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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:
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