JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS VOLUME 12 -2014 0649913201, 0644701007

Citation preview

VOLUME 12, NUMBERS 1-2 APRIL 2014

JANUARY-

ISSN:1548-5390 PRINT, 1559-176X ONLINE

JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS EUDOXUS PRESS,LLC

1

SCOPE AND PRICES OF THE JOURNAL Journal of Concrete and Applicable Mathematics A quartely international publication of Eudoxus Press,LLC Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis Memphis, TN 38152, U.S.A. [email protected] Assistant to the Editor: Dr.Razvan Mezei,Lenoir-Rhyne University,Hickory,NC 28601, USA. The main purpose of the "Journal of Concrete and Applicable Mathematics" is to publish high quality original research articles from all subareas of Non-Pure and/or Applicable Mathematics and its many real life applications, as well connections to other areas of Mathematical Sciences, as long as they are presented in a Concrete way. It welcomes also related research survey articles and book reviews.A sample list of connected mathematical areas with this publication includes and is not restricted to: Applied Analysis, Applied Functional Analysis, Probability theory, Stochastic Processes, Approximation Theory, O.D.E, P.D.E, Wavelet, Neural Networks,Difference Equations, Summability, Fractals, Special Functions, Splines, Asymptotic Analysis, Fractional Analysis, Inequalities, Moment Theory, Numerical Functional Analysis,Tomography, Asymptotic Expansions, Fourier Analysis, Applied Harmonic Analysis, Integral Equations, Signal Analysis, Numerical Analysis, Optimization, Operations Research, Linear Programming, Fuzzyness, Mathematical Finance, Stochastic Analysis, Game Theory, Math.Physics aspects, Applied Real and Complex Analysis, Computational Number Theory, Graph Theory, Combinatorics, Computer Science Math.related topics,combinations of the above, etc. In general any kind of Concretely presented Mathematics which is Applicable fits to the scope of this journal. Working Concretely and in Applicable Mathematics has become a main trend in many recent years,so we can understand better and deeper and solve the important problems of our real and scientific world. "Journal of Concrete and Applicable Mathematics" is a peer- reviewed International Quarterly Journal. We are calling for papers for possible publication. The contributor should send via email the contribution to the editor in-Chief: TEX or LATEX (typed double spaced) and PDF files. [ See: Instructions to Contributors]

Journal of Concrete and Applicable Mathematics(JCAAM) ISSN:1548-5390 PRINT, 1559-176X ONLINE. is published in January,April,July and October of each year by EUDOXUS PRESS,LLC, 1424 Beaver Trail Drive,Cordova,TN38016,USA, Tel.001-901-751-3553 [email protected] http://www.EudoxusPress.com. Visit also www.msci.memphis.edu/~ganastss/jcaam.

2

Annual Subscription Current Prices:For USA and Canada,Institutional:Print $500,Electronic $250,Print and Electronic $600.Individual:Print $200, Electronic $100,Print &Electronic $250.For any other part of the world add $60 more to the above prices for Print. Single article PDF file for individual $20.Single issue in PDF form for individual $80. No credit card payments.Only certified check,money order or international check in US dollars are acceptable. Combination orders of any two from JoCAAA,JCAAM,JAFA receive 25% discount,all three receive 30% discount. Copyright©2014 by Eudoxus Press,LLC all rights reserved.JCAAM is printed in USA. JCAAM is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JCAAM and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers. JCAAM IS A JOURNAL OF RAPID PUBLICATION

PAGE CHARGES: Effective 1 Nov. 2009 for current journal page charges, contact the Editor in Chief. Upon acceptance of the paper an invoice will be sent to the contact author. The fee payment will be due one month from the invoice date. The article will proceed to publication only after the fee is paid. The charges are to be sent, by money order or certified check, in US dollars, payable to Eudoxus Press, LLC, to the address shown on the Eudoxus homepage.

3

Editorial Board Associate Editors of Journal of Concrete and Applicable Mathematics

Editor in -Chief: George Anastassiou Department of Mathematical Sciences The University Of Memphis Memphis,TN 38152,USA tel.901-678-3144,fax 901-678-2480 e-mail [email protected] www.msci.memphis.edu/~ganastss Areas:Approximation Theory, Probability,Moments,Wavelet, Neural Networks,Inequalities,Fuzzyness. Associate Editors: 1)Ravi P. Agarwal Chairman Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 Office: 361-593-2600

Email: [email protected] Differential Equations,Difference Equations,Inequalities

2) Carlo Bardaro Dipartimento di Matematica & Informatica Universita' di Perugia Via Vanvitelli 1 06123 Perugia,ITALY tel.+390755855034, +390755853822, fax +390755855024 [email protected] , [email protected] Functional Analysis and Approximation Th., Summability,Signal Analysis,Integral Equations, Measure Th.,Real Analysis 3) Francoise Bastin Institute of Mathematics University of Liege 4000 Liege BELGIUM [email protected] Functional Analysis,Wavelets

21) Gustavo Alberto Perla Menzala National Laboratory of Scientific Computation LNCC/MCT Av. Getulio Vargas 333 25651-075 Petropolis, RJ Caixa Postal 95113, Brasil and Federal University of Rio de Janeiro Institute of Mathematics RJ, P.O. Box 68530 Rio de Janeiro, Brasil [email protected] and [email protected] Phone 55-24-22336068, 55-21-25627513 Ext 224 FAX 55-24-22315595 Hyperbolic and Parabolic Partial Differential Equations, Exact controllability, Nonlinear Lattices and Global Attractors, Smart Materials 22) Ram N.Mohapatra Department of Mathematics University of Central Florida Orlando,FL 32816-1364 tel.407-823-5080 [email protected] Real and Complex analysis,Approximation Th., Fourier Analysis, Fuzzy Sets and Systems 23) Rainer Nagel Arbeitsbereich Funktionalanalysis Mathematisches Institut Auf der Morgenstelle 10 D-72076 Tuebingen Germany tel.49-7071-2973242 fax 49-7071-294322 [email protected] evolution equations,semigroups,spectral th., positivity 24) Panos M.Pardalos Center for Appl. Optimization University of Florida 303 Weil Hall P.O.Box 116595 Gainesville,FL 32611-6595 tel.352-392-9011 [email protected] Optimization,Operations Research

4

4) Yeol Je Cho Department of Mathematics Education College of Education Gyeongsang National University Chinju 660-701 KOREA tel.055-751-5673 Office, 055-755-3644 home, fax 055-751-6117 [email protected] Nonlinear operator Th.,Inequalities, Geometry of Banach Spaces

25) Svetlozar (Zari) Rachev, Professor of Finance,

College of Business,and Director of Quantitative Finance Program, Department of Applied Mathematics & Statistics Stonybrook University 312 Harriman Hall, Stony Brook, NY 11794-3775 Phone: +1-631-632-1998, Email : [email protected];

26) John Michael Rassias University of Athens Pedagogical Department 5) Sever S.Dragomir School of Communications and Informatics Section of Mathematics and Infomatics 20, Hippocratous Str., Athens, 106 80, Greece Victoria University of Technology PO Box 14428 Address for Correspondence Melbourne City M.C 4, Agamemnonos Str. Victoria 8001,Australia Aghia Paraskevi, Athens, Attikis 15342 Greece tel 61 3 9688 4437,fax 61 3 9688 4050 [email protected] [email protected], [email protected] [email protected] Math.Analysis,Inequalities,Approximation Approximation Theory,Functional Equations, Inequalities, PDE Th., Numerical Analysis, Geometry of Banach 27) Paolo Emilio Ricci Spaces, Universita' degli Studi di Roma "La Sapienza" Information Th. and Coding Dipartimento di Matematica-Istituto "G.Castelnuovo" 6) Oktay Duman P.le A.Moro,2-00185 Roma,ITALY TOBB University of Economics and tel.++39 0649913201,fax ++39 0644701007 Technology, [email protected],[email protected] Department of Mathematics, TR-06530, Orthogonal Polynomials and Special functions, Ankara, Turkey, [email protected] Numerical Analysis, Transforms,Operational Classical Approximation Theory, Calculus, Summability Theory, Differential and Difference equations Statistical Convergence and its Applications 28) Cecil C.Rousseau Department of Mathematical Sciences The University of Memphis 7) Angelo Favini Memphis,TN 38152,USA Università di Bologna tel.901-678-2490,fax 901-678-2480 Dipartimento di Matematica [email protected] Piazza di Porta San Donato 5 Combinatorics,Graph Th., 40126 Bologna, ITALY Asymptotic Approximations, tel.++39 051 2094451 Applications to Physics fax.++39 051 2094490 [email protected] 29) Tomasz Rychlik Partial Differential Equations, Control Institute of Mathematics Theory, Polish Academy of Sciences Differential Equations in Banach Spaces Chopina 12,87100 Torun, Poland [email protected] 8) Claudio A. Fernandez Mathematical Statistics,Probabilistic Facultad de Matematicas Inequalities Pontificia Unversidad Católica de Chile Vicuna Mackenna 4860 Santiago, Chile

5

tel.++56 2 354 5922 fax.++56 2 552 5916 [email protected] Partial Differential Equations, Mathematical Physics, Scattering and Spectral Theory 9) A.M.Fink Department of Mathematics Iowa State University Ames,IA 50011-0001,USA tel.515-294-8150 [email protected] Inequalities,Ordinary Differential Equations 10) Sorin Gal Department of Mathematics University of Oradea Str.Armatei Romane 5 3700 Oradea,Romania [email protected] Approximation Th.,Fuzzyness,Complex Analysis 11) Jerome A.Goldstein Department of Mathematical Sciences The University of Memphis, Memphis,TN 38152,USA tel.901-678-2484 [email protected] Partial Differential Equations, Semigroups of Operators 12) Heiner H.Gonska Department of Mathematics University of Duisburg Duisburg,D-47048 Germany tel.0049-203-379-3542 office [email protected] Approximation Th.,Computer Aided Geometric Design 13) Dmitry Khavinson Department of Mathematical Sciences University of Arkansas Fayetteville,AR 72701,USA tel.(479)575-6331,fax(479)575-8630 [email protected] Potential Th.,Complex Analysis,Holomorphic PDE, Approximation Th.,Function Th.

30) Bl. Sendov Institute of Mathematics and Informatics Bulgarian Academy of Sciences Sofia 1090,Bulgaria [email protected] Approximation Th.,Geometry of Polynomials, Image Compression 31) Igor Shevchuk Faculty of Mathematics and Mechanics National Taras Shevchenko University of Kyiv 252017 Kyiv UKRAINE [email protected] Approximation Theory 32) H.M.Srivastava Department of Mathematics and Statistics University of Victoria Victoria,British Columbia V8W 3P4 Canada tel.250-721-7455 office,250-477-6960 home, fax 250-721-8962 [email protected] Real and Complex Analysis,Fractional Calculus and Appl., Integral Equations and Transforms,Higher Transcendental Functions and Appl.,q-Series and q-Polynomials, Analytic Number Th. 33) Stevo Stevic Mathematical Institute of the Serbian Acad. of Science Knez Mihailova 35/I 11000 Beograd, Serbia [email protected]; [email protected] Complex Variables, Difference Equations, Approximation Th., Inequalities 34) Ferenc Szidarovszky Dept.Systems and Industrial Engineering The University of Arizona Engineering Building,111 PO.Box 210020 Tucson,AZ 85721-0020,USA [email protected] Numerical Methods,Game Th.,Dynamic Systems, Multicriteria Decision making, Conflict Resolution,Applications in Economics and Natural Resources Management

14) Virginia S.Kiryakova 35) Gancho Tachev Institute of Mathematics and Informatics Dept.of Mathematics

6

Bulgarian Academy of Sciences Sofia 1090,Bulgaria [email protected] Special Functions,Integral Transforms, Fractional Calculus

Univ.of Architecture,Civil Eng. and Geodesy 1 Hr.Smirnenski blvd BG-1421 Sofia,Bulgaria [email protected] Approximation Theory

15) Hans-Bernd Knoop Institute of Mathematics Gerhard Mercator University D-47048 Duisburg Germany tel.0049-203-379-2676 [email protected] Approximation Theory,Interpolation

36) Manfred Tasche Department of Mathematics University of Rostock D-18051 Rostock Germany [email protected] Approximation Th.,Wavelet,Fourier Analysis, Numerical Methods,Signal Processing, Image Processing,Harmonic Analysis

16) Jerry Koliha Dept. of Mathematics & Statistics University of Melbourne VIC 3010,Melbourne Australia [email protected] Inequalities,Operator Theory, Matrix Analysis,Generalized Inverses 17) Robert Kozma Dept. of Mathematical Sciences University of Memphis Memphis, TN 38152, USA [email protected] Mathematical Learning Theory, Dynamic Systems and Chaos, Complex Dynamics. 18) Mustafa Kulenovic Department of Mathematics University of Rhode Island Kingston,RI 02881,USA [email protected] Differential and Difference Equations 19) Gerassimos Ladas Department of Mathematics University of Rhode Island Kingston,RI 02881,USA [email protected] Differential and Difference Equations

37) Chris P.Tsokos Department of Mathematics University of South Florida 4202 E.Fowler Ave.,PHY 114 Tampa,FL 33620-5700,USA [email protected],[email protected] Stochastic Systems,Biomathematics, Environmental Systems,Reliability Th. 38) Lutz Volkmann Lehrstuhl II fuer Mathematik RWTH-Aachen Templergraben 55 D-52062 Aachen Germany [email protected] Complex Analysis,Combinatorics,Graph Theory.

20) Rupert Lasser Institut fur Biomathematik & Biomertie,GSF -National Research Center for environment and health Ingolstaedter landstr.1 D-85764 Neuherberg,Germany [email protected] Orthogonal Polynomials,Fourier Analysis,Mathematical Biology

7

Instructions to Contributors Journal of Concrete and Applicable Mathematics A quartely international publication of Eudoxus Press, LLC, of TN.

Editor in Chief: George Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152-3240, U.S.A.

1. Manuscripts files in Latex and PDF and in English, should be submitted via email to the Editor-in-Chief: Prof.George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis,TN 38152, USA. Tel. 901.678.3144 e-mail: [email protected] Authors may want to recommend an associate editor the most related to the submission to possibly handle it. Also authors may want to submit a list of six possible referees, to be used in case we cannot find related referees by ourselves.

2. Manuscripts should be typed using any of TEX,LaTEX,AMS-TEX,or AMS-LaTEX and according to EUDOXUS PRESS, LLC. LATEX STYLE FILE. (Click HERE to save a copy of the style file.)They should be carefully prepared in all respects. Submitted articles should be brightly typed (not dot-matrix), double spaced, in ten point type size and in 8(1/2)x11 inch area per page. Manuscripts should have generous margins on all sides and should not exceed 24 pages. 3. Submission is a representation that the manuscript has not been published previously in this or any other similar form and is not currently under consideration for publication elsewhere. A statement transferring from the authors(or their employers,if they hold the copyright) to Eudoxus Press, LLC, will be required before the manuscript can be accepted for publication.The Editor-in-Chief will supply the necessary forms for this transfer.Such a written transfer of copyright,which previously was assumed to be implicit in the act of submitting a manuscript,is necessary under the U.S.Copyright Law in order for the publisher to carry through the dissemination of research results and reviews as widely and effective as possible.

8

4. The paper starts with the title of the article, author's name(s) (no titles or degrees), author's affiliation(s) and e-mail addresses. The affiliation should comprise the department, institution (usually university or company), city, state (and/or nation) and mail code. The following items, 5 and 6, should be on page no. 1 of the paper. 5. An abstract is to be provided, preferably no longer than 150 words. 6. A list of 5 key words is to be provided directly below the abstract. Key words should express the precise content of the manuscript, as they are used for indexing purposes. The main body of the paper should begin on page no. 1, if possible. 7. All sections should be numbered with Arabic numerals (such as: 1. INTRODUCTION) . Subsections should be identified with section and subsection numbers (such as 6.1. Second-Value Subheading). If applicable, an independent single-number system (one for each category) should be used to label all theorems, lemmas, propositions, corrolaries, definitions, remarks, examples, etc. The label (such as Lemma 7) should be typed with paragraph indentation, followed by a period and the lemma itself. 8. Mathematical notation must be typeset. Equations should be numbered consecutively with Arabic numerals in parentheses placed flush right, and should be thusly referred to in the text [such as Eqs.(2) and (5)]. The running title must be placed at the top of even numbered pages and the first author's name, et al., must be placed at the top of the odd numbed pages. 9. Illustrations (photographs, drawings, diagrams, and charts) are to be numbered in one consecutive series of Arabic numerals. The captions for illustrations should be typed double space. All illustrations, charts, tables, etc., must be embedded in the body of the manuscript in proper, final, print position. In particular, manuscript, source, and PDF file version must be at camera ready stage for publication or they cannot be considered. Tables are to be numbered (with Roman numerals) and referred to by number in the text. Center the title above the table, and type explanatory footnotes (indicated by superscript lowercase letters) below the table. 10. List references alphabetically at the end of the paper and number them consecutively. Each must be cited in the text by the appropriate Arabic numeral in square brackets on the baseline. References should include (in the following order): initials of first and middle name, last name of author(s) title of article,

9

name of publication, volume number, inclusive pages, and year of publication. Authors should follow these examples: Journal Article 1. H.H.Gonska,Degree of simultaneous approximation of bivariate functions by Gordon operators, (journal name in italics) J. Approx. Theory, 62,170-191(1990).

Book 2. G.G.Lorentz, (title of book in italics) Bernstein Polynomials (2nd ed.), Chelsea,New York,1986.

Contribution to a Book 3. M.K.Khan, Approximation properties of beta operators,in(title of book in italics) Progress in Approximation Theory (P.Nevai and A.Pinkus,eds.), Academic Press, New York,1991,pp.483-495.

11. All acknowledgements (including those for a grant and financial support) should occur in one paragraph that directly precedes the References section.

12. Footnotes should be avoided. When their use is absolutely necessary, footnotes should be numbered consecutively using Arabic numerals and should be typed at the bottom of the page to which they refer. Place a line above the footnote, so that it is set off from the text. Use the appropriate superscript numeral for citation in the text. 13. After each revision is made please again submit via email Latex and PDF files of the revised manuscript, including the final one. 14. Effective 1 Nov. 2009 for current journal page charges, contact the Editor in Chief. Upon acceptance of the paper an invoice will be sent to the contact author. The fee payment will be due one month from the invoice date. The article will proceed to publication only after the fee is paid. The charges are to be sent, by money order or certified check, in US dollars, payable to Eudoxus Press, LLC, to the address shown on the Eudoxus homepage. No galleys will be sent and the contact author will receive one (1) electronic copy of the journal issue in which the article appears.

15. This journal will consider for publication only papers that contain proofs for their listed results.

10

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.'S 1-2, 11-21, 2014, COPYRIGHT 2014 EUDOXUS PRESS

ORTHOGONAL STABILITY OF AN ADDITIVE-QUADRATIC FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN SPACES CHOONKIL PARK, MADJID ESHAGHI GORDJI, HASSAN AZADI KENARY, AND JUNG RYE LEE∗ Abstract. Using the fixed point method, we prove the Hyers-Ulam stability of an orthogonally additive-quadratic functional equation in non-Archimedean normed spaces.

1. Introduction and preliminaries Assume that X is a real inner product space and f : X → R is a solution of the orthogonally Cauchy functional equation f (x + y) = f (x) + f (y), ⟨x, y⟩ = 0. By the Pythagorean theorem f (x) = ∥x∥2 is a solution of the conditional equation. Of course, this function does not satisfy the additivity equation everywhere. Thus orthogonally Cauchy equation is not equivalent to the classic Cauchy equation on the whole inner product space. G. Pinsker [39] characterized orthogonally additive functionals on an inner product space when the orthogonality is the ordinary one in such spaces. K. Sundaresan [50] generalized this result to arbitrary Banach spaces equipped with the Birkhoff-James orthogonality. The orthogonally Cauchy functional equation f (x + y) = f (x) + f (y),

x ⊥ y,

in which ⊥ is an abstract orthogonality relation, was first investigated by S. Gudder and D. Strawther [18]. They defined ⊥ by a system consisting of five axioms and described the general semi-continuous real-valued solution of conditional Cauchy functional equation. In 1985, J. R¨atz [47] introduced a new definition of orthogonality by using more restrictive axioms than of S. Gudder and D. Strawther. Moreover, he investigated the structure of orthogonally additive mappings. J. R¨atz and Gy. Szab´o [48] investigated the problem in a rather more general framework. Let us recall the orthogonality in the sense of J. R¨atz; cf. [47]. Suppose X is a real vector space (algebraic module) with dim X ≥ 2 and ⊥ is a binary relation on X with the following properties: (O1 ) totality of ⊥ for zero: x ⊥ 0, 0 ⊥ x for all x ∈ X; (O2 ) independence: if x, y ∈ X − {0}, x ⊥ y, then x, y are linearly independent; (O3 ) homogeneity: if x, y ∈ X, x ⊥ y, then αx ⊥ βy for all α, β ∈ R; (O4 ) the Thalesian property: if P is a 2-dimensional subspace of X, x ∈ P and λ ∈ R+ , 2010 Mathematics Subject Classification. Primary 39B55, 46S10, 47H10, 39B52, 47S10, 30G06, 46H25, 12J25. Key words and phrases. Hyers-Ulam stability, fixed point, orthogonally additive-quadratic functional equation, non-Archimedean normed space, orthogonality space. ∗ Corresponding author.

11

C. PARK, M. ESHAGHI GORDJI, H.A. KENARY, AND J. LEE

which is the set of nonnegative real numbers, then there exists y0 ∈ P such that x ⊥ y0 and x + y0 ⊥ λx − y0 . The pair (X, ⊥) is called an orthogonality space. By an orthogonality normed space we mean an orthogonality space having a normed structure. Some interesting examples are (i) The trivial orthogonality on a vector space X defined by (O1 ), and for non-zero elements x, y ∈ X, x ⊥ y if and only if x, y are linearly independent. (ii) The ordinary orthogonality on an inner product space (X, ⟨., .⟩) given by x ⊥ y if and only if ⟨x, y⟩ = 0. (iii) The Birkhoff-James orthogonality on a normed space (X, ∥.∥) defined by x ⊥ y if and only if ∥x + λy∥ ≥ ∥x∥ for all λ ∈ R. The relation ⊥ is called symmetric if x ⊥ y implies that y ⊥ x for all x, y ∈ X. Clearly examples (i) and (ii) are symmetric but example (iii) is not. It is remarkable to note, however, that a real normed space of dimension greater than 2 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. There are several orthogonality notions on a real normed space such as Birkhoff-James, Boussouis, Singer, Carlsson, unitary-Boussouis, Roberts, Phythagorean, isosceles and Diminnie (see [1]– [3], [7, 14, 23, 24, 35]). The stability problem of functional equations originated from the following question of Ulam [52]: Under what condition does there exist an additive mapping near an approximately additive mapping? In 1941, Hyers [20] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces. In 1978, Th.M. Rassias [41] extended the theorem of Hyers by considering the unbounded Cauchy difference ∥f (x + y) − f (x) − f (y)∥ ≤ ε(∥x∥p + ∥y∥p ), (ε > 0, p ∈ [0, 1)). The result of Th.M. Rassias has provided a lot of influence in the development of what we now call generalized Hyers-Ulam stability or Hyers-Ulam stability of functional equations. During the last decades several stability problems of functional equations have been investigated in the spirit of Hyers-Ulam-Rassias. The reader is referred to [10, 11, 21, 25, 46] and references therein for detailed information on stability of functional equations. R. Ger and J. Sikorska [17] investigated the orthogonal stability of the Cauchy functional equation f (x+y) = f (x)+f (y), namely, they showed that if f is a mapping from an orthogonality space X into a real Banach space Y and ∥f (x + y) − f (x) − f (y)∥ ≤ ε for all x, y ∈ X with x ⊥ y and some ε > 0, then there exists exactly one orthogonally additive mapping g : X → Y such that ∥f (x) − g(x)∥ ≤ 16 ε for all x ∈ X. 3 The first author treating the stability of the quadratic equation was F. Skof [49] by proving that if f is a mapping from a normed space X into a Banach space Y satisfying ∥f (x + y) + f (x − y) − 2f (x) − 2f (y)∥ ≤ ε for some ε > 0, then there is a unique quadratic mapping g : X → Y such that ∥f (x) − g(x)∥ ≤ 2ε . P.W. Cholewa [8] extended the Skof’s theorem by replacing X by an abelian group G. The Skof’s result was later generalized by S. Czerwik [9] in the spirit of Hyers-Ulam-Rassias. The stability problem of functional equations has been extensively investigated by some mathematicians (see [38], [42]–[45]). The orthogonally quadratic equation f (x + y) + f (x − y) = 2f (x) + 2f (y), x ⊥ y

12

ORTHOGONAL STABILITY OF FUNCTIONAL EQUATION

was first investigated by F. Vajzovi´c [53] when X is a Hilbert space, Y is the scalar field, f is continuous and ⊥ means the Hilbert space orthogonality. Later, H. Drljevi´c [15], M. Fochi [16], M.S. Moslehian [31, 32] and Gy. Szab´o [51] generalized this result. In 1897, Hensel [19] introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications (see [12, 27, 28, 34]). Definition 1.1. By a non-Archimedean field we mean a field K equipped with a function (valuation) | · | : K → [0, ∞) such that for all r, s ∈ K, the following conditions hold: (1) |r| = 0 if and only if r = 0; (2) |rs| = |r||s|; (3) |r + s| ≤ max{|r|, |s|}. Definition 1.2. ([33]) Let X be a vector space over a scalar field K with a nonArchimedean non-trivial valuation | · | . A function || · || : X → R is a non-Archimedean norm (valuation) if it satisfies the following conditions: (1) ||x|| = 0 if and only if x = 0; (2) ||rx|| = |r|||x|| (r ∈ K, x ∈ X); (3) The strong triangle inequality (ultrametric); namely, ||x + y|| ≤ max{||x||, ||y||},

x, y ∈ X.

Then (X, ||.||) is called a non-Archimedean space. Note that ||xn − xm || ≤ max{||xj+1 − xj || : m ≤ j ≤ n − 1}

(n > m).

Definition 1.3. A sequence {xn } is Cauchy if and only if {xn+1 − xn } converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent. Let X be a set. A function m : X × X → [0, ∞] is called a generalized metric on X if m satisfies (1) m(x, y) = 0 if and only if x = y; (2) m(x, y) = m(y, x) for all x, y ∈ X; (3) m(x, z) ≤ m(x, y) + m(y, z) for all x, y, z ∈ X. We recall a fundamental result in fixed point theory. Theorem 1.4. [4, 13] Let (X, m) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x ∈ X, either m(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) m(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; n (2) the sequence {J x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | m(J n0 x, y) < ∞}; 1 (4) m(y, y ∗ ) ≤ 1−α m(y, Jy) for all y ∈ Y .

13

C. PARK, M. ESHAGHI GORDJI, H.A. KENARY, AND J. LEE

In 1996, G. Isac and Th.M. Rassias [22] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [5, 6, 26, 30, 36, 37, 40]). In this paper, we prove the Hyers-Ulam stability of the following orthogonally additivequadratic functional equation (

)

(

)

3f (x) f (−x) f (y) f (−y) x+y x−y 2f + 2f = − + + (1.1) 2 2 2 2 2 2 in non-Archimedean normed spaces by using the fixed point method. Throughout this paper, assume that (X, ⊥) is an orthogonality space and that (Y, ∥.∥Y ) is a non-Archimedean Banach space. Assume that |2| ̸= 1. 2. Hyers-Ulam stability of the orthogonally additive-quadratic functional equation (1.1) For a given mapping f : X → Y , we define (

)

(

)

x+y x−y 3f (x) f (−x) f (y) f (−y) + 2f − + − − 2 2 2 2 2 2 for all all x, y ∈ X with x ⊥ y, where ⊥ is the orthogonality in the sense of R¨atz. Let f : X → Y be an even mapping f (0) = 0 and (1.1). Then f is a ( ) ( satisfying ) x+y x−y quadratic mapping, i.e., 2f 2 + 2f 2 = f (x) + f (y) holds. Using the fixed point method and applying some ideas from [17, 21], we prove the orthogonal Hyers-Ulam stability of the additive-quadratic functional equation Df (x, y) = 0 in non-Archimedean Banach spaces. Df (x, y) : = 2f

Theorem 2.1. Let φ : X 2 → [0, ∞) be a function such that there exists an α < 1 with (

x y φ(x, y) ≤ |4|αφ , 2 2

)

(2.1)

for all x, y ∈ X with x ⊥ y. Let f : X → Y be an even mapping satisfying f (0) = 0 and ∥Df (x, y)∥Y ≤ φ(x, y)

(2.2)

for all x, y ∈ X with x ⊥ y. Then there exists a unique orthogonally quadratic mapping Q : X → Y such that α ∥f (x) − Q(x)∥Y ≤ φ(x, 0) (2.3) 1−α for all x ∈ X. Proof. Letting y = 0 in (2.2), we get

( )

x

4f

− f (x)

2

Y

14

≤ φ(x, 0)

(2.4)

ORTHOGONAL STABILITY OF FUNCTIONAL EQUATION

for all x ∈ X, since x ⊥ 0. Thus



1

f (x) − f (2x)

4

Y

≤

1 |4|α φ(2x, 0) ≤ φ(x, 0) |4| |4|

(2.5)

for all x ∈ X. Consider the set S := {h : X → Y } and introduce the generalized metric on S: m(g, h) = inf{µ ∈ R+ : ∥g(x) − h(x)∥Y ≤ µφ(x, 0), ∀x ∈ X}, where, as usual, inf ϕ = +∞. It is easy to show that (S, m) is complete (see [29, Lemma 2.1]). Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 4 for all x ∈ X. Let g, h ∈ S be given such that m(g, h) = ε. Then ∥g(x) − h(x)∥Y ≤ φ(x, 0) for all x ∈ X. Hence ∥Jg(x) − Jh(x)∥Y



1

1

= g (2x) − h (2x)

≤ αφ(x, 0) 4 4 Y for all x ∈ X. So m(g, h) = ε implies that m(Jg, Jh) ≤ αε. This means that m(Jg, Jh) ≤ αm(g, h) for all g, h ∈ S. It follows from (2.5) that m(f, Jf ) ≤ α. By Theorem 1.4, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., Q (2x) = 4Q(x)

(2.6)

for all x ∈ X. The mapping Q is a unique fixed point of J in the set M = {g ∈ S : m(h, g) < ∞}. This implies that Q is a unique mapping satisfying (2.6) such that there exists a µ ∈ (0, ∞) satisfying ∥f (x) − Q(x)∥Y

≤ µφ(x, 0)

for all x ∈ X; (2) m(J n f, Q) → 0 as n → ∞. This implies the equality lim

n→∞

1 f (2n x) = Q(x) 4n

for all x ∈ X;

15

C. PARK, M. ESHAGHI GORDJI, H.A. KENARY, AND J. LEE

(3) m(f, Q) ≤

1 m(f, Jf ), 1−α

which implies the inequality α m(f, Q) ≤ . 1−α This implies that the inequality (2.3) holds. It follows from (2.1) and (2.2) that 1 ∥Df (2n x, 2n y)∥Y ∥DQ(x, y)∥Y = lim n→∞ |4|n 1 |4|n αn n n ≤ lim φ(2 x, 2 y) ≤ lim φ(x, y) = 0 n→∞ |4|n n→∞ |4|n for all x, y ∈ X with x ⊥ y. So DQ(x, y) = 0 for all x, y ∈ X with x ⊥ y. Hence Q : X → Y is an orthogonally quadratic mapping, as desired.  Corollary 2.2. Assume that (X, ⊥) is an orthogonality non-Archimedean normed space. Let θ be a positive real number and p a real number with p > 2. Let f : X → Y be an even mapping satisfying f (0) = 0 and ∥Df (x, y)∥Y ≤ θ(∥x∥p + ∥y∥p )

(2.7)

for all x, y ∈ X with x ⊥ y. Then there exists a unique orthogonally quadratic mapping Q : X → Y such that |2|p θ ∥f (x) − Q(x)∥Y ≤ ∥x∥p |4| − |2|p for all x ∈ X. Proof. Taking φ(x, y) = θ(∥x∥p + ∥y∥p ) for all x, y ∈ X with x ⊥ y and choosing α = |2|p−2 in Theorem 2.1, we get the desired result.  Theorem 2.3. Let φ : X 2 → [0, ∞) be a function such that there exists an α < 1 with α φ(x, y) ≤ φ (2x, 2y) |4| for all x, y ∈ X with x ⊥ y. Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.2). Then there exists a unique orthogonally quadratic mapping Q : X → Y such that 1 ∥f (x) − Q(x)∥Y ≤ φ(x, 0) 1−α for all x ∈ X. Proof. Let (S, m) be the generalized metric space defined in the proof of Theorem 2.1. Now we consider the linear mapping J : S → S such that ( ) x Jg(x) := 4g 2 for all x ∈ X. It follows from (2.4) that m(f, Jf ) ≤ 1. The rest of the proof is similar to the proof of Theorem 2.1. 

16

ORTHOGONAL STABILITY OF FUNCTIONAL EQUATION

Corollary 2.4. Assume that (X, ⊥) is an orthogonality non-Archimedean normed space. Let θ be a positive real number and p a real number with 0 < p < 2. Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.7). Then there exists a unique orthogonally quadratic mapping Q : X → Y such that |2|p θ ∥f (x) − Q(x)∥Y ≤ p ∥x∥p |2| − |4| for all x ∈ X. Proof. Taking φ(x, y) = θ(∥x∥p + ∥y∥p ) for all x, y ∈ X with x ⊥ y and choosing α = |2|2−p in Theorem 2.3, we get the desired result.  Let (f : X) → Y( be an ) odd mapping satisfying (1.1). Then f is an additive mapping, x+y x−y i.e., f 2 + f 2 = f (x) holds. Theorem 2.5. Let φ : X 2 → [0, ∞) be a function such that there exists an α < 1 with ( ) x y φ(x, y) ≤ |2|αφ , 2 2 for all x, y ∈ X with x ⊥ y. Let f : X → Y be an odd mapping satisfying (2.2). Then there exists a unique orthogonally additive mapping A : X → Y such that α ∥f (x) − A(x)∥Y ≤ φ(x, 0) |2| − |2|α for all x ∈ X. Proof. Letting y = 0 in (2.2), we get

( )

x

4f

≤ φ(x, 0) − 2f (x) (2.8)

2 Y for all x ∈ X, since x ⊥ 0. Thus



1 |2|α 1

f (x) − f (2x) ≤ φ(2x, 0) ≤ φ(x, 0) (2.9)

2 |4| |4| Y for all x ∈ X. Let (S, m) be the generalized metric space defined in the proof of Theorem 2.1. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 2 for all x ∈ X. α It follows from (2.9) that m(f, Jf ) ≤ |2| . The rest of the proof is similar to the proof of Theorem 2.1.  Corollary 2.6. Assume that (X, ⊥) is an orthogonality non-Archimedean normed space. Let θ be a positive real number and p a real number with p > 1. Let f : X → Y be an odd mapping satisfying (2.7). Then there exists a unique orthogonally additive mapping A : X → Y such that |2|p θ ∥f (x) − A(x)∥Y ≤ ∥x∥p |2|(|2| − |2|p )

17

C. PARK, M. ESHAGHI GORDJI, H.A. KENARY, AND J. LEE

for all x ∈ X. Proof. Taking φ(x, y) = θ(∥x∥p + ∥y∥p ) for all x, y ∈ X with x ⊥ y and choosing α = |2|p−1 in Theorem 2.5, we get the desired result.  Theorem 2.7. Let φ : X 2 → [0, ∞) be a function such that there exists an α < 1 with α φ(x, y) ≤ φ (2x, 2y) |2| for all x, y ∈ X with x ⊥ y. Let f : X → Y be an odd mapping satisfying (2.2). Then there exists a unique orthogonally additive mapping A : X → Y such that 1 ∥f (x) − A(x)∥Y ≤ φ(x, 0) |2| − |2|α for all x ∈ X. Proof. Let (S, m) be the generalized metric space defined in the proof of Theorem 2.1. Now we consider the linear mapping J : S → S such that ( ) x Jg(x) := 2g 2 for all x ∈ X. 1 It follows from (2.8) that m(f, Jf ) ≤ |2| .  The rest of the proof is similar to the proof of Theorem 2.1. Corollary 2.8. Assume that (X, ⊥) is an orthogonality non-Archimedean normed space. Let θ be a positive real number and p a real number with 0 < p < 1. Let f : X → Y be an odd mapping satisfying (2.7). Then there exists a unique orthogonally additive mapping A : X → Y such that ∥f (x) − A(x)∥Y ≤

|2|p θ ∥x∥p |2|(|2|p − |2|)

for all x ∈ X. Proof. Taking φ(x, y) = θ(∥x∥p + ∥y∥p ) for all x, y ∈ X with x ⊥ y and choosing α = |2|1−p in Theorem 2.7, we get the desired result.  (−x) Let f : X → Y be a mapping satisfying f (0) = 0 and (1.1). Let fe (x) := f (x)+f 2 (−x) and fo (x) = f (x)−f . Then fe is an even mapping satisfying (1.1) and fo is an odd 2 mapping satisfying (1.1) such that f (x) = fe (x) + fo (x). So we obtain the following.

Theorem 2.9. Assume that (X, ⊥) is an orthogonality non-Archimedean normed space. Let θ be a positive real number and p a positive real number with p ̸= 1. Let f : X → Y be a mapping satisfying f (0) = 0 and (2.7). Then there exist an orthogonally additive mapping A : X → Y and an orthogonally quadratic mapping Q : X → Y such that (

∥f (x) − A(x) − Q(x)∥Y ≤

)

|2|p |2|p + θ||x||p p p |2| · | |2| − |2| | | |4| − |2| |

for all x ∈ X.

18

ORTHOGONAL STABILITY OF FUNCTIONAL EQUATION

References [1] J. Alonso and C. Ben´ıtez, Orthogonality in normed linear spaces: a survey I. Main properties, Extracta Math. 3 (1988), 1–15. [2] J. Alonso and C. Ben´ıtez, Orthogonality in normed linear spaces: a survey II. Relations between main orthogonalities, Extracta Math. 4 (1989), 121–131. [3] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935), 169–172. [4] L. C˘adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). [5] L. C˘adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [6] L. C˘adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). [7] S.O. Carlsson, Orthogonality in normed linear spaces, Ark. Mat. 4 (1962),297–318. [8] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [9] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [10] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, London, Singapore and Hong Kong, 2002. [11] S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Palm Harbor, Florida, 2003. [12] D. Deses, On the representation of non-Archimedean objects, Topology Appl. 153 (2005), 774–785. [13] J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [14] C.R. Diminnie, A new orthogonality relation for normed linear spaces, Math. Nachr. 114 (1983), 197–203. [15] F. Drljevi´c, On a functional which is quadratic on A-orthogonal vectors, Publ. Inst. Math. (Beograd) 54 (1986), 63–71. [16] M. Fochi, Functional equations in A-orthogonal vectors, Aequationes Math. 38 (1989), 28–40. [17] R. Ger and J. Sikorska, Stability of the orthogonal additivity, Bull. Polish Acad. Sci. Math. 43 (1995), 143–151. [18] S. Gudder and D. Strawther, Orthogonally additive and orthogonally increasing functions on vector spaces, Pacific J. Math. 58 (1975), 427–436. [19] K. Hensel, Ubereine news Begrundung der Theorie der algebraischen Zahlen, Jahresber. Deutsch. Math. Verein 6 (1897), 83–88. [20] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [21] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [22] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [23] R.C. James, Orthogonality in normed linear spaces, Duke Math. J. 12 (1945), 291–302. [24] R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265–292. [25] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Florida, 2001. [26] Y. Jung and I. Chang, The stability of a cubic type functional equation with the fixed point alternative, J. Math. Anal. Appl. 306 (2005), 752–760. [27] A.K. Katsaras and A. Beoyiannis, Tensor products of non-Archimedean weighted spaces of continuous functions, Georgian Math. J. 6 (1999), 33–44.

19

C. PARK, M. ESHAGHI GORDJI, H.A. KENARY, AND J. LEE

[28] A. Khrennikov, Non-Archimedean analysis: quantum paradoxes, dynamical systems and biological models, Mathematics and its Applications 427, Kluwer Academic Publishers, Dordrecht, 1997. [29] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [30] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [31] M.S. Moslehian, On the orthogonal stability of the Pexiderized quadratic equation, J. Difference Equat. Appl. 11 (2005), 999–1004. [32] M.S. Moslehian, On the stability of the orthogonal Pexiderized Cauchy equation, J. Math. Anal. Appl. 318, (2006), 211–223. [33] M.S. Moslehian and Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.–TMA 69 (2008), 3405–3408. [34] P.J. Nyikos, On some non-Archimedean spaces of Alexandrof and Urysohn, Topology Appl. 91 (1999), 1–23. [35] L. Paganoni and J. R¨atz, Conditional function equations and orthogonal additivity, Aequationes Math. 50 (1995), 135–142. [36] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007). [37] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). [38] C. Park and J. Park, Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping, J. Difference Equat. Appl. 12 (2006), 1277–1288. [39] A.G. Pinsker, Sur une fonctionnelle dans l’espace de Hilbert, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 20 (1938), 411–414. [40] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [41] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [42] Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babe¸s-Bolyai Math. 43 (1998), 89–124. [43] Th.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352–378. [44] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 23–130. [45] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. [46] Th.M. Rassias (ed.), Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003. [47] J. R¨atz, On orthogonally additive mappings, Aequationes Math. 28 (1985), 35–49. [48] J. R¨atz and Gy. Szab´o, On orthogonally additive mappings IV , Aequationes Math. 38 (1989), 73–85. [49] F. Skof, Propriet` a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [50] K. Sundaresan, Orthogonality and nonlinear functionals on Banach spaces, Proc. Amer. Math. Soc. 34 (1972), 187–190. [51] Gy. Szab´o, Sesquilinear-orthogonally quadratic mappings, Aequationes Math. 40 (1990), 190–200. [52] S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960. ¨ [53] F. Vajzovi´c, Uber das Funktional H mit der Eigenschaft: (x, y) = 0 ⇒ H(x + y) + H(x − y) = 2H(x) + 2H(y), Glasnik Mat. Ser. III 2 (22) (1967), 73–81.

20

ORTHOGONAL STABILITY OF FUNCTIONAL EQUATION

Choonkil Park Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea E-mail address: [email protected] Madjid Eshaghi Gordji Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran E-mail address: [email protected] Hassan Azadi Kenary Department of Mathematics, College of Science, Yasouj University, Yasouj 75914-353, Iran E-mail address: [email protected] Jung Rye Lee Department of Mathematics, Daejin University, Kyeonggi 487-711, Korea E-mail address: [email protected]

21

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.'S 1-2, 22-46, 2014, COPYRIGHT 2014 EUDOXUS PRESS LLC

STABILITY OF THE LEIBNIZ ADDITIVE-QUADRATIC FUNCTIONAL EQUATION IN QUASI-BETA NORMED SPACE: DIRECT AND FIXED POINT METHODS MATINA J. RASSIAS1 , M. ARUNKUMAR2 , S. RAMAMOORTHI3

1

Department of Statistical Science , University College London, 1-19 Torrington Place, #140, London, WC1E 7HB, UK. E-mail: [email protected] 2 Department of Mathematics, Government Arts College, Tiruvannamalai - 606 603, TamilNadu, India. E-mail: [email protected] 3 Department of Mathematics, Arunai Engineering College, Tiruvannamalai - 606 603, TamilNadu, India. E-mail:3 [email protected]

Abstract. In this paper, the authors introduced the Leibniz type additivequadratic functional equation of the form     x+y+z 2x − y − z f (x − t) + f (y − t) + f (z − t) = 3f −t +f 3 3     −x + 2y − z −x − y + 2z +f +f 3 3 and obtained its general solution and generalized Ulam - Hyers stability of Leibniz AQ - mixed type functional equation in quasi-beta normed space using direct and fixed point methods.

1. INTRODUCTION The study of stability problems for functional equations is related to a question of Ulam [26] concerning the stability of group homomorphisms was affirmatively answered for Banach spaces by Hyers [9]. It was further generalized via excellent results obtained by a number of authors [2, 6, 18, 21, 23]. Over the last six or seven decades, the above Ulam problem was tackled by numerous authors who provided solutions in various forms of functional equations like 2010 Mathematics Subject Classification. :39B52, 32B72, 32B82 . Key words and phrases. : Additive functional equations, quadratic functional equation, Mixed type AQ functional equation, Ulam - Hyers stability, Leibniz Theorem.

22

2

MATINA J. RASSIAS, M. ARUNKUMAR, S. RAMAMOORTHI

additive, quadratic, cubic, quartic, mixed type functional equations involving only these types of functional equations were discussed. We refer the interested readers for more information on such problems to the monographs [1, 5, 8, 10, 13, 15, 17, 19, 20, 22, 24, 25, 27, 28, 29]. In 2006, K.W. Jun and H.M. Kim [11] introduced the following generalized additive and quadratic type functional equation ! n n X X X f xi + (n − 2) f (xi ) = f (xi + xj ) (1.1) i=1

1 ≤ i 0 , we deduce



fo (2n · 2m x)

fo (2n+m x) fo (2m x) 1 m



− f (2 x) o

2(n+m) − 2m = 2m 2n Y Y n−1 K n−1 X α(2k+m+1 x, 2k+m x, 0, 0) ≤ 2β k=0 2k+m ∞ K n−1 X α(2k+m+1 x, 2k+m x, 0, 0) ≤ 2β k=0 2k+m

→ 0 as m → ∞ for all x ∈ U. Thus it follows that a sequence   fo (2n x) , 2n

27

STABILITY OF THE LEIBNIZ ADDITIVE-QUADRATIC . . .

7

is a Cauchy in Y and so it converges. Therefore we see that a mapping A : X → Y defined by fo (2n x) A(x) = lim n→∞ 2n is well defined for all x ∈ X. In addition it is clear from (4.1) that the following inequality 1 kDfo (2n x, 2n y, 2n z, 2n t)kpY n→∞ 2pn 1 ≤ lim pn α(2n x, 2n y, 2n z, 2n t)p n→∞ 2 → 0 as n → ∞

kDA(x, y, z, t)kpY = lim

holds for all x, y, z, t ∈ X and so the mapping A is additive. Letting n → ∞ in (4.9) and using the definition of A(x) we see that (4.3) holds for all x ∈ U . To prove uniqueness, we assume now that there is another function A0 : X → Y which satisfies (1.5) and the inequality (4.3) then it follows that A(2x) = 2A(x), A0 (2x) = 2A0 (x) for all x ∈ X and all n ∈ N . Thus 1 p p kA(x) − A0 (x)kY = βpn kA(2n x) − A0 (2n x)kY 2 Kp  p = βpn kA(2n x) − fo (2n x)kpY + kfo (2n x) − A0 (2n x)kY 2 ! ∞ K p 2p K p(n−1) X α(2k+n+1 x, 2k+n x, 0, 0)p ≤ βn 2 2pβ 2p(k+n) k=0 → 0 as n → ∞ for all x ∈ X. Hence A is unique. For j = −1, we can prove a similar stability result. This completes the proof of the theorem.  The following Corollary is an immediate consequence of Theorem 4.1 concerning the stability of (1.5). Corollary 4.2. Let fo : X → Y be an odd mapping and there exits real numbers λ and s such that kDfo (x, y, z, t)kY  λ,      λ {||x||s + ||y||s + ||z||s + ||t||s } , s < 1 or s > 1; ≤  s s s s 4s 4s  λ {||x|| ||y|| ||z|| ||t|| + {||x|| + ||y|| + ||z||4s + ||t||4s }} ,    s < 1 or s > 1 ; 4

28

4

(4.10)

8

MATINA J. RASSIAS, M. ARUNKUMAR, S. RAMAMOORTHI

for all x, y, z, t ∈ U , then there exists a unique additive function A : X → Y such that    (n−1) p 2λK   ,   2β    p  2(2s + 1)λK (n−1) ||x||s p kfo (x) − A(x)kY ≤ (4.11) , β |2 − 2s |  2  p  4s    2(2 + 1)λK (n−1) ||x||4s   , 2β |2 − 24s | for all x ∈ X. Theorem 4.3. Let j = ±1. Let fe : X → Y be an even mapping for which there exists a function α : X 4 → [0, ∞) with the condition 1 α(2nj x, 2nj y, 2nj z, 2nj t) = 0 nj n→∞ 4 lim

(4.12)

such that the functional inequality kDfe (x, y, z, t)kY ≤ α(x, y, z, t)

(4.13)

for all x, y, z, t ∈ X. Then there exists a unique quadratic mapping A : X → Y which satisfies (1.5) and the inequality kfe (x) −

Q(x)kpY

∞ K p(n−1) X α(2k+1 x, 2k x, 0, 0)p ≤ 4pβ k=0 4pk

(4.14)

for all x ∈ X. Proof. Replacing (x, y, z, t) by (2x, x, 0, 0) in the functional inequality (4.12), we get kfe (2x) − 3fe (x) − fe (−x)kY ≤ α(2x, x, 0, 0)

(4.15)

for all x ∈ X. Using evenness of fe in (4.15), we obtain kfe (2x) − 4fe (x)kY ≤ α(2x, x, 0, 0) for all x ∈ X. It follows from (4.16) that

fe (2x)

≤ 1 α(2x, x, 0, 0) − f (x) e

4

4β Y for all x ∈ X. Replacing x by 2x and dividing by 2 in (4.17), we get

fe (22 x) fe (2x) 1 2

42 − 4 ≤ 4β · 2 α(2 x, 2x, 0, 0) Y

29

(4.16)

(4.17)

(4.18)

STABILITY OF THE LEIBNIZ ADDITIVE-QUADRATIC . . .

9

for all x ∈ X. From (4.17) and (4.18), we have



 

fe (22 x)

fe (22 x) fe (2x)

fe (2x)





42 − fe (x) ≤ K 4 − fe (x) + 42 − 4 Y Y Y   2 α(2 x, 2x, 0, 0) K (4.19) ≤ β α(2x, x, 0, 0) + 4 4 for all x ∈ U . Proceeding further and using induction on a positive integer n , we get

p n−1

fe (2n x)

K p(n−1) X α(2k+1 x, 2k x, 0, 0)p

(4.20)

4n − fe (x) ≤ 4pβ pk 4 Y k=0 ∞ K p(n−1) X α(2k+1 x, 2k x, 0, 0)p ≤ 4pβ k=0 4pk

for all x ∈ U . In order to prove the convergence of the sequence   fe (2n x) , 4n replacing x by 2m x and dividing by 4m in (4.20), for any m, n > 0 , we deduce



fe (2n · 2m x)

fe (2n+m x) fe (2m x) 1 m



− f (2 x) e

4(n+m) − 4m = 4m 4n Y Y n−1 K n−1 X α(2k+m+1 x, 2k+m x, 0, 0) ≤ 4β k=0 4k+m ∞ K n−1 X α(2k+m+1 x, 2k+m x, 0, 0) ≤ 4β k=0 4k+m

→ 0 as m → ∞ for all x ∈ U. Thus it follows that a sequence   fe (2n x) , 4n is a Cauchy in Y and so it converges. Therefore we see that a mapping Q : X → Y defined by fe (2n x) Q(x) = lim n→∞ 4n is well defined for all x ∈ X. To show that Q satisfies (1.5) and it is unique the proof is similar to that of Theorem 4.1. For j = −1, we can prove a similar stability result. This completes the proof of the theorem. 

