The set of regularity of a multivalued mapping in a space with a vector-valued metric

We consider multivalued mappings acting in spaces with a vector-valued metric. A vector-valued metric is understood as a mapping satisfying the axioms “of an ordinary metric” with values in the cone of a linear normed space. The concept of the regularity set of a multivalued mapping is defined. A set of regularity is used in the study of inclusions in spaces with a vector-valued metric.

ON SETS OF METRIC REGULARITY OF MAPPINGS IN SPACES WITH VECTOR-VALUED METRIC

Spaces with vector-valued metric are considered. The values of a vectorvalued metric are elements of a cone in some linear normed space. The concept of the set of metric regularity for mapping in spaces with vector-valued metric is formulated. A statement on the stability of the set of metric regularity of a given mapping for its Lipschitz perturbations in spaces with vector-valued metric is obtained.

On the Maksa-Volkmann functional inequality |f (x + y)| ≥ |f (x) + f (y)| when the range of f is a space of functions

P. Volkmann functional inequality |f (x + y)| ≥ |f (x) + f (y)| is extended to functions f : G → F (X, E) where G is an additive group and F (X, E) is the space of functions from a set X to a linear normed space E. As a corollary one proves that an operator T : C (X, K) → C (X, K) which satisfies the functional inequality |T (f + g)| ≥ |T (f) + T (g)| , f, g ∈ C (X, K) is additive. Here we denoted by X a compact topological space, K is R or C and C (X, K) is the linear space of continuous functions defined on X with values in K.