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

Title & Authors
NONTRIVIAL SOLUTIONS FOR BOUNDARY-VALUE PROBLEMS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS
Guo, Yingxin;

Abstract
In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem(BVP) $\small{-D_0^{\alpha}+u(t)=\lambda[f(t, u(t))+q(t)]}$, 0 < t < 1 u(0) = u(1) = 0, where $\small{\lambda}$ > 0 is a parameter, 1 < $\small{\alpha}$ $\small{\leq}$ 2, $\small{D_{0+}^{\alpha}}$ is the standard Riemann-Liouville differentiation, f : [0, 1] $\small{{\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}}$ is continuous, and q(t) : (0, 1) $\small{\rightarrow}$ [0, $\small{+\infty}$] is Lebesgue integrable. We obtain serval sufficient conditions of the existence and uniqueness of nontrivial solution of BVP when $\small{\lambda}$ 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.
Keywords
standard Riemann-Liouville differentiation;fractional differential equation;boundary-value problem;nontrivial solution;Leray-Schauder nonlinear alternative;
Language
English
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