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 dF(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 dF(x), whereas the second one β:[0,+∞)→[0,1) estimates dF(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.