scholarly journals An instability theorem for a certain fifth-order delay differential equation

Filomat ◽  
2011 ◽  
Vol 25 (3) ◽  
pp. 145-151 ◽  
Author(s):  
Cemil Tunç
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Delay Differential Equation ◽  
Functional Approach ◽  
Constant Delay ◽  
Order Nonlinear Differential Equation ◽  
Delay Differential ◽  
Fifth Order

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.

scholarly journals Instability of a Fifth-Order Nonlinear Vector Delay Differential Equation with Multiple Deviating Arguments

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Cemil Tunç
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Zero Solution ◽  
Functional Approach ◽  
Previous Literature ◽  
Deviating Arguments ◽  
Multiple Deviating Arguments ◽  
Nonlinear Vector ◽  
Delay Differential ◽  
Fifth Order

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.

Variation formulas of solution for a delay differential equation taking into account delay perturbation and the continuous initial condition

2011 ◽  
Vol 18 (2) ◽  
pp. 345-364
Author(s):  
Tamaz Tadumadze
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Delay Differential Equation ◽  
Initial Function ◽  
Initial Moment ◽  
Initial Condition ◽  
Constant Delay ◽  
Non Linear ◽  
Linear Differential ◽  
Delay Differential

Abstract Variation formulas of solution are proved for a non-linear differential equation with constant delay. In this paper, the essential novelty is the effect of delay perturbation in the variation formulas. The continuity of the initial condition means that the values of the initial function and the trajectory always coincide at the initial moment.

scholarly journals Upper semi continuity of attractors of delay differential equations in the delay

2006 ◽  
Vol 73 (2) ◽  
pp. 299-306 ◽  
Author(s):  
Peter E. Kloeden
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Global Attractor ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Constant Delay ◽  
Nonzero Constant ◽  
Nonempty Compact ◽  
Delay Differential ◽  
Compact Subsets

It is shown that if a retarded delay differential equation has a global attractor in the space C ([—τ0, ], ℝd) for a given nonzero constant delay τ0, then the equation has an attractor Aτ in the space C ([—τ, 0], ℝd) for nearby constant delays τ. Moreover the attractors Aτ converge upper semi continuously to in C ([—τ0, 0], ℝd) in the sense that they are identified through corresponding segments of entire trajectories in ℝd with nonempty compact subsets of C ([—τ0, 0], ℝd) which converge upper semi continuously to in C ([—τ0, 0], ℝd).

scholarly journals Oscillatory behavior of a second order nonlinear advanced differential equation with mixed neutral terms

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hongwei Shi ◽  
Yuzhen Bai
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Oscillation Criteria ◽  
Order Nonlinear Differential Equation

AbstractIn this paper, we present several new oscillation criteria for a second order nonlinear differential equation with mixed neutral terms of the form $$ \bigl(r(t) \bigl(z'(t)\bigr)^{\alpha }\bigr)'+q(t)x^{\beta } \bigl(\sigma (t)\bigr)=0,\quad t\geq t_{0}, $$(r(t)(z′(t))α)′+q(t)xβ(σ(t))=0,t≥t0, where $z(t)=x(t)+p_{1}(t)x(\tau (t))+p_{2}(t)x(\lambda (t))$z(t)=x(t)+p1(t)x(τ(t))+p2(t)x(λ(t)) and α, β are ratios of two positive odd integers. Our results improve and complement some well-known results which were published recently in the literature. Two examples are given to illustrate the efficiency of our results.

scholarly journals Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 105
Author(s):  
Lokesh Singh ◽  
Dhirendra Bahuguna
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Invariant Manifold ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Stable Manifold ◽  
Exponential Dichotomy ◽  
Delay Differential ◽  
Stable Invariant ◽  
Nonuniform Exponential Dichotomy

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

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