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Periodic solutions of discontinuous impulsive differential systems. (English) Zbl 0753.34027
The authors study the existence of periodic solutions for a system of impulsive differential equations of second order with discontinuous (Carathéodory) functions. More precisely, it is considered the following periodic boundary value problem $$(*)$$ $$-x''=f(t,x,x')$$ for a.e. $$t\in J$$, $$t\neq t_ k,0,T$$, $$x(t^ +_ k)=I_ k(x(t_ k))$$, $$x'(t^ +_ k)=N_ k(x'(t_ k))$$, $$x(0)=x(T)$$, $$x'(0)=x'(T)$$, where $$J=[0,T]$$, $$t_ k\in(0,T)$$, $$k\in A=\{1,2,\ldots,p\}$$, $$f:J\times\mathbb{R}^ n\times\mathbb{R}^ n\to\mathbb{R}^ n$$ is a Carathéodory function, and $$I_ k,N_ k:\mathbb{R}^ n\to\mathbb{R}^ n$$ for each $$k\in A$$. The main result is given by Theorem 2.1: Let $$\alpha,\beta$$ be lower and upper solutions of $$(*)$$, respectively, with $$\alpha\leq\beta$$ on $$J$$. Suppose that $$f$$ is a Carathéodory function for each interval $$(t_ i,t_{i+1})$$, $$i=0,1,\ldots,p$$; it is quasimonotone nondecreasing and satisfies a Nagumo condition with respect to $$\alpha$$ and $$\beta$$. Moreover, it is assumed that $$I_ k,N_ k:\mathbb{R}^ n\to\mathbb{R}^ n$$ are continuous and nondecreasing for every $$k\in A$$. Then, the problem $$(*)$$ has at least one solution $$x$$ such that $$\alpha(t)\leq x(t)\leq\beta(t)$$ and $$| x'(t)|\leq C$$ for every $$t\in J$$.

##### MSC:
 34C25 Periodic solutions to ordinary differential equations 34A37 Ordinary differential equations with impulses 34B15 Nonlinear boundary value problems for ordinary differential equations
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