Fixed points of weakly compatible mappings satisfying a generalized common limit range property


Authors

Aziz Khan - Department of Mathematics, University of Peshawar, P. O. Box 25000, Khyber Pakhtunkhwa, Pakistan. Hasib Khan - College of Engineering, Mechanics and Materials, Hohai University, 210098, Nanjing, P. R. China. - Shaheed Benazir Bhutto University Sheringal, Dir Upper, 18000, Khyber Pakhtunkhwa, Pakistan. Dumitru Baleanu - Department of Mathematics, Cankaya University, 06530 Ankara, Turkey. - Institute of Space Sciences, P. O. BOX, MG-23, 76900 Magrrele-Bucharest, Romania. Erdal Karapinar - Department of Mathematics, Atilim University, 06586 Incek, Ankara, Turkey. Tahir Saeed Khan - Department of Mathematics, University of Peshawar, P. O. Box 25000, Khyber Pakhtunkhwa, Pakistan.


Abstract

In this paper, we produce new fixed point theorems for \(2n\) self-mappings \(\wp^a_1,\wp^a_2,\ldots,\wp^a_n\), \(\gamma^b_1,\gamma^b_2,\ldots,\gamma^b_n:\mathcal{X}\rightarrow\mathcal{X}\) on a metric space \((\mathcal{X},\rho )\), satisfying a generalized common limit range (CLR) property or CLR\(_{\wp^a_k\gamma^b_l}\) for \(k,l=2,\ldots,n\). Along with the newly introduced property CLR\(_{\wp^a_k\gamma^b_l}\) for \(k,l=2,\ldots,n\) for the \(2n\) self-mappings, we also assume that the pairs \((\wp^a_1,\gamma^b_1),(\wp^a_2,\gamma^b_2),\ldots,(\wp^a_n,\gamma^b_n)\) are weakly compatible. From the main result, we produce three more corollaries as its special cases. These results generalize the work of Sarwar et al. [M. Sarwar, M. Bahadur Zada, I. M. Erhan, Fixed Point Theory Appl., \({\bf 2015}\) (2015), 15 pages] and many others in the available literature. Two examples are also presented for the applications of our new FPTs.


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