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In this paper, the authors obtain new sufficient conditions for stability (S) and uniform stability (US) of solutions of the first order retarded Volterra integro-differential equations (VIDEs) in the formx′=A(t)x+∫t-τtC(t,s)ϕ(s,x(s))ds+f(t,x,x(t-τ)).The analysis of the obtained (S) and (US) results mainly depend on the definition of an appropriate Lyapunov functional (LF). An example is provided to illustrate the effectiveness of the proposed results. MATLAB-Simulink is applied to show the behaviors of the paths of solutions of the considered (VIDEs) for a particular case. Keywords: Non-linear, (VIDE), First order, Stability, Uniformly stability, Lyapunov functional, 2010 AMS subject classifications: 34D05, 34K20, 45J05

In this article, we construct new specific conditions for the asymptotic stability (AS) and boundedness (B) of solutions to nonlinear Volterra integro-differential equations (VIDEs) of first order with a constant retardation. Our analysis is based on the successful construction of suitable Lyapunov–Krasovskii functionals (LKFs). The results of this paper are new, and they improve and complete that can be found in the literature. Keywords: Volterra integro-differential equation, First order, Retardation, Asymptotic stability, Boundedness, Lyapunov–Krasovskii functional

In this article, the authors obtain some clear assumptions for the asymptotic stability (AS) and boundedness (B) of solutions of non-linear retarded Volterra integro-differential equations (VIDEs) of first order by constructing a new Lyapunov functional (LF). The results obtained are new and differ from those found in the literature, and they also contain and improve a result found in the literature under more less restrictive conditions. We establish an example and give a discussion to indicate the applicability of the weaker conditions obtained. We also employ MATLAB-Simulink to display the behaviors of the orbits of the (VIDEs) considered. Keywords: Nonlinear, Volterra integro-differential equations, First order, Asymptotic stability, Boundedness, Lyapunov functional, MSC: 34D05, 34K20, 45J05

In this article, the non-integer equations of the coupled mass-spring system with Atangana Baleanu fractional derivatives is offered. The physical entities of the structure are well-preserved by presenting an supplementary stricture chi. A nonlinear model with damping factor is considered. The existence and uniqueness problem to related model are scanned by fixed point principle. Our consequences spectacle that the mechanical components reveal viscoelastic behaviors generating temporal fractality at diverse scales and exhibit the existence of material heterogeneities in the mechanical modules. The comparison Jajarmi predictor corrector and Caputo methods is also given. (C) 2021 Published by Elsevier Ltd.

This paper establishes certain sufficient conditions to guarantee the non-existence of periodic solutions for a class of nonlinear vector differential equations of fifth order. With this work, we extend and improve two related results in the literature from scalar cases to vectorial cases. An example is given to illustrate the theoretical analysis made in this paper.DOI : http://dx.doi.org/10.22342/jims.22.1.249.83-91

In this paper, we consider a certain non-linear Volterra integro-differential equations with delay. We study stability and boundedness of solutions. The technique of proof involves defining suitable Lyapunov functionals. Our results improve and extend the results obtained in literature.

The purpose of this paper is to study the global existence and boundedness of solutions to a kind of nonlinear integro-differential equations of second order by integral inequalities. Some illustrative examples are also provided. Our results complement and improve some ones in the literature.

By making use of a special Lyapunov-Krasovskii functional and applying Krasovskii’s properties, we prove instability of zero solution of a modified vector Lienard equation with multiple constant delays that includes Van der Pol, Rayleigh and Lienard equations, widely encountered in applications. Usando un funcional especial de Lyapunov-Krasovskii y aplicando propiedades de Krasovskii, probamos la inestabilidad de la solución nula de una ecuación de Liénard vectorial modificada con retardos constantes múltiples que incluyen a las ecuaciones de Van der Pol, Rayleigh y Liénard ampliamente encontradas en las aplicaciones.

