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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 (*).

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