14 pages
English
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Oscillation of impulsive functional differential equations with oscillatory potentials and Riemann-Stieltjes integrals

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14 pages
English

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This paper addresses the oscillation problem of a class of impulsive differential equations with delays and Riemann-Stieltjes integrals that cover many equations in the literature. In the case of oscillatory potentials, both El-Sayed type and Kamenev type oscillation criteria are established by overcoming the difficulty caused by impulses and oscillatory potentials in the estimation of the delayed argument. The main results not only generalize some existing results but also drop a restrictive condition imposed on impulse constants. Finally, two examples are presented to illustrate the theoretical results. MSC: 34K11.

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Publié le 01 janvier 2012
Nombre de lectures 7
Langue English

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Liu and SunAdvances in Difference Equations2012,2012:175 http://www.advancesindifferenceequations.com/content/2012/1/175
R E S E A R C HOpen Access Oscillation of impulsive functional differential equations with oscillatory potentials and Riemann-Stieltjes integrals * Zhi Liu and Yuangong Sun
* Correspondence: sunyuangong@yahoo.cn School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P.R. China
Abstract This paper addresses the oscillation problem of a class of impulsive differential equations with delays and Riemann-Stieltjes integrals that cover many equations in the literature. In the case of oscillatory potentials, both El-Sayed type and Kamenev type oscillation criteria are established by overcoming the difficulty caused by impulses and oscillatory potentials in the estimation of the delayed argument. The main results not only generalize some existing results but also drop a restrictive condition imposed on impulse constants. Finally, two examples are presented to illustrate the theoretical results. MSC:34K11 Keywords:oscillation; impulse; delay; Riemann-Stieltjes integral
1 Introduction Recent years have witnessed a rapid progress in the theory of impulsive differential equa-tions which provide a natural description of the motion of several real world processes subject to short time perturbations. Due to many applications in physics, chemistry, pop-ulation dynamics, ecology, biological systems, control theory,etc.[–], the theory of im-pulsive differential equations has been extensively studied in [–]. We are here concerned with the oscillation problem of impulsive functional differential equations. Compared to equations without impulses, the oscillation of impulsive differen-tial equations receives less attention [–]. In this paper, we investigate the oscillation of the following impulsive differential equation with delay and Riemann-Stieltjes integral: h  α(s) (r(t)x(t)) +q(t)x(t) +p(t,s)|x(τ(t,s))|sgnx(τ(t,s))dξ(s) =e(t),t=tk, () + –+x(t) =ckx(t),x(t) =dkx(t), k kk k h wherett,  <h<;f(s)dξ(s) denotes the Riemann-Stieltjes integral of the function fon [,h] with respect toξ, andξ: [,h]Ris nondecreasing;α(s) is a strictly increasing continuous function on [,h] satisfying α() <  <α(h);{tk}denotes the sequence k= of impulse moments satisfying t<t<t<∙ ∙ ∙<tk<∙ ∙ ∙andlimk→∞tk= +;ck, dkare positive impulse constants, andck;rC[t,) withr(t) > ,q,eC[t,), andpC([t,)×[,h]); the time delayτ(t,s) : [t,)×[,h][σ,) withσtis continuous,τ(t,s)tandlimt+τ(t,s) = +fors[,h].
©2012 Liu and Sun; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.