**Deepak Singh**
- Department of Applied Sciences, NITTTR, Under Ministry of HRD, Govt. of India, Bhopal, (M.P.), India,462002

**Varsha Chauhan**
- Department of Mathematics, NRI Institute of Research & Technology, Bhopal M.P,India

**Mehdi Asadi**
- Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran.

The purpose of this paper is three fold. Firstly, we establish a tripled coincidence fixed point theorem for a sequence of mappings involving Geraghty contraction using compatibility and weakly reciprocally continuous maps in the structure of partially ordered metric spaces. The technique used in A. Roldan et al. \cite{les24} and in S. Radenovic \cite{les25} are not applicable on the presented theorems, we show that our results can not be obtained from the existing results in this field of study and thus our results are completely new and give rise a new dimension. Secondly, the notable works due to V. Berinde \cite{les20}, V. Lakshmikantam and L. Ciric \cite{les8} and Babu and Subhashini \cite{les22} are generalized and extended. Finally, some sufficient conditions are given for the uniqueness of a tripled common fixed point. Consequently, we point out some slip-ups in the main results of R. Vats et al.\cite{les19} and present a furnished version of the same. Some illustrative examples to highlight the realized improvements are also furnished. Moreover, existence and uniqueness for the solution of an initial-boundary-value problem is discussed. On the other hand, as an application to establish existence and uniqueness for the system of integral equations our results are utilized.

Partially ordered metric spaces, compatible mappings, weakly reciprocally maps, tripled coincidence point

- [1] G. V. R. Babu, P. Subhashini, Coupled common fixed points for a pair of compatible maps satisfying geraghty contraction in partially ordered metric spaces, Int. J. Math. Sci. Comput. , 2 (2012), 41--48
- [2] T. G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379–1393
- [3] V. Berinde, Coupled coincidence point theorems for mixed monotone nonlinear operators, Comput. Math. Appl., 64 (2012), 1770–1777
- [4] V. Berinde, M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal., 74 (2011), 4889–4897
- [5] M. Borcut, V. Berinde, Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces, Appl. Math. Comput., 218 (2012), 5929–5936
- [6] M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604–608
- [7] M. E. Gordji, M. Ramezani, Y. J. Cho, S. Pirbavafa, A generalization of Geraghty’s theorems in partially ordered metric spaces and applications to ordinary differential equations, Fixed Point Theory Appl., 2012 (2012), 9 pages
- [8] V. Lakshmikantham, L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341–4349
- [9] A. Roldan, J. Martinez-Moreno, C. Roldan, E. Karapinar, Some remarks on multidimensional fixed point theorems, Fixed Point Theory, 14 (2014), 545–558
- [10] S. Radenovic, A note on tripled coincidence and tripled common fixed point theorems in partially ordered metric spaces, Appl. Math. Comput., 236 (2014), 367–372
- [11] Y. Sang, Q. Meng, Fixed point theorems with generalized alternating distance functions in partially ordered metric spaces via w-distance and applications, Fixed Point Theory Appl., 2015 (2015), 25 pages
- [12] R. Vats, K. Tas, V. Sihag, A. Kumar, Tripled fixed point theorems via α-series in partially ordered metric spaces, J. Ineq. Appl., 2014 (2014), 12 pages