30

10

MATINA J. RASSIAS, M. ARUNKUMAR, S. RAMAMOORTHI

The following Corollary is an immediate consequence of Theorem 4.3 concerning the stability of (1.5). Corollary 4.4. Let fe : X → Y be an even mapping and there exits real numbers λ and s such that kDfe (x, y, z, t)kY  λ,      λ {||x||s + ||y||s + ||z||s + ||t||s } , s < 2 or s > 2; ≤  s s s s 4s 4s 4s  λ {||x|| ||y|| ||z|| ||t|| + {||x|| + ||y|| + ||z|| + ||t||4s }} ,    1 s < or s > 1 ; 2

(4.21)

2

for all x, y, z, t ∈ U , then there exists a unique additive function A : X → Y such that    (n−1) p 4λK   ,  β  3 · 4   p   s 4(2 + 1)λK (n−1) ||x||s p kfe (x) − Q(x)kY ≤ (4.22) ,  4β |4 − 2s |     p   4(24s + 1)λK (n−1) ||x||4s   , 4β |4 − 24s | for all x ∈ X. Now we are ready to prove our main theorem. Theorem 4.5. Let j ∈ {−1, 1} and α : X 4 → [0, ∞) be a function satisfying (4.1) and (4.12) for all x, y, z, t ∈ X. Let f : X → Y be a function satisfying the inequality kDf (x, y, z, t)ky ≤ α (x, y, z, t)

(4.23)

for all x, y, z, t ∈ X. Then there exists a unique additive mapping A : X → Y and a unique quadratic mapping Q : X → Y such that kf (x) − A(x) − Q(x)kpY "  ∞  K p K p(n−1) X α(2k+1 x, 2k x, 0, 0)p α(−2k+1 x, −2k x, 0, 0)p + ≤ p 2 2pβ k=0 2pk 2pk # ∞  p(n−1) X k+1 k p k+1 k p K α(2 x, 2 x, 0, 0) α(−2 x, −2 x, 0, 0) + + (4.24) pβ pk 4 4 4pk k=0 for all x ∈ X.

31

STABILITY OF THE LEIBNIZ ADDITIVE-QUADRATIC . . .

11

fo (x) − fo (−x) for all x ∈ x. Then fa (0) = 0 and fa (−x) = 2 −fa (x) for all x ∈ X. Hence Proof. Let fa (x) =

kDfa (x, y, z, t)kY ≤

α(x, y, z, t) α(−x, −y, −z, −t) + 2 2

(4.25)

By Theorem 4.1, we have  ∞  1 K p(n−1) X α(2k+1 x, 2k x, 0, 0)p α(−2k+1 x, −2k x, 0, 0)p kfa (x) − A(x)kY ≤ + 2 2pβ k=0 2pk 2pk (4.26) fe (x) + fe (−x) for all x ∈ X. Also, let fq (x) = for all x ∈ X. Then fq (0) = 0 and 2 fq (−x) = fq (x) for all x ∈ x. Hence kDfq (x, y, z, t)kY ≤

α(x, y, z, t) α(−x, −y, −z, t) + 2 2

(4.27)

By Theorem 4.3, we have  ∞  1 K p(n−1) X α(2k+1 x, 2k x, 0, 0)p α(−2k+1 x, −2k x, 0, 0)p + kfq (x) − Q(x)kY ≤ 2 4pβ k=0 4pk 4pk (4.28) for all x ∈ X. Define f (x) = fa (x) + fq (x)

(4.29)

for all x ∈ x. From (5.24),(5.26) and (5.27), we arrive kf (x) − A(x) − Q(x)kpy = kfa (x) + fq (x) − A(x) − Q(x)kpY ≤ kfa (x) − A(x)kpY + kfq (x) − Q(x)kpY "  ∞  K p K p(n−1) X α(2k+1 x, 2k x, 0, 0)p α(−2k+1 x, −2k x, 0, 0)p + ≤ p 2 2pβ k=0 2pk 2pk # ∞  K p(n−1) X α(2k+1 x, 2k x, 0, 0)p α(−2k+1 x, −2k x, 0, 0)p + + 4pβ k=0 4pk 4pk for all x ∈ X. Hence the theorem is proved.



Using Corollaries 4.2 and 4.4 we have the following Corollary concerning the stability of (1.5).

32

12

MATINA J. RASSIAS, M. ARUNKUMAR, S. RAMAMOORTHI

Corollary 4.6. Let λ and s be nonnegative real numbers. Let a function f : X → Y satisfies the inequality kDf (x, y, z, t)kY  λ,      λ {||x||s + ||y||s + ||z||s + ||t||s } , s < 1 or s > 1; ≤  s s s s 4s 4s  λ {||x|| ||y|| ||z|| ||t|| + {||x|| + ||y|| + ||z||4s + ||t||4s }} ,    s < 1 or s > 1 ; 4

(4.30)

4

for all x, y, z, t ∈ X. Then there exists a unique additive function A : X → Y and a unique quadratic function Q : X → Y such that kf (x) − A(x) − Q(x)kpY p  p    4λK (n−1) 2λK (n−1)   + ,   2β 3 · 4β    p   p  4(2s + 1)λK (n−1) ||x||s 2(2s + 1)λK (n−1) ||x||s ≤ + , β |2 − 2s | β |4 − 2s |  2 4   4s p  4s p     2(2 + 1)λK (n−1) ||x||4s 4(2 + 1)λK (n−1) ||x||4s   + 2β |2 − 24s | 4β |4 − 24s |

(4.31)

for all x ∈ X. 5. STABILITY RESULTS: FIXED METHOD In this section, the generalized Ulam - Hyers - Rassias stability of the Leibniz AQ - functional equation (1.5) is given by the Fixed point method . For notational convenience, we denote for a given mapping f : X → Y and define the difference operator Df : X → Y by     x+y+z 2x − y − z Df (x, y, z, t) = f (x − t) + f (y − t) + f (z − t) − 3f −t −f 3 3     −x + 2y − z −x − y + 2z +f +f 3 3 for all x, y, z, t ∈ X . Now we will recall the fundamental results in fixed point theory. Theorem 5.1. (Banach’s contraction principle) Let (X, d) be a complete metric space and consider a mapping T : X → X which is strictly contractive mapping, that is (A1) d(T x, T y) ≤ Ld(x, y) for some (Lipschitz constant) L < 1. Then, (i) The mapping T has one and only fixed point x∗ = T (x∗ );

33

STABILITY OF THE LEIBNIZ ADDITIVE-QUADRATIC . . .

13

(ii)The fixed point for each given element x∗ is globally attractive, that is (A2) limn→∞ T n x = x∗ , for any starting point x ∈ X; (iii) One has the following estimation inequalities: 1 (A3) d(T n x, x∗ ) ≤ 1−L d(T n x, T n+1 x), ∀ n ≥ 0, ∀ x ∈ X; 1 (A4) d(x, x∗ ) ≤ 1−L d(x, x∗ ), ∀ x ∈ X. Theorem 5.2. [16](The alternative of fixed point) Suppose that for a complete generalized metric space (X, d) and a strictly contractive mapping T : X → X with Lipschitz constant L. Then, for each given element x ∈ X, either (B1) d(T n x, T n+1 x) = ∞ ∀ n ≥ 0, or (B2) there exists a natural number n0 such that: (i) d(T n x, T n+1 x) < ∞ for all n ≥ n0 ; (ii)The sequence (T n x) is convergent to a fixed point y ∗ of T (iii) y ∗ is the unique fixed point of T in the set Y = {y ∈ X : d(T n0 x, y) < ∞}; 1 (iv) d(y ∗ , y) ≤ 1−L d(y, T y) for all y ∈ Y. In this section, let us assume V be a vector space and B Banach space respectively. Theorem 5.3. Let fo : V → B be a mapping for which there exists a function α : V 4 → [0, ∞) with the condition α(µni x, µni y, µni z, µni t) lim =0 (5.1) n→∞ µni where µi = 2 if i = 0 and µi =

1 2

if i = 1 such that the functional inequality with

kDfo (x, y, z, t)kY ≤ α(x, y, z, t)

(5.2)

for all x, y, z, t ∈ V . If there exists L = L(i) such that the function  x  x → γ(x) = α x, , 0, 0 , 2 has the property   x γ(x) ≤ L µi γ (5.3) µi for all x ∈ V . Then there exists unique additive function A : V → B satisfying the functional equation (1.5) and  1−i p L p k fa (x) − A(x) kY ≤ γ(x)p (5.4) 1−L holds for all x ∈ V . Proof. Consider the set Ω = {g/g : V → B, g(0) = 0} and introduce the generalized metric on Ω, d(g, h) = inf{M ∈ (0, ∞) :k g(x) − h(x) kY ≤ M γ(x), x ∈ V }.

34

14

MATINA J. RASSIAS, M. ARUNKUMAR, S. RAMAMOORTHI

It is easy to see that (Ω, d) is complete. Define T : Ω → Ω by T g(x) =

1 g(µi x), µi

f or all x ∈ V.

Now g, h ∈ Ω, d(g, h) ≤ M ⇒ k g(x) − h(x) kY ≤ M γ(x), x ∈ V.



1 1 1

⇒

µi g(µi x) − µi h(µi x) ≤ µi M γ(µi x), x ∈ V,

Y

1

1

≤ L M γ(x), x ∈ V, ⇒ g(µ x) − h(µ x) i i

µi

µi Y ⇒ k T g(x) − T h(x) kY ≤ LM γ(y), x ∈ V, ⇒d(T g, T h) ≤ LM. This implies d(T g, T h) ≤ Ld(g, h), for all g, h ∈ Ω . i.e., T is a strictly contractive mapping on Ω with Lipschitz constant L. It follows form (4.6) that,

fo (2x)

≤ 1 α(2x, x, 0, 0) − f (x) o

2

2β Y

(5.5)

for all y ∈ V . Using (5.3) for the case i = 0 it reduces to

fo (2x)



2 − fo (x) ≤ Lγ(x) Y for all x ∈ V . i.e.,

d(fo , T fo ) ≤ L =

1 ⇒ d(fo , T fo ) ≤ L = L1 < ∞. β 2

x 2

in (5.5), we get

 x   x 

fo (x) − 2fo

≤ α x, , 0, 0 2 Y 2 for all x ∈ V . Using (5.3) for the case i = 1 it reduces to

 x 

fo (x) − 2fo

≤ γ(x) 2 Y for all X ∈ V . Again replacing x =

i.e.,

(5.6)

d(fo , T fo ) ≤ 1 ⇒ d(fo , T fo ) ≤ 1 = L0 < ∞.

In both cases, we have d(fo , T fo ) ≤ L1−i Therefore (B1 (i)) holds.

35

(5.7)

STABILITY OF THE LEIBNIZ ADDITIVE-QUADRATIC . . .

15

By (B1 (ii)), it follows that there exists a fixed point A of T in Ω such that A(x) = lim

n→∞

fo (µni y) µni

∀ x ∈ V.

(5.8)

In order to prove A : V → B is Additive. Replacing (x, y, z, t) by (µni x, µni y, µni z, µni t) in (5.2) and dividing by µni , it follows from (5.1) and (5.8), A satisfies (1.5) for all x, y, z, t ∈ V . By (B1 (iii)), A is the unique fixed point of T in the set ∆ = {fo ∈ X : d(fo , A) < ∞}, such that kfo (x) − A(x)kY ≤ M β(x) for all x ∈ V and M > 0. Finally, by (B1 (iV )), we obtain d(fo , A) ≤

1 d(fo , T fo ) 1−L

this implies d(fo , A) ≤

L1−i . 1−L

Hence we conclude that k fo (x) − A(x)

kpY ≤



L1−i 1−L

p

γ(x)p .

for all x ∈ V . This completes the proof of the theorem.



From Theorem 5.3, we obtain the following corollary concerning the Hyers-UlamRassias stability for the functional equation (1.5). Corollary 5.4. Let fo : X → V be a mapping and there exits real numbers λ and s such that kDfo (x, y, z, t)kY  λ {||x||s + ||y||s + ||z||s + ||t||s } ,    s < 1 or s > 1; ≤ s s s s 4s 4s λ {||x|| ||y|| ||z|| ||t|| + {||x|| + ||y|| + ||z||4s + ||t||4s }} ,    s < 1 or s > 1 ; 4

(5.9)

4

for all x, y, z, t ∈ U , then there exists a unique additive function A : X → Y such that   s p (2 + 1)λ||x||s   ,  2β |2 − 2s | p   kfo (x) − A(x)kY ≤ (5.10) p (24s + 1)λ||x||4s   ,  2β |2 − 24s | for all x ∈ X.

36

16

MATINA J. RASSIAS, M. ARUNKUMAR, S. RAMAMOORTHI

Proof. Setting  α(x, y, z, t) =

s λ {||x|| + ||y||s + ||z||s +||t||s },  λ ||x||s ||y||s ||z||s ||t||s + ||x||4s + ||y||4s + ||z||4s + ||t||4s

for all x, y, z, t ∈ X. Then,for s < 1 if i = 0 and for s > 1 if i = 1, we get α(µni x, µni y, µni z, µni t) µni  λ    n {||µni x||s + ||µni y||s + ||µni z||s + ||µni t||s }, µi = o  n 4s λ n n s  n s n s n s n 4s n 4s n 4s   n ||µi x|| ||µi y|| ||µi z|| ||µi t|| ||µi x|| + ||µi y|| + ||µi z|| + ||µi w|| µi  → 0 as n → ∞, = → 0 as n → ∞. Thus, (5.1) is holds.
But we have γ(x) = α x, x2 , 0, 0 has the x ∈ X. Hence     1  x γ(x) = β α x, , 0, 0 =  2 2 

property γ(x) ≤ L · µi γ (µi x) for all λ 2β λ 2β

x  ||x||s + || ||s , 2  x 4s  4s ||x|| + || || . 2 

Now, λ n µi x s o s ||µ x|| + || || , i βµ 1 2 2 i γ(µi x) = λ n µi x 4s o  µi 4s  ||µ x|| + || || . i 2β µi 2    s λ 1 + 2  s  ||x||s ,  β µi s 2 µi 2   = 4s λ 1 + 2  4s  ||x||4s .  β µi 4s 2 µi 2    1 + 2s  s−1 λ  ||x||s ,  µi 2β  2s  = 1 + 24s  4s−1 λ  ||x||4s .  µi 2β 24s  s−1 µi γ(x), = µ4s−1 γ(x). i   

Hence the inequality (5.3) holds either, L = 2s−1 for s < 1 if i = 0 and L = for s > 1 if i = 1.

37

1 2s−1

STABILITY OF THE LEIBNIZ ADDITIVE-QUADRATIC . . .

17

Now from (5.4), we prove the following cases for condition (i). Case:1 L = 2s−1 for s < 1 if i = 0
1−0   2(s−1) 1 + 2s λ kfo (x) − A(x)kY ≤ ||x||s (s−1) s 1−2 2 2β   1 + 2s λ 2s ≤ ||x||s 2 − 2s 2s 2β
1+2s λ||x||s s ≤ 2β 2 (2 − 2s ) Case:2 L =

1 2s−1

for s > 1 if i = 1
1−1 1 2(s−1) 1 1 − 2(s−1)  s

 λ 1 + 2s kfo (x) − A(x)kY ≤ ||x||s s 2 2β  2 1 + 2s λ ≤ s ||x||s s 2 −2 2 2β (1 + 2s ) λ||x||s ≤ 2β (2s − 2) 

Again, the inequality (5.3) holds either, L = 24s−1 for s < 2 if i = 0 and L = for s > 2 if i = 1. Now from (5.4), we prove the following cases for condition (ii). Case:1 L = 24s−1 for s < 1 if i = 0
1−0   2(4s−1) 1 + 24s λ kfo (x) − A(x)kY ≤ ||x||4s (4s−1) 4s β 1−2 2 2   4s 4s 2 1+2 λ ≤ ||x||4s 4s 4s 2−2 2 2β (1 + 24s ) λ||x||4s ≤ 2β (2 − 24s ) Case:2 L =

1 24s−1

1 24s−1

for s > 1 if i = 1
1−1 1 2(4s−1) 1 1 − 2(4s−1)  4s

 1 + 24s λ kfo (x) − A(x)kY ≤ ||x||4s 4s β 2 2  4s 2 λ 1+2 ≤ 4s ||x||4s 4s 2 −2 2 2β (1 + 24s ) λ||x||4s ≤ 2β (24s − 2) Hence the proof is complete





38

18

MATINA J. RASSIAS, M. ARUNKUMAR, S. RAMAMOORTHI

Theorem 5.5. Let fe : V → B be a mapping for which there exists a function α : V 4 → [0, ∞) with the condition α(µni x, µni y, µni z, µni t) lim =0 n→∞ µni where µi = 2 if i = 0 and µi =

1 2

(5.11)

if i = 1 such that the functional inequality with

kDfe (x, y, z, t)kY ≤ α(x, y, z, t)

(5.12)

for all x, y, z, t ∈ V . If there exists L = L(i) such that the function   x x → γ(x) = α x, , 0, 0 , 2 has the property   x 2 γ(x) ≤ L µi γ (5.13) µi for all x ∈ V . Then there exists unique quadratic function Q : V → B satisfying the functional equation (1.5) and  1−i p L p k fa (x) − Q(x) kY ≤ γ(x)p (5.14) 1−L holds for all x ∈ V . Proof. Consider the set Ω = {g/g : V → B, g(0) = 0} and introduce the generalized metric on Ω, d(g, h) = inf{M ∈ (0, ∞) :k g(x) − h(x) kY ≤ M γ(x), x ∈ V }. It is easy to see that (Ω, d) is complete. Define T : Ω → Ω by T g(x) =

1 g(µi x), µ2i

f or all x ∈ V.

Now g, h ∈ X, d(g, h) ≤ M ⇒ k g(x) − h(x) kY ≤ M γ(x), x ∈ V.

1

1 1

≤ 2 M γ(µi x), x ∈ V, ⇒ 2 g(µi x) − h(µi x)

µ µi µi

i

Y

1

1

≤ L M γ(x), x ∈ V, ⇒ g(µ x) − h(µ x) i i

µ2

µi i Y ⇒ k T g(x) − T h(x) kY ≤ LM γ(y), x ∈ V, ⇒d(T g, T h) ≤ LM. This implies d(T g, T h) ≤ Ld(g, h), for all g, h ∈ Ω . i.e., T is a strictly contractive mapping on Ω with Lipschitz constant L.

39

STABILITY OF THE LEIBNIZ ADDITIVE-QUADRATIC . . .

It follows form (4.17) that,

fe (2x)

≤ 1 α(2x, x, 0, 0) − f (x) e

4

4β Y

19

(5.15)

for all y ∈ V . Using (5.13) for the case i = 0 it reduces to



fe (2x)

≤ Lγ(x)

− f (x) e

4 Y

for all x ∈ V . 1 ⇒ d(fe , T fe ) ≤ L = L1 < ∞. 2β Again replacing x = x2 in (5.15), we get

 x   x 

fe (x) − 4fe

≤ α x, , 0, 0 2 Y 2 for all x ∈ V . Using (5.13) for the case i = 1 it reduces to

 x 

fe (x) − 4fe

≤ γ(x) 2 Y for all X ∈ V . i.e.,

i.e.,

d(fe , T fe ) ≤ L =

(5.16)

d(fe , T fe ) ≤ 1 ⇒ d(fe , T fe ) ≤ 1 = L0 < ∞.

In both cases, we have d(fe , T fe ) ≤ L1−i

(5.17)

Therefore (B1 (i)) holds. By (B1 (ii)), it follows that there exists a fixed point Q of T in Ω such that Q(x) = lim

n→∞

fe (µni y) µni

∀ x ∈ V.

(5.18)

In order to prove Q : V → B is quadratic. Replacing (x, y, z, t) by (µni x, µni y, µni z, µni t) in (5.12) and dividing by µ2n i , it follows from (5.11) and (5.18), Q satisfies (1.5) for all x, y, z, t ∈ V . By (B1 (iii)), Q is the unique fixed point of T in the set ∆ = {fe ∈ X : d(fe , Q) < ∞}, such that kfe (x) − Q(x)k ≤ M β(x) for all x ∈ V and M > 0. Finally, by (B1 (iV )), we obtain 1 d(fe , A) ≤ d(fe , T fe ) 1−L this implies L1−i d(fe , A) ≤ . 1−L

40

20

MATINA J. RASSIAS, M. ARUNKUMAR, S. RAMAMOORTHI

Hence we conclude that k fe (x) − Q(x)

kpY ≤



L1−i 1−L

p

γ(x)p .

for all x ∈ V . This completes the proof of the theorem.



From Theorem 5.5, we obtain the following corollary concerning the Hyers-UlamRassias stability for the functional equation (1.5). Corollary 5.6. Let fe : X → V be a mapping and there exits real numbers λ and s such that kDfe (x, y, z, t)ky  λ {||x||s + ||y||s + ||z||s + ||t||s } ,    s < 2 or s > 2; ≤ s s s s 4s 4s 4s λ {||x|| ||y|| ||z|| ||t|| + {||x|| + ||y|| + ||z|| + ||t||4s }} ,    s < 1 or s > 1 ; 2

(5.19)

2

for all x, y, z, t ∈ U , then there exists a unique quadratic function Q : X → Y such that   s p (2 + 1)λ||x||s   ,  2β |4 − 2s |   4s kfe (x) − Q(x)kpY ≤ (5.20) p (2 + 1)λ||x||4s   ,  2β |4 − 24s | for all x ∈ X. Proof. Setting  α(x, y, z, t) =

s λ {||x|| + ||y||s + ||z||s +||t||s },  λ ||x||s ||y||s ||z||s ||t||s + ||x||4s + ||y||4s + ||z||4s + ||t||4s

for all x, y, z, t ∈ X. Then,for s < 1 if i = 0 and for s > 1 if i = 1, we get α(µni x, µni y, µni z, µni t) µ2n i  λ    2n {||µni x||s + ||µni y||s + ||µni z||s + ||µni t||s }, µi =  n 4s o λ n n s  n s n s n s n 4s n 4s n 4s   2n ||µi x|| ||µi y|| ||µi z|| ||µi t|| ||µi x|| + ||µi y|| + ||µi z|| + ||µi w|| µi  → 0 as n → ∞, = → 0 as n → ∞. Thus, (5.11) is holds.

41

STABILITY OF THE LEIBNIZ ADDITIVE-QUADRATIC . . .

21


But we have γ(x) = α x, x2 , 0, 0 has the property γ(x) ≤ L · µi γ (µi x) for all x ∈ X. Hence  λ  x s  s   ||x|| + || || , 1 x 4β  2 γ(x) = β α x, , 0, 0 = λ x 4s   4 2 4s  ||x|| + || || . 4β 2 

Now,    

µi x s o λ n s ||µ x|| + || || , i 1 4β µ2i 2 γ(µ x) = n o i λ µi x 4s  µ2i  ||µi x||4s + || || .  β 2 4 µi 2    s λ 1 + 2  s  ||x||s ,  β 2 µi s 4 µi 2   = 4s λ 1 + 2  4s  ||x||4s .  β 2 µi 4s 4 µi 2    1 + 2s  s−2 λ  ||x||s ,  µi 4β  2s  = 1 + 24s  4s−2 λ  ||x||4s .  µi 4β 24s  s−2 µi γ(x), = µ4s−2 γ(x). i

Hence the inequality (5.13) holds either, L = 2s−2 for s < 2 if i = 0 and L = for s > 2 if i = 1. Now from (5.14), we prove the following cases for condition (i). Case:1 L = 2s−2 for s < 2 if i = 0
1−0   2(s−2) 1 + 2s λ kfe (x) − Q(x)kY ≤ ||x||s (s−2) s 1−2 2 4β   2s 1 + 2s λ ≤ ||x||s s s 4−2 2 4β (1 + 2s ) λ||x||s ≤ 4β (4 − 2s )

42

1 2s−2

22

Case:2 L =

MATINA J. RASSIAS, M. ARUNKUMAR, S. RAMAMOORTHI 1 2s−1

for s > 1 if i = 1 1


1−1 

 λ 1 + 2s ||x||s kfe (x) − Q(x)kY ≤ 1 s 2 4β 1 − 2(s−2)   2s 1 + 2s λ ≤ s ||x||s s 2 −4 2 4β (1 + 2s ) λ||x||s ≤ 4β (2s − 4) 2(s−2)

Again, the inequality (5.13) holds either, L = 24s−2 for s < 21 if i = 0 and L = for s > 12 if i = 1. Now from (5.14), we prove the following cases for condition (ii). Case:1 L = 24s−1 for s < 12 if i = 0
1−0   2(4s−2) 1 + 24s λ kfe (x) − Q(x)kY ≤ ||x||4s (4s−2) 4s 1−2 2 4β   24s 1 + 24s λ ≤ ||x||4s 4s 4s 4−2 2 4β (1 + 2s ) λ||x||4s ≤ 4β (4 − 24s ) Case:2 L =

1 24s−1

for s >

1 2

1 24s−2

if i = 1
1−1 1 2(4s−2) 1 1 − 2(4s−2)  4s

 λ 1 + 24s ||x||4s kfe (x) − Q(x)kY ≤ 4s β 2 4  4s 2 1+2 λ ≤ 4s ||x||4s 2 −4 24s 4β (1 + 2s ) λ||x||4s ≤ 4β (24s − 4) 

Hence the proof is complete



Theorem 5.7. Let fo : V → B be a mapping for which there exist a function α : V 4 → [0, ∞) with the conditions (5.1) and (5.11) where µi = 2 if i = 0 and µi = 21 if i = 1 such that the functional inequality with kDf (x, y, z, t)kY ≤ α(x, y, z, t)

(5.21)

for all x, y, z, t ∈ V . If there exists L = L(i) such that the function  x  x → γ(x) = α x, , 0, 0 , 2 has the properties (5.3) and (5.13) for all x ∈ V . Then there exists unique additive function A : V → B and unique quadratic function Q : V → B satisfying the

43

STABILITY OF THE LEIBNIZ ADDITIVE-QUADRATIC . . .

23

functional equation (1.5) and k f (x) − A(x) − Q(x)

kpY ≤

K

p



L1−i 1−L

p

[γ(x)p + γ(−x)p ]

(5.22)

holds for all x ∈ V . fo (x) − fo (−x) for all x ∈ x. Then fa (0) = 0 and fa (−x) = 2 −fa (x) for all x ∈ X. Hence Proof. Let fa (x) =

kDfa (x, y, z, t)kY ≤

α(x, y, z, t) α(−x, −y, −z, −t) + 2 2

(5.23)

By Theorem 5.3, we have 1 kfa (x) − A(x)kY ≤ 2



L1−i 1−L

 [γ(x) + γ(−x)]

(5.24)

fe (x) + fe (−x) for all x ∈ X. Then fq (0) = 0 and 2 fq (−x) = fq (x) for all x ∈ x. Hence for all x ∈ X. Also, let fq (x) =

kDfq (x, y, z, t)kY ≤

α(x, y, z, t) α(−x, −y, −z, t) + 2 2

(5.25)

By Theorem 4.3, we have 1 kfq (x) − Q(x)kY ≤ 2



L1−i 1−L

 [γ(x) + γ(−x)]

(5.26)

for all x ∈ X. Define f (x) = fa (x) + fq (x)

(5.27)

for all x ∈ x. From (5.24),(5.26) and (5.27), we arrive kf (x) − A(x) − Q(x)kpy = kfa (x) + fq (x) − A(x) − Q(x)kpY ≤ K p kfa (x) − A(x)kpY + kfq (x) − Q(x)kpY  1−i p L p ≤K [γ(x)p + γ(−x)p ] 1−L for all x ∈ X. Hence the theorem is proved.

44



24

MATINA J. RASSIAS, M. ARUNKUMAR, S. RAMAMOORTHI

Corollary 5.8. Let λ and s be nonnegative real numbers. Let a function f : X → Y satisfies the inequality kDf (x, y, z, t)kY  λ,      λ {||x||s + ||y||s + ||z||s + ||t||s } , s < 1 or s > 1; ≤  s s s s 4s 4s 4s  λ {||x|| ||y|| ||z|| ||t|| + {||x|| + ||y|| + ||z|| + ||t||4s }} ,    s < 1 or s > 1 ; 4

(5.28)

4

for all x, y, z, t ∈ X. Then there exists a unique additive function A : X → Y and a unique quadratic function Q : X → Y such that kf (x) − A(x) − Q(x)kpY   p 1 1   + β (2s + 1)p λp ||x||ps ,  β |2 − 2s | s| 2 4 |4 − 2 p  ≤ 1 1   + (24s + 1)p λp ||x||4ps ,  2β |2 − 24s | 4β |4 − 24s |

(5.29)

for all x ∈ X.

References [1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ, Press, 1989. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66. [3] M. Arunkumar, S. Karthikeyan, Solution and stability of n−dimensional mixed Type additive and quadratic functional equation, Far East Journal of Applied Mathematics, Volume 54, Number 1, 2011, 47-64. [4] I.S. Chang, E.H. Lee, H.M. Kim, On the Hyers-Ulam-Rassias stability of a quadratic functional equations, Math. Ineq. Appl., 6(1) (2003), 87-95. [5] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002. [6] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings , J. Math. Anal. Appl., 184 (1994), 431-436. [7] M. Eshaghi Gordji, H. Khodaie, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, arxiv: 0812. 2939v1 Math FA, 15 Dec 2008. [8] M. Eshaghi Gordji, H. Khodaei, J.M. Rassias, Fixed point methods for the stability of general quadratic functional equation, Fixed Point Theory 12 (2011), no. 1, 71-82. [9] D.H. Hyers, On the stability of the linear functional equation, Proc.Nat. Acad.Sci.,U.S.A.,27 (1941) 222-224. [10] D.H. Hyers, G. Isac, Th.M. Rassias, Stability of functional equations in several variables,Birkhauser, Basel, 1998. [11] K.W. Jun, H.M. Kim, On the stability of an n-dimensional quadratic and additive type functional equation, Math. Ineq. Appl 9(1) (2006), 153-165.

45

STABILITY OF THE LEIBNIZ ADDITIVE-QUADRATIC . . .

25

[12] S.M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126-137. [13] S.M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001. [14] Pl. Kannappan, Quadratic functional equation inner product spaces, Results Math. 27, No.3-4, (1995), 368-372. [15] Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, 2009. [16] B.Margoils, J.B.Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull.Amer. Math. Soc. 126 74 (1968), 305-309. [17] M.M. Pourpasha, J. M. Rassias, R. Saadati, S.M. Vaezpour, A fixed point approach to the stability of Pexider quadratic functional equation with involution J. Inequal. Appl. 2010, Art. ID 839639, 18 pp. [18] J.M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, 46, (1982) 126-130. [19] J.M. Rassias, H.M. Kim, Generalized Hyers-Ulam stability for general additive functional equations in quasi-β-normed spaces J. Math. Anal. Appl. 356 (2009), no. 1, 302-309. [20] J.M. Rassias, E. Son, H.M. Kim, On the Hyers-Ulam stability of 3D and 4D mixed type mappings, Far East J. Math. Sci. 48 (2011), no. 1, 83-102. [21] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc.Amer.Math. Soc., 72 (1978), 297-300. [22] Th.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Acedamic Publishers, Dordrecht, Bostan London, 2003. [23] K. Ravi, M. Arunkumar and J.M. Rassias, On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation, International Journal of Mathematical Sciences, Autumn 2008 Vol.3, No. 08, 36-47. [24] K. Ravi, J.M. Rassias, M. Arunkumar, R. Kodandan, Stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation, J. Inequal. Pure Appl. Math. 10 (2009), no. 4, Article 114, 29 pp. [25] S.M. Jung, J.M. Rassias, A fixed point approach to the stability of a functional equation of the spiral of Theodorus, Fixed Point Theory Appl. 2008, Art. ID 945010, 7 pp. [26] S.M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, NewYork, 1964. [27] T.Z. Xu, J.M. Rassias, W.X Xu, Generalized Ulam-Hyers stability of a general mixed AQCQ-functional equation in multi-Banach spaces: a fixed point approach, Eur. J. Pure Appl. Math. 3 (2010), no. 6, 1032-1047. [28] T.Z. Xu, J.M. Rassias, M.J. Rassias, W.X. Xu, A fixed point approach to the stability of quintic and sextic functional equations in quasi-β-normed spaces, J. Inequal. Appl. 2010, Art. ID 423231, 23 pp. [29] T.Z. Xu, J.M Rassias, W.X. Xu, A fixed point approach to the stability of a general mixed AQCQ-functional equation in non-Archimedean normed spaces, Discrete Dyn. Nat. Soc. 2010, Art. ID 812545, 24 pp.

46

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.'S 1-2, 47-62, 2014, COPYRIGHT 2014 EUDOXUS PRESS LLC

Random Hybrid Proximal Point Algorithm for Fuzzy Nonlinear Set Valued Inclusions Salahuddin Department of Mathematics Jazan University, Jazan K. S. A. [email protected] Abstract The main purpose of this paper is to introduced and studied a new class of fuzzy nonlinear set valued random variational inclusions involving random nonlinear (At , ηt )-monotone mapping in Hilbert spaces. Using the random hybrid proximal point operator associated with random nonlinear (At , ηt )-monotone mapping and random relaxed co-coercive mappings, we proved an existence theorem for the iterative sequences generated by the proposed algorithm. Keywords: Fuzzy mappings, Hilbert spaces, fuzzy nonlinear set valued random variational inclusions, random relaxed cocoercive mapping, existence theorem, iterative sequences, algorithm. Mathematics Subject Classification: 47H09, , 47J20, 47J25, 49J40.

1

Introduction

The set valued inclusion problem, which was introduced and studied by De Bella [5], Huang et al. [17] is a useful extension of the mathematical analysis. It provides us with unified, natural, novel, innovative and general technique to study a wide range of problem arising in different branches of mathematics, engineering and financial sciences. Ding and Luo [10], Verma [30], Huang [16] and Lan et al. [21] introduced the concept of η-subdifferential operators, maximal η-monotone operators, H-monotone operators, A-monotone operators, (H, η)-monotone operators, (A, η)-accretive mappings, (G, η)monotone operators and defined resolvent operators associated with them respectively. Recently Verma [31] has developed a hybrid version of the Eckstein and Bertsekas [12] proximal point algorithm based on the (A, η)-maximal monotonicity framework [31] and studied convergence of the algorithm. A fuzzy set introduced in the seminal article written by Zadeh [33] is an existence of a crisp set by enlarging the true valued set {0, 1} to the real unit interval [0, 1]. Fuzzy set theory is a powerful hand set for modeling, uncertainty and vagueness in various problems arising in the field of science and engineering. It has also very useful applications in various field to all aspects of fuzzyness from theoretical to practical in almost all sciences, technology, networking and industry, in our real world, we mostly perform fuzzy approximations. In 1989 Chang and Zhu [9] introduced the concepts of variational inequalities with fuzzy mappings and extended some results of Lassando [20] in the fuzzy setting. Later, they were developed by Agarwal et al. [1], Ahmad et al. [2], Ding et al. [11], Lee et al. [23, 24], Huang [15], Lan et al. [21] and Anastassiou et al. [4] etc.

47

SALAHUDDIN: SET VALUED INCLUSIONS

On the other hand, random variational inequality problems and random quasi variational inequality problems have been considered by Chang [6, 7], Chang and Huang [8], Husain et al. [18], Tan [29], Yuan [32], Salahuddin and Ahmad [28], Khan and Salahuddin [19] and Salahuddin [27] etc. Inspired and motivated by recent research works [3, 13, 15, 22, 25, 32, 34], in this paper we proposed a general nonlinear framework for a random hybrid proximal point algorithm using the notion of (At , ηt )-monotonicity in fuzzy environment. The existence and convergence analysis for the algorithm of solving a fuzzy nonlinear set valued random variational inclusion problems are explored along with some results on the resolvent operator corresponding to (At , ηt )-monotonicity mappings. The results of random sequences {xn (t)} generated by the random algorithm converges linearly to a solution of fuzzy nonlinear set valued random variational inclusion problems as the convergence rate θ is proved.

2

Preliminaries

Let H be a real Hilbert space with k · k and inner product h·, ·i, respectively. Let F(H) be a collection of all fuzzy sets over H. A mapping F from H into F(H) is called a fuzzy mapping on H. If F is a fuzzy mapping on H, then F (x) (denote it by Fx , in the sequel) is a fuzzy set on H and Fx (y) is the membership function of y in Fx . Let S ∈ F(H), q ∈ [0, 1]. Then the set (S)q = {u ∈ H : S(u) ≥ q} is called a q-cut set of S. In this communication, we denote by (Ω, Σ) a measurable space, where Ω is a set and Σ is ˆ ·), the class of Borel σ-field a σ-algebra of subsets of Ω and by B(H), 2H , CB(H) and H(·, in H, the family of all nonempty subset of H, the family of all nonempty closed bounded subsets of H and the Hausdorff metric on CB(H) respectively. A mapping x : Ω → H is said to be measurable if for any B ∈ B(H), {t ∈ Ω : x(t) ∈ B} ∈ Σ. A mapping f : Ω × H → H is called a random operator if for any x ∈ H, f (t, x) = x(t) is a measurable. A random operator f is said to be continuous if for any t ∈ Ω, the mapping f (t, ·) : H → H is continuous. A set valued mapping T : Ω → 2H is said to be measurable if for any B ∈ B(H), T −1 (B) = {t ∈ Ω : T (t) ∩ B 6= ∅} ∈ Σ. A mapping u : Ω → H is called a measurable selection of a set valued measurable mapping T : Ω → 2H , if u is a measurable and for any t ∈ Ω, u(t) ∈ T (t). A mapping T : Ω × H → 2H is called a random set valued mapping if for any x ∈ H, T (·, x) is measurable. A random set valued mapping ˆ T : Ω × H → CB(H) is said to be H-continuous if for any t ∈ Ω, T (t, ·) is continuous in the Hausdorff metric. Definition 2.1 A fuzzy mapping F : Ω → F(H) is called measurable if for any α ∈ (0, 1], (F (·))α : Ω → 2H is a measurable set valued mapping. Definition 2.2 A fuzzy mapping F : Ω × H → F(H) is called a random fuzzy mapping, if for any x ∈ H, F (·, x) : Ω → F(H) is a measurable fuzzy mapping. 2

48

SALAHUDDIN: SET VALUED INCLUSIONS

Let T, P, Q : Ω × H → F(H) be the three random fuzzy mappings satisfying the following condition (C) : (C) : there exist three mappings a, b, c : H → [0, 1] such that (Tt,x(t) )a(x(t)) ∈ CB(H), (Pt,x(t) )b(x(t)) ∈ CB(H), (Qt,x(t) )c(x(t)) ∈ CB(H), ∀(t, x) ∈ Ω × H. By using the random fuzzy mappings T, P, Q, we can define three random set valued ˜ as follows: mappings T˜, P˜ and Q T˜ : Ω × H → CB(H), x → (Tt,x )a(x) ∀(t, x) ∈ Ω × H, P˜ : Ω × H → CB(H), x → (Pt,x )b(x) ∀(t, x) ∈ Ω × H, ˜ : Ω × H → CB(H), x → (Qt,x )c(x) ∀(t, x) ∈ Ω × H and Tt,x = T (t, x(t)). Q ˜ are called the random set valued mappings induced by random In the sequel T˜, P˜ and Q fuzzy mappings T, P and Q, respectively. Let η, N : Ω × H × H → H be two random mappings. Let f, g, p : Ω × H → H be the three random single valued mappings and M : Ω×H ×H → 2H the random set valued T mapping with for each t ∈ Ω, u ∈ H, M (t, ·, u) is a maximal η-monotone with Range (g) DomM (t, ·, u) 6= ∅. we consider the following problem for finding u, x, y, z : Ω → H such that for all t ∈ Ω, u(t) T ∈ H, Tt,u(t) (x(t)) ≥ a(u(t)), Pt,u(t) (y(t)) ≥ b(u(t)), Qt,u(t) (z(t)) ≥ c(u(t)) and g(t, u(t)) Dom(M (t, ·, z(t))) 6= ∅ for t ∈ Ω, such that 0 ∈ ft (x(t)) + Nt (pt (u(t)), y(t)) + Mt (gt (u(t)), z(t)).

(2.1)

The problem (2.1) is called fuzzy nonlinear set valued random variational inclusions. It is known that a number of problems involving the nonmonotone, nonconvex and nonsmooth mapping arising in structural engineering, mechanics, economics and optimization theory can be reduced to study this type of variational inclusions.

3

Random Iterative Algorithm

The following definitions and results are needed to prove the main results. ˆ Lemma 3.1 [6] Let T : Ω × H → CB(H) be a H-continuous random set valued mapping. Then for any measurable mapping w : Ω → H, the set valued mapping T (·, w(t)) : Ω → CB(H) is measurable. Lemma 3.2 [6] Let P, T : Ω → CB(H) be the two measurable set valued mappings,  ≥ 0 be a constant and v : Ω → H be a measurable selection of P. Then there exists a measurable selection w : Ω → H of T such that for all t ∈ Ω, ˆ (t), T (t)). kv(t) − w(t)k ≤ (1 + )H(P Definition 3.3 A random operator A : Ω × H → H is said to be (i) randomly monotone, if hAt (u1 (t)) − At (u2 (t)), u1 (t) − u2 (t)i ≥ 0, ∀u1 (t), u2 (t) ∈ H, t ∈ Ω, 3

49

SALAHUDDIN: SET VALUED INCLUSIONS

(ii) randomly rt -strongly monotone, if there exists a measurable mapping r : Ω → (0, ∞) such that hAt (u1 (t)) − At (u2 (t)), u1 (t) − u2 (t)i ≥ rt ku1 (t) − u2 (t)k2 , ∀u1 (t), u2 (t) ∈ H, t ∈ Ω, (iii) randomly rt -relaxed monotone, if there exists a measurable mapping r : Ω → (0, ∞) such that hAt (u1 (t)) − At (u2 (t)), u1 (t) − u2 (t)i ≥ −rt ku1 (t) − u2 (t)k2 , ∀u1 (t), u2 (t) ∈ H, t ∈ Ω, (iv) randomly ξt -cocoercive if hAt (u1 (t)) − At (u2 (t)), u1 (t) − u2 (t)i ≥ ξt ku1 (t) − u2 (t)k2 , ∀u1 (t), u2 (t) ∈ H, t ∈ Ω, (v) randomly (αt , ξt )-relaxed cocoercive, if there exists measurable mappings α, ξ : Ω → (0, ∞) such that hAt (u1 (t))−At (u2 (t)), u1 (t)−u2 (t)i ≥ −αt kAt (u1 (t))−At (u2 (t))k2 +ξt ku1 (t)−u2 (t)k2 , ∀u1 (t), u2 (t) ∈ H, t ∈ Ω. Definition 3.4 Let N : Ω × H × H → H and p : Ω × H → H be the two single valued ˜ : Ω × H → CB(H) the random mapping, then mappings, and Q (i) Nt is said to be randomly (αt , t )-p-relaxed cocoercive with respect to first variable of Nt if hNt (pt (u(t)), ·)−Nt (pt (v(t)), ·), u(t)−v(t)i ≥ −αt kNt (pt (u(t)), ·)−Nt (pt (v(t)), ·)k2 +t ku(t)−v(t)k2 ∀u(t), v(t) ∈ H, t ∈ Ω. (ii) Nt is said to be randomly (ϕt , ψt )-Qt -relaxed cocoercive with respect to second variable of Nt if hNt (·, y1 (t)) − Nt (·, y2 (t)), u(t) − v(t)i ≥ −ϕt kNt (·, y1 (t)) − Nt (·, y2 (t))k2 + ψt ku(t) − v(t)k2 ˜ t (u(t)), y2 (t) ∈ Q ˜ t (v(t)), u(t), v(t) ∈ H, t ∈ Ω. ∀y1 (t) ∈ Q Definition 3.5 Let η : Ω × H × H → H be a single valued mapping. The map ηt is called randomly τt -Lipschitz continuous if there is a measurable mapping τ : Ω → (0, ∞) such that kηt (u(t), v(t))k ≤ τt ku(t) − v(t)k, ∀u(t), v(t) ∈ H, t ∈ Ω. Definition 3.6 Let η : Ω×H×H → H be a single valued mapping and let M : Ω×H → 2H be a random set valued mapping. The random map Mt is said to be 4

50

SALAHUDDIN: SET VALUED INCLUSIONS

(i) randomly (rt , ηt )-strongly monotone if hu∗ (t) − v ∗ (t), ηt (u(t), v(t))i ≥ rt ku(t) − v(t)k2 , ∀(u(t), u∗ (t)), (v(t), v ∗ (t)) ∈ Graph(M ); (ii) randomly ηt -pseudomonotone if hv ∗ (t), ηt (u(t), v(t))i ≥ 0 =⇒ hu∗ (t), ηt (u(t), v(t))i ≥ 0 ∀(u(t), u∗ (t)), (v(t), v ∗ (t)) ∈ Graph(M ); (iii) randomly (rt , ηt )-relaxed monotone if there exists a measurable mapping r : Ω → (0, ∞) such that hu∗ (t) − v ∗ (t), ηt (u(t), v(t))i ≥ −rt ku(t) − v(t)k2 , ∀(u(t), u∗ (t)), (v(t), v ∗ (t)) ∈ Graph(M ). Definition 3.7 A random mapping M : Ω × H → 2H is said to be random maximal (mt , ηt )-relaxed monotone if (i) Mt is random (Mt , ηt )-monotone (ii) for (u(t), u∗ (t)) ∈ H × H and hu∗ (t) − v ∗ (t), ηt (u(t), v(t))i ≥ −mt ku(t) − v(t)k2 , ∀(v(t), v ∗ (t)) ∈ Graph(M ) we have u∗ (t) ∈ Mt (u(t)). Definition 3.8 Let A : Ω × H → H and η : Ω × H × H → H be two random single valued mappings, the random mapping M : Ω × H → 2H is said to be randomly (At , ηt )-monotone if (i) Mt is randomly (Mt , ηt )-relaxed monotone, (ii) R(At + ρt Mt ) = H for a measurable mapping ρ : Ω → (0, 1). Note that alternatively, the random mapping M : Ω×H → 2H is said to randomly (At , ηt )monotone if (i) Mt is randomly (Mt , ηt )-relaxed monotone, (ii) At + ρt Mt is randomly ηt -pseudomonotone for a measurable mapping ρ : Ω → (0, 1). Proposition 3.9 Let a random mapping A : Ω × H → H be randomly (rt , ηt )-strongly monotone, M : Ω × H → 2H be a randomly (At , ηt )-monotone mapping, and η : Ω × H × H → H be the randomly τt -Lipschitz continuous, then Mt is randomly (mt , ηt )-relaxed rt monotone and (At + ρt Mt )H = H for 0 < ρt < m . t 5

51

SALAHUDDIN: SET VALUED INCLUSIONS

Proposition 3.10 Let a map A : Ω × H → H be the randomly (rt , ηt )-strongly monotone and M : Ω×H → 2H be a randomly (At , ηt )-monotone mapping. Let η : Ω×H ×H → H be the randomly τt -Lipschitz continuous. Then (At +ρt Mt ) is randomly maximal ηt -monotone rt for 0 < ρt < m . t Proof. Given that At is randomly (rt , ηt )-strongly monotone and Mt is randomly (At , ηt )maximal monotone, then (At + ρt Mt ) is randomly (rt − mt ρt , ηt )-strongly monotone. This in turn implies that (At + ρt Mt ) is randomly ηt -pseudomonotone and hence (At + ρt Mt ) is randomly ηt -monotone under given conditions. Proposition 3.11 Let A : Ω × H → H be a randomly (rt , ηt )-strongly monotone mapping and M : Ω × H → 2H be the randomly (At , ηt )-monotone mapping. If in addition, η : Ω × H × H → H is randomly τt -Lipschitz continuous, then the operator (At + ρt Mt )−1 is rt randomly single valued for 0 < ρt < m . t Lemma 3.12 Let H be a real Hilbert space and η : Ω × H × H → H be a randomly τt -Lipschitz continuous nonlinear mapping. Let A : Ω × H → H be a randomly (rt , ρt )strongly monotone and M : Ω × H × H → 2H be randomly (At , ηt )-monotone in first argument in Mt . Then the generalized resolvent operator associated with Mt (·, v(t)) for a fixed v(t) ∈ H and defined by η ,M (·,v(t))

Jρtt,Att

(u(t)) = (At + ρt Mt (·, v(t)))−1 (u(t)), ∀u(t) ∈ H

τt is randomly ( rt −ρ )-Lipschitz continuous. t mt

Definition 3.13 A random set valued mapping T : Ω×H → CB(H) is said to be random ˆ H-Lipschitz continuous if there exists a measurable mapping λHˆ T : Ω → (0, ∞) such that t

ˆ t (u1 (t)), Tt (u2 (t))) ≤ λ ˆ ku1 (t) − u2 (t)k, ∀u1 (t), u2 (t) ∈ H. H(T t,Ht Lemma 3.14 The set of measurable mappings u, x, y, z : Ω → H is a random solution of problem (2.1) if and only if for all t ∈ Ω, u(t) ∈ H, x(t) ∈ T˜t (u(t)), y(t) ∈ P˜t (u(t)), z(t) ∈ ˜ t (u(t)) and Q η ,M (·,z(t))

gt (u(t)) = Jρtt,Att

[At (gt (u(t))) − ρt (ft (x(t)) + Nt (pt (u(t)), y(t)))]

(3.1)

where ρ : Ω → (0, ∞) is a measurable mapping. η ,M (·,z(t))

Proof. The proof directly follows from the definition of Jρtt,Att . Based on Lemma 3.14 and Nadler [26], developed a fuzzy random iterative algorithm for solving the problem (3.1) as follows Algorithm 3.15 Suppose that T, P, Q : Ω × H → F(H) be three fuzzy random mappings ˜ : Ω × H → CB(H) be the H-continuous ˆ satisfying the condition (C). Let T˜, P˜ , Q random set valued mappings induced by T, P, Q, respectively. Let A, f, g, p : Ω × H → H be the single valued random mappings and η, N : Ω × H × H → H be the two random 6

52

SALAHUDDIN: SET VALUED INCLUSIONS

bifunctions. Let M : Ω × H × H → 2H be a set valued random mapping such that for each fixed t ∈ Ω, M (t, ·, ·) : H × H → 2H is randomly At -monotone mapping with Im(gt ) ∩ domMt (·, ·) 6= ∅. For any given measurable mapping u0 : Ω → H, the set valued ˜ t (u0 (t)) : Ω → CB(H) are measurable by Lemma random mappings T˜t (u0 (t)), P˜t (u0 (t)), Q 3.1. Hence there exists measurable selections x0 : Ω → H of T˜t (u0 (t)), y0 : Ω → H of ˜ t (u0 (t)). By Himmelberg [14], let P˜t (u0 (t)), and z0 : Ω → H of Q η ,M (·,z0 (t))

u1 (t) = u0 (t)−gt (u0 (t))+Jρtt,Att

[At (gt (u0 (t)))−ρt {ft (x0 (t))+Nt (pt (u0 (t)), y0 (t))}]+e0 (t).

where ρt is same as in Lemma 3.14, 1 > t > 0 is a constant, and e0 (t) : Ω → H is a measurable function which is a random error to take into account a possible inexact computation of random hybrid proximal point. Then, it is easy to know that u1 : Ω → H is a measurable. By Lemma 3.14, there exists a measurable selections x1 : Ω → H of ˜ t (u1 (·)) such that for all t ∈ Ω, T˜t (u1 (·)), y1 : Ω → H of P˜t (u1 (·)) and z1 : Ω → H of Q ˆ T˜t (u0 (t)), T˜t (u1 (t))), kx0 (t) − x1 (t)k ≤ (1 + 1)H( ˆ P˜t (u0 (t)), P˜t (u1 (t))), ky0 (t) − y1 (t)k ≤ (1 + 1)H( ˆ Q ˜ t (u0 (t)), Q ˜ t (u1 (t))). kz0 (t) − z1 (t)k ≤ (1 + 1)H( Let η ,M (·,z1 (t))

u2 (t) = u1 (t)−gt (u1 (t))+Jρtt,Att

[At (gt (u1 (t)))−ρt {ft (x1 (t))+Nt (pt (u1 (t)), y1 (t))}]+e1 (t).

The u2 (t) is a measurable. Continuing the above process inductively, we can define the following random iterative sequences for fuzzy mappings {un (t)}, {xn (t)}, {yn (t)} and {zn (t)} for solving (2.1) as follows η ,M (·,zn (t))

un+1 (t) = un (t)−gt (un (t))+Jρtt,Att

[At (gt (un (t)))−ρt {ft (xn (t))+Nt (pt (un (t)), yn (t))}]+en (t)

˜ t (un (t)), xn (t) ∈ T˜t (un (t)), yn (t) ∈ P˜t (un (t)), zn (t) ∈ Q ˆ T˜t (un (t)), T˜t (un+1 (t))), kxn (t) − xn+1 (t)k ≤ (1 + (1 + n)−1 )H( ˆ P˜t (un (t)), P˜t (un+1 (t))), kyn (t) − yn+1 (t)k ≤ (1 + (1 + n)−1 )H( ˆ Q ˜ t (un (t)), Q ˜ t (un+1 (t))), kzn (t) − zn+1 (t)k ≤ (1 + (1 + n)−1 )H( for any 0 < t < 1 and n = 0, 1, 2, · · · ; en (t) : Ω → H(n ≥ 0) is a random error to take into account a possible inexact computation of the proximal point.

