Garai, Hiranmoy; Dey, Lakshmi Kanta; Senapati, Tanusri On Kannan-type contractive mappings. (English) Zbl 1489.54122 Numer. Funct. Anal. Optim. 39, No. 13, 1466-1476 (2018). 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. Cited in 16 Documents MSC: 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems 54E40 Special maps on metric spaces Keywords:boundedly compact set; fixed point property; Kannan-type mapping; metric space; T-orbitally compact space Citations:Zbl 1490.54059 PDFBibTeX XMLCite \textit{H. Garai} et al., Numer. Funct. Anal. Optim. 39, No. 13, 1466--1476 (2018; Zbl 1489.54122) Full Text: DOI arXiv 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 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.