Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy: Proceedings of the Third International Conference, MMCITRE 2022 9811999058, 9789811999055

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Advances in Intelligent Systems and Computing 1440

Manoj Sahni · José M. Merigó · Walayat Hussain · Ernesto León-Castro · Raj Kumar Verma · Ritu Sahni   Editors

Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy Proceedings of the Third International Conference, MMCITRE 2022

Advances in Intelligent Systems and Computing Volume 1440

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Advisory Editors Nikhil R. Pal, Indian Statistical Institute, Kolkata, India Rafael Bello Perez, Faculty of Mathematics, Physics and Computing, Universidad Central de Las Villas, Santa Clara, Cuba Emilio S. Corchado, University of Salamanca, Salamanca, Spain Hani Hagras, School of Computer Science and Electronic Engineering, University of Essex, Colchester, UK László T. Kóczy, Department of Automation, Széchenyi István University, Gyor, Hungary Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA Chin-Teng Lin, Department of Electrical Engineering, National Chiao Tung University, Hsinchu, Taiwan Jie Lu, Faculty of Engineering and Information Technology, University of Technology Sydney, Sydney, NSW, Australia Patricia Melin, Graduate Program of Computer Science, Tijuana Institute of Technology, Tijuana, Mexico Nadia Nedjah, Department of Electronics Engineering, University of Rio de Janeiro, Rio de Janeiro, Brazil Ngoc Thanh Nguyen , Faculty of Computer Science and Management, Wrocław University of Technology, Wrocław, Poland Jun Wang, Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong

The series “Advances in Intelligent Systems and Computing” contains publications on theory, applications, and design methods of Intelligent Systems and Intelligent Computing. Virtually all disciplines such as engineering, natural sciences, computer and information science, ICT, economics, business, e-commerce, environment, healthcare, life science are covered. The list of topics spans all the areas of modern intelligent systems and computing such as: computational intelligence, soft computing including neural networks, fuzzy systems, evolutionary computing and the fusion of these paradigms, social intelligence, ambient intelligence, computational neuroscience, artificial life, virtual worlds and society, cognitive science and systems, Perception and Vision, DNA and immune based systems, self-organizing and adaptive systems, e-Learning and teaching, human-centered and human-centric computing, recommender systems, intelligent control, robotics and mechatronics including human-machine teaming, knowledge-based paradigms, learning paradigms, machine ethics, intelligent data analysis, knowledge management, intelligent agents, intelligent decision making and support, intelligent network security, trust management, interactive entertainment, Web intelligence and multimedia. The publications within “Advances in Intelligent Systems and Computing” are primarily proceedings of important conferences, symposia and congresses. They cover significant recent developments in the field, both of a foundational and applicable character. An important characteristic feature of the series is the short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by DBLP, INSPEC, WTI Frankfurt eG, zbMATH, Japanese Science and Technology Agency (JST). All books published in the series are submitted for consideration in Web of Science. For proposals from Asia please contact Aninda Bose ([email protected]).

Manoj Sahni · José M. Merigó · Walayat Hussain · Ernesto León-Castro · Raj Kumar Verma · Ritu Sahni Editors

Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy Proceedings of the Third International Conference, MMCITRE 2022

Editors Manoj Sahni Pandit Deendayal Energy University Gandhinagar, India Walayat Hussain Peter Faber Business School Australian Catholic University North Sydney, NSW, Australia Raj Kumar Verma School of Economics and Business Universidad de Talca Talca, Chile

José M. Merigó School of Computer Science University of Technology Sydney Ultimo, NSW, Australia Ernesto León-Castro Universidad Católica de la Santísima Concepcion, Chile Ritu Sahni Pandit Deendayal Energy University Gandhinagar, India

ISSN 2194-5357 ISSN 2194-5365 (electronic) Advances in Intelligent Systems and Computing ISBN 978-981-19-9905-5 ISBN 978-981-19-9906-2 (eBook) https://doi.org/10.1007/978-981-19-9906-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This proceedings features contributions from mathematicians, physicists, and engineers from all over the world including numerous Indian states who attended the Third International Conference on Mathematical Modeling, Computational Intelligence Techniques, and Renewable Energy (MMCITRE 2022). The preceding two conferences were held at Pandit Deendayal Energy University in Gandhinagar, Gujarat, India. This time the conference is held online at the University of Technology Sydney, Australia, located at the southern border of Sydney’s central business district, along with the associated universities: Pandit Deendayal Energy University, Gandhinagar, Gujarat, India; Victoria University, Melbourne, Australia; Universidad Autónoma Baja California (UABC), Mexico; Universidad Católica de la Santsima Concepción (UCSC), Chile; and Forum for Interdisciplinary Mathematics (FIM). The purpose of this conference is the exchange of knowledge, which can lead to research collaboration with scholars and researchers and the development of numerous new ideas for future research. Keeping researchers and scientists up to date on the most recent discoveries and innovations, as well as on the solutions to the practical challenges that are helpful not only for academicians but also for industry persons, is the purpose of this series of MMCITRE conferences, the goal of which is to provide a plethora of usable content that is based on mathematical modeling, artificial intelligence techniques, renewable energy, and a variety of other topics that are interrelated. From the bottom of our hearts, we want to extend a hearty greeting to all of the world’s leading scientists, academics, young researchers, business delegates, and students who have participated in this international conference. Our deepest appreciation also extends to everyone who wished us well and provided encouraging notes to help us pull off a smooth event. Academicians and professionals from a number of nations, as well as different parts of India, have attended the conference in order to give the academic community the opportunity to learn about their knowledge, research findings, and educational practices. A substantial number of research articles came from all around the world and were submitted to this international conference. It has been taken into account that these articles cover the most recent mathematical techniques that are valuable not only for academics but also for students studying at the undergraduate and postgraduate v

vi

Preface

levels in a variety of fields. Only the finest papers have been selected to be presented orally after undergoing a rigorous peer-review procedure by the experts in their respective fields. Finally, on the basis of the quality of the work of the experts from many fields, young researchers, academics, and students who presented papers, a total of 36 articles have been chosen to be published in this proceedings. New fundamental mathematical results with a wide range of applications are presented in the proceedings. It helps aspiring engineers and computer scientists develop logic, creativity, and critical thinking skills that apply to almost all of the world’s challenges. The proceedings covers recent mathematical breakthroughs, rigorous mathematical methodologies, and unique mathematical modeling of reallife occurrences in education, medical, business, and marketing for society’s benefit. We wish this proceedings will help graduate students in mathematics, physics, engineering, and other subjects find new mathematical tools. Gandhinagar, India Ultimo, Australia Concepcion, Chile North Sydney, Australia Gandhinagar, India Talca, Chile

Dr. Manoj Sahni Prof. José M. Merigó Dr. Ernesto León-Castro Dr. Walayat Hussain Dr. Ritu Sahni Dr. Raj Kumar Verma

Contents

Theoretical Advancements for Applied Mathematics A New Result Using Quasi-β-Power Increasing Sequence . . . . . . . . . . . . . Smita Sonker and Rozy Jindal

3

(C, 1, 1)-Quasinormal Convergence of Double Sequence of Functions . . . Smita Sonker and Priyanka

13

Fixed Point Theorems in Neutrosophic Soft Metric Space . . . . . . . . . . . . . Vishal Gupta and Aanchal

25

Existence of Best Proximity Points on (ψ, φ) Contractions in RMS . . . . . S Arul Ravi

39

Approximation of Signals by El1 El1 Product Summability Means of Fourier–Laguerre Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Smita Sonker and Neeraj Devi

49

Approximation of Signal Belongs to Generalized W  (L r , ξ (t)) Class by (C, α, η) A -Matrix Summability of Fourier Series . . . . . . . . . . . . Smita Sonker and Paramjeet Sangwan

59

A New Connection on Generalised Tangent Bundle . . . . . . . . . . . . . . . . . . . Rashmirekha Patra

69

Mathematical Modelling Analysis of Third-Order Resonant Periodic Orbits in Perturbed Circular Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bhavika M. Patel, Niraj M. Pathak, and Elbaz I. Abouelmagd Exergy Optimization in Closed-Loop Spray Drying . . . . . . . . . . . . . . . . . . . Zexin Lei and Timothy Langrish

77 91

vii

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Contents

Simulation and Modeling of Linear and Nonlinear PID Controller . . . . . 103 Ramendra Singh, Vikas Pandey, Meenakshi Sharma, Shashikant, and Manoj Sahni Evaluation of Transitional and Plastic Stresses in Transversely Isotropic Disk Made of Piezoelectric Material Subjected to Internal Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Richa Sharma Comparison of Numerical Solution of Consolidation Equation in One Dimension by Finite Difference Methods and Finite Element Method with Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Sanjay L. Gosiya, Mansi S. Palav, and Vikas H. Pradhan Existence of Solutions for Stochastic Fractional Differential Equations Driven by Lévy Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Gunjan Rani, Arpit Dwivedi, and Ganga Ram Gautam Mathematical Modelling of Failure Process for Repairable Systems . . . . 163 Sunil Bhardwaj, Sandeep Kumar Mogha, and Vikas Garg Mathematical Analysis of a Prey–Predator Model in Presence of Two Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Sudhakar Yadav and Vivek Kumar An Investigation of a Novel Design of Savonius Wind Turbine Along Highway and for Small-scale Power Generation Applications . . . . 185 Udit Mittal, Gunjan Varshney, Satyajeet, Ujjwal Gupta, and Pranshul Agarwal Numerical and Economical Investigations on Thermal Enhanced Oil Recovery Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Jainish Shingala, Jhanvi Desai, Vrutang Shah, Yashraj Sonara, Harsh Dhangar, Harsh Shah, and Vivek Ramalingam Medical and Statistical Applications Applications of Augmented Reality in Medical Training . . . . . . . . . . . . . . . 215 Riya Garg, Kirti Aggarwal, and Anuja Arora Application of Piezoelectric Material in Surgery . . . . . . . . . . . . . . . . . . . . . . 229 Meenakshi Sharma, Shashikant, Ramendra Singh, Vikas Pandey, and Manoj Sahni A Mathematical Analysis on the Obesity and Diabetic Model . . . . . . . . . . 237 V. S. V. Naga Soundarya Lakshmi and A. Sabarmathi Numerical Simulations for Human Liver Model with Caputo Fractional Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 A. S. V. Ravi Kanth and Sangeeta Devi

Contents

ix

Physical Exercise: Effective Aspect in Diabetes Management . . . . . . . . . . 261 Ankit Sharma, Harendra Pal Singh, and Nilam Single Variable Regression Model with Error Analysis for Evolution of Periodic Orbit in Formation Satellite . . . . . . . . . . . . . . . . . 275 Mitali J. Doshi, Niraj Pathak, and Elbaz I. Abouelmagd Predicting Tissue Chloride Through FTIR-ATR Spectroscopy—Application of Exhaustive Feature Extraction Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Rahul Sreekumar, Nanjappa Ashwath, Vijayalaxmi Beeravalli, Phul Subedi, and Kerry Walsh A Phenomenological Approach to Mellin Moments of Parton Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Akbari Jahan and Diptimonta Neog Estimation of Parameters of Some Continuous Distributions Using Frequency Ratio Method Based on Local Information . . . . . . . . . . . . . . . . 319 V. Vipin and Santanu Koley Impact of Information and Communication Technology on Tourist Spending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Ana Laura Herrera-Prado, Elizabeth Olmos-Martínez, and Ernesto León-Castro Fluid Modelling Mathematical Modelling of Mie Scattering for Magnetic Spheres Surrounded by Magnetic Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Keyur Khatsuriya and Jaysukh Markana A Non-Newtonian Fluid Model for Blood Accounting for the Haematological Disorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 A. Karthik, K. Ketana, and T. S. L. Radhika Numerical Solution of Non-dimensional Contaminant Transport Equation with Varying Coefficients (Temporal) by Haar Wavelet Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Asmita C. Patel and V. H. Pradhan Computational Analysis of Single-Pass Roughened Solar Air Heater Using Design of Experiment Approach . . . . . . . . . . . . . . . . . . . . . . . 389 Yogeshkumar D. Khimsuriya, D. K. Patel, Vivek Patel, and Hitesh Panchal Semi-analytic Numerical Methods to Solve One-Dimensional Dispersion Phenomenon of Miscible Fluid Flow Through Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Aruna Sharma and Amit K. Parikh

x

Contents

The Influence of Magnetic Effect in a Channel Partially Filled with Porous Material: A Numerical Investigation . . . . . . . . . . . . . . . . . . . . . 415 Nitish Gupta and D. Bhargavi A Substantial Model for Blood Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 A. Karthik, M. Shriya, B. Nithin Reddy, K. Ketana, and T. S. L. Radhika Analytical Study of Quasi-One-Dimensional Cylindrical Weak Shock Wave Problem Under the Action of Magnetic Field at Stellar Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 Akmal Husain, S. A. Haider, and V. K. Singh Numerical Solution for System of Nonlinear Time-Fractional Boussinesq–Burgers Equations in Propagation of Shallow Water Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 L. Meenatchi and M. Kaliyappan Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

About the Editors

Dr. Manoj Sahni is a dedicated and experienced Mathematics teacher and researcher with more than 18 years of experience and currently serving as an Associate Professor and the head of the Department of Mathematics, School of Technology, Pandit Deendayal Energy University, Gandhinagar, Gujarat, India. He has published more than 70 research papers in peer-reviewed journals, conference proceedings and book chapters with reputed publishers like Springer, Elsevier, etc. He also serves as an advisory board member, a technical Committee member and a reviewer for many international journals of repute. He organized the 1st and 2nd International Conferences on Mathematical Modeling, Computational Techniques, and Renewable Energy on February 21–23, 2020, and February 06–08, 2021, respectively. He has organized many seminars, workshops, and short-term training programs at PDEU and various other universities. In addition, he is a member of many international professional societies, including the American Mathematical Society, Forum for Interdisciplinary Mathematics, IAENG and many more. Prof. José M. Merigó is a professor in the School of Information, Systems and Modelling at the Faculty of Engineering and Information Technology at the University of Technology Sydney (Australia) and a part-time full professor in the Department of Management Control and Information Systems at the School of Economics and Business at the University of Chile. He holds master’s and Ph.D. degrees in Business Administration from the University of Barcelona. He also holds B.Sc. and M.Sc. degrees from Lund University (Sweden). He has published more than 500 articles in journals, books and conference proceedings, including journals such as Information Sciences, IEEE Computational Intelligence Magazine, European Journal of Operational Research. He has also published several books with Springer and with World Scientific. Recently, Thomson and Reuters (Clarivate Analytics) have distinguished him as a highly cited researcher in Computer Science (2015–present).

xi

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About the Editors

Dr. Walayat Hussain is a professional academic, a researcher and a practitioner in computer science and information system with over 15 years of academic and industry experience. He worked as a lecturer with the Faculty of Engineering and IT, University of Technology Sydney, Australia, for four years. Walayat’s research areas are Computational Intelligence, Soft Computing, Forecasting, Cloud Computing, SLA Management and Decision Support System. His work has been published in different top-ranked reputable journals and conferences such as Future Generation Computer Systems, The Computer Journal (Oxford University), Information Systems, Computer and Industrial Engineering, IEEE Access, Journal of Ambient Intelligence and Humanized Computing, MONET, IEEE TGCN, IEEE TETCI, IJCS, FUZZ-IEEE, ICONIP. He is the associate editor of IET-Communications, an international journal, and has been a guest editor for several other international journals such as—Computer Networks, Forecasting, IEEE TETCI, WCMC, IJCS. Dr. Ernesto León-Castro is a full professor at Universidad Autonoma of Baja California (Mexico) and an adjunct research professor at Universidad Católica de la Santísima Concepcion (Chile). He has Ph.D. in Management from University of Occidente. He also has a M.Sc. in Management from University of Occidente and another one in Laws from University Autonoma de Sinaloa. His interest in research is in aggregation operators, decision making, finance, economics, computational intelligence and other related topics. He has published many papers in different journals, books and proceedings, including journals such as Applied Soft Computing, Technological and Economic Development of Economy, International Journal of Intelligent Systems, International Journal of Machine Learning and Cybernetics and Economic Computation and Economic Cybernetics Studies and Research. Also, he has been the guest editor in Special Issues in Technological and Economic Development of Economy, Journal of Intelligent and Fuzzy Systems, Computational and Mathematical Organization Theory and Springer Book Series. Dr. Raj Kumar Verma received a M.Sc. degree in mathematics from Chaudhary Charan Singh University, Meerut (U.P.), India, in 2006 and a Ph.D. degree in applied mathematics with a specialty in information theory and computational intelligence techniques from Jaypee Institute of Information Technology (Deemed University), Noida (U.P.), India, in 2014. He has authored over 70 research articles published in refereed international journals, including the International Journal of Intelligent Systems, Kybernetika, Journal of Intelligent and Fuzzy Systems, International Journal of Machine Learning and Cybernetics, Neural Computing and Applications, Informatica, Soft Computing, Cybernetics and Systems, Iranian Journal of Fuzzy Systems, Granular Computing, Applied Intelligence Review. Dr. Verma has also been a member of the scientific advisory committee of several international conferences and a reviewer in a wide range of refereed international journals. He is currently interested in information measures, aggregation operators, multiple attribute group decision making, computational intelligence techniques and computing with words.

