**Title:** On Ulam-Hyers-Rassias Stability for Mild Solutions of a Two-time Dynamical System

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-20-00004; **Volume 2 / Issue 1 / Year 2020**, Pages 69-80

**Document Type:** Research Paper

**Author(s):** Mohamed Akkouchi ^{a} , Mohamed Amine Ighachane ^{b} , My, Hicham, Lalaoui Rhali ^{c}

^{a}Department of Mathematics, Faculy of Sciences-Semlalia, University Cadi Ayyad, Av. Prince My. Abdellah, BP: 2390, Marrakesh (40.000-Marrakech), Morocco (Maroc)

^{b}Department of Mathematics, Faculy of Sciences-Semlalia, University Cadi Ayyad, Av. Prince My. Abdellah, BP: 2390, Marrakesh (40.000-Marrakech), Morocco (Maroc)

^{c}Department of Mathematics, Faculy of Sciences-Semlalia, University Cadi Ayyad, Av. Prince My. Abdellah, BP: 2390, Marrakesh (40.000-Marrakech), Morocco (Maroc)

Received: 13 February 2020, Accepted: 26 March 2020, Available online: 9 May 2020.

**Corresponding Author:** Mohamed Akkouchi (Email address: akkm555@yahoo.fr)

**Full Text:** PDF

**Abstract**

In this work, we introduce the concept of mild solution for an abstract Cauchy problem of non-homogeneous type governed by the generator of a two-parameter *C*_{0}-semigroup on a real or complex Banach space *X*. Precisely we are concerned by the following two-time dynamical system:

for all (*s*, *t*)∈[*s*_{0}, *S*]×[*t*_{0}, *T*], where *s*_{0} ≤ *S* ≤ +∞ and *t*_{0} ≤ *T* ≤ +∞. Under certain conditions on the functions *u*_{1}, *u*_{2}, *F*_{1} and *F*_{2}, we investigate the generalized stability in the sense of Ulam, Hyers and Rassias of these mild solutions. Our approach to stablity is based on the fixed point method.

**Keywords:** Two-parameter semigroups, Two-time dynamical systems, Mild solutions, Stability, Ulam-Hyers-Rassias

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