MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS Nonlinear Functional Analysis in Banach Spaces and Banach Algebras Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications Aref Jeribi University of Sfax Tunisia Bilel Krichen University of Sfax Tunisia MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS Series Editors John A. Burns Thomas J. Tucker Miklos Bona Michael Ruzhansky Published Titles Application of Fuzzy Logic to Social Choice Theory, John N. Mordeson, Davender S. Malik and Terry D. Clark Blow-up Patterns for Higher-Order: Nonlinear Parabolic, Hyperbolic Dispersion and Schrödinger Equations, Victor A. Galaktionov, Enzo L. Mitidieri, and Stanislav Pohozaev Cremona Groups and Icosahedron, Ivan Cheltsov and Constantin Shramov Difference Equations: Theory, Applications and Advanced Topics, Third Edition, Ronald E. Mickens Dictionary of Inequalities, Second Edition, Peter Bullen Iterative Optimization in Inverse Problems, Charles L. 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Moll Forthcoming Titles Actions and Invariants of Algebraic Groups, Second Edition, Walter Ferrer Santos and Alvaro Rittatore Analytical Methods for Kolmogorov Equations, Second Edition, Luca Lorenzi Complex Analysis: Conformal Inequalities and the Bierbach Conjecture, Prem K. Kythe Computational Aspects of Polynomial Identities: Volume l, Kemer’s Theorems, 2nd Edition Belov Alexey, Yaakov Karasik, Louis Halle Rowen Forthcoming Titles (continued) Geometric Modeling and Mesh Generation from Scanned Images, Yongjie Zhang Groups, Designs, and Linear Algebra, Donald L. Kreher Handbook of the Tutte Polynomial, Joanna Anthony Ellis-Monaghan and Iain Moffat Lineability: The Search for Linearity in Mathematics, Juan B. Seoane Sepulveda, Richard W. Aron, Luis Bernal-Gonzalez, and Daniel M. Pellegrinao Line Integral Methods and Their Applications, Luigi Brugnano and Felice Iaverno Microlocal Analysis on Rˆn and on NonCompact Manifolds, Sandro Coriasco Practical Guide to Geometric Regulation for Distributed Parameter Systems, Eugenio Aulisa and David S. Gilliam Reconstructions from the Data of Integrals, Victor Palamodov Special Integrals of Gradshetyn and Ryzhik: the Proofs – Volume ll, Victor H. Moll Stochastic Cauchy Problems in Infinite Dimensions: Generalized and Regularized Solutions, Irina V. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To my mother Sania, my father Ali, my wife Fadoua, my children Adam and Rahma, my brothers Sofien and Mohamed Amin, my sister Elhem, my mother-in-law Zineb, my father-in-law Ridha, and all members of my extended family .... Aref Jeribi To the memory of my mother Jalila, my father Hassan, my wife Nozha, and my children Mohamed and Zaineb. Bilel Krichen Contents Preface xi Symbol Description xv I Fixed Point Theory 1 Introduction 3 1 Fundamentals 19 1.1 Basic Tools in Banach Spaces . . . . . . . . . . . . . . . . . . 19 1.1.1 Normed vector spaces . . . . . . . . . . . . . . . . . . 19 1.2 Contraction Mappings . . . . . . . . . . . . . . . . . . . . . . 21 1.2.1 The contraction mapping principle . . . . . . . . . . . 22 1.3 Weak Topology . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.3.1 Weakly compact linear operators . . . . . . . . . . . . 30 1.3.2 The Dunford–Pettis property (DP property) . . . . . 34 1.4 Measure of Weak Noncompactness (MNWC) . . . . . . . . . 35 1.5 Basic Tools in Banach Algebras . . . . . . . . . . . . . . . . 40 1.6 Elementary Fixed Point Theorems . . . . . . . . . . . . . . . 43 1.7 Positivity and Cones . . . . . . . . . . . . . . . . . . . . . . . 47 2 Fixed Point Theory under Weak Topology 51 2.1 Fixed Point Theorems in DP Spaces and Weak Compactness 51 2.1.1 Schauder’s fixed point theorem in DP spaces . . . . . 52 2.1.2 Krasnosel’skii’s fixed point theorem in DP spaces . . . 53 2.2 Banach Spaces and Weak Compactness . . . . . . . . . . . . 54 2.2.1 Schauder’s fixed point theorem . . . . . . . . . . . . . 54 2.2.2 Krasnosel’skii’s fixed point theorem . . . . . . . . . . 56 2.3 Fixed Point Theorems and MNWC . . . . . . . . . . . . . . 57 2.3.1 Sum of two weakly sequentially continuous mappings . 58 vii viii Contents 2.3.2 Leray–Schauder’s alternatives for weakly sequentially continuous mappings . . . . . . . . . . . . . . . . . . . 