Advances in Metric Fixed Point Theory and Applications

Preface
Contents
Editors and Contributors
1 The Relevance of a Metric Condition on a Pair of Operators in Common Fixed Point Theory
1.1 Introduction and Preliminaries
1.1.1 The Purpose of the Paper
1.1.2 Notations
1.1.3 Picard Operators
1.1.4 Graphic Contractions That are Picard Operators
1.2 A Variant of Kannan's Common Fixed Point Theorem: A New Research Direction
1.3 Pairs of Operators on a Set with Two Metrics
1.4 Contraction Pairs of Operators on Ordered Metric Spaces
1.5 Pairs of Operators on mathbbR+m-Metric Spaces
1.6 Data Dependence for the Common Fixed Point Problem
1.7 Other Problems
1.7.1 Common Fixed Point Set as a Fixed Point Set
1.7.2 Lipschitz Pairs on Compact Convex Subsets
1.8 Conclusion
References
2 Some Convergence Results of the Kast Iteration Process in CAT(0) Spaces
2.1 Introduction
2.2 Preliminaries and Lemmas
2.3 Some Convergence Results for the Class of Mappings Satisfying the Condition ( Eµ)
2.4 Some Convergence Results for Total Asymptotically Nonexpansive Mappings
2.5 Conclusions
References
3 Split Variational Inclusion Problem and Fixed Point Problem for Asymptotically Nonexpansive Semigroup with Application to Optimization Problem
3.1 Introduction
3.2 Preliminaries
3.3 Main Results
3.4 Applications
3.4.1 Applications to Split Optimization Problems
3.4.2 Applications to Split Variational Inequality Problems
3.5 Conclusions
References
4 Convergence Theorems and Convergence Rates for the General Inertial Krasnosel'skiǐ–Mann Algorithm
4.1 Introduction
4.2 Preliminaries
4.3 The General Inertial Krasnosel'skiǐ–Mann Algorithms with Negative Inertial Parameters
4.3.1 αnin[0,1] and βnin(-infty,0]
4.3.2 αnin[-1,0] and βnin[0,+infty)
4.4 Linear Convergence
4.5 Numerical Examples
4.6 Conclusions
References
5 Digital Space-Type Fixed Point Theory and Its Applications
5.1 Introduction
5.2 Preliminaries
5.3 Some Categories Associated with the Digital Topological Structures
5.4 Digital Versions of the Banach Contraction Principle
5.5 Relationships Between the MA-Contractibility of an MA-Space X and the FPP and the AFPP of X
5.6 FPP and AFPP for Compact (or Finite) Digital Planes
5.7 Product Properties of the FPP and the AFPP for Digital Spaces
5.8 Digital Topological Invariants of the FPP and the AFPP
5.9 Concluding Remarks and a Further Work
References
6 Existence and Approximations for Order-Preserving Nonexpansive Semigroups over CAT(κ) Spaces
6.1 Introduction
6.2 Preliminaries
6.3 Existence Theorems
6.3.1 Some Introductory Notes
6.3.2 An Existence Theorem
6.4 Explicit Approximation Scheme
6.5 Implicit Approximation Scheme
6.6 An Example
6.7 Conclusions and Remarks
References
7 A Solution of the System of Integral Equations in Product Spaces via Concept of Measures of Noncompactness
7.1 Introduction
7.2 Preliminaries
7.3 Main Results
7.4 Systems of Ordinary and Fractal Integral Equations
7.5 Conclusion
References
8 Fixed Points That Are Zeros of a Given Function
8.1 Introduction
8.2 (F,)-Contractions
8.3 mathcalZ-Contractions
8.4 Fixed Points for S-F-Contractions
8.5 Fixed Points for Ordered S-F-Contractions
8.6 Consequences
8.7 Conclusions
References
9 A Survey on Best Proximity Point Theory in Reflexive and Busemann Convex Spaces
9.1 Introduction
9.1.1 Kannan Contractions
9.1.2 Kannan Nonexpansive Mappings
9.2 Geodesic Metric Spaces
9.3 Best Proximity Points
9.3.1 Cyclic Relatively Nonexpansive Mappings
9.3.2 Cyclic Kannan Contractions
9.3.3 Cyclic Relatively Kannan Nonexpansive Mappings
9.4 Structure of Minimal Sets and Min-Max Property
9.5 On Dropping of PQNS for Cyclic Relatively Kannan Nonexpansive Mappings
