UNIFORM LIPSCHITZ STABILITY AND ASYMPTOTIC BEHAVIOR FOR PERTURBED DIFFERENTIAL SYSTEMS

2016 ◽  
Vol 99 (3) ◽  
pp. 393-412 ◽  
Author(s):  
Yoon Hoe Goo
Keyword(s):  
Asymptotic Behavior ◽  
Lipschitz Stability ◽  
Differential Systems

scholarly journals New Results on Qualitative Behavior of Second Order Nonlinear Neutral Impulsive Differential Systems with Canonical and Non-Canonical Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 934
Author(s):  
Shyam Sundar Santra ◽  
Khaled Mohamed Khedher ◽  
Kamsing Nonlaopon ◽  
Hijaz Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Discrete Equation ◽  
Qualitative Behavior ◽  
Differential Systems ◽  
Impulsive Differential Systems

The oscillation of impulsive differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of impulsive differential equations. In this work, several sufficient conditions are established for oscillatory or asymptotic behavior of second-order neutral impulsive differential systems for various ranges of the bounded neutral coefficient under the canonical and non-canonical conditions. Here, one can see that if the differential equations is oscillatory (or converges to zero asymptotically), then the discrete equation of similar type do not disturb the oscillatory or asymptotic behavior of the impulsive system, when impulse satisfies the discrete equation. Further, some illustrative examples showing applicability of the new results are included.

Asymptotic analysis of positive solutions of first-order cyclic functional differential systems

2017 ◽  
Vol 24 (1) ◽  
pp. 63-80
Author(s):  
Jaroslav Jaroš ◽  
Kusano Takaŝi
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Positive Solutions ◽  
Continuous Functions ◽  
Regularly Varying Functions ◽  
Differential Systems ◽  
First Order ◽  
Functional Differential Systems ◽  
Functional Differential ◽  
Scalar Equations

AbstractThe structure and the asymptotic behavior of positive increasing solutions of functional differential systems of the form$x^{\prime}(t)=p(t)\varphi_{\alpha}\bigl{(}y(k(t))\bigr{)},\quad y^{\prime}(t)=% q(t)\varphi_{\beta}\bigl{(}x(l(t))\bigr{)}$are investigated in detail, where α and β are positive constants,${p(t)}$and${q(t)}$are positive continuous functions on${[0,\infty)}$,${k(t)}$and${l(t)}$are positive continuous functions on${[0,\infty)}$tending to${\infty}$witht, and${\varphi_{\gamma}(u)=\lvert u\rvert^{\gamma}\operatorname{sgn}u}$,${\gamma>0}$,${u\in\mathbb{R}}$. An extreme class of positive increasing solutions, calledrapidly increasing solutions, of the system above is analyzed by means of regularly varying functions. The results obtained find applications to systems of the form$x^{\prime}(g(t))=p(t)\varphi_{\alpha}\bigl{(}y(k(t))\bigr{)},\quad y^{\prime}(% h(t))=q(t)\varphi_{\beta}\bigl{(}x(l(t))\bigr{)},$and to scalar equations of the type$\Bigl{(}p(t)\varphi_{\alpha}\bigl{(}x^{\prime}(g(t))\bigr{)}\Bigr{)}^{\prime}=% p(t)\varphi_{\beta}\bigl{(}x(l(t))\bigr{)}.$

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