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A fixed-point theorem of Krasnoselskii. (English) Zbl 1127.47318

Krasnosel’skij’s fixed-point theorem asks for a convex set \(M\) and a mapping \(Pz=Bz+Az\) such that (i) \(Bx+Ay\in M\) for each \(x,y\in M\), (ii) \(A\) is continuous and compact, (iii) \(B\) is a contraction. Then \(P\) has a fixed point. A careful reading of the proof reveals that (i) need only ask that \(Bx+Ay\in M\) when \(x=Bx+Ay\). The proof also yields a technique for showing that such \(x\) is in \(M\).

MSC:

47H10 Fixed-point theorems
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References:

[1] Krasnoselskii, M. A., Amer. Math. Soc. Transl., 10, 2, 345-409 (1958) · Zbl 0080.10403
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