈ X such that j (v) − j (u) ≥ (ξ, v − u)X

.

∀ v ∈ X.

This implies that the element .τ0 = f − ξ ∈ X satisfies the inequality (τ0 , v − u)X + j (v) − j (u) ≥ (f, v − u)X

.

∀ v ∈ X.

(4.132)

We now take .v = 2u and .v = 0X in this inequality and use assumption (4.121) to deduce that (τ0 , u)X + j (u) = (f, u)X .

.

(4.133)

170

4 Variational Inequalities

Next, we combine (4.132) and (4.133) to see that (τ0 , v)X + j (v) ≥ (f, v)X

.

∀ v ∈ X,

which shows that .τ0 ∈ Σ. This regularity allows us to test with .τ = τ0 in (4.131) to deduce that (τ0 , u)X + j (u) ≥ (Au, u)X + j (u),

.

and using (4.133) yields (f, u)X ≥ (Au, u)X + j (u).

.

Since the converse inequality follows from (4.130), we deduce that (f, u)X = (Au, u)X + j (u).

.

(4.134)

We now use (4.130) and (4.134) to see that u satisfies inequality (4.124), and therefore, u is a solution to Problem .P. We conclude from above that condition (c) in Definition 2.6 is satisfied, too. This

shows that problems .P and .Q are dual of each other. In the study of the well-posedness of these problems, we consider the Tykhonov triples .TP = (I, ΩP , C) and .TQ = (I, ΩQ , C) defined as follows: .

I = R+ = [0, +∞),   C = {θn } ⊂ I : θn → 0 as n → ∞ ,  ΩP (θ ) = u ∈ X : (Au − f, v − u)X + j (v) − j (u)  +θ u − vX ≥ 0 ∀ v ∈ X ,   ΩQ (θ ) = σ ∈ Σ : (A−1 σ, τ − σ )X + θ σ − τ X ≥ 0 ∀ τ ∈ Σ ,

for each .θ > 0. Our second result in this section is the following. Theorem 4.6 Assume (4.120), (4.121), and (4.122). Then Problem .P is .TP -wellposed and Problem .Q is .TQ -well posed. Proof The .TP -well-posedness of Problem .P follows from Theorem 1.18 on page 40. We now move to the study of Problem .Q, and to this end, we claim that the set .Σ is not empty. Indeed, the subdifferentiability of j on X allows us to consider an element .ξ ∈ ∂c j (0X ). Since .j (0X ) = 0, the inclusion .ξ ∈ ∂jc (0X ) yields .j (v) ≥ (ξ, v)X for any .v ∈ X, which implies that the element .τ = f − ξ belongs to .Σ

4.3 Split and Dual Variational Inequalities

171

and proves the claim. On the other hand, it is easy to see that .Σ is a convex closed subset of X. So, since .A−1 : X → X is a strongly monotone Lipschitz continuous operator, we are in a position to use Theorem 1.18 in order to see that Problem .Q is .TQ -well-posed.

We end this subsection with the following remarks. Remark 4.2 The triples .TP and .TQ are not dual Tykhonov triples in the sense of Definition 2.7, i.e., with operator .D = A : X → X. Indeed, for simplicity, assume in what follows that j vanishes. Then, using (4.123), it follows that in this case the set .Σ is a singleton since .Σ = {f }. Therefore, .ΩQ (θ ) = {f }, for each .θ ≥ 0. Arguing by contradiction, assume that .TP and .TQ are dual Tykhonov triples with operator .D = A. Then, by Definition 2.7 for any .θ ≥ 0, we have .u ∈ ΩP (θ ) iff .σ = Au ∈ ΩQ (θ ) or, equivalently, u ∈ ΩP (θ )

.

∀θ ≥ 0

⇐⇒

Au = f.

(4.135)

Let .θ > 0 and .g ∈ X such that .g = 0X and .gX ≤ θ and let . u = A−1 (f + g). Then, .A u = f + g, and it is easy to check that (A u, v −  u)X + θ v −  uX ≥ (f, v −  u)X

.

∀ v ∈ X,

which shows that . u ∈ ΩP (θ ). Then, (4.135) implies that .A u = f , and since on the other hand we have .A u = f + g, it follows that .g = 0X . This is in contradiction with assumption .g = 0X . We conclude from here that the triples .TP and .TQ are not dual Tykhonov triples, as claimed.  (θ ) by equalities Remark 4.3 For every .θ ≥ 0, define the sets .ΩP (θ ) and .ΩQ

.

  ΩP (θ ) = u ∈ X : (Au − f, v − u)X + j (v) − j (u) + θ ≥ 0 ∀ v ∈ X ,    ΩQ (θ ) = σ ∈ Σ : (A−1 σ, τ − σ )X + θ ≥ 0 ∀ τ ∈ Σ .

Then, it is easy to see that the Tykhonov triples .TP = (I, ΩP , C) and .TQ =  , C) are dual triples (with operator .D = A : X → X). Moreover, Problem .P (I, ΩQ  is .TP -well-posed and Problem .Q is .TQ -well posed. The proofs of this statements follow by using arguments similar to those used in the proofs of Theorems 4.5 and 4.6. Remark 4.4 We now follow the construction on page 77 and consider the canonical split problem .M associated with the dual problems .P and .Q, together with the corresponding Tykhonov triple .TM . Then, as explained on page 77, it follows that Problem .M is .TM -well-posed. Nevertheless, Remark 4.2 shows that .TP and .TQ are not dual Tykhonov triples. This example represents an evidence of the statement in Remark 2.9 on page 78. Indeed, it shows that the canonical split

172

4 Variational Inequalities

problem .M associated with two dual problems could be .TM -well-posed, even if the corresponding triples .TP and .TQ are not dual Tykhonov triples.

4.4 Mixed Variational Inequalities We now move to the study of well-posedness of an elliptic mixed variational problem. The functional framework, which we assume everywhere in this section, is the following. First, X and Y represent real Hilbert spaces endowed with the inner products .(·, ·)X and .(·, ·)Y and the associate norms . · X and . · Y , respectively. Moreover, .X × Y denotes their product space endowed with the canonical inner product. A typical element of .X × Y will be denoted by .(u, λ). In addition, we assume that .A : X → X, .b : X × Y → R, .Λ ⊂ Y , .f, h ∈ X, and finally, .X × Λ represents the product of the sets X and .Λ.

4.4.1 An Existence and Uniqueness Result The mixed variational problem we consider in this section can be formulated as follows. Problem .P Find .(u, λ) ∈ X × Λ such that .

(Au, v)X + b(v, λ) = (f, v)X b(u, μ − λ) ≤ b(h, μ − λ)

∀ v ∈ X, . ∀ μ ∈ Λ.

(4.136) (4.137)

0 the set of solutions to Problem .P. Moreover, we As usual, we denote by .SP associate with this problem the triple .T = (I, Ω, C) defined as follows:

I =. R+ = [0, +∞),   C = {θn } ⊂ I : θn → 0 as n → ∞ , (u, λ) ∈ Ω(θ )

⇐⇒

u ∈ X, λ ∈ Λ and, moreover,

(Au, v)X + b(v, λ) ≤ (f, v)X + θ vX b(u, μ − λ) ≤ b(h, μ − λ) + θ μ − λY

∀ v ∈ X, . ∀ μ ∈ Λ,

(4.138) (4.139)

0 = ∅, then .Ω(θ ) = ∅ for each .θ ∈ I , for each .θ ≥ 0. It is easy to see that if .SP and therefore, .T is a Tykhonov triple. We shall use this remark in the proof of Theorem 4.8 below. Our aim in this section is to study the .T -well-posedness of the mixed problem (4.136)–(4.137) under different hypotheses on the operator A and the form b. To this end, we consider the following assumptions:

4.4 Mixed Variational Inequalities

.

173

⎧ A : X → X is a strongly monotone Lipschitz continuous ⎪ ⎪ ⎨ operator, i.e., there exist m > 0 and L > 0 such that: . 2 ⎪ ⎪ (a) (Au − Av, u − v)X ≥ mu − vX ∀ u, v ∈ X. ⎩ (b) Au − AvX ≤ L u − vX ∀ u, v ∈ X.  A : X → X is a demicontinuous operator, i.e., . un → u in X ⇒ Aun  Au in X. ⎧ ⎨ b : X × Y → R is a bilinear continuous form, . i.e., there exists M > 0 such that ⎩ |b(v, μ)| ≤ MvX μY ∀v ∈ X, μ ∈ Y. ⎧ b : X × Y → R is a bilinear form which satisfies ⎪ ⎪ ⎨ the inf − supcondition, i.e., there exists α > 0 such that b(v, μ) ⎪ ⎪ inf ≥ α. sup ⎩ μ∈Y,μ=0Y v∈X,v=0X vX μY Λ is a closed . convex subset of Y such that 0Y ∈ Λ.

(4.140)

(4.141)

(4.142)

(4.143)

(4.144)

Moreover, recall that f ∈ X,

.

h ∈ X.

(4.145)

The following existence and uniqueness result guarantees the unique solvability of Problem .P. Theorem 4.7 Assume (4.140) and (4.142)–(4.145). Then, Problem .P has a unique solution .(u, λ) ∈ X × Λ. Theorem 4.7 was proved in [144] (under assumption that .Λ is an unbounded subset of Y ) and [143] (under assumption that .Λ ⊂ Y is bounded). In both references, the proofs are carried out in several steps, based on arguments of saddle points and the Banach fixed point theorem.

4.4.2 Well-Posedness Results Our main results in this subsection are gathered in Theorems 4.8, 4.9, and 4.10 that we state and prove below. Theorem 4.8 Assume (4.140), (4.143), and (4.145). Then, Problem .P has a unique solution if and only if .Ω(θ ) = ∅ for each .θ ∈ I and .P is .T -well-posed. 0 = ∅, and Proof Assume that Problem .P has a unique solution .(u, λ). Then .SP therefore, .Ω(θ ) = ∅ for each .θ ∈ I , which shows that .T is a Tykhonov triple. Let

174

4 Variational Inequalities

{(un , λn )} be a .T -approximating sequence. Then, there exists a sequence .0 ≤ θn → 0 such that, for each .n ∈ N, the following inequalities hold:

.

.

(Aun , v)X + b(v, λn ) ≤ (f, v)X + θn vX

∀ v ∈ X, .

b(un , μ − λn ) ≤ b(h, μ − λn ) + θn μ − λn Y

∀ μ ∈ Λ.

(4.146) (4.147)

We subtract (4.136) from (4.146) to see that (Aun − Au, v)X + b(v, λn − λ) ≤ θn vX

.

∀ v ∈ X,

(4.148)

which implies that b(v, λn − λ) ≤ θn vX + (Au − Aun , v)X

.

∀ v ∈ X.

Moreover, the use of assumption (4.140)(b) yields b(v, λn − λ) ≤ θn vX + Lun − uX vX

.

∀ v ∈ X.

Assume that .λn = λ. Then the previous inequality implies that .

 b(v, λn − λ) 1 θn + Lun − uX ∀ v ∈ X, v = 0X , ≤ λn − λY vX λn − λY

and therefore, .

sup

v∈X,v=0X

 b(v, λn − λ) 1 θn + Lun − uX . ≤ λn − λY vX λn − λY

We conclude from here that .

inf

sup

μ∈Y,μ=λ v∈X,v=0X

 b(v, μ − λ) 1 θn + Lun − uX . ≤ vX μ − λY λn − λY

This inequality combined with assumption (4.143) implies that λn − λY ≤

.

L θn + un − uX . α α

(4.149)

Note that, obviously, inequality (4.149) holds if .λn = λ, too. On the other hand, we take .μ = λ in (4.147), and then we take .μ = λn in (4.137) and add the resulting inequalities to obtain that b(un − u, λ − λn ) ≤ θn λn − λY .

.

(4.150)

4.4 Mixed Variational Inequalities

175

Next, we take .v = un − u in (4.148) and use (4.150) to deduce that .

(Aun − Au, un − u)X ≤ b(un − u, λ − λn ) + θn un − uX ≤ θn λn − λY + θn un − uX .

Therefore, using assumption (4.140)(a), we find that mun − u2X ≤ θn λn − λY + θn un − uX .

.

(4.151)

We now substitute inequality (4.149) in (4.151) to obtain that un − u2X ≤

.

θn  L θn2 + + 1 un − uX . αm m α

Then, using inequality (1.33), we find that un − uX ≤

.

θn θn  L +1 + √ . m α αm

We now combine this inequality with the bound (4.149) to deduce that there exists a constant .C > 0, which depends only on m, L, and .α and does not depend on n, such that un − uX + λn − λY ≤ Cθn .

.

We now use the convergence .θn → 0 to see that .(un , λn ) → (u, λ) in .X × Y , which shows that Problem .P is .T -well-posed. Conversely, assume that .Ω(θ ) = ∅ for each .θ ∈ I , and moreover, Problem .P is .T -well-posed. Then Definition 2.1(c) guarantees that Problem .P has a unique solution, which concludes the proof.

We now proceed our analysis with a second equivalence result, obtained under different assumptions on the data. Theorem 4.9 Assume (4.141), (4.142), (4.144), and (4.145). Then, the following statements are equivalent: (a) .Ω(θ ) =  ∅ for each .θ ∈ I and Problem .P is .T -well–posed. 0 = ∅ and .diam(Ω(θ )) → 0 as 0 ≤ θ → 0. (b) .SP n n (c) .Ω(θ ) =  ∅ for each .θ ∈ I and .diam(Ω(θn )) → 0 as 0 ≤ θn → 0. Proof First, we remark that condition .Ω(θ ) = ∅ for each .θ ∈ I is needed in order to make sure that .T is a Tykhonov triple. Next, the equivalence (a) .⇐⇒ (b) follows from Theorem 2.2 on page 58 since, 0 ⊂ Ω(θ ) is satisfied, for each .θ ∈ I . obviously, the condition .SP

176

4 Variational Inequalities

Finally, to prove the equivalence (a) .⇐⇒ (c), we use Theorem 2.3 on page 60, and to this end, we prove that the Tykhonov triple .T is regular with Problem .P, in the sense of Definition 2.4. Note that condition (a) in this definition is satisfied since for any .θ1 , .θ2 ≥ 0, inequalities (4.138) and (4.139) show that either .Ω(θ1 ) ⊂ Ω(θ2 ) or .Ω(θ2 ) ⊂ Ω(θ1 ). Consider now a convergent .T -approximating sequence .{(un , λn )}. Then, there exist .(u, λ) ∈ X × Y and a sequence .{θn } ⊂ R+ such that .

(un , λn ) → (u, λ)

in

X × Y, .

(4.152)

(un , λn ) ∈ Ω(θn ) ∀ n ∈ N, .

(4.153)

θn → 0.

(4.154)

This implies that for each .n ∈ N we have .λn ∈ Λ, and therefore, assumption (4.144) implies that .λ ∈ Λ. On the other hand, by (4.153) and the definition of .Ω(θn ), it follows that inequalities (4.146) and (4.147) hold. Therefore, passing to the limit in these inequalities and using the convergences (4.152) and (4.154) combined with assumptions (4.141) and (4.142), we deduce that .(u, λ) satisfies (4.136) and (4.137). 0 . It follows from This shows that .(u, λ) is a solution to Problem .P, i.e., .(u, λ) ∈ SP above that condition (b) in Definition 2.4 is satisfied. We are now in a position to use Theorem 2.3 in order to deduce that the statements (a) and (c) are equivalent, which concludes the proof.

Note that Theorems 4.8 and 4.9 provide only equivalence results and do not guarantee the well-posedness of the mixed variational problem .P. The next theorem provides sufficient conditions that guarantee the well-posedness of this problem. Theorem 4.10 Assume (4.140) and (4.142)–(4.145). Then, the statements (a)–(c) in Theorem 4.9 hold. Proof We use Theorem 4.7 to see that Problem .P has a unique solution. Then, Theorem 4.8 shows that .Ω(θ ) =  ∅ for each .θ ∈ I and, moreover, Problem .P is .T -well-posed. We conclude the proof by using Theorem 4.9.

We end this section with the remark that mixed variational formulation for various elastic, viscoelastic, and viscoplastic contact problems has been considered in [9, 93, 105, 106], [148], and [18], respectively. Moreover, results on optimal control problems for mixed variational inequalities can be found in [221].

Chapter 5

Hemivariational Inequalities

In this chapter we present well-posedness results for hemivariational and variational–hemivariational inequalities in reflexive Banach spaces. We start with an elliptic hemivariational inequality for which we prove the well-posedness with various Tykhonov triples, together with the corresponding convergence results. We then move to the study of a second elliptic hemivariational inequality for which we prove generalized well-posedness results. Finally, we consider a variational– hemivariational inequality for which we prove its well-posedness and show its link with a corresponding minimization problem. Below in this chapter X is a real reflexive Banach space and .·, · denotes the duality pairing between X and its dual ∗ .X .

5.1 A First Elliptic Hemivariational Inequality Everywhere in this section, .K ⊂ X, .A : X → X∗ , .j : X → R, and .f ∈ X∗ . We assume that j is a locally Lipschitz function, and as usual, we denote by .j 0 (u; v) the Clarke directional derivative of j at the point u in the direction v.

5.1.1 Well-Posedness Results In the functional framework described above, we consider the following problem. Problem .P Find u such that u ∈ K,

.

Au, v − u + j 0 (u; v − u) ≥ f, v − u

∀ v ∈ K.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Sofonea, Well-Posed Nonlinear Problems, Advances in Mechanics and Mathematics 50, https://doi.org/10.1007/978-3-031-41416-9_5

(5.1)

177

178

5 Hemivariational Inequalities

Assume (1.17), (1.18), (1.20), (1.21), and (1.25). Then, using Corollary 1.3, we deduce that Problem .P has a unique solution, denoted in what follows by u. Next, we consider a function .h : [0, +∞) × X → R such that: ⎧ ⎪ (a)h(ε, u) ≥ 0 ∀ u ∈ X, ε ≥ 0. ⎪ ⎪ ⎪ ⎪ ⎨ (b)h(εn , 0X ) → 0 whenever 0 ≤ εn → 0. . (5.2) (c) There exists Lh : [0, +∞) → R such that ⎪ ⎪ ⎪ (c1 ) |h(ε, u) − h(ε, v)| ≤ Lh (ε) u − v X ∀ u, v ∈ X, ε > 0. ⎪ ⎪ ⎩ (c2 ) Lh (εn ) → 0 whenever 0 ≤ εn → 0. A typical example of a function that satisfies condition (5.2) is the function h(ε, u) = ε( u X + 1) for each .ε ≥ 0 and .u ∈ X. Moreover, we assume the following additional condition on the function j :

.

⎧ ⎨ For all sequences {un }, {vn } ⊂ X such that . u  u in X, vn → v in X, we have ⎩ n lim sup j 0 (un ; vn − un ) ≤ j 0 (u; v − u).

(5.3)

Note that this condition can be avoided in the proof of Theorem 5.1 below. Nevertheless, we keep it for two reasons: first, it allows us to simplify the proof of this theorem, and second, it is satisfied in the example we present in Sect. 9.4 below. Moreover, we mention that examples of functions j which satisfy these conditions are given in [222]. Next, we consider the Tykhonov triples .T1 = (I1 , Ω1 , C1 ), .T2 = (I2 , Ω2 , C2 ), and .T3 = (I3 , Ω3 , C3 ) defined as follows: ⎧ = [0, +∞), I1 = R+  ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Ω1 (ε) = u ∈ K : Au, v − u + j 0 (u; v − u) .

⎪ ⎪ ⎪ ⎪ ⎪ ⎩



+h(ε, u) v − u X ≥ f, v − u ∀ v ∈ K

C1 = {εn } : εn ∈ I1



 ∀ n ∈ N, εn → 0 as n → ∞ .

∀ ε ∈ I1 ,

⎧ I2 = X∗ ,  ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Ω2 (g) = u ∈ K : Au, v − u + j 0 (u; v − u)  . ⎪ ≥ g, v − u ∀ v ∈ K ∀ g ∈ I2 , ⎪ ⎪   ⎪ ⎪ ⎩ C2 = {gn } : gn ∈ I2 ∀ n ∈ N, gn → f in X∗ as n → ∞ .

(5.4)

(5.5)

5.1 A First Elliptic Hemivariational Inequality

179

  ⎧ :K  is a nonempty closed convex subset of X , ⎪ I = K ⎪ 3 ⎪  ⎪ ⎪ ⎪  = u∈K  : Au, v − u + j 0 (u; v − u) ⎨ Ω3 (K)  .   ∈ I3 , ⎪ ≥ f, v − u ∀ v ∈ K ∀K ⎪ ⎪ ⎪   ⎪ M ⎪ ⎩ C3 = {Kn } : Kn ∈ I3 ∀ n ∈ N, Kn − → K as n → ∞ .

(5.6)

With the notation above, we consider an additional Tykhonov triple .T (I, Ω, C), defined by ⎧  : ε ∈ I1 , g ∈ I2 , K  ∈ I3 }, I = I1 ×I2 × I3 = { θ = (ε, g, K) ⎪ ⎪ ⎪ ⎪ 0 ⎪  : Au, v − u + j (u; v − u) + h(ε, u) v − u X ⎪ Ω(θ ) = u ∈ K ⎪ ⎪  ⎨   ∈ I, ≥ g, v − u ∀ v ∈ K ∀ θ = (ε, g, K) .  ⎪ ⎪ ⎪ ⎪ C = C1 × C2 × C3 = {θn } : θn = (εn , gn , Kn ), ⎪ ⎪  ⎪ ⎪ ⎩ {εn } ∈ C1 , {gn } ∈ C2 , {Kn } ∈ C3 .

=

(5.7)

Our main result in this section is the following. Theorem 5.1 Assume (1.17), (1.18), (1.20), (1.21), (1.25), (5.2), and (5.3). Then, (a) .T1 ≤ T , .T2 ≤ T and .T3 ≤ T . (b) Problem .P is .T -well-posed. Proof (a) We use Theorem 2.1 on page 52. To this end, we consider the sequences .θ 1 = {θn1 }, .θ 2 = {θn2 }, and .θ 3 = {θn3 } defined by θn1 = 0,

.

θn2 = f,

θn3 = K

∀ n ∈ N.

Using definitions (5.4)–(5.6) and assumptions (1.17) and (1.21), we see that 0 ∈ I1 , .f ∈ I2 , and .K ∈ I3 , and moreover, .θ 1 ∈ C1 , .θ 2 ∈ C2 , and .θ 3 ∈ C3 . We conclude from here that condition (2.11) is satisfied with .p = 3, .c1 = 0, .c2 = f , and .c3 = K. On the other hand, using assumption (5.2) and (5.4)–(5.7), it is easy to see that the multifunction .Ω : I1 × I2 × I3 → 2X has the following properties: .

⎧ ⎨ Ω1 (ε) = Ω(ε, f, K) . Ω (g) ⊂ Ω(0, g, K) ⎩ 2   Ω3 (K) ⊂ Ω(0, f, K)

∀ ε ∈ I1 , ∀ g ∈ I2 ,  ∈ I3 . ∀K

This shows that condition (2.12) is satisfied, too. It follows from above that we are in a position to apply Theorem 2.1. Therefore, we deduce that the Tykhonov triple .T = (I, Ω, C) satisfies the

180

5 Hemivariational Inequalities

inequalities .T1 ≤ T , .T2 ≤ T , and .T3 ≤ T , which concludes the proof of the first part of the theorem. (b) We now move to the proof of the second part of the theorem. First, we recall that the unique solvability of Problem .P is a consequence of Corollary 1.3. Next, we consider a .T -approximating sequence of Problem .P, denoted by .{un }. Then, Definition 2.1(b) and (5.7) show that there exists a sequence .θ n = {θn } ∈ C with .θn = (εn , gn , Kn ) for each .n ∈ N such that un . ∈ Kn ,

Aun , v − un  + j 0 (un ; v − un ) + h(εn , un ) v − un X (5.8) ≥ gn , v − un  ∀ v ∈ Kn ,

for each .n ∈ N. Recall also that inclusion .θ n = {θn } ∈ C implies the following convergences: .

εn → 0, . gn → f

(5.9) in

X∗ , .

(5.10)

M

Kn − → K.

(5.11)

We shall prove that .un → u in X, and to this end, we divide the proof in three steps, described below: Step (i) The sequence .{un } is bounded in X. M

Let .v ∈ K be a given element. Then, the convergence .Kn − → K as .n → ∞, guaranteed by (5.11), implies that there exists a sequence .{vn } ⊂ X such that .vn ∈ Kn for all .n ∈ N and .vn → v in X. Using inequality (5.8), we have .

Aun , vn − un  + j 0 (un ; vn − un ) + h(εn , un ) vn − un X

(5.12)

≥ gn , vn − un . We now use assumption (1.18)(b) to see that .

mA vn − un 2X ≤ Avn − Aun , vn − un  = Avn , vn − un  − Aun , vn − un ,

and therefore, inequality (5.12) yields .

mA vn − un 2X ≤ Avn − gn X∗ vn − un X

(5.13)

+j 0 (un ; vn − un ) + h(εn , un ) vn − un X . Moreover, using assumption (1.20) and the properties of the Clarke directional derivative, we have

5.1 A First Elliptic Hemivariational Inequality

181

j 0 (un ; vn − un ) = j 0 (un ; vn − un ) + j 0 (vn ; un − vn ) − j 0 (vn ; un − vn )

.

≤ j 0 (un ; vn − un ) + j 0 (vn ; un − vn ) + |j 0 (vn ; un − vn )|   ≤ αj un − vn 2X + | max ξ, un − vn  : ξ ∈ ∂j (vn ) | ≤ αj un − vn 2X + (c0 + c1 vn X ) un − vn X . Therefore, j 0 (un ; vn − un ) ≤ αj un − vn 2X + (c0 + c1 vn X ) un − vn X .

.

(5.14)

On the other hand, note that condition (5.2)(c.1 ) implies that h(εn , un ) ≤ Lh (εn ) vn − un X + h(εn , vn ).

.

(5.15)

We now combine inequalities (5.13)–(5.15) to see that (mA − αj − Lh (εn )) un − vn X ≤ Avn − gn X∗ + h(εn , vn ) + c0 + c1 vn X ,

.

and using assumptions (5.2)(c.2 ) and (1.25), we find that there exists a positive constant .C0 which does not depend on n such that un − vn X ≤ C0 ( Avn − gn X∗ + h(εn , vn ) + c0 + c1 vn X ),

.

(5.16)

for n large enough. We now use assumption (5.2)(c.1 ), again, to see that h(εn , vn ) ≤ h(εn , 0X ) + Lh (εn ) vn X .

.

(5.17)

Recall that the sequences .{vn } and .{gn } are bounded in X and .X∗ , respectively, and A is a bounded operator. Therefore, using the convergence (5.9) and assumptions (5.2)(b), (c.2 ) together with inequalities (5.16) and (5.17), we deduce that the sequence .{un − vn } is bounded in X. This implies that .{un } is a bounded sequence in X, which concludes the proof of this step. Step (ii) The sequence .{un } converges weakly to the solution u of Problem .P. Using step (i) and the reflexivity of the space X, we deduce that, passing to a subsequence, if necessary, we have that un   u in X,

.

as

n → ∞,

(5.18)

with some . u ∈ X. Our aim in what follows is to prove that . u is a solution to Problem P. To this end, we remark that the convergences (5.11) and (5.18) imply that

.

 u ∈ K.

.

(5.19)

182

5 Hemivariational Inequalities

Consider now an arbitrary element .v ∈ K and a sequence .{vn } ⊂ X such that vn ∈ Kn for all .n ∈ N and .vn → v in X. Note that the existence of such sequence is guaranteed by the convergence (5.11). Then, we use inequality (5.12) to see that

.

Aun , un − vn  ≤ j 0 (un ; vn − un ) + h(εn , un ) vn − un X + gn , un − vn .

.

Passing to the upper limit in this inequality, we find that .

lim sup Aun , un − vn  ≤ lim sup j 0 (un ; vn − un )

(5.20)

+ lim sup h(εn , un ) vn − un X + lim sup gn , un − vn . On the other hand, recall that assumption (5.2)(c.1 ) implies that h(εn , un ) ≤ h(εn , 0X ) + Lh (εn ) un X .

.

(5.21)

We now use the convergences (5.10), (5.18), .vn → v in X, and assumption (5.3) to deduce that .

lim sup j 0 (un ; vn − un ) ≤ j 0 ( u; v −  u), .

(5.22)

gn , un − vn  → f,  u − v.

(5.23)

Moreover, since .{un } is a bounded sequence of X, (5.9), (5.2)(b), (5.2)(c.2 ), and (5.21) imply that .h(εn , un ) → 0. Therefore, since .{un − vn } is a bounded sequence of X, we deduce that .

h(εn , un ) vn − un X → 0.

(5.24)

We now combine relations (5.20) and (5.22)–(5.24) to find that lim sup Aun , un − vn  ≤ j 0 ( u; v −  u) + f,  u − v.

.

(5.25)

Next, we write Aun , un − vn  = Aun , un − v + Aun , v − vn ,

.

(5.26)

and since the operator A is bounded, the convergence .vn → v in X implies that Aun , v − vn  → 0. Therefore, using (5.26), we deduce that

.

.

lim sup Aun , un − vn  = lim sup Aun , un − v.

(5.27)

We now combine (5.25) and (5.27) to obtain that .

lim sup Aun , un − v ≤ j 0 ( u; v −  u) + f,  u − v.

(5.28)

5.1 A First Elliptic Hemivariational Inequality

183

Recall that this inequality holds for each .v ∈ K. Next, we take .v =  u in (5.28) and use the property .j 0 ( u; 0X ) = 0 of the Clarke directional derivative to deduce that .

lim sup Aun , un −  u ≤ 0.

(5.29)

Exploiting now the pseudomonotonicity of the operator A, from (5.18) and (5.29), we have A u,  u − v ≤ lim inf Aun , un − v

.

∀ v ∈ X.

(5.30)

u is a solution to Problem .P, Next, from (5.19), (5.30), and (5.28), we obtain that . as claimed. Thus, by the uniqueness of the solution, we find that . u = u. A careful analysis of the results presented above indicates that every subsequence of .{un } which converges weakly in X has the same weak limit u. On the other hand, .{un } is bounded in X. Therefore, using Theorem 1.3, we deduce that the whole sequence .{un } converges weakly to u in X, as .n → ∞, which concludes the proof of this step. Step (iii) The sequence .{un } converges strongly to the solution of Problem .P. We use assumption (1.18)(b) and Proposition 1.2 on page 6 to see that the operator .A : X → X∗ is an operator of type .(S+ ). Therefore, inequality (5.29), u = u, and the convergence .un  u in X imply that .un → u in X, which equality . ends the proof of this step. To conclude, we proved that any .T -approximating sequence converges to the solution of Problem .P. Therefore, using Definition 2.1(c), it follows that Problem .P is .T -well-posed. 

We end this subsection with the following result which represents a direct consequence of Theorem 5.1. Corollary 5.1 Assume (1.17), (1.18), (1.20), (1.21), (1.25), (5.2), and (5.3), and let T1 , .T2 , and .T3 be the Tykhonov triples defined by (5.4), (5.5), and (5.6), respectively. Then, Problem .P is .Ti -well-posed, for each .i = 1, 2, 3.

.

5.1.2 Convergence Results In this subsection we provide some consequences of Theorem 5.1. To this end, besides the data K, A, j , and f in Problem .P, for each .n ∈ N, we consider a set .Kn , an operator .An , a function .ϕn , and an element .fn such that the following hold: Kn is nonempty, closed, and convex subset of X and . (5.31) M Kn − → K as n → ∞.

184

5 Hemivariational Inequalities

⎧ ⎪ An : X → X∗ and there exist T : X → X∗ , ωn ≥ 0 such that: ⎪ ⎪ ⎪ ⎪ ⎨ (a) An v = Av + ωn T v ∀ v ∈ X. . (b) T u − T v X∗ ≤ LT u − v X ∀ u, v ∈ X with LT > 0. . ⎪ ⎪ ⎪ (c) T u − T v, u − v ≥ 0 ∀ u, v ∈ X. ⎪ ⎪ ⎩ (d) ωn → 0 as n → ∞. ⎧ ϕn : X × X → R satisfies condition (1.19) with αϕn = αn ≥ 0 ⎪ ⎪ ⎨ and there exists δn ≥ 0 such that: ⎪ (a) ϕn (η, v1 ) − ϕn (η, v2 ) ≤ δn η X v1 − v2 X ∀ η, v1 , v2 ∈ X. ⎪ ⎩ (b) δn → 0 as n → ∞. .

αj + αn < mA .. ∗

(5.32)

(5.33)

(5.34) ∗

fn ∈ X and fn → f in X as n → ∞..

(5.35)

αn → 0 as n → ∞.

(5.36)

With these data, for each .n ∈ N, we consider the following problem. Problem .Pn Find .un ∈ X such that .

un ∈ Kn ,

An un , v − un  + ϕn (un , v) − ϕn (un , un ) + j 0 (un ; v − un ) ≥ fn , v − un  ∀ v ∈ Kn .

(5.37)

Our main result in this section is as follows. Theorem 5.2 Assume (1.17), (1.18), (1.20), (1.21), (5.3), and (5.31)–(5.36). Then, the following statements hold: (a) There exists a unique solution u to Problem .P and, for each .n ∈ N, there exists a unique solution .un to Problem .Pn . (b) The sequence .{un } converges to u in X. Proof (a) Note that, since .αn ≥ 0, assumption (5.34) implies (1.25). Therefore, the existence of the unique solution to Problem .P is a direct consequence of Corollary 1.3. Let .n ∈ N. It follows from Proposition 1.1(b), (c) that a monotone Lipschitz continuous operator is pseudomonotone and the sum of two pseudomonotone operators is pseudomonotone. Therefore, assumptions (1.18) and (5.32) show that the operator .An is pseudomonotone. Moreover, it is strongly monotone with the constant .mA . It follows from here that operator .An satisfies condition (1.18), too. On the other hand, assumption (5.33) shows that the function .ϕn satisfies condition (1.19) and (5.34) implies (1.22), both with constant .αn instead of .αϕ . Recall also that .Kn is a nonempty, closed, and convex subset of X, .fn ∈ X∗ ,

5.1 A First Elliptic Hemivariational Inequality

185

and j satisfies (1.20). All these ingredients show that we are in a position to use Theorem 1.7 with .Kn , .An , .ϕn , and .fn instead of K, A, .ϕ, and f , respectively. In this way, we deduce the unique solvability of Problem .Pn , which concludes the proof of the first part of the theorem. (b) For the second part of the proof, we use the Tykhonov triple .T = (I, Ω, C) defined by (5.7). Let .n ∈ N and .v ∈ X. We use assumption (5.32)(a) and (5.37) to see that .

un ∈ Kn ,

Aun , v − un  + ωn T un , v − un 

(5.38)

+ϕn (un , v) − ϕn (un , un ) + j (un ; v − un ) ≥ fn , v − un  ∀ v ∈ Kn . 0

Then, we write .

ωn T un , v − un  ≤ ωn T un X∗ v − un X

 ≤ ωn T un − T 0X X∗ + T 0X X∗ v − un X ,

and using assumption (5.32)(b), we find that

 ωn T un , v − un  ≤ ωn LT un X + T 0X X∗ v − un X .

.

(5.39)

Next, assumption (5.33)(a) implies that ϕn (un , v) − ϕn (un , un ) ≤ δn un X v − un X .

.

(5.40)

We now denote εn = max {ωn , δn },

.

(5.41)

and then we combine (5.38)–(5.41) to see that .un satisfies inequality (5.8) with gn = fn and .h : [0, +∞) × X → R given by

.



h(ε, u) = ε (LT + 1) u X + T 0X X∗

.

∀ ε ∈ R+ , u ∈ X.

(5.42)

Note that the function h satisfies assumption (5.2) with .Lh (ε) = ε(LT + 1). On the other hand, (5.32)(d), (5.33)(c), and (5.41) show that .εn → 0, and therefore, (5.9) holds. Moreover, (5.35) implies (5.10) with .gn = fn and (5.31) implies (5.11). It follows from here that the sequence .{θn } with .θn = (εn , fn , Kn ) belongs to the set .C defined by (5.7), and therefore, .{un } is a .T -approximating sequence for Problem .P. Finally, since (1.25) holds, the convergence .un → u in X follows from the .T -well-posedness of Problem .P, guaranteed by Theorem 5.1. 

186

5 Hemivariational Inequalities

Next, we consider the following particular versions of inequality (5.37): 0 un ∈ . Kn , Aun , v − un  + j (un ; v − un ) ≥ f, v − un  ∀ v ∈ Kn , . (5.43)

un ∈ K, Aun , v − un  + j 0 (un ; v − un ) ≥ fn , v − un  ∀ v ∈ K, .

(5.44)

un ∈ K, An un , v − un  + j 0 (un ; v − un ) ≥ f, v − un  ∀ v ∈ K.

(5.45)

Then, using Theorem 5.2(b), we can obtain the convergence of the solution of each of the inequalities (5.43)–(5.45) to the solution of inequality (5.1), under appropriate assumptions. The convergence of the solutions of (5.44) and (5.45) to the solution of inequality (5.1) stands for a continuous dependence result of the solution of Problem .P with respect to the element f and the operator A, respectively. The convergence of the solution of inequality (5.43) to the solution of (5.1) shows the continuous dependence of the solution of Problem .P with respect to the convex set K. This result is important in the numerical analysis of the hemivariational M

inequality (5.1). There, the assumption .Kn − → K shows that .Kn represents an approximation of the set K in the sense used in [68, 69]. The approximation is external if .Kn ⊂ K and is internal if .Kn ⊂ K. The internal approximation of hemivariational inequalities with the choice .Kn = Xn ∩ K where .Xn is a finite dimensional subspace of X was used in [88], for instance. More details on abstract approximations of hemivariational inequalities can be found in [78, 85]. There, besides the convergence of the solution of the discrete scheme (5.37) to the solution of Problem .P, various error estimates have been obtained. We end this section with the remark that inequality problems of the form (5.1), (5.37) will be considered in Sect. 9.4, in the study of a nonsmooth contact problem. There, we shall exemplify the abstract results provided by Theorems 5.2, and we shall provide the corresponding mechanical interpretation.

5.2 A Second Elliptic Hemivariational Inequality For the hemivariational inequality we consider in this section, besides the reflexive space X, we consider a reflexive Banach space Y . Let K be a nonempty subset of X, .A : X → X∗ , .j : Y → R, .γ : X → Y , and .f ∈ X∗ . We use . γ to represent the norm of the operator .γ , assumed to satisfy condition (1.30).

5.2.1 Existence, Uniqueness, and Compactness Results The hemivariational inequality we consider in this section is as follows. Problem .P Find .u ∈ K such that Au, v − u + j 0 (γ u; γ v − γ u) ≥ f, v − u

.

∀ v ∈ K.

(5.46)

5.2 A Second Elliptic Hemivariational Inequality

187

Our first result in the study of Problem .P is the following. Theorem 5.3 Assume (1.28)–(1.32). Then the solution set of Problem .P is nonempty and weakly compact in X. Proof Note that the solvability of the hemivariational inequality (5.46) follows 0, from Theorem 1.8. Therefore, the set of solutions of Problem .P, denoted by .SP is nonempty. We now claim that this set is bounded. 0 is unbounded. Then there exists a Arguing by contradiction, suppose that .SP sequence .{un } such that . un X → +∞ as .n → ∞, and for all .n ∈ N, the inequality below holds: .

Aun , v − un  + j 0 (γ un ; γ v − γ un ) ≥ f, v − un  ∀ v ∈ K.

(5.47)

Let .n ∈ N. Testing with .v = 0X ∈ K in (5.47), we find that Aun , un  − j 0 (γ un ; −γ un ) ≤ f, un ,

.

which implies that .

Aun , un  − j 0 (γ un ; −γ un ) ≤ f X∗ . un X

Next, using assumptions (1.29)(b) and (1.31)(c)(i), we find that

.

cA un 2X − dA un X − eA − cj γ un 2Y − dj γ un Y − ej ≤ f X∗ . un X

Passing to the limit as .n → ∞ in this inequality and using assumption (1.31)(c)(ii), we deduce that .

+ ∞ ≤ f X∗ ,

0 is bounded, which represents a contradiction. We conclude from here that the set .SP as claimed. 0 is weakly closed in X. To this end, we consider a We now prove that the set .SP 0 sequence .{un } ⊂ SP such that .un  u in X as .n → ∞, for some .u ∈ X. Since K is a closed convex subset of X, we deduce that .u ∈ K. Now, for each .n ∈ N, inequality (5.47) holds, and taking .v = u in this inequality, we obtain that

Aun , un − u ≤ j 0 (γ un ; γ u − γ un ) + f, un − u.

.

Then, passing to the upper limit as .n → ∞ and using the convergence .un  u in X, the compactness of the operator .γ , Proposition 1.3, and the upper semicontinuity of the function .(u, v) → j 0 (u; v) (guaranteed by Proposition 1.9(ii)), we find that

188

5 Hemivariational Inequalities .

lim sup Aun , un − u ≤ lim sup j 0 (γ un ; γ u − γ un ) + lim sup f, un − u ≤ j 0 (u; 0X ).

Therefore, the equality .j 0 (u; 0X ) = 0 (guaranteed by Proposition 1.9(i)) shows that .

lim sup Aun , un − u ≤ 0.

(5.48)

Next, using (5.48), the convergence .un  u in X as .n → ∞, and the pseudomonotonicity of A, we deduce that .

lim inf Aun , un − v ≥ Au, u − v

∀ v ∈ X.

(5.49)

On the other hand, using (5.47), we find that .

Aun , un − v ≤ j 0 (γ un ; γ v − γ un ) + f, un − v ∀ v ∈ K.

(5.50)

Passing to the upper limit as .n → ∞ in (5.50) and using arguments similar to those used in the proof of inequality (5.48), we find that lim. sup Aun , un − v ≤ j 0 (γ u; γ v − γ u) + f, u − v ∀ v ∈ K.

(5.51)

We now combine inequalities (5.49) and (5.51) to see that .

Au, u − v ≤ j 0 (γ u; γ v − γ u) + f, u − v

∀ v ∈ K.

0 , and therefore, the set .S 0 is weakly closed in X. This implies that .u ∈ SP P 0 is nonempty, bounded, and weakly The analysis above shows that the set .SP 0 is weakly compact closed in X. Therefore, the reflexivity of X guarantees that .SP in X, which concludes the proof of the theorem. 

We now complete Theorem 5.3 with the following result. Theorem 5.4 Under the assumptions of Theorem 5.3 if, in addition, A is an operator of type .(S+ ), then the set of solutions of Problem .P is a nonempty compact set of X. 0 is nonempty and weakly compact. Proof Recall that Theorem 5.3 entails that .SP 0 . Then, there exist a subsequence of .{u }, Let .{un } be any sequence included in .SP n 0 such that still denoted by .{un }, and an element .u ∈ SP

un  u in X, as n → ∞.

.

Moreover, using the arguments in the proof of Theorem 5.3, it follows that (5.48) holds. Therefore, the convergence .un  u in X and the .(S+ )-property of A imply that

5.2 A Second Elliptic Hemivariational Inequality

189

un → u in X, as n → ∞.

.

0 is compact in X, which concludes the proof. This proves that the set .SP



Corollary 5.2 Assume (1.28), (1.29)(a), (1.30), (1.31) (a), (b), (c)(i), and (1.32). In addition, assume that A is a strongly monotone operator with constant .mA and, moreover, cj γ 2 < mA .

(5.52)

.

Then the set of solutions of Problem .P is a nonempty compact set of X. Proof Since A is strongly monotone with constant .mA > 0, we have Au − Av, u − v ≥ mA u − v 2X

.

∀ u, v ∈ X.

(5.53)

Let .u ∈ X. We use (5.53) to see that Au, u = Au − A0X , u + A0X , u ≥ mA u 2X − A0X X∗ u X ,

.

which proves that condition (1.29)(b) holds with .cA = mA . Therefore, inequality (5.52) shows that condition (1.31)(c) holds, too. We now use Proposition 1.2 to see that the operator A is of type .(S+ ). Corollary 5.2 is now a direct consequence of Theorem 5.4. 

5.2.2 Generalized Well-Posedness Results In this subsection, we study the generalized well-posedness of Problem .P. To this end, we consider a function h which satisfies condition (5.2). We also consider the triple .T = (I, Ω, C) defined as follows: .

I = R+ ,  Ω(θ ) = u ∈ K : Au, v − u + j 0 (γ u; γ v − γ u) + h(θ, u) v − u X  ≥ f, v − u ∀ v ∈ K ∀ θ ∈ I,   C = {θn } ⊂ I : θn → 0 .

Note that assumption (5.2)(a) implies that h(θ, u) v − u X ≥ 0

.

∀ θ ≥ 0, u, v ∈ X,

190

5 Hemivariational Inequalities

which shows that 0 SP ⊂ Ω(θ )

.

∀ θ ≥ 0.

This implies that, under the assumption of Theorem 5.3, .T is a Tykhonov triple. Our first result in this section is the following. Theorem 5.5 Assume (1.28)–(1.32), and moreover, assume that condition (5.2) holds. Then Problem .P is weakly generalized .T -well-posed. Proof We use Definition 2.3(c) on page 52. First, we recall that Theorem 1.8 0 is not empty. The rest of the proof is guarantees that the set of solutions .SP structured in two steps, as follows: Step (i) Any .T -approximating sequence of Problem .P is bounded in X. Let .{un } ⊂ X be a .T -approximating sequence of Problem .P. Then, there exists a sequence .{θn } ⊂ R+ such that θn → 0 as n → ∞ and un ∈ Ω(θn ) for all n ∈ N.

.

We argue by contradiction, and we assume that the sequence .{un } is not bounded in X. Then, without loss of generality, we may assume that un X → +∞ as n → ∞.

.

(5.54)

Let .n ∈ N be fixed. We have Aun , v − un  + j 0 (γ un ; γ v − γ un )

.

(5.55)

+h(θn , un ) v − un X ≥ f, v − un  for all .v ∈ K. We now test with .v = 0X ∈ K in (5.55) to see that Aun , un  ≤ j 0 (γ un ; −γ un ) + h(θn , un ) un X + f, un .

.

(5.56)

Moreover, we recall inequality (5.21), i.e., .

h(θn , un ) ≤ Lh (θn ) un X + h(θn , 0X ).

(5.57)

On the other hand, using condition (1.29)(b), we have Aun , un  ≥ cA un 2X − dA un X − eA ,

.

(5.58)

and in addition, the growth condition (1.31)(c) yields j 0 (γ un ; −γ un ) ≤ cj γ un 2Y + dj γ un Y + ej .

.

(5.59)

5.2 A Second Elliptic Hemivariational Inequality

191

We now combine inequalities (5.56)–(5.59), and then we use inequalities γ un Y ≤ γ un X ,

.

f, un  ≤ f X∗ un X

to see that .

cA un 2X ≤ cj γ 2 un 2X + Lh (θn ) un 2X + dj γ un X + dA un X +eA + ej + (h(θn , 0X ) + f X∗ ) un X .

Therefore,

.

 cA − cj γ 2 − Lh (θn ) un X ≤ dA + dj γ + f X∗ + h(θn , 0X ) +

eA + ej . un X

This inequality, (5.54), and assumptions .cj γ 2 < cA , and (5.2)(c.2 ) imply that .

+ ∞ = lim (cA − cj γ 2 − Lh (θn )) un X ≤ dA + dj γ + f X∗ , n→∞

which represents a contradiction. We conclude from above that the sequence .{un } is bounded. Step (ii) Any .T -approximating sequence has a subsequence which converges weakly 0. to some point of .SP Let .{un } ⊂ X be a .T -approximating sequence of Problem .P. Using Step (i), we deduce that there exists .u ∈ X such that passing to a subsequence, if necessary, we have un  u in X as n → ∞.

.

(5.60)

Since .K ⊂ X is closed and convex, it follows from (5.60) that .u ∈ K. Let .n ∈ N. Taking .v = u in (5.55) gives .

Aun , un − u ≤ j 0 (γ un ; γ u − γ un )

(5.61)

+h(θn , un ) u − un X + f, un − u. It follows from (5.61) and (5.57) that .

Aun , un − u ≤ j 0 (γ un ; γ u − γ un )

 + Lh (θn ) un X + h(θn , 0X ) u − un X + f, un − u.

(5.62)

192

5 Hemivariational Inequalities

Passing to the upper limit as .n → ∞ in (5.62) and taking into account the convergence (5.60), the compactness of the operator .γ , the properties of the function .(u, v) → j 0 (u; v), the boundedness of the sequence .{un }, and assumption (5.2)(b),(c.2 ), one obtains .

lim sup Aun , un − u ≤ 0.

(5.63)

Therefore, the pseudomonotonicity of A yields Au, u − v ≤ lim inf Aun , un − v ∀ v ∈ X,

.

which shows that .

lim sup Aun , v − un  ≤ Au, v − u

∀ v ∈ X.

(5.64)

We now pass to the upper limit as .n → ∞ in (5.55) and use (5.60), (5.64), and (5.57) together with the properties of the functions h and j to see that .

f, v − u = lim sup f, v − un 

 ≤ lim sup Aun , v − un  + j 0 (γ un ; γ v − γ un ) + h(θn , un ) v − un X ≤ lim sup Aun , v − un  + lim sup j 0 (γ un ; γ v − γ un ) + lim sup h(θn , un ) v − un X ≤ Au, v − u + j 0 (γ u; γ v − γ u)

 + lim sup Lh (θn ) un X + h(θn , 0X ) v − un X ≤ Au, v − u + j 0 (γ u; γ v − γ u)

∀ v ∈ K.

0. It follows from above that .u ∈ SP To conclude, we proved that every .T -approximating sequence of Problem 0 .P has a subsequence which converges weakly to some point of .S . Therefore, P using Definition 2.3(c), we deduce that Problem .P is weakly generalized .T -wellposed. 

We complete Theorem 5.5 with the following generalized .T -well-posedness results. Theorem 5.6 Keep the assumptions from Theorem 5.5, and in addition, assume that A is an operator of type .(S+ ). Then Problem .P is strongly generalized .T -wellposed. 0 is a direct consequence Proof We use Definition 2.3(b). The nonemptiness of .SP of Theorem 1.8. We now show that any .T -approximating sequence has at least a 0. subsequence which converges strongly to some point of .SP

5.3 A Variational–Hemivariational Inequality

193

Let .{un } ⊂ X be any .T -approximating sequence. Then there exists a sequence {θn } ⊂ R+ such that

.

θn → 0 as n → ∞ and un ∈ Ω(θn ) for each n ∈ N,

.

that is, (5.55) holds, for each .n ∈ N. Moreover, the proof of Theorem 5.5 points out that passing to a sequence, if necessary, the convergence (5.60) holds with some 0 .u ∈ S , and in addition, (5.63) holds, too. This inequality combined with the .(S+ )P property of A implies that un → u in X as n → ∞,

.

which means that Problem .P is strongly generalized .T -well-posed.



Corollary 5.3 Keep the assumption of Corollary 5.2 and, in addition, assume that (5.2) holds. Then Problem .P is strongly generalized .T -well-posed. Proof It follows from the proof of Corollary 5.2 that conditions (1.28)–(1.32) are satisfied, and moreover, A is an operator of type .(S+ ). Corollary 5.3 is now a direct 

consequence of Theorem 5.6. We end this section with the mention that more details in the study of hemivariational inequalities of the form (5.46) can be found in [36].

5.3 A Variational–Hemivariational Inequality In contrast with the inequality problems studied in Sects. 5.1 and 5.2, in this section we deal with a variational–hemivariational inequality to which, in addition, we associate a minimization problem. We work in a reflexive Banach space X. Let ∗ ∗ .K ⊂ X, .A : X → X , .ϕ : X × X → R, .j : X → R, and .f ∈ X . We assume that 0 .j : X → R is a locally Lipschitz function, and we denote by .j (u; v) the Clarke directional derivative of j at the point u in the direction v.

5.3.1 Well-Posedness Results The variational–hemivariational inequality we study in this section is the following. Problem .V Find an element u such that .

u ∈ K,

Au, v − u + ϕ(u, v) − ϕ(u, u) +j (u; v − u) ≥ f, v − u ∀ v ∈ K. 0

(5.65)

194

5 Hemivariational Inequalities

We now associate with Problem .V the gap function .g : X → R ∪ {+∞} defined by g(v) = sup

.



 Av − f, v − w + ϕ(v, v) − ϕ(v, w) − j 0 (v; w − v)

(5.66)

w∈K

for each .v ∈ X, together with the following minimization problem. Problem .M Find an element u such that u ∈ K,

.

g(u) ≤ g(v) ∀ v ∈ K.

For the solvability of Problems .V and .M, we assume that (1.17)–(1.22) hold. Then, Theorem 1.7 on page 16 guarantees the unique solvability of Problem .V. The solvability of Problem .M follows now from the following result. Proposition 5.1 An element u is a solution to Problem .V if and only if u is a solution to Problem .M and .g(u) = 0. Proof For any .w ∈ K, define the function .kw : X → R by kw (v) = Av − f, v − w + ϕ(v, v) − ϕ(v, w) − j 0 (v; w − v)

.

∀ v ∈ X,

(5.67)

and note that (5.66) and (5.67) imply that g(v) = sup kw (v)

.

∀ v ∈ X.

(5.68)

w∈K

Let .v ∈ K. Then, (5.68) implies that .g(v) ≥ kv (v), and since .kv (v) = 0, we find that .g(v) ≥ 0. Therefore, v ∈ K ⇒ g(v) ≥ 0.

.

(5.69)

Assume that u is a solution to Problem .V. This implies that .u ∈ K, and moreover, kw (u) = Au − f, u − w + ϕ(u, u) − ϕ(u, w) − j 0 (u; w − u) ≤ 0 ∀ w ∈ K.

.

Then, (5.68) implies that .g(u) ≤ 0, and since (5.69) guarantees that .g(u) ≥ 0, we deduce that .g(u) = 0. We combine this equality with (5.69) to see that .g(u) ≤ g(v) for all .v ∈ K, which shows that u is a solution of Problem .M. Conversely, assume that u is a solution of Problem .M and .g(u) = 0. Then .u ∈ K, and using (5.66), we find that

5.3 A Variational–Hemivariational Inequality

.

195

  inf Au − f, v − u + ϕ(u, v) − ϕ(u, u) + j 0 (u; v − u)

v∈K

  = − sup Au − f, u − v + ϕ(u, u) − ϕ(u, v) − j 0 (u; v − u) v∈K

= −g(u) = 0. This shows that u is a solution of problem (5.65), which ends the proof.



We now turn to the well-posedness of Problems .V and .M. To this end, we assume (1.17)–(1.22), (5.3), and in addition, we assume that the following conditions are satisfied: For each u ∈ K there exists cϕ (u) ≥ 0 such that . . (5.70) ϕ(u, v1 ) − ϕ(u, v2 ) ≤ cϕ (u) v1 − v2 X ∀ v1 , v2 ∈ X. ⎧ ⎪ ⎨ For any sequences {ηn } ⊂ X, {un } ⊂ X such that ηn  η ∈ X, un  u ∈ X one has (5.71) ⎪ ⎩ lim sup [ϕ(ηn , v) − ϕ(ηn , un )] ≤ ϕ(η, v) − ϕ(η, u) ∀ v ∈ X. n→∞

Moreover, we consider a function h which satisfies the assumption (5.2). We use the notation .d(x, K) for the distance between an element .x ∈ X and a set .K ⊂ X, that is, d(x, K) = inf x − v X ,

.

v∈K

(5.72)

M

and in addition, the notation .Kn − → K will represent a short-hand notation for the following property: ⎧ ⎨ For any v ∈ X and any sequence {vn } ⊂ X such that . v ∈ Kn for each n ∈ N, ⎩ n the convergence vn  v in X implies that v ∈ K.

