Existence and stability analysis for $\Phi$-Caputo generalized proportional fractional differential Langevin equation

Abstract

This paper aims to investigate the existence, uniqueness and stability results for a new class of $\Phi$-Caputo generalized proportional fractional $(\mathsf{GPF})$ differential Langevin equation. We present and discuss some of the characteristics of the generalized proportional fractional derivative which can be considered as generalization and modification of the fractional conformable derivative by generating $\Phi$-Caputo generalized proportional fractional derivatives involving exponential functions in it's kernel also this kind of fractional derivative generalize the well-known fractional derivatives, for different values of function $\Phi$. Utilizing Krasnoselskii's fixed point theorem and the Banach contraction principle, we established results on existence and uniqueness, we also examine various types of stability, including Ulam-Hyers stability and generalized Ulam-Hyers stability. As an application, we provide an example to illustrate our theoretical result.

Keywords

• $\Phi$-Caputo generalized proportional fractional derivative
• $\Phi$-Caputo fractional differential Langevin equations
• fixed point theorems
• Ulam-Hyers stability

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