A two-parameter control for contractive-like multivalued mappings

We propose a general approach to defining a contractive-like multivalued mappings $F$ which avoids any use of the Hausdorff distance between the sets $F(x)$ and $F(y)$. Various fixed point theorems are proved under a two-parameter control of the distance function $d_{F}(x)=dist(x,F(x))$ between a point $x \in X$ and the value $F(x) \ss X$. Here, both parameters are numerical functions. The first one $\a\,:[0,+\i)\rightarrow [1,+\i)$ controls the distance between $x$ and some appropriate point $y \in F(x)$ in comparison with $d_{F}(x)$, whereas the second one $\b\,:[0,+\i)\rightarrow [0,1)$ estimates $d_{F}(y)$ with respect to $d(x,y)$. It appears that the well harmonized relations between $\a$ and $\b$ are sufficient for the existence of fixed points of $F$. Our results generalize several known fixed-point theorems.