Table Of ContentMONOGRAPHS 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. Byrne
Modeling and Inverse Problems in the Presence of Uncertainty, H. T. Banks, Shuhua Hu,
and W. Clayton Thompson
Monomial Algebras, Second Edition, Rafael H. Villarreal
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 and Bilel Krichen
Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative
Analysis, Vicenţiu D. Rădulescu and Dušan D. Repovš
A Practical Guide to Geometric Regulation for Distributed Parameter Systems,
Eugenio Aulisa and David Gilliam
Signal Processing: A Mathematical Approach, Second Edition, Charles L. Byrne
Sinusoids: Theory and Technological Applications, Prem K. Kythe
Special Integrals of Gradshetyn and Ryzhik: the Proofs – Volume l, Victor H. 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. Melnikova and Alexei Filinkov
Symmetry and Quantum Mechanics, Scott Corry
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2016 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20150721
International Standard Book Number-13: 978-1-4987-3389-2 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable
efforts have been made to publish reliable data and information, but the author and publisher cannot
assume responsibility for the validity of all materials or the consequences of their use. The authors and
publishers have attempted to trace the copyright holders of all material reproduced in this publication
and apologize to copyright holders if permission to publish in this form has not been obtained. If any
copyright material has not been acknowledged please write and let us know so we may rectify in any
future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information stor-
age or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copy-
right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222
Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro-
vides licenses and registration for a variety of users. For organizations that have been granted a photo-
copy license by the CCC, a separate system of payment has been arranged.
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
