ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

# Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

## New Conditions on the Solutions of a Certain Third Order Delay Differential Equations with Multiple Deviating Arguments

### Author(s):

Department of Mathematics, Faculty of Science
Obafemi Awolowo University, Post Code 220005 Ile-Ife, Nigeria

S. Moyo

Institute for Systems Science & Research and Postgraduate Support Directorate
Durban University of Technology, Durban 4000, South Africa

moyos@dut.ac.za

B. S. Ogundare

Department of Mathematics, Faculty of Science
Obafemi Awolowo University, Post Code 220005 Ile-Ife, Nigeria

bogunda@oauife.edu.ng

M. O. Ogundiran

Department of Mathematics, Faculty of Science
Obafemi Awolowo University, Post Code 220005 Ile-Ife, Nigeria

mogundiran@oauife.edu.ng

Department of Mathematics, Faculty of Science
Obafemi Awolowo University, Post Code 220005 Ile-Ife, Nigeria

### Abstract:

In this work we consider a class of third order delay differential equations, where the nonlinear functions, especially the first two restoring terms, are sum of n multiple deviating arguments, the forcing term depends explicitly on the independent variable t for all i in [1..n], the last restoring term has variable coefficient, and deviating arguments vary for all i. By employing the direct technique of Lyapunov, where a complete Lyapunov functional is constructed and used, we obtain sufficient conditions that guarantee the existence of solutions which are periodic, uniformly asymptotically stable, uniformly ultimately bounded. The behaviour of solutions as t tends to infinity is studied. The obtained results are new and include many recent results in the literature. Finally, two examples are given to show the feasibility of our results.

### Keywords

• periodic solutions
• Third order nonlinear differential equation
• uniform stability
• uniform ultimate boundedness

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