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Keywords:
Krasnoselskii's fixed points; periodic solution; large contraction
Krasnoselskii's fixed points; periodic solution; large contraction
Summary:
The fixed point theorem of Krasnoselskii and the concept of large contractions are employed to show the existence of a periodic solution of a nonlinear integro-differential equation with variable delay \begin{equation*} x'(t)= -\int_{t-\tau (t)}^{t}a(t,s) g(x(s))\,ds + \frac{d}{dt}Q (t,x(t-\tau (t)))+ G(t,x(t),x(t-\tau (t))). \end{equation*} We transform this equation and then invert it to obtain a sum of two mappings one of which is completely continuous and the other is a large contraction. We choose suitable conditions for $\tau $, $g$, $a$, $Q$ and $G$ to show that this sum of mappings fits into the framework of a modification of Krasnoselskii's theorem so that existence of nonnegative T-periodic solutions is concluded.
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