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Fixed point theorems for $$\alpha$$-$$\psi$$-contractive type mappings. (English) Zbl 1242.54027
Let $$\Psi$$ be the family of nondecreasing functions $$\psi:[0,+\infty)\to[0,+\infty)$$ such that $$\sum_{n=1}^{\infty}\psi^n(t)<+\infty$$ for each $$t\geq0$$, where $$\psi^n$$ is the $$n$$-th iterate of $$\psi$$. Let $$(X,d)$$ be a metric space and $$T:X\to X$$. The authors call the mapping $$T$$ $$\alpha$$-$$\psi$$-contractive if there exist functions $$\alpha:X\times X\to[0,+\infty)$$ and $$\psi\in\Psi$$ such that $$\alpha(x,y)\,d(Tx,Ty)\leq\psi(d(x,y))$$ for all $$x,y\in X$$. Furthermore, $$T$$ is called $$\alpha$$-admissible if $$\alpha(x,y)\geq1$$ implies that $$\alpha(Tx,Ty)\geq1$$.
The authors prove that, if the space $$(X,d)$$ is complete and the mapping $$T$$ is continuous, $$\alpha$$-admissible and $$\alpha$$-$$\psi$$-contractive, then $$T$$ has a fixed point in $$X$$, provided that $$\alpha(x_0,Tx_0)\geq1$$ for some $$x_0\in X$$. Continuity of $$T$$ can be omitted if, for each sequence $$\{x_n\}$$ in $$X$$, $$\alpha(x_n,x_{n+1})\geq1$$ for all $$n$$ and $$x_n\to x\in X$$ as $$n\to\infty$$ imply that $$\alpha(x_n,x)\geq1$$ for all $$n$$. The uniqueness of the fixed point holds if, for all $$x,y\in X$$, there is some $$z\in X$$ such that $$\alpha(x,z)\geq1$$ and $$\alpha(y,z)\geq1$$.
Several known (and some new) fixed and coupled fixed point theorems in metric, as well as in ordered metric spaces can be obtained from these results. Some examples are presented, showing how the obtained theorems can be used. Also, some applications to ordinary differential equations are given.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 54E40 Special maps on metric spaces 54E50 Complete metric spaces
##### Keywords:
fixed point; coupled fixed point; contractive mapping
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