Existence and stability of a positive solution for nonlinear hybrid fractional differential equations with singularity

Abstract

In this paper, we will study solution existence and its stability for hybrid fractional DE with fractional integral, fractional differential derivative and Φp-operator in Caputo sense. Our problem deals with two basic types of fractional order derivatives, that is, Riemann-Liouville derivative of order $\delta $ and Caputo fractional derivative of order $\lambda , $ where $n - 1 \lt \lambda ,\sigma \le n, $ and $n \ge 3. $ We will transform the problem into an integral equation by using Green function and examine whether it is increasing or decreasing and positive or negative function. Some fixed point theorems (Krasnoselskii Theorem) are utilized for the existence of a positive solution (EPS). Addition to studying HU-stability technique for our suggested problem. An example is included to apply the results.

Keywords

• fractional derivative
• caputo fractional derivative
• riemann-liouville
• existence of positive solution
• green function
• hyers-ulam stability
• hybrid fractional differential equation