A two-parameter control for contractive-like multivalued mappings

We propose a general approach to defining a contractive-like multivalued mapping 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 ∈ X and the value F ( x ) ⊂ X . Here, both parameters are numerical functions. The first one α : [ 0 , + ∞ ) → [ 1 , + ∞ ) controls the distance between x and some appropriate point y ∈ F ( x ) in comparison with d F ( x ) , whereas the second one β : [ 0 , + ∞ ) → [ 0 , 1 ) estimates d F ( y ) with respect to d ( x , y ) . It appears that the well harmonized relations between α and β are sufficient for the existence of fixed points of F. Our results generalize several known fixed point theorems.