On Sandwich Theorem for Delta-Subadditive and Delta-Superadditive Mappings

Home , Additive map

Results Math 72 (2017), 385–399 c 2016 The Author(s). This article is published with open access at Springerlink.com 1422-6383/17/010385-15 published online December 5, 2016 Results in Mathematics DOI 10.1007/s00025-016-0627-7

Andrzej Olbry´s

Abstract. In the present paper, inspired by methods contained in Gajda and Kominek (Stud Math 100:25–38, 1991) we generalize the well known sandwich theorem for subadditive and superadditive functionals to the case of delta-subadditive and delta-superadditive mappings. As a consequence we obtain the classical Hyers–Ulam stability result for the Cauchy functional equation. We also consider the problem of supporting delta- subadditive maps by additive ones.

Mathematics Subject Classiﬁcation. 39B62, 39B72.

Keywords. Functional inequality, additive mapping, delta-subadditive mapping, delta-superadditive mapping.

1. Introduction We denote by R, N the sets of all reals and positive integers, respectively, moreover, unless explicitly stated otherwise, (Y,·) denotes a real normed space and (S, ·) stands for not necessary commutative semigroup. We recall that a functional f : S → R is said to be subadditive if f(x · y) ≤ f(x)+f(y),x,y∈ S. A functional g : S → R is called superadditive if f := −g is subadditive or, equivalently, if g satisfies g(x)+g(y) ≤ g(x · y),x,y∈ S. If a : S → R is at the same time subadditive and superadditive then we say that it is additive, in this case a satisfies the Cauchy functional equation a(x · y)=a(x)+a(y),x,y∈ S. The generalizations of the celebrated separation theorem of Rodé[19] (cf. also Köning [16]) which represents a far-reaching generalization of the classical

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Hahn–Banach theorem has been studied by many mathematicians. A survey of results of this type can be found for instance in the book of Buskes [5]. The problem reads as follow: Suppose f,g : S → R are maps with f subadditive, g superadditive, and g(x) ≤ f(x),x,y∈ S. Does there exist an additive map a : S → R separating g from f, that is, satisfying g(x) ≤ a(x) ≤ f(x), for every x ∈ S ? Results of this type, for commutative semigroups were ﬁrst obtained by Kaufman [14] and Kranz [17]. The paper of Gajda and Kominek [9] on separations theorems and the paper of Chaljub-Simon and Volkmann [7]onthe non-commutative version of Rode’s theorems are examples of papers where the assumption of the commutativity is replaced by some essentially weaker conditions. Recall that a semigroup (S, ·) is said to be weakly commutative if

n n n (x · y)2 = x2 · y2 ,x,y∈ S. This implies that for any x, y ∈ S there exists a sequence of positive integers nk (depending on x and y) such that nk →∞as k →∞and

n n n (x · y)2 k = x2 k · y2 k ,x,y∈ S. The definition of weakly commutative semigroups was introduced by Józef Tabor in [21]. It is clear that every Abelian semigroup is weakly commutative, but there exist non-Abelian semigroups and even groups which are weakly commutative. The multiplicative group consisting of the quaternions 1, −1,i,−i, j, −j, k, −k, with i, j and k being the quaternion imaginary units is an example of non-Abelian weakly commutative group. The aim of the present paper is to prove some version of separation and support theorem for delta-subadditive mappings. In the second section following Ger [9] we introduce and study a basic properties of delta-subadditive and delta-superadditive maps. Section 3 deals with the definition of Lorenz cone and a partial order generated by this cone. It turns out that there is a close relationship between the above-mentioned order and the concept of delta- subadditivity. The fourth section contains the main result of the paper. We prove the separation theorem for delta-subadditive mappings. Using this theorem we give an easy proof of the Hyers–Ulam stability result for the Cauchy equation. In the last section we give a necessary and sufficient conditions under which a delta-subadditive map can be support at a given point by an additive map. We recall in this place the following well-known definition.

