Common fixed points for expansive mappings in cone metric spaces (Q2906118)

The article deals with coincidence theorems for pairs of expansive operators in cone metric space. More precisely, in the cone metric space \((X,d)\) with the cone metric \(d\) taking values in a solid cone \(K\), it is considered a pair of operators \(f, g: X \to X\) and the problem of the existence and uniqueness a point \(x_*\) such that \(fx_* = gx_*\). It is assumed that \(fX \supset gX\) and that one of the subsets \(fX\) and \(gX\) is complete in the natural sense. Then there exists a point of coincidence for \(f\) and \(g\) provided that one of the following conditions holds: NEWLINE\[NEWLINEd(fx,fy) \geq \alpha d(gx,gy) \quad (x,y \in X) \quad \text{with} \quad \alpha > 1;NEWLINE\]NEWLINE NEWLINE\[NEWLINEd(fx,fy) \geq \beta[d(gx,fx) + d(gy,fy)] \quad (x,y \in X) \quad \text{with} \quad \frac12 \beta < 1;NEWLINE\]NEWLINE NEWLINE\[NEWLINEd(fx,fy) \geq \alpha d(gx,fx) + \beta d(gy,fx) + \gamma d(gx,gy) \quad (x,y \in X)NEWLINE\]NEWLINE NEWLINE\[NEWLINE\text{with} \quad \alpha + \beta + \gamma > 1, \quad \text{and} \quad \alpha < 1 \;\;\text{or} \;\;\beta < 1.NEWLINE\]NEWLINE In the case of two continuous and surjective maps \(f, g: X \to X\), there exists a coincidence point if, for some \(\alpha > 1\), NEWLINE\[NEWLINEd(fx,gy) \geq \alpha \, u(x,y), \quad \text{where} \quad u(x,y) \in \{d(x,fx),d(y,gy),d(x,y)\}.NEWLINE\]NEWLINE Some uniqueness results are also presented.