A note on Krasnosel’skii fixed point theorem. (English) Zbl 1370.47053
Summary: In this note, a couple of unclear and unnecessary points made in the two existing papers by Y. Liu and Z. Li [Proc. Am. Math. Soc. 136, No. 4, 1213–1220 (2008; Zbl 1134.47040)] and the authors [ibid. 139, No. 3, 1033–1044 (2011; Zbl 1223.47061)] are first pointed out and clarified. Second, a few additional remarks are observed. Upon these observations, three corresponding refined and unified Krasnosel’skii fixed point theorems in strong topology setup are formulated. As an illustration, several new classes of Krasnosel’skii fixed point theorems are obtained, which expand and complement some known related results by R. P. Agarwal et al. [Fixed Point Theory Appl. 2010, Article ID 243716, 20 p. (2010; Zbl 1218.47078)] and D. E. Edmunds [Math. Ann. 174, 233–239 (1967; Zbl 0152.34701)].
MSC:
| 47H10 | Fixed-point theorems |
| 47H08 | Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc. |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
Krasnosel’skii fixed point theorem; noncompact mapping; multi-valued mapping; nonexpansive mappingReferences:
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