9.6 More on Minimal Invariant Pairs for Strongly Cyclic Relatively Kannan Nonexpansive Mappings
References
10 On Monotone Mappings in Modular Function Spaces
10.1 Introduction
10.2 Preliminaries
10.3 Monotone Nonexpansive Mappings
10.4 Γρ Nonexpansive Mappings
10.5 Synopsis
References
11 Contributions to Fixed Point Theory of Fuzzy Contractive Mappings
11.1 Introduction
11.2 Fuzzy Metric Spaces
11.3 Fuzzy Contractive Mappings
11.4 Fuzzy Ψ-Contractive Mappings
11.5 α-φ-Fuzzy Contractive Mappings
11.6 β-ψ-Fuzzy Contractive Mappings
11.7 Fuzzy mathcalH-Contractive Mappings and α Type Fuzzy mathcalH-Contractive Mappings
11.8 Fuzzy mathcalZ-Contractive Mappings
11.9 Conclusions
References
12 Common Fixed Point Theorems for Four Maps
12.1 Introduction and Preliminaries
12.2 Main Results
12.3 Results in Ordered Metric Spaces
12.4 Results in Metric Spaces Endowed with a Graph
12.5 Application
12.6 Conclusion
References
13 Measure of Noncompactness in Banach Algebra and Its Application on Integral Equations of Two Variables
13.1 Introduction
13.2 Measure of Noncompactness
13.2.1 Preliminaries
13.2.2 Kuratowski Measure of Noncompactness
13.2.3 Hausdorff Measure of Noncompactness
13.3 Existence of Solution of a Functional Integral Equation with Two Variables in C(I timesI)
13.4 Existence of Solution of a Functional Integral Equation with Two Variables in BC(mathbbR+ timesmathbbR+)
13.5 Conclusion
References
14 Generalization of Darbo-Type Fixed Point Theorem and Applications to Integral Equations
14.1 Introduction and Preliminaries
14.2 Generalized Darbo-Type Fixed Point Theorems
14.3 Darbo-Type n-Tupled Fixed Point Theorems
14.4 Application I
14.5 Combination of Some Effective Modified Methods to Solve Volterra Nonlinear Singular Mixed Integral Equations (14.14)
14.6 Application II
14.7 Conclusion
References
15 Approximating Fixed Points of Suzuki (α,β)-Nonexpansive Mappings in Ordered Hyperbolic Metric Spaces
15.1 Introduction and Preliminaries
15.2 Existence Results on Picard Iterations
15.3 Existence Results on the CR-Iteration
15.4 Convergence Results
15.5 Conclusions
References
16 Generalized JS-Contractions in b-Metric Spaces with Application to Urysohn Integral Equations
16.1 Introduction
16.2 Preliminaries
16.3 Main Results
16.3.1 Discussion on Coincidence Point Results
16.3.2 Discussion on Common Fixed Point Results
16.3.3 Uniqueness of Common Fixed Point
16.3.4 Example
16.3.5 Periodic Point Results
16.4 Application
16.5 Conclusion
References
17 Unified Multi-tupled Fixed Point Theorems Involving Monotone Property in Ordered Metric Spaces
17.1 Introduction
17.2 Extended Notions Upto Product Sets
17.3 Auxiliary Results
17.4 Multi-tupled Coincidence Theorems for Compatible Mappings
17.5 Multi-tupled Coincidence Theorems Without Compatibility of Mappings
17.6 Multi-tupled Fixed Point Theorems
17.7 Conclusion
References
18 Convergence Analysis of Solution Sets for Minty Vector Quasivariational Inequality Problems in Banach Spaces
18.1 Introduction
18.2 Preliminaries
18.3 Main Results
18.4 Conclusion
References
19 Common Solutions for a System of Functional Equations in Dynamic Programming Passing Through the JCLR-Property in Sb-Metric Spaces
19.1 Introduction and Preliminaries
19.2 Main Results
19.3 Applications to the Dynamic Programming
19.4 Conclusions
References
20 A General Approach on Picard Operators
20.1 Introduction and Preliminaries
20.2 Some Generalized Metric Spaces
20.2.1 ν-Generalized Metric Space
20.2.2 mathfrakD-Generalized Metric Spaces
20.2.3 Asymmetric Metric Space
20.3 ρ-Metric Spaces
20.3.1 Contractions in an Implicit Form
20.3.2 Definition of ρ-Metric Space and Its Properties
20.3.3 Fixed Point Theorems in ρ-Metric Spaces
20.3.4 Example
References