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On Kannan-type contractive mappings. (English) Zbl 1489.54122

Summary: In this article, we consider Kannan-type contractive self-map \(T\) on a metric space \((X,d)\) such that \[ d(Tx,Ty) < \frac{1}{2} \{ d(x,Tx)+ d(y,Ty)\} \text{ or all } x\neq y\in X \] and establish some new fixed point results without taking the compactness of \(X\) and also relaxing the continuity of \(T\). Further, we anticipate a result ensuring the completeness of the space \(X\) via fixed point property of this map. Finally, we give an affirmative answer to the open question posed by J. Górnicki [J. Fixed Point Theory Appl. 19, No. 3, 2145–2152 (2017; Zbl 1490.54059)]. Apart from these, our manuscript consists of several nontrivial examples which signify the motivation of our investigations.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
54E40 Special maps on metric spaces

Citations:

Zbl 1490.54059
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References:

[1] Banach, S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. Math, 3, 133-181 (1922) · JFM 48.0201.01
[2] Kannan, R., Some results on fixed points, Bull. Calcutta Math. Soc, 60, 71-76 (1968) · Zbl 0209.27104
[3] Subrahmanyam, V., Completeness and fixed points, Monatsh. Math, 80, 4, 325-330 (1975) · Zbl 0312.54048
[4] Fisher, B., A fixed point theorem for compact metric spaces, Publ. Inst. Math, 25, 193-194 (1978) · Zbl 0389.54030
[5] Khan, M. S., On fixed point theorems, Math. Japonica, 23, 201-204 (1978) · Zbl 0398.54029
[6] Chen, H. Y.; Yeh, C. C., A note on fixed point in compact metric spaces, Indian J. Pure Appl. Math, 11, 3, 297-298 (1980) · Zbl 0436.54041
[7] Górnicki, J., Fixed point theorems for Kannan type mappings, J. Fixed Point Theory Appl, 19, 3, 2145-2152 (2017) · Zbl 1490.54059
[8] Edwards, R. E., Functional Analysis: Theory and Applications (1965), New York, NY: Holt, Rinehart and Winston · Zbl 0182.16101
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