Positive solutions for boundary-value problems of nonlinear fractional differential equations. (English) Zbl 1096.34016

Zhang, Shuqin

Electron. J. Differ. Equ. 2006, Paper No. 36, 12 p. (2006).

Summary: We consider the existence and multiplicity of positive solutions for the nonlinear fractional differential equation boundary value problem \[ \mathbf{D}_{0+}^\alpha u(t)=f(t,u(t)),\quad 0<t<1,\quad u(0)+u'(0)=0,\quad u(1)+u'(1)=0, \] where \(1<\alpha\leq 2\) is a real number, and \(\mathbf{D}_{0+}^\alpha\) is the Caputo 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 on positive solutions are obtained.

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
26A99	Functions of one variable

Keywords:

Caputo’s fractional derivative; fractional differential equation; boundary value problem; positive solution; fractional Green’s function; fixed-point theorem