Burton, T. A. A fixed-point theorem of Krasnoselskii. (English) Zbl 1127.47318 Appl. Math. Lett. 11, No. 1, 85-88 (1998). 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 \(M\). Cited in 3 ReviewsCited in 117 Documents MSC: 47H10 Fixed-point theorems Keywords:fixed points; integral equation; periodic solutions PDF BibTeX XML Cite \textit{T. A. Burton}, Appl. Math. Lett. 11, No. 1, 85--88 (1998; Zbl 1127.47318) Full Text: DOI OpenURL References: [1] Krasnoselskii, M.A, Amer. math. soc. transl., 10, 2, 345-409, (1958) [2] Smart, D.R, Fixed point theorems, (1980), 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 New York · Zbl 0368.46014 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.