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A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type. (English) Zbl 1113.54027
The article deals with fixed points for a pair of mappings \(S\) and \(T\) of a complete metric space \((X,d)\). It is assumed that \[ \int_0^{d(Sx,Ty)} \varphi(t) \, dt \leq \psi\biggl(\int_0^{M(x,y)} \varphi(t) \, dt\biggr)\eqno(*) \] \[ \biggl(M(x,y) = \max \;\biggl\{d(x,y),d(x,Sx),d(y,Ty),\frac{| (d(x,Ty) + d(y,Tx)| }2\biggr\}\biggr), \] where \(k \in [0,1)\), \(\varphi: \;{\mathbb R}_+ \to {\mathbb R}\) is nonnegative and \(\int_0^\epsilon \varphi(t) \, dt > 0\) for each \(\epsilon > 0\), \(\psi: \;{\mathbb R}_+ \to {\mathbb R}_+\) is nondecreasing, \(\psi(t) < t\), \(\sum_{n=1}^\infty \psi^n(t) < \infty\). Under these assumptions the authors prove that \(S\) and \(T\) have a unique common fixed point. Remark: there are some vague places in the article; among others, the role of the number \(k\) that is absent in (*).

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
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