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

Dmitrienko, Vladimir T.; Zvyagin, Viktor G.

Topol. Methods Nonlinear Anal. 21, No. 1, 53-80 (2003).

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.

Reviewer: Sotiris K. Ntouyas (Ioannina)

MSC:

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

Keywords:

index of solution set; \(f\)-condensing map; measure of noncompactness; delay differential equation