Existence of Symmetric Positive Solutions for a Caputo Fractional Singular Boundary Value Problem

TL;DR

This paper proves the existence of symmetric positive solutions for a Caputo fractional boundary value problem with singularities at the boundary points, expanding understanding of fractional differential equations with singular nonlinearities.

Contribution

It establishes the existence of symmetric positive solutions for a specific Caputo fractional boundary value problem with boundary singularities, which was not previously addressed.

Findings

Existence of symmetric positive solutions proven.

Applicable to fractional differential equations with boundary singularities.

Provides new methods for handling singular nonlinearities in fractional calculus.

Abstract

In this article, we establish the symmetric positive existence for the following Caputo fractional boundary value problem \begin{align*} {}^{C}D_{0}^{\,\mu}x(t)+f(t,x(t))&=0,\hspace{1cm}t\in(-1,\,1),\hspace{1cm}1<\mu\leq2,\\ x(\pm1)=x'(0^{\pm})&=0, \end{align*} where ${}^{C}D_{0}^{\,\mu}x(t)={}^{C}D_{0^{+}}^{\,\mu}x(t)$ for $t\geq0$, ${}^{C}D_{0}^{\,\mu}x(t)={}^{C}D_{0^{-}}^{\,\mu}x(t)$ for $t\leq0$. Moreover, $f:(-1,\,1)\times(0,\infty)\rightarrow\mathbb{R}$ is continuous and singular at $t=-1$, $t=1$ and $x=0$. Here, ${}^{C}D_{0^{+}}^{\,\mu}$ and ${}^{C}D_{0^{-}}^{\,\mu}$, respectively, are Caputo fractional left and right derivatives of order $\mu$.