POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

Abstract

We investigate the positive solution of nonlinear fractional differential equation with semi-positive nonlinearity \[ \begin{cases} D^\alpha_{0^+}u(t) + f(t, u(t)) = 0,\,\,\,\,\, 0 < t < 1,\\ u(0) = u'(1) = u''(0) = 0 \end{cases} \] where \(2 < \alpha\leq 3\) is a real number, \(D^\alpha_{0^+}\) is the Caputo's differentiation, and \(f : [0; 1] \times [0, \infty) \rightarrow (-\infty , \infty)\). By use of Krasnosel'skii fixed point theorem, the existence results of positive solution are obtained.