New results on exponential stability of nonlinear Volterra integro-differential equations with constant time-lag

Abstract

In the present work, we pay attention to a number of nonlinear Volterra integro-differential equations (VIDEs) with constant timelag. We define three new Lyapunov functionals (LFs) and employ them to get specific conditions guaranteeing the uniform exponential asymptotic stability (UEAS) of the trivial solutions of the (VIDEs) considered. The results obtained generalize, compliment and improve the existing results in the literature from the cases of the without delay to the more general cases with time-lag.Subjclass [2010] : 34D05, 34K20, 45J05.

New Exponential Stability Criteria for Certain Neutral Integro-Differential Equations

“…Therefore, qualitative behaviors of solutions NIDEs, stability, boundedness, convergence, instability, integrability, globally existence of solutions, etc., have been extensively investigated in the literature by this time. For a comprehensive treatment of these qualitative properties of solutions of NIDEs and some applications, we refer the readers to the papers or books of Hale et al (1993), Boyd et al (1994), Agarwal and Grace (2000), El-Morshedy and Gopalsamy (2000), Fridman (2001), Fridman (2002), Gu et al (2003), Park (2004), Kwon and Park (2006), Sun and Wang (2006), Kwon and Park (2008), Park and Kwon (2008), Deng et al (2009), Liao et al (2009), Li New Exponential Stability Criteria for Certain Neutral Integro-Differential Equations 120 (2009), Nam and Phat (2009), Rojsiraphisal and Niamsup (2010), Chen et al (2011), Chen and Huabin (2012), Tunç and Altun (2012), Tunç (2013), Pinjai and Mukdasai (2013), Li and Fu (2013), Keadnarmol et al (2014), Chatbupapan et al (2016), Gözen and Tunç (2017a and b), Tunç and Mohammed (2017), Gözen and Tunç (2018), Tunç and Tunç (2018), Hristova and Tunç (2019), Slyn'ko and Tunç (2019) and the references can be found in these sources. In a particular case, we should mention the following related paper on the qualitative behaviors of the solutions of NIDEs .…”

A comparison between applications of the Lyapunov’s second (direct) method and fixed point theory

“…There are many doings achieved on the specific conduct of mentioned differential equations, According to Lyapunov second method fixed point theories, readers can judge Becker and Burton's books or papers [1], Burton [11,2,3,4,5,6,7], Burton and Furumochi [8], Burton and Hering [9], Burton and Townsend [10], Chen et al [12], Cherkas and Malysheva [13], Hou and Wu [14], Jin [15], Jitsuro and Yusuke [16], Liu and Huang [18], Npoles Valds [36], Krasovskii [17], Pi [19], Sugie and Amano [20], Tun [21, 22, 23, 24? , 25, 26, 27, 28], Tun and mohammed [29,30,31,32,33,34], Tun and Tun [35], Yoshizawa [31] Zhang ([38], [40]) and Zhou and Jiang [39]. Separate text sections with In 2005, said Burton [5] equation type Linard with a record number with constant delay, (L > 0) :…”

Qualitative analysis of integro-differential equations with variable retardation

“…In our results concerning (1), there are consistencies with respect to the facts, but they are not of direct relevance to the results presented here. Next, we note that the nonlinear system of RIDEs (2) considered here is different from the integro-differential equations in [43][44][45][46][47][48][49][50][51]. In [43][44][45][46][47][48][49][50][51], some integrodifferential equations have constant delay(s), some equations are without delay, some of them have scalar forms, and some other ones are in the form of systems.…”

“…Next, we note that the nonlinear system of RIDEs (2) considered here is different from the integro-differential equations in [43][44][45][46][47][48][49][50][51]. In [43][44][45][46][47][48][49][50][51], some integrodifferential equations have constant delay(s), some equations are without delay, some of them have scalar forms, and some other ones are in the form of systems. In the papers [43][44][45][46][47][48][49][50][51], the considered equations do not include any variable delay and terms such as BF (x(t − τ (t))) and t t−τ (t) Ω(t, s)F (x(s))ds.…”

“…In the papers [43][44][45][46][47][48][49][50][51], the considered equations do not include any variable delay and terms such as BF (x(t − τ (t))) and t t−τ (t) Ω(t, s)F (x(s))ds. This fact shows that the nonlinear system of RIDEs (2) improves the results in [43][44][45][46][47][48][49][50][51] for some particular cases. This is an improvement from the cases without delay and the cases with constant delay to the case of variable delay.…”