A singular boundary value problem for nonlinear differential equations of fractional order

Abstract

We are concerned with the nonlinear differential equation of fractional order $$\mathcal{D}^{\alpha}_{0+}u(t)=f(t,u(t),u'(t)),\quad \mbox{a.\,e.}\ t\in (0,1),$$ where \(\mathcal{D}^{\alpha}_{0+}\) is the Riemann-Liouville fractional order derivative, subject to the boundary conditions $$u(0)=u(1)=0.$$ We obtain the existence of at least one solution using the Leray-Schauder Continuation Principle.