Search

Our article proves inequalities for interval optimization and shows that feasible and descent directions do not intersect in constrained cases. Mainly, we establish some new interval inequalities for interval-valued functions by defining LC-partial order. We use LC-partial order to study Karush–Kuhn–Tucker (KKT) conditions and expands Gordan’s theorems for interval linear inequality systems. By applying Gordan’s theorem, we can determine the best outcomes for interval optimization problems (IOPs) that have constraints, such as Fritz John and KKT conditions. The optimality conditions are observed with inclusion relations rather than equality. We can use the KKT condition for binary classification with interval data and support vector machines(SVMs). We present some examples to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider a non-linear Volterra integro-differential equation (NVIDE) of first order with multiple constant time lags. They obtain new sufficient conditions on stability (S), boundedness (B), global asymptotic stability (GAS) of solutions, and in addition, every solution of the considered NVIDE belong to and The authors prove five new theorems on S, B, GAS, and properties of solutions. The technique of the proofs involves the construction of suitable Lyapunov functionals. The obtained conditions are nonlinear generalizations and extensions of those of Becker [Uniformly continuous L1 solutions of Volterra equations and global asymptotic stability. Cubo 11(3);2009:1-24], Graef et al. [Behavior of solutions of non-linear functional Voltera integro-differential equations with multiple delays. Dynam Syst Appl. 25(1-2);2016:39-46] and Tunç [A note on the qualitative behaviors of non-linear Volterra integro-differential equation. J Egyptian Math Soc. 24(2);2016:187-192; New stability and boundedness results to Volterra integro-differential equations with delay. J Egyptian Math Soc. 24(2);2016:210-213] and they improve some results can found in the literature. The results of this paper are new, and they have novelty and complete some results exist in the literature. [ABSTRACT FROM AUTHOR]

We investigate a modified Lienard equation (LE) with multiple variable time lags and find assumptions for the Hyers-Ulam stabil- ity (HUS) and Hyers-Ulam -Rassias stability (HURS) of the modified (LE) considered. We obtain two results on (HUS) and (HURS) of the equation considered where we use Banach's contraction principle (BCP). [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear integro-vector differential equation of the second order. We establish sufficient conditions that guarantee the global existence and boundedness of solutions of the equation considered. The method of proof involves constructing a suitable Lyapunov functional that gives meaningful results for the problem to be investigated. The result obtained is new and complements that found in the literature. We give an example to verify the result obtained and for illustration purposes. Using MATLABSimulink, the behaviors of the orbits of the equation considered are clearly shown. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this paper is first to investigate the stability of the zero solution to a new Liénard type equation with multiple variable delays by two different methods. The methods to be used in the proofs involve the Lyapunov-Krasovskiĭ functional approach and the fixed point technique under an exponentially weighted metric, respectively. We make a comparison between the applications of these methods with the established conditions on the same stability problems. Then, we obtain three new results for uniformly stability and boundedness/uniformly boundedness of the solutions to the considered equation by the Lyapunov-Krasovskiĭ functional approach. An example is given to verify the results obtained by the Lyapunov-Krasovskiĭ functional approach. Our results complement and improve some recent ones in the literature. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this work, we get assumptions that guaranteeing the global exponential stability (GES) of the zero solution of a neutral differential equation (NDE) with time-lags. By help of the Liapunov-Krasovskii functional (LKF) approach, we obtain a new result related (GES) of the zero solution of the studied (NDE). An example is given to illustrate the applicability and correctness of the obtained result by MATLAB-Simulink. The obtained result includes and improves the results found in the literature. [ABSTRACT FROM AUTHOR]

The author gives sufficient conditions for non-existence of periodic solutions of two higher order nonlinear delay differential systems. Our technical approach is based on the construction of two suitable Lyapunov type functionals. An example is given to illustrate the obtained results. The main results here improve recent results found on the topic in the literature from the case of without delay to the delay case. [ABSTRACT FROM AUTHOR]

This paper considers a class of scalar and vector nonlinear Volterra integro-differential equations of the first order with a constant delay. We demonstrate the stability, uniform stability, boundedness, convergence and square integrability of the solutions. The technique of proof involves defining appropriate Lyapunov functionals. Our results improve the results obtained in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundedness and square integrability of solutions in certain non-linear systems of differential equations of second order. We establish two new theorems, which include suitable sufficient conditions guaranteeing the boundedness and square integrability of solutions to the considered systems. The presented proofs simplify previous works since the Gronwall inequality is avoided which is the usual case. The technique of proof involves the integral test, and two examples are included to illustrate the results. [ABSTRACT FROM AUTHOR]

