Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness

This book offers a comprehensive treatment of the theory of measures of noncompactness. It discusses various applications of the theory of measures of noncompactness, in particular, by addressing the results and methods of fixed-point theory. The concept of a measure of noncompactness is very useful for the mathematical community working in nonlinear analysis. Both these theories are especially useful in investigations connected with differential equations, integral equations, functional integral equations and optimization theory. Thus, one of the book⛨QAةs central goals is to collect and present sufficient conditions for the solvability of such equations. The results are established in miscellaneous function spaces, and particular attention is paid to fractional calculus Preface 5 Contents 8 Editors and Contributors 10 1 Measures of Noncompactness in the Space of Continuous and Bounded Functions Defined on the Real Half-Axis 13 1.1 Introduction 13 1.2 Notation, Definitions, and Auxiliary Facts 15 1.3 Measures of Noncompactness in the Space BC ( mathbbR+ ) 17 1.4 Measures of Noncompactness in the Space of Continuous Tempered Functions 23 1.5 Existence of Solutions of a Quadratic Hammerstein Integral Equation in the Class of Functions Vanishing at Infinity 26 1.6 Existence of Solutions of a Neutral Differential Equation with Deviating Argument in the Space of Continuous Tempered Functions 36 1.7 Solvability of a Functional Integral Equation of Fractional Order in the Class of Functions Having Limits at Infinity 45 1.8 Attractivity and Asymptotic Stability of Solutions of Functional Integral Equations 58 References 67 2 Measures of Noncompactness and Their Applications 71 2.1 Introduction 71 2.2 Standard Measures of Noncompactness 72 2.2.1 The Kuratowski Measure of Noncompactness 72 2.2.2 The Hausdorff Measure of Noncompactness 73 2.2.3 The Istrǎtescu Measure of Noncompactness 75 2.2.4 Inner Hausdorff Measure of Noncompactness 75 2.3 Measures of Noncompactness in Some Spaces 76 2.3.1 Hausdorff Measure of Noncompactness in the Space C([a,b];mathbbR) 77 2.3.2 Hausdorff Measure of Noncompactness in the Space Lp([a,b];mathbbR) 77 2.3.3 Hausdorff Measure of Noncompactness in Banach Spaces with Schauder Bases 78 2.3.4 Inner Measure of Noncompactness in Paranormed Spaces 81 2.4 Constructing Measures of Noncompactness 85 2.4.1 Measure of Noncompactness in C([a,b];mathbbR) 86 2.4.2 Some Measures of Noncompactness in BC(mathbbR+;mathbbR) 86 2.4.3 Measure of Noncompactness with Kernel 87 2.5 Fixed Point Theorems Involving a Measure of Noncompactness 88 2.5.1 The Class of μ-Contractive Mappings and Darbo's Fixed Point Theorem 89 2.5.2 A Fixed Point Theorem for (ψ,)-μ-Contractive Mappings 89 2.5.3 A Fixed Point Theorem for -μ-Contractive Mappings 91 2.5.4 A Fixed Point Theorem for an Implicit μ-Contraction 93 2.5.5 A Fixed Point Theorem for θ-μ-Contractive Mappings 95 2.5.6 Meir--Keeler Generalization of Darbo's Theorem 96 2.5.7 A Fixed Point Result in a Banach Algebra 100 2.6 Some Applications of the Measure of Noncompactness Concept 102 2.6.1 An Existence Result for a Class of Nonlinear Integral Equations of Fractional Orders 103 2.6.2 Solvability of an Implicit Fractional Integral Equation 113 2.6.3 q-Integral Equations of Fractional Orders 121 References 135 3 On Some Results Using Measures of Noncompactness 138 3.1 Introduction 138 3.2 Notation and Preliminaries 139 3.3 The Kuratowski Measure of Noncompactness 140 3.4 The Hausdorff Measure of Noncompactness 142 3.5 The Istrǎţesku Measure of Noncompactness 146 3.6 The Axiomatic Approach of Banaś and Goebel 147 3.7 Operators 150 3.8 An Equivalence Problem 153 3.9 Fredholm and Semi-Fredholm Operators 154 3.10 Applications of the Hausdorff Measure of Noncompactness to Operators Between BK Spaces 160 3.10.1 FK and BK Spaces 161 3.10.2 A Characterization of the Classes B(c) and K(c) 166 3.10.3 Matrix Transformations on Matrix Domains 169 3.10.4 Matrix Transformations on Spaces of Strongly Summable Sequences 176 3.10.5 Operators on the Spaces sα,sα°, sα (c) ,ellαp 178 References 186 4 Space of Functions with Growths Tempered by a Modulus of Continuity 192 4.1 Space of Functions with Growths Tempered by a Modulus of Continuity 192 4.2 Application to the Solvability of a Class of Fractional Boundary Value Problems 195 4.3 Measure of Noncompactness in the Space of Functions with Growths Tempered by a Modulus of Continuity 204 4.4 Sadovskii-Type Theorems in the Space of Functions with Growths Tempered by a Modulus of Continuity 206 4.5 Existence of Solutions of a Nonlinear Quadratic Integral Equation 211 References 225 5 Measure of Noncompactness in Functional Fractional Calculus 226 5.1 Introduction 226 5.2 The Basic Idea and Some Auxiliary Facts 227 5.2.1 Riemann--Liouville Integrals 228 5.2.2 Measure of Noncompactness 230 5.3 Cauchy Problem for Fractional Differential Equations in Banach Spaces 232 5.3.1 Imposed Conditions for Cauchy Problem 234 5.4 Attractivity of Solutions for Fractional Differential Equations 242 5.4.1 Attractivity of Solutions with Schauder Fixed Point Principle 244 5.5 Uniform Local Attractivity