About the Editors

xiii

Dr. Ritu Sahni is a dedicated and experienced Mathematics teacher and a researcher with more than 15 years who is actively involved with both undergraduate and postgraduate-level students. Presently, she is working as a visiting faculty at Pandit Deendayal Energy University, Gandhinagar, Gujarat, India. She worked with her students on various research-related problems and published their work in reputed journals such as Elsevier, Springer. She has published more than 40 research papers in peer-reviewed SCI, ESCI and Scopus indexed international journals and conferences. She has delivered talks and presented many research articles in various international conferences in India and abroad. Her research field includes Fixed Point Theory and its Applications, Numerical Methods, Fuzzy Decision-Making Systems and other allied areas. She is a devoted faculty member, a committed researcher, and enthusiastic about striving to improve the institution’s educational offerings and research fields for societal growth.

Theoretical Advancements for Applied Mathematics

A New Result Using Quasi-β-Power Increasing Sequence Smita Sonker and Rozy Jindal

Abstract Two generalized results were established, concerning the absolute matrix summability. Bor [2–10] worked on many interesting results dealing with |N , pn |k absolute Riesz summability. Özarslan and many other authors have been worked on matrix summability. In [14–21], they gave new and advanced results on matrix summability and generalize many theorems of Bor. Sonker and Jindal [23, 24] worked on triple product summability means and absolute matrix summability. Özarslan and Yavuz [16] proved two results on |U, pl |q summability factors. Here, we generalized both the results for ϕ − |U, pl |q matrix summability. Further, we develop new and arbitrary previous findings from the main theorems. Keywords Matrix summability Abel’s theorem · Matrix transformation · Quasi-monotone sequences

1 Introduction Let {sl } denotes the sequence of partial sum of transform of {sl } is given by u l , where ul =

∞ 



al . The lth sequence to sequence

u lk sk .

(1)

k=0

Definition 1 If lim u l = s

l→∞

S. Sonker (B) · R. Jindal National Institute of Technology Kurukshetra, Kurukshetra 136119, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1440, https://doi.org/10.1007/978-981-19-9906-2_1

3

4

S. Sonker and R. Jindal

and

∞ 

|u l − u l−1 | < ∞,

(2)

l=1

then



al is called absolute summable.

Definition 2 [11] If

∞ 

l 1−k |tl − tl−1 |k < ∞,

(3)

l=1

then



al is s.t.b. summable |C, 1|k .

Definition 3 Let { ps } be of +ive numbers and Ps =

s 

pr → ∞,

(4)

r =0

where (P−s = p−s = 0, s ≥ 1). If σs defines the (N , ps ) mean [12], i.e., σs = and lims→∞ σs = s, then



s 1  pq sq , Ps q=0

Ps = 0, s ∈ N

(5)

as is s.t.b. (N , ps ) summable.

Further, if {σs } is of bounded variation (BV) with q ≥ 1 [1], i.e.,  ∞   Ps q−1 s=1

ps

|σs − σs−1 |q < ∞,

(6)

 then as is s.t.b. |N , ps |q summable. Let U = (u lw ) be a lower triangular matrix with diagonal terms nonzero, i.e., it is a normal matrix. Then, the transformation of sequence s = {sl } to U s = {Ul (s)} by U is given by: l  u lw sw , n = 0, 1, . . . (7) Ul (s) = w=0

If

 ∞   Pl δq+q−1 l=1

pl

|Ul (s)|q < ∞,

(8)

A New Result Using Quasi-β-Power Increasing Sequence

5

 then al is s.t.b. |U, pl , δ|q summable [22], q ≥ 1. Let {ϕl } be of +ive real numbers. If ∞  q−1 ϕl |Ul (s)|q < ∞, (9) l=1

 then al is s.t.b. ϕ − |U, pl |q summable, q ≥ 1 and Ul (s) = Ul (s) − Ul−1 (s). Taking ϕl = Pl / pl in condition (9), ϕ − |U, pl |q changes to |U, pl |q summability. Also, if we take δ = 0 in condition (8), then |U, pl , δ|q changes to |U, pl |q summability. Now, we introduce other notations used in the main result as follows. We are given with a normal matrix U = (u lw ). Two lower semi-matrices U = (u lw ) and Uˆ = (uˆ lw ) are defined as: u lw =

l 

u li ; l, w = 0, 1, 2, . . .

(10)

i=w

and uˆ 00 = u 00 = u 00 , uˆ lw = u lw − u l−1,w ; l = 1, 2, . . . Then, we have: Ul (s) =

l 

u lw sw =

w=0

and Ul (s) =

l 

u lw aw

(11)

(12)

w=0

l 

uˆ lw aw .

(13)

w=0

2 Known Result Özarslan and Yavuz [16] have proved the two theorems as given below.

2.1 Theorem [16] Let U = (u lw ) be a +ive normal matrix with u l0 = 1, where l = 0, 1, 2, . . .

(14)

u l−1,w ≥ u lw for l ≥ w + 1,

(15)

6

S. Sonker and R. Jindal



and u ll = O

pl Pl

 .

(16)

Let {Yl } be quasi-β-power increasing and ∃ {Bl } and {λl } satisfying: |λl | ≤ Bl , ∀ l,

(17)

Bl → 0 as l → ∞,

(18)

∞ 

l|Bl |Yl < ∞,

(19)

l=1

|λl |Yl = O(1),

(20)

(λl ) ∈ BV,

(21)

l  |z w |q

and

 m   pl Pl

l=1

Then,



= O(Yl ),

(22)

|zl |q = O(Ym ).

(23)

w

w=1

al λl is |U, pl |q summable, q ≥ 1.

2.2 Theorem [16] Let U = (u lw ) and {Yl } be as defined in Theorem 2.1, and satisfying (17)–(21), (23) with ∞ 

Pl |Bl |Yl < ∞

(24)

l=1

and

m  |zl |q l=1

Then,



Pl

= O(Ym ).

al λl is |U, pl |q summable, q ≥ 1.

(25)

A New Result Using Quasi-β-Power Increasing Sequence

7

3 Main Result A sequence {λl } is of bounded variation, if ∞ 

|λl | = |λl − λl−1 | < ∞.

l=1

Our purpose is to generalize both the known results one by one by finding the least conditions. So, the theorems were stated as follows.

3.1 Theorem Let the matrix U, {Yl } and {Bl } be as defined in the known result such that the conditions (14)–(21) with l 

 ϕwq−1

w=1 m 

pw Pw

 q−1

ϕl

l=1

and

m+1 

 q

ϕl

l=w+1

are satisfied. Then,



pl Pl

pl Pl

q−1

|z w |q = O(Yl ), w

(26)

|zl |q = O(Ym ),

(27)

q−1

q

 |w uˆ lw | = O

pw Pw

q−1 as m → ∞,

(28)

al λl is ϕ − |U, pl |q summable, q ≥ 1.

3.2 Theorem Let the matrix U, {Yl } and {Bl } be as defined in the known result such that the conditions (14)–(21), (23)–(24) with m 

 q−1 ϕl

l=1

are satisfied. Then,



pl Pl

q−1

|zl |q = O(Ym ) as m → ∞, Pl

al λl is ϕ − |U, pl |q summable, q ≥ 1.

(29)

8

S. Sonker and R. Jindal

4 Lemmas The following lemmas are needed to prove the main result.

4.1 Lemma [13] By using the conditions of result 3.1, we have: l Bl Yl = O(1) as l → ∞ and

∞ 

(30)

Bl Yl < ∞.

(31)

l=1

4.2 Lemma Let {Yl } and {Bl } be as defined in Theorem 3.2. Then, we have: Pl Bl Yl = O(1) and

∞ 

(32)

pl Yl Bl < ∞.

(33)

l=1

5 Proof of the Main Results Now, we prove all these theorems one by one. Proof of Theorem 3.1 Let K l represents the A-transform of

K l =

l 

uˆ lw λw aw

w=0

=

l−1  w=1

w uˆ lw λw

w  k=1

ak + uˆ ll λl

l  w=1

aw



al λl . Then, we obtain

A New Result Using Quasi-β-Power Increasing Sequence

=

l−1 

9

(uˆ lw λw − uˆ l,w+1 λw+1 )z w + u ll λl zl

w=1

=

l−1 

(uˆ lw λw − uˆ l,w+1 λw+1 − uˆ l,w+1 λw + uˆ l,w+1 λw )z w + u ll λl zl

w=1

= =

l−1 

w (uˆ lw )λw z w +

w=1 K l(1)

l−1 

uˆ l,w+1 λw z w + u ll λl zl

w=1

+ K l(2) + K l(3) .

(34)

To prove the main result, it is enough to show that:   |K l(1) + K l(2) + K l(3) |q ≤ 3q |K l(1) |q + |K l(2) |q + |K l(3) |q , then Eq. (34) reduces to: ∞ 

q−1

ϕl

|K l(v) |q = Jv < ∞ for v = 1, 2, 3.

(35)

l=1

Now, we have

J1 = O(1)

m+1 

q−1 ϕl

= O(1)

q−1 ϕl

 l−1 

m 

|w (uˆ lw )||λw ||z w |

|w (uˆ lw )||λw | |z w | q

w=1

l=2

= O(1)

q

w=1

l=2 m+1 

×

 l−1 

|λw ||λw |q−1 |z w |q ×

w=1

= O(1) = O(1)

 m   ϕw pw q−1

w=1 m 

Pw  ϕwq−1

w=1

pw Pw

m+1  l=w+1



ϕl pl Pl

m−1 

|λw |

w 

|λw ||z w |q 

q−1

q−1 |w (uˆ lw )|

w=1

|w (uˆ lw )| m+1  l=w+1

q

ϕi

 l−1 

q−1

|λw ||λw |q−1 |z w |q ×

 pi q |z i |q P i w=1 i=1   m q  pw + O(1)|λm | ϕwq−1 |z w |q P w w=1

= O(1)

×

q

|w (uˆ lw )|

10

S. Sonker and R. Jindal

= O(1)

m−1 

Bw Yw + O(1)Ym |λm |

w=1

= O(1),

(36)

using the conditions of result 3.1 and Lemma 4.1. Now, using (λw ) ∈ BV, we have

J2 = O(1)

m+1 

q−1 ϕl

= O(1)

q−1 ϕl

 l−1 

×



|(uˆ l,w+1 )||λw ||z w | q−1

|(uˆ l,w+1 )||λw |

w=1

= O(1)

|λw ||(uˆ l,w+1 )||z w | q

w=1

l=2

 l−1 

q

w=1

l=2 m+1 

×

 l−1 

m+1  l=2

ϕl pl Pl

q−1 ×

 l−1 

w|w (uˆ lw )|Bw |z w |q

w=1

 ϕl pl q−1 |w (uˆ lw )| Pl w=1 l=w+1   m m+1  ϕw pw q−1  v Bw |z w |q |w (uˆ lw )| = O(1) Pw w=1 l=w+1  q m  pw q−1 = O(1) w Bw ϕw |z w |q P w w=1  q m−1 w   pi q−1 = O(1) (w Bw ) ϕi |z i |q P i w=1 i=1  δq−1 m  pw |z w |q + O(1)m Bm ϕwq−1 Pw w w=1 = O(1)

= O(1)

m 

m−1 

w Bw |z w |q

m+1 



w|Bw |Yw + O(1)

w=1

m−1 

Bw+1 Yw+1

w=1

+ O(1)m Bm Ym = O(1), using the conditions of result 3.1 and Lemma 4.1.

(37)

A New Result Using Quasi-β-Power Increasing Sequence

11

Again, using the concept in J1 , finally we have J3 = O(1)

m 

q−1

ϕl

l=1

= O(1)

m 

 q−1 ϕl

l=1

= O(1)

m 

|u ll |q |λl |q |zl |q



pl Pl

q

w=1

pw Pw

m−1 

w 

ϕwq−1

|λl ||λl |q−1 |zl |q q |λw ||z w |q 

 pi q = O(1) |λw | |z i |q P i w=1 i=1  q m  pw + O(1)|λm | ϕwq−1 |z w |q P w w=1 q−1 ϕi

= O(1) as m → ∞,

(38)

using the conditions of result 3.1 and Lemma 4.1. So, collecting conditions (36)–(38), condition (35) holds. Hence, Theorem 3.1 has been proved. Proof of Theorem 3.2 Using Lemma as in Theorem 3.1 and put back m4.2 and doing  m q q B P | /P B |z | , Theorem 3.2 can be easily proved. in place of (|z ) v v v v v v v=1 v=1

6 Corollaries Here, we derive some new results and previous known results as follows. Corollary 1 If we take ϕl = Pl / pl in the main result 3.1 and 3.2, then known results can be easily derived. Corollary 2 If we take pn = 1 and in the main result ϕl = Pl / pl 3.1 and 3.2, then we get two new results on |U |q summability factors. Corollary 3 Finally, if we take (Yl ) and ϕl = Pl / pl as almost increasing sequence in the main result 3.1 and 3.2, we get two different results on |U, pl |q summability.

7 Conclusion The main objective is to get negligible set of equations for absolute matrix summability. Here, we get two generalized results on ϕ − |U, pl |q summability, from which several previous results can be derived.

12

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Acknowledgements The authors were highly thankful for the financial support to the Science and Engineering Research Board through Project No.: EEQ/2018/000393.

