A Singular Behaviour of a Set-Valued Approximate Orthogonal Additivity

Abstract

Abstract. We show that unlikely to the single-valued case, the set-valued orthogonally additive equation is unstable. After presenting an example showing this phenomenon, we provide some special cases where a set-valued approximately orthogonally additive function can be approximated by the one which satisfies the equation of orthogonal additivity exactly.Mathematics Subject Classification. 41A65, 54C60, 26E25, 39B82.

On a class of abstract convex cone valued functional equations

“…In case of a positive answer we say that (1) is stable (the reader interested in learning more on stability is referred, e.g., to [1,5]). We will show, however, that without any additional assumption imposed on the class of functions for which (5) is considered, the equation is not stable (cf., [13]).…”

An extension of the Rådström cancellation theorem to cornets

“…This lemma turned out to be a basic tool in various fields and hundreds of papers have used it by now. For instance, in nonsmooth analysis [7][8][9]14,[18][19][20]34], optimization theory [15,36,38], theory of convex sets and functions [10,12,16,17,[23][24][25][26][27][28][29][30][31][32][33]35,40,59,60], set-valued analysis [2,13,37,39,41,[44][45][46]48], set-valued differential equations [3,4,11,22,43,49], set-valued functional equations [6,42,[51][52][53]…”

An extension of the Rådström Cancellation Theorem to Cornets

“…This lemma turned out to be a basic tool in various fields and hundreds of papers have used it by now. For instance, in nonsmooth analysis [7][8][9]14,[18][19][20]34], optimization theory [15,36,38], theory of convex sets and functions [10,12,16,17,[23][24][25][26][27][28][29][30][31][32][33]35,40,59,60], set-valued analysis [2,13,37,39,41,44,45,47,48], set-valued differential equations [3,4,11,22,43,49], set-valued functional equations [6,42,[51][52]…”