Fixed Points of Weakly Compatible Mappings Satisfying a Generalized Common Limit Range Property

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.