A fixed point theorem on multiplicative metric space with integral-type inequality


Authors

Aziz Khan - Department of Mathematics, University of Peshawar, P. O. Box 25000, Khybar Pakhtunkhwa, Pakistan Hasib Khan - College of Engineering, Mechanics and Materials, Hohai University, 211100, Nanjing, P. R. China Dumitru Baleanu - Department of Mathematics, Cankaya University, 06530 Ankara, Turkey Hossein Jafari - Department of Mathematics, University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran Tahir Saeed Khan - Department of Mathematics, University of Peshawar, P. O. Box 25000, Khybar Pakhtunkhwa, Pakistan Maysaa Alqurashi - College of Science, Department of Mathematics, King Saud University, Riyad, Saudi Arabia


Abstract

In this paper, we prove fixed point theorems (FPTs) on multiplicative metric space (MMS) (\(\mathcal{X},\blacktriangle\)) by the help of integral-type contractions of self-quadruple mappings (SQMs), i.e., for \(\wp_1,\wp_2,\wp_3,\wp_4:\mathcal{X}\rightarrow \mathbb{R}\). For this, we assume that the SQMs are weakly compatible mappings and the pairs \(\big(\wp_1,\wp_3\big)\) and \(\big(\wp_2,\wp_4\big)\) satisfy the property \((CLR_{\wp_3\wp_4})\). Further, two corollaries are produced from our main theorem as special cases. The novelty of these results is that for the unique common fixed point (CFP) of the SQMs \(\wp_1,\wp_2,\wp_3,\wp_4\), we do not need to the assumption of completeness of the MMS \((\mathcal{X},\blacktriangle)\). These results generalize the work of Abdou, [A. A. N. Abdou, J. Nonlinear Sci. Appl., \({\bf 9}\) (2016), 2244--2257], and many others in the available literature.


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