Subadditive Multifunctions and Hyers-Ulam Stability

A Property of a Functional Inclusion Connected with Hyers-Ulam Stability

“…Ulam in 1940 in a talk to a conference at Wisconsin University (see [5], [6]). W. Smajdor [16] and R. Ger, Z. Gajda [3] observed that if f is a solution of (1.1), then the set-valued map F : X → P 0 (Y ) defined by the relation…”

The properties of functional inclusions and Hyers–Ulam stability

“…The subject was later strongly developed by many authors, see for example: [1,4,11,12,16]. An interesting connection between the stability of the Cauchy equation and subadditive set-valued functions was established by Smajdor [17] and Gajda and Ger [6]. They observed that if f satisfies (1), then the set-valued function F : X → n(Y ) (n(Y ) denotes the family of all nonempty subsets of Y ) given by…”

“…An answer to this equation can be found in [6]. Next, the previous result was extended by Nikodem and Popa to set-valued functions satisfying general linear inclusions: [10,[13][14][15]).…”

On Selections of Set-Valued Inclusions in a Single Variable with Applications to Several Variables

“…there exists a unique additive function g : X → Y such that Smajdor [18] and Gajda and Ger [4] observed that if f satisfies (1), then the set-valued function F : X → n(Y ) (n(Y ) denotes the family of all nonempty subsets of Y ) given by…”

“…Now one may ask under what conditions a subadditive set-valued function admits an additive selection. We recall the result of Gajda and Ger [4] (δ(F (x)) denotes the diameter of the set F (x)).…”