An Introduction to Nonlinear Analysis and Fixed Point Theory PDF
Preview An Introduction to Nonlinear Analysis and Fixed Point Theory
Hemant Kumar Pathak An Introduction to Nonlinear Analysis and Fixed Point Theory An Introduction to Nonlinear Analysis and Fixed Point Theory Hemant Kumar Pathak An Introduction to Nonlinear Analysis and Fixed Point Theory 123 Hemant KumarPathak Schoolof Studies in Mathematics Pandit Ravishankar ShuklaUniversity Raipur, Chhattisgarh India ISBN978-981-10-8865-0 ISBN978-981-10-8866-7 (eBook) https://doi.org/10.1007/978-981-10-8866-7 LibraryofCongressControlNumber:2018935212 ©SpringerNatureSingaporePteLtd.2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. 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 authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. partofSpringerNature Theregisteredcompanyaddressis:152BeachRoad,#21-01/04GatewayEast,Singapore189721, Singapore Dedicated to my parents who assigned me this heavenly body with a soul to think for humanity, enlightened my mind with wisdom of thought to think righteously; and endowed me with a capacity to render service for mankind. Hemant Kumar Pathak Foreword Thebookentitled“AnIntroductiontoNonlinearAnalysisandFixedPointTheory” by Prof. H. K. Pathak covers both the area of nonlinear analysis and fixed point theory in great detail. The book begins with the fundamentals of nonlinear func- tional analysis, e.g. geometry of Banach spaces, differential calculus in Banach spaces,monotone operator theory,accretive operatorsandtheirvariants.The book laysspecialemphasisonapplyingtechniquesofnonlinearanalysistomodelandto treat nonlinear phenomena with which nature confronts us. Coverage of applica- tions includes many branches of science and technology such as control theory, nonlinear stochastic operator equations, variational methods in Hilbert spaces, degreetheory,k-setcontractionandcondensingoperators,variationalmethodsand optimization.ApplicationsofmonotoneoperatortheorytoODE,integralequations and solution of nonlinear equations by computational schemes and strong con- vergenceresultsarepresented.Applicationsoffixedpointtheoremstogeometryof Banachspaces,systemoflinearequations,controltheory,gametheory,differential equations, nonlinear integral equations, abstract Volterra integrodifferential equa- tions,surjectivityproblems,simultaneouscomplementarityproblemsandproblems of integral inclusion for multifunctions are thoroughly discussed. In my opinion, the book should be very useful to mathematics students in their final semester course of master’s degree and also for the first semester course for Ph.D.students,enhancingtheircapabilitytogainthedesiredinsightintononlinear analysis and fixed point theory. Jhusi, Allahabad, India Prof. Satya Deo, Ph.D., FNASc. Formerly Vice Chancellor APS University Rewa, and RD University, Jabalpur NASI Senior Scientist, Harish-Chandra Research Institute vii Preface Nonlinear analysis is the fascinating emerging field of the twenty-first century characterized by a remarkable mixture of nonlinear functional analysis, nonlinear operator theory, topology, mathematical modelling and applications. Its scope of enquiries not only encompasses the geometric theory of infinite dimensional function spaces and operator-theoretic real-world problems but also widens the range of interdisciplinary fields ranging from engineering to space science, hydromechanics to astrophysics, chemistry to biology, theoretical mechanics to biomechanics and economics to stochastic game theory. The deep-rooted concepts andtechniquesprovidethetoolsfordevelopingmorerealisticandaccuratemodels for a variety of phenomena encountered in various applied fields. This gives nonlinearanalysisaratherinterdisciplinarycharacter.Today,themoretheoretically inclined nonmathematician (engineer, economist, geologist, pharmacologist, biol- ogist or chemist) needs a working knowledge of at least a part of the nonlinear analysisinordertobeabletoconductacompletequalitativeanalysisofhismodels. Thissupportsahighdemandforbooksonnonlinearanalysis.Moreover,thesubject has become so vast that no single book can cover all its theoretical and applied parts.Inthisvolume,wehavefocusedonthosetopicsofnonlinear analysis which arepertinent tooperator-theoreticfixedpoint results, especiallymetric, topological and lattice-based fixed point theorems and their applications to control theory, dynamic programming, matrix theory, differential and integral equations, calculus