References 1. Bor, H.: On two summability methods. Math. Proc. Camb. Philos. Soc. 97, 147–149 (1985) 2. Bor, H.: On the relative strength of two absolute summability methods. Proc. Am. Math. Soc. 113, 1009–1012 (1991) 3. Bor, H.: A note on |N , pn |k summability factors. Bull. Calcutta Math. Soc. 82, 357–362 (1990) 4. Bor, H.: Absolute summability factors for infinite series. Math. Jpn. 36, 215–219 (1991) 5. Bor, H.: Factors for |N , pn |k summability of infinite series. Ann. Acad. Sci. Fenn. Ser. A 1 Math. 16, 151–154 (1991) 6. Bor, H.: On absolute summability factors for |N , pn |k summability. Comment. Math. Univ. Carol. 32(3), 435–439 (1991) 7. Bor, H.: On the |N , pn |k summability factors of infinite series. Proc. Indian Acad. Sci. 101, 143–146 (1991) 8. Bor, H.: A note on |N , pn |k summability factors. Rend. Mat. Appl. 12(7), 937–942 (1992) 9. Bor, H.: On absolute summability factors. Proc. Am. Math. Soc. 118, 71–75 (1993) 10. Bor, H.: On absolute Riesz summability factors. Rocky Mt. J. Math. 24, 1263–1271 (1994) 11. Flett, T.M.: On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. Lond. Math. Sci. 3(1), 113–141 (1957) 12. Hardy, G.H.: Divergent Series. Oxford University Press, Oxford (1949) 13. Leindler, L.: A new application of quasi-power increasing sequences. Publ. Math. (Debr.) 58, 791–796 (2001) 14. Özarslan, H.S., Keten, A.: On a new application of almost increasing sequences. J. Inequal. Appl. 2013, 13 (2013) 15. Özarslan, H.S., Kartal, B.: A generalization of a theorem of Bor. J. Inequal. Appl. 2017, 179 (2017) 16. Özarslan, H.S., Yavuz, E.: A new note on absolute matrix summability. J. Inequal. Appl. 2013, 474 (2013) 17. Özarslan, H.S., Ari, T.: Absolute matrix summability methods. Appl. Math. Lett. 24, 2102– 2106 (2011) 18. Özarslan, H.S.: A new study on generalized absolute matrix summability methods. Maejo Int. J. Sci. Technol. 19. Özarslan, H.S.: A new study on generalized absolute matrix summability. Commun. Math. Appl. 7(4), 303–309 (2016) 20. Özarslan, H.S.: An application of δ-quasi monotone sequence. Int. J. Appl. 14(2), 134–139 (2017) 21. Özarslan, H.S., Sakar, M.Ö.: A new application of absolute matrix summability. Math. Sci. Appl. E-Notes 3, 36–43 (2003) 22. Sulaiman, W.T.: Inclusion theorems for absolute matrix summability methods of an infinite series (IV). Indian J. Pure Appl. Math. 34(11), 1547–1557 (2003) 23. Sonker, S., Jindal, R.: Approximation of signals by the triple product summability means of the Fourier series. Soft Comput. Theor. Appl. 425, 169–179 (2022) 24. Sonker, S., Jena, B.B., Jindal, R., Paikray, S.K.: A generalized theorem on double absolute factorable matrix summability. Appl. Math. Inf. Sci. 16(2), 315–322 (2022)

(C, 1, 1)-Quasinormal Convergence of Double Sequence of Functions Smita Sonker and Priyanka

Abstract In this present paper, we have considered the extension of the concept of strong Ces`aro-type quasinormal convergence, strong lacunary-quasinormal convergence, statistical-quasinormal convergence, lacunary-statistical-quasinormal convergence for double sequence of functions and derived some inclusive relation between these notions. Keywords Statistical convergence · Quasinormal convergence · Lacunary strong quasinormal convergence · (C, 1, 1)-quasinormal convergence

1 Introduction Fast [9] introduced a generalization of ordinary convergences of sequences named as statistical convergence. And later some basic properties of statistical convergence were established by Schoenberg [20] for real and complex sequences. G¨okhan and G¨ung¨or [12] introduced the notion of pointwise statistical convergence. Also the notion of pointwise and uniform statistical convergence of α order, λ-statistical convergence of α order and pointwise lacunary statistical convergence of α order are studied in [4, 7, 8], respectively. The concept of statistical convergence of double sequences is introduced by Mursaleen and Edely in [17]. Let’s define some basic definitions from the literature. For a subset X of IN × IN, let X (i, j) be the numbers of (k, l) in X such that k ≤ i and l ≤ j. Then double natural density of X ⊂ IN × IN is defined as δ2 (X ) = lim

i, j→∞

provided

X (i, j) ij

X (i, j) , ij

has a limit in Pringsheim’s sense.

S. Sonker (B) · Priyanka National Institute of Technology Kurukshetra, Kurukshetra, Haryana 136119, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1440, https://doi.org/10.1007/978-981-19-9906-2_2

13

14

S. Sonker and Priyanka

The sequence {xr s } of reals is known as statistically convergent [10] to the number x0 if for each δ > 0, the set δ2 ({(r, s) : |xr s − x0 | ≥ δ}) = 0 A double sequence {gr s (u)} of functions is known as statistically convergent to g(u) if for every δ > 0 and u ∈ E,  1  {(i, j) : i ≤ r, j ≤ s, |gi j (u) − g(u)| > δ} = 0 r,s→∞ r s lim

A real sequence xs is called Ces`aro summable [11] to a number x0 if 1 xk = x0 s→∞ s k=1 s

lim

and we will write this as (C, 1)-lim xs = x0 . Bukovsk`a [1] introduced the notion of quasinormal convergence. Also the same concept with the name as equal convergence is introduced by Cs`asz`ar and Laczkovich in [5, 6]. More details about this are given in [2, 3, 19]. In the present work, we extended the concept to double sequences of functions. For the real valued mappings gr s , g defined on a non empty set E, the double sequence {gr s (u)} is known as quasinormal convergent to g(u) if we have a double sequence δr s of positive reals tending to 0 so that for every u ∈ E, ∃ n 0 = n 0 (u) with |gr s (u) − g(u)| < δr s ∀ n ≥ n 0 . The uniform convergence of double sequence {gr s } to g implies the quasinormal convergence.

2 Preliminaries From [18], we have the following results: Theorem 1 For real valued mappings gk , g defined on a non empty set E, the following inclusive relation holds: 1. If gk (u) is strongly Ces`aro convergent to g(u), then gk (u) is statistically quasinormal convergent to g(u). 2. If gk (u) is uniformly bounded and statistical-quasinormal convergent to g(u), gk (u) is strongly Ces`aro convergent to g(u).

(C, 1, 1)-Quasinormal Convergence of Double Sequence …

15

Theorem 2 For real valued functions gk , g on a non empty domain E. θ be the lacunary sequence, we have 1. If {gk (u)} is strong lacunary convergent to g(u), then {gk (u)} is lacunaryquasinormal statistical convergent to g(u). 2. If {gk (u)} is uniformly bounded and statistically quasinormal convergent to g(u), then {gk (u)} is lacunary strongly convergent to g(u).

3 Main Result Throughout this paper let gr s (u), g(u) be real valued mappings on a non empty set E and the limit of double sequences is taken in Pringsheim’s sense.

3.1 (C, 1, 1)-Quasinormal Convergence of Double Sequence of Functions As defined in [21], the sequence {gr s (u)} is called Ces`aro (C, 1, 1)-summable to g(u) if for every u ∈ E, r,s 1  lim gkl (u) = g(u) r,s→∞ r s k=1,l=1 Also the sequence {gr s (u)} is known as strongly Ces`aro (C, 1, 1)-summable to g(u) if for u ∈ E, r,s 1  |gkl (u) − g(u)| = 0 lim r,s→∞ r s k=1,l=1 Definition 1 The double sequence {gr s (u)} is known as (C, 1, 1)-Quasinormal summable to g(u) if we have a sequence {δr s } of positive real numbers tending to 0 as r, s → ∞ such that for every u ∈ E, ∃ n 0 = n 0 (u) with r,s 1  gkl (u) − g(u) < δr s ∀ r, s ≥ n 0 r s k=1,l=1

Definition 2 The double sequence {gr s (u)} is known as strongly Ces`aro (C, 1, 1)quasinormal summable to g(u) if there is a positive sequence {δr s } of real numbers tending to zero as r, s → ∞ such that for every u ∈ E, ∃ n 0 = n 0 (u) with r,s 1  |gkl (u) − g(u)| < δr s ∀ r, s ≥ n 0 . r s k=1,l=1

16

S. Sonker and Priyanka

3.2 Statistical-Quasinormal Convergence of Double Sequences of Functions Definition 3 The double sequence {gr s (u)} is said to be statistical-quasinormal convergent to a real valued function g(u) if there is a double sequence δr s → 0 (as r, s → ∞) such that for each u ∈ E, lim

r,s→∞

1 |{(k, l) : k ≤ r, l ≤ s, |gkl (u) − g(u)| ≥ δkl }| = 0. rs

Let QNS2 be the space of all statistical-quasinormal convergent double sequences of functions. Theorem 3 For the double sequence {gr s (u)}, following relations holds 1. If {gr s (u)} is strongly Ces`aro (C, 1, 1) convergent to g(u), then {gr s (u)} is statistically quasinormal convergent to g(u). 2. If {gr s (u)} is uniformly bounded and statistical-quasinormal convergent to g(u), then {gr s (u)} is strongly Ces`aro (C, 1, 1) convergent to g(u). Proof 1. As {gr s (u)} is strongly Ces`aro (C, 1, 1) convergent to g(u), so for δr s → 0 and every u in E, we’ve 1 rs

r,s 

|g pq (u) − g(u)| =

p=1,q=1

1 rs

+ ≥

r,s 

1 rs

1 rs

|g pq (u) − g(u)|

p=1,q=1,|g pq (u)−g(u)|≥δ pq r,s 

|g pq (u) − g(u)|

p=1,q=1,|g pq (u)−g(u)|1 ls

ls−1

>∞

20

S. Sonker and Priyanka

3. QNS2 = QNS2θφ if and only if 1 < lim inf r and 1 < lim inf s

ls ls−1

≤ lim sups

ls ls−1

< ∞.

Proof 1. Sufficient condition: Let lim inf r

kr kr −1

kr kr −1

≤ lim supr

kr kr −1

1 and lim inf s

ls ls−1

> 1 and let

= pr and ls−1 = qs . So there exists δ1 > 0 and δ2 > 0 such that 1 + δ1 ≤ kr −1 pr ∀r ≥ 1 and 1 + δ2 ≤ qs ∀ s ≥ 1. Then for {gkl (u)} ∈ QNS2 , we have ls

kr

1 |{(k, l) : k ∈ Ir , l ∈ Js , |gkl (u) − g(u)| ≥ δkl }| ≤ mr ns



1 + δ1 δ1



 1 + δ2 τr s , δ2

where τr s = kr1ls |{(k, l) : 1 ≤ k ≤ kr , 1 ≤ l ≤ ls and |gkl (u) − g(u)| ≥ δkl }| → 0 as r, s → 0. So lim

r,s→∞

1 |{(k, l) : k ∈ Ir , l ∈ Js , |gkl (u) − g(u)| ≥ δkl }| = 0 mr ns

i.e. {gkl (u)} ∈ QNS2θφ . Necessary condition: On the contrary, let lim inf r pr = lim inf s qs = 1. Let θ and φ be lacunary sequences. Select the subsequence {kr j } of θ and {lsi } of φ satisfying kr j −1 kr j 1 < 1 + and > j; where r j ≥ r j−1 + 2 kr j −1 j mr j and

lsi lsi −1

i; where si ≥ si−1 + 2. i n si

Also define a double sequence of functions {gkl (u)} on E  gkl (u) =

1, if k ∈ Ir j and l ∈ Jsi for some i and j g(u), otherwise



Let u ∈ E. Then consider 1 m r j n si 1 = m r j n si

τr j si =

  {(k, l) : k ∈ Ir , l ∈ Js , |gkl (u) − g(u)| ≥ δkl } j i   {(k, l) : k ∈ Ir , l ∈ Js , |1 − g(u)| ≥ δkl } j i

Now since δkl → 0 as k, l → ∞, so there exists i 0 and j0 such that ∀ i ≥ i 0 and j ≥ j0 , we obtain

(C, 1, 1)-Quasinormal Convergence of Double Sequence …

τr j si =

21

1 |{(k, l) : k ∈ Ir j , l ∈ Jsi }| ≥ 1 m r j n si

Thus {gkl (u)} ∈ / QNS2θφ . Now ∀ k and l there is Ir j and Jsi such that k ∈ Ir j and l ∈ Jsi . This leads us to 1 |{(m, n) : m ≤ k, n ≤ l, |gmn (u) − g(u)| ≥ δmn }| kl kr j − kr j −1 lsi − lsi −1 . ≤ kr j −1 lsi −1 1 ≤ ji which implies {gkl (u)} ∈ QNS2 . 2. Sufficient condition: Let lim supr pr < ∞ and lim sups qs < ∞, where pr = ls kr and qs = ls−1 . So, we have K > 0 and L > 0 such that pr < K ∀ r ≥ kr −1 1 and qs < L ∀ s ≥ 1. Let R and S be positive numbers such that supk,l≥R τkl < δ and τkl < S ∀ k, l = 1, 2, 3, . . .. Then for k ∈ Ir and l ∈ Js , consider  1  {(i, j) : i ≤ k, j ≤ l, |gi j (u) − g(u)| ≥ δi j } kl r,s    1 {(i, j) : i ∈ I p , j ∈ Jq , |gi j (u) − g(u)| ≥ δi j } ≤ k l p=1,q=1 r −1 s−1 =

r,s 

m p nq · τ pq ; where m 1 = n 1 k r −1 l s−1 p=1,q=1 r,s 

=

p=1,q=1, p,q≥R

m p nq · τ pq + kr −1ls−1

r,s  p=1,q=1, p,q j and qsi > i. Define  gkl =

1 if kr j −1 < k < 2kr j −1 and lsi −1 < l < lsi −1 for i, j = 1, 2, 3, . . . g(u) otherwise

22

S. Sonker and Priyanka

By proceeding in a similar way as in the part (1), it can be easily proved that / QNS2 . {gkl (u)} ∈ QNSQNS2θφ but {gkl (u)} ∈ 3. Combining the results of (1) and (2), we get the result (3), QNS2 = QNS2θφ ls r r if and only if 1 < lim inf r krk−1 ≤ lim supr krk−1 < ∞ and 1 < lim inf s ls−1 ≤ lim sups

ls ls−1

< ∞.

4 Conclusion From the results of this paper, we can conclude that the inclusive relations given in the preliminaries also holds for sequences of double functions, provided the limit is considered in Pringsheim’s sense.