of Solutions with Measure of Noncompactness 247 5.6 Concrete Examples 253 References 256 6 Measures of Weak Noncompactness and Fixed Points 258 6.1 Introduction 258 6.2 The Axiomatic Measure of Weak Noncompactness 265 6.3 The De Blasi Measure of Weak Noncompactness 268 6.4 Fixed Point Theorems Under Weak Topology Features 277 6.5 Application 291 6.6 Fixed Point Theorems in Banach Algebras Relative to the Weak Topology 297 References 305 7 The Class of F-Contraction Mappings with a Measure of Noncompactness 308 7.1 Introduction 308 7.2 Notation and Preliminaries 309 7.3 F-Contractions of Hardy-Rogers-type 314 7.4 F-Weak Contractions 320 7.5 F-Contractions in Ordered Metric Spaces 325 7.6 F-Contractions of Darbo-Type 329 7.7 Solvability of Functional Equations via F-Contraction Mappings 332 7.8 Solvability of Integral Equations via F-Contraction Mappings 335 References 340 8 On the Measure of Noncompactness in Banach Spaces and Application to the Theory of Differential and Integral Equations 343 8.1 Kuratowski and Hausdorff Measures of Noncompactness in Banach Spaces 343 8.1.1 Properties and Examples of Kuratowski and Hausdorff MNC 344 8.2 Axiomatic Definition of MNC in Banach Spaces 345 8.2.1 Examples of MNCs in the Space of BC (mathbbR+) 346 8.3 Darbo's Fixed-Point Theorem and Its Generalizations 347 8.3.1 Common Fixed-Point Theorems via Measure of Noncompactness 350 8.4 Application of MNC in Integral and Differential Equations 354 8.5 Existence and Attractivity of Solutions of a Class of Nonlinear Integral Equations with Feedback Control 362 8.6 Positive Periodic Solution for a Nonlinear Neutral Delay Population Equation with Feedback Control 371 References 382 9 Partial Hadamard-Stieltjes Fractional Integral Equations in Banach Spaces 384 9.1 Introduction 384 9.2 Preliminaries 385 9.3 Existence Results 389 9.3.1 Examples 397 References 399 10 On the Aronszajn Property for Differential Equations of Fractional Order in Banach Spaces 401 10.1 Introduction 401 10.2 Notation, Definitions, and Auxiliary Facts 402 10.3 Aronszajn Type Theorems 404 10.4 On the Structure of Solution Sets of an Implicit Differential Equation of Fractional Order 410 10.5 Aronszajn Property for a Differential Equation of Order βin(0,infty) 414 10.6 On the Aronszajn Property for an Integro-differential Equation of Fractional Order in Banach Spaces 417 10.7 On the Convergence of Successive Approximations for a Fractional Differential Equation in Banach Spaces 421 References 425 11 On the Qualitative Behaviors of Nonlinear Functional Differential Systems of Third Order 428 11.1 Introduction 428 11.2 Stability 431 11.3 Boundedness 441 11.4 Ultimately Boundedness 443 References 444 12 On the Approximation of Solutions to a Fixed Point Problem with Inequality Constraints in a Banach Space Partially Ordered by a Cone 447 12.1 Introduction and Preliminaries 447 12.2 Main Results 449 12.3 Some Consequences 458 12.3.1 A Fixed Point Problem with One Equality Constraint 458 12.3.2 A Common Fixed Point Result 459 References 461 13 A Short Survey on Dislocated Metric Spaces via Fixed-Point Theory 462 13.1 Introduction and Preliminaries 462 13.1.1 (c)-Comparison Functions 467 13.1.2 (b)-Comparison Functions 467 13.1.3 Admissible Mappings 468 13.1.4 Simulation Functions 469 13.2 Fixed Point of α-ψ Contractive Mapping on Dislocated b-Metric Spaces 470 13.3 Consequences 473 13.3.1 Fixed-Point Theorems on Dislocated b-Metric Spaces Endowed with a Partial Order 474 13.3.2 Fixed-Point Theorems for Cyclic Contractive Mappings 475 13.3.3 Consequences on Standard b-Metric Spaces 476 13.3.4 Consequences on Standard Metric Spaces 477 13.3.5 Consequences on Standard Dislocated Metric Spaces 478 13.4 Generalized α-Admissible mathcalZ-Contraction 479 13.4.1 More Consequences on Dislocated Metric Spaces 484 13.4.2 Consequences on Standard Partial Metric Spaces 485 References 486 Index 489 Front Matter....Pages i-xiii Measures of Noncompactness in the Space of Continuous and Bounded Functions Defined on the Real Half-Axis....Pages 1-58 Measures of Noncompactness and Their Applications....Pages 59-125 On Some Results Using Measures of Noncompactness....Pages 127-180 Space of Functions with Growths Tempered by a Modulus of Continuity....Pages 181-214 Measure of Noncompactness in Functional Fractional Calculus....Pages 215-246 Measures of Weak Noncompactness and Fixed Points....Pages 247-296 The Class of F-Contraction Mappings with a Measure of Noncompactness....Pages 297-331 On the Measure of Noncompactness in Banach Spaces and Application to the Theory of Differential and Integral Equations....Pages 333-373 Partial Hadamard-Stieltjes Fractional Integral Equations in Banach Spaces....Pages 375-391 On the Aronszajn Property for Differential Equations of Fractional Order in Banach Spaces....Pages 393-419 On the Qualitative Behaviors of Nonlinear Functional Differential Systems of Third Order....Pages 421-439 On the Approximation of Solutions to a Fixed Point Problem with Inequality Constraints in a Banach Space Partially Ordered by a Cone....Pages 441-455 A Short Survey on Dislocated Metric Spaces via Fixed-Point Theory....Pages 457-483 Back Matter....Pages 485-487