66 2.4 Fixed Point Theorems for Multi-Valued Mappings . . . . . . 68 2.4.1 Multi-valued maps with a weakly sequentially closed graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.4.2 Leray–Schauder’s and Furi–Pera’s types of fixed point theorems . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.5 Some Leray–Schauder’sAlternatives . . . . . . . . . . . . . . 79 2.5.1 Leray–Schauder’s alternatives involving nonlinear con- traction mappings . . . . . . . . . . . . . . . . . . . . 80 2.5.2 Leray–Schauder’salternativesforthesumoftwoweakly sequentially continuous mappings . . . . . . . . . . . . 83 2.5.3 Furi–Pera’s fixed point theorem for the sum of two weakly sequentially continuous mappings . . . . . . . 88 3 Fixed Point Theory in Banach Algebras 91 3.1 Fixed Point Theorems Involving Three Operators . . . . . . 91 3.1.1 Fixed point theorems for -Lipschitzian mappings . . 92 D 3.1.2 Fixed point theorems in Banach algebras satisfying the condition ( ) . . . . . . . . . . . . . . . . . . . . . . . 102 P 3.1.3 Existence of positive solutions. . . . . . . . . . . . . . 105 3.1.4 Fixed point theorems in Banach algebras and MNWC 106 3.2 WC–Banach Algebras . . . . . . . . . . . . . . . . . . . . . . 118 3.2.1 Fixed point theorems in WC–Banach algebras. . . . . 118 3.3 Leray–Schauder’s Alternatives in Banach Algebras Involving Three Operators . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.4 Convex-PowerCondensing Operators . . . . . . . . . . . . . 131 3.5 ws-Compact and ω-Convex-PowerCondensing Maps . . . . . 137 4 Fixed Point Theory for BOM on Banach Spaces and Banach Algebras 143 4.1 Some Variants of Schauder’s and Krasnosel’skii’s Fixed Point Theorems for BOM . . . . . . . . . . . . . . . . . . . . . . . 143 4.1.1 One of the diagonal entries of is invertible . . . 143 I−L 4.1.2 None of the diagonal entries of is invertible. . . 146 I−L 4.2 Fixed Point Theory under Weak Topology Features . . . . . 148 4.2.1 One of the diagonal entries of is invertible . . . 151 I−L 4.2.2 None of the diagonal entries of is invertible. . . 154 I−L Contents ix 4.3 Fixed Point Theorems for BOM in Banach Algebras . . . . . 155 4.3.1 Banach algebras satisfying the condition ( ) . . . . . 164 P 4.4 Fixed Point Results in a Regular Case . . . . . . . . . . . . . 170 4.5 BOM with Multi-Valued Inputs . . . . . . . . . . . . . . . . 175 4.5.1 Fixed point theorems of multi-valued mappings . . . 176 II Applications in Mathematical Physics and Biology 187 5 Existence of Solutions for Transport Equations 189 5.1 Transport Equations in the Kinetic Theory of Gas . . . . . . 189 5.1.1 Leakage of energy at the boundary of the slab. . . . . 189 5.1.2 Case where (x,v,ψ(x,v))=σ(x,v)ψ(x,v) . . . . . . 191 V 5.1.3 Positive solutions of the boundary value problem . . 200 5.1.4 Existence of solutions for a generalnonlinear boundary value problem . . . . . . . . . . . . . . . . . . . . . . 202 5.2 Transport Equations Arising in Growing Cell Population . . 208 5.2.1 A particular case . . . . . . . . . . . . . . . . . . . . . 209 5.2.2 Regular collision and weak compactness results . . . . 214 5.2.3 The general case . . . . . . . . . . . . . . . . . . . . . 224 6 Existence of Solutions for Nonlinear Integral Equations 229 6.1 Existence of Solutions for Hammerstein’s Integral Equation . 229 6.1.1 Hammerstein’s integral equation . . . . . . . . . . . . 229 6.1.2 A general Hammerstein’s integral equation . . . . . . 233 6.2 A Study of Some FIEs in Banach Algebras . . . . . . . . . . 237 6.2.1 The weak sequential continuity and the weak compact- ness in FIEs . . . . . . . . . . . . . . . . . . . . . . . . 237 6.2.2 Regular maps in FIEs . . . . . . . . . . . . . . . . . . 243 6.2.3 ω-condensing mappings in FIEs . . . . . . . . . . . . . 254 6.2.4 ω-convex-power-condensingmappings in FIEs . . . . . 257 6.3 Existence Results for FDEs in Banach Algebras . . . . . . . 263 6.4 An Application of Leray–Schauder’sTheorem to FIEs . . . . 266 7 Two-Dimensional Boundary Value Problems 273 7.1 A System of Transport Equations in L (1<p< ) . . . . 273 p ∞ 7.1.1 Non-dependence of σ on the density of the population 274 i 7.1.2 Dependence of σ on the density of the population . . 284 i 7.2 A Study of a Biological Coupled System in L . . . . . . . . 287 1