Search

In this research, two systems of nonlinear Volterra integral equation and Volterra integro‐differential equation were considered. New results in sense of Ulam stabilities in relation to these two systems were proved on a finite interval. The proof of the results on the Ulam stabilities of that classes of the equations are based on the nonlinear alternative related to Banach's contraction principle. The outcomes of this research give new contribution to the theory of Ulam stabilities. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the sufficient conditions that guarantee the stability, continuity, and boundedness of solutions for a type of second-order stochastic delay integro-differential equation (SDIDE). To demonstrate the main results, we employ a Lyapunov functional. An example is provided to demonstrate the applicability of the obtained result, which includes the results of this paper and obtains better results than those available in the literature. [ABSTRACT FROM AUTHOR]

In this work, we delve into a nonlinear integral equation (IEq) with multiple variable time delays and a nonlinear integro-differential equation (IDEq) without delay. Global existence and uniqueness (GEU) of solutions of that IEq with multiple variable time delays and IDEq are investigated by the fixed point method using progressive contractions, which are due to T.A. Burton. We prove four new theorems including sufficient conditions with regard to GEU of solutions of the equations. The results generalize and improve some related published results of the relevant literature. [ABSTRACT FROM AUTHOR]

This paper deals with the uniformly stability (US), globally uniformly asymptotically stability (GUAS) and instability of zero solution as well as integrability and boundedness of nonzero solutions of certain nonlinear integro‐differential equations (IDEs). We prove five new theorems, which include sufficient conditions related to these fundamental qualitative properties of solutions to the IDEs considered. The main tools used in the proof are two new and suitable Lyapunov–Krasovskiiˇ functionals (LKFs). In particular cases, two numerical examples are given and solved via the fourth order Runge–Kutta method (RKM) in MATLAB to illustrate the theoretical results of this paper. Compared with the existing results on the fundamental qualitative behaviors of scalar IDEs, our results are new, original, more effective and convenient for tests and applications. [ABSTRACT FROM AUTHOR]

In this article, a class of scalar nonlinear integro-differential equations of first order with fading memory is investigated. For the considered fading memory problem, we discuss the effects of the memory over all the values of the parameter in the kernel of the equations. Using the Lyapunov–Krasovski functional method, we give various sufficient conditions of stability, asymptotic stability, uniform stability of zero solution, convergence and boundedness, and square integrability of nonzero solutions in relation to the considered scalar nonlinear integro-differential equations for various cases. As the novel contributions of this article, the new scalar nonlinear integro-differential equation with the fading memory is firstly investigated in the literature, and seven theorems, which have novel sufficient qualitative conditions, are provided on the qualitative behaviors of solutions called boundedness, convergence, stability, integrability, asymptotic stability and uniform stability of solutions. The novel outcomes and originality of this article are that the considered integro-differential equations are new mathematical models, they include former mathematical models in relation to the mathematical models of this paper as well as the given main seven qualitative results are also new. The outcomes of this paper enhance some present results and provide new contributions to the relevant literature. The results of the article have complementary properties for the symmetry of integro-differential equations. [ABSTRACT FROM AUTHOR]

In this article, a scalar nonlinear integro-differential equation of second order and a non-linear system of integro-differential equations with infinite delays are considered. Qualitative properties of solutions called the global asymptotic stability, integrability and boundedness of solutions of the second-order scalar nonlinear integro-differential equation and the nonlinear system of nonlinear integro-differential equations with infinite delays are discussed. In the article, new explicit qualitative conditions are presented for solutions of both the second-order scalar nonlinear integro-differential equations with infinite delay and the nonlinear system of integro-differential equations with infinite delay. The proofs of the main results of the article are based on two new Lyapunov–Krasovskiĭ functionals. In particular cases, the results of the article are illustrated with three numerical examples, and connections to known tests are discussed. The main novelty and originality of this article are that the considered integro-differential equation and system of integro-differential equations with infinite delays are new mathematical models, the main six qualitative results given are also new. [ABSTRACT FROM AUTHOR]

In the present article, we study existence of periodic solutions (EPSs) of a nonlinear neutral integro-differential equation (NIDE) with multiple variable delays using Krasnoselskiis fixed point theorem. Transforming the considered NIDE to an equivalent integral equation, we prove the EPSs using a fixed point mapping, which is defined as a sum of a contraction and a compact map. The result of this paper has contributions to the topic of the EPSs of NIDEs. [ABSTRACT FROM AUTHOR]

This work studies certain perturbed and un-perturbed nonlinear systems of continuous and discrete integro-delay differential equations (IDDEs). Using the Lyapunov–Krasovskii functional (LKF) method and the Lyapunov–Razumikhin method (LRM), uniform asymptotic stability (UAS), uniform stability (US), integrability and boundedness of solutions as well as exponential stability (ES) and instability of solutions are discussed. In this paper, five new theorems and a corollary are given and three numerical applications are provided with their simulations. With this work, we aim to make new contributions to the theory of the continuous and discrete integro-differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the stability (S), asymptotic stability (AS), uni- form stability (US) of null solution and boundedness of solutions of non-linear Volterra integro-differential equations (IDEs) are investi- gated by means of the Lyapunov-Krasovskii method (LKM). Four new theorems are provided on the mentioned concepts for the considered (IDEs). Compared with past related results in the literature, our conditions are simple, weaker and convenient for testing and applica- tions. The obtained results also improve and extend the past related results. Finally, two numerical examples are provided to illustrate the applications of our results. [ABSTRACT FROM AUTHOR]

This paper is concerned with certain non-linear unperturbed and perturbed systems of integro-delay differential equations (IDDEs). We investigate fundamental properties of solutions such as uniformly stability (US), uniformly asymptotically stability (UAS), integrability and instability of the un-perturbed system of the IDDEs as well as the boundedness of the perturbed system of IDDEs. In this paper, five new and improved fundamental qualitative results, which have less conservative conditions, are obtained on the mentioned fundamental properties of solutions. The technique used in the proofs depends on Lyapunov-Krasovski functionals (LKFs). In particular cases, three examples and their numerical simulations are provided as numerical applications of this paper. This paper provides new, extensive and improved contributions to the theory of IDDEs. [ABSTRACT FROM AUTHOR]

In this paper, a non-linear fractional order integro-singular system (FrOISS) is considered. Explicit criteria for asymptotic stability of that FrOISS are given via two new theorems. The studies of the paper are based on applications of two Lyapunov- Krasovskiĭ functionals (LKFs). In particular case, two examples and their simulations are included as applications of the results. By this work, some existing results are improved and generalized. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate some qualitative properties of solutions of nonlinear fractional retarded Volterra integro-differential equations (FrRIDEs) with Caputo fractional derivatives. These properties include uniform stability, asymptotic stability, Mittag–Leffer stability and boundedness. The presented results are proved by defining an appropriate Lyapunov function and applying the Lyapunov–Razumikhin method (LRM). Hence, some results that are available in the literature are improved for the FrRIDEs and obtained under weaker conditions via the advantage of the LRM. In order to illustrate the results, two examples are provided. [ABSTRACT FROM AUTHOR]