Some significant remarks on multivalued Perov type contractions on cone metric spaces with a directed graph. (English) Zbl 1485.54058
Summary: Using the approach of so-called c-sequences introduced by the fifth author in his recent work (cf. [S. Aleksić et al., J. Int. Math. Virtual Inst. 9, No. 1, 93–101 (2019; Zbl 1474.54102)]), we give much simpler and shorter proofs of multivalued Perov’s type results [A. I. Perov, Priblizhen. Metody Reshen. Differ. Uravn. 2, 115–134 (1964; Zbl 0196.34703)] with respect to the ones presented in [M. Abbas et al., Bull. Belg. Math. Soc. - Simon Stevin 25, No. 3, 331–354 (2018; Zbl 1489.54034)]. Our proofs improve, complement, unify and enrich the ones from the recent papers. Further, in the last section of this paper, we correct and generalize the well-known Perov’s fixed point result. We show that this result is, in fact, equivalent to Banach’s contraction principle.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54C60 | Set-valued maps in general topology |
| 54E40 | Special maps on metric spaces |
| 54E50 | Complete metric spaces |
Keywords:
common fixed point; Perov’s type results; multivalued mapping; directed graph; graphic contraction; cone metric space; c-sequenceReferences:
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