Existence of solution for boundary value problem of nonlinear fractional differential equation

Abstract

This paper is concerned with the boundary value problem of a nonlinear fractional differential equation. By means of Schauder fixed-point theorem, an existence result of solution is obtained. §1 Introduction Fractional differential equations have received considerable attention in the recent years due to their wide applications in engineering, economy and other fields. Many papers on fractional calculus, fractional differential equations have appeared. Most of them are devoted to the solvability of nonlinear fractional differential equations.Bai and Lü [1] discussed the existence and the multiplicity of positive solutions for boundary value problem of nonlinear fractional differential equationwhere 1 < α ≤ 2 is a real number, D α 0 + is the standard Riemann-Liouville differentiation, and f : [0, 1] × [0, ∞) → [0, ∞) is a continuous function. When the boundary values are not zero, Riemann-Liouville fractional derivative is not suitable. Therefore, in [2], the author investigated the existence and the multiplicity of positive solutions of the following problem:It is worthwhile to mention that the nonlinear term f in papers [1] and [2] is independent of fractional derivative of unknown function u(t). But the opposite case is more difficult and Received: 2006-11-15. MR Subject Classification: 34B15.

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