Fixed Point Theory and Graph Theory provides an intersection between the theories of fixed point theorems that give the conditions under which maps (single or multivalued) have solutions and graph theory which uses mathematical structures to illustrate the relationship between ordered pairs of objects in terms of their vertices and directed edges.
This edited reference work is perhaps the first to provide a link between the two theories, describing not only their foundational aspects, but also the most recent advances and the fascinating intersection of the domains.
The authors provide solution methods for fixed points in different settings, with two chapters devoted to the solutions method for critically important non-linear problems in engineering, namely, variational inequalities, fixed point, split feasibility, and hierarchical variational inequality problems. The last two chapters are devoted to integrating fixed point theory in spaces with the graph and the use of retractions in the fixed point theory for ordered sets.
1. 2.1 Introduction - p. 28 2. 2.2 Fixed Point Theorems for Single-valued Self Almost Contractions - p. 31 3. 2.3 Implicit Almost Contractions - p. 50 4. 2.4 Common Fixed Point Theorems for Almost Contractions - p. 58 5. 2.5 Almost Contractive type Mappings on Product Spaces - p. 65 6. 2.6 Fixed Point Theorems for Single-valued Nonself Almost Contractions - p. 81 7. Chapter 3: Approximate Fixed Points - p. 97 8. Abstract - p. 97 9. 3.1 Introduction - p. 97 10. 3.2 Approximate Fixed Points of Mappings in Banach Spaces - p. 101 11. 3.3 Approximate Fixed Points of Mappings in Hyperbolic Spaces - p. 107 12. 3.4 Approximate Fixed Points of Monotone Mappings - p. 117 13. 3.5 Approximate Fixed Points of Nonlinear Semigroups - p. 124 14. 3.6 Approximate Fixed Points of Monotone Nonlinear Semigroups - p. 128 15. Chapter 4: Viscosity Methods for Some Applied Nonlinear Analysis Problems - p. 138 16. Abstract - p. 138 17. 4.1 Introduction - p. 138 18. 4.2 Viscosity Method for Fixed Point Problems - p. 142 19. 4.3 Viscosity Method for Split Common Fixed Point Problems - p. 154 20. 4.4 Viscosity Method for Split Equilibrium Problems - p. 170 21. Chapter 5: Extragradient Methods for Some Nonlinear Problems - p. 185 22. Abstract - p. 185 23. 5.1 Introduction - p. 185 24. 5.2 Preliminaries - p. 189 25. 5.3 Projection Gradient Method - p. 194 26. 5.4 Extragradient Method for Nonexpansive Mappings and Variational Inequalities - p. 196 27. 5.5 Modified Extragradient Method for Nonexpansive Mappings and Variational Inequalities - p. 200 28. 5.6 Extragradient Method for Hierarchical Variational Inequalities - p. 206 29. 5.7 Extragradient Methods for Split Feasibility Problems - p. 218 30. Acknowledgments - p. 227 31. Chapter 6: Iterative Methods for Nonexpansive Type Mappings - p. 230 32. Abstract - p. 230 33. 6.1 Introduction and Preliminaries - p. 230 34. 6.2 Nonexpansive Mappings in Uniformly Convex Banach Spaces - p. 235 35. 6.3 Nonexpansive Mappings in CAT(0) Spaces - p. 243 36. 6.4 An Algorithm of Asymptotically Nonexpansive Mappings - p. 253 37. 6.5 Existence and Approximation of Fixed Points - p. 265 38. 6.6 Viscosity Method for Generalized Asymptotically Nonexpansive Mappings - p. 272 39. Acknowledgments - p. 283 40. Chapter 7: Metric Fixed Point Theory in Spaces with a Graph - p. 285 41. Abstract - p. 285 42. 7.1 Introduction - p. 285 43. 7.2 Banach Contraction Principle - p. 290 44. 7.3 Basic Definitions and Properties - p. 294 45. 7.4 Banach Contraction Principle in Metric Spaces with a Graph - p. 305 46. 7.5 Caristi’s Fixed Point Theorem - p. 305 47. 7.6 The Contraction Principle in Modular Metric Spaces with a Graph - p. 319 48. 7.7 Monotone Pointwise Contractions in Banach Spaces with a Graph - p. 322 49. 7.8 Monotone Ćirić Quasi-Contraction Mappings - p. 331 50. 7.9 Monotone Nonexpansive Mappings in Banach Spaces with a Graph - p. 337 51. 7.10 Monotone Nonexpansive Mappings in Hyperbolic Metric Spaces with a Graph - p. 340 52. 7.11 The Contraction Principle for Monotone Multivalued Mappings - p. 343 53. 7.12 Monotone Nonexpansive Multivalued Mappings - p. 355 54. Acknowledgments - p. 361 55. Chapter 8: The Use of Retractions in the Fixed Point Theory for Ordered Sets - p. 361 56. Abstract - p. 361 57. 8.1 Introduction - p. 366 58. 8.2 Chain-Complete Ordered Sets - p. 368 59. 8.3 The Abian-Brown Theorem - p. 370 60. 8.4 Comparative Retractions - p. 373 61. 8.5 Irreducible Points - p. 376 62. 8.6 Constraint Propagation - p. 379 63. 8.7 Retractable Points - p. 383 64. 8.8 Verifying Connected Collapsibility Directly - p. 387 65. 8.9 Graphs - p. 392 66. 8.10 Simplicial Complexes - p. 399 67. 8.11 Topological Realizations - p. 405 68. 8.12 Iterated Clique Graphs - p. 406 69. 8.13 K-Null Comparability Graphs - p. 413 70. Acknowledgments - p. 416 71. Index - p. 417
Fixed Point Theory and Graph Theory provides an intersection between the theories of fixed point theorems that give the conditions under which maps (single or multivalued) have solutions and graph theory which uses mathematical structures to illustrate the relationship between ordered pairs of objects in terms of their vertices and directed edges.
