Some stability and boundedness conditions for non-autonomous differential equations with deviating arguments

In this article, the author studies the stability and boundedness of solutions for the non-autonomous third order differential equation with a deviating argument, $r$: \begin{equation*} \begin{array}{c} x^{\prime \prime \prime }(t)+a(t)x^{\prime \prime }(t)+b(t)g_{1}(x^{\prime}(t-r))+g_{2}(x^{\prime}(t))+h(x(t-r)) \\ =p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime }(t-r),x^{\prime \prime }(t)), \end{array} \end{equation*} where $r>0$ is a constant. Sufficient conditions are obtained; a stability result in the literature is improved and extended to the preceding equation for the case $p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime}(t-r),x^{\prime \prime }(t))=0,$ and a new boundedness result is also established for the case $p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime}(t-r),x^{\prime \prime }(t))\neq 0.

UNIFORM STABILITY AND BOUNDEDNESS OF SOLUTIONS OF NONLINEAR DELAY DIFFERENTIAL EQUATIONS OF THE THIRD ORDER

In this paper, a complete Lyapunov functional was con- structed and used to obtain criteria (when p = 0) for uniform asymptotic stability of the zero solution of the nonlinear delay differential equation (1.1). When p ≠ 0, sufficient conditions are also established for uni- form boundedness and uniform ultimate boundedness of solutions of this equation. Our results improve and extend some well known results in the literature

Ultimate boundedness and periodicity results for a certain system Of third-order nonlinear Vector delay differential equations

In the last years, there has been increasing interest in obtaining the sufficient conditions for stability, instability, boundedness, ultimately boundedness, convergence, etc. For instance, in applied sciences some practical problems concerning mechanics, engineering technique fields, economy, control theory, physical sciences and so on are associated with third, fourth and higher order nonlinear differential equations. The problem of the boundedness and stability of solutions of vector differential equations has been widely studied by many authors, who have provided many techniques especially for delay differential equations. In this work a class of third order nonlinear non-autonomous vector delay differential equations is considered by employing the direct technique of Lyapunov as basic tool, where a complete Lyapunov functional is constructed and used to obtain sufficient conditions that guarantee existence of solutions that are periodic, uniformly asymptotically stable, uniformly ultimately bounded and the behavior of solutions at infinity. In addition to being for a more general equation, the obtained results here are new even when our equation is specialized to the forms previously studied and include many recent results in the literature. Finally, an example is given to show the feasibility of our results

Boundedness and Square Integrability in Neutral Differential Systems of Fourth Order

The aim of this paper is to study the asymptotic behavior of solutions to a class of fourth-order neutral differential equations. We discuss the stability, boundedness and square integrability of solutions for the considered system. The technique of proofs involves defining an appropriate Lyapunov functional. Our results obtained in this work improve and extend some existing well-known related results in the relevant literature which were obtained for nonlinear differential equations of fourth order with a constant delay. The obtained results here are new even when our equation is specialized to the forms previously studied and include many recent results in the literature. Finally, an example is given to show the feasibility of our results

Boundedness criteria for a class of second order nonlinear differential equations with delay

summary:We consider certain class of second order nonlinear nonautonomous delay differential equations of the form $$a(t)x^{\prime \prime } + b(t)g(x,x^\prime ) + c(t)h(x(t-r))m(x^\prime ) = p(t,x,x^\prime )$$ and $$(a(t)x^\prime )^\prime + b(t)g(x,x^\prime ) + c(t)h(x(t-r))m(x^\prime ) = p(t,x,x^\prime ),$$ where $a$, $b$, $c$, $g$, $h$, $m$ and $p$ are real valued functions which depend at most on the arguments displayed explicitly and $r$ is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovski\v ı functional to establish our results. This work extends and improve on some results in the literature

On stability, boundedness, and square integrability of solutions of certain third order neutral differential equations

summary:The authors establish some new sufficient conditions under which all solutions of a certain class of nonlinear neutral delay differential equations of the third order are stable, bounded, and square integrable. Illustrative examples are given to demonstrate the main results

On the stability, boundedness, and square integrability of solutions of third order neutral delay differential equations

In this paper, sufficient conditions are established for the stability, boundedness and square integrability of solutions for some non-linear neutral delay differential equations of third order. Lyapunov’s direct method is used to obtain the results

On the Global Existence and Boundedness of Solutions of Nonlinear Vector Differential Equations of Third Order

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\u27s 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

Global properties of an age-structured virus model with saturated antibody immune response, multi-target cells and general incidence rate

Some viruses, such as human immunodeficiency virus, can infect several types of cell populations. The age of infection can also affect the dynamics of infected cells and production of viral particles. In this work, we study a virus model with infection-age and different types of target cells which takes into account the saturation effect in antibody immune response and a general non-linear infection rate. We construct suitable Lyapunov functionals to show that the global dynamics of the model is completely determined by two critical values: the basic reproduction number of virus and the reproductive number of antibody response