4

Convergence Results

In this section, we shall give some existence and convergence theorem for fuzzy nonlinear set valued inclusions. 7

53

SALAHUDDIN: SET VALUED INCLUSIONS

Theorem 4.1 Let a random mapping η : Ω × H × H → H be randomly (mt , ηt )-relaxed monotone and Lipschitz continuous with constant τt . Let M : Ω × H × H → H be a random set valued mapping such that for each fixed t ∈ Ω, M (t, ·, ·) : Ω × H × H → 2H be the randomly (At , ηt )-monotone mapping in the first argument in Mt and A : Ω × H → H be the randomly (rt , ηt )-strongly monotone and χt -Lipschitz continuous with constant χt . Let T, P, Q : Ω × H → F(H) be the fuzzy random mappings satisfies the condition (C) ˜ : Ω × H → CB(H) be the H-continuous ˆ and T˜, P˜ , Q random set valued mappings induced ˆ by T, P, Q, respectively. Suppose that T, P, Q are randomly H-Lipschitz continuous with random variables ιt , υt , dt , respectively. Let pt , ft : Ω × H → H be the Lipschitz continuous random mappings with constants st , ωt , respectively. Let N : Ω×H ×H → H be the bilinear random mapping which is Lipschitz continuous with first variable with constant βt and second variable with γt . Assume that Nt (·, ·) is randomly (αt , t )-p-relaxed cocoercive with respect to first argument. A random mapping g : Ω×H → H is random strongly monotone with constant νt and random Lipschitz continuous with constant ξt and Aog is randomly (ςt , κt )-relaxed cocoercive. Let Nt (·, ·) be the randomly (ϕt , ψt )-Qt -relaxed cocoercive with respect to the second argument. Let M : Ω × H × H → 2H be a set valued mapping such that for each fixed t ∈ Ω, v(t) ∈ H, Mt (·, v(t)) : H → 2H be the randomly (At , ηt )-monotone random mapping and range (gt ) ∩ domMt (·, v(t)) 6= ∅. For any t ∈ Ω, u(t), v(t), w(t) ∈ H there exists a random real valued variable δt > 0 such that η ,M (·,zn (t))

kJρtt,Att

η ,M (·,zn−1 (t))

w(t) − Jρtt,Att

w(t)k ≤ δt kzn (t) − zn−1 (t)k

(4.1)

and 4 |< |ρ−  4>

p √

42 − ` 

`

D(t)τt2 > τt G(t) + mt (1 − B(t)) τt > rt (1 − B(t)) − τt C(t) p E(t)τt > τt G(t) + mt (1 − B(t)) where  = E 2 (t)τt2 − (τt G(t) + mt (1 − B(t)))2 4 = D(t)τt2 − (τt G(t) + mt (1 − B(t))) ` = τt2 − (rt (1 − B(t)) − τt C(t))2 and lim ken (t)k = 0,

n→∞

∞ X

ken (t) − en−1 (t)k < ∞, ∀t ∈ Ω.

(4.2)

n=1

The random variable iterative sequences {un (t)}, {xn (t)}, {yn (t)} and {zn (t)} : Ω → H generated by Algorithm 3.15, converge strongly to random variables u∗ (t), x∗ (t), y ∗ (t) and z ∗ (t) : Ω → H respectively and (u∗ (t), x∗ (t), y ∗ (t), z ∗ (t)) is a solution set of problem (2.1). 8

54

SALAHUDDIN: SET VALUED INCLUSIONS

Proof. From Algorithm 3.15, for any t ∈ Ω, we have η ,M (·,zn (t))

kun+1 (t) − un (t)k = kun (t) − gt (un (t)) + Jρtt,Att

[At (gt (un (t))) − ρt {ft (xn (t)) η ,M (·,zn−1 (t))

+Nt (pt (un (t)), yn (t))}] + en (t) − un−1 (t) + gt (un−1 (t)) − Jρtt,Att

[At (gt (un−1 (t)))

− ρt {ft (xn−1 (t)) + Nt (pt (un−1 (t)), yn−1 (t))}] − en−1 (t)k ≤ kun (t) − un−1 (t) − (gt (un (t)) − gt (un−1 (t)))k η ,M (·,zn (t))

+kJρtt,Att

η ,M (·,zn−1 (t))

[wn (t)] − Jρtt,Att

[wn−1 (t)]k + ken (t) − en−1 (t)k

≤ kun (t) − un−1 (t) − (gt (un (t)) − gt (un−1 (t)))k η ,M (·,zn (t))

+kJρtt,Att η ,M (·,zn (t))

+kJρtt,Att

η ,M (·,zn (t))

[wn (t)] − Jρtt,Att η ,M (·,zn−1 (t))

[wn−1 (t)] − Jρtt,Att

[wn−1 (t)]k

[wn−1 (t)]k + ken (t) − en−1 (t)k

≤ kun (t) − un−1 (t) − (gt (un (t)) − gt (un−1 (t)))k +

τt kwn (t) − wn−1 (t)k rt − ρt mt

+δt kzn (t) − zn−1 (t)k + ken (t) − en−1 (t)k

(4.3)

where wn (t) = At (gt (un (t))) − ρt (ft (xn (t)) + Nt (pt (un (t)), yn (t))). Now kwn (t) − wn−1 (t)k = kAt (gt (un (t))) − ρt (ft (xn (t)) + Nt (pt (un (t)), yn (t))) −At (gt (un−1 (t))) + ρt (ft (xn−1 (t)) + Nt (pt (un−1 (t)), yn−1 (t)))k = kAt (gt (un (t))) − At (gt (un−1 (t))) −ρt (ft (xn (t)) − ft (xn−1 (t)) + Nt (pt (un (t)), yn (t)) − Nt (pt (un−1 (t)), yn−1 (t)))k = kun (t) − un−1 (t) − (At (gt (un (t))) − At (gt (un−1 (t))))k +kun (t) − un−1 (t) − ρt (Nt (pt (un (t)), yn (t)) − Nt (pt (un−1 (t)), yn−1 (t)))k +ρt kft (xn (t)) − ft (xn−1 (t))k.

(4.4)

From (4.3) and (4.4), we obtain kun+1 (t) − un (t)k ≤ kun (t) − un−1 (t) − (gt (un (t)) − gt (un−1 (t)))k +

τt [kun (t) − un−1 (t) − (At (gt (un (t))) − At (gt (un−1 (t))))k rt − ρt mt

+kun (t) − un−1 (t) − ρt (Nt (pt (un (t)), yn (t)) − Nt (pt (un−1 (t)), yn−1 (t)))k +ρt kft (xn (t)) − ft (xn−1 (t))k] + δt kzn (t) − zn−1 (t)k + ken (t) − en−1 (t)k. 9

55

(4.5)

SALAHUDDIN: SET VALUED INCLUSIONS

˜ t are randomly H-Lipschitz ˆ Since Nt , gt , pt , ft are random Lipschitz continuous and T˜t , P˜t , Q continuous, we have kgt (un (t)) − gt (un−1 (t))k ≤ ξt kun (t) − un−1 (t)k,

(4.6)

kpt (un (t)) − pt (un−1 (t))k ≤ st kun (t) − un−1 (t)k,

(4.7)

kft (xn (t)) − ft (xn−1 (t))k ≤

ωt kxn (t) − xn−1 (t)k ˆ T˜t (un (t)), T˜t (un−1 (t))) ωt H( 1 )ιt kun (t) − un−1 (t)k, ωt (1 + n+1

≤ ≤

(4.8)

1 ˆ P˜t (un (t)), P˜t (un−1 (t))) ≤ (1+ 1 )υt kun (t)−un−1 (t)k, )H( n+1 n+1 1 ˆ Q ˜ t (un (t)), Q ˜ t (un−1 (t))) ≤ (1+ 1 )dt kun (t)−un−1 (t)k )H( kyn (t)−yn−1 (t)k ≤ (1+ 1+n n+1 and kzn (t)−zn−1 (t)k ≤ (1+

kNt (pt (un (t)), yn (t)) − Nt (pt (un−1 (t)), yn−1 (t))k ≤ βt kpt (un (t)) − pt (un−1 (t))k + γt kyn (t) − yn−1 (t)k ≤ βt st kun (t) − un−1 (t)k + γt (1 +

1 )dt kun (t) − un−1 (t)k n+1

1 )dt )kun (t) − un−1 (t)k. n+1 Since gt is random strongly monotone and random Lipschitz continuous, we have ≤ (βt st + γt (1 +

(4.9)

kun (t) − un−1 (t) − (gt (un (t)) − gt (un−1 (t)))k2 ≤ kun (t) − un−1 (t)k2 −2hgt (un (t)) − gt (un−1 (t)), un (t) − un−1 (t)i + kgt (un (t)) − gt (un−1 (t))k2 ≤ kun (t) − un−1 (t)k2 − 2νt kun (t) − un−1 (t)k2 + ξt2 kun (t) − un−1 (t)k2 ≤ (1 − 2νt + ξt2 )kun (t) − un−1 (t)k2 .

(4.10)

Since At and gt are randomly Lipschitz continuous with χt and ξt respectively, and randomly (ςt , κt )- relaxed cocoercive and from Algorithm 3.15, we obtain kun (t) − un−1 (t) − (At (gt (un (t))) − At (gt (un−1 (t))))k2 ≤ kun (t) − un−1 (t)k2 −2hAt (gt (un (t))) − At (gt (un−1 (t))), un (t) − un−1 (t)i + kAt (gt (un (t))) − At (gt (un−1 (t)))k2 ≤ kun (t) − un−1 (t)k2 + χ2t ξt2 kun (t) − un−1 (t)k2 +2ςt kAt (gt (un (t))) − At (gt (un−1 (t)))k2 − 2κt kun (t) − un−1 (t)k2 ≤ kun (t) − un−1 (t)k2 + χ2t ξt2 kun (t) − un−1 (t)k2 + 2ςt χ2t ξt2 kun (t) − un−1 (t)k2 10

56

SALAHUDDIN: SET VALUED INCLUSIONS

−2κt kun (t) − un−1 (t)k2 ≤ ((1 − 2κt ) + (2ςt + 1)χ2t ξt2 )kun (t) − un−1 (t)k2 .

(4.11)

Since Nt (·, ·) is randomly (αt , t )-p-relaxed cocoercive with respect to the first argument of Nt . Again Nt (·, ·) is randomly (ϕt , ψt )-Qt -relaxed cocoercive with respect to the second argument of Nt ; Nt and pt are randomly Lipschitz continuous, we have kun (t)−un−1 (t)−ρt (Nt (pt (un (t)), yn (t))−Nt (pt (un−1 (t)), yn−1 (t)))k2 = kun (t)−un−1 (t)k2 −2ρt hNt (pt (un (t)), yn (t)) − Nt (pt (un−1 (t)), yn−1 (t)), un (t) − un−1 (t)i +ρ2t kNt (pt (un (t)), yn (t)) − Nt (pt (un−1 (t)), yn−1 (t))k2 ≤ kun (t) − un−1 (t)k2 − 2ρt hNt (pt (un (t)), yn (t)) − Nt (pt (un−1 (t)), yn (t)), un (t) − un−1 (t)i −2ρt hNt (pt (un−1 (t)), yn (t)) − Nt (pt (un−1 (t)), yn−1 (t)), un (t) − un−1 (t)i +ρ2t kNt (pt (un (t)), yn (t)) − Nt (pt (un−1 (t)), yn−1 (t))k2 ≤ kun (t) − un−1 (t)k2 − 2ρt (−αt kNt (pt (un (t)), yn (t)) − Nt (pt (un−1 (t)), yn (t))k2 +t kun (t) − un−1 (t)k2 ) − 2ρt (−ϕt kNt (pt (un−1 (t)), yn (t)) − Nt (pt (un−1 (t)), yn−1 (t))k2 +ψt kun (t) − un−1 (t)k2 ) + ρ2t (βt kpt (un (t)) − pt (un−1 (t))k + γt kyn (t) − yn−1 (t)k)2 ≤ kun (t) − un−1 (t)k2 + 2ρt αt βt2 s2t kun (t) − un−1 (t)k2 − 2ρt t kun (t) − un−1 (t)k2 +2ρt ϕt γt2 kyn (t) − yn−1 (t)k2 − 2ρt ψt kun (t) − un−1 (t)k2 + ρ2t (βt st kun (t) − un−1 (t)k +γt kyn (t) − yn−1 (t)k)2 ≤ kun (t) − un−1 (t)k2 + 2ρt αt βt2 s2t kun (t) − un−1 (t)k2 − 2ρt t kun (t) − un−1 (t)k2 1 2 2 ) d kun (t) − un−1 (t)k2 − 2ρt ψt kun (t) − un−1 (t)k2 n+1 t 1 +ρ2t (βt st + γt (1 + )dt )2 kun (t) − un−1 (t)k2 n+1 1 2 2 1 ≤ [1 + 2ρt (αt βt2 s2t − t + ϕt γt2 (1 + ) dt − ψt ) + ρ2t (βt st + γt (1 + )dt )2 ] n+1 n+1 +2ρt ϕt γt2 (1 +

kun (t) − un−1 (t)k2 .

(4.12)

From (4.5),(4.8), (4.9), (4.10), (4.11) and (4.12), we have q kun+1 (t) − un (t)k ≤ 1 − 2νt + ξt2 kun (t) − un−1 (t)k + r

q τt [ (1 − 2κt ) + (2ςt + 1)χ2t ξt2 kun (t) − un−1 (t)k rt − ρt mt

1 2 1 ) − ψt ) + ρ2t (βt st + γt dt (1 + ))2 1+n 1+n 1 kun (t) − un−1 (t)k + ρt ωt (1 + )ιt kun (t) − un−1 (t)k] 1+n

+ 1 + 2ρt (αt βt2 s2t − t + ϕt γt2 d2t (1 +

11

57

SALAHUDDIN: SET VALUED INCLUSIONS

1 )υt kun (t) − un−1 (t)k + ken (t) − en−1 (t)k 1+n q q 1 τt 2 [ (1 − 2κt ) + (2ςt + 1)χ2t ξt2 ≤ [ 1 − 2νt + ξt + δt (1 + )υt + n+1 rt − ρt mt r 1 1 2 + (1 − 2ρt (−αt βt2 s2t + t − ϕt γt2 d2t (1 + ) + ψt ) + ρ2t (βt st + γt dt (1 + ))2 n+1 1+n +δt (1 +

1 )ιt ]kun (t) − un−1 (t)k + ken (t) − en−1 (t)k n+1 q {C(t) + 1 − 2ρt Dn (t) + ρ2t En2 (t) + Gn (t)ρt }]kun (t) − un−1 (t)k

+ρt ωt (1 + ≤ [Bn (t) +

τt rt − ρt mt

+ken (t) − en−1 (t)k ≤ θn (t)kun (t) − un−1 (t)k + ken (t) − en−1 (t)k where q τt [C(t) + 1 − 2ρt Dn (t) + ρ2t En2 (t) + Gn (t)ρt ] rt − ρt mt q 1 )υt , Bn (t) = 1 − 2νt + ξt2 + δt (1 + 1+n q C(t) = (1 − 2κt ) + (2ςt + 1)χ2t ξt2

θn (t) = Bn (t) +

Dn (t) = −αt βt2 s2t + t − ϕt γt2 d2t (1 + En (t) = βt st + γt dt (1 + Gn (t) = ωt ιt (1 +

1 2 ) + ψt 1+n

1 ) 1+n

1 ). 1+n

Letting τt θ(t) = B(t) + [C(t) + rt − ρt mt and

q

1 − 2ρt D(t) + ρ2t E 2 (t) + G(t)ρt ]

q B(t) = 1 − 2νt + ξt2 + δt υt , q C(t) = (1 − 2κt ) + (2ςt + 1)χ2t ξt2 , D(t) = −αt βt2 s2t + t − ϕt γt2 d2t + ψt , E(t) = βt st + γt dt , Gn (t) = ωt ιt . 12

58

(4.13)

SALAHUDDIN: SET VALUED INCLUSIONS

We have that θn (t) → θ(t) as n → ∞. It follows from condition (4.2) and 0 < θ(t) < 1, hence there exists N0 > 0 and θ∗ (t) ∈ (θ(t), 1) such that θn (t) < θ∗ (t) for all n ≥ N0 . Therefore, from (4.13), we have kun+1 (t) − un (t)k ≤ θ∗ (t)kun (t) − un−1 (t)k + ken (t) − en−1 (t)k, ∀n ≤ N0 . Without loss of generality, we may assume kun+1 (t) − un (t)k ≤ θ∗ (t)kun (t) − un−1 (t)k + ken (t) − en−1 (t)k, ∀n ≤ 1. Hence, for any m > n > 0, we have kum (t) − un (t)k ≤

m−1 X

kui+1 (t) − ui (t)k

i=n

≤

m−1 X

θi∗ (t)ku1 (t)

− u0 (t)k +

i=1

m−1 i XX

∗ θi−j (t)kej (t) − ej−1 (t)k.

i=1 j=1

It follows from condition (4.3) that kum (t) − un (t)k → 0 as n → ∞ and so {un (t)} is a Cauchy sequence in H. Let un (t) → u(t) as n → ∞. By the random ˜ t (·), we obtain Lipschitz continuity of T˜t (·), P˜t (·) and Q kxn+1 (t) − xn (t)k ≤ (1 +

1 ˆ T˜t (un+1 (t)), T˜t (un (t))) )H( 1+n

1 )kun+1 (t) − un (t)k, n+1 1 ˆ Q ˜ t (un+1 (t)), Q ˜ t (un (t))) kyn+1 (t) − yn (t)k ≤ (1 + )H( 1+n 1 )kun+1 (t) − un (t)k, ≤ dt (1 + n+1 1 ˆ P˜t (un+1 (t)), P˜t (un (t))) kzn+1 (t) − zn (t)k ≤ (1 + )H( 1+n 1 ≤ υt (1 + )kun+1 (t) − un (t)k. n+1 It follows that {un (t)}, {xn (t)}, {yn (t)} and {zn (t)} are also Cauchy sequences in H. We can assume that un (t) → u∗ (t), xn (t) → x∗ (t), yn (t) → y ∗ (t) and zn (t) → z ∗ (t) respectively. Note that xn (t) ∈ T˜t (un (t)), we have ≤ ιt (1 +

d(x∗ (t), T˜t (u∗ (t))) ≤ ≤

kx∗ (t) − xn (t)k + d(xn (t), T˜t (u∗ (t))) ˆ T˜t (un (t)), T˜t (u∗ (t))) kx∗ (t) − xn (t)k + H(

≤

kx∗ (t) − xn (t)k + ιt kun (t) − u∗ (t)k → 0 as n → ∞. 13

59

(4.14)

SALAHUDDIN: SET VALUED INCLUSIONS

Hence d(x∗ (t), T˜t (u∗ (t))) = 0 and therefore x∗ (t) ∈ T˜t (u∗ (t)). Similarly we can prove that ˜ t (u∗ (t)), and z ∗ (t) ∈ P˜t (u∗ (t)). By the random Lipschitz continuity of T˜t (·), Q ˜ t (·) y ∗ (t) ∈ Q ˜ and Pt (·) and Lemma 3.14, condition (4.2) and limn→∞ ken (t)k = 0, we have η ,M (·,z ∗ (t))

u∗ (t) = u∗ (t) − gt (u∗ (t)) + Jρtt,Att

[At (gt (u∗ (t))) − ρt {ft (x∗ (t)) + Nt (pt (u∗ (t)), y ∗ (t))}].

By Lemma 3.14 we know that (u∗ (t), x∗ (t), y ∗ (t), z ∗ (t)) is a solution of problem (2.1). This completes the proof.

References [1] R. P. Agarwal, M. F. Khan, D. O’. Regan and Salahuddin, On generalized multivalued nonlinear variational like inclusions with fuzzy mappings, Advances in Nonlinear Variational Inequalities, 8 (2005) 41-55. [2] R. Ahmad, F. F. Bazan, An iterative algorithm for random generalized nonlinear mixed variational inclusions for random fuzzy mappings, Applied Mathematics Computation, 167 (2005) 1400-1411. [3] G. A. Anastassiou, Fuzzy Mathematics: ApproximationTheory, Memphis University, Memphis, USA. [4] G. A. Anastassiou, M. K. Ahmad and Salahuddin, Fuzziffied random generalized nonlinear variational inequalities, J. Concrete Applicable Mathematics, 10(3) (2012) 186-206. [5] B. D. Bella, An existence theorem for a class of inclusions, Applied Mathematics Letters, 13(3) (2000) 15-19. [6] S. S. Chang, Variational Inequality and Complementarity Problem, Theory with Applications, Shanghai Scientific and Tech. Literature Publishing House, Shanghai, 1991. [7] S. S. Chang, Fixed Point Theory with Applications, Chongqing Publishing House, Chongqing, 1984. [8] S. S. Chang and N. J. Huang, Generalized random multivalued quasi complementarity problems, Indian J. Mathematics, 35 (1993) 305-320. [9] S. S. Chang and Y. Zhu, On Variational inequalities for fuzzy mappings, Fuzzy Sets and Systems, 32 (1989) 356-367. [10] X. P. Ding and C. L. Luo, Perturbed proximal point algorithms for general quasi variational like inclusions, J. Computational and Applied Mathematics, 113(1-2)(2000) 153-165. 14

60

SALAHUDDIN: SET VALUED INCLUSIONS

[11] X. P. Ding, M. K. Ahmad and Salahuddin, Fuzzy generalized vector variational inequalities and complementarity problems, Nonlinear Functional Analysis and Applications, Vol. 13, No. 2 (2008) 253-263. [12] J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, Vol 55, No., 3 (1992) 293-318. [13] Paul R. Halemous, Measure theory, Springer-Verlag, New York, 1974. [14] C. J. Himmelberg, Measurable relations, Fund. Math., Vol. 87, (1975) 53-72. [15] N. J. Huang, Random generalized nonlinear variational inclusions for fuzzy mappings, Fuzzy Sets Systems, Vol. 105, (1999) 437-444. [16] N. J. Huang, Nonlinear implicit quasi variational inclusions involving generalized m-accretive mappings Arch. Inequalities and Applications, Vol. 2, No. 4, (2004) 413-425. [17] N. J. Huang, Y. Y. Tang and Y. P. Liu, Some new existence theorem for nonlinear inclusion with an application, Nonlinear Functional Analysis and Applications, Vol. 6, No. 3 (2001) 341-350. [18] T. Hussain, E. Tarafdar and X. Z. Yuan, Some results on random generalized games and random quasi variational inequalities, Far East J. of Mathematical Society, Vol. 2, (1994), 35-55. [19] M. F. Khan and Salahuddin, Completely generalized nonlinear random variational inclusions, South East Asian Bulletin of Mathematics, Vol. 30, (2006) 261-276. [20] M. Lassando, fixed points for Kakutani factorizable multifunctions, J. Mathematical Analysis and Applications, Vol. 152 (1990) 146-160. [21] H. Y. Lan, Y. J. Cho and R. U. Verma, Nonlinear relaxed cocoercive variational inclusions involving (A, η)-accretive mappings in Banach spaces, Computer Mathematics with Applications, Vol. 51, No. 9-10, (2006) 1529-1538. [22] H. Y. Lan and R. U. Verma, Iterative algorithms for nonlinear fuzzy variational inclusion systems with (A, η)−accretive mappings in Banach spaces, Advances in Nonlinear Variational Inequalities, Vol. 11, Issue 1, (2008), 15-30. [23] B. S. Lee, M. F. Khan and Salahuddin, Fuzzy generalized nonlinear mixed random variational like inclusions, Pacific J. Optimization, Vol. 6, No. 3, (2010), 573-590. [24] B. S. Lee, M. F. Khan and Salahuddin, fuzzy nonlinear setvalued variational inclusions, Computer Mathematics with Applications, Vol. 60, No. 6, (2010), 1768-1775. [25] H. G. Li, Generalized fuzzy random set valued mixed variational inclusions involving random nonlinear (At , ηt )−accretive mappings in Banach spaces, J. Nonlinear Science and Applications, Vol. 3, No. 1, (2010), 63-77. 15

61

SALAHUDDIN: SET VALUED INCLUSIONS

[26] Jr. S. B. Nadler, Multivalued contraction mappings, Pacific J. Mathematics, Vol. 30 (1969) 475-488. [27] Salahuddin, Some Aspects of Variational Inequalities, Ph.D. Thesis AMU, India 2000. [28] Salahuddin and M. K. Ahmad, Stable perturbed algorithms for a new class of generalized nonlinear implicit quasi variational inclusions in Banach spaces, Advances in Pure Mathematics, Vol. 2, No. 2, (2012), 139-148. [29] N. X. Tan, Random quasi-variational inequalities, Math. Nachr., 125 (1986) 319-328. [30] R. U. Verma, Approximation-solvability of a class of A-monotone variational inclusion problems, J. KSIAM, Vol. 8, No. 1, (2004) 55-66. [31] R. U. Verma, A Hybrid proximal point algorithm based on the (A, η)-maximal monotonicity framework, Applied Mathematics Letters, Vol. 21, No. 2, (2008) 142-147. [32] X. Z. Yuan, Noncompact random generalized games and random quasi variational inequalities, J. Math. Stoch. Anal., Vol. 7, (1994) 467-486. [33] L. A. Zadeh, Fuzzy sets, Inform. Control, Vol. 8, (1965) 335-353. [34] C. Zhang and Z. S. Bi, Random generalized nonlinear variational inclusions for random fuzzy mappings, J. Sichuan Univer. (Natural Science Edition) 6,(2007), 499502.

16

62

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.'S 1-2, 63-85, 2014, COPYRIGHT 2014 EUDOXUS PRESS LLC

Hyperbolic expressions of polynomial sequences and parametric number sequences defined by linear recurrence relations of order 2 Tian-Xiao He,

∗

Peter J.-S. Shiue,

†

and Tsui-Wei Weng

February 9, 2013

Abstract A sequence of polynomial {an (x)} is called a function sequence of order 2 if it satisfies the linear recurrence relation of order 2: an (x) = p(x)an−1 (x) + q(x)an−2 (x) with initial conditions a0 (x) and a1 (x). In this paper we derive a parametric form of an (x) in terms of eθ with q(x) = B constant, inspired by Askey’s and Ismail’s works shown in [2] [6], and [18], respectively. With this method, we give the hyperbolic expressions of Chebyshev polynomials and Gegenbauer-Humbert Polynomials. The applications of the method to construct corresponding hyperbolic form of several well-known identities are also discussed in this paper. AMS Subject Classification: 05A15, 12E10, 65B10, 33C45, 39A70, 41A80. ∗

Department of Mathematics and Computer Science, Illinois Wesleyan University, Bloomington, Illinois 61702. † Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, Nevada, 89154-4020. ‡ Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 106. The last two authors would like to thank The Institute of Mathematics, Academia Sinica, Taiwan for its financial support during the summer of 2009 during which the research in this paper was carried out.

1 63

‡

2

T. X. He, P. J.-S. Shiue and T.-W. Weng Key Words and Phrases: sequence of order 2, linear recurrence relation, Fibonacci sequence, Chebyshev polynomial, the generalized Gegenbauer-Humbert polynomial sequence, Lucas number, Pell number.

1

Introduction

In [2, 6, 18], a type of hyperbolic expressions of Fibonacci polynomials and Fibonacci numbers are given using parameterization. We shall extend the idea to polynomial sequences and number sequences defined by linear recurrence relations of order 2. Many number and polynomial sequences can be defined, characterized, evaluated, and/or classified by linear recurrence relations with certain orders. A number sequence {an } is called a sequence of order 2 if it satisfies the linear recurrence relation of order 2: an = aan−1 + ban−2 ,

n ≥ 2,

(1)

for some non-zero constants p and q and initial conditions a0 and a1 . In Mansour [21], the sequence {an }n≥0 defined by (1) is called Horadam’s sequence, which was introduced in 1965 by Horadam [14]. [21] also obtained the generating functions for powers of Horadam’s sequence. To construct an explicit formula of its general term, one may use a generating function, characteristic equation, or a matrix method (see Comtet [8], Hsu [15], Strang [24], Wilf [26], etc.) In [5], Benjamin and Quinn presented many elegant combinatorial meanings of the sequence defined by recurrence relation (1). For instance, an counts the number of ways to tile an n-board (i.e., board of length n) with squares (representing 1s) and dominoes (representing 2s) where each tile, except the initial one has a color. In addition, there are p colors for squares and q colors for dominoes. In particular, Aharonov, Beardon, and Driver (see [1]) have proved that the solution of any sequence of numbers that satisfies a recurrence relation of order 2 with constant coefficients and initial conditions a0 = 0 and a1 = 1, called the primary solution, can be expressed in terms of Chebyshev polynomial values. For instance, the authors show Fn = i−n Un (i/2) and Ln = 2i−n Tn (i/2), where Fn and Ln respectively are Fibonacci numbers and Lucas numbers, and Tn and Un are Chebyshev polynomials of the first kind and the second kind, respectively (see also in [2, 3]). Some identities drawn from those rela-

64

Sequences of numbers and Polynomials

3

tions were given by Beardon in [4]. Marr and Vineyard in [22] use the relationship to establish an explicit expression of five-diagonal Toeplitz determinants. In [12], the first two authors presented a new method to construct an explicit formula of {an } generated by (1). For the sake of the reader’s convenience, we cite this result as follows. Proposition 1.1 ([12]) Let {an } be a sequence of order 2 satisfying linear recurrence relation (1), and let α and β be two roots of of quadratic equation x2 − ax − b = 0. Then    (  a1 −βa0 a1 −αa0 n α − α−β β n , if α 6= β; α−β (2) an = na1 αn−1 − (n − 1)a0 αn , if α = β. If the coefficients of the linear recurrence relation of a function sequence {an (x)} of order 2 are real or complex-value functions of variable x, i.e., an (x) = p(x)an−1 (x) + q(x)an−2 (x), (3) we obtain a function sequence of order 2 with initial conditions a0 (x) and a1 (x). In particular, if all of p(x), q(x), a0 (x) and a1 (x) are polynomials, then the corresponding sequence {an (x)} is a polynomial sequence of order 2. Denote the solutions of t2 − p(x)t − q(x) = 0 by α(x) and β(x). Then p p 1 1 α(x) = (p(x) + p2 (x) + 4q(x)), β(x) = (p(x) − p2 (x) + 4q(x)). 2 2 (4) Similar to Proposition 1.1, we have Proposition 1.2 [12] Let {an } be a sequence of order 2 satisfying the linear recurrence relation (3). Then    (  a1 (x)−β(x)a0 (x) a1 (x)−α(x)a0 (x) n α (x) − β n (x), if α(x) 6= β(x); α(x)−β(x) α(x)−β(x) an (x) = na1 (x)αn−1 (x) − (n − 1)a0 (x)αn (x), if α(x) = β(x), (5) where α(x) and β(x) are shown in (4).

65

4

T. X. He, P. J.-S. Shiue and T.-W. Weng

In this paper, we shall consider the polynomial sequence defined by (3) with q(x) = B, a constant, to derive a parametric form of function sequence of order 2 by using the idea shown in [18]. Our construction will focus on four type Chebyshev polynomials and the following Gegenbauer-Humbert polynomial sequences although our method is limited by those function sequences. A sequence of the generalized Gegenbauer-Humbert polynomials λ,y,C {Pn (x)}n≥0 is defined by the expansion (see, for example, [8], Gould [10], Lidl, Mullen, and Turnwald[20], the first two of authors with Hsu [11]) X Φ(t) ≡ (C − 2xt + yt2 )−λ = Pnλ,y,C (x)tn , (6) n≥0

where λ > 0, y and C 6= 0 are real numbers. As special cases of (6), we consider Pnλ,y,C (x) as follows (see [11]) Pn1,1,1 (x) = Un (x), Chebyshev polynomial of the second kind, Pn1/2,1,1 (x) = ψn (x), Legendre polynomial, Pn1,−1,1 (x) = Pn+1 (x), P ell polynomial, x = Fn+1 (x), F ibonacci polynomial, Pn1,−1,1  x2 = Φn+1 (x), F ermat polynomial of the f irst kind, Pn1,2,1 2 Pn1,2a,2 (x) = Dn (x, a), Dickson polynomial of the second kind, a 6= 0, (see, f or example, [20]), where a is a real parameter, and Fn = Fn (1) is the Fibonacci number. In particular, if y = C = 1, the corresponding polynomials are called Gegenbauer polynomials (see [8]). More results on the GegenbauerHumbert-type polynomials can be found in [16] by Hsu and in [17] by the second author and Hsu, etc. Similarly, for a class of the generalized Gegenbauer-Humbert polynomial sequences defined by λ + n − 1 λ,y,C 2λ + n − 2 λ,y,C Pn−1 (x) − y Pn−2 (x) Cn Cn for all n ≥ 2 with initial conditions Pnλ,y,C (x) = 2x

P0λ,y,C (x) = Φ(0) = C −λ , P1λ,y,C (x) = Φ0 (0) = 2λxC −λ−1 ,

66

(7)

Sequences of numbers and Polynomials

5

the following theorem has been obtained in [12] √ Theorem 1.3 ([12]) Let x 6= ± Cy. The generalized GegenbauerHumbert polynomials {Pn1,y,C (x)}n≥0 defined by expansion (6) can be expressed as  Pn1,y,C (x) = C −n−2

x+

n+1  n+1 p p x2 − Cy − x − x2 − Cy p . (8) 2 x2 − Cy

We may use recurrence relation (6) to define various polynomials that were defined using different techniques. Comparing recurrence relation (6) with the relations of the generalized Fibonacci and Lucas polynomials shown in Example 4, with the assumption of P01,y,C = 0 and P11,y,C = 1, we immediately know 1,1,1 1,1,1 Pn1,1,1 (x) = 2xPn−1 (x) − Pn−2 (x) = Un (2x; 0, 1)

defines the Chebyshev polynomials of the second kind, and 1,−1,1 1,−1,1 Pn1,−1,1 (x) = 2xPn−1 (x) + Pn−2 (x) = Pn (2x; 0, −1)

defines the Pell polynomials. In addition, in [20], Lidl, Mullen, and Turnwald defined the Dickson polynomials are also the special case of the generalized GegenbauerHumbert polynomials, which can be defined uniformly using recurrence relation (6), namely Dn (x; a)) = xDn−1 (x; a) − aDn−2 (x; a) = Pn1,2a,2 (x) with D0 (x; a) = 2 and D1 (x; a) = x. Thus, the general terms of all of above polynomials can be expressed using (8). For λ = y = C = 1, using (8) we obtain the expression of the Chebyshev polynomials of the second kind: √ √ (x + x2 − 1)n+1 − (x − x2 − 1)n+1 √ , Un (x) = 2 x2 − 1 where x2 6= 1. Thus, U2 (x) = 4x2 − 1.

67

6

T. X. He, P. J.-S. Shiue and T.-W. Weng

For λ = C = 1 and y = −1, formula (8) gives the expression of a Pell polynomial of degree n + 1: √ √ (x + x2 + 1)n+1 − (x − x2 + 1)n+1 √ Pn+1 (x) = . 2 x2 + 1 Thus, P2 (x) = 2x. Similarly, let λ = C = 1 and y = −1, the Fibonacci polynomials are √ √ (x + x2 + 4)n+1 − (x − x2 + 4)n+1 √ , Fn+1 (x) = 2n+1 x2 + 4 and the Fibonacci numbers are ( √ !n ) √ !n 1− 5 1 1+ 5 − , Fn = Fn (1) = √ 2 2 5 which has been presented in Example 1. Finally, for λ = C = 1 and y = 2, we have Fermat polynomials of the first kind: √ √ (x + x2 − 2)n+1 − (x − x2 − 2)n+1 √ Φn+1 (x) = , 2 x2 − 2 where x2 6= 2. From the expressions of Chebyshev polynomials of the second kind, Pell polynomials, and Fermat polynomials of the first kind, we may get a class of the generalized Gegenbauer-Humbert polynomials with respect to y defined by the following which will be parameterized. Definition 1.4 The generalized Gegenbauer-Humbert polynomials with (y) respect to y, denoted by Pn (x) are defined by the expansion X (1 − 2xt + yt2 )−1 = Pn(y) (x)tn , n≥0

or by (y)

(y)

Pn(y) (x) = 2xPn−1 (x) − yPn−2 (x), or equivalently, by Pn(y) (x)

=

(x +

p

x2 − y)n+1 − (x − p 2 x2 − y

68

p

x2 − y)n+1

Sequences of numbers and Polynomials (y)

7

(y)

with P0 (x) = 1 and P1 (x) = 2x, where x2 6= y. In particular, (−1) (1) (2) Pn (x), Pn (x) and Pn (x) are respectively Pell polynomials, Chebyshev polynomials of the second kind, and Fermat polynomials of the first kind. In the next section, we shall parameterize the function sequences defined by (3) and number sequences defined by (1) by using the idea of [18]. The application of the parameterization will be applied to construct the corresponding hyperbolic form of several well-known identities.

2

Hyperbolic expressions of parametric polynomial sequences

Suppose q(x) = b, a constant, and re-write (5) as

an (x) a1 (x) − α(x)a0 (x) n a1 (x) − β(x)a0 (x) n α (x) − β (x) = α(x) − β(x) α(x) − β(x) a0 (x)(αn+1 (x) − β n+1 (x)) + (a1 (x) − a0 (x)p(x))(αn (x) − β n (x)) = , α(x) − β(x) (9) where we assume α(x) 6= β(x) due to the reason shown below. Inspired by [18], we now set  √ θ(x) √ −θ(x) ), f or b > 0, (√be , −√be (α(x), β(x)) = ( −beθ(x) , −be−θ(x) ), f or b < 0,

(10)

for some real or complex value function θ ≡ θ(x). Thus one may have α(x) · β(x) = −b and  √ 2√b sinh(θ(x)), f or b > 0, α(x) + β(x) = p(x) = 2 −b cosh(θ(x)), f or b < 0, which implies

69

(11)

8

T. X. He, P. J.-S. Shiue and T.-W. Weng

    sinh−1 p(x) √ , f or b > 0 2 b  θ(x) =  cosh−1 p(x) √ , f or b < 0. 2 −b

(12)

For b > 0, substituting expressions (10) into the last formula of (9) yields √ a0 (x) b cosh((n + 1)θ)  √ +(a1 (x) − 2a0 (x) b sinh(θ)) sinh(nθ) , f or even n,  an (x) = √ b(n−1)/2  a (x) b sinh((n + 1)θ)  0 cosh(θ)    √    +(a1 (x) − 2a0 (x) b sinh(θ)) cosh(nθ) , f or odd n, (13) √ −1 where θ = sinh (p(x)/(2 b)). Still in the case of b > 0, substituting (10) into the formula before the last one shown in (9), we obtain an equivalent expression:        

b(n−1)/2 cosh(θ)



an (x)    √  b(n−1)/2 a1 (x) sinh nθ + ba0 (x) cosh(n − 1)θ , f or even n; cosh θ   = √  b(n−1)/2 a1 (x) cosh nθ + ba0 (x) sinh(n − 1)θ , f or odd n, cosh θ (14) √

where θ = sinh−1 (p(x)/(2 b)). Similarly, for b < 0 we have √ (−b)(n−1)/2  a0 (x) −b sinh((n + 1)θ) an (x) = sinh(θ)  √ +(a1 (x) − 2a0 (x) −b cosh(θ)) sinh(nθ) ,

(15)

or equivalently, √ (−b)(n−1)/2 (a1 (x) sinh nθ − a0 (x) −b sinh(n − 1)θ), sinh θ √ where θ = cosh−1 (p(x)/(2 −b)). We survey the above results as follows. an (x) =

70

(16)

Sequences of numbers and Polynomials

9

Theorem 2.1 Let function sequence an (x) be defined by an (x) = p(x)an−1 (x) + ban−2 (x)

(17)

with initials a0 (x) and a1 (x), and let function θ(x) be defined by (12). Then the roots of the characteristic function t2 − p(x)t − b can be shown as (10), and there hold the hyperbolic expressions of functions an (x) shown in (13) and (14) for b > 0 and (15) and (16) for b < 0. Let us consider some special cases of Theorem 2.1: Corollary 2.2 Suppose {an (x)} is the function sequence defined by (17) with initials a0 (x) = 0 and a1 (x), then sinh(2n)θ ; cosh θ cosh(2n + 1)θ a2n+1 (x) = bn a1 (x) cosh θ √ for b > 0, where θ = sinh−1 (p(x)/(2 b)); and a2n (x) = b(2n−1)/2 a1 (x)

sinh nθ an (x) = (−b)(n−1)/2 a1 (x) sinh θ √ −1 for b < 0, where θ = cosh (p(x)/(2 −b)).

(18)

(19)

Example 2.1 Let {Fn (kx)} be the sequence of the generalized Fibonacci polynomials defined by Fn+2 (kx) = kxFn+1 (kx) + Fn (kx),

k ∈ R\{0},

with initials F0 (kx) = 0 and F1 (kx) = 1. From Corollary 2.2, we have sinh 2nθ , cosh θ cosh(2n + 1)θ F2n+1 (kx) = F2n+1 (2 sinh θ) = , cosh θ

F2n (kx) = F2n (2 sinh θ) =

when k = 2 which are (6) and (7) shown in [6]. Obviously, from the above formulas and the identity cosh x + cosh y = 2 cosh((x + y)/2) cosh((x − y)/2), there holds

71

10

T. X. He, P. J.-S. Shiue and T.-W. Weng

F2n+1 (kx) + F2n−1 (kx) = 2 cosh(2nθ), which was given in [6] as (8) when k = 2. Identity (9) in [6] is clearly the recurrence relation of {Fn (2x)}. The expressions of F2n and F2n+1 can also be found in [13] with a general complex form Fn (x) = in−1

sinh nz , sinh z

where x = 2i cosh z. Corollary 2.3 Suppose {an (x)} is the function sequence defined by (17), an (x) = p(x)an−1 (x) + ban−2 (x) (b > 0), with initials a0 (x) = c, a constant, and a1 (x) = p(x), then cosh(2n − 1)θ cosh θ sinh 2nθ a2n+1 (x) = 2bn+1/2 sinh(2n + 1)θ + (c − 2)bn+1/2 , (20) cosh θ √ where θ(x) = sinh−1 (p(x)/(2 b). If {an (x)} is the function sequence defined by (17), an (x) = p(x)an−1 (x) + ban−2 (x) (b < 0), with initials a0 (x) = c, a constant, and a1 (x) = p(x), then a2n (x) = 2bn cosh(2nθ) + (c − 2)bn

√ (−b)(n−1)/2 (2 cosh θ sinh nθ − c −b sinh(n − 1)θ), sinh θ √ where θ(x) = cosh−1 (p(x)/(2 −b)). an (x) =

(21)

√ Proof. Substituting a0 (x) = c, a1 (x) = p(x) = 2 b sinh θ into (14) yields bn [2 sinh θ sinh(2nθ) + c cosh(2n − 1)θ] , cosh θ bn a2n+1 (x) = [2 sinh θ cosh(2n + 1)θ + c sinh(2nθ)] . cosh θ

a2n (x) =

Then in the above equations using the identities

72

Sequences of numbers and Polynomials

11

cosh θ cosh(2nθ) − sinh θ sinh(2nθ) = cosh(2n − 1)θ, cosh θ sinh(2n + 1)θ − sinh θ cosh(2n + 1)θ = sinh(2nθ), respectively, we obtain (20). Similarly, using (16) one may obtain (21).

Example 2.2 Since the generalized Lucas polynomials are defined by Ln (kx) = kxLn−1 (kx) + Ln−2 (kx) with the initials L0 (x) = 2 and L1 (x) = kx, from Corollary 2.3 we have L2n (kx) = L2n (2 sinh θ) = 2 cosh(2nθ), L2n+1 (kx) = L2n+1 (2 sinh θ) = 2 sinh(2n + 1)θ. [13] also presented a general complex form of Ln (x) as Ln (x) = 2in cosh nz, where x = 2i cosh z. Example 2.3 In 1959, Morgan-Voyce discovered two large families of polynomials, bn (x) and Bn (x), in his study of electrical ladder networks of resistors [23]. The recurrence relations of the polynomials were presented in [19] as follows. Bn (x) = (x + 2)Bn−1 (x) − Bn−2 (x),

n ≥ 2,

where B0 (x) = 1 and B1 (x) = x + 2, while bn (x) = (x + 2)bn−1 (x) − bn−2 (x),

n ≥ 2,

where b0 (x) = 1 and b1 (x) = x + 1. It can be found that bn (x) = Bn (x) − Bn−1 (x), xBn (x) = bn+1 (x) − bn (x). Using Corollary 2.3, it is easy to obtain the hyperbolic expressions of Bn (x) and bn (x). From (21) in the corollary and noting B1 (x) = x+2 = 2 cosh θ and B0 (x) = 1, we have

73

12

T. X. He, P. J.-S. Shiue and T.-W. Weng

sinh(n + 1)θ , x = 2 cosh θ − 2. sinh θ Similarly, substituting b1 (x) = x + 1 = 2 cosh θ − 1 and b0 (x) = 1 into (16) yields Bn (x) =

bn (x) =

sinh(n + 1)θ − sinh nθ cosh(2n + 1)θ/2 = , sinh θ cosh θ/2

x = 2 cosh θ − 2.

We now consider the generalized Gegenbauer-Humbert polynomial (y) sequences defined by (7) with λ = C = 1 and denoted by Pn (x) ≡ Pnλ,y,C (x). Thus (y)

(y)

Pn(y) (x) = 2xPn−1 (x) − yPn−2 (x), (y)

(22)

(y)

P0 (x) = 1 and P1 (x) = 2x. We use the similar parameterization shown above to present the hyperbolic expression of those generalized Gegenbauer-Humbert polynomial sequences. (y)

(y)

Corollary 2.4 Let Pn (x) be defined by (22) with initials P0 (x) = 1 (y) and P1 (x) = 2x. If y < 0, then cosh(2n + 1)θ , cosh θ sinh(2n + 2)θ (y) , P2n+1 (x) = (−y)n+1/2 cosh θ √ where θ(x) = sinh−1 (p(x)/(2 −y). If y > 0, then (y)

P2n (x) = (−y)n

Pn(y) (x) = y n/2 √ where θ(x) = cosh(−1) (p(x)/(2 y).

sinh(n + 1)θ , sinh θ

(23)

(24)

Proof. A similar argument in the proof of (20) with b = −y and c = 1 can be used to prove (23): cosh(2n − 1)θ cosh θ sinh 2nθ (y) , P2n+1 (x) = 2(−y)n+1/2 sinh(2n + 1)θ − (−y)n+1/2 cosh θ (y)

P2n (x) = 2(−y)n cosh(2nθ) − (−y)n

74

Sequences of numbers and Polynomials

13

√ where θ(x) = sinh−1 (p(x)/(2 −y), which implies (23) due to the identities cosh(2n + 1)θ + cosh(2n − 1)θ = 2 cosh(2nθ) cosh θ and sinh(2n + 2)θ + sinh(2nθ) = 2 sinh(2n + 1)θ cosh θ. To prove (24), we substitute √ −b = y, and a1 (x) = 2x = 2 y cosh θ, and a0 (x) = 1 into (16). Thus y n/2 (2 cosh θ sinh nθ − sinh(n − 1)θ) sinh θ sinh(n + 1)θ = y n/2 , sinh θ √ where θ(x) = cosh−1 (x/ y) and the identity sinh(n + 1)θ + sinh(n − 1)θ = 2 sinh nθ cosh θ is applied in the last step. Pn(y) (x) =

Example 2.4 Using Corollary 2.4 one may obtain the following hyper(−1) bolic expressions of Pell polynomials Pn (x) = Pn (x) and the Cheby(1) shev polynomials of the second kind Un (x) = Pn (x): cosh(2n + 1)θ , cosh θ sinh(2n + 2)θ , P2n+1 (x) = cosh θ P2n (x) =

where θ(x) = sinh−1 (x), and Un (x) =

sinh(n + 1)θ , sinh θ

(25)

where θ(x) = cosh−1 (x). Example 2.5 Finally, we consider the Chebyshev class of polynomials including the polynomials of the first kind, second kind, third kind, and fourth kind, denoted by Tn (x), Un (x), Vn (x), and Wn (x), respectively, which are defined by an (x) = 2xan−1 (x) − an−2 (x),

n ≥ 2,

(26)

with a0 (x) = 1 and a1 (x) = x, 2x, 2x−1, 2x+1 for an (x) = Tn (x), Un (x), Vn (x), and Wn (x), respectively. Noting among those four polynomial sequences only {Un (x)} is in the generalized Gegenbauer-Humbert class, which has been presented in Example 2.3. From (16) there holds

75

14

T. X. He, P. J.-S. Shiue and T.-W. Weng

Tn (x) =

1 (x sinh nθ − sinh(n − 1)θ), sinh θ

where x = cosh θ due to θ = cosh−1 x. By using this substitution and the identity sinh(n − 1)θ = sinh nθ cosh θ − cosh nθ sinh θ we immediately obtain Tn (x) = Tn (cosh θ) = cosh nθ. Similarly, cosh(n + 1/2)θ , cosh(θ/2) sinh(n + 1/2)θ . Wn (x) = Wn (cosh θ) = sinh(θ/2)

Vn (x) = Vn (cosh θ) =

√ A simple transformation θ 7→ iθ, i = −1, leads cos(iθ) = cosh θ and sin(iθ) = − sinh θ. Thus from the trigonometric expressions of Tn (x), Un (x), Vn (x), and Wn (x) shown below, one may also obtain their corresponding hyperbolic expressions by simply transforming θ 7→ iθ, respectively.

Tn (cos θ) = cos nθ, Vn (cos θ) =

3

sin(n + 1)θ , sin θ sin(n + 1/2)θ Wn (cos θ) = . sin(θ/2)

Un (cos θ) =

cos(n + 1/2)θ , cos(θ/2)

Hyperbolic expressions of parametric number sequences

Suppose {an } is a number sequence defined by (1), i.e. an = aan−1 + ban−2 ,

n ≥ 2,

(27)

with the given initials a0 and a1 . From [12] (see Proposition 1.1), the sequence defined by (27) has the expression

76

Sequences of numbers and Polynomials

a0 (αn+1 − β n+1 ) + (a1 − a0 a)(αn − β n ) α−β a1 − βa0 n a1 − αa0 n α − β , n ≥ 2, = α−β α−β

15

an =

(28)

where α and β are two distinct roots of characteristic polynomial t2 − at − b. Similar to (10) we denote  √ θ √ −θ f or b > 0,  (√be , −√be ) θ −θ (α(θ), β(θ)) = −be , −be ( √ √ ) −θ f or b < 0, a > 0,  θ (− −be , − −be ) f or b < 0, a < 0,

(29)

for some real or complex number θ. Thus we have  √ f or b > 0,  2√b sinh(θ) a(θ) = α + β = 2 √ −b cosh(θ) f or b < 0, a > 0,  −2 −b cosh(θ) f or b < 0, a < 0,

(30)

and define a parameter generalization of {an (θ)} as  √ f or b > 0,  2√b sinh(θ)an−1 (θ) + ban−2 (θ) an (θ) = −b cosh(θ)an−1 (θ) + ban−2 (θ) f or b < 0, a > 0, 2 √  −2 −b cosh(θ)an−1 (θ) + ban−2 (θ) f or b < 0, a < 0 (31) with initials a0 (θ) = a0 and a1 (θ) = a1 when a0 = 0 or a1 (θ) = when a0 6= 0. Obviously, if    −1 a  √ sinh f or b > 0,    2 b  θ= cosh−1 2√a−b f or b < 0, a > 0, (32)      −a  cosh−1 √ f or b < 0, a < 0, 2 −b {an (θ)} is reduced to {an }. For b > 0, substituting expressions (29) into the second expression of an in (28), we obtain

77

16

T. X. He, P. J.-S. Shiue and T.-W. Weng

an (θ)

√ nθ n −nθ a (e − (−1) e ) + ba0 (e(n−1)θ + (−1)n e−(n−1)θ ) 1 = b(n−1)/2 eθ + e−θ (33)    √  b(n−1)/2 a1 sinh nθ + ba0 cosh(n − 1)θ , if n is even, cosh θ   (34) = √  b(n−1)/2 a1 cosh nθ + ba0 sinh(n − 1)θ , if n is odd, cosh θ √ where θ = sinh−1 (a/(2 b)). Similarly, for b < 0 we have

=

an         

√ a0 −b sinh((n + 1)θ) √
−b cosh(θ)) sinh(nθ) f or a > 0, +(a − 2a 1 0 √ √ (− −b)n−1 −a0 −b sinh((n + 1)θ) sinh(θ) √
+(a1 + 2a0 −b cosh(θ)) sinh(nθ) f or a < 0, (−b)(n−1)/2 sinh(θ)

(35)

or equivalently,

an (

nθ

−nθ

√

(n−1)θ

−(n−1)θ

−b(e −e ) (−b)(n−1)/2 a1 (e −e )−a0eθ√−e −θ = √ nθ −nθ −b(e(n−1)θ −e−(n−1)θ ) (− −b)n−1 a1 (e −e )+a0eθ −e −θ ( √ (n−1)/2 (−b) (a1 sinh nθ − a0 −b sinh(n − 1)θ) sinh θ √ = √ (− −b)n−1 (a1 sinh nθ + a0 −b sinh(n − 1)θ) sinh θ

f or a > 0, f or a < 0, f or a > 0, (36) f or a < 0,

√ √ where θ = cosh−1 (a/(2 −b)) when a > 0 and cosh−1 (−a/(2 −b)) when a < 0. If the characteristic polynomial t2 −at−b has the same roots α = β, √ √ then a = ±2 −b, α = β = ± −b, and √ √ an = na1 (± −b)n−1 − (n − 1)a0 (± −b)n . We summarize the above results as follows.

