On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument

Abstract

This paper investigates stability and boundedness of solutions to third order nonlinear differential equation with retarded argument:By the use of the Lyapunov functional, sufficient conditions for stability and boundedness of solutions to the considered equations are obtained. Examples are introduced throughout the paper for illustrations.Keywords Stability · Boundedness · Lyapunov functional · Third order nonlinear differential IntroductionIt is well known that the systems with after-effect, with time lag or with delay are of great theoretical interest and form an important class, as regards their applications. This class of systems is described by functional differential equations, which are also called differential equations with deviating arguments.C. Tunç ( )

Bound of solutions to third-order nonlinear differential equations with bounded delay

“…Especially, despite the great number of results found on the bound of solutions to third-order nonlinear differential equations without delay, only a few results are available in the literature on the same subject related to third-order nonlinear delay differential equations. For relevant contributions to this study, the reader can refer to papers such as ( [12,13,16,17,18,19,21]) and the references cited in these papers. However, especially, since the 1960s many good books were published on delay differential equations (see, for example, books such as [1][2][3][4][5][6][7][8]20] and the references cited in these sources).…”

Stability and bounded of solutions to non-autonomous delay differential equations of third order

“…However, with respect to our observations in the literature, until now, many good results especially dealt with the stability and bounded of solutions of nonlinear third order differential equations without delay (see, for example, the book of Reissig et al [6], the papers of Tunç [8][9][10][11], Tunç and Ateş [19], and the references thereof). Besides, it is worth mentioning that to the best of our knowledge, in spite of the existence of many papers on the stability and bounded of solutions of nonlinear third order differential equations without delay, we found the papers of Sadek [7] and Tunç [12,[14][15][16][17][18] on the stability and boundedness of solutions to autonomous and non-autonomous third order differential equations with delay. Namely, first, in 2005, Sadek [7] discussed the asymptotic stability of the solution x = 0 to the following non-autonomous third order differential equation with constant delay, r :…”

“…This work is the first attempt in the literature on the bounded of solutions of non-autonomous third order differential equation with delay, (2), when a(t) = 1, b(t) = 1 and c(t) = 1. The motivation for the present work has been inspired basically by the papers of Sadek [7] and Tunç [12][13][14][15][16][17][18][19]. Throughout all foregoing papers, the Liapunov's second (or direct) method [5] has been used as a main tool to carry out the proofs of the main results thereof.…”

On the stability and boundedness of a class of higher order delay differential equations

“…It also follows that the equation studied here, Eq. (1), is different from that investigated in [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Finally, throughout all the above papers, the coefficient of the second term x 00 (t) in Eq.…”