scholarly journals Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under(μ,ν)-Dichotomies

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Lijun Pan
Keyword(s):  
Differential Equations ◽  
Nonlinear Equation ◽  
Delay Differential Equations ◽  
Invariant Manifolds ◽  
Parameter Dependence ◽  
Nonlinear Perturbations ◽  
Linear Delay ◽  
Stable Invariant Manifolds ◽  
Delay Differential ◽  
Stable Invariant

We obtain the existence of stable invariant manifolds for the nonlinear equationx′=L(t)xt+f(t,xt,λ)provided that the linear delay equationx′=L(t)xtadmits a nonuniform(μ,ν)-dichotomy andfis a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent on parameterλ. Namely, the stable invariant manifolds are Lipschitz in the parameterλ. In addition, we also show that nonuniform(μ,ν)-contraction persists under sufficiently small nonlinear perturbations.

scholarly journals An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Süleyman Cengizci
Keyword(s):  
Differential Equations ◽  
Hybrid Method ◽  
Delay Differential Equations ◽  
Numerical Procedure ◽  
Taylor Series Expansion ◽  
Expansion Method ◽  
Singularly Perturbed ◽  
Convection Term ◽  
Linear Delay ◽  
Delay Differential

In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.

scholarly journals Notes on oscillation of linear delay differential equations

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Božena Dorociaková ◽  
Radoslav Chupáč ◽  
Rudolf Olach
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Linear Delay ◽  
Delay Differential

Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms

2019 ◽  
Vol 69 (5) ◽  
pp. 1117-1126
Author(s):  
Said R. Grace ◽  
John R. Graef ◽  
Irena Jadlovská
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Oscillation Criteria ◽  
Linear Delay ◽  
Delay Differential

Abstract This article concerns the oscillatory behavior of solutions to second-order half-linear delay differential equations with mixed neutral terms. The authors present new oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are illustrated with examples.

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