Asymptotic Behavior by Krasnoselskii Fixed Point Theorem for Nonlinear Neutral Differential Equations with Variable Delays

In this paper, we consider a neutral differential equation with two variable delays. We construct new conditions guaranteeing the trivial solution of this neutral differential equation is asymptotic stable. The technique of the proof based on the use of Krasnoselskii’s fixed point Theorem. An asymptotic stability theorem with a necessary and sufficient condition is proved. In particular, this paper improves important and interesting works by Jin and Luo. Moreover, as an application, we also exhibit some special cases of the equation, which have been studied extensively in the literature

New sufficient conditions for global asymptotic stability of a kind of nonlinear neutral differential equations

summary:This paper addresses the stability study for nonlinear neutral differential equations. Thanks to a new technique based on the fixed point theory, we find some new sufficient conditions ensuring the global asymptotic stability of the solution. In this work we extend and improve some related results presented in recent works of literature. Two examples are exhibited to show the effectiveness and advantage of the results proved

New criteria for global asymptotic stability of linear neutral differential equations by a fixed point approach

New criteria ensuring global asymptotic stability of the zero solution for a class of linear neutral differential equations in C 1 are proved, by using two auxiliary functions on a contraction condition. Necessary and sufficient conditions for the stability of our equation which also improves recent results on this field are shown. Finally, an example is provided to illustrate the feasibility and advantage of our result

Existence of periodic positive solutions to a nonlinear Lotka-Votlerra competition systems

We investigate the existence of positive periodic solutions of a nonlinear Lotka-Volterra competition system with deviating arguments. The main tool we use to obtain our result is the Krasnoselskii fixed point theorem. In particular, this paper improves important and interesting work [X.H. Tang, X. Zhou, On positive periodic solution of Lotka–Volterra competition systems with deviating arguments, Proc. Amer. Math. Soc. 134 (2006), 2967–2974]. Moreover, as an application, we also exhibit some special cases of the system, which have been studied extensively in the literature

Stability results for neutral stochastic functional differential equations via fixed point methods

In this paper we prove some results on the mean square asymptotic stability of a class of neutral stochastic differential systems with variable delays by using a contraction mapping principle. Namely, a necessary and sufficient condition ensuring the asymptotic stability is proved. The assumption does not require neither boundedness or differentiability of the delay functions, nor do they ask for a fixed sign on the coefficient functions. In particular, the results improve some previous ones proved by Guo, Y., Xu, C., & Wu, J. [(2017). Stability analysis of neutral stochastic delay differential equations by a generalisation of Banach’s contraction principle. International Journal of Control, 90, 1555–1560]. Finally, an example is exhibited to illustrate the effectiveness of the proposed results

Existence of solutions and stability for impulsive neutral stochastic functional differential equations

In this paper we prove some results on the existence of solutions and the mean square asymptotic stability for a class of impulsive neutral stochastic differential systems with variable delays by using a contraction mapping principle. Namely, a sufficient condition ensuring the asymptotic stability is proved. The assumptions do not impose any restrictions neither on boundedness nor on the differentiability of the delay functions. In particular, the results improve some previous ones in the literature. Finally, an example is exhibited to illustrate the effectiveness of the results.Ministerio de Economía y Competitividad (MINECO). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Junta de AndalucíaEuropean Mathematical Societ

On the existence of positive periodic solutions for n−species Lotka-Volterra competitive systems with distributed delays and impulses

In this paper, we investigate the existence of positive periodic solutions for an n-species Lotka-Volterra system with distributed delays and impulsive effect. In the process we use integrating factors and convert the given Lotka-Volterra differential equation into an equivalent integral equation. Then we construct appropriate mappings and use Krasnoselskii’s fixed point theorem to show the existence of a positive periodic solution of this system. In particular, the results improve some previous ones in the literature. Finally, as an application, we exhibit an example to illustrate the effectiveness of our abstract result

New sufficient conditions for global asymptotic stability of a kind of nonlinear neutral differential equations

This paper addresses the stability study for nonlinear neutral differ ential equations. Thanks to a new technique based on the fixed point theory, we find some new sufficient conditions ensuring the global asymp totic stability of the solution. In this work we extend and improve some related results presented in recent works of literature. Two examples are exhibited to show the effectiveness and advantage of the proved results. AMS Subject Classifications: 34K20, 34K13, 92B20

Existence of periodic positive solutions to nonlinear Lotka-Volterra competition

We investigate the existence of positive periodic solutions of a nonlinear Lotka-Volterra competition system with deviating arguments. The main tool we use to obtain our result is the Krasnoselskii fixed point theorem. In particular, this paper improves important and interesting work [X.H. Tang, X. Zhou, On positive periodic solution of Lotka-Volterra competition systems with deviating arguments, Proc. Amer. Math. Soc. 134 (2006), 2967-2974]. Moreover, as an application, we also exhibit some special cases of the system, which have been studied extensively in the literature