Nontrivial solutions for boundary-value problems of nonlinear fractional differential equations
Bull. Korean Math. Soc. 2010 Vol. 47, No. 1, 81-87
Printed January 1, 2010
Yingxin Guo
Qufu Normal University
Abstract : In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem (BVP) $$\begin{aligned}
-{\bf D}_{0+}^\alpha u(t)=\lambda [f(t,u(t))+q(t)],\quad 0 < t < 1 \\
\end{aligned}$$ where $\lambda >0$ is a parameter, $1 < \alpha \leq 2$, $\mathbf{D}_{0+}^\alpha$ is the standard Riemann-Liouville differentiation, $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous, and $q(t):(0,1)\to [0, +\infty)$ is Lebesgue integrable. We obtain serval sufficient conditions of the existence and uniqueness of nontrivial solution of BVP when $\lambda$ 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
MSC numbers : 34B15, 34B05, 26A33
Downloads: Full-text PDF  

Copyright © Korean Mathematical Society. All Rights Reserved.
The Korea Science Technology Center (Rm. 411), 22, Teheran-ro 7-gil, Gangnam-gu, Seoul 06130, Korea
Tel: 82-2-565-0361  | Fax: 82-2-565-0364  | E-mail:   | Powered by INFOrang Co., Ltd