On the qualitative properties of differential equations of third order with retarded argument

Abstract

By using the standard Liapunov-Krasovskii functional approach, in this paper, new stability, boundedness and ultimately boundedness criteria are established for a class of vector functional differential equations of third order with retarded argument.

Stability and boundedness in delay system of differential equations of third order

“…(1) using the Lyapunov-functional approach. By this paper, we will extend and improve the results of Omeike (2014), Tunc (2009b), Tunc and Mohammed (2014), Zhang and Yu (2013). This is the novelty of this work.…”

“…During the last years, many good results have been obtained on the qualitative behaviors in ordinary and functional differential equations of third order without and with delay. In particular, for some works on the stability and boundedness in scalar ordinary and functional differential equations of third order without and with delay, we referee the interested reader to the papers of Ademola et al (2015), Ademola and Arawomo (2011), Afuwape and Castellanos (2010), , Graef and Tunc (2015), Mehri and Shadman (1999), Meng (1993), Omeike (2014), Omeike and Afuwape (2010), Qian (2000), Remili and Oudjedi (2014), Tunc (2004Tunc ( , 2005aTunc ( ,b,c, 2007Tunc ( , 2009aTunc ( ,b, 2010aTunc ( ,b, 2013aTunc ( ,b, 2014Tunc ( , 2015, Tunc and Mohammed (2014), Tunc and Ates ß (2006), Zhang and Yu (2013) and their references. However, to the best of our knowledge from the literature, by this time, little attention was given to the investigation into the stability/boundedness/ultimately boundedness in vector functional differential equations of third order with delay (see Tunc and Mohammed (2014)).…”

“…Recently, the authors in Tunc and Mohammed (2014) discussed the stability and boundedness in non-linear vector differential equation of third order with constant delay s 1 > 0:…”

“…The motivation of this paper comes from the results established in Datko (1994), De la Sen (1988a,b), De la Sen and Luo (2004), Omeike and Afuwape (2010), Qian (2000), Tunc and Mohammed (2014), Zhang and Yu (2013), the mentioned papers and their references. The main purpose of this paper is to get some new stability/boundedness/ultimately boundedness results in Eq.…”

“…We say that V : C ! R is a Lyapunov function on a set G & C relative to f if V is continuous on G, the closure of G, _ V is defined on G, and _ V 6 0 on G. The following form of the LaSalle's invariance principle can be found in Tunc and Mohammed (2014).…”