Nontrivial solutions for boundary-value problems of nonlinear fractional differential equations
Bull. Korean Math. Soc. 2010 Vol. 47, No. 1, 81-87
https://doi.org/10.4134/BKMS.2010.47.1.81
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) $$\begin{aligned}
-{\bf D}_{0+}^\alpha u(t)=\lambda [f(t,u(t))+q(t)],\quad 0 < t < 1 \\
u(0)=u(1)=0,
\end{aligned}$$ 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
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