An existence results for a fractional differential equation with Φ-fractional derivative

Abstract

In this article, we establish certain sufficient conditions to show the existence of solutions of a fractional differential equation with the ?-Riemann-Liouville and ?-Caputo fractional derivative in a special Banach space. Our approach is based on fixed point theorems for Meir-Keeler condensing operators via measure of non-compactness. Also an example is given to illustrate our approach.

Mathematical analysis of impulsive fractional differential inclusion of pantograph type

“…1 There exist several approaches to fractional derivatives such as, Caputo fractional derivatives, Reimann-Liouville, Hadamard-Grunwald-Lethikov, and Weyl. [2][3][4][5] One of the most important classes of fractional differential equation is impulsive fractional order differential equation. Various phenomena can be modeled through it such as in physics, engineering, chemistry, and biology.…”

“…It plays a vital role in modeling of different fields such as electronics, electromagnetism, electrochemistry, thermal engineering, mechanics, biophysics, rheology, automatic control, mechatronics, telecommunications, robotics, signal processing, biology, economics, electrical engineering, image processing, and physics 1 . There exist several approaches to fractional derivatives such as, Caputo fractional derivatives, Reimann–Liouville, Hadamard–Grunwald–Lethikov, and Weyl 2–5 …”