In this article, we give sufficient conditions for the boundedness, uniformly asymptotic stability and square integrability of the solutions to a fourth-order non-autonomous differential equation with bounded delay by us- ing Lyapunov's second method. [ABSTRACT FROM AUTHOR]

This paper is concerned with non-linear Volterra integro-differential equation (VIDE) with constant time-lag, τ : x'(t) = P(t)f(x(t)) - ∫t t-τ K(t, s)f(x(s))ds. Via Lyapunov functionals and basic inequalities, sufficient conditions are given for the exponential stability (ES) and instability (I) of the trivial solution of the former (VIDE). We introduce two new results for the above topics for the trivial solution of that (VIDE). Our conditions involve the nonlinear generalization and extensions of those found in the literature. The results to be obtained are new and complements that in the literature. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we give some criteria to ensure the global existence and boundedness of solutions to a kind of third order nonlinear vector differential equations. By using the Lyapunov's direct method, we obtain a new result on the topic and give an example for the illustrations. Our result includes, completes and improves some earlier results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, by defining Lyapunov functionals, we investigate proper sufficient conditions for the uniform stability of the zero solution, and also for the uniform boundedness and uniform ultimate boundedness of all solutions of a certain third-order nonlinear vector delay differential equation of the type X˙˙˙ + A X˙˙ + G( X˙) + H( X( t - r)) = P( t). [ABSTRACT FROM AUTHOR]

In this paper, certain system of linear homogeneous differential equations of second-order is considered. By using integral inequalities, some new criteria for bounded and L 2 [ 0 , ∞ ) -solutions, upper bounds for values of improper integrals of the solutions and their derivatives are established to the considered system. The obtained results in this paper are considered as extension to the results obtained by Kroopnick (2014) [1]. An example is given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Fractal analogue of Newton, Lagrange, Hamilton, and Appell's mechanics are suggested. The fractal α -velocity and α -acceleration are defined in order to obtain the Langevin equation on fractal curves. Using the Legendre transformation, Hamilton's mechanics on fractal curves is derived for modeling a non-conservative system on fractal curves with fractional dimensions. Fractal differential equations have solutions that are non-differentiable in the sense of ordinary derivatives and explain space and time with fractional dimensions. The illustrated examples with graphs present the details. [ABSTRACT FROM AUTHOR]

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 -t )). Our results improve some related results in the literature. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of first order differential iterative equations. Under proper conditions, we employ a novel argument to establish a criterion on the existence of pseudo almost periodic solutions. The obtained result complements with some existing ones. [ABSTRACT FROM AUTHOR]

By using the standard Liapunov-Krasovskii functional approach, in this paper, new stability, boundedness and ultimately boundedness criteria are established for a class of vector functional differential equations of third order with retarded argument. [ABSTRACT FROM AUTHOR]

In this research, we take into account the Kirchhoff type equation with variable exponent. The Kirchhoff type equation is known as a kind of evolution equations, namely, PDEs, where t is an independent variable. This type problem can be extensively used in many mathematical models of various applied sciences such as flows of electrorheological fluids, thin liquid films, and so on. This research, we investigate the upper bound for blow up time under suitable conditions. [ABSTRACT FROM AUTHOR]

This paper investigates the instability of the zero solution of a certain vector differential equation of the sixth order with delay. Using the Lyapunov-Krasovskii functional approach, we obtain a new result on the topic and give an example for the related illustrations [ABSTRACT FROM AUTHOR]

We give some sufficient conditions to guarantee convergence of solutions to a nonlinear vector differential equation of third order. We prove a new result on the convergence of solutions. An example is given to illustrate the theoretical analysis made in this paper. Our result improves and generalizes some earlier results in the literature. [ABSTRACT FROM AUTHOR]

We discuss the stability of solutions to a kind of scalar Liénard type equations with multiple variable delays by means of the fixed point technique under an exponentially weighted metric. By this work, we improve some related results from one delay to multiple variable delays. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the instability of solutions to a certain class of nonlinear vector functional differential equations of the eighth order with n-deviating arguments. We employ the Lyapunov- Krasovskii functional approach and base on the Krasovskii criteria to prove two new theorems on the topic. Our results improve certain results in the literature from scalar functional differential equations to their vectorial forms. [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]

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. [ABSTRACT FROM AUTHOR]

This paper deals with a class of fourth-order nonlinear differential equations with multiple deviating arguments, and some sufficient conditions are established for the existence and exponential stability of anti-periodic solutions of the equation. [ABSTRACT FROM AUTHOR]

This paper deals with the boundedness of solutions to a nonlinear differential equation of fourth order. Using the Cauchy formula for the particular solution of nonhomogeneous differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded. [ABSTRACT FROM AUTHOR]

