# Article ID: MTJPAM-D-20-00004

## 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

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

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

cDepartment 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 C0-semigroup on a real or complex Banach space X. Precisely we are concerned by the following two-time dynamical system:

$(ACP(2)): \begin{cases} \frac{\partial\psi(s,t)}{\partial s} =A_{1}\psi(s,t)+u_{1}(s,t)F_{1}(s,\psi(s,t)),\,\, \\ \frac{\partial\psi(s,t)}{\partial t} =A_{2}\psi(s,t)+u_{2}(s,t)F_{2}(t,\psi(s,t)),\,\, \\ \psi(s_{0},t_{0})=x_{0}\in X, \end{cases}$

for all (s, t)∈[s0, S]×[t0, T], where s0 ≤ S ≤ +∞ and t0 ≤ T ≤ +∞. Under certain conditions on the functions u1, u2, F1 and F2, 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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