An Introduction to Nonlinear Analysis and Fixed Point Theory

845 Pages·2018·8.466 MB·English

by  Hemant Kumar Pathak

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
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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