NONTRIVIAL SOLUTIONS FOR BOUNDARY-VALUE PROBLEMS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

Abstract

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem(BVP) <TEX>$-D_0^{\alpha}+u(t)=\lambda[f(t, u(t))+q(t)]$</TEX>, 0 < t < 1 u(0) = u(1) = 0, where <TEX>$\lambda$</TEX> > 0 is a parameter, 1 < <TEX>$\alpha$</TEX> <TEX>$\leq$</TEX> 2, <TEX>$D_{0+}^{\alpha}$</TEX> is the standard Riemann-Liouville differentiation, f : [0, 1] <TEX>${\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$</TEX> is continuous, and q(t) : (0, 1) <TEX>$\rightarrow$</TEX> [0, <TEX>$+\infty$</TEX>] is Lebesgue integrable. We obtain serval sufficient conditions of the existence and uniqueness of nontrivial solution of BVP when <TEX>$\lambda$</TEX> in some interval. Our approach is based on Leray-Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity which was essential for the technique used in almost all existed literature on f.