(5.73)

We use this notation since below we shall consider sequences .{Kn } with .K ⊂ Kn , and in this case, property (5.73) is equivalent with the convergence of .{Kn } to the set K in the sense of Mosco, see Definition 1.2. We now introduce the set of parameters .

  ε) : K  is closed convex subset of X such that I = θ = (K,   ε≥0 , K ⊂ K,

(5.74)

 ε), we consider the approximating sets .ΩV (θ ) and .ΩM (θ ) and for each .θ = (K, defined as follows:

196

5 Hemivariational Inequalities

 and u ∈ ΩV. (θ ) ⇐⇒ u ∈ K

(5.75)

.

Au − f, v − u + ϕ(u, v) − ϕ(u, u) + j 0 (u; v − u) +h(ε, u) v − u X ≥ 0

∀ v ∈ K.

 and g(u) ≤ h(ε, u) d(u, K). u ∈ ΩM (θ ) ⇐⇒ u ∈ K

(5.76)

Finally, we introduce the set of sequences .C defined as follows: {θn } ∈ .C ⇐⇒ θn = (Kn , εn ) ∈ I for all n ∈ N,

(5.77)

M

→ K and εn → 0. Kn − Note that, under assumptions (1.17)–(1.22), the solution u of Problems .V and .M belongs to the sets .ΩV (θ ) and .ΩM (θ ), for each .θ ∈ I . We conclude that .ΩV (θ ) =  ∅ and .ΩM (θ ) =  ∅, for each .θ ∈ I . Therefore, according to Definition 2.1(a), the triples .TV = (I, ΩV , C) and .TM = (I, ΩM , C) are Tykhonov triples. Our main result in this subsection is the following. Theorem 5.7 Assume (1.17)–(1.22), (5.2), (5.3), (5.70), and (5.71). Then, Problem V is .TV -well-posed and Problem .M is .TM -well posed.

.

The proof of Theorem 5.7 is based on some intermediate results that we present in what follows. To this end, everywhere below we assume that (1.17)–(1.22), (5.2), (5.3), (5.70), and (5.71) hold, even if we do not mention it explicitly. Lemma 5.1 Any .TM -approximating sequence is a .TV -approximating sequence, i.e., .STM ⊂ STV . Proof Let .{un } ⊂ X be a .TM -approximating sequence. Then, using Definition 2.1(b), (5.74), and (5.76), we deduce that there exists a sequence .{θn } ∈ C such that, for each .n ∈ N, the following hold: θn = (Kn , εn ),

.

un ∈ Kn ,

g(un ) ≤ h(εn , un ) d(un , K).

Using now definitions (5.68) and (5.72), we deduce that kv (un ) ≤ g(un ) ≤ h(εn , un ) d(un , K) ≤ h(εn , un ) v − un X

.

for all .v ∈ K, .n ∈ N. This inequality combined with definition (5.67) of the function kv implies that

.

.

Aun − f, v − un  + ϕ(un , v) − ϕ(un , un ) + j 0 (un ; v − un ) +h(εn , un ) v − un X ≥ 0

∀ v ∈ K,

and since .un ∈ Kn , we deduce from (5.75) that .un ∈ ΩV (θn ). Therefore, Definition 2.1(b) guarantees that .{un } is a .TV - approximating sequence. 

5.3 A Variational–Hemivariational Inequality

197

Lemma 5.2 Problem .V is .TV -well-posed. Proof Recall that the unique solvability of Problem .V is guaranteed by Theorem 1.7. Now, consider a .TV -approximating sequence, denoted by .{un }. Then, Definition 2.1(b) and (5.75) show that there exists a sequence .{θn } ∈ C with .θn = (Kn , εn ) such that un ∈ K. n ,

Aun , v − un  + ϕ(un , v) − ϕ(un , un ) + j 0 (un ; v − un ) +h(εn , un ) v − un X ≥ f, v − un 

(5.78)

∀ v ∈ K.

Recall also that the inclusion .{θn } ∈ C implies the following convergences: M

.

Kn − → K, .

(5.79)

εn → 0.

(5.80)

We shall prove that .un → u in X, and to this end, we use arguments similar to those used in the proof of Theorem 5.1(b). Note that there, the hemivariational inequality considered was of the form (5.65) with .ϕ ≡ 0. Therefore, to avoid repetition, we present only the main differences in proof, which are generated by the additional terms .ϕ(u, v) − ϕ(u, u) and .ϕ(un , v) − ϕ(un , un ) in (5.65) and (5.78), respectively. The rest of the proof is structured in three steps. Step (i) The sequence .{un } is bounded in X. Let .v ∈ K be a given element. We use assumption (1.18)(b) to see that .

mA v − un 2X ≤ Av − Aun , v − un  = Av, v − un  − Aun , v − un ,

and therefore, inequality (5.78) yields .

mA v − un 2X ≤ ( Av − f X∗ + h(εn , un )) v − un X

(5.81)

+ϕ(un , v) − ϕ(un , un ) + j 0 (un ; v − un ). Next, we write .

ϕ(un , v) − ϕ(un , un ) = ϕ(un , v) − ϕ(un , un ) + ϕ(v, un ) − ϕ(v, v) + ϕ(v, v) − ϕ(v, un ),

and then we use assumptions (1.19)(b) and (5.70) to see that .

ϕ(un , v) − ϕ(un , un ) ≤ αϕ v − un 2X + cϕ (v) v − un X .

(5.82)

Now, using arguments similar to those used to deduce inequalities (5.15) and (5.14), we have

198

5 Hemivariational Inequalities .

h(εn , un ) ≤ Lh (εn ) v − un X + h(εn , v), . αj un − v 2X

j (un ; v − un ) ≤ 0

+ (c0 + c1 v X ) un − v X .

(5.83) (5.84)

We now combine inequalities (5.81)–(5.84) to see that .

(mA − αϕ − αj − Lh (εn )) un − v X ≤ Av − f X∗ + cϕ (v) + h(εn , v) + c0 + c1 v X ,

and using assumptions (5.2)(c.2 ) and (1.22), we find that there exists a positive constant .C0 which does not depend on n such that un − v X ≤ C0 ( Av − f X∗ + cϕ (v) + h(εn , v) + c0 + c1 v X ),

.

(5.85)

for n large enough. We now use (5.85), the inequality h(εn , v) ≤ h(εn , 0X ) + Lh (εn ) v X ,

.

the convergence (5.80), and assumptions (5.2)(b),(c.2 ) to see that the sequence {un − v} is bounded in X. This implies that .{un } is a bounded sequence in X, which concludes the proof of this step.

.

Step (ii) The sequence .{un } converges weakly to the solution u of Problem .V. Using the step (i) and the reflexivity of the space X, we deduce that, passing to a subsequence, if necessary, we have that un   u as

.

n → ∞,

(5.86)

with some . u ∈ X. Moreover, assumption (5.79) and convergence (5.86) imply that  u ∈ K.

.

(5.87)

Consider now an arbitrary element .v ∈ K and let .n ∈ N. We use inequality (5.78) to see that .

Aun , un − v ≤ ϕ(un , v) − ϕ(un , un ) + j 0 (un ; v − un )

(5.88)

+h(εn , un ) v − un X + f, un − v. Then, assumptions (5.3), (5.71), and (5.2) and the convergences (5.86) and (5.80) imply that .

lim sup j 0 (un ; v − un ) ≤ j 0 ( u; v −  u), .

 lim sup ϕ(un , v) − ϕ(un , un ) ≤ ϕ( u, v) − ϕ( u,  u), .

(5.89)

h(εn , un ) v − un X → 0.

(5.91)

(5.90)

5.3 A Variational–Hemivariational Inequality

199

Therefore, passing to the upper limit in (5.88) and using (5.89)–(5.91), we find that lim sup. Aun , un − v ≤ ϕ( u, v) − ϕ( u,  u) + j 0 ( u; v −  u) + f,  u − v.

(5.92)

u in (5.92) and use the property .j 0 ( u; 0X ) = 0 of the Clarke Next, we take .v =  directional derivative to deduce that .

lim sup Aun , un −  u ≤ 0.

(5.93)

Exploiting now the pseudomonotonicity of the operator A and from (5.86) and (5.93), we have A u,  u − v ≤ lim inf Aun , un − v

.

∀ v ∈ X.

(5.94)

Next, from (5.87), (5.94), and (5.92), we obtain that . u is a solution to Problem .V, as claimed. Thus, by the uniqueness of the solution, we find that . u = u. We deduce from here that the whole sequence .{un } converges weakly to u in X, as .n → ∞, which concludes the proof of this step. Step (iii) The sequence .{un } converges strongly to the solution of Problem .V. We use assumption (1.18)(b) and Proposition 1.2 to see that the operator .A : X → X∗ is an operator of type .(S+ ). Therefore, (5.93), (5.86), and the equality . u = u imply that .un → u in X, which ends the proof of this step. To conclude, we proved that any .TV -approximating sequence converges to the solution of Problem .V, and moreover, this problem has a unique solution. Therefore, using Definition 2.1(c), it follows that the variational–hemivariational inequality .V is well-posed with the Tykhonov triple .TV , which ends the proof of the lemma.  We are now in a position to provide the proof of Theorem 5.7. Proof The .TV -well-posedness of Problem .V follows from Lemma 5.2. Let u be the solution of Problem .V. Then Proposition 5.1 guarantees that u is a solution of Problem .M and .g(u) = 0. It follows from here that u is the unique solution of Problem .M. Indeed, if .u is another solution of .M, then .u ∈ K and .g(u ) ≤ g(u) = 0. On the other hand, (5.69) shows that .g(u ) ≥ 0. We conclude from here that .g(u ) = 0, and using Proposition 5.1, again, it follows that .u is a solution to Problem .V. Since this problem has a unique solution, we find that .u = u. Assume now that .{un } ⊂ X is a .TM -approximating sequence. Then Lemma 5.1 implies that .{un } is a .TV -approximating sequence, and using the .TV -well-posedness of Problem .V, we deduce that .un → u in X. Therefore, using Definition 2.1(c), we obtain that Problem .M is .TM -well-posed, which concludes the proof. 

200

5 Hemivariational Inequalities

5.3.2 Some Consequences In this subsection we provide several consequences of Theorem 5.7. This allows us to recover some well-posedness results previously obtained in the literature and, in addition, to deduce some additional convergence results concerning the solution of the variational–hemivariational inequality (5.65). Everywhere below we use the Tykhonov triples .TV = (I, ΩV , C) and .TM = (I, ΩM , C) defined in Sect. 5.3.1, with various function h which will be specified. (a) Classical Tykhonov well-posedness of Problem .V. Our first result in this section is the following. Corollary 5.4 Keep the assumptions (1.17)–(1.22), (5.3), (5.70), and (5.71). Also, assume that .{un } is a sequence with the following property: for each .n ∈ N, there exists .εn ≥ 0 such that un . ∈ K,

Aun , v − un  + ϕ(un , v) − ϕ(un , un ) + j 0 (un ; v − un ) +εn un − v X ≥ f, v − un 

(5.95)

∀v ∈ K

and, moreover, .εn → 0. Then .un → u in X, with u being the solution to Problem .V. Proof For each .n ∈ N, define .θn ∈ I by the equality .θn = (K, εn ). Moreover, consider the function .h : R+ × X → R+ defined by .h(ε, u) = ε, for all .ε ∈ R+ , .u ∈ X, which, obviously, satisfies condition (5.2). Then it is easy to see that .{θn } ∈ C, and moreover, (5.95) shows that the sequence .{un } represents a .TV -approximating sequence. The convergence .un → u in X is now a direct consequence of the .TV well-posedness of Problem .V, guaranteed by Theorem 5.7. 

Note that Corollary 5.4 represents a well-posedness result for the variationalhemivariational inequality (5.65), in the sense of Tykhonov. It extends Theorem 1.18 formulated in the particular case of variational inequality (1.86) and corresponds to the well-posedness of Problem .V with respect to the Tykhonov triple .T1V = (I1 , Ω1V , C1 ) where I1 =. R+ = [0, +∞), u ∈ Ω1V (θ ) ⇐⇒ u ∈ K

and

Au − f, v − u + ϕ(u, v) − ϕ(u, u) +j 0 (u; v − u) + θ v − u X ≥ 0 ∀ v ∈ K for any θ ∈ I1 , {θn } ∈ C1 ⇐⇒ {θn } ∈ I1 and θn → 0.

5.3 A Variational–Hemivariational Inequality

201

Obviously, the proof we presented above illustrates the strategy described on page 93. Indeed, it is based on the inclusion .ST1 ⊂ STV combined with the .TV -wellposedness of Problem .V, guaranteed by Theorem 5.7. (b) Levitin-Polyak well-posedness of problems .V and .M. We now proceed with the following result. Corollary 5.5 Keep the assumptions (1.17)–(1.22), (5.3), (5.70), and (5.71), and moreover, assume that .{un } is a sequence with the following property: for each .n ∈ N, there exist two sequences .{wn } ⊂ X and .{εn } ⊂ R+ such that .wn → 0X in X, .εn → 0, and in addition, .

un + wn ∈ K,

Aun , v − un  + ϕ(un , v) − ϕ(un , un )

+j 0 (un ; v − un ) + εn v − un X ≥ f, v − un 

(5.96) ∀ v ∈ K,

for each .n ∈ N. Then, .un → u in X, with u being the solution to Problem .V. Proof For each .ω ≥ 0, define the set .Kω by equality .Kω = K + B(0X , ω), where B(0X , ω) represents the ball of radius .ω centered at .0X , see (2.83). Using (1.17), it is easy to see that .Kω is a nonempty, closed, convex subset of X. Next, for each .n ∈ N, we denote by .θn the element of I given by the equality .θn = (K wn X , εn ). Then, since .wn → 0X in X, using the convergence (2.84) proved on page 85, it is easy to see that .{θn } ∈ C. Also, consider the function .h : R+ × X → R+ defined by .h(ε, u) = ε, for all .ε ∈ R+ , .u ∈ X, which, obviously, satisfies condition (5.2). We now use inequality (5.96) to see that for each .n ∈ N we have .un ∈ ΩV (θn ) with .θn = (K wn X , εn ). Next, since .{θn } ∈ C, we deduce that the sequence .{un } represents a .TV -approximating sequence. The convergence .un → u in X is now a direct consequence of the .TV -well-posedness of Problem .V, guaranteed by Theorem 5.7. 

.

Recall that Corollary 5.5 represents a well-posedness result for the variational– hemivariational inequality (5.65), in the sense of Levitin-Polyak. This concept extends Definition 1.22 formulated in the particular case of variational inequality (1.86). It corresponds to the well-posedness of Problem .V with respect to the Tykhonov triple .T2V = (I2 , Ω2V , C2 ) where   I2 .= θ = (ω, ε) : ω ≥ 0, ε ≥ 0, = [0, +∞) × [0, +∞), u ∈ Ω2V (θ ) ⇐⇒ u ∈ Kω

and

Au − f, v − u + ϕ(u, v) − ϕ(u, u) +j 0 (u; v − u) + ε v − u X ≥ 0

∀ v ∈ K,

for any θ = (ω, θ ) ∈ I2 , {θn } ∈ C2 ⇐⇒ θn = (ωn , εn )

and

ωn → 0, εn → 0.

202

5 Hemivariational Inequalities

Note that, obviously, .ST2V ⊂ STV and, therefore, .T2V ≤ TV . We turn now to the following result concerning the well-posedness of Problem .M. Corollary 5.6 Keep the assumptions (1.17)–(1.22), (5.3), (5.70), and (5.71), and moreover, assume that .{un } is a sequence with the following property: for each .n ∈ N, there exist two sequences .{wn } ⊂ X and .{εn } ⊂ R+ such that .wn → 0X in X, .εn → 0, and in addition, .

un + wn ∈ K,

g(un ) ≤ εn d(un , K),

for each .n ∈ N. Then, .un → u in X, where, again, u denotes the solution to Problem M.

.

Proof Using arguments similar to those used in the proof of Corollary 5.5, we deduce that .{un } is a .TM -approximating sequence. The convergence .un → u in X is now a direct consequence of the .TM -well-posedness of Problem .V, guaranteed 

by Theorem 5.7. (c) A convergence result. We now present a convergence result to the solution of Problem .V. To this end, besides the data K, A, j , and f , for each .n ∈ N we consider a set .Kn , an operator .An , a function .ϕn , and an element .fn such that the following hold: .

Kn is a closed convex subset of X M

such that K ⊂ Kn and Kn − → K as n → ∞.

⎧ An : X → X∗ satisfies condition (1.18) with mn > 0 ⎪ ⎪ ⎨ and there exists an ≥ 0 such that: ⎪ (a) An v − Av X∗ ≤ an ( v X + 1) ∀ v ∈ X, n ∈ N. ⎪ ⎩ (b) an → 0 as n → ∞. ⎧ ⎪ ϕn : X × X → R satisfies condition (1.19) with αn ≥ 0 ⎪ ⎪ ⎪ ⎪ ⎨ and there exists bn ≥ 0 such that: (a) ϕn (u, v) − ϕn (u, u) + ϕ(u, u) − ϕ(u, v) ⎪ ⎪ ⎪ ≤ bn ( u X + 1) u − v X ∀ u, v ∈ X, n ∈ N. ⎪ ⎪ ⎩ (b) b → 0 as n → ∞. n .

αn + αj < mn .. ∗

(5.97)

.

.

(5.98)

(5.99)

(5.100) ∗

fn ∈ X and fn → f in X as n → ∞..

(5.101)

αn → 0 as n → ∞.

(5.102)

5.3 A Variational–Hemivariational Inequality

203

With these data, we consider the following perturbation of the variational– hemivariational inequality .V. Problem .Vn Find .un ∈ X such that .

un ∈ Kn ,

An un , v − un  + ϕn (un , v) − ϕn (un , un )

(5.103)

+j 0 (un ; v − un ) ≥ fn , v − un  ∀ v ∈ Kn . Our next result in this subsection is the following. Corollary 5.7 Assume (1.17)–(1.22), (5.3), (5.70), (5.71), and (5.97)–(5.102). Then, for each .n ∈ N, there exists a unique solution .un to Problem .Vn . Moreover, the sequence .{un } converges in X to the solution u of Problem .V. Proof The existence of the unique solution to Problem .Vn is a direct consequence of Theorem 1.7. Let .n ∈ N and .v ∈ K. We write 0 Au . n , v − un  + ϕ(un , v) − ϕ(un , un ) + j (un ; v − un ) − f, v − un 

= Aun , v − un  − An un , v − un  + An un , v − un  +ϕ(un , v) − ϕ(un , un ) − ϕn (un , v) + ϕn (un , un ) + ϕn (un , v) − ϕn (un , un ) +j 0 (un ; v − un ) + fn − f, v − un  − fn , v − un . Then, since .K ⊂ Kn , inequality (5.103) implies that 0 Au . n , v − un  + ϕ(un , v) − ϕ(un , un ) + j (un ; v − un ) − f, v − un 

≥ Aun − An un , v − un  + ϕ(un , v) − ϕ(un , un ) − ϕn (un , v) + ϕn (un , un ) +fn − f, v − un . Therefore, 0 Au . n , v − un  + ϕ(un , v) − ϕ(un , un ) + j (un ; v − un ) − f, v − un 

≥ − Aun − An un X∗ v − un X +ϕ(un , v) − ϕ(un , un ) − ϕn (un , v) + ϕn (un , un ) − fn − f X∗ v − un X , and using assumptions (5.98)(a) and (5.99)(a), we obtain 0 Au . n , v − un  + ϕ(un , v) − ϕ(un , un ) + j (un ; v − un ) (5.104)

 + (an + bn )( un X + 1) + fn − f X∗ v − un X ≥ f, v − un .

204

5 Hemivariational Inequalities

We now denote εn = max {an + bn , fn − f X∗ },

.

(5.105)

and then we use (5.104) to see that .un satisfies inequality (5.78) with .h : [0, +∞) × X → R defined by

 h(ε, u) = ε u X + 2

.

∀ ε ∈ [0, +∞), u ∈ X.

Note that this function satisfies condition (5.2) with .Lh (ε) = ε. On the other hand, (5.105), (5.98)(b) (5.99)(b), and (5.101) show that .εn → 0, and therefore, (5.80) holds. Moreover, using (5.97), we deduce that the sequence .{θn } with .θn = (Kn , εn ) belongs to the set .C defined in (5.77), which implies that .{un } is a .TV -approximating sequence. The convergence .un → u in X follows from the .TV well-posedness of Problem .V, guaranteed by Theorem 5.7. 

Remark 5.1 Assume that .A satisfies condition (1.18), and for each .n ∈ N, assume that the operator .An satisfies condition (5.32). Then, it is easy to see that .An satisfies condition (5.98), too. Next, we provide an example of functions .ϕ and .ϕn which satisfy conditions (1.19), (5.99). Example 5.1 Let .ϕ : X × X → R be the function defined by .ϕ(u, v) = p(u)q(v), where .p : X → R and .q : X → R. Assume that p is a Lipschitz continuous function with Lipschitz constant .Lp and q is a convex Lipschitz continuous function with Lipschitz constant .Lq . Then, it is easy to see that .ϕ satisfies condition (1.19) with .αϕ = Lp Lq . Next, for each .n ∈ N, consider the function .ϕn : X × X → R defined by .ϕn (u, v) = pn (u)q(v), where .pn : X → R is a Lipschitz continuous function with Lipschitz constant .Ln . Moreover, assume that there exists .cn ≥ 0 such that |pn (u) − p(u)| ≤ cn ( u X + 1)

.

for all u ∈ X,

and in addition, .cn → 0. Then, it is easy to see that .ϕn satisfies condition (5.99) with .bn = cn Lq .

Chapter 6

Inclusions

In this chapter we study the well-posedness of inclusions. We start with a stationary inclusion for which we prove the well-posedness with various Tykhonov triples, together with several convergence results, including a convergence criterion. We extend a part of these results to a history-dependent inclusion. The proofs are based on the results obtained in Chap. 3, in the study of fixed point problems. Finally, we consider a history-dependent variational inequality with time-dependent constraints and perform its analysis by using arguments of duality introduced in Sect. 2.2. Given a Tykhonov triple, everywhere in this chapter we use the notation and definitions introduced in Sect. 2.1.2.

6.1 Stationary Inclusions In this section X represents a real Hilbert space endowed with the inner product (·, ·)X and its associated norm . · X . We denote by .0X the zero element of X, by .IX the identity map on X, and by .2X the set of parts of X. The symbols “.” and “.→” represent the weak and the strong convergence in X, respectively. All the limits, upper and lower limits, will be considered as .n → ∞, even if we do not mention it explicitly.

.

6.1.1 Problem Statement and Preliminaries The inclusion problem we consider in this section is governed by the data K, A, and f , assumed to satisfy the following conditions: .

K is a nonempty, closed, convex subset of X.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Sofonea, Well-Posed Nonlinear Problems, Advances in Mechanics and Mathematics 50, https://doi.org/10.1007/978-3-031-41416-9_6

(6.1) 205

206

6 Inclusions

⎧ ⎪ A : X → X is a strongly monotone Lipschitz continuous ⎪ ⎪ ⎪ ⎪ ⎨ operator, i.e., there exist mA > 0 and LA > 0 such that: .

⎪ (a) (Au − Av, u − v)X ≥ mA u − v2X ∀ u, v ∈ X. ⎪ ⎪ ⎪ ⎪ ⎩ (b) Au − AvX ≤ LA u − vX ∀ u, v ∈ X.

(6.2)

.

f ∈ X.

(6.3)

We denote by .PK : X → K the projection operator on K and by .NK : X → 2X the outward normal cone of K and recall that equivalences (1.38) and (1.45) hold. Then, we consider the following inclusion. Problem .P. Find an element .u ∈ X such that .

 − u ∈ NK Au + f ).

(6.4)

The unique solvability of Problem .P was provided in Theorem 3.4 on page 112. For the convenience of the reader, we recall the following ingredients which have been used in the proof of this theorem: ⎧ ⎪ ⎨ The operator A : X → X is invertible and its inverse A−1. : X → X is strongly monotone and Lipschitz continuous ⎪ ⎩ with constants m = m2A and L = 1 , respectively. mA

(6.5)

.

LA

⎧ ⎨ The element u ∈ X is a solution of Problem P if and only if σ := Au + f is a fixed point of the operator Λρ : X → X   ⎩ defined by Λρ ξ = PK ξ − ρA−1 (ξ − f ) ∀ ξ ∈ X, for any ρ > 0. ⎧ The operator Λρ defined in (6.6) is a contraction on X, ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎨ for any real number ρ such that 0 < ρ < 2m 2 = 2m2A , that is, L ⎪ Λρ ξ − Λρ ηX ≤ k(ρ)ξ − ηX ∀ ξ, η ∈ X ⎪ ⎪ ⎪  ⎪ ⎩ with k(ρ) = 1 − 2ρm + ρ 2 L 2 < 1.

LA

.

(6.6)

(6.7)

Recall that the statements (6.5) and (6.7) correspond to Proposition 1.12 and Remark 3.4, respectively. Moreover, (6.6) follows from the equivalence (3.56) on page 113. Everywhere below we denote by u the solution of Problem .P provided by Theorem 3.4. Consider a sequence of elements .{un } ⊂ X. Our aim in what follows is to provide conditions that guarantee the convergence un → u

.

in X, as n → ∞.

(6.8)

6.1 Stationary Inclusions

207

In order to provide an answer to the question above, we use the two step strategy (a) described on page 93, namely: (i) First, we identify a Tykhonov triple .T such that Problem .P is well-posed with .T ; this means that each sequence in the set of .T -approximating sequences, denoted by .ST , converges in X to the solution u of Problem .P. (ii) Second, we prove that the given sequence .{un } belongs to .ST , and using step i), we conclude that the convergence (6.8) holds. Our interest is to apply this strategy in order to establish convergence results of the solution u with respect to the data K, A, and f . To this end, we consider three sequences .{Kn }, .{An } and .{fn } such that, for each .n ∈ N, .Kn , .An , and .fn represent perturbations of K, A, and f , respectively, assumed to satisfy conditions (6.1)– (6.3). Then, using Theorem 3.4, it follows that, for each .n ∈ N, there exists a unique solution to the following inclusion problem. Problem .Pn . Find an element .un ∈ X such that .

 − un ∈ NKn An un + fn ).

(6.9)

In Sects. 6.1.2 and 6.1.3 below, we shall deduce the convergence of the solution un to u, under various assumptions on the data. We end this introductory subsection with the following elementary result which will be used below.

.

Proposition 6.1 Let K be a closed convex nonempty subset of X and let .A = IX . Then, for each .f ∈ X, the solution of the inclusion (6.4) is given by u = PK f − f.

(6.10)

.

In addition, if K is the ball of radius 1 centered at .0X , then u=

.

⎧ ⎨ f1 − 1 f X ⎩

0

if f X > 1, (6.11) if f X ≤ 1.

Proof We use (1.45) to see that, in the particular case when .A = IX , u is a solution to (6.4) if and only if u + f ∈ K,

.

(u + f − v, u)X ≤ 0

∀v ∈ K

or, equivalently, u + f ∈ K,

.

((u + f ) − v, (u + f ) − f )X ≤ 0

∀ v ∈ K.

(6.12)

We now combine (6.12) and (1.38) to see that .u + f = PK f which proves (6.10).

208

6 Inclusions

Assume now that K is the closed ball of radius 1 centered at .0X , i.e.,  .K = v ∈ X : vX ≤ 1 . Then, using (1.38), it is easy to see that PK f =

.

⎧ f ⎨ f X ⎩

if f X > 1,

f

if f X ≤ 1, 

and using (6.10), we deduce (6.11).

6.1.2 Some Well-Posedness Results We now introduce three relevant Tykhonov triples associated with Problem .P. To this end, we use equivalence (1.45) to see that an element .u ∈ X is a solution to Problem .P if and only if Au + f ∈ K,

.

(Au + f − v, u)X ≤ 0

∀ v ∈ K.

(6.13)

Relaxing this inequality suggests us to consider the approximating sets  Ω1 (θ ) =. u˜ ∈ X : Au˜ + f ∈ K, (Au˜ + f − v, u) ˜ X ≤ θ ∀ v ∈ K ,.  Ω2 (θ ) = u ∈ X : A u + f ∈ K,   (A u + f − v, u)X ≤ θ  uX + 1 ∀ v ∈ K ,  Ω3 (θ ) = u ∈ X : A u + f ∈ K, .   (A u + f − v, u)X ≤ θ A u + f − vX + 1 ∀ v ∈ K ,

(6.14) (6.15)

(6.16)

for all .θ ≥ 0. Moreover, we consider the sets I and .C defined by I = R+ = . [0, +∞), .

(6.17)

 C = {θn } ⊂ I : θn → 0, as n → ∞ .

(6.18)

With these ingredients, we introduce the triples .T1 , .T2 , and .T3 defined by T1 = (I, Ω1 , C),

.

T2 = (I, Ω2 , C),

T3 = (I, Ω3 , C).

6.1 Stationary Inclusions

209

Recall that, since we assume (6.1)–(6.3), Theorem 3.4 guarantees that Problem .P has a unique solution u. Then, using (6.13), it is easy to see that .u ∈ Ω1 (θ ), .u ∈ Ω2 (θ ), and .u ∈ Ω3 (θ ), for each .θ ∈ I . This implies that .Ω1 (θ ) =

∅, .Ω2 (θ ) =

∅, and .Ω3 (θ ) =

∅ for each .θ ∈ I , and therefore, .T1 , .T2 , and .T3 are Tykhonov triples in the sense of Definition 2.1 a). The properties of these triples can be summarized as follows. Proposition 6.2 Assume (6.1)–(6.3). Then, the following statements hold: (a) The Tykhonov triples .T1 and .T2 are equivalent, i.e., .T1 ∼ T2 . (b) The Tykhonov triples .T1 and .T2 are smaller than the Tykhonov triple .T3 , i.e., .T1 ≤ T3 and .T2 ≤ T3 . Moreover, unless additional assumptions, these inequalities are strict, i.e., .T1 < T3 and .T2 < T3 . (c) If .K ⊂ X is bounded, then the Tykhonov triples .T1 , .T2 , and .T3 are equivalent, i.e., .T1 ∼ T2 ∼ T3 . Proof (a) Let .{un } be a .T2 -approximating sequence, i.e., .{un } ⊂ ST2 . This implies that there exists a sequence .{θn } ⊂ R+ such that .θn → 0 and, moreover, Aun + f ∈ K,

.

  (Aun + f − v, un )X ≤ θn un X + 1 ∀ v ∈ K,

(6.19)

for each .n ∈ N. We fix .n ∈ N and .v ∈ K. Then, using (6.2)(a) and (6.19), we find that .

mA un 2X ≤ (Aun − A0X , un )X = (Aun , un )X − (A0X , un )X   ≤ θn un X + 1 + (v − f, un )X − (A0X , un )X   ≤ θn + v − f X + A0X X un X + θn .

This implies that there exists .D > 0, which does not depend on n, such that un  ≤ D.

.

(6.20)

θn = θn (D + 1), and We now use (6.19) and (6.20) to see that .un ∈ Ω1 ( θn ) with . since .θn → 0, using notation (2.2), we deduce that .{un } ⊂ ST1 . It follows from here that .ST2 ⊂ ST1 . On the other hand, it is easy to see that .Ω1 (θ ) ⊂ Ω2 (θ ) for each .θ ∈ I , which implies that .ST1 ⊂ ST2 . We deduce from above that .ST = ST and, therefore, .T1 ∼ T2 . 1 2 (b) Note that .Ω1 (θ ) ⊂ Ω3 (θ ) for each .θ ∈ I , which shows that .ST1 ⊂ ST3 . We deduce from here that .T1 ≤ T3 , and since .T1 ∼ T2 , we obtain that .T2 ≤ T3 , too. In order to prove that these inequalities are strict, we consider the following counter-example. Let .K = X, .A = IX , .f = 0X , .u0 ∈ X, and .u0 = 0X and let 1 1 .un = n u0 and .θn = n u0 X , for each .n ∈ N. Then, using the inequality

210

6 Inclusions .

(un − v, un )X ≤ un − vX un X =

1 u0 X un − vX n

∀ v ∈ X, n ∈ N,

we deduce that .un ∈ Ω3 (θn ) for each .n ∈ N, which shows that .{un } is a .T3 approximating sequence. Assume now that .{un } is a .T1 -approximating sequence. Then Definition 2.1(b) guarantees that there exists a sequence .{ θn } ∈ C such that (un − v, un )X ≤ θn

.

∀ v ∈ X, n ∈ N

or, equivalently, .

1 1 u0 2X − (v, u0 )X ≤ θn 2 n n

∀ v ∈ X, n ∈ N.

(6.21)

We now choose .n ∈ N arbitrary and take .v = −nu0 in inequality (6.21) to  θn ≥ 1 + n12 u0 2X for each .n ∈ N. This shows that the sequence deduce that . .{ θn } does not converge to zero which is in contradiction with the inclusion .{ θn } ∈ C. We conclude from above that there exist .T3 -approximating sequences which are not .T1 -approximating sequences, and therefore, .T1 < T3 . Moreover, since .T1 ∼ T2 , we deduce that .T2 < T3 , too. (c) Assume now that the set K is bounded, and let .{un } be a .T3 -approximating sequence. Then there exists a sequence .{θn } ⊂ R+ such that .θn → 0, and moreover,   Aun. + f ∈ K, (Aun + f − v, un )X ≤ θn Aun + f − vX + 1 (6.22) ∀ v ∈ K, n ∈ N. Now, since K is a bounded set, the inclusions .Aun +f ∈ K and .v ∈ K in (6.22) show that there exists a constant .E > 0 such that Aun + f − vX ≤ E

.

∀ v ∈ K, n ∈ N.

(6.23)

We now combine (6.22) and (6.23) to see that Aun + f ∈ K,

.

(Aun + fn − v, un )X ≤ θn (E + 1)

∀ v ∈ K, n ∈ N,

which shows that .{un } is a .T1 -approximating sequence. We conclude from here that .ST3 ⊂ ST1 , and since we already proved that .ST1 = ST2 ⊂ ST3 , we deduce that .ST1 = ST2 = ST3 , which completes the proof. 

6.1 Stationary Inclusions

211

We now state and prove the following well-posedness result. Theorem 6.1 Assume (6.1)–(6.3). Then Problem .P is .T1 -, .T2 -, and .T3 -well-posed. Proof Let .{un } be a .T3 -approximating sequence. Then there exists a sequence {θn } ⊂ R+ such that .θn → 0, and moreover, (6.22) holds. We fix .n ∈ N, and then we take .v = Aun + f in (6.13) and .v = Au + f in (6.22). Adding the resulting inequalities, we find that

.

  (Aun − Au, un − u)X ≤ θn Aun − AuX + 1 .

.

Next, we use the properties (6.2) of the operator A to deduce that un − u2X ≤

.

θn LA θn un − uX + . mA mA

We now use the elementary inequality (1.33) to see that  θn LA .un − uX ≤ + mA

θn , mA

and therefore, the convergence .θn → 0 implies that .un → u in X. It follows from here that .ST3 ⊂ SP , where, recall, .SP is the set of sequences defined by (2.1). We now use the equivalence (2.3) to deduce that Problem .P is .T3 -well-posed. On the other hand, using Proposition 6.2, it follows that .ST1 = ST2 ⊂ ST3 , and therefore, .ST1 = ST2 ⊂ SP . This implies the .T1 - and .T2 -well-posedness of Problem .P and concludes the proof.  We now wonder if the Tykhonov triples .T1 , .T2 , and .T3 can be used to prove the convergence of the solution of the perturbed inclusion (6.9) to the solution of the original inclusion (6.4). To this end, we start with the following consequence of Theorem 6.1. Corollary 6.1 Assume (6.1)–(6.3). For each .n ∈ N, let .fn ∈ X, denote by .un the solution of the inclusion (6.4) with .fn instead of f , and assume that .Aun + f ∈ K. Then, the convergence .fn → f in X implies the convergence (6.8). Proof Let .n ∈ N be fixed. We have .−un ∈ NK (Aun + fn ), and using (1.45), we deduce that (Aun + fn − v, un )X ≤ 0

.

∀ v ∈ K.

(6.24)

Therefore, (Aun + f − v, un )X ≤ (f − fn , un )X

.

∀ v ∈ K.

(6.25)

212

6 Inclusions

We now prove that the sequence .{un } is bounded in X. To this end, we fix an element .v ∈ K, and using (6.24), we have (Aun − A0X , un )X ≤ (v − A0X − fn , un )X .

.

Therefore, using assumption (6.2)(a), it follows that mA un 2X ≤ (Aun − A0X , un )X ≤ (v − A0X − fn , un )X

.

≤ A0X + fn − vX un X . This shows that there exists .D > 0, which does not depend on n, such that (6.20) holds. Next, we use (6.25) and (6.20) to see that (Aun + f − v, un )X ≤ Dfn − f X

.

∀ v ∈ K.

(6.26)

Inequality (6.26), the convergence .fn → f in X, and the regularity .Aun + f ∈ K guarantee that .{un } is a .T1 -approximating sequence. We now use Theorem 6.1 and Definition 2.1(c) to deduce the convergence (6.8), which concludes the proof.  Note that Corollary 6.1 provides the convergence of the solution .un of the inclusion .Pn with .Kn = K and .An = A to the solution u of the inclusion .P, under the very restrictive condition .Aun + f ∈ K, for each .n ∈ N. This condition is satisfied in Example 6.1 but fails to be satisfied in Example 6.2 we present below. Example 6.1 Assume (6.1), .A = IX , and .f ∈ int(K), where .int(K) represents the interior of K in the strong topology of X. We claim that if .fn → f in X, then the regularity condition .Aun + f ∈ K is satisfied. Indeed, since .f ∈ int(K), the convergence .fn → f in X implies that, for n large enough, we have .fn ∈ K. Therefore, Proposition 6.1 implies that .un = PK fn − fn = 0X , and hence .Aun + f = un + f = f ∈ int(K) ⊂ K. Example 6.2 Assume that K is the ball of radius 1 centered at .0X , .A = IX , and .f ∈ X, .f X = 2. Then, using (6.11), we obtain  that the  solution of the inclusion (6.4) is .u = − f2 . Next, for each .n ≥ 2, let .fn = 1 − n1 f . Using again (6.11), it is easy   to see that the solution of the inclusion (6.4) with .fn instead of f is .un = n1 − 12 f . 1 1 This equality implies that .un +f = n + 2 f , and therefore, .un +f X = 1+ n2 > 1. We conclude from here that .un + f ∈ K, which shows that the regularity condition .Aun + f ∈ K is not satisfied in this case, as claimed. This implies that the sequence .{un } is neither a .T1 -approximating sequence nor a .T2 -approximating sequence and nor a .T3 -approximating sequence. Thus, the well-posedness of Problem .P with the Tykhonov triples .T1 , .T2 , and .T3 cannot be used in order to prove the continuous dependence of the solution of Problem .P with respect to the data. Nevertheless, note that .fn → f in X and .un → u in X.

6.1 Stationary Inclusions

213

Motivated by Example 6.2, in the next section we consider an additional Tykhonov triple, .TM , such that the sequence .{un } in Example 6.2 is a .TM approximating sequence. This will allow us to extend the convergence result in Corollary 6.1 by describing the continuous dependence of the solution u with respect to the data K, A, and f .

6.1.3 An Additional Well-Posedness Result We now consider the triple .TM = (IM , ΩM , CM ) defined as follows: 

ε) : K

is closed nonempty convex subset of X,. IM =. θ = (K, ε≥0 , 

(A

. ΩM (θ ) = u ∈ X : A u + f ∈ K, u + f − v, u )X ≤ ε ∀ v ∈ K

(6.27)

(6.28)

ε) ∈ IM , ∀ θ = (K,  M CM = {θn } ⊂ IM : θn = (Kn , εn ), Kn − → K in X, εn → 0 .

(6.29)

Note that Theorem 3.4 implies that .ΩM (θ ) = ∅ for each .θ ∈ IM , and therefore, TM is a Tykhonov triple in the sense of Definition 2.1(a). Our main result, based on the pseudomonotonicity and Mosco convergence ingredients, is the following.

.

Theorem 6.2 Assume (6.1)–(6.3). Then Problem .P is .TM -well-posed. Proof Denote by u the unique solution of Problem .P obtained in Theorem 3.4, and consider a .TM -approximating sequence, denoted by .{un }. Then, Definition 2.1(b) and (6.28) show that there exists a sequence .{θn } ∈ CM with .θn = (Kn , εn ) such that Aun + f. ∈ Kn ,

(Aun + f − v, un )X ≤ εn

∀ v ∈ Kn ,

(6.30)

for each .n ∈ N. Recall also that the inclusion .{θn } ∈ CM implies the following convergences: M

.

Kn − →K εn → 0.

in X, .

(6.31) (6.32)

214

6 Inclusions

We shall prove that .un → u in X, and to this end, we divide the proof in three steps, described below. The sequences .{un } and .{Aun } are bounded in X.

Step (i)

Let .v ∈ K be a given element. Then, (6.31) implies that there exists a sequence {vn } ⊂ X such that .vn ∈ Kn for all .n ∈ N and .vn → v in X. Let .n ∈ N. We write

.

.

(Aun − Avn , un − vn )X + (Avn + f − vn , un − vn )X +(Aun + f − vn , vn )X = (Aun + f − vn , un )X ,

and then we use (6.30) with .v = vn to see that .

(Aun − Avn , un − vn )X + (Avn + f − vn , un − vn )X +(Aun + f − vn , vn )X ≤ εn .

Therefore, assumption (6.2)(a) yields .

mA un − vn 2X ≤ Avn + f − vn X un − vn X +Aun + f − vn X vn X + εn

and, since .Aun +f −vn = (Aun −Avn )+(Avn +f −vn ), using assumption (6.2)(b), we find that .

mA un − vn 2X ≤ Avn + f − vn X un − vn X +LA un − vn X vn X + Avn + f − vn X vn X + εn .

Now, since the sequence .{vn } is bounded in X and A is a bounded operator, the convergence (6.32) and the previous inequality imply that there exist two positive constants .C1 and .C2 which do not depend on n such that un − vn 2X ≤ C1 un − vn X + C2 .

.

This inequality combined with (1.33) shows that the sequence .{un − vn } is bounded in X and, therefore, .{un } is a bounded sequence in X, too. We conclude this step by using the property (6.2)(b) of the operator A. Step (ii)

The sequence .{un } converges weakly to the solution u of Problem .P.

Using the step (i) and the reflexivity of the space X, we deduce that, passing to a subsequence, if necessary, we have

6.1 Stationary Inclusions

215

un  u in X,

.

in X,

Aun  z

as

n → ∞, .

as

(6.33)

n → ∞,

(6.34)

u, z ∈ X. Our aim in what follows is to prove that . u is a solution of with some . Problem .P. To this end, we remark that the regularity .Aun + f ∈ Kn combined with the convergences (6.34) and (6.31) implies that z + f ∈ K.

(6.35)

.

Next, we use (6.30) to see that .

(Aun + f − vn , un )X ≤ εn

∀ n ∈ N.

Here we assume that .v ∈ K is given, .{vn } ⊂ X is a sequence such that .vn ∈ Kn for each .n ∈ N, and .vn → v in X. Then, passing to the upper limit and using the convergence (6.32), we find that .

lim sup (Aun + f − vn , un )X ≤ 0.

(6.36)

We now use the convergences (6.33), .vn → v in X, and inequality (6.36) to deduce that .

lim sup (Aun , un )X ≤ (v − f, u )X

∀ v ∈ K.

(6.37)

On the other hand, (6.35) allows us to take .v = z + f in (6.37) in order to find that .

lim sup (Aun , un )X ≤ (z, u )X .

(6.38)

Inequality (6.38) and the convergence (6.34) yield .

lim sup (Aun , un − u)X ≤ 0

and, therefore, Proposition 1.14 implies that (A u, u − v)X ≤ lim inf (Aun , un − v)X

.

∀ v ∈ X.

(6.39)

Moreover, the convergence (6.34) yields .

lim sup (Aun , un − v)X = lim sup (Aun , un )X − (z, v)X

∀v ∈ X

and, therefore, inequality (6.38) shows that .

lim sup (Aun , un − v)X ≤ (z, u − v)X

∀ v ∈ X.

(6.40)

216

6 Inclusions

We now combine inequalities (6.39) and (6.40) to find that .

(A u, u − v)X ≤ (z, u − v)X

∀ v ∈ X,

which implies that A u = z.

(6.41)

.

Next, we use (6.34) and (6.41) to see that .

lim sup (Aun , un − v)X = lim sup (Aun , un )X − (A u, v)X

∀v ∈ X

and, therefore, (6.37) yields .

lim sup (Aun , un − v)X ≤ (v − f, u)X − (A u, v)X

∀ v ∈ K.

(6.42)

We now combine (6.39) and (6.42) to find that (A u, u − v)X ≤ (v − f, u)X − (A u, v)X

.

∀v ∈ K

or, equivalently, (A u + f − v, u)X ≤ 0

.

∀ v ∈ K.

(6.43)

Next, from (6.35), (6.41), and (6.43), we obtain that . u is a solution to Problem P, as claimed. Thus, by the uniqueness of the solution of Problem .P, guaranteed by Theorem 3.4, we find that . u = u. A careful analysis of the results presented above indicates that every subsequence of .{un } which converges weakly in X has the same weak limit u. On the other hand, Step (i) guarantees that .{un } is bounded in X. Therefore, we deduce that the whole sequence .{un } converges weakly to u in X, as .n → ∞, which concludes the proof of this step.

.

The sequence .{un } converges strongly to the solution u of Problem .P.

Step (iii)

We take .v = u in (6.39) and (6.40), and then we use the equality . u = u to obtain 0 ≤ lim inf (Aun , un − u)X ≤ lim sup (Aun , un − u)X ≤ 0,

.

which shows that .(Aun , un − u)X → 0, as .n → ∞. Therefore, using the strong monotonicity of the operator A and the convergence .un  u in X, we have mA un − u2X ≤ (Aun − Au, un − u)X = (Aun , un − u)X − (Au, un − u)X → 0,

.

as .n → ∞. Hence, it follows that .un → u in X, which concludes the proof of this step.

6.1 Stationary Inclusions

217

To summarize, we proved that any .TM -approximating sequence converges in X to the unique solution of Problem .P. Therefore, using Theorem 3.4 and Definition 2.1(c), it follows that Problem .P is .TM -well-posed, which concludes the proof of the theorem.  We end this subsection with a consequence of Theorem 6.2 which extends Corollary 6.1, since it represents a continuous dependence result of the solution of Problem .P with respect to the data K, A, and f . So, consider three sequences .{Kn }, .{An }, and .{fn } such that, for each .n ∈ N, the following hold: Kn is a. nonempty, closed, convex subset of K..

(6.44)

An : X → X satisfies condition (6.2) with constants mn and Ln ..

(6.45)

fn ∈ X.

(6.46)

Then, using Theorem 3.4 it follows that for each .n ∈ N there exists a unique solution to the inclusion problem .Pn . Consider now the following additional assumptions:

.

⎧ For each n ∈ N, there exists an ≥ 0 such that: ⎪ ⎪ ⎨ (a) An v − AvX ≤ an (vX + 1) for all v ∈ X. ⎪ ⎪ ⎩ (b) an → 0 as n → ∞. 

There exist m0 > 0 and L0 > 0 such that m 0 ≤ m n ≤ Ln ≤ L0 M

Kn − →K fn → f

∀ n ∈ N.

.

.

(6.47)

(6.48)

in X..

(6.49)

in X.

(6.50)

We have the following result. Corollary 6.2 Assume (6.1)–(6.3) and (6.44)–(6.50). Then the solution .un of Problem .Pn converges to the solution u of Problem .P, that is, .un → u in X. Proof Let .n ∈ N. Then, inclusion (6.9) implies that An un + fn ∈ Kn ,

(An un + fn − v, un )X ≤ 0

.

∀ v ∈ Kn .

(6.51)

We first prove that the sequence .{un } is bounded in X. To this end, we denote wn = Aun − An un + f − fn ,

.

218

6 Inclusions

which implies that Aun + f = An un + fn + wn .

.

(6.52)

Moreover, using assumptions (6.47), we have   wn X ≤ Aun − An un X + fn − f X ≤ an un X + 1 + fn − f X ,

.

and using the notation

.an = an + fn − f X ,

(6.53)

  wn X ≤ an un X + 1 .

(6.54)

we find that .

We now fix an element .v ∈ K, and using (6.49), we know that there exists a sequence .{vn } such that vn ∈ Kn

∀ n ∈ N,

.

vn → v in X.

(6.55)

Moreover, (6.51) implies that (An un + fn − vn , un )X ≤ 0

.

or, equivalently, (An un − An 0X , un )X ≤ (vn − fn − An 0X , un )X .

.

Therefore, using assumptions (6.45) and (6.48) it follows that m0 un 2X ≤ mn un 2X ≤ (An un − An 0X , un )X ≤ (vn − fn − An 0X , un )X ,

.

which implies that m0 un X ≤ An 0X + fn − vn X .

.

(6.56)

We now use assumption (6.47) to see that .An 0X → A0X in X, which combined with (6.50) and (6.55) shows that the sequence .{An 0X + fn − vn } is bounded in X. Therefore, the bound (6.56) implies that there exists .D > 0, which does not depend on n, such that (6.20) holds. Let δn = an (D + 1),

.

(6.57)

6.1 Stationary Inclusions

219

n be the subset of denote by .Bn the closed ball of radius .δn centered at .0X , and let .K X given by

n = Kn + Bn . K

.

(6.58)

We use (6.54) and (6.20) and notation (6.57) to see that .wn ∈ Bn , and since (6.51) shows that .An un + fn ∈ Kn , (6.52), and (6.58) yield

n . Aun + f ∈ K

.

(6.59)

n . Using (6.58), we can write Let .v ∈ K .v = v + z, where .v ∈ Kn and .z ∈ Bn . Then, using (6.52) and (6.51) and inequalities .wn X ≤ δn and .zX ≤ δn , we find that .

(Aun + f − v , un )X = (An un + fn + wn − v − z, un )X = (An un + fn − v, un )X + (wn − z, un )X ≤ (wn − z, un )X   ≤ wn X + zX un X ≤ 2δn un X .

Now, using the bound (6.20), it follows that (Aun + f − v , un )X ≤ 2δn D.

.

(6.60)

We now gather relations (6.59) and (6.60) to conclude that

n , 2δn D). with θn = (K

un ∈ ΩM (θn )

.

(6.61)

On the other hand, assumptions (6.47)(b), (6.50), and (6.53) show that .an → 0 and, therefore, (6.57) implies that .δn → 0. In addition, using this convergence, M

n − definition (6.58), and assumption (6.49), it is easy to see that .K → K. It follows from here that {θn } ∈ CM .

.

(6.62)

We now use (6.61) and (6.62) to see that .{un } is a .TM -approximating sequence. Therefore, the .TM -well-posedness of Problem .P, guaranteed by Theorem 6.2,  implies that .un → u in X, which concludes the proof. We end this section with two comments. First, Corollary 6.2 corresponds to Theorem 3.5, proved on page 114 by using the fixed point structure of the inclusion (6.4). Note that the proof we present here is based on a different argument. Second, Corollary 6.2 (which represents a consequence of the .TM -wellposedness of inclusion (6.4)) can be used to provide the convergence of the sequence

220

6 Inclusions

{un } in Example 6.2 to the solution of the corresponding Problem .P. Indeed, it is easy to see that in this case .−un ∈ NK (un + fn ) for all .n ≥ 2, and, therefore, .un → u in X. Recall that the Tykhonov triples .T1 , .T2 , and .T3 defined on page 208 cannot be used in order to prove this convergence. This illustrates the importance of the choice of the Tykhonov triple in employing the strategy presented on page 207. .

6.1.4 A Convergence Criterion In this subsection we construct two Tykhonov triples in the study of the inclusion .P, which are equivalent with the Tykhonov triple .TP introduced on page 50. Therefore, as explained in Remark 2.4, the class of equivalence of these triples represents the ˆ and this will allow us to formulate a criterion greatest element of the set .(AP , ≤), of convergence to the solution of the inclusion (6.4). To introduce these triples, we need some preliminaries. First, everywhere in this section, we assume (6.1)–(6.3) and

short-hand use the , assumed to notation .Λ for the operator .Λρ defined in (6.7) with any .ρ ∈ 0, 2m L 2 be fixed. Moreover, we introduce the approximating sets  Ωa (θ ) =. u ∈ X : A u + f − Λ(A u + f )X ≤ θ , .

(6.63)

   Ωb (θ ) = u ∈ X : A u + f − Λ(A u + f )X ≤ θ A u + f X + 1 ,

(6.64)

for all .θ ≥ 0. In addition, with notation (6.17) and (6.18) for I and .C, respectively, we introduce the triples .Ta and .Tb defined by Ta = (I, Ωa , C),

.

Tb = (I, Ωb , C).

Recall that Theorem 3.4 guarantees that Problem .P has a unique solution u. Then, using (6.6), it is easy to see that .u ∈ Ωa (θ ) and .u ∈ Ωb (θ ), for each .θ ∈ I . This implies that .Ωa (θ ) = ∅ and .Ωb (θ ) = ∅ for each .θ ∈ I and, therefore, .Ta and .Tb are Tykhonov triples, in the sense of Definition 2.1. Our main result in this section is the following. Theorem 6.3 Assume (6.1)–(6.3). Then the Tykhonov triples .Ta , .Tb , and .TP are equivalent. Proof We use the notation (2.1) and (2.2). We start by proving the inclusion S P ⊂ S Ta .

.

(6.65)

Assume that .{un } is a sequence which converges to u in X, that is, .{un } ∈ SP . Denote by .σ and .σn the elements of X given by

6.1 Stationary Inclusions

221 .

σ = Au + f, .

(6.66)

σn = Aun + f

∀ n ∈ N.

(6.67)

Then, using (6.5), it follows that .

u = A−1 (σ − f ), . un = A−1 (σn − f )

(6.68) ∀ n ∈ N,

(6.69)

and moreover, σn → σ

.

in X.

(6.70)

Fix .n ∈ N. Using (6.7), it follows that there exists .k ∈ [0, 1) such that Λτ − ΛωX ≤ k τ − ωX

.

∀ τ, ω ∈ X

(6.71)

and using (6.6), we deduce that Λσ = σ.

(6.72)

.

We now write σn − Λσn X ≤ σn − σ X + σ − Λσn X ,

.

and then we use (6.72) and (6.71) to deduce that σn − Λσn X ≤ (1 + k)σn − σ X .

.

This implies that .un ∈ Ωa (θn ) with .θn = (1 + k)σn − σ X , and since (6.70) guarantees that .θn → 0, we deduce that .{un } is a .Ta -approximating sequence, that is, .{un } ∈ STa . It follows from above that the inclusion (6.65) holds. Next, we prove the inclusion S Tb ⊂ S P .

(6.73)

.