The aims and objectives of this manuscript are devoted to investigate some adequate results for the existence of solution to a nonlinear coupled system of fractional order differential equations (FDEs). Appropriate conditions for the existence of at least one solution are developed by using a fixed point theorem of Leray Schauder type to the considered problem. Further, we also investigate the Hyers Ulam stability results for the proposed problem. Appropriate examples are given to demonstrate the established results. (C) 2017 The Authors. Production and hosting by Elsevier B.V. on behalf of Taibah University. This is an open access article under the CC BY-NC-ND license

In this paper, we investigate the stability of the zero solution of an integro-differential equation of the second order with variable delay. By means of the fixed point theory and an exponential weighted metric, we find sufficient conditions under which the zero solution of the equation considered is stable.

In this work, we are concerned with the investigation of the qualitative behaviors of certain systems of non-linear differential equations of second order. We make a comparison between applications of the integral test and Lyapunov’s function approach on some recent stability and boundedness results in the literature. An example is furnished to illustrate the hypotheses and main results in this paper.

By this work, we construct certain specific assumptions guaranteeing the asymptotic stability (AS) of trivial solution to a retarded linear neutral differential equation with periodic coefficients, and we estimate the decay rate of the solutions of the considered equation. To reach the desirable results, we benefit from a Lyapunov functional. We give an example to show applicability of the constructed assumptions and use MATLAB-Simulink to show the behaviors of the paths of the solutions of the considered equation.

In this paper, the authors give sufficient conditions for the boundedness and global asymptotic stability of solutions to certain nonlinear multi-delay functional differential equations of the third order. The technique of proof involves defining an appropriate Lyapunov-Krasovskii functional and applying LaSalle's invariance principle. An example is included to illustrate the results. Copyright (C) 2014 JohnWiley & Sons, Ltd.

We study the boundedness of the solutions to a non-autonomous and non-linear differential equation of second order with two constant deviating arguments. We give two examples to illustrate the main results. By this work, we extend some boundedness results obtained for a differential equation with a constant deviating argument in the literature to the boundedness of the solutions of a differential equation with two constant deviating arguments. (C) 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

In this paper, we give sufficient conditions to guarantee the asymptotic stability and bounded ness of solutions to a kind of fourth-order functional differential equations with multiple delays. By using the Lyapunov-Krasovskii functional approach, we establish two new results on the stability and bounded ness of solutions, which include and improve some related results in the literature. © 2014, University of Nis. All rights reserved.

We give certain sufficient conditions for the existence of periodic solutions to a Rayleigh-type equation with state-dependent delay. With this work we extend and improve some known results in the literature.

In this paper, we consider the vector Lienard equation with the constant deviating argument, τ>0, $$X''(t) + F\bigl(X(t),X'(t) \bigr)X'(t) + H\bigl(X(t - \tau)\bigr) = P(t) $$ in two cases: (i) P(.)≡0, (ii) P(.)≠0. Based on the Lyapunov–Krasovskii functional approach, the asymptotic stability of the zero solution and the boundedness of all solutions are discussed for these cases. We give an example to illustrate the theoretical analysis made in this work and to show the effectiveness of the method utilized here.

We study a fifth-order nonlinear vector delay differential equation with multiple deviating arguments. Some criteria for guaranteeing the instability of zero solution of the equation are given by using the Lyapunov-Krasovskii functional approach. Comparing with the previous literature, our result is new and complements some known results.

This paper is concerned with a non-autonomous delayed Nicholson's blowflies model with a linear harvesting term. Under proper conditions, by suing the exponential dichotomy theory and fixed point theorem, we employ a novel argument to establish a criterion on the global exponential stability of positive pseudo almost periodic solutions of the model. The obtained result complements with some existing ones.

In this article, we apply the exp(-Phi(xi))-expansion method to construct many families of exact solutions of nonlinear evolution equations (NLEEs) via the nonlinear diffusive predator-prey system and the Bogoyavlenskii equations. These equations can be transformed to nonlinear ordinary differential equations. As a result, some new exact solutions are obtained through the hyperbolic function, the trigonometric function, the exponential functions and the rational forms. If the parameters take specific values, then the solitary waves are derived from the traveling waves. Also, we draw 2D and 3D graphics of exact solutions for the special diffusive predator-prey system and the Bogoyavlenskii equations by the help of programming language Maple. (C) 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

The present article investigates the existence of solutions of the following nonlinear second order coupled system with nonlinear coupled boundary conditions (CBCs) where f1, f2 : [0, 1] × R → R, µ : R6 → R2 and ν : R2 → R2 are continuous functions. The results presented in [7, 11] are extended in our article. Coupled lower and upper solutions, Arzela-Ascoli theorem and Schauder’s fixed point theorem play an important role in establishing the arguments. Some examples are taken to ensure the validity of the theoretical results

In this paper, we give sufficient conditions to guarantee the asymptotic stability and boundedness of solutions to a kind of fourth-order functional differential equations with multiple retardations. By using the Lyapunov-Krasovskii functional approach, we establish two new results on the stability and boundedness of solutions, which include and improve some related results in the literature. © 2016, Wilmington Scientific Publisher. All Rights Reserved.