References 1. Bukovsk´a, Z.: Quasinormal convergence. Math. Slov. 41(2), 137–146 (1991) 2. Bukovsk´y, L., Reclaw, I., Repick´y, M.: Spaces not distinguishing pointwise and quasinormal convergence of real functions. Topol. Appl. 41(1–2), 25–40 (1991) 3. Bukovsk´y, L., Haleˆs, J.: QN-space, wQN-space and covering properties. Topol. Appl. 154(4), 848–858 (2007) 4. Çinar, M., Karaka¸s, M., Et, M.: On pointwise and uniform statistical convergence of order α for sequences of functions. Fixed Point Theory Appl. 33, 11 (2013) ´ Laczkovich, M.: Discrete and equal convergence. Stud. Sci. Math. Hungar. 10, 5. Cs´asz´ar, A., 463–472 (1975) ´ Laczkovich, M.: Some remarks on discrete Baire classes. Acta Math. Acad. Sci. 6. Cs`asz`ar, A., Hungar. 33, 51–70 (1979) 7. Et, M., Çinar, M., Karaka¸s, M.: On λ-statistical convergence of order of sequences of function. J. Inequal. Appl. 204, 8 (2013) 8. Et, M., Seng¨ ¸ ul, H.: On pointwise lacunary statistical convergence of order of α sequences of function. Proc. Natl. Acad. Sci. India Sect. A 85(2), 253–258 (2015) 9. Fast, H.: Sur la convergence statistique. Cooloq. Math. 2, 241–244 (1951) 10. Fridy, J.A.: On statistical convergence. Analysis 5, 301–313 (1985) 11. Freedman, A.R., Sember, J.J., Raphael, M.: Some Ces`aro-type summability spaces. Proc. Lond. Math. Soc. 37(3), 508–520 (1978) 12. G¨okhan, A., G¨ung¨or, M.: On pointwise statistical convergence. Indian J. Pure Appl. Math. 33(9), 1379–1384 (2002) 13. G¨okhan, A., G¨ung¨or, M., Bulut, Y.: On the strong lacunary convergence and strong Ces`aro summability of sequences of real-valued functions. Appl. Sci. 8, 70–77 (2006) 14. G¨okhan, A.: Lacunary statistical convergence of sequences of real-valued functions. J. Math. 4 (2013) 15. Fridy, J.A., Orhan, C.: Lacunary statistical summability. J. Math. Anal. Appl. 173, 497–504 (1993) 16. Fridy, J.A., Orhan, C.: Lacunary statistical convergence. Pac. J. Math. 160, 43–51 (1993) 17. Mursaleen, Edely, O.H.H.J.: Statistical convergence of double sequences. Math. Appl. 288, 223–231 (2003) 18. Nuray, F.: Some Ces`aro-type quasinormal convergences. Creat. Math. Inform. 30(1), 75–80 (2021)

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23

19. Prokaj, V.: A note on equal convergence. Acta Math. Hungar. 73(1–2), 155–158 (1996) 20. Schoenberg, I.J.: The integrability of certain functions and related summability methods. Am. Math. Mon. 66, 361–375 (1959) 21. Ta¸s, E., Orhan, C.: Ces`aro means of subsequences of double sequences. Sarajevo J. Math. 15(2), 169–179 (2019)

Fixed Point Theorems in Neutrosophic Soft Metric Space Vishal Gupta

and Aanchal

Abstract Our work deals with introducing neutrosophic contractions and some new notions in neutrosophic soft metric space (NSMS). Additionally, we have stated and proved some new fixed point theorems for our newly defined contractions including Banach fixed point theorem. Keywords Soft set · Neutrosophic soft metric space · Fixed point AMS Subject Classification 54H25 · 47H10

1 Introduction Zadeh [1] in 1975 developed the fuzzy set concept to handle uncertainties. Further, the metric space notion was applied to fuzzy sets and fuzzy metric space was introduced by Kaleva and Seikkala [2] in 1984. George and Veeramani [3] moulded the earlier definition and hence gave their own version of fuzzy metric space. The notion of intuitionistic fuzzy metric space was given by Park [4] along with some fixed point theorems. It was found that fuzzy sets and intuitionistic fuzzy sets do not tell anything about the degree of indeterminacy. To overcome this, Smarandache [5] introduced neutrosophic sets (NS) that includes degree of indeterminacy (I) as well. Then, NS was extended to metric space by Kirisci and Simsek [6] and hence neutrosophic metric space (NMS) was introduced. Molodstov [7] developed the soft set theory to deal with the uncertainties in case of data consisting of parameters. The soft set theory was then applied to different spaces and thus their soft versions were introduced. Karaaslan [8] established neutrosophic soft sets (NSS) by applying the soft set theory to NS. Gupta and Gondhi [9] V. Gupta (B) · Aanchal Department of Mathematics, MMEC, Maharishi Markandeshwar (Deemed to be University), Mullana, Ambala, Haryana, India e-mail: [email protected]; [email protected] Aanchal Department of Mathematics, M.D.S.D. Girls College, Ambala, Haryana, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1440, https://doi.org/10.1007/978-981-19-9906-2_3

25

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generalize the metric space theory to NSS and hence gave neutrosophic soft metric space (NSMS) along with some basic notions. Simsek and Kirisci [10] gave neutrosophic contractions and maps. They have also given fixed point theorems in NMS. After that, in 2020, Sowndrarajan et al. [11] also investigated some fixed point results in the new setting of neutrosophic metric spaces. In this paper, we are going to extend fixed point theorems given in [10] in the setting of NSMS.

2 Preliminaries Now, we are going to give some basic results already present in the literature that will be used for the development of our new results. Definition 2.1 [5] Consider a non-empty set χ , then a NS is given as a set ϒ = {(ι, ϒ (ι), ϒ (ι), ϒ (ι)) : ι ∈ χ}, where ϒ (ι), ϒ (ι) and ϒ (ι) depict the belongingness degree, indeterminacy degree and non-belongingness degree, respectively, of every ι ∈ χ . Definition 2.2 [6] Consider a non-empty set χ , then a NMS is given by (χ , ϒ, ∗, ♦), where ϒ = {(ι, ϒ (ι), ϒ (ι), ϒ (ι)) : ι ∈ χ } is the neutrosophic set and ∗ and ♦ are continuous t-norms and continuous t-conorms respectively, so that for all ι, κ, γ ∈ χ and , μ > 0, the following assertions are satisfied: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

0 ≤ (ι, κ, ) ≤ 1, 0 ≤ (ι, κ, ) ≤ 1, 0 ≤ (ι, κ, ) ≤ 1; (ι, κ, ) + (ι, κ, ) + (ι, κ, ) ≤ 3; (ι, κ, ) = 1 ⇔ ι = κ; (ι, κ, ) = (κ, ι, ); (ι, κ, ) ∗ (κ, γ , μ) ≤ (ι, γ , + μ); (ι, κ, .) : [0, ∞) → [0, 1] is continuous; lim →∞ (ι, κ, ) = 1; (ι, κ, ) = 0 ⇔ ι = κ; (ι, κ, ) = (κ, ι, ); (ι, κ, ) ♦ (κ, γ , μ) ≥ (ι, γ , + μ); (ι, κ, .) : [0, ∞) → [0, 1] is continuous; lim →∞ (ι, κ, ) = 0; (ι, κ, ) = 0 ⇔ ι = κ; (ι, κ, ) = (κ, ι, ); (ι, κ, ) ♦ (κ, γ , μ) ≥ (ι, γ , + μ); (ι, κ, .) : [0, ∞) → [0, 1] is continuous; lim →∞ (ι, κ, ) = 0; (ι, κ, ) = 0, (ι, κ, ) = 1 and (ι, κ, ) = 1 if < 0.

Fixed Point Theorems in Neutrosophic Soft Metric Space

27

Definition 2.3 [8] A NSS over a non-empty set χ is given by a map ς : U → (χ ), where U is the set of parameters and (χ ) is the collection of all NS on χ . Now, ς is written as ς=

   

, ι, ς( ) (ι), ς( ) (ι), ς( ) (ι) : ι ∈ χ : ∈ U .

= Definition 2.4 [9] Consider χ be any absolute soft set and  

,  ( ) (ι),  ( ) (ι),  ( ) (ι) : ι ∈ χ be NSS, so that  : SP(χ ) × SP(χ) × R(U )∗ → [0, 1]. Consider ∗ be the continuous t-norm and ♦ defines the continuous t-conorm, then the space (χ , , ∗, ♦) is called NSMS if for all α e1 , β e2 , γ e3 ∈ SP(χ) and , μ ∈ R(U )∗ , the given conditions hold:     0 ≤  ( ) α e1 , β e2 , ≤ 1, 0 ≤  ( ) α e1 , β e2 , ≤ 1, 1.   0 ≤  ( ) α e1 , β e2 , ≤ 1;       2.  ( ) α e1 , β e2 , +  ( ) α e1 , β e2 , +  ( ) α e1 , β e2 , ≤ 3;   3.  ( ) α e1 , β e2 , = 1 ⇔ α e1 = β e2 ;     4.  ( ) α e1 , β e2 , =  ( ) β e2 , α e1 , ;       5.  ( ) α e1 , β e2 , ∗  ( ) β e2 , γ e3 , μ ≤  ( ) α e1 , γ e3 , + μ ;   6.  ( ) α e1 , β e2 , . : R(U )∗ → [0, 1] is continuous;   7. lim →∞  ( ) α e1 , β e2 , = 1;   8.  ( ) α e1 , β e2 , = 0 ⇔ α e1 = β e2 ;     9.  ( ) α e1 , β e2 , =  ( ) β e2 , α e1 , ;       10.  ( ) α e1 , β e2 , ♦  ( ) β e2 , γ e3 , μ ≥  ( ) α e1 , γ e3 , + μ ;   11.  ( ) α e1 , β e2 , . : R(U )∗ → [0, 1] is continuous;   12. lim →∞  ( ) α e1 , β e2 , = 0;   13.  ( ) α e1 , β e2 , = 0 ⇔ α e1 = β e2 ;     14.  ( ) α e1 , β e2 , =  ( ) β e2 , α e1 , ;       15.  ( ) α e1 , β e2 , ♦  ( ) β e2 , γ e3 , μ ≥  ( ) α e1 , γ e3 , + μ ;   16.  ( ) α e1 , β e2 , . : R(U )∗ → [0, 1] is continuous;   17. lim →∞  ( ) α e1 , β e2 , = 0;       18.  ( ) α e1 , β e2 , = 0,  ( ) α e1 , β e2 , = 1,  ( ) α e1 , β e2 , = 1 if ≤ 0.   The neutrosophic soft metric is given as  =  ( ) ,  ( ) ,  ( ) and        ( ) α e1 , β e2 , ,  ( ) α e1 , β e2 , ,  ( ) α e1 , β e2 , denotes the nearness degree, the neutralness degree and non-nearness degree between α e1 and β e2 with respect to , respectively. Example 2.1 [9] Consider (χ, ϑ) be soft metric space. Let ∗ and ♦ be t-norm and t-conorm defined as ι ∗ κ = min{ι, κ} and ι ♦ κ = max{ι, κ} and    ( ) ψ, ω, =

 ;

+ ϑ ψ, ω

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  ϑ ψ, ω  ;

+ ϑ ψ, ω     ϑ ψ, ω ,  ( ) ψ, ω, =

   ( ) ψ, ω, =

for all ψ, ω ∈ χ and > 0. Then, (χ , , ∗, ♦) is NSMS, where  : SP(χ ) × SP(χ) × R(U )∗ → [0, 1] is the NSM. be NSMS, 0  < Definition 2.5 [9] Consider (χ , , ∗, ♦)  < 1, λ > 0 and ι ∈ χ . The set, B ι, , λ =         κ ∈ χ :  (e) ι, κ, λ > 1 − ,  (e) ι, κ, λ < ,  (e) ι, κ, λ <  is defined as the open ball having center ι and radius  with respect to λ.   Theorem 2.1 [9] Every open ball B ι, , λ in a NSMS (χ , , ∗, ♦) is an open set. Definition 2.6 [9] Consider (χ , , ∗, ♦) be NSMS, then the topologyon χ is given as τ = {ϒ ⊂ χ : for every ι ∈ ϒ, there exists > 0 and  ∈ 0, 1 so that B(ι, , ) ⊆ ϒ}. Theorem 2.2 [9] Every NSMS is a Hausdorff space. Definition 2.7 [9] Any subset ϒ ⊆ χ in a NSMS (χ, , ∗, ♦) is known  as Neutrosophic bounded, if for all ψ, ω ∈ χ there exists > 0 and  ∈ 0, 1 , so that        (e) ψ, ω, > 1 − ,  (e) ψ, ω, <  and  (e) ψ, ω, < . Definition  2.8 [9] Consider (χ, , ∗, ♦) be NSMS and ϒ ⊆ χ and ε = {U : ϒ ⊆ U ∈ε U }, then the collection ε is known as open cover of ϒ. Definition 2.9 [9] A subspace Y of NSMS (χ , , ∗, ♦) is compact, if every open cover of Y has a finite subcover. Theorem 2.3 [9] A compact subset of NSMS (χ , , ∗, ♦) is Neutrosophic bounded. Remark 2.1 [9] In a NSMS (χ, , ∗, ♦), Y ⊆ χ is Neutrosophic bounded if and only if it is bounded. Theorem 2.4 [9] Consider (χ, , ∗, ♦) be NSMS and τ be the topology on ιen in χ is convergent  induced by NSM, then a sequence    to ι if and only if  ( ) ιen , ι, → 1,  ( ) ιen , ι, → 0 and  ( ) ιen , ι, → 0 when n → ∞.   ιen in χ is Definition 2.10 [9] Consider (χ, , ∗, ♦) be NSMS. A sequence 

called Cauchy if for every  > 0 and n o ∈ N so that  ( ) ιem , ιen , 2 > 1 − ,      ( ) ιem , ιen , 2 <  and  ( ) ιem , ιen , 2 < , for all n, m ≥ n o . Definition 2.11 [9] A NSMS (χ , , ∗, ♦) is complete if every Cauchy sequence is convergent in it.

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29

3 Fixed Point Results Before proving our main theorems, we are going to give some definitions. Definition 3.1 Consider (χ , , ∗, ♦) be NSMS, then for 0 <  < 1, let  : SP(χ) × SP(χ) × R(U )∗ → [0, 1] defined as       (ι, κ) = inf λ > 0 :  ( ) ι, κ, λ > 1 − ,  ( ) ι, κ, λ < ,     ( ) ι, κ, λ <  . Then 1. (χ,  ) is the generating space for soft quasi metric family, 2.  is compatible soft symmetric for topology τ . Definition 3.2 Consider (χ , , ∗, ♦) be NSMS, then map φ : χ → χ is known as neutrosophic contraction if (3.1) holds for all ι, κ ∈ χ and λ > 0 where 0 < s < 1, 1



1



   −1≤s  ,  ( ) φι, φκ, λ  ( ) ι, κ, λ − 1      ( ) φι, φκ, λ ≤  ( ) ι, κ, λ ,      ( ) φι, φκ, λ ≤  ( ) ι, κ, λ .

(3.1)

Proposition 3.1 Consider φ be a neutrosophic contraction, then φ n is also a neutrosophic contraction. Also, if s is the constant of φ, then s n is the constant for φn .    Proposition 3.2 For a neutrosophic contraction φ and ι ∈ χ, φ B ι, , λ ⊂  n  B φ (ι),  ∗ , λ , where  ∗ = s n × .     Definition 3.3 Consider two NSMS χ 1 , 1 , ∗, ♦ and χ 2 , 2 , ∗, ♦ and Bi be the uniformly generated balls by i , then a map φ : χ 1 → χ 2 is known as uniformly continuous with respect to B1 and B2 if given 0 <  2 < 1 and λ2 > 0 implies the existence of 0 <  1 < 1 and λ1 > 0 and vice versa, so that (3.2) and (3.3) hold for all ι, κ ∈ χ 1 ,        ( ) ι, κ, λ1 ≥ 1 −  1 ,  ( ) ι, κ, λ1 ≤  1 ,  ( ) ι, κ, λ1 ≤  1 ,

(3.2)

that implies        ( ) φι, φκ, λ2 ≥ 1 −  2 ,  ( ) φι, φκ, λ2 ≤  2 ,  ( ) φι, φκ, λ2 ≤  2 . (3.3)

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Definition 3.4 Consider (χ , , ∗, ♦) be NSMS, then the map φ : χ → χ is known as λ-uniformly continuous if any given 0 <  < 1 implies the existence of 0 < ζ < 1, so that (3.4) and (3.5) holds for all ι, κ ∈ χ and λ > 0,        ( ) ι, κ, λ ≥ 1 − ,  ( ) ι, κ, λ ≤ ,  ( ) ι, κ, λ ≤ ,

(3.4)

that implies        ( ) φι, φκ, λ ≥ 1 − ζ ,  ( ) φι, φκ, λ ≤ ζ ,  ( ) φι, φκ, λ ≤ ζ .

