On nonlinear pantograph fractional differential equations with Atangana–Baleanu–Caputo derivative

Abstract

In this paper, we obtain sufficient conditions for the existence and uniqueness results of the pantograph fractional differential equations (FDEs) with nonlocal conditions involving Atangana–Baleanu–Caputo (ABC) derivative operator with fractional orders. Our approach is based on the reduction of FDEs to fractional integral equations and on some fixed point theorems such as Banach’s contraction principle and the fixed point theorem of Krasnoselskii. Further, Gronwall’s inequality in the frame of the Atangana–Baleanu fractional integral operator is applied to develop adequate results for different kinds of Ulam–Hyers stabilities. Lastly, the paper includes an example to substantiate the validity of the results.

Existence and uniqueness of periodic solutions for some nonlinear fractional pantograph differential equations with $$\psi $$-Caputo derivative

“…For υ ∈ Img L, there exists u ∈ Dom L, such that υ = Lu ∈ Y. From Lemma 2.5, we obtain that for every Lemma 3.2 Let L be defined by (3). Then, L is a Fredholm operator of index zero, and the linear continuous projector operators Q : Y → Y and P : X → X can be written as…”

“…Pantograph equations have been widely used in the fields of quantum mechanics and dynamical system [20,21]. Several researchers have investigated some new existence and uniqueness results for NFDE pantograph models and others by applying fixed point theorems, the nonlinear alternative on cones, or coincidence degree theory [3,[8][9][10][11][12].…”

Analysis of a fractional boundary value problem involving Riesz-Caputo fractional derivative

“…For the development of FC, there are sundry common denitions of fractional derivatives and integrals, such as Rimann-Liouville type, Caputo type, Hadamard type, Hilfer type, ψ-Caputo, ψ-Hilfer type, Caputo-Fabrizio type, Atangana-Baleanu type, conformable type, and Erdelyi-Kober type, etc, (see [11,15,16,25,32,33,37,3]). Some recent contributions have been investigated the existence and uniqueness of solutions for dierent kinds of nonlinear fractional dierential equations (FDEs) and inclusion (FDIs) by using various types of xed point theorems, which can be found in [13,6,17,7,39,38,8,19,20,21,4,5,1,2,18], and the references cited therein. The study of FDEs or FDIs with anti-periodic boundary conditions, that are applied in numerous dierent elds, like chemical engineering, physics, economics, dynamics, etc., has received much attention recently, (see [23,27,40,10,26]) and the papers mentioned therein.…”

Stability of solutions for generalized fractional differential problems by applying significant inequality estimates

“…A generalized fractional derivative with respect to function t ρ is a novel sort of fractional derivatives, which has been presented by Kilbas et al [4], and then modified by Katugampola [5] and Almeida et al [6]. In a series of papers [7][8][9][10][11][12][13][14][15][16][17][18] the authors studied the qualitative analysis for some classes of fractional differential equations involving this generalized fractional derivative.…”