In this paper, we establish some results for the stability and the boundedness of solutions of non-autonomous differential equations of second order with a variable deviating argument. We make a comparison between a stability result verified via fixed points and the Lyapunov–Krasovskii functional approach, and we discuss the uniform stability and the boundedness of solutions. An example is given to illustrate the feasibility of the proposed results.

This paper deals with a certain third-order non-linear differential equation with multiple deviating arguments. Some sufficient conditions are set up for all solutions and their derivatives to be uniformly bounded.En este artículo se estudia un tipo de ecuaciones diferenciales no lineales de tercer orden con argumentos de desviación múltiple. Se establecen algunas condiciones suficientes para que todas las soluciones y sus derivadas sean uniformemente acotadas.

In this paper, we establish some new sufficient conditions which guaranteethe boundedness and the uniform boundedness of solutions of certain nonlinearand nonautonomous differential equations of second order. By defining appropriateLiapunov functions, we obtain some new results on the subject, and examples arealso provided to illustrate the feasibility of the proposed results. By this work, weextend and improve some boundedness results in the literature.DOI : http://dx.doi.org/10.22342/jims.16.2.35.115-126

We establish some sufficient conditions to guarantee the stability of the zero solution and the boundedness of all solutions of certain nonlinear differential equations of third order with multiple deviating arguments. In the proofs, the Lyapunov functional approach is employed as main tool and an example is given to illustrate the feasibility of the established results.

In this paper, we study the instability properties of solutions of a kind of functional differential equations of the fifth order with constant delay. Using the Lyapunov-Krasovskii functional approach, we obtain certain sufficient conditions to guarantee that the zero solution of the equation is unstable.

We study the boundedness of the solutions to a non-autonomous differential equation of second order with finitely many deviating arguments. We give two examples to illustrate the main results. By this work, we improve some boundedness results obtained for a differential equation with a deviating argument in the literature to the boundedness of the solutions of a differential equation with finitely many deviating arguments.

Sucient conditions are obtained for the instability of the zero solutionof a certain sixth order nonlinear functional dierential equation by the Lyapunov-Krasovskii functional approach. DOI : http://dx.doi.org/10.22342/jims.17.2.7.123-128

This paper is devoted to the mathematical analysis of the stability, boundedness and uniform boundedness of solutions of a class of nonlinear differential equations of second order with multiple constant deviating arguments. We use Lyapunov functionals to verify the stability and boundedness of the solutions and some examples are given to illustrate the theoretical analysis in this work.

The main purpose of this paper is to introduce a new instability theorem related to a fifth order nonlinear differential equation with a constant delay. By means of the Lyapunov-Krasovskii ([8], [13]) functional approach, we obtain a new result on the topic.

By defining some appropriate Liapunov functions, we discuss boundedness of solutions to a class of non-autonomous and nonlinear differential equations of second order. By this work, we prove some results established in the literature by Liapunov's second method instead of the integral test. We give six examples to illustrate the theoretical analysis in this work and effectiveness of the method utilized here.

We discuss the existence of solutions with oblique asymptotes to a class of second order nonlinear ordinary differential equations by means of Lyapunov functions. The approach is new in this field and allows for simpler proofs of general results regarding Emden-Fowler like equations. (C) 2009 Elsevier Inc. All rights reserved.

In this paper, it is investigated the existence of periodic solutions to the nonlinear third order differential equation : x'" + c2(t)x" + c1(f)x' + f(t, x) = p(t, x, x', x"). The Leray-Schauder principle is used to show the existence of periodic solutions of this equation.

