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Distance in cone metric spaces and common fixed point theorems. (English) Zbl 1230.54048

Let (X,d) be a cone metric space over a normal cone P in a real Banach space E in the sense of L.-G. Huang and X. Zhang [J. Math. Anal. Appl. 332, No. 2, 1468–1476 (2007; Zbl 1118.54022)]. Let q:X×XE satisfy the following properties: (q1) θq(x,y) for all x,yX; (q2) q(x,z)q(x,y)+q(y,z) for all x,y,zX; (q3) if {y n } is a sequence in (X,d) converging to yX and for some xX and u=u x P, q(x,y n )u for each n1, then q(x,y)u; (q4) for each cE with θc, there exists eE with θe, such that q(z,x)e and q(z,y)e imply d(x,y)c. The function q is called a c-distance in (X,d) (this notion is a cone metric version of the notion of ω-distance of O. Kada, T. Suzuki and W. Takahashi [Math. Japon. 44, No. 2, 381–391 (1996; Zbl 0897.54029)]).

The authors prove the following common fixed point result in terms of a c-distance. Let a i (0,1), i=1,2,3,4 be constants with a 1 +2a 2 +a 3 +a 4 <1 and let f,g:XX be two mappings satisfying the condition q(fx,fy)a 1 q(gx,gy)+a 2 q(gx,fy)+a 3 q(gx,fx)+a 4 q(gy,fy) for all x,yX. Suppose that gXfX and gX is a complete subset of X. If f and g satisfy inf{q(fx,y)+q(gx,fy)+q(gx,fx):xX}>0 for all yX with yfy or ygy, then f and g have a common fixed point in X. No compatibility assumptions have to be used.

MSC:
54H25Fixed-point and coincidence theorems in topological spaces
References:
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