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Two fixed-point theorems for mappings satisfying a general contractive condition of integral type. (English) Zbl 1052.47052
The author proves fixed point theorems for mappings satisfying a general contractive inequality of integral type.
Theorem: Let \((X,d)\) be a complete metric space, \(k\in [0,1)\), let \(f: X\to X\) and \(\varphi: \mathbb{R}^+\to \mathbb{R}^+\) be a Lebesgue-integrable mapping which is summable, nonnegative and such that \[ \int^\varepsilon_0 \varphi(t)\,dt> 0\quad\text{for each }\varepsilon> 0. \] Case A. Let \[ m(x,y)= \max\{d(x,y),\,d(x,fx),\,d(y, fy),\,\textstyle{{1\over 2}}[d(x,fy)+ d(y,fx)]\}. \] If \(f\) is a mapping such that for each \(x,y\in X\) \[ \int ^{d(fx,fy)}_0 \varphi(t)\,dt\leq k\, \int^{m(x,y)}_0 \varphi(t)\,dt, \] then \(f\) has a unique fixed point \(z\in X\), and for each \(x\in X\), \(\lim_{n\to\infty} f^n x= z\).
Case B. Let \[ M(x,y)= \max\{d(x,y),\, d(x,fx),\, d(y, fy),\, d(x,fy),\, d(y, fx)\}. \] If \(f\) is a mapping such that for each \(x,y\in X\) \[ \int^{d(fx, fy)}_0 \varphi(t)\,dt\leq k \int^{M(x,y)}_0 \varphi(t)\,dt \] and for some \(x\in X\) the orbit is bounded, then \(f\) has a unique fixed point \(z\in X\).

MSC:
47H10 Fixed-point theorems
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