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Accepted Manuscript1.0 Nonlinear World – Journal of Interdisciplinary Nature Published by GVP – Prof. V. Lakshmikantham Institute for Advanced Studies and GVP College of Engineering (A) About Journal: Nonlinear World is published in association with International Federation of Nonlinear Analysts (IFNA), which promotes collaboration among various disciplines in the world community of nonlinear analysts. The journal welcomes all experimental, computational and/or theoretical advances in nonlinear phenomena, in any discipline – especially those that further our ability to analyse and solve the nonlinear problems that confront our complex world. Nonlinear World will feature papers which demonstrate multidisciplinary nature, preferably those presented in such a way that other nonlinear analysts can at least grasp the main results, techniques, and their potential applications. In addition to survey papers of an expository nature, the contributions will be original papers demonstrating the relevance of nonlinear techniques. Manuscripts should be submitted to: Dr. J Vasundhara Devi, Associate Director, GVP-LIAS, GVP College of Engineering (A), Madhurawada, Visakhapatnam – 530048 Email: [email protected] Subscription Information 2017: Volume 1 (1 Issue) USA, India Online Registration at www.nonlinearworld.com will be made available soon. c Copyright 2017 GVP-Prof. V. Lakshmikantham Insitute of Advanced Studies ISSN 0942-5608 Printed in India by GVP – Prof. V. Lakshmikantham Institute for Advanced Studies, India Contributions should be prepared in accordance with the ‘‘ Instructions for Authors} 1 Nonlinear World Honorary Editors Dr. Chris Tsokos President of IFNA, Executive Director of USOP, Editor in Chief, GJMS, IJMSM, IJES, IJMSBF Dr. S K Sen Director, GVP-LIAS, India Editor in Chief Dr. J Vasundhara Devi Dept. of Mathematics, GVPCE(A) and Associate Director, GVP-LIAS, India Editorial Board Dr. R. P. Agarwal Department of Mathematics,Texas A& M University,USA Dr. V. Berinde Department of Mathematics & Comp Sc,North University Center At Baia Maretechnical University of Cluj-Napoca, Romania Dr. Eyup Cetin School of Transportation & Logistics,Istanbul University, Turkey Dr. Siva Chittajallu R& D Manager, Math & Statistics Department, Roche Diabetes Care, Indianapolis, USA Dr. J. K. Kim Department of Mathematics Education, Kyungnam University, South Korea Dr. Irena Lasiecka Department of Mathematical Sciences University Of Memphis, USA Dr. Xinzhi Liu Dept. Of Applied Mathematics, University of Waterloo, Canada Dr. A.A. Martynyuk Ukraine Dr. S. Raha Indian Institute of Science, Bangalore, India Dr. A. B. Koteswara Rao Department of Mechanical Engineering, GVPCE(A), India Dr. M. Sambandham Professor,Morehouse College, USA Dr. Pankaj Srivastava Department of Mathematics, MNNIT, Allahabad, India Dr. S. Sundar Department of Mathematics, IIT Madras, India Dr. A.S. Vatsala Department of Mathematics,University of Louisiana at Lafayette, USA Associate Editors Dr. PVS Anand Department of Mathematics, Rayalaseema University, India Dr. Z. Drici Department of Mathematics, Illinois Wesleyan University, USA Dr. F. A. Mcrae Department of Mathematics, Catholic University of America, USA Dr. B. Srinivas Department of Chemical Engineering, GVPCE(A), India Dr. G. Sudheer Department of Mathematics, GVPCEW, India Dr. Vishwesh Vyavahare Department of Electronics Engineering, RAIT, Navi Mumbai, India Dr. Doo Young Department of Mathematics and Statistics, Arkansas State University, Jonesboro, USA 2 Contents From the Editor’s Desk J Vasundhara Devi :::::::::::::::::::::::::::::: .::::::::::::::::::::::::::::::::::::::: .:::::::::::: .4 Differential Inequalities and the Comparison Principle: The Core of Professor V. Lakshmikantham’s Research Z. Drici , J. Vasundhara Devi , and F. A. McRae ::: .....:::::::::::::::::::::::::::::::::::::::::::::::::::::: ..5 Fallacy: Category and Role of Environment/Assumption Syamal K. Sen ::: .....:::::::::::::::::::::::::::::::::::: .:::::::::::::::::::::::::::::::::::::::::: ..10 Nonlinear Flow-Structure Interaction: A Survey Igor Chueshov and Irena Lasiecka and Justin Webster :::::::::::::::::::::::::::::::::::::::::::::::::::::: 32 Stability in Two Measures with Applications to Impulsive Control Systems Xinzhi Liu :::::::::::::::::::::::::::::::::::::::::: .:::::::::::::::::::::::::::::::::::::::::::::::: 52 A Shrinking Projection Method for a Common Zero of a Finite Family of Monotone Operators Jong Kyu Kim and Truong Minh Tuyen:::::::::::::::::::::::: .:::::::::::::::::::::::::::::::::::::::::: 73 3 From the Editor’s Desk Dear Reader, We are proud to relaunch "Nonlinear World", at the International Conference on Recent Advances in Mathematical Sciences and Applications, held from 19th-22nd December,2017, at GVP College of