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Zbl 0896.47042
Burton, T.A.; Kirk, Colleen
A fixed point theorem of Krasnoselskii-Schaefer type.
(English)
[J] Math. Nachr. 189, 23-31 (1998). ISSN 0025-584X; ISSN 1522-2616/e

The authors focus on three fixed point theorems and an integral equation. Schaefer's fixed point theorem yields a $T$-periodic solution of $$x(t)= a(t)+ \int^t_{t- h}D(t, s)g(s, x(s))ds\tag 1$$ if $D$ and $g$ satisfy certain sign conditions independent of their magnitude. A combination of the contraction mapping theorem and Schauder's theorem (known as Krasnoselskii's theorem) yields a $T$-periodic solution of $$x(t)= f(t, x(t))+ \int^t_{t- h}D(t,s)g(s, x(s))ds\tag 2$$ if $f$ defines a contraction and if $D$ and $g$ are small enough.\par The authors prove a fixed point theorem which is a combination of the contraction mapping theorem and Schaefer's theorem which yields a $T$-periodic solution of (2) when $f$ defines a contraction mapping, while $D$ and $g$ satisfy the aforementioned sign conditions.
[J.Appell (Würzburg)]
MSC 2000:
*47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
45M15 Periodic solutions of integral equations

Keywords: fixed point theorems; integral equation; Schaefer's fixed point; $T$-periodic solution; contraction mapping theorem; Schauder's theorem; Krasnoselskii's theorem; sign conditions

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