To this end, we consider a .Tb -approximating sequence .{un } ∈ STb . We keep the notation (6.66) and (6.67) and use (6.64) to see that there exists a sequence .{θn } ⊂ R+ such that .θn → 0, and moreover,   σn − Λσn X ≤ θn σn X + 1

.

∀ n ∈ N.

(6.74)

222

6 Inclusions

Let .n ∈ N. We write σn − σ X ≤ σn − Λσn X + Λσn − σ X ,

.

and then we use (6.74), (6.72) and (6.71) to deduce that   σn − σ X ≤ θn σn X + 1 + k σn − σ X .

.

So,   (1 − k)σn − σ X ≤ θn σn X + 1 .

.

(6.75)

On the other hand, writing σn X ≤ σn − Λσn X + Λσn − Λ0X X + Λ0X X

.

and using (6.74) and (6.71), we obtain that   σn X ≤ θn σn X + 1 + k σn X + Λ0X X .

.

Therefore, (1 − k − θn )σn X ≤ θn + Λ0X X .

.

(6.76)

Now, since .θn → 0 and .k ∈ [0, 1), for n large enough, we may assume that .θn ≤ 1−k 1−k 1 2 , which implies that .1 − k − θn ≥ 2 and .θn ≤ 2 . So, inequality (6.76) shows that σn X ≤

.

1 (1 + 2Λ0X X ). 1−k

(6.77)

We now combine inequalities (6.75) and (6.77) and then use the convergence .θn → 0 to deduce that .σn → σ in X. This convergence, equalities (6.69) and (6.68), and the property (6.5) of the operator A imply that .un → u in X and, therefore, .{un } ∈ SP . We conclude from above that inclusion (6.73) holds. On the other hand, it is easy to see that .Ωa (θ ) ⊂ Ωb (θ ) for each .θ ∈ I , which implies that S Ta ⊂ S Tb .

.

(6.78)

We now gather the inclusions (6.65), (6.78), and (6.73) to see that .STa = STa = SP , which concludes the proof.  We end this subsection with the following remark.

6.2 History-Dependent Inclusions

223

Remark 6.1 Note that Theorem 6.3 states that .STa = STb = SP , and therefore, the definition (2.1) shows that a sequence .{un } converges to u in X if and only if .{un } ∈ STa = ST or, equivalently, if and only if there exists a sequence .{θn } ⊂ R+ b such that, for any .n ∈ N, one of the following two inequalities holds: Aun + f − Λ(Aun + f )X ≤ θn ,

.

  Aun + f − Λ(Aun + f )X ≤ θn Aun + f X + 1 . We conclude from here that Theorem 6.3 provides necessary and sufficient conditions that guarantee the convergence (6.8), i.e., it represents a convergence criterion. This criterion is intrinsic since no reference to the solution u of Problem .P is made on the inequalities above. It can be used in order to provide a different proof of Corollary 6.2, as shown in [210].

6.2 History-Dependent Inclusions In this section we introduce a history-dependent inclusion and provide existence, uniqueness, and well-posedness results. To this end, everywhere in this section, we assume that X is a Hilbert space and .A : X → X, .S : C(R+ ; X) → C(R+ ; X). Moreover, we assume that .K : R+ → 2X is a multivalued mapping, and for any nonempty, closed, convex set .M ⊂ V , we denote by .PM : V → M and .NM : V → 2V the projection operator on M and the outward normal cone of M, respectively.

6.2.1 An Existence and Uniqueness Result The inclusion problem we consider in this section is stated as follows. Problem .P. Find .u : R+ → X such that .

  − u(t) ∈ NK(t) Au(t) + Su(t)

∀ t ∈ R+ .

(6.79)

In the study of this problem, we assume the following:

.

⎧ (a) For each t ∈ R+ the setK(t) ⊂ X is ⎪ ⎪ ⎪ ⎪ nonempty closed and convex. ⎪ ⎪ ⎨ (b) For each t ∈ R+ and each sequence {tn } ⊂ R+ ⎪ ⎪ ⎪ converging to t one has ⎪ ⎪ ⎪ ⎩ PK(tn ) u − PK(t) uX → 0 ∀ u ∈ X.

(6.80)

224

6 Inclusions

.

⎧ ⎪ ⎨ A : X → X is a strongly monotone Lipschitz continuous operator, i.e., it satisfies inequalities ⎪ ⎩ (6.2)(a) and (6.2)(b) with mA , LA > 0. 

.

S : C(R+ ; X) → C(R+ ; X) satisfies inequality (1.6) with lm > 0 and Lm > 0, for any m ∈ N.

(6.81)

(6.82)

Remark 6.2 Note that Proposition 1.17 on page 23 guarantees that condition (6.80) M

is equivalent with the convergence .K(tn ) −→ K(t) for each .t ∈ R+ and each sequence .{tn } ⊂ R+ converging to t. The unique solvability of Problem .P follows from the following existence and uniqueness result. Theorem 6.4 Assume (6.80)–(6.82). Moreover, assume that (6.83)

lm < m A ,

.

for any .m ∈ N. Then there exists a unique function .u ∈ C(R+ ; X) such that (6.79) holds. The proof of Theorem 6.4 is carried out in several steps that we describe in what follows. In the first step, we consider a function .η : R+ → X such that η ∈ C(R+ ; X),

(6.84)

.

and we introduce the following auxiliary problem. Problem .Pη . Find .uη : R+ → X such that   − uη (t) ∈ NK(t) Auη (t) + η(t)

.

∀ t ∈ R+ .

(6.85)

For Problem .Pη , we have the following existence and uniqueness result. Lemma 6.1 Assume (6.80), (6.81), and (6.84). Then, there exists a unique function uη ∈ C(R+ ; X) such that (6.85) holds.

.

Proof First, we use Theorem 3.4 on page 112 to deduce that inclusion (6.85) has a unique solution .uη (t) ∈ X, for each .t ∈ R+ . We now prove the continuity of the function .t → uη (t) : R+ → X. To this end, we fix .ρ > 0 to be specified later, and we consider an element .t0 ∈ R+ as well as a sequence .{tn } ⊂ R+ converging to .t0 . For simplicity, we denote .uη (tn ) = un , .K(tn ) = Kn , and .η(tn ) = ηn , for .n = 0, 1, . . .. Then, using (6.85), we find that .

 − un ∈ NKn Aun + ηn )

∀ n = 0, 1, . . . .

6.2 History-Dependent Inclusions

225

Moreover, (1.45) implies that Aun + ηn ∈ Kn , (Aun + ηn − v, ρun )X ≤ 0

.

∀ v ∈ Kn

or, equivalently, .

Aun + ηn ∈ Kn , (Aun + ηn − v, (Aun + ηn − ρun ) − (Aun + ηn ))X ≥ 0

∀ v ∈ Kn .

This inequality and (1.38) yield Aun + ηn = PKn (Aun + ηn − ρun )

.

∀ n = 0, 1, . . . .

(6.86)

We now introduce the notation σn = Aun + ηn

.

∀ n = 0, 1, . . . ,

(6.87)

and using assumption (6.81) together with Proposition 1.12, we find that un = A−1 (σn − ηn )

.

∀ n = 0, 1, . . . .

(6.88)

We now combine (6.86)–(6.88) to deduce that   σn = PKn σn − ρA−1 (σn − ηn )

.

∀ n = 0, 1, . . . .

(6.89)

Let .n ∈ N. Then, (6.89) implies that .

σn − σ0 X

(6.90)

    ≤ PKn σn − ρA−1 (σn − ηn ) − PKn σ0 − ρA−1 (σ0 − η0 ) X     +PKn σ0 − ρA−1 (σ0 − η0 ) − PK0 σ0 − ρA−1 (σ0 − η0 ) X . We now estimate the first term in (6.90). Using the nonexpansivity of the projection operator, guaranteed by inequality (1.40), together with Propositions 1.13 and 1.12, we find that .

    PKn σn − ρA−1 (σn − ηn ) − PKn σ0 − ρA−1 (σ0 − η0 ) X     ≤  σn − ρA−1 (σn − ηn ) − σ0 − ρA−1 (σ0 − η0 ) X     ≤  (σn − ηn ) − ρA−1 (σn − ηn ) − (σ0 − η0 ) − ρA−1 (σ0 − η0 ) X +ηn − η0 X ≤ kρ (σn − ηn ) − (σ0 − η0 )X + ηn − η0 X .

226

6 Inclusions

Here, recall, .kρ :=

 1 − 2ρm + ρ 2 L 2 with .m =

we conclude that

mA L2A

and .L =

1 mA .

    P.Kn σn − ρA−1 (σn − ηn ) − PKn σ0 − ρA−1 (σ0 − η0 ) X

Therefore,

(6.91)

≤ kρ σn − σ0 X + (kρ + 1)ηn − η0 X . Next, taking .0 < ρ
0. Nevertheless, in contrast with the inequality studied in Sect. 4.2, we assume that the set of constraints depends on t, i.e., .K = K(t). Our aim is to study the wellposedness of this inequality by using its dual problem, which is in the form of a history-dependent inclusion. Everywhere in this section, V will be a real Hilbert space endowed with the inner product .(·, ·)V and the associated norm . · V . For any nonempty, closed, convex set .M ⊂ V , we denote by .PM : V → M and V the projection operator on M and the outward normal cone of M, .NM : V → 2 respectively. Moreover, on occasion, we use the notation .X = C([0, T ]; V ) for the space of continuous functions on .[0, T ] with values in V , equipped with the norm of the uniform convergence. Finally, we mention that, unless stated otherwise, all the limits below are considered as .n → ∞, even if we do not mention it explicitly.

6.3.1 Well-Posedness Results Let .K : [0, T ] → 2V , .A : V → V , .S : C([0, T ]; V ) → C([0, T ]; V ), and .f : [0, T ] → V . With these data, we consider the following history-dependent inequality. Problem .P. Find a function .u ∈ C([0, T ]; V ) such that the following inequality holds: .

u(t) ∈ K(t),

(Au(t), v − u(t))V + (Su(t), v − u(t))V ≥ (f (t), v − u(t))V

∀ v ∈ K(t),

(6.102)

t ∈ [0, T ].

Our aim in what follows is to associate Problem .P with a dual problem .Q and to deduce its well-posedness with respect to a specific Tykhonov triple. To this end, we consider the following hypotheses:

6.3 A Dual History-Dependent Inclusion

.

.

231

⎧ ⎪ ⎪ (a) For each t ∈ [0, T ], the setK(t) ⊂ X is ⎪ ⎪ nonempty closed and convex. ⎪ ⎪ ⎨ (b) For each t ∈ [0, T ] and each sequence {tn } ⊂ [0, T ] ⎪ ⎪ ⎪ converging to t, one has ⎪ ⎪ ⎪ ⎩ PK(tn ) u − PK(t) uX → 0 ∀ u ∈ X.

.

(6.103)

⎧ ⎪ ⎨ A : V → V is a linear operator and there exist LA , mA > 0 such that ⎪ ⎩ AuV ≤ LA uV , (Au, u)V ≥ mA u2V ∀ u ∈ V .

(6.104)

⎧ S : C([0, T ]; V ) → C([0, T ]; V ) is a history-dependent ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ operator, i.e., there exists LS > 0 such that  t ⎪ ⎪ Su(t) − Sv(t)V ≤ LS u(s) − v(s)V ds ⎪ ⎪ ⎪ 0 ⎩ ∀ u, v ∈ C([0, T ]; V ), t ∈ [0, T ].

(6.105)

f ∈ C([0, T ]; V ).

.

(6.106)

Remark 6.3 Concerning assumption (6.103), we recall our comment in Remark 6.2 on page 224. Moreover, note that assumption (6.104) implies that the operator A is continuous and invertible and its inverse .A−1 : V → V satisfies the inequalities A−1 uV ≤

.

1 uV , mA

(A−1 u, u)V ≥

mA u2V L2A

∀ u ∈ V.

(6.107)

We now use Theorem 1.5 on page 9 to deduce the following result. Proposition 6.3 Assume (6.104)–(6.106) and denote Du(t) = Au(t) + Su(t) − f (t)

.

∀ u ∈ C([0, T ]; V ), t ∈ [0, T ].

(6.108)

Then the operator .D : C([0, T ]; V ) → C([0, T ]; V ) is bijective and its inverse is of the form .A−1 + R, with .R : C([0, T ]; V ) → C([0, T ]; V ) being a historydependent operator with some constant .LR > 0. Based on Proposition 6.3, we consider the following problem. Problem .Q. Find a function .σ ∈ C([0, T ]; V ) such that the following inclusion holds: .

− σ (t) ∈ NK(t) (A−1 σ (t) + Rσ (t)) ∀ t ∈ [0, T ].

(6.109)

232

6 Inclusions

In the study of problems .P and .Q, we consider the triples .TP = (I, ΩP , C) and TQ = (I, ΩQ , C) defined as follows:

.

I= . R+ = [0, +∞),  C = {θn } : θn ∈ I ∀ n ∈ N,

θn → 0 as

n→∞



and, for each .θ ≥ 0, ⎧ ⎪ u ∈ ΩP (θ ) if and only if u ∈ C([0, T ]; V ) and u(t) ∈ K(t), ⎪ ⎨ . (Au(t), v − u(t))V + (Su(t), v − u(t))V + θ ≥ (f (t), v − u(t))V ⎪ ⎪ ⎩ ∀ v ∈ K(t), t ∈ [0, T ], ⎧ ⎪ σ ∈ ΩQ (θ ) if and only if σ ∈ C([0, T ]; V ) and ⎪ ⎨ A−1 σ. (t) + Rσ (t) ∈ K(t), ⎪ ⎪ ⎩ −1 (A σ (t) + Rσ (t) − v, σ (t))V ≤ θ

(6.110)

(6.111) ∀ v ∈ K(t), t ∈ [0, T ].

∅ and .ΩQ (θ ) = ∅ for any .θ ∈ I follow Note that the conditions .ΩP (θ ) = from the solvability of Problems .P and .Q, proved in Theorem 6.6(b) below. For this reason, as already mentioned in Remark 2.2, we refer in what follows to triples .TP and .TQ as Tykhonov triples. Our main result in this subsection is the following. Theorem 6.6 Assume (6.103)–(6.106). Then, (a) Problems .P and .Q are dual problems (with operator (6.108)) and the triples .TP and .TQ are dual Tykhonov triples (with operator (6.108)). (b) Problem .P is .TP -well-posed and Problem .Q is .TQ -well-posed. Proof (a) We start with the remark that any history-dependent operator defined on the space .C([0, T ]; V ) with values in .C([0, T ]; V ) is continuous. Therefore, Proposition 6.3 implies that the operator D defined by (6.108) satisfies conditions (a) and (b) in Definition 2.6 with .X = Y = C([0, T ]; V ). Assume now that .u ∈ C([0, T ]; V ) and let .σ = Du. Then, using Proposition 6.3, it follows that .

σ (t) = Au(t) + Su(t) − f (t), . −1

u(t) = A

σ (t) + Rσ (t)

(6.112) (6.113)

for all .t ∈ [0, T ]. It is easy to see that u is a solution to Problem .P if and only if u(t) ∈ K(t),

.

(f (t) − Au(t) − Su(t), v − u(t))V ≤ 0

(6.114)

6.3 A Dual History-Dependent Inclusion

233

for all .t ∈ [0, T ] and .v ∈ K(t). Thus, using (6.112) and (6.113), we deduce that (6.114) can be rewritten, equivalently, as A−1 σ (t) + Rσ (t) ∈ K(t),

(A−1 σ (t) + Rσ (t) − v, σ (t))V ≤ 0

.

(6.115)

for all .t ∈ [0, T ] and .v ∈ K(t). On the other hand, inequality (6.115) is valid for any .t ∈ [0, T ] and .v ∈ K(t) if and only if .σ is a solution to Problem .Q. We deduce from here that condition (c) in Definition 2.6 is satisfied, too. Therefore, problems .P and .Q are dual problems (with operator (6.108)). Let .θ ∈ I , .u ∈ C([0, T ]; V ) and let .σ = Du. We shall prove the equivalence u ∈ ΩP (θ )

.

⇐⇒

σ ∈ ΩQ (θ ).

(6.116)

To this end, we note that .u ∈ ΩP (θ ) if and only if u(t) ∈ K(t),

.

(f (t) − Au(t) − Su(t), v − u(t))V ≤ θ

(6.117)

for all .t ∈ [0, T ] and .v ∈ K(t), and using (6.112), (6.113), we deduce that (6.117) can be rewritten, equivalently, as A−1 σ (t) + Rσ (t) ∈ K(t),

(A−1 σ (t) + Rσ (t) − v, σ (t))V ≤ θ

.

(6.118)

for all .t ∈ [0, T ] and .v ∈ K(t). On the other hand, (6.118) is equivalent to the inclusion .σ ∈ ΩQ (θ ). We conclude from above that the equivalence (6.116) holds, and following Definition 2.7, we deduce that the triples .TP and .TQ are dual Tykhonov triples (with operator (6.108)). (b) First, we recall that the existence of a unique solution to Problem .Q follows from Theorem 3.8 on page 125. Let .σ ∈ C([0, T ]; V ) be the solution of Problem .Q. Then (6.115) holds, for each .t ∈ [0, T ] and .v ∈ K(t). Let .t ∈ [0, T ] and let .{σn } ⊂ C([0, T ]; V ) be a .TQ -approximating sequence. Then there exists a sequence .{θn } ∈ C such that .σn ∈ ΩQ (θn ), for each .n ∈ N. We now use (6.111), (6.115), and standard arguments to deduce that .

(A−1 σ (t) − A−1 σn (t), σ (t) − σn (t))V ≤ θn + (Rσ (t) − Rσn (t), σn (t) − σ (t))V .

Using now (6.107) and the history-dependence of the operator .R, we find that .

mA−1 σ (t) − σn (t)2V ≤ θn + LR



t 0

σ (s) − σn (s)V ds σ (t) − σn (t)V ,

234

6 Inclusions mA L2A

where, here and below, .mA−1 = yields

and .LR > 0. Therefore, inequality (1.33)

 σ (t) − σn (t)V ≤

.

θn LR + mA−1 mA−1



t

σ (s) − σn (s)V ds,

0

and using Lemma 1.1 on page 18, we find that  σ (t) − σn (t)V ≤

.

θn mLR−1 t e A . mA−1

Next, since .θn → 0, we deduce that .σn → σ in .C([0, T ]; V ). We conclude from here that Problem .Q is .TQ -well-posed. We are now in a position to use Theorem 2.9 in order to conclude the proof of Theorem 6.6. 

6.3.2 Convergence Results In this section we use the well-posedness of Problem .P with respect to the Tykhonov triple .TP to deduce a continuous dependence result of the solution with respect to the data. To this end, we assume that (6.103) holds and we consider three sequences .{An }, .{Sn }, and .{fn } such that, for each .n ∈ N, the following conditions hold: 

An : V → V satisfies condition (6.104) . with constants Ln and mn > 0.

(6.119)

.

⎧ Sn : C([0, T ]; V ) → C([0, T ]; V ) is a history-dependent operator, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ i.e., there exists Ln > 0 such  that t

⎪ Sn u(t) − Sn v(t)V ≤ Ln u(s) − v(s)V ds ⎪ ⎪ ⎪ 0 ⎪ ⎩ ∀ u, v ∈ C([0, T ]; V ), t ∈ [0, T ]. fn ∈ C([0; T ]; V ).

.

(6.120)

(6.121)

Moreover, we consider the following variational problem. Problem .Pn . Find a function .un ∈ C([0, T ]; V ) such that the following inequality holds: un (t) . ∈ K(t),

(An un (t), v − un (t))V + (Sn un (t), v − un (t))V

≥ (fn (t), v − un (t))V

∀ v ∈ K(t),

t ∈ [0, T ].

(6.122)

6.3 A Dual History-Dependent Inclusion

235

Then, the arguments in the previous subsection imply that Problem .Pn has a unique solution, for each .n ∈ N. Assume now that: ⎧ ⎪ (a) For each n ∈ N, there exists αn > 0 such that ⎪ ⎨ An u − AuV ≤ αn (uV + 1) ∀ u ∈ V . . . (6.123) ⎪ ⎪ ⎩ (b) αn → 0 as n → ∞. ⎧ (a) For each n ∈ N, there exists βn > 0 such that ⎪ ⎪ ⎪ t

⎪ ⎪ ⎨ Sn u(t) − Su(t)V ≤ βn u(s)V ds + 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

0

∀ u ∈ C([0, T ]; V ), t ∈ [0, T ].

.

(6.124)

(b) βn → 0 as n → ∞. fn → f

in C([0, T ]; V )..

There exists u0 ∈ V such that u0 ∈ K(t) ∀ t ∈ [0, T ].

(6.125) (6.126)

Our main result of this subsection is the following. Theorem 6.7 Assume (6.103)–(6.106), (6.119)–(6.121), and (6.123)–(6.126). Then, the solution .un of Problem .Pn converges to the solution u of Problem .P, i.e., un → u in C([0, T ]; V ).

.

(6.127)

Proof The proof is carried out in three steps, as follows: Step (i).

We prove that there exists .M > 0 such that un (t)V ≤ M

.

∀ n ∈ N, t ∈ [0, T ].

(6.128)

Let .n ∈ N and .t ∈ [0, T ]. Then, using (6.126) and (6.122), we find that .

(An un (t), un (t) − u0 )V ≤ (Sn un (t), u0 − un (t))V + (fn (t), un (t) − u0 )V .

We write (An u . n (t), un (t) − u0 )V = (An un (t) − Aun (t), un (t) − u0 )V +(Aun (t) − Au0 , un (t) − u0 )V + (Au0 , un (t) − u0 )V ,

(6.129)

236

6 Inclusions

and then we make use of assumptions (6.104) and (6.123)(a) to see that (A.n un (t), un (t) − u0 )V ≥ −(αn (un (t)V + 1))un (t) − u0 V

(6.130)

+mA un (t) − u0 2V − Au0 V un (t) − u0 V . Note also that (Sn u. n (t), u0 − un (t))V = (Sn un (t) − Sun (t), u0 − un (t))V +(Sun (t) − Su0 (t), u0 − un (t))V + (Su0 (t), u0 − un (t))V . Then, assumptions (6.105) and (6.124)(a) imply that (Sn u. n (t), u0 − un (t))V ≤ βn



t

un (s)V ds + 1 un (t) − u0 V

(6.131)

0

+LS



t

un (s) − u0 V ds un (t) − u0 V + Su0 (t)V un (t) − u0 V .

0

Moreover, we have .

(fn (t), un (t) − u0 )V

(6.132)

= (fn (t) − f (t), un (t) − u0 )V + (f (t), un (t) − u0 )V ≤ fn (t) − f (t)V un (t) − u0 V + f (t)V un (t) − u0 V . We now combine inequalities (6.129)–(6.132) to see that .

mA un (t) − u0 V ≤ βn



t

un (s)V ds + 1

0



t

+LS

un (s) − u0 V ds + Su0 (t)V + fn (t) − f (t)V

0

+f (t)V + αn (un (t)V + 1) + Au0 V , and using inequalities .

un (t)V − u0 V ≤ un (t) − u0 V , un (s) − u0 V ≤ un (s)V + u0 V

∀ s ∈ [0, T ],

6.3 A Dual History-Dependent Inclusion

237

we find that .

(mA − αn )un (t)V ≤ βn



t

un (s)V ds + 1

0



t

+LS

un (s)V ds + LS T u0 V + Su0 (t)V + fn (t) − f (t)V

0

+f (t)V + αn + Au0 V + mA u0 V . Using now assumptions (6.123)(b), (6.124)(b), and (6.125), we find that there exist N0 ∈ N and two positive constants .C0 and .C1 which do not depend on t such that

.



t

un (t)V ≤ C0 + C1

.

un (s)V ds

0

for all .n ≥ N0 . It follows from the Gronwall inequality that un (t)V ≤ C0 eC1 t

.

for all .n ≥ N0 . This inequality concludes the proof of (6.128). Step (ii). We prove that the sequence .{un } ⊂ C([0, T ]; V ) is a .TP -approximating sequence. Let .n ∈ N and .t ∈ [0, T ]. We use assumption (6.126) and write .

(Aun (t), u0 − un (t))V + (Sun (t), u0 − un (t))V − (f (t), u0 − un (t))V = (An un (t), u0 − un (t))V + (Sn un (t), u0 − un (t))V −(fn (t), u0 − un (t))V + (Aun (t) − An un (t), u0 − un (t))V +(Sun (t) − Sn un (t), u0 − un (t))V + (fn (t) − f (t), u0 − un (t))V ,

and then we use inequality (6.122) to find that .

(Aun (t), u0 − un (t))V + (Sun (t), u0 − un (t))V ≥ (f (t), u0 − un (t))V − Aun (t) − An un (t)V u0 − un (t)V −Sun (t) − Sn un (t)V u0 − un (t)V − fn (t) − f (t)V u0 − un (t)V .

238

6 Inclusions

Therefore, (6.123)(a), (6.124)(a), and (6.128) imply that (Au . n (t), u0 − un (t))V + (Sun (t), u0 − un (t))V

(6.133)

  + αn (M + 1) + βn (MT + 1) + fn (t) − f (t)V (u0 V + M) ≥ (f (t), u0 − un (t))V . Take   θn = α. n (M + 1) + βn (MT + 1) + fn (t) − f (t)V (u0 V + M),

(6.134)

And combine (6.133), (6.134), and (6.110) to see that .un ∈ ΩP (θn ). Moreover, (6.123)(b), (6.124)(b), and (6.125) imply that .θn → 0 as .n → ∞. It follows from here that .{un } is a .TP -approximating sequence for Problem .P. Step (iii).

End of proof.

We use Theorem 6.6 and Definition 2.1(c) to deduce the convergence (6.127), which concludes the proof.  We end this section with the following example in which .V = R. Example 6.3 Consider Problem .P in the particular case when .

K(t) = [0, 2 − e−t ] 

t

Su(t) =

u(s) ds

∀ t ∈ [0, T ],

Au = u

∀ u ∈ V,

∀ t ∈ [0, T ], u ∈ C([0, T ]; V ),

f (t) = t

∀ t ∈ [0, T ].

0

Note that in this case inequality (6.102) becomes .u(t) ∈ K(t),  t



. u(t) + u(s) ds − t v − u(t) ≥ 0

∀ v ∈ K(t), t ∈ [0, T ].

0

Assume now that .

An u =

n+1 u n

n+1 Sn u(t) = n fn (t) = t +

1 n

∀ u ∈ V, 

t

u(s) ds 0

∀ t ∈ [0, T ],

∀ t ∈ [0, T ], u ∈ C([0, T ]; V ),

(6.135)

6.3 A Dual History-Dependent Inclusion

239

for each .n ∈ N. Then, inequality (6.122) becomes .un (t) ∈ K(t), n + 1 n

.

un (t) +

n+1 n



t

un (s) ds − t −

0

1 v − un (t) ≥ 0 n

(6.136)

for all .v ∈ K(t), t ∈ [0, T ]. It is easy to see that in this particular case assumptions (6.103)–(6.106), (6.119)–(6.121), and (6.123)–(6.126) are satisfied. Therefore, Theorems 6.6 and 6.7 guarantee the unique solvability of inequalities (6.135) and (6.136), as well as the convergence (6.127). This convergence can be proved directly. Indeed, consider the integral equation 

t

u(t) +

.

u(s) ds = t

∀ t ∈ [0, T ].

0

The solution of this equation is u(t) = 1 − e−t

.

∀ t ∈ [0, T ],

(6.137)

and since .0 ≤ 1 − e−t ≤ 2 − e−t for all .t ∈ [0, T ], we deduce that the function (6.137) is the solution of the history-dependent variational inequality (6.135), too. Next, using a similar argument based on the solvability of the integral equation n+1 n+1 . un (t) + n n



t

un (s) ds = t +

0

1 n

∀ t ∈ [0, T ],

we find that the solution of the history-dependent variational inequality (6.136) is given by un (t) =

.

n − 1 −t n − e n+1 n+1

∀ t ∈ [0, T ].

Then, a simple calculation shows that |un (t) − u(t)| ≤

.

3 n+1

which implies the convergence (6.127).

∀ t ∈ [0, T ],

Chapter 7

Minimization and Optimal Control Problems

We start this chapter with the well-posedness of a class of minimization problems. Thereby, under specific assumptions, we deduce their weak and strong generalized well-posedness in the sense of Hadamard. Moreover, we show that, under strict and strong convexity assumptions, these results provide the weak and strong wellposedness of the corresponding problems. Next, we extend our study to optimal control problems for which we prove well-posedness and convergence results. We exemplify the abstract results obtained in this chapter in Sects. 9.2, 9.3, in the study of static contact problems with elastic materials, as well as in Sect. 10.1, in the study of a time-dependent contact problem with viscoelastic materials.

7.1 Minimization Problems Everywhere in this section, X is assumed to be a reflexive Banach space, unless stated otherwise. We use . · X and .0X for the norm and the zero element of X, respectively. All the limits, upper and lower limits, below are considered as .n → ∞, even if we do not mention it explicitly. The symbols “.” and “.→” denote the weak and the strong convergence in the space X, respectively.

7.1.1 Problem Statement Let K be a nonempty subset of X and let .J : X → R be a given functional. We use notation .θ = (K, J ), and we consider the following problem.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Sofonea, Well-Posed Nonlinear Problems, Advances in Mechanics and Mathematics 50, https://doi.org/10.1007/978-3-031-41416-9_7

241

242

7 Minimization and Optimal Control Problems

Problem .Pθ . Find u such that u ∈ K,

.

J (u) = min J (v). v∈K

(7.1)

We denote by .Sθ0 the set of solutions to Problem .Pθ , i.e.,  Sθ0 = u ∈ K : J (u) ≤ J (v)

.

 ∀v ∈ K .

Our aim in what follows is to study the well-posedness of the family of Problems (Pθ ) in the sense of Hadamard, introduced in Sect. 2.3.2. So, we investigate the dependence of the solution with respect to the data K and J , i.e., with respect to  ⊂ X and .J : X → R, we use the notation .θ = (K, J ). To this end, for any .K    .θ = (K, J ), and we consider Problem .P θ defined as follows. .

Problem .P θ . Find u such that  u ∈ K,

.

J(u) = min J(v).  v∈K

0 the set of solutions to Problem .P . We denote by .S  θ θ Next, we introduce the following notation:

.

  ⊂X : K  is nonempty and weakly closed .. K= K

(7.2)

  ⊂X : K  is nonempty, closed, and convex .. K = K

(7.3)

  J = J : X → R : J is weakly l.s.c. and coercive ..

(7.4)

  J = J : X → R : J is strictly convex, l.s.c., and coercive ..

(7.5)

  J = J : X → R : J is strongly convex and l.s.c. .

(7.6)

We now use Theorems 1.6 and 1.1, Proposition 1.4, and Remark 1.1 to obtain the following result. Theorem 7.1 Let X be a reflexive Banach space. Then, the following statements hold:  J) ∈ K × J , there exists at least one solution to Problem .P (a) For any . θ = (K, θ, 0 i.e., .S =

∅. θ  J) ∈ K × J , there exists a unique solution to Problem .P (b) For any . θ = (K, θ, 0 i.e., .S is a singleton. θ

7.1 Minimization Problems

243

We now consider the triple .Tθ = (I, Ω, Cθ ), defined as follows: I =K . × J ,. 0 Ω( θ ) = S θ

(7.7) ∀ θ ∈ I, .

(7.8)

  Cθ = {θn } = {(Kn , Jn )} ⊂ I : conditions (c1 )–(c4 ) below hold .  (c1 )

.

 (c2 )

.

 (c3 )

.

(c4 )

.

(7.9)

For any sequence {un } ⊂ X such that un X → +∞, one has Jn (un ) → +∞. For any weakly convergent sequence {un } ⊂ X, one has Jn (un ) − J (un ) → 0. For any sequence {un } ⊂ X such that

u → u in X, one has Jn (un ) − Jn (u) → 0. ⎧ n The sequence {Kn } converges to K in the sense of Mosco, i.e., ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ the following two properties hold: ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (a) For each v ∈ K, there exists a sequence {v } ⊂ X such that n

⎪ vn ∈ Kn for each n ∈ N and vn → v in X. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (b) For each sequence {vn } ⊂ X such that ⎪ ⎪ ⎪ ⎩ vn ∈ Kn for each n ∈ N and vn  v in X, one has v ∈ K.

θ ∈ K × J the set .Ω( Note that Theorem 7.1 guarantees that for each . θ ) is not empty. Therefore, .Tθ is a Tykhonov triple in the sense of Definition 2.1(a). Next, using Theorem 1.1, it follows that .K ⊂ K. Moreover, using Propositions 1.7 and 1.4, we deduce that .J ⊂ J ⊂ J . Then, with notation I = K × J ,

.

I = K × J ,

(7.10)

we deduce that I ⊂ I ⊂ I.

.

(7.11)

In the next subsection, we shall study the .Tθ -well-posedness of Problem .Pθ in the cases when .θ ∈ I , .θ ∈ I , and .θ ∈ I , respectively. We end this subsection with the following remark. Remark 7.1 For any sequence .{θn } = {(Kn , Jn )} ⊂ I and .θ = (K, J ) ∈ I , we write .θn → θ in I if the conditions (.c1 )–(.c4 ) above hold. Therefore, (7.9) shows that   .Cθ = {θn } = {(Kn , Jn )} ⊂ I : θn → θ in I . (7.12)

244

7 Minimization and Optimal Control Problems

Then, it is easy to see that equalities (7.8) and (7.12) are of the form (2.88) and (2.89), respectively, and, according to Remark 2.13 on page 90, the Tykhonov triple .Tθ represents a Tykhonov–Hadamard-type triple.

7.1.2 Well-Posedness Results Our first result in this subsection is the following. Theorem 7.2 Let X be a reflexive Banach space and let .θ ∈ I . Then Problem .Pθ is weakly generalized well-posed with the Tykhonov triple .Tθ . Proof Recall that the existence of at least one solution to Problem .Pθ follows from Theorem 7.1(a). Next, let .{un } be a .Tθ -approximating sequence. Then, there exists a sequence .{θn } ⊂ Cθ such that .un ∈ Ω(θn ) for all .n ∈ N. Therefore, using (7.9), it follows that .θn = (Kn , Jn ) ∈ I , and moreover, conditions (.c1 )–(.c4 ) hold. We claim that the sequence .{un } is bounded in X. Indeed, if .{un } is unbounded, we can find a subsequence of the sequence .{un }, again denoted by .{un }, such that .un X → +∞. Therefore, using condition .(c1 ), we deduce that Jn (un ) → +∞.

(7.13)

.

Let v be a given element in K, and note that condition .(c4 ) (a) implies that there exists a sequence .{vn } such that .vn ∈ Kn for each .n ∈ N and vn → v

.

in X.

(7.14)

Moreover, since .un is a solution of Problem .Pθn , we have .Jn (un ) ≤ Jn (vn ) and, therefore, Jn (un ) ≤ [Jn (vn ) − Jn (v)] + [Jn (v) − J (v)] + J (v)

.

∀ n ∈ N.

(7.15)

On the other hand, the convergence (7.14) and condition .(c3 ) imply that .Jn (vn ) − Jn (v) → 0, while condition .(c2 ) shows that .Jn (v) − J (v) → 0. Thus, inequality (7.15) implies that the sequence .{Jn (un )} is bounded in .R, which contradicts (7.13). We conclude from above that the sequence .{un } is bounded in X, and therefore, there exists a subsequence of the sequence .{un }, again denoted by .{un }, and an element .u ∈ X, such that un  u

.

in X.

(7.16)

We now prove that u is a solution of Problem .Pθ , i.e., .u ∈ Sθ0 . To this end, we use condition (.c4 ) (b) and the convergence (7.16) to deduce that

7.1 Minimization Problems

245

u ∈ K.

.

(7.17)

Next, we consider an arbitrary element .v ∈ K, and again, using condition (.c4 ) (a), we know that there exists a sequence .{vn } such that .vn ∈ Kn for each .n ∈ N, and moreover, (7.14) holds. Since .un is a solution to Problem .Pθn , we have .Jn (un ) ≤ Jn (vn ), which implies that .

0 ≤ [Jn (vn ) − Jn (v)] + [Jn (v) − J (v)] + [J (v) − J (u)]

(7.18)

+[J (u) − J (un )] + [J (un ) − Jn (un )]. We now use the convergences (7.14) and (7.16) combined with conditions .(c3 ) and (c2 ) and the weak lower semicontinuity of J to deduce that

.

.

Jn (vn ) − Jn (v) → 0, .

(7.19)

Jn (v) − J (v) → 0, .

(7.20)

lim sup [J (u) − J (un )] ≤ 0, .

(7.21)

J (un ) − Jn (un ) → 0.

(7.22)

Therefore, passing to the upper limit in inequality (7.18) and using (7.19)–(7.22), we find that 0 ≤ J (v) − J (u).

.

(7.23)

We now combine (7.17) and (7.23) to deduce that u is a solution of Problem Pθ . Finally, we use the convergence (7.16) and Definition 2.3(c) to conclude the proof.



.

Our second result in this subsection is the following. Theorem 7.3 Let X be a reflexive Banach space and let .θ ∈ I . Then, Problem .Pθ is weakly well-posed with the Tykhonov triple .Tθ . Proof First, since .θ ∈ I , Theorem 7.1(b) shows that there exists a unique solution to Problem .Pθ , i.e., .Sθ0 is a singleton. Moreover, inclusion (7.11) shows that .θ ∈ I , and therefore, the results presented in Theorem 7.2 hold in this case, too. Let .{un } ⊂ X be a .Tθ -approximating sequence. It follows from the proof of Theorem 7.2 that .{un } is bounded and any weakly convergent subsequence of .{un } converges weakly to the solution of Problem .Pθ which, recall, is unique. We now use Theorem 1.3 to deduce that the whole sequence .{un } converges weakly to the unique solution of Problem .Pθ . Therefore, using Definition 2.3(a) on page 52, we conclude the proof of the theorem.

We now proceed with our third result in this subsection.

246

7 Minimization and Optimal Control Problems

Theorem 7.4 Let X be a reflexive Banach space and let .θ ∈ I . Then, Problem .Pθ is strongly well-posed with the Tykhonov triple .Tθ . Proof Since .θ ∈ I , it follows from inclusion (7.11) that .θ ∈ I . Therefore, we are in a position to use Theorem 7.3 to get the unique solvability of Problem .Pθ and the weak convergence of any .Tθ -approximating sequence to its solution. Denote by u the unique solution of Problem .Pθ and let .{un } be a .Tθ approximating sequence. Then, there exists a sequence .{θn } ⊂ Cθ such that .un ∈ Ω(θn ), for all .n ∈ N. Therefore, using (7.9), it follows that .θn = (Kn , Jn ) ∈ I , conditions .(c1 ) − −(c4 ) on page 243 hold, and moreover, un  u

.

in X.

(7.24)

Let .{ un } be a sequence such that . un ∈ Kn for each .n ∈ N and  un → u

.

in X.

(7.25)

Recall that the existence of such sequence follows from condition .(c4 )(a). Then, using the strong convexity of the function J with .t = 12 , we find that .

  m 1

un + un 1

un + un  un −un 2X ≤ J ( un )−J + J (un )−J 4 2 2 2 2

(7.26)

with some .m > 0. We write J ( un ) − J

.

  un + un un + un = [J ( un ) − J (u)] + J (u) − J , 2 2

(7.27)

and using the convergences (7.24) and (7.25), Proposition 1.6, and the weak lower semicontinuity of the function J , we obtain that .

J ( un ) − J (u) → 0,

 un + un ≤ 0. lim sup J (u) − J 2

Therefore, (7.27) implies that .

 un + un ≤ 0. lim sup J ( un ) − J 2

On the other hand, we write

(7.28)

7.1 Minimization Problems

.

J (un ) − J

247

 un + un = [J (un ) − Jn (un )] 2

(7.29)

 

un + un un + un un + un   + Jn −J . + Jn (un ) − Jn 2 2 2 Using the convergences (7.24) and (7.25) and condition .(c2 ), we get that .

J (un ) − Jn (un ) → 0, Jn

  un + un un + un −J → 0, 2 2

and moreover, since .un is a solution to Problem .Pθn , we have Jn (un ) − Jn

.

 un + un ≤ 0. 2

Therefore, with these three ingredients, equality (7.29) yields .

 un + un ≤ 0. lim sup J (un ) − J 2

(7.30)

We now combine inequalities (7.26), (7.28), and (7.30) to deduce that .

lim sup  un − un 2X = 0,

which implies that un −  un → 0X

.

in X.

(7.31)

Finally, using the convergences (7.25) and (7.31), we deduce that .un → u in X, which concludes the proof.

We end this subsection with the following comments. Remark 7.2 Remark 7.1, Definition 2.10 on page 91, and Theorem 7.2 show that the family of problems .(Pθ )θ∈I is weakly generalized well-posed in the sense of Hadamard. Similarly, Definition 2.9 and Theorem 7.3 show that the family of problems .(Pθ )θ∈I is weakly well-posed in the sense of Hadamard. Finally, Definition 2.8 and Theorem 7.4 show that the family of problems .(Pθ )θ∈I is wellposed in the sense of Hadamard. Note also that Theorems 7.2, 7.3, and 7.4 hold under the assumptions .θ ∈ I , .θ ∈ I , and .θ ∈ I , respectively, and, in addition, we recall that the inclusion (7.11) holds. Recall also that the well-posedness implies the weak well-posedness, and the weak well-posedness implies the weak generalized well-posedness. We conclude

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7 Minimization and Optimal Control Problems

from above that we proved a regularity result concerning the Hadamard wellposedness of the family of problems .(Pθ ): the more .θ is regular, the stronger is the well-posedness of the family .(Pθ ).

7.1.3 Convergence Results The well-posedness result in Sect. 7.1.1 can be used to deduce several convergence results. Even if various examples can be constructed, in this subsection we restrict ourselves to present only two relevant ones. To this end, we assume in what follows that .K ⊂ X and .J : X → R, and we consider the following minimization problem. Problem .P. Find u such that u ∈ K,

.

J (u) = min J (v). v∈K

Consider also two sequences .{Kn } and .{Jn } such that, for each .n ∈ N, .Kn ⊂ X and .Jn : X → R. With these data, for each .n ∈ N, we consider the following problems. Problem .PnK . Find u such that u ∈ K,

.

Jn (u) = min Jn (v). v∈K

Problem .PnJ . Find u such that u ∈ Kn ,

.

J (u) = min J (v). v∈Kn

In the study of these problems, we use notations (7.2)–(7.10). Our first result concerns Problem .P and is as follows. Theorem 7.5 Let X be a reflexive Banach space. The following statements hold: (a) If .K ∈ K and .J ∈ J , then the set of solutions of Problem .P is nonempty and weakly compact in X. (b) If .K ∈ K and .J ∈ J , then Problem .P has a unique solution. Proof (a) Denote .θ = (K, J ) ∈ K × J . Then, using (7.7), we deduce that .θ ∈ I . The existence of at least one solution to Problem .P is now a direct consequence of Theorem 7.1(a). Consider now a sequence .{un } of solutions to Problem .P. Since J is coercive, we deduce that .{un } is bounded in X. This implies that .{un } has a subsequence which converges weakly to some element .u ∈ X. It is easy to see that .u ∈ K, and since .J (u) ≤ lim inf J (un ), we deduce that u is a solution

7.1 Minimization Problems

249

of Problem .P. It follows now from Definition 1.1(d) that the set of solutions of Problem .P is weakly compact in X. (b) The unique solvability of Problem .P follows from Theorem 7.1(b) since .θ = (K, J ) ∈ K × J = I .

In the rest of this subsection, we assume that .K ∈ K and .J ∈ J , and we denote by u the solution of Problem .P obtained in Theorem 7.5(b). We have the following result which establishes the link between the solutions of Problems .P and .PnK . Theorem 7.6 Let X be a reflexive Banach space, .K ∈ K , .J ∈ J , and .{Jn } ⊂ J , and assume that conditions .(c1 )–.(c3 ) on page 243 hold. Then, for each .n ∈ N, Problem .PnK has a unique solution .un ∈ K, and moreover, .un  u in X. In addition, if .J ∈ J , then .un → u in X. Proof Let .θ = (K, J ) ∈ K × J = I , and .n ∈ N and let .θn = (K, Jn ) ∈ K × J = I . The unique solvability of Problem .PnK is now a direct consequence of Theorem 7.1(b). Denote by .un the solution of this problem. It follows from (7.9) that .{θn } ∈ Cθ , and moreover, (7.8) implies that .un ∈ Ω(θn ), for each .n ∈ N. We conclude from here that .{un } is a .Tθ -approximating sequence. Then, using Theorem 7.3 and Definition 2.3(a), we deduce that the sequence .{un } converges weakly in X to u. Assume now that, in addition, .J ∈ J , which implies that .θ = (K, J ) ∈ K × J = I . Then, using Theorem 7.4 and Definition 2.1(c), we deduce that .un → u in X, which concludes the proof.

We now provide a result that establishes the link between the solutions of Problems .P and .PnJ . Theorem 7.7 Let X be a reflexive Banach space, .K ∈ K , .J ∈ J , and .{Kn } ⊂ K , and assume that condition .(c4 ) on page 243 holds. Then, for each .n ∈ N, Problem J .Pn has a unique solution .un ∈ K and, moreover, .un  u in X. In addition, if .J ∈ J , then .un → u in X. Proof Let .θ = (K, J ) ∈ K × J = I and .n ∈ N and let .θn = (Kn , J ) ∈ K × J = I . The unique solvability of Problem .PnJ is now a direct consequence of Theorem 7.1(b). Denote by .un the solution of this problem. It follows from (7.9) that .{θn } ∈ Cθ , and moreover, (7.8) implies that .un ∈ Ω(θn ), for each .n ∈ N. We conclude from here that .{un } is a .Tθ -approximating sequence. Then, using Theorem 7.3 and Definition 2.3(a), we deduce that the sequence .{un } converges weakly in X to the solution u of Problem .P. Assume now that, in addition, .J ∈ J . Then, using Theorem 7.4 and Definition 2.1(c), we deduce that .un → u in X, which concludes the proof.

We end this section with the remark that the results presented in this section will be used in Sect. 9.3 in the study of two representative frictionless contact problems with elastic materials.

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7 Minimization and Optimal Control Problems

7.2 Optimal Control Problems In this section we complete our results in Sect. 7.1 with the study of optimal control problems. Our aim is to provide results on the existence and convergence of optimal pairs for optimal control problems associated with an abstract problem. We do not describe this problem explicitly since we need only its well-posedness properties. The material presented in this section lays the background necessary to obtain existence and convergence results for a large number of optimal control problems and show that these results can be presented in a simple, unified, and elegant functional framework.

7.2.1 Problem Statement We consider two real normed spaces .(X, ·X ) and .(Y, ·Y ) and a reflexive Banach space .(Z, ·Z ). We also denote by .X×Z and .Y ×Z the product of the spaces X and Z, on the one hand, and Y and Z, on the other hand. A typical element in .X × Z will be denoted by .(u, f ) and a typical element in .Y × Z will be denoted by .(θ, f ). The symbols .“ → and .“  will represent the strong and weak convergence in various normed spaces which will be specified. All the limits, upper limits and lower limits, will be considered when .n → ∞, even if we do not mention it explicitly. Finally, U and V will represent two given nonempty sets such that .U ⊂ Y and .V ⊂ Z. We consider an abstract problem .Pθf which depends on two parameters .θ ∈ U and .f ∈ V . Problem .Pθf could be an equation, a minimization problem, a fixed point problem, an inclusion, or an inequality problem. Its rigorous statement varies from example to example. We associate with Problem .Pθf the concept of solution which follows from the context. We assume that the solution is unique and belongs to the space X, and to point out its dependence on .θ and f , we denote it by .u = u(θ, f ). To resume, Problem .Pθf can be formulated as follows. Problem .Pθf . Given .(θ, f ) ∈ U × V , find an element .u = u(θ, f ) ∈ X. Next, for each .θ ∈ U , we consider a subset .K(θ ) ⊂ V , which depends on .θ. With these notation, define the set of admissible pairs for Problem .Pθf by the equality   Vad (θ ) = (u, f ) ∈ X × Z : f ∈ K(θ ), u = u(θ, f ) .

.

(7.32)

In other words, a pair .(u, f ) belongs to .Vad (θ ) if and only if .f ∈ K(θ ), and moreover, u is the solution of Problem .Pθf . Consider also a cost functional .L : X×V → R. Then, the optimal control problem we are interested in is the following. Problem .Qθ . Given .θ ∈ U , find .(u∗ , f ∗ ) ∈ Vad (θ ) such that L(u∗ , f ∗ ) =

.

min

(u,f )∈Vad (θ)

L(u, f ).

(7.33)

7.2 Optimal Control Problems

251

Note that the solutions of the optimal control Problem .Qθ , if any, belong to the space .X × Z. Moreover, they depend on the parameter .θ . Nevertheless, when no confusion arises, we do not mention explicitly this dependence. Our aim in what follows is twofold. The first one is to formulate sufficient conditions on Problem .Pθf , the cost functional .L, and the set .K(θ ) which guarantee the solvability and/or the unique solvability of Problem .Qθ , for any .θ ∈ U . Our second aim is to study the dependence of the solution of Problem .Qθ with respect to .θ and to establish various convergence results.

7.2.2 Well-Posedness Results Note that Problem (7.33) is of the form (7.1) with .K = Vad (θ ) and .J = L. Therefore, the study of this optimal control problem could be made by using the results presented in Sect. 7.1. Nevertheless, this would lead to cumbersome notation which makes it difficult to check the validity of the conditions in the corresponding theorems. On the other hand, on the cost functional .L, we shall make specific assumptions that are slightly different to those made on the functional J in Sect. 7.1. For all these reasons, we present in what follows direct proofs of our results, even if they are based on arguments similar to those already used in the previous section. In the study of the optimal control problem .Qθ , we start by assuming that 

Problem Pθf has a unique solution u = u(θ, f ), . for each (θ, f ) ∈ U × V .

(7.34)

Assumption (7.34) allows us to consider the following Tykhonov triple, for any element .(θ, f ) ∈ U × V : ⎧ Tθf = (I, Ω, Cθf ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨I = U × V, .

  X , Ω(ξ, g) = u(ξ, g) ⎪ ∀ (ξ, g) ∈ U × V , Ω : U × V → 2 ⎪ ⎪ ⎪   ⎪ ⎪ ⎩ C = {(θ , f )} ⊂ I : θ → θ in Y, f  f in Z . θf n n n n

(7.35)

Recall that in (7.35) and below .u(ξ, g) denotes the solution of Problem .Pξg , for any (ξ, g) ∈ U × V . Next, we consider the following additional assumptions:

.



Problem Pθf is well-posed with the Tykhonov triple . (7.35), for each (θ, f ) ∈ U × V .

.

(7.36)

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7 Minimization and Optimal Control Problems

 

K(θ ) ⊂ V is a nonempty weakly closed subset of Z, for each θ ∈ U.

For each θ ∈ U and each sequence {θn } ⊂ U such that θn → θ

M

in Y, we have K(θn ) −→ K(θ ) in Z.

(7.37)

.

(7.38)

.

⎧ For all sequences {un } ⊂ X and {fn } ⊂ V such that ⎪ ⎪ ⎪ ⎪ ⎨ un → u in X, fn  f in Z, we have ⎪ ⎪ ⎪ ⎪ inf L(un , fn ) ≥ L(u, f ). ⎩ lim n→∞

(7.39)

⎧ The function L : X × V → R is continuous, i.e., ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ for all sequences {un } ⊂ X and {fn } ⊂ V such that .

⎪ un → u in X, fn → f in Z, we have ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lim L(u , f ) = L(u, f ), n n

(7.40)

.

n→∞

⎧ The subset V ⊂ Z is unbounded and there exists h : Z → R ⎪ ⎪ ⎪ ⎪ ⎨ such that: ⎪ (a) L(u, f ) ≥ h(f ) ∀ u ∈ X, f ∈ V . ⎪ ⎪ ⎪ ⎩ (b) h(fn ) → +∞ ∀ {fn } ∈ V such that fn Z → +∞. V is a bounded subset of Z.

.

(7.41)

(7.42)

Remark 7.3 Note that assumption (7.36) and Definition 2.1(a) imply (7.34). Nevertheless, in order to formulate assumption (7.36), we need to define the Tykhonov triple (7.35), which, in turn, requires assumption (7.34). For this reason, in the statements of Theorems 7.8 and 7.9 below, we keep both assumptions (7.34) and (7.36). Remark 7.4 Using Remark 2.13, it is easy to see that the triple .Tθf defined by (7.35) is a Tykhonov–Hadamard triple. Therefore, assumption (7.36) and Theorem 2.12 show that the family .(Pθf )(θ,f )∈U ×V is well-posed in the sense of Hadamard. Remark 7.5 Assumption (7.36) implies that for each .(θ, f ) ∈ U ×V , Problem .Pθf has a unique solution .u = u(θ, f ). Moreover, it also shows that if .{(θn , fn )} ∈ U × V is a sequence such that .θn → θ in Y and .fn  f in Z, then .u(θn , fn ) → u(θ, f ) in X. We shall use this convergence in various places, below in this section.

7.2 Optimal Control Problems

253

We shall prove in Theorem 7.8 below that, under the previous assumptions, Problem .Qθ has at least one solution, for each .θ ∈ U . Therefore, under these assumptions we are in a position to consider the following Tykhonov triple related to Problem .Qθ , for any .θ ∈ U : ⎧ Tθ = (I, Ω, Cθ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ I = U, ⎪ ⎪ ⎨   . Ω( θ ) = (u∗ , f ∗ ) : (u∗ , f ∗ ) is a solution to Problem Q θ ⎪ ⎪ ⎪ ⎪ ∀ θ ∈ U, ⎪ ⎪ ⎪   ⎪ ⎪ ⎩ C = {θ } : θ → θ in Y . θ n n

(7.43)

Our first result in this section is the following. Theorem 7.8 Assume (7.34), (7.36)–(7.40), and either (7.41) or (7.42). Then, for each element .θ ∈ U , the optimal control problem .Qθ is weakly generalized wellposed with the Tykhonov triple (7.43). Proof Let .θ ∈ U be fixed, and consider the function .Sθ : K(θ ) → R defined by Sθ (f ) = L(u(θ, f ), f )

.

∀ f ∈ K(θ ).

(7.44)

Moreover, consider the following auxiliary minimization problem: find f ∗ ∈ K(θ ) such that Sθ (f ∗ ) = min Sθ (f ).

.

f ∈K(θ)

(7.45)

We claim that this problem has at least one solution .f ∗ and, to this end, we use the Weierstrass-type argument provided by Theorem 1.6. Consider a sequence .{fn } ⊂ K(θ ) such that fn  f

.

in Z,

(7.46)

with f being a given element in Z. Then, (7.35) shows that the sequence {(θ, fn )} belongs to .Cθf , and moreover, Definition 2.1(b) implies that the sequence .{u(θ, fn )} is a .Tθf -approximating sequence. We now use assumption (7.36) and Definition 2.1(c) to deduce that .

u(θ, fn ) → u(θ, f ) in X.

.

(7.47)

Therefore, the convergences (7.46) and (7.47) combined with assumption (7.39) yield .

lim inf L(u(θ, fn ), fn ) ≥ L(u(θ, f ), f ). n→∞

(7.48)

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7 Minimization and Optimal Control Problems

Next, inequality (7.48) and definition (7.44) imply that lim inf Sθ (fn ) ≥ Sθ (f ),

.

n→∞

which shows that the function .Sθ : K(θ ) → R is weakly lower semicontinuous. Assume now that (7.41) holds. Then, for any sequence .{fn } ⊂ K(θ ), condition (7.41)(a) shows that Sθ (fn ) = L(u(θ, fn ), fn ) ≥ h(fn )

.

