Periodic solutions of discontinuous impulsive differential systems
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 \(t\in J\).
impulsive differential equations of second order
scientific article; zbMATH DE number 57720
scientific article