It is proved that an (α, β)-subadditive set valued map with close, convex and equibounded values in a Banach space has exactly one additive selection if α, β are positive numbers and α + β ≠ 1

Dorian Popa

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GLASNIK MATEMATICKIVol. 36(56)(2001), 11 { 16ADDITIVE SELECTIONS OF (; )-SUBADDITIVE SETVALUED MAPSDorian PopaTechnical University, Dept. MathematicsAbstract. It is proved that an (; )-subadditive set valued mapwith close, convex and equibounded values in a Banach space has exactlyone additive selection if ;  are positive numbers and  +  6= 1.The study of subadditive set valued maps (s.v. maps) is related to theclassical Hyers-Ulam stability problem for the Cauchy functional equation.This problem is cf.[1] and [4] the next one:If f : R! R is a measurable solution of the inequality(0.1) jf(x+ y)  f(x)  f(y)j < "where " > 0 then there exists a linear function a(x) = mx, m 2 R, such thatjf(x)  a(x)j < " for every x 2 R.The inequality (1) can be written on the formf(x+ y)  f(x)  f(y) 2 B(0; ")where B(0; ") is the ball with centre 0 and radius ".Hence we havef(x+ y) +B(0; ")  f(x) +B(0; ") + f(y) + B(0; ")and denoting by F (x) = f(x) +B(0; "), x 2 R, we get(0.2) F (x+ y)  F (x) + F (y); x; y 2 Randa(x) 2 F (x); x 2 R;which means that the s.v. map F has the additive selection a.2000 Mathematics Subject Classication. (1991): 28B20, 54C65.Key words and phrases. set valued map, subadditive, selection.1112 DORIAN POPANow one may ask in what conditions a subadditive s.v. map (i.e. a s.v.map that satises relation (2) admits an additive selection. An answer at thisquestion is given in [1]. It is also proved that there exists subadditive s.v.maps that have not additive selection (see [1]).In the sequel we shall generalize the notion of subadditive s.v. map andwe shall give for such s.v. maps an existence and uniqueness theorem of anadditive selection.Let X be a real vector space. We shall denote by P(X) and P0(X) thefamily of all subsets, respectively nonempty subsets of X . If Y is a realnormed space the family of all close and convex subsets of Y will be denotedby ccl(Y ).If A;B are nonempty subsets of the real vector space X and ;  are realnumbers we dene(0.3)A+B = fxj x = a+ b; a 2 A; b 2 BgA = fxj x = a; a 2 Ag:The next properties will be often used in what follows(0.4)(A +B) = A+ B(+ )A  A+ A:If A is a convex set and   0 we have (cf.[3])(0.5) (+ )A = A+ A:A set A  X is said to be a cone if K + K  K and K  K for all  > 0.If the zero vector from X belongs to K we say that K is a zero cone.Definition 1. Let X;Y be real vector spaces, K  X a cone and  > 0, > 0. A s.v. map F : K ! P0(Y ) is called (; )-subadditive if(0.6) F (x+ y)  F (x) + F (y)for all x; y 2 X.For  =  = 1, F is called subadditive s.v. map.Let us remark that there exists (; )-subadditive s.v. map that are notsubadditive.The s.v. map F : [0;+1)! P0(R) given byF (x) =px;+1is12;12-subadditive but is not subadditive.For all x; y  0 we haveFx+ y2="rx+ y2;+1!ADDITIVE SELECTIONS OF (; )-SUBADDITIVE SET VALUED MAPS 13and12F (x) +12F (y) =px+py2;+1:Taking account of rx+ y2px+py2it results thatFx+ y2 F (x) + F (y)2that means F is12;12-subadditive, but it is not subadditive becauseF (x+ y) =px+ y;+1 6 px+py;1 = F (x) + F (y)for x > 0, y > 0.In the following theorem we shall give a sucient condition for an (; )-subadditive multifunction to admit an additive selection. Let us recall thatif F : X ! P(Y ) is a s.v. map a function f : X ! Y with the propertyf(x) 2 F (x) for all x 2 X is said to be a selection of F .Theorem 2. Let X be a real vector space, K  X a zero cone, Y aBanach space and  > 0,  > 0.If F : K ! ccl(Y ) is (; )-subadditive s.v. map which satises theconditions:1) +  6= 12) supfdiamF (x) : x 2 Xg < +1then F admits exactly one additive selection.Proof. Using (; )-subadditivity of F and convexity of its values wehave for all x 2 K(0.7) F ((+ )x)  F (x) + F (y) = (+ )F (x)and replacing x by (+ )nx, n 2 N, in (7) we obtainF ((+ )n+1x)  (+ )F (( + )nx)andF ((+ )n+1x)( + )n+1 F ((+ )nx)(+ )n:I. Let +  > 1.Denoting