A common fixed point for weak \(\varphi \)-contractions on \(b\)-metric spaces (Q2873955)

A \(b\)-metric space satisfies the first two properties of a metric space and the triangular inequality is replaced by \(d(x, y) \leq s[d(x, z) + d(z, y)]\) for some \(s \geq 1\). For \((X, d)\) a complete \(b\)-metric space, let \(P_{cl,b}(X)\) denote the nonempty collection of closed, bounded subsets of \(X\) and \(\Phi\) the set of lower semi-continuous functions \(\phi : [0, \infty) \to [0, \infty)\) that satisfy \(\phi(0) = 0\) and \(\phi(t) >(1 - 1/s^2)t\) for each \(t > 0\). Let \(T\) be a selfmap of \(X\), \(S : X \to P_{cl,b}(X)\), and let \(H\) denote the Hausdorff distance. The authors prove that, if \(H(\{Tx\}, Sy) \leq M(x, y) - \phi(M(x, y))\), where \(M(x, y) :- \max\{d(x, y), D(x, Tx), D(y, Sy) ,[D(x, Sy) + D(y, Tx)]/2s\}\), then \(T\) and \(S\) have a unique common fixed point in \(X\).