On the Existence and Uniqueness of Solution of A Nonlinear Fractional Differential Equations

Abstract

In this paper, we investigate the existence and uniqueness of solution for fractional boundary value problem for nonlinear fractional differential equation

D-0+(alpha) u(t) = f(t,u(t)), 0 < t < 1, 2 < alpha <= 3,

with the integral boundary conditions

{u(0) - gamma(1) u(1) = lambda(1) integral(1)(0) g(1) (s, u(s))ds,

u'(0) - gamma(2)u'(1) = lambda(2) integral(1)(0) g(2) (s, u(s))ds,

u ''(0) - gamma(2)u ''(1) = 0,

where D-0+(alpha) denotes Caputo derivative of order alpha. by using the fixed point theory. We apply the contraction mapping principle and Krasnoselskii's fixed point theorem to obtain some new existence and uniqueness results. Two examples are given to illustrate the main results.