of variations and many real-world problems such as stochastic modelling of physical and biological sciences. The first half of the twentieth century witnessed an extensive theoretical investigation pertaining to linear functional analysis which deals with infinite dimensional topological vector spaces that provide a suitable pathway to mix in a fruitful way the linear (algebraic) structure with topological one and the linear operators acting between them. This investigation facilitated extending standard results of the linear analysis to an infinite dimensional context paving the way for rigorous treatment of linear mathematical models. Systematic efforts to extend the lineartheorytovarioustypesofnonlinearoperatorswerestartedintheearly1960s. This marks the beginning of what is known today as “nonlinear analysis”. ix x Preface However,itisquiteinterestingtonotethatwellbeforethisperiodtherewasashort periodduringthe1930swhenthenotionsofthecompactoperatorandtheextension ofBrouwer’sdegreetheoryinfinitedimensiontoLeray–Schauderdegreetheoryin infinite dimensional context came into existence. On the one hand, with the advent of nonlinear analysis, several theories have been developed simultaneously in this respect, and today some of them are well established approaching their limits, while others are still the object of intense research activity. The appearance of set-valued analysis, nonsmooth analysis, dif- ferential topology, combinatorics, geometry of manifolds and of computational mathematicsallofwhichweremotivatedbyconcreteneedsinappliedareassuchas controltheory,optimization,game theoryandeconomicsisevidencetothis effect. Their development provided nonlinear analysis with new concepts, tools and the- ories that enriched the subject considerably. On the other hand, fixed point theory hasanenormousnumberofapplicationsinvariousfieldsofmathematics.Keeping in view the above facts, it is legitimate to introduce the book with the title “An IntroductiontoNonlinearAnalysisandFixedPointTheory”.Inthisbook,Itriedto present most of the significant results in the field of nonlinear analysis, especially monotone operator, fixed point theory, topological degree theory, variational methods and optimization, and then to present various related applications. Chapter 1 is an introduction to some fundamental concepts needed for the development of the theory of nonlinear functional analysis. We deal with certain large classes of nonlinear operatorswhich arise often in applications. In particular, we examine the Nemytskii, Hammerstein and Urysohn operators and their conti- nuityproperties.Alltheseoperatorsareencounteredinalmostallproblems.Finally, weintroducetheconceptofSobolevspaces(thesuitablespacesforweaksolutions of elliptic equations) needed for the development of differential equations. Chapter2dealswithgeometricalstructuressuchasconvexityandsmoothnessof Banach spaces and of certain broad classes of nonsmooth functions. This chapter also deals with useful properties of duality mappings that interplay with these geometrical structures of Banach spaces. We show that the subdifferential of norm functional is precisely the duality mapping. Chapter 3 deals with calculus in real Banach spaces. We start with the Gâteaux and Fréchet derivatives. We discuss the generic differentiability of continuous convex functions. This chapter also deals with an important concept of nonlinear analysis—subdifferential of convex functionals. Properties of the derivative are discussed, and some fundamental theorems of calculus are presented, especially Taylor’s theorem, inverse function theorem and implicit function theorem. Chapter 4 deals with monotone and maximal monotone operators and their properties. We give some results regarding the approximate solvability of operator equations involving monotone operators with the hope this will help develop computeralgorithmsfortheapproximatesolutionofoperatorequations.Monotone properties of the subdifferential of convex functionals are discussed. We close this chapter by introducing various generalizations of monotonicity concepts—pseu- domonotonicity, generalized pseudomonotonicity, etc. Preface xi Chapter 5 deals with metric, topological and lattice-based fixed point theory alongwithcommonfixedpointtheoremsforafamilyofcommutingmappings.We present some fixed point theorems for multifunctions motivated by their applica- tions to integral inclusions. In this chapter, we also present common fixed point