This edited reference work is perhaps the first to provide a link between the two theories, describing not only their foundational aspects, but also the most recent advances and the fascinating intersection of the domains.
The authors provide solution methods for fixed points in different settings, with two chapters devoted to the solutions method for critically important non-linear problems in engineering, namely, variational inequalities, fixed point, split feasibility, and hierarchical variational inequality problems. The last two chapters are devoted to integrating fixed point theory in spaces with the graph and the use of retractions in the fixed point theory for ordered sets.
1. 2.1 Introduction - p. 28 2. 2.2 Fixed Point Theorems for Single-valued Self Almost Contractions - p. 31 3. 2.3 Implicit Almost Contractions - p. 50 4. 2.4 Common Fixed Point Theorems for Almost Contractions - p. 58 5. 2.5 Almost Contractive type Mappings on Product Spaces - p. 65 6. 2.6 Fixed Point Theorems for Single-valued Nonself Almost Contractions - p. 81 7. Chapter 3: Approximate Fixed Points - p. 97 8. Abstract - p. 97 9. 3.1 Introduction - p. 97 10. 3.2 Approximate Fixed Points of Mappings in Banach Spaces - p. 101 11. 3.3 Approximate Fixed Points of Mappings in Hyperbolic Spaces - p. 107 12. 3.4 Approximate Fixed Points of Monotone Mappings - p. 117 13. 3.5 Approximate Fixed Points of Nonlinear Semigroups - p. 124 14. 3.6 Approximate Fixed Points of Monotone Nonlinear Semigroups - p. 128 15. Chapter 4: Viscosity Methods for Some Applied Nonlinear Analysis Problems - p. 138 16. Abstract - p. 138 17. 4.1 Introduction - p. 138 18. 4.2 Viscosity Method for Fixed Point Problems - p. 142 19. 4.3 Viscosity Method for Split Common Fixed Point Problems - p. 154 20. 4.4 Viscosity Method for Split Equilibrium Problems - p. 170 21. Chapter 5: Extragradient Methods for Some Nonlinear Problems - p. 185 22. Abstract - p. 185 23. 5.1 Introduction - p. 185 24. 5.2 Preliminaries - p. 189 25. 5.3 Projection Gradient Method - p. 194 26. 5.4 Extragradient Method for Nonexpansive Mappings and Variational Inequalities - p. 196 27. 5.5 Modified Extragradient Method for Nonexpansive Mappings and Variational Inequalities - p. 200 28. 5.6 Extragradient Method for Hierarchical Variational Inequalities - p. 206 29. 5.7 Extragradient Methods for Split Feasibility Problems - p. 218 30. Acknowledgments - p. 227 31. Chapter 6: Iterative Methods for Nonexpansive Type Mappings - p. 230 32. Abstract - p. 230 33. 6.1 Introduction and Preliminaries - p. 230 34. 6.2 Nonexpansive Mappings in Uniformly Convex Banach Spaces - p. 235 35. 6.3 Nonexpansive Mappings in CAT(0) Spaces - p. 243 36. 6.4 An Algorithm of Asymptotically Nonexpansive Mappings - p. 253 37. 6.5 Existence and Approximation of Fixed Points - p. 265 38. 6.6 Viscosity Method for Generalized Asymptotically Nonexpansive Mappings - p. 272 39. Acknowledgments - p. 283 40. Chapter 7: Metric Fixed Point Theory in Spaces with a Graph - p. 285 41. Abstract - p. 285 42. 7.1 Introduction - p. 285 43. 7.2 Banach Contraction Principle - p. 290 44. 7.3 Basic Definitions and Properties - p. 294 45. 7.4 Banach Contraction Principle in Metric Spaces with a Graph - p. 305 46. 7.5 Caristi’s Fixed Point Theorem - p. 305 47. 7.6 The Contraction Principle in Modular Metric Spaces with a Graph - p. 319 48. 7.7 Monotone Pointwise Contractions in Banach Spaces with a Graph - p. 322 49. 7.8 Monotone Ćirić Quasi-Contraction Mappings - p. 331 50. 7.9 Monotone Nonexpansive Mappings in Banach Spaces with a Graph - p. 337 51. 7.10 Monotone Nonexpansive Mappings in Hyperbolic Metric Spaces with a Graph - p. 340 52. 7.11 The Contraction Principle for Monotone Multivalued Mappings - p. 343 53. 7.12 Monotone Nonexpansive Multivalued Mappings - p. 355 54. Acknowledgments - p. 361 55. Chapter 8: The Use of Retractions in the Fixed Point Theory for Ordered Sets - p. 361 56. Abstract - p. 361 57. 8.1 Introduction - p. 366 58. 8.2 Chain-Complete Ordered Sets - p. 368 59. 8.3 The Abian-Brown Theorem - p. 370 60. 8.4 Comparative Retractions - p. 373 61. 8.5 Irreducible Points - p. 376 62. 8.6 Constraint Propagation - p. 379 63. 8.7 Retractable Points - p. 383 64. 8.8 Verifying Connected Collapsibility Directly - p. 387 65. 8.9 Graphs - p. 392 66. 8.10 Simplicial Complexes - p. 399 67. 8.11 Topological Realizations - p. 405 68. 8.12 Iterated Clique Graphs - p. 406 69. 8.13 K-Null Comparability Graphs - p. 413 70. Acknowledgments - p. 416 71. Index - p. 417
