On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives

Abstract

Our main interest in this manuscript is to explore the main positive solutions (PS) and the first implications of their existence and uniqueness for a type of fractional pantograph differential equation using Caputo fractional derivatives with a kernel depending on a strictly increasing function $ \Psi $ (shortly $ \Psi $-Caputo). Such function-dependent kernel fractional operators unify and generalize several types of fractional operators such as Riemann-Liouvile, Caputo and Hadamard etc. Hence, our investigated qualitative concepts in this work generalise and unify several existing results in literature. Using Schauder's fixed point theorem (SFPT), we prove the existence of PS to this equation with the addition of the upper and lower solution method (ULS). Furthermore using the Banach fixed point theorem (BFPT), we are able to prove the existence of a unique PS. Finally, we conclude our work and give a numerical example to explain our theoretical results.

Keywords

• fractional differential equations
• fixed point theorem
• positive solution
• $ \psi $-caputo fractional derivative