Positive solutions for boundary-value problems of nonlinear fractional differential equations

Zhang, Shuqin

Electronic Journal of Differential Equations, Tome 2006 (2006).

Résumé

Summary: In this paper, we consider the existence and multiplicity of positive solutions for the nonlinear fractional differential equation boundary-value problem $$\displaylines{ \hbox{\bf D}_{0+}^\alpha u(t)=f(t,u(t)),\quad 0 less than t less than 1\cr u(0)+u'(0)=0,\quad u(1)+u'(1)=0 }$$ where $1 less than \alpha\leq 2$ is a real number, and $\hbox{\bf D}_{0+}^\alpha$ is the Caputo's fractional derivative, and $f:[0,1]\times[0,+\infty)\to [0,+\infty)$ is continuous. By means of a fixed-point theorem on cones, some existence and multiplicity results of positive solutions are obtained.

Détail

Classification : 34B15
Keywords: Caputo's fractional derivative, fractional differential equation, boundary-value problem, positive solution, fractional Green's function, fixed-point theorem