by Fn(x) =F (( + )nx)(+ )n, x 2 K, n 2 N, it results that(Fn(x))n0 is a decreasing sequence of closed subsets of the Banach space Y .We have alsodiamFn(x) =1(+ )ndiamF ((+ )nx)14 DORIAN POPAand taking account of the condition 2) we obtain(0.8) limn!1diamFn(x) = 0:Using the Cantor theorem for the sequence (Fn(x))n0 the intersection\n0Fn(x) is a singleton and we denote this intersection by a(x) for all x 2 K.Thus we obtain a function a : K ! Y which is a selection of F becausea(x) 2 F0(x) = F (x) for all x 2 K.We shall show that a is additive. We haveFn(x + y) =F ((+ )n(x + y))(+ )n=F (( + )nx+ (+ )ny)(+ )n F (( + )nx) + F (( + )ny)(+ )n= F (( + )nx)(+ )n+ F ((+ )ny)(+ )n== Fn(x) + Fn(y)for all x; y 2 K and n 2 N.From the last relation it results thata(x+ y) =\n0Fn(x+ y) \n0(Fn(x) + Fn(y))andka(x+ y)  (a(x) + a(y))k  diam(Fn(x) + Fn(y))  diamFn(x) + diamFn(y)! 0;when n!1, cf.(8).So(0.9) a(x+ y) = a(x) + a(y);for all x; y 2 K.Putting x = y = 0 in (9) it results a(0) = 0 and for y = 0 and x = 0 in(9) we obtain(0.10)a(x) = a(x)a(y) = a(y)for all x; y 2 K. Using (10) we havea(x+ y) = ax+ y= ax+ ay= a(x) + a(y)for all x; y 2 K, thus a is additive.II. For  +  < 1, we replace x in (7) byx(+ )n+1, n 2 N, and weobtain multiplying this relation by (+ )n(0.11) (+ )nFx(+ )n (+ )n+1Fx(+ )n+1:ADDITIVE SELECTIONS OF (; )-SUBADDITIVE SET VALUED MAPS 15The sequence (F 0n(x))n0,F 0n(x) = ( + )nFx(+ )n;is increasing and it follows that the sequence of positive numbers(diamF 0n(x))n0 is increasing too. We havediamF 0n(x) = (+ )ndiamFx(+ )nandlimn!1diamF 0n(x) = 0;hence F 0n(x) is single valued for all x 2 K. The s.v. map F is single valuedF (x+ y) = F (x) + F (y)for all x; y 2 K, and using the same methods as in case I, we obtain theadditivity of F . We conclude that F is its own selection.We have proved the existence of an additive selection of F and let usprove the uniqueness.Suppose that F has two additive selection a1; a2 : K ! Y . We havenai(x) = ai(nx) 2 F (nx)for all n 2 N, x 2 K, 1  i  2. We havenka1(x)  a2(x)k = kna1(x)   na2(x)k = ka1(nx)   a2(nx)k  diamF (nx)for all x 2 K, n 2 N, and taking account of the condition 2) from Theorem1, it results a1(x) = a2(x) for all x 2 K.Corollary 3. ([1]). Let X be a real vector space, K  X a cone withzero and Y a Banach space. If F : K ! ccl(Y ) is a subadditive s.v. map withsupfdiamF (x) : x 2 Kg < +1then F admits exactly one additive selection.Proof. We take  =  = 1 in Theorem 1.Corollary 4. Let X be a real vector space, K  X a cone with zero, Ya normed space. If F : K ! P0(Y ) is (; )-subadditive with  +  < 1 andconvex values andsupfdiamF (x) : x 2 Kg < +1;then F is single valued and additive.Proof. It results from the proof of Theorem 1, part II.16 DORIAN POPARemark 5. There exists (; )-subadditive s.v. maps with  +  = 1,which satises the conditions of Theorem 1 and have not additive selection.Concave s.v. map are (; )-subadditive s.v. maps with +  = 1. It isknown that if f; g : [0;1)! R are two functions such that f is concave and gis convex and f(x)  g(x), for each x 2 R then the s.v. map F : R! P0(R),F (x) = [f(x); g(x)] is concave (see [3]). Using this result it follows thatF (x) =xx+ 1; 1; x 2 [0;1)is concave without additive selection, because if a : R ! R is an additiveselection of F we must have a(x) = mx for each x 2 [0;+1) \ Q (m 2 R)andxx+ 1 mx  1; 8 x 2 [0;+1) \Q:The last inequality is impossible because taking the limit it resultslimx!1mx = 1.References[1] Gajda Z., Ger R., Subadditive multifunction and Hyers-Ulam stability, NumericalMathematics, Vol.80, 1987, Birkhauser Verlag Basel.[2] Kuczma M., An introduction to the theory of functional equations and Jensen inequal-ity, PWN Universitet Slaski, Warszwa-Krakow-Katovice, 1985.[3] Nikodem K., K-convex and K-concave set valued functions, Zeszyty Naukowe Nr.559,Lodz 1989.[4] Smajdor W., Subadditive set valued functions, Glasnik Matematicki, Vol.21(41), 1986,No.2, 281-291.Technical University,Dept. MathematicsStr. Emil Isac nr.153400 Cluj-Napoca, RomaniaReceived : 21.05.1998.

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