78

(37)

Sequences of numbers and Polynomials

17

Theorem 3.1 Suppose {an }n≥0 is a number sequence defined by (27) with characteristic polynomial t2 − at − b. If the characteristic polynomial has the same roots, then there holds an expression of an shown in (37). If the characteristic polynomial has distinct roots, there hold hyperbolic extensions (51) or (52) for b > 0 and (36) or (36) for b < 0. Example 3.1 [18] gave the hyperbolic expression of the generalized Fibonacci number sequence {Fn (θ)} defined by Fn (θ) = 2 sinh θFn−1 (θ) + Fn−2 (θ),

n ≥ 2,

with initials F0 (θ) = 0 and F1 (θ) = 1. From Theorem 3.1, one may obtain the same result as that in [18]: enθ − (−1)n e−nθ eθ + e−θ  sinh nθ , if n is even; cosh θ = cosh nθ , if n is odd, cosh θ

Fn (θ) =

(38)

Similarly, for the generalized Lucas number sequence {Ln (θ)} defined by Ln (θ) = 2 sinh θLn−1 (θ) + Ln−2 (θ),

n ≥ 2,

with initials L0 (θ) = 2 and L1 (θ) = 2 sinh θ, we have nθ

Ln (θ) = e

n −nθ

+ (−1) e

 =

2 cosh(nθ), if n is even; 2 sinh(nθ), if n is odd.

(39)

Example 3.2 [9] defined the following generalization of Fibonacci numbers and Lucas numbers: cn − d n , `n = cn + dn , (40) c−d where c and d are two roots of√t2 − st − 1, s ∈ N. Denote ∆ = s2 + 4 and α = ln c, where c = (s + s2 + 4)/2. Then the above expressions are equivalent to fn =

1 eαn − (−1)n e−αn √ , fn = 2 2 ∆

79

1 eαn + (−1)n e−αn `n = . 2 2

18

T. X. He, P. J.-S. Shiue and T.-W. Weng

It is obvious that by transferring c 7→ eθ and d 7→ −e−θ that two expressions in (40) are equivalently (38) and (39), respectively, shown in Example 3.1, which are obtained using Theorem 3.1 with (a, b, a0 , a1 ) = (s, 1, 0, 1) and (s, 1, 2, s) for fn and `n , respectively. Hence, the corresponding identities regarding fn and `n obtained in [9] can be established similarly. However, we may derive more new identities as follows. For instance, there holds `n + sfn = 2fn+1 ,

(41)

which can be proved by substituting s = eθ − e−θ = 2 sinh θ into the left-hand side. Indeed, for even n, from Example 3.1

`n + sfn = 2 cosh(nθ) + 2 sinh θ

cosh(n + 1)θ sinh nθ =2 , cosh θ cosh θ

and similarly, for odd n, `n + sfn = 2 sinh(n + 1)θ/ cosh θ, which brings (41). When s = 1, (41) reduces to the classical identity Ln + Fn = 2Fn+1 . From the above examples, we find many identities relevant to Fibonacci numbers and Lucas numbers can be proved using hyperbolic identities. Here are more examples. Example 3.3 In the identity sinh 2nθ = 2 sinh nθ cosh nθ substituting (38) and (39), namely, sinh 2nθ = cosh θ F2n (θ) and  sinh nθ =  cosh nθ =

cosh θ Fn (θ), if n iseven, 1 L (θ), if n is odd, 2 n 1 L (θ), 2 n

if n is even, cosh θ Fn (θ), if n is odd,

we immediately obtain F2n (θ) = Fn (θ)Ln (θ). Similarly, since sinh(m + n)θ = cosh θ Fm+n (θ) when m + n is even,

80

Sequences of numbers and Polynomials

19



1 2 1 2

cosh θ Fm (θ)Ln (θ), if m and n are even, cosh θ Fn (θ)Lm (θ), if m and n are odd,



1 2 1 2

cosh θ Fn (θ)Lm (θ), if m and n are even, cosh θ Fm (θ)Ln (θ), if m and n are odd,

sinh mθ cosh nθ = and

cosh mθ sinh nθ = from identity

sinh(m + n)θ = sinh mθ cosh nθ + cosh mθ sinh nθ

(42)

we have 2Fm+n (θ) = Fm (θ)Ln (θ) + Fn (θ)Lm (θ) for even m + n. When m + n is odd, sinh(m + n)θ = Lm+n (θ)/2, from (42), 

cosh2 θ Fm (θ)Fn (θ), if m is even and n is odd, 1 L (θ)Ln (θ), if m is odd and n is even, 4 m



1 L (θ)Ln (θ), 4 m cosh2 θ Fm (θ)Fn (θ),

sinh mθ cosh nθ = and

cosh mθ sinh nθ =

if m is even and n is odd, if m is odd and n is even,

we obtain 2Lm+n (θ) = Fm (θ)Fn (θ) + Lm (θ)Ln (θ). More examples can be found in [25]. Our scheme may also extend some well-know identities to their hyperbolic setting. Example 3.4 [7] considers equation t2 −at+b = 0 (b 6= 0) with distinct roots t1 and t2 , i.e., ∆2 = a2 − 4b 6= 0, and defines a sequence {gn } by gn = agn−1 − bgn−2 (n ≥ 2) with initials g0 and g1 . If the initials

81

20

T. X. He, P. J.-S. Shiue and T.-W. Weng

are 0 and 1, the corresponding sequence is denoted by {rn }. Denote sn = tn1 + tn2 and ∆ = a2 − 4b. Then [7] gives identities rn = −bn r−n , sn = bn s−n , s2n = ∆rn2 + 4bn , sn sn+1 = ∆rn rn+1 + 2abn , , 2bn rj−n = rj sn − rn sj , , rj+n = rn sj + bn rj−n .,

(43) (44) (45) (46) (47) (48)

We now show all the above identities can be extended to the hyperbolic setting. For b > 0, from (36) there holds rn = (b)(n−1)/2

enθ − e−nθ (b)(n−1)/2 = sinh nθ, eθ − e−θ sinh θ

(49)

and similarly, sn = 2bn/2 cosh nθ, √ where θ = cosh−1 (a/(2 b)). For b > 0, substituting expressions (29) into (28), we obtain an (θ) = b

=

√ − (−1)n e−nθ ) + ba0 (e(n−1)θ + (−1)n e−(n−1)θ ) eθ + e−θ (51)   √ a1 sinh nθ + ba0 cosh(n − 1)θ , if n is even;   (52) √ a1 cosh nθ + ba0 sinh(n − 1)θ , if n is odd,

(n−1)/2 a1 (e

 

b(n−1)/2 cosh θ



b(n−1)/2 cosh θ

(50)

nθ

√ where θ = sinh−1 (a/(2 b)). Similarly, for b < 0 we have

an

(−b)(n−1)/2  √ = a0 −b sinh((n + 1)θ) sinh(θ)  √ +(a1 − 2a0 −b cosh(θ)) sinh(nθ) ,

82

(53)

Sequences of numbers and Polynomials

21

or equivalently, − e−nθ ) −

√

−ba0 (e(n−1)θ − e−(n−1)θ ) (54) eθ − e−θ √ (−b)(n−1)/2 (a1 sinh nθ − a0 −b sinh(n − 1)θ), = (55) sinh θ √ where θ = cosh−1 (a/(2 −b)). an = (−b)

(n−1)/2 a1 (e

nθ

Acknowledgments We wish to thank the referees for their helpful comments and suggestions.

References [1] D. Aharonov, A. Beardon, and K. Driver, Fibonacci, Chebyshev, and orthogonal polynomials, Amer. Math. Monthly. 122 (2005) 612–630. [2] R.Askey, Fibonacci and Related Sequences, Teacher, (2004), 116-119.

Mathematics

[3] R.Askey, Fibonacci and Lucas Numbers, Mathematics Teacher, (2005), 610-614. [4] A. Beardon, Fibonacci meets Chebyshev, The Mathematical Gaz. 91 (2007), 251-255. [5] A. T. Benjamin and J. J. Quinn, Proofs that really count. The art of combinatorial proof. The Dolciani Mathematical Expositions, 27. Mathematical Association of America, Washington, DC, 2003. [6] P. S. Bruckman, Advanced Problems and Solutions H460, Fibonacci Quart. 31 (1993), 190-191. [7] P. Bundschuh and P. J.-S. Shiue, A generalization of a paper by D.D.Wall, Atti della Accademia. Nazionale dei Lincei, Vol. LVI (1974), 135-144.

83

22

T. X. He, P. J.-S. Shiue and T.-W. Weng

[8] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. [9] E. Ehrhart, Associated Hyperbolic and Fibonacci identities, Fibonacci Quart. 21 (1983), 87-96. [10] H. W. Gould, Inverse series relations and other expansions involving Humbert polynomials, Duke Math. J. 32 (1965), 697–711. [11] T. X. He, L. C. Hsu, P. J.-S. Shiue, A symbolic operator approach to several summation formulas for power series II, Discrete Math. 308 (2008), no. 16, 3427–3440. [12] T. X. He and P. J.-S. Shiue, On sequences of numbers and polynomials defined by linear recurrence relations of order 2, Intern. J. of Math. Math. Sci., Vol. 2009 (2009), Article ID 709386, 1-21. [13] V. E. Hoggatt, Jr. and M. Bicknell, Roots of Fibonacci polynomials. Fibonacci Quart. 11 (1973), no. 3, 271274. [14] A. F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quart. 3 (1965), 161–176. [15] L. C. Hsu, Computational Combinatorics (Chinese), First edition, Shanghai Scientific & Technical Publishers, Shanghai, China, 1983. [16] L. C. Hsu, On Stirling-type pairs and extended GegenbauerHumbert-Fibonacci polynomials. Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 367–377, Kluwer Acad. Publ., Dordrecht, 1993. [17] L. C. Hsu and P. J.-S. Shiue, Cycle indicators and special functions. Ann. Comb. 5 (2001), no. 2, 179–196. [18] M. E.H. Ismail, One parameter generalizations of the Fibonacci and lucas numbers, The Fibonacci Quart. 46/47 (2008/09), No. 2 ,167-179. [19] T. Koshy, Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2001.

84

Sequences of numbers and Polynomials

23

[20] R. Lidl, G. L. Mullen, and G. Turnwald, Dickson polynomials. Pitman Monographs and Surveys in Pure and Applied Mathematics, 65, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. [21] T. Mansour, A formula for the generating functions of powers of Horadam’s sequence, Australas. J. Combin. 30 (2004), 207–212. [22] R. B. Marr and G. H. Vineyard, Five-diagonal Toeplitz determinants and their relation to Chebyshev polynomials, SIAM Matrix Anal. Appl. 9 (1988), 579-586. [23] A. M. Morgan-Voyce, Ladder network analysis using Fibonacci numbers, IRE, Trans. on Circuit Theory, CT-6 (1959, Sept.), 321322. [24] G. Strang, Linear algebra and its applications. Second Edition, Academic Press (Harcourt Brace Jovanovich, Publishers), New York-London, 1980. [25] S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section, John Wiley, New York, 1989. [26] H. S. Wilf, Generatingfunctionology, Academic Press, New York, 1990.

85

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.'S 1-2, 86-93, 2014, COPYRIGHT 2014 EUDOXUS PRESS LLC

On a system of nonlinear differential equations for the model of totally connected traffic ∗ Alexander P. Buslaeva, Valery V. Kozlovb February 16, 2013

a

Moscow Automobile and Road Technical University, Russia; E-mail: [email protected]; b

Steklov Mathematical Institute of RAS, Russia; E-mail: [email protected] Abstract

In the paper the qualitative properties solutions of the system nonlinear equations, describing one-way movement of particles chain on a line with follower velocity defined by some function of distance from the leader, are researched. In the case when the given function is the velocity of the first particle (leader) in the chain, the model is called a model of leader. If the given function is the velocity of the last particle (outsider), the model is called a model of “shepherd”. The sufficient conditions for the existence of the chain with the given constraints on the velocity and acceleration are obtained.

AMS 2000 Mathematics Subject Classification: 34A34, 46E35 Keywords: systems of nonlinear ordinary differential equations; follow-theleader model; interpretation for traffic

1

Introduction

One of the basic models of traffic flow is a model of follow the leader [1]-[4]. This model reduce to the next differential equations: xn+1 − xn = f(x˙ n ),

(1)

where xn (t) is a vehicle coordinate, xn(t) < xn+1 (t), ∗ The

n = 1, 2, ...

paper was supported by Grant of RFBR No.11-01-12140-ofi-m

1

86

(2)

BUSLAEV-KOZLOV: TOTALLY CONNECTED TRAFFIC

Flow satisfying (1)-(2) is called totally connected. The function f in (1) is a parabola with positive coefficients in classic case,[1]-[2], f(x) = a + bx + cx2, where a is static distance, b is driver reaction delay and c is braking distance coefficient. Function f by condition x ≥ 0 is continuous with several successive derivatives, positive, monotone and convex. For simplify, we set f(0) = 1. Let us denote the inverse of this function f by g and obtain a system of differential equations x˙ n = g(xn+1 − xn ), n = 1, ..., N − 1.

2

Follow - the - leader problem statement: the cluster with front wheel drive

We consider a system of ordinary differential equations (ODE) x˙ n = g(xn+1 − xn), n = 1, 2, .., N − 1,

(3)

supp(g) = [1, ∞),

(4)

g(1) = 0,

(5)

g0 (x) > 0, ∀ x ≥ 1,

(6)

g00 (x) ≤ 0, ∀ x ≥ 1.

(7)

where

g has enough smoothness and,

Let the initial conditions are x1(0) = x1,0, x2(0) = x2,0, ..., xN −1(0) = xN −1,0 such that xn+1,0 − xn,0 > 1, n = 1, ..., N − 1,

(8)

and boundary condition is xN (t) = r(t) ∀ t ≥ 0.

(9)

We associate problem (3)-(9) with follow the leader models. For function r(t) let assume the following. 1. The function r(t) ˙ is absolutely continuous for t ≥ 0;

2

87

BUSLAEV-KOZLOV: TOTALLY CONNECTED TRAFFIC

2. There is the speed boundaries 0 ≤ r(t) ˙ ≤ M1 , ∀ t ≥ 0;

(10)

3. There is acceleration boundaries |¨ r(t)| ≤ M2

(11)

almost everywhere by ∀ t ≥ 0. Conditions (10)-(11) define functions of the Sobolev class , [8], 1 (R+ ) = {h ∈ L∞ (R+ ), h˙ ∈ L0∞ (R+ )}, h(t) ∈ W∞

where h(t) = r(t) ˙ − M1 /2. −1 Main purpose is to investigate properties of the functions cluster {xn}N n=1 , followed the leader xN (t).

2.1

An elementary case N = 2.

We have equation x˙ = g(r − x). Lemma 1. If x is solution of (12),(8)-(9), then x˙ > 0

(12) ∀ t > 0.

Proof. If x˙ → 0, then g(r − x) → 0, and it’s equivalent to r − x → 1 + 0. If r(T ) − x(T ) = 1, it is true at a moment of time T then x(T ˙ ) = 0 and x ¨(T ) = g0 (1)(r(T ˙ ) − x(T ˙ )), ˙ ) > 0, which contradicts with (7). So, r(t) − x(t) − 1 whence x ¨(T ) = g0 (1)r(T can’t go to null in a finite time. Lemma 2. The following inequality is true ||x|| ˙ C(R+) ≤ max(||r|| ˙ C(R+), x(0)). ˙

(13)

x ¨ = g0 × (r˙ − x). ˙

(14)

Proof. From (12) ¨(t0) = 0, whence Suppose x˙ reaches local maximum at some point t0. Then x and from (14) r(t ˙ 0 ) = x(t ˙ 0 ). (15) If x˙ monotonically increases on R+ , then from (12) (r − x)(t) monotonically increases too, whence r(t) ˙ − x(t) ˙ ≥ 0, ie r(t) ˙ ≥ x(t) ˙ ≥ 0. 3

88

(16)

BUSLAEV-KOZLOV: TOTALLY CONNECTED TRAFFIC

Analogously if x˙ monotonically decreases, then r˙ − x˙ ≤ 0, and 0 ≤ x(t) ˙ ≤ x(0). ˙

(17)

Inequality (13) follows from (15)-(17). Lemma 3. The following inequality is true ||¨ x||C(R+) ≤ max(||¨ r||C(R+), g(1)|| ˙ r|| ˙ C(R+) , g(1)g(r(0) ˙ − x(0))). Proof. From (14) we have ... x = g00 × (r˙ − x) ˙ 2 + g0 × (¨ r−x ¨).

(18)

(19)

Because g0 > 0 and g00 < 0 for admissible values of arguments, then ... x = 0 ⇐⇒ r¨ − x ¨ ≥ 0, from where follow x ¨(t) ≤ r¨(t)

(20)

at those points t, where x ¨(t) has a local extremum. On the other hand from (14) it follows that |¨ x(t)| ≤ |g(r(t) ˙ − x(t))||r˙ − x|, ˙ whence with Lemma 2 and monotonically decreasing g, ˙ we have |¨ x(t)| ≤ |g(1)|max(|| ˙ r|| ˙ C(R+) , x(0)). ˙

(21)

Statement of Lemma 3 follows from (20) and (21) . Lemma 4. Let suppose g(r(0) − x(0)) ≤ ||r|| ˙ C(R+) ,

(22)

and max(g(1)|| ˙ r|| ˙ C(R+) , g(1)g(r(0) ˙ − x(0))) ≤ ≤ ||¨ r||C(R+) .

(23)

Then the following inequalities are true ||x|| ˙ C(R+) ≤ ||r|| ˙ C(R+) ,

(24)

||¨ x||C(R+) ≤ ||¨ r||C(R+) .

(25)

Proof. It follows from previous lemmas. Theorem 1. Solution x(t) of problem (3) - (11) with conditions (22)-(23) and N = 2 exists and belongs to the same set of functions(10)-(11) with the leader function r(t). 4

89

BUSLAEV-KOZLOV: TOTALLY CONNECTED TRAFFIC

2.2

Cluster follows the leader length N.

Applying the obvious considerations of induction can be established that Theorem 2. Solution of problem (3)-(11) with conditions g(xn+1 (0) − xn(0)) ≤ ||x˙ n+1||C(R+),

(26)

max(x˙ n+1(1)||x˙ n+1||C(R+), x˙ n+1(1)g(xn+1 (0) − xn(0))) ≤ ≤ ||¨ xn+1||C(R+) ,

(27)

n = 1, ..., N − 1 exists for any natural N. In this case, all links are infinitely differentiable functions, if functions g and r are those.

2.3

Uniform movement of the leader

Let us consider some of the specific behaviors of the leader. Suppose that begining from some moment t0 we have r(t) = r(t0 ) + M1 (t − t0), t ≥ t0 ≥ 0.

(28)

Then if t > t0 is true then we have ˙ − x˙ N −1 (t)) = f 0 × (M1 − x˙ N −1 (t)). x ¨N −1 (t) = g0 × (r(t) So far as M1 − x˙ N −1 ≥ 0, then x ¨N −1 > 0, x˙ N −1 monotonically increases and is limited by the constant M1 , i.e. x˙ N −1 → M1 , xN −1 → M1 (t − t0 ) + C from the top. Thus the movement of the follower also converges to the uniform movement. In general case if r(t) = r(t0 ) + M (t − t0), t ≥ t0,

(29)

where M isn’t necessarily the maximum constant, then x ¨N −1 (t) = (a − x˙ N −1 )g0 (M t + M0 ).

(30)

From (30) it follows that if M > x˙ N −1 , then x˙ N −1 is increasing, and if M < x˙ N −1 , then x˙ N −1 decreases. Moreover it follows from the concavity of g that 0 < g0 (x) ≤ g0 (0), which implies that |x˙ N −1(t) − M | → 0 monotonically and from equation (29) r(t) − xN −1(t) → Const. 5

90

BUSLAEV-KOZLOV: TOTALLY CONNECTED TRAFFIC

Discoursing by induction, we get Theorem 3.If in chain (3)-(9),(26)-(27) of N links the leader r(t) converges in norm C1[t0, ∞) to uniform traffic, then next links converge to uniform traffics in this metric too.

2.4

Generalized cluster traffic

In constraints of statements (3)-(11) function g depends on the numbers of managers, i.e. x˙ n = gn (xn+1 − xn), n = 1, 2, .., N − 1, (30) Lemma 5. Let suppose k = 1, 2, N − 1 gk (xk+1(0) − xk (0)) ≤ ||x˙ k+1||C(R+) ,

(31)

and max(g˙ k (1)||x˙ k+1||C(R+) , g˙ k (1)gk (xk+1(0) − xk (0))) ≤ ≤ ||¨ xk||C(R+) .

(32)

||x˙ k||C(R+) ≤ ||x˙ k+1||C(R+),

(33)

||¨ xk||C(R+) ≤ ||¨ xk+1||C(R+).

(34)

Then next relations

are true. Theoreme 4. Solution of problem (3’)-(11), (31)-(32) exist for any natural N. In this case, if functions gk , 1 ≤ k ≤ N −1 and xN are infinite differentiable, then all links are infinite differentiable functions too.

2.5

Generalized traffic - cluster with random dynamic dimensions.

Functions fk , k = 1, .., N are a family of functions depending on a finite number of random variables such as linear or quadratic. In this case, the chains are finite random functions. The conditions (29) - (30) are probability and sufficient conditions of a connected traffic hold with a certain probability, which should be evaluated.

2.6

Cluster with rear wheel drive

We consider the problem (1)-(11) xn+1 − xn = f(x˙ n ), n = 1, ..., N − 1, where instead (9) we assume 6

91

(35)

BUSLAEV-KOZLOV: TOTALLY CONNECTED TRAFFIC

x1(t) = r(t) ∀ t ≥ 0.

(90)

¨nf 0 (x˙ n(t)), x˙ n+1(t) = x˙ n(t) + x

(36)

... ¨n (t) + x n f 0 (x˙ n (t)) + (¨ xn)2 f 00 (x˙ n (t)). x ¨n+1 (t) = x

(37)

From (35) it follows

Let us assume x1(t) = r(t) is admissible operating regime of traffic, which satisfied (10)-(11). If the traffic is not totally accelerating, i.e. monotonically increasing acceleration, what can’t be subject to the speed limit, then there will be time t∗ , when the acceleration (deceleration) has local (global) maximum. ... From (37)it follows that since at that moment x n (t∗ ) = 0, then x ¨n+1 (t∗ ) > x ¨n (t∗ ).

(38)

So, if a moment exists when acceleration xn peaks, then from (36)it follows that xn+1 isn’t satisfies admissible conditions. Theorem 5. For solution of problem (1)-(9’)-(11) ||¨ xk||C(R+) < ||¨ xk+1||C(R+),

(39)

k = 1, 2, ..., is true. It means gap connected traffic in the link of the chain, where the corresponding rate of acceleration is a maximum.

7

92

BUSLAEV-KOZLOV: TOTALLY CONNECTED TRAFFIC

References 1. Morisson R.B. The traffic Flow Analogy to Compressible Fluid Flow. Advanced Res. Eng. Bull., 1964 2. Inosse H.,.Hamada ., Road Traffic Control. Univ. of Tokio Press,1975 3. Rothery R.W. Car Following Models in Traffic Flow Theory. Transportation research board, ed. Gartner N , Special report, 165, 1992, p. 4.1 - 4.42 4.Pipes L.A. An operational Analysis of Traffic Dynamics. Journal of Applied Physics, 1953, v. 24, p. 271-281. 5. Buslaev A.P., Gasnikov A.V., Yashina M.V. Mathematical Problems of Traffic Flow Theory. Proceed. of the 2010 International Conference on Computational and Mathematical Methods in Science and Engineering, ed J.Vigo Aguar, Almeria, Spain, 26-30.06.2010, v.1, p.307-313 6. Buslaev A.P., Gasnikov A.V., Yashina M.V. Selected Mathematical Problems of Traffic Flow Theory. International Journal of Computer Mathematics Vol. 89, No. 3, 2012, p.409-432 7. Buslaev A.P., Provorov A.V., Yashina M.V. Recently approach to investigation of connected flow of particle with motivation ,T-Com: Telecommunications and Transport, No. 2, 2011, . 61-62 (in Russian) 8. Tikhomirov V.M. Some problems of approximation theory , Nauka, 1976 (in Russian)

8

93

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.'S 1-2, 94-101, 2014, COPYRIGHT 2014 EUDOXUS PRESS LLC

REMOTALITY OF EXPOSED POINTS R. KHALIL1 , S. HAYAJNEH, M. HAYAJNEH AND M. SABABHEH2 Abstract. In this article, we discuss the problem of remotality of exposed points of bounded sets in certain Banach spaces. Indeed, we present a full characterization of a class of exposed points that are remotal points.

1. Introduction and preliminaries Let X be a Banach space, and E be a closed bounded convex subset of X. For x ∈ X, let D(x, E) = sup kx − ek e∈E

be the maximum distance from x to E. If an e ∈ E exists such that D(x, E) = kx − ek, then e is said to be a remotal, or farthest, point in E for x, and we define F (x, E) = {e ∈ E : D(x, E) = kx − ek}. If F (x, E) 6= φ for all x ∈ X, then E is said to be a remotal set. The theory of remotal sets in Banach spaces is not as well as developed as that of proximinal sets; where the minimum distance is required to be attained. In [3], the authors proposed and discussed the following problem: Problem 1: When is a boundary point of E a remotal point? This seems to be a tough question and more general than Problem 2: When is an extreme point of E a remotal point? Recall that a point e ∈ E is said to be an extreme point of the convex set E, if e is not the middle point of any two other points of E. A special type of extreme points are exposed points. A point e ∈ E is said to be an exposed point of E, if there exists a linear functional f ∈ X ∗ , the dual space of the normed space X, such that f (y) < f (e) for all y ∈ E\{e}. Recall that, in this case, the set H := {x ∈ X : f (x) = f (e)} is called a supporting hyperplane of E at e; see [4]. In [1], it is proved that any normed linear space contains a bounded convex set whose exposed points are not necessarily remotal points. This is why we study here the problem: Problem 3: When is an exposed point of E a remotal point? We refer the reader to [3] and [1] for some results on this problem. The object of this paper is to address problem 3 above, where we give necessary and sufficient conditions for a class of exposed points to be remotal points in certain Banach spaces. In the sequel, X ∗ denotes the dual space of the normed space X, S(m, r) denotes the sphere centered at m with radius r and B(m, r) denotes the ball centered 2000 Mathematics Subject Classification. 46B20, 41A50, 41A65. Key words and phrases. Remotal sets, Approximation theory in Banach spaces. 1

94

2

R. KHALIL, S. HAYAJNEH, M. HAYAJNEH, M. SABABHEH,

at m with radius r. If E is a subset of the normed space X, and x ∈ X, then P (x, E) denotes the set of closest elements of E from X. For 1 < p < ∞, we define the conjugate exponent of p to be the number q that satisfies 1/p+1/q = 1. For 1 < p < ∞, we define the spaces p

` := {(xi ) : xi ∈ C,

∞ X

|xi |p < ∞}

i=1

and Z

p

L [a, b] := {f : [a, b] → C :

b

|f (t)|p dt < ∞}.

a

For (xi ) ∈ `p and f ∈ Lp [a, b], the following norms are defined !1/p Z b 1/p ∞ X p p k(xi )k = |xi | and kf k = |f (t)| dt . a

i=1

Recall that (`p )∗ = `q and (Lp [a, b])∗ = Lq [a, b] where p and q are conjugate exponents. For p = ∞, `∞ := {(xi ) : xi ∈ C, sup |xi | < ∞} and L∞ [a, b] := {f : [a, b] → C : ess supf < ∞}. It is known that (`1 )∗ = `∞ and (L1 [a, b])∗ = L∞ [a, b]. Finally, c0 is defined to be {(xi ) : xi ∈ C, xi → 0}. It is known that (c0 )∗ = `∞ . We refer the reader to any standard book in functional analysis as a reminder of these concepts; see [2]. 2. Basic Results Definition 2.1. A differentiable strictly convex function defined on [0, ∞) will be called a nice convex function if it satisfies the following properties: (1) ϕ ≥ 0. (2) ϕ(0) = 0. (3) lim ϕ0 (x) = ∞.

x→∞

(4) ϕ0 (0) = 0.

95

REMOTALITY OF EXPOSED POINTS

3

It follows that if ϕ is a nice convex function, then ϕ is strictly increasing. Moreover, since ϕ is strictly convex and increasing, it is unbounded, hence ϕ(x) lim = lim ϕ0 (x) = ∞. x→∞ x x→∞ This observation will be used in the sequel. Observe that for any p > 1, ϕ(t) = tp is a nice convex function. Now let X be a Banach space and let X ∗ be its dual space. Definition 2.2. The pair (X, X ∗ ) is called a strictly convex pair if there exists a nice convex function ϕ such that for each x ∈ X, there exists fx ∈ X ∗ with the property fx (x) = kfx k kxk = ϕ(kxk). It should be noted that the first equality in the above definition always holds, for a certain f , according to the Hahn-Banach theorem. So, in fact, our interest is the second equality. Example 2.3. The pairs (`p , `q ), 1 < p < ∞, are strictly convex pairs, with ϕ(t) = tp . Indeed, for x = (xn ) ∈ `p , define ∞ X
fx (y) = |xn |p−1 sgn xn yn . n=1

Then, clearly, fx ∈ (`p )∗ , kf k = kxkp−1 and fx (x) = kxkp = kfx k kxk = ϕ(kxk). Example 2.4. The pairs (c0 , `1 ) and (`1 , `∞ ) are not strictly convex pairs. Example 2.5. The pairs (Lp [0, 1], Lq [0, 1]), 1 < p < ∞ are strictly convex pairs, but (L1 [0, 1], L∞ [0, 1]) is not. Definition 2.6. Let (X, X ∗ ) be a strictly convex pair, ϕ be the corresponding nice convex function, and let H be a subspace of X. We shall say that H is a ϕ− summand subspace of X if there exists a subspace W such that X = H ⊕ W in such a way that x = h + w ⇒ ϕkxk = ϕkhk + ϕkwk. Example 2.7. If X is a Hilbert space, then (X, X ∗ ) is a strictly convex pair. This can be seen by letting ϕ(t) = t2 . In this case, fx (y) = < x, y > . Let H be a nontrivial subspace of X, then H is a ϕ−summand of X, with W = H ⊥ . Example 2.8. If 1 < p < ∞, a subspace H of `p is p−summand if, and only if, there exists J ⊂ N such that H = {(xn ) : xn = 0, ∀n 6∈ J}. By p−summand, we mean ϕ−summand with ϕ(t) = tp . Similarly, a subspace H of Lp [a, b] is p−summand if, and only if, there exists E ⊂ [a, b] such that 0 < µ(E) < 1 and H = {f ∈ Lp [a, b] : f (t) = 0, a.e. on E c }. Definition 2.9. Let (X, X ∗ ) be a strictly convex pair. An exposed point e ∈ E ⊂ X is called a ϕ−exposed point if the kernel of the linear functional that supports E uniquely at e is a ϕ−summand subspace.

96

4

R. KHALIL, S. HAYAJNEH, M. HAYAJNEH, M. SABABHEH,

Example 2.10. Let X be a Hilbert space, E be a closed bounded convex subset of X. Then every exposed point of E is a 2-exposed point, since every subspace of a Hilbert space is a 2-summand subspace. Proposition 2.11. Let 1 < p < ∞ and let E be a closed bounded convex subset of `p . Then, an exposed point e of E is a p−exposed point if, and only if, there exists an index j such that hj = 0 for all h ∈ H. Here h = (hi ). 3. Main Results Let (X, X ∗ ) be a strictly convex pair, and let ϕ be the associated nice convex function. Let E be a closed bounded convex subset of X, and e be a ϕ−exposed point of E, and H be the supporting hyperplane of E uniquely at e. Let x ∈ X\H, and denote the minimum distance from x to H by d(x, H), then the ratio ϕkx − ek R(x, e) = d(x, H) will be called the remotality ratio of E at e with respect to x. Lemma 3.1. Let (X, X ∗ ) be a strictly convex space, E be a closed bounded convex subset of E, and e be a ϕ− exposed point of E. If a sphere S(m, r) exists such that S(m, r) ∩ E = {e} and E ⊂ B(m, r), and if H is the supporting hyperplane of S(m, r) at e, then sup R(x, e) ≤ x∈E

sup R(x, e). x∈S(m,r)

Proof. Without loss of generality, we may assume e = 0. Let x ∈ E, and θ be the closest element in [m] := {αm : α ∈ R} from x. Let x0 be the intersection of the array [θ, x, −] and S(m, r). Clearly, θ is the closest element in [m] from x0 . Now, let H be the supporting hyperplane of S(m, r) at e := 0. We assert that kxk ≤ kx0 k. Since [m] and H are ϕ− summands in X, and ϕ is strictly convex, then both are proximinal, and if x = y1 + z1 then y1 ∈ P (x, [m]) and z1 ∈ P (x, H). Similarly, if x0 = y2 + z2 , then y2 ∈ P (x0 , [m]) and z2 ∈ P (x0 , H). Hence, y1 = y2 = θ. Consequently, kz1 k = kx − θk and kz2 k = kx0 − θk. But, by our choice of x0 , it can be easily seen that kx − θk ≤ kx0 − θk, and hence, kz1 k ≤ kz2 k. This implies that ϕkxk ≤ ϕkx0 k. Since ϕ is increasing, we infer that kxk ≤ kx0 k. Moreover, d(x, H) = d(x0 , H) follows from the fact that x0 ∈ [θ, x, −]. Hence, kxk ≤ kx0 k ⇒

ϕkxk ϕkx0 k ≤ ; x ∈ E, x0 ∈ S(m, r). d(x, H) d(x0 , H)

Thus, we have shown that for every x ∈ E, there exists x0 ∈ S(m, r) such that R(x, e) ≤ R(x0 , e). This completes the proof of the lemma.  Lemma 3.2. Let X be a Banach space and S(m, r) a sphere in X containing 0. If 0 is a ϕ− exposed point of S(m, r) and H is the hyperplane supporting S(m, r)

97

REMOTALITY OF EXPOSED POINTS

5

uniquely at 0, then ϕkuk < ∞. u∈S(m,r) d(u, H) sup

Proof. Observe first that if u ∈ S(m, r), then u = x + m where x ∈ H, and 0 ≤  ≤ 2. Then, ϕkuk = ϕkxk + ϕ(r). Now, u − m = x + m − m = x + ( − 1)m ⇒ ϕku − mk = ϕ(r) = ϕkxk + ϕ(| − 1|r). Now, ϕkuk ϕkxk + ϕkmk = d(u, H) kmk ϕ(r) − ϕ(| − 1|r) + ϕ(r) = := g(). r It is clear that the function g() is continuous on (0, 2] and that lim→0 g() = ϕ0 (r). Consequently, g is a bounded function. This completes the proof.  For the proof of the main theorem of this paper, we need the following Lemma. But first, recall from [3] that a nice exposed point of E is an exposed point, where the functional that determines the hyperplane supporting E at e attains its norm. It is worth to remark that exposed points of convex sets in any reflexive space are nice exposed points. Lemma 3.3. Let e be a nice exposed point of the convex bounded subset E in a normed space X. If H is the hyperplane that supports E uniquely e, then there exists a sequence of spheres S(mk , rk ) which lie in the same side of H as E, and such that H is a supporting hyperplane of S(mk , rk ) for all k ∈ N. Proof. Without loss of generality, assume that e = 0, and that f (y) > 0 for all y ∈ E\{0}. Here f ∈ X ∗ is the functional that determines H, and kf k = 1. If a ∈ X is such that f (a) > 0 and f (a) = kf k, where such an a exists since f attains its norm, then the spheres S(ka, kf (a)) satisfies the required properties.  Now, we prove the main theorem of the paper. Theorem 3.4. Let e be a ϕ−nice exposed point of the closed bounded convex subset E of the strictly convex space (X, X ∗ ). Then e is a remotal point of E if, and only if, sup R(x, e) < ∞. x∈E

Proof. Suppose that e is a remotal point. We assert that supx∈E R(x, e) < ∞. Again, assume e = 0. Being a remotal point, there exists a sphere S(m, r) such that E ∩ S(m, r) = {0} and E ⊂ B(m, r). Let H be the supporting hyperplane of S(m, r) uniquely at 0. By Lemma 3.1, it is enough to prove that R (u, 0) < u∈S(m,r)

∞. But this follows from lemma 3.2

sup R(u, 0) < ∞. u∈S(m,r)

Conversely, suppose that the remotality ratio R(x, e) is bounded for x ∈ E. To show that e is a remotal point. Suppose on the way of contrary that e is not remotal. Assuming e = 0, there exists a sequence of spheres S(mk , k), by virtue of

98

6

R. KHALIL, S. HAYAJNEH, M. HAYAJNEH, M. SABABHEH,

Lemma 3.3 such that 0 ∈ S(mk , k) and E\B(mk , k) 6= φ, for each k ∈ N. Observe that all these spheres are still supported by the same hyperplane supporting E at 0. Let uk ∈ E\B(mk , k), hence kuk − mk k ≥ k, ∀k ∈ N. But then, following the same ideas in the beginning of the Lemma 3.2, we find that R(uk , 0) ≥

ϕ(k) − ϕ(|1 − k |k) + ϕ(k k) . k k

Here we have two cases: Case 1: If 0 < k ≤ 1, then ϕ(k) − ϕ(|1 − k |k) + ϕ(k k) k k ϕ(k) − ϕ(k − k k) + ϕ(k k) = k k ϕ(k k) , = ϕ0 (ck ,k ) + k k

R(uk , 0) ≥

where k − k k < ck ,k < k, by the mean value theorem. Case 2: If 1 < k ≤ 2, then ϕ(k) − ϕ(|1 − k |k) + ϕ(k k) k k ϕ(k) − ϕ(k k − k) + ϕ(k k) = k k ϕ(k k) , ≥ k k

R(uk , 0) ≥

where the last inequality is a consequence of the fact that ϕ is increasing. Now, since we have infinitely many values of k, we also have infinitely many values of k . Consequently, we either have infinitely many values of k which are less than or equal to 1, or infinitely many values of k which are greater than 1. Let us treat these two cases: Case I: If there are infinitely many values of k which are greater than 1, then there is a corresponding subsequence of the radii, say (kn ), in which kn → ∞. But then, R(ukn , 0) is unbounded because kn k → ∞ and R(ukn , 0) ≥

ϕ(kn kn ) → ∞,  kn k n

where we have used the assumption that ϕ(x) = ∞. x→∞ x Case II: If there are infinitely many values of k which are less than or equal to 1, then there is a corresponding sequence (kn ) such that kn → ∞ and lim

R(ukn , 0) ≥ ϕ0 (ckn ,kn ) +

ϕ(kn kn ) , kn − kn kn < ckn ,kn < kn .  kn kn

99

REMOTALITY OF EXPOSED POINTS

7

Now two subcases of this case are available: Case II-i If the sequence (kn kn ) is bounded. Then ckn ,kn → ∞, and hence R(ukn , 0) → ∞ where we have used the assumption that limx→∞ ϕ0 (x) = ∞. Case II-ii If the sequence (kn kn ) is unbounded, then R(ukn , 0) → ∞ where we have used the fact that ϕ(x) lim = ∞. x→∞ x Thus, we have shown that if 0 is not a remotal point of E then the ration R(u, 0) is unbounded, contradicting our assumption. This shows that 0 is a remotal point, and completes the proof.  4. Miscellaneous Remarks In this section we a remark and an example in inner product spaces. Proposition 4.1. Let H be an inner product space, S(m, r) a sphere in H and e be a ϕ−exposed point of S(m, r). Then the ratio R(u, e) = 2r for u ∈ S(m, r). Proof. . Here ϕ(t) = t2 . Assuming e = 0, for simplicity and following the computations above, we see that ϕ(r) − ϕ(| − 1|r) + ϕ(r) R(u, 0) = r r2 − 2 r2 + 2r2 − r2 + 2 r2 = r = 2r.  The following example was shown in [3] for the purpose of giving an example of an exposed point which is not a remotal point in an inner product space. In the following example, we show that the remotality ratio R(x, e) is unbounded, explaining why e is not a remotal point of E. Example 4.2. Let X = R2 endowed with the standard norm, and let    1 1 E0 = ± , 3 :n∈N . n n Let E be the closed convex hull of E0 , then clearly 0 is a 2-exposed point of E. It was shown that 0 is not a remotal point of E, [3]. Easy computations show that   1 1 1 = n + 3, R ( , 3 ), (0, 0) n n n and hence   1 1 R ( , 3 ), (0, 0) → ∞. n n We conclude our paper with the problem: Problem Describe exposed points which are necessarily remotal points. In this paper, we have answered the question for ϕ−nice exposed points in strictly convex spaces (X, X ∗ ).

100

8

R. KHALIL, S. HAYAJNEH, M. HAYAJNEH, M. SABABHEH,

References [1] Edelstein, M., and Lewis, J., On exposed and farthest points in normed linear spaces, J. Aust. Math. Soc, 12(1971), pp.301-308. 367-373. 1 [2] Rudin, W., Real and complex analysis, McGraw-Hill, 1970. 1 [3] Sababheh, M. and Khalil, R., Remotal Points and a Krein-Milman Type Theorem, Journal of Nonlinear and Convex Analysis, Vol.(12), Number 1, 2011, pp.5-15. 1, 3, 4, 4.2 [4] Singer, I., Best approximation in normed linear spaces by elements of linear subspaces, Springer-Verlag Berlin, 1970. 1 1

Department of Mathematics, Jordan University, Al Jubaiha, Amman 11942, Jordan. E-mail address: [email protected] 2

Department of Basic Sciences, Princess Sumaya University For Technology, Al Jubaiha, Amman 11941, Jordan. E-mail address: [email protected]

101

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.'S 1-2, 102-115, 2014, COPYRIGHT 2014 EUDOXUS PRESS LLC

The dual reciprocity boundary element method for two-dimensional Burgers’ equations with inverse multiquadric approximation scheme

M. Sarboland, and A. Aminataei∗ Department of Mathematics, K. N. Toosi University of Technology, P.O. Box: 16315-1618, Tehran, Iran

Abstract The two-dimensional Burgers’ equation is a mathematical model to describe various kinds of phenomena such as turbulence and viscous fluid. In this paper, the dual reciprocity boundary element method (DRBEM) is used for solving this problem. In DRBEM, the fundamental solution of the Laplace equation is applied for the integral equation formulation and hence a domain integral arises in the boundary integral equation. Further, the time derivative is approximated by the forward divided difference of it, and the domain integral also appears from these approximations. The domain integral is transformed into boundary integral by using the dual reciprocity method (DRM). This method is applied on some test experiments and the numerical results have been compared with the exact solutions and the solutions in [1, 25]. Root-mean-square error (RMSE) of the solutions show the efficiency and the accuracy of the method. Keywords: Nonlinear two-dimensional Burgers’ equation; Dual reciprocity boundary element method; Radial basis function. 2010 Mathematics Subject Classification: 35K55; 65M99; 33E99.

1

Introduction

The nonlinear coupled Burgers’ equation is a special form of incompressible NavierStokes equation without having pressure term and continuity equation. Burgers’ equation is a fundamental partial differential equation (PDE) from fluid mechanics. It is used in various areas of applied mathematics and physics, such as modeling of gas dynamics and turbulence, heat conduction, and acoustic waves [2, 5, 15, 18]. The exact solution of the Burgers’ equations can be obtained for simple geometry using the Hopf-Cole transformation [8, 11]. Using the Hopf-Cole transformation, the exact solution of the Burgers’ equations was given by Fletcher [9]. The numerical solutions were obtained by Jain and Hola [12] using two algorithms based on cubic spline function ∗ Corresponding author. E-mail addresses: [email protected] (M. Sarboland), [email protected] (A. Aminataei).

1

102

The dual reciprocity boundary element method for two-dimensional Burgers’ equations with inverse multiquadric approximation scheme

technique, Fletcher [10] who discussed the comparison of a number of different numerical approaches, Wubs and Goede [23] using an explicit-implicit method, Bahadir [1] using a fully implicit finite-difference scheme, Zhu et al. [25] using the discrete Adomian decomposition method and Young et al. [24] using the Eulerian-Lagrangian method. Boundary element method (BEM) is attractive and important computational techniques for solving problems in applied sciences and engineering. The main idea in this method is to convert the original PDE to an equivalent boundary integral equation by using Green’s theorem and a fundamental solution. Consequently the main advantage in this method over the classical domain methods such as finite element method (FEM) and finite difference method (FDM), is that only boundary discretization is required due to dimension reduction [6]. But there are some difficulties in extending the method to applications such as nonhomogeneous, nonlinear and time dependent problems. The main drawback in these cases is the need to discretize the domain into a series of internal cells to deal with the terms taken to the boundary by application of the fundamental solution. This additional discretization destroys some of the attraction of the method. Several methods have been suggested for the resolution of these problems that in these methods, the DRM is the most efficient method. This method was introduced by Brebbia and Nardini [4] and Partridge and Brebbia [16]. The main idea behind this approach is to expand the inhomogeneous, nonlinear and time dependent terms in terms of its values at the nodes which lie in domain and boundary. These terms are approximated by interpolation in terms of some well-known functions φ(r), called radial basis functions (RBFs), where r is the distance between a source point and the field point. These functions are a powerful tool for scattered data interpolation problem [17, 22]. By applying the DRM, the problem will be reduced to a boundary only formulation, thus we do not have any domain integration in the boundary integral equation. The DRBEM is used by Chino and Tosaka [7] for the one-dimensional time independent Burgers’ equation. Kakuda and Tosaka [13] adopted the generalized BEM to treat the Burgers’ equations. The organization of this paper is as follows. In Section 2, we describe the DRBEM for the nonlinear two-dimensional Burgers’ equations. The results of three numerical experiments are presented in Section 3 and are compared with the analytical solutions and the results in [1,25]. Finally, a brief discussion and conclusion is presented in Section 4.

2

The dual reciprocity boundary element method Consider the coupled two-dimensional Burgers’ equations: 1 (uxx + uyy ), R 1 vt + uvx + vvy = (vxx + vyy ), R

ut + uux + vuy =

(1)

with the initial conditions: u(x, y, 0) = f1 (x, y),

(x, y) ∈ Ω,

v(x, y, 0) = f2 (x, y),

(x, y) ∈ Ω,

and the boundary conditions: u(x, y, t) = g1 (x, y, t),

103

(x, y) ∈ Γ,

(2)

M. Sarboland and A. Aminataei (3) v(x, y, t) = g2 (x, y, t),

(x, y) ∈ Γ,

where Ω = {(x, y)|a 6 x 6 b, c 6 y 6 d} and Γ is its boundary. u(x, y, t) and v(x, y, t) are the two unknown variables which can be regarded as the velocities in fluid-related problems. f1 (x, y), f2 (x, y), g1 (x, y, t) and g2 (x, y, t) are all known functions. R is the Reynolds number. In order to implement the dual reciprocity method, we consider the time derivative and the nonlinear terms in Eqs. (1), with b1 (x, y, t) and b2 (x, y, t) in the following forms: R(ut + uux + vuy ) = b1 (x, y, t), R(vt + uvx + vvy ) = b2 (x, y, t). Thus, Eqs. (1) convert to the following system: ∇2 u = b1 (x, y, t), (4) 2

∇ v = b2 (x, y, t), ∂ ∂ where ∇2 = ∂x 2 + ∂y 2 . Now, we approximate b1 (x, y, t) and b2 (x, y, t) as a linear combination of interpolation functions for each of them. Therefore, we choose N +L collocation points where N is the number of boundary points and L is the number of internal points. The collocation points are denoted by (xi , yi ) for i = 1, 2, . . . , N + L. The approximation of b1 and b2 can be written over domain Ω in the following forms: N +L X b1 (x, y, t) = ϕi (x, y)αi (t), i=1 (5)

b2 (x, y, t) =

N +L X

ϕi (x, y)βi (t),

i=1

where the interpolation function, ϕi is a radial basis function (RBF). In this work, we use the inverse multiquadric (IMQ) approximation scheme ϕi (x, y) = (ri 2 + ε2 )−2 , p where ri = (x − xi )2 + (y − yi )2 and ε is a shape parameter. Toutip [21] used a linear function ϕi (r) = 1 + ri in the DRBEM. Now, if the function ψi be the particular solution of Laplace’s equation ∇2 ψ i = ϕ i , then, Eqs. (5) convert to the following expressions b1 (x, y, t) =

N +L X

∇2 ψi (x, y)αi (t), (6)

i=1

b2 (x, y, t) =

N +L X

∇2 ψi (x, y)βi (t).

i=1

104

The dual reciprocity boundary element method for two-dimensional Burgers’ equations with inverse multiquadric approximation scheme

For IMQ-RBF, the function ψi is given as follows: 1 ln(ri ). 2ε2

ψi (x, y) =

The above function is a combination of logarithmic RBF and multiquadic (MQ) RBF. Initially, this combination of RBFs used by Mazarei and Aminataei [14] for the solution of Possions’ equation. Substituting Eqs. (6) into Eqs. (4), and writing the weight residual formulation of Eq. (4) with using the second Green’s theorem [19], lead to: δ k uk +

δk vk +

Z

Γ

Z

Γ

∂u∗k udΓ − ∂n

Z

∂u∗k vdΓ − ∂n

Z

∂u u∗k dΓ ∂n

Γ

u∗k Γ

=

N +L X

[δk ψki +

i=1

Z

Γ

∂u∗k ψi dΓ − ∂n

Z

u∗k Γ

∂ψi dΓ]αi (t), ∂n

Z Z N +L X ∂v ∂ψi ∂u∗k dΓ = [δk ψki + ψi dΓ − u∗k dΓ]βi (t), ∂n ∂n Γ Γ ∂n i=1

θk 1 ln(rk ), δk = 2π ; θk is the interior angle at the for k = 1, 2, . . . , N + L, where u∗k = − 2π ∂ψi point k, and ψki = ψi (xk , yk ). The term ∂n is the normal derivative of ψi and can be written as

qˆi =

∂ψi ∂x ∂ψi ∂y ∂ψi = · + · . ∂n ∂x ∂n ∂y ∂n

At this step, the boundary Γ is discretized into N elements, thus we rewrite the above equations in the following expressions δ k uk +

N X

Hki ui −

i=1

δk vk +

N X

N X

Gki q1 i =

N X

Gki q2 i =

i=1

i=1

for k = 1, 2, . . . , N + L, where q1 i =

N X

N +L X

(7) Ski βi (t),

i=1

∂u ∂n (xi , yi , t),

Ski = δk ψki +

Ski αi (t),

i=1

i=1

Hki vi −

N +L X

q2 i =

Hki ψi −

i=1

N X

∂v ∂n (xi , yi , t),

Gki qˆi ,

i=1

and the definition of the terms of Hki and Gki are defined as in [21]. From Eqs. (5), we obtain αi (t) =

N +L X

Fij b1 (xj , yj , t) =

j=1

βi (t) =

N +L X

N +L X

Fij b1 j (t),

N +L X

Fij b2 j (t),

j=1

Fij b2 (xj , yj , t) =

j=1

j=1

105

(8)

M. Sarboland and A. Aminataei

where F = Φ−1 , Φ is a (N + L) × (N + L) matrix that Φ(k, i) = ϕi (xk , yk ). Substituting Eqs. (8) into the right hand side of Eqs. (7), lead to: N +L X

Ski αi (t) =

Ski

i=1

i=1

N +L X

N +L X

Ski βi (t) =

i=1

N +L X

N +L X

Fij b1 j (t) =

N +L X

Fij b2 j (t) =

j=1

Ski

i=1

N +L X

Pkj b1 j (t),

N +L X

Pkj b2 j (t),

N +L X

Pkj b1 j (t),

j=1

j=1

(9)

j=1

where Pkj =

N +L X

Ski Fij .

i=1

By combining Eqs. (7) and (9), we have δk uk (t) +

N X

Hki ui (t) −

i=1

δk vk (t) +

N X

N X

Gki q1 i (t) =

i=1

Hki vi (t) −

i=1

j=1

N X

Gki q2 i =

i=1

N +L X

(10)

Pkj b2j (t),

j=1

for k = 1, 2, . . . , N + L. we note that b1 j (t) = R(ut (xj , yj , t) + u(xj , yj , t)ux (xj , yj , t) + v(xj , yj , t)uy (xj , yj , t)), b2 j (t) = R(vt (xj , yj , t) + u(xj , yj , t)vx (xj , yj , t) + v(xj , yj , t)vy (xj , yj , t)). For the time derivatives, we use forward difference method to approximate the time derivatives ut (xj , yj , t) and vt (xj , yj , t). Thus, we obtain ut (xj , yj , t) =

un+1 − unj j , ∆t

vt (xj , yj , t) =

vjn+1 − vjn , ∆t

(11)

where unj = u(xj , yj , n4t) and vjn = v(xj , yj , n4t). Also, we approximate ux , uy , vx and vy as described in [21]. Therefore, we obtain ux (xj , yj , t) = vx (xj , yj , t) =

N +L X

ˆ i (xj , yj )ui (t), L

i=1 N +L X

ˆ i (xj , yj )vi (t), L

uy (xj , yj , t) = vy (xj , yj , t) =

N +L X

i=1 N +L X

ˇ i (xj , yj )ui (t), L ˇ i (xj , yj )vi (t), L

i=1

i=1

where ˆ i (x, y) = L

N +L X i=1

Fij

∂ϕi (x, y), ∂x

ˇ i (x, y) = L

N +L X i=1

106

Fij

∂ϕi (x, y). ∂y

The dual reciprocity boundary element method for two-dimensional Burgers’ equations with inverse multiquadric approximation scheme

Substituting the above approximations in Eqs. (10), we obtain the following expressions: δk un+1 + k

N X

Hki un+1 − i

N X

Gki q1 n+1 = i

i=1

i=1

N +L X

Pkj [λun+1 − λunj + u ˜j j

j=1

N +L X

ˆ ji L un+1 i

i=1

+˜ vj

N +L X

ˇ ji ], un+1 L i

(12)

i=1

δk vkn+1 +

N X i=1

Hki vin+1 −

N X

Gki q2 n+1 = i

i=1

N +L X

Pkj [λvjn+1 − λvjn + u ˜j

j=1

+˜ vj

N +L X

ˆ ji vin+1 L

i=1

N +L X

ˇ ji ], vin+1 L

(13)

i=1

R ˆ ji = L ˆ i (xj , yj ) and L ˇ ji = L ˆ i (xj , yj ). u ,L ˜j and v˜j for k = 1, 2, . . . , N + L, where λ = ∆t are given by the known approximations of uj (t) and vj (t), respectively, as described in the below. Using the boundary conditions (2), we have

unj = g1 (xj , yj , n∆t),

vjn = g2 (xj , yj , n∆t),

j = 1, 2, . . . , N,

in each time step. At first time step, when n = 0, the initial conditions (2) give u0j = f1 (xj , yj ) and vj0 = f2 (xj , yj ). In each time step, at first, we put u ˜j = unj and v˜j = vjn . Having these, Eqs. (12) and (13) are solved as a system of linear algebraic equations for unknowns u n+1 j and q2 n+1 for j = 1, . . . , N . Recompute and vjn+1 for j = N + 1, . . . , N + L and q1 n+1 j j u ˜j = un+1 and v˜j = vjn+1 , where un+1 and vjn+1 are obtained from solving Eqs. (12) and j j (13). We iterate between calculating u ˜j and v˜j and solving the approximation values of the unknowns, until the solutions of un+1 and vjn+1 satisfy the condition of the iteration j method in each time step. Here, we use the following criteria for stopping the iterations in each time step, max

|un+1,l − un+1,l−1 | 6 ζ, j j

max

|vjn+1,l − vjn+1,l−1 | 6 ζ,

L6j6N +L

and L6j6N +L

where ζ is a fixed number. Also, un+1,l and vjn+1,l are the values of the un+1 and vjn+1 j j at the l − th iteration. When this condition is satisfied, we put un+1 = un+1,l , j j

vjn+1 = vjn+1,l ,

and go ahead to the next time step. This iteration method is namely called as predictorcorrector method.