∀ n ∈ N.

Therefore, if .fn Z → +∞, (7.41)(b) implies that .Sθ (fn ) → +∞, which shows that the function .Sθ is coercive. Recall also the assumption (7.37) and the reflexivity of the space Z. The existence of at least one solution to problem (7.45) is now a direct consequence of Theorem 1.6. On the other hand, if (7.42) holds, we are still in a position to apply Theorem 1.6 since now the set of constraints .K(θ ) is bounded. We deduce from here that if either (7.41) or (7.42) holds, then problem (7.45) has at least a solution. Finally, using the definitions (7.44) and (7.32), it is easy to see that  .

(u∗ , f ∗ ) is a solution of Problem Qθ if and only if f ∗ is a solution of problem (7.45) and u∗ = u(θ, f ∗ ).

(7.49)

We now combine the equivalence (7.49) with the solvability of the optimization problem (7.45) to deduce that there exists at least one solution .(u∗ , f ∗ ) to Problem .Qθ . Next, to prove the weak generalized .Tθ -well-posedness of Problem .Qθ , we follow Definition 2.3(c) and consider a .Tθ -approximate sequence .{(u∗n , fn∗ )}. Then, (7.43) implies that there exists a sequence .{θn } ⊂ Y such that θn → θ

.

in Y,

(7.50)

and therefore, assumption (7.38) shows that M

K(θn ) −→ K(θ ) in Z.

(7.51)

.

Moreover, (7.43) also implies that, for each .n ∈ N, .(u∗n , fn∗ ) is a solution to Problem .Qθn , i.e., (u∗n , fn∗ ) ∈ Vad (θn ),

.

L(u∗n , fn∗ ) =

min

(u,f )∈Vad (θn )

L(u, f )

(7.52)

where, recall,   Vad (θn ) = (u, f ) ∈ X × Z : f ∈ K(θn ), u = u(θn , f ) .

.

(7.53)

7.2 Optimal Control Problems

255

Let .n ∈ N, and consider the function .Sθn : K(θn ) → R defined by Sθn (f ) = L(u(θn , f ), f )

∀ f ∈ K(θn ),

.

(7.54)

together with the following minimization problem: find fn∗ ∈ K(θn ) such that Sθn (fn∗ ) = min Sθn (f ).

.

f ∈K(θn )

(7.55)

It follows from (7.52)–(7.54) that fn∗ is a solution of problem (7.55) and u∗n = u(θn , fn∗ ).

.

(7.56)

We now proceed in three steps that we describe in what follows. Step (i) We prove that there exist a subsequence of the sequence .{fn∗ }, again denoted by .{fn∗ }, and an element .f ∗ ∈ Z, such that fn∗  f ∗

.

in

(7.57)

Z.

To prove this statement, we show that the sequence .{fn∗ } is bounded in Z. This boundedness is obvious if we assume that (7.42) holds. Assume in what follows that (7.41) holds. If .{fn∗ } is not bounded in Z, then we can find a subsequence of the sequence .{fn∗ }, again denoted by .{fn∗ }, such that .fn∗ Z → +∞. Therefore, using condition (7.41), we deduce that L(u(θn , fn∗ ), fn∗ ) ≥ h(fn∗ ) → +∞,

.

which implies that L(u(θn , fn∗ ), fn∗ ) → +∞.

(7.58)

.

Let s be a given element in .K(θ ) and note that (7.51) implies that there exists a sequence .{sn } such that .sn ∈ K(θn ) for each .n ∈ N and sn → s

.

in Z.

(7.59)

Moreover, since .fn∗ is a solution of problem (7.55), we have .Sθn (fn∗ ) ≤ Sθn (sn ), and therefore, L(u(θn , fn∗ ), fn∗ ) ≤ L(u(θn , sn ), sn )

.

∀ n ∈ N.

(7.60)

On the other hand, the convergences (7.50) and (7.59) combined with assumption (7.36) show that u(θn , sn ) → u(θ, s)

.

in X,

256

7 Minimization and Optimal Control Problems

and using (7.40), we get .

lim L(u(θn , sn ), sn ) = L(u(θ, s), s).

n→∞

(7.61)

We now use (7.60) and (7.61) to see that the sequence .{L(u(θn , fn∗ ), fn∗ )} is upper bounded, which contradicts (7.58). We conclude from above that the sequence .{fn∗ } is bounded in Z, and therefore, there exist a subsequence of the sequence .{fn∗ }, again denoted by .{fn∗ }, and an element .f ∗ ∈ Z, such that (7.57) holds. Step (ii)

We now prove that .f ∗ obtained in Step (i) is a solution of problem (7.45).

To prove this statement, we recall that .fn∗ ∈ K(θn ), for all .n ∈ N. Therefore, using the convergences (7.51) and (7.57), we deduce that .f ∗ ∈ K(θ ). Next, with the notation in Step (i), we use (7.60) and (7.61) to see that .

lim inf L(u(θn , fn∗ ), fn∗ ) ≤ L(u(θ, s), s). n→∞

(7.62)

On the other hand, the convergences (7.50) and (7.57) and Remark 7.5 imply that u(θn , fn∗ ) → u(θ, f ∗ )

.

in X,

(7.63)

and therefore, assumption (7.39) yields L(u(θ, f ∗ ), f ∗ ) ≤ lim inf L(u(θn , fn∗ ), fn∗ ).

.

n→∞

(7.64)

Finally, inequalities (7.64) and (7.62) imply that L(u(θ ∗ , f ∗ ), f ∗ ) ≤ L(u(θ, s), s).

.

Therefore, using notation (7.44) and regularity .f ∗ ∈ K(θ ), we deduce that .f ∗ is a solution of problem (7.45), which concludes the proof of this step. Step (iii)

End of proof.

Let .u∗ = u(θ, f ∗ ). Then, equivalence (7.49) shows that .(u∗ , f ∗ ) is a solution of Problem .Qθ . Moreover, (7.56) implies that .u∗n = u(θn , fn∗ ), and therefore, (7.63) shows that u∗n = u(θn , fn∗ ) → u∗ = u(θ, f ∗ )

.

in X.

(7.65)

We now use the convergences (7.65) and (7.57) and Definition 2.3(c) to conclude the proof of the theorem.

Remark 7.6 A careful analysis of the proof of Theorem 7.8 shows that it provides more than the weakly generalized well-posedness of the control problem .Qθ with the Tykhonov triple (7.43). Indeed, Definition 2.3(c) requires only the weak convergence of the .Tθ -approximating sequences, i.e., with the notation above,

7.2 Optimal Control Problems

u∗n = u(θn , fn∗ )  u∗ = u(θ, f ∗ )

.

257

in X,

fn∗  f ∗

in Z.

(7.66)

And, obviously, the convergences (7.65) and (7.57) obtained in the proof of Theorem 7.8 provide more information than the convergences in (7.66). We now reinforce the assumptions of the previous theorem by considering the following hypotheses on the set .K(θ ) and the functions .Sθ and .L where, recall, .Sθ is defined by equality (7.44): .

For each θ ∈ U , the set K(θ ) is a convex subset of Z.. ⎧ ⎪ ⎨ For each θ ∈ U , the function Sθ is strictly convex, i.e., . (1 − t)Sθ (f ) + tSθ (g) − Sθ ((1 − t)f + tg) > 0 ⎪ ⎩ for all f, g ∈ K(θ ), f = g, t ∈ (0, 1). ⎧ For each θ ∈ U , the function Sθ is strongly convex, i.e., ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ there exists mθ > 0 such that ⎪ ⎨ (1 − t)Sθ (f ) + tSθ (g) − Sθ ((1 − t)f + tg) ⎪ ⎪ ⎪ ⎪ ≥ mθ t (1 − t)f − g2Z ⎪ ⎪ ⎪ ⎪ ⎩ for all f, g ∈ K(θ ), t ∈ [0, 1]. ⎧ ⎪ For all sequences {un } ⊂ X and{fn } ⊂ V such that ⎪ ⎪ ⎪ ⎨ u. n → u in X, fn  f in Z, we have ⎪   ⎪ ⎪ lim L(un , fn ) − L(u, fn ) = 0. ⎪ ⎩ n→∞

(7.67)

(7.68)

(7.69)

(7.70)

Our second result in this section is the following. Theorem 7.9 Assume (7.34), (7.36)–(7.40) and either (7.41) or (7.42). Then, for each .θ ∈ U , the following statements hold: (a) Under assumptions (7.67) and (7.68), Problem .Qθ is weakly well-posed with the Tykhonov triple (7.43). (b) Under assumptions (7.67), (7.69), and (7.70), Problem .Qθ is (strongly) wellposed with the Tykhonov triple (7.43). Proof (a) Let .θ ∈ U be fixed and let .{(u∗n , fn∗ )} be a .Tθ -approximating sequence. It follows from the proof of Theorem 7.8 that .{fn∗ } ⊂ Z is bounded and any weakly convergent subsequence of .{fn∗ } converges in Z to the solution of the optimization problem (7.45). On the other hand, assumptions (7.67) and (7.68) and Remark 1.1 on page 12 guarantee that the solution of this problem is unique. We now use Theorem 1.3 to deduce that the whole sequence .{fn∗ } converges

258

7 Minimization and Optimal Control Problems

weakly in Z to the solution of problem (7.45). Therefore, the arguments in the proof of Theorem 7.8 show that the whole sequence .{(u∗n , fn∗ )} converges weakly in .X ×Z to the solution of Problem .Qθ . We now use Definition 2.3(b) to conclude that Problem .Qθ is weakly well-posed with the Tykhonov triple (7.43). (b) Assume now that (7.67), (7.69), and (7.70) hold, and let .θ ∈ U be fixed. Since condition (7.69) implies (7.68), we are in a position to use the part (a) of the theorem, which guarantees the unique solvability of Problem .Qθ and the weak convergence (in .X × Z) of any .Tθ -approximating sequence to its solution. Denote by .(u∗ , f ∗ ) the solution of Problem .Qθ and let .{(u∗n , fn∗ )} be a .Tθ -approximating sequence. Then, (7.43) implies that there exists a sequence .{θn } ⊂ Y such that the convergence (7.50) holds. Therefore, assumption (7.38) implies that the convergence (7.51) holds, too. Moreover, for each .n ∈ N, ∗ ∗ .(un , fn ) satisfies (7.52) with .Vad (θn ) being the set given by (7.53). In addition, the proof of Theorem 7.8 shows that the convergences (7.57) and (7.65) hold. Let .{fn∗ } be a sequence such that .fn∗ ∈ K(θn ) for each .n ∈ N and fn∗ → f ∗

.

in Z.

(7.71)

Recall that the existence of such sequence follows from the Mosco convergence (7.51). Moreover, the convergences (7.57) and (7.71) imply that .

fn∗ + fn∗  f∗ 2

in

Z.

(7.72)

Next, (7.57), (7.65) and assumption (7.70) yield L(u(θn , fn∗ ), fn∗ ) − L(u(θ, f ∗ ), fn ∗ ) → 0.

.

(7.73)

A similar argument (based on the convergence (7.72) and Remark 7.5) shows that    f∗ + fn∗ f∗ + fn∗ fn∗ + fn∗ , − L u(θ, f ∗ ), n → 0. L u θn , n 2 2 2

.

(7.74)

We now use the notation (7.44) to see that .

Sθ (fn∗ ) − Sθn (fn∗ ) = L(u(θ, fn∗ ), fn∗ ) − L(u(θn , fn∗ ), fn∗ )   = L(u(θ, fn∗ ), fn∗ ) − L(u(θ, f ∗ ), fn∗ )   + L(u(θ, f ∗ ), fn∗ ) − L(u(θn , fn∗ ), fn∗ )

for all .n ∈ N, and therefore, Remark 7.5, assumption (7.70), and the convergence (7.73) yield Sθ (fn∗ ) − Sθn (fn∗ ) → 0.

.

(7.75)

7.2 Optimal Control Problems

259

A similar argument, based on the convergence (7.74), shows that Sθ

.

 f∗ + f ∗ n n 2

− Sθn

 f∗ + f ∗ n n 2

→ 0.

(7.76)

On the other hand, the convergence (7.71), Remark 7.5, and assumption (7.40) imply that L(u(θn , fn∗ ), fn∗ ) − L(u(θ, f ∗ ), f ∗ ) → 0,

.

(7.77)

while (7.57), Remark 7.5, and (7.39) yield .

lim inf L(u(θn , fn∗ ), fn∗ ) ≥ L(u(θ, f ∗ ), f ∗ )

or, equivalently, .

lim sup [L(u(θ, f ∗ ), f ∗ ) − L(u(θn , fn∗ ), fn∗ )] ≤ 0.

(7.78)

We now use (7.44) and write .

Sθ (fn∗ ) − Sθn (fn∗ ) = L(u(θ, fn∗ ), fn∗ ) − L(u(θn , fn∗ ), fn∗ ) = [L(u(θ, fn∗ ), fn∗ ) − L(u(θ, f ∗ ), f ∗ )] +[L(u(θ, f ∗ ), f ∗ ) − L(u(θn , fn∗ ), fn∗ )]

for all .n ∈ N, and therefore, (7.77) and (7.78) yield .

lim sup [Sθ (fn∗ ) − Sθn (fn∗ )] ≤ 0.

(7.79)

To proceed, we fix .n ∈ N and note that, since .fn∗ is a solution of the optimization problem (7.55), we have Sθn (fn∗ ) − Sθn

.

 f∗ + f ∗ n n

Next, we use assumption (7.69) with .t = .

2 1 2

≤ 0.

(7.80)

to find that

mθ ∗ fn − fn∗ 2Z (7.81) 4  f∗ + f ∗  f∗ + f ∗ 1

1

n n + Sθ (fn∗ ) − Sθ n . Sθ (fn∗ ) − Sθ n ≤ 2 2 2 2

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7 Minimization and Optimal Control Problems

We now write  f∗ + f ∗ n = [Sθ (fn∗ ) − Sθn (fn∗ )] Sθ (fn∗ ) − Sθ n 2  f∗ + f ∗  f∗ + f ∗  f∗ + f ∗

n n n + Sθn n − Sθ n , + Sθn (fn∗ ) − Sθn n 2 2 2

.

and using (7.79), (7.80),and (7.76), we find that .

 f∗ + f ∗

n ≤ 0. lim sup Sθ (fn∗ ) − Sθ n 2

(7.82)

On the other hand, we write .

Sθ (fn∗ ) − Sθ

 f∗ + f ∗ n n 2

= [Sθ (fn∗ ) − Sθn (fn∗ )]

 f∗ + f ∗  f∗ + f ∗  f∗ + f ∗

n n n − Sθ n , + Sθn n + Sθn (fn∗ ) − Sθn n 2 2 2 and using (7.75), (7.80), and (7.76), again, we deduce that .

 f∗ + f ∗

n ≤ 0. lim sup Sθ (fn∗ ) − Sθ n 2

(7.83)

Next, we combine inequalities (7.81), (7.82), and (7.83) to deduce that .

lim sup fn∗ − fn∗ 2Z = 0,

which implies that fn∗ − fn∗ → 0Z

.

in Z.

(7.84)

Finally, we use (7.71) and (7.84) to see that .fn∗ → f ∗ in Z. This convergence

combined with the convergence (7.65) concludes the proof. We now present the following consequence of Theorems 7.8 and 7.9. Corollary 7.1 Assume (7.34), (7.36)–(7.40) and either (7.41) or (7.42). Then, for each .θ ∈ U , the set of solutions of Problem .Qθ is weakly compact in X. Moreover, if (7.67) and (7.68) hold, it is a singleton. Proof Let .θ ∈ U and let .{(u∗n , fn∗ )} be a sequence of solutions to Problem .Qθ . Then, using (7.43), it follows that for each .n ∈ N we have .(u∗n , fn∗ ) ∈ Ω(θ ), and therefore, .{(u∗n , fn∗ )} is a .Tθ -approximating sequence. We now use Theorem 7.8 and Definition 2.3(c) to see that .{(u∗n , fn∗ )} contains a subsequence which converges weakly in .X × Z to a solution of Problem .Qθ . This concludes the proof of the first

7.2 Optimal Control Problems

261

part of the corollary. The second part is a direct consequence of Theorem 7.9(a) and Definition 2.3(a) on page 52.

The results above give rise to various convergence results. In order to avoid repetitions, we shall restrict ourselves to mention the following one, which represents a direct consequence of Theorem 7.8. Corollary 7.2 Assume (7.34), (7.36)–(7.40) and either (7.41) or (7.42). Let .θ ∈ U and let .{θn } ⊂ U be a sequence such that M

θn → θ in Y and K(θn ) −→ K(θ ) in Y.

.

For each .n ∈ N, let .(u∗n , fn∗ ) be a solution of Problem .Qθn . Then, there exist a subsequence of the sequence .{(u∗n , fn∗ )}, again denoted by .{(u∗n , fn∗ )}, and an element .(u∗ , f ∗ ) ∈ X × Z, such that the convergences (7.66) hold. Moreover, ∗ ∗ .(u , f ) is a solution of Problem .Q. θ ) and .Cθ in (7.43) are defined by Remark 7.7 It is easy to see that the sets .Ω( equalities of the form (2.88) and (2.89), respectively. Therefore, using Remark 2.13, we deduce that the triple (7.43) represents a Tykhonov–Hadamard triple. In addition, Definition 2.10 on page 91 shows that, under the assumptions of Theorem 7.8, the family of problems .(Qθ )θ∈U is weakly generalized well-posed in the sense of Hadamard. It is weakly well-posed in the sense of Hadamard under the assumptions of Theorem 7.9(a), and it is well-posed in the sense of Hadamard, under the assumptions of Theorem 7.9(b). Recall that all these results have been obtained under assumption (7.36), among others, which requires that the family .(Pθf )(θf )∈U ×V is well-posed in the sense of Hadamard, as explained in Remark 7.4 on page 252.

7.2.3 An Example In this subsection we apply the results obtained in Sects. 7.2.1 and 7.2.3 in the study of the stationary inclusion. To this end, we use the notation introduced there, unless stated otherwise. The space X is assumed to be a real Hilbert space endowed with the inner product .(·, ·)X and the associated norm .·X , .(Y, ·Y ) is a normed space, and .(Z, ·Z ) is a reflexive Banach space. In addition, .(W, ·W ) is a normed space, X .U ⊂ Y , and .V ⊂ Z. Moreover, .Σ : Y × W → 2 , .B : Z → W , .A : X → X, and .η ∈ X. Then, for each .(θ, f ) ∈ U × V , we consider the following inclusion problem. Problem .Pθf . Given .(θ, f ) ∈ U × V , find an element .u = u(θ, f ) ∈ X such that .

 − u ∈ NΣ(θ,Bf ) Au + η).

(7.85)

262

7 Minimization and Optimal Control Problems

Inclusion (7.85) will play the role of Problem .Pθf on page 250. Let .L : X × V → R, and for each .θ ∈ U , consider a subset .K(θ ) ⊂ V , the set .Vad (θ ) defined by (7.32), together with Problem .Qθ defined by (7.33). In the study of this optimal control problem, we consider the following assumptions:  For each (θ, f ) ∈ U × V the setΣ(θ, Bf ) ⊂ X is . . (7.86) nonempty closed and convex. A is a strongly monotone Lipschitz continuous operator..

(7.87)

η ∈ X.

(7.88)

Then, using Theorem 3.4 on page 112, we deduce that condition (7.34) is satisfied. Therefore, for each element .(θ, f ) ∈ U × V , we are in a position to consider the Tykhonov triple (7.35). In what follows we assume (7.37)–(7.40), either (7.41) or (7.42) and, under these assumptions, we construct the triple (7.43). Moreover, we consider the following additional assumptions: ⎧ For each ∈ X, (θ, w) ∈ U × W and all ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ sequences {(θn , wn )} ⊂ U × W such that .

⎪ ⎪ θn → θ in Y, wn → w in W, one has ⎪ ⎪ ⎪ ⎩ PΣ(θn ,wn ) u → PΣ(θ,w) u in X. 

.

B : Z → X is a completely continuous operator, i.e., fn  f in Z ⇒ Bfn → Bf in W.

(7.89)

(7.90)

Our main result in this section is the following. Theorem 7.10 Assume (7.86)–(7.90), (7.37)–(7.40), and either (7.41) or (7.42). Then, for each .θ ∈ U , the triple .Tθ defined by (7.43) is a Tykhonov triple. Moreover, the optimal control problem .Qθ associated with the inclusion (7.85) is weakly generalized .Tθ -well-posed. Proof We start by proving that condition (7.36) is satisfied. To this end, we fix a pair .(θ, f ) ∈ U × V , and we recall that the unique solvability of Problem .Pθf is guaranteed by Theorem 3.4. Assume now that .{un } is a .Tθf -approximating sequence. Then, there exist two sequences .{θn } ⊂ U and .{fn } ⊂ V such that .

θn → θ

in Y, .

(7.91)

fn  f

in Z,

(7.92)

7.2 Optimal Control Problems

263

and moreover, .

 − un ∈ NΣ(θn ,Bfn ) Au + η)

∀ n ∈ N.

(7.93)

We now use (7.92) and (7.90) to see that Bfn → Bf

.

in W,

(7.94)

and combining the convergences (7.91), (7.94) with assumption (7.89), we find that PΣ(θn ,Bfn ) u → PK(θ,Bf ) u in X,

.

∀ u ∈ X.

(7.95)

The convergence (7.95) and Proposition 1.17 on page 23 guarantee that M

Σ(θn , Bfn ) −→ Σ(θ, Bf ) in Z.

.

We now use Corollary 6.2 with .Kn = Σ(θn , Bfn ), .K = Σ(θ, Bf ), .An = A, and fn = f = η to deduce that .un → u in X. This shows that Problem .Pθ is .Tθf -wellposed, i.e., condition (7.36) is satisfied. Finally, a careful analysis of assumptions in Theorem 7.10 reveals that we are in a position to use Theorem 7.8. Theorem 7.10 is now a direct consequence of Theorem 7.8, used in the particular setting of the inclusion problem (7.85).



.

We end this subsection with the remark that the abstract results in Sects. 7.2.1 and 7.2.2 can be applied in the study of various nonlinear problems, including variational and hemivariational inequalities and minimization problems, as well. For instance, in Sect. 9.2.3, we illustrate the use of Theorem 7.8 in the study of a variational inequality which describes the frictionless contact of an elastic body, and in Sect. 10.1.5, we shall use the same theorem in the study of a Volterra-type equation which describes the contact of a viscoelastic body with a deformable foundation. Based on the proof of Theorem 7.10, we underline that when Theorem 7.8 is used in various applications, besides checking the validity of the assumptions (7.37) and (7.38) on the set of constraints .K(·) and the assumptions (7.39)–(7.41) on the cost functional .L, the key point of the proof consists in checking the validity of condition (7.36), which, roughly speaking, represents the Hadamard well-posedness of the considered problem.

Part III

Well-Posed Contact Problems

Chapter 8

Mathematical Modeling in Contact Mechanics

In this chapter, we present preliminary material needed in modeling and analysis of contact problems. This concerns the function spaces, the balance equations, the constitutive laws, and the interface laws. We start with the description of the physical setting, and then we introduce the function spaces we use for the displacement, the stress and the strain fields, for which we present their main properties. Next, we move to the modeling of static contact problems. We introduce the linear and nonlinear elastic constitutive laws we use to describe the material’s behavior and the various boundary conditions, with or without unilateral constraints, needed to describe the contact with a rigid, a deformable, or a rigid–deformable foundation. The friction is modeled with versions of Coulomb’s law of dry friction. We extend our modeling to quasistatic process of contact in which the material’s behavior is described with a viscoelastic or viscoplastic constitutive law. We present relevant one-dimensional examples of contact models, in both static and quasistatic cases. Below in this chapter, all variables are assumed to have sufficient degree of smoothness consistent with developments they are involved in.

8.1 Preliminaries In this section, we present a general physical setting in which most of the real-world contact problems can be cast. Then, we discuss the various ingredients needed in their modeling as well as the functional spaces in which their solution is sought.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Sofonea, Well-Posed Nonlinear Problems, Advances in Mechanics and Mathematics 50, https://doi.org/10.1007/978-3-031-41416-9_8

267

268

8 Mathematical Modeling in Contact Mechanics

8.1.1 Physical Setting and Mathematical Models The general physical setting we consider is as follows. A deformable body occupies, in the reference configuration, an open bounded connected set .Ω ⊂ Rd (.d = 1, 2, 3) with Lipschitz continuous boundary .Γ . We denote by .Ω = Ω ∪ Γ the closure of d .Ω in .R . We use boldface letters for vectors and tensors, such as the outward unit normal on .Γ , denoted by .ν. A typical point in .Rd is denoted by .x = (xi ). Here and below the indices i, j , k, l run between 1 and d and, unless stated otherwise, the summation convention over repeated indices is used. Finally, .Γ is decomposed into three parts .Γ 1 , .Γ 2 , and .Γ 3 , with .Γ1 , .Γ2 , and .Γ3 being relatively open and mutually disjoint, and, moreover, the .d − 1 measure of .Γ1 , denoted by .meas (Γ1 ), is positive. We assume that the body is clamped on .Γ1 , surface tractions of density .f 2 act on .Γ2 , and body forces of density (per unit volume) .f 0 act in .Ω. Moreover, the body is, or can arrive, in contact on .Γ3 with an obstacle, the so-called foundation. This physical setting is depicted in Fig. 8.1. We are interested in mathematical models that describe the equilibrium of the mechanical state of the body, in the physical setting above, in the framework of small strain theory. Here and everywhere in this part of the book by a mathematical model, we understand a system of partial differential equations, associated to boundary conditions and to initial conditions, eventually, which describes a specific contact process. To construct various mathematical models of contact, we denote by .u, .σ , and .ε = ε(u) the displacement vector, the stress tensor, and the linearized strain tensor, respectively. These are functions that depend on the spatial variable .x and, eventually, on the time variable t. Nevertheless, in what follows, we do not indicate explicitly the dependence of these quantities on .x and t, i.e., for instance, we write .u instead of .u(x) or .u(x, t). Recall that the components of the linearized strain tensor .ε(u) are given by

Fig. 8.1 Physical setting

8.1 Preliminaries

269

εij (u) =

.

1 (ui,j + uj,i ). 2

(8.1)

Here and below, an index that follows a comma denotes the partial derivative with respect to the corresponding component of .x, i.e., .ui,j = ∂ui /∂xj . The time interval of interest will be denoted by .I0 and will be either bounded, i.e., .I0 = [0, T ] with .T > 0, or unbounded, i.e., .I0 = R+ = [0, +∞). Moreover, a dot above a variable will represent the derivative with respect to time, i.e., for instance, u˙ =

.

∂u , ∂t

u¨ =

∂ 2u . ∂t 2

The mathematical models we study in this book are constructed based on the following ingredients: the constitutive law, the balance equation, the boundary conditions, the interface laws, and, eventually, the initial conditions. A short description of these ingredients follows. A constitutive law represents a relationship between the stress .σ , the strain .ε, and their derivatives, eventually, which characterizes a specific material. It describes the deformations of the body resulting from the action of forces and tractions. Though the constitutive laws must satisfy some basic axioms and invariance principles, they originate mostly from experiments. We refer the reader to [51, 57, 84] for a general description of several diagnostic experiments that provide information needed in constructing constitutive laws for specific materials. The balance equation is deduced from the fundamental principle of Mechanics of Continua and is given by the Cauchy equation of motion or its simplified version, the equilibrium equation. The boundary conditions are the displacement and traction conditions imposed on .Γ1 and .Γ2 , respectively. They reflect the fact that the body is fixed on the part .Γ1 of its boundary and surface tractions act on the part .Γ2 . The interface laws fall naturally into condition in the normal direction (called also contact conditions) and those in the tangential directions (called friction laws). Finally, the initial conditions are used in order to prescribe the displacement field and/or the stress field at .t = 0, if necessary.

8.1.2 Function Spaces In order to introduce the mathematical models that describe various contact processes, we need to precise the function spaces to which the data and the unknown belong. The aim of this subsection is to introduce these spaces together with their basic properties. We denote by .Rd the d-dimensional real linear space, and the symbol .Sd stands for the space of second -order symmetric tensors on .Rd or, equivalently, the space of symmetric matrices of order d. The canonical inner products and the corresponding norms on .Rd and .Sd are given by

270

8 Mathematical Modeling in Contact Mechanics

u · v = ui vi ,

v = (v · v)1/2

.

σ · τ = σij τij ,

∀ u = (ui ), v = (vi ) ∈ Rd ,

τ  = (τ · τ )1/2

∀ σ = (σij ), τ = (τij ) ∈ Sd ,

respectively. Notation .I d will represent the identity operator on .Rd or, equivalently, the unit matrix of order d. The zero elements of .Rd and .Sd will be denoted by .0Rd and .0Sd , respectively. We use standard notation for the Lebesgue and Sobolev spaces associated to .Ω and .Γ , and we assume that the reader has some basic knowledge on the properties of these spaces. Moreover, if X represents a vectorial space, we use the notation .Xd and .Xsd×d for the spaces .

Xd =



Xsd×d =

 x = (xi ) : xi ∈ X, 1 ≤ i ≤ d , 

x = (xij ) : xij = xj i ∈ X, 1 ≤ i, j ≤ d



and, in particular, we will frequently use the spaces   d L2 (Ω) . = v = (vi ) : vi ∈ L2 (Ω), 1 ≤ i ≤ d , .

(8.2)

  2 = τ = (τ ) : τ = τ ∈ L (Ω), 1 ≤ i, j ≤ d . Q = L2 (Ω)d×d ij ij ji s

(8.3)

The spaces (8.2) and (8.3) are real Hilbert spaces with the canonical inner products  .

(u, v)L2 (Ω)d =

Ω

 (σ , τ )Q =

 ui vi dx =

u · v dx



σij τij dx =

σ · τ dx

Ω

∀ u = (ui ), v = (vi ) ∈ L2 (Ω)d ,

Ω

∀ σ = (σij ), τ = (τij ) ∈ Q

(8.4)

Ω

and the associated norms denoted by . · L2 (Ω)d and . · Q , respectively. For an element .v ∈ H 1 (Ω)d , we usually write .v for the trace .γ v ∈ L2 (Γ )d of .v on .Γ . Moreover, we denote by .vν and .v τ the normal and tangential component of .v on the boundary, given by vν = v · ν,

.

v τ = v − vν ν.

(8.5)

We also use the space   V = v ∈ H 1 (Ω)d : v = 0 a.e. on Γ1 ,

.

(8.6)

endowed with the canonical inner product  (u, v)V =

ε(u) · ε(v) dx

.

Ω

∀ u, v ∈ V

(8.7)

8.1 Preliminaries

271

and the associated norm . · V . Recall that the operator .ε : V → Q in (8.7) represents the deformation operator, i.e., .ε(u) = (εij (u)), where the components .εij (u) are given by (8.1). Since .meas Γ1 > 0, it follows from Korn’s inequality that .(V , (·, ·)V ) is a real Hilbert space. Details can be found in [84, 177]. Equalities (8.7) and (8.4) show that (u, v)V = (ε(u), ε(v))Q

.

vV = ε(v)Q

∀ u, v ∈ V , .

∀ v ∈ V,

(8.8) (8.9)

which, in particular, imply that the deformation operator .ε : V → Q is continuous. Moreover, the following result holds. Theorem 8.1 Assume that meas. (Γ1 ) > 0, let .W ⊂ V be a closed subspace of V , and denote by .ε(W ) the range of the deformation operator .ε : W → Q, i.e.,   ε(W ) = ε(v) : v ∈ W .

.

Then .ε(W ) is a closed subspace of Q. Proof Let .{τ n } be a sequence of elements of .ε(W ) that converges in Q to an element .τ ∈ Q, i.e., τn → τ

.

in Q,

n → ∞.

as

(8.10)

Then there exists a sequence .{wn } ⊂ W such that τ n = ε(w n )

.

∀ n ∈ N.

(8.11)

It follows from (8.10) that .{τ n } is a Cauchy sequence in Q and, therefore, (8.9) implies that .{wn } is a Cauchy sequence in V . Next, since W is a closed subspace of V , there exists an element .w ∈ W such that wn → w

.

in V ,

as

n → ∞.

(8.12)

We use now the convergence (8.12) to see that ε(wn ) → ε(w)

.

in Q,

as

n → ∞.

(8.13)

Then, we combine (8.10), (8.11), and (8.13) to deduce that .τ = ε(w). We deduce from here that .τ ∈ ε(W ), which concludes the proof. 

We now proceed with additional properties of the space V . First, it follows from the Sobolev trace theorem that there exists a constant .c0 > 0 such that vL2 (Γ3 )d ≤ c0 vV

.

∀ v ∈ V.

(8.14)

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8 Mathematical Modeling in Contact Mechanics

Inequality (8.14) implies the continuity of the trace operator .γ : V → L2 (Γ3 )d . Moreover, it is well known that this operator is compact, and, using Proposition 1.3, we deduce that the following implication holds: un u

.

in V ⇒ un → u in

L2 (Γ3 )d .

Besides the space V defined in (8.6), we need its subspace   V1 = v ∈ V : vν = 0 a.e. on Γ3 .

.

(8.15)

Over .V1 , we use the inner product .(·, ·)V and the associated norm . · V of the space V . Since .V1 is a closed subspace of V , we see that .(V1 , (·, ·)V ) is itself a real Hilbert space. Next, we recall that if .σ is a regular function, say .σ ∈ C 1 (Ω)d×d , then the s normal component and the tangential part of the stress field .σ on the boundary are defined by σν = (σ ν) · ν,

.

σ τ = σ ν − σν ν.

Moreover, the following Green’s formula holds: 





σ · ε(v) dx +

.

Ω

Div σ · v dx = Ω

σ ν · v da

∀ v ∈ H 1 (Ω)d .

(8.16)

Γ

A proof of this formula is based on a standard density argument. First, it follows from the classical Green–Gauss formula that (8.16) is valid for all .v ∈ C ∞ (Ω)d ; then we use the density of the space .C ∞ (Ω)d in .H 1 (Ω)d to see that the equality in (8.16) is valid for all .v ∈ H 1 (Ω)d . In the study of various contact problems, we shall use the space of fourth order tensor fields .Q∞ given by   ∞ Q . ∞ = E = (eij kl ) : eij kl = ej ikl = eklij ∈ L (Ω), 1 ≤ i, j, k, l ≤ d . (8.17) It is easy to see that .Q∞ is a real Banach space with the norm EQ∞ =

.

max

0≤i,j,k,l≤d

eij kl L∞ (Ω)

and, moreover, Eτ Q ≤ d EQ∞ τ Q

.

∀ E ∈ Q∞ , τ ∈ Q.

(8.18)

We note that most of the function spaces introduced above (the spaces V , .V1 , and Q, for instance) are Hilbert space and, therefore, are reflexive. Nevertheless,

8.2 Modeling of Static Contact Problems

273

the space .Q∞ as well some .L∞ and .L1 -type spaces we shall use in this part of the book are not reflexive. Finally, we recall that in the analysis of time-dependent problems we shall use the notation .C(I0 ; X) for the space of continuous functions defined on .I0 with values in X, where X is one of the spaces mentioned above in this subsection.

8.2 Modeling of Static Contact Problems Static contact problems represent an idealization of contact problems in which both the data and the unknown do not depend on time. They are simplified versions of more complex contact problems, i.e., quasistatic and dynamic problems. In this section, we present the main ingredients in the modeling of static contact problems with elastic materials.

8.2.1 Basic Equations and Boundary Conditions We start by introducing the elastic constitutive laws, the equilibrium equation, and the displacement–traction boundary conditions. Elastic constitutive laws A general elastic constitutive law is given by σ = Fε(u),

.

(8.19)

where .F is the elasticity operator, which can be nonlinear. We allow .F to depend on the location of the point, and we use the short-hand notation .Fε(u) for .F(x, ε(u)). In the study of contact problems involving elastic materials, we assume that the operator .F satisfies the following conditions: ⎧ (a) F : Ω × Sd → Sd . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (b) There exists LF > 0 such that ⎪ ⎪ ⎪ ⎪ F(x, ε1 ) − F(x, ε 2 ) ≤ LF ε 1 − ε 2  ⎪ ⎪ ⎪ ⎪ ∀ ε 1 , ε 2 ∈ Sd , a.e. x ∈ Ω. ⎪ ⎪ ⎪ ⎪ ⎨ (c) There exists mF > 0 such that . ⎪ (F(x, ε 1 ) − F(x, ε 2 )) · (ε 1 − ε 2 ) ≥ mF ε 1 − ε 2 2 ⎪ ⎪ ⎪ ⎪ ∀ ε 1 , ε 2 ∈ Sd , a.e. x ∈ Ω. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (d) The mapping x → F(x, ε) is measurable on Ω, ⎪ ⎪ ⎪ ⎪ ⎪ for any ε ∈ Sd . ⎪ ⎪ ⎪ ⎩ (e) The mapping x → F(x, 0Sd ) belongs to Q.

(8.20)

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8 Mathematical Modeling in Contact Mechanics

Recall that, here and below, Q represents the space given by (8.3). Note that assumptions (8.20)(b) and (c) show that the mapping .F(x, ·) is Lipschitz continuous and strongly monotone on .Sd , for almost every .x ∈ Ω, respectively. Conditions (8.20)(d) and (e) are introduced for mathematical reasons since, combined with condition (8.20)(b), guarantee that the mapping .x → F(x, ε(x)) belongs to Q whenever .ε ∈ Q. Moreover, it is easy to see that if (8.20) holds, then the mapping .ε → Fε defines a strongly monotone Lipschitz continuous operator on the space Q. This property of the elasticity operator will be crucial in the analysis of static contact problems we consider in the next chapter. In particular, if .F = (fij kl ) ∈ Q∞ , equation (8.19) leads to the constitutive law of linearly elastic materials, σij = fij kl εkl (u),

.

where .σij are the components of the stress tensor .σ . Assume that there exists .mF > 0 such that fij kl εij εkl ≥ mF ε2

.

∀ ε = (εij ) ∈ Sd .

Then, it follows that, in this particular case, condition (8.20) is satisfied. Let .d = 3. When the material is linear and isotropic, it can be proved that the elasticity tensor .F is characterized by only two coefficients, the so-called Lamé coefficients, .λ > 0 and .μ > 0. More precisely, the constitutive law of a linearly elastic isotropic material is given by σ = 2 μ ε(u) + λ (tr ε(u)) I 3 ,

.

where .tr ε(u) denotes the trace of the tensor .ε(u), tr ε(u) = εii (u),

.

and, recall, .I 3 denotes the identity tensor on .R3 . In components, we have σij = 2 μ ε ij (u) + λ εkk (u) δij ,

.

where .δij is the Kronecker symbol, i.e., .δij are the components of the unit matrix 3 × 3.

.

Besides the linear case described above, a second example of elastic constitutive law of the form (8.19) is provided by σ = Fε(u) + α (ε(u) − PB ε(u)).

.

(8.21)

Here .F is a linear or nonlinear operator that satisfies condition (8.20), .α is an elasticity coefficient, and B is a given set such that

8.2 Modeling of Static Contact Problems .

α ∈ L∞ (Ω),

275

α(x) ≥ 0 a.e. x ∈ Ω..

B ⊂ Sd is closed, convex and 0Sd ∈ B.

(8.22) (8.23)

Moreover, .PB : Sd → B denotes the projection operator. Since the projection operator is nonexpansive, it is easy to see that the elasticity operator ε → Fε(u) + α (ε(u) − PB ε(u))

.

(which governs the constitutive law (8.21)) satisfies condition (8.20). An example of convex subset B that satisfies the condition (8.23) is the von Mises convex defined by B=



 ε ∈ Sd : ε D  ≤ k .

.

Here, .k > 0 is a given yield and .ε D denotes the deviatoric part of the tensor .ε, that is, εD = ε −

.

1 (tr ε) I d . d

Elastic constitutive laws of the form (8.21) have been used in [128, 218, 238], for instance. A third family of elasticity operators satisfying the condition (8.20) is provided by nonlinear Hencky materials, see, for instance, [84, p. 125] for details. Equilibrium equation In the static case, the mechanical state of the body is governed by the equilibrium equation Div σ + f 0 = 0.

.

(8.24)

Here .f 0 is the density of applied forces, such as gravity or electromagnetic forces and “Div” is the divergence operator, that is, .Div σ = (σij,j ), where, recall, .σij,j = ∂σij ∂xj . Equation (8.24) is obtained from the equation of motion under the assumption that the accelerations in the system are rather small and can be neglected. Since in the static case the time variable does not appear, equation (8.24) is valid in .Ω. Boundary conditions As already mentioned on page 268, we assume that the body is held fixed on .Γ1 and, therefore, u=0

.

on

Γ1 .

(8.25)

This represents the displacement boundary condition. Known tractions of density f 2 act on the portion .Γ2 ; thus,

.

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8 Mathematical Modeling in Contact Mechanics

σν = f 2

.

on

(8.26)

Γ2 .

This condition is called the traction boundary condition. We also remark that assumption .meas (Γ1 ) > 0 is essential in the study of the mechanical problems presented in this book. Without this assumption, mathematically, the problem becomes noncoercive, and many of the results presented in this part of the book do not hold. This accurately reflects the physical situation, since when .Γ1 = ∅ the body is not held in place, and it may move freely in space as a rigid body.

8.2.2 Interface Laws In this subsection, we present various interface laws describing the contact process of an elastic body and a foundation. The equalities and inequalities we write below are valid on .Γ3 even if, for the sake of simplicity, we do not mention it explicitly. We start with the interface laws in the normal direction, the contact conditions, and consider three different physical settings. In the first one, the foundation is a rigid body, in the second one it is deformable, and in the third one it consists of a rigid body covered by a layer of deformable material, which may be another material or just the surface asperities. Contact conditions with a rigid body Although there are no perfect rigid bodies, the conditions below turn out to be useful in many applied settings. We consider three situations, as follows: (i) First, we assume that in the reference configuration there is no gap between the deformable body and the foundation, as shown in Fig. 8.2a. Here and in the pictures below in this section, for simplicity, we consider the case of a rectangular body. A popular contact condition that describes this setting, used in both the engineering literature and mathematical publications, is the Signorini contact condition, formulated as follows:

a)

b)

Fig. 8.2 Physical setting: (a) contact with a rigid obstacle without gap; (b) contact with a rigid obstacle with gap

8.2 Modeling of Static Contact Problems

277

Fig. 8.3 The Signorini condition: (a) without gap; (b) with gap

uν ≤ 0,

.

σν ≤ 0,

σν uν = 0.

(8.27)

This condition was first introduced in [203] and then used in many papers, see, e.g., [200] and the references therein. Note that condition (8.27) does not allow interpenetration. When .uν < 0, there is separation between the body and the foundation, and (8.27) implies that .σν = 0, i.e., the normal stress vanishes. When .uν = 0, there is contact. Therefore, (8.27) implies that .σν ≤ 0, i.e., the reaction of the foundation is toward the body. A graphic representation of the Signorini condition (8.27) is provided in Fig. 8.3a. (ii) Second, we assume that in the reference configuration there is a gap .g > 0 between the body and the foundation, see Fig. 8.2b. Then, the Signorini condition reads as uν ≤ g,

.

σν ≤ 0,

σν (uν − g) = 0.

(8.28)

The mechanical interpretation of (8.28) is very similar to the case without gap, i.e., to the case .g = 0. A graphic representation of condition (8.28) is depicted in Fig. 8.3b. As usual in the literature, we say that conditions (8.27) and (8.28) model an unilateral contact. (iii) Third, we mention that in many machines with moving parts and components of mechanical equipment there is no separation between the body and the foundation. In this situation, we say that the contact is bilateral, and we model it with condition uν = 0.

.

(8.29)

Its graphic representation is given by the vertical axis of the orthogonal system in Fig. 8.3. Contact conditions with a deformable body Consider now the case when the foundation is deformable. We can imagine the situation when the behavior of the

278

8 Mathematical Modeling in Contact Mechanics

Fig. 8.4 The normal compliance condition: (a) without gap; (b) with gap

foundation is elastic, rigid–plastic, and rigid–elastic, respectively. Therefore, we consider in what follows three types of contact conditions, described below: (i) Assume that the foundation has an elastic behavior, and there is no initial gap, in the reference configuration. Then, for the normal stress, we use the so-called normal compliance contact condition, i.e., .

− σν = p(uν ),

(8.30)

where p is a nonnegative regular function that vanishes for a negative argument. Indeed, when .uν < 0, there is no contact and the normal pressure vanishes. When .uν ≥ 0, there is contact, and .uν represents a measure of the penetration into the deformable body. Then, condition (8.30) shows that the foundation exerts on the body a pressure that depends on the penetration. We conclude from here that this condition is appropriate when the behavior of the foundation is elastic. Its graphic representation is given in Fig. 8.4a. Finally, note that in the case when there is an initial gap between the body and the foundation, then condition (8.30) is replaced by equality .

− σν = p(uν − g).

(8.31)

Its graphic representation is given in Fig. 8.4b. The normal compliance contact condition was first introduced in [180], and since then used in many publications, see, e.g., [114, 117, 118, 140] and the references therein. (ii) Next, we consider the case when the foundation has a rigid–plastic behavior and, for simplicity, there is no initial gap. In this case, we assume that .

− F ≤ σν ≤ 0,

σν =

0 if uν < 0, −F

if uν > 0.

(8.32)

8.2 Modeling of Static Contact Problems

279

Here, F is a given positive traction threshold that may depend on the spatial variable .x. Using (8.32), we have .

−F < σν ≤ 0 σν = −F

⇒

⇒

uν ≤ 0,

uν ≥ 0.

This shows that the foundation does not allow penetration and, therefore, behaves as a rigid body, as far as the inequality .−F < σν ≤ 0 holds. It allows penetration only when the threshold is reached, .σν = −F , and then, it offers no additional resistance, as surface plastic flow commences. We conclude from here that condition (8.32) describes well situations when the foundation has a rigid–plastic behavior and F could be interpreted as its yield limit. The graphic representation of the contact condition (8.32) is given in Fig. 8.5a. (iii) Finally, we consider the case when the foundation has a rigid–elastic behavior and, again, there is no initial gap. In this case, we assume that σν = 0 .

−F ≤ σν ≤ 0 −σν = F + p(uν )

⎫ if uν < 0 ⎪ ⎪ ⎪ ⎬ if uν = 0 . ⎪ ⎪ ⎪ ⎭ if uν > 0

(8.33)

Condition (8.33) represents a combination of conditions (8.30) and (8.32) in which F is a positive function and p is the normal compliance function; it is positive when the argument is positive and vanishes for a negative argument. Arguments similar to those above show that condition (8.33) is convenient to describe the contact with an rigid–elastic obstacle. Here, F could be interpreted as the yield limit of the obstacle, while the normal compliance function p describes its elastic properties. The graphic representation of the contact condition (8.33) is given in Fig. 8.5b.

Fig. 8.5 Two contact conditions: (a) condition (8.32); (b) condition (8.33)

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8 Mathematical Modeling in Contact Mechanics

Fig. 8.6 Contact with a rigid obstacle covered by a deformable layer

Contact conditions with a rigid body covered by a deformable layer We now consider more complex conditions, when the foundation is made of a rigid body covered with a layer of deformable material of thickness .g > 0. This layer may be just the asperities or a softer material, as is shown in Fig. 8.6. Then, since the rigid obstacle is impenetrable, we have uν ≤ g.

.

(8.34)

Moreover, using the principle of superposition, we assume that the normal stress has an additive decomposition of the form σν = σνD + σνR ,

.

(8.35)

in which .σνD describes the reaction of the deformable layer and .σνR describes the reaction of the rigid body. In function of the behavior of the deformable layer, we distinguish three cases, as follows: (i) Assume that the deformable layer has an elastic behavior. Then, for the part .σνD of the normal stress, we use the normal compliance contact condition (8.30), that is, .

− σνD = p(uν ).

(8.36)

Here, again, p is a nonnegative regular function that vanishes for a negative argument. Indeed, when .uν < 0, there is no contact and the normal pressure vanishes. When .0 ≤ uν ≤ g, there is contact and .uν represents a measure of the penetration into the elastic layer. Then, condition (8.36) shows that the layer exerts on the body a pressure that depends on the penetration. In addition, when .uν = g, this layer is completely squeezed and the normal pressure it exerts is .p(g). On the other hand, for the rigid part of the obstacle, we use the Signorini contact condition with gap (8.28). Therefore, σνR ≤ 0,

.

σνR (uν − g) = 0,

(8.37)

8.2 Modeling of Static Contact Problems

281

where, recall, .g > 0 represents the thickness of the deformable layer. We now gather conditions (8.34)–(8.37), and, in this way, we obtain the contact condition ⎫ σν = 0 if uν < 0 ⎪ ⎪ ⎬ . uν ≤ g, (8.38) −σν = p(uν ) if 0 ≤ uν < g . ⎪ ⎪ ⎭ −σν ≥ p(g) if uν = g A careful analysis of (8.38) reveals that this contact condition can be formulated, equivalently, as follows: uν ≤ g,

.

σν + p(uν ) ≤ 0,

(uν − g)(σν + p(uν )) = 0.

(8.39)

A graphic depiction of the contact condition (8.38) (or, equivalently, (8.39)) is provided in Fig. 8.7a. (ii) Next, we consider the case when the deformable layer has a rigid–plastic behavior. In this case, in addition to (8.34), (8.35), and (8.37), we use condition (8.32) for .σνD , that is, - v

- v

F g

O

uv

O

a)

g

uv

b)

- v

F O

g

uv

c)

Fig. 8.7 Three contact conditions with unilateral constraint: (a) condition (8.38); (b) condition (8.41); (c) condition (8.43)

282

8 Mathematical Modeling in Contact Mechanics

.

− F ≤ σνD ≤ 0,

σνD =

0 if uν < 0, −F

(8.40)

if uν > 0.

Therefore, gathering (8.34), (8.35), (8.37), and (8.40), we deduce the contact condition ⎫ if uν < 0 σν = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ −F ≤ σν ≤ 0 if uν = 0 . .uν ≤ g, (8.41) ⎪ if 0 < uν < g ⎪ σν = −F ⎪ ⎪ ⎪ ⎭ σν ≤ −F if uν = g We may justify condition (8.41) as follows: (a) If .uν < 0, there is no contact, and then (8.40) implies that .σνD = 0, (8.37) implies that .σνR = 0, and, therefore, equality (8.35) shows that .σν = 0. Thus, the contact traction vanishes, as expected. (b) If .uν = 0, the contact has just been established (or about to be lost), and then (8.40) implies that .−F ≤ σνD ≤ 0, (8.37) implies that .σνR = 0, and, therefore, equality (8.35) shows that .−F ≤ σν ≤ 0. Thus, the layer behaves as a rigid surface. (c) If .0 < uν < g, there is penetration into the layer, and then (8.40) implies that .σνD = −F , (8.37) implies that .σνR = 0, and, therefore, equality (8.35) shows that .σν = −F . In this case, the layer is in the plastic flow regime. (d) If .uν = g, the layer is completely squeezed and then (8.40) implies that D = −F , (8.37) implies that .σ R ≤ 0, and, therefore, equality (8.35) .σν ν shows that .σν ≤ −F . The contact condition (8.41) is depicted in Fig. 8.7b. It was used in a number of papers, see, e.g., [222], and the references therein. (iii) Finally, we consider the case when the deformable layer has a rigid–elastic behavior. In this case, in addition to (8.34), (8.35), and (8.37), we assume that D .σν satisfies condition (8.33), i.e., σνD = 0 .

−F ≤ σνD ≤ 0 −σνD = F + p(uν )

⎫ if uν < 0 ⎪ ⎪ ⎬ if uν = 0 . ⎪ ⎪ ⎭ if uν > 0

(8.42)

We now gather (8.34), (8.35), (8.37), (8.42) and use arguments similar to those used above to obtain the following contact condition:

8.2 Modeling of Static Contact Problems

.

uν ≤ g,

283

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

σν = 0

if uν < 0

−F ≤ σν ≤ 0

if uν = 0

−σν = F + p(uν )

⎪ if 0 < uν < g ⎪ ⎪ ⎪ ⎪ ⎭ if uν = g

−σν ≥ F + p(uν )

.

(8.43)

This condition is depicted in Fig. 8.7c. We now proceed with some comments on the contact conditions (8.27), (8.28), (8.38), (8.41), and (8.43). First, these conditions are expressed in terms of unilateral constraints and are governed by the data .g, p, and F . Moreover, all of them are described by multivalued relations between the normal displacement and the compressive normal stress. In addition, there exists a hierarchy among these contact conditions as follows: (a) Condition (8.41) can be obtained from condition (8.43) when the normal compliance function p vanishes, i.e., .p ≡ 0. (b) Condition (8.38) is obtained from condition (8.43) when the yield limit F vanishes, i.e., .F = 0. (c) Condition (8.28) can be recovered from condition (8.38) when .p ≡ 0, from condition (8.41), when .F = 0, and from condition (8.43) when .p ≡ 0 and .F = 0. (d) The Signorini contact condition (8.27) is obtained from conditions (8.28), (8.38), (8.41), and (8.43) when .g = 0. We conclude that among the above conditions, condition (8.43) is the most general one. Finally, we note that, on occasion, we shall use regularization of the contact conditions presented above in this subsection. Friction laws We turn now to the conditions in the tangential directions, called also frictional conditions or friction laws. The simplest one is the so-called frictionless condition in which the tangential part of the stress vanishes, i.e., σ τ = 0 on

.

Γ3 .

(8.44)

This is an idealization of the process, since even completely lubricated surfaces generate shear resistance to tangential motion. However, the frictionless condition (8.44) is a sufficiently good approximation of the reality in some situations, and, for this reason, it was used in several publications, see [84, 200] and the references therein. In the case when the friction force does not vanish on the contact surface, the contact is frictional. Frictional contact between solid surfaces is usually modeled with a number of variants of the Coulomb law of dry friction. The static version of this law, commonly used in frictional contact problems describing the equilibrium of elastic bodies, is formulated as follows:

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8 Mathematical Modeling in Contact Mechanics

σ τ  ≤ Fb ,

.

uτ uτ 

σ τ = − Fb

uτ = 0

if

on

Γ3 .

(8.45)

Here, .uτ is the tangential displacement or the slip, .Fb is the friction bound, and, again, .σ τ represents the tangential shear or the friction force. On a nonhomogeneous surface, .Fb depends explicitly on the position .x on the surface. We note here that the “friction force” is not a force in the usual sense, since friction is only resistance to motion and cannot initiate motion, unlike a “real” force. Although we use the term friction force, “frictional resistance force” is the more accurate term in physics, since it just opposes motion. The friction law (8.45) shows that during the contact process the magnitude of the friction force is bounded by the positive function .Fb , the friction bound. This is the maximal strength that friction resistance can provide, and above it, the surfaces undergo a relative motion. It indicates that the points on the contact surface where the inequality .σ τ  < Fb holds are in the stick state since there .uτ = 0. The points of the contact surface where .uτ = 0 are in the slip state. There, the friction force .σ τ is opposite to the slip .uτ , and, moreover, its magnitude equals the magnitude of the friction bound since, in this case, (8.45) implies that .σ τ  = Fb . In certain applications, especially where the bodies are light or the friction is very large, the friction bound .Fb does not depend on the process variables and behaves like a function that depends only on the position .x on the contact surface. The choice Fb = Fb (x)

.

in (8.45) leads to the static Tresca friction law and simplifies considerably the analysis of the corresponding contact problem. Often, especially in engineering literature, the friction bound .Fb is chosen as Fb = Fb (σν ) = μ |σν |,

.

