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

Volume 18, Issue 1, pp 18--28 Publication Date: December 27, 2017       Article History
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### 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.

### Keywords

• Multiplicative metric space
• fractional integral inequalities
• fixed point theorems

•  47H10
•  54H25

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