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Deﬁnition 1. Let (S, ·) and (Y,+) be semigroups and let x0 be a ﬁxed element of S. A mapping F : S → Y satisfying n F (x0 )=nF (x0) (1)

for some n ∈ N is said to be n-homogeneous at x0.If(1) holds for every n ∈ N, then F is N-homogeneous at x0. Moreover, if F is n-(resp. N)-homogeneous at every point of S, then we simply say that it is n-(resp. N)-homogeneous. In the sequel we will need the following result from [9]. Lemma 1. Let (S, ·) be a semigroup and let f : S → R be a subadditive or superadditive function. If f is 2-homogeneous then it is N-homogeneous, moreover, f(x · y)=f(y · x),x,y∈ X.

2. Delta-Subadditive Maps In 1989 L. Veselý and L. Zaji˘cek introduced an interesting generalization of functions which are representable as a difference of two convex functions. In the paper [20] the authors have introduced the following definition. Definition 2. Given two real normed spaces (X, ·), (Y,·) and a non-empty open and convex subset D ⊆ X we say that a map F : D → Y is delta-convex if there exists a continuous and convex functional f : D → R such that f +y∗ ◦F is continuous and convex for any member y∗ of the space Y ∗ dual to Y with y∗ = 1. If this is the case then we say that F is a delta-convex mapping with a control function f. It turns out that a continuous function F : D → Y is a delta-convex mapping controlled by a continuous function f : D → R if and only if the functional inequality x + y F (x)+F (y) f(x)+f(y) x + y F − ≤ − f , (2) 2 2 2 2 is satisfied for all x, y ∈ D. (Corollary 1.18 in [20]) The inequality (2) may obviously be investigated without any regular- ity assumption upon F and f which, additionally considerably enlarges the class of solutions. Note that the notion of delta-convex mappings has many nice properties (see [20]) and seems to be the most natural generalization of functions which are representable as a difference of two convex functions. Motivated by the concept of delta-convexity Ger in [10] introduced the following definition. Definition 3. A map F : S → Y is called delta-subadditive with a control function f : S → R if the following inequality F (x)+F (y) − F (x · y)≤f(x)+f(y) − f(x · y), (3) holds for all x, y ∈ S.

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In a natural way, this allows us to the following deﬁnition.

Deﬁnition 4. A map F : S → Y is said to be a delta superadditive, if −F is a delta-subadditive i.e. there exists a control function f : S → R such that the inequality

F (x)+F (y) − F (x · y)≤f(x · y) − f(x) − f(y),

is satisﬁed for all x, y ∈ S.

Observe that, if F is at the same time delta-subadditive and delta- superadditive with a control function f then both maps F and f are additive. The following result establishes the necessary and suﬃcient conditions for a given map to be delta-subadditive.

Proposition 1. For the mappings F : S → Y and f : S → R the following statements are equivalent:

(i) y∗ ◦ F + f is subadditive for any y∗ ∈ Y ∗, y∗ =1, (ii) F (x)+F (y) − F (x · y)≤f(x)+f(y) − f(x · y), for all x, y ∈ S, (iii)

n n n n F (xi) − F xi ≤ f(xi) − f xi , i=1 i=1 i=1 i=1 for all x1,...,xn ∈ S, n ∈ N. (4)

Proof. (i) implies (ii). For every y∗ ∈ Y ∗, y∗ =1wehave

y∗(F (x · y)) + f(x · y) ≤ y∗(F (x)) + f(x)+y∗(F (y)) + f(y),

and, consequently,

F (x)+F (y) − F (x · y) =sup{y∗(F (x)+F (y) − F (x · y)) : y∗ ∈ Y ∗, y∗ =1} ≤ f(x)+f(y) − f(x · y),x,y∈ S.

