Bulletin of the
Korean Mathematical Society BKMS

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. Lakshmikantham, {\it Fixed point theory in partially ordered metric spaces and applications}, Nonlinear Anal. {\bf 65} (2006), 1379--1393; V. Berinde and M. Borcut, {\it Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces}, Nonlinear Anal. {\bf 74} (2011), no. 15, 4889--4897; E. Karapinar and N. V. Luong, {\it 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.

Keywords: contractive mapping, quadruple fixed point, mixed monotone mapping, fixed point

MSC numbers: 47H10, 54H25