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Zbl 1225.54014
Berinde, Vasile; Borcut, Marin
Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 15, 4889-4897 (2011). ISSN 0362-546X

Let $(X,d,\le)$ be a partially ordered complete metric space, and $F:X^3\to X$ a continuous mixed monotone map. Assume that {\bf i)} there exist $j,k,l\in [0,1)$ with $j+k+l< 1$ for which $d(F(x,y,z),F(u,v,w))\le jd(x,u)+kd(y,v)+ld(z,w)$, for all $(x,y,z),(u,v,w)\in X^3$ with $x\ge u$, $y\le v$, $z\ge w$, {\bf ii)} there exists $(x_0,y_0,z_0)\in X^3$ such that $x_0\le F(x_0,y_0,z_0)$, $y_0\ge F(y_0,x_0,y_0)$, $z_0\le F(z_0,y_0,x_0)$. 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.
[Mihai Turinici (Iaşi)]
MSC 2000:
*54H25 Fixed-point theorems in topological spaces
54F05 Ordered topological spaces

Keywords: Metric space; order; contraction; triple fixed point.

Citations: Zbl 1106.47047

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