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Impulsive periodic boundary value problem and topological degree. (English) Zbl 1048.34061

Summary: The paper deals with the impulsive periodic boundary value problem \[ u''=f(t,u,u'),\;u(t_1+)= J\bigl(u(t_1)\bigr),\;u'(t_1+)= M\bigl(u'(t_1-) \bigr), \]
\[ u(0)=u(T),\;u'(0)=u'(T). \] The problem is reformulated as an operator equation \(u-Fu=0\). Our main results are obtained by determining the Leray-Schauder degree of the operator \(I-F\) with respect to certain open sets \(\Omega_1\) or \(\Omega_2\) which are given in terms of a strict lower function \(\sigma_1\) and a strict upper function \(\sigma_2\). We do not restrict ourselves to the well ordered \(\sigma_1<\sigma_2\) on \([0,T]\) but we study the nonordered \(\sigma_1\) and \(\sigma_2\) as well as the reversely ordered \(\sigma_2<\sigma_1\) on \([0,T]\). These results are substantially used in our forthcoming papers, where we get rid of the assumption on the strictness of lower and upper functions and obtain new existence criteria for the given problem.

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators
47N20 Applications of operator theory to differential and integral equations
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