(3.5)

Definition 3.5 Consider (χ , , ∗, ♦) be NSMS, then a map φ : χ → χ is λuniformly continuous if any given > 0 implies the existence of 0 < ζ < 1 and vice versa, so that (3.6) and (3.7) holds for all ι, κ ∈ χ and λ > 0,      ( ) ι, κ, λ  ( ) ι, κ, λ 1     − 1 ≤ ζ,  ≤ ζ,  ≤ ζ , (3.6)  ( ) ι, κ, λ 1 −  ( ) ι, κ, λ 1 −  ( ) ι, κ, λ that implies      ( ) φι, φκ, λ  ( ) φι, φκ, λ     − 1 ≤ ,  ≤ ,  ≤ .  ( ) φι, φκ, λ 1 −  ( ) φι, φκ, λ 1 −  ( ) φι, φκ, λ (3.7) 1

Proposition 3.3 Consider (χ , μ) be soft metric space, then a self map φ on χ is metric contractive on soft metric space (χ, μ) with contractive constant s if φ is neutrosophic contractive with the same contractive constant s in NSMS (χ , , ∗, ♦) set up by μ and vice versa.   Remark 3.1 A sequence ιen in soft metric  space  is called contractive if  (χ, μ) there exists 0 < s < 1 so that μ ιen+1 , ιen+2 ≤ sμ ιen , ιen+1 . Proposition 3.4 Consider (χ, , ∗, ♦) be NSMS set up by soft metric μ defined  on χ, then the sequence ιen in χ is contractive in (χ , μ) if and only if ιen is neutrosophic contractive in NSMS. Now, we are going to prove Banach Fixed Point Theorem for neutrosophic contractions in NSMS. Theorem 3.1 Consider (χ , , ∗, ♦) be complete NSMS having neutrosophic contractive sequences as Cauchy. Consider F : χ → χ be a neutrosophic contraction, then F owns a unique fixed point. Proof Consider ι ∈ χ and ιen = F n (ι), then for every λ > 0, we have

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31



1    −1≤s  −1 ,  ( ) F(ι), F 2 (ι), λ  ( ) ι, ιe1 , λ

1 1 1    −1≥  −1 , s  ( ) ι, ιe1 , λ  ( ) F(ι), F 2 (ι), λ

1 1 1    −1≥  −1 . s  ( ) ι, ιe1 , λ  ( ) F(ι), F 2 (ι), λ 1

By applying induction, we get (3.8) for every m ∈ N ,

1



1

   −1≤s  −1 ,  ( ) ιem+1 , ιem+2 , λ  ( ) ιem , ιem+1 , λ

1 1 1    −1≥  −1 , s  ( ) ιem , ιem+1 , λ  ( ) ιem+1 , ιem+2 , λ

1 1 1    −1≥  −1 . s  ( ) ιem , ιem+1 , λ  ( ) ιem+1 , ιem+2 , λ

(3.8)

  Thus, ιen is a neutrosophic sequence. Hence, it is a Cauchy sequence.   contractive As χ is complete, sequence ιen converges to some κ ∈ χ . Claim that F possess κ as its fixed point. As, we have

1 1    −1≤s  −1 ,  ( ) Fκ, Fιem , λ  ( ) κ, ιem , λ    

 ( ) κ, ιem , λ  ( ) Fκ, Fιem , λ  ≤s  ,   1 −  ( ) Fκ, Fιem , λ 1 −  ( ) κ, ιem , λ    

 ( ) κ, ιem , λ  ( ) Fκ, Fιem , λ    ≤s  . 1 −  ( ) Fκ, Fιem , λ 1 −  ( ) κ, ιem , λ Taking m → ∞, we get 1  ( ) ( Fκ,Fιem ,λ)

− 1 → 0, 1− ( ) ( 

 ( )

Fκ,Fιem ,λ) ( Fκ,Fιem ,λ)

→ 0, 1− ( ) ( 

 ( )

Fκ,Fιem ,λ) ( Fκ,Fιem ,λ)

→ 0.

Hence, for every λ > 0, we have     lim  ( ) Fκ, Fιem , λ = 1, lim  ( ) Fκ, Fιem , λ = 0, m→∞ m→∞   lim  ( ) Fκ, Fιem , λ = 0. m→∞

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  Thus, limm→∞ F ιem = F(κ), that is limm→∞ ιem+1 = F(κ) and thus κ = F(κ). Uniqueness: Consider F(σ ) = σ , for some σ ∈ χ, then for λ > 0, we have

1 1 1     −1=  −1≤s  −1  ( ) κ, σ , λ  ( ) Fκ, Fσ , λ  ( ) κ, σ , λ



1 1    − 1 ≤ s2  −1 =s  ( ) Fκ, Fσ , λ  ( ) κ, Fσ , λ

1 n   − 1 → 0, ≤ ··· ≤ s  ( ) κ, σ , λ as n → ∞. Also for n → ∞, we get      

 ( ) κ, σ , λ  ( ) Fκ, Fσ , λ  ( ) κ, σ , λ    =  ≤s  1 −  ( ) κ, σ , λ 1 −  ( ) Fκ, Fσ , λ 1 −  ( ) κ, σ , λ   



 ( ) Fκ, Fσ , λ  ( ) κ, σ , λ 2    ≤s  =s 1 −  ( ) Fκ, Fσ , λ 1 −  ( ) κ, σ , λ  

 κ, σ , λ  ( )  → 0,  ≤ · · · ≤ sn 1 −  ( ) κ, σ , λ and      

 ( ) κ, σ , λ  ( ) κ, σ , λ  ( ) Fκ, Fσ , λ    =  ≤s  1 −  ( ) κ, σ , λ 1 −  ( ) Fκ, Fσ , λ 1 −  ( ) κ, σ , λ 



   ( ) Fκ, Fσ , λ  κ, σ , λ  ( )    ≤ s2  =s 1 −  ( ) Fκ, Fσ , λ 1 −  ( ) κ, σ , λ  

 ( ) κ, σ , λ n   → 0. ≤ ··· ≤ s 1 −  ( ) κ, σ , λ       Therefore,  ( ) κ, σ , λ = 1,  ( ) κ, σ , λ = 0 and  ( ) κ, σ , λ = 0, thus κ = σ. Now, we will give corollary to Theorem 3.1: Corollary 3.1 Consider (χ , , ∗, ♦) be a complete NSMS and map F : χ → χ be a neutrosophic contraction, then F owns a unique fixed point.   Definition 3.6 A sequence ιen is an s-increasing sequence if there exists n o ∈ N so that ιem + 1 ≤ ιem+1 , where m ≥ n o .

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33

Definition 3.7 An infinite product  in a NSMS is defined as ∞

         ( ) ι, κ, λi =  ( ) ι, κ, λ1  ( ) ι, κ, λ2  . . .  ( ) ι, κ, λn  . . . ,

i=1 ∞

         ( ) ι, κ, λi =  ( ) ι, κ, λ1  ( ) ι, κ, λ2  . . .  ( ) ι, κ, λn  . . . ,

i=1 ∞

         ( ) ι, κ, λi =  ( ) ι, κ, λ1  ( ) ι, κ, λ2  . . .  ( ) ι, κ, λn  . . . .

i=1

Theorem 3.2 (χ, , ∗, ♦) be a complete NSMS so that for ζ > 0 and for   Consider a sequence λn which is s-increasing, there exists m o ∈ N , so that (3.9) holds for all ι, κ ∈ χ .



       ( ) ι, κ, λi > 1 − ζ ,  ( ) ι, κ, λi < ζ ,  ( ) ι, κ, λi < ζ .

n≥m o

n≥m o

n≥m o

(3.9) Consider, 0 < s < 1 and a self map F on χ that satisfies (3.10) for all η, θ ∈ χ      ( ) Fη, Fθ , sλ ≥  ( ) η, θ , λ ,      ( ) Fη, Fθ , sλ ≤  ( ) η, θ , λ ,      ( ) Fη, Fθ , sλ ≤  ( ) η, θ , λ .

(3.10)

Then, F owns a fixed point which is unique as well. Proof Consider ι ∈ χ and ιen = F n (ι), then for λ > 0 we have



λ λ  ( ) ιe1 , ιe2 , λ =  ( ) Fι, F ι, λ ≥  ( ) ι, Fι, =  ( ) ι, ιe1 , , s s



    λ λ 2  ( ) ιe1 , ιe2 , λ =  ( ) Fι, F ι, λ ≤  ( ) ι, Fι, =  ( ) ι, ιe1 , , s s



    λ λ 2  ( ) ιe1 , ιe2 , λ =  ( ) Fι, F ι, λ ≤  ( ) ι, Fι, =  ( ) ι, ιe1 , . s s 





2





By applying induction, we get (3.11) for n ∈ N ,

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  λ  ( ) ιen , ιen+1 , λ >  ( ) ι, ιe1 , n , s

  λ  ( ) ιen , ιen+1 , λ <  ( ) ι, ιe1 , n , s

  λ  ( ) ιen , ιen+1 , λ <  ( ) ι, ιe1 , n . s

(3.11)

Now, suppose t < r , for r, t ∈ N and let μe(i) > 0, (i = t, . . . , r − 1) that satisfies μe(t) + · · · + μe(r −1) ≤ 1, then we have        ( ) ιet , ιer , λ >  ( ) ιet , ιet+1 , μet λ  . . .  ( ) ιer −1 , ιer , μer −1 λ



μer −1 λ μet λ ≥  ( ) ι, ιe1 , t  . . .  ( ) ι, ιe1 , r −1 , s s        ( ) ιet , ιer , λ <  ( ) ιet , ιet+1 , μet λ  . . .  ( ) ιer −1 , ιer , μer −1 λ



μer −1 λ μet λ ≤  ( ) ι, ιe1 , t  . . .  ( ) ι, ιe1 , r −1 , s s        ( ) ιet , ιer , λ <  ( ) ιet , ιet+1 , μet λ  . . .  ( ) ιer −1 , ιer , μer −1 λ



μer −1 λ μet λ ≤  ( ) ι, ιe1 , t  . . .  ( ) ι, ιe1 , r −1 . s s As,

∞

1 r =1 r (r +1) ,

take μe( j) =

   ( ) ιet , ιer , λ ≥  ( ) ι, ιe1 ,

1 , j j( j+1) (

= t, . . . , r − 1), then we have



λ λ  . . .  ( ) ι, ιe1 , t(t + 1)s t r (r − 1)s r −1

∞

λ  ( ) ι, ιe1 , ≥ , t(t + 1)s t t=1



  λ λ  ( ) ιet , ιer , λ ≤  ( ) ι, ιe1 ,  . . .  ( ) ι, ιe1 , t(t + 1)s t r (r − 1)s r −1

∞

λ  ( ) ι, ιe1 , ≤ , t(t + 1)s t t=1



  λ λ  ( ) ιet , ιer , λ ≤  ( ) ι, ιe1 ,  . . .  ( ) ι, ιe1 , t(t + 1)s t r (r − 1)s r −1

Fixed Point Theorems in Neutrosophic Soft Metric Space

≤

∞

35

 ( )

t=1

λ ι, ιe1 , . t(t + 1)s t

  n λ → ∞, when n → ∞. Thus, Take λ = n(n+1)s n , it is trivial that λn+1 − λn   λn is an s-increasing sequence. So, there exists m o ∈ N so that (3.12) holds

λ  ( ) ι, ιe1 , > 1 − ζ, n(n + 1)s n n=m o

∞

λ  ( ) ι, ιe1 , < ζ, n(n + 1)s n n=m o

∞

λ  ( ) ι, ιe1 , < ζ. n(n + 1)s n n=m ∞



(3.12)

o

      Hence,  ( ) ιen , ιem , λ > 1 − ζ ,  ( ) ιen , ιem , λ < ζ and  ( ) ιen , ιem , λ <   ζ for m, n ≥ m o . Thus, ιen is Cauchy and as NSMS is complete, there exists κ ∈ χ, so that limn→∞ ιen = κ. Claim that F possess κ as its fixed point. For n → ∞, we have



  λ λ  ( ) F(κ), κ, λ ≥  ( ) F(κ), F ιen , ∗  ( ) ιen+1 , κ, 2 2



λ λ ≥  ( ) κ, ιen , ∗  ( ) ιen+1 , κ, → 1 ∗ 1, 2s 2



   λ  λ  ( ) F(κ), κ, λ ≤  ( ) F(κ), F ιen , ♦  ( ) ιen+1 , κ, 2 2



λ λ ≤  ( ) κ, ιen , ♦  ( ) ιen+1 , κ, → 0 ♦ 0, 2s 2



    λ λ  ( ) F(κ), κ, λ ≤  ( ) F(κ), F ιen , ♦  ( ) ιen+1 , κ, 2 2



λ λ ≤  ( ) κ, ιen , ♦  ( ) ιen+1 , κ, → 0 ♦ 0. 2s 2 





    = 1,  ( ) F(κ), κ, λ Therefore,  ( ) F(κ), κ, λ    ( ) F(κ), κ, λ = 0, thus we have F(κ) = κ. Uniqueness: Consider σ ∈ χ so that F(σ ) = σ , then we have

=

0 and

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V. Gupta and Aanchal



    λ 1 ≥  ( ) σ , κ, λ =  ( ) F(σ ), F(κ), λ ≥  ( ) σ , κ, s



λ λ ≥  ( ) σ , κ, 2 ≥ · · · ≥  ( ) σ , κ, n ; s s

    λ 0 ≤  ( ) σ , κ, λ =  ( ) F(σ ), F(κ), λ ≤  ( ) σ , κ, s



λ λ ≤  ( ) σ , κ, 2 ≤ · · · ≤  ( ) σ , κ, n ; s s

    λ 0 ≤  ( ) σ , κ, λ =  ( ) F(σ ), F(κ), λ ≤  ( ) σ , κ, s



λ λ ≤  ( ) σ , κ, 2 ≤ · · · ≤  ( ) σ , κ, n . s s   λ sn

is an s-increasing sequence, thus for given 0 < ζ < 1, there       exists m o ∈ N , so that n≥m o  ( ) σ , κ, sλn ≥ 1 − ζ , n≥m o  ( ) σ , κ, sλn ≤      λ λ ≤ = 1, ζ,  σ , κ, ζ and thus lim  σ , κ, n n  n→∞  ( ) ( ) n≥m o s  s   limn→∞  ( ) σ , κ, sλn = 0 and limn→∞  ( ) σ , κ, sλn = 0. Therefore,        ( ) σ , κ, λ = 1,  ( ) σ , κ, λ = 0 and  ( ) σ , κ, λ = 0 and thus σ = κ. It is trivial that

4 Conclusion In this work, we have given new fixed point theorems in the setting of NSMS. We have extended Banach fixed point theorem in NSMS that can be the basis of some more new results in this new space. Our new results will surely be the foundation of some new notions for the researchers to explore.

