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Existence of solutions to boundary value problems for impulsive second order differential inclusions. (English) Zbl 0784.34012

The authors consider a second order differential inclusion ${y}^{\text{'}\text{'}}\in F\left(t,y,{y}^{\text{'}}\right)$ subject to a set of nonlinear boundary constraints

$y\left({t}_{k}^{+}\right)={l}_{k}\left(y\left({t}_{k}\right)\right)\phantom{\rule{2.em}{0ex}}{y}^{\text{'}}\left({t}_{k}^{+}\right)={N}_{k}\left(y\left({t}_{k}\right),{y}^{\text{'}}\left({t}_{k}\right)\right)$

where ${l}_{k}$ are given homeomorphisms and ${N}_{k}$ are continuous, moreover

${G}_{i}\left(y\left({a}_{0}\right),{y}^{\text{'}}\left({a}_{0}\right),y\left({a}_{1}\right),{y}^{\text{'}}\left({a}_{1}\right)\right),\phantom{\rule{1.em}{0ex}}i=1,2$

where ${a}_{0}={t}_{0}<{t}_{1}<\cdots <{t}_{m}<{t}_{m+1}={a}_{1}$. Their main existence result, Theorem 2.2, is proved by means of the topological transversality method of Granas based on the existence of a priori bounds for the solutions of the above boundary value problem which is suitably modified to deal with the impulsive nature of the problem. Two motivating examples involving impulses are presented.

##### MSC:
 34A60 Differential inclusions 34A37 Differential equations with impulses 34B15 Nonlinear boundary value problems for ODE