This paper investigates stability and boundedness of solutions to third order nonlinear differential equation with retarded argument: $$\begin{array}{l}x'''(t)+\varphi\bigl(x(t-r),x'(t-r),x''(t-r)\bigr)x''(t)\\\qquad{}+\psi\bigl(x'(t-r)\bigr)+h\bigl(x(t-r)\bigr)\\\quad=p\bigl(t,x(t),x(t-r),x'(t),x'(t-r),x''(t)\bigr).\end{array}$$ By the use of the Lyapunov functional, sufficient conditions for stability and boundedness of solutions to the considered equations are obtained. Examples are introduced throughout the paper for illustrations.

By constructing a Lyapunov function, a new instability result is established, which guarantees that the trivial solution of a certain nonlinear vector differential equation of the fifth order is unstable. An example is also given to illustrate the importance of the result obtained. By this way, our findings improve an instability result related to a scalar differential equation in the literature to instability of the trivial solution to the afore-mentioned differential equation. Copyright (C) 2008 C. Tunc and M. Karta.

In this paper, the second order non-linear differential equation x ¨ + a ( t ) f ( x , x ˙ ) x ˙ + b ( t ) g ( x ) = p ( t , x , x ˙ ) is considered, and Lyapunov's second method is used to show that uniform boundedness and convergence to zero of all solutions of this equation together with their derivatives of the first order.

The paper gives sufficient conditions to ensure the asymptotic stability of the zero solution of a certain fourth-order delay differential equation. By constructing a Lyapunov functional, we obtain a new result which includes and improves some related results existing in the relevant literature.

In this paper, the authors discuss the continuability and boundedness of solutions of a second order functional integro-differential equation with multiple delays. The proof involves the construction of a Lyapunov-Krasovskii type functional.

We study the asymptotic behavior of solutions to a nonlinear differential equation of the second order whose coefficient of nonlinearity is a bounded function for arbitrarily large values of \(x\) in \(R\). We obtain certain sufficient conditions which guarantee boundedness of solutions, their convergence to zero as \(x\rightarrow \infty\) and their unboundedness.

This paper is concerned with a nonlinear Duffing system with pseudo almost periodic coefficients and delay. Under proper conditions, by using theory of exponential dichotomies and contraction mapping principle, some sufficient conditions are established to ensure the existence and uniqueness of pseudo almost periodic solutions, which improve and extend some known relevant results. Moreover, an example is given to illustrate the theoretical results.

In this paper, on the basis of a new Lyapunov–Krasovskii functional and inequality technique, a set of novel sufficient conditions is derived for all solutions of a kind of nonlinear neutral differential equations of first order to approach zero as \(t\rightarrow \infty \). The derived conditions are delay independent. An example is discussed to illustrate the efficiency of the result. Our result extends and improves some related ones in the literature.

In this paper, we consider a class of high-order cellular neural networks (HCNNs) model with time-varying delays in the leakage terms. We give some sufficient conditions which guarantee the exponential stability of pseudo almost periodic solutions for the model. The obtained results complement with some recent ones in the literature.The technique of proof involves the exponential dichotomy theory and the fixed point theorem. An illustrative example is given with an application.

In this article, the existence of solution for the first-order nonlinear coupled system of ordinary differential equations with nonlinear coupled boundary condition (CBC for short) is studied using a coupled lower and upper solution approach. Our method for a nonlinear coupled system with nonlinear CBC is new and it unifies the treatment of many different first-order problems. Examples are included to ensure the validity of the results.

In this paper, cellular neural networks with leakage delays and complex deviating arguments are considered. Some criteria are established for the existence of pseudo almost periodic solutions for this model by using the exponential dichotomy theory, contraction mapping fixed point theorem and inequality analysis technique. The results of this paper are new and complement previously known results.

This paper is devoted to the mathematical analysis of stability of a scalar Li ´ enard type equation with variable delay. We use the fixed point technique under an exponentially weighted me tric to prove the stability of the zero solution. By this work , we extend and improve some related results in the literature

In this paper, by using the fixed point method, we prove two new results on the Hyers-Ulam-Rassias and the Hyers-Ulam stability for the first order delay differential equation of the form y′(t) = F(t, y(t), y(t −τ )). Our results improve some related results in the literature.