Existence of positive solutions for singular fractional differential equations. (English) Zbl 1172.34313

Qiu, Tingting; Bai, Zhanbing

Electron. J. Differ. Equ. 2008, Paper No. 146, 9 p. (2008).

From the introduction: We discuss the existence of a positive solution to boundary-value problems of the nonlinear fractional differential equation
\[ D_{0^+}^\alpha u(t)+f(t,u(t))=0, \quad 0<t<1, \qquad u(0)=u'(1)=u''(0)=0, \]
where \(2<\alpha\leq 3\), \(D_{0^+}^\alpha\) is the Caputo’s differentiation, and \(f:(0,1]\times[0,1)\to [0,1)\) with \(\lim_{t\to0^+}f(t,\cdot)=+\infty\) (that is \(f\) is singular at \(t=0\)). We obtain two results about this boundary-value problem, by using Krasnoselskii’s fixed point theorem and nonlinear alternative of Leray-Schauder type in a cone.

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34B16	Singular nonlinear boundary value problems for ordinary differential equations
26A33	Fractional derivatives and integrals
47N20	Applications of operator theory to differential and integral equations

Keywords:

boundary value problem; positive solution; singular fractional differential equation; fixed-point theorem