(8.46)

where .μ > 0 is the coefficient of friction. The choice (8.46) in (8.45) leads to the classical version of Coulomb’s law that was intensively studied in the literature, see, for instance, the references in [200]. In many geophysical publications, the motion of tectonic plates is modeled with the Coulomb law (8.45) in which the friction bound is assumed to depend on the magnitude of the tangential displacement, that is, .Fb = Fb (uτ ). Details can be found in [192, 198] and the references therein. We now return to the static friction law (8.45) and note that, using definition (1.44), it can be written in the subdifferential form .

− σ τ ∈ ∂c jτ (uτ ),

(8.47)

where jτ (ξ ) = Fb ξ 

.

∀ ξ ∈ Rd .

(8.48)

8.2 Modeling of Static Contact Problems

285

An extension of (8.47) is given by .

− σ τ ∈ ∂jτ (uτ ),

(8.49)

in which .∂jτ represents the Clarke subdifferential of a given locally Lipschitz function .jτ . Various examples of friction laws used in the literature can be cast in the form (8.47) and/or (8.49), i.e., expressed in terms of the subdifferential of a convex or nonconvex function. Details, comments, and mechanical interpretation can be found in [157, 184] and many other recent references. Nevertheless, in the next two chapters of this book, we shall restrict ourselves either to frictionless contact models, or to frictional models constructed with Coulomb’s law. Conclusion To conclude, a mathematical model that describes the equilibrium of an elastic body in the physical setting introduced above consists of finding the unknown functions .u and .σ that satisfy a constitutive law of the form (8.19), the equilibrium equation (8.24), the displacement boundary condition (8.25), and the traction boundary condition (8.26) together with one of the contact conditions and one of the friction laws described above. Such a mathematical model is represented by a system of partial differential equations associated to linear or nonlinear boundary conditions. As a result of the above, combining the various contact conditions and friction laws with various choices of the friction bound, we obtain several mathematical models of contact with elastic materials. The analysis of such models, selected by taking into account their relevance from a physical or mathematical point of view, is presented in Chap. 9.

8.2.3 A One-Dimensional Example In this subsection, we consider a representative one-dimensional example that describes the static contact of an elastic rod with a two-layered foundation. We choose it since it is easier to explain the main ideas of this part of the book and without the complications that arise in two or three dimensions. Below, for simplicity, we do not indicate the units associated to the data and unknowns. So, we consider an elastic rod of length .l = 1 that is rigidly attached at .x = 0 and may come in contact with a foundation at .x = 1, under the action of a force density (per unit length) f . The foundation has a deformable layer of the rigid–plastic type of thickness g, which is attached to a rigid body underneath. In the notation above, we have .d = 1, .Ω = (0, 1), .Γ1 = {0}, .Γ2 = ∅, .Γ3 = {1}. The setting is depicted in Fig. 8.8. We denote by .u = u(x) the displacement, and then, the linearized strain field is given by .ε(u) = u , where, here and below, the prime represents the derivative with respect to .x ∈ [0, 1]. We also denote by E the Young modulus of the material. The stress in the rod is given by .σ (x), and within linearized elasticity, .σ = Eu . We

286

8 Mathematical Modeling in Contact Mechanics

O

O

l =1

g

foundation

x

f

rigid-plastic layer rigid body

Fig. 8.8 A rod in contact with a two-layered foundation

have .E > 0, .g > 0, and, for the sake of simplicity, we assume that .f ∈ R does not depend on the spatial variable. Then, the statement of the problem of static contact between an elastic rod and a rigid–plastic foundation described above is the following. Problem .P 1d . Find a displacement field .u : [0, 1] → R and a stress field .σ : [0, 1] → R, such that .

σ (x) = E u (x)

∀ x ∈ [0, 1], .

(8.50)

σ  (x) + f = 0

∀ x ∈ [0, 1], .

(8.51)

u(0) = 0, .

(8.52) σ (1) = 0

u(1) ≤ g,

if

u(1) < 0

−F ≤ σ (1) ≤ 0 if u(1) = 0 σ (1) = −F σ (1) ≤ −F

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ if 0 < u(1) < g ⎪ ⎪ ⎪ ⎪ ⎭ if u(1) = g

.

(8.53)

Note that (8.50) represents the elastic constitutive law, (8.51) is the equilibrium equation, (8.52) is the displacement boundary condition, and (8.53) is the onedimensional version of the contact condition (8.41) in which F represents the rigid–plastic material’s yield limit, assumed to be positive. Since the example is “simple”, direct calculations allow us to solve Problem .P 1d and to obtain closed-form solutions. These calculations could be used to calibrate and verify numerical algorithms for realistic engineering problems. It is found that there are four different possible cases that depend on the relationship between f , F , E, and g. We describe each of them and its corresponding mechanical interpretation: (a) The case .f < 0. The body force acts away from the foundation, and then the solution of Problem .P 1d is given by

8.2 Modeling of Static Contact Problems

287

⎧ ⎨ σ (x) = f (1 − x),

 . ⎩ u(x) = f 1 − 1 x x, E 2

∀ x ∈ [0, 1].

(8.54)

In this case, as is to be expected since there is no contact, .u(1) < 0 and .σ (1) = 0. Since there is separation between the rod’s end and the foundation, there is no reaction at .x = 1. This case corresponds to Fig. 8.9a, in which the deformed configuration of the rod is depicted. (b) The case .0 ≤ f < 2F . The force pushes the rod toward the foundation, and the solution of Problem .P 1d is given by .

σ (x) = u(x) =

f 2 (1 − 2x), f 2E (1 − x) x,

∀ x ∈ [0, 1].

(8.55)

We have .u(1) = 0 and .−F < σ (1) ≤ 0, which show that the rod is in contact with the foundation, just touching it, and the reaction of the foundation is toward the rod. Nevertheless, there is no penetration, since the magnitude of the stress

O

l =1

foundation

O

l =1

f

g

rigid-plastic layer foundation

f

g

rigid-plastic layer rigid body

rigid body

a)

b)

O

O

l =1

foundation

l =1

f

g

rigid-plastic layer foundation

f

g

rigid-plastic layer

rigid body

c)

rigid body

d)

Fig. 8.9 The deformed configuration of the rod in contact with a two-layered foundation: (a) The case .f < 0; (b) The case .0 ≤ f < 2F ; (c) The case .2F ≤ f < 2Eg + 2F ; (d) The case .2Eg + 2F ≤ f

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8 Mathematical Modeling in Contact Mechanics

at .x = 1 is under the yield limit F and, therefore, the rigid–plastic layer behaves like a rigid layer. This case is depicted in Fig. 8.9b. (c) The case .2F ≤ f < 2Eg + 2F . In this case, the force is sufficiently large to cause the penetration of the rod’s end into the rigid–plastic layer. The solution of Problem .P 1d is given by .

σ (x) = f (1 − x) − F, u(x) =

f 2E

(2 − x) x −

F E

x,

∀ x ∈ [0, 1].

(8.56)

We have .0 ≤ u(1) < g and .−σ (1) = F . This, indeed, shows that the stress at .x = 1 reached the yield limit and, therefore, there is penetration into the rigid–plastic layer that now behaves plastically. Nevertheless, the penetration is partial, since .u(1) < g. This case is shown in Fig. 8.9c. (d) The case .2Eg +2F ≤ f . Here, the applied force is sufficient to make the whole layer plastic. The solution of Problem .P 1d is given by .

σ (x) = u(x) =

f 2 (1 − 2x) + Eg, f 2E (1 − x) x + gx,

∀ x ∈ [0, 1].

(8.57)

We have .u(1) = g and .σ (1) ≤ −F that show that the rigid–plastic layer is completely penetrated and the rod’s end .x = 1 reaches the rigid body. The magnitude of the reaction in this point is larger than the yield limit F since, besides the reaction of the rigid–plastic layer, there is also the reaction of the rigid body, which becomes active in this case. This case is depicted in Fig. 8.9d. The analytic forms (8.54)–(8.57) of the solution in the four cases show clearly the continuous dependence of the solution on the data f , F , and g. More precisely, if .σf,F,g and .uf,F,g represent the solution of Problem .P 1d with the data .f ∈ R, .F ≥ 0, and .g ≥ 0, then the functions (f, F, g) → σf,F,g (x)

.

and

(f, F, g) → uf,F,g (x)

defined on .R × R+ × R+ with values in .R are continuous, for each .x ∈ [0, 1]. In particular, it follows from here that σf,F,g (x) → σf,F,0 (x)

.

and

uf,F,g (x) → uf,F,0 (x)

as 0 < g → 0, (8.58)

for any .f ∈ R, .F ∈ R+ , and .x ∈ [0, 1]. Note that .σf,F,0 =  σf and .uf,F,0 =  uf for σf : [0, 1] → R and . uf : [0, 1] → R are the functions defined any .F ≥ 0, where . by

8.3 Modeling of Quasistatic Contact Problems

⎧ ⎪ σf (x) = f (1 − x), ⎨ . 

⎪ f ⎩ 1 − 12 x x, uf (x) = E

289

∀ x ∈ [0, 1]

if .f < 0, and ⎧ σf (x) = ⎨ .

⎩

 uf (x) =

f 2 (1 − 2x), f 2E

∀ x ∈ [0, 1]

(1 − x) x,

if .f ≥ 0. On the other hand, it is easy to see that the couple .( uf ,  σf ) is the solution to the Signorini problem without gap, that is: Problem .PS1d . Find a displacement field .u : [0, 1] → R and a stress field .σ : [0, 1] → R that satisfy (8.50)–(8.52) and, moreover, .u(1) ≤ 0, .σ (1) ≤ 0, .σ (1)u(1) = 0. Therefore, the convergence (8.58) shows that we can approach the solution of the contact problem with a rigid obstacle by the solution of a contact problem with a rigid obstacle covered by a layer of rigid–plastic material, as the thickness of this layer converges to zero. Besides the continuous dependence of the solution with respect to the data, the one-dimensional example we considered in this subsection clearly illustrates the link between various contact models, constructed based on different mechanical assumptions. The analysis of such convergences in the study of some relevant contact models represents one of the main aims of the next two chapters. It is obtained by using results on .T -well-posedness we presented in Part II of the book.

8.3 Modeling of Quasistatic Contact Problems The quasistatic models of contact are studied in a time interval of interest .I0 . This will be either bounded, i.e., .I0 = [0, T ] with .T > 0, or unbounded, i.e., .I0 = R+ = [0, +∞). The equations and boundary conditions we write below in this section are valid et each time moment .t ∈ I0 . Moreover, a dot above a variable will denote the derivative of that variable with respect to the time.

8.3.1 Basic Equations and Boundary Conditions We start with a short description of the viscoelastic constitutive laws we shall use in the next chapters.

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8 Mathematical Modeling in Contact Mechanics

Viscoelastic constitutive laws A general viscoelastic constitutive law with short memory is given by ˙ + Bε(u). σ = Aε(u)

.

(8.59)

We allow the viscosity operator .A and the elasticity operator .B to depend on the location of the point. Consequently, all that follows is valid for nonhomogeneous materials. In the study of mechanical problems involving viscoelastic materials with short memory, we assume that the operators .A and .B satisfy the following conditions: ⎧ (a) A : Ω × Sd → Sd . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (b) There exists LA > 0 such that ⎪ ⎪ ⎪ ⎪ ⎪ A(x, ε1 ) − A(x, ε 2 ) ≤ LA ε 1 − ε 2  ⎪ ⎪ ⎪ ⎪ ∀ ε 1 , ε 2 ∈ Sd , a.e. x ∈ Ω. ⎪ ⎪ ⎪ ⎪ ⎨ (c) There exists mA > 0 such that . ⎪ (A(x, ε 1 ) − A(x, ε 2 )) · (ε1 − ε 2 ) ≥ mA ε 1 − ε 2 2 ⎪ ⎪ ⎪ ⎪ ∀ ε 1 , ε 2 ∈ Sd , a.e. x ∈ Ω. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (d) The mapping x → A(x, ε) is measurable on Ω, ⎪ ⎪ ⎪ ⎪ ⎪ for any ε ∈ Sd . ⎪ ⎪ ⎪ ⎩ (e) The mapping x → A(x, 0Sd ) belongs to Q.

(8.60)

⎧ ⎪ (a) B : Ω × Sd → Sd . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (b) There exists LB > 0 such that ⎪ ⎪ ⎪ ⎪ B(x, ε1 ) − B(x, ε2 ) ≤ LB ε 1 − ε 2  ⎪ ⎨ ∀ ε 1 , ε 2 ∈ Sd , a.e. x ∈ Ω. . ⎪ ⎪ ⎪ ⎪ (c) The mapping x → B(x, ε) is measurable on Ω, ⎪ ⎪ ⎪ ⎪ for any ε ∈ Sd . ⎪ ⎪ ⎪ ⎪ ⎩ (d) The mapping x → B(x, 0Sd ) belongs to Q.

(8.61)

On assumptions (8.60) and (8.61), we have similar comments as those made on assumption (8.20) on page 273. In particular, we note that these assumptions allow us to consider .A and .B as operators defined on the space Q with values in Q. Moreover, .A is a strongly monotone Lipschitz continuous operator and .B is a Lipschitz continuous operator. Next, if .A and .B are linear operators, then (8.59) leads to the Kelvin–Voigt constitutive law ˙ + bij kl εkl (u), σij = aij kl εkl (u)

.

8.3 Modeling of Quasistatic Contact Problems

291

where .σij , .aij kl , and .bij kl denote the components of the stress tensor .σ , the viscosity tensor .A, and the elasticity tensor .B, respectively. Clearly, assumption (8.60) is satisfied if .A = (aij kl ) ∈ Q∞ , and there exists .mA > 0 such that aij kl εij εkl ≥ mA ε2

∀ ε = (εij ) ∈ Sd .

.

Moreover, assumption (8.61) is satisfied if .B = (bij kl ) ∈ Q∞ . A second example of viscoelastic constitutive law of the form (8.59) is provided by the nonlinear viscoelastic constitutive law ˙ + α (ε(u) − PB ε(u)). σ = Aε(u)

.

Here .A is a linear or nonlinear operator that satisfies condition (8.60), .α is a positive coefficient that satisfies condition (8.22), B is a set that satisfies (8.23), and, as usual, d .PB : S → B denotes the projection operator on B. Note that, since the derivative of the displacement field appears in the constitutive law (8.59), in solving contact problems with such type of viscoelastic materials, we have to prescribe the displacement field at the initial moment .t = 0. Therefore, we supplement (8.59) with the initial condition u(0) = u0

in Ω,

.

(8.62)

in which .u0 is a given initial displacement. A general viscoelastic constitutive law with long memory is given by 

t

σ (t) = Fε(u(t)) +

R(t − s)ε(u(s)) ds.

.

(8.63)

0

Here .F represents the elasticity operator, assumed to satisfy condition (8.20), and R is the relaxation tensor, assumed to have the regularity

.

R ∈ C(I0 ; Q∞ ).

.

(8.64)

In the linear case, the stress tensor .σ = (σij ) that satisfies (8.63) is given by  σij (t) = fij kl εkl (u(t)) +

t

.

rij kl (t − s) εkl (u(s)) ds,

0

where .fij kl and .rij kl denote the components of the elasticity tensor .F and relaxation tensor .R, respectively. Finally, note that in various publications the behavior of a viscoelastic material is described with a constitutive law of the form  t ˙ ˙ .σ (t) = Aε(u(t)) + Bε(u(t)) + R(t − s)ε(u(s)) ds. (8.65) 0

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8 Mathematical Modeling in Contact Mechanics

Here .A is the viscosity operator, .B is the elasticity operator, and .R is the relaxation tensor. The corresponding assumptions used in the study of such constitutive law are (8.60), (8.61), and (8.64), respectively. Note also that, since the derivative of the displacement field appears in the constitutive law (8.65), in solving contact problems with such materials, we have to prescribe the initial condition (8.62). Examples, details, and mechanical interpretation for (8.65) can be found in [12– 14], for instance. In addition, note that when the relaxation tensor vanishes, (8.65) becomes the viscoelastic constitutive law with short memory (8.59). Rate-type constitutive laws A general rate-type viscoelastic or viscoplastic constitutive law may be written as ˙ + G(σ , ε(u)), σ˙ = Eε(u)

(8.66)

.

where .E is a fourth order tensor and .G is a constitutive function. We assume in what follows that .E and .G satisfy the following conditions:

.

⎧ ⎪ ⎨ (a) E = (eij kl ) ∈ Q∞ . (b) There exists mE > 0 such that ⎪ ⎩ E(x, τ ) · τ ≥ mE τ 2 ∀ τ ∈ Sd , a.e. x in Ω.

.

⎧ ⎪ (a) G : Ω × Sd × Sd → Sd . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (b) There exists LG > 0 such that ⎪ ⎪ ⎪ ⎪ G(x, σ 1 , ε 1 ) − G(x, σ 2 , ε 2 ) ⎪ ⎨ ≤ LG (σ 1 − σ 2  + ε 1 − ε 2 ) ⎪ ⎪ ∀ σ 1 , σ 2 , ε 1 , ε 2 ∈ Sd , a.e. in Ω. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (c) For any σ , ε ∈ Sd , x → G(x, σ , ε) is measurable. ⎪ ⎪ ⎪ ⎪ ⎩ (d) The mapping x → G(x, 0Sd , 0Sd ) belongs to Q.

(8.67)

(8.68)

Examples of viscoplastic laws of this form can be found in [59, 84, 218, 222], for instance. They have been used to model the properties of real materials such as metals, rubbers, polymers, and rocks, among others. It follows from assumptions (8.67) that .E is a fourth order invertible tensor, for almost every .x ∈ Ω. ˙ can be interchanged with that of .σ˙ in (8.66) to obtain Therefore, the role of .ε(u)  , ε(u)), ˙ = E −1 σ˙ + G(σ ε(u)

.

 , ε(u)) = −E −1 G(σ , ε(u)). This where .E −1 denotes the inverse of .E and .G(σ ˙ also denoted as .ε˙ , can be equality shows that the rate of deformation .ε(u), decomposed into two parts: the elastic part .ε˙ e = E −1 σ˙ and the anelastic one given  , ε(u)). by .ε˙ p = G(σ

8.3 Modeling of Quasistatic Contact Problems

293

A well-known example of constitutive laws of the form (8.66) is Perzyna’s law given by ˙ = E −1 σ˙ + ε(u)

.

1 (σ − PB σ ). μ

Here .E is a fourth order tensor that satisfies (8.67), .μ > 0 is a viscosity constant, B is a set that satisfies condition (8.23), and, as usual, .PB represents the projection operator. Note that in this case the function .G does not depend on .ε and is given by G(σ , ε) = −

.

1 E(σ − PK σ ). μ

Moreover, it is easy to see that this function satisfies condition (8.68). A second example of rate-type constitutive law of the form (8.66) is given by ˙ + α(σ − B(ε(u)), σ˙ = Eε(u)

.

in which .α is a viscosity coefficient assumed to satisfy condition (8.22) and .B is a linear or nonlinear operator that satisfies (8.61). We shall use such constitutive law in Sect. 10.3, in the study of two frictionless models of contact. Note that, since the derivatives of the displacement and the stress fields appear in the constitutive law (8.66), in solving contact problems with such kind of ratetype materials, we have to prescribe the displacement field and the stress field at the initial moment .t = 0. Therefore, we supplement (8.66) with the initial conditions u(0) = u0 ,

.

σ (0) = σ 0

in Ω,

(8.69)

in which .u0 and .σ 0 represent the initial displacement and the initial stress, respectively. Equilibrium equation and boundary conditions In the quasistatic case, the mechanical state of the body is governed by the equilibrium equation (8.24). Nevertheless, in contrast with the setting in Sect. 8.2, here we assume that the density of body forces is time-dependent. Therefore, the equilibrium equation becomes Div σ (t) + f 0 (t) = 0,

.

(8.70)

and we recall that this equation is valid in .Ω, at any time moment .t ∈ I0 . Now, since we assume that the body is held fixed on .Γ1 during the contact process, the displacement field vanishes there. Hence, the displacement boundary condition we use is u = 0 on

.

Γ1 × I0 .

(8.71)

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8 Mathematical Modeling in Contact Mechanics

Moreover, the traction boundary condition becomes σν = f 2

.

on

Γ2 × I0 .

(8.72)

Note that here we assume that the density of surface tractions depends on time, i.e., f 2 = f 2 (t).

.

8.3.2 Interface Laws The interface laws we use in the study of quasistatic contact are similar to those used in the study of static contact. Nevertheless, a first difference arises from the fact that the conditions in the normal direction are now valid at any time moment .t ∈ I0 . For instance, the bilateral contact condition (8.29) becomes uν = 0

.

on Γ3 × I0 ,

(8.73)

while the time-dependent version of the contact conditions with normal compliance (8.30) and (8.31) is given by .

−σν = p(uν )

on Γ3 × I0 , .

(8.74)

−σν = p(uν − g)

on Γ3 × I0 ,

(8.75)

respectively. In Chap. 10, we shall use the contact conditions (8.73)–(8.75) as well as the time-dependent version of condition (8.39), that is, .

uν ≤ g,

σν + p(uν ) ≤ 0,

(uν − g)(σν + p(uν )) = 0

on Γ3 × I0 .

(8.76)

A second difference arises from the friction law. Thus, on occasion, we shall use the frictionless contact condition (8.44) that, in the quasistatic case, becomes στ = 0

.

on

Γ3 × I0 .

(8.77)

For the frictional contact, we shall use the quasistatic version of Coulomb’s law of dry friction. Therefore, we replace (8.45) by condition σ τ  ≤ Fb ,

.

σ τ = − Fb

u˙ τ u˙ τ 

if u˙ τ = 0 on

Γ3 × I0 .

(8.78)

Here, .u˙ τ is the tangential velocity or the slip rate, .σ τ represents the tangential shear or the friction force, and, again, .Fb is the friction bound.

8.3 Modeling of Quasistatic Contact Problems

295

For the quasistatic contact models, we consider in Chap. 10, besides the frictionless condition (8.77) we shall use the quasistatic version of the Tresca friction law, i.e., the law (8.78) in which the friction bound .Fb is assumed to be given. Moreover, for the contact model, we introduce on page 379 we shall use the Coulomb law (8.78) in the case when the friction bound depends on various process variables. Conclusion To conclude, a mathematical model that describes the equilibrium of a viscoelastic or viscoplastic body in the physical setting introduced on page 268 consists of finding the unknown functions .u and .σ that satisfy one of the constitutive laws (8.59), (8.63), (8.65), (8.66), the equilibrium equation (8.70), the displacement boundary condition (8.71), and the traction boundary condition (8.72), together with one of the contact conditions (8.73), (8.74), (8.75), (8.76), and one of the friction laws (8.77) or (8.78). Except the particular case when both the constitutive law (8.63) and the frictionless condition (8.77) are used, such a mathematical model is evolutionary, and, therefore, there is a need to include appropriate initial conditions for the displacement field or the displacement and stress fields. The wellposedness of such models, selected by taking into account their relevance from physical or mathematical point of view, is presented in Chap. 10.

8.3.3 Two One-Dimensional Examples In this section, we consider two one-dimensional examples of an elastic or viscoelastic rods in quasistatic contact with an obstacle, the so-called foundation. We choose these examples since they allow us to compute the solution when two different constitutive laws are used and to compare the corresponding solutions, without the complications that arise in two or three dimensions. As in Sect. 8.2.3, for simplicity, we do not indicate the units associated to the data and unknowns. So, we consider a rod of length .l = 1 that is rigidly attached at .x = 0 and may come in contact with a foundation at .x = 1, under the action of a time-dependent body force of density f t. In the reference configuration, there is a gap .g > 0 between the end .x = 1 of the rod and a foundation. In the notation above, we have .Ω = (0, 1), .Γ1 = {0}, .Γ2 = ∅, .Γ3 = {1}. The setting is depicted in Fig. 8.10. We denote by .u = u(x, t) the displacement field, and then the linearized strain field is given by .ε(u) = u , where, here and below, the prime denotes the derivative with respect to .x ∈ [0, 1]. As usual, the stress field will be denoted by .σ . For the sake of simplicity, we assume that .f ∈ R does not depend on the spatial variable and, moreover, .f > 0. Then, the rod will lengthen in time, and its end will touch the foundation at a moment .tc , characterized by condition .u(1, tc ) = g. Before the contact, the rod’s end .x = 1 is traction free and, therefore, on .[0, tc ] the evolution of the mechanical state of the rod is described by a pure displacement–traction boundary value problem.

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8 Mathematical Modeling in Contact Mechanics

O

l =1

O

ft

x

g obstacle

Fig. 8.10 A rod in potential contact with an obstacle

We model the rod with both a viscoelastic constitutive law with long memory and an elastic constitutive law. Based on these ingredients, the time-dependent problem that describes the deformation of the elastic rod before the contact is the following. Problem .Pe1d . Find a time moment .tc > 0, a displacement field .u : [0, 1]×[0, tc ] → R, and a stress field .σ : [0, 1] × [0, tc ] → R, such that .

σ (x, t) = E u (x, t)

∀ (x, t) ∈ [0, 1] × [0, tc ], .

(8.79)

σ  (x, t) + f t = 0

∀ (x, t) ∈ [0, 1] × [0, tc ], .

(8.80)

u(0, t) = 0

∀ t ∈ [0, tc ], .

(8.81)

σ (1, t) = 0

∀ t ∈ [0, tc ], .

(8.82)

u(1, tc ) = g.

(8.83)

The time-dependent problem that describes the deformation of the viscoelastic rod before the contact is the following. 1d . Find a time moment .t > 0, a displacement field .u : [0, 1]×[0, t ] → Problem .Pve c c R, and a stress field .σ : [0, 1] × [0, tc ] → R, such that 



t

σ (x,. t) = E u (x, t) + k

u (x, s) ds

∀ (x, t) ∈ [0, 1] × [0, tc ], .

(8.84)

0

σ  (x, t) + f t = 0

∀ (x, t) ∈ [0, 1] × [0, tc ], .

(8.85)

u(0, t) = 0

∀ t ∈ [0, tc ], .

(8.86)

σ (1, t) = 0

∀ t ∈ [0, tc ], .

(8.87)

u(1, tc ) = g.

(8.88)

8.3 Modeling of Quasistatic Contact Problems

297

Note that equation (8.79) represents the elastic constitutive law of the rod in which E denotes the Young modulus of the material. In contrast, equation (8.84) represents the viscoelastic constitutive law of the rod in which E is the Young modulus and k is a relaxation coefficient, assumed to be positive. Our aims in what follows are: 1d (a) To find the solutions of problems .Pe1d and .Pve 1d as (b) To compare these solutions and to study the behavior of the solution of .Pve .k → 0 (c) To provide the corresponding mechanical interpretations

In order to accomplish these aims, we start with some direct calculations: 1d . We use the equilibrium equation (8.80) (a) Solution of problems .Pe1d and .Pve and the traction boundary condition (8.82) to deduce that

σ (x, t) = f t (1 − x)

∀ (x, t) ∈ [0, 1] × [0, tc ].

.

(8.89)

Then, using the constitutive law (8.79) and the displacement boundary condition (8.81), we find that u(x, t) =

.

x2  ft x− 2 E

∀ (x, t) ∈ [0, 1] × [0, tc ].

It follows from here that .u(1, t) = tion (8.83) implies that tc =

.

ft 2E

(8.90)

and, therefore, the contact condi-

2Eg . f

(8.91)

We now gather equalities (8.89)–(8.91) to conclude that the solution of Problem Pe1d , denoted in what follows by .(σe , ue , tce ), is given by

.

σe (x, t) = f t (1 − x),

.

ue (x, t) =

x2  ft , x− 2 E

tce =

2Eg , f

(8.92)

for all .(x, t) ∈ [0, 1] × [0, tce ]. 1d . To this end, we use the Next, we move to the solution of Problem .Pve equilibrium equation (8.85) and the traction boundary condition (8.87) to deduce that the stress field .σ is given by (8.89). Then, differentiating the constitutive law (8.84) with respect to t and, using notation .ε = u , it follows that

σ˙ (x, t) = E ε˙ (x, t) + kε(x, t)

.

∀ (x, t) ∈ [0, 1] × [0, tc ].

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8 Mathematical Modeling in Contact Mechanics

So, in view of (8.89), we have k f (1 − x) ε(x, t) = E E

ε˙ (x, t) +

.

∀ (x, t) ∈ [0, 1] × [0, tc ].

(8.93)

We now take .t = 0 in (8.84) to see that 1 σ (x, 0) E

ε(x, 0) =

.

∀ x ∈ [0, 1],

(8.94)

and, on the other hand, (8.89) guarantees that σ (x, 0) = 0

∀ x ∈ [0, 1].

.

(8.95)

We now integrate the differential equation (8.93) with respect to t, and then we use the initial condition (8.94) and equality (8.95) to obtain that ε(x, t) =

.

 k f (1 − x) 1 − e− E t k

∀ (x, t) ∈ [0, 1] × [0, tc ].

(8.96)

Finally, we integrate equation (8.96) with respect to x and use the displacement boundary condition (8.86) to deduce that u(x, t) =

.

 k x 2  f 1 − e− E t x− 2 k

It follows from here that .u(1, t) =

f 2k

k

∀ (x, t) ∈ [0, 1] × [0, tc ].

 k 1 − e− E t and, therefore, (8.88) yields

e − E tc =

.

(8.97)

f − 2kg . f

This equation has a solution if and only if .f > 2kg. Moreover, under this assumption, we have tc = ln

.

E f k . f − 2kg

(8.98)

We now gather equalities (8.89), (8.97), and (8.98) to conclude that the solution 1d , denoted in what follows by .(σ , u , t ve ), is given by of Problem .Pve ve ve c .

σve (x, t) = f t (1 − x), . uve (x, t) =

 k x 2  f x− 1 − e− E t , . k 2

(8.99) (8.100)

8.3 Modeling of Quasistatic Contact Problems

tcve = ln

299

E f k , f − 2kg

(8.101)

for all .(x, t) ∈ [0, 1] × [0, tcve ]. (b) Comparison of the solutions and convergence results. We use the elementary equality .1 + y < ey , valid for all .y > 0, to see that k

1 − e− E t
tce .

(8.105)

.

Indeed, arguing by contradiction, if .tcve ≤ tce , then, taking .t = tce in (8.104)) and using equalities .ue (1, tce ) = uve (1, tcve ) = g, we deduce that uve (1, tce ) < uve (1, tcve ).

(8.106)

.

k

f (1 − e− E t ) is increasing for .t ∈ Now, since the function .t →  uve (1, t) = 2k ve e (0, +∞), assumption .tc ≤ tc implies that .uve (1, tcve ) ≤ uve (1, tce ), which contradicts (8.106).

Moreover, an elementary calculus based on (8.100), (8.92), (8.101), and (8.91) shows that .

lim uve (x, t) = ue (x, t)

k→0

lim tcve = tce .

k→0

∀ (x, t) ∈ [0, 1] × [0, tcve ], .

(8.107) (8.108)

300

8 Mathematical Modeling in Contact Mechanics

(c) Mechanical interpretation. The previous results lead us to the following mechanical interpretation: (i) Under the action of the time-dependent body force of density f t, the elastic rod arrives in contact with the foundation, whatever the initial gap g might be. In contrast, under the action of the same force, the viscoelastic rod arrives in contact with the foundation if and only if inequality .f > 2kg holds. In fact, this restriction represents a smallness condition for the gap g or, alternatively, for the relaxation coefficient k. It shows that the contact arises if the foundation is sufficiently close to the end .x = 1 of the viscoelastic rod or, alternatively, if the relaxation coefficient is small enough. (ii) If condition .f > 2kg is satisfied, then inequality (8.105) shows that the elastic rod hits the foundation before the viscoelastic rod. Moreover, inequality (8.102) shows that at each time moment .t ≤ tce the displacement field of the viscoelastic rod is smaller than the displacement field of the elastic rod, at each point .x ∈ (0, 1]. In addition, inequality (8.103) shows that at each time moment .t ≤ tce the strain field on the viscoelastic rod is smaller than the strain field of the elastic rod, at each point .x ∈ (0, 1]. This behavior is due to the relaxation coefficient k, assumed to be positive. Due to this coefficient, an internal force pulls the rod in the opposite sense of the applied body force and, as a result, the deformation becomes smaller. (iii) The stress field in the elastic and viscoelastic rods is the same, as it results from equalities (8.92) and (8.99). Moreover, the convergence result (8.107) shows that the solution of the elastic rod can be approached by the solution of the viscoelastic rod, for a small relaxation coefficient k. The same comment holds for the contact time .tce , which can be approached by the contact time .tcve for k small, as it results from equality (8.108).

Chapter 9

Static Contact Problems

In this chapter, we study the well-posedness of several static mathematical models of contact. For each model, we introduce a classical formulation that gathers the corresponding equations and boundary conditions, and then we list the assumptions on the data and derive a variational formulation, which is either in a form of a variational or hemivariational inequality, an inclusion, or a minimization problem. Next, we use the results presented in Part II of the book to prove the well-posedness of each problem, with appropriate Tykhonov triples. As a consequence, besides the unique solvability of the problems, we obtain various convergence results. This allows us to deduce the continuous dependence of the solution with respect to the data as well as to establish the link between various models of contact. For part of the models, we also consider optimal control problems for which we prove the existence of optimal pairs and provide the corresponding mechanical interpretations. Everywhere in this chapter, we refer to the physical setting described in Sect. 8.1, and we use the function spaces introduced there.

9.1 A Contact Problem with Unilateral Constraints In this section, we consider a mathematical model that describes the frictional contact with a rigid obstacle covered by an elastic layer. We apply the abstract results presented in Sect. 4.1 to deduce the unique solvability of the model as well as various convergence results.

9.1.1 The Model The classical formulation of the problem is the following. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Sofonea, Well-Posed Nonlinear Problems, Advances in Mechanics and Mathematics 50, https://doi.org/10.1007/978-3-031-41416-9_9

301

302

9 Static Contact Problems

Problem .P. Find a displacement field .u : Ω → Rd and a stress field .σ : Ω → Sd such that σ = Fε(u)

in Ω, .

(9.1)

Div σ + f 0 = 0

in Ω, .

(9.2)

u=0

on Γ1 , .

(9.3)

σν = f 2

on Γ2 , .

(9.4)

on Γ3 , .

(9.5)

on Γ3 .

(9.6)

.

uν ≤ g,

⎫ σν + p(uν ) ≤ 0, ⎬

(uν − g)(σν + p(uν )) = 0 ⎭ σ τ  ≤ μ p(uν ), −σ τ = μ p(uν ) uuττ 

⎫ ⎬ if uτ = 0 ⎭

We now provide a description of the equations and boundary conditions in Problem .P. First, Eq. (9.1) represents the elastic constitutive law of the material in which .F is assumed to be a nonlinear operator. Equation (9.2) is the equilibrium equation. Conditions (9.3) and (9.4) represent the displacement and traction boundary conditions, respectively. Condition (9.5) represents the normal compliance condition with unilateral constraint (see (8.39)). Here, .g > 0 is a given bound that limits the normal displacement and p is a given positive function that will be described below. As explained in Sect. 8.2.2, this condition describes the contact with a rigid body covered by a layer of deformable material with thickness g. Condition (9.6) represents a static version of Coulomb’s law of dry friction in which .μ denotes the coefficient of friction and .μp(uν ) is the friction bound. The coupling of boundary conditions (9.5) and (9.6) describes a contact with normal compliance, as far as the normal displacement satisfies the condition .uν < g, associated to the classical Coulomb’s law of dry friction. When .uν = g, the contact follows a Signorini-type condition and is associated to the Tresca friction law with the friction bound .μp(g). It follows from here that conditions (9.5), (9.6) describe a natural transition from the Coulomb law of dry friction (which is valid as far as .0 ≤ uν < g) to the Tresca law (which is valid when .uν = g). In the study of the mechanical problem (9.1)–(9.6), we assume that the elasticity operator .F satisfies condition (8.20), and the normal compliance function p satisfies the following conditions:

9.1 A Contact Problem with Unilateral Constraints

⎧ (a) p : Γ3 × R → R+ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (b) There exists Lp > 0 such that ⎪ ⎪ ⎪ ⎪ |p(x, r1 ) − p(x, r2 )| ≤ Lp |r1 − r2 | ⎪ ⎪ ⎪ ⎪ ∀ r1 , r2 ∈ R, a.e. x ∈ Γ3 . ⎪ ⎪ ⎪ ⎨ . (c) (p(x, r1 ) − p(x, r2 )) (r1 − r2 ) ≥ 0 ⎪ ⎪ ⎪ ∀ r1 , r2 ∈ R, a.e. x ∈ Γ3 . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (d) The mapping x → p(x, r) is measurable on Γ3 , ⎪ ⎪ ⎪ ⎪ for any r ∈ R. ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (e) p(x, r) = 0 for all r ≤ 0, a.e. x ∈ Γ3 .

303

(9.7)

The densities of body forces and surface tractions have the regularity f 0 ∈ L2 (Ω)d ,

.

f 2 ∈ L2 (Γ2 )d ,

(9.8)

μ(x) ≥ 0 a.e. x ∈ Γ3 ,

(9.9)

the coefficient of friction is such that μ ∈ L∞ (Γ3 ),

.

and recall that g > 0.

(9.10)

c02 Lp μL∞ (Γ3 ) < mF ,

(9.11)

.

Moreover, we assume that .

where .c0 , .mF , and .Lp are the constants that appear in (8.14), (8.20)(c), and (9.7)(b), respectively. Note that inequality (9.11) could be interpreted as a smallness condition on the coefficient of friction .μ. We now derive a variational formulation of Problem .P and, to this end, we consider the set K defined by

K = v ∈ V : vν ≤ g a.e. on Γ3 .

.

(9.12)

Assume that .u and .σ are sufficiently regular functions that satisfy (9.1)–(9.6). Then, using (9.5) and (9.12), it follows that u ∈ K.

.

(9.13)

Let .v ∈ K. We use Green’s formula (8.16) and equalities (9.2)–(9.4) to see that

304

9 Static Contact Problems



 σ · (ε(v) − ε(u)) dx =

.

Ω

Ω



f 0 · (v − u) dx

(9.14)



+ Γ2

f 2 · (v − u) da +

σ ν · (v − u) da. Γ3

Moreover, using the boundary conditions (9.5) and (9.6), it is easy to find that σν (vν − uν ) ≥ p(uν )(uν − vν )

.

a.e. on Γ3 ,

σ τ (v τ − uτ ) ≥ μ p(uν )(uτ  − v τ )

a.e. on Γ3 .

Therefore, since σ ν · (v − u) = σν (vν − uν ) + σ τ · (v τ − uτ )

.

a.e. on Γ3 ,

we deduce that  . σ ν · (v − u) da

(9.15)

Γ3



 p(uν )(uν − vν ) da +

≥

μ p(uν )(uτ  − v τ ) da.

Γ3

Γ3

Next, we combine equality (9.14) with inequality (9.15), and then we use the constitutive law (9.1) and the regularity (9.13). As a result, we find that  .

u ∈ K,

Fε(u) · (ε(v) − ε(u)) dx

(9.16)

Ω





+

p(uν )(vν − uν ) da + Γ3

μ p(uν )(v τ  − v τ ) da Γ3



 ≥ Ω

f 0 · (v − u) dx +

Γ2

f 2 · (v − u) da

∀ v ∈ K.

We now consider the operator .A : V → V , the function .ϕ : V × V → R, and the element .f ∈ V defined by 



(Au, .v)V =

Fε(u) · ε(v) dx + Ω

p(uν )vν da

∀ u, v ∈ V , .

(9.17)

Γ3

 μ p(uν )v τ  da

ϕ(u, v) = Γ3

∀ u, v ∈ V , .

(9.18)

9.1 A Contact Problem with Unilateral Constraints



305



(f , v)V = Ω

f 0 · v dx +

Γ2

f 2 · v da

∀ v ∈ V.

(9.19)

Then, using (9.16)–(9.19), we find the following variational formulation of Problem .P. Problem .PV . Find a displacement field .u such that u ∈ K, (Au, v − u)V + ϕ(u, v) − ϕ(u, u) ≥ (f , v − u)V ∀ v ∈ K.

.

(9.20)

Note that Problem .PV is formulated in terms of the displacement field. Once the displacement field is known, the stress field can be easily obtained by using the constitutive law (9.1). A couple .(u, σ ) that satisfies (9.1) and (9.20) is called a weak solution to the contact problem .P.

9.1.2 A Well-Posedness Result We now illustrate our abstract results of Sect. 4.1 in the study of the variational inequality (9.20). To this end, we consider a function q and an element . g ∈ R+ ∪ {+∞} that satisfy the following conditions: ⎧ (a) q : Γ3 × R → R+ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (b) There exists Lq > 0 such that ⎪ ⎪ ⎪ ⎪ |q(x, r1 ) − q(x, r2 )| ≤ Lq |r1 − r2 | ⎪ ⎪ ⎪ ⎪ ∀ r1 , r2 ∈ R, a.e. x ∈ Γ3 . ⎪ ⎪ ⎪ ⎨ (c) (q(x, r1 ) − q(x, r2 )) (r1 − r2 ) ≥ 0 . ⎪ ⎪ ∀ r1 , r2 ∈ R, a.e. x ∈ Γ3 . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (d) The mapping x → q(x, r) is measurable on Γ3 , ⎪ ⎪ ⎪ ⎪ for any r ∈ R. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (e) q(x, r) = 0 if and only if r ≤ 0, a.e. x ∈ Γ3 . g ≥ g.

(9.21)

(9.22)

.

and the operator .G : V → V defined by equalities Next, we introduce the set .K .

= K



g on Γ3 , . v ∈ V : vν ≤

(9.23)

 (Gu, v)V =

q(uν − g)vν da Γ3

∀ u, v ∈ V .

(9.24)

306

9 Static Contact Problems

Moreover, for each .θ = (λ, ε) ∈ (0, +∞) × [0, +∞), we consider the variational inequality u. ∈ K,

(Au, v − u)V +

1 (Gu, v − u)V + ϕ(u, v) − ϕ(u, u) λ

+εv − uV ≥ (f , v − u)V

(9.25)

∀ v ∈ K.

With the previous ingredients, we consider the triple .T = (I, Ω, C), defined as follows: .

I = (0, +∞) × [0, +∞), .

Ω(θ ) = u such that (9.25) holds ∀ θ = (λ, ε) ∈ I, .

C = {θn } ⊂ I : θn = (λn , εn ) ∀ n ∈ N, λn → 0, εn → 0 .

(9.26) (9.27) (9.28)

Remark 9.1 Note that the arguments in Theorem 9.1 below allow us to use Proposition 4.1 in order to see that, under the assumptions above, inequality (9.20) has a unique solution .u ∈ K. This implies that .u ∈ Ω(θ ) for each .θ ∈ I . Therefore, .Ω(θ ) = ∅ for each .θ ∈ I , which shows that .T is a Tykhonov triple in the sense of Definition 2.1(a). Our main result in this section is the following. Theorem 9.1 Assume (8.20), (9.7)–(9.11), (9.21), and (9.22). Then, Problem .PV is well-posed with respect to the Tykhonov triple (9.26)–(9.28). Proof We use Theorem 4.1 in the Hilbert space .X = V . To this end, we have to check the validity of conditions (1.47)–(1.51), (4.2), (4.3), (4.9)–(4.12), and this is what we shall do in what follows. Obviously, the set K is a convex nonempty subset of V . Moreover, using the properties of the trace map, we deduce that K is closed and, therefore, (1.47) holds. Next, we use assumptions (8.20), (9.7) and the trace inequality (8.14) to see that .

(Au − Av, u − v)V ≥ mF u − v2V , Au − AvV ≤ (LF + c02 Lp ) u − vV

for all .u, v ∈ V . Therefore, condition (1.48) holds with .mA = mF . Condition (1.49)(a) is obviously satisfied. On the other hand, an elementary calculation based on the definition (9.18) and assumptions (9.7), (9.9) yields

9.1 A Contact Problem with Unilateral Constraints .

307

ϕ(u1 , v 2 ) − ϕ(u1 , v 1 ) + ϕ(u2 , v 1 ) − ϕ(u2 , v 2 )  ≤ Lp μL∞ (Γ3 ) u1 − u2 v 1 − v 2  da Γ3

for all .u1 , u2 , v 1 , v 2 ∈ V . Hence, the trace inequality (8.14) shows that condition (1.49)(b) holds with .αϕ = c02 Lp μL∞ (Γ3 ) . Therefore, assumption (9.11) implies that condition (1.50) holds, too. Recall also that condition (1.51) is satisfied by the definition (9.19). Next, assumptions (9.21) and (9.22) guarantee that conditions (4.2), (4.3), (4.9) are satisfied. and .v ∈ K. Then, using the properties (9.21) of the function Assume that .u ∈ K q and assumption (9.22), it follows that q(uν − g)(vν − uν ) ≤ 0 a.e. on Γ3

.

and, therefore, definition (9.24) implies that .(Gu, v − u)V ≤ 0. We conclude from here that condition (4.10) is satisfied. = V . Let .u ∈ V . If Assume now that . g = +∞ and note that in this case .K .Gu = 0V , then  . q(uν − g)uν da = 0 Γ3

and, since .q(uν − g)uν ≥ 0 a.e. on .Γ3 , using the implication  f da = 0,

.

f ≥ 0 a.e. on ω

⇒

f = 0 a.e. on ω,

(9.29)

ω

we deduce that .q(uν − g)uν = 0 a.e. on .Γ3 . We now use condition (9.21)(e) to deduce that .uν ≤ g a.e. on .Γ3 , which shows that .u ∈ K. On the other hand, if and .(Gu, v − u)V = 0 for any .v ∈ K, we have . g < +∞, .u ∈ K  q(uν − g)(uν − vν ) da = 0

.

∀ v ∈ K.

Γ3

Now, since .

q(uν − g)(uν − vν ) = (q(uν − g) − q(vν − g))((uν − g) − (vν − g)) ≥ 0

a.e. on .Γ3 , using the implication (9.29), we deduce that .q(uν − g)(uν − vν ) = 0 a.e. on .Γ3 , for any .v ∈ K. We now use condition (9.21)(e) to find that .uν ≤ g. We conclude from above that condition (4.11) is satisfied.

308

9 Static Contact Problems

Finally, using assumptions (9.7)(b), (e) and (9.9), it is easy to see that  .

ϕ(u, v 1 ) − ϕ(u, v 2 ) ≤

μp(uν )v 1 − v 2  da Γ3

 ≤ Lp μL∞ (Γ3 )

uv 1 − v 2  da Γ3

for all .u, v 1 , v 2 ∈ V . Therefore, the trace inequality (8.14) shows that condition (4.12) is satisfied with .cϕ (u) = c02 Lp μL∞ (Γ3 ) uV . It follows from above that we are in a position to use Theorem 4.1 to conclude the proof.  

9.1.3 Convergence Results The well-posedness result in Theorem 9.1 provides various information in the study of Problem .PV and, implicitly, in the study of the contact Problem .P. First, it shows the unique solvability of Problem .PV , which implies the existence of a unique weak solution to Problem .P. On the other hand, it provides a number of convergence results to the solution .u that could be naturally divided into the following two classes: (a) Convergence results for penalty methods (b) Continuous dependence results with respect to the data Besides the mathematical interest in these convergence results, they lead to an interesting interpretation and, therefore, they are important from the mechanical point of view. We present in what follows these convergences as a consequence of Corollaries 4.1–4.3 in Sect. 4.1.2, applied with .X = V : g = +∞ and, therefore, (a) Penalty methods. Our first convergence result is when . = V . For each .n ∈ N, let .λn > 0 and consider the following problem. .K n . Find an element .u ∈ V such that Problem .P1V n

(Au . n , v − un )V +

1 (Gun , v − un )V + ϕ(un , v) − ϕ(un , un ) λn

≥ (f , v − un )V

(9.30)

∀ v ∈ V.

The following result represents a direct consequence of Corollary 4.1. Corollary 9.1 Assume (8.20), (9.7)–(9.11), (9.21) and let .{λn } ⊂ (0, +∞). Then n has a unique solution, for each .n ∈ N. In addition, .u → u in V as Problem .P1V n .λn → 0.

9.1 A Contact Problem with Unilateral Constraints

309

Fig. 9.1 Two penalty contact conditions: (a) condition (9.31); (b) condition (9.33)

Fig. 9.2 Physical setting: (a) contact with a deformable obstacle covered by an elastic layer; (b) contact with a rigid obstacle covered by two elastic layers

A careful analysis based on arguments similar to those used on pages 303–305 shows that (9.30) represents the variational formulation of an elastic contact problem of the form .P in which the boundary condition (9.5) is penalized, i.e., is replaced by the condition .

− σν = p(uν ) +

1 q(uν − g) λn

on Γ3 .

(9.31)

The graphic representation of condition (9.31) is given in Fig. 9.1a. It represents a contact condition with normal compliance and describes the contact with a foundation made of a deformable obstacle covered by an elastic layer of thickness g, as shown in Fig. 9.2a. The penetration into the deformable obstacle is allowed but penalized. The coefficient . λ1n in (9.31) can be interpreted as a stiffness coefficient of the deformable obstacle. Corollary 9.1 guarantees the convergence of the sequence of solutions of the n to the solution of Problem .P as the penalty parameter penalty problems .P1V V converges to zero. Its mechanical interpretation is the following: the solution of the frictional contact problem between an elastic body and a rigid obstacle covered by an elastic layer can be approached by the solution of the frictional contact problem between the elastic body and a deformable obstacle covered by an elastic layer, as the stiffness of the deformable obstacle converges to infinity.

310

9 Static Contact Problems

= V . For Our second convergence result is when . g < +∞ and, therefore, .K each .n ∈ N, let .λn > 0 and consider the following problem. n . Find an element .u ∈ K such that Problem .P2V n

(Au . n , v − un )V +

1 (Gun , v − un )V + ϕ(un , v) − ϕ(un , un ) λn

≥ (f , v − un )V

(9.32)

∀ v ∈ K.

The following result represents a direct consequence of Corollary 4.2. Corollary 9.2 Assume (8.20), (9.7)–(9.11), (9.21), (9.22) and let .{λn } ⊂ (0, +∞). n has a unique solution, for each .n ∈ N. In addition, .u → u in Then Problem .P2V n V as .λn → 0. A careful analysis shows that (9.32) represents the variational formulation of an elastic contact problem of the form .P in which the boundary condition (9.5) is partially penalized, i.e., is replaced by the condition ⎫ 1 ⎪ q(uν − g) ≤ 0, ⎪ ⎬ λn ⎪ 1 ⎭ (uν − g )(σν + p(uν ) + q(uν − g)) = 0 ⎪ λn

uν ≤ g, .

σν + p(uν ) +

on Γ3 .

(9.33)

This condition represents a contact condition with normal compliance and unilateral constraint, as depicted in Fig. 9.1b. It describes the contact with a foundation made of a rigid obstacle covered by a first elastic layer of thickness . g − g and a second elastic layer of thickness g. The physical setting is depicted in Fig. 9.2b. Here . λ1n can be interpreted as a stiffness coefficient of the first layer, i.e., the layer of thickness . g − g. Indeed, the penetration of this layer is allowed but penalized. Corollary 9.2 guarantees the convergence of the sequence of solutions of the n to the solution of Problem .P , as the penalty parameter penalty problems .P2V V converges to zero. Its mechanical interpretation is similar to that of Corollary 9.1. The difference arises in the fact that now the stiffness of the layer of thickness . g−g converges to infinity. (b) A continuous dependence result. We restrict ourselves to present a continuous dependence result of the solution with respect to the data .f 0 and .f 2 . Assume that .f 0n and .f 2n are functions with regularity (9.8), that is, f 0n ∈ L2 (Ω)d ,

.

f 2n ∈ L2 (Γ2 )d ,

(9.34)

for each .n ∈ N. Moreover, for each .n ∈ N, denote by .f n ∈ V the element given by

9.2 A Contact Problem with Bilateral Constraints

 .

(f n , v)V =

Ω

311

 f 0n · v dx +

Γ2

f 2n · v da

∀v ∈ V

(9.35)

and consider the following problem. n . Find an element .u such that Problem .P3V n

u. n ∈ K,

(Aun , v − un )V + ϕ(un , v) − ϕ(un , un ) ≥ (f n , v − un )X

∀ v ∈ K.

We now use Corollary 4.3 to deduce the following existence, uniqueness, and convergence result. n has a Corollary 9.3 Assume (8.20), (9.7)–(9.11), and (9.34). Then Problem .P3V unique solution, for each .n ∈ N. In addition, if

f 0n → f 0

.

in L2 (Ω)d ,

f 2n → f 2

in L2 (Γ2 )d ,

(9.36)

then .un → u in V . Note that Corollary 9.3 provides a continuous dependence result of the weak solution to Problem .P with respect to the densities of body forces and surface tractions. Various other convergence results can be obtained by considering perturbations of the elasticity operator .F, the coefficient of friction .μ, and the thickness g.

9.2 A Contact Problem with Bilateral Constraints In this section, we consider a model of bilateral contact. We present three variational formulations of the model, in terms of displacement, stress, and strain fields, respectively. We prove that these formulations are pairwise dual and have a unique solution. Then, based on a Hadamard well-posedness result, we study an associated optimal control problem.

9.2.1 The Model The classical formulation of the problem is the following. Problem .P. Find a displacement field .u : Ω → Rd and a stress field .σ : Ω → Sd such that

312

9 Static Contact Problems

 σ = Fε(u) + α ε(u) − PB ε(u)

in Ω, .

(9.37)

Div σ + f 0 = 0

in Ω, .

(9.38)

u=0

on Γ1 , .

(9.39)

σν = f 2

on Γ2 , .

(9.40)

uν = 0

on Γ3 , .

(9.41)

uτ if uτ = 0 uτ 

on Γ3 .

(9.42)

.

σ τ  ≤ Fb (θ ),

σ τ = −Fb (θ )

We now provide a short description of the equations and boundary conditions in Problem .P. First, equation (9.37) represents the elastic constitutive law of the material (8.21). Equation (9.38) is the equilibrium equation, while conditions (9.39), (9.40) represent the displacement and traction boundary conditions, respectively. Equality (9.41) shows that there is no separation between the body and the obstacle, i.e., the contact is bilateral. The condition (9.42) represents the static version of the Tresca friction law, in which .Fb (θ ) ≥ 0 denotes the friction bound, assumed to depend on a parameter .θ , say the temperature. In the study of the contact problem (9.37)–(9.42), we assume that the elasticity operator .F and the set B satisfy the conditions (8.20) and (8.23), respectively. Moreover, we assume that the elasticity coefficient, the densities of body forces and tractions, the parameter .θ , and the friction bound are such that: .

α ≥ 0..

(9.43)

f 0 ∈ L2 (Ω)d ..

(9.44)

f 2 ∈ L2 (Γ2 )d ..

(9.45)

θ ∈ L2 (Γ3 )..

