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Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. (English) Zbl 1225.54014
Let $$(X,d,\leq)$$ be a partially ordered complete metric space, and $$F:X^3\to X$$ a continuous mixed monotone map. Assume that i) there exist $$j,k,l\in [0,1)$$ with $$j+k+l< 1$$ for which $$d(F(x,y,z),F(u,v,w))\leq jd(x,u)+kd(y,v)+ld(z,w)$$, for all $$(x,y,z),(u,v,w)\in X^3$$ with $$x\geq u$$, $$y\leq v$$, $$z\geq w$$, ii) there exists $$(x_0,y_0,z_0)\in X^3$$ such that $$x_0\leq F(x_0,y_0,z_0)$$, $$y_0\geq F(y_0,x_0,y_0)$$, $$z_0\leq F(z_0,y_0,x_0)$$. Then, there exists $$(x,y,z)\in X^3$$ with the triple fixed point property: $$x=F(x,y,z)$$, $$y=F(y,x,y)$$, $$z=F(z,y,x)$$. Sufficient conditions guaranteeing the uniqueness of this tripled fixed point or its diagonal properties are also given.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
##### Keywords:
Metric space; order; contraction; triple fixed point.
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##### References:
 [1] Gnana Bhaskar, T.; Lakshmikantham, V., Fixed point theorems in partially ordered metric spaces and applications, Nonlinear anal., 65, 7, 1379-1393, (2006) · Zbl 1106.47047 [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), Art. ID 125426 · Zbl 1179.54069 [3] Beg, I.; Abbas, M., Fixed points and invariant approximation in random normed spaces, Carpathian J. math., 26, 1, 36-40, (2010) · Zbl 1212.47038 [4] Berinde, V., () [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, 5, 1435-1443, (2004) · Zbl 1060.47056 [6] Rus, I.A.; Petruşel, A.; Petruşel, G., Fixed point theory, (2008), Cluj University Press Cluj-Napoca · Zbl 1171.54034 [7] Abbas, M.; Ali Khan, M.; Radenović, S., Common coupled fixed point theorems in cone metric spaces for $$w$$-compatible mappings, Appl. math. comput., 217, 1, 195-202, (2010) · Zbl 1197.54049 [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), Article ID 132367 · Zbl 1213.54070 [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 [10] Sedghi, S.; Altun, I.; Shobe, N., Coupled fixed point theorems for contractions in fuzzy metric spaces, Nonlinear anal., 72, 3-4, 1298-1304, (2010) · Zbl 1180.54060 [11] Altun, I.; Simsek, H., Some fixed point theorems on ordered metric spaces and application, Fixed point theory appl., 2010, 17, (2010), Article ID 621469 · Zbl 1197.54053 [12] Karapinar, E., Coupled fixed point theorems for nonlinear contractions in cone metric spaces, Comput. math. appl., 59, 12, 3656-3668, (2010) · Zbl 1198.65097 [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 [14] Rus, I.A., Generalized contractions and applications, (2001), Cluj University Press Cluj-Napoca · Zbl 0968.54029 [15] Samet, B., Coupled fixed point theorems for a generalized meir – keeler contraction in partially ordered metric spaces, Nonlinear anal., 72, 12, 4508-4517, (2010) · Zbl 1264.54068
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