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Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. (English) Zbl 1225.54014
Let $\left(X,d,\le \right)$ be a partially ordered complete metric space, and $F:{X}^{3}\to X$ a continuous mixed monotone map. Assume that i) there exist $j,k,l\in \left[0,1\right)$ with $j+k+l<1$ for which $d\left(F\left(x,y,z\right),F\left(u,v,w\right)\right)\le jd\left(x,u\right)+kd\left(y,v\right)+ld\left(z,w\right)$, for all $\left(x,y,z\right),\left(u,v,w\right)\in {X}^{3}$ with $x\ge u$, $y\le v$, $z\ge w$, ii) there exists $\left({x}_{0},{y}_{0},{z}_{0}\right)\in {X}^{3}$ such that ${x}_{0}\le F\left({x}_{0},{y}_{0},{z}_{0}\right)$, ${y}_{0}\ge F\left({y}_{0},{x}_{0},{y}_{0}\right)$, ${z}_{0}\le F\left({z}_{0},{y}_{0},{x}_{0}\right)$. Then, there exists $\left(x,y,z\right)\in {X}^{3}$ with the triple fixed point property: $x=F\left(x,y,z\right)$, $y=F\left(y,x,y\right)$, $z=F\left(z,y,x\right)$. Sufficient conditions guaranteeing the uniqueness of this tripled fixed point or its diagonal properties are also given.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 54F05 Linearly, generalized, and partial ordered topological spaces
##### Keywords:
Metric space; order; contraction; triple fixed point.
##### References:
 [1] Bhaskar, T. Gnana; Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications, Nonlinear anal. 65, No. 7, 1379-1393 (2006) · Zbl 1106.47047 · doi:10.1016/j.na.2005.10.017 [2] Sabetghadam, F.; Masiha, H. P.; Sanatpour, A. H.: Some coupled fixed point theorems in cone metric spaces, Fixed point theory. Appl. 2009, 8 (2009) · Zbl 1179.54069 · doi:10.1155/2009/125426 [3] Beg, I.; Abbas, M.: Fixed points and invariant approximation in random normed spaces, Carpathian J. Math. 26, No. 1, 36-40 (2010) · Zbl 1212.47038 [4] Berinde, V.: Iterative approximation of fixed points, Lecture notes in mathematics 1912 (2007) [5] Ran, A. C. M.; Reurings, M. C. B.: A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. amer. Math. soc. 132, No. 5, 1435-1443 (2004) · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4 [6] Rus, I. A.; Petruşel, A.; Petruşel, G.: Fixed point theory, (2008) [7] Abbas, M.; Khan, M. Ali; Radenović, S.: Common coupled fixed point theorems in cone metric spaces for w-compatible mappings, Appl. math. Comput. 217, No. 1, 195-202 (2010) · Zbl 1197.54049 · doi:10.1016/j.amc.2010.05.042 [8] Nashine, H. K.; Altun, I.: Fixed point theorems for generalized weakly contractive condition in ordered metric spaces, Fixed point theory appl. 2011, 20 (2011) · Zbl 1213.54070 · doi:10.1155/2011/132367 [9] Luong, N. V.; Thuan, N. X.: Coupled fixed points in partially ordered metric spaces and application, Nonlinear anal. 74, 983-992 (2011) · Zbl 1202.54036 · doi:10.1016/j.na.2010.09.055 [10] Sedghi, S.; Altun, I.; Shobe, N.: Coupled fixed point theorems for contractions in fuzzy metric spaces, Nonlinear anal. 72, No. 3–4, 1298-1304 (2010) · Zbl 1180.54060 · doi:10.1016/j.na.2009.08.018 [11] Altun, I.; Simsek, H.: Some fixed point theorems on ordered metric spaces and application, Fixed point theory appl. 2010, 17 (2010) · Zbl 1197.54053 · doi:10.1155/2010/621469 [12] Karapinar, E.: Coupled fixed point theorems for nonlinear contractions in cone metric spaces, Comput. math. Appl. 59, No. 12, 3656-3668 (2010) · Zbl 1198.65097 · doi:10.1016/j.camwa.2010.03.062 [13] Lakshmikantham, V.; Ćirić, L.: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear anal. 70, 4341-4349 (2009) · Zbl 1176.54032 · doi:10.1016/j.na.2008.09.020 [14] Rus, I. A.: Generalized contractions and applications, (2001) [15] Samet, B.: Coupled fixed point theorems for a generalized Meir–Keeler contraction in partially ordered metric spaces, Nonlinear anal. 72, No. 12, 4508-4517 (2010)