Boundary Value Problems for Fractional Differential Equations and Systems

Description

This book is devoted to the study of existence of solutions or positive solutions for various classes of Riemann-Liouville and Caputo fractional differential equations, and systems of fractional differential equations subject to nonlocal boundary conditions. The monograph draws together many of the authors' results, that have been obtained and highly cited in the literature in the last four years. In each chapter, various examples are presented which support the main results. The methods used in the proof of these theorems include results from the fixed point theory and fixed point index theory. This volume can serve as a good resource for mathematical and scientific researchers, and for graduate students in mathematics and science interested in the existence of solutions for fractional differential equations and systems. Contents Preface Notations Chapter 1 Preliminaries 1.1 Fractional Integral and Fractional Derivatives 1.2 Fixed Point Theorems Chapter 2 Riemann–Liouville Fractional Differential Equations with Nonlocal Boundary Conditions 2.1 Singular Fractional Differential Equations with Parameters and Multi-Point Boundary Conditions 2.1.1 Auxiliary results 2.1.2 Existence of positive solutions 2.1.3 Examples 2.2 A Fractional Differential Equation with Integral Terms and Multi-Point Boundary Conditions 2.2.1 Existence of nonnegative solutions 2.2.2 An example 2.3 Semipositone Singular Fractional Boundary Value Problems with Integral Boundary Conditions 2.3.1 Preliminary results 2.3.2 Existence and multiplicity of positive solutions 2.3.3 An example 2.4 Singular Fractional Differential Equations with General Integral Boundary Conditions 2.4.1 Auxiliary results 2.4.2 Existence of multiple positive solutions 2.4.3 An example 2.5 On a Singular Fractional Boundary Value Problem with Parameters 2.5.1 Existence of positive solutions 2.5.2 Some remarks on a related semipositone problem 2.5.3 Examples 2.6 A Singular Fractional Differential Equation with Integral Boundary Conditions 2.6.1 Preliminary results 2.6.2 Existence and multiplicity of positive solutions 2.6.3 An example Chapter 3 Systems of Two Riemann–Liouville Fractional Differential Equations with Multi-Point Boundary Conditions 3.1 Systems of Fractional Differential Equations with Uncoupled Multi-Point Boundary Conditions 3.1.1 Auxiliary results 3.1.2 Existence and multiplicity of positive solutions 3.2 Systems of Fractional Differential Equations with Coupled Multi-Point Boundary Conditions 3.2.1 Preliminary results 3.2.2 Nonsingular nonlinearities 3.2.3 Singular nonlinearities 3.2.4 Examples Chapter 4 Systems of Two Riemann–Liouville Fractional Differential Equations with p-Laplacian Operators, Parameters and Multi-Point Boundary Conditions 4.1 Systems of Fractional Differential Equations with Uncoupled Multi-Point Boundary Conditions 4.1.1 Auxiliary results 4.1.2 Existence of positive solutions 4.1.3 Nonexistence of positive solutions 4.1.4 An example 4.1.5 A relation between two supremum limits 4.2 Systems of Fractional Differential Equations with Coupled Multi-Point Boundary Conditions 4.2.1 Preliminary results 4.2.2 Existence of positive solutions 4.2.3 Nonexistence of positive solutions 4.2.4 An example Chapter 5 Systems of Three Riemann–Liouville Fractional Differential Equations with Parameters and Multi-Point Boundary Conditions 5.1 Systems of Fractional Differential Equations with Uncoupled Multi-Point Boundary Conditions 5.1.1 Auxiliary results 5.1.2 Existence of positive solutions 5.1.3 Nonexistence of positive solutions 5.1.4 Examples Chapter 6 Existence of Solutions for Riemann–Liouville Fractional Boundary Value Problems 6.1 Riemann–Liouville Fractional Differential Equations with Nonlocal Boundary Conditions 6.1.1 Preliminary results 6.1.2 Existence of solutions 6.1.3 Examples 6.2 Systems of Riemann–Liouville Fractional Differential Equations with Uncoupled Boundary Conditions 6.2.1 Auxiliary results 6.2.2 Existence of solutions 6.2.3 Examples 6.3 Systems of Riemann–Liouville Fractional Differential Equations with Coupled Boundary Conditions 6.3.1 Preliminary results 6.3.2 Existence of solutions 6.3.3 Examples Chapter 7 Existence of Solutions for Caputo Fractional Boundary Value Problems 7.1 Sequential Caputo Fractional Differential Equations and Inclusions with Nonlocal Boundary Conditions 7.1.1 Auxiliary results 7.1.2 Existence of solutions for problem (7.1), (7.3) 7.1.3 Existence of solutions for problem (7.2), (7.3) 7.1.3.1 The upper semicontinuous case 7.1.3.2 The Lipschitz case 7.1.4 Examples 7.2 Sequential Caputo Fractional Integro-Differential Systems with Coupled Integral Boundary Conditions 7.2.1 Preliminary results 7.2.2 Existence of solutions 7.2.3 Examples 7.3 Caputo Fractional Differential Systems with Coupled Nonlocal Boundary Conditions 7.3.1 Auxiliary results 7.3.2 Existence of solutions 7.3.3 Examples Bibliography Index