Uniform stability and boundedness of solutions of nonlinear delay differential equations of the third order (original) (raw)

Stability and boundedness analysis for a system of two nonlinear delay differential equations

In this paper, the stability and boundedness analysis of a certain system of two nonlinear delay differential equations with variable delay ρ(t) is carried out. By using the Lyapunov's second method and Lyapunov-Krasovskii's functional derived from the differential equations describing the system which yielded a better stability and boundedness estimate to establish sufficient conditions for the uniform asymptotic stability of the trivial solution and uniform ultimate boundedness of solution. These new results improve and generalize some results that can be found in the literature.

On Some Qualitative Behaviors of Solutions to a Kind of Third Order Nonlinear Delay Differential Equations

Electron. J. Qual. Theory Differ. Equ, 2010

Sufficiency criteria are established to ensure the asymptotic stability and boundedness of solutions to third-order nonlinear delay differential equations of the form ... x (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.

STABILITY AND BOUNDEDNESS OF SOLUTIONS OF CERTAIN VECTOR DELAY DIFFERENTIAL EQUATIONS

is considered where X ∈ R n , 0 ≤ r(t) ≤ γ and A is a real constant, symmetric positive definite n×n matrix. By using the second method of Lyapunov and Lyapunov-Krasovskii's funtion we established sufficient conditions for the asymptotic stability of the zero solution when P (t, X,Ẋ) = 0 and boundedness of all solutions when P (t, X,Ẋ) = 0. The results obtained here are generalizations of some of the results obtained for R 1 .

Global stability of a class of scalar nonlinear delay differential equations

2003

The problem of global stability in scalar delay differential equations of the form x (t) = f 1 (x(t − h))g 2 (x(t)) − f 2 (x(t − h))g 1 (x(t)) is studied. Functions f i and g i , i = 1, 2, are continuous and such that the equation assumes a unique positive equilibrium. Two types of sufficient conditions for the global asymptotic stability of the unique equilibrium are established: (i) delay independent, and (ii) conditions involving the size h of the delay. Delay independent conditions make use of the global stability in the limiting (as h → ∞) difference equation f 1 (x n)g 2 (x n+1) = f 2 (x n)g 1 (x n+1): the latter always implying the global stability in the differential equation for all values of the delay h ≥ 0. The delay dependent conditions involve the global attractivity property in specially constructed one-dimensional maps (difference equations) that include the nonlinearities f i and g i , and the delay h.

Stability, Boundedness and periodic solutions to certain second order delay differential equations

Proyecciones, 2017

Stability, boundedness and existence of a unique periodic solution to certain second order nonlinear delay differential equations is discussed. By employing Lyapunov's direct (or second) method, a complete Lyapunov functional is constructed and used to establish sufficient conditions, on the nonlinear terms, that guarantee uniform asymptotic stability, uniform ultimate boundedness and existence of a unique periodic solution. Obtained results complement many outstanding recent results in the literature. Finally, examples are given to show the effectiveness of our method and correctness of our results.