(9.46)

⎧ ⎪ (a) Fb : Γ3 × R → R+ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (b) There exists Lb > 0 such that ⎪ ⎪ ⎪ ⎪ |Fb (x, θ1 ) − Fb (x, θ2 )| ≤ Lb |θ1 − θ2 | ⎪ ⎨ for all θ1 , θ2 ∈ R, a.e. x ∈ Γ3 . ⎪ ⎪ ⎪ ⎪ (c) The mapping x → Fb (x, θ ) is measurable on Γ3 , ⎪ ⎪ ⎪ ⎪ for all θ ∈ R. ⎪ ⎪ ⎪ ⎪ ⎩ (d) Fb (x, 0) = 0 a.e. x ∈ Γ3 .

(9.47)

9.2 A Contact Problem with Bilateral Constraints

313

Below in this section we keep assumptions (8.20), (8.23), and (9.43)–(9.47), even if we do not mention this explicitly. We also note that, since the contact is bilateral, the displacement field will be sought in the space .V1 (see (8.15)).

9.2.2 Dual Variational Formulations In order to deduce variational formulations of Problem .P, we introduce the : Q → Q defined by the equalities operators .A : V1 → V1 and .A  (Au, v). V =

Fε(u) · ε(v) dx



(9.48)

Ω

+α

 ε(u) − PB ε(u) · ε(v) dx

∀ u, v ∈ V1 ,

Ω



τ. )Q = (Aω,



 ω − PB ω · τ dx

Fω · τ dx + α Ω

∀ ω, τ ∈ Q.

(9.49)

Ω

: ε(V1 ) → Q, i.e., We also denote by .Q1 the range of the operator .A

Q1 = Aε(v) : v ∈ V1 .

.

(9.50)

: Q → Q Using the assumptions on the data .F, .α, and B, it is easy to see that .A is a strongly monotone Lipschitz continuous operator. Therefore, arguments similar to those used in the proof of Theorem 8.1 show that .Q1 is a closed subspace of −1 : Q → Q the inverse of the operator .A, it follows that Q, and denoting by .A −1 σ ∈ ε(V1 ) for all .σ ∈ Q1 . .A Next, we consider the function .j (θ, ·) : V1 → R, the element .f ∈ V1 , and the set .Σ(θ, f ) defined by the equalities  j (θ,. v) =

Fb (θ )v τ  da

∀ v ∈ V1 , .

(9.51)

Γ3





(f , v)V = Ω

f 0 · v dx +

Γ2

f 2 · v da

∀ v ∈ V1 , .

Σ(θ, f ) = τ ∈ Q1 : (τ , ε(v))Q + j (θ, v) ≥ (f , v)V ∀ v ∈ V1 .

(9.52)

(9.53)

Assume now that .u and .σ are sufficiently regular functions that satisfy Problem P. We use (9.41) to see that

.

314

9 Static Contact Problems

u ∈ V1 .

(9.54)

ω = ε(u).

(9.55)

.

Denote .

Then, using the constitutive law (9.37) and the definitions (9.48), (9.49), (9.55), we find that .

(σ , ε(v) − ε(u))Q = (Au, v − u)V σ = Aω,

∀ v ∈ V1 , .

−1 σ , ω=A

(9.56) (9.57)

which imply that σ ∈ Q1 .

(9.58)

.

Let .v ∈ V1 . Then, using standard arguments, we deduce that 





σ · (ε(v) − ε(u)) dx +

.

Ω

Fb (θ )v τ  da − Γ3

 ≥ Ω



f 0 · (v − u) dx +

Fb (θ )uτ  da

(9.59)

Γ3

Γ2

f 2 · (v − u) da.

Next, we use notations (9.51) and (9.52) to obtain that (σ , ε(v) − ε(u))Q + j (θ, v) − j (θ, u) ≥ (f , v − u)V .

.

(9.60)

We now test in (9.60) with .v = 2u and .v = 0V1 to see that (σ , ε(u))Q + j (θ, u) = (f , u)V .

.

(9.61)

Therefore, using (9.60) and (9.61), we find that (σ , ε(v))Q + j (θ, v) ≥ (f , v)V .

.

This inequality combined with the regularity (9.58) implies that σ ∈ Σ(θ, f ).

.

To proceed, we use (9.53), (9.54), and (9.61) to see that (τ − σ , ε(u))Q ≥ 0

.

∀ τ ∈ Σ(θ, f ),

(9.62)

9.2 A Contact Problem with Bilateral Constraints

315

and using (9.55), we find that (τ − σ , ω)Q ≥ 0

.

∀ τ ∈ Σ(θ, f ).

(9.63)

We now combine (9.54), (9.60), and (9.56) to deduce the following variational formulation of Problem .P, in terms of the displacement field. Problem .P1V . Find a displacement field .u such that u ∈ V1 ,

.

(Au, v − u)V + j (θ, v) − j (θ, u) ≥ (f , v − u)V

∀ v ∈ V1 .

(9.64)

On the other hand, using (9.62), (9.63), and (9.57), we obtain the following variational formulation of Problem: .P, in terms of the stress field. Problem .P2V . Find a stress field .σ such that σ ∈ Σ(θ, f ),

.

−1 σ , τ − σ )Q ≥ 0 (A

∀ τ ∈ Σ(θ, f ).

(9.65)

Finally, using (9.62), (9.57), and (9.63), it follows that ∈ Σ(θ, f ), Aω

ω)Q ≥ 0 (τ − Aω,

.

∀ τ ∈ Σ(θ, f )

and, with notation .NΣ(θ,f ) for the outward normal cone of the set .Σ(θ, f ) ⊂ Q1 , we obtain the following variational formulation of Problem .P, in terms of the strain field. Problem .P3V . Find a strain field .ω ∈ Q1 such that .

− ω ∈ NΣ(θ,f ) (Aω).

(9.66)

Our first result in the study of these problems is the following. Theorem 9.2 Assume (8.20), (8.23), and (9.43)–(9.47). Then, problems .P1V , .P2V , and .P3V are pairwise duals of each other, in the sense of Definition 2.6. Proof We start by proving that problems .P1V and .P2V are dual with the operator .D : V1 → Q1 given by Du = Aε(u)

.

∀ u ∈ V1 .

(9.67)

on page 313, it is easy to see that D is bijective Using the properties of the operator .A and both D and its inverse .D −1 : Q1 → V1 are continuous operators. Therefore, conditions (a) and (b) in Definition 2.6 are satisfied. Next, let .u be a solution to Problem .P1V and let .σ = Du, i.e., σ = Aε(u) ∈ Q1 .

.

(9.68)

316

9 Static Contact Problems

Then, (9.49) and (9.48) imply (9.56), and, therefore, using (9.64), we deduce that (9.60) holds, for each .v ∈ V1 . Then, using the arguments on page 314, we deduce that (9.65) holds, too. Therefore, .σ is a solution to Problem .P2V . Conversely, assume now that .σ is a solution to Problem .P2V , and denote .u = −1 which shows D σ ∈ V1 . This equality and (9.67) imply that .σ = Du = Aε(u), that −1 σ , ε(u) = A

(9.69)

.

and, using (9.65), we get σ ∈ Σ(θ, f ),

(τ − σ , ε(u))Q ≥ 0

.

∀ τ ∈ Σ(θ, f ).

(9.70)

Recall now that Proposition 1.19 implies that the function .j (θ, ·) : V1 → R is subdifferentiable on X, which allows us to consider an element .ξ ∈ ∂c j (θ, u). Let .τ 0 = ε(f − ξ ) ∈ Q1 . Then using (8.8) and (1.44), it is easy to see that (τ 0 , ε(v) − ε(u))Q + j (θ, v) − j (θ, u) ≥ (f , v − u)V

.

∀ v ∈ V1 .

(9.71)

We now test in (9.71) with .v = 2u and .v = 0V1 to deduce that (τ 0 , ε(u))Q + j (θ, u) = (f , u)V .

.

(9.72)

Therefore, using (9.71) and (9.72), we find that (τ 0 , ε(v))Q + j (θ, v) ≥ (f , v)V

.

∀ v ∈ V1 ,

which implies that .τ 0 ∈ Σ(θ, f ). This regularity allows us to test with .τ = τ 0 in (9.70) in order to see that (τ 0 , ε(u))Q + j (θ, u) ≥ (σ , ε(u))Q + j (θ, u)

.

and, using (9.72), we deduce that (f , u)V ≥ (σ , ε(u))Q + j (θ, u).

.

(9.73)

On the other hand, since .σ ∈ Σ(θ, f ), we find that (σ , ε(v))Q + j (θ, v) ≥ (f , v)V

.

∀ v ∈ V1 ,

(9.74)

∀ v ∈ V1 .

(9.75)

which, in particular, implies that (σ , ε(u))Q + j (θ, u) ≥ (f , u)V

.

9.2 A Contact Problem with Bilateral Constraints

317

We now combine inequalities (9.73) and (9.75) to obtain that (σ , ε(u))Q + j (θ, u) = (f , u)V

.

∀ v ∈ V1 ,

(9.76)

and then we use (9.74) and (9.76) to find that (9.60) holds, for each .v ∈ V1 . and A, Moreover, note that (9.69) and definitions (9.49), (9.48) of the operators .A respectively, imply that (9.56) holds, too. Finally, we use (9.60) and equality (9.56), valid for any .v ∈ V1 , in order to see that .u satisfies inequality (9.64), which implies that .u is a solution of Problem .P1V . We conclude from above that condition (c) in Definition 2.6 is satisfied, too. Therefore, Problems .P1V and .P2V are dual of each other. −1 : Q1 → We now prove the duality of Problems .P2V and .P3V with operator .A Q1 that, obviously, satisfies conditions (a) and (b) in Definition 2.6. This result follows from equalities (9.57) and the following equivalences: .

σ is a solution to Problem P2V −1 σ , τ − σ )Q ≥ 0 (A

⇐⇒

σ ∈ Σ(θ, f ),

⇐⇒

∈ Σ(θ, f ), Aω

⇐⇒

−ω ∈ NΣ(θ,f ) (Aω)

⇐⇒

ω is a solution to Problem P3V .

∀ τ ∈ Σ(θ, f )

(τ − σ , ω)Q ≥ 0 ∀ τ ∈ Σ(θ, f )

−1 σ and the Recall that these equivalences are obtained by using the notation .ω = A definition of the normal cone .NΣ(θ,f ) . Finally, by transitivity, it follows that Problems .P1V and .P3V are dual with operator .v → ε(v) : V1 → Q1 .   Our second result in this subsection is the following. Theorem 9.3 Assume (8.20), (8.23), and (9.43)–(9.47). Then, problems .P1V , .P2V , and .P3V have a unique solution. Moreover, the solution depends Lipschitz continuously on the data .(θ, f ) ∈ L2 (Γ3 ) × V1 . Proof It is easy to see that the operator .A : V1 → V1 is a strongly monotone Lipschitz continuous operator. Moreover, the function .j (θ, ·) : V1 → R is a continuous seminorm and, therefore, it is convex and lower semicontinuous. The existence of a unique solution to Problem .P1V is a direct consequence of Corollary 1.4. Finally, Theorem 9.2 guarantees the unique solvability of Problems V V V .P 2 and .P3 . More precisely, if .u represents the solution of Problem .P1 , then, using represents Theorem 9.2 and Definition 2.6, it follows that the function .σ = Aε(u) V the unique solution of Problem .P2 and the function .ω = ε(u) is the unique solution of Problem .P3V .

318

9 Static Contact Problems

Assume now that .(θ1 , f 1 ) ∈ L2 (Γ3 ) × V1 , .(θ2 , f 2 ) ∈ L2 (Γ3 ) × V1 and denote by .ui ∈ V1 the solution of inequality (9.64) for .θ = θi and .f = f i , .i = 1, 2. Then, for any .v ∈ V1 , we have .

(Au1 , v − u1 )V + j (θ1 , v) − j (θ1 , u1 ) ≥ (f 1 , v − u1 )V , .

(9.77)

(Au2 , v − u2 )V + j (θ2 , v) − j (θ2 , u2 ) ≥ (f 2 , v − u2 )V .

(9.78)

We take .v = u2 in (9.77), .v = u1 in (9.78), and then we add the resulting inequalities to find that .

(Au1 − Au2 , u1 − u2 )V ≤ j (θ1 , u2 ) − j (θ1 , u1 ) + j (θ2 , u1 ) − j (θ2 , u2 ) + (f 1 − f 2 , u1 − u2 )V .

Next, the strong monotonicity of A, definition (9.51), and assumption (9.47) yield .

mF u1 − u2 2V  ≤ Lb Γ3

|θ1 − θ2 |u1 − u2  da + f 1 − f 2 V u1 − u2 V .

We now use the trace inequality (8.14) to deduce that u1 − u2 V ≤

.

 1 c0 Lb θ1 − θ2 L2 (Γ3 ) + f 1 − f 2 V , mF

(9.79)

which shows that the solution .u depends Lipschitz continuously on the data .(θ, f ) ∈ L2 (Γ3 ) × V1 . The Lipschitz continuity of the solutions .σ and .ω follows now from : Q1 → Q1 and .A −1 : Q1 → Q1 are equalities (9.68) and (9.69) since, recall, .A Lipschitz continuous operators.   We end this subsection with the remark that, with the terminology introduced on pages 26 and 112, relations (9.64), (9.65) and (9.66) represent an elliptic variational inequality of the second kind, an elliptic variational inequality of the first kind, and a stationary inclusion, respectively, in which the unknowns are the displacement field, the stress field, and the strain field, correspondingly. Despite the fact that these problems have different structure, each of them can be interpreted as a variational formulation of the contact Problem .P. We conclude from here that the variational formulation of contact models is not unique and could lead to different mathematical problems that, in fact, are dual of each other. Moreover, anyone among the displacement, the stress and the strain field can be considered as main unknown, provided that an appropriate variational formulation is used. Finally, note that Theorem 9.3 provides the unique weak solvability of these problems as well as

9.2 A Contact Problem with Bilateral Constraints

319

the Lipschitz continuous dependence of the weak solution with respect to the data f and the parameter .θ .

.

9.2.3 Optimal Control In this subsection, we study an optimal control problem associated to the contact problem .P. We underline that any of the variational formulations .P1V , .P2V , and .P3V can be used to govern the control problem. For instance, the choice of the variational formulation .P3V leads to an optimal control problem governed by an inclusion for which Theorem 7.10 can be applied. Nevertheless, here we shall consider the variational formulation .P1V and, therefore, the optimal control problem we consider below is governed by an elliptic variational inequality of the second kind. Our aim below is to use Theorem 7.8 in Sect. 7.2, and, to this end, we need to accommodate the notation we used above in this current section with the notation used in Theorem 7.8. Therefore, we assume in what follows that (8.20), (8.23), and (9.43)–(9.47) hold, and we denote by Z the product space .Z = L2 (Ω)d × L2 (Γ2 )d endowed with the canonical Hilbertian structure. Moreover, we define the operator .T : Z → V1 by the equality 



(Tf, v)V =

.

Ω

f 0 · v dx +

Γ2

f 2 · v da

∀ v ∈ V1 ,

(9.80)

for all .f = (f 0 , f 2 ) ∈ Z. It is easy to see that .T : Z → V1 is a linear continuous operator. In fact, T is a linear completely continuous operator, as we shall see in the proof of Theorem 9.4 below. Next, for each .θ ∈ L2 (Γ3 ) and .f = (f 0 , f 2 ) ∈ Z, we restate Problem .P1V as follows. V . Find a displacement field .u such that Problem .Pθf

u ∈ V1 ,

.

(Au, v − u)V + j (θ, v) − j (θ, u) ≥ (Tf, v − u)V

∀ v ∈ V1 .

(9.81)

Concerning this problem, we have the following comment. V has a unique solution .u = Remark 9.2 Theorem 9.3 implies that Problem .Pθf u(θ, f ), for each .(θ, f ) ∈ L2 (Γ3 ) × Z. Moreover, (9.79) shows that there exists .C > 0 such that



u(θ1 , f1 ) − u(θ2 , f2 )V ≤ C θ1 − θ2 L2 (Γ3 ) + Tf1 − Tf2 V

.

(9.82)

for all .(θ1 , f1 ), (θ2 , f2 ) ∈ L2 (Γ3 ) × Z. We now consider a function .k : R+ → R+ that satisfies the following properties:

320

9 Static Contact Problems

⎧ (a) There exists Lk ≥ 0 such that ⎪ ⎪ ⎪ ⎪ |k(r1 ) − k(r2 )| ≤ Lk |r1 − r2 | ⎪ ⎨ .

∀ r1 , r2 ∈ R+ .

(b) There exists Mk ≥ 0 such that k(r) ≤ Mk ∀ r ∈ R+ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (c) k(r) = 0 if and only if r = 0.

(9.83)

This allows us to consider the subset .K(θ ) of Z defined by .

K(θ ) (9.84)

= f = (f 0 , f 2 ) ∈ Z : f 0 L2 (Ω)d + f 2 L2 (Γ2 )d ≤ k(θL2 (Γ3 ) ) .

V With the notation above, we define the set of admissible pairs for Problem .Pθf by the equality



Vad (θ ) = (u, f ) ∈ V1 × Z : f ∈ K(θ ), u = u(θ, f ) .

.

In other words, a pair .(u, f ) belongs to .Vad (θ ) if and only if .f ∈ K(θ ) and .u satisfies the variational inequality (9.81). Consider also an element .σ 0 such that σ 0 ∈ Q.

(9.85)

.

Then, the optimal control problem we are interested in is the following. Problem .Oθ . Given .θ ∈ L2 (Γ3 ), find .(u∗ , f ∗ ) ∈ Vad (θ ) such that ∗ ) − σ 0 Q ≤ Aε(u) Aε(u − σ 0 Q ,

.

for all .(u, f ) ∈ Vad (θ ). Our main result in this section is the following. Theorem 9.4 Assume (8.20), (8.23), and (9.43)–(9.47), (9.83), and (9.85). Then, for each .θ ∈ L2 (Γ3 ), there exists at least one solution .(u∗ , f ∗ ) to Problem .Oθ . Moreover, if for each .n ∈ N the pair .(u∗n , fn∗ ) is a solution to Problem .Oθn with 2 2 .θn ∈ L (Γ3 ) and .θn → θ in .L (Γ3 ), then there exists a subsequence of the sequence ∗ ∗ ∗ ∗ ∗ ∗ .{(un , fn )}, again denoted by .{(un , fn )}, and a solution .(u , f ) of Problem .Oθ , such that un → u

.

in V1

and

fn∗  f ∗

in Z.

(9.86)

Proof Consider the functional .L : V1 × Z → R defined by L(u, f ) = Aε(u) − σ 0 Q

.

∀ (u, f ) ∈ V1 × Z.

(9.87)

9.2 A Contact Problem with Bilateral Constraints

321

Then, it is easy to see that Problem .Oθ may be recovered from Problem .Qθ studied in Sect. 7.2.2. Therefore, we shall use Theorem 7.8 with .X = V1 , .Y = U = L2 (Γ3 ),

V = f = (f 0 , f 2 ) ∈ Z : f 0 L2 (Ω)d + f 2 L2 (Γ2 )d ≤ Mk ,

.

(9.88)

and, to this end, we check in what follows the validity of the conditions of this theorem. First, it follows from Remark 9.2 that Problem .Pθf satisfies condition (7.34) and, obviously, condition (7.37) holds, too. We now move to condition (7.36), and, to this end, we prove that the operator .T : Z → V1 is completely continuous. Indeed, assume that .fn = (f 0n , f 2n ) ∈ Z, .f = (f 0 , f 2 ) ∈ Z and .fn  f in Z. Then .

f 0n  f 0

in L2 (Ω)d , .

(9.89)

f 2n  f 2

in L2 (Γ2 )d ,

(9.90)

and, therefore, the definition (9.80) shows that .T fn  Tf in .V1 . We now use the compactness of the embedding .V1 ⊂ L2 (Ω)d , the compactness of the trace map (defined on .V1 with valued in .L2 (Γ2 )d ) and Proposition 1.3 on page 6 to deduce that .

T fn → Tf

in L2 (Ω)d , .

(9.91)

T fn → Tf

in L2 (Γ2 )d .

(9.92)

Then, using again definition (9.80), we find that  .

T fn − Tf 2V

= Ω

(f 0n − f 0 ) · (T fn − Tf ) dx

(9.93)

 + Γ2

(f 2n − f 2 ) · (T fn − Tf ) da

and, using the convergences (9.89)–(9.92), we deduce that the two integrals above converge to zero. We conclude from (9.93) that .Tfn → Tf in .V1 , which shows that T is a completely continuous operator. Assume now that .θn → θ in .L2 (Γ3 ) and .fn  f in Z. Then, it follows from above that .Tfn → Tf in .V1 , and using inequality (9.82), we deduce that .u(θn , fn ) → u(θ, f ) in .V1 . This convergence shows that condition (7.36) is satisfied. Next, assume that .θ ∈ L2 (Γ3 ) is fixed and .{θn } ⊂ L2 (Γ3 ) is a sequence such that 2 .θn → θ in .L (Γ3 ). Let .f = (f 0 , f 2 ) ∈ K(θ ), and, for each .n ∈ N, define

322

9 Static Contact Problems

⎧ ⎨ k(θn L2 (Γ3 ) ) (f , f ) 0 2 k(θL2 (Γ ) ) .fn = 3 ⎩ f

if θ = 0L2 (Γ3 ) , if θ = 0L2 (Γ3 ) .

Then, using assumption (9.83), it follows that .fn ∈ K(θn ), and moreover,     k(θn L2 (Γ3 ) ) − k(θ L2 (Γ3 ) ) ≤ Lk θn − θ L2 (Γ3 ) .

.

This inequality and the convergence .θn → θ in .L2 (Γ3 ) imply that k(θn L2 (Γ3 ) ) → k(θ L2 (Γ3 ) )

.

(9.94)

and, moreover, .fn → f in Z. On the other hand, let .fn = (f 0n , f 2n ) ∈ K(θn ) and let .f = (f 0 , f 2 ) be an element of Z such that .fn  f in Z. Then, it follows that the convergences (9.89) and (9.90) hold, which imply that .

f 0 L2 (Ω)d ≤ lim inf f 0n L2 (Ω)d , f L2 (Γ2 )d ≤ lim inf f 2n L2 (Γ2 )d .

We use these inequalities and the regularity .fn = (f 0n , f 2n ) ∈ K(θn ) to see that .

f 0 L2 (Ω)d + f 2 L2 (Γ2 )d ≤ lim inf f 0n L2 (Ω)d + lim inf f 2n L2 (Γ2 )d ≤ lim inf (f 0n L2 (Ω)d + f 2n L2 (Γ2 )d ) ≤ lim inf k(θn )L2 (Γ3 ) ).

Therefore, using (9.94), we find that f 0 L2 (Ω)d + f 2 L2 (Γ2 )d ≤ k((θ )L2 (Γ3 ) ).

.

It follows from here that .f Definition 1.2 show that

∈ K(θ ). The above properties combined with M

K(θn ) −→ K(θ ) in L2 (Γ2 )d .

.

We conclude from here that condition (7.38) is satisfied, too. In addition, it is easy to see that the functional (9.87) satisfies conditions (7.39) and (7.40). Finally, note that assumption (9.88) implies that condition (7.42) holds. It follows from above that we are in a position to apply Theorem 7.8 that guarantees the weakly generalized well-posedness of Problem .Oθ with the Tykhonov triple

9.3 Two Frictionless Contact Problems

323

⎧ Tθ = (I, Ω, Cθ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ I = L2 (Γ3 ),

. ⎪ ⎪ Ω(ξ ) = (u∗ , f ∗ ) : (u∗ , f ∗ ) is a solution to Problem Oξ ∀ ξ ∈ L2 (Γ3 ), ⎪ ⎪

⎪ ⎪ ⎩ C = {θ } ⊂ R : θ → θ , θ n + n for any .θ ∈ L2 (Γ3 ). This proves the existence part of Theorem 9.4. For the convergence part, we consider a function .θ ∈ L2 (Γ3 ), assume that for each .n ∈ N the pair .(u∗n , fn∗ ) represents a solution to Problem .Oθn with .θn ∈ L2 (Γ3 ), and, in addition, we assume that .θn → θ in .L2 (Γ3 ). Then, Definition 2.1(b) implies that .{(u∗n , fn∗ )} is a .Tθ -approximating sequence. The existence of a subsequence of the sequence .{(u∗n , fn∗ )}, again denoted by .{(u∗n , fn∗ )}, such that (9.86) holds, is now a direct consequence of weakly generalized well-posedeness of Problem .Oθ with   respect to .Tθ , guaranteed by Theorem 7.8, combined with Remark 7.6. We end this section with the following comment. Remark 9.3 The mechanical interpretation of the optimal control Problem .Oθ is the following: given an equilibrium process governed by the variational inequality (9.81) with the data .F, B, .α, .f 0 , .f 2 , which satisfy conditions (8.20), (8.23), and (9.43)–(9.47), respectively, and given a temperature field .θ ∈ L2 (Γ3 ), we are looking for a density of applied forces .f = (f 0 , f 2 ) ∈ L2 (Ω)d × L2 (Γ2 )d such that the corresponding stress field given by (9.68) is as close as possible to the “desired” stress .σ 0 ∈ Q. Theorem 9.4 guarantees the existence of at least one couple of densities .f ∗ = (f ∗0 , f ∗2 ), which solves this problem. Moreover, the optimal solutions depend continuously on the temperature field, in the sense given in the statement of the above theorem.

9.3 Two Frictionless Contact Problems In this section, we consider two frictionless contact models that, in a variational formulation, lead to minimization problems. We illustrate the use of the abstract results presented in Sect. 7.1 in the study of these problems. In this way, we deduce existence, uniqueness, and convergence results. Everywhere in this section, V will denote the space (8.6) and .c0 > 0 will represent the constant defined in (8.14).

9.3.1 The Models The first contact model we consider in this section is as follows.

324

9 Static Contact Problems

Problem .P. Find a displacement field .u : Ω → Rd and a stress field .σ : Ω → Sd such that σ = Eε(u) + α(ε(u) − PB ε(u))

in Ω, .

(9.95)

Div σ + f 0 = 0

in Ω, .

(9.96)

u=0

on Γ1 , .

(9.97)

σν = f 2

on Γ2 , .

(9.98)

on Γ3 , .

(9.99)

on Γ3 .

(9.100)

.

σν = 0 uν ≤ g,

if uν < 0

−F ≤ σν ≤ 0 if uν = 0 σν = −F σν ≤ −F

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ if 0 < uν < g ⎪ ⎪ ⎪ ⎪ ⎭ if uν = g στ = 0

We now provide a short description of the equations and boundary conditions in Problem .P. First, equation (9.95) represents an elastic constitutive law of the form (8.21). In contrast with the constitutive law used in Sect. 9.2, here we assume that .E is a fourth order tensor. Equation (9.96) is the equilibrium equation, and conditions (9.97), (9.98) represent the displacement and traction boundary conditions, respectively. Condition (9.99) represents the contact condition (8.41) depicted in Fig. 8.7b. It models the contact with a rigid body covered with a rigid– plastic layer of thickness g, say asperities. The function F is assumed to be positive and could be interpreted as the yield limit of the rigid–plastic material. Finally, condition (9.100) represents the frictionless contact condition. The second contact model we consider in this section is governed by a parameter .ε > 0 and is stated as follows. Problem .Pε . Find a displacement field .u : Ω → Rd and a stress field .σ : Ω → Sd such that σ =. Eε(u) + αε (ε(u) − PB ε(u))

in Ω, .

(9.101)

Div σ + f 0ε = 0

in Ω, .

(9.102)

u=0

on Γ1 , .

(9.103)

σ ν = f 2ε

on Γ2 , .

(9.104)

9.3 Two Frictionless Contact Problems

325

σν = 0 uν ≤ gε ,

if uν < 0

−σν =  σν ≤ − 

F u+ ν 2 2 (u+ ν ) + kε F gε

gε2 + kε2

if 0 ≤ uν < gε if uν = gε

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

στ = 0

on Γ3 , .

(9.105)

on Γ3 .

(9.106)

The equations and boundary conditions (9.101)–(9.106) have a similar meaning as the corresponding ones in Problem .P. A brief comparison between the two contact models shows that in Problem .Pε we replaced the data .α, .f 0 , .f 2 , and g with their perturbations .αε , .f 0ε , .f 2ε , and .gε , respectively. The second difference arises from the contact condition (9.105) that represents a regularization of the contact condition (9.99). It models the contact with a rigid body covered by an elastic layer of thickness .gε . Here .kε represents a deformability coefficient and .r + denotes the positive part of r, i.e., .r + = max {0, r}. Note that, formally, (9.99) is recovered from (9.105) in the limit as .kε → 0. The graphic representation of the regularized contact condition (9.105) is depicted in Fig. 9.3. In the study of the contact problem (9.95)–(9.100), we assume that the elasticity tensor .E and the set B satisfy the conditions (8.67) and (8.23), respectively. For the rest of the data, we assume that: .

α > 0, . f 0 ∈ L2 (Ω)d ,

(9.107) f 2 ∈ L2 (Γ2 )d , .

(9.108)

F ∈ L2 (Γ3 ), F (x) ≥ 0 a.e. x ∈ Γ3 , .

(9.109)

g > 0.

(9.110)

Under these assumptions, we introduce the set .K ⊂ V and the functional .J : V → R defined by

Fig. 9.3 The regularized contact condition (9.105)

326

9 Static Contact Problems

.



K = v ∈ V : vν ≤ g a.e. on Γ3 , .

(9.111)

  1 α J (v) = Eε(v) · ε(v) dx + ε(v) − PB ε(v)2 dx (9.112) 2 Ω 2 Ω    f 0 · v dx − f 2 · v da + F vν+ da ∀ v ∈ V. − Ω

Γ2

Γ3

Assume now that .(u, σ ) represents a regular solution of Problem .P. Then, using standard arguments, it can be shown that, for any .v ∈ K, the following inequalities hold: .

Eε(u) · ε(v) − ε(u)) ≤

1 1 Eε(v) · ε(v) − Eε(u) · ε(u) 2 2

a.e. in Ω,

(ε(u) − PB ε(u)) · (ε(v) − ε(u)) ≤

1 1 ε(v) − PB ε(v)2 − ε(u) − PB ε(u)2 2 2

−σν (vν − uν ) ≤ F vν+ − F u+ ν

a.e. in Ω,

a.e. on Γ3 .

Using these inequalities, notation (9.111), (9.112), and the Green formula (8.16), we deduce that u ∈ K,

.

J (u) ≤ J (v)

∀ v ∈ K.

(9.113)

We now denote .θ = (K, J ), and, using this notation together with inequality (9.113), we consider the following variational formulation of Problem .P. Problem .Pθ . Find a displacement field .u such that u ∈ K,

.

J (u) = min J (v). v∈K

(9.114)

Next, in the study of Problem .Pε , we consider the following additional assumptions, for each .ε > 0: .

αε > 0, .

(9.115)

kε ≥ 0, .

(9.116)

f 0ε ∈ L2 (Ω)d , gε > 0.

f 2ε ∈ L2 (Γ2 )d , .

(9.117) (9.118)

9.3 Two Frictionless Contact Problems

327

Under these assumptions, we introduce the set .Kε ⊂ V and the functional .Jε : V → R defined by .



Kε = v ∈ V : vν ≤ gε a.e. on Γ3 , .

Jε (v) =

1 2

 Eε(v) · ε(v) dx + Ω

Ω

 ε(v) − PB ε(v)2 dx

f 0ε · v dx −

Γ2

(9.120)

Ω



 −

αε 2

(9.119)

 f 2ε · v da +

Γ3

 F (vν+ )2 + kε2 da

∀ v ∈ V.

Assume now that .(u, σ ) represents a regular solution of Problem .Pε . Then, it is easy to check that, for any .v ∈ Kε , the following inequality holds: .

  + 2 2 2 2 −σν (vν − uν ) ≤ F (vν ) + kε − F (u+ ν ) + kε

a.e. on Γ3 .

Using this inequality, the notation (9.119), (9.120), and arguments similar to those used to obtain inequality (9.113), we deduce that u ∈ Kε ,

.

Jε (u) ≤ Jε (v)

∀ v ∈ Kε .

(9.121)

We now denote .θε = (Kε , Jε ), .u = uε and, using inequality (9.121), we consider the following variational formulation of Problem .Pε . Problem .Pθε . Find a displacement field .uε such that uε ∈ K,

.

Jε (uε ) = min Jε (v). v∈Kε

(9.122)

The analysis of problems .Pθ and .Pθε , including existence, uniqueness, and various convergence results, will be provided in the next subsection. Here we restrict ourselves to mention that a function .u that satisfies (9.114) is called a weak solution of the elastic contact problem (9.95)–(9.100) and a function .uε that satisfies (9.122) is called a weak solution of the elastic contact problem (9.101)–(9.106).

9.3.2 Existence and Uniqueness Everywhere in this subsection as well as in the next one, we use the notation (7.3)– (7.6) and (7.10) introduced in Sect. 7.1, in the case when .X = V . Our main result in this section is the following.

328

9 Static Contact Problems

Theorem 9.5 (a) Under assumptions (8.23), (8.67), (9.107)–(9.110), there exists a unique solution to Problem .Pθ . (b) Under assumptions (8.23), (8.67), (9.115)–(9.118), there exists a unique solution to Problem .Pθε . Proof (a) We shall use Theorem 7.1 on page 242, and, to this end, we shall prove that  .θ ∈ I . First, a simple calculation shows that the function .G : V → R defined by 

α 2

Gv =

.

ε(v) − PB ε(v)2 dx

∀v ∈ V

Ω

is Gâteaux differentiable on V and its gradient .∇G : V → V , given by  (∇Gu, v)V = α

.

 ε(u) − PB ε(u) · ε(v) dx

∀ u, v ∈ V ,

Ω

is a monotone operator. The details of proof can be found in [128]. These properties guarantee that this function is convex (see for instance Proposition 1.32 in [218]). In addition, using the convexity of the function .r → r + , it follows that the function    .v → − f 0 · v dx − f 2 · v da + F vν+ da Ω

Γ2

Γ3

is a convex function on V , too. Moreover, a simple calculation based on the properties of the tensor .E shows that  .



(1 − t)

Eε(u) · ε(u) dx + t Ω

Eε(v) · ε(v) dx

(9.123)

Ω

 −

Eε((1 − t)u + tv) · ε((1 − t)u + tv) dx Ω

 Eε(v − u) · ε(v − u) dx

= t (1 − t) Ω

≥ mE t (1 − t) u − v2V for all .u, v ∈ V and .t ∈ [0, 1]. We gather all these ingredients to see that the function J is strongly convex. Moreover, using the properties of the elasticity, projection, and trace operators,

9.3 Two Frictionless Contact Problems

329

it follows that J is continuous and, therefore, is lower semicontinuous. This implies that .J ∈ J  . Finally, since .g > 0, it is clear that the set K defined by (9.111) is nonempty, closed, and convex and, therefore, .K ∈ K . It follows from above that .θ = (K, J ) ∈ K ×J  = I  ⊂ K ×J  . The unique solvability of Problem .Pθ is now a consequence of Theorem 7.1(b). (b) The unique solvability of Problem .Pθε follows from arguments similar to those used above. First, we use the continuity of the function  v →

.

Γ3

 F (vν+ )2 + kε2 da

(9.124)

to see that .Jε is continuous. On the other hand, inequality (9.123) combined with the convexity of the function (9.124) shows that .Jε is strongly convex. We conclude from here that .Jε ∈ J  . Since the inclusion .Kε ∈ K is obviously satisfied, we deduce that .θε = (Kε , Jε ) ∈ K × J  = I  . The existence of a unique solution to Problem .Pθε follows from Theorem 7.1(b).   Remark 9.4 Using the trace inequality (8.14), we deduce that there exists .c > 0, which does not depend on .ε such that Jε (v) ≥

.

mE v2V − c (f 0ε L2 (Ω)d + f 2ε L2 (Γ2 )d )vV 2

(9.125)

for all .v ∈ V . This shows that the function .Jε is coercive.

9.3.3 Convergence Results For the results we present in this subsection, we need some convergence conditions on the data, gathered in the following assumptions: αε → . α

as

ε → 0, .

(9.126)

kε → 0

as

ε → 0, .

(9.127)

f 0ε  f 0 in L2 (Ω)d , gε → g

as

f 2ε  f 2 in L2 (Γ2 )d

ε → 0.

as

ε → 0, .

(9.128) (9.129)

Our main result in this section is the following. Theorem 9.6 Assume (8.23), (8.67), (9.107)–(9.110), (9.115)–(9.118), and (9.126)–(9.129). Then, the solution .uε of Problem .Pθε converges in V to the solution of Problem .Pθ , as .ε → 0.

330

9 Static Contact Problems

Proof We use Theorem 7.4. To this end, we consider a sequence .{εn } ⊂ R+ such that .εn → 0. For each .n ∈ N, we use the short-hand notation .Kn = Kεn , .Jn = Jεn , and .θn = (Kn , Jn ). We shall prove that .{θn } ⊂ Cθ where, recall, .Cθ is the set defined by (7.9). First, it follows from the proof of Theorem 9.5 that .θn ∈ I  ⊂ I , for any .n ∈ N. Moreover, using inequality (9.125) combined with the convergence (9.128), it is easy to see that condition (.c1 ) on page 243 is satisfied. Assume now that .{un } ⊂ V is a weakly convergent sequence. Then, an elementary calculation shows that Jn (u . n ) − J (un ) =

1 (αε − α) 2 n

 ε(un ) − PB ε(un )2 dx

(9.130)

Ω



 − Ω

(f 0εn − f 0 ) · un dx −

 + Γ3

Γ2

(f 2εn − f 2 ) · un da

 + 2 2 F ( (u+ nν ) + kεn − unν ) da.

We now use the properties of the projection operator .PB , the compactness of the embedding .V ⊂ L2 (Ω)d , the compactness of the trace operator .γ : V → L2 (Γ )d , the inequality    2 + k 2 − u+  ≤ k )  (u+ εn nν nν εn

.

a.e. on Γ3 ,

and the convergences (9.126)–(9.129) to see that each term on the right-hand side of equality (9.130) converges to zero. We deduce from here that .Jn (un ) − J (un ) → 0 and, therefore, condition (.c2 ) on page 243 is satisfied, too. Next, assume that .un → u in V . Then, for each .n ∈ N, we have    1 Eε(un ) · ε(un ) − Eε(u) · ε(u) dx Jn (u. n ) − Jn (u) = (9.131) 2 Ω    αεn ε(un ) − PB ε(un )2 − ε(u) − PB ε(u)2 dx + 2 Ω   − f 0εn · (un − u) dx − f 2εn · (un − u) da Ω

Γ2

 +

F Γ3



 +  2 2 (unν )2 + kε2n − (u+ ν ) + kεn da.

We now use the continuity of the bilinear form

9.3 Two Frictionless Contact Problems

331

 (u, v) →

Eε(u) · ε(v) dx

.

∀ u, v ∈ V ,

Ω

the properties of the projection operator .PB , the embedding .V ⊂ L2 (Ω)d , and the trace operator .γ : V → L2 (Γ )d , the inequality     + 2 + + 2  2 2  (u+ nν ) + kεn − (unν ) + kεn  ≤ |unν − uν | ≤ un − u a.e. on Γ3 ,

.

and the convergences (9.126)–(9.129). In this way, we see that each term of the right-hand side of equality (9.131) converges to zero. We deduce from here that .Jn (un ) − Jn (u) → 0, which shows that condition (.c3 ) on page 243 holds. Finally, using definitions (9.111), (9.119) combined with assumptions (9.110), g (9.118), we see that .Kn = gεn K for all .n ∈ N. On the other hand, (9.129) guarantees that .gεn → g. Based on these ingredients, it is easy to see that condition (.c4 ) on page 243 is satisfied, too. It follows from above that {θn } ∈ Cθ .

(9.132)

.

Moreover, with the notation .un = uεn for the solution of Problem .Pθn , (7.8) shows that un ∈ Ω(θn )

.

∀ n ∈ N.

(9.133)

We now use (9.132) and (9.133) to deduce that .{un } is a .Tθ -approximating sequence, where the Tykhonov triple .Tθ is defined by (7.7)–(7.9) with .X = V . On the other hand, since .θ ∈ I  , it follows from Theorem 7.4 that Problem .Pθ is strongly well-posed with respect to the Tykhonov triple .Tθ . This implies that .un → u in V , which concludes the proof.   Note that Theorem 9.5(a) and (b) provides the unique weak solvability of Problems .Pθ and .Pθε , respectively. Next, in order to provide the mechanical interpretation of the convergence result given by Theorem 9.6, we denote in what follows by .uε (αε , kε , f 0ε , f 2ε , gε ) the weak solution of Problem .Pε constructed with the data .αε , .kε , .f 0ε , .f 2ε , .gε that satisfy (9.115)–(9.118). In addition, we denote by .u(α, f 0 , f 2 , g) the weak solution of Problem .P constructed with the data .α, .f 0 , .f 2 , g that satisfy (9.107)–(9.110). It follows from Theorem 9.6 that, if the convergences (9.126)–(9.129) hold, then uε (αε , kε , f 0ε , f 2ε , gε ) → u(α, f 0 , f 2 , g)

.

in V ,

as

ε → 0.

(9.134)

On the other hand, a careful analysis based on the definitions (9.112) and (9.120) of the functionals J and .Jε reveals that if .kε = 0, then Problem .Pθε reduces to a problem of the form .Pθ constructed with the data .αε , .f 0ε , .f 2ε , .gε . Therefore,

332

9 Static Contact Problems

uε (αε , 0, f 0ε , f 2ε , gε ) = u(αε , f 0ε , f 2ε , gε ).

(9.135)

.

We now take .kε = 0 in (9.134) and use equality (9.135) to deduce that, if (9.126), (9.128), and (9.129) hold, then u(αε , f 0ε , f 2ε , gε ) → u(α, f 0 , f 2 , g)

.

in V ,

as

ε → 0.

(9.136)

In addition to the mathematical interest in the convergence result (9.136), it is important from the mechanical point of view, since it shows that the weak solution of the contact problem .P depends continuously on the elasticity coefficient .α, the densities of the applied forces, and the thickness of the rigid–plastic layer of the foundation. Finally, the convergence result (9.134) shows that if (9.127) holds, then uε (α, kε , f 0 , f 2 , g) → u(α, f 0 , f 2 , g)

.

in V ,

as

ε → 0.

(9.137)

This convergence result can be interpreted as follows: the weak solution of the contact problem with a rigid body covered by a layer of rigid–plastic material can be approached by the solution of the contact problem with a rigid body covered by a layer of elastic material as the deformability coefficient of this material is small enough. We end this section with the remark that (9.136) concerns only Problem .P and represents a continuous dependence result for its weak solution. In contrast, the convergence result (9.137) concerns both Problems .P and .Pε . It establishes a link between the weak solutions of these contact problems that model two different physical settings.

9.4 A Nonsmooth Contact Problem The contact model we consider in this section represents the frictionless version of the contact model outlined in Sect. 9.1. Nevertheless, we now use different assumptions on the normal compliance function p, and, consequently, the weak formulation of the model is now obtained in the form of a hemivariational inequality for the displacement field. This allows us to illustrate the results of Sect. 5.1 in the study of the corresponding contact problem.

9.4.1 The Model The classical formulation of the frictionless contact problem we consider in this section is as follows.

9.4 A Nonsmooth Contact Problem

333

Problem .P. Find a displacement field .u : Ω → Rd and a stress field .σ : Ω → Sd such that σ = Fε(u)

in Ω, .

(9.138)

Div σ + f 0 = 0

in Ω, .

(9.139)

u=0

on Γ1 , .

(9.140)

σν = f 2

on Γ2 , .

(9.141)

on Γ3 , .

(9.142)

on Γ3 .

(9.143)

.

uν ≤ g,

⎫ σν + p(uν ) ≤ 0, ⎬

(uν − g)(σν + p(uν )) = 0 ⎭ στ = 0

We assume that the elasticity operator satisfies conditions (8.20) and the densities of body forces and surface tractions have the regularity (9.8). Moreover, we assume that (9.10) holds and the normal compliance function is such that ⎧ (a) p : R → R is a continuous function. ⎪ ⎪ ⎪ ⎪ ⎨ (b) |p(r)| ≤ c0 + c1 |r| ∀ r ∈ R, with c0 , c1 ≥ 0. . ⎪ ⎪ ⎪ (c) (p(r1 ) − p(r2 ))(r2 − r1 ) ≤ αp |r1 − r2 |2 ⎪ ⎩ ∀ r1 , r2 ∈ R, with αp ≥ 0.

(9.144)

Here, for simplicity, we assume that p does not depend on the spatial variable .x ∈ Γ3 . Note also that, in contrast with assumption (9.7) in Sect. 9.1, no monotonicity condition is imposed in (9.144). Below, we use the space (8.6) for the displacement field, together with its dual ∗ .V and the duality pairing mapping .·, ·. Let .q : R → R be the function defined by  q(r) =

r

∀r ∈R

p(s) ds

.

(9.145)

0

and let .



K = v ∈ V : vν ≤ g a.e. on Γ3 , . ∗

A: V → V ,

 Au, v = 

j : V → R,

(9.146)

j (v) =

Fε(u) · ε(v) dx, .

q(vν ) da, . Γ3

(9.147)

Ω

(9.148)

334

9 Static Contact Problems



f ∈ V ∗,



f , v = Ω

f 0 · v dx +

Γ2

f 2 · v da,

(9.149)

for all .u, v ∈ V . Note that the function q is continuously differentiable and, therefore, it is regular in the sense of Definition 1.10. Moreover, .q 0 (r; s) = p(r)s for all .r, s ∈ R and, in addition, assumption (9.144) guarantees that the function .j : V → R is locally Lipschitz and satisfies the relations  j 0 (u; v) =

 q 0 (uν ; vν ) da =

.

Γ3

(9.150)

p(uν )vν da Γ3

for all .u, v ∈ V . These properties of j (including equalities (9.150)) represent a direct consequence of Theorem 3.47 in [157] and Lemma 8 in [222] and, therefore, we omit their proof. Assume that .u and .σ are sufficiently regular functions that satisfy (9.138)– (9.143). Then, using arguments similar to those used in the proof of inequality (9.16), we deduce that  .

u ∈ K,

 Fε(u) · (ε(v) − ε(u)) dx +

Ω

p(uν )(vν − uν ) da Γ3



 ≥ Ω

f 0 · (v − u) dx +

Γ2

f 2 · (v − u) da

∀v ∈ K

and, therefore, definitions (9.147) and (9.149) yield  u ∈ K,

.

Au, v − u +

p(uν )(vν − uν ) da ≥ f , v − u

∀ v ∈ K.

Γ3

We now use equality (9.150) to deduce the following variational formulation of Problem .P, in terms of displacements. Problem .P V . Find a displacement field .u ∈ K such that Au, v − u + j 0 (u; v − u) ≥ f , v − u

.

∀ v ∈ K.

Note that Problem .P V represents an example of hemivariational inequality studied in Sect. 5.1. There, the well-posedness of abstract inequalities of this form with the Tykhonov triple (5.7) was proved, and a convergence result was obtained (see Theorem 5.2). Our aim in what follows is to illustrate the abstract result of Theorem 5.2 in the study of Problem .P V and to deduce the corresponding mechanical interpretation.

9.4 A Nonsmooth Contact Problem

335

9.4.2 A Convergence Result In order to illustrate Theorem 5.2, we need to introduce perturbations of the data K, A, and f . To this end, we consider a set B that satisfies condition (8.23). Moreover, for each .n ∈ N, we assume that .αn , .μn , .f 0n , .f 2n , .kn are given and satisfy the conditions .

αn > 0,

αn → 0, .

(9.151)

μn > 0,

μn → 0, .

(9.152)

f 0n ∈ L2 (Ω)d , f 2n ∈ L2 (Γ2 )d , .

(9.153)

f 0n → f 0 in L2 (Ω)d , f 2n → f 2 in L2 (Γ2 )d , .

(9.154)

gn ≥ 0,

gn → g, .

(9.155)

c02 (αp + μn ) < mF .

(9.156)

Note that here we consider only the homogeneous case, for simplicity. Nevertheless, we remark that the results below can be easily extended to the case when the function p as well as the coefficients .αn , .μn depend on the spatial variable .x ∈ Ω ∪ Γ . Moreover, we recall that, as usual, the constant .c0 in (9.156) represents the positive constant in the trace inequality (8.14). We now introduce the following notation: .

Kn = v ∈ V : vν ≤ gn a.e. on Γ3 , .

(9.157)

An : V → V ∗ ,

(9.158)

An u, v = Au, v. 

 +αn ε(u) − PB ε(u) · ε(v) dx, Ω

ϕn : V × V → R, f n ∈ V ∗,



ϕn (u, v) = μn Γ3

 f n , v =

Ω

f 0n · v dx +

u+ ν v τ  da, .

(9.159)

 Γ2

f 2n · v da,

(9.160)

for all .u, v ∈ V . Here and below, as usual, .PB : Sd → B denotes the projection operator on the set B and .r + represents the positive part of r, i.e., .r + = max {r, 0}. Then, for each .n ∈ N, we consider the following problem.

336

9 Static Contact Problems

Problem .PnV . Find a displacement field .un ∈ Kn such that .

An un , v − un  + ϕn (un , v) − ϕn (un , un ) +j 0 (un ; v − un ) ≥ f n , v − un 

∀ v ∈ Kn .

We use the terminology on page 15 to note that, in contrast to Problem .P V , which represents a pure hemivariational inequality, Problem .PnV represents a variational– hemivariational inequality. Our main result in the study of Problems .P V and .PnV is the following. Theorem 9.7 Assume (8.20), (8.23), (9.8), (9.10), (9.144), (9.151)–(9.156). Then, the following statements hold: (a) There exists a unique solution .u to Problem .P V and, for each .n ∈ N, there exists a unique solution .un to Problem .PnV . (b) The sequence .{un } converges to .u, i.e., .un → u in V . Proof We use Theorem 5.2 with .X = V and, to this end, we have to check the validity of conditions (1.17), (1.18) (1.20), (1.21), (5.3), and (5.31)–(5.36). First, we remark that condition (1.17) is obviously satisfied. Moreover, the operator A defined by (9.147) satisfies condition (1.18). Indeed, using assumption (8.20)(a), we find that Au − Av, w ≤ LF u − vV wV

.

for all .u, .v, .w ∈ V . This implies that Au − AvV ∗ ≤ LF u − vV

.

for all .u, .v ∈ V , and shows that A is Lipschitz continuous. On the other hand, assumption (8.20)(c) yields Au − Av, u − v ≥ mF u − v2V

.

for all .u, .v ∈ V . This shows that condition (1.18)(b) is satisfied with .mA = mF . Since A is Lipschitz continuous and monotone, it follows that A is pseudomonotone and, therefore, (1.18)(a) holds. On the other hand, using (9.150) and the properties (9.144) of the function p, it follows that the function j given by (9.148) satisfies condition (1.20) with 2 .αj = c αp . In addition, condition (1.21) is guaranteed by assumption (9.8) and 0 definition (9.149). Next, we remark that condition (5.31) is a direct consequence of definitions (9.146), (9.157) and assumptions (9.10), (9.155). Consider now the operator .T : V → V ∗ given by  T u, v =

.

Ω

 ε(u) − PB ε(u) · ε(v) dx

9.4 A Nonsmooth Contact Problem

337

for all .u, .v ∈ V . Then, the nonexpansivity of the projection operator yields T u − T vV ∗ ≤ 2 u − vV

.

T u − T v, u − v ≥ 0

and

for all .u, .v ∈ V . Therefore, assumption (9.151) and notation (9.158) imply that condition (5.32) is satisfied. Next, we apply the trace inequality (8.14) to see that the function .ϕn defined by (9.159) satisfies condition (5.33) with .αn = δn = c02 μn and, therefore, assumption (9.156) shows that condition (5.34) holds. Moreover, we note that assumptions (9.153) and (9.154) imply condition (5.35) and, since .αn = c02 μn , (9.152) shows that (5.36) holds, too. Finally, condition (5.3) follows from the compactness of the trace operator 2 d .γ : V → L (Γ3 ) . Indeed, if .un  u and .v n → v in V , using (9.150) and the convergences .p(unν ) → p(uν ), .unν → un , .vnν → vn in .L2 (Γ3 ), we deduce that  .

lim sup j 0 (un ; v n − un ) = lim sup

p(unν )(vnν − unν ) da Γ3



p(uν )(vν − uν ) da = j 0 (u; v − u),

= Γ3

which shows that condition (5.3) holds, as claimed. It follows from above that we are in a position to use Theorem 5.2 in order to conclude the proof of Theorem 9.7.   Note that Problem .PnV represents the variational formulation of the following mathematical model of contact. Problem .Pn . Find a displacement field .un : Ω → Rd and a stress field .σ n : Ω → Sd such that σ n = Fε(un ) + αn (ε(un ) − PB ε(un ))

in Ω,

Div σ n + f 0n = 0

in Ω,

un = 0

on Γ1 ,

σ n ν = f 2n

on Γ2 ,

.

unν ≤ gn , σnν + p(unν ) ≤ 0, (unν − gn )(σnν σ nτ  ≤ μu+ nν ,

.

⎫ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎭ + p(unν )) = 0 ⎪

σ nτ = −μu+ nν

on Γ3 ,

unτ if unτ = 0 unτ 

on Γ3 .

338

9 Static Contact Problems

This problem is obtained from Problem .P by operating the following changes: first, the constitutive law .σ = Fε(u) is perturbed by using the elasticity coefficient .αn and the projection on the convex set B; second, the densities of body forces and surface tractions are replaced by their perturbations .f 0n and .f 2n , respectively; third, the thickness of the deformable material is replaced by its perturbation .gn ; finally, the frictionless condition was replaced by a version of Coulomb’s law, governed by a coefficient of friction .μn and the friction bound .μn u+ nν . We end this section with the following mechanical interpretations: first, Theorem 9.7 provides the unique weak solvability of the contact models .P and .Pn ; second, it establishes the link between the weak solutions of these models that, recall, are constructed by using different mechanical assumptions; third, it provides the continuous dependence of the weak solution of the first model with respect to the densities of body forces, the surface tractions, and the thickness of the deformable layer. All these ingredients show that, in addition to the mathematical interest in Theorem 9.7, it is also important from the mechanical point of view.

Chapter 10

Quasistatic Contact Problems

In this chapter, we study the well-posedness of several quasistatic mathematical models of contact. For each model, we introduce a classical formulation that gathers the corresponding equations, boundary, and initial conditions. Then we list the assumptions on the data and derive a variational formulation, which is either in the form of a fixed point problem, a Volterra-type integral equation, a history-dependent inclusion, or a history-dependent variational inequality. Next, we use the results of Part II of the book to deduce existence, uniqueness, and convergence results. This allows us to prove the continuous dependence of the solution with respect to the data, to study an associated optimal control problem, and to establish the link between various models of contact, as well. Finally, we present a quasistatic problem for rate-type materials for which we carry out a direct well-posedness analysis. Everywhere in this chapter, we use the notations V , Q, and .Q∞ for the spaces given by (8.6), (8.3), and (8.17), respectively.

