scholarly journals Oscillation of Second-Order Sublinear Impulsive Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
A. Zafer
Keyword(s):  
Differential Equations ◽  
Impulsive Differential Equations ◽  
Second Order ◽  
Oscillation Criteria ◽  
Impulsive Perturbations

Oscillation criteria obtained by Kusano and Onose (1973) and by Belohorec (1969) are extended to second-order sublinear impulsive differential equations of Emden-Fowler type:x″(t)+p(t)|x(τ(t))|α-1x(τ(t))=0,t≠θk;Δx'(t)|t=θk+qk|x(τ(θk))|α-1x(τ(θk))=0;Δx(t)|t=θk=0,  (0<α<1)by considering the casesτ(t)≤tandτ(t)=t, respectively. Examples are inserted to show how impulsive perturbations greatly affect the oscillation behavior of the solutions.

scholarly journals Interval Oscillation Criteria of Second Order Mixed Nonlinear Impulsive Differential Equations with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Zhonghai Guo ◽  
Xiaoliang Zhou ◽  
Wu-Sheng Wang
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Second Order ◽  
Oscillation Criteria ◽  
Nonnegative Constant ◽  
Interval Oscillation ◽  
Equations With Delay

We study the following second order mixed nonlinear impulsive differential equations with delay(r(t)Φα(x′(t)))′+p0(t)Φα(x(t))+∑i=1npi(t)Φβi(x(t-σ))=e(t),t≥t0,t≠τk,x(τk+)=akx(τk),x'(τk+)=bkx'(τk),k=1,2,…, whereΦ*(u)=|u|*-1u,σis a nonnegative constant,{τk}denotes the impulsive moments sequence, andτk+1-τk>σ. Some sufficient conditions for the interval oscillation criteria of the equations are obtained. The results obtained generalize and improve earlier ones. Two examples are considered to illustrate the main results.

scholarly journals Oscillation of Certain Second-Order Sub-Half-Linear Neutral Impulsive Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Yuangong Sun
Keyword(s):  
Differential Equations ◽  
Impulsive Differential Equations ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Oscillation Criteria ◽  
Auxiliary Functions

By introducing auxiliary functions, we investigate the oscillation of a class of second-order sub-half-linear neutral impulsive differential equations of the form[r(t)ϕβ(z′(t))]′+p(t)ϕα(x(σ(t)))=0,  t≠θk,Δϕβ(z′(t))|t=θk+qkϕα(x(σ(θk)))=0,Δx(t)|t=θk=0,whereβ>α>0,  z(t)=x(t)+λ(t)x(τ(t)).  Several oscillation criteria for the above equation are established in both the case0≤λ(t)≤1and the case-1<-μ≤λ(t)≤0,which generalize and complement some existing results in the literature. Two examples are also given to illustrate the effect of impulses on the oscillatory behavior of solutions to the equation.

scholarly journals Interval Oscillation Criteria for Super-Half-Linear Impulsive Differential Equations with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Zhonghai Guo ◽  
Xiaoliang Zhou ◽  
Wu-Sheng Wang
Keyword(s):  
Differential Equations ◽  
Impulsive Differential Equations ◽  
Second Order ◽  
Riccati Transformation ◽  
Oscillation Criteria ◽  
Nonnegative Constant ◽  
Interval Oscillation ◽  
Equations With Delay

We study the following second-order super-half-linear impulsive differential equations with delay[r(t)φγ(x′(t))]′+p(t)φγ(x(t-σ))+q(t)f(x(t-σ))=e(t),t≠τk,x(t+)=akx(t),x′(t+)=bkx′(t),t=τk, wheret≥t0∈ℝ,φ*(u)=|u|*-1u,σis a nonnegative constant,{τk}denotes the impulsive moments sequence withτ1<τ2<⋯<τk<⋯,lim k→∞τk=∞, andτk+1-τk>σ. By some classical inequalities, Riccati transformation, and two classes of functions, we give several interval oscillation criteria which generalize and improve some known results. Moreover, we also give two examples to illustrate the effectiveness and nonemptiness of our results.

scholarly journals Second-order impulsive differential systems with mixed and several delays

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shyam Sundar Santra ◽  
Apurba Ghosh ◽  
Omar Bazighifan ◽  
Khaled Mohamed Khedher ◽  
Taher A. Nofal
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Oscillation Theory ◽  
Second Order ◽  
Neutral Delay ◽  
Differential Systems ◽  
Impulsive Differential Systems ◽  
Several Delays ◽  
Necessary And Sufficient

AbstractIn this work, we present new necessary and sufficient conditions for the oscillation of a class of second-order neutral delay impulsive differential equations. Our oscillation results complement, simplify and improve recent results on oscillation theory of this type of nonlinear neutral impulsive differential equations that appear in the literature. An example is provided to illustrate the value of the main results.

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