3

The numerical experiments

Three experiments are studied to investigate the robustness and the accuracy of the proposed method. We compare the numerical results of the two-dimensional Burgers’

107

M. Sarboland and A. Aminataei

equations by using this scheme with the analytical solutions and solutions in [1]. The RMSE which is defined by s PN 2 i=1 (unum (Xi ) − uexa (Xi )) , RM SE = N is used to measure the accuracy of our scheme wherein Xi is the collocation points. We perform the computations associated with our experiments in Maple 16 on a PC with a CPU of 2.4 GHZ.

Experiment 1. In this experiment, we consider the two-dimensional Burgers’ equations (1) with exact solutions u(x, y, t) =

1 3 − , 4 4[1 + exp(−4x + 4y − t)/(32µ)]

(14)

1 3 v(x, y, t) = + . 4 4[1 + exp(−4x + 4y − t)/(32µ)] Above solutions obtained using a Hopf-Cole transformation in [9]. The initial conditions are obtained from (14) at t = 0, and the boundary conditions in (3) can be obtained from the exact solutions. In this experiment, the Reynolds number R = 80, time step size ∆t = 10−4 , shape parameter ε = 1.5 and ζ = 10−18 are used. The computational domain for this problem is Ω = {(x, y)|0 6 x 6 1, 0 6 y 6 1}. The numerical computation were performed using 13 internal points and 12 boundary points. Tables 1 and 2 give the numerical and exact solutions of u and v at internal points at time levels t = 0.01, 0.1 and t = 0.3. Table 1 Comparison of numerical solutions with the exact solutions of u at t = 0.01, 0.1 and t = 0.3 with R = 80 of experiment 1.

Points (0.1,0.1) (0.5,0.1) (0.9,0.1) (0.3,0.3) (0.7,0.3) (0.1,0.5) (0.5,0.5) (0.9,0.5) (0.3,0.7) (0.7,0.7) (0.1,0.9) (0.5,0.9) (0.9,0.9)

t = 0.01 Numerical 0.62359 0.50424 0.50055 0.62391 0.50411 0.74527 0.62403 0.50390 0.74518 0.62394 0.74996 0.74511 0.62381

Exact 0.62344 0.50439 0.50008 0.62344 0.50439 0.74539 0.62344 0.50439 0.74539 0.62344 0.74991 0.74539 0.62344

t = 0.1 Numerical 0.61058 0.50252 0.50442 0.61352 0.50183 0.74356 0.61488 0.49917 0.74275 0.61418 0.74975 0.74284 0.61310

108

Exact 0.60946 0.50352 0.50006 0.60946 0.50352 0.74426 0.60946 0.50352 0.74426 0.60946 0.74989 0.74426 0.60946

t = 0.3 Numerical 0.57821 0.50278 0.50873 0.58623 0.50334 0.74417 0.59220 0.49488 0.73881 0.59092 0.74624 0.73990 0.58856

Exact 0.58021 0.50214 0.50004 0.58021 0.50214 0.74067 0.58021 0.50214 0.74067 0.58021 0.74982 0.74067 0.58021

The dual reciprocity boundary element method for two-dimensional Burgers’ equations with inverse multiquadric approximation scheme

Table 2 Comparison of numerical solutions with the exact solutions of v at t = 0.01, 0.1 and t = 0.3 with R = 80 of experiment 1.

Points (0.1,0.1) (0.5,0.1) (0.9,0.1) (0.3,0.3) (0.7,0.3) (0.1,0.5) (0.5,0.5) (0.9,0.5) (0.3,0.7) (0.7,0.7) (0.1,0.9) (0.5,0.9) (0.9,0.9)

t = 0.01 Numerical 0.87658 0.99572 0.99947 0.87607 0.99589 0.75470 0.87596 0.99614 0.75482 0.87609 0.75001 0.75497 0.87604

Exact 0.87656 0.99561 0.99992 0.87656 0.99561 0.75461 0.87656 0.99561 0.75461 0.87656 0.75009 0.75461 0.87656

t = 0.1 Numerical 0.89148 0.99726 0.99588 0.88668 0.99800 0.75619 0.88480 1.00115 0.75712 0.88633 0.75009 0.75788 0.88580

Exact 0.89054 0.99648 0.99994 0.89054 0.99648 0.75574 0.89054 0.99648 0.75574 0.89054 0.75011 0.75574 0.89054

t = 0.3 Numerical 0.92685 1.00091 0.99318 0.91727 0.99674 0.75505 0.90690 1.00420 0.75977 0.90973 0.75474 0.75917 0.90979

Exact 0.91979 0.99786 0.99996 0.91979 0.99786 0.75933 0.91979 0.99786 0.75933 0.91979 0.75018 0.75933 0.91979

Table 3 Comparison of absolute errors of u(x, y, t) between the numerical solution using our method and the solution in [1, 25] at t = 0.01 for R = 100 of experiment 1.

Points (0.1,0.1) (0.5,0.1) (0.9,0.1) (0.3,0.3) (0.7,0.3) (0.1,0.5) (0.5,0.5)

Proposed method 1.76859E-4 6.50996E-5 5.75592E-4 7.88296E-4 3.92464E-4 2.76094E-4 9.79140E-4

Bahadir [1] 7.24132E-5 2.42869E-5 8.39751E-6 8.25331E-5 8.25331E-5 8.25331E-5 7.32522E-5

Zhu et al. [25] 5.91368E-5 4.84030E-6 3.41000E-8 5.91368E-5 4.84030E-6 1.64290E-6 5.91368E-5

Exact 0.62305 0.50162 0.50001 0.62305 0.50162 0.74827 0.62305

Table 4 Comparison of absolute errors of v(x, y, t) between the numerical solution using our method and the solution in [1, 25] at t = 0.01 for R = 100 of experiment 1.

Points (0.1,0.1) (0.5,0.1) (0.9,0.1) (0.3,0.3) (0.7,0.3) (0.1,0.5) (0.5,0.5)

Proposed method 8.72333E-6 2.10136E-5 5.49827E-4 8.10210E-4 3.86695E-4 2.40453E-4 9.86737E-4

Bahadir [1] 8.35601E-5 5.13642E-5 7.03298E-6 6.15201E-5 5.41000E-5 7.35192E-5 8.51040E-5

Zhu et al. [25] 5.91368E-5 4.84030E-6 3.41000E-8 5.91368E-5 4.84030E-6 1.64290E-6 5.91368E-5

Exact 0.87695 0.99838 0.99999 0.87695 0.99838 0.75173 0.87695

We compare the absolute error of our scheme with the absolute errors of Bahadir method [1] and Zhu et al. method [25] in Tables 3 and 4. In [1, 25], points are uniformly distributed and their number is 400 whereas in our scheme, points are scattered and their number is 25. Tables 5 and 6 show RMSEs of u and v at t = 0.05, 0.1 and t = 0.2 for different Reynolds numbers, respectively. We also plot the graphs of the numerical and exact solutions of u and v at internal points at time level t = 0.05 for R = 100 in Fig. 1.

109

M. Sarboland and A. Aminataei

Figure 1: Comparison of numerical and exact solutions of u and v for R = 100 at time level t = 0.05 of experiment 1.

110

The dual reciprocity boundary element method for two-dimensional Burgers’ equations with inverse multiquadric approximation scheme

Table 5 RMSE of u at different times for different Reynolds numbers of experiment 1.

Reynolds number 50 80 100

t=0.05 6.39710 × 10−4 1.70407 × 10−3 2.60059 × 10−3

t=0.1 1.26354 × 10−3 3.15403 × 10−3 4.91042 × 10−3

t=0.2 3.29974 × 10−3 5.25154 × 10−3 8.74585 × 10−3

Table 6 RMSE of v at different times for different Reynolds numbers of experiment 1.

Reynolds number 50 80 100

t=0.05 1.35422 × 10−4 1.76704 × 10−3 2.66745 × 10−3

t=0.1 1.25054 × 10−3 3.24367 × 10−3 5.02683 × 10−3

t=0.2 3.29974 × 10−3 5.34885 × 10−3 8.98673 × 10−3

Table 7 Comparison of numerical solutions with the exact solutions of u at t = 0.01, 0.2 and t = 0.4 of experiment 2.

Points (0.125,0.125) (0.125,0.250) (0.125,0.375) (0.250,0.125) (0.250,0.250) (0.250,0.375) (0.375,0.125) (0.375,0.250) (0.375,0.375)

t = 0.01 Numerical 0.24760 0.37264 0.49758 0.37009 0.49511 0.62003 0.49259 0.61760 0.74257

Exact 0.24755 0.37257 0.49760 0.37007 0.49510 0.62012 0.49260 0.61762 0.74265

t = 0.2 Numerical 0.21872 0.35425 0.48721 0.29956 0.43510 0.56762 0.37977 0.51538 0.64956

Exact 0.21739 0.35326 0.48913 0.29891 0.43478 0.57065 0.38043 0.51630 0.65217

t = 0.4 Numerical 0.22270 0.39152 0.55193 0.25950 0.43837 0.59603 0.28571 0.47267 0.64687

Exact 0.22059 0.40441 0.58824 0.25735 0.44118 0.62500 0.29412 0.47794 0.66176

Experiment 2. In this experiment, we consider the two-dimensional Burgers’ equations (1) with the initial conditions (2) at t = 0 are given by f1 (x, y) = x + y,

f2 (x, y) = x − y.

The exact solutions are given by [3] u(x, y, t) =

x + y − 2xt , 1 − 2t2

v(x, y, t) =

x − y − 2yt , 1 − 2t2

and the boundary functions g1 (x, y, t) and g2 (x, y, t) can be obtained from the exact solutions. In this experiment, we consider ∆t = 10−4 , ε = 1.5, ζ = 10−18 and Ω = {(x, y)|0 6 x 6 0.5, 0 6 y 6 0.5}. The numerical computations were performed using 25 points that distributed uniformly. The numerical solutions compared with the exact solutions at internal points at time levels t = 0.01, 0.2 and t = 0.4 for arbitrary Reynolds number R are listed in Tables 7 and 8.

111

M. Sarboland and A. Aminataei

Table 8 Comparison of numerical solutions with the exact solutions of v at t = 0.01, 0.2 and t = 0.4 of experiment 2.

Points (0.125,0.125) (0.125,0.250) (0.125,0.375) (0.250,0.125) (0.250,0.250) (0.250,0.375) (0.375,0.125) (0.375,0.250) (0.375,0.375)

t = 0.01 Numerical -0.00248 -0.13000 -0.25756 0.12252 -0.00500 -0.13257 0.24756 0.12004 -0.00752

t = 0.2 Numerical -0.05405 -0.24421 -0.43538 0.08167 -0.10862 -0.30054 0.21726 0.02716 -0.16435

Exact -0.00250 -0.13003 -0.25755 0.12252 -0.00500 -0.13253 0.24755 0.12002 -0.00750

Exact -0.05435 -0.24457 -0.43478 0.08152 -0.10870 -0.29891 0.21739 0.02717 -0.16304

t = 0.4 Numerical -0.14888 -0.47729 -0.80380 0.03891 -0.29433 -0.63517 0.21796 -0.11364 -0.45597

Exact -0.14706 -0.47794 -0.80882 0.03677 -0.29412 -0.62500 0.22059 -0.11029 -0.44118

Table 9 Comparison of numerical solutions with the exact solutions of u at t = 1, 1.5 and t = 2 with R = 1000 of experiment 3.

Points (0.25,0.25) (0.25,0.50) (0.25,0.75) (0.50,0.25) (0.50,0.50) (0.50,0.75) (0.75,0.25) (0.75,0.50) (0.75,0.75)

t=1 Numerical 0.00205 0.00244 0.00366 0.00658 0.01110 0.00961 0.00033 0.00015 0.00274

Exact 0.00000 0.00000 0.00000 0.00637 0.01141 0.00637 0.00000 0.00000 0.00000

t = 1.5 Numerical 0.00272 0.00320 0.00481 0.00647 0.01060 0.01056 0.00045 0.00023 0.00381

t=2 Numerical 0.00322 0.00376 0.00564 0.00637 0.01020 0.00113 0.00055 0.00031 0.00476

Exact 0.00000 0.00000 0.00000 0.00614 0.01089 0.00614 0.00000 0.00000 0.00000

Exact 0.00000 0.00000 0.00000 0.00592 0.01040 0.00592 0.00000 0.00000 0.00000

Experiment 3. In the following experiment, we consider the two-dimensional Burgers’ equation with the initial conditions: u(x, y, 0) =

−4π cos(2πx) sin(πy) , R(2 + sin(2πx) + sin(πy)

v(x, y, 0) =

−2π sin(2πx) cos(πy) , R(2 + sin(2πx) + sin(πy)

and the exact solutions are as follows [20]: u(x, y, t) =

−4πe R(2 + e

v(x, y, t) =

−2πe R(2 + e

−5π2 t R −5π2 t R

−5π2 t R −5π2 t R

cos(2πx) sin(πy)

,

sin(2πx) + sin(πy) sin(2πx) cos(πy)

.

sin(2πx) + sin(πy)

The boundary conditions are taken from the exact solutions and the computational domain is Ω = {(x, y)|0 6 x 6 1, 0 6 y 6 1}. The numerical computations were performed

112

The dual reciprocity boundary element method for two-dimensional Burgers’ equations with inverse multiquadric approximation scheme

using ∆t = 10−3 , ε = 1.5, ζ = 10−18 , R = 1000 and 25 points that distributed uniformly. Tables 9 and 10 show the numerical solutions and the exact solutions of u and v at time levels t = 1, 1.5 and t = 2. Table 10 Comparison of numerical solutions with the exact solutions of v at t = 1, 1.5 and t = 2 with R = 1000 of experiment 3.

Points (0.25,0.25) (0.25,0.50) (0.25,0.75) (0.50,0.25) (0.50,0.50) (0.50,0.75) (0.75,0.25) (0.75,0.50) (0.75,0.75)

4

t=1 Numerical -0.00208 -0.00007 0.00196 -0.00008 0.00001 0.00008 0.00212 0.00000 -0.00214

Exact -0.00211 0.00000 0.00000 0.00000 0.00000 0.00000 0.00211 0.00000 -0.00211

t = 1.5 Numerical -0.00202 -0.00114 0.00182 -0.00012 0.00002 0.00012 0.00207 0.00000 -0.00211

Exact -0.00206 0.00000 0.00000 0.00000 0.00000 0.00000 0.00206 0.00000 -0.00206

t=2 Numerical -0.00197 -0.00017 0.00167 -0.00015 0.00003 0.00015 0.00202 0.00000 -0.00208

Exact -0.00201 0.00000 0.00201 0.00000 0.00000 0.00000 0.00201 0.00000 -0.00201

conclusions

In this paper, we apply DRBEM with IMQ-RBF for solving the nonlinear two-dimensional Burgers’ equations. The numerical results which are given in the previous section show that the proposed method is a reliable tool for Burgers’ equations. We may improve the solutions of such problems by linearization and using optimization value of shape parameter. The results have very close relation to the shape parameter ε. The choice of the shape parameter is still a pendent question. Advantage of the presented scheme is that we could use the scattered points for interpolation of nonhomogeneous, nonlinear and time dependent terms in DRM. Therewith, we would like to emphasize that, the scheme introduced in this paper can be studied for any other nonlinear PDEs.

References [1] A. R. Bahadir, A fully implicit finite-difference scheme for two-dimensional Burgers’ equations, Appl. Math. comput., 137, 131-137 (2003). [2] M. Basto, V. Semiao, F. Calheiros, Dynamics and sychronization of numerical solutions of the Burgers’ equation, Comput. Appl. Math., 231, 793-806 (2009). [3] J. Biazar, H. Aminikhah, Exact and numerical solutions for non-linear Burgers’ equation by VIM, Math. Comput. Modelling, 49, 1394-1400 (2009). [4] C.A. Brebbia, D. Nardini, Dynamic analysis in solid mechanics by an alternative boundary element procedure, Int. J. Soil Dyn. Earthquake Engrg., 2, 228-233 (1983). [5] J. M. Burger, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1, 171-199 (1948). [6] R.D. Ciskawski, C.A. Brebbia, Boundary element method in acoustics, AddisonWesley, 1991.

113

M. Sarboland and A. Aminataei

[7] E. Chino, N. Toska, Dual reciprocity boundary element analysis of time-independent Burgers’ equation, Eng. Anal. bound. Elem., 21, 261-270 (1998). [8] J. D. Cole, on a quasi-linear parabolic equation occurring in aerodynamic, Q. Appl. Math., 19, 225-236 (1951). [9] C. A. J. Fletcher, Generating exact solutions of the two-dimensional Burgers’ equation, Int. J. Numer. Methods Fluids, 3, 213-216 (1983) . [10] C. A. J. Fletcher, A comparsion of finite element and finite difference solution of the one- and two-dimensional Burgers’ equations, Int. J. Comput. Phys., 51, 159-188 (1983). [11] E. Hopf, The partial differential equation ut + uux = µuxx , Commun. Pure Appl. Math., 3, 201-230 (1950). [12] P.C. Jain, D. N. Hola, Numerical solution of coupled Burgers’ equations, Int. J. Numer. Meth. Eng., 12, 213-222 (1978). [13] K. Kakuda, N. Tosaka, The generalized boundary element approach to Burgers’ equation, Int. J. Numer. Methods Eng., 29, 245-261 (1990). [14] M. M. Mazarei, A. Aminataei, Numerical solution of Poisson’s equation using a combination of logarithmic and multiquadric radial basis function networks, J. of Applied Mathematics, doi: 10.1155/2012/286391. [15] W. M. Moslem, R. Sabry, Zakharov-Kuznetsov-Burgers equation for dust ion acoustic waves, Chaos Solitons Fractals, 36, 628-634 (2008). [16] P.W. Partridge, C.A. Brebbia, The dual reciprocity boundary element method for the Helmholtz equation, in: C.A. Brebbia, A. Choudouet- Miranda (Eds.), Proceedings of the International Boundary Elements Symposium, Computational Mechanics Publications/ Springer, Berlin, 1990, pp. 543-555. [17] M. Powell, The theory of radial basis function approximation in 1990. Oxford, Oxford: Clarendon, 1992. [18] M. M. Rashidi, E. Erfani, New analytical method for solving Burger and nonlinear heat transfer equations and comparsion with HAM, Comput. Phys. Commun., 180, 1539-1544 (2009). [19] K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical methods for physics and engineering, Cambridge University Press, 2010. [20] M. Tamsir, V.K. Srivastava, A semi-implicit finite-difference approach for twodimensional coupled Burgers’ equations, International Journal of Scientific and Engineering Research, 2, 1-6 (2011). [21] W. Toutip, The dual reciprocity boundary element method for linear and nonlinear problems, PhD thesis, University of Hertfordshire, England, 2001. [22] H. Wendland, Scattered data approximation. New York: Cambridge University Press, 2005. [23] F. W. Wubs, E. D. de Goede, An explicit-implicit method for a class of timedependent partial differential equations, Appl. Numer. Math., 9, 157-181 (1992).

114

The dual reciprocity boundary element method for two-dimensional Burgers’ equations with inverse multiquadric approximation scheme

[24] D. L. Young, C. M. Fan, S. P. Hu, S. N. Atluri, The Eulerian-Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations, Eng. Anal. bound. Elem., 32, 395-412 (2008). [25] H. Zhu, H. Shu, M. Ding, Numerical solution of two-dimensional Burgers’ equation by discrete Adomian decomposition method, Comput. and Math. with Appl., 60, 840-848 (2010).

115

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.'S 1-2, 116-123, 2014, COPYRIGHT 2014 EUDOXUS PRESS LLC

ON ASYMPTOTICALLY ALMOST AUTOMORPHIC C-SEMIGROUPS ´ EKATA ´ G. M. N’GUER

Abstract. We introduce the concepts of complete trajectory, rest point and translation invariant set in the context of C-semigroups and prove that the principal part of an asymptotically almost automorphic C-semigroup is a complete trajectory and describe some of their properties.

1. Introduction It is well-known that the concepts of C0 -semigroups and abstract dynamical systems are equivalent (see for instance [7] Theorem 2.7.2). We studied for the first time (topological and dynamical) properties of asymptotically almost automorphic C0 -semigroups in [7] Section 2.7. In this paper, we prove that some of the properties can be extended to C-semigroups, a generalization of C0 -semigroups introduced by Da Prato ([2]). C-semigroups have the advantage to be applied to many differential and integral equations that may be written as abstract Cauchy problems on a Banach space when C0 -semigroups cannot be used directly. For instance backward heat equations, Shr¨ odinger equations on Lp , with p ̸= 2, the Laplace equation, etc...See for instance [4, 9] and references therein for recent developments. In this paper, X will denote a Banach space with norm ∥ · ∥. For a given linear operator A : X → X, D(A), R(A) will represent respectively the domain and the range of A. C0 (R+ , X) will denote the space of all continuous functions f : R+ → X such that limt→∞ ∥f (t)∥ = 0.

2. Asymptotically Almost automorphic functions Definition 2.1. (S. Bochner) Let f : R 7→ X be a bounded continuous function. We say that f is almost automorphic if for every sequence of real numbers {sn }∞ n=1 , we can extract a sub∞ sequence {τn }n=1 such that: g(t) = lim f (t + τn ) n→∞

1991 Mathematics Subject Classification. 34C27; 34C99. Key words and phrases. almost automorphic, C-semigroups, complete trajectory, ω-limit set. 1

116

2

´ EKATA ´ G. M. N’GUER

is well-defined for each t ∈ R, and lim g(t − τn ) = f (t)

n→∞

for each t ∈ R. Denote by AA(X) the set of all such functions. Remark 2.2. Clearly when the convergence above is uniform in t ∈ R, f is almost periodic. Thus the class of almost automorphic functions is larger than the one of almost periodic functions. Remark 2.3. The function g is measurable, but not continuous in general. As one can see with the example below, almost automorphic functions may not be uniformly continuous. But if the function g in the above definition is continuous, then f is uniformly continuous ([8].) 1 √ ) is almost automorphic. But Example The function ψ(t) := sin( 2+cost+cos 2t since it is not uniformly continuous, it is not almost periodic. Denote by AA(X), the set of all almost automorphic functions f : R → X. With the sup norm supt∈R ∥f (t)∥, this space turns out to be a Banach space.

Definition 2.4. A bounded continuous function f : R+ → X is said to be asymptotically almost automorphic, if there exists g ∈ AA(X) and h ∈ C0 (R+ , X) such that f (t) = g(t) + h(t) for every t ≥ 0. Denote by AAA(X) the linear space of all functions f : R+ → X which are asymptotically almost automorphic. Then it turns out to be a Banach space when equipped with the norm |f | = sup ∥g(t)∥ + sup ∥h(t)∥. t≥0

t∈R

Moreover AAA(X) = AA(X) ⊕ C0 (R+ ; X). Remark 2.5. Note that AAA(X) can also be equipped with the equivalent norm ∥f ∥ := supt∈R+ ∥f (t)∥; (cf. Lemma 1.8 [3]). Moreover the range of any asymptotically almost automorphic function is relatively compact (cf. Lemma 1.9 [3]). Remark 2.6. If f ∈ AAA(X) with f = g + h then {g(t) : t ∈ R} ⊂ {f (t) : t ∈ R} (Lemma 1.7 [3]).

|f | = sup ∥g(t)∥ + sup ∥h(t)∥. t≥0

t∈R

Moreover AAA(X) = AA(X) ⊕ C0 (R ; X). +

Remark 2.7. Note that AAA(X) can also be equipped with the equivalent norm ∥f ∥ := supt∈R+ ∥f (t)∥; (cf. Lemma 1.8 [3]). Moreover the range of any asymptotically almost automorphic function is relatively compact (cf. Lemma 1.9 [3]).

117

ALMOST AUTOMORPHIC FUNCTIONS

3

3. C-semigroups Definition 3.1. Let S be a Banach space and C be an injective operator in L(X). A family of bounded linear operators S := (S(t))t≥0 is called an exponentially bounded C-semigroup if the following are satisfied: • • • •

(i) S(0) = C, (ii) S(t + s)C = S(t)S(s); ∀t, s ≥ 0, (iii) S(·)x : [0, ∞) → X is continuous for any x ∈ X, (iv) There exists M ≥ 0 and δ ∈ R such that ∥S(t)∥ ≤ M eδt for t ≥ 0.

Remark 3.2. C = I, then S is a C0 -semigroup. We define an operator A as follows: D(A) := {x ∈ X/ lim+ h→0

S(t)x − Cx ∈ R(C)} h

S(t)x − Cx , ∀x ∈ D(A)}. h h→0 This operator is called the generator of S. It is well-known that A is closed, but not necessarily densely defined. Ax := C −1 lim+

Lemma 3.3. Let C be an injective linear operator and S := (S(t))t≥0 be a Csemigroup with generator A. Then the following properties hold: • • • •

(i) S(t)S(s) = S(s)S(t), for all t, s, ≥ 0, (ii) If x ∈ D(A), then S(t)x ∈ D(A), AS(t)x = S(t)Ax, and ∫t (iii) 0 S(ξ)Axdξ = S(t)x − Cx, ∀t ≥ 0, ∫t ∫t (iv) 0 S(ξ)xdξ ∈ D(A) and A 0 S(ξ)xdξ = S(t)x − Cx, ∀x ∈ X, and t ≥ 0, • (v) A is closed and satisfies C −1 AC = A, • (vi) R(C) ⊂ D(D).

3.1. Complete trajectories. In what follows we assume that X = D(C) = R(C). Let S := (S(t))t≥0 be a C-semigroup. Then C and C −1 will commute with S(t) on X. Definition 3.4. Let x ∈ X. The set γ + (x) := {S(t)x/t ∈ R+ } is called the trajectory (or orbit) of S(t)x. Definition 3.5. A function φ : R → X is said to be a complete trajectory of S if Cφ(t) = S(t − a)φ(a) for all a ∈ R and all t ≥ a. Theorem 3.6. If S(t)x ∈ AAA(X) for some x ∈ X, then the principal term of S(t)x is a complete trajectory of S.

118

´ EKATA ´ G. M. N’GUER

4

Proof. Let S(t)x = f (t) + h(t), t ∈ R+ where f ∈ AA(X) and h ∈ C0 (R+ , X). Then there exists (nk ) ⊂ (n) = N such that g(t) :=

lim

k→inf ty

f (t + nk )

exists for each t ∈ R and lim g(t − nk ) = f (t)

k→∞

for each t ∈ R. Define Cφ(t) := S(t)x; then Cφ(0) = S(0)x = Cx. Therefore φ(0) = x. Let y = C −1 x. Fix a ∈ R and choose k large enough such that a + nk ≥ 0. If s ≥ 0, we have Cφ(a + s + nk ) = S(a + s + nk )x = S(a + s + nk )Cy = S(s)S(a + nk )y = S(s)S(a + nk )C −1 x = S(s)C −1 S(a + nk )x = S(s)φ(a + nk ). Therefore for s ≥ 0 and a + nk ≥ 0, we get f (a + s + nk ) + h((a + s + nk ) = S(a + s + nk )x = S(s)φ(a + nk ). Since lim f (a + s + nk ) = g(a + s)

k→∞

and lim h(a + s + nk ) = 0,

k→∞

then lim φ(a + s + nk ) = lim C −1 S(s)φ(a + nk ) = C −1 g(a + s).

k→∞

k→∞

It is also clear that lim φ(a + nk ) = C −1 g(a).

k→∞

Therefore in view of the continuity of S(s) we obtain lim S(s)φ(a + nk ) = S(s)C −1 g(a).

k→∞

It follows immediately that S(s)C −1 g(a) = g(a + s), ∀a ∈ R, ∀s ≥ 0. On the other hand, since lim g(t − nk ) = f (t)

k→∞

for each t ∈ R and g(a + s − nk ) = S(s)C −1 g(a − nk ), ∀a ∈ R, ∀s ≥ 0, it follows that lim g(a + s − nk ) = S(s)C −1 f (a), ∀a ∈ R, ∀s ≥ 0.

k→∞

119

ALMOST AUTOMORPHIC FUNCTIONS

5

Therefore f (a + s) = S(s)C −1 f (a), ∀a ∈ R, ∀s ≥ 0. Finally let’s put s = t − a with t ≥ 0. Then we obtain Cf (t) = S(t − a)f (a), ∀a ∈ R, ∀s ≥ 0, which proves that f is a complete trajectory.  3.2. ω-limit sets. Definition 3.7. Given x ∈ X and f the principal term of S(t)x, the set ω + (x) := {y ∈ X/∃0 ≤ tn → ∞, lim S(tn )x = Cy} n→∞

will be called the ω-limit set of S(t)x, and the set ωf+ (x) := {y ∈ X/∃0 ≤ tn → ∞, lim f (tn ) = y} n→∞

is the ω-limit set of f . We now describe some topological properties of the above ω-limit sets. Theorem 3.8. ω + (x) ̸= ∅ Proof. Since f ∈ AA(X), there exists (nk ) ⊂ (n) = N such that lim f (nk ) = g(0).

k→∞

But we have lim S(nk )x = lim f (nk ).

k→∞

k→∞

Therefore lim S(nk )x = g(0).

k→∞ +

Now take ξ = C −1 g(0). Then ξ ∈ ω (x). This completes the proof.  Theorem 3.9. ω + (x) = ωf+ (x) Proof. This follows immediately from the fact that lim S(t)x = lim f (t).

t→∞

t→∞

 Let’s now recall this definition Definition 3.10. A set A ⊂ X is said to be invariant under S if S(t)y ∈ CA for every y ∈ A and t ∈ R+ . We can prove the following

120

´ EKATA ´ G. M. N’GUER

6

Theorem 3.11. ω + (x) is invariant under S. Proof. Let y ∈ ω + (x). Then there exists 0 ≤ tn → ∞ such that limn→∞ S(tn )x = Cy. Fix t ∈ R+ and consider sn := t + tn , n = 1, 2, ... Obviously limn→∞ sn = ∞. Since S(sn )Cx = S(t)S(tn )x, n = 1, 2, ..., in using continuity of S(t), we get lim S(sn )Cx = lim S(t)S(tk )x = S(t)Cy = CS(t)y.

n→∞

n→∞

which proves that S(t)y ∈ Cω + (x). 

The proof is complete. Theorem 3.12. ω + (x) is a closed subset of X.

Proof. It suffices to prove that ω + (x) ⊂ ω + (x). Let y ∈ ω + (x). Then there exists a sequence ym ∈ ω + (x) such that limm→∞ ym = y. Now for each ym , there exists 0 ≤ tm,n → ∞ such that lim S(tm,n )x = Cym .

n→∞

Now define recursively a sequence tk,nk as follows. Choose t1,n1 > 1 such that ∥Cy1 − S(t1,n1 )x∥ < 12 , t2,n2 > max(2, t1,n1 ) such that ∥Cy2 − S(t2,n2 )x∥
max(k, tk−1,nk−1 ) such that ∥Cyk − S(tk,nk )x∥
0. ¯h Ω

(24)

Denote by w = max |ε2N1 j | and let the ω ¯ ij – be the solution of 4 Γh

Λ′h [¯ ωij ] = 0 in Ωh , ω ¯ N1 j = w in Γ4h , ω ¯ ij = 0 in σh , (1)

lh [¯ ω0j ] = 0 in Γ2h . Lemma 1 implies that ¯ h, |ε2ij | ≤ ω ¯ ij in Ω ω ¯ ij ≤ τi w, 0 < τi < 1 in Ωh .

(25) (26)

On the other hand lh [ε2N1 j ] = −lh [ε1N1 j ] + O(h2 ) in Γ4h . Hence, respectively to (25), (26) we have l − xk 2 xk+1 − l 2 l − xk 1 xk+1 − l 1 αj |ε2N1 j | ≤ |εk+1j | + |εkj | + |εk+1j | + |εkj | + C2 h2 h1 h1 h1 h1 or αj w ≤ τ w + C1 h + C2 h, where τ = max{τk+1 , τk }. Hence we have C3 h w≤ ≤ C4 h, (27) αj − κi where C3 C4 = . min(αj − τ ) j

Then from (25)-(27) we have max |ε2ij | ≤ C5 h, C5 = max τi C4 . ¯h Ω

i

(28)

Based on (20), (24) and (28) we have max |εij | ≤ C6 h, ¯h Ω

(29)

where C6 = C1 + C5 . Theorem 1 is proved. Below we show that by imposing additional conditions on the function β(y), δ(y) the order of accuracy with in h2 can be improved.

208

THE NUMERICAL SOLUTION OF NON-LINEAR NON-LOCAL PROBLEMS

5

As can be seen from the above, it is sufficient to increase the order of approxi(1) mation of the operator lh . Assume, that h1 = wh2 (0 < w ≤ 1) and β(y), δ(y) satisfy one of the following conditions |β(y)| < w, (30) ′ |β(y)| ≥ w, δ (y) ≤ 0, (31) ′ |β(y)| ≤ −w, δ (y) ≥ 0. (32) Consider the operators U1j − U0j U0j+1 U0j−1 (1) l1h [U0j ] ≡ + βj + δj U0j , (33) h1 2h2 U1j − U0j U0j+1 U0j (1) l2h [U0j ] ≡ + βj + δj U0j , (34) h1 h2 U0j − U0j−1 U1j − U0j (1) + βj + δj U0j . (35) l3h [U0j ] ≡ h1 h2 Let p p ∂ u0j ∂ u0j (p) , ≤ Mj , (p ≥ 1). ∂xp ∂y p (0,j) (0,j) Taking into account (3), (7), (33) and applying the Taylor formula is easy to see that ˜(1) (36) l1h u0j − (l(1) u)(0,j) ≤ c(1) h22 , where

h1 u0j+1 − 2u0j + u0j−1 (1) ˜l(1) u0j ∼ − = l1h u0j + 1h 2 h22 h1 − f {0, yj , u0j , Dh1 x [u0j ], Dh2 y [u0j ]} , 2 ui+1j−uij uij+1 − uij Dh1 x [uij ] = , Dh2 y [uij ] = , h1 h2 { } 2(w2 + w + β) + h1 M (3) w2 M (2) C (1) = max Mj + Mj . j 12 4 Indeed, from (33) we have: h1 ∂ 2 u (1) (1) (1) l1h u0j = (l u)(0,j) + + Rj , 2 ∂x2 (0,j) [ ] h21 ∂ 3 u h22 ∂ 3 u ∂ 3 u (1) Rj = + + βj . 6 ∂x3 (ξ(1) ,j) 12 ∂y 3 (0,η(1) ) ∂y 3 (0,η(2) ) 0

j

j

From (3) we have: ( ) u0j+1 − 2u0j + u0j−1 ∂ 2 u = − ], D − + f 0, y , u , D [u ◦ [u0j ] 0j j 0j h x 1 2 h2 y ∂x2 (0,j) h2 ] [ h2 ∂ 3 u ∂ 3 u ∂ 2 u h1 ′ − − + f (0, y , u , p , q ) + j 0j j j p 6 ∂y 3 (0,η(3) ) ∂y 3 (0,η(4) ) ∂x2 (ξ(2) ,j ) 2 0 j j   3 3 2 ∂ u ∂ u (  h2 . +fq′ (0, yj , u0j , pj , qj )  3 ( )+ ) 3 ∂y 0,η(3) ∂y 0,η(4) 12 j

209

j

6

A.Y.ALIYEV (1)

Taking into account this l1h uij , we get: h1 u0j+1 −2u0j +u0j−1 + 2 h22 (1) h1 ′ ˜ + 2 f (0, yj , u0j , Dh1 x [u0j ], D y◦ [u0j ]) + Rj , h (1)

l1h u0j = (l(1) u)(0,j) −

2

where ˜ (1) R j

  3 3 2 ∂ u ∂ u ∂ u h  βj − + = + 2 6 ∂x3 (ξ(1) ,j) 12 ∂y 3 (0,η(1) ) ∂y 3 (0,η(2) ) h21

3

i





j

j

h1 h2  ∂ 3 u ∂ 3 u h21 ′ ∂ 2 u  − − + fp (0, yj , u0j , pj , qj ) + 12 ∂y 3 (0,η(3) ) ∂y 3 (0,η(4) ) 4 ∂x2 (ξ(2) ,j ) 0 j j   3 3 h1 h22 ′ ∂ u ∂ u . + f (0, yj , u0j , pj , qj )  3 ( + 24 q ∂y 0,η(3) ) ∂y 3 (0,η(4) ) j

j

Hence we find that ˜l(1) u0j = (l(1) u)(0,j) + R ˜ (1) , j 1h consequently,

˜(1) ˜ (1) l1h u0j − (l(1) u)(0,j) ≤ R j .

And this implies (36). Now we prove that

˜(1) l2h u0j − (l(1) u)(0,j) ≤ C (2) h22 ,

(37)

where ˜l(1) u0j ≡ l(1) u0j + βj h2 − h1 Dh h xy [u0j ] + δj (βj h2 − h1 )Dh y [u0j ]+ 1 2 2 2h 2h 2βj βj +

δj′ γj′ h1 (βj h2 − h1 )u0j − (βj h2 − h1 ) − f (0, yj , u0j , Dh1 x [u0j ], Dh2 y [u0j ]), 2βj 2βj 2 {[ ] 2 βj + w (βj − w)(1 − w) h1 M (3) (2) C = max + + Mi + j 6 4βj 12 [ ] } δj + β ′ (βj − w) w2 M j (2) + + Mj , 4βj 4

Dh1 h2 xy [u0j ] = Dh1 x {Dh2 y [u0j ]} . Suppose that β(y) ̸= 0. Then from (7) we have: δ(y) + βj′ ∂u(0, y) γ ′ (y) ∂ 2 u(0, u) 1 ∂ 2 u(0, y) δ ′ (y) = − − u(0, y) − + . ∂y 2 β(y) ∂x∂y β(y) β(y) ∂y β(y) Obviously (1)

l2h u0j = (l3 u)(0,j) + where (2) Rj

h1 ∂ 2 u h2 ∂ 2 u (2) + β + Rj , j 2 ∂x2 (0,j) 2 ∂y 2 (0,j)

h21 ∂ 3 u h22 ∂ 3 u = . + βj 6 ∂x3 (ξ(1) ,j) 6 ∂y 3 (0,η(1) ) 0

210

j

(38)

THE NUMERICAL SOLUTION OF NON-LINEAR NON-LOCAL PROBLEMS

7

From (3) we get: ( ) ∂ 2 u ∂u ∂ 2 u ∂u = − 2 + f 0, yj , u0j , , . ∂x2 (0,j) ∂y (0,j) ∂x (0,j) ∂y (0,j) ∂ 2 u 1 (1) + l2h u0j = (l(1) u)(0,j) + (βj h2 − h1 ) 2 ∂y 2 (0,j) ( ) h1 ∂u ∂u (2) + f 0, yj , u0j , , + Rj . 2 ∂x (0,j) ∂y (0,j)

Then

Taking into account (38) 1 (1) l2h u0j = (l(1) u)(0,j) + (βj h2 − h1 )× 2 ] [ ′ 2 δj + βj′ ∂u δj 1 ∂ u γj + × − − u0j − + βj ∂x∂y (0,j) βj βj ∂y (0,j) βj ( ) h1 ∂u ∂u (2) + f 0, yj , u0j , , + Rj = 2 ∂x (0,j) ∂y (0,j) ∂ 2 u 1 (1) (βj h2 − h1 ) = (l u)(0,j) − − 2βj ∂x∂y (0,j) γj′ δj′ ∂u δ j + βj (βj h2 − h1 ) (βj h2 − h1 )u0j + (βj h2 − h1 )+ − − 2βj ∂y (0,j) 2βj 2βj ( ) h1 ∂u ∂u (2) + f 0, yj , u0j , , + Rj = (l(1) u)(0,j) − 2 ∂x (0,j) ∂y (0,j) δj + βj′ βj h 2 − h 1 Dh1 h2 xy [u0j ] − − (βj h2 − h1 )Dh2 y [u0j ]− 2βj βj δj′ γj′ − (βj h2 − h1 )u0j + (βj h2 − h1 )+ 2βj 2βj h1 ˜ (2) , + f (0, yj , u0j , Dh1 x [u0j ], Dh2 y [u0j ]) + R j 2 where [ 3 ] ∂ u ∂3u ˜ (2) = R(2) − βj h2 − h1 R h − h 1 2 − j j 4βj ∂x2 ∂y ∂x∂y 2 δj + βj′ ∂ 2 u ∂ 2 u h21 ′ − (βj h2 − h1 )h2 f (0, y , u , p , q ) + + j 0j j j 4βj ∂y 2 (0,η(2) ) 4 p ∂x2 (ξ(2) ,j ) 0 j   3 3 ∂ u ∂ u h1 h22 ′ . f (0, yj , u0j , pj , qj )  3 ( + )+ (2) 24 q ∂y ∂y 3 ( (3) ) −

0,ηj

Then This implies (37). Finally, we prove that

0,ηj

˜(1) ˜ (2) l2h u0j − (l3 u)(0,j) ≤ R j . ˜(1) l3h u0j − (l3 u)(0,j) ≤ C (2) h22 ,

211

(39)

8

A.Y.ALIYEV

( w 2 + β

) |w + β| (w + 1) h M 1 C (3) = + + M3 + 6 2 |β| 12 ( ) |w + β| |δ + β ′ | w2 M + + M2 , 4 |β| 4 ˜l(1) u0j ≡ l(1) u0j − βj h2 − h1 Dh h xy [u0j ] − δj (βj h2 + h1 )Dh y [u0j ]− 1 2 2 3h 3h 2βj βj δj′ γj′ h1 − (βj h2 + h1 )u0j + (βj h2 − h1 ) + f (0, yj , u0j , Dh1 x [u0j ], Dh2 y [u0j ]), 2βj 2βj 2 Dh1 h2 xy [u0j ] = Dh1 x {Dh2 y [u0j ]} . Indeed, h1 + h2 βj ∂ 2 u (1) l3h u0j = (l(1) u)(0,j) − − 2 ∂y 2 (0,j) ( ) ∂u h1 ∂u (3) − f 0, yj , u0j , , + Rj , 2 ∂x (0,j) ∂y (0,j) h22 ∂ 3 u h2 ∂ 3 u (3) + β Rj = 1 . j 6 ∂x3 (ξ(1) ,j) 6 ∂y 3 (0,η(1) )

where

0

j

Taking into account (38) h1 + h2 βj ∂ 2 u + 2βj ∂x∂y h1 + h2 βj δj′ h1 + h2 βj δj + βj′ ∂u + u0j + − 2 βj 2 βj ∂y (0,j) ( ) h1 ∂u h1 + h2 βj γj′ ∂u (3) − f 0, yj , u0j , − , + Rj , 2 βj 2 ∂x (0,j) ∂y (0,j) (1)

l3h u0j = (l(1) u)(0,j) +

(1)

h1 + h2 βj (h1 + h2 βj )(δj + β ′ ) Dh1 h2 xy [u0j ] + Dh2 y [u0j ]+ 2βj 2βj (h1 + h2 βj )γj′ h1 ˜ (3) , − − f (0, yj , u0j , Dh1 x [u0j ], Dh2 y [u0j ]) + R j 2βj 2

l3h u0j = (l(1) u)(0,j) +

(h1 + h2 βj )δj′ u0j 2βj where [ 3 ] ∂ u ∂3u ˜ (3) = R(3) + h1 + h2 βj R h + h 1 2 + j j 2βj ∂x2 ∂y ∂x∂y 2 (h1 + h2 βj )(δj + βj′ ) 2 ∂ 2 u h21 ′ ∂ 2 u + h − fp (0, yj , u0j , pj , qj ) − 4βj ∂y 2 (0,η(2) ) 4 ∂x2 (ξ(2) ,j ) 0 j   3 3 h1 h22 ′ ∂ u ∂ u . − f (0, yj , u0j , pj , qj )  3 ( + 24 q ∂y 0,η(3) ) ∂y 3 (0,η(4) ) +

j

Consequently,

j

˜(1) ˜ (3) l3h u0j − (l(1) u)(0,j) ≤ R j ,

which was required to prove. We now state the difference problem corresponding to the problem (3)-(7).

212

THE NUMERICAL SOLUTION OF NON-LINEAR NON-LOCAL PROBLEMS

9

(k) ¯h It is required to find a discrete function Uij (k = 1, 2, 3) determined in Ω satisfying the properties (11) - (13), and one of the following conditions

˜l(1) U0j = γj (j = 1, N2 − 1, k = 1, 2, 3) kh

(40)

respectively, when one of the conditions (30), (31) and (32) is satisfied. The solution of the difference scheme (11) - (13), with one of the conditions (40) will be taken ¯ h. as an approximate solution of the problem (3) - (7) at the points Ω Consider the following linear difference operators:  ˜ h [Uij ],   L (k) lh [UN1 j ], Λh [Uij ] =   l(1) [U ], (k = 1, 2, 3), 0j

kh

where ˜ h [Uij ] ≡ ∆h [Uij ] + ξij D L h

◦

1

x

[Uij ] + ηij D

◦

h2 y

[Uij ] − µij Uij ,

h1 U0j+1 − 2U0j + U0j−1 (1) (1) l1h [U0j ] ≡ l1h [U0j ] + − 2 h22 [ ] h1 − ξ0j Dh1 x [U0j ] + η0j D ◦ [U0j ] − µ0j U0j , h2 y 2 β h (1) j 2 − h1 (1) l2h [U0j ] ≡ l2h [U0j ] + Dh1 h2 xy [U0j ]+ 2βj δj′ δj + (βj h2 − h1 )Dh2 y [U0j ] + (βj h2 − h1 )U0j − βj 2βj ] h1 [ ξ0j Dh1 x [U0j ] + η0j Dh2 y [U0j ] − µ0j U0j , − 2 βj h2 − h1 (1) (1) l3h [U0j ] ≡ l3h [U0j ] − Dh1 h2 xy [U0j ]− 2βj δj′ δj − (βj h2 + h1 )Dh2 y [U0j ] + (βj h2 + h1 )U0j + βj 2βj ] h1 [ + ξ0j Dh1 x [U0j ] + η0j Dh2 y [U0j ] − µ0j U0j . 2 We assume that if (30) is satisfied, then M h2 < 2(1 − sup |β(x)|),

(41)

and if the (31), (32) are satisfied, then M h2 < 1, where

(

  M = max

(42)

M ,  1 + (sup |β|)−1

M + sup

|β|+1 |β|

) |β ′ + δ| 

inf |β| + (sup |β|)−1



.

