Analysis of positive continuous solutions for semi-linear fractional Navier boundary value problems

Abstract

In this paper, we study the existence, uniqueness, and behavior of asymptotic nonnegative  continuous solutions to the  following  fractional Navier boundary-value problem:  $$ \begin{gathered}
        ^cD^\beta\left(^cD^\nu u\right)(x)=-q (x) u^\gamma, \quad x\in(0,1), \\
        \lim _{x \rightarrow 0^+} { }^cD^\nu u(x)=0, \quad u(1)=0,
    \end{gathered} $$  where $\nu, \beta \in(0,1]$ such that $\nu+\beta>1,{ }^cD^\beta$ and $^cD^\alpha$ denote  the Caputo fractional derivatives. The exponent  $\gamma \in(-1,1)$ and $q$ is a nonnegative continuous function in $(0,1)$, which may exhibit a singularity at $x=0$ and satisfies certain conditions derived from Karamata's theory of regular variation. The main results are established via the application of the Schäuder  fixed point theorem.