Application of interval analysis to impulsive differential equations (Q1184815)

Starting with the definition of an interval operator \(K\) associated with a given operator \(k\) on a subset \(D\) of an ordered Banach space, the author proves a criterion which yields the smallest and the largest fixed point of \(k\) within some given interval \(U\subseteq D\). This criterion is applied to initial value problems of the form (*) \(y'=f(t,y(t))\), \(t\in[0,T]\setminus\{t_ 1,\dots,t_ p\}\); \(y(t^ +_ k)=I_ k(y(t_ k))\), \(t_ k\in(0,T)\), \(k=1,\dots,p\); \(y(0)=y_ 0\), with \(f: [0,T]\times\mathbb{R}^ n\to\mathbb{R}^ n\) being continuous for \(t\neq t_ k\) and \(I_ k:\mathbb{R}^ n\to\mathbb{R}^ n\). A solution \(x\) of (*) is required to fulfill (*), to be continuously differentiable for \(t\neq t_ k\) and to be left sided continuous for \(t=t_ k\) with existing \(x(t^ +_ k)\). Introducing the definition of lower and upper solutions \(\alpha\), \(\beta\) of (*) and restricting \(f\) and \(I_ k\) slightly, an infinite sequence of intervals is constructed starting with \([\alpha,\beta]\) and tending to some interval \(U_ 0=[\underline{U}_ 0,\overline {U}_ 0]\subseteq[\alpha,\beta]\) which contains all solutions of (*) from \([\alpha,\beta]\) and for which \(\underline{U}_ 0\), \(\overline{U}_ 0\) are, respectively, the smallest and the largest solution of (*) within \([\alpha,\beta]\). In a final section (*) is considered for Lebesgue integrable right hand sides \(f(t,x(t))\). The existence of an interval \(U_ 0\) with the properties above is proved.