¯ h , that satisfies the inequality Lemma 2. Let V ̸= const be a function defined in Ω (k) (k) Λh [Vij ] ≥ 0 (Λh [Vij ] ≤ 0) k = 1, 2, 3. Then V may take the greatest positive (least negative) value only at the points σh . Proof. It’s obvious that (1)

(1)

(1)

(1)

(1)

l1h [Uij ] ≡ A1j U1j + A2j U0j−1 + A3j U0j+1 − A0j U0j ,

213

10

A.Y.ALIYEV (1)

(2)

(2)

(2)

(2)

(1)

(3)

(3)

(3)

(3)

l2h [Uij ] ≡ A1j U1j + A2j U1j+1 + A3j U0j+1 − A0j U0j , l3h [Uij ] ≡ A1j U1j + A2j U0j−1 + A3j U0j−1 − A0j U0j , ( ) ξj h1 h1 1 1 h1 (1) − δj + + µj , A1j = 1 − ξj , = 2+ h2 h1 2 2 h1 2 ( ) ( ) h1 h1 h1 h1 1 1 (1) (1) − βj − ηj , A3j = + βj + ηj , A2j = 2h2 h2 2 2h2 h2 2 ( ) 1 1 βj h 2 − h 1 δj (2) A0j = βj + − δj − + (βj h2 − h1 )− h1 h2 2βj h2 h1 h2 βj δj′ ξj h1 h1 − (βj h2 − h1 ) − − ηj − µj , 2βj 2 2h2 2 βj h 2 − h 1 ξj βj βj h 2 − h 1 (2) (2) A1j = − − , A2j = , h1 2βj h2 h1 2 2βj h2 h1 βj h 2 − h 1 δj h1 βj (2) − + (βj h2 − h1 ) − ηj , A3j = h2 2βj h2 h1 h2 βj 2h2 1 βj βj h 2 − h 1 δj (βj h2 + h1 ) (3) A0j = − − δj − + + h1 h2 2βj h1 h2 h2 βj δj′ ξj h1 h1 + (βj h2 + h1 ) + − ηj + µj , 2βj 2 2h2 2 1 β h − h ξ j 2 1 j (3) A1j = − + , h1 2βj h1 h2 2 βj βj h2 − h1 δj h1 (3) A2j = − − + (βj h2 + h1 ) − ηj , h2 2βj h1 h2 h 2 βj 2h2 βj h2 − h1 (3) A3j = . 2βj h1 h2 All these coefficients are positive and satisfy the following conditions: h1 (1) (1) (1) (1) A0j − A1j − A2j − A3j = −δj + µj ≥ 0, 2 δj′ h1 (2) (2) (2) (2) A0j − A1j − A2j − A3j = −δj − µj − (βj h2 − h1 ) ≥ 0, 2 2βj δj′ h1 (3) (3) (3) (3) A0j − A1j − A2j − A3j = −δj + (βj h2 + h1 ) + µj ≥ 0. 2βj 2 Taking into account these properties of the coefficients, applying Lemma 1 we obtain Lemma 2. Corollary. Lemma 2 implies that the solution of (11)-(13) (40) is unique. Theorem 2. Let u the exact solution of the problem (3)-(7) limited the fourth derivatives and continued in the third derivative Ω. Then the error εij = uij − Uij , where Uij - the approximate solution of (11)-(13), (40), the estimate ε = O(h2 ). Proof. With the help of Taylor’s formula for the error εij = uij − Uij we have:  ˜ Lh [εij ] = O(h2 ) in Ωh ,    l [ε ] = O(h2 ) in Γ4 , h N1j h (43) εij = 0 in σh ,    (1) lkh [ε0j ] = O(h2 ), k = 1, 2, 3 in Γ2h . where

(1) A0j

214

THE NUMERICAL SOLUTION OF NON-LINEAR NON-LOCAL PROBLEMS

11

As in the proof of Theorem 1, we represent the solution of the system (43) of the form εij = ε1ij + ε2ij , where  ˜ h [ε1 ] = O(h2 ) in Ωh ,  L ij    ε1 = 0 in Γ4h , N1 j (44) ε1ij = 0 in σh ,     l(1) [ε1 ] = O(h2 ), k = 1, 2, 3 in Γ2 , kh ij h  2 ˜  Lh [εij ] = 0 in Ωh ,    l [ε2 N j] = − l [ε1 ] + O(h2 ) in Γ4 , h 1 h N1 j h (45) 2 ε = 0 in σh ,  ij   (1)  l [ε2 ] = 0, k = 1, 2, 3 in Γ2 . kh h 1 0j An estimate of max εh ≤ c7 h2 for the solutions system of (44) is obtained on the Ωh

basis of Lemma 2, due to scheme of proof of Theorem 1 by the majorant function 1 g(x, y) = (ev0 a − ev0 x ), k and the parameters k and ν0 are selected as follows: { } k = µ0 v0 , µ0 = min α0 , M β 0 , { sup |β| if |β| < 1, α0 = 1−θ if |β| ≥ 1, { 2 sup |β| if |β| < 1, β0 = 1−θ if |β| ≥ 1, ( ) 2 2M 2δ − δ v0 = arcth , 2 δ { 1 − sup |β| if |β| < 1, ¯ δ= θ if |β| ≥ 1. An estimate of max ε2h ≤ c8 h2 for the solutions of the system (45) is obtained Ωh

by the same way as the estimate of the solution of system (22) in the proof of Theorem 1. Theorem 2 is proved. References [1] N.I.Ionkin, On finding the numerical solution of a non-classical problem, Herald of the Moscow University, Computational Mathematics and Cybernetics, 1, 64-68 (1979) (Russian). [2] V.L. Makarov, D.T. Kuliev, The method of lines for quasi-linear parabolic equation with a non-classical boundary condition, Ukrainian Mathematical Journal, 37 (1), 42-48 (1985) (Russian). [3] R.J. Ciegis, The study of two-dimensional heat conduction problem with non-local condition, Differential equations and their applications, Vilnius, IMC Academy Lit.SSR, 35, 74-82 (1984) (Russian). [4] M.P. Sapagovas, Numerical methods for two-dimensional problem with non-local condition, J. Differential Equations, 20(7), 1258-1266 (1984) (Russian). [5] D.G.Gordeziani , On a class of non-local boundary value problems in the theory of elasticity and the theory of shells, Proceedings of the theory and numerical methods for the calculation of plates and shells. Proceedings of the Seminar, Tblisi, 106-127 (1984) (Russian).

215

12

A.Y.ALIYEV

[6] A.Y.Aliyev, The applicability of the grid method to solve a non-local problem for elliptic equations, Thematic collection of scientific papers ”Approximate methods for solving operator equations”. Publishing House of the Baku State University, Baku, 3-9 (1991) (Russian). [7] A.Y.Aliyev, A.A.Dosiyev, An approximation method for solutions of non-local problems for the Laplace equation, Proceedings of the International Science and Technology. Conference ”Actual problems of basic sciences,” the Soviet Union, ed. Moscow State Technical University, Moscow, 2, 115-117 (1991) (Russian). [8] A.Y.Aliyev, G.Y.Mehdiyeva, Numerical solution one non-local problem, Problems of cybernetics and informatics, Proceedings IV International conference, Baku, 3, 115-118 (2010). [9] A.Y.Aliyev, G.Y.Mehdiyeva, Numerical solution of a non-local boundary value problem for partial differential equations, Mathematical science and applications: Abstracts book International conference, Abu Dhabi, 7 (2012). [10] A.Y.Aliyev, On numerical solution non-local boundary values problems for elliptic equations, Ph. D. thesis, Baku, 1992 (Russian).

(A.Y. Aliyev) Baku State University, Baku, Azerbaijan E-mail address : aydin [email protected]

216

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.'S 3-4, 217-228, 2014, COPYRIGHT 2014 EUDOXUS PRESS LLC

SOME GENERATING RELATIONS FOR GENERALIZED EXTENDED HYPERGEOMETRIC FUNCTIONS INVOLVING GENERALIZED FRACTIONAL DERIVATIVE OPERATOR RAKESH K.PARMAR

Abstract. Very recently, Lee et al.[10] have established generalization of the extended beta function, hypergeometric function and confluent hypergeometric function introduced by earlier researchers in this area. The aim of this research paper is to obtain some linear and bilinear generating relations for generalized extended Gauss, Appell and Lauricella hypergeometric functions in one, two and three variables by defining the further generalization of the extended fractional derivative operator. Some properties and Mellin transform of the generalized extended fractional derivative operator are also obtained.

1. Introduction Several extensions of well known special functions have been obtained recently by several authors (see, for example [1, 2, 3, 4, 5] ). Especially, Chaudhry et al.[4] introduced the following extension of classical Beta function : Z1 Bp (x, y) = B(x, y; p) =

 tx−1 (1 − t)y−1 exp −

 p dt t(1 − t)

0

( 0, 0, 0)

(1)

and proved that this extension has connection with Macdonald, error and Whittaker’s functions. It is obvious, that B0 (x, y) = B(x, y; 0) = B(x, y) More recently, Chaudhry et al.[6] considered the extension of Gauss hypergeometric functions as follows: Fp (a, b; c; z) =

∞ X B(b + n, c − b; p) zn (a)n B(b, c − b) n! n=0

(p ≥ 0, | z |< 1, 0)

(2)

and (α)k , denotes Pochhammer’s symbol or ascending factorial,defined by  α(α + 1)...(α + k − 1) , k ≥ 1 Γ(α+k) (α)k = Γ(α) = 1 , k = 0, α 6= 0 2010 Mathematics Subject Classification. Primary 26A33, 33C05; Secondary 33C20. Key words and phrases. Gamma and Beta functions; Eulerian integrals;Gauss’s hypergeometric function, generating functions,Appell–Lauricella hypergeometric function, fractional derivative operator, Mellin transform. 1

217

2

RAKESH K.PARMAR

They obtained the corresponding Euler type integral representation : 1 Fp (a, b; c; z) = B(b, c − b)

Z1 t

b−1

c−b−1

(1 − t)

−a

(1 − zt)

 exp −

 p dt t(1 − t)

0

(p ≥ 0 and | arg(1 − z) |< π, 0)

(3)

Clearly, F0 (a, b; c; z) = 2 F1 (a, b; c; z). Very recently, Lee et al.[10] introduced further generalization of extended Beta function and extended Gauss’s hypergeometric function as: Z1 Bp;k (x, y) = B(x, y; p; k) =

 tx−1 (1 − t)y−1 exp −

 p dt tk (1 − t)k

0

( 0, 0, 0, 0)

(4)

∞ X Bp;k (b + n, c − b) zn (a)n B(b, c − b) n! n=0

Fp (a, b; c; z; k) = Fp;k (a, b; c; z) =

(p ≥ 0, 0; | z |< 1; 0)

(5)

They called these functions as generalized extended beta function (GEBF) and generalized extended hypergeometric functions (GEGHF) and obtained the Euler type integral representation :

1 Fp;k (a, b; c; z) = B(b, c − b)

Z1

b−1

t

c−b−1

(1 − t)

−a

(1 − zt)

 exp −

 p dt tk (1 − t)k

0

(p > 0, p = 0, 0and | arg(1 − z) |< π, 0)

(6)

Clearly, it is seen that for k = 1, it gives the Chaudhry et al.[6] results and for p = 0 , it reduces to original functions. They also obtained the various integral representations, some properties, differentiation formulas,transformations formulas, recurrence relations , summation formulas,Beta distribution and Mellin transforms of these functions. Very recently, using the well-known Riemann-Liouville integral representation for fractional derivative Zz 1 µ Dz f (z) = f (t)(z − t)−µ−1 dt (7) Γ(−µ) 0

which is valid for Re(µ) < 0, where the integration path is a line from 0 to z in the complex t− plane and where the case m − 1 < Re(µ) < m(m = 1, 2, 3, ...) yields   Zz  m m  d d 1 Dzµ f (z) = m Dzµ−m f (z) = m f (t)(z − t)−µ+m−1 dt  dz dz  Γ(−µ + m) 0

218

SOME GENERATING RELATIONS FOR GENERALIZED EXTENDED HYPERGEOMETRIC FUNCTIONS3

¨ Ozarslan and Ozergin [9] defined the following extended Riemann-Liouville fractional derivative by adding a new parameter. Explicitly, they considered   Zz 1 −pz 2 µ,p Dz f (z) = f (t)(z − t)−µ−1 exp dt (8) Γ(−µ) t(z − t) 0

with a, (1.1) Ia+ ϕ (x) = Γ(α) (x − t)1−α a

are well studied both in weighted H¨older spaces or in generalized H¨older spaces. A non-weighted statement on action of the fractional integral operator from H0λ into H0λ+α is due to Hardy and Littlewood ([1], see [11], Theorems 3.1 and α 3.2), and it is known that the operator Ia+ with 0 < α < 1 establishes an λ isomorphism between the H¨older spaces H0 ([a, b]) and H0λ+α ([a, b]) of function vanishing at the point x = a, if λ + α < 1. The weighted results with power weights were obtained in [9], [10](see their presentation in [11], Theorems 3.3, 3.4 and 13.13). For weighted generalized H¨older spaces H0ω (ρ) of function ϕ with a given dominant of continuity modulus of ρϕ, mapping properties in the case of power weight were studied in [7], [8], [12] (see also their presentation in [11], Section 13.6). Different proofs were suggested in [3], [4], where the case of 1

272

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

complex fractional orders was also considered, the shortest proof being given in [3]. The case of weights more general than power ones, including in particular power-logarithmic type weights, in the spaces H0ω (ρ) was considered in [13], where operators more general than just fractional integrals were treated. We refer also to paper [2] where the mapping properties of fractional integration operators were reconsidered in terms of the Matuszewska-Orlich indices of the characteristic ω defining the generalized H¨older space H ω . Finally, we mention also the papers [5], [6], where fractional integrals were studied in spaces of Nikolsky type. In the multidimensional case, statements on mapping properties in generalized H¨older spaces are known [14] for the Riesz fractional integrals (see also this presentation in [11], Theorem 25.5). Mixed Riemann-Liouville fractional integrals of order (α, β): ³

´

α,β I0+,0+ ϕ

1 (x, y) = Γ(α)Γ(β)

Zx Zy 0

0

ϕ(t, τ ) dtdτ, (x − t)1−α (y − τ )1−β

(1.2)

and mixed fractional differentiation operators in the form Marchaud of order (α, β):  Zy ´ ³ f (x, β f (x, y) − f (x, τ ) 1 y) α,β  + α dτ + Da+,c+ f (x, y) = Γ(1 − α)Γ(1 − β) xα y β x (y − τ )1+β 0

µ +

α yβ

Zx 0

f (x, y) − f (t, y) dt + αβ (x − t)1+α

Zx 0

Zy 0

1,1 ∆ x−t,y−τ

(x −

¶ f

t)1+α (y

−

 (t, τ )

τ )1+β

 dtdτ  ,

(1.3)

where x > 0, y > 0,were not studied either in the usual H¨older spaces, or in the H¨older spaces defined by mixed differences. Meanwhile, there arise ”points of interest” related to the investigation of the above mixed differences of fractional integrals (1.2) and differentials (1.3). For operators (1.2) and (1.3) in H¨older spaces of mixed order there arise some questions to be answered in relation to the usage of these or Those differences in the definition of H¨older spaces. Such mapping properties in H¨older spaces of mixed order were not studied. This paper is aimed to fill in this gap. We deal with non-weighted spaces. We consider the operators (1.2) and (1.3) in the rectangle Q = {(x, y) : 0 < x < a, 0 < y < d} .

2. Preliminaries 2.1. Notation and some properties of H¨ older spaces

2

273

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

For a continuous function ϕ(x, y) on R2 we introduce the notation µ ¶ µ ¶ 1,0 0,1 ∆ h ϕ (x, y) = ϕ(x + h, y) − ϕ(x, y), ∆ η ϕ (x, y) = ϕ(x, y + η) − ϕ(x, y), µ

1,1 ∆ h,η

¶ ϕ (x, y) = ϕ(x + h, y + η) − ϕ(x + h, y) − ϕ(x, y + η) + ϕ(x, y),

so that

µ ϕ(x + h, y + η) =

1,1 ∆ h,η

µ

¶

+

0,1 ∆η

¶ µ ¶ 1,0 ϕ (x, y) + ∆ h ϕ (x, y)+

ϕ (x, y) + ϕ(x, y).

(2.1)

everywhere in the sequel by C1 , C2 , C3 , C etc we denote positive constants which may different values in different occurrences, and even in the same line. We introduce two types of H¨older spaces by the following definitions. Definition 2.1. I. Let λ, γ ∈ (0, 1]. We say that ϕ ∈ H λ,γ (Q), if |ϕ(x1 , y1 ) − ϕ(x2 , y2 )| ≤ C1 |x1 − x2 |λ + |y1 − y2 |γ for all (x1 , y1 ), (x2 , y2 ) ∈ Q. Condition separate conditions ¯µ ¯ ¶ ¯ 1,0 ¯ ¯ ∆ h ϕ (x, y)¯ ≤ C1 |h|λ , ¯ ¯

(2.2)

(2.2) is equivalent to the couple of the ¯µ ¯ ¶ ¯ 0,1 ¯ ¯ ∆ η ϕ (x, y)¯ ≤ C2 |η|γ ¯ ¯

uniform with respect to another variable. By H0λ,γ (Q) we define a subspace of functions f ∈ H λ,γ (Q), vanishing at the boundaries x = 0 and y = 0 of Q. II. Let λ = 0 and/or γ = 0. We put H 0,0 (Q) = L∞ (Q) and ¯µ ¯ ¶ ¯ 1,0 ¯ H λ,0 (Q) = {ϕ ∈ L∞ (Q) : ¯¯ ∆ h ϕ (x, y)¯¯ ≤ C1 |h|λ }, λ ∈ (0, 1], H

0,γ

¯µ ¯ ¶ ¯ 0,1 ¯ ¯ (Q) = {ϕ ∈ L (Q) : ¯ ∆ h ϕ (x, y)¯¯ ≤ C2 |h|γ }, ∞

γ ∈ (0, 1].

˜ λ,γ (Q), where λ, γ ∈ (0, 1], if Definition 2.2. We say that ϕ(x, y) ∈ H ¯µ ¯ ¶ ¯ 1,1 ¯ ϕ ∈ H λ,γ (Q) and ¯¯ ∆ h,η ϕ (x, y)¯¯ ≤ C3 |h|λ |η|γ . (2.3) we say that ϕ(x, y) ∈

˜ λ,γ (Q), H 0

˜ λ,γ

if ϕ(x, y) ∈ H

¯ ¯ ¯ (Q) and ϕ(x, y)¯ ¯

= 0. x=0,y=0

These spaces become Banach spaces under the standard definition of the norms: ¯µ ¯ ¯µ ¯ ¶ ¶ ¯ 0,1 ¯ ¯ 1,0 ¯ ¯ ∆ η ϕ (x, y)¯ ¯ ∆ h ϕ (x, y)¯ ° ° ° ° ¯ ¯ ¯ ¯ ° ° ° ° °ϕ° ° ° + sup , sup + sup sup ° ° λ,γ := °ϕ° λ γ |h| |η| x∈[0,b] y,y+η∈[0,d] H C(Q) x,x+h∈[0,b] y∈[0,d] 3

274

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

° ° ° ° ° ° ° ° °ϕ° ° ° ° ° ˜ λ,γ := °ϕ°

+

H λ,γ

H

note that ϕ∈H

λ,γ

sup

sup

¯µ ¯ ¶ ¯ 1,1 ¯ ¯ ∆ h,η ϕ (x, y)¯ ¯ ¯

x,x+h∈[0,b] y,y+η∈[0,d]

|h|λ |η|γ

.

¯µ ¯ ¶ ¯ 1,1 ¯ ¯ ⇒ ¯ ∆ h,η ϕ (x, y)¯¯ ≤ Cθ |h|θλ |η|(1−θ)γ

(2.4)

for any θ ∈ [0, 1], where Cθ = 2C1θ C21−θ , so that \ ˜ λ,γ (Q) ,→ H λ,γ (Q) ,→ ˜ θλ,(1−θ)γ (Q), H H

(2.5)

0≤θ≤1

where ,→ stands for the continuous embedding, and the norm for

T

˜ θλ,(1−θ)γ (Q) H

0≤θ≤1

˜ θλ,(1−θ)γ (Q). Since θ ∈ [0, 1] is introduced as the maximum in θ of norms for H is arbitrary, it isn’t hard to see that the inequality in (2.4) is equivalent (up to the constant factor C) to ¯µ ¯ ¶ ¯ 1,1 ¯ ¯ ∆ h,η ϕ (x, y)¯ ≤ C min{|h|λ |, η|γ } (2.6) ¯ ¯ 2.2. A one-dimensional statements The following statements are known, being fist proved in [1], see also the presentations of these proofs in [11], p.57 and 190. We use the schemes of the proofs to make the presentation easier for the two-dimensional case. Theorem 2.3. Let ϕ(x) ∈ H λ ([0, b]), 0 < λ < 1, 0 < α < 1 and λ + α < 1. α f )(x) representation Then for the fractional operator (I0+ ¡ α ¢ I0+ ϕ (x) =

ϕ(0) xα + ψ(x), Γ(1 + α)

(2.7)

holds, where ψ(x) ∈ H α+λ and |ψ(x)| ≤ Cxλ+α . The proof of the theorem is the same as in [11], pp. 54-55. Lemma 2.4. If f (x) ∈ H λ+α ([0, b]) and 0 < λ, 0 < α + λ < 1, then ° ° ° ° ° ° ° ° f (x) − f (0) λ ° ° ° ∈ H ([0, b]), and °z ° ≤ C° z(x) = °f ° λ+α , |x|α Hλ H where C doesn’t depend from f (x). Proof. Let h > 0; x, x + h ∈ [0, b]. We consider the difference |z(x + h) − z(x)| ≤

(x + h)α − xα |f (x + h) − f (x)| + |f (x) − f (0)| . (x + h)α xα (x + h)α

4

275

(2.8)

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

Since f ∈ H λ+α , we have |f (x + h) − f (x)| ≤ C1 hλ+α ,

|f (x) − f (0)| ≤ C2 xλ+α .

(2.9)

Using these inequalities we obtain |z(x + h) − z(x)| ≤ C1

α α hλ+α λ (x + h) − x + C x = Z1 + Z2 . 2 (x + h)α (x + h)α

For Z1 , we have

µ Z1 = C1

h x+h

¶α hλ ≤ Chλ .

Let’s estimate Z2 , here we shall consider two cases: x ≤ h and x > h. In the first case, we use inequality |σ1µ − σ2µ | ≤ |σ1 − σ2 |µ , (σ1 6= σ2 ) and obtain Z2 ≤ xλ

hα ≤ Chλ . (x + h)α

In second case, using (1 + t)α − 1 ≤ αt, t > 0 we have ¯µ ¯ ¶α ¯ ¯ h xλ ¯ 1+ − 1¯¯ ≤ Chxλ−1 ≤ Chλ , Z2 = C2 ¯ α (x + h) x which completes the proof. The Marchaud fractional differentiation operator has a form: ¡

Dα 0+ f

¢

α f (x) + (x) = α x Γ(1 − α) Γ(1 − α)

Zx 0

f (x) − f (t) dt, (x − t)1+α

(2.10)

where 0 < α < 1. Theorem 2.5. If f (x) ∈ H λ+α ([a, b]), 0 < α + λ < 1, that ¡ α ¢ D0+ f (x) =

f (0) + χ(x), xα Γ(1 − α) ° ° ° ° ° ° ° ° ° °f ° where χ(x) ∈ H λ ([0, b]) and χ(0) = 0, thus ° χ ≤ C ° ° λ ° ° λ+α . H H ¡ α ¢ Proof. We present D0+ f (x) as ¡ α ¢ D0+ f (x) =

f (x) − f (0) α f (0) + + xα Γ(1 − α) Γ(1 − α)xα Γ(1 − α)

Zx 0

f (x) − f (t) dt, (x − t)1+α

receive equality (2.11), where α f (x) − f (0) + χ(x) = χ1 (x) + χ2 (x) = α Γ(1 − α)x Γ(1 − α) 5

276

Zx 0

(2.11)

f (x) − f (t) dt. (x − t)1+α

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

Here χ1 (x) ∈ H λ ([0, b]) by Lemma 2.4. It is enough to show χ2 (x) ∈ H ([0, b]). Let h > 0, x, x + h ∈ [0, b]. Let’s consider the difference λ

Zx

f (x + h) − f (x) dt + (x + h − t)1+α

χ2 (x + h) − χ2 (x) = 0

Zx +

x+h Z

x

f (x + h) − f (t) dt+ (x + h − t)1+α

£ ¤ [f (x) − f (t)] (x + h − t)−α−1 − (x − t)−1−α dt = I1 + I2 + I3 .

0

Since f ∈ H λ+α ([0, b]), then we have for I1 Zx |I1 | ≤ Chλ+α

(t + h)−1−α dt ≤ C1 hλ . 0

Let’s estimate I2 . We have x+h Z

(x + h − t)λ−1 dt = C2 hλ .

|I2 | ≤ C x

For I3 , we have Zx |I3 | ≤ C

¯ ¯ (x − t)λ+α ¯(x + h − t)−1−α − (x − t)−1−α ¯ dt =

0 x

Zh = Chλ

¯ ¯ tλ ¯(1 + t)−1−α − t−1−α ¯ dt ≤ C3 hλ ,

0

where

Z∞ C3 = C

¯ ¯ tλ ¯(1 + t)−1−α − t−1−α ¯ dt < ∞.

0

Finally, it remains to note that χ2 (0) = 0, since Zx tλ−1 dt.

|χ2 (x)| ≤ C 0

3. Mapping properties of the mixed fractional integration operator in the H¨ older spaces

6

277

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

Lemma 3.1. Let ϕ(x, y) ∈ H λ,γ (Q), 0 ≤ λ, γ ≤ 1, 0 < α, β < 1. Then for the mixed fractional integral operator (1.2) the representation ³

´ α,β I0+,0+ ϕ (x, y) =

ψ1 (x)y β xα ψ2 (y) ϕ(0, 0)xα y β + + + ψ(x, y) (3.1) Γ(1 + α)Γ(1 + β) Γ(1 + β) Γ(1 + α)

holds, where 1 ψ1 (x) = Γ(α)

Zx 0

ϕ(t, 0) − ϕ(0, 0) dt, (x − t)1−α

1 ψ2 (y) = Γ(β) µ

ψ(x, y) = and

1 Γ(α)Γ(β)

0

∆ t,τ ϕ (0, 0)

(x − t)1−α (y − τ )1−β

0

|ψ1 (x)| ≤ C1 xλ+α , y

θ∈[0,1]

dtdτ,

|ψ2 (y)| ≤ C2 y γ+β ,

α+θλ β+(1−θ)γ

|ψ(x, y)| ≤ C min x

0

ϕ(0, τ ) − ϕ(0, 0) dτ, (y − τ )1−β

¶

1,1

Zx Zy

Zy

α β

(3.2) λ

γ

= Cx y min{x , y }.

(3.3)

Proof. Representation (3.1) itself is easily obtained by means of (2.1). Since ϕ ∈ H λ,γ (Q), inequalities (3.2) are obvious. Estimate (3.3) is obtained by means of (2.4) and (2.6). α,β Theorem 3.2. Let 0 ≤ λ, γ < 1. The operator I0+,0+ is bounded from λ,γ λ+α,γ+β H0 (Q) to H0 (Q), if λ + α < 1 and γ + β < 1. Proof. Sice ϕ(x, y) ∈ H0λ,γ (Q), by (3.1) we have ³ ´ α,β I0+,0+ ϕ (x, y) = ψ(x, y). We denote

µ g(t, τ ) =

¶ ∆ t,τ ϕ (0, 0)

1,1

(3.4)

for brevity. Note that µ

1,1

¶

∆ t,τ ϕ (0, 0) = ϕ(t, τ )

for ϕ ∈ H0λ,γ , but we prefer to keep the notation for g(t, τ ) via the mixed difference as in (3.4). By (2.4) we have |g(t, τ )| ≤ Ctθλ τ (1−θ)γ ≤ C min{tλ , τ γ }. For h > 0, x, x + h ∈ Q1 = [0, b], we consider the difference (x + h)α − xα ψ(x + h, y) − ψ(x, y) = Γ(1 + α)Γ(β) 7

278

Zy 0

g(x, y − τ ) + τ 1−β

(3.5)

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

1 + Γ(α)Γ(β)

+

1 Γ(α)Γ(β)

Zh Zy 0

0

g(x + t, y − τ ) − g(x, y − τ ) dtdτ + (h − t)1−α τ 1−β

Zx Zy [g(x − t, y − τ ) − g(x, y − τ )] [(t + h)α−1 − tα−1 ]τ β−1 dtdτ = 0

0

= ∆ 1 + ∆2 + ∆3 .

(3.6)

We make use of (3.5) with θ = 1 and obtain |∆1 | ≤ C|(x + h)α − xα |xλ ≤ Chα+λ . For ∆2 in view of (2.4), we have ¯µ ¯ ¶ ¯ 1,1 ¯ ¯ |g(x − t, y − τ ) − g(x, y − τ )| = ¯ ∆ −t,y−τ ϕ (x, 0)¯¯ ≤ C|t|λ , and then

(3.7)

∆2 ≤ Chλ+α .

For ∆3 by (3.7) and (2.4) we have Zx

Z∞ λ α−1

∆3 ≤ C

t |t

−(t+h)

α−1

|dt ≤ C0 h

λ+α

,

tλ |tα−1 −(t+1)α−1 |dt < ∞.

C0 =

0

0

Gathering the estates for ∆1 , ∆2 , ∆3 we obtain |ψ(x + h, y) − ψ(x, y)| ≤ Chλ+α . Rearranging symmetrically representation (3.6), we can similarly obtain that |ψ(x, y + η) − ψ(x, y)| ≤ Cη γ+β , which proves the theorem. α,β Theorem 3.3. The mixed fractional integral operator I0+,0+ is bounded ˜ λ,γ (Q), 0 ≤ λ, γ ≤ 1 into the space H ˜ λ+α,γ+β (Q), if λ + α ≤ 1 from the space H 0 0 and γ + β ≤ 1. ˜ λ,γ (Q). By Theorem 3.2 and embedding (2.5), for Proof. Let ϕ ∈ H 0 µ ¶ ³ ´ 1,1 α,β f (x, y) = I0+,0+ ϕ (x, y) it satisfies to estimate the difference ∆ h,η f (x, y). ¯ ¯ ¯ Since ϕ(x, y)¯ = 0, according to (3.1) we have f (x, y) = ψ(x, y), where ¯ x=0,y=0

ψ(x, y) is the function from (3.1). The main moment in the estimations is to find the corresponding splitting which allows to derive the best information in

8

279

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

each variable not losing the corresponding information in another variable. The suggested splitting runs as follows µ

¶

1,1 ∆ h,η

f

µ

1,1 ∆ h,η

(x, y) =

¶ ψ (x, y) =

9 X

Tk :=

k=1

:=

g(x, y) [(x + h)α − xα ] [(y + η)β − y β ]+ Γ(1 + α)Γ(1 + β) Z0

(y + η)β − y β + Γ(α)Γ(1 + β)

g(x − t, y) − g(x, y) dt+ (t + h)1−α

−h

(x + h)α − xα + Γ(1 + α)Γ(β) (y + η)β − y β Γ(α)Γ(1 + β)

+

+

Zx

(x + h)α − xα Γ(1 + α)Γ(β)

+

0

Zy

£ ¤ [g(x, y − τ ) − g(x, y)] (τ + η)β−1 − τ β−1 dτ +

0

µ

µ +

1 Γ(α)Γ(β)

+

1 Γ(α)Γ(β)

1 + Γ(α)Γ(β)

Zx Zy µ

Z0

0 −η

Z0

Z0

1,1 ∆ −t,−τ

¶ g (x, y)

(h + t)1−α (η + τ )1−β

−h −η

dtdτ +

¶

1,1

∆ −t,−τ g (x, y) £ ¤ (τ + η)β−1 − τ β−1 dtdτ + 1−α (h + t)

µ

1,1 ∆ −t,−τ

(η +

¶ g (x, y)

τ )1−β

£ ¤ (t + h)α−1 − tα−1 dtdτ +

¶

£ ¤£ ¤ α−1 − tα−1 (τ + η)β−1 − τ β−1 dtdτ, ∆ −t,−τ g (x, y) (t + h)

1,1

0

Zy

−h 0

Zx

−η

g(x, y − τ ) − g(x, y) dτ + (τ + η)1−β

¤ £ [g(x − t, y) − g(x, y)] (t + h)α−1 − tα−1 dt+

1 Γ(α)Γ(β) Z0

Z0

0

where h > 0, η > 0; x, x + h ∈ Q1 ; y, y + η ∈ Q2 and g(x, y) is the function from (3.4). The validity of this representation may µ ¶ be checked directly. ˜ λ,γ , we have |g(x, y)| = | Since ϕ ∈ H

1,1

∆ x,y ϕ (0, 0)| ≤ Cxλ y γ and then

¯ ¯ |T1 | ≤ Cxλ y γ |(x + h)α − xα | ¯(y + η)β − y β ¯ , 9

280

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

¯ ¯ |T2 | ≤ Cy ¯(y + η)β − y β ¯

Z0

γ

|t|λ dt, (t + h)1−α

−h

Z0 λ

α

α

|T3 | ≤ Cx |(x + h) − x | −η

¯ ¯ |T4 | ≤ Cy γ ¯(y + η)β − y)β ¯

Zx

|τ |γ dτ, (τ + η)1−β

¯ ¯ |t|λ ¯(t + h)α−1 − tα−1 ¯ dt,

0

Zy |T5 | ≤ Cxλ |(x + h)α − xα |

¯ ¯ |τ |γ ¯(τ + η)β−1 − τ β−1 ¯ dτ.

0

For T6 − T9 we similarly, make use of ¯µ ¯ ¯µ ¯ ¶ ¶ ¯ 1,1 ¯ ¯ 1,1 ¯ ¯ ∆ −t,−τ g (x, y)¯ = ¯ ∆ −t,−τ ϕ (x, y)¯ ≤ C|t|λ |η|γ . ¯ ¯ ¯ ¯ and obtain

Z0 Z0 |T6 | ≤ C −h −η

Z0 Zy |T7 | ≤ C −h 0

Zx Z0 |T8 | ≤ 0 −η

Zx Zy |T9 | ≤ 0

(h +

|t|λ |τ |γ dtdτ, + τ )1−β

t)1−α (η

¯ |t|λ |τ |γ ¯¯ β−1 β−1 ¯ (η + τ ) − τ dtdτ, (h + t)1−α

¯ |t|λ |τ |γ ¯¯ (h + t)α−1 − tα−1 ¯ dtdτ, (η + τ )1−β

¯ ¯¯ ¯ |t|λ |τ |γ ¯(h + t)α−1 − tα−1 ¯ ¯(η + τ )β−1 − τ β−1 ¯ dtdτ,

0

after which every term is estimated in the standard way, and we get ¯µ ¯ ¶ ¯ 1,1 ¯ ¯ ∆ h,η f (x, y)¯ ≤ C3 hλ+α η γ+β . ¯ ¯ This completes the proof.

4. Mapping properties of the mixed fractional differentiation operator in the H¨ older spaces

10

281

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

e λ+α,γ+β (Q), 0 < α + λ < 1, 0 < β + γ < 1. Lemma 4.1. Let f (x, y) ∈ H Then for the mixed fractional differential operator (1.3) the representation " # ³ ´ Γ−1 (1 − α) f (0, 0) χ1 (x) χ2 (y) α,β + + + χ(x, y) , (4.1) D0+,0+ f (x, y) = Γ(1 − β) xα y β yβ xα holds, where f (x, 0) − f (0, 0) +α χ1 (x) = xα

Zx

f (0, y) − f (0, 0) +β χ2 (y) = yβ µ

¶

1,1

∆ x, y f

χ(x, y) =

+

β xα

0

+

xα y β

µ Zy

(0, 0)

α yβ

¶

1,1

∆ x,y−τ f

0

Zy 0

f (0, τ ) − f (0, 0) dτ, (y − τ )1+β µ

Zx

∆ x−t, y f

|χ1 (x)| ≤ C1 xλ ,

(t, 0) dt+

(x − t)1+α

0

µ

dτ + αβ 0

¶

1,1

0

¶

1,1

Zx Zy

(0, τ )

(y − τ )1+β

and

f (t, 0) − f (0, 0) dt, (x − t)1+α

∆ x−t, y−τ f

(t, τ )

(x − t)1+α (y − τ )1+β

dtdτ

|χ2 (y)| ≤ C2 y γ ,

(4.2)

λ γ

|χ(x, y)| ≤ C3 x y .

(4.3)

Proof. Representation (4.1) itself is easily obtained by means of (2.1). Since f ∈ H λ+α,γ+β (Q), inequalities (4.2) are obvious. Estimate (4.3) is obtained by means of (2.4), i.e. ·

Zy

Zx λ γ

χ(x, y) ≤ C x y + αy

γ

λ−1

(x − t)

dt + βx

0

0

¸

Zx Zy λ−1

+αβ

(x − t) 0

(y − τ )γ−1 dτ +

λ

γ−1

(y − τ )

dtdτ .

0

It is easy to receive · ¸ Z1 Z1 Z1 Z1 χ(x, y) ≤ Cxλ y γ 1 + sλ−1 ds + ξ γ−1 dξ + sλ−1 ξ γ−1 dsdξ ≤ C3 xλ y γ . 0

0

0

0

e λ+α,γ+β (Q), 0 < λ + α < 1, 0 < γ + β < 1. Theorem 4.2. Let f (x) ∈ H 0 α,β e λ+α,γ+β (Q) into H e λ,γ (Q). Then the operator D0+,0+ continuously maps H 0 0 11

282

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

e λ+α,γ+β (Q), by (4.1) we have Proof. Since f (x, y) ∈ H 0 ³ ´ ϕ(x, y) = Dα,β 0+,0+ f (x, y) = χ(x, y). Let h > 0; x, x + h ∈ [0, b]. We consider the difference µ ¶ 1,1 10 ∆ h, y f (0, 0) X 1 + χ(x + h, y) − χ(x, y) = Φk := β y (x + h)α k=0

µ +

1 yβ

µ

¶

1,1 ∆ x, y

f

£ ¤ α (0, 0) (x + h)−α − x−α + β y

µ α + β y

1,1 ∆ x+h−t, y

x+h Z

α + β y

Zx µ

¶

1,1

∆ x−t, y f

β dt + (x + h)α

0

µ

µ +αβ 0

0

x

1,1

0

∆ x,y−τ

0

f

dτ +

µ

(0, τ )

(y − τ )1+β

dτ +

¶ (x, τ )

(x + h − t)1+α (y − τ )1+β

x+h Z Zy

(0, τ )

¶

1,1

∆ h, y−τ f

0

+αβ

Zy

1,1

Zx Zy 0

¶ f

(y − τ )1+β

0

µ

+αβ

1,1 ∆ h,y−τ

dt+

£ ¤ (t, 0) (x + h − t)−1−α − (x − t)−1−α dt+

£ ¤ +β (x + h)−α − x−α

Zx Zy

Zy

(x, 0)

(x + h − t)1+α

µ

(t, 0)

(x + h − t)1+α

x

∆ h, y f

0

¶ f

¶

1,1

Zx

1,1 ∆ x+h−t, y−τ

dtdτ +

¶ f

(t, τ )

(x + h − t)1+α (y − τ )1+β

dtdτ +

¶

∆ x−t, y−τ f

(t, τ )

(y − τ )1+β

£ ¤ (x + h − t)−1−α − (x − t)−1−α dtdτ. (4.4)

e λ+α,γ+β , we have Since f ∈ H 0 |Φ1 | ≤ Cy γ

hλ+α hλ+α ≤ C , 1 (x + h)α (x + h)α

¯ ¯ |Φ2 | ≤ Cy γ xλ+α ¯(x + h)−α − x−α ¯ ≤ C2

12

283

xλ [(x + h)α − xα ] , (x + h)α

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

Zx γ λ+α

|Φ3 | ≤ Cα y h

0

dt ≤ C3 hλ+α (x + h − t)1+α

(x + h − t)λ−1 dt ≤ C4

Zx |Φ6 | ≤ Cα y γ

(x + h − t)λ−1 dt, x

x

|Φ5 | ≤ C

0

dt , (x + h − t)1+α

x+h Z

x+h Z

|Φ4 | ≤ Cα y γ

Zx

hλ+α β (x + h)α

Zy (y − τ )γ−1 dτ ≤ C5 0

hλ+α , (x + h)α

¯ ¯ (x − t)λ+α ¯(x + h − t)−1−α − (x − t)−1−α ¯ dt ≤

0

Zx ≤ C6

¯ ¯ (x − t)λ+α ¯(x + h − t)−1−α − (x − t)−1−α ¯ dt,

0

|Φ7 | ≤ Cβx

¯ ¯ ¯(x + h)−α − x−α ¯

λ+α

Zy

xλ dτ ≤ C [(x + h)α − xα ] , 7 (y − τ )1−γ (x + h)α

0

Zx |Φ8 | ≤ Cαβh

λ+α 0

dt (x + h − t)1+α

x+h Z

Zx γ−1

(y−τ )

dτ ≤ C8 h

0

Zy

x λ+α

(x − t)

dt , (x + h − t)1+α

x+h Z

(x + h − t)λ−1 dt

(y − τ )γ−1 dτ ≤ C9 x

0

Zx

λ+α

0

(x + h − t)λ−1 dt

|Φ9 | ≤ Cαβ

|Φ10 | ≤ Cαβ

Zy

¯ ¯ ¯(x + h − t)−1−α − (x − t)−1−α ¯ dt

0

Zy (y − τ )1+β dτ ≤ 0

Zx ≤ C10

¯ ¯ (x − t)λ+α ¯(x + h − t)−1−α − (x − t)−1−α ¯ dt,

0

where

Zy (y − τ )γ−1 dτ < ∞. 0

Using estimations Z1 , Z2 of the proof of Lemma 2.4 and estimations Ii , i = 1, 2, 3 of the proof of the Theorem 2.5, it is easily possible to receive an estimation |χ(x + h, y) − χ(x, y)| ≤ Chλ . Rearranging symmetrically representation (4.4), we can similarly obtain that |χ(x, y + h) − χ(x, y)| ≤ Chγ . 13

284

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

The main moment in the estimations is to find the corresponding splitting which allows to derive the best information in each variable not losing the corresponding information in another variable. Let h, η > 0; x, x + h ∈ [0, b], y, y + η ∈ [0, d]. We consider the difference µ

1,1

∆ h, η

¶ 25 X χ (x, y) = Pk := k=1

µ :=

µ

¶

1,1 ∆ h, η

f

1,1 ∆ h, y

¤ (x, 0) £ (y + η)β − y β + (x + h)α (y + η)β y β

(x, y) +

(x + h)α (y + η)β µ

1,1 ∆ x, η

¶

¶ f

f

(0, y)

[(x + h)α − xα ] + (y + η)β (x + h)α xα £ ¤ µ ¶ 1,1 [(x + h)α − xα ] (y + η)β − y β + + ∆ x, y f (x, y) (x + h)α xα (y + η)β y β µ µ ¶ ¶ 1,1 1,1 y+η y Z Z f (x, τ ) f (x, y) ∆ h, y+η−τ ∆ h, η β dτ + dτ + (y + η − τ )1+β (x + h)α (y + η − τ )1+β +

+

β (x + h)α

y

0

µ y+η Z

£ ¤ +β x−α − (x + h)−α

∆ x, y+η−τ

Zy µ

1,1 ∆ h, y−τ

¶ f

µ

µ +

α (y + η)β

x+h Z

x

0

1,1

∆ x+h−t, η f

dτ +

¶ f

¶ f

(0, y) dτ +

(y + η − τ )1+β

0 1,1 ∆ x, y−τ

1,1 ∆ x, η

Zy

£ ¤ +β x−α − (x + h)−α £ ¤ +β x−α − (x + h)−α

(0, τ )

£ ¤ (x, τ ) (y − τ )−1−β − (y + η − τ )−1−β dτ +

0

Zy µ

f

(y + η − τ )1+β

y

β + (x + h)α

¶

1,1

£ ¤ (0, τ ) (y − τ )−1−β − (y + η − τ )−1−β dτ + µ

¶ (t, y) dt +

(x + h − t)1+α

£ ¤ +α y −β − (y + η)−β

µ x+h Z

α (y + η)β

1,1 ∆ x+h−t, y

Zx 0

f

1,1

14

285

(x, y)

(x + h − t)1+α ¶ (t, 0)

(x + h − t)1+α

x

¶

∆ h, η f

dt+

dt+

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

Zx µ

α + (y + η)β

¶

1,1

∆ x−t, η f

£ ¤ (t, 0) (x − t)−1−α − (x + h − t)−1−α dt+

0

µ

¤ £ +α y −β − (y + η)−β

Zx

£ ¤ +α y −β − (y + η)−β

∆ h, η f

+ 0

0

∆ h, y−τ f

(x, τ )

(x + h − t)1+α

0

µ

0

0

∆ x+h−t, η f

x

Zx

(x + h − µ Zy

+ 0

¶

∆ x+h−t, y−τ f

0

1,1 ∆ x−t, η

(t, τ )

t)1+α

x

y

1,1

f

(t, y)

τ )1+β

∆ x+h−t, y+η−τ f

(t, τ )dtdτ

(x + h − t)1+α (y + η − τ )1+β

£ ¤ (y − τ )−1−β − (y + η − τ )−1−β dtdτ +

£ ¤ (x − t)−1−α − (x + h − t)−1−α dtdτ +

µ ¶ 1,1 Zx y+η Z f (t, τ ) ∆ x−t, y+η−τ £ ¤ (x − t)−1−α − (x + h − t)−1−α dtdτ + + 1+β (y + η − τ ) 0

y

Zx Zy µ

1,1

+

¶

∆ x−t, y−τ f

0

0

+

¶

¶

(y + η −

0

(x + h − t)1+α (y + η − τ )1+β

x+h Z y+η Z

(t, y)dtdτ

1,1

+

(x, τ )dtdτ

µ

µ

x+h Z Zy

y

∆ h, y+η−τ f

¶

£ ¤ (y − τ )−1−β − (y + η − τ )−1−β dtdτ +

+ (x + h − t)1+α (y + η − τ )1+β

+

1,1

¶

1,1

x+h Z Zy

+

¶

1,1

+

x

(x, y)dtdτ

µ Zx Zy

dt+

µ Zx y+η Z

(x + h − t)1+α (y + η − τ )1+β

0

(x, 0)

£ ¤ (t, 0) (x − t)−1−α − (x + h − t)−1−α dt+

0

1,1

Zy

f

¶

µ Zx

1,1 ∆ x−t, y

¶

∆ h, y f

(x + h − t)1+α

0

Zx µ

¶

1,1

£ ¤ (t, τ ) (x − t)−1−α − (x + h − t)−1−α ×

£ ¤ × (y − τ )−1−β − (y + η − τ )−1−β dtdτ.

The validity of this representation may be checked directly. e λ+α,γ+β (Q), we have Since f (x, y) ∈ H 0 |P1 | ≤ C

hλ+α η γ+β , (x + h)α (y + η)β 15

286

+

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

¯ ¯ hλ+α y γ ¯(y + η)β − y β ¯ , |P2 | ≤ C (x + h)α (y + η)β xλ η γ+β |(x + h)α − xα | , (y + η)β (x + h)α ¯ ¯ |(x + h)α − xα | ¯(y + η)β − y β ¯

|P3 | ≤ C |P4 | ≤ Cxλ y γ

(x + h)α

(y + η − τ )γ−1 dτ, y

Zy

hλ+α η γ+β |P6 | ≤ C (x + h)α

|P7 | ≤ Cx

,

y+η Z

hλ+α |P5 | ≤ C (x + h)α

λ+α

(y + η)β

0

dτ , (y + η − τ )1+β

¯ −α ¯ ¯x − (x + h)−α ¯

y+η Z

(y + η − τ )γ−1 dτ, y

hλ+α |P8 | ≤ C (x + h)α

Zy

¯ ¯ (y − τ )γ+β−1 ¯(y − τ )−1−β − (y + η − τ )−1−β ¯ dτ,

0

λ+α γ+β

|P9 | ≤ x

η

¯ −α ¯ ¯x − (x + h)−α ¯

Zy 0

dτ , (y + η − τ )1+β

¯ Zy ¯ ¯ ¯ ¯(y − τ )−1−β − (y + η − τ )−1−β ¯ λ+α ¯ −α −α ¯ dτ, |P10 | ≤ Cx x − (x + h) (y − τ )−γ−β 0

η γ+β |P11 | ≤ C (y + η)β

x+h Z

(x + h − t)λ−1 dt, x

hλ+α η γ+β |P12 | ≤ C (y + η)β

|P13 | ≤ y

γ+β

Zx 0

dt , (x + h − t)1+α

¯ −β ¯ ¯y − (y + η)−β ¯

x+h Z

(x + h − t)λ−1 dt, x

|P14 | ≤ C

η γ+β (y + η)β

|P15 | ≤ Ch

Zx

¯ ¯ (x − t)λ+α ¯(x − t)−1−α − (x + h − t)−1−α ¯ dt,

0

λ+α γ+β

y

¯ −β ¯ ¯y − (y + η)−β ¯

Zx 0

16

287

dt dt, (x + h − t)1+α

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

|P16 | ≤ y

¯ −β ¯ ¯y − (y + η)−β ¯

γ+β

¯ Zx ¯¯ (x − t)−1−α − (x + h − t)−1−α ¯ (x − t)−λ−α

0

Zx Zy |P17 | ≤ Ch

λ+α γ+β

η

0

0

dtdτ , (x + h − t)1+α (y + η − τ )1+β

Zx y+η Z |P18 | ≤ Ch

λ+α 0

Zx

Zy

0

0

|P19 | ≤ Chλ+α

dt,

y

(y + η − τ )γ−1 dtdτ , (x + h − t)1+α

¯ (y − τ )γ+β ¯¯ (y − τ )−1−β − (y + η − τ )−1−β ¯ dtdτ, 1+α (x + h − t) x+h Z Zy

|P20 | ≤ Cη

γ+β x

0

(x + h − t)λ−1 dtdτ (y + η − τ )1+β

x+h Z y+η Z

(x + h − t)λ−1 (y + η − τ )γ−1 dtdτ,

|P21 | ≤ C x x+h Z Zy

|P22 | ≤ C x

0

¯ (y − τ )γ+β ¯¯ (y − τ )−1−β − (y + η − τ )−1−β ¯ dtdτ, (x + h − t)1−λ

Zx Zy |P23 | ≤ Cη

γ+β 0

y

0

¯ (x − t)λ+α ¯¯ (x − t)−1−α − (x + h − t)−1−α ¯ dtdτ, 1+β (y + η − τ )

Zx y+η Z

¯ ¯ (x − t)λ+α (y + η − τ )γ−1 ¯(x − t)−1−α − (x + h − t)−1−α ¯ dtdτ,

|P24 | ≤ C 0

y

Zx Zy |P25 | ≤ C 0

0

¯ ¯ (x − t)λ+α (y − τ )γ+β ¯(x − t)−1−α − (x + h − t)−1−α ¯ × ¯ ¯ × ¯(y − τ )−1−β − (y + η − τ )−1−β ¯ dtdτ,

after which every term is estimated in the standard way, and we get ¯µ ¯ ¶ ¯ 1,1 ¯ ¯ ∆ h, η ϕ (x, y)¯ ≤ C3 hλ η γ . ¯ ¯ This completes the proof.

References 17

288

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

1. H.G. Hardy and J.E. Littlewood, Some properties of fractional integrals. I. Math. Z. 27, No 4 (1928), 565-606. 2. N.K. Karapetiants and N.G. Samko, Weighted theorems on fractional integrals in the generalized H¨older spaces H0ω (ρ) via the indices mω and Mω . Fract. Calc. Appl. Anal. 7, No 4 (2004), 437-458. 3. N.K. Karapetians and L.D. Shankishvili, A short proof of Hardy-Littlewoodtype theorem for fractional integrals in weighted H¨older spaces. Fract. Calc. Appl. Anal. 2, No 2 (1999), 177-192. 4. N.K. Karapetians and L.D. Shankishvili, Fractional integro-differentiation of the complex order in generalized H¨older spaces H0ω ([0, 1], ρ). Integral Transforms Spec. Funct. 13, No 3 (2003), 199-209. 5. N.K. Karapetians, Kh.M. Murdaev and A.Ya. Yakubov, The isomorphism realized by fractional integrals in generalized H¨older classes. Dokl. Akad. Nauk SSSR 314, No 2 (1990), 288-21. 6. N.K. Karapetians, Kh.M. Murdaev and A.Ya. Yakubov, On isomorphism provided by fractional integrals in generalized Nikolskiy classes. Izv. Vuzov. Matematika (9), (1992), 49-58. 7. Kh.M. Murdaev and S.G. Samko, Mapping properties of fractional integrodifferentiation in weighted generalized H¨older spaces H0ω (ρ) with the weight ρ(x) = (x − a)µ (b − x)ν and given continuity modulus (Russian), Deponierted in VINITI, Moscow, 1986: No 3350-B, 25 p. 8. Kh.M. Murdaev and S.G. Samko, Weighted estimates of continuity modulus of fractional integrals of function having a prescribed continuity modulus with weight (Russian). Deponierted in VINITI, Moscow, 1986: No 3351-B, 42 p. 9. B.S. Rubin, Fractional integrals in H¨older spaces with weight, and operators of potential type. Izv. Akad. Nauk Armjan. SSR Ser. Mat. 9, No 4 (1974), 308-324. 10. B.S. Rubin, Fractional integrals and Riesz potentials with radial density in spaces with power weight. Izv. Akad. Nauk Armjan. SSR Ser. Mat. 21, No 5 (1986), 488-503. 11. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach. Sci. Publ., N. York - London, 1993, 1012 pp. (Russian Ed. - Fractional Integrals and Derivatives and Some of Their Applications, Nauka i Texnika, Minsk, 1987.) 12. S.G. Samko and Kh.M. Murdaev, Weighted Zygmund estimates for fractional differentiation and integration and their applications. Trudy Matem. Inst. Steklov 180 (1987), 197- 198 p.; English transl. in: Proc. Steklov Inst. Math. (AMS) 1989, Issue 3 (1989), 233-235. 18

289

MAMATOV TULKIN: FRACTIONAL INTEGRO-DIFFERENTIATION

13. S.G. Samko and Z. Mussalaeva, Fractional type operators in weighted generalized H¨older spaces. Proc. Georgian Acad. Sci., Math. 1, No 5 (1993), 601-626. 14. B.G. Vakulov, Potential type operator on a sphere in generalized H¨older classes. Izv. Vuzov. Matematika (11) (1986), 66-69; English transl.: Soviet Math. (Izv. VUZ) 30, No 11 (1986), 90-94. e-mail: [email protected] Samarkand State University, Mathematics Department University Boulevard 15, Samakand, 703004 - UZBEKISTAN

19

290

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.'S 3-4, 291-301, 2014, COPYRIGHT 2014 EUDOXUS PRESS LLC

Some fixed point theorems of set-valued increasing operators∗ Jin-Ming Wang, Xiong-Jun Zheng, Hui-Sheng Ding† College of Mathematics and Information Science, Jiangxi Normal University Nanchang, Jiangxi 330022, People’s Republic of China

Abstract In this paper, we study two kinds of set-valued increasing operators in partially ordered Banach spaces and partially ordered topological spaces respectively. We obtain three fixed point theorems, which generalize and improve some earlier results. Keywords: set-valued, increasing operator, partially ordered, fixed point, weakly compact set.

