## 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

### Keywords:

fixed points; integral equation; periodic solutions
Full Text:

### References:

  Krasnoselskii, M.A, Amer. math. soc. transl., 10, 2, 345-409, (1958)  Smart, D.R, Fixed point theorems, (1980), Cambridge University Press Cambridge · Zbl 0427.47036  O’regan, D, Fixed-point theory for the sum of two operators, Appl. math. lett., 9, 1, 1-8, (1996) · Zbl 0858.34049  Reinermann, J, Fixpunktsätze vom krasnoselski-typ, Math. Z., 119, 339-344, (1971) · Zbl 0204.45802  Sadovskii, B.N, A fixed-point principle, Func. anal. and applications, 1, 151-153, (1967) · Zbl 0165.49102  Kreyszig, E, Introductory functional analysis with applications, (1978), Wiley New York · Zbl 0368.46014
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