10.1 Two Frictionless Contact Problems In this section, we study two models that describe the contact of a body with a deformable foundation. In both models, the material is assumed to be viscoelastic with long memory, and the contact is frictionless and is modeled with normal compliance. The time interval of interest is .R+ = [0, +∞) for the first model and .[0, T ] with .T > 0 for the second one. The study of the first model is made by using a fixed point argument, while the study of the second one is carried out using arguments of Volterra-type integral equations. In this way, we illustrate the abstract results presented in Sects. 3.5 and 3.4, respectively.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Sofonea, Well-Posed Nonlinear Problems, Advances in Mechanics and Mathematics 50, https://doi.org/10.1007/978-3-031-41416-9_10

339

340

10 Quasistatic Contact Problems

10.1.1 The Models The classical formulation of the first contact problem we study in this section is the following. Problem .Q. Find a displacement field .u : Ω × R+ → Rd and a stress field .σ : Ω × R+ → Sd such that, for any .t ∈ R+ , the following equalities hold: 

t

σ (t) = Eε(u(t)) +

R(t − s)ε(u(s)) ds

in Ω, .

(10.1)

Div σ (t) + f 0 (t) = 0

in Ω, .

(10.2)

u(t) = 0

on Γ1 , .

(10.3)

σ (t)ν = f 2 (t)

on Γ2 , .

(10.4)

−σν (t) = kp(uν (t) − g) on Γ3 , .

(10.5)

σ τ (t) = 0

(10.6)

.

0

on Γ3 .

We now provide a short description of the equations and boundary conditions in Problem .Q. First, Eq. (10.1) represents the viscoelastic constitutive law in which .E is a fourth order elasticity tensor and .R is the relaxation tensor. Equation (10.2) represents the equilibrium equation in which .f 0 denotes the density of body forces, (10.3) is the displacement boundary condition, and (10.4) is the traction boundary condition in which .f 2 denotes the density of surface tractions. Condition (10.5) is the contact condition with normal compliance in which p is a given function, g represents the gap function, and k is a positive stiffness coefficient. Moreover, as usual, .uν and .σν denote the normal components of the displacement and stress fields, respectively. Finally, condition (10.6) is the frictionless condition in which .σ τ represents the tangential shear. The classical formulation of the second contact problem is the following. Problem .M. Find a displacement field .u : Ω × [0, T ] → Rd and a stress field d .σ : Ω × [0, T ] → S such that, for any .t ∈ [0, T ], the following equalities hold:  σ (t) = Aε(u(t)) +

.

t

k(θ )e−(t−s) ε(u(s)) ds

in Ω, .

(10.7)

Div σ (t) + f 0 (t) = 0

in Ω, .

(10.8)

u(t) = 0

on Γ1 , .

(10.9)

σ (t)ν = f

on Γ2 , .

(10.10)

−σν (t) = p(uν (t))

on Γ3 , .

(10.11)

σ τ (t) = 0

on Γ3 .

(10.12)

0

10.1 Two Frictionless Contact Problems

341

The equation and boundary conditions in Problem .M have the same meaning as those in Problem .Q. Nevertheless, despite their similarities, there are some differences between the problems .Q and .M that we underline in what follows. First, in contrast with the viscoelastic constitutive law (10.1), the relaxation operator in (10.7) has a particular form and is governed by the relaxation coefficient .k(·), assumed to depend on a parameter .θ , say the temperature. Second, we assume that the density of surface tractions in (10.10) is time-independent. Moreover, there is no gap in the contact condition (10.11), and the stiffness coefficient that appears in (10.5) is assumed to be equal to 1. Finally, note that the assumptions we shall made on the data of Problem .M are slightly different from those made on the data of Problem .Q since, for instance, the elasticity operator .E in (10.1) is assumed to be linear and, in contrast, the elasticity operator .A in (10.7) is nonlinear. We included these differences in the models above since our aims in the study of Problems .Q and .M are different. Indeed, our aim in the study of Problem .Q is to illustrate the abstract results of Sect. 3.5 concerning the fixed point problems with almost history-dependent operators. In contrast, our aim in the study of Problem .M is to apply the abstract results of Sect. 7.2 to an optimal control problem associated to the Volterra-type equation studied in Sect. 3.4.1. And, according to these aims, we chose an appropriate statement of the models, together with convenient assumptions on the data.

10.1.2 A Fixed Point Formulation We now list the assumptions on the data of the contact problem .Q. The elasticity tensor .E is symmetric and positively definite, i.e., it satisfies the condition (8.67). The relaxation tensor .R has the regularity R ∈ C(R+ ; Q∞ ),

.

(10.13)

and the normal compliance function p is such that

.

⎧ ⎪ ⎪ (a) p : Γ3 × R → R+ . ⎪ ⎪ ⎪ ⎪ ⎪ (b) There exists Lp > 0 such that ⎪ ⎪ ⎪ ⎨ |p(x, r1 ) − p(x, r2 )| ≤ Lp |r1 − r2 | for all r1 , r2 ∈ R, a.e. x ∈ Γ3 . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (c) p(·, r) is measurable on Γ3 , for all r ∈ R. ⎪ ⎪ ⎪ ⎪ ⎩ (d) p(x, r) = 0 for all r ≤ 0, a.e. x ∈ Γ3 .

(10.14)

342

10 Quasistatic Contact Problems

We also assume that the densities of the body forces and surface tractions, the stiffness coefficient, and the gap function are such that .

f 0 ∈ C(R+ ; L2 (Ω)d )..

(10.15)

f 2 ∈ C(R+ ; L2 (Γ2 )d )..

(10.16)

k ∈ L∞ (Γ3 ), k(x) ≥ 0

a.e. x ∈ Γ3 ..

(10.17)

g ∈ L2 (Γ3 ), g(x) ≥ 0 a.e. x ∈ Γ3 .

(10.18)

We now use assumption (8.67) to endow the space V with the inner product  (u, v)E =

Eε(u) · ε(v) dx

.

(10.19)

Ω

and the associated norm . · E . Then, using inequalities (8.18) and (8.67)(b), it follows that  √ mE vV ≤ vE ≤ d EQ∞ vV

.

∀ v ∈ V,

(10.20)

which shows that the norms . · V and . · E are equivalent norms on the space V . Therefore, V is a Hilbert space endowed with the inner product .(·, ·)E , too. Moreover, the Sobolev trace theorem and the continuity of the embedding .V ⊂ L2 (Ω)d imply that there exist some constants  .c0 > 0, .d0 > 0 and .e0 > 0 such that .

vL2 (Γ3 )d ≤  c0 vE , .

(10.21)

vL2 (Γ2 )d ≤ d0 vE ,

vL2 (Ω)d ≤ e0 vE ,

(10.22)

for all .v ∈ V . Finally, we assume the smallness condition 2  .c0 Lp kL∞ (Γ3 ) < 1,

(10.23)

where  .c0 and .Lp are the positive constants in (10.21) and (10.14), respectively. We now turn to construct a fixed point weak formulation of Problem .Q. Let .v ∈ V and .t ∈ R+ . We use standard arguments based on the Green formula to see that if .(u, σ ) is a smooth solution of Problem .Q, then     . σ (t) · ε(v) dx = f 0 (t) · v dx + f 2 (t) · v da − kp(uν (t) − g)vν da. Ω

Ω

Γ2

Γ3

We combine this equality with the constitutive law (10.1) to see that

10.1 Two Frictionless Contact Problems

 .

Eε(u(t)) · ε(v) dx +

Ω

343

 

t

R(t − s)ε(u(s)) ds · ε(v) dx



 = Ω

f 0 (t) · v dx +

(10.24)

0

Ω



Γ2

f 2 (t) · v da −

kp(uν (t) − g)vν da. Γ3

Next, we use Riesz’s representation theorem to define the operator .S : C(R+ ; V ) → C(R+ ; V ) by the equality  (Su(t), v)E =

.

Ω

 f 0 (t) · v dx +

 −

kp(uν (t) − g)vν da − Γ3

Γ2

f 2 (t) · v da

  Ω

t

(10.25)

R(t − s)ε(u(s)) ds · ε(v) dx,

0

for all .u ∈ C(R+ ; V ), .t ∈ R+ , and .v ∈ V . We now combine equalities (10.19), (10.24), and (10.25) to deduce that (u(t), v)E = (Su(t), v)E .

.

This equality leads us to consider the following fixed point weak formulation of the Problem .Q, in terms of displacements. Problem .QV . Find a displacement field .u ∈ C(R+ ; V ) such that .Su(t) = u(t) for any .t ∈ R+ . The unique solvability of Problem .QV is provided by the following existence and uniqueness result. Theorem 10.1 Assume (8.67), (10.13)–(10.18), and (10.23). Then Problem .QV has a unique solution .u ∈ C(R+ ; V ). Proof Let .u, v ∈ C(R+ ; V ), .m ∈ N, .t ∈ [0, m], and .w ∈ V . We use the definition (10.25) to see that  . |(Su(t) − Sv(t), w)E | ≤ k|p(uν (t) − g) − p(vν (t) − g)||wν | da  + 0

Γ3

t

  R(t − s)(ε(u(s)) − ε(v(s))) ds · ε(w)Q Q

and, therefore, assumption (10.14)(b) and inequality (8.18) yield .

|(Su(t) − Sv(t), w)E | ≤ Lp kL∞ (Γ3 ) u(t) − v(t)L2 (Γ3 )d wL2 (Γ3 )d  t

+d R(t − s)Q∞ ε(u(s)) − ε(v(s))Q ds ε(w)Q . 0

344

10 Quasistatic Contact Problems

We now use the trace inequality (10.21) to deduce that |(Su(t) . − Sv(t), w)E | ≤  c02 Lp kL∞ (Γ3 ) u(t) − v(t)E wE  t

+d max R(r)Q∞ u(s)) − v(s)V ds wV . r∈[0,m]

(10.26)

0

Finally, we use (10.20) to see that 1 u(s) − v(s)V ≤ √ u(s) − v(s)E mE

.

∀ s ∈ R+ ,

1 wV ≤ √ wE , mE and then we substitute these inequalities in (10.26) and take .w = Su(t) − Su(t) in the resulting inequality to find that .

Su(t) − Sv(t)E ≤  c02 Lp kL∞ (Γ3 ) u(t) − v(t)E  t d + max R(r)Q∞ u(s)) − v(s)E ds. mE r∈[0,m] 0

(10.27)

Inequality (10.27) combined with the smallness assumption (10.23) and Definition 1.5 on page 7 shows that the operator defined by (10.25) is an almost historydependent operator. Theorem 10.1 is now a direct consequence of Theorem 1.4. 

10.1.3 Convergence Results We now prove a continuous dependence result for the solution with respect to the data. To this end, we consider the sequences .{Rn }, .{f 0n }, .{f 2n }, .{kn } such that, for each .n ∈ N, the following conditions hold: .

Rn ∈ C(R+ , Q∞ )..

(10.28)

f 0n ∈ C(R+ ; L2 (Ω)d )..

(10.29)

f 2n ∈ C(R+ ; L2 (Γ2 )d )..

(10.30)

kn ∈ L∞ (Γ3 ), kn (x) ≥ 0 a.e. x ∈ Γ3 ..

(10.31)

 c02 Lp kn L∞ (Γ3 ) < 1.

(10.32)

10.1 Two Frictionless Contact Problems

345

With these data, we define the operator .Sn : C(R+ ; V ) → C(R+ ; V ) by the equality  .

(Sn u(t), v)E =

Ω

 f 0n (t) · v dx +

 −

kn p(uν (t) − g)vν da −

Γ2

f 2n (t) · v da

 

Γ3

t

(10.33)

Rn (t − s)ε(u(s)) ds · ε(v) dx

0

Ω

for all .u ∈ C(R+ ; V ), .t ∈ R+ , and .v ∈ V . Then, we consider the following fixed point problem. Problem .QVn . Find a displacement field .un ∈ C(R+ ; V ) such that .Sn un (t) = un (t) for any .t ∈ R+ . Using Theorem 10.1, it follows that Problem .QVn has a unique solution, for each .n ∈ N. Consider now the following assumptions: .

Rn → R

in C(R+ , Q∞ ) as n → ∞..

(10.34)

f 0n → f 0

in C(R+ ; L2 (Ω)d ) as n → ∞..

(10.35)

f 2n → f 2

in C(R+ ; L2 (Γ3 )d ) as n → ∞..

(10.36)

kn → k

in L∞ (Γ3 )

as n → ∞.

(10.37)

We have the following convergence result. Theorem 10.2 Assume (8.67), (10.13)–(10.18), (10.23), (10.28)–(10.32), and (10.34)–(10.37). Then, the solution .un of Problem .QVn converges to the solution V .u of Problem .Q , i.e., un → u in C(R+ , V ) as n → ∞.

.

(10.38)

Proof Let .n, m ∈ N, .t ∈ [0, m], and .v ∈ C(R+ ; V ). Then, a simple calculation (based on the definitions (10.25), (10.33), inequalities (10.22), and arguments similar to those used in the proof of inequality (10.27)) implies that there exists a constant .C0 > 0 that does not depend on m, n, and t, such that .

Sn v(t) − Sv(t)E ≤ C0



(10.39)

max f 0n (r) − f 0 (r)L2 (Ω)d + max f 2n (r) − f 2 (r)L2 (Γ2 )d

r∈[0,m]

r∈[0,m]

+ c02 Lp kn − kL∞ (Γ3 ) v(t)E +

d max Rn (r) − R(r)Q∞ mE r∈[0,m]



t 0

v(s)E ds.



346

10 Quasistatic Contact Problems

Denote .

αnm = max f 0n (r) − f 0 (r)L2 (Ω)d , .

(10.40)

βnm = max f 2n (r) − f 2 (r)L2 (Γ2 )d , .

(10.41)

γnm = max Rn (r) − R(r)Q∞ , .

(10.42)

d m θnm = max C0 (αnm + βnm ), γ . c02 Lp kn − kL∞ (Γ3 ) , mE n

(10.43)

r∈[0,m]

r∈[0,m]

r∈[0,m]

Then, inequality (10.39) shows that .

 t

Sn v(t) − Sv(t)E ≤ θnm 1 + v(t)E + v(s)E ds .

(10.44)

0

Moreover, assumptions (10.34)–(10.37) and notation (10.40)–(10.43) imply that θnm → 0 as n → ∞, ∀ m ∈ N.

.

(10.45)

It follows now from (10.44) and (10.45) that condition (3.133) is satisfied with .Λn = Sn and .Λ = S. Theorem 10.2 is now a direct consequence of Theorem 3.10(c).  We end this section with the following comments: (1) We refer to the solution .u of Problem .QV as the weak solution of the contact problem .Q. We conclude from above that Theorem 10.1 provides the unique weak solvability of this contact problem, and Theorem 10.2 shows the continuous dependence of the weak solution with respect to the data. (2) In addition to the mathematical interest in the convergence (10.38), it is important from the mechanical point of view since it shows that small perturbations of the relaxation tensor .R, the density of body forces .f 0 , the density of traction forces .f 2 , and the stiffness coefficient k lead to a small perturbation of the weak solution of the contact problem .Q.

10.1.4 A Volterra-Type Variational Formulation We now turn to the variational formulation of Problem .M, and, to this end, below in this section, we use the notation .X = C([0, T ]; V ). Moreover, besides assumption (8.60) on the operator .A and assumption (10.14) on the normal compliance function, we consider the following hypotheses:

10.1 Two Frictionless Contact Problems

 .

347

k : R+ → R+ and there exists Lk > 0 such that |k(r1 ) − k(r2 )| ≤ Lk |r1 − r2 |



∀ r1 , r2 ∈ R+ .

(p(x, r1 ) − p(x, r2 ))(r1 − r2 ) ≥ 0 ∀ r1 , r2 ∈ R+ , a.e. x ∈ Γ3 .

.

(10.46)

(10.47)

.

f 0 ∈ C([0, T ], L2 (Ω)d ).

(10.48)

Let .θ ≥ 0, .f ∈ L2 (Γ2 )d and assume that .(u, σ ) is a regular solution of Problem .M. Let .t ∈ [0, T ] and .v ∈ V . Then, following a standard approach based on the Green formula, we obtain that  Aε(u(t)) . · ε(v) dx + Ω

 

t

(k(θ )e−(t−s) ε(u(s)) ds · ε(v) dx







p(uν (t))vν da =

+

(10.49)

0

Ω

Γ3

Ω

f 0 (t) · v dx +

f · v da. Γ2

We now introduce the operators .A : V → V , .Rθ : X → X, and the element Bf ∈ V defined by the equalities

.





(Au, . v)V =

Aε(u) · ε(v) dx + Ω

(Rθ u(t), v)V =

p(uν )vν da

∀ u, v ∈ V , .

(10.50)

Γ3

  Ω

t

k(θ )e−(t−s) ε(u(s))ds · ε(v) dx.

(10.51)

0

 − Ω

f 0 (t) · v dx

∀ v ∈ V , u ∈ X, t ∈ [0, T ],

 (Bf , v)V =

f · v da

∀ v ∈ V.

(10.52)

Γ2

Then, using (10.49)–(10.52), we derive the following variational formulation of Problem .M, in terms of displacements. Problem .Mθf . Given .(θ, f ) ∈ R+ × L2 (Γ2 )d , find a displacement field .u = u(θ, f ) : [0, T ] → V such that Au(t) + Rθ u(t) = Bf

.

∀ t ∈ [0, T ].

(10.53)

In the study of this problem, we have the following existence, uniqueness, and continuous dependence result.

348

10 Quasistatic Contact Problems

Theorem 10.3 Assume (8.60), (10.14), (10.46)–(10.48). Then, for each .(θ, f ) ∈ R+ × L2 (Γ2 )d , Problem .Mθf has a unique solution with regularity .u = u(θ, f ) ∈ X. Moreover, if .θn ∈ R+ , .f n ∈ L2 (Γ2 )d , .un = u(θn , f n ) ∈ X for each .n ∈ N and θn → θ

.

in R,

fn f

in L2 (Γ2 )d ,

(10.54)

then un → u

.

in X.

(10.55)

Proof Let .θ ∈ R+ and .f ∈ L2 (Γ2 )d . We use Theorem 3.7 on page 123, and, to this end, we check the validity of the conditions in this theorem. First, it is easy to see that assumptions (8.60), (10.14), (10.47) imply that .

(Au − Av, u − v)V ≥ mA u − v2V , Au − AvV ≤ (LA + c02 Lp )u − vV

for all .u, v ∈ V , where, recall, .c0 is the constant in (8.14). Therefore, condition (3.86) is satisfied. Next, assumptions (3.87) and (3.88) are satisfied with .bn (t) = k(θn )e−t , .b(t) = k(θ )e−t , .fn (t) = Bf n , .f (t) = Bf , for each .t ∈ [0, T ] and .n ∈ N. Moreover, it is easy to check that assumption (10.46) and the convergence .θn → θ in (10.54) imply that (3.89) holds. Finally, arguments similar to those used on pages 321–321 show that the operator .B : L2 (Γ2 )d → V is completely continuous. Then, the convergence .f n f in .L2 (Γ2 )d implies the convergence .Bf n → Bf in V that guarantees that, with the notation above, condition (3.90) holds, too. We deduce from above that we are in a position to use Theorem 3.7 in order to conclude the proof.  We end this section with the following remarks. Remark 10.1 The unique solvability of the history-dependent equation (10.53) allows us to consider the following Tykhonov triple, for any element .(θ, f ) ∈ R+ × L2 (Γ2 )d : ⎧ Tθf = (I, Ω, Cθf ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ I = R+ × L2 (Γ2 )d , ⎪ ⎪ ⎨

. Ω : R+ × L2 (Γ2 )d → 2X , Ω(ξ, g) = u(ξ, g) ⎪ ⎪ ⎪ ⎪ ∀ (ξ, g) ∈ R+ × L2 (Γ2 )d , ⎪ ⎪ ⎪

⎪ ⎪ ⎩ C = {(θ , f )} : θ → θ, f f in L2 (Γ ) . θf n n 2 n n

(10.56)

10.1 Two Frictionless Contact Problems

349

Recall that in (10.56) notation .u(ξ, g) represents the solution of equation (10.53) with the data .θ = ξ and .f = g. Moreover, a careful analysis based on the convergence result in the statement of Theorem 10.3 combined with Definition 2.1(c) shows that, under assumptions (8.60), (10.14), (10.46)–(10.48), Problem .Mθf is .Tθf -well-posed. Remark 10.2 In addition to its mathematical interest, Theorem 10.3 is important from the mechanical point of view. Indeed, it provides the existence of a unique weak solution to the viscoelastic contact problem .M, and, moreover, it shows that small perturbations of the density of surface tractions and the temperature field lead to a small perturbation of the weak solution.

10.1.5 An Optimal Control Problem In this subsection, we study an optimal control problem associated to the history-dependent problem .Mθf . To this end, we assume in what follows that (8.60), (10.14), (10.46)–(10.48) hold, and, for each .θ ∈ R+ , we consider the subset .K(θ ) of .L2 (Γ2 )d defined by

K(θ ) = f ∈ L2 (Γ2 )d : f L2 (Γ2 )d ≤ F (θ ) .

.

(10.57)

Here .F : R+ → R+ is a given function assumed to satisfy the following properties:  .

(a) F is continuous. (b) F (r) = 0 if and only if r = 0.

(10.58)

With this notation, we define the set of admissible pairs for Problem .Mθf by the equality

Vad (θ ) = (u, f ) ∈ X × L2 (Γ2 )d : f ∈ K(θ ), u = u(θ, f ) .

.

(10.59)

In other words, a pair .(u, f ) belongs to .Vad (θ ) if and only if .f ∈ K(θ ), .u ∈ X = C([0, T ]; V ), and, moreover, .u satisfies (10.53). Consider also an element .u0 such that u0 ∈ V

(10.60)

a, b > 0.

(10.61)

.

and let .

Then, the optimal control problem we are interested in is the following.

350

10 Quasistatic Contact Problems

Problem .Oθ . Given .θ ∈ R+ , find .(u∗ , f ∗ ) ∈ Vad (θ ) such that  .

a

(u∗ (T ) − u0 )2 dx + b



Ω

f ∗2 da Γ2

 ≤a

 (u(T ) − u0 )2 dx + b

Ω

f 2 da Γ2

for all .(u, f ) ∈ Vad (θ ). Our main result in this subsection is the following. Theorem 10.4 Assume (8.60), (10.14), (10.46)–(10.48), (10.58), (10.60), and (10.61). Then, for each .θ ≥ 0, there exists at least one solution .(u∗ , f ∗ ) to Problem .Oθ . Moreover, if for each .n ∈ N the pair .(u∗n , f ∗n ) is a solution to Problem .Oθn with .θn ≥ 0 and .θn → θ , then there exists a subsequence of the sequence .{(u∗n , f ∗n )}, again denoted by .{(u∗n , f ∗n )}, and a solution .(u∗ , f ∗ ) of Problem .Oθ , such that un → u

.

in X

and

f ∗n f ∗

in L2 (Γ2 )d .

(10.62)

Proof Consider the functional .L : X × L2 (Γ2 )d → R defined by L(u, f ) = a u(T ) − u0 2V + b f 2L2 (Γ

.

2)

d

∀ (u, f ) ∈ X × L2 (Γ2 )d .

(10.63)

Then, it is easy to see that Problem .Oθ may be recovered from Problem .Qθ studied in Sect. 7.2.2 by replacing Problem .Pθf with Problem .Mθf . Therefore, we shall use Theorem 7.8 with .X = C([0, T ]; V ), .Y = R, .U = R+ , .V = Z = L2 (Γ2 )d , and, to this end, we check in what follows the validity of the conditions of this theorem. First, we note that Remark 10.1 guarantees that Problem .Mθf satisfies conditions (7.34) and (7.36). Moreover, condition (7.37) is obviously satisfied. Next, assume that .θ ∈ R+ is fixed and .{θn } ⊂ R+ is a sequence such that .θn → θ. Let .f ∈ K(θ ) and, for each .n ∈ N, define fn =

.

⎧ F (θ ) n f ⎨ F (θ) ⎩

f

if if

θ = 0, θ = 0.

Then, using assumption (10.58), it follows that .f n ∈ K(θn ) and, moreover, .f n → f in .L2 (Γ2 )d . On the other hand, if .f n ∈ K(θn ) and .f n f in .L2 (Γ2 )d , using the convergence .θn → θ and (10.58), we have that f L2 (Γ2 )d ≤ lim inf f n L2 (Γ2 )d ≤ lim inf F (θn ) = F (θ ),

.

which shows that .f ∈ K(θ ). The above properties combined with Definition 1.2 show that

10.1 Two Frictionless Contact Problems

351

M

K(θn ) −→ K(θ ) in L2 (Γ2 )d .

.

We conclude from here that condition (7.38) is satisfied, too. In addition, it is easy to see that the functional (10.63) satisfies conditions (7.39), (7.40), and (7.41). It follows from above that we are in a position to apply Theorem 7.8 that guarantees the generalized well-posedness of Problem .Oθ with the Tykhonov triple ⎧ ⎪ Tθ = (I, Ω, Cθ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ I = R+ = [0, +∞),

. ∗ , f ∗ ) : (u∗ , f ∗ ) is a solution to Problem O ⎪ ∀ ξ ∈ R+ , Ω(ξ ) = (u ⎪ ξ ⎪ ⎪ ⎪

⎪ ⎪ ⎩ C = {θ } ⊂ R : θ → θ , θ n + n for each .θ ∈ R+ . In particular, this proves the solvability of Problem .Oθ , for each θ ∈ R+ . Let .θ ∈ R+ , assume that for each .n ∈ N the pair .(u∗n , f ∗n ) represents a solution to Problem .Oθn with .θn ∈ R+ , and, in addition, assume that .θn → θ . Then, Definition 2.1(c) implies that .{(u∗n , f ∗n )} is a .Tθ -approximating sequence. The existence of a subsequence of the sequence .{(u∗n , f ∗n )}, again denoted by .{(u∗n , f ∗n )}, such that (10.62) holds, is now a direct consequence of the weakly generalized wellposedness of Problem .Oθ with the triple .Tθ , guaranteed by Theorem 7.8, combined  with Remark 7.6.

.

We end this section with the following remarks. Remark 10.3 The mechanical interpretation of the optimal control problem .Oθ is the following: given an equilibrium process governed by the history-dependent equation (10.53) with the data .A, k, p, .f 0 , F that satisfy conditions (8.60), (10.46)– (10.48), and (10.58), respectively, and given a constant temperature field .θ ∈ R+ , we are looking for a density of surface tractions .f ∈ L2 (Γ2 )d such that, at the end of the time interval of interest, the solution is as close as possible to a “desired” displacement .u0 ∈ V . Furthermore, this choice has to fulfill a minimum expenditure condition that is taken into account by the last term in the structure of the cost function (10.63). In fact, a compromise policy between the two aims (“.u(T ) close to .u0 ” and “minimal data .f ”) has to be found, and the relative importance of each criterion with respect to the other is expressed by the choice of the weight coefficients .a, b > 0. Theorem 10.4 guarantees the existence of at least one optimal density .f ∗ that solves this problem. Moreover, the optimal solutions depend continuously on the temperature field, in the sense given in the statement of this theorem. Remark 10.4 Results similar to those obtained in Theorems 10.4 can be obtained if in the statement of Problem .Oθ we replace the functional (10.63) with the functional

352

10 Quasistatic Contact Problems

L(u, f ) = u(T ) − u0 2L2 (Ω)d

∀ (u, f ) ∈ X × L2 (Γ2 )d .

.

Note that in this case condition (7.41) is not satisfied. Nevertheless, we can use condition (7.42) under the assumption that .F : R+ → R+ is a bounded function.

10.2 A Frictional Contact Problem The mathematical model we consider in this section describes a bilateral contact associated to the quasistatic version of Tresca’s friction law. The time interval of interest is .R+ . We provide a variational formulation of the model in the form of a history-dependent inclusion for the strain-rate field, which represents the trait of novelty of this section. Then, we provide a well-posedness result with an appropriate Tykhonov triple and deduce a continuous dependence result.

10.2.1 The Model The classical formulation of the problem is as follows. Problem .P. Find a displacement field .u : Ω × R+ → Rd and a stress field .σ : Ω × R+ → Sd such that 

t

˙ σ (t) = Aε(u(t)) + Bε(u(t)) +

˙ C(t − s)ε(u(s)) ds

in Ω, .

(10.64)

Div σ (t) + f 0 (t) = 0

in Ω, .

(10.65)

u(t) = 0

on Γ1 , .

(10.66)

σ (t)ν = f 2 (t)

on Γ2 , .

(10.67)

uν (t) = 0

on Γ3 , .

(10.68)

on Γ3

(10.69)

.

0

σ τ (t) ≤ Fb ,

−σ τ (t) = Fb

u˙ τ (t) u˙ τ (t)

if u˙ τ (t) = 0

for all .t ∈ R+ and, moreover, u(0) = u0

.

in Ω.

(10.70)

10.2 A Frictional Contact Problem

353

The operators .A and .B in the constitutive law (10.64) are the viscosity and elasticity operators, respectively. Moreover, .C denotes the relaxation tensor. Equation (10.65) is the equation of equilibrium, while conditions (10.66), (10.67) are the displacement and traction conditions, respectively. Condition (10.68) represents the bilateral contact condition, and (10.69) represents the quasistatic version of Tresca’s friction law in which .Fb denotes the friction bound. Finally, condition (10.70) is the initial condition in which .u0 denotes a given initial displacement field. In the study of the mechanical problem (10.64)–(10.70), we use the spaces .V1 and Q, together with their inner products and associated norms, as well as the space .Q∞ , introduced in Sect. 8.1.2. We assume that the viscosity operator .A and the elasticity operator .B satisfy the conditions (8.60) and (8.61), respectively. We also assume that the relaxation tensor .C and the densities of body forces and tractions are such that .

C ∈ C(R+ ; Q∞ )..

(10.71)

f 0 ∈ C(R+ ; L2 (Ω)d )..

(10.72)

f 2 ∈ C(R+ ; L2 (Γ2 )d ).

(10.73)

Finally, for the friction bound and the initial displacement, we assume that .

Fb ∈ L2 (Γ3 )

and

Fb (x) ≥ 0 a.e. x ∈ Γ3 ..

u0 ∈ V1 .

(10.74) (10.75)

We shall keep these assumptions everywhere in this section, even if we do not mention it explicitly.

10.2.2 Variational Formulation The previous assumptions allow us to consider the operators .A : Q → Q, .S : C(R+ , Q) → C(R+ , Q), the functions .j : V1 → R, .f : R+ → V1 , the family of sets .{Σ(t)}t∈R+ , and the element .ω0 ∈ Q defined by  (Aω, .τ )Q =

Aω · τ dx

∀ ω, τ ∈ Q, .

(10.76)

Ω

(Sω(t), τ )Q = (B



 0

t

 ω(s)) ds + ω0 +



t

C(t − s)ω(s)) ds, τ )Q.

0

∀ t ∈ R+ , ω ∈ C(I ; Q), τ ∈ Q,

(10.77)

354

10 Quasistatic Contact Problems

 j (v) =

Fb v τ  da

∀ v ∈ V1 , .

(10.78)

Γ3





(f (t), v)V = Ω

f 0 (t) · v dx +

Γ2

f 2 (t) · v da

∀ v ∈ V1 , t ∈ R+ , . (10.79)

Σ(t) = τ ∈ Q : (τ , ε(v))Q + j (v) ≥ (f (t), v)V ∀ v ∈ V1 .

(10.80)

∀ t ∈ R+ , ω0 = ε(u0 ).

(10.81)

Assume in what follows that .(u, σ ) represents a regular solution of Problem .P and let .v ∈ V1 , .t ∈ R+ be arbitrary fixed. Then, using standard arguments based on the Green formula (8.16), we find that    ˙ . σ (t) · (ε(v) − ε(u(t))) dx + Fb v τ (s) da − Fb u˙ τ (s) da Ω

Γ3

Γ3



 ≥ Ω

˙ f 0 (t) · (v − u(t)) dx +

Γ2

˙ f 2 (t) · (v − u(t)) da.

We now use the notation (10.78) and (10.79) to see that ˙ ˙ ˙ (σ (t), ε(v) − ε(u(t))) ≥ (f (t), v − u(t)) Q + j (v) − j (u(t)) V

.

(10.82)

˙ and .v = 0V1 in this inequality, we obtain that and, taking successively .v = 2u(t) ˙ ˙ ˙ (σ (t), ε(u(t))) = (f (t), u(t)) Q + j (u(t)) V.

(10.83)

.

Then, the use of (10.82), (10.83), and (10.80) yields ˙ (τ − σ (t), ε(u(t))) Q ≥0

σ (t) ∈ Σ(t),

.

∀ τ ∈ Σ(t),

˙ and, with the notation .ε(u(t)) = ω(t) for the strain-rate field, we find that σ (t) ∈ Σ(t),

.

(τ − σ (t), ω(t))Q ≥ 0

∀ τ ∈ Σ(t).

(10.84)

On the other hand, the initial condition (10.70) and notation (10.81) show that 

t

ε(u(t)) =

.

ω(s) ds + ω0 .

0

So, using the constitutive law (10.64) and definitions (10.76), (10.77), we deduce that

10.2 A Frictional Contact Problem

355

(σ (t), τ )Q = (Aω(t) + Sω(t), τ )Q

.

∀τ ∈ Q

and, therefore, σ (t) = Aω(t) + Sω(t).

(10.85)

.

It follows now from (10.84), (10.85) that Aω(t) + Sω(t) ∈ Σ(t),

.

(τ − Aω(t) − Sω(t), ω(t))Q ≥ 0

∀ τ ∈ Σ(t)

and, using the definition (1.45) of the outward normal cone, we obtain the following variational formulation of Problem .P. Problem .PV . Find a strain-rate field .ω : R+ → Q such that .

  − ω(t) ∈ NΣ(t) Aω(t) + Sω(t)

∀ t ∈ R+ .

Note that Problem .PV represents a history-dependent inclusion in which the unknown is the strain-rate field. To the best of our knowledge, this problem is nonstandard since, usually, the variational formulation of a frictional viscoelastic contact problem is provided by a history-dependent variational inequality for the displacement or the velocity field, as shown in [218] and the references therein. We now state and prove the following existence and uniqueness result. Theorem 10.5 Assume (8.60), (8.61), (10.71)–(10.75). Then Problem .PV has a unique solution .ω ∈ C(R+ ; Q). Proof We use Corollary 6.3 with .X = Q and .K(t) = Σ(t) for all .t ∈ R+ and, to this end, we start by checking the validity of conditions (6.80) and (6.81). First, we introduce the set

.Σ0 = τ ∈ Q : (τ , ε(v))Q + j (v) ≥ 0 ∀ v ∈ V1 , and note that this set is a closed convex subset of Q such that .0Q ∈ Σ0 . We also note that assumptions (10.72) and (10.73) imply that the function .f given by (10.79) has the regularity f ∈ C(R+ ; V1 ).

(10.86)

.

Moreover, since .(f (t), v)V = (ε(f (t)), ε(v))Q for all .v ∈ V1 , we deduce that Σ(t) = Σ0 + ε(f (t))

.

∀ t ∈ R+ .

(10.87)

Let .t ∈ R+ . It follows from above that .Σ(t) is a nonempty convex closed subset in Q, which shows that condition (6.80)(a) is valid. Next, using (10.87), it is easy to see that

356

10 Quasistatic Contact Problems

  PΣ(t) ω = PΣ0 ω − ε(f (t) + ε(f (t))

.

∀ ω ∈ Q.

This equality and the nonexpansivity of the projection operator .PΣ0 imply that PΣ(tn ) ω − PΣ(t) ωQ ≤ 2 ε(f (tn )) − ε(f (t))Q = 2f (tn ) − f (t)V

.

for any sequence .{tn } ⊂ R+ and any .n ∈ N. We now use the regularity (10.86) to deduce that if .tn → t, then PΣ(tn ) ω − PΣ(t) ωQ → 0

∀ ω ∈ Q,

.

which guarantees that condition (6.80)(b) is valid, too. Next, note that assumption (8.60) implies that the operator (10.76) satisfies conditions (6.81) with .mA = mA and .LA = LA . Finally, assumptions (8.61), (10.71) and inequality (8.18) imply that .

Sω1 (t) − Sω2 (t)Q

(10.88) 

≤ (LB + d max C(r)Q∞ ) r∈[0,m]

t

ω1 (s) − ω2 (s)Q ds

0

for any .m ∈ N, .t ∈ [0, m], and any functions .ω1 , ω2 ∈ C(R+ ; Q). This proves that the operator .S is a history-dependent operator. It follows from above that we are in a position to apply Corollary 6.3 to conclude the proof.  Remark 10.5 A careful analysis of the proof of Corollary 10.1 below reveals that the solution .ω obtained in Theorem 10.5 has the regularity .ω ∈ C(R+ ; ε(V1 )) where, recall, .ε(V1 ) denotes the range of the operator .ε : V1 → Q. We complete Theorem 10.5 with the following existence and uniqueness result. Corollary 10.1 Assume (8.60), (8.61), (10.71)–(10.75). Then, there exists a unique couple of functions .u ∈ C 1 (R+ ; V1 ), .σ ∈ C(R+ ; Q) such that ˙ ˙ σ (t) . = Aε(u(t)) + Sε(u(t)) σ (t) ∈ Σ(t),

∀ t ∈ R+ , .

˙ (τ − σ (t), ε(u(t))) Q ≥0

(10.89) ∀ τ ∈ Σ(t), t ∈ R+ , . (10.90)

u(0) = u0 .

(10.91)

Proof Let .ω ∈ C(R+ ; Q) be the solution of Problem .PV obtained in Theorem 10.5 and denote by .σ the function given by σ (t) = Aω(t) + Sω(t)

.

∀ t ∈ R+ .

(10.92)

10.2 A Frictional Contact Problem

357

Then, .σ ∈ C(R+ , Q) and .

− ω(t) ∈ NΣ(t) (σ (t))

∀ t ∈ R+ .

(10.93)

Therefore, (1.45) yields σ (t) . ∈ Σ(t),

(τ − σ (t), ω(t))Q ≥ 0

∀ τ ∈ Σ(t), t ∈ R+ .

(10.94)

Let .t ∈ R+ and let .z ∈ Q be such that (z, ε(v))Q = 0

.

∀ v ∈ V1 .

(10.95)

Then, it is easy to see that .τ = σ (t) ± z ∈ Σ(t), and testing with these elements in (10.94) implies that (z, ω(t))Q = 0.

(10.96)

.

Equalities (10.95), (10.96) and Theorem 8.1 show that .ω(t) ∈ ε(V1 )⊥⊥ = ε(V1 ), where .M ⊥ represents the orthogonal of M in Q and .M ⊥⊥ = (M ⊥ )⊥ . Therefore, the regularity .ω ∈ C(R+ ; Q) implies that .ε −1 ω ∈ C(R+ ; V1 ) where .ε −1 : ε(V1 ) → V1 represents the inverse of the operator .ε : V1 → ε(V1 ). Consider now the function .u : R+ → V given by  u(t) =

.

0

t

ε −1 ω(s) ds + u0

∀ t ∈ R+

(10.97)

˙ = ω and, thereand note that, obviously, .u ∈ C 1 (R+ ; V1 ). Moreover, .ε(u) fore, (10.92) and (10.94) imply that (10.89) and (10.90) hold. In addition, equality (10.91) is a direct consequence of equality (10.97). This proves the existence part in Corollary 10.1. To prove the uniqueness part, consider two couples of functions .(ui , σ i ) with regularity .ui ∈ C 1 (R+ ; V ), .σ i ∈ C(R+ ; Q) such that (10.89)–(10.91) hold, with ˙ i ). Then, it is easy to see that .ωi is a solution of Problem .PV , .i = 1, 2. Let .ω i = ε(u with regularity .ωi ∈ C(R+ ; Q). We now use the uniqueness part in Theorem 10.5 to see that .ω1 = ω2 , which, in turn, implies that .u1 = u2 and .σ 1 = σ 2 .  We refer to a couple .(u, σ ) that satisfies (10.89)–(10.91) as a weak solution to the contact problem .P. We conclude from above that, under assumptions (8.60), (8.61), (10.71)–(10.75), Problem .P has a unique weak solution, with regularity .u ∈ C 1 (R+ ; V ), .σ ∈ C(R+ ; Q).

358

10 Quasistatic Contact Problems

10.2.3 A Well-Posedness Result In this subsection, we study the well-posedness of Problem .PV with a specific Tykhonov triple, associated to a variational formulation of the contact problem (10.64)–(10.70) expressed in terms of velocity. We start with the remark that for any function .η ∈ C(R+ , ε(V1 )) there exists a unique function .w ∈ C(R+ , V1 ) such that .η(t) = ε(w(t)), for all .t ∈ R+ . We have −1 (η(t)) for all .t ∈ R , and, for simplicity, we shall use the short-hand .w(t) = ε + notation .w = w η . To conclude, wη = ε −1 (η)

.

∀ η ∈ C(R+ , ε(V1 )).

(10.98)

Next, everywhere below in this subsection, we assume (8.60), (8.61), (10.71)– (10.75) and denote by .ω ∈ C(R+ ; Q) the solution of Problem .PV obtained in Theorem 10.5. Moreover, let .(u, σ ) be the weak solution obtained in Corollary 10.1 ˙ for all .t ∈ R+ . and let .w ∈ C(R+ ; V1 ) denote the velocity field, that is, .w(t) = u(t) Then (10.97) and (10.98) show that w = ε−1 (ω) = wω .

(10.99)

.

 : V1 → V1 and .S : C(R+ ; V1 ) → C(R+ , V1 ) Next, we consider the operators .A defined by . v)V = (Aε(u), ε(v))Q (Au,

∀ u, v ∈ V1 , .

 (Su(t), v)V = (Sε(u(t)), ε(v))Q

(10.100) (10.101)

∀ t ∈ R+ , u ∈ C(R+ ; V1 ), v ∈ V1 , where, recall, A and .S are given by (10.76) and (10.77), respectively. Our preliminary result in this subsection is the following. Lemma 10.1 The velocity field .w satisfies the following inequality:   . (Aw(t), v − w(t))V + (Sw(t), v − w(t))V + j (v) − j (w(t)) ≥ (f (t), v − w(t))V

(10.102)

∀ v ∈ V1 , t ∈ R+ .

Proof Let .t ∈ R+ . Then, the subdifferentiability of the function j at the point .w(t), guaranteed by Proposition 1.19, implies that there exists .ξ (t) ∈ V1 such that j (v) − j (w(t)) ≥ (ξ (t), v − w(t))V

.

∀ v ∈ V1 .

(10.103)

10.2 A Frictional Contact Problem

359

Let .τ 0 (t) = ε(f (t) − ξ (t)). Then, (8.8) and (10.103) imply that .

(τ 0 , ε(v) − ε(w(t)))Q + j (v) − j (w(t))

(10.104)

∀ v ∈ V1

≥ (f (t), v − w(t))V

and, taking .v = 2w(t), .v = 0V1 in the previous inequality, we find that (τ 0 (t), ε(w(t)))Q + j (w(t)) = (f (t), w(t))V .

.

(10.105)

We now combine (10.105) and (10.104) to obtain that (τ 0 (t), ε(v))Q + j (v) ≥ (f (t), v)V

∀ v ∈ V1 ,

.

and then we use (10.80) to see that τ 0 (t) ∈ Σ(t).

(10.106)

.

The regularity (10.106) allows us to test with .τ = τ 0 (t) in (10.90). Therefore, ˙ = w(t), we obtain that using the equality .u(t) (τ 0 (t) − σ (t), ε(w(t)))Q ≥ 0,

.

and combining this inequality with (10.105), we find that (f (t), w(t))V ≥ (σ (t), ε(w(t)))Q + j (w(t)).

.

(10.107)

Now, the regularity .σ (t) ∈ Σ(t) in (10.90) guarantees that (σ (t), ε(v))Q + j (v) ≥ (f (t), v)V

∀ v ∈ V1

(10.108)

(σ (t), ε(w(t)))Q + j (w(t)) ≥ (f (t), w(t))V .

(10.109)

.

and, therefore, .

We now use (10.107) and (10.109) to see that (σ (t), ε(w(t)))Q + j (w(t)) = (f (t), w(t))V ,

.

(10.110)

and then we use (10.108) and (10.110) to deduce that .

(σ (t), ε(v) − ε(w(t)))Q + j (v) − j (w(t)) ≥ (f (t), v − w(t))V

∀ v ∈ V1 .

(10.111)

360

10 Quasistatic Contact Problems

Finally, we substitute the right-hand side of equality (10.89) in (10.111), and then we use the definitions (10.100), (10.101) and the equality .u˙ = w to obtain (10.102), which concludes the proof.  Consider now the Tykhonov triple .T = (I, Ω, C), where I .= R+ = [0, +∞), C = {θn }n : θn ∈ I ∀ n ∈ N,

θn → 0

as

n→∞

and, for each .θ ∈ I , the set .Ω(θ ) is defined as follows: ⎧ ⎪ η ∈ Ω(θ ) if and only if η ∈ C(R+ ; ε(V1 )) and ⎪ ⎪ ⎪ ⎪ ⎪ ⎨   η (t), v − w η (t))V (Aw η (t), v − wη (t))V + (Sw . ⎪ ⎪ ⎪ +j (v) − j (wη (t)) + θ v − wη (t)V ⎪ ⎪ ⎪ ⎩ ≥ (f (t), v − w η (t))V ∀ v ∈ V1 , t ∈ R+ .

(10.112)

Using Remark 10.5, (10.99), and Lemma 10.1, it follows that .ω ∈ Ω(θ ) for each θ ∈ I and, therefore, .Ω(θ ) =  ∅, for each .θ ∈ I . Our main result in this subsection is the following.

.

Theorem 10.6 Assume (8.60), (8.61), (10.71)–(10.75). Then Problem .PV is .T well-posed. Proof The unique solvability of Problem .PV follows from Theorem 10.5. Assume now that .{ωn } represents a .T -approximating sequence. Then, using Definition 2.1(b) and (10.112), it follows that there exists a sequence .{θn }n ⊂ R+ such that .θn → 0 and, denoting .wωn = w n , we have .

 n (t), v − wn (t))V  n (t), v − w n (t))V + (Sw (Aw +θn v − wn (t)V ≥ (f (t), v − w n (t))V

(10.113)

∀ v ∈ V1 , t ∈ R+ , n ∈ N.

Let .n, m ∈ N and let .t ∈ [0, m]. Then, taking .v = wn (t) in (10.102), .v = w(t) in (10.113) and adding the resulting inequalities, we find that .

 n (t) − Aw(t),  (Aw wn (t) − w(t))V  n (t) − Sw(t),  ≤ (Sw w(t) − w n (t))V + θn wn (t) − w(t)V

and, therefore, .

 n (t) − Aw(t),  (Aw wn (t) − w(t))V

(10.114)

  n (t) − Sw(t) ≤ Sw V w(t) − w n (t)V + θn w(t) − w n (t)V .

10.2 A Frictional Contact Problem

361

 : V1 → On the other hand, (10.100), (10.101), and (8.9) show that the operator .A V1 is strongly monotone and Lipschitz continuous with the same constants as the operator .A : Q → Q. Moreover, the operator .S : C(R+ ; V1 ) → C(R+ ; V1 ) is history-dependent with the same constants as the operator .S : C(R+ ; Q) → C(R+ ; Q). Based on these remarks and using (10.88), (10.114), we deduce that .

mA wn (t) − w(t)V  ≤ θn + (LB + d max C(r)Q∞ ) r∈[0,m]

t

wn (s) − w(s)V ds.

0

Next, we use Gronwall’s inequality to see that there exists a constant .C > 0, which depends on .A, .B, .C, and m but does not depend on n, such that .

max w n (t) − w(t)V ≤ Cθn .

t∈[0,m]

We now use (8.9) and the equalities .ε(w n ) = ωn , .ε(w) = ω to deduce that .

max ωn (t) − ω(t)Q ≤ Cθn .

t∈[0,m]

Finally, the convergence .θn → 0 and (1.3) imply that .ωn → ω in .C(R+ ; Q), which shows that Problem .PV is .T -well-posed.  We now proceed with the following convergence result. Corollary 10.2 Keep the assumption of Theorem 10.6. Then, the solution .ω of Problem .PV depends continuously on the friction bound, i.e., if .ωn represents the solution of Problem .PV with the data .Fbn that satisfies (10.74) for each .n ∈ N and, moreover, Fbn → Fb

.

in L2 (Γ3 ),

(10.115)

in C(R+ , Q).

(10.116)

then ωn → ω

.

Proof Let .v ∈ V1 , .t ∈ R+ , .n ∈ N and denote .wn = wωn . It follows from Lemma 10.1 that .

 n (t), v − w n (t))V  n (t), v − wn (t))V + (Sw (Aw +jn (v) − jn (w n (t)) ≥ (f (t), v − wn (t))V

(10.117) ∀ v ∈ V1 , t ∈ R+ ,

362

10 Quasistatic Contact Problems

where, here and below,  jn (v) =

.

Γ3

Fbn v τ  da

∀ v ∈ V1 .

(10.118)

We now use (10.118), (10.78) and the trace inequality (8.14) to see that .

jn (v) − jn (wn (t)) = j (v) − j (wn (t))  + Γ3

(Fbn − Fb )(v τ  − wnτ (t)) da 

≤ j (v) − j (wn (t)) + Γ3

|Fbn − Fb | v − wn (t) da

≤ j (v) − j (wn (t)) + c0 Fbn − Fb L2 (Γ3 ) v − w n (t)V and, using the notation θn = c0 Fbn − Fb L2 (Γ3 ) ,

.

we find that jn (v) − jn (wn (t)) ≤ j (v) − j (wn (t)) + θn v − wn (t)V .

.

(10.119)

We now combine (10.117), (10.119), and (10.112) to deduce that .ωn ∈ Ω(θn ). Next, assumption (10.115) shows that .θn → 0 and, therefore, .{ωn } is a .T approximating sequence. The convergence (10.116) is now a direct consequence of Theorem 10.6.  We end this section with the following remarks: (1) Corollary 10.2 can be used to prove the continuous dependence of the weak solution .(u, σ ) of the contact problem .P with respect to the friction bound. In addition to the mathematical interest in this result, it is important from the mechanical point of view since, in particular, it shows that we can approach the solution of the frictionless contact problem by the solution of the frictional contact problem, for a small friction bound. (2) The method presented above in this subsection allows us to provide the continuous dependence of the weak solution .(u, σ ) with respect to the densities of body forces and surface tractions. Nevertheless, in this case, we need to work with a different Tykhonov triple, since the data .f 0 and .f 2 are time-dependent.

10.3 A Contact Problem for Rate-Type Materials

363

10.3 A Contact Problem for Rate-Type Materials In this section, we study a mathematical model that describes the frictionless contact with a foundation made of a rigid body covered by a deformable layer of thickness g. We model the material’s behavior with a rate-type constitutive law and the contact with normal compliance and unilateral constraint. The time interval of interest is .R+ = [0, +∞). The analysis of the model is carried out by using a fixed point argument and arguments on history-dependent variational inequalities, as well.

10.3.1 The Model The classical formulation of the model is the following. Problem .P. Find a displacement field .u : Ω × R+ → Rd and a stress field .σ : Ω × R+ → Sd such that .

˙ + α(σ − B(ε(u)) σ˙ = Eε(u)

in Ω × R+ , .

(10.120)

Div σ + f 0 = 0

in Ω × R+ , .

(10.121)

u=0

on Γ1 × R+ , .

(10.122)

σν = f 2

on Γ2 × R+ , .

(10.123)

on Γ3 × R+ , .

(10.124)

on Γ3 × R+.

(10.125)

in Ω.

(10.126)

uν ≤ g,

σν + p(uν ) ≤ 0,



(uν − g)(σν + p(uν )) = 0 στ = 0 σ (0) = σ 0 ,

u(0) = u0

The novelty of Problem .P arises from the fact that now we use the constitutive law (10.120), in which .α represents a viscosity coefficient and .B is a nonlinear relaxation operator. As a consequence, we need the initial conditions (10.126) in which the functions .u0 and .σ 0 are some initial displacement and initial stress, respectively. In the analysis of Problem .P, we use the spaces V and Q defined by (8.6) and (8.3), respectively, together with their Hilbertian structures. We assume that the elasticity operator .E satisfies condition (8.67), the relaxation operator .B satisfies condition (8.61), and the viscosity coefficient .α is such that (8.22) holds. Moreover, the densities of body forces and surface traction have the regularity (10.72) and (10.73), respectively. The thickness g satisfies condition (10.18), the normal

364

10 Quasistatic Contact Problems

compliance function is such that (10.14) and (10.47) hold, and, finally, the initial data have the regularity σ 0 ∈ Q,

u0 ∈ V .

.

(10.127)

We also recall that the following inequalities hold: .

vL2 (Ω)d ≤ c0 vV

∀ v ∈ V,.

(10.128)

c0 vV vL2 (Γ2 )d ≤ 

∀ v ∈ V,.

(10.129)

vL2 (Γ3 )d ≤ c0 vV

∀ v ∈ V,

(10.130)

.c0 , and .c0 are positive constants that depend on .Ω, .Γ2 , and .Γ3 , where .c0 ,  respectively.

10.3.2 A Fixed Point Argument We now turn to construct a weak formulation of the problem. To this end, we consider the set .K ⊂ V , the operators .A : V → V , .Λ : C(R+ ; Q × Q) → C(R+ ; Q) and the function .f : R+ → V defined by

K . = v ∈ V : vν ≤ g a.e. on Γ3 , . 

(10.131)



(Au, v)V =

Eε(u) · ε(v) dx + Ω

p(uν )vν da

∀ u, v ∈ V , . (10.132)

Γ3

 Λ(σ , τ )(t) =

t

α(σ (s) − B(τ (s))) ds + σ 0 − Eε(u0 ).

(10.133)

0

∀ σ , τ ∈ C(R+ ; Q), t ∈ R+ ,  (f (t), v)V = Ω

 f 0 (t) · v dx +

Γ3

f 2 (t) · v da

(10.134)

∀ v ∈ V , t ∈ R+ . Assume now that .(u, σ ) is a smooth solution of Problem .P. Let .v ∈ K, .t ∈ R+ and note that

10.3 A Contact Problem for Rate-Type Materials

365

u(t) ∈ K.

(10.135)

.

We integrate the constitutive law (10.120) with the initial conditions (10.126) to find that  t .σ (t) = Eε(u(t)) + α(σ (s) − B(ε(u(s))) ds + σ 0 − Eε(u0 ). (10.136) 0

Moreover, we use standard arguments based on the Green formula to see that 

 σ (t) · (ε(v) − ε(u(t))) dx +

.

Ω



p(uν (t))(vν − uν ) da



≥ Ω

f 0 (t) · (v − u) dx +

(10.137)

Γ3

Γ2

f 2 (t) · (v − u) da.

Let  η(t) =

.

t

α(σ (s) − B(ε(u(s))) ds + σ 0 − Eε(u0 )

(10.138)

0

and note that .η(t) represents the anelastic part of the stress field at the moment t. We now combine relations (10.135)–(10.138) and then use definitions (10.132)– (10.134) to deduce the following weak formulation of Problem .P. Problem .P V . Find a displacement field .u ∈ C(R+ ; V ), a stress field .σ ∈ C(R+ ; Q), and an anelastic stress field .η ∈ C(R+ ; Q) such that, for any .t ∈ R+ , the following hold: .