1

Introduction

The fixed point theory for various set-valued operators has been of great interest for many authors. Recently, there is a larger literature on fixed point theory of set-valued operators. We refer the reader to [1–8, 10–13] and references therein for some contributions on this topic. Especially, several authors have studied the fixed point theory for set-valued increasing operators in partially ordered spaces (see, e.g., [2, 5, 6, 8, 10–12] and references therein). In this paper, we will make further study on the fixed point theory of set-valued increasing operators in partially ordered spaces. More specifically, we will consider two m P kinds of set-valued increasing operators A = CB and A = Ci Bi , where C, Ci are singlei=1

valued increasing operators and B, Bi are set-valued increasing operators. For some earlier ∗

The work was supported by the Natural Science Foundation of Jiangxi Province (No.

20122BAB201008) and Science and Technology Plan of Education Department of Jiangxi Province (No. GJJ08169). † E-mail addresses:

math wang [email protected]

[email protected] (H. Ding).

291

(J.

Wang),

[email protected]

(X.

Zheng),

WANG ET AL: FIXED POINT THEOREMS

works on these operators, we refer the reader to [10–12]. As one will see, our main results are generalizations and improvements of [11, 12]. Let E be a real Banach space and P be a cone in E which defines a partial ordering in E by x ≤ y iff y − x ∈ P . For D ⊂ E, the weak closure of D is denoted by D

W

and

the complement set of D is denoted by CD. co(D) denotes the closed convex hull of D. If W

{xn } ⊂ D converges weakly to x ∈ E then we write xn −→ x. Definition 1.1. [10] Let X, Y be partially ordered sets, M be a subset of X and A : M → 2Y be a set-valued operator. The operator A is called a set-valued increasing operator if for any x ∈ M , y ∈ M , x ≤ y and any u ∈ Ax, then there exists v ∈ Ay such that u ≤ v. Definition 1.2. [11] Let X be an additive group with an ordering structure. X is called an ordered additive group if x, y, z, w ∈ X and x ≤ y, z ≤ w imply x + z ≤ y + w. Remark 1.3. Let S1 , S2 are two nonempty sets in X. We define S1 + S2 as follows: S = S1 + S2 = {x1 + x2 ∈ X|x1 ∈ S1 , x2 ∈ S2 }. Since X is an ordered additive group, we have S ⊂ X. Definition 1.4. [10] Let X be a Hausdorff topological space with a partially ordered structure. X is said to be a partially topological space if for any two directed sequences {xτ |τ ∈ T } and {yτ |τ ∈ T } in X, xτ ≤ yτ (∀τ ∈ T ), {xτ } is a net converging to x, {yτ } is a net converging to y imply x ≤ y. Lemma 1.5. [6] Let (E, P ) be a partially ordered Banach space, W be a nonempty subset of E and y ∈ E. If z ≤ y (or y ≤ z) for all z ∈ W , then for all z ∈ co(W ), z ≤ y (or y ≤ z). Lemma 1.6. [9] Let X be a Banach space. Suppose that M ⊂ X is closed and convex. If W

{xn } is a sequence in M with xn −→ x, then x ∈ M . Lemma 1.7. [12] If X is a partially ordered topological space, then for any α ∈ X, {y ∈ X|y ≥ α} is a closed set in X.

2

Main results

Theorem 2.1. Let X be a partially ordered set, D be a nonempty subset of X and (Y, P ) be a partially ordered Banach space. U is a convex closed set in Y . If the operator A : D → 2X satisfies the following conditions

292

WANG ET AL: FIXED POINT THEOREMS

S

(i) There exists a set-valued increasing operator B : D → 2Y with B(D) =

Bx ⊂ U

x∈D

and an increasing operator C : U → D such that A = CB. (ii) There exists x0 ∈ D and u ∈ Ax0 such that x0 ≤ u.

(iii) Any totally ordered subset of B(D) is a relatively weakly compact subset in Y . (iv) For any x ∈ D, Bx is a weakly compact set in Y . Then A has a fixed point in D, i.e. there exists x∗ ∈ D such that x∗ ∈ Ax∗ . Proof. Set G = {x ∈ D| there exists u ∈ Ax, such that x ≤ u}. From condition (ii) we have x0 ∈ G , so G is nonempty. Suppose that N is any totally ordered set of G. In what follows, we now show that N has an upper bound in G. Since B is a set-valued increasing operator, for any x ∈ N , y ∈ N , x ≤ y and any u ∈ Bx, there exists v ∈ By such that u ≤ v, so there exists a totally ordered set D1 ⊂ B(N ) in Y and for any T W x ∈ N , D1 Bx 6= Ø. By the hypothesis (iii), we get D1 is a weakly compact set. W

Then, it follows from the Krein-Smulian theorem that co(D1 ) is also weakly compact. So W

co(D1 ) ⊂ co(D1 ) implies co(D1 ) is a weakly compact set. For any y ∈ D1 , set T (y) = {z ∈ Y |z ≥ y}. Since P is a convex closed set, T (y) is T also a convex closed set. Let J(y) = {z ∈ co(D1 )|z ≥ y} = co(D1 ) T (y), then J(y) is a convex closed set, thus J(y) is a weakly closed set. Obviously, J(y) 6= Ø for y ∈ J(y). For y1 , y2 , · · · , yn ∈ D1 , we assume y ∗ = max{yi |i = 1, 2, · · · , n} . Since D1 is a totally n T ordered set, y ∗ makes sense and yi ≤ y ∗ which implies y ∗ ∈ J(yi ), then we get i=1

n \

J(yi ) 6= Ø.

(2.1)

i=1

Now we claim

T y∈D1

J(y) 6= Ø. If we assume otherwise, then we get co(D1 ) ⊂

S

CJ(y).

y∈D1

Evidently, {CJ(y)|y ∈ D1 } is an open cover of co(D1 ) in weak topology. As co(D1 ) is a 0

0

0

weakly compact set, co(D1 ) has a finite subcover, that is, there exists y1 , y2 , · · · , ym ∈ D1 m S 0 0 CJ(yi ). Note that J(yi ) ⊂ co(D1 ), we have such that co(D1 ) ⊂ i=1

m \

0

J(yi ) ⊂ co(D1 ) ⊂

i=1

0

CJ(yi ).

i=1

m T 0 0 CJ(yi ) implies J(yi ) = Ø contradicting (2.1). Hence, our claim i=1 i=1 i=1 T holds, i.e. there exists y ∈ J(y). This means that for any y ∈ D1

Then

m T

m [

0

J(yi ) ⊂

m S

y∈D1

y≤y∈

\ y∈D1

293

J(y) ⊂ co(D1 ).

(2.2)

WANG ET AL: FIXED POINT THEOREMS

By B(D) ⊂ U and the fact that U is a convex closed subset of Y , we have y ∈ co(D1 ) ⊂ co(B(N )) ⊂ co(B(D)) ⊂ co(U ) = U. Then x = Cy ∈ D is well defined. In order to show that x is an upper bound of N in G, we will divide it into two steps. Step 1. x is an upper bound of N . In fact, for any x1 ∈ N there exists x2 ∈ N such that x1 ≤ x2 . Since B is a set-value T 0 0 increasing operator, for any y ∈ Bx1 , there exists y ∈ Bx2 D1 such that y ≤ y ≤ y. Moreover, from monotonicity of C, we know Cy ≤ Cy = x.

(2.3)

As a result of x1 ∈ G, there exists u0 ∈ Ax1 such that x1 ≤ u0 . Since u0 ∈ Ax1 , there exists y0 ∈ Bx1 such that u0 = Cy0 , then by (2.3) we have u0 = Cy0 ≤ Cy = x. Therefore, we get x1 ≤ x , i.e. x is an upper bound of N . Step 2. x ∈ G. As B is a set-valued increasing operator, for any x ∈ N , x ≤ x, and any y ∈ D1

T

Bx,

there exists vy ∈ Bx such that y ≤ vy . From the hypothesis (iv), we know Bx is a weakly compact set which implies that there exists a subset {vyk } of the following set {vy |y ≤ vy , vy ∈ Bx, y ∈ D1

\

Bx}

such that {vyk } converges weakly to some v ∈ Bx. Since y ≤ vy , i.e. vy − y ∈ P , we have W

vyk − y ∈ P . By Lemma 1.6 with vyk − y −→ v − y, we can get v − y ∈ P . Thus for all y ∈ D1 , y ≤ v. By Lemma 1.5 with y ∈ co(D1 ), we have y ≤ v. Furthermore, as C is an 0

0

increasing operator, we can obtain x = Cy ≤ Cv = v , where v ∈ CBx = Ax. We have proved that for x ∈ D, there exists v 0 ∈ Ax such that x ≤ v 0 . Thus, x ∈ G. The two steps show that any totally ordered subset of G has an upper bound in G. It follows from Zorn’s lemma that G has a maximal element denoted by x∗ . Since x∗ ∈ G, there exists u∗ ∈ Ax∗ such that x∗ ≤ u∗ . As C is an increasing operator and B is a set-value increasing operator, we know A is also a set-value increasing operator. So there exists v ∗ ∈ Au∗ such that u∗ ≤ v ∗ which implies u∗ ∈ G. Since x∗ is a maximal element, we get x∗ = u∗ ∈ Ax∗ , that is, x∗ is a fixed point of A in D.

294

WANG ET AL: FIXED POINT THEOREMS

Remark 2.2. In the case of B being a single-valued operator, the condition (iv) is obviously true. Thus, Theorem 2.1 generalizes [6, Theorem 1]. But, here we use a different approach. Theorem 2.3. Let X be an ordered additive group, D be a nonempty subset in X, (Yi , Pi ) (i = 1, 2) be partially ordered Banach spaces, U1 and U2 be convex closed subsets of Y1 and Y2 respectively. If the operator A : D → 2X satisfies the following conditions S (I) There exists set-valued increasing operators Bi : D → 2Yi with Bi (D) = Bi x ⊂ x∈D

Ui (i=1,2) and increasing operators Ci : Ui → D (i = 1, 2) such that A = C1 B1 + C2 B2 . (II) There exists x0 ∈ D and u ∈ Ax0 such that x0 ≤ u. (III) Any totally ordered subset of Bi (D) is a relatively weakly compact subset in Yi . (IV) For any x ∈ D, Bi x are weakly compact sets in Yi . Then A has a fixed point in D, that is, there exists x∗ ∈ D such that x∗ ∈ Ax∗ . Proof. Set K = {x ∈ D| there exists u ∈ Ax such that x ≤ u}. By the condition (II), we know x0 ∈ K, so K is nonempty. Suppose that N is any totally ordered set of K. We want to show that N has an upper bound in K. Since B1 is a set-value increasing operator, for any x ∈ N , y ∈ N , x ≤ y and any u1 ∈ B1 x, there exists v1 ∈ B1 y such that u1 ≤ v1 . Thus there exists a totally ordered set S1 ⊂ B1 (N ) in Y1 and for any x ∈ N , T S1 B1 x 6= Ø. Similarly, there exists a totally ordered set S2 ⊂ B2 (N ) in Y2 and for T W W any x ∈ N , S2 B2 x 6= Ø. From the condition (III), we know S 1 and S 2 are weakly W

W

compact sets. Then, it follows from the Krein-Smulian Theorem that co(S 1 ) and co(S 2 ) W

W

are also weakly compact. Moreover, co(S 1 ) ⊂ co(S 1 ) and co(S 2 ) ⊂ co(S 2 ) imply that co(S1 ) and co(S2 ) are weakly compact sets. For any p ∈ S1 and q ∈ S2 , set T1 (p) = {y1 ∈ Y1 |y1 ≥ p} and T2 (q) = {y2 ∈ Y2 |y2 ≥ q} respectively. Since P1 , P2 are convex closed sets, T1 (p), T2 (q) are also convex closed sets. T T Let J1 (p) = co(S1 ) T1 (p), J2 (q) = co(S2 ) T2 (q), then J1 (p), J2 (q) are convex closed sets. So J1 (p), J2 (q) are weakly closed sets. Obviously, J1 (p) 6= Ø and J2 (q) 6= Ø for p ∈ J1 (p) and q ∈ J2 (q). For p1 , p2 , · · · , pn ∈ S1 , we set p∗ = max{pi |i = 1, 2, · · · , n}. n T Since S1 is a totally ordered set, p∗ makes sense and pi ≤ p∗ , which implies p∗ ∈ J1 (pi ), i=1

then we get n \

J1 (pi ) 6= Ø.

(2.4)

i=1

T J1 (p) 6= Ø. If we assume otherwise, then we get co(S1 ) ⊂ Now we claim that p∈S 1 S CJ1 (p), this means that {CJ1 (p)|p ∈ S1 } is an open cover of co(S1 ) in weak topology.

p∈S1

295

WANG ET AL: FIXED POINT THEOREMS

0

0

As co(S1 ) is a weakly compact set, co(S1 ) has a finite subcover, i.e. there exists p1 , p2 ,..., m S 0 0 0 CJ1 (pi ). Note that J1 (pi ) ⊂ co(S1 ), we obtain pm ∈ S1 such that co(S1 ) ⊂ i=1 m \

0

J1 (pi ) ⊂ co(S1 ) ⊂

i=1 m T

m [

0

CJ1 (pi ).

i=1

0

J1 (pi ) = Ø which contradicts the previous result (2.4). This means our claim i=1 T T J1 (p). Again for every p ∈ S1 , p ≤ p ∈ J1 (p) ⊂ co(S1 ). holds, so there exists p ∈ p∈S1 p∈S1 T J2 (q) and for every q ∈ S2 , Using the same method we can prove that there exists q ∈ q∈S2 T q≤q∈ J2 (q) ⊂ co(S2 ). Hence

q∈S2

By the fact that U1 and U2 are convex closed sets in Y1 and Y2 respectively, we get p ∈ co(S1 ) ⊂ co(B1 (N )) ⊂ co(B1 (D)) ⊂ co(U1 ) = U1 , q ∈ co(S2 ) ⊂ co(B2 (N )) ⊂ co(B2 (D)) ⊂ co(U2 ) = U2 . Then C1 p, C2 q are well defined. Setting x = C1 p + C2 q, we have x ∈ D. In order to show that x is an upper bound of N in K, we will divide it into two steps. Step 1. x is an upper bound of N . Indeed, for any x1 ∈ N there exists x2 ∈ N such that x1 ≤ x2 . Since B1 is a set-value T increasing operator , for any z1 ∈ B1 x1 there exists z2 ∈ B1 x2 S1 such that z1 ≤ z2 ≤ p. Besides, by monotonicity of C1 , we have C1 z1 ≤ C1 p. As B2 is a set-valued increasing operator, for any w1 ∈ B2 x1 there exists w2 ∈ B2 x2

(2.5) T

S2

such that w1 ≤ w2 ≤ q, then C2 w1 ≤ C2 q.

(2.6)

Since X is an ordered additive group, by (2.5) and (2.6), we get C1 z1 + C2 w1 ≤ C1 p + C2 q = x.

(2.7)

As result of x1 ∈ K, there exists u0 ∈ Ax1 = C1 B1 x1 + C2 B2 x1 such that x1 ≤ u0 , where u0 = C1 z0 + C2 w0 for some z0 ∈ B1 x1 and w0 ∈ B2 x1 . By (2.7), then we obtain x1 ≤ u0 ≤ x, i.e. x is an upper bound of N .

296

WANG ET AL: FIXED POINT THEOREMS

Step 2. x ∈ K. Since B1 is a set-valued increasing operator, for any x ∈ N , x ≤ x, and any y ∈ T S1 B1 x, there exists uy ∈ B1 x such that y ≤ uy . From the condition (IV), we know B1 x is a weakly compact set which implies that there exists a subset {uyk } of the following set {uy |y ≤ uy , uy ∈ B1 x, y ∈ S1

\

B1 x}

0

such that {uyk } converges weakly to some u ∈ B1 x. So we have uyk − y ∈ P1 and W

0

0

uyk − y −→ u − y. By Lemma 1.6, we can get u − y ∈ P1 . Thus 0

∀y ∈ S1 , y ≤ u . T

In a similar way, we can obtain that for any z ∈ S2

(2.8) B2 x, there exists vz ∈ B2 x such that

z ≤ vz . Then B2 x is a weakly compact set implies that there exists a subset {vzi } of the following set {vz |z ≤ vz , vz ∈ B2 x, z ∈ S2

\

B2 x}

0

such that {vzi } converges weakly to some v ∈ B2 x. By Lemma 1.6 we have 0

∀z ∈ S2 , z ≤ v .

(2.9)

By (2.8), (2.9), Lemma 1.5 with p ∈ co(S1 ) and q ∈ co(S2 ), we get 0

0

p ≤ u ,q ≤ v . 0

Since C1 , C2 are increasing operators, C1 p ≤ C1 u0 , C2 q ≤ C2 v . From the hypothesis (I), since X is an ordered additive group, 0

x = C1 p + C2 q ≤ C1 u0 + C2 v ∈ C1 B1 x + C2 B2 x = Ax. Consequently, x ∈ K. From the two steps, we have showed that any totally ordered subset of K has an upper bound in K. It follows from Zorn’s lemma that K has a maximal element denoted by x∗ . Since x∗ ∈ K, there exists u∗ ∈ Ax∗ such that x∗ ≤ u∗ . Again as A is a set-value increasing operator, there exists v ∗ ∈ Au∗ such that u∗ ≤ v ∗ . By the definition of K, u∗ ∈ K. But x∗ is a maximal element which implies x∗ = u∗ ∈ Ax∗ , i.e. x∗ is a fixed point of A in D. From Theorem 2.3, we can obtain the following corollary:

297

WANG ET AL: FIXED POINT THEOREMS

Corollary 2.4. If in Theorem 2.3 we substitute the operator A = C1 B1 + C2 B2 by the setm P valued increasing operator A = Ci Bi , we can also obtain a fixed point for the operator i=1

A.

Theorem 2.5. Let X be an ordered additive group, D be a nonempty subset in X, and Yi (i = 1, 2, · · · , m) be partially ordered topological spaces. If the operator A : D → 2X satisfies the following conditions (a) There exists x0 ∈ D and u ∈ Ax0 such that x0 ≤ u. (b)There exists set-valued increasing operators Bi : D → 2Yi and increasing operators m P Ci : Bi (D) → X(i = 1, 2, · · · , n) such that A = Ci Bi . i=1

(c) Any totally ordered subset of Bi (D) is a relatively compact set. (d)For any x ∈ D, Bi x are compact sets in Yi . Then A has a fixed point x∗ in D, i.e. x∗ ∈ Ax∗ . Proof. Set R = {x ∈ D| there exists u ∈ Ax such that x ≤ u}. By the condition (a), we have x0 ∈ R, so R 6= Ø. Let N be any totally ordered subset of R. We want to show that N has an upper bound in R. Let i(1 ≤ i ≤ m) be fixed. Since Bi : D → 2Yi is a set-value increasing operator, for any x ∈ N , y ∈ N , x ≤ y and any ui ∈ Bi x, there exists vi ∈ Bi y such that ui ≤ vi . Thus there exists Si ⊂ Bi (N ) where Si is a totally ordered set in Yi and for any x ∈ N , T Si Bi x 6= Ø. From the hypothesis (c), S i is a compact set in Yi . For any pi ∈ Si , set U (pi ) = {z ∈ S i |z ≥ pi } = S i

\

{z ∈ Yi |z ≥ pi }.

Since Yi is a partially ordered topological space, by Lemma 1.7, we know U (pi ) is a closed set in Yi . Now we consider the closed subset family {U (pi )|pi ∈ Si } of S i where {U (pi,j )|pi,j ∈ Si , j = 1, 2, · · · , n} are finite members given arbitrarily. Set p∗i = max{pi,j |j = 1, 2, · · · , n}. Since Si is a totally ordered set, p∗i makes sense and pi,j ≤ p∗i , j = 1, 2, · · · , n which implies n n T T p∗i ∈ U (pi,j ), so U (pi,j ) is nonempty. Note that S i is a compact set, by virtue of j=1

j=1

finite intersection property of compact sets, we have \

U (pi ) 6= Ø.

pi ∈Si

Let pi ∈

T pi ∈Si

U (pi ). Then for any pi ∈ Si , pi ≤ pi ∈

T pi ∈Si

U (pi ) ⊂ S i , thus there

exists {pi,α |α ∈ Λ} ⊂ Si such that {pi,α |α ∈ Λ} is a net converging to pi ∈ S i . Since

298

WANG ET AL: FIXED POINT THEOREMS

pi ∈ S i ⊂ Bi (N ) ⊂ Bi (D), Ci pi is well defined. Set x = to show that x is an upper bound of N in R.

m P i=1

Ci pi . In what follows, we want

First, for any x1 ∈ N , there exists x2 ∈ N such that x1 ≤ x2 . Since Bi is a setT value increasing operator, for any yi,1 ∈ Bi x1 there exists yi,2 ∈ Bi x2 Si such that yi,1 ≤ yi,2 ≤ pi . Again by monotonicity of Ci , we get Ci yi,1 ≤ Ci pi .

(2.10)

As x1 ∈ R, there exists some u0 ∈ Ax1 such that x1 ≤ u0 . Since u0 ∈ Ax1 =

m P i=1

Ci Bi x1 , there exists di ∈ Bi x1 such that u0 =

for X is an ordered additive group, we have x1 ≤ u0 =

m X

Ci di ≤

m X

i=1

m P i=1

Ci di . By (2.10),

Ci pi = x.

i=1

Therefore, x is an upper bound of N . Second, for any x ∈ N , x ≤ x and any y ∈ Si

T

Bi x, there exists uy ∈ Bi x such that

y ≤ uy . By the condition (d), we know Bi x is a compact set, so there exists a subset {uyτ } of the following set {uy |y ≤ uy , uy ∈ Bi x, y ∈ Si

\

Bi x}

such that {uyτ } is a net converging to some ui ∈ Bi x. As Yi are partially ordered topological spaces, by Definition 1.4, we get y ≤ ui . At this time, we have pi,β ≤ ui where {pi,β } is a subsequence of {pi,α |α ∈ Λ} ⊂ Si . Since S i is a compact set and {pi,α |α ∈ Λ} is a net converging to pi , then the subsequence {pi,β } is also a net converging to pi . Since Yi are partially ordered topological spaces with pi,β ≤ ui , we know pi ≤ ui . Again by m P monotonicity of Ci , we get Ci pi ≤ Ci ui . Set u = Ci ui . The fact X is ordered additive i=1

group implies x=

m X

Ci pi ≤

i=1

and u∈

m X

Ci ui = u

i=1 m X

Ci Bi x = Ax.

i=1

with ui ∈ Bi x ⊂ Yi and Ci ui ∈ Ci Bi x ⊂ X. Consequently, x ∈ R. This shows that x is an upper bound of N in R. It follows from Zorn’s lemma that R has a maximal element denoted by x∗ . Since x ∈ R, there exists u∗ ∈ Ax∗ such

299

WANG ET AL: FIXED POINT THEOREMS

that x∗ ≤ u∗ . Again by the fact that A is a set-valued increasing operator, there exists y ∗ ∈ Au∗ such that u∗ ≤ y ∗ , which means u∗ ∈ R. Since x∗ is a maximal element of R, x∗ = u∗ ∈ Ax∗ , i.e. x∗ is a fixed point of A.

References [1] A. Amini-Harandi, Fixed and coupled fixed points of a new type set-valued contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012, 2012:215. [2] I. Beg, A. R. Butt, Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces, Nonlinear Anal. 71 (2009), 3699–3704. [3] I. Beg, A. R. Butt, Fixed point of set-valued graph contractive mappings, J. Inequal. Appl. 2013, 2013:252. [4] L. Khan, M. Imdad, Meir and Keeler type fixed point theorem for set-valued generalized contractions in metrically convex spaces, Thai J. Math. 10 (2012), 473–480. [5] B. Y. Li, S. S. Chang, Y. J. Cho, Fixed points for set-valued increasing operators and applications, J. Korean Math. Soc. 31 (1994), 325–331. [6] X. Liu, C. Wu, Fixed point of discontinuous weakly compact increasing operators and its application to initial value problem in Banach spaces (in Chinese), J. Systems Sci. Math. Sci., 20, (2000), 175–180. [7] B. D. Pant, B. Samet, S. Chauhan, Coincidence and common fixed point theorems for single-valued and set-valued mappings, Commun. Korean Math. Soc. 27 (2012), 733–743. [8] N. Petrot, J. Balooee, Fixed point theorems for set-valued contractions in ordered cone metric spaces, J. Comput. Anal. Appl. 15 (2013), 99–110. [9] B. P. Rynne, M. A. Youngson, Linear Functional Analysis, Springer Undergraduate Mathematics Series, London, 2008. [10] J. Sun, Fixed point and generalized fixed point of the increasing operator, Acta Math. Sinica, 32, (1989), 457–463. [11] J. Sun, Z. Zhao, Fixed point theorems of increasing operators and applications to nonlinear integro-differential equatios with discontinous terms, J. Math. Anal. Appl., 175, (1993), 33–45.

300

WANG ET AL: FIXED POINT THEOREMS

[12] X. Zheng, J. Sun, Fixed point theorems of discontinuous increasing operators in partly ordered space (in Chinese), J. Jiangxi Norm. Univ. Nat. Sci. Ed., 32, (2008), 597–600. [13] X. Zhu, J. Xiao, Minimum selections and fixed points of set-valued operators in Banach spaces with some uniform convexity, Appl. Math. Comput., 217 (2011), 6004– 6010.

301

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.'S 3-4, 302-320, 2014, COPYRIGHT 2014 EUDOXUS PRESS LLC

DYNAMICS AND APPROXIMATIONS FOR 2D GENERALIZED NAVIER-STOKES EQUATION WITH PIECEWISE DISTRIBUTED CONTROLS DE G. AKMEL

AND

L. C. BAHI

Abstract. We study the dynamics of a piecewise (in time) distributed optimal control problem for Generalized Navier-Stokes equation. The long-time behavior of solutions for an optimal distributed control problem associated with the tracking of the velocity of the Generalized Navier Stockes equations is studied. The existence of a solution of optimal control problem is proved also optimality system is derived. The long-time decay properties for the optimal solutions is established. We also study the dynamics of semidiscrete and fully discrete approximations of this problem. Some computational results are presented, which reinforces the theoretical results derived.

1. Introduction The control of viscous flows is very crucial to many technological and scientific applications. We are motivated to study the asymptotic behaviors and dynamics of solutions for the controlled Generalized Navier-Stokes equation. Several treatments of similar optimal control problems can be found in literature. Indeed, the optimal control with the systems governed by Navier-Stokes, Boussinesq and MHD equations was studying by L. Hou and Y. Yan [8], by H. Chun Lee and B. Chun Shin [4] and in [9], respectively. The existence of solutions of Generalized Navier-Stokes equation in Besov spaces was studied by Wu [11] and by Cheskidev and Dai [3]. We formulate here a controllability problem for the Generalized Navier-Stokes equation: find a (u, f ) such that the functional Z Z Z Z α +∞ β +∞ 2 2 (1.1) J(0;+∞) (u, f ) = |u − U | dxdt + |f − F | dxdt 2 0 2 0 Ω Ω is minimized subject to the 2-D Generalized Navier-Stokes equation: (1.2)

∂u + ν(−4)r u + (u.O)u + Op = f in Ω × (0, ∞) ∂t

1991 Mathematics Subject Classification. 52B10, 65D18, 68U05, 68U07. Key words and phrases. Optimal control, Generalized Navier-Stokes equation, Long-time behavior. 1

302

2

DE G. AKMEL

(1.3)

AND

L. C. BAHI

in Ω × (0, ∞)

O.u = 0

u = 0, 4u = 0, ..., 4r−1 u = 0 on ∂Ω × (0, ∞)

(1.4) and (1.5)

u(. , 0) = u0

in Ω

where r ≥ 1 is an integer and n is an outward normal vector of Ω, also ν > 0 is the kinematic viscosity. Here α, β > 0 are given constants, Ω is a bounded, sufficiently smooth domain in R2 with ∂Ω denoting its boundary; U and F are a given desired velocity field, a given desired body force, respectively. Also, f is a distributed control (body force), u and p denote the velocity field and the pressure field, respectively. We choose the fixed body force F as F := ∂t U + ν(−4)r U + (U.∇) U + ∇P

(1.6)

We make the following regularity assumptions on the prescribed data U and F : (
U ∈ L∞ 0, ∞; H2 (Ω) ∩ Vr (A1)
F ∈ L∞ 0, ∞; L2 (Ω) . Thus one application of the optimal control problem is to match a steady state flows field through the control of external forces. Observe that U is not an optimal solution because U in general does not satisfy the initial conditions. For technical reasons, we will need the following assumption 2

(A2)

|k∇U k| >

ν 2 λ1 8

1 − 4λ2r−2 1




Our plan of the paper is as follows: Section 2 is devoted to preliminary material. In Section 3 we construct a quasi-optimal control solution and some preliminary estimates for all solutions of the Generalized Navier-Stokes equation. In Section 4 we prove the existence of an optimal solution on the finite time interval. In Section 5 and Section 6 we will analyze semidiscrete and fully discrete approximations, respectively. Finally, in Section 7 the results of some computational experiments are presented. 2. Notation and formulation of the optimal control problem Throughout this work, C denotes a generic constant depending only on the physical domain Ω, the viscosity constant ν. We will use the standard notations for the function spaces Lp (Ω) with the norm denoted by k.kLp (Ω) and the Sobolev spaces H m (Ω) with the norm denoted by k.km . We simply denote by k.k the norm of L2 (Ω). The space H0m (Ω) is consisting of functions in H m (Ω) which vanish

303

DYNAMICS AND APPROXIMATIONS FOR 2D GENERALIZED NAVIER-STOKES EQUATION3

on boundary ∂Ω. The vector valued counterparts of these spaces are denoted by Lp (Ω), Hm (Ω) and Hm 0 (Ω). We now introduce the solenoidal spaces  Wr = u ∈ Hr−1 (Ω), ∇.u = 0 and u.n|∂Ω = 0  Vr = u ∈ Hr0 (Ω), ∇.u = 0 and 4u = ... = 4r−1 u = 0 on ∂Ω We identify the dual space of Wr with Wr itself under the L2 (Ω) inner product and the dual space of Vr is denoted by (Vr )∗ . We have Vr ⊂ (Vr )∗ , where the injections are continuous and each space is dense in the following one. Next, we introduce the temporal-spatial function spaces Lr (0, T ; Hm (Ω)) defined on QT = Ω × (0, T ) equipped with the norm !1/p Z T p kukLp (0,T ;Hm ) = ku(t)km dt , where p ∈ [1, ∞) . 0

We simply denote Q∞ by Q. The solenoidal temporal-spatial function space Hr (QT )

=



u ∈ L2 (0, T ; Vr ) ; ∂t u ∈ L2 (0, T ; (Vr )∗ )



that associated norm is given by 2

kvkHr

2

2

= kvkL2 (0,T ;Vr ) + k∂t vkL2 (0,T ;(Vr )∗ ) .

We denote by k|.|k the simplified norm notations of k.kL∞ (0,T ;L2 (Ω)) . This norm will be applied solely to U , ∇U and 4U . For a function u in a temporal-spatial space, we often use the notation u(t) := u(., t) to stand for the restriction of u at time t as a function defined over the spatial domain Ω. We introduce some standard continuous linear, bilinear and trilinear forms: Z k(u, p) = − p∇.ϕdx ∀ϕ ∈ Hr (Ω) ∀p ∈ L20 (Ω) ZΩ a2k (u, ϕ) = ν ((−4)k u).((−4)k ϕ)dx, k ∈ N∗ , ∀u, ϕ ∈ H2k (Ω), ZΩ a(2k+1) (u, ϕ) = ν ∇((−4)k u) : ∇((−4)k ϕ)dx, k ∈ N, ∀u, ϕ ∈ H2k+1 (Ω), Z Ω c(u, v, w) = (u.∇)v.wdx ∀u, v, w ∈ Hr (Ω) Ω

where the colon notation : denotes the inner product on R2×2 . Also, we denote by h., .i the duality pairing between a Banach space and its dual. Note that for all

304

4

DE G. AKMEL

AND

L. C. BAHI

u, v, w ∈ H1 (Ω), c have the following continuity properties (see [10]) |c(u, v, w)| ≤ 21/4 . kuk

(2.1)

1/2

. k∇uk

1/2

1/2

. k∇vk . kwk

1/2

. k∇wk

.

Also the trilinear form c have followings properties (2.2)

c(u, v, w) = −c(u, w, v) and c(u, v, v) = 0 for all u, v, w ∈ H1 (Ω).

Let λ1 > 0 be the greatest real number satisfying the Poincar´e inequality, ∀ϕ ∈ Hr (Ω) λ1 kϕk ≤ k∇ϕk .

(2.3)

Let Π : L2 (Ω) → Wr be the Leray operator (i.e., the orthogonal projection with respect to the L2 (Ω)−norm), it is well known (see [5] and [6]) that there are constants γ1 > 0 and γ2 > 0 depending only on Ω such that γ1 kΠ∆ϕk ≤ k∆ϕk ≤ γ2 kΠ∆ϕk ,

∀ϕ ∈ H2 (Ω) ∩ Hr0 (Ω).

So that kΠ∆.k is equivalent to the H2 (Ω)-norm on H2 (Ω) ∩ Hr (Ω)
Definition 2.1. Given T ∈ (0, ∞), u0 ∈ Wr and f ∈ L2 0, T ; L2 (Ω) , u is said to be a solution of the Generalized Navier-Stokes equation on (0, T ) if and only if u ∈ Hr (QT ) and u satisfies (2.4)

h∂t u(t), ϕi + ar (u(t), ϕ) + c (u(t), u(t), ϕ)

(2.5)

+k(ϕ, p(t)) = hf (t), ϕi

k(u(t), r) = 0

∀ϕ ∈ Vr a.e. t ∈ (0, ∞),

∀r ∈ L20 (Ω)

and (2.6)

lim u(t) = u0

t→0+

in Wr .

We point out that u ∈ Hr (QT ) implies u ∈ C ([0, T ]; Wr ). Hence, (2.6) makes sense. Now for T = ∞, we define a solution for the Generalized Navier-Stokes equation as follows.
Definition 2.2. Given u0 ∈ Wr and f ∈ L2loc 0, T ; L2 (Ω) , u is said to be a solution of the Generalized Navier-Stokes equation on (0, ∞) if and only if u ∈ L2loc (0, ∞; Vr ) ∩ L∞ (0, ∞; Wr ) , ∂t u ∈ L2loc (0, ∞; (Vr )∗ ) and u satisfies (2.4) − (2.6)with T = ∞.

305

DYNAMICS AND APPROXIMATIONS FOR 2D GENERALIZED NAVIER-STOKES EQUATION5

We define the admissible elements as follows with XT and YT denoting respectively the functional spaces as follows: XT X∞

= Hr (QT ) for T ∈ (0, ∞)  = u ∈ L2loc (0, ∞; Vr ) ∩ L∞ (0, ∞; Wr ) ; ∂t u ∈ L2loc (0, ∞; (Vr )∗ )

YT

= L2 (0, T ; (Vr )∗ ) for T ∈ (0, ∞),

Y∞

= L2loc (0, ∞; (Vr )∗ ) .

Definition 2.3. For a given T ∈ (0, ∞] , a pair (u, f ) ∈ XT × YT is called an admissible element if JT (u, f ) < ∞ and (u, f ) satisfies (2.4) − (2.6) . The set of all admissible elements are denoted by Uad (T ). Now for each T ∈ (0, ∞] , we state the optimal control problem on (0, T ) as follows: (2.7)

find a (u, f )

∈

Uad (T ) such that

JT (u, f ) ≤ JT (ω, h) ∀(ω, h) ∈ Uad (T ). We point out that in general, the initial state u0 is at a certain distance away from the desired flow, or u0 6= U (t) for all t, the cost functional generally has a positive minimum. We give following Lemma which we are proved in [2]. Lemma 2.4. For all u ∈ Vr , we have ^ r (2.8) || u||L2 ≥ λ2r−1 ||∇u||L2 1 where λ1 is a constant that appears in the Poincarr´e inequality and

V

= (−4).

The use of the Lemma 2.4, the Schwarz inequality and r integrations by parts give ∀u ∈ Vr , (2.9)

2r−2)

aνr (u, u) ≥ νλ1

k∇uk2 .

Also

2

 aν2k (v(t), −Π∆v(t)) = ν(−∆)2k v(t), −Π∆v(t) = ν Π∇(−∆)k v(t) and

2

 aν(2k+1) (v(t), −Π∆v(t)) = −ν∆(−∆)2k v(t), −Π∆v(t) = ν Π(−∆)k+1 v(t) . Throughout this paper we denote by v = u − U and g = f − F unless we specify them. Then (2.4) − (2.6) are equivalent to

306

6

DE G. AKMEL

v ∈ XT ∩ L2 (0, ∞; Vr ) ,

AND

L. C. BAHI


g ∈ YT ∩ L2 0, T ; L2 (Ω)

(2.10) h∂t v(t), ϕi + ar (v(t), ϕ) + c (v(t), v(t), ϕ) + c (v(t), U (t), ϕ) + c (U (t), v(t), ϕ) = hg(t), ϕi ,

∀ϕ ∈ Vr a.e. t ∈ (0, ∞)

and lim v(t) = u0 − U0 in Wr

(2.11)

t→0+

3. Preliminary estimates for the dynamics 3.1. A quasi optimizer. To estimate the dynamics of the optimal control solution, we need to find a sharp bound for the value of inf (u,f )∈Uad (T ) JT (u, f ). It is important that this bound is uniform in T . We now construct a quasi-optimizer (e u, fe) ∈ Uad (∞) for J∞ (.,.). We can in turn derive some preliminary estimates for the optimal solutions. By a quasi-optimizer we mean an element (e u, fe) ∈ Uad (∞) satisfying ke u(t) − U (t)k → 0 as t → ∞. The following Theorem asserts the existence of such an element. Theorem 3.1. that the assumptions (A1) and (A2) hold. Then there  Assume  e exists a pair u e, f ∈ Uad (∞) satisfying ∀t ≥ 0 (3.1)

2

2

ke u(t) − U (t)k ≤ ku0 − U0 k e−t

and ∀T ∈ (0, ∞] (3.2)

α ku0 − U0 k JT (e u, fe) ≤ 2

2

1 − e−T




with (3.3)

 := 2νλ2r−2 − 1

ν 2

−

4 νλ1

2

|k∇U k|

Remark 3.2. It follows from Theorem 3.1 that lim

min

JT (e u, fe) = 0. We see

T →∞(e u,fe)∈Uad (T )

that a quasi optimizer (e u, fe) has been created in the sense that ke u(t) − U (t)k → 0 e as t → ∞ and J∞ (e u, f ) is bounded. In fact, ke u(t) − U (t)k → 0 exponentially as t → ∞. The true optimizer is expected to have the property ke u(t) − U (t)k → 0 as t → ∞ and at the same time, minimize the work involved to realize and maintain the optimizer flow. 3.2. Estimate for the dynamics of admissible elements. In this section, we will derive some estimates for the dynamics of all solutions of (1.2) − (1.4). These estimates in turn will allow us to derive preliminary estimates for the dynamics of
the optimal solutions. First we consider the L∞ 0, T ; L2 (Ω) estimates in terms of the initial data and the functional values.

307

DYNAMICS AND APPROXIMATIONS FOR 2D GENERALIZED NAVIER-STOKES EQUATION7

Theorem 3.3. Let T ∈ (0, ∞]. Assume that the assumptions (A1) and (A2) hold. If (u, f ) ∈ Uad (T ), then ∀t ∈ [0, T ] , 2 2 2 ku(t) − U (t)k ≤ ku0 − U0 k + √ JT (u, f ). αβ

(3.4) If in addition,

JT (u, f ) ≤ JT (e u, fe) then 2

2

ku(t) − U (t)k ≤ K0 ku0 − U0 k

(3.5)

 where  and (e u, fe) are defined in Theorem 3.1 and K0 = 1 +

1 2

q  α β

.

Proof. Setting ϕ = v in (2.10) and applying the Schwarz and the Young inequalities we find d 2 dt kv(t)k

(3.6)

+ kv(t)k2 ≤

√1 (αk(v)(t)k2 αβ

+ βkg(t)k2 )

Multiplying both sides of this inequality by et and then integrating in t over (0, t), lead us to 2

kv(t)k

Z t  1 2 2 ≤ kv(0)k e +√ α kv(s)k + β kg(s)k e(s−t) ds αβ 0 2 2 −t ≤ kv(0)k e + √ JT (u, f ). αβ 2 −t

This yields the inequality (3.4) . Moreover combining the condition JT (u, f ) ≤ JT (e u, fe) with the inequality (3.4) and the Theorem 3.1 we find the inequality (3.5) .  Now, using the uniform Gronwall’s inequality we derive L∞ (0, T ; Hr ) estimates. Theorem 3.4. Let T ∈ (0, ∞] and (u, f ) ∈ Uad (T ). Assume that the assumptions (A1) and (A2) hold and assume further that JT (u, f ) ≤ JT (e u, fe). Then for each ε > 0, we have u − U ∈ L2 (0, T ; Hr (Ω)) ∩ L∞ (ε, T ; Hr (Ω)) ∩ C ([ε, T ] ; Hr (Ω)) , with Z

T

2

k∇u(s) − ∇U (s)k ds ≤ K1 ku0 − U0 k

(3.7)

2

0

and (3.8)

2

2

k∇u(t) − ∇U (t)k ≤ K2 ku0 − U0 k , ∀t ≥ ε,

where K1 =

λ1 ε

 r  1 α 1+ ε β

308

8

DE G. AKMEL

AND

L. C. BAHI

and  K2 = 2CK0

1 2 + ν3 θν



2

ku0 − U0 k

and K3 = 2 (C5 + C6 ) . The following preliminary estimates for the optimal solutions is an immediate consequence of Theorems 3.3 and 3.4 Theorem 3.5. Assume that the assumptions (A1) and (A2) hold. Let T ∈ (0, ∞] and (b u, fb) ∈ Uad (T ) be an optimal solution for (2.7) . Then 2

2

kb u (t) − U (t)k ≤ K0 ku0 − U0 k

(3.9) Z

T

2

k∇b u(s) − ∇U (s)k ds ≤ K1 ku0 − U0 k

(3.10)

2

0

and 2

2

k∇b u(t) − ∇U (t)k ≤ K2 (ε) ku0 − U0 k

(3.11)

∀t ≥ ε, where all constants are as defined in Theorem 3.3 and Theorem 3.4. 3.3. Existence of solution and dynamics of optimal controls. The existence results are similar to the results from Generalized MHD equations [2], in both case, finite time interval and infinite time interval. The following Theorem gives the results. Theorem 3.6. • Let T ∈ (0, ∞) . Then there exists an optimal solution b (b u, f ) ∈ Uad (T ) for the problem (2.7) , i.e. there exists at least an element
b f ∈ L2 0, T ; L2 (Ω) and u b ∈ C ([0, T ]; Wr ) ∩ L2 (0, T ; Vr ) such that the functional JT (u, f ) attains its minimum at (b u, fb) and u b satisfies (2.4)−(2.6) with fb = f. • There exists an optimal solution (b u, fb) ∈ Uad (T ) for (2.7) with T = ∞. For many feedback control models, the controlled flow exponentially decays to the desired flow. For our optimal control system, Theorem 3.4 and Theorem 3.5 gave some preliminary results as ku (t) − U (t)k stays bounded. Lemma 3.7. Let T ∈ (0, ∞) . Assume that (u, f ) ∈ Uad (T ) and λ1 > 1. If k(u, b) (t) − U (t)k > 0 for all t ∈ (t1 , t2 ) ⊂ [0, T ] , then √ 1/2 ku (t2 ) − U (t2 )k ≤ ku (t1 ) − U (t1 )k + K4 t2 − t1 (JT (u, f ))

309

DYNAMICS AND APPROXIMATIONS FOR 2D GENERALIZED NAVIER-STOKES EQUATION9

 with K4 =

1 α


2 2 ν



2

|k∇U k|

2

+

1 β

1/2 .

If in addition, the assumptions (A1) and (A2) hold and JT (u, f ) ≤ JT (e u, fe), where (e u, fe) is defined in Theorem 3.1, then √ ku (t2 ) − U (t2 )k ≤ ku (t1 ) − U (t1 )k + K4 t2 − t1 ku0 − U0 k

r

α . 2

Proof. By setting ϕ = v(t) in (2.10) we obtain 2

2

d kv(t)k + 1 kv(t)k ≤ C0 . kv(t)k + kg(t)k . kv(t)k kv(t)k dt

where   2(2k−1) 1 = νλ1 λ1 − 1 and C0 =

1 ν

2

|k∇U k| .

If kv(t)k > 0 for all t ∈ (t1 , t2 ) , then we may divide this inequality by kv(t)k, multiplying by e1 t and then integrating over (t1 , t2 ), we are led to kv(t2 )k e

1 t2

≤ kv(t1 )k e

1 t1

+



1 2 α C0

+

1 β

1/2 Z

t2



2

2

α kv(t)k + β kg(t)k

1/2

e1 t dt

t1

we have kv(t2 )k ≤ kv(t1 )k e−1 (t2 −t1 ) 1/2 Z  1/2 Z t2 2 2 1 1 2 + α C0 + β (α kv(t)k + β kg(t)k )dt t1

t2

e

−2ε1 (t2 −t)

1/2 , dt

t1

with e−1 (t2 −t1 ) < 1 1/2 Z t2 1/2 1 2 1 1/2 C0 + (JT (u, f )) . e−2ε1 (t2 −t) dt α β t1  1/2 √ 1 2 1 1/2 kv(t1 )k + t2 − t1 C + (JT (u, f )) , α 0 β 

kv(t2 )k

≤ ≤

kv(t1 )k +

where we have used the fact that 1 − e−y ≤ y for y ≥ 0. Hence, we have shown (3.12) and (3.12) simply follows from the bound (3.2) so that applying the mean value theorem to the last factor we have the result.



We give the asymptotic decay property of ku(t) − U (t)k as t → ∞ for any (u, f ) ∈ Uad (∞). Theorem 3.8. Assume that (u, f ) ∈ Uad (T ). Then (3.12)

lim ku(t) − U (t)k = 0.

t→∞

310

10

DE G. AKMEL

AND

L. C. BAHI

4. Semidiscrete approximations of the piecewise optimal control problem We semidiscretize the functional J(tn ,tn+1 ) (u, f ) by the right-endpoint rectangle rule Z tn+1 ϕ(t)dt ≈ (tn+1 − tn ) ϕ(tn+1 ) = δϕ(tn+1 ) so that the semidiscretized functn

tional becomes J n+1 (u, f ) =



δα

u(n) − U n+1 2 + δβ f − F n+1 2 , 2 2

∀u ∈ Vr , ∀f ∈ L2 (Ω) ,

where U n+1 = U n+1 (x) = U (x, tn+1 ) and F n+1 = F n+1 (x) = F (x, tn+1 ) with tn = δn for n = 0, 1, 2, . . . For convenience, we define Ln+1 (u, f ) =



α

u − U n+1 2 + β f − F n+1 2 , 2 2

so that the minimization of the functional J n+1 (u, f ) is equivalent to the minimization of the functional Ln+1 (u, f ) . Using the techniques of [7] concerning optimal control problems for the steady-state Navier-Stokes equations, we can show the existence of a solution (b u, pb, fb)n+1 for the (n + 1)th optimal control problem. The remainder of this Section will be devoted to the study of u bn as n → ∞. We now study the behavior of the semidiscrete solutions u bn as n → ∞. By finite difference approximation formula ∂t U (x, t) =

1 (U (x, t + ∆t) − U (x, t)) − ∂tt U (x, t + α∆t).∆t, ∆t

def

where α = α(x, t) with |α| < 1, we have that

(4.1)



1 n+1  1 U , ϕ + ar U n+1 , ϕ + c U n+1 , U n+1 , ϕ = hU n , ϕi ∆t ∆t

n+1  n+1  + f ,ϕ − τ , ϕ , ∀ϕ ∈ Vr

and (4.2)


k U n+1 , r = 0,

∀r ∈ L20 (Ω)

where (4.3)

τ n+1 = ∆t.∂tt U (x, tn + α(xn , tn )∆t)∆t).

Lemma 4.1. Assume that hypotheses (A1)-(A2) and (A4)

∂t U ∈ C [0, ∞) ; H1






∂tt U ∈ L∞ 0, ∞; L2 (Ω) ∩ C [0, ∞) ; L2 (Ω)

311

DYNAMICS AND APPROXIMATIONS FOR 2D GENERALIZED NAVIER-STOKES EQUATION 11

hold. Assume further that



n+1 u b, pb, fb is a solution of the (n + 1)th semidiscrete

optimal control problem for n = 1, 2, . . .. Then ! 3  α kb  n n 2 u − U k C (∆t) 6 n+1 n+1 ≤ (b u, fb) + (4.4) L 2 1 + C5 ∆t 1 + C5 ∆t where (4.5)

def

C5 = C5 (ν, Ω) =

Proof. Let (e u, pe) (4.6)

(4.7)

λ1 2

def

and C6 = C6 (ν, Ω, U ) =

2

2 |k∂tt U k| . λ1

n+1

be a solution of the equations

1 n+1 he u , ϕi + ar u en+1 , ϕ + c u en+1 , u en+1 , ϕ ∆t


1 n hb u , ϕi + f n+1 , ϕ , ∀ϕ ∈ Vr + k ϕ, pen+1 = ∆t
k u en+1 , r = 0, ∀r ∈ L20 (Ω)

(The existence of such a (e u, pe)

n+1

can be proved by using the techniques for

proving the existence of a solution for the steady-state Navier-Stokes equations).  n+1 Set fen+1 = F n+1 ; then we see that u e, fe, pe satisfies the semidiscrete NavierStokes equations (4.6) − (4.7). Let ven+1 = u en+1 − U n+1 , vbn = u bn − U n and qen+1 = pen+1 . Then by subtracting (4.1) − (4.2) from (4.6) − (4.7), we obtain (4.8)


1 n+1  ve , ϕ + ar ven+1 , ϕ + e c ven+1 , ven+1 , ϕ + e c U n+1 , ven+1 , ϕ ∆t



1 hb v n , ϕi − τ n+1 , ϕ , ∀ϕ ∈ Hr0 (Ω) +e c ven+1 , U n+1 , ϕ + k ϕ, pen+1 = ∆t and
(4.9) k ven+1 , r = 0, ∀r ∈ L20 (Ω) . Setting ϕ = ven+1 in (4.8), we have by Young‘s inequality

2 

1  2

ven+1 2 − kb v n k + vbn+1 − ven 2∆t (4.10)



 n+1

2 ≤ λ1 ven+1 2 + 1 τ n+1 2 . + ∇e v 2 4 λ1

2 Dropping the term vbn+1 − ven , applying Poincar´e inequality and rearranging, we have  λ

1  1 n+1 2

2 ≤ 1 τ n+1 2 ,

ven+1 2 − kb vn k + ve 2∆t 4 λ1 so that using the estimates

n+1

≤ ∆t| k∂tt U k | and | k∂tt U k | = k∂tt U k

τ , ∞

(4.11)

L

we are led to

2 2 3 (1 + C5 ∆t) ven+1 ≤ kb v n k + C6 (∆t) ,

312

(Ω)

12

DE G. AKMEL

AND

L. C. BAHI

where C5 and C6 are defined by (4.5). Hence, we arrive at n+1

L



 α

2 α (e u, fe)n+1 = ven+1 ≤ 2 2

2

kb vn k C6 3 + (∆t) (1 + C5 ∆t) (1 + C5 ∆t)

! ,

(b u, pb, fb)n+1 being a solution for the (n + 1)th optimal control problem, the desired estimate follows trivially from this last inequality.



