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Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. (English) Zbl 1225.54014
Let (X,d,) be a partially ordered complete metric space, and F:X 3 X a continuous mixed monotone map. Assume that i) there exist j,k,l[0,1) with j+k+l<1 for which d(F(x,y,z),F(u,v,w))jd(x,u)+kd(y,v)+ld(z,w), for all (x,y,z),(u,v,w)X 3 with xu, yv, zw, ii) there exists (x 0 ,y 0 ,z 0 )X 3 such that x 0 F(x 0 ,y 0 ,z 0 ), y 0 F(y 0 ,x 0 ,y 0 ), z 0 F(z 0 ,y 0 ,x 0 ). Then, there exists (x,y,z)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.

54H25Fixed-point and coincidence theorems in topological spaces
54F05Linearly, generalized, and partial ordered topological spaces
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