Engineering (A), India. This issue takes it shape, as it was envisioned by late Prof. V. Lakshmikantham, as a compendium of advances in nonlinear research in a diverse range of fields, and to serve as a guiding torch of knowledge. My sincere thanks to the International Federation of Nonlinear Analysts, for their continued support over the years, as well as to all the honored editors, knowledgable contributors and esteemed readers, all without whom this journal would not have been possible. Dr.J.Vasundhara Devi Professor, Dept of Mathematics and Associate Director - GVP - Prof. V. Lakshmikantham Institute for Advanced Studies GVP College of Engineering Madhurawada Viakhapatnam 530 048 India 4 Differential Inequalities and the Comparison Principle: the Core of Professor V. Lakshmikantham’s Research Z. Drici , J. Vasundhara Devi , F. A. McRae Department of Mathematics, Illinois Wesleyan University, Bloomington, IL 61702-2900, USA. G. V. P. Prof. Lakshmikantham Institute for Advanced Studies, G.V.P. College of Engineering, Visakhapatnam, India. Department of Mathematics, Catholic University of America, Washington, DC 20064, USA. ABSTRACT: In this article we give a brief exposition of how Professor V. Lakshmikantham used the principle of comparison in his work. A review of the literature indicates that by the time Professor V. Lakshmikantham began his research the concepts of differential inequalities, upper and lower solutions and the idea of comparison were known. With his insight, he recognized the importance of these ideas and exploited them to open a whole new area of research. In this article we illustrate how his research was dominated by the theory of differential inequalities and the comparison principle. In his first paper [1], which appeared in the Proceedings of the American Mathematical Society, Professor V. Lakshmikantham recognized that using differential inequalities instead of the commonly used integral inequalities to obtain qualitative results in differential equations, such as convergence of successive approximations, and uniqueness, boundedness and asymptotic behavior of solutions, one could dispense with the required monotonicity property of the functions involved. Consider the differential system dy = f(t; x); x(t ) = x ; (1) dx 0 0 n n where f 2 C[R+ × R ; R ] satisfies the inequality jjf(t; x)jj ≤ g(t; jjxjj); (2) + where g 2 C[R × R ; R]. Letting m(t) = jjx(t)jj, it follows that dm(t) ≤ g(t; m(t)) a:e: (3) dt Using the theory of differential inequalities Professor V. Lakshmikantham showed that one can obtain from (3) the following estimate m(t) ≤ r(t; t0; u0); (4) provided m(t0) ≤ u0; where r(t; t0; u0) is the maximal solution of the scalar differential equation du = g(t; u); u(t ) = u ; (5) dt 0 0 existing on [t0; ), without requiring the function g to be monotone nondecreasing. He further observed that in order to obtain the estimate (4), it is sufficient to have 1 Dm(t) ≤ g(t; m(t)); (6) where Dm(t) is any one of the Dini Derivatives. This observation led him to conclude that all that was needed is an assumption of the following type jjx + hf(t; x)jj ≤ jjxjj + hg(t; jjxjj) + o(h); (7) instead of (2) in order to obtain the desired inequality (6). Furthermore, he also observed that the advantage of assumption (7) is that g is not even required to be nonnegative, which results in a better estimate for jjx(t)jj. 5 From equations (6) and (7), he noted that instead of jjx(t)jj, one could use a nonnegative function with some properties of norm as follows V(t + h; x + hf(t; x)) ≤ V(t; x) + hg(t; V(t; x)) + o(h); n n where V 2 C[R+ × R ;R ] and V(t; x) is locally Lipschitz in x. Setting m(t) = V(t; x(t)) leads to the differential inequality (6) from which the estimate (4) is obtained. Thus, given an initial-value problem, an important technique in the theory of differential equations consists of first obtaining a scalar differential equation, such as equation (4) and then estimating the solution of the differential system (1) in terms of the extremal solution of the scalar differential equation (4). This development of the theory of differential and integral inequalities using norms or Lyapunov-like functions led to a number of results, such as convergence of successive approximations and stability and boundedness of solutions. The central theme in all generalizations, extensions, and refinements is the use of the comparison method in terms of Lyapunov-like functions. Professor V. Lakshmikantham recognized the advantage of using vector Lyapunov functions, which were first introduced by Bellman [2] and Matrosov [3], to investigate and refine the Lyapunov theory of stability. This intense research activity by Professor V. Lakshmikantham resulted in a two-volume monograph [4,5], entitled Differential and Integral
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