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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 $M$.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
[1] Krasnoselskii, M. A.: Amer. math. Soc. transl.. 10, No. 2, 345-409 (1958)
[2] Smart, D. R.: Fixed point theorems. (1980) · Zbl 0427.47036
[3] O’regan, D.: Fixed-point theory for the sum of two operators. Appl. math. Lett. 9, No. 1, 1-8 (1996)
[4] Reinermann, J.: Fixpunktsätze vom krasnoselski-typ. Math. Z. 119, 339-344 (1971) · Zbl 0204.45802
[5] Sadovskii, B. N.: A fixed-point principle. Func. anal. And applications 1, 151-153 (1967)
[6] Kreyszig, E.: Introductory functional analysis with applications. (1978) · Zbl 0368.46014