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The paper deals with the problems of the global existence, boundedness and integrability, and solutions of an integro-differential equation (IDE) of second order with n-constant retardations. Three new theorems, Theorems 2.1–2.3 and a corollary, Corollary 2.1, have been proved, which consist of new sufficient criteria on global existence, stability, asymptotic stability, boundedness of solutions and square integrability of the first derivatives of solutions of the considered nonlinear retarded IDE of second order (RIDE). The techniques used in the proofs are based on the construction of a suitable Lyapunov–Krasovskii functional (LK functional). The results of this work include and get better results that are available in the literature. The given criteria have very simple forms, and they can be easily verified with applications. A numerical example is given and the orbits of the exact solutions are drawn with assigned initial conditions to demonstrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a non-linear stochastic differential delay equation (SDDE) of second order. We derive new sufficient conditions which guarantee stochastically stability and stochastically asymptotically stability of the zero solution of that SDDE. Here, the technique of the proof is based on the definition of a suitable Lyapunov-Krasovskii functional, which gives meaningful results for the problem under consideration. The derived results extend and improve some result of in the relevant literature, which are related to the qualitative properties of solutions of a SDDE of second order. The results of this paper are new and have novelty, and they do a contribution to the topic and relevant literature. As an application, an example is given to show the effectiveness and applicability of the obtained results. Finally, by the results of this paper, we extend and improve some recent results that can be found in the relevant literature. [ABSTRACT FROM AUTHOR]

This paper analyzes the stability, uniformly stability, asymptotically stability, boundedness, uniformly boundedness and square integrability of solutions of a system of differential equations of second order with variable delay by applying the direct method of Lyapunov- Krasovskii. By means of a new Lyapunov-Krasovskii functional, we simplify and extend some previous work that is found in the recent literature. Finally, the validity and applicability of the proceeded results are indicated by some numerical examples applying MATLAB-Simulink. By the results of this paper, we can obtain the results of Omeike et al. [Stability and boundedness of solutions of certain vector delay differential equations. J Nigerian Math Soc. 2018;37(2):77–87], Theorem 1.1 and Theorem 1.2, under weaker conditions. In addition, we establish two new results on the uniformly stability and integrabilty of solutions the considered equation. Finally, in particular cases, the applicability of the results of this paper can be shown by two new examples. These are the contribution of this paper to the subject and the relevant literature. [ABSTRACT FROM AUTHOR]

The aim of paper is to analyse some qualitative properties of solutions of nonlinear Volterra integro-differential equations (VIDDEs) with constant delay and Volterra integro-differential equations (VIDEs) without delay by means of the Razumikhin method. A suitable Lyapunov function is defined and then applied to that IDEs such that some former results can be obtained under weaker conditions and additionally some new results are given on the qualitative properties of that IDEs such as uniformly stability and integrability of solutions. For the VIDDEs with constant delay, the improved less conservative conditions of the stability, asymptotically stability and the boundedness of solutions and the new conditions of uniformly stability, integrability of solutions are all derived subject to the functions appeared in the considered IDEs. Moreover, the established stability, asymptotically stability and boundedness conditions of VIDDEs with constant delay simplify, extend and improve some previous works can be found in the literature and remove some unnecessary conditions. Finally, the validity of the presented results is indicated by some numerical examples using MATLAB-Simulink. [ABSTRACT FROM AUTHOR]