(ii) implies (iii). The proof runs by induction on n. The case n =1is trivial, while for n = 2 the inequality (4) is identical with (3). Now suppose (4)tobetrueforann ∈ N,n>2. Take arbitrary x1,...,xn+1 ∈ S. By (ii) and the induction hypothesis we obtain

Content courtesy of Springer Nature, terms of use apply. Rights reserved. Vol. 72 (2017) On Sandwich Theorem for Delta-Subadditive 389 ⎛ ⎞ ⎛ ⎞ n+1 n+1 n n ⎝ ⎠ ⎝ ⎠ F (xj ) − F xj = F (xj )+F (xn+1) − F xj · xn+1 j=1 j=1 j=1 j=1 ⎛ ⎞ ⎛ ⎞ n n n ⎝ ⎠ ⎝ ⎠ = F (xj ) − F xj + F xj + F (xn+1) j=1 j=1 j=1 ⎛ ⎞ ⎛ ⎞ n n n ⎝ ⎠ ⎝ ⎠ −F xj · xn+1 ≤ F (xj ) − F xj j=1 j=1 j=1 ⎛ ⎞ ⎛ ⎞ n n+1 ⎝ ⎠ ⎝ ⎠ + F (xn+1)+F xj − F xj j=1 j=1 ⎛ ⎞ n n ⎝ ⎠ ≤ f(xj ) − f xj j=1 j=1 ⎛ ⎞ ⎛ ⎞ n n+1 ⎝ ⎠ ⎝ ⎠ +f(xn+1)+f xj − f xj j=1 j=1 ⎛ ⎞ n+1 n+1 ⎝ ⎠ = f(xj ) − f xj . j=1 j=1

(iii) implies (ii). Trivial. (ii) implies (i). Let y∗ ∈ Y ∗, y∗ = 1 be arbitrary. For x, y ∈ S, we have y∗(F (x)+F (y) − F (x · y)) ≤F (x)+F (y) − F (x · y) ≤ f(x)+f(y) − f(x · y), or, equivalently, y∗(F (x · y)) + f(x · y) ≤ y∗(F (x)) + f(x)+y∗(F (y)) + f(y), which completes the proof.

Immediately from the above proposition we obtain the following result.

Corollary 1. Under the assumptions of the previous proposition if F : S → Y is a delta-subadditive map with a control function f : S → R then (i) nF (x) − F (xn)≤nf(x) − f(xn),x∈ S, n ∈ N, (ii) If e ∈ S is a neutral element of semigroup S, then F (e)≤f(e).

In the sequel we will use the following notation:

Ds(S):={F := (F, f):F : S → Y is delta-subadditive with control function f : S → R}

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3. A Partially Order connected with the Notion of Delta-Subadditivity In this place we give the definition and basic properties of so-called the Lorenz cone and partially order generated by this cone. As we will see this partial order appears in a natural way in the connection of the concept of delta- subadditivity. Let us recall that a nonempty subset C of a vector space is said to be a pointed, convex cone if it satisfies the following properties: (i) C + C⊆C, (ii) αC⊆C, for all α ≥ 0, (iii) C∩(−C)={0}. An arbitrary pointed, convex cone C of a vector space Y induces a vector ordering C letting x C y whenever y−x ∈C. This partial order is compatible with the linear structure of Y in the sense that if x C y, then (1) x + z C y + z,foreachz ∈ Y , (2) αx C αy, for all α ≥ 0. Now, for a normed space (Y,·) consider the linear space Y := Y × R, where as usual, the addition and scalar multiplication are defined coordinate- wise. Given a positive number ε, the convex cone defined by the formula

Cε := {(x, t) ∈ Y : εx≤t} is called the Lorenz cone or ice cream cone. More informations about the Lorenz cone can be found for instance in the book [1]. It is easy to check that this cone is closed, pointed and defines a vector ordering on Y in the following manner ⇔ − ≤ − (x1,t1) Cε (x2,t2) ε x2 x1 t2 t1. In 1962 Bishop and Phelps [4] introduced a slightly different order in functional analysis. The proof of the Bishop–Phelps lemma on the existence of certain minimal elements uses this order concept. As a consequence of this lemma one obtains the celebrated Bishop–Phelps theorem which says that the set of support functionals of a non-empty closed, bounded and convex subset of a real Banach space is dense in its dual space. Now, for an arbitrary map F : S → Y and a function f : S → R let define a map F : S → Y via the formula: F (x):=(F (x),f(x)),x∈ S. Observe that we can rewrite the inequality defining the notion of delta-subadditivity of F in the form · ∈ F (x y) C1 F (x)+F (y),x,yS,

where C1 = {(x, t) ∈ Y : x≤t} is the Lorenz cone. In the sequel for ∈ Y1,Y2 Y we will write Y1 Y2 instead of Y1 C1 Y2.

