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This paper deals with a perturbed nonlinear system of fractional order differential equations (FrODEs) with Caputo derivative. The purpose of the paper is to discuss uniform stability (US), asymptotic stability (AS), Mittag-Leffer stability (MLS) of zero solution and boundedness at infinity of non-zero solutions of this perturbed nonlinear system of FrODEs with Caputo derivative. We obtain four new theorems on these mathematical concepts via a Lyapunov function (LF) and its Caputo derivative. For illustration, an example is provided which satisfies assumptions of the four new results and, in particular, shows their applications. The new results of this paper generalize and improve some recent ones in the literature and they have contributions to theory of FrODEs. [ABSTRACT FROM AUTHOR]

The concept of P-perturbed systems with respect to given system of ordinary differential equations is introduced. Certain sufficient conditions for uniform-ultimate Poisson boundedness of solutions of P-perturbed systems are obtained by using the method of Lyapunov functions. In conclusion, an interesting application of the above sufficient conditions to the standard closed electric RLC circuit with nonlinear elements is given. [ABSTRACT FROM AUTHOR]

Linear impulsive switched differential equations in Hilbert space are studied. We propose a new method of stability investigation, which is based in constructing an equivalent impulsive system without switching. Using operator-valued and scalar Lyapunov functions, sufficient conditions for Lyapunov stability of linear switched impulsive equations in a Banach space are established. The examples of studies of this class of equations with periodic switching are given. [ABSTRACT FROM AUTHOR]

This work is devoted to investigate the stochastic asymptotically stability of the zero solution for a kind of third-order stochastic differentials equation with variable and constant delays by a suitable Lyapunov functional. Our results improve and form a complement to some results that can be found in the literature. In the last section, we give an example to illustrate our main result. [ABSTRACT FROM AUTHOR]

We consider a linear functional differential equation of first order and neutral type with multiple constant delays and present a functional Lyapunov approach to square integrability and asymptotic analysis of solutions of the equation considered. Consequently explicit criteria are derived for square integrability and asymptotic analysis of solutions of the equation considered. A discussion of the result obtained and an illustrative example are given. The result given extends and improves some found in the literature. [ABSTRACT FROM AUTHOR]

This paper is devoted to the instability of Set Differential Equations (SDEs). Using the geometric inequalities of Brunn–Minkowski and A.D. Aleksandrov, we propose new methods for constructing Lyapunov functions. In combination with the known methods of stability theory, the Lyapunov's direct method, the comparison method and the vector-function method, we establish conditions for the collapse of the solutions of the SDEs. Estimates of the collapse time of solutions are also obtained. Examples of SDEs in spaces of dimension 2 and 3 illustrating general theorems are given. [ABSTRACT FROM AUTHOR]

For a linear periodic system with impulsive action and delay, new approaches to the study of stability were proposed on the basis of the methods of spectral theory of linear operators, direct Lyapunov method, and N.G. Chetaev method for construction of the Lyapunov functions for the periodic linear systems, as well as the perturbation method for construction of the Lyapunov functions. These methods underlie the sufficient conditions for asymptotic stability of the linear periodic systems with impulsive action and delay. We gave some illustrative examples of studying stability of such systems under different assumptions about the dynamic properties of the continuous and discrete components of the impulsive system. [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]

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]

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, 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]

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]

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 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]

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]

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 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]

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]

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]

In this paper, we are concerned with the existence of periodic solutions to nonlinear third order delay differential equation: x‴(t) +ψ (x′(t))x″(t) + g(x′(t - r)) + f (x(t)) = p(t, x(t), x(t - r), x′(t), x′(t - r), x″(t)), when p(t, x(t), x(t - r), x′(t), x′(t - r), x″(t)) is a periodic function of period T , T > 0. With use of Lyapunov's functional approach, we establish some new sufficient conditions which guarantee that there exists a periodic solution of this equation of period T , T > 0. For illustrations, an example is also given on the existence of periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundedness of solutions to fourth order nonlinear delay differential equation x(4) (t) + f1 (xll(t))xlll(t) + f2(xl(t),xll(t))xll(t)+ g(xl(t - r)) + h(x(t - r)) = p(t,x(t),x(t - r),xl(t),xl(t - r),xll(t),xll(t - r),xlll(t)), where r > 0 is a constant delay. In proving our main result, we make use of the Lyapunov's second method by constructing a Lyapunov functional. [ABSTRACT FROM AUTHOR]