theoremsforafamilyofcommutingmappings.Weconcludethischapterbygiving a discussion on fixed point theorems in ordered Banach spaces. Our treatment is briefandismotivatedbytheirapplicationstothesystemoflinearequations,matrix theory, control theory, differential and integral equations. Chapter6dealswithdegreetheory,k-setcontractionsandcondensingoperators. This chapter is motivated with the fact that topological degree theory has an important advantage over the fixed point theory in the sense that it gives infor- mationaboutthenumberofdistinctsolutions,continuousfamiliesofsolutionsand stabilityofsolutions.Leray–Schauderdegreetheoryispresented.Inthesequel,we presentageneralizationofLeray–Schauderdegreethatextendedtheconceptofthe degree to the class of limit-compact operators. Subsequently, this theory was used to discuss k-set contractions and condensing mapping. Chapter 7 provides an introduction and use of variational methods and opti- mization in nonlinear analysis. Chapter 8discusses integralequations inthemostgeneralsetting.Weconclude the chapter by giving computational scheme for the solvability of nonlinear equations. Chapter 9 provides applications offixed point theorems to the system of linear equations, nonlinear matrix equations, control theory, dynamic programming, stochastic game theory and existence theorems for nonlinear differential and inte- gral equations. In most of the illustrated problems, the differential equations are transformed into equivalent operator equations involving integral operators and then appropriate fixed point theorems or degree theoretic methods are invoked to prove the existence of desired solutions by recasting the operator equations into fixed point equations. Chapter 10 deals with applications offixed point theorems for multifunction to integral inclusions. Aglimpseoffundamentals,expositionofarichvarietyoftopics,boththeoretical and applied, make nonlinear analysis useful to graduate students and researchers, working in analysis or its applications to control theory, variational inequalities, theoretical mechanics, or dynamical systems. An appendix contains requisite background material needed, and a detailed bibliography facilitates further study. This book evolved from classes taught by the author at Pt. Ravishankar Shukla University, Raipur, India, in a course of Master of Philosophy entitled “Nonlinear AnalysisandTopologicalStructures”.Moreover,thebookisself-containedandthe presentation is detailed, to avoid irritating readers by frequent references to details inotherbooks.Theexamplesaresimple,tomakethebookteachable.Wehopethat this book will be extremely useful to students having a background in nonlinear functional analysis, operator theory and topological properties. xii Preface I would like to express my profound thanks and gratitude to my friends— Prof.RaviP.Agarwal,USA;Prof.BillyE.Rhoades,USA;Prof.G.Jungck,USA; Prof.BrianFisher,England;Prof.YeolJeCho,SouthKorea;Prof.ShinMinKang, South Korea; Prof. J. K. Kim, South Korea; Prof. J. S. Jung, South Korea; Prof. S. K. Ntouyas, Greece; Prof. Rosana Rodriguez-Lopez, Spain; Prof. Donal O’Regan, Ireland; Prof. S. N. Mishra, South Africa; Prof. V. Popa, Romania; Prof.N.Shahzad,SaudiArabia;Prof.N.Hussain,SaudiArabia;Prof.M.S.Khan, Sultanate of Oman; Prof. S. S. Chang, P.R. China; Prof. Zeqing Liu, P. R. China; Prof.T.Suzuki,Japan;Prof.V.Kannan,Prof.SatyaDeoTripathi,Prof.S.L.Singh, Prof. M. Imdad, Prof. Q. H. Ansari, Prof. P. Veeramani, Prof. M. T. Nair, Prof.D.R.SahuallfromIndia—fortheirkindencouragementatdifferentoccasion duringthepreparationofthisbook.Iwouldalso liketoexpressmysincere thanks to my colleagues—Prof. B. K. Sharma and Prof. B. S. Thakur—for their constant moralsupport.IalsowanttothankmyprojectfellowEktaTamrakarforhersupport in typing some part of the book in LATEX. Theauthorisindebtedtotheanonymousreviewersfortheirpreciouscomments andhelpfulsuggestionstowardstheimprovementoftheoriginaldraftofthebook. Finally,theauthorisverygratefultoDr.LynnBrandonfromSpringerUK,Shamim Ahmad and Shubham Dixit from Springer Nature for their indefatigable coopera- tion,patienceandunderstandingthroughoutourcommunication.Itisapleasureto acknowledge the great help and technical support given to us by N. S. Pandian, Praveenkumar Vijayakumar, Krati Shrivastav and their technical team of Springer Nature in their rapid and meticulous publication of the book. Raipur, India Hemant Kumar Pathak July 2017
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