Existence of Solutions for Nonlocal Boundary Value Problems of Higher-Order Nonlinear Fractional Differential Equations

Bashir Ahmad,Juan J Nieto

doi:10.1155/2009/494720

Abstract

We study some existence results in a Banach space for a nonlocal boundary value problem involving a nonlinear differential equation of fractional orderqgiven bycDqx(t)=f(t,x(t)),0<t<1,q∈(m−1,m],m∈ℕ,m≥2, x(0)=0, x′(0)=0, x′′(0)=0,…,x(m−2)(0)=0,x(1)=αx(η). Our results are based on the contraction mapping principle and Krasnoselskii's fixed point theorem.

Highlights

Fractional differential equations involve derivatives of fractional order
For a function g : 0, ∞ → R, the Caputo derivative of fractional order q is defined as cDqg t t t − s n−q−1g n s ds, n − 1 < q < n, n q 1, 2.1
We remark that the Caputo derivative becomes the conventional nth derivative of the function as q → n and the initial conditions for fractional differential equations retain the same form as that of ordinary differential equations with integer-order derivatives

Summary

Introduction

Fractional differential equations involve derivatives of fractional order. They arise in many engineering and scientific disciplines such as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electro-dynamics of complex medium, and polymer rheology. For a function g : 0, ∞ → R, the Caputo derivative of fractional order q is defined as cDqg t t t − s n−q−1g n s ds, n − 1 < q < n, n q 1, 2.1 The Riemann-Liouville fractional integral of order q is defined as The Riemann-Liouville fractional derivative of order q for a function g t is defined by

Results

Conclusion