Stability and square integrability of derivatives of solutions of nonlinear fourth order differential equations with delay

Erdal Korkmaz

doi:10.1186/s13660-017-1353-8

Abstract

In this paper, we give sufficient conditions for the boundedness, uniform asymptotic stability and square integrability of the solutions to a certain fourth order non-autonomous differential equations with delay by using Lyapunov’s second method. The results obtained essentially improve, include and complement the results in the literature.

Highlights

In mathematical literature, ordinary differential equations have been studied for more than years since the seventeenth century after the concepts of differentiation and integration were formulated by Newton and Leibniz
The question concerning the stability of ordinary differential equations has been originally raised by the general problem of the stability of motion [ ]
In many applications, it can be seen that physical or biological background of a modeling system shows that the change rate of the system’s current status often depends on the current state and on the history of the system. This usually leads to the so-called retarded functional differential equations [ ]

Summary

Introduction

Ordinary differential equations have been studied for more than years since the seventeenth century after the concepts of differentiation and integration were formulated by Newton and Leibniz. Of ordinary or functional differential equations of fourth order, see the book of Reissig et al [ ] as a good survey for the works done by and the papers of Burton [ ], Cartwright [ ], Ezeilo [ – ], Harrow [ , ], Tunç [ – ], Remili et al [ – ], Wu [ ] and others and the references therein. Our results differ from those obtained in the literature (see, [ – ] and the references therein) By this way, we mean that this paper has a contribution to the subject in the literature, and it may be useful for researchers working on the qualitative behaviors of solutions of functional differential equations of higher order. The zero solution of equation ( ) is uniformly asymptotically stable

The main results

Conclusion