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 $F(x)\text{ss}X$. Here, both parameters are numerical functions. The first one $\text{a}:[0,+\text{i})\to [1,+\text{i})$ controls the distance between $x$ and some appropriate point $y\in F(x)$ in comparison with $d_F(x)$, whereas the second one $\text{b}:[0,+\text{i})\to [0,1)$ estimates $d_F(y)$ with respect to $d(x,y)$. It appears that the well harmonized relations between $\text{a}$ and $\text{b}$ are sufficient for the existence of fixed points of $F$. Our results generalize several known fixed-point theorems.