References 1. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965) 2. Kaleva, O., Seikkala, S.: On fuzzy metric spaces. Fuzzy Sets Syst. 12, 225–229 (1984) 3. George, A., Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 64, 395–399 (1994) 4. Park, J.H.: Intuitionistic fuzzy metric spaces. Chaos Solitons Fract. 22, 1039–1046 (2004)

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5. Smarandache, F.: Neutrosophy: Neutrosophic Probability, Set and Logic, pp. 1–105. ProQuest Information and Learning, Ann Arbor, MI (1998) 6. Kirisci, M., Simsek, N.: Neutrosophic metric spaces. Math. Sci. 14(3), 1–11 (2020) 7. Molodtsov, D.: Soft set theory—first results. Comput. Math. Appl. 37(4–5), 19–31 (1999) 8. Karaaslan, F.: Neutrosophic soft sets with applications in decision making. Int. J. Inf. Sci. Intell. Syst. 4(2), 1–20 (2015) 9. Gupta, V., Gondhi, A.: Neutrosophic Soft Metric Space, Communicated 10. Simsek, N., Kirisci, M.: Fixed Point Theorems in Neutrosophic Metric Spaces. 10, 221–230 (2019) 11. Sowndrarajan, S., Jeyaraman, M., Smarandache, F.: Fixed point results for contraction theorems in neutrosophic metric spaces. Neutrosophic Sets Syst. 36(1), 1–12 (2020)

Existence of Best Proximity Points on (ψ, φ) Contractions in RMS S Arul Ravi

Abstract During the last few years, FPT was one of the most developing disciplines in analysis. The development of this idea has sped up research, resulting in a massive rise in publishing [1–8]. The traditional idea of MS has developed in several other areas by partially modifying the metric requirements in a vast area of works. Mathews [6, 7] introduced PMS, which is among these generalizations. We see RMS described by Branciari [12] in [1, 3, 9–12]. Branciari defined an RMS and also demonstrated a Banach contraction principle analogy. The nature of these concepts and FPT for several RMS contractions has been produced in [13–18]. Boyd and Wong [19] introduced CM called φ-contractions. In [20] a concept of weak φ-contractions was used and generalized by Alber and Guerre, A map T on a MS (M, d) is called WC if a map φ : [0, ∞) → [0, ∞) having φ(0) = 0 and φ(t) > 0 for every t > 0 s.t. d(T a, T b) ≤ d(a, b) − φ(d(a, b)) for all a, b ∈ M  . The above contractions were discussed by many [20–23]. This type of (ψ − φ) WCM has been an area of interest in [10, 20–23]. In the recent development in [23] we see that FPT was got by using (ψ − φ) WCM on CRMS. Here, we try to extend the result of Erhan. IM, and his companions [23] for the existence of BPP of (ψ − φ) contractions on RMS. Existence of BPP of a GC (ψ, φ)-contractive mappings on CRMS is discussed and an example is enumerated. Keywords RMS · BPP · p-property · Hausdroff

S. Arul Ravi (B) St Xavier’s College (Autonomous), Palayamkottai, Tamil Nadu 627002, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1440, https://doi.org/10.1007/978-981-19-9906-2_4

39

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S. Arul Ravi

1 Introduction and Preliminaries We wish to recall some definitions in order to discuss the result. Definition 1 [12] Let M be a set and d : M × M → [0, ∞) that satisfies the axioms for every a, b ∈ M and each is distinct a, b ∈ M varies from c and d. (i) d(a, b) = 0 iff a = b (ii) d(a, b) = d(b, a) (ii) d(a, b) ≤ d(a, c) + d(c, d) + d(d, b). Then d is called RM and (M, d) is called RMS. Definition 2 A sequence {an } is convergent to a limit a iff d(an , a) → 0 as n → ∞ (denoted by an → a). (ii) A sequence {an } is a C iff for every ε > 0 there is a PI N such that d(an , am ) < ε for all n, m > N . (iii) A RMS is complete if every CS in M converges in M. (i)

The modified notation of Samet and Lakzian [9] is vividly seen and let ψ be the set of CF. ψ : [0, ∞) → [0, ∞) where ψ(t) = 0 iff t = 0 NDF that is known as ADF [23]. Definition 3 [2] A0 = {a ∈ A : d(a, b) = d(A, B), for b ∈ B B0 = {b ∈ B : d(a, b) = d( A, B), for a ∈ A where d(A, B) = inf{d(a, b) : a ∈ A, b ∈ B}. Definition 4 [24] Let (C, D) be a pair of sets of MS (M, d) with C0 = 0. The pair (C, D) is said to have a pp if and only if for any c1 , c2 ∈ C0 and d1 , d2 ∈ D0 , d(c1 , d1 ) = d(C, D) = d(c2 , d2 ).

Existence of Best Proximity Points on (ψ, φ) Contractions in RMS

41

2 Known Results Branciari [12] has proved the following theorem as given below.

2.1 Theorem [12] Let (M, d) be a HS and CRMS and let T : M → M be a self-map that satisfies ψ(d(T a, T b)) ≤ ψ(d(a, b)) − φ(d(a, b)) for all a, b ∈ M where ψ, φ ∈  and ψ is considered as ND and continuous. Then T has a UFP.

3 Main Results 3.1 Theorem Let (M, d) be a HS and CRMS and let (A, B) be a pair of closed subsets of MS s.t. A0 is NE. T : A → B be a map that satisfies T (A0 ) ⊂ B0 . Suppose ψ(d(T a, T b)) ≤ ψ(M(a, b) − d(A, B)) − φ(M(a, b) − d(A, B))

(1)

for all a ∈ A, b ∈ B and ψ, φ ∈  where L > 0, and ψ is considered as ND and M(a, b) = max{d(a, b), d(a, T a), d(b, T b)} m(a, b) = min{d(a, T a), d(b, T b), d(a, T b), d(a, T a)} Then T has a BPP. Proof Choose a0 ∈ A. For T a0 ∈ T (A0 ) ⊂ B0 , there exists a1 ∈ A0 s.t. d(a1 , T a0 ) = d(A, B). Similarly, regarding the assumption, T a1 ∈ T (A0 ) ⊂ B0 , we determine a2 ∈ A0 such that d(a2 , T a1 ) = d(A, B). We get a sequence by recursion {an } in A0 satisfying d(an+1 , T an ) = d(A, B) for all n ∈ N Claim d(an , an+1 ) → 0 If a N = a N +1 , then a N is a BPP. By the p-property, we get d(an+1 , an+2 ) = d(T an , T an+1 )

(2)

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we suppose that an = an+1 for all n ∈ N . Since d(an+1 , T an ) = d( A, B), from (3), we have for all n ∈ N . ψ(d(an+1 , an+2 )) = ψ(d(T an , T an+1 )) ≤ ψ(M(an , an+1 ) − d(A, B)) − φ(M(an , an+1 ) − d( A, B)

(3)

where M(an , an+1 ) = max{d(an , an+1 ), d(an , T an ), d(an+1 , T an+1 )}. If M(an+1 , an+2 ) = d(an , an+1 ), then we get ψ(d(an+1 , an+2 )) ≤ ψ(d(an , an+1 ) − d(A, B)) − φ(d(an , an+1 ) − d( A, B)), where φ(d(an , an+1 )) = 0 and hence d(an , an+1 ) = 0, contradicts our assumption. Therefore d(an+1 , an+2 ) < d(an , an+1 ) for any n ∈ N and hence {d(an , an+1 )} is MDS of NNRN and there is s ≥ 0 such that limn→∞ d(an , an+1 ) = s. From (3), for any n ∈ N , we get ψ(d(an+1 , an+2 )) ≤ ψ(M(an , an+1 ) − d(A, B)) − φ(M(an , an+1 ) − d( A, B)) As n → ∞ in the above equation and using ψ and φ we get ψ(s) ≤ ψ(s) − φ(s) which implies φ(s) = 0. Hence lim d(an , an+1 ) = 0

n→∞

(4)

Next we show that {an } is a CS. Otherwise there is ε > 0, for which we can get two  sequences  of PI (m k ) and ε and d a , a (n k ) s.t. for allPI m k >n k >k, d am k , an k ≥   m k n k−1 < ε. Now ε ≤ d am k , an k ≤ d am k , an k−1 + d an k−1 , an k ,     ε ≤ d am k , an k < ε + d an k−1 , an k As k → ∞ in the above equation and using (5) we get   lim d am k , an k = ε

k→∞

(5)

        Again d am k , an k ≤ d am k , am k+1 + d am k+1 , an k+1 + d an k+1 , an k . As k → ∞ in the above equations and using (5) and (6) we get   lim d am k+1 , an k+1 = ε

k→∞

      Again d am k , an k ≤ d am k , an k+1 + d an k+1 , an k . Letting k → ∞ in the above equations and using (5) and (6) we get

(6)

Existence of Best Proximity Points on (ψ, φ) Contractions in RMS

43

  lim d am k , an k+1 = ε

(7)

  lim d an k , am k+1 = ε

(8)

k→∞

k→∞

For a = am k , b = an k we have       d am k , T am k − d(A, B) ≤ d am k , am k+1 + d am k+1 , T an k − d( A, B)   = d am k , am k+1     Similarly, d an k , T an k − d(A, B) = d am k , an k+1 and     d am k , T am k − d(A, B) = d an k , am k+1       From (1) we get ψ d am k+1 , an k+1 = ψ d T am k , T an k         ≤ ψ M am k+1 , an k+1 − d(A, B) − φ M am k+1 , an k+1 − d(A, B)   − Lm am k+1 , an k+1 − d(A, B)

(9)

         where M am k+1 , an k+1 = max d am k , am k+1 , d am k , T am k , d am k+1 , an k+1 ,          m am k+1 , an k+1 = min d am k , am k+1 , d am k , T am k , d am k+1 , an k+1 . Recalling (5), (6), (7) and (8) and let k → ∞ in the above equations and using ψ and φ, we get ψ(ε) ≤ ψ(ε) − φ(ε) + 0

(10)

This leads to φ(ε) = 0 and hence ε = 0. Contradicting our assumption. Hence {an } is a CS. Since {an } ⊂ A and A is a closed subsetf the CMS (M, d), there is a ∗ in A s.t. an → a ∗ . Taking a = an and b = a ∗ and since       d an , T a ∗ ≤ d an , a ∗ + d a ∗ , T an and       d a ∗ , T an ≤ d a ∗ , T a ∗ + d T a ∗ , T an We get ψ(d(an+1 , T a ∗ ) − d(A, B) ≤ ψd(T an , T a ∗ ))           ≤ ψ M an , a ∗ − d(A, B) − φ M an , a ∗ − d( A, B) + Lm an , a ∗ − d( A, B)

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As the limit n → ∞ in equations and using ψ and φ, we get             ψ d a ∗ , T a ∗ − d(A, B) ≤ ψ d a ∗ , T a ∗ − d( A, B) − φ d a ∗ , T a ∗ − d( A, B)   + Ld a ∗ , T a ∗ − d(A, B) This implies that d(a ∗ , T a ∗ ) = d( A, B). Hence a ∗ is a BPP of T . To prove uniqueness Let c and d be two BPP and assume that c = d, Taking a = c and b = d in (1) we get ψ(d(Tc , Td )) ≤ ψ(d(c, d) − d(A, B)) − φ(d(c, d) − d( A, B)) + Lm(c, d) − d(A, B) By ψ and φ there is a contradiction. Therefore c = d.

4 Corollaries Here we derive some corollaries using the main result.

4.1 Corollary Let (M, d) be a HS and CRMS and let (C, D) be a pair of closed subsets of MS s.t. C0 is nonempty. Let T : C → D be a map that satisfies T (C0 ) ⊂ D0 . Suppose ψ(d(T c, T d)) ≤ ψ(M(c, d) − d(C, D)) − φ(M(c, d) − d(C, D))

(11)

for all c ∈ C, d ∈ D where ψ, φ ∈  and ψ is taken as ND and M(c, d) = max{d(c, d), d(c, T c), d(d, T d)} M(c, d) = max{d(c, d), d(c, T c), d(d, T d)} Then T has a BPP. Proof Observe ψ(d(T c, T d)) ≤ ψ(M(c, d) − d(C, D)) − φ(M(c, d) − d(C, D))

Existence of Best Proximity Points on (ψ, φ) Contractions in RMS

45

≤ ψ(M(c, d) − d(C, D)) − φ(M(c, d) − d(C, D)) + Lm(c, d) − d(C, D) for some L > 0. Then by the Result 2.1, T has a BPP.

4.2 Corollary Let (M, d) be a HS and CRMS and let (C, D) be a pair of subsets of MS s.t. C0 is nonempty. Let T : C → D be a map satisfy T (C0 ) ⊂ D0 . Suppose d(T c, T d) ≤ k max{d(c, d), d(c, T c), d(d, T d)}

(12)

for all c ∈ C, d ∈ D for 0 ≤ k < 1. Then T has a BPP. Proof Let ψ(t) = t and φ(t) = (1 − k)t. By using the Result 2.2 T has a BPP.

4.3 Corollary Let (M, d) be a HS and CRMS and let (C, D) be a pair of subsets of MS s.t. C0 is nonempty. Let T : C → D be a map satisfy T (C0 ) ⊂ D0 . Suppose d(T c, T d) ≤ k[{d(c, d) + d(c, T c) + d(d, T d)} − d(C, D)] + L min[d(c, d) + d(c, T c) + d(d, T d)] for all c ∈ C, d ∈ D and 0 ≤ k < Then T has a BPP.

1 3

(13)

and L > 0.

Proof Clearly, k[{d(c, d) + d(c, T c) + d(d, T d)} − d(C, D)] + L min{d(c, T c), d(d, T d), d(c, T d), d(d, T c)} ≤ 3k max{d(c, d), d(c, T c), d(d, T d)} + L min{d(c, T c), d(d, T d), d(d, T c)}. Taking ψ(t) = t and φ(t) = (1 − 3k)t. By the Result 2.1, T has a BPP.

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4.4 Corollary Let (M, d) be a HS and CRMS and let (C, D) be a pair of subsets of MS s.t. C0 is nonempty. Let T : C → D be a map satisfy T (C0 ) ⊂ D0 . Suppose d(T c, T d) ≤ ψ(M(c, d) − d(C, D)) − φ(M(c, d) − d(C, D)) + Lm(c, d) − d(C, D) for all c ∈ C, d ∈ D where M(c, d) = max{d(c, d), d(c, T c), d(d, T d)} m(c, d) = min{d(c, d), d(c, T c), d(d, T d)} Then T has a BPP. Proof Let ψ(t) = t. By the Result 2.1, T has a BPP.

5 Example Here we enumerate an example to justify the main result.

5.1 Example   Let A ∪ B = M, where A = 41 , 15 , 16 , 17 , B = {1, 2}. Define the GMS M as follows:     1 1 1 1 =d , = 0.05 d , 4 5 6 7     1 1 1 1 d , =d , = 0.03 4 7 5 6     1 1 1 1 d , =d , = 0.08 4 6 5 7     1 1 1 1 d , =d , =0 5 5 6 6 and d(a, b) = |a − b| − d( A, B) if a, b ∈ B (or) a ∈ A, b ∈ B (or) a ∈ B, b ∈ A.