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4. The Separation Theorem The following lemma corresponds to the Lemma 2 from [9]. Lemma 2. Let (S, ·) be a weakly commutative semigroup, let (Y,·) be a real Banach space and assume that F =(F, f) ∈ Ds(S) and −G =(−G, −g) ∈ Ds(S).If G(x) F (x),x∈ S, (5) then there exist F 1 =(F1,f1), G1 =(G1,g1):D → Y such that (a) G(x) G1(x) F1(x) F (x),x∈ S, (b) F 1, −G1 ∈ Ds(S), (c) F 1, G1 are N − homogeneous, (d) F 1(x · y)=F 1(y · x) and G1(x · y)=G1(y · x),x,y∈ S, (e) moreover, if f (resp. g)isN-homogeneous at a point x0 ∈ X, then F 1(x0)=F (x0) (resp. G1(x0)=G(x0)). Proof. On account of condition (i) from Corollary 1 we have n n 2nF (x) − F (x2 )≤2nf(x) − f(x2 ),x∈ S, n ∈ N, (6) and n n 2nG(x) − G(x2 )≤g(x2 ) − 2ng(x),x∈ S, n ∈ N. (7) Deﬁne the sequences F n, Gn : S → Y by the formulas

F n(x):=(Fn(x),fn(x)), Gn(x):=(Gn(x),gn(x)), where 1 2n 1 2n Fn(x):= F (x ),fn(x):= f(x ), 2n 2n and,

1 2n 1 2n Gn(x):= G(x ),gn(x):= g(x ). 2n 2n By virtue of (5), (6), (7)wehave

G(x) Gn(x) F n(x) F (x),x∈ S.

Observe that the sequence {F n}n∈N is decreasing, {Gn}n∈N is increasing in the sense of an order generated by a Lorenz cone. Indeed, for the sequence {F n}n∈N we get n+1 n − 1 2 − 1 2 Fn+1(x) Fn(x) = n+1 F (x ) n F (x ) 2 2 n n 1 2 2 − 2 = n+1 F ((x ) ) 2F (x ) 2 1 n n+1 ≤ 2f(x2 ) − f(x2 ) 2n+1 1 n 1 n+1 = f(x2 ) − f(x2 ) 2n 2n+1 = fn(x) − fn+1(x),

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and similarly for the sequence {Gn}n∈N. Therefore

G(x) Gn(x) Gn+1(x) F n+1(x) F n(x) F (x),x∈ S, n ∈ N. In particular, we have

g(x) ≤ gn(x) ≤ gn+1(x) ≤ fn+1(x) ≤ fn(x) ≤ f(x),x∈ S, n ∈ N.

For each, ﬁxed x ∈ S, the sequences {fn(x)}n∈N and {gn(x)}n∈N being monotone and bounded, are convergent in R. Therefore we may deﬁne f0,g0 : S → R by

f0(x) := lim fn(x),g0(x) := lim gn(x). n→∞ n→∞

Since the sequences {fn(x)}n∈N and {gn(x)}n∈N are convergent, in particular, they are the Cauchy sequences. Observe that {Fn(x)}n∈N and {Gn(x)}n∈N also are the Cauchy sequences. Indeed, for arbitrary n, k ∈ N we have 1 2n+k 1 2n Fn+k(x) − Fn(x) = F (x ) − F (x ) 2n+k 2n 1 2n 2k − k 2n = n+k F ((x ) ) 2 F (x ) 2 1 n n+k ≤ 2kf(x2 ) − f(x2 ) 2n+k = fn(x) − fn+k(x).

