Existence, uniqueness, and Mittag-Leffler-Ulam stability results for Cauchy problem involving \(\psi \)-Caputo derivative in Banach and Fréchet spaces
type: Item
instance of: scholarly article
title: Existence, uniqueness, and Mittag-Leffler-Ulam stability results for Cauchy problem involving \(\psi \)-Caputo derivative in Banach and Fréchet spaces
zbMATH Open document ID: 1470.34012
DOI: 10.1155/2020/6383916
author: Z. Baitiche
author: Mouffak Benchohra
author: Choukri Derbazi
author: G. M. N'Guérékata
published in: International Journal of Differential Equations
publication date: 2021-05-19T00:00:00Z
review text: Summary: Our aim in this paper is to investigate the existence, uniqueness, and Mittag-Leffler-Ulam stability results for a Cauchy problem involving \(\psi \)-Caputo fractional derivative with positive constant coefficient in Banach and Fréchet Spaces. The techniques used are a variety of tools for functional analysis. More specifically, we apply Weissinger's fixed point theorem and Banach contraction principle with respect to the Chebyshev and Bielecki norms to obtain the uniqueness of solution on bounded and unbounded domains in a Banach space. However, a new fixed point theorem with respect to Meir-Keeler condensing operators combined with the technique of Hausdorff measure of noncompactness is used to investigate the existence of a solution in Banach spaces. After that, by means of new generalizations of Gronwall's inequality, the Mittag-Leffler-Ulam stability of the proposed problem is studied on a compact interval. Meanwhile, an extension of the well-known Darbo's fixed point theorem in Fréchet spaces associated with the concept of measures of noncompactness is applied to obtain the existence results for the problem at hand. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility of the main theorems.
Mathematics Subject Classification ID: 34A08
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 34D10
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 7349543
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2020/6383916
OpenAlex ID: W3097630799
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