1st Edition

# Fixed Point Theorems with Applications

437 Pages
by Chapman & Hall

437 Pages
by Chapman & Hall

Also available as eBook on:

As a very important part of nonlinear analysis, fixed point theory plays a key role in solvability of many complex systems from mathematics applied to chemical reactors, neutron transport, population biology, infectious diseases, economics, applied mechanics, and more.

The main aim of Fixed Point Theorems with Applications is to explain new techniques for investigation of different classes of ordinary and partial differential equations. The development of the fixed point theory parallels the advances in topology and functional analysis. Recent research has investigated not only the existence but also the positivity of solutions for various types of nonlinear equations. This book will be of interest to those working in functional analysis and its applications.

Combined with other nonlinear methods such as variational methods and the approximation methods, the fixed point theory is powerful in dealing with many nonlinear problems from the real world.

The book can be used as a textbook to develop an elective course on nonlinear functional analysis with applications in undergraduate and graduate programs in mathematics or engineering programs.

Preface

1 Preliminaries

1.1 Normed Linear Spaces

1.2 Banach Spaces

1.3 Linear Operators in Normed Vector Spaces

1.4 Inverse Operators

1.5 Measures of Noncompactness

1.5.1 The General Setting

1.5.2 Case of Normed Spaces

1.6 Related Maps

1.6.1 k-Set Contractions

1.6.2 Nonexpansive and Expansive Maps

1.7 Ascoli-Arzelà Theorem

1.7.1 Ascoli-Arzelà Theorem　: a First Version

1.7.2 Applications

1.7.3 General Forms of Ascoli-Arzelà Theorem

1.8 Corduneanu-Avramescu Compactness Criterion in C

1.8.1 Main Results

1.8.2 Application 1

1.8.3 Application 2

1.9 Przeradzki’s Compactness Criterion in BC (RY)

1.9.1 Main Result

1.9.2 Application

1.10 Compactness Criterion in C ([0+Y)Rn)

1.11 Compactness Criteria in BC (XR)

1.11.1 The Stone-Cech Compactification

1.11.2 Bartles’s Compactness Criterion and Consequences

1.12 Higher-order Derivative Spaces

1.12.1 The Compact Case

1.12.2 The Noncompact Case

1.13 Zima’s Compactness Criterion

1.13.1 Main Result

1.13.2 Application 1

1.13.3 Application 2

1.14 Cones and Partial Order Relations

2 Fixed Point Index for Sums of Two Operators

2.1 Auxiliary Results

2.2 Fixed Point Index on Cones

2.2.1 The Case Where T is an h-Expansive Mapping and F is a k-Set Contraction with 0<= k <h-1

2.2.2 The Case Where T is an h-Expansive Mapping and F is an (h-1)-Set Contraction

2.2.3 The Case Where T is a Nonlinear Expansive Mapping and F is an k-Set Contraction

2.2.4 The Case Where (I-T) is a Lipschitz Invertible Mapping and F is a k-Set Contraction

2.3 Fixed Point Index on Translates of Cones

3 Positive fixed points for sums of two operators

3.1 Krasnosel’skii’s Compression-Expansion Fixed Point Theorems Type

3.1.1 Fixed Point in Conical Annulus

3.1.2 Vector Version

3.1.3 Extensions

3.1.4 Case of Translates of Cones

3.1.5 Further Extensions

3.2 Leggett-Williams Compression-Expansion Fixed Point Theorem Type

3.3 Fixed Point Theorems on Open Sets of Cones for Special Mappings.

4 Applications to ODEs

4.1 Periodic Solutions for First Order ODEs

4.2 Systems of ODEs

4.3 Existence of Positive Solutions for a Class of BVPs in Banach Spaces

4.4 A Nonlinear IVP

4.5 Existence of Positive Solutions for a Class of nth-Order ODEs

4.6 Boundary Value Problems with p-Laplacian in Banach Spaces

4.7 A Three-Point Fourth-Order Eigenvalue BVP

5 Applications to Parabolic Equations

5.1 Existence of Solutions of a Class IBVPs for Parabolic Equations

5.2 Existence of Classical Solutions for Burgers-Fisher Equation

5.3 IBVPs for Nonlinear Parabolic Equations

5.3.1 Some Preliminary Results

5.3.2 Proof of the Main Result

6 Applications to Hyperbolic Equations

6.1 Applications to One Dimensional Hyperbolic Equations

6.2 Applications to IVPs for a Class Two-Dimensional Nonlinear Wave Equations

6.3 An IVP for Nonlinear Wave Equations in any Spaces Dimension

Bibliography

Index

### Biography

Karima Mebarki is a professor in the Department of Mathematics, Bejaia University, Algeria. Her research interests are: fixed point theory, fixed point index theory, nonlinear ordinary differential equations, and boundary value problems.

Svetlin G. Georgiev is a mathematician who has worked in various aspects of mathematics. Currently, he focuses on harmonic analysis, ordinary differential equations, partial differential equations, time scale calculus, integral equations, and nonlinear analysis. He has published several books with Taylor & Francis/CRC Press.

Smail Djebali works on fixed point theory and applications in differential equations. He is currently a professor at Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia,

Khaled Zennir earned his PhD in mathematics in 2013 from Sidi Bel Abbès University, Algeria. In 2015, he received his highest diploma in Habilitation in mathematics from Constantine University, Algeria. He is currently an assistant professor at Qassim University in the Kingdom of Saudi Arabia. His research interests lie in the subjects of nonlinear hyperbolic partial differential equations: global existence, blowup, and long time behavior.