σ (t) = Eε(u(t)) + η(t), u(t) ∈ K,

(Au(t), v − u(t))V + (η(t), ε(v) − ε(u(t)))Q

≥ (f (t), v − u(t))V

∀ v ∈ K,

η(t) = Λ(σ (t), ε(u(t))). We now state and prove the following existence and uniqueness result. Theorem 10.7 Assume (8.22), (8.61), (8.67), (10.14), (10.18), (10.47), (10.72), (10.73), and (10.127). Then Problem .P V has a unique solution. Proof The proof of Theorem 10.7 is carried out in four steps, based on a fixed point argument, already used in various references, including [218, 222]. For this reason, we restrict ourselves to the following brief description of the steps:

366

10 Quasistatic Contact Problems

Step (i)

An intermediate stress–displacement problem.

We claim that for each .η ∈ C(R+ ; Q) there exists a unique couple of functions (σ η , uη ) such that .uη ∈ C(R+ ; V ), .σ η ∈ C(R+ ; Q), and

.

σ η (t) . = Eε(uη (t)) + η(t), .

(10.139)

uη (t) ∈ K,

(10.140)

(Auη (t), v − uη (t))V + (η(t), ε(v) − ε(uη (t)))Q ≥ (f (t), v − uη (t))V

∀ v ∈ K,

for any .t ∈ R+ . Indeed, let .η ∈ C(R+ ; Q). We use assumptions (8.67), (10.14), and (10.47) to see that the operator .A : V → V defined by (10.132) is strongly monotone and Lipschitz continuous. Moreover, assumption (10.18) implies that the set K is a closed nonempty convex subset of V , and the regularities (10.72), (10.73) guarantee that .f ∈ C(R+ ; V ). Then, the existence of a unique function .uη ∈ C(R+ ; V ) that solves (10.140) follows from Corollary 1.6. We now use equality (10.139) to obtain the existence and uniqueness part of this step. Step (ii)

A Lipschitz continuous dependence result.

We claim that if .σ i = σ ηi and .ui = uηi for .ηi ∈ C(R+ ; Q), .i = 1, 2, then there exists a constant .C0 > 0 such that σ 1 (t) − σ 2 (t)Q + u1 (t) − u2 (t)V ≤ C0 η1 (t) − η2 (t)Q

.

(10.141)

for all .t ∈ R+ . The estimate (10.141) follows from standard arguments applied to the system (10.139), (10.140) and, therefore, we skip its proof. Step (iii)

A fixed point problem for the anelastic stress field.

Note that Step (i) allows us to introduce the operator .S : C(R+ ; Q) → C(R+ ; Q) defined by Sη(t) = Λ(σ η (t), ε(uη (t)))

.

∀ η ∈ C(R+ ; Q), t ∈ R+ .

(10.142)

We now consider the fixed point problem of finding an anelastic stress field .η∗ ∈ C(R+ ; Q) such that Sη∗ (t) = η∗ (t)

.

∀ t ∈ R+ ,

(10.143)

and claim that this problem has a unique solution .η∗ ∈ C(R+ ; Q). For the proof, we consider two elements .η1 , η2 ∈ C(R+ ; Q). Let .σ i = σ ηi , .ui = uηi for .i = 1, 2 and let .t ∈ R+ . We use equalities (10.142), (10.133) and assumptions (8.22), (8.61) to see that

10.3 A Contact Problem for Rate-Type Materials .

367

Sη1 (t) − Sη2 (t)Q 

t

≤ αL∞ (Ω) 0

(10.144) (σ 1 (s) − σ 2 (s)Q + LB u1 (s) − u2 (s)V ) ds.

Next, we use inequality (10.141) to see that .S is a history-dependent operator. Finally, we end the proof of this step by using Theorem 1.4. Step (iv) End of proof. It follows from (10.143), (10.142) that η∗ (t) = Λ(σ η∗ (t), ε(uη∗ (t)))

.

∀ t ∈ R+ .

(10.145)

We now write (10.139) and (10.140) for .η = η∗ , and then we combine the resulting relations with (10.145) to see that the triple (.uη∗ ,.σ η∗ .η∗ ) represents a solution of Problem .P V . This proves the existence part in Theorem 10.7. The uniqueness part results from the uniqueness of the fixed point of the historydependent operator (10.142), guaranteed by Theorem 1.4. 

10.3.3 Convergence Results In this subsection, we prove the continuous dependence of the solution with respect to part of the data. To this end, we keep assumptions of Theorem 10.7, and, in addition, we consider the sequences .{αn }, .{σ 0n }, .{u0n } such that, for each .n ∈ N, the following conditions hold: .

αn ∈ L∞ (Ω), αn (x) ≥ 0 σ 0n ∈ Q,

a.e. x ∈ Ω..

u0n ∈ V .

(10.146) (10.147)

With these data, we define the operator .Λn : C(R+ ; Q × Q) → C(R+ ; Q) by the equality 

t

Λn (σ . , τ )(t) =

αn (σ (s) − B(τ (s))) ds + σ 0n − Eε(u0n )

(10.148)

0

∀ σ , τ ∈ C(R+ ; Q), t ∈ R+ . Then, we consider the following variational problem. Problem .PnV . Find a displacement field .un ∈ C(R+ ; V ), a stress field .σ n ∈ C(R+ ; Q), and an anelastic stress field .ηn ∈ C(R+ ; Q) such that, for any .t ∈ R+ , the following hold:

368

10 Quasistatic Contact Problems

σ n. (t) = Eε(un (t)) + ηn (t), un (t) ∈ K,

(Aun (t), v − un (t))V + (ηn (t), ε(v) − ε(un (t)))Q

≥ (f (t), v − un (t))V

∀ v ∈ K,

ηn (t) = Λn (σ n (t), ε(un (t))). Using Theorem 10.7, it follows that Problem .PnV has a unique solution, for each .n ∈ N. Consider now the following additional assumptions: .

αn → α in L∞ (Ω), as n → ∞.. σ 0n → σ 0

u0n → u0

in Q,

(10.149) in V , as n → ∞.

(10.150)

We have the following convergence result. Theorem 10.8 Assume (8.22), (8.61), (8.67), (10.14), (10.18), (10.47), (10.72), (10.73), (10.127), (10.146), (10.147), (10.149), and (10.150). Then, the solution V V .(un , σ n , η n ) of Problem .Pn converges to the solution .(u, σ , η) of Problem .P , i.e., .

un → u

in C(R+ , V ), as n → ∞..

(10.151)

σn → σ

in C(R+ , Q), as n → ∞..

(10.152)

ηn → η

in C(R+ , Q), as n → ∞.

(10.153)

Proof Let .n ∈ N and let .Sn : C(R+ ; Q) → C(R+ ; Q) be the operator defined by Sn η(t) = Λn (σ η (t), ε(uη (t)))

.

∀ η ∈ C(R+ ; Q), t ∈ R+ .

(10.154)

Recall that, here and below, we use the notation .(σ η , uη ) for the solution of problem (10.139), (10.140). Let .η ∈ C(R+ ; Q), .m ∈ N, and .t ∈ [0, m]. Then, a simple calculation based on the definitions (10.154), (10.142), (10.148), (10.133) combined with the bound (8.18) and arguments similar to those used in the proof of inequality (10.144) show that .

Sn η(t) − Sη(t)Q

(10.155) 

≤ αn − αL∞ (Ω)

t

  σ η (s)Q + B(ε(uη (s)))Q ds

0

+σ 0n − σ 0 Q + d EQ∞ u0n − u0 V .

10.3 A Contact Problem for Rate-Type Materials

369

Denote

εn = max. αn − αL∞ (Ω) , σ 0n − σ 0 Q + d EQ∞ u0n − u0 V .

(10.156)

Then, inequality (10.155) implies that Sn η(t) − Sη(t)Q

.

(10.157)

 t  

σ η (s)Q + B(ε(uη (s)))Q ds . ≤ εn 1 + 0

Next, let .(u0 , σ 0 ) be the solution obtained in Step (i) of the proof of Theorem 10.7 corresponding to the function .η(t) = 0Q for all .t ∈ R+ . Then, using assumption (8.61) and inequality (10.141), we find that .

σ η (s)Q + B(ε(uη (s)))Q ≤ σ η (s) − σ 0 (s)Q

(10.158)

+B(ε(uη (s))) − B(ε(u0 (s)))Q + σ 0 (s)Q + B(ε(u0 (s)))Q ≤ C0 max {1, LB }η(s)Q + σ 0 (s)Q + B(ε(u0 (s)))Q

∀ s ∈ [0, t].

Let   Bm = max σ 0 (s)Q + B(ε(u0 (s)))Q .

.

s∈[0,m]

(10.159)

Then, combining (10.157)–(10.159), we deduce that 



Sn η(t) − Sη(t)Q ≤ εn 1 + C0 max {1, LB }

.

t

η(s)Q ds + mBm



0

and, using the notation

θnm = εn max 1 + mBm , C0 max {1, LB } ,

(10.160)

 t

Sn η(t) − Sη(t)Q ≤ θnm 1 + η(s)Q ds .

(10.161)

.

we find that .

0

Moreover, assumptions (10.149), (10.150) and notation (10.156), (10.160) show that θnm → 0 as n → ∞.

.

(10.162)

370

10 Quasistatic Contact Problems

It follows now from (10.161) and (10.162) that condition (3.133) on page 132 is satisfied with .Λn = Sn and .Λ = S. We now use Theorem 3.10(c) to deduce the convergence (10.153). Finally, inequalities (10.141) and (10.153) imply the convergences (10.151) and (10.152), which concludes the proof.  We end this subsection with the following comments: (1) The solution of Problem .P V is called a weak solution for the contact Problem .P. We conclude from above that Theorem 10.7 provides the unique weak solvability of this contact problem and Theorem 10.8 represents a continuous dependence result of the weak solution with respect to the data. (2) In addition to the mathematical interest in the convergences (10.151)–(10.153), they are important from the mechanical point of view since they show that small perturbations on the viscosity coefficient .α, the initial stress field .σ 0 , and the initial displacement field .u0 lead to a small perturbation of the weak solution of the contact problem .P.

10.3.4 A Well-Posedness Result We now present a different approach in the study of Problem .P. To this end, we keep the assumptions of Theorem 10.7 as well as notation (10.131), (10.134). Moreover, we introduce the short-hand notation Y for the set defined by Y = C(R+ , K) × C(R+ , Q).

.

In addition, we note that assumption (8.67) implies that there exists .LE > 0 such that Eε(u) − Eε(v)Q ≤ LE u − vV

.

∀ u, v ∈ V

(10.163)

and, moreover, (Eε(u) − Eε(v), ε(u) − ε(v))Q ≥ mE u − v2V

.

∀ u, v ∈ V .

(10.164)

Then, using (10.136) and inequality (10.137), we deduce the following variational formulation of Problem .P. V . Find a displacement field .u ∈ C(R+ , K) and a stress field .σ : Problem .P C(R+ , Q) such that 

t

σ (t) . = Eε(u(t)) + 0

α(σ (s) − Bε(u(s))) ds + σ 0 − Eε(u0 ), . (10.165)

10.3 A Contact Problem for Rate-Type Materials

371

 (σ (t), ε(v) − ε(u(t)))Q +

p(uν (t))(vν − uν (t)) da

(10.166)

Γ3

≥ (f (t), v − u(t))V for all .v ∈ K and .t ∈ R+ . Note that, in contrast to the variational formulation .P V (in which the unknowns are the displacement field, the stress field, and the anelastic stress field), Problem V is formulated in terms of the displacement and stress fields. Nevertheless, it is .P easy to see that if .(u, σ , η) is a solution of Problem .P V , then .(u, σ ) is a solution of V . Conversely, if .(u, σ ) is a solution of Problem .P V and .η is the function Problem .P defined by (10.138), then .(u, σ , η) is a solution of Problem .P V . This remark allows V . However, in order to us to use Theorems 10.7 and 10.8 in the study of Problem .P introduce a well-posedness result, we present in what follows a direct approach in the analysis of this problem. V is given by the following existence and uniqueness The unique solvability of .P result. Theorem 10.9 Assume (8.22), (8.61), (8.67), (10.14), (10.18), (10.47), (10.72), V has a unique solution .(u, σ ) ∈ Y . (10.73), and (10.127). Then Problem .P Proof We use arguments similar to those used in the proof of Theorem 114 in [222] and, for this reason, we skip the details. We restrict ourselves to mention that the proof is structured in four steps, as follows: Step (i) First, we use Theorem 34 in [222] to see that there exists an operator .R : C(R+ ; V ) → C(R+ ; Q) such that, for all functions .u ∈ C(R+ ; V ) and .σ ∈ C(R+ ; Q) and .t ∈ R+ , equality (10.165) holds if and only if σ (t) = Eε(u(t)) + Rε(u(t)),

.

(10.167)

for all .t ∈ R+ . Moreover, the operator .R : C(R+ ; V ) → C(R+ ; Q) is a historydependent operator, i.e., ⎧ for every m ∈ N there exists dm > 0 such that ⎪ ⎪ ⎪ ⎪ ⎪  t ⎨ . Ru1 (t) − Ru2 (t)Q ≤ dm u1 (s) − u2 (s)V ds ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩ ∀ u1 , u2 ∈ C(R+ ; V ), ∀ t ∈ [0, m].

(10.168)

Step (ii) Next, we consider the auxiliary problem of finding a function .u ∈ C(R+ ; K) such that (Eε(u(t)), . ε(v) − ε(u(t)))Q + (Rε(u(t)), (ε(v) − ε(u(t)))Q (10.169)  p(uν (t))(vν − uν (t)) da ≥ (f (t), v − u(t))V

+ Γ3

372

10 Quasistatic Contact Problems

for all .v ∈ K and .t ∈ R+ . This problem is obtained by substituting the stress field given by equality (10.167) in the variational inequality (10.166). Then, it is easy to see that a pair .(u, σ ) with regularity .(u, σ ) ∈ Y is a solution to Problem V if and only if both equality (10.167) and inequality (10.169) hold, for all .P .v ∈ K and .t ∈ R+ . Step (iii) Note that K is a nonempty closed convex subset of the space V . Therefore, the property (10.168) of the operator .R combined with assumptions (8.67), (10.14), (10.47), among others, allows us to apply Corollary 1.6 in order to deduce that inequality (10.169) has a unique solution .u ∈ C(R+ ; K). Step (iv) Let .σ be the function defined by equality (10.167). Then, it is easy to see that .σ ∈ C(R+ ; Q). Moreover, using the arguments in Step ii) above, it V , with regularity follows that the pair .(u, σ ) is the unique solution of Problem .P .(u, σ ) ∈ Y .  V . We now provide several Tykhonov triples in the study of Problem .P Example 10.1 Take .T = (I, Ω, C) where .

I = θ = {θ m }m : θ m ∈ R+

∀m ∈ N ,

C = {θ n }n : θ n = {θnm } ⊂ I ∀ n ∈ N, θnm → 0

as

n → ∞,

∀m ∈ N

and, for each .θ = {θ m }m ∈ I , the set .Ω(θ) is defined as follows: .(u, σ ) ∈ Ω(θ) if and only if (u,. σ ) ∈ Y,  t       α σ (s) − Bε(u(s)) ds − σ 0 + Eε(u0 ) ≤ θ m σ (t) − Eε(u(t)) − Q

0

∀ t ∈ [0, m], m ∈ N,  (σ (t), ε(v) − ε(u(t)))Q +

p(uν (t))(vν − uν (t)) da Γ3

+θ m v − u(t)V ≥ (f (t), v − u(t))V ∀ v ∈ K, t ∈ [0, m], m ∈ N. Note that the solution .(u, σ ) in Theorem 10.9 is such that .(u, σ ) ∈ Ω(θ ) for each .θ ∈ I and, therefore, .Ω(θ ) = ∅ for each .θ ∈ I .

10.3 A Contact Problem for Rate-Type Materials

373

Example 10.2 Take .T = (I, Ω, C), where .

I = R+ = [0, +∞), C = {θn }n ⊂ I : θn → 0

as

n→∞

and, for each .θ ∈ I , the set .Ω(θ ) is defined as follows: .(u, σ ) ∈ Ω(θ ) if and only if the following conditions hold: (u,. σ ) ∈ Y,  t     α(σ (s) − Bε(u(s))ds − σ 0 + Eε(u0 ) ≤ θ σ (t) − Eε(u(t)) − Q

0

∀ t ∈ R+ ,  (σ (t), ε(v) − ε(u(t)))Q +

p((uν (t))(vν − uν (t)) da

(10.170)

Γ3

+θ v − u(t)V ≥ (f (t), v − u(t))V

∀ v ∈ K, t ∈ R+ .

Note that, again, .Ω(θ) =  ∅, for each .θ ∈ I . Our main result in this subsection is the following. V is wellTheorem 10.10 Keep the assumptions of Theorem 10.9. Then Problem .P posed with the Tykhonov triples introduced in Examples 10.1 and 10.2. Proof We start by recalling that the existence of a unique solution .(u, σ ) ∈ Y to V was provided in Theorem 10.9. Problem .P To proceed, we focus on the Tykhonov triple .T from Example 10.1. Consider a .T -approximating sequence, denoted by .{(un , σ n )}. Then, according to Definition 2.1(b), it follows that there exists a sequence .{θ n }n ∈ C with .θ n = {θnm }m ∈ I such that θnm → 0

.

as

n → ∞,

∀m ∈ N

(10.171)

and, moreover, for each .n ∈ N, the following properties hold: (un , σ n. ) ∈ Y, .

(10.172)

 t       α σ n (s) − Bε(un (s) ds − σ 0 + Eε(u0 ) σ n (t) − Eε(un (t)) − 0

≤ θnm ∀ t ∈ [0, m], m ∈ N,

.

Q

(10.173)

374

10 Quasistatic Contact Problems

 (σ n (t), ε(v) − ε(un (t)))Q +

p(unν (t))(vν − unν (t)) da

(10.174)

Γ3

+θnm v − un (t)V ≥ (f (t), v − un (t))V ∀ v ∈ K, t ∈ [0, m], m ∈ N. Let .n ∈ N be fixed. We introduce the functions .ηn : R+ → Q and .η : R+ → Q defined by  ηn (t) =

.

t

  α σ n (s) − Bε(un (s)) ds + σ 0 − Eε(u0 ), .

(10.175)

0

 η(t) =

t

  α σ (s) − Bε(u(s)) ds + σ 0 − Eε(u0 ),

(10.176)

0

for all .t ∈ R+ . Consider now .m ∈ N and let .t ∈ [0, m]. Then, using (10.173) and (10.175), it follows that σ n (t) − Eε(un (t)) − ηn (t)Q ≤ θnm .

.

(10.177)

Moreover, using (10.165) and (10.176), we have σ (t) = Eε(u(t)) + η(t).

.

(10.178)

Then, we use (10.178) to write .

  σ n (t) − σ (t) = σ n (t) − Eε(un (t)) − ηn (t)

(10.179)

    + Eε(un (t)) − Eε(u(t)) + ηn (t) − η(t) , which implies that .

σ n (t) − σ (t)Q ≤ σ n (t) − Eε(un (t)) − ηn (t)Q +Eε(un (t)) − Eε(u(t))Q + ηn (t) − η(t)Q .

Next, we use inequalities (10.177) and (10.163) to find that σ n (t) − σ (t)Q ≤ θnm + LE un (t) − u(t)V + ηn (t) − η(t)Q .

.

(10.180)

On the other hand, we test in (10.174) with .v = u(t), then in (10.166) with .v = un (t), both in K, and add the resulting inequalities to obtain that

10.3 A Contact Problem for Rate-Type Materials

375

(σ n (t) − σ (t), ε(un (t)) − ε(u(t)))Q  ≤ (p(unν (t) − p(uν (t))(uν (t)) − unν (t)) da

.

(10.181)

Γ3

+θnm u(t) − un (t)V . We now substitute (10.179) to (10.181) and use (10.164), (10.47) to find that .

mE un (t) − u(t)2V ≤ (σ n (t) − Eε(un (t)) − ηn (t), ε(u(t)) − ε(un (t)))Q +(ηn (t) − η(t), ε(u(t)) − ε(un (t)))Q + θnm u(t) − un (t)V .

Next, exploiting inequality (10.177), we deduce that .

mE un (t) − u(t)V ≤ 2 θnm + ηn (t) − η(t)Q

and, therefore, un (t) − u(t)V ≤

.

2 m 1 θ + η (t) − η(t)Q . mE n mE n

(10.182)

To proceed, we use definitions (10.175) and (10.176) together with property (8.61)(b) of the operator .B to see that  .

ηn (t) − η(t)Q ≤ αL∞ (Ω)  +αL∞ (Ω) LB

t

t

σ n (s) − σ (s)Q ds

(10.183)

0

un (s) − u(s)V ds.

0

We now combine inequalities (10.180), (10.182), and (10.183), use inequality .t ≤ m, and, after some algebra, we find that there exist two positive constants .C1 > 0 and .C2 > 0 that depend on .E, .α, and .B but are independent of t, m, and n such that  ηn (t) − η(t)Q ≤ C1 mθnm + C2

t

.

0

ηn (s) − η(s)Q ds.

Therefore, using the Gronwall argument, we deduce that ηn (t) − η(t)Q ≤ C1 meC2 t θnm

.

and, moreover, .

max ηn (t) − η(t)Q ≤ C1 meC2 m θnm .

t∈[0,m]

(10.184)

376

10 Quasistatic Contact Problems

We now use inequalities (10.182), (10.184) and the convergence (10.171) to find that .

max un (t) − u(t)V → 0

t∈[0,m]

as

n → ∞.

(10.185)

Finally, inequalities (10.180), (10.184) and the convergences (10.171), (10.185) guarantee that .

max σ n (t) − σ (t)Q → 0

t∈[0,m]

as

n → ∞.

(10.186)

Next, (1.5) and the convergences (10.185), (10.186) yield .

un → u

in C(R+ , V ), as n → ∞..

(10.187)

σn → σ

in C(R+ , Q), as n → ∞.

(10.188)

V These convergences and Definition 2.1(c) imply the well-posedness of Problem .P with the Tykhonov triple defined in Example 10.1. V with the Tykhonov triple in Example 10.2 The well-posedness of Problem .P follows from similar arguments and, therefore, we skip its proof. Note that in this case some estimates are simpler since various quantities do not depend on m. It also follows from the proof above, since, obviously, the Tykhonov triple introduced in Example 10.2 is smaller than the Tykhonov triple from Example 10.1.  We end this section by recalling that a pair of functions .(u, σ ) ∈ Y that satisfies (10.165) and (10.166) is called a weak solution to Problem .P.

10.3.5 Additional Convergence Results V depends on the data .f 0 and .f 2 . Its continuous The solution of Problem .P dependence with respect to these data is provided by the following convergence result, which completes the statement of Theorem 10.9. Theorem 10.11 Assume (8.22), (8.61), (8.67), (10.14), (10.18), (10.47), V . (10.72), (10.73), (10.127) and denote by .(u, σ ) the solution of Problem .P V  for Moreover, for each .n ∈ N, denote by .(un , σ n ) the solution of Problem .P the data .f 0n and .f 2n that satisfy f 0n ∈ .C(R+ ; L2 (Ω)d ),

f 2n ∈ C(R+ ; L2 (Γ2 )d ).

(10.189)

In addition, assume that f 0n →. f 0 in C(R+ ; L2 (Ω)d ), f 2n → f 2 in C(R+ ; L2 (Γ2 )d ),

(10.190)

10.3 A Contact Problem for Rate-Type Materials

377

as .n → ∞. Then, .

un → u in C(R+ ; V ), σ n → σ in C(R+ ; Q),

(10.191)

as .n → ∞. Proof Let .n, m ∈ N, .t ∈ [0, m], and .v ∈ K. Then, using the statement of Problem V , it follows that .(un , σ n ) ∈ Y and P

.

 σ n (t) = . Eε(un (t)) +

t

α(σ n (s) − Bε(un (s)) ds + σ 0 − Eε(u0 ), .

(10.192)

0

 (σ n (t), ε(v) − ε(un (t)))Q +

p(unν (t))(vν − unν (t)) da

(10.193)

Γ3

≥ (f n (t), v − un (t))V , where  .

(f n (t), v)V =

 Ω

f 0n (t) · v dx +

Γ2

f 2n (t) · v da

(10.194)

∀ v ∈ V , t ∈ R+ . Inequality (10.193) implies that  .

(σ n (t), ε(v) − ε(un (t)))Q +

p(unν (t)(vν − unν (t)) da Γ3

+(f (t) − f n (t), v − un (t))V ≥ (f (t), v − un (t))V , which, combined with (10.134), (10.194), (10.128), and (10.129), yields  .

(σ n (t), ε(v) − ε(un (t)))Q +

p(unν (t))(vν − unν (t)) da

(10.195)

Γ3

c0 f 2n (t) − f 2 (t)L2 (Γ )d v − un (t)V + c0 f 0n (t) − f 0 (t)L2 (Ω)d +  ≥ (f (t), v − un (t))V . Below we use the notation introduced in Example 10.1 and consider the sequence {θ n } defined as follows: .θ n = {θnm } ∈ I ,

.

 θnm = max f 0n (t) − f 0 (t)L2 (Ω)d +  c0 f 2n (t) − f 2 (t)L2 (Γ )d ,

.

t∈[0,m]

(10.196)

378

10 Quasistatic Contact Problems

for each .n, m ∈ N. Then, (10.192), (10.195), and (10.196) show that (10.173), (10.174) hold, which implies that .(un , σ n ) ∈ Ω(θ n ). On the other hand, notation (10.196) and assumption (10.190) show that .θnm → 0 as .n → ∞, for any .m ∈ N. We conclude from Definition 2.1(b) that .{(un , σ n )} is a .T -approximating V with the Tykhonov triple sequence. Therefore, the well-posedness of Problem .P in Example 10.1 guarantees that the convergences (10.191) hold, which concludes the proof.  In addition to the mathematical interest in the convergence (10.191), it is important from the mechanical point of view since it shows that small perturbations on the densities of body forces and surface tractions give rise to a small perturbation of the weak solution to the frictionless contact problem .P. We now remark that Theorem 10.11 cannot be proved by using the wellV with the Tykhonov triple in Example 10.2. A counterposedness of Problem .P example that proves this statement is the following. V and the Tykhonov triple .T in Example 10.2 Example 10.3 Consider Problem .P in the particular case when .Γ3 = ∅. Note that in this particular case .K = V and, moreover, inequalities (10.166) and (10.170) become (σ (t), ε(v) . − ε(u(t)))Q = (f (t), v − u(t))V , .

(10.197)

(σ (t), ε(v) − ε(u(t)))Q + θ v − u(t)V ≥ (f (t), v − un (t))V

(10.198)

for all .v ∈ V , .t ∈ R+ , respectively. Let .f 0 ∈ L2 (Ω)d , .f 2 ∈ L2 (Γ2 )d and note that in this case the function .f defined by (10.134) does not depend on t. Moreover, assume .f 0 = 0L2 (Ω)d that implies that .f = 0V and, for each .n ∈ N, consider the functions .f 0n and .f 2n defined by f 0n . (t) = f 0 +

t f , n 0

f 2n (t) = f 2 +

t f n 2

∀ t ∈ R+ . (10.199)

Then, it is easy to see that conditions (10.189) and (10.190) are satisfied. Denote in V with the data .f 0n , .f 2n , .u0 , what follows by .(un , σ n ) the solution of Problem .P .σ 0 . Then, using (10.197), we deduce that (σ n (t),. ε(v) − ε(un (t)))Q = (f n (t), v − un (t))V

(10.200)

for each .v ∈ V , .t ∈ R+ and .n ∈ N, where, recall, .f n : [0, T ] → V is the function defined by (10.194). We claim that the sequence .{(un , σ n )} is not a .T -approximating sequence. Indeed, arguing by contradiction, assume that .{(un , σ n )} is a .T -approximating sequence. Then, using (10.198), we deduce that there exists a sequence .{θn } ⊂ R+ such that .θn → 0 and, for each .n ∈ N, the couple .(un , σ n ) satisfies the inequality (σ n (t),. ε(v) − ε(un (t)))Q + θn v − un (t)V ≥ (f , v − un (t))V ,

(10.201)

10.3 A Contact Problem for Rate-Type Materials

379

for each .v ∈ V and .t ∈ R+ . We now combine (10.200) and (10.201) to see that (f − f n (t), v − un (t))V ≤ θn v − un (t)V ,

.

(10.202)

for each .v ∈ V , .t ∈ R+ and .n ∈ N. On the other hand, using (10.134), (10.194), and (10.199), we find that t (f − f n (t), v − un (t))V = − (f , v − un (t))V , n

.

(10.203)

for each .v ∈ V , .t ∈ R+ and .n ∈ N. We now combine (10.202) and (10.203), and then we test with .v = −f + un (t) in the resulting inequality to deduce that t≤

.

nθn , f V

for each .t ∈ R+ and .n ∈ N. We now fix n, and then we pass to the limit as .t → ∞ and obtain a contradiction. We conclude from above that the sequence .{(un , σ n )} V is not a .T -approximating sequence. Therefore, the well-posedness of Problem .P with the Tykhonov triple .T in Example 10.2, guaranteed by Theorem 10.10, cannot be used to obtain the convergences (10.191). On the other hand, since condition (10.190) holds, it follows from Theorem 10.11 that the convergences (10.191) hold, too. We conclude from here that there exist sequences .{(un , σ n )} ⊂ Y that are not .T -approximating sequences but which V . This shows that the choice of converge to the solution .(u, σ ) of Problem .P the Tykhonov triple plays a crucial role to deduce convergence results, as already mentioned in various places in this book. We now present a second additional convergence result that represents a direct consequence of Theorem 10.9. To this end, we start by introducing a frictional version of the contact problem .P. Its statement is as follows. Problem .Q. Find a displacement field .u : Ω ×R+ → Rd and a stress field .σ : Ω × R+ → Sd such that (10.120)–(10.124), (10.126) hold and, moreover, σ τ  ≤ Fb ,

.

−σ τ = Fb

u˙ τ if u˙ τ = 0 on Γ3 × R+ . u˙ τ 

(10.204)

Note that Problem .Q is obtained from Problem .P by replacing the frictionless condition (10.125) by the Coulomb’s law of dry friction (10.204). Here .Fb represents a positive function, the friction bound, which can depend on the spatial variable and other process variables, denoted by r and assumed to take values in a set Z. Thus, .Fb = Fb (x, r), where .x ∈ Γ3 and .r ∈ Z. For instance, Problem .Q was considered in [222] with .Fb = Fb (x, uν (x)), and in this case .Z = R. A dependence of the form .Fb = Fb (x, uν (x), uτ (x)) can also be considered and, in this case, .Z = R × R+ .

380

10 Quasistatic Contact Problems

In the study of Problem .Q, we keep the assumption made in Theorem 10.9, and, in addition, we assume that there exists .ω ≥ 0 such that 0 ≤ Fb (x, r) ≤ ω

∀ r ∈ Z, a.e. x ∈ Γ3 .

.

(10.205)

Under these hypotheses, using the constant .c0 defined in (10.130), we consider the following variational problem. V . Find a displacement field .u ∈ C(R+ , K) and a stress field .σ : Problem .Q C(R+ , Q) such that 

t

σ (t) . = Eε(u(t)) +

α(σ (s) − Bε(u(s))) ds + σ 0 − Eε(u0 ), . (10.206)

0

 (σ (t), ε(v) − ε(u(t)))Q +

p(uν (t))(vν − uν (t)) da

(10.207)

Γ3

+c0 ωv − u(t)V ≥ (f (t), v − u(t))V for all .v ∈ K and .t ∈ R+ . We have the following result. Theorem 10.12 Assume (8.22), (8.61), (8.67), (10.14), (10.18), (10.47), (10.72), (10.73), (10.127), and (10.205). Then, the following statements hold: (a) If .(u, σ ) is a regular solution to Problem .Q, then .(u, σ ) is a solution to Problem V . .Q (b) Problem .QV has at least one solution, for each .ω > 0. Moreover, if, for each V with .ω = ωn and .ωn → 0, then .n ∈ N, .(un , σ n ) is a solution to Problem .Q V , i.e., (10.191) holds. .(un , σ n ) converges to the solution .(u, σ ) of Problem .P Proof (a) Let .(u, σ ) be a regular solution to Problem .Q and let .v ∈ K, .t ∈ R+ . Then, using the bound .σ τ  ≤ Fb (see (10.204)), we have 





σ τ ·(v τ −uτ (t)) da ≥ −

.

Γ3

σ τ v τ −uτ (t) da ≥ − Γ3

Fb v−u(t) da. Γ3

We combine this inequality with assumption (10.205) and use the trace inequality (10.130) to deduce that 

1

σ τ · (v τ − uτ (t)) da ≥ −c0 ω meas(Γ3 ) 2 v − u(t)V .

.

Γ3

We now use this inequality, the Green formula, and standard arguments to deduce that (10.207) holds. Finally, note that (10.206) follows by integrating the constitutive law (10.120) with the initial conditions (10.126).

10.3 A Contact Problem for Rate-Type Materials

381

V obtained in Theorem 10.9. Then it is (b) Let .(u, σ ) be the solution to Problem .P V for any .ω > 0, which proves easy to see that .(u, σ ) is a solution to Problem .Q the first part of the statement. In order to prove the second part, we consider the Tykhonov triple .T = (I, Ω, C) in Example 10.2 as well as a sequence .{ωn } ⊂ R+ such that .ωn → 0 as .n → ∞. Let .{(un , σ n )} be a sequence of couples such V with .ω = ωn , for each .n ∈ N, and that .(un , σ n ) is a solution to Problem .Q 1 let .θ = {θn } be the sequence defined by .θn = c0 ωn meas(Γ3 ) 2 , for each .n ∈ N. Then, using (10.206) and (10.207), it is easy to see that .(un , σ n ) ∈ Ω(θ n ), for each .n ∈ N. Now, since .ωn → 0 as .n → ∞, we deduce that .θn → 0 as .n → ∞, which implies that .{θn } ∈ C and, therefore, the sequence .{(un , σ n )} is a .T -approximating sequence. Its convergence to the unique solution of Problem V is a direct consequence of Theorem 10.10 and Definition 2.1(c). .P  We end this section with the following comments and mechanical interpretation of Theorem 10.12: V as a variational (i) First, Theorem 10.12(a), (b) allows us to consider Problem .Q V can formulation of Problem .Q and, consequently, any solution of Problem .Q be considered as a weak solution of this problem. Nevertheless, this variational formulation is very weak, since it only partially takes into account the boundary conditions of the corresponding contact model. For instance, to obtain the variational inequality (10.207), it was enough to use the bound in (10.204) and assumption (10.205). The rest of the conditions in the Coulomb law of dry friction (10.204) have not been used. (ii) Theorem 10.12(b) guarantees the weak solvability of the contact problems .Q. Nevertheless, it does not guarantee its unique weak solvability. The reason V considered is very weak, as explained is that the variational formulation .Q above. (iii) The mechanical interpretation of the convergence result in Theorem 10.12(b) is the following: any weak solution of the frictional contact problem .Q converges to the weak solution of the frictionless contact problem .P as the friction bound .Fb converges to zero. This result is interesting from the mechanical point of view since the nature of Problem .P, on one hand, and that of Problem .Q, on the other hand, are different. It shows that the solution of the frictionless problem .P can be approached by the weak solutions of a different contact problem, which could be frictional. The results above underline the importance of the concept of .T -well-posedness in the study of mathematical models of contact. Indeed, besides the unique weak solvability and the continuous dependence of the weak solution with respect to the data, the concept provides a functional framework in which different convergence results can be established, illustrating in this way the link between different models of contact.

Bibliographical Notes

In this additional section, we provide more information on the references we used in writing this book. We mention both our previous works and references which deal with similar nonlinear problems, as well as general references for various wellknown results we used in the proofs. We are conscious that the list of references and citations below is far to be complete since the number of works which deal with the nonlinear problems considered in this book is very large and, moreover, it impressively increased in the last two decades. Nevertheless, we believe that the comments below could be useful for the reader interested in the topics we deal with. The material presented in Chap. 1 is standard. More details on the elements of nonsmooth analysis we presented in Sect. 1.1 can be found in [44, 52, 157, 184, 254]. Comprehensive references on convex analysis in Banach and Hilbert spaces which complete the results we mentioned in Sect. 1.2 can be found in [23, 54, 61, 94, 109, 186]. For the survey of the classical well-posedness concepts we presented in Sect. 1.3, we followed [56] as well as [126, 127]. For additional results on wellposedness of optimization problems, we refer the reader to [45, 66, 67, 102, 103, 256]. Chapter 2 was written based on our original research. Some preliminary results have been obtained in [230], where a general concept of well-posedness in the sense of Tykhonov for abstract problems in metric spaces has been defined. The results there have been extended in [250], where the concepts of Tykhonov triple and .T well-posedness have been introduced. These latter have been used in [207, 224, 226] in the study of a viscoplastic constitutive law, an antiplane contact problem with elastic materials, and a transfer problem with unilateral constraints, respectively. The results presented in Sect. 2.1.5 have been obtained in [250]. The concept of split problem has been introduced in [38] in the framework of Hilbert spaces. Such kind of problems have been used in various mathematical models arising from phase retrievals and medical reconstruction, as shown in [30]. Results on split variational inequalities have been obtained in [37, 39]. They have been extended in [101, 172] to split variational–hemivariational inequalities and split inclusions, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Sofonea, Well-Posed Nonlinear Problems, Advances in Mechanics and Mathematics 50, https://doi.org/10.1007/978-3-031-41416-9

383

384

Bibliographical Notes

respectively. Applications arising from signal recovery, image processing, and radiation therapy can be found in [37, 91, 141, 172]. A metric characterization of the well-posedness of split hemivariational inequalities has been provided in [201]. An example of variational inequalities that are dual to each other was considered in [213]. The inequalities considered there arise from Contact Mechanics and model the equilibrium of an elastic body in frictional contact with a foundation. Fixed point theory represents an important topic of Nonlinear Analysis which has gone through substantial development in the last decades. Based on arguments of topology, it represents a powerful tool in proving existence and uniqueness results for the solution to various nonlinear problems. Comprehensive references in the field of fixed point theory and their applications are [7, 25, 70, 73, 111, 116, 253]. In Chap. 3, we restricted ourselves to the study of well-posedness of fixed point problems with contractive, history-dependent, and almost history-dependent operators. The results we presented there are based on our original work. The material in Sects. 3.1–3.4 is published here for the first time. The results in Sects. 3.5 have been published in our recent paper [211]. Interest in variational inequalities originates in mechanical problems. General references in the field are [11, 28, 29, 95, 113, 115, 133]. Convergence results for variational inequalities with maximal monotone operators have been considered in [10]. History-dependent variational inequalities of the form (1.54) with applications in Mechanics can be found in [24, 110, 217]. Existence and uniqueness results for evolutionary variational inequalities can be found in [82, 83, 247] as well as in the books [84, 216]. There, error estimates and convergence results for spatially discrete and fully discrete schemes have been provided, together with various applications in Contact Mechanics. An evolutionary variational inequality with Volterra integral term was considered in [196]. Existence and uniqueness results for mixed variational problems can be found in [143, 144, 204] as well as in [219, 220]. The first systematic and comprehensive treatment of variational inequalities with random data is provided in the recent book [75]. In writing Chap. 4, we used some of our previous works. Thus, Sect. 4.1 was written following some ideas in [225]. The study of well-posedness of variational inequality (4.39) in Sect. 4.2 was written following [208]. The results on the split and dual variational inequalities in Sect. 4.3 extend our previous results in [202]. There, well-posedness results for a split variational–hemivariational inequality, an elliptic variational inequality, and a history-dependent variational inequality have been proved. Finally, the abstract results in Sect. 4.4 have been inspired by our paper [32]. There, the well-posedness of a mixed variational problem governed by a nonlinear operator and a set of constraints has been proved. Moreover, existence and equivalence results have been obtained and used in the study of a frictionless contact model with elastic materials. The notion of hemivariational and variational–hemivariational inequality originates in the pioneering works of Panagiotopoulos [182]. It is closely related to the development of the concept of generalized gradient of a locally Lipschitz function. General references in the field are the books [175, 184] and, more recently, [157, 222]. Results on hemivariational inequalities include [31, 72, 79, 134, 152, 153,

Bibliographical Notes

385

155, 156, 158, 214], for instance. The edited volume [80], divided in three parts, is devoted to the analysis and numerical approximation of nonlinear inclusions and hemivariational inequalities. It also contains results in the study of a number of inequality problems issued from Contact Mechanics. The topics in this volume are completed by the works [85–87, 90], dedicated to the numerical approximation of various classes of hemivariational inequalities. Results in the study of a hyperbolic variational–hemivariational inequality have been obtained in [185]. The concept of well-posedness for hemivariational inequalities was first introduced in [71], and then it was extended to variational–hemivariational inequalities (see [229, 246] and the references therein). Chapter 5 was devoted to the study of well-posedness of hemivariational inequalities. Section 5.1 was written following our papers [98, 99]. There, additional results in the study of hemivariational inequality of the form (5.1) have been obtained, including a characterization of the well-posedness in terms of the metric properties of the family of approximating sets and various convergence results. These abstract results have been illustrated in [98] in the study of a mathematical model which describes the equilibrium of a rod-spring system with unilateral constraints. The well-posedness results for hemivariational inequalities in Sect. 5.2 have been inspired by our paper [36]. There, results on the strong and weak wellposedness in the generalized sense for inequalities of the form (5.46) have been obtained, under various assumptions on the data. Moreover, two perturbations of (5.46) have been considered, for which convergence results in the sense of Kuratowski have been proved. Section 5.3 was written based on our recent paper [100]. An additional reference in the field is the paper [96], where the Levitin– Polyak well-posedness of a variational–hemivariational inequality is studied and an example arising from Contact Mechanics is given. Stationary, time-dependent, and evolutionary inclusions represent an interesting mathematical topic useful in Nonsmooth Analysis, Mechanics, Physics, and Engineering Science. In particular, variational and hemivariational inequalities are strongly related to particular classes of inclusions associated with the subdifferential in the sense of convex analysis and the Clarke subdifferential operator, respectively. Moreover, sweeping processes represent a special class of differential inclusions, associated with a family of time-dependent convex sets. Introduced in early 70s in the pioneering works of Moreau [165–168], such kind of inclusions arise in the study of unilateral problems in Sold Mechanics, where the convex set is related to the elastic-viscoplastic constitutive law or, alternatively, to the frictional unilateral contact conditions. References in the field of inclusions and sweeping processes include [4, 5, 119, 122, 163, 251]. Results on optimal control for evolutionary inclusions involving subdifferential operators can be found in [22], for instance. The results in Chap. 6 are based on our original research, too. Thus, in writing Sect. 6.1, we followed [210]. Related references are [209] and [173]. A timedependent inclusion of the form (6.79) has been studied in [6, 174], under different assumptions. A preliminary version of the results in Sect. 6.3 on the dual historydependent inclusions has been obtained in [97].

386

Bibliographical Notes

Basic references for the optimal control of systems governed by partial differential equations are the books [20, 130, 178, 179]. Application of the optimal control theory in Mechanics can be found in [1, 2, 154], for instance. Optimal control problems for variational inequalities have been discussed in several works, including [27, 65, 150, 151, 240]. The literature concerning optimal control problems in the study of mathematical models of contact is quite limited. The reason is the strong nonlinearities that arise from the boundary conditions included in such models. Results on optimal control for various contact problems with elastic materials can be found in [8, 19, 26, 33–35, 145–147, 206, 241] and the references therein. The results in Chap. 7 are based on our original research. Section 7.1 has been written inspired by [232]. There, various well-posedness results for the minimization problem (7.1) have been obtained, under different assumptions on the data. Convergence results for optimal control of variational inequalities, similar to those proved in Sect. 7.2, can be found in [205, 225]. Results on the optimal control of hemivariational inequalities have been obtained in [187]. General results on Mechanics of Continua can be found in [55, 74, 112, 239]. A comprehensive reference on three-dimensional elasticity is the book [42]. For various results concerning the theory of viscoelasticity, we refer the reader to [190], and for numerical methods in plasticity, we refer the reader to [81, 89]. We also mention the book [51] (in which various viscoelastic and viscoplastic rate-type constitutive laws are introduced) as well as the book [107] (in which functional methods in the study of displacement-traction boundary value problems for viscoplastic materials are considered). References on Contact Mechanics include [15, 124, 237, 252]. Frictional contact with unilateral constraints for elastic materials leads to very difficult mathematical problems and requires a regularization operator. Some physical motivations can be found in [181]. The analysis of the corresponding models can be found in [46–48, 58]. Recent results on optimal control problems in Contact Mechanics can be found in [228, 234, 236]. Numerical solution for various contact problems with elastic materials can be found in [16–18, 41]. More results on the functional spaces used in Part III of the book can be found in [84, 114, 157, 177, 222]. The properties of these spaces arise from the well-known properties of the Lebesgue and Sobolev spaces associated with a smooth domain d .Ω ⊂ R , for which we refer the reader to [3, 131, 132, 176]. The material in Chap. 8 is standard. More details on the modelling of contact problems (including a description of the constitutive laws and the interface conditions) can be found in the books [84, 157, 183, 222], for instance. Chapter 9 was devoted to static contact problems. The study of well-posedness of contact problems similar to that in Sect. 9.1 has been performed in [223], within the framework of linear elasticity. There, various convergence results have been obtained and the corresponding mechanical interpretations have been provided. Results that parallel the results presented in Sect. 9.2 can be found in [209, 225, 235]. The contents of Sects. 9.3 and 9.4 follow our papers [232] and [96], respectively. A model of contact with normal compliance for elastic materials was considered in [108]. There, besides the unique solvability of the problem, a dual variational formulation of the contact model in terms of stress has been considered and various

Bibliographical Notes

387

equivalence results have been proved. Additional existence and uniqueness results for elliptic and evolutionary variational inequalities with applications in the study of static and quasistatic contact problems with elastic materials can be found in [170, 171]. Mixed variational formulations for elastic and elastic–plastic contact problems have been considered in [32] and [194], respectively. The material we presented in Chap. 10 is based on our original work. Thus, Sect. 10.1 was written following the ideas in [211]. Variational formulations for quasistatic contact models in terms of inclusions, similar to that presented in Sect. 10.2, can be found in [233, 234]. Part of the results in Sect. 10.3 represents a version of some results obtained in [231]. Reference in the field include [160, 195, 197] where existence and uniqueness results for various viscoelastic contact problems have been obtained. Additional results in the study of quasistatic contact problems can be found in [40, 149]. The models considered in [40] deal with viscoplastic materials and nonmonotone boundary conditions. In contrast, the model in [149] is based on convex-type boundary conditions, and therefore, its analysis is based on monotonicity arguments. There, the interest was to present a dual variational formulation of the problem, in terms of the stress field.

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Index

A Admissible pairs, 250 Almost history-dependent operator, 7, 8 Approximating sequence, 35, 36, 38, 40 generalized, 42 LP, 42 Approximating sets, 47

B Balance equation Cauchy equation, 269 equilibrium equation, 269 Banach fixed point theorem, 18 Bilateral contact, 277 Boundary condition displacement, 269, 275, 294 traction, 269, 276, 294

C Clarke directional derivative, 14 generalized gradient, 14 regular function, 14 subdifferential, 14 Coefficient deformability, 325 elasticity, 274 friction, 284 Lamé, 274 stiffness, 309, 340 Constitutive law, 269 elastic, 273

Kelvin-Voigt, 290 Perzyna, 293 rate-type, 292 viscoelastic, 290, 291 Contact bilateral, 277 frictional, 283 frictionless, 283 unilateral, 277 Contact condition, 269 deformable body, 277 elastic body, 278 elastic layer, 280 normal compliance, 278 regularized, 325 rigid-elastic body, 279 rigid-elastic layer, 282 rigid-plastic body, 278 rigid-plastic layer, 281 rigid body, 276 Signorini, 276 Contact model frictional, 301, 311, 337, 352 frictionless, 323, 340, 363 nonsmooth, 332 one-dimensional, 285, 295 regularized, 324 Contraction, 18 Convergence criterion, 94, 102, 223 Convergence results elastic contact, 308, 329 elliptic variational inequality, 144 fixed point problem, 103, 119, 131 hemivariational inequality, 183

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Sofonea, Well-Posed Nonlinear Problems, Advances in Mechanics and Mathematics 50, https://doi.org/10.1007/978-3-031-41416-9

401

402 Convergence results (cont.) history-dependent inclusion, 234 history-dependent variational inequality, 152, 234 minimization problem, 248 rate-type viscoelastic contact, 367, 376 variational-hemivariational inequality, 202 viscoelastic contact, 344 Convex function, 10 set, 4 subdifferential, 11, 24 von Mises, 275 Coulomb friction law classical, 284 quasistatic, 294 static, 283

D Deformability coefficient, 325 Deviatoric part, 275 Diameter, 58 Distance function, 28 Divergence operator, 275 Dual problem, 76 history-dependent inclusion, 230 history-dependent variational inequality, 230 Dual Tykhonov triple, 77

E Elastic constitutive law, 273 Elasticity operator, 273, 275, 291, 292 tensor, 291 Equilibrium equation, 275, 293

F Fixed point problem almost history-depenent operator, 127 contractive mapping, 99 history-dependent operator, 116 Fréchet space, 8 Friction bound, 284, 294 coefficient, 284 force, 284, 294 law, 269, 283 Frictional condition, 283 contact, 283

Index Friction law Coulomb, 284 quasistatic, 294 static, 283 Tresca, 284, 295 Frictionless condition, 283 contact, 283 Function coercive, 10 convex, 10 convex subdifferential, 11, 24 directional derivative, 14 distance, 28 effective domain, 9 gap, 194 generalized gradient, 14 indicator, 11 locally Lipschitz, 13 lower semicontinuous, 10 proper, 10 regular in the sense of Clarke, 14 strictly convex, 10 strongly convex, 10 subdifferential in the sense of Clarke, 14 G Gap function, 194 Generalized gradient, 14 Generalized well-posedness results hemivariational inequality, 189 optimal control, 262 Green formula, 272 Gronwall inequality, 18 H Hadamard well-posedness, 90 generalized, 91 minimization problem, 35, 55 variational inequality, 57 weakly, 91 weakly generalized, 91 Hemivariational inequality, 15 History-dependent inclusion, 124, 223 operator, 7, 8 variational inequality, 27 I Inclusion history-dependent, 124, 223 stationary, 112, 206

Index Indicator function, 11 Inf-sup condition, 173 Initial condition, 269 Interface law contact condition, 269 friction law, 269 quasistatic process, 294 static process, 276 Internal approximation, 66 Iterations Krasnoselskii, 103 Mann, 103 Picard, 103, 119 K Kelvin-Voigt constitutive law, 290 Krasnoselskii iterations, 103 Kronecker symbol, 274 L Lamé coefficients, 274 Levitin-Polyak well-posedness extended, 89 minimization problem, 30, 54, 88, 201 variational-hemivariational inequality, 201 variational inequality, 42, 57, 88 Lipschitz constant near x, 13 Lower semicontinuous (l.s.c.), 10 M Mann iterations, 103 Mathematical model of contact, 268, 285, 295 Mazur theorem, 4 Minimizing sequence, 29 generalized, 31 LP, 31 Mosco convergence, 5, 23 Multivalued operator domain, 20 graph, 20 maximal monotone, 21 monotone, 21 range, 20 strongly monotone, 21 N Normal component, 270, 272 O Operator almost history-dependent, 7, 8

403 bounded, 5 compact, 6 completely continuous, 6 contraction, 18 demicontinuous, 5 divergence, 275 elasticity, 273, 290–292 hemicontinuous, 5, 19 history-dependent, 7, 8 Lipschitz continuous, 20 monotone, 5, 19 nonexpansive, 20 projection, 22 proximality, 24 pseudomonotone, 5, 19 resolvent, 21 strongly monotone, 5, 19 symmetric, 20 trace, 272 of type .(S+ ), 5 viscosity, 290, 292 Optimal control admissible pairs, 250 cost functional, 250 elastic contact, 319 stationary inclusion, 261 viscoelastic contact, 349 Outward normal cone, 24

P Perzyna law, 293 Picard iterations, 103, 119 Problem dual, 76 ill-posed, 29 optimal control, 250 split, 71 well-posed with .T , 47 Projection, 22 lemma, 22 operator, 22 Proximal element, 23 Proximality operator, 23, 24

Q Quasivariational inequality, 25

R Relaxation tensor, 291, 292 Resolvent operator, 21

404 S Sequence approximating, 35, 38, 40 minimizing, 29 .T -approximating, 47 Set of parameters, 47 Signorini contact condition, 276 Slip, 284 rate, 294 state, 284 Smulyan theorem, 5 Sobolev trace theorem, 271 Split problem, 71 regular Tykhonov triple, 72 Stationary inclusion, 112 Stick state, 284 Stiffness coefficient, 309, 340 Strain tensor, 268 deviatoric part, 275 trace, 274 Subdifferential Clarke, 14 convex function, 24 Subgradient convex function, 11 Subset closed, 4 compact, 4 convex, 4 diameter, 58 weakly closed, 4 weakly compact, 4

T Tangential component, 270 Tangential part, 272 .T -approximating sequence, 47 Tensor elasticity, 291 relaxation, 291, 292 stain, 268 trace, 274 viscosity, 291 Theorem Banach fixed point, 18 Mazur, 4 Sobolev trace, 271 Smulyan, 5 Weierstrass, 12 Trace poperator, 272 tensor, 274 Traction boundary condition, 276, 294

Index Tresca friction law quasistatic, 295 static, 284 Triple Tykhonov, 46, 47 Tykhonov-Hadamard, 90 Tykhonov-Levitin-Polyak, 84, 89 .T -well-posedness, 47 dual problems, 77 generalized, 52 metric caracterization, 58 split problem, 70, 72 weakly, 52 weakly generalized, 52 Tykhonov-Hadamard triple, 90 Tykhonov-Levitin-Polyak triple, 84, 89 Tykhonov triple, 46, 47 dual, 77 greatest element, 51 regular, 59, 72 relation of equivalence, 49 relation of order, 50 Tykhonov well-posedness fixed point problem, 36, 55 inclusion, 38, 56 minimization problem, 29, 54 variational-hemivariational inequality, 200 variational inequality, 40, 56 U Unilateral contact, 277 V Variational-hemivariational inequality, 15 Variational inequality, 15 elliptic dual, 167 first kind, 26 penalty method, 144, 145 second kind, 26 well-posedness, 137 history-dependent, 27, 159 split, 162 well-posedness, 147 mixed, 172 Viscoelastic constitutive law, 291, 292 Kelvin-Voigt, 290 long memory, 291 short memory, 290 Viscosity operator, 290, 292 tensor, 291 Volterra-type integral equation, 123

Index W Weak solution, 305, 327, 346, 349, 357, 370, 376, 381 Weierstrass theorem, 12 Well-posedness results dual variational inequalities, 167 elastic contact, 305 elliptic variational inequality, 137 fixed point problem, 99, 116, 126 hemivariational inequality, 177 history-dependent inclusion, 228, 230 history-dependent variational inequality, 147, 149, 230 minimization problem, 244 mixed variational inequality, 173 nonsmooth elastic contact, 335

405 optimal control problem, 251 rate-type viscoelastic contact, 370 split variational inequality, 165 stationary inclusion, 208, 213 variational-hemivariational inequality, 193 viscoelastic contact, 358 Well-posed problem extended Hadamard, 89 extended Levitin-Polyak, 83 generalized, 91 Hadamard, 33, 43, 55, 57, 90 Levitin-Polyak, 30, 42, 54, 57, 88 Tykhonov, 29, 35, 38, 40, 54–56 weakly, 91 weakly generalized, 91