Global Existence for an Implicit Hybrid Differential Equation of Arbitrary Orders with a Delay

Abstract

In this paper, we present a qualitative study of an implicit fractional differential equation involving Riemann–Liouville fractional derivative with delay and its corresponding integral equation. Under some sufficient conditions, we establish the global and local existence results for that problem by applying some fixed point theorems. In addition, we have investigated the continuous and integrable solutions for that problem. Moreover, we discuss the continuous dependence of the solution on the delay function and on some data. Finally, further results and particular cases are presented.

Keywords

• fractional differential equations
• hybrid differential equations
• Rothe fixed-point theorem
• nonlinear alternative of Leray–Schauder type
• Kolmogorov compactness criterion
• continuous dependence