Fractional Integro-Differential Equations With Nonlocal Conditions and Ψ–hilfer Fractional Derivative

Abstract

Considering a fractional integro-differential equation with nonlocal conditions involving a general form of Hilfer fractional derivative with respect to another function. We show that weighted Cauchy-type problem is equivalent to a Volterra integral equation, we also prove the existence, uniqueness of solutions and Ulam-Hyers stability of this problem by employing a variety of tools of fractional calculus including Banach fixed point theorem and Krasnoselskii's fixed point theorem. An example is provided to illustrate our main results.

Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses

“…(2.1), and we prove that x also satisfies the fractional system (1.4). The following proof process is similar to the relevant conclusion, and we can refer to Lemma 3.1 in [46]. The proof of this lemma is completed.…”

“…[44,46]) The ψ-Riemann-Liouville fractional integral and fractional derivative of order α (n -1 < α < n) for an integrable function Φ : [a, b] → R with respect to another function ψ : [a, b] → R that is an increasing differentiable function such that ψ (t) = 0, for all t ∈ [a, b] (-∞ ≤ a < b ≤ +∞) are defined as follows:…”

“…[44,46]) Let n -1 < α < n with n ∈ N + , I = [a, b] be the interval such that -∞ ≤ a < b ≤ +∞ and ψ ∈ C n ([a, b], R) be a function such that ψ is increasing and ψ (t) = 0, for all t ∈ I. The ψ-Hilfer functional derivative of the function Φ ∈ C n ([a, b], R) of order α and the type 0 ≤ β ≤ 1 is defined by H D α,β;ψ t…”

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

“…The main advantages of these operator is the freedom of choice of the function ψ and its merge and acquire the properties of the aforementioned fractional operators. Results based on these setting can be found in [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. The Ulam-Hyers stability point of view, is the vital and special type of stability that attracts many researchers in the field of mathematical analysis.…”

“…Motivated by the papers [21,47,48] and some familiar results on fractional pantograph differential equations [16,52,55,58]. We discuss the existence and uniqueness of the solution of the implicit pantograph fractional differential equations involving φ-Hilfer fractional derivatives.…”