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Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces. (English) Zbl 1176.54028

Let (X,d;) be a complete partially ordered metric space and F,G:B(x 0 ,r)C(X) be two maps with bounded values, fulfilling

T(D(Fx,Gy),d(x,y),d(x,Fx),d(y,Gy),d(x,Gy),d(y,Gx))0 for all x,yB(x 0 ,r), where T: + 6 + satisfies some mild conditions,

for each xX, there exists yLx with xy such that d(x,y)d(x,Lx)+ε, where L{F,G},

if (x n )B(x 0 ,r) fulfills x n x n+1 for all n and x n x as n, then x n x for all n.

In addition, assume that there exists a continuous strictly increasing function Φ: + + with Φ(t)<t for all t>0, such that

d(x 0 ,x 1 )<r-Φ(r) for some x 1 Fx 0 with x 0 x 1 ,

n Φ n (r-Φ(r))Φ(r).

Then there exists in B(x 0 ,r) a common fixed point for F and G.

Reviewer’s remark: The seminal 2004 fixed point result of A. C. M. Ran and M. C. B. Reurings [Proc. Am. Math. Soc. 132, No. 5, 1435–1443 (2004; Zbl 1060.47056)] was obtained in 1986 by the reviewer [J. Math. Anal. Appl. 117, 100–127 (1986; Zbl 0613.47037)].

54H25Fixed-point and coincidence theorems in topological spaces
54F05Linearly, generalized, and partial ordered topological spaces
54C60Set-valued maps (general topology)