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Index of solution set for perturbed Fredholm equations and existence of periodic solutions for delay differential equations. (English) Zbl 1055.34133

The authors consider the topological index of the solution set of Fredholm equations with \(f\)-condensing-type perturbations and apply this to investigate periodic solutions of delay differential equations of the form \[ a(t,x(t),x(t-\tau), x'(t),x'(t-\tau))=b(t,x(t),x(t-\tau),x'(t),x'(t-\tau)), \] where the functions \(a, b\) are \(\omega\)-periodic in the first variable and the delay \(\tau\) is commensurable with \(\omega,\) that is there exists \(\tau_ 0>0\) such that \(p\tau_ 0=\omega\) and \(k\tau_ 0=\tau\) for some integers \(p\) and \(k,\) \(p>k.\) As an application, the existence of a periodic solution for a nonlinear differential equation is considered.

MSC:

34K13 Periodic solutions to functional-differential equations
47H11 Degree theory for nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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