• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.993-1005
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• 2013

In this paper, to include more generalized cases, the authors present a modified concept for the tripled and quadruple fixed point of the mixed monotone mappings. Also, they investigate the existence and uniqueness of fixed point of the ordered monotone operator with the Matkowski contractive conditions in the partial ordered metric spaces. As the direct consequences, the existence of coupled fixed point, tripled fixed point and quadruple fixed point are explored at the common framework and some previous results in [T. G. Bhaskar and V. Lakshmikan-tham, Fixed point theory in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), 1379-1393; V. Berinde and M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011), no. 15, 4889-4897; E. Karapinar and N. V. Luong, Quadruple fixed point theorems for nonlinear contractions, Computers and Mathematics with Applications (2012), doi:10.1016/j.camwa.2012.02061] are improved. Finally, some fixed point theorems are proved.

• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.789-807
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• 2015

We propose coincidence and common fixed point results for a quadruple of self mappings satisfying common limit range property and weakly compatibility under generalized ${\Phi}$-contractive conditions i Non-Archimedean Menger PM-spaces. As examples we exhibit different types of situations where these conditions can be used. A common fixed point theorem for four finite families of self mappings is presented as an application of the proposed results. The existence and uniqueness of solutions for certain system of functional equations arising in dynamic programming are also presented as another application.