Theorem 4.2. Assume that the hypotheses (A1)-(A2) and (A4) hold and 0 < ∆t ≤ 1. Then there are positive constants ξ1 and ρ1 such that

2

n+1 2 3

u un − U n k + C6 (∆t) b − U n+1 ≤ (1 − ξ1 ∆t) kb with 1 − ξ1 ∆t > 0 and (4.12)

2

2

2

kb un − U n k ≤ ku0 − U0 k .e−ξ1 tn + ρ1 (∆t)

where (4.13)

ξ1 =

C5 −

p α/β

and

2

(1 + C5 ∆t)

ρ1 =

C6 . ξ1

In the semidiscretization of the Navier-Stokes equations we used the first-order backward Euler scheme. Therefore, the appearance of the term O(∆t) in the last estimate is expected. If we use higher-order approximation scheme, we expect to obtain improved estimates. However, the analysis in the context of semidiscrete piecewise optimal control with more sophisticated schemes becomes complicated. The proof of Theorem 4.2 gives a rough estimate of k∇b un − ∇U n k = k∇b vn k . Proposition 4.3. Assume that the conditions of Theorem 4.2 hold. Then  r  2 α −ξ1 δ 2 n n 2 ∆tk∇b u − ∇U k ≤ 1+ e ku0 − U0 k e−ξ1 tn  β  r  (4.14) 2 α 2 + 1+ (ρ1 + C6 ) (∆t) .  β We now derive an improved bound for the eventual error in H0n norm. We first observe the following direct consequence of (4.14). Lemma 4.4. Assume that the conditions of Theorem 4.2 hold. Then for any constant σ > 0, there exist constants 0 = 0 (Ω, ν; σ) > 0 and e t=e t (Ω, ν, u0 , U0 ; σ) > 0 such that (4.15)

2

∆t k∇b v n k ≤ σ,

∀tn ≥ e t, ∀∆t ∈ (0, 0 ) .

We also need a stronger version of Proposition 4.3.

313

DYNAMICS AND APPROXIMATIONS FOR 2D GENERALIZED NAVIER-STOKES EQUATION 13

Proposition 4.5. Assume that the conditions of Theorem 4.2 hold. then for each n ≥ 1, (4.16)



Ln+1 (b u, fb)n+1



α ≤ 2

2

2

3

ku0 − U0 k e−ξ1 tn + ρ1 (∆t) + C6 (∆t) 1 + C5 ∆t

! .

Moreover, for all n2 ≥ n1 ≥ 1, (4.17)   n2 ∆t P 2 2 2 2 k∇b v n k ≤ kb v n1 k + C7 (tn2 − tn1 ) ku0 − U0 k e−ξ1 tn1 + (∆t) 2 n=n1 +1 where r C7 := C7 (ν, Ω, U ) =

α β

r ! β 1 + ρ1 + C6 . α

Proof. Combining (4.4) and (4.12) yields (4.16). By using (4.16)together with (4.12), we obtain that r

2 r α

n+1 2 ∆t α 2 3 2

− kb

vb

∇b vn k + v n+1 ≤ ku0 − U0 k e−ξ1 tn ∆t + (∆t) 2 β β Summing up n over n1 ≤ n ≤ n2 − 1, we have (4.17).

r ! β ρ1 + C 6 . α 

Theorem 4.6. Assume that the hypotheses (A1)-(A4) hold. Then there exist constants 0 = 0 (Ω, ν; σ) > 0 and e t=e t (Ω, ν, u0 ; σ) > 0 such that  
2 2 2 k∇b un − ∇U n k ≤ C8 τ1 + 1 + τ ku0 − U0 k e−ξ1 (tn −τ ) + (∆t) o n  (4.18) 4 4 . exp C9 (1 + τ ) ku0 − U0 k e−2ξ1 (tn −τ ) + (∆t) , ∀ ∆t ∈ (0, 0 ) and ∀ tn ≥ e t, where ξ1 is as in Theorem 4.2 and C8 , C9 are constants depending only on Ω, ν, U and B. 5. Fully discrete approximations of the piecewise optimal control problem Let Xh ⊂ Hr0 (Ω) and Sh ⊂ L20 (Ω) be two families of the finite-dimensional subspaces. First, we have the approximation properties: there exist an integer k ≥ 1 and a constant C 0 > 0, independent of h, u and p such that for 1 ≤ m ≤ k inf ku − uh k1 ≤ C 0 hm kukm+1 ∀u ∈ Hm+1 (Ω) ∩ Hr0 (Ω),

uh ∈Xh

inf kp − ph k0 ≤ C 0 hm kpkm

ph ∈Sh

∀p ∈ H m (Ω) ∩ L20 (Ω).

Next, we assume the inf-sup condition, or Ladyzhenskaya-Babuska-Brezzi condition there exists a constant C 00 , independent of h, such that (5.1)

k (uh , ph ) ≥ C 00 . 06=ph ∈Sh 06=uh ∈Vh,0r kuh k1 kph k0 inf

sup

314

14

DE G. AKMEL

AND

L. C. BAHI

This condition assures the stability of finite-element discretizations of the Navier def

Stokes equations. For each n ≥ 0, we define the affine space Yhn+1 = {fh = yh + Fhn+1 : yh ∈ Xh }, for the approximate distributed controls, where Fhn+1 is the L2 projection of F n+1 onto Xh . In order to preserve the antisymmetric property of the trilinear form c(., ., .), we introduce the form c (u, v, w) = 21 {c (u, v, w) − c (u, w, v)}

(5.2)

It can be easily verified that c (u, v, w) = c (u, v, w) ,

c (u, v, w) = −c (u, w, v)

and c (u, v, v) = 0

on all H10 (Ω) × H10 (Ω) × H10 (Ω). We also have (5.3)

|c (u, v, w) | ≤ C 0 k∇uk . kvkL∞ . k∇wk ,

(5.4)

|c (u, v, w) | ≤ C 1 k∇uk . k∇vk . k∇wk

(5.5)

|c (u, v, w) | ≤ C 2 kuk2 . kvk . k∇wk

for all u ∈ H2 (Ω) ∩ H10 (Ω) and v, w ∈ H10 (Ω), where C 0 , C 1 and C 2 are positive reals. We define the fully discrete approximations of the piecewise optimal control problem. • Set ∆t = δ. • Define u b0h = u0,h where u0,h is the L2 (Ω)-projection (or interpolation) of u0 onto Xh . • The (n+1)th fully discrete optimal control problem:  n+1 for n = 0, 1, 2, . . . , find u b, pb, fb ∈ Xh × Sh × Zhn+1 such that the functional def

Ln+1 (un+1 , fhn+1 ) = h h



α

un+1 − U n+1 2 + β f n+1 − F n+1 2 h h 2 2

∀un+1 ∈ Xh , ∀fhn+1 ∈ Zhn+1 h

is minimized subject to the fully discrete Generalized Navier Stokes equations 

1 n+1 uh , ψh + ar un+1 , ϕh + c un+1 , vhn+1 , ϕh h h ∆t


 1 n + k ϕh , pn+1 = hb uh , ϕh i + fhn+1 , ϕh , ∀ϕh ∈ Xh h ∆t

(5.6)

and (5.7)


k un+1 , rh = 0, h

315

∀rh ∈ Sh .

DYNAMICS AND APPROXIMATIONS FOR 2D GENERALIZED NAVIER-STOKES EQUATION 15

Using the techniques of [7] concerning finite element approximations of optimal control problems for the steady-state Navier-Stokes equations, we can show the existence of a solution u bn+1 , pbn+1 , fbn+1 for the (n + 1)th fully discrete optimal h

h

h

control problem. We now study the behavior of the fully discrete solutions u bnh as n → ∞. For every t, we introduce an auxiliary element Uh (t), Ph (t) ∈ Xh × Sh determined by (5.8)

ar (Uh (t), ϕh ) + k (ϕh , Ph (t)) = ar (U (t), ϕh )

∀ϕh ∈ Xh

and (5.9)

k (Uh (t), rh ) = 0 ∀rh ∈ Sh .

The existence of such a (Uh (t), Ph (t)) follows from the well-known results for the finite element approximations of the steady-state Navier-Stokes equations. Furthermore, under the assumption that there is a k ≥ 1 such that  
(A6) U ∈ C [0, ∞); Hk+1 (Ω) ∩ L∞ 0, ∞; Hk+1 (Ω) . The following error estimates hold: (5.10) kUh (t) − U (t)k1 + kPh (t)k ≤ C 3 hk kU (t)kk+1 ≤ C 3 hk kU kL∞ (0,∞;Hk+1 (Ω)) and (5.11)

kUh (t) − U (t)k ≤ C 4 hk+1 kU (t)kk+1 ≤ C 4 hk+1 kU kL∞ (0,∞;Hk+1 (Ω))

where C 3 and C 4 are constant depending on Ω only; see, e.g. [8], By differentiating (5.8), (5.9) with respect t, we see that (∂t Uh (t), ∂t Ph (t)) satisfies a system of equations similar to (5.8), (5.9) so that under the assumption  
(A7) ∂t U ∈ C [0, ∞); Hk+1 (Ω) ∩ L∞ 0, ∞; Hk+1 (Ω) , we have the error estimates (5.12) k∂t Uh (t) − ∂t U (t)k1 + k∂t Ph (t)k ≤ C 3 hk k∂t U (t)kk+1 ≤ C 3 hk k∂t U kL∞ (0,∞;Hk+1 (Ω)) and (5.13) k∂t Uh (t) − ∂t U (t)k ≤ C 4 hk+1 k∂t U (t)kk+1 ≤ C 4 hk+1 k∂t U kL∞ (0,∞;Hk+1 (Ω))

∀s ∈ [0, 2].

By differentiating (5.8), (5.9) twice with respect t, we see that (∂tt Uh (t), ∂tt Ph (t)) also satisfies a system of equations similar to (5.8), (5.9) so that under the assumption (A8)

 
∂tt U ∈ C [0, ∞); Hk+1 (Ω) ∩ L∞ 0, ∞; Hk+1 (Ω)

316

16

DE G. AKMEL

AND

L. C. BAHI

we have the error estimates (5.14) k∂tt Uh (t) − ∂tt U (t)k1 + k∂tt Ph (t)k ≤ C 3 hk k∂tt U (t)k1 ≤ C 3 hk k∂tt U kL∞ (0,∞;H1 (Ω)) and (5.15) k∂tt Uh (t) − ∂tt U (t)k ≤ C 4 hs k∂tt U (t)ks ≤ C 4 hs k∂tt U kL∞ (0,∞;Hs (Ω))

∀s ∈ [0, 1]

in particular, (5.16)

k∂tt Uh (t) − ∂tt U (t)k ≤ C 4 k∂tt U (t)ks ≤ C 4 k∂tt U kL∞ (0,∞;H1 (Ω)) .

Note that the regularity assumption (A8) for ∂tt U is weaker than the assumption (A6) for U or (A7) for ∂t U . The proof of the following Lemma is same as [8]. Lemma 5.1. Assume that hypotheses (A1), (A2), (A4), (A6), (A7) and (A8) hold. Assume that further kU kL∞ (0,∞;L4 (Ω))
0 and constants K 1 , K 2 and K 3 such that for all h ≤ h0 and all n, (5.17) Ln+1 (b un+1 , fbhn+1 ) h h

kb un+1 − Uhn+1 k2 K 2 h2k+2 ∆t K 3 (∆t)3 h + + ≤α 1 + λ1 K 1 ∆t 1 + λ1 K 1 ∆t 1 + λ1 K 1 ∆t 2 + αC 4 h2k+2 kU k2L∞ (0,∞;Hk+1 (Ω)) 



where def

(5.18)

(5.19)

h0 = min

def

K1 =

 

ε − C 0 kU kL∞ (0,∞;L4 (Ω))

!1/k

 C 1 C 3 kU kL∞ (0,∞;Hk+1 (Ω))

,1

  

 1 ε − C 1 C 3 hk kU kL∞ (0,∞;Hk+1 (Ω)) − C 0 kU kL∞ (0,∞;L4 (Ω)) 2

(5.20) def

K2 =

4  2 2 2k+2

U n+1 2 kU k2 ∞ 4C 2 C 4 h L (0,∞;Hk+1 (Ω)) 2 K1 ! 2 C 4 h2k 2 4 4k 4 + C 1 C 3 h kU kL∞ (0,∞;Hk+1 (Ω)) + k∂t U k2L∞ (0,∞;Hk+1 (Ω)) λ1

(5.21)

def

K3 =

2 2 2 (C + 1) k∂tt U kL∞ (0,∞;Hk+1 (Ω)) λ1 4

with the constants C 0 , C 1 , C 2 , C 3 and C 4 defined by (5.3), (5.4), (5.5), (5.10) and (5.11), respectively .

317

DYNAMICS AND APPROXIMATIONS FOR 2D GENERALIZED NAVIER-STOKES EQUATION 17

Theorem 5.2. Assume that the hypotheses of Lemma 5.1 hold. Assume further that u0 ∈ Hk+1 (Ω) and α (λ1 K 1 )2 < . β 8

(A10)

where K 1 is defined by (5.7). Let h0 be defined by (5.18). Then there are positive constants δ0 , K 4 , K 5 , K 6 and κ such that for all h ≤ h0 and all ∆t ≤ δ0 , (5.22)

kb un+1 − Uhn+1 k2 ≤ (1 − K 4 ∆t)kb unh − Uhn k2 K 5 (∆t)3 + K 6 h2k+2 (∆t) h

and (5.23)

kb unh − Uhn k2 ≤ 3e−K 4 tn kb u0 − U 0 k2 + κ[(∆t)2 + h2k+2 ].

As a consequence of Theorem 5.2 and the triangle inequality 2

2

2

kb u(tn ) − u bnh k ≤ 2 kb u(tn ) − U n k + 2 kU n − u bnh k

we obtain an estimate for the difference between the continuous and fully discrete solutions of the piecewise optimal control problem. Remark 5.3. In order to solve the (n + 1)th fully discrete optimal control problem for each n, we need to introduce a Lagrange multiplier (ξbn+1 , π bn+1 ) to convert the h

h

(n + 1)th fully discrete optimal control problem into a discrete optimality system of equations (similar to the semi discrete case). 6. Computational example Thanks to GNU licence, we have implemented the following algorithm. (a) initialization: • Chose a (sufficiently small) δ > 0 and set ∆t = δ. Choose h (sufficiently small). • Define u0h = Uh0 where Uh0 is the L2 (Ω) projection of U 0 on to Xh . (b) solving the (n + 1)th fully discrete optimal control problem: For n = 0, 1, 2, ..., find a (un+1 , pn+1 , ξhn+1 , πhn+1 ) ∈ Xh × Sh × Xh × Sh such that h h (6.1)

(6.2)

(6.3)




1 n+1 uh , ϕh + ar un+1 , ϕh + c un+1 , un+1 , ϕh + k ϕh , pn+1 h h h h ∆t

 1 = Fhn+1 − β −1 ξhn+1 , ϕh + hun , ϕh i , ∀ϕh ∈ Xh , ∆t h
k un+1 , qh = 0, ∀qh ∈ Sh , h 


1 n+1 ξh , ϕh + ar ξhn+1 , ϕh + k ϕh , πhn+1 + c ϕh , un+1 , ξhn+1 h ∆t


 +c un+1 , ϕh , ξhn+1 = α un+1 − U n+1 , ϕh , ∀ϕh ∈ Xh , h h

318

18

DE G. AKMEL

AND

(6.4)


k ξhn+1 , rh = 0,

L. C. BAHI

∀rh ∈ Sh ,

(c) Set fhn+1 = Fhn+1 − β −1 ξhn+1 We use a gradient method to implement this algorithm. The finite elements are chosen to be the Taylor-Hood elements; i.e., the finite element space Vh is chosen to be piecewise biquadratic elements (for uh and ξh ) and Sh is chosen to be piecewise linear elements (for ph and πh ). Newton s method is used to solve the finitedimensional nonlinear system of equations. We choose the domain Ω = (0, 1) × (0, 1). The desired velocity is given by U (x, t) = (U1 (x, y), U2 (x, y)) where U1 =

d U2 = − dx φ(t, x)φ(t, y)

d dy φ(t, x)φ(t, y)

with φ(t, z) = (1 − z)2 (1 − cos(2kπzt)),

z ∈ [0, 1].

The integer parameter k involved in U adjusts the number of eddies of circulation presented in the desired flow, thus determines the complexity of the desired flow. We choose the kinematic viscosity ν = 1/Re = 0.01, the time step ∆t = 0.1, h = 1/16, α = 10 and β = 0.1 For the initial velocity we choose U10 = (cos(2πx) − 1) sin(2πy)

and

U20 = sin(2πx)(1 − cos(2πy))

Fig. 1: Controlled (first row) and target (second row) at t = 0.0, t = 0.15, t = 0.5 and t = 1. In our numerical computations, we observed that the graphics for the decreasing of the error ku − U k does’nt change enough when we pass from the case r = 1 to the case r = 2.

319

DYNAMICS AND APPROXIMATIONS FOR 2D GENERALIZED NAVIER-STOKES EQUATION 19

Fig. 2: The error graphics for β = 0.1 and β = 0.001, respectively. More over the quickness of the decreasing of the error ku−U k between the controlled flow u and the target flow U depends on β. Indeed the more β becomes small, more the decreasing is rapid. References [1] F. Abergel and R. Temam, On some optimal control problems in fluid mechanics, Theoret. Compt. Fluid Dynamics, 1 (1990) pp. 303-325 [2] D´ e G. Akmel and L. Bahi, Dynamics for controlled 2D Generalized MHD systems with distributed controls, J. Part. Diff. Eq., Vol. 26, No. 1, pp. 48-75,(2013). [3] A. Cheskidev and M. Dai, Norm inflation for Generalised Navier-Stokes Equations,... (2013) [4] H. Chun Lee and B. Chun Shin, Dynamics for controlled 2-D Boussinesq systems with distributed controls, J. Math. Anal. Appl. 273 (2002) 57-479 [5] P. Constantin and C. Foias, Navier-Stockes Equations, University of Chicago, Chicago, 1988. [6] V. Girault and P. Raviart, Finite element Methods for Navier-Stockes Equations, Springer-Verlag, Berlin, 1986. [7] M. Gunzburger, L. Hou, and T. Svobodny, Analysis and finite element approximation of optimal control problems for stationary Navier-Stockes equations with distributed and Neumann controls, Math. Comp. 57 (1991), pp. 123-161 [8] L.S. Hou and Y. Yan, Dynamics for controlled Navier–Stokes systems with distributed control, SIAM J. Control Optim. 35 No. 2,(1997) 654–677. [9] S. S. Ravindran, On the Dynamics of controlled magnatohydrodynamic system, Nonlinear Analysis Modelling and Control, 2008, vol 13 No 3, 351-377. [10] R. Temam, Navier-Stockes Equations, Theory and Numerical Methods, North-Holland, Amsterdam, (1980) [11] J. Wu, The Generalized Incompressible Navier-Stokes Equations in Besov Spaces, Dynamics of PDE, Vol.1, No.4, (2004), 381-400. ´ FHB, UFR de Mathe ´matiques et Informatique: 22 BP 582 Abidjan, Ivory Universite Coast. E-mail address: [email protected] (De G. Akmel),[email protected] (L. C. Bahi)

320

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.'S 3-4, 321-330, 2014, COPYRIGHT 2014 EUDOXUS PRESS LLC

Some Applications on Generating Functions Ali Boussayoud, Mohamed Kerada, Rokiya Sahali and Wahiba Rouibah August 31, 2013 In this paper, we calculate the generating functions by using the concepts of symmetric functions. Although the methods cited in previous works are in principle constructive, we are concerned here only with the question of manipulating combinatorial objects, known as symmetric operators. The proposed generalized symmetric functions can be used to find explicit formulas of the Fibonacci numbers, and of the Tchebychev polynomials of first and second kinds. Moreover, we give new results for the product of Hadamard.

1

Introduction

By studying the Fibonacci sequence (Fn +2 = Fn +1 + Fn with F0 = F1 = 1), we note its close connection with the equation x2 = x + 1, whose roots are the golden numbers Φ1 and Φ2 . It is also noticed that the eigenvalues of the symmetric matrix   1 1 M= (1) 1 0 represent the two golden numbers Φ1 and Φ2 of Fibonacci sequence [3]. Consequently, we obtain the following Vieta’s formulas σ 1 = λ1 + λ2 = 1 and σ 2 = λ1 λ2 = −1

(2)

where σ 1 , σ 2 are called elementary symmetric functions of real roots λ1 , λ2 , respectively. So, the eigenvectors of matrix M are multiples of     λ1 λ2 → − − v1 = and → v2 = (3) 1 1 If we assume that |λ1 | > |λ2 |, then for any positive integer n, we have [3]   Sn (λ1 + λ2 ) σ 1 Sn−1 (λ1 + λ2 ) n M = Sn−1 (λ1 + λ2 ) σ 2 Sn−2 (λ1 + λ2 ) λn+1 −λn+1

(4)

where Sn (λ1 + λ2 ) = 1 λ1 −λ22 . In this paper, we are interested in the use of symmetric functions to generate the well-knwon Fibonacci numbers and Tchebychev polynomiales of first and 1

321

BOUSSAYOUD ET AL: GENERATING FUNCTIONS

second kinds. In this framework, some necessary preliminaries and definitions are given in Section 2. In Section 3, we propose a new theorem which allows the determination of the generating functions. The proposed theorem is based on symmetric functions and a new proposition on the symmetric operators. In Section 4, some applications are given for the generating functions of Fibonacci numbers and Tchebychev polynomials. The products of Hadamard are given in Section 5.

2 2.1

Preliminaries Definition of symmetric functions in several variables

Consider an equation of degree n of the form (x − λ1 )(x − λ2 ) · · · (x − λn ) = 0

(5)

with λ1 , λ2 , . . . , λn being real roots. If we expand the left hand side, we obtain xn − σ 1 xn−1 + σ 2 xn−2 − σ 3 xn−3 + · · · + (−1)n σ n = 0

(6)

where σ 1 , σ 2 , . . . , σ n are homogeneous and symmetrical polynomials in λ1 , λ2 , . . . , λn . To be more accurate, these polynomials can be denoted as σ i (λ1 , λ2 , . . . , λn ) (n) with i = 1, 2, . . . , n, or simply as σ i . (n) The general formula of the polynomials σ i are given by [9] X (n) mn 1 m2 λm (7) σi = 1 λ2 . . . λn m1 +m2 +···+mn =i

with m1 , m2 , ..., mn = 0 or 1. (n) The polynomials σ i can be considered as the sum of all distinct products (n) that can be formed by monomial polynomials Cni . It is noticed that σ i = 0 for i > n.

2.2

Symmetric functions

Let A and B be two alphabets, we denote by Sn (A − B) the coefficients of the rational sequence of poles A and zeros B as follows [2] Q (1 − bz) ∞ X b∈B n (8) Sn (A − B)z = Q (1 − az) n=0 a∈A

Equation (8) can be rewritten in the following form ! ! ∞ ∞ ∞ X X X Sn (A − B)z n = Sn (A)z n × Sn (−B)z n n=0

n=0

n=0

2

322

(9)

BOUSSAYOUD ET AL: GENERATING FUNCTIONS

with Sn (A − B) =

n X

Sn−j (−B)Sj (A)

(10)

j=0

The polynomial whose roots are B is written as Sn (x − B) =

n X

Sn−j (−B)z n , with card(B) = n

(11)

j=0

On the other hand, if A has cardinality equal to 1, i.e., A = {x} , then equality (8) can be rewritten as follows [1] Q (1 − bz) ∞ X Sn (x − B) n b∈B n = 1 + · · · + Sn−1 (x − B)z n−1 + z Sn (x − B)z = (1 − xz) (1 − xz) n=0 (12) where Sn+k (x − B) = xk Sn (x − B) for all k ≥ 0. The summation is actually limited to a finite number of terms since S−k (·) = 0 for all k > 0. In particular, we have Y (x − b) = Sn (x−B) = S0 (−B)xn +S1 (−B)xn −1 +S2 (−B)xn −2 +· · · (13) b∈B

where Sk (−B) are the coefficients of the polynomials Sn (x − B) for 0 ≤ k ≤ n. This coefficients are zero for k > n. For example, if all b ∈ B are equal, i.e., B = nb, then we have Sn (x − nb) = (x − b)n     By choosing b = 1, i.e., B = 1, 1, ...1 , we obtain | {z }

(14)

n

    n n+k−1 Sk (−n) = (−1)k and Sk (n) = k k

(15)

By combining (10) and (15), we obtain the following expression       n n n n Sn (A − nx) = Sn (A) − Sn−1 (A)x + Sn−2 (A)x2 − · · · + (−1)n x 1 2 n (16) For any pair (x, y) we can associate the divided difference ∂xy defined by [8] ∂xy(f ) =

f (x, y, z, . . .) − f (y, x, z, . . .) x−y

3

323

(17)

BOUSSAYOUD ET AL: GENERATING FUNCTIONS

3

The major formulas

In this section, we provide some definitions and a new propostion which will be useful for the next theorem. P∞ P∞ Definition 1 The inverse of the sequence n=0 Sn (A)z n is the sequence n=0 Sn (−A)z n , that is ∞ X 1 Sn (A)z n = P (18) ∞ n n=0 Sn (−A)z n=0

Definition 2 The symmetric operator π nxy is defined by [7] π nxy f (x) =

xn f (x) − y n f (y) x−y

(19)

Proposition 1 Given an alphabet E2 = {e1 , e2 }, then for any positive integer k, the operator π ke1 e2 satisfied the following formula π ke1 e2 f (e1 ) = f (e1 )Sk−1 (e1 + e2 ) + ek2 ∂e1 e2 (f )

(20)

Proof. From (19) we have π ke1 e2 f (e1 )

ek1 f (e1 ) − ek2 f (e2 ) e1 − e2 ek1 f (e1 ) − ek2 f (e1 ) + ek2 f (e1 ) − ek2 f (e2 ) e − e2   1 f (e1 ) ek1 − ek2 + ek2 [f (e1 ) − f (e2 )] e1 − e2

= = =

Using the formulas (4) and (17) we obtain π ke1 e2 f (e1 ) = f (e1 )Sk−1 (e1 + e2 ) + ek2 ∂e1 e2 (f ) This completes the proof of proposition 1. Theorem 2 Given two alphabets E2 = {e1 , e2 } and A = {a1 , a2 , ...} , then ∞ X

Sn (A)Sk+n−1 (e1 + e2 )z n

n=0 k−1 X

=

Sn (−A)en1 en2 Sk−n−1 (e1 + e2 )z n − ek1 ek2 z k+1

n=0

∞ X

Sn+k+1 (−A)Sn (e1 + e2 )z n

n=0



∞ P n =0

Sn (A)en1 z n



∞ P

n =0

Sn (A)en1 z n

 (21)

4

324

BOUSSAYOUD ET AL: GENERATING FUNCTIONS

Proof. Let f (e1 ) = can be written as

P∞

π e1 e2 f (e1 )

n n n=0 e1 Sn (A)z ,

∞ X

= π e1 e2

then the left hand side of formula (21) ! Sn (A)en1 z n

n =0

ek1 = = =

∞ P n=0

Sn (A)en1 z n − ek2

∞ P n=0

Sn (A2 )en1 z n

e1 − e2 ∞ X n=0 ∞ X

en+k 1

− en+k 2 e1 − e2

Sn (A)

! zn

Sn (A)Sn+k−1 (e1 + e2 )z n

n=0

and the right hand side of this formula can be written as Sk−1 (e1 + e2 )f (e1 ) + ek2 ∂e1 e2 f (e1 ) =

Sk−1 (e1 + e2 ) 1 + ek2 ∂e1 e2 P ∞ ∞ P n Sn (−A)e1 z n Sn (−A)en1 z n

n=0

n=0

∞ P

=

Sn (−A)Sn−1 (e1 + e2 )z n Sk−1 (e1 + e2 ) n=0  ∞  −∞ ∞ P P P n n n n n n Sn (−A)e1 z Sn (−A)e1 z Sn (−A)e2 z

n=0 ∞ P

=

n=0

n=0

  Sn (−A) en2 Sk−1 (e1 + e2 ) − ek2 Sn−1 (e1 + e2 ) z n j=0 ∞  ∞  P P n n n n Sn (−A)e1 z Sn (−A)e2 z n=0

n=0

k−1 P

=

  Sn (−A) en2 Sk−1 (e1 + e2 ) − ek2 Sn−1 (e1 + e2 ) z n j=0 ∞  ∞  P P n n n n Sn (−A)e1 z Sn (−A)e2 z n=0

n=0

∞ P

  Sn (−A) en2 Sk−1 (e1 + e2 ) − ek2 Sn−1 (e1 + e2 ) z n j=k+1  ∞  ∞ + P P n n n n Sn (−A)e1 z Sn (−A)e2 z n=0

n=0

∞ P Sn (−A)en1 en2 Sk−n−1 (e1 + e2 )z n − ek1 ek2 z k+1 Sn+k+1 (−A)Sn (e1 + e2 )z n n=0 n=0 ∞  ∞  P P n n n n Sn (−A)e1 z Sn (−A)e2 z k−1 P

=

n=0

n=0

This completes the proof.of Theorem 2. 5

325

BOUSSAYOUD ET AL: GENERATING FUNCTIONS

4

Applications to the generating functions

In this section, we attempt to give results for some well-known generating functions. In fact, we will use Theorem 2 to derive Fibonacci numbers and Tchebychev polynomials of second kind. Moreover, the generating functions for some special cases of Fibonacci numbers and Tchebychev polynomials are given. Then Theorem 2 can be written Corollary 3 If A2 = {a1 , a2 } and k = 1 then ∞ X

Sn (A2 )Sn (e1 + e2 )z n = 

n=0

1 − e1 e2 a1 a2 z 2  ∞  ∞ P P n n n n Sn (−A2 )e1 z Sn (−A2 )e2 z

n=0

(22)

n=0

Case 1: For a1 = 1 and a2 = 0, one can apply Corollary 3 to arrive at [3] ∞ X

Sn (e1 + [−e2 ])z n =

n=0

1 (1 − e1 z)(1 − e2 z)

(23)

In (23) replace e2 by (−e2 ), and choose e1 , e2 such that: e1 −e2 = 1, e1 e2 = 1 to obtain ∞ X

Sn (e1 + [−e2 ])z n =

n=0

1 , with Fn = Sn (e1 + [−e2 ]) 1 − z − z2

(24)

where Fn are Fibonacci numbers. Also, if we replace e1 by (2e1 ), e2 by (−2e2 ) with the condition 4e1 e2 = −1, then there follows that ∞ X

1 , with Un (e1 −e2 ) = Sn (2e1 +[−2e2 ]) 1 − 2(e − e2 )z + z 2 1 n=0 (25) where Un are the Tchebychev polynomials of second kind. Sn (2e1 +[−2e2 ])z n =

By using the previous formula (25), we can deduce that ∞ X

1 − (e1 − e2 )z 1 − 2(e1 − e2 )z + z 2 n=0 (26) Then the Tchebychev polynomials of first kind can be derived directly as follows [3] [Sn (2e1 + [−2e2 ]) − (e1 − e2 )Sn−1 (2e1 + [−2e2 ])] z n =

Tn (e1 − e2 ) = [Sn (2e1 + [−2e2 ]) − (e1 − e2 )Sn−1 (2e1 + [−2e2 ])]

6

326

(27)

BOUSSAYOUD ET AL: GENERATING FUNCTIONS

Case 2: For a1 = 1, a2 = x, and e1 = 1, e2 = y, in an application of Corollary 3 yields the following result [6] ∞ X

[1 + x + · · · + xn ] [1 + y + · · · + y n ] z n =

n=0

1 − xyz 2 [(1 − z)(1 − xz)(1 − yz)(1 − xyz)] (28)

Case 3: By replacing e2 by (−e2 ) and a2 by (−a2 ), we obtain ∞ X

1 − e1 e2 a1 a2 z 2 (1 − a1 e1 z) (1 + a2 e1 z) (1 + a1 e2 z) (1 − a2 e2 z) n=0 (29) This case consists of three related parts. Sn (a1 +[−a2 ])Sn (e1 +[−e2 ])z n =

Firstly, by making the following restrictions: a1 − a2 = 1, a1 a2 = 1, and e1 − e2 = 1, e1 e2 = 1 in (29) we gives ∞ X

∞ X 1 − z2 = Fn2 z n 2 − z3 + z4 1 − z − 4z n=0 n=0 (30) This corresponds to the square of Fibonacci numbers [5] given by

Sn (a1 + [−a2 ])Sn (e1 + [−e2 ])z n =

Fn2 = Sn (a1 + [−a2 ])Sn (e1 + [−e2 ])

(31)

Secondly, by making the following restrictions:e1 − e2 = 1, e1 e2 = 1, a1 a2 = −1, and by replacing (a1 − a2 ) by 2(a1 − a2 ) in (29), we get the identity of Foata [5], involving the product of Fibonacci numbers with Tchebychev polynomial of second kind as follows 1 + z2 2

1 − 2 (a1 − a2 ) z + (3 − 4 (a1 − a2 ) )z 2 + 2 (a1 − a2 ) z 3 + z 4

=

∞ X

Fn Un (a1 −a2 )z n

n=0

(32) In the last case, choose ai and ei such that e1 e2 = −1, a1 a2 = −1, and by replace (a1 −a2 ) by 2(a1 −a2 ), and (e1 −e2 ) by 2(e1 −e2 ) in (29), to obtain the identity of Foata [5], involving the square of Tchebychev polynomials of second kind given by ∞ X

Un (e1 − e2 )Un (a1 − a2 )z n

n=0

=

1 − z2 1 − 4(e1 − e2 )(a1 − a2 )z + (4(a1 − a2 )2 + 4(e1 − e2 )2 − 2)z 2 − 4(e1 − e2 )(a1 − a2 )z 3 + z 4 (33)

Notice that, under the same restrictions and by using (25) and (27), and the fact that n

Sn−1 (2a1 + [−2a2 ]) = 7

327

n

(2a1 ) − (−2a2 ) 2a1 + 2a2

(34)

BOUSSAYOUD ET AL: GENERATING FUNCTIONS

we obtain the identity of Foata [4], involving the product of Tchebychev polynomials of second kind with Tchebychev polynomials of first kind: ∞ X

Un (e1 − e2 )Tn (a1 − a2 )z n

n=0

=

1 − 2(e1 − e2 )(a1 − a2 )z + (2(a1 − a2 )2 − 1)z 2 1 − 4(e1 − e2 )(a1 − a2 )Z + (4(a1 − a2 )2 + 4(e1 − e2 )2 − 2)z 2 − 4(e1 − e2 )(a1 − a2 )z 3 + z 4 (35)

and also the identity of Foata [5], involving the square of Tchebychev polynomials of first kind: ∞ X

Tn (e1 − e2 )Tn (a1 − a2 )z n

n=0

=

5

1 − 3(e1 − e2 )(a1 − a2 )z + (2(a1 − a2 )2 + 2(e1 − e2 )2 − 1)z 2 − (e1 − e2 )(a1 − a2 )z 3 1 − 4(e1 − e2 )(a1 − a2 )Z + (4(a1 − a2 )2 + 4(e1 − e2 )2 − 2)z 2 − 4(e1 − e2 )(a1 − a2 )z 3 + z 4 (36)

The product of Hadamard

In this section, we show the efficiency of the proposed method by determining the product of Hadamard. In fact, by taking A = Φ in (8), we obtain ∞ X

Y

Sn (−B)z n =

n=0

(1 − bz)

(37)

b∈B

For the special case where a1 = a2 = 1 in (37), we have ∞ X

(n + 1)z n =

n=0

1 (1 − z)2

(38)

1 (1 − e1 z)2

(39)

By replacing z by e1 z in (38), we get ∞ X

(n + 1)en1 z n =

n=0

Use Corollary 3 with the action of the operator π e1 e2 on both sides of the identity (39) to obtain ∞ X

(n + 1)Sn (e1 + e2 )z n =

n=0

1 − e1 e2 z 2 (1 − e1 z)2 (1 − e2 z)2

(40)

1+z . (1 − z)3

(41)

By taking e1 = 1 and e2 = 1, we have ∞ X

(n + 1)2 z n =

n=0

8

328

BOUSSAYOUD ET AL: GENERATING FUNCTIONS

On the other hand, using formula (22) with the action of the operator π e1 e2 on both sides of (41), and by replacing z by e1 z leads to ∞ X

(n + 1)2 Sn (e1 + e2 )z n = π e1 e2

n=0

e1 1 + zπ e1 e2 (1 − e1 z)3 (1 − e1 z)3

(42)

Using formulas (15), (19) and (21),it follows that   1 P 3 n+2 1 − e1 e2 z 2 (−1) Sn (e1 + e2 )z n 1 n+2 n=0 π e1 e2 = (43) (1 − e1 z)3 (1 − e1 z)3 (1 − e2 z)3       1 1 P P 3 n+3 n 3 n n n n 2 2 3 (−1) Sn (E2 )z (−1) e e S1−n (E2 )z − e1 e2 z e1 n+3 n 1 2 n=0 n=0 π e1 e2 = (1 − e1 z)3 (1 − e1 z)3 (1 − e2 z)3 (44) Notice that, for e1 = 1 and e2 = 1, we have        1 P 3 n+1 n n+2 2 1 − z (−1) z +   n+2 n     n=0      1 0  P P 2−n n 3 n+1 n  n 3 n+3 z (−1) z − z3 (−1) z ∞ X n 1−n n+3 n n=0 n=0 3 n (n+1) z = (1 − z)6 n=0 (45) which gives after simplification ∞ X

(n + 1)3 z n =

n=0

1 + 4z + z 2 (1 − z)4

(46)

Using the same procedure, we deduce, for instance, the following identities ∞ X

1 + 11z + 11z 2 + z 3 (1 − z)5

(47)

1 + 26z + 66z 2 + 26z 3 + z 4 (1 − z)6

(48)

1 + 57z + 302z 2 + 302z 3 + 57z 4 + z 5 (1 − z)7

(49)

(n + 1)4 z n =

n=0 ∞ X

(n + 1)5 z n =

n=0 ∞ X

(n + 1)6 z n =

n=0 ∞ X

(n + 1)7 z n =

n=0 ∞ P

(n+1)8 z j =

j=0

1 + 120z + 1191z 2 + 2416z 3 + 1191z 4 + 120z 5+ z 6 (1 − z)8

(50)

1 + 247z + 4293z 2 + 15619z 3 + 15619z 4 + 4293z 5 + 247z 6 + z 7 (1 − z)9 (51) 9

329

BOUSSAYOUD ET AL: GENERATING FUNCTIONS

∞ P

(n+1)9 z j =

n=0

∞ P

(n + 1)10 z j

1 + 502z + 14608z 2 + 88234z 3 + 156190z 4 + 88234z 5 + 14608z 6 + 502z 7 + z 8 (1 − z)10 (52)

=

n=0

6

1 + 1013z + 47840z 2 + 455192z 3 + 1310354z 4 + 1310354z 5 + 455192z 6 + (1 − z)11 47840z 7 + 1013z 8 + z 9 (53) (1 − z)11

Conclusion

In this paper, a new theorem has been proposed in order to determine the generating functions. The proposed theorem is based on the symmetric fonctions. The obtained results agree with the results obtained in some previous works.

References [1] Abderrezzak, A.: G´en´eralisation d’identit´es de Carlitz, Howard et Lehmer, Aequationes Mathematicae 49, 36-46 (1995) [2] Abderrezzak, A.: Quelques Formules d’Inversion ` a Plusieurs Variables, Eur. J. Comb 14, 507-512 (1993) [3] Boussayoud, A.; Kerada, M.; Abderrezzak, A. : A Generalization of some orthogonal polynomials, Advances in Applied Mathematics and Approximation Theory, Springer Proceedings in Mathematics & Statistics 41, 229-235, (2013) [4] Foata, D.; Han, G-N.: Calcul basique des permutations sign´ees.1, Longueur et nombre d’inversions, Adv. in Appl. Math 18, 489-509 (1997) [5] Foata, D.; Han, G-N.: Nombres de Fibonacci et polynˆ omes orthogonaux, Leonardo Fibonacci: il tempo, le opere, l’eredit scientifica [Pisa. 23-25 Marzo 1994, Marcello Morelli e Marco Tangheroni, ed.], 179-200( 1990) [6] Lascoux, A.: Addition of ±1 : application to arithmetic, S´eminaire lotharingien de combinatoire 52, 1-9 (2004) [7] Lascoux, A.: Inversion des matrices de Hankel, Linear Algebra and its Applications 129, 77-102 (1990) [8] Macdonald, I.G.: Symmetric functions and Hall polynomias, second edition, Oxford Mathematical Monographs, (1995) [9] Manivel, L.: Cours sp´ecialis´ees, fonctions sym´etriques, polynˆomes de Schuet et lieux de d´egcn´erexence, N3, Soci´et´e Math´ematiques de France, (1998)

10

330

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.'S 3-4, 331-335, 2014, COPYRIGHT 2014 EUDOXUS PRESS LLC

New Expansions for Two Trigonometric Functions Demetrios P. Kanoussis1 and Vassilis G. Papanicolaou2 Department of Mathematics National Technical University of Athens Zografou Campus 157 80, Athens, GREECE 1 2 [email protected] [email protected] Abstract We introduce a new type expansions for the functions sin (πx) and cot (πx), 0 < x < 1. In particular, the sin (πx) is expressed as an infinite product (different from the Euler’s product for the sine function), while the cot (πx) is expressed as an infinite series of terms involving the logarithmic function. The resulting formulas lead to some product expansions for eγ , ϕ (the golden ratio), as well as eλπ , where λ takes some specific real, algebraic values.

2010 Mathematics Subject Classification. 00A08; 00A99. Key words and phrases: Sine; cotangent; golden ratio.

1

Introduction

In a recent paper [4] a product type expansion for the Gamma function Γ(x) was obtained:   1 k+1 ∞ k Y Y √ k j (x+j)( j )(−1)  −x  Γ(x) = 2eπe (x + j) , x > 0. (1.1) k=0

j=0

In the same paper [4] it was shown that the Psi (or Digamma) function, Ψ(x) :=

Γ0 (x) d ln Γ(x) = dx Γ(x)

(1.2)

admits the following representation Ψ(x) =

∞ X k=0

  k 1 X j k (−1) ln(x + j), k+1 j j=0

331

x > 0.

(1.3)

KANOUSSIS-PAPANICOLAOU: TRIGONOMETRIC FUNCTIONS

The expression (1.3) has been also derived by J. Guillera and J. Sondow (see [3]), with the help of the so-called Lerch transcendent. In [4], expressions (1.1) and (1.3) are derived by a fundamentally different approach, that is they result as a solution of an appropriate difference equation. Expression (1.1) for the Γ(x), x > 0, is obtained as a solution of the difference equation ln Γ(x + 1) − ln Γ(x) = ln x,

x > 0,

(1.4)

while expression (1.3) for the Ψ(x), x > 0, is obtained as a solution of the difference equation Ψ(x + 1) − Ψ(x) =

2

1 , x

x>0

(1.5)

An expansion for the function sin(πx), 0 < x < 1

Making use of the well known reflection formula for the Gamma function (see [1], Th. 2.12) Γ(x) Γ(1 − x) =

π , sin(πx)

0 < x < 1,

(2.1)

and taking into consideration (1.1), the following product type expansion for sin(πx) is obtained   1 k+1 ∞ k n k j+1 o Y Y (−1) ( ) 1 j (x+j) (1−x+j)   sin(πx) = (x + j) (1 − x + j) , 2 k=0

j=0

(2.2) i.e. sin(πx) = #1  1 " 3 1 1 (x + 1)x+1 (2 − x)2−x 2 (x + 1)2(x+1) (2 − x)2(2−x) · x · · · 2 x (1 − x)1−x xx (1 − x)1−x xx (x + 2)x+2 (1 − x)1−x (3 − x)3−x "

(x + 1)3(x+1) (x + 3)x+3 (2 − x)3(2−x) (4 − x)4−x xx (x + 2)3(x+2) (1 − x)1−x (3 − x)3(3−x)

#1

4

··· .

(2.3)

This product formula for sin(πx), 0 < x < 1, which expresses sin(πx) in terms of x alone, is very different from the well known Euler’s product expansion of the sine function and, as far we know, is new.

2

332

KANOUSSIS-PAPANICOLAOU: TRIGONOMETRIC FUNCTIONS

3

An expansion for the function cot(πx), 0 < x < 1

With the aid of the reflection formula for the Psi function (see [2]) we have Ψ(1 − x) − Ψ(x) = π cot(πx)

(3.1)

and using (3.1), the following expression for the cot(πx) is obtained:   (k)(−1)j ∞ k  X Y j 1  1−x−j 1 , (3.2) ln cot(πx) = π k+1 x+j k=0

j=0

i.e.      1−x 1 1 (1 − x)(1 + x) (1 − x)(3 − x)(1 + x)2 π cot(πx) = ln + ln + ln + x 2 (2 − x)x 3 (2 − x)2 x(2 + x)   1 (1 − x)(3 − x)3 (1 + x)3 (3 + x) ln + .... (3.3) 4 (2 − x)3 (4 − x)x(2 + x)3 

In the next paragraph we show some rather interesting applications of the expansions, just derived.

4

Applications

1. Setting x = 1 in (1.3) and recalling that Ψ(1) = −γ (see [2]), where γ is the Euler’s constant, an expression for eγ is obtained, i.e.  1/2  2 1/3  3 1/4  4 4 1/5 2 2 2 ·4 2 ·4 e = ··· . 1 1·3 1 · 33 1 · 36 · 5 γ

(4.1)

This expression was first derived by J. Ser [5] and subsequently rederived by J. Sondow. √
π 2. Let ϕ be the golden ratio, namely ϕ = 1+2 5 = 12 csc 10 . Applying 1 (2.2)–(2.3) at x = 10 , the following product for eϕ is obtained: 1

ϕ= 1 ·9


9 1/10





11 · 99 1111 · 1919

1/20 

11 · 99 · 2121 · 2929 1122 · 1938

19 · 99 · 2163 · 2987 1133 · 1957 · 3131 · 3939

1/30 ·

1/40 ··· .

(4.2)


Knowing that ϕ can also be expressed as ϕ = 2 sin 3π 10 , another product 3 expression can be obtained if we set x = 10 in (2.2)–(2.3):

3

333

KANOUSSIS-PAPANICOLAOU: TRIGONOMETRIC FUNCTIONS

 ϕ=

1 3 3 · 77

1/10 

1313 · 1717 33 · 77

1/20 

1326 · 1734 33 · 77 · 2323 · 2727 1/40  39 13 · 1751 · 3333 · 3737 ··· . 33 · 77 · 2369 · 2781

1/30 ·

(4.3)

3. It may be of interest to notice that (3.2) can be used to find fancy product expansions of numbers of the form eλπ , where λ is a real algebraic number of a certain kind. We present some examples. (i) By setting x = 14 in (3.2)–(3.3) and recalling that cot( π4 ) = 1, one easily obtains an expression for eπ :  1/1    1/3  1/4 3 3 · 5 1/2 3 · 52 · 11 3 · 53 · 113 · 13 e = ··· . 1 1·7 1 · 72 · 9 1 · 73 · 93 · 15 π

(4.4)

This product expansion for eπ has also been derived by J. Guillera and J. Sondow in [3]. (ii) By setting x = 13 in (3.2)–(3.3) one obtains e

π √ 3

  1/3  1/4  1/1  2 · 4 1/2 2 · 42 · 8 2 · 43 · 83 · 10 2 ··· , = 1 1·5 1 · 52 · 7 1 · 53 · 73 · 11

while for x =

1 6

(4.5)

we obtain

 1/1    1/3  1/4 5 5 · 7 1/2 5 · 72 · 11 5 · 73 · 173 · 19 ··· . 1 1 · 11 1 · 112 · 13 1 · 113 · 133 · 22 (4.6)

p √ 1 π π 2 (iii) The formula ϕ = 2 csc 10 also implies cot 10 = 4ϕ − 1 = 4ϕ + 3 1 (since ϕ2 = ϕ + 1). Making use of (3.2)–(3.3), at x = 10 , we obtain the following expression, which involves e, π, and ϕ: eπ

√

3

=

√ π 4ϕ+3

e

 1/1    1/3  1/4 9 9 · 11 1/2 9 · 112 · 29 9 · 113 · 293 · 31 = ··· . 1 1 · 19 1 · 192 · 21 1 · 193 · 213 · 39 (4.7)

References [1] W.W. Bell, Special Functions for Scientists and Engineers, Dover Publications Inc., Mineola, New York, 1967. [2] G. Boros and V.H. Moll, Irresistible Integrals. Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge 2004.

4

334

KANOUSSIS-PAPANICOLAOU: TRIGONOMETRIC FUNCTIONS

[3] J. Guillera and J. Sondow, Double Integrals and Infinite Products for some classical constants via analytic continuations of Lerch’s transcendent, Ramanujan J., 16, 247–270 (2008). [4] D.P. Kanoussis and V.G. Papanicolaou, On the Inverse of the Taylor operation, Scientia, Series A: Mathematical Sciences, 24 (to appear in 2013). [5] J. Ser, Sur une expression de la function j(s) de Riemman (in French), C.R. Acad. Sci. Paris Ser. I Math., 182, 1075–1077 (1926).

5

335

TABLE OF CONTENTS, JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS, VOL. 12, NO.’S 3-4, 2014 Mechanical Models with Internal Body Forces, Igor Neygebauer,……………………………181 A New Comprehensive Class of Analytic Functions Defined by Ruscheweyh Derivative and Multiplier Transformations, Alina Alb Lupaș, and Adriana Cătaș,……………………………201 The Numerical Solution of Non-Linear Non-Local Problems for Elliptic Equations, Aydin Y. Aliyev,…………………………………………………………………………………………..205 Some Generating Relations for Generalized Extended Hypergeometric Functions Involving Generalized Fractional Derivative Operator, Rakesh K.Parmar,……………………………….217 An Equivalent Reformulation of Absolute Weighted Mean Methods, Mehmet Ali Sarigol,….229 On the Effectiveness of the Exponential Ruscheweyh Differential Operator Product Sets in Cn, M.A. Abul-Dahab, M. A. Saleem, and Z. G. Kishka,…………………………………………234 Normality, Regularity and compactness of sb*-closed sets in Topological spaces, A. Poongothai, and R. Parimelazhagan,…………………………………………………………………………249 New Results on Harmonious Labeling, Abdullah Aljouiee,………………………………...….257 Mapping Properties of Mixed Fractional Integro-Differentiation in Hölder Spaces, Mamatov Tulkin,…………………………………………………………………………………………..272 Some Fixed Point Theorems of Set-Valued Increasing Operators, Jin-Ming Wang, Xiong-Jun Zheng, and Hui-Sheng Ding,…………………………………………………………………...291 Dynamics and Approximations for 2D Generalized Navier-Stokes Equation with Piecewise Distributed Controls, De G. Akmel, and L. C. Bahi,…………………………………………..302 Some Applications on Generating Functions, Ali Boussayoud, Mohamed Kerada, Rokiya Sahali, and Wahiba Rouibah,…………………………………………………………………………..321 New Expansions for Two Trigonometric Functions, Demetrios P. Kanoussis, and Vassilis G. Papanicolaou,…………………………………………………………………………………...331