## 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

### Keywords:

fixed point theorems; contractive inequality
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