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 [2] Smart, D. R., Fixed Point Theorems (1980), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0427.47036 [3] O’regan, D., Fixed-point theory for the sum of two operators, Appl. Math. Lett., 9, 1, 1-8 (1996) · Zbl 0858.34049 [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) · Zbl 0165.49102 [6] Kreyszig, E., Introductory Functional Analysis with Applications (1978), Wiley: Wiley New York · Zbl 0368.46014
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