(14)

Existence of Best Proximity Points on (ψ, φ) Contractions in RMS

47

Obviously d does not hold T on A. Indeed 

1 1 0.08 = d , 4 6





1 1 ≥d , 4 5





1 1 +d , 5 6

 = 0.08

RMS holds, and d is a RM. Let T : A → B be defined as ⎧1 ⎨ 7 , if a ∈ [1,  1 2] 1 T a = 6 , if a ∈ 4 , 15 , 16 ⎩1 , if a = 17 5 Taking (t) = t and φ(t) = 7t . T Satisfies the Result 2.1 and has a BPP d(a, b) = d(A, B). Acknowledgements I would to acknowledge and thank the MMCITRE2022 Conference for having given me space and chance to quench my intellectual thirst and to bring out my research aptitude.

References 1. Abedelijawad, T., Karapinar, E., Tas, K.: Existence and uniqueness of a common fixed point on partial metric spaces. Appl. Math. Lett. 24, 1900–1904 (2011) 2. Arul Ravi, S.: Best proximity point theorem for (ψ, φ)-contractions. Int. J. Math. Arch. 11(9), 1–4 (2020) 3. Aydi, H., Karapinar, E., Shatanawi, W.: Coupled fixed point results for (ψ, φ) weakly contractive condition in ordered partial metric spaces. Comput. Math. Appl. 62, 4449–4460 (2011) 4. Radenovic, S., Rakocavic, V., Rasapour, S.: Common fixed points for (g, f ) type maps in cone metric spaces. Appl. Math. Comput. 218, 480–491 (2011) 5. Borinda, V.: A common fixed point theorem for compatible quasi contractive self-mappings in metric spaces. Appl. Math. Comput. 213, 348–354 (2009) 6. Cric, I., Abbas, M., Saadati, R., Hussain, N.: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput. 217, 5784–5789 (2011) 7. Mathews, S.G.: Partial Metric Topology, Research Report 212. Department of Computer Science, University of Warwick (1992) 8. Mathews, S.G.: Partial metric topology. In: Proceedings of 8th Summer Conference on General Topology and Applications. Ann. N. Y. Acad. Sci. 728, 183–197 (1994) 9. Karapinar, E., Erhan, I.M.: Fixed point theorems for operation on partial metric spaces. Appl. Math. Lett. 24, 1894–1899 (2011) 10. Karapinar, E.: Generalization Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. 2011, 4 (2011). https://doi.org/10.1186/1687-1812-2011-4 11. Karapinar, E., Erhan, I.M., Yildiz Ulus, A.: A fixed point theorem for cyclic maps on partial metric spaces. Appl. Math. Inf. Sci. 6, 239–244 (2012) 12. Branciari, A.: A fixed point theorem of Banach Caccippoli type on a class of generalized metric spaces. Publ. Math. (Debr.) 57, 31–37 (2000) 13. Das, P.: A fixed point theorem on a class of generalized metric spaces. Korean J. Math. Sci. 9, 29–33 (2002) 14. Das, P.: A fixed point theorem in a generalized metric spaces. Soochow J. Math. 33, 33–39 (2007)

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´ c’s quasi contraction mapping in a generalized 15. Das, P., Lahiri, B.K.: Fixed point of Ljubomir Ciri´ metric spaces. Publ. Math. [Debr.] 61, 589–594 (2002) 16. Das, P., Lahiri, B.K.: Fixed point of contractive mappings in generalized metric spaces. Math. Slov. 59, 499–504 (2009) 17. Azam, A., Arshad, M.: Kannan fixed point theorems on generalized metric spaces. J. Nonlinear Sci. Appl. 1, 45–48 (2008) 18. Azam, A., Arshad, M., Beg, I.: Banach contraction principle on cone rectangular metric spaces. Appl. Anal. Discrete Math. 3, 236–241 (2009) 19. Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969) 20. Alber, Y.I., Guerre, D.S.: Principles of weakly contractive maps in Hilbert spaces. In: Gohberg, I., Lyubich, Y. (eds.) New Results in Operator Theory and Its Applications. Operator Theory, Advances and Applications, vol. 98, pp. 7–22. Brikhouser, Basel (1997) 21. Karapinar, E.: Fixed point theory for cyclic weak φ-contractive. Appl. Math. Lett. 24, 822–825 (2011) 22. Rhodes, B.E.: Some theorems on weakly contractive maps. Nonlinear Anal. Theory Methods Appl. 47, 2683–2693 (2001). In: Proceedings of the Third World Congress of Nonlinear Analysis, Part 4, Catania (2000) 23. Karapinar, E., Sadarangani, K.: Fixed point theory for cyclic (ψ, φ)-contractions. Fixed Point Theory Appl. 2011, 69 (2011). https://doi.org/10.1186/1687-1812-2011-69 24. Sankar Raj, V.: A best proximity point theorem for weakly contractive non-self mappings. Nonlinear Anal. 74, 4804–4808 (2011)

Approximation of Signals by El1 El1 Product Summability Means of Fourier–Laguerre Expansion Smita Sonker and Neeraj Devi

Abstract A lot of work has been done in the field of finding approximation of signals of Fourier–Laguerre series. Many researchers studied approximation of signals of Fourier–Laguerre series using Ces`aro means, harmonic means, Euler summability means, (C, 2)(E, q) means. Also some researchers worked on approximation of signals of Fourier–Laguerre series using product summability, but upto now no work is done on approximation of signals of Fourier–Laguerre series using double summability. In present problem, we will prove a theorem on approximation of signals of Fourier Laguerre series by El1 El1 product Summability. Keywords Approximation of signals · El1 summability · El1 El1 product summability · Fourier–Laguerre series

1 Introduction Approximation of signals of its Fourier–Laguerre series using different summability techniques is well discussed by many researchers such as Gupta [1], Khatri and Mishra [3], Kubiak et al. [5], Mittal and Singh [6], Sahani et al. [8], Sonker [12], Singh and Saini [10], Krasniqi [4], Singroura [11], and Zygmund [14] studied approximation of signals by using Ces`aro summability, Harmonic-Euler summability, pointwise summability, Matrix-Euler operators, uniform approximation by Ces`aro summability and Ces`aro–Euler summability respectively. Here, we will discuss approximation of signals of Fourier–Laguerre series using Double Euler summability. ∞ cl . If Let the sequence {sl } be the lth partial sum of the series l=0 l   1  l sl−h = s, l→∞ 2l h h=0

lim El1 = lim

l→∞

S. Sonker (B) · N. Devi National Institute of Technology Kurukshetra, Kurukshetra, Haryana 136119, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1440, https://doi.org/10.1007/978-981-19-9906-2_5

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then {sl } converges to a definite value ‘s’ by El1 means (by Hardy [2]), and we write it as, sl → s(El1 ). The product summability (E, 1)(E, 1) defines El1 El1 summability. Thus the El1 El1 means is given by l   h   1  l 1  h sv . 2l h=0 h 2h v=0 v

tlE E =

A signal k(x) ∈ L(0, ∞) is expanded by Fourier–Laguerre method as k(x) ≡

∞ 

al L l η (x),

(1)

l=0

where al =

1 (η + 1)

∞ l+η l

e−t t η k(t)L l η (t)dt,

(2)

0

and L l η (x) denotes the lth Laguerre polynomial of order η ≥ −1, defined by generating function ∞  −xw L l η (x)wl = (1 − w)−η−1 e( 1−w ) , (3) l=0

and the integral (2) exists. Also, ψ(t) =

1 e−t t η [k(t) − k(0)]. (η + 1)

(4)

Szegö’s [13] provided us Theorem (1) by using much lighter condition than the Gupta [1] who approximated the signals by Ces`aro means.

2 Known Results Theorem 1 If z F(z) = 0

  1+ p |k(t)| 1 , −1 < p < ∞, z → 0, dt = O log t z

Approximation of Signals by El1 El1 Product …

∞

e−t/2 t (3η − 3h − 1)/3|k(t)|dt < ∞,

51

(5)

1

then σlh (0) = O(log l)1+ p where h > η + 1/2, η > −1, with σlh (0) be lth Ces`aro mean. Singh [9] approximated the signals by harmonic means with lighter condition than Theorem (1) given as follows: Theorem 2 For η ∈ [− 5/6, − 1/2], zl (0) − k(0) = O(log l)1+ p

(6)

given with δ

|ψ(t)| dt = O{log(1/z)}1+ p , −1 < p < ∞, t → 0, δ > 0 t η+1

z

l δ

et/2 t −(2η+3)/4 |ψ(t)|dt = O{l −(2η+1)/4 (log l)1+ p }, ∞

et/2 t −1/3 |ψ(t)|dt = O{(log l)1+ p }, l → ∞.

(7)

l

Also, Nigam and Sharma [7] approximated the signals using Euler means which is completely dissimilar to (C, h) and harmonic means. They used a weaker condition than (7) and the range of η increased to (− 1, − 1/2) that is of more use for further applications. They gave us the following result: Theorem 3 If El1 =

l   1  l sh → s, l → ∞, 2l h=0 h

then the approximation of signals of Fourier–Laguerre expansion (1) by El1 means at point x=0 is El1 (0) − k(0) = O(ζ (l)) given with

z (z) = 0

|ψ(t)|dt = O(z η+1 ζ (1/z)), z → 0,

52

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et/2 t −(2η+3/4) |ψ(t)|dt = O(l −(2η+1)/4 ζ (l)),

δ

∞

et/2 t −1/3 |ψ(t)|dt = O(ζ (l)), l → ∞,

l

where δ is a fixed positive constant, η ∈ (− 1, − 1/2) and ζ (z) is a monotonically increasing positive signal of z such that ζ (z) → ∞ as l → ∞. Therefore, product summability is more powerful than the individual methods. To reach this goal, we need an additional result as follows:

3 Lemma Let η is arbitrary real number, a and b are fixed positive constant, and let l → ∞. Then η (8) L l (x) = O(l η ) if 0 < x < a/l, and

η

L l = O(x −(2η+1)/4 l (2η−1)/4 ) if a/l < x < b.

(9)

4 Main Result Theorem 4 The approximation of signals of Fourier–Laguerre expansion (1) by [(E, 1)(E, 1)]l means at point x = 0 is [(E, 1)(E, 1)]l (0) − k(0) = O(ζ (l)) given with

z (z) =

|ψ(t)|dt = O(z η+1 ζ (1/z)), z → 0,

(10)

(11)

0

l δ

et/2 t −(2η+3)/4 |ψ(t)|dt = O(l (−2η+1)/4 ζ (l)),

(12)

Approximation of Signals by El1 El1 Product …

and

∞

53

et/2 t −1/3 |ψ(t)|dt = O(ζ (l)), l → ∞,

(13)

l

where ζ (z) is a monotonically increasing positive signal of z such that ζ (l) → ∞ as l → ∞. Proof Based on the equality   l +η , η

η

L l (0) = we obtain sl (0) =

l 

η

ah L h (0)

h=0

=

=

1 (η + 1) 1 (η + 1)

∞ 0 ∞

e−t t η k(t)

l 

η

L h (t)dt

h=0 (η+1)

e−t t η k(t)L l

(t)dt.

0

Now, ∞ l   h   1 1  l 1  h (η+1) e−t t η k(t)L l (t)dt. [(E, 1)(E, 1)]l (0) = l 2 h=0 h 2h v=0 v (η + 1) 0

Using (4), we have l   h   ∞ 1  l 1  h ψ(t)L (η+1) (t)dt [(E, 1)(E, 1)]l (0) − k(0) = l v 2 h=0 h 2h v=0 v 0 ⎛ ⎞ 1/l δ  l ∞ l   ⎜ ⎟ 1  l =⎝ + + + ⎠ l 2 h=0 h 0

1/l

δ

l

h   1  h ψ(t)L (η+1) · h (t)dt v 2 v=0 v

= i1 + i2 + i3 + i4 .

(14)

54

S. Sonker and N. Devi

Using orthogonal property (11) and condition (8) of lemma, we get 1/l l   h   1  l 1  h η+1 O(v ) |ψ(t)|dt i1 = l 2 h=0 h 2h v=0 v =

1 2l

l   h=0



=O  =O

1 2l 1 2l



h  

0



  l 1 h O l η+1 O h h 2 v=0 v  l   h    l 1  h ζ (l) h 2h v=0 v h=0  l    l ζ (l) h h=0



1

l



ζ (l) η+1

= O(ζ (l))

(15)

since l    l h=0

h

= 2l .

Further using the orthogonal property (12) and condition (9) of lemma, we get l   h     δ (2η+1) 1  l 1  h O h 4 |ψ(t)|t −(2η+3)/4 dt. i2 = l 2 h=0 h 2h v=0 v

(16)

1/l

Now, h    h v=0

v

h

(2η+1) 4

=

⎧ ⎨[h/2]  ⎩

=h

v=0

(2η+1) 4

+

v=[h/2]+1 [h/2] 

 v=0

≤h h    h v=0

v

(2η+1) 4

(2η+1) 4

=h

(2η+1) 4

h v



l    h v=0

h

⎫ h ⎬   (2η+1)  h h 4 ⎭ v

2h +

v 



 (2η+5) h h 4 [h/2]



 (2η+5) h h 4 [h/2]

+ +

 (2η+5) h h 4 , [h/2]

(17)

Approximation of Signals by El1 El1 Product …

55

using h    h

2 = h

v

v=0

after some calculations, we get  h/2

h [h/2]

 ≤ 2l .

Using these results in Eq. (17), we get h    h v=0

v

h

(2η+1) 4

≤h

(2η+1) 4

2h + 2(2h )h

(2η+1) 4

 (2η+1)  = O h 4 2h .

(18)

Using result (18) in Eq. (16), we get i2 =

δ l    (2η+1)/4  1  l O h t −(2η+3)/4 |ψ(t)|dt, 2l h=0 h 1/l

again using the same result, we get   i 2 = O l (2η+1)/4

δ

t −(2η+3)/4 |ψ(t)|dt,

1/l

integrating by parts, we get ⎡  ⎢  δ i 2 = O l (2η+1)/4 ⎣ t −(2η+3)/4 (t) 1/l +

δ

⎤ (2η + 3) −(2η+7)/4 ⎥ t (t)dt ⎦ 4

1/l

#

  −(2η+3)/4   t O(t η+1 ζ (1/t)) = O l (2η+1)/4 δ + 1/l

t

−(2η+7)/4

O(t

η+1

$ ζ (1/t))dt

56

S. Sonker and N. Devi

⎧ ⎪ ⎨ δ

⎫⎤ ⎪ ⎬  (2η+1)/4 δ ⎥ (2η+1)/4 ⎢ (2η−3)/4 = O(l ) ⎣O l ζ (1/t) 1/l + O y ζ (1/t)dt ⎦ ⎪ ⎪ ⎩ ⎭ 1/l ⎧ ⎫⎤ ⎡ ⎪ ⎪ δ ⎨ ⎬  −(2η+1)/4  ⎥ (2η+1)/4 ⎢ (2η−3)/4 = O(l ) ⎣ O(1) + O l ζ (l) + ζ (l)O t dt ⎦ ⎪ ⎪ ⎩ ⎭ 1/l ⎧ ⎫ ⎪ δ ⎪ ⎬  ⎨ = O(ζ (l)) + O l (2η+1)/4 ζ (l) t (2η−3)/4 dt ⎪ ⎪ ⎩ ⎭ ⎡

1/l

  = O(ζ (l)) + O l (2η+1)/4 ζ (l)



t (2η−3)/4+1 (2η − 3)/4 + 1  (2η+1)/4   −(2η+1)/4  = O(ζ (l)) + O l ζ (l) l = O(ζ (l)) + O(ζ (l))

δ 1/l

= O(ζ (l)).

