## Remarks on contractive conditions of integral type.(English)Zbl 1215.54021

In this excellent paper, the author shows that a number of contractive conditions of integral type are equivalent to corresponding contractive conditions of non-integral type. I shall illustrate this by stating Theorem 1 of the paper. Let $$\mathfrak{F} := \{ F : \mathbb{R_+} \to \mathbb{R_+} : F$$ is increasing, continuous and satisfies $$F^{-1}(0) = \{0\}\}$$ and $$\Phi := \{\varphi : \mathbb{R_+} \to \mathbb{R_+} : \varphi$$ is nondecreasing, right upper continuous, and satisfies $$\varphi(t) < t \text{ for each } t > 0\}$$.
Let $$A, B, S,$$ and $$T$$ be selfmaps of a metric space $$(X, d)$$. For $$x,y \in X$$, set $$M(x,y) :=\max\{d(Ax,by), d(Ax,Sx), d(By,Ty), [d(Ax,By) + d(By,Sx)]/2\}.$$ Then the following are equivalent:
(i)
there exist $$\varphi \in \Phi$$ and $$f \in \mathfrak{F}$$ such that $\int_0^{d(Sx,Ty)}f(s)\,ds \leq \varphi\Big(\int_0^{M(x,y)}f(s)\,ds\Big)\text{ for all }x,y \in X;$
(ii)
there exist $$F \in \mathfrak{F}$$ and $$\varphi \in \Phi$$ such that $$F(d(Sx,Ty)) \leq \varphi(F(M(x,y))$$ for all $$x,y \in X$$;
(iii)
for every $$\alpha \in (0,1)$$, there exists $$F \in\mathfrak{F}$$ such that $$F(d(Sx,Ty)) \leq \alpha(F(M(x,y))$$ for all $$x,y \in X$$;
(iv)
there exists $$\varphi \in\Phi$$ such that $$d(Sx, Ty) \leq \varphi(M(x,y))$$ for all $$x,y \in X$$.
The author also proves the following fixed point theorem. Theorem 8. Let $$T$$ be a selfmap of a complete metric space $$(X,d)$$ such that, for some $$\varphi \in \Phi$$ and $$F \in \mathfrak{F}$$, $$F(d(Tx,Ty)) \leq \varphi(F(d(x,y)))$$ for all $$x, y \in X$$. Then $$T$$ has a unique fixed point $$x_{\epsilon}$$ and, for any $$x \in X, \lim_nT^nx = x_{\epsilon}$$.

### MSC:

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

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