By constructing a Lyapunov function, a new result is given, which guarantees the non-existence of nontrivial periodic solutions to nonlinear vector differential equation of eighth order: ... An example is also established for the illustrations of topic. By this way, our findings raise a new result for the nonexistence of nontrivial periodic solutions related to this nonlinear vector differential equation of eighth order. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, by using Liapunov’s second method, we establish some new results for stability and boundedness of solutions of nonlinear vector differential equations of third order. By constructing a Liapunov function, sufficient conditions for stability and boundedness of solutions of equations considered are obtained. Concerning to the subject, some explanatory examples are also given. Our results improve and include a result existing in the literature. [Copyright &y& Elsevier]

In this paper, a new instability criteria for nonlinear sixth-order vector differential equations of the form X(6) + AX(5) + BX(4) + E(X, Ẋ, Ẍ, X⃛, X(4), X(5)) X⃛ + F(Ẋ) Ẍ +G(X,Ẋ, Ẍ, X⃛, X(4), X(5)) Ẋ + H(X) = 0 is given. Our result includes and improves the result established in [6] [ABSTRACT FROM AUTHOR]

By constructing a Lyapunov function, a new instability result is established, which guarantees thatthe 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, ourfindings improve an instability result related to a scalar differential equation in the literature toinstability of the trivial solution to the afore-mentioned differential equation.

By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established result is less restrictive than those reported in the literature. [ABSTRACT FROM AUTHOR]

The sufficient conditions are given for all solutions of certain non-autonomous differential equation to be uniformly bounded and convergence to zero as t→∞. The result given includes and improves that result obtained by Abou-El-Ela & Sadek. [ABSTRACT FROM AUTHOR]

Sufficient conditions have been obtained so that all solutions of a different fourth order differential equation are bounded and uniformly bounded. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions are established under which all solutions of some non-autonomous differential equations of the fourth order are bounded and uniformly bounded. [ABSTRACT FROM AUTHOR]

In this paper, we consider two different models of nonlinear ordinary differential equations (ODEs) of second order. We construct two new Lyapunov functions to investigate boundedness of solutions of those nonlinear ODEs of second order. By using the Lyapunov direct or second method and inequality techniques, we prove two new theorems on the boundedness solutions of those ODEs of second order as t → ∞ . When we compare the conditions of the theorems of this paper with those of Meng in (J. Syst. Sci. Math. Sci. 15(1):50–57, 1995) and Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002), we can see that our theorems have less restrictive conditions than those in (Meng in J. Syst. Sci. Math. Sci. 15(1):50–57, 1995) and Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002) because of the two new suitable Lyapunov functions. Next, in spite of the use of the Lyapunov second method here and in (Meng in J. Syst. Sci. Math. Sci. 15(1):50–57, 1995; Sun and Meng in Ann. Differ. Equ. 18(1):58–64 2002), the proofs of the results of this paper are proceeded in a very different way from that used in the literature for the qualitative analysis of ODEs of second order. Two examples are given to show the applicability of our results. At the end, we can conclude that the results of this paper generalize and improve the results of Meng in (J. Syst. Sci. Math. Sci. 15(1):50–57, 1995), Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002), and some other that can be found in the literature, and they have less restrictive conditions than those in these references. [ABSTRACT FROM AUTHOR]

This paper proposes two models of a general nonlinear within-host Chikungunya virus (CHIKV) dynamics. The production, incidence, proliferation and removal rates of all compartments are modeled by general nonlinear functions that satisfy a set of reasonable conditions. The second model takes into consideration two forms of infected host cells: (i) latently infected cells which do not produce the CHIKV, (ii) actively infected cells which generate the CHIKV particles. We show that all the solutions of the models are nonnegative and bounded. The global stability of the steady states of the models is proven by applying Lyapunov method and LaSalle’s invariance principle. We perform numerical simulations to complement the obtained theoretical results. [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]