(19)

Now, we consider l   h   l 1  h 1  l i3 = l et/2 t −(2η+3)/4 |ψ(t)| 2 h=0 h 2h v=0 v δ

−t/2 (2η+3)/4 η+1 ·e t |L l |dt

l l   h   1  l 1  h (2η+1)/4 O(l ) et/2 t −(2η+3)/4 |ψ(t)|dt = l 2 h=0 h 2h v=0 v δ

l   1  l O(h (2η+1)/4 )O{l −(2η+1)/4 ζ (l)} (using (18)) = l 2 h=0 h

= O(l (2η+1)/4 )O{l −(2η+1)/4 ζ (l)} (again using (18)) = O(ζ (l)) i.e. i 3 = O(ζ (l)).

(20)

Approximation of Signals by El1 El1 Product …

57

Finally, l   h   ∞ 1  h 1  l et/2 t −(3η+5)/6 |ψ(t)| i4 = l 2 h=0 h 2h v=0 v l

η+1 · e−t/2 t (3η+5)/6 |L l |dt ⎫ ⎧ ∞ t/2 −1/3 l  ⎨ h     l h e t |ψ(t)| ⎬ 1 1 (η+1)/2 O(v dt ) = l ⎭ 2 h=0 h ⎩ 2h v=0 v t(η + 1)/2

=

1 2l

l   h=0



l O h

# 

1 2h





l

(2h ) {h (η+1)/2 l −(η+1)/2 }O{ζ (l)}

$

= O(1)O(ζ (l)) = O(ζ (l)) i.e. i 4 = O(ζ (l)).

(21)

Combining (14), (15), (19), (20) and (21), we get [(E, 1)(E, 1)]l (0) − k(0) = O(ζ (l)). Hence, the proof is complete.

5 Conclusion Here, we come to know that the Fourier–Laguerre series can be approximated by a simple signal k(x) by using Euler double product summability. The product summability is more powerful than the individuals as the series which can not be approximated using simple summability can be approximated using product summability.

References 1. Gupta, D.P.: Degree of approximation by Cesaro means of Fourier–Laguerre expansions. Acta Sci. Math. 32(3–4), 255–259 (1971) 2. Hardy, G.H.: Divergent Series. Oxford University Press, New York (1959) 3. Khatri, K., Mishra, V.N.: Approximation of functions belonging to L[0, ∞] by product summability means of its Fourier–Laguerre series. Cogent Math. 3(1), e-1250854 (2016) 4. Krasniqi, X.Z.: On the degree of approximation of a function by (C, 1)(E, q) means of its Fourier–Laguerre series. Int. J. Anal. Appl. 1(1), 33–39 (2013)

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5. Kubiak, M., Lenski, W., Szal, B.: Pointwise summability of Fourier–Laguerre series of integrable functions (2019). arXiv preprint arXiv:1904.08642 6. Mittal, M.L., Singh, M.V.: Error estimation of functions by Fourier–Laguerre polynomials using matrix-Euler operators. Int. J. Anal. 1–4 (2015) 7. Nigam, H.K., Sharma, A.: A study on degree of approximation by (E, 1) summability means of the Fourier–Laguerre expansion. Int. J. Math. Math. Sci. (1–2) (2010) 8. Sahani, S.K., Mishra, V.N., Pahari, N.P.: On the degree of approximation of a function by Nörlund means of its Fourier Laguerre series. Nepal J. Math. Sci. 1, 65–70 9. Singh, T.: Degree of approximation by harmonic means of Fourier–Laguerre expansions. Publ. Math. Debr. 24(1–2), 53–57 (1977) 10. Singh, U., Saini, S.: Uniform approximation in L[0, ∞)-space by Cesàro means of Fourier– Laguerre series. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 1–7 (2021) 11. Singroura, A.N.S.: On Ces`aro summability of Fourier–Laguerre series. Proc. Jpn. Acad. 39(4), 208–210 (1963) 12. Sonker, S.: Approximation of functions by means of its Fourier–Laguerre series. In: Proceeding of ICMS-2014, pp. 125–128 (2014) 13. Szegö, G.: Orthogonal Polynomials. Colloquium Publications American Mathematical Society, New York, NY (1959) 14. Zygmund, A.: Trigonometric Series, vol. I. Cambridge University Press, Cambridge (1959)

Approximation of Signal Belongs to Generalized W  (Lr , ξ (t)) Class by (C, α, η) A-Matrix Summability of Fourier Series Smita Sonker and Paramjeet Sangwan

Abstract Analysis of periodic signals is crucial because it conveys characteristics of diverse phenomena and has numerous applications in engineering fields like signal analysis, mechanical engineering, etc. Scientists and engineers use the features of Fourier approximation to construct digital filters. The present article determines a new theorem which belongs to W  (Lr , ξ (t)), (r ≥ 1), (t > 0) class by (C, α, η) Amatrix summability of Fourier series. Keywords Signal approximation · Generalized weighted Lipschitz class · A-matrix summability · Hölder’s inequality, Fourier series

1 Introduction Let



um be a given infinite series and {sm } be the sequence of its mth partial sum.

(I) If A ≡ (aim ) is a lower triangular matrix, its sequence-to-sequence transformation will be m  aij sj tm = j=0

gives the sequence {tm } of matrix A-mean of {sm } generated by (aim ). If tm =

m 

aij sj → s as m → ∞

j=0

 then um is A-summable to the sum s [1]. We know that matrix summability is regular, provided

S. Sonker (B) · P. Sangwan National Institute of Technology Kurukshetra, Kurukshetra 136119, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Sahni et al. (eds.), Mathematical Modeling, Computational Intelligence Techniques and Renewable Energy, Advances in Intelligent Systems and Computing 1440, https://doi.org/10.1007/978-981-19-9906-2_6

59

60

S. Sonker and P. Sangwan

(i) limi→∞ a im = 0 ∞ (ii) limi→∞ ∞ m=0 aim = 1 (iii) supi m=0 |aim | < K, where K is constant. (α,η) α−1 η 1 m (II) If Cm(α,η) = Bα+η denotes mth h=0 Bm−h Bh sh → s as m → ∞, where Cm m   α+η α+η α+η , α + η > −1 and B0 = Cesàro  means of order (α, η) with Bm = O m 1, then um is (C, α, η)-summable to ‘s’.   α−1 η α−1 η h 1 m 1 m (III) If C (α,η) .A m = Bα+η Bm−h Bh th = Bα+η Bm−h Bh j=0 ahj sj → s as h=0 h=0 m  m m → ∞ then um is (C, α, η) A-product means to ‘s’. Let a signal denoted by g having 2π as periodic time and is integrable in the same way as of Lebesgue for the limit (− π, π ). Let sm (g; x) ∼

 a0  + (ah cosh x) + (bh sinh x) 2 m

m

h=1

h=1

(1)

is (1 + m)th partial sum and is known as trigonometric polynomial of degree m of (1). Definition 1 The signal g, in L∞ -norm, is represented by g∞ and given by g∞ = sup {|g (x) | : x ∈ R}

(2)

whereas in Lr -norm by gr and is

gr =

⎧ 2π ⎨ ⎩

|g (x) |r dx

⎫1/r ⎬ ⎭

, r≥1

(3)

0

Definition 2 The degree of approximation of g by tm (x) under .∞ is presented by Zygmund [2] with tm − g∞ = sup {|tm (x) − g (x) | : x ∈ R}

(4)

and Em (g) of g ∈ Lr is defined as Em (g) = min tm (g; x) − g (x) r tm

(5)

Definition 3 The signal g of Lipschitz class generally represented as g ∈ Lip β if   |g (x + t) − g (x) | = O |t|β , 0 < β ≤ 1, t > 0

Approximation of Signal Belongs to Generalized …

61

whereas, g ∈ Lip (β, r) if ⎛ ⎝

2π

⎞ 1r

  |g (x + t) − g (x) |r dx⎠ = O |t|β for 0 < β ≤ 1, r ≥ 1, t > 0.

(6)

0

For ξ (t) (positive increasing function) and r ≥ 1, t > 0, g ∈ Lip (ξ(t), r), if ⎛ 2π ⎞ 1r ⎝ |g (x + t) − g (x) |r dx⎠ = O (ξ (t)) .

(7)

0

For γ ≥ 0, r ≥ 1, t > 0, a real-valued signal g ∈ W (Lr , ξ (t)), provided ⎛ ⎝

2π

⎞ 1r |g (x + t) − g (x) |r sinγ r (x) dx⎠ = O (ξ (t))

(8)

0

and g ∈ W  (Lr , ξ (t)) if sinγ r (x) dx in Eq. (8) is replaced by sinγ r

x 2

dx.

Notations  (α,η)  C .A m (t) =

1

m 

α+η

Bm

h=0

η

α−1 Bm−h Bh

⎧ h ⎨ ⎩

φx (t) = g (x + t) − g (x − t)

j=0

 ⎫  sin j + 21 t ⎬ ahj . 2 sin 2t ⎭

(9) (10)

Summability theory is used for study of functional analysis and having various applications. A theorem of Weierstrass is used to originate the approximation theory. Thereafter this study was further carried out using trigonometric polynomials. The nth partial sum is a better estimate for approximating the signal, and if the nth trigonometric polynomial’s coefficients are same as Fourier coefficients, then the error is minimum. Many researchers such as Hardy [3], Das and Dutta [4], Lal and Kushwaha [5], Krasniqi [6] carried out their research work by different kinds of Lipschitz classes. Chandra [7] did his research work on Lp -norms. Some researchers focused on weighted Lipschitz classes. Qureshi [8], Padhy et al. [9] worked out on the approximation by (E, q) A product means. Rhoades [10] worked on Hausdorff means and completed his study on functions by using Lipschitz class. Mittal et al. [11] used linear operator and worked on the approximation of signals which belongs to Lipschitz (α, p) class. Mishra [12] used (E, r)(N , p, q) means for their study. Various researches focused on Lip classes like Lal and Nigam [13] on Lip (ξ(t), p), Mishra et al. [14] on Lip (ξ(t), r), Lal and Mishra [15] on Lip (α, r). Recently, Krasniqi and

62

S. Sonker and P. Sangwan

Deepmala [16] carried out their research for conjugate series using some classes by (N , pn , qn )(E, α) means. Our theorem generalizes several known results of Krasniqi [6], Tiwari and Upadhyay [17] for approximating the signal.

2 Main Theorem Theorem 1 Let A ≡ (aim ) be an infinite matrix. If g denotes a signal with period 2π which can be integrated same as Lebesgue for (−π, π ) and of class W  (Lr , ξ (t)), (r ≥ 1), by using (C, α, η) A-product means of (1), the degree of approximation is tmC

(α,η)

.A

   1 − gr = O (1 + m)γ + r ξ (1 + m)−1

(11)

for 

 ξ (t) . t −1 is a non-increasing sequence, ⎛ π ⎞ 1r r (1+m)  |φx (t) | ⎜ ⎟ sinγ r (t/2) dt ⎠ = O (1) ⎝ ξ (t)

(12)

(13)

0

⎛ ⎜ ⎝

π 

π (1+m)

|φx (t) | ξ (t) . t δ

r

⎞ 1r ⎟ dt ⎠ = O



1 (1 + m)−δ

 (14)

where, (i) δ = arbitrary number with (1 − δ) s − 1 > 0, (ii) 1r + 1s = 1, 1 ≤ r ≤ ∞, (α,η) (iii) tmC .A = (C, α, η) A-summable of the series (1), (iv) Equations (13) and (14) behave uniformly in x.

3 Lemmas   Lemma 1 | C (α,η) .A m (t) | = O (1 + m), for 0 < t ≤

π ; (1+m)

sin mt ≤ m sin mt.

Approximation of Signal Belongs to Generalized …

63

Proof  ⎧  ⎫   m h  α−1 η ⎨ sin j + 21 t ⎬   Bm−h Bh ahj t α+η  ⎩ ⎭ 2 sin π.Bm  h=0  2 j=0  ⎧   ⎫  m h  α−1 η ⎨ (2j + 1) sin 2t ⎬ 1   ≤ Bm−h Bh ahj t α+η  ⎩ ⎭ sin 2π.Bm  h=0  2 j=0 ⎧  ⎫  h ⎨ ⎬  m α−1 η 1  ≤ Bm−h Bh (2h + 1) ahj  α+η ⎩ ⎭ 2π.Bm  h=0 j=0

 (α,η)    C .A m (t) ≤

1

K. (2m + 1)  m

≤

α+η

2π.Bm

η

α−1 Bm−h Bh

h=0 m 



= O (1 + m) ·

 α−1 η Bm−h Bh

= Bmα+η

h=0

      π Lemma 2 | C (α,η) .A m (t) | = O 1t , for (1+m) < t ≤ π ; t ≤ π sin 2t and sin mt ≤ 1. Proof  ⎧  ⎫   h  m α−1 η ⎨ sin j + 21 t ⎬   Bm−h Bh ahj t α+η  ⎩ ⎭ 2 sin π.Bm  h=0  2 j=0  ⎧ ⎫   h  m α−1 η ⎨ 1 1 ⎬  ≤ Bm−h Bh ahj t  α+η ⎩ ⎭ 2π.Bm  h=0 π j=0     h  m α−1 η   1  ≤ Bm−h Bh ahj  α+η  2t.Bm  h=0  j=0

 (α,η)    C .A m (t) ≤

1

K

m 

η

α−1 Bm−h Bh α+η 2t.Bm h=0     m 1 η α−1 · =O Bm−h Bh = Bmα+η t

=

h=0

64

S. Sonker and P. Sangwan

4 Proof of Theorem If partial sum of (1) is denoted by sm (g; x), then 1 sm (g; x) − g (x) = 2π

π 0

  sin m + 21 t φx (t) dt. sin 2t

A-transform of sm (g; x) is π

1 tm − g (x) = 2π

0

  sin h + 21 t φx (t) amh dt. sin 2t h=0 m 

(C, α, η) A-transform of sm (g; x) is  π ⎧    ⎫   m h 1  ⎨ ⎬  (α,η)     sin j + t 1   C .A α−1 η 2  φx (t)  − g = B B a tm hj m−h h ⎩ t α+η  ⎭ sin 2 2π.Bm  j=0 h=0 0

π =

   |φx (t)| .  C (α,η) .A m (t) dt.

(15)

0

By using the assumptions of theorem and on taking φx (t) as φx , it is to be shown that π

     1 |φx | | C (α,η) .A m (t) | dt = O (1 + m)γ + r ξ (1 + m)−1

0

Now, |tmC

(α,η)

.A

π

  |φx || C (α,η) .A m (t) |dt

− g| = 0

⎡ ⎢ =⎣

π

(1+m)

π |φx | +

⎤  ⎥  |φx |⎦ | C (α,η) .A m (t) |dt

π (1+m)

0

= |J1.1 | + |J1.2 | (say) Let

π

(1+m) |J1.1 | ≤ 0

  |φx |.| C (α,η) .A m (t) |dt

(16)

Approximation of Signal Belongs to Generalized …

65

Using Lemma 1, condition (13), (sin t/2)−1 ≤ ⎛

π

(1+m) 

⎜ |J1.1 | ≤ ⎝

0

|φx | ξ (t)

⎞ 1r ⎡

r

⎟ ⎢ sinγ r (t/2) dt ⎠ ⎣

π , t

and Hölder’s inequality

(1+m) " π

#s   ξ (t) | C (α,η) .A m (t) | sinγ (t/2)

0

⎡ $ %1 ⎢ = O (1) ess sup ξ (t)s s ⎣ π 0