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Existence of solutions of periodic boundary value problems for impulsive functional Duffing equations at nonresonance case. (English) Zbl 1158.34337

Summary: This paper deals with the existence of solutions of the periodic boundary value problem of the impulsive Duffing equation:
\[ x''(t)+\alpha x'(t)+\beta x(t)=f(t,x(t),x(\alpha_1(t)),\dots,x(\alpha_n(t))),\text{ a.e. }t\in[0,T], \]
\[ \begin{aligned}\Delta x(t_k)=I_k(x(t_k),\;x'(t_k)),&\quad k=1,\dots,m,\\ \Delta x'(t_k)=J_k(x(t_k),x'(t_k)),&\quad k=1,\dots,m,\\ x^{(i)}(0)=x^{(i)}(T),&\quad i=0,1. \end{aligned} \]
Sufficient conditions are established for the existence of at least one solution of above-mentioned boundary value problem. Our method is based upon Schaefer’s fixed-point theorem. Examples are presented to illustrate the efficiency of the obtained results.

MSC:

34K10 Boundary value problems for functional-differential equations
34K45 Functional-differential equations with impulses
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References:

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