Combined effects in some initial value problems involving Riemann–Liouville fractional derivatives in bounded domains

Abstract

We consider the following semilinear fractional initial value problem

$$D^{\alpha }u(x)=a_{1}(x)u^{\sigma _{1}}(x)+a_{2}(x)u^{\sigma _{2}}, \quad x\in (0,1) \quad and \quad \lim_{x\longrightarrow 0^{+}}x^{1-\alpha }u(x)=0,$$

where ${0 < \alpha < 1}$, ${\sigma _{1},\sigma _{2} \in (-1,1)}$ and ${a_{1},a_{2}}$ are positive measurable functions on ${(0,1]}$ satisfying appropriate assumptions related to Karamata regular variation theory. We establish the existence and the uniqueness of a positive solution in the space of weighted continuous functions. We also give the boundary behavior of such solution, where appear the combined effects of singular and sublinear terms in the nonlinearity.