Stability Results for Implicit Fractional Pantograph Differential Equations via ϕ-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition

Abstract

This paper presents a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition. A certain class of generalized fractional derivative is used to set the problem. The existence and uniqueness of the problem is obtained using Schaefer’s and Banach fixed point theorems. In addition, the Ulam-Hyers and generalized Ulam-Hyers stability of the problem are established. Finally, some examples are given to illustrative the results.

Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

“…Using the Banach and Krasnoselskii's fixed point theorems, we have established the existence and uniqueness of the solution for fractional pantograph differential equation with impulsive and generalized anti-periodic boundary conditions. We note that it would be interesting to study this kind of problem for a certain kind of generalized fractional derivatives and integrals [7,25]. In addition, this is the first paper, to the best of our knowledge, dealing with a fractional pantograph differential equation with an impulsive and generalized anti-periodic boundary conditions.…”

Existence and uniqueness results for Φ-Caputo implicit fractional pantograph differential equation with generalized anti-periodic boundary condition

“…Case 4: Suppose α 1 = α 2 > 1 and (t) = t, then the Green's function defined in (3.30) takes the form (3.36), if there exists a solution of (1.5), it is unique on J . Comparing (1.4) with (3.37) yields r = 7 3 , p = 1 3 , q = 5 3 , ξ = 1, and σ = 2 7 . Also from the boundary conditions we can see that α 1 = 2 and α 2 = 4.…”

“…Recently, Idris et al [7] examined the implicit fractional pantograph differential equation given by Banach's fixed point theorem techniques. Besides, two different kinds of stability were discussed in the context of Ulam-Hyers and the generalized Ulam-Hyers theory.…”

Study on Krasnoselskii’s fixed point theorem for Caputo–Fabrizio fractional differential equations

“…But recently another form of derivative, called nonsingular type, has attracted much attention from the researchers. The existence theory, together with stability results, has been very well investigated for other FODEs; for details, see [8][9][10]. The considered differential operator has been introduced in 2015 by Caputo and Fabrizio [11] (in short, we write it as (CFFD)), which replaces the singular kernel by a nonsingular kernel of exponential type.…”