Using Lyapunov-Krasovskii functional approach, we establish a new result to guarantee the existence of periodic solutions of a certain multidelay nonlinear functional differential equation of second order. By this work, we extend and improve some earlier result in the literature. [ABSTRACT FROM AUTHOR]

Consideration of the observabilities of linear hybrid descriptor systems implies the distinguishability of these systems to be imperative. We have obtained some results related to the distinguishability of the descriptor systems. Also, we have attained equivalent criteria for input distinguishability of descriptor systems with a regular pencil. [ABSTRACT FROM AUTHOR]

This paper considers a non-autonomous differential equation of second order with multiple variable deviating arguments. By using time-varying delay Lyapunov-Krasovskii functional, we propose to find some sufficient conditions for the solutions of the equation considered are quadratic integrable. [ABSTRACT FROM AUTHOR]

This paper considers a non-autonomous differential equation of second order with multiple variable deviating arguments. By using time-varying delay Lyapunov-Krasovskii functional, we propose to find some sufficient conditions for the solutions of the equation considered are quadratic integrable. [ABSTRACT FROM AUTHOR]

We establish sufficient conditions for the existence of periodic solutions for a Rayleigh-type equation with state-dependent delay. Our approach is based on the continuation theorem in degree theory, and some analysis techniques. An example illustrates that our approach to this problem is new. [ABSTRACT FROM AUTHOR]

We establish certain new sufficient conditions which guarantee the existence of periodic solutions for a nonlinear differential equation of the third order with multiple deviating arguments. Using the Lyapunov functional approach, we prove a specific theorem and provide an example to illustrate the theoretical analysis in this work and the effectiveness of the method utilized here. [ABSTRACT FROM AUTHOR]

This paper establishes some new sufficient conditions to ensure the stability and boundedness of solutions of certain second order non-autonomous and non-linear differential equations. By defining two appropriate Liapunov functions, we obtain three new results on the subject and give some examples to illustrate the theoretical analysis in this work. By this work, as well as two new results, we also extend and improve an important result from the literature. [ABSTRACT FROM AUTHOR]

In this paper, we give some sufficient conditions on the instability of the zero solution of a kind of eighth order nonlinear differential equations of retarded type by using the Lyapunov direct method. The obtained sufficient conditions improve an existing result in the literature. [ABSTRACT FROM AUTHOR]

We establish some new sufficient conditions which guarantee the boundedness and the uniform ultimate boundedness of the solutions of a kind of nonlinear differential equations of third order. By defining an appropriate Lyapunov function, we prove two new theorems on the subject. Our results generalize and are complement some known boundedness results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear differential equation of third order. By the Lyapunov function approach, we discuss the convergence of the solutions of the equation considered. Our findings generalize some well known results in the literature. [ABSTRACT FROM AUTHOR]

By defining some appropriate Liapunov functions, we discuss boundedness of solutions to a class of non-autonomous and nonlinear differential equations of second order. In 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. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to obtain some new sufficient conditions which guarantee the instability of the trivial solution of some certain nonlinear vector differential equations of fourth order. Our results improve and include some well known instability results, which were established on the instability of the trivial solution of certain scalar differential equations of fourth order, to certain nonlinear vector differential equations of fourth order. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a kind of Rayleigh equation with finitely many deviating arguments of the form By using the coincidence degree theory, we establish the results on the existence and uniqueness of periodic solutions for the above equation. The results we obtained include and improve the results existing in the literature. [Copyright &y& Elsevier]

This paper presents sufficient conditions for all solutions as well as their derivatives up to second order to x‴ +a(t)f (x′)x″ + b(t, x, x′)+c(t)g(x) = e(t, x, x′, x″) to be bounded. What is more, it shows that all the solutions are in LP[0, ∞) under somewhat more restrictive conditions. [ABSTRACT FROM AUTHOR]

Sufficiency criteria are established to ensure the asymptotic stability and boundedness of solutions to third-order nonlinear delay differential equations of the form x⃛ (t) + e(x(t), ẋ(t), ẍ(t))ẍ(t) + g(x(t - r), ẋ(t - r)) + ψ(x(t - r)) = p(t, x(t), x(t - r), x'(t), x'(t - r), x"(t)). By using Lyapunov's functional approach, we obtain two new results on the subject, which include and improve some related results in the relevant literature. Two examples are also given to illustrate the importance of results obtained. [ABSTRACT FROM AUTHOR]

This paper presents sufficient conditions for all solutions as well as their derivatives up to second order to x‴+a(t)f(x′)x″+b(t, x, x′)+c(t)g(x) = e(t, x, x′ , x″) to be bounded. What is more, it shows that all the solutions are in Lp[0,∞) under somewhat more restrictive conditions. [ABSTRACT FROM AUTHOR]