Qualitative analysis of integro-differential equations with variable retardation

Martin Bohner; Osman Tunç

doi:10.3934/dcdsb.2021059

Article Contents

2022, Volume 27, Issue 2: 639-657. Doi: 10.3934/dcdsb.2021059

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Qualitative analysis of integro-differential equations with variable retardation

Martin Bohner^1, , and
Osman Tunç^2,

1.
Missouri S & T, Department of Mathematics and Statistics, Rolla, MO 65409-0020, USA
2.
Van Yuzuncu Yil University, Department of Mathematics, Faculty of Sciences, Van, 65080, Turkey

^* Corresponding author: Martin Bohner
^* Corresponding author: Martin Bohner

Received: May 31, 2020

Revised: December 31, 2020

Early access: February 2021

Published: February 2022

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Abstract

The paper is concerned with a class of nonlinear time-varying retarded integro-differential equations (RIDEs). By the Lyapunov–Krasovski$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ı} $ functional method, two new results with weaker conditions related to uniform stability (US), uniform asymptotic stability (UAS), integrability, boundedness, and boundedness at infinity of solutions of the RIDEs are given. For illustrative purposes, two examples are provided. The study of the results of this paper shows that the given theorems are not only applicable to time-varying linear RIDEs, but also applicable to time-varying nonlinear RIDEs.

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Mathematics Subject Classification: 34D05, 34K20, 45J05.

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Figure 1. Solution $ x_1 $ of system of RIDEs (9) for different initial values

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Figure 2. Solution $ x_2 $ of system of RIDEs (9) for different initial values

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Figure 3. Bounded solution $ x_1 $ of system of RIDEs (11) for different initial values

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Figure 4. Bounded solution $ x_2 $ of system of RIDEs (11) for different initial values

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