The proof for the sequence {Gn}n∈N is similar. By the completeness of the space Y we can deﬁne functions F0,G0 as

F0(x) := lim Fn(x),G0(x) := lim Gn(x). n→∞ n→∞

Observe that (F0,f0), (−G0, −g0) ∈ Ds(S). To see it, ﬁx x, y ∈ S. Using the week commutativity of S we can ﬁnd a sequence {nk}k∈N of positive integers such that nk →∞as k →∞and n n n (x · y)2 k = x2 k · y2 k ,k∈ N. Then we get

Fn (x)+Fn (y) − Fn (x · y) k k k 1 nk 1 nk 1 nk nk = F (x2 )+ F (y2 ) − F (x2 · y2 ) nk nk nk 2 2 2 1 nk nk nk nk ≤ f(x2 )+f(y2 ) − f(x2 · y2 ) 2nk − · = fnk (x)+fnk (y) fnk (x y). Tending to the limit with n →∞we obtain

F0(x)+F0(y) − F0(x · y)≤f0(x)+f0(y) − f0(x · y),x,y∈ S,

which means that F0 is delta-subadditive with a control function f0. A similar argument ensures the delta-superadditivity of G0. Moreover, since

Fn(x) − Gn(x)≤fn(x) − gn(x),x∈ S, n ∈ N,

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then

F0(x) − G0(x)≤f0(x) − g0(x),x∈ S, which means that G(x) F (x),x∈ S. Further, observe that for each x ∈ S, one has 2 1 2 2n 1 2n+1 1 2n+1 f0(x )= lim f((x ) )= lim f(x )=2 lim f(x )=2f0(x), n→∞ 2n n→∞ 2n n→∞ 2n+1 and analogously, 2 g0(x )=2g0(x),x∈ S. On account of Lemma 1 and condition (i) from Corollary 1 the last two identi- ties guarantee the N-homogeneity of F 0 and G0, whereas Lemma 1 also implies (d). Finally, if f is N-homogeneous at x0 ∈ S then fn(x0)=f(x0),n∈ N. The same argument works for g0. We apply the above lemma to the proof of the following theorem. Theorem 1. Let (S, ·) be a weakly commutative semigroup,and let (Y,·) be a real Banach space. Assume that F : S → Y is a delta-subadditive map with a control function f : S → R and G : S → Y is a delta-superadditive map with a control function g : S → R. Suppose that (G, g) (F, f), i.e. F (x) − G(x)≤f(x) − g(x),x∈ S. If, moreover, sup{f(x) − g(x): x ∈ S} < ∞, then there exist unique additive mappings A : S → Y and a : S → R such that (G(x),g(x)) (A(x),a(x)) (F (x),f(x)),x∈ S.

Proof. Let F1,G1 : S → Y and f1,g1 : S → R be the mappings associated with F, G : S → Y and f,g : S → R according to Lemma 2. Using assertions (a) and (c) of that lemma, we obtain n n n n nF1(x) − G1(x) = F1(x ) − G1(x )≤f1(x ) − g1(x ) ≤ M, for every x ∈ S and n ∈ N, where M =sup{f(x) − g(x):x ∈ S}. Whence M F1(x) − G1(x)≤f1(x) − g1(x) ≤ , n for x ∈ S, n ∈ N, consequently passing to the limit as n →∞, we infer that F1 = G1,f1 = g1. Obviously A := F1 = G1 and a := f1 = g1 are additive, moreover, (G, g) (A, a) (F, f).

Now, suppose that A1 : S → Y and a1 : S → R are another mappings that A1 := (A1,a1) separate (F, f) and (G, g). Then n n n n nA(x) − A1(x) = A(x ) − A1(x )≤f(x ) − g(x ) ≤ M,

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for each x ∈ S and n ∈ N. Dividing by n and passing to the limit as n →∞ we obtain

A(x)=A1(x), and a(x)=a1(x). This proves the uniqueness of A and a.

As an application of this theorem we obtain an easy proof of the classical Hyers–Ulam stability result for the Cauchy equation. For the theory of the stability of functional equations see Hyers et al. [13]. Corollary 2. Let (S, ·) be a weakly commutative semigroup, and let (Y,·) be a real Banach space. If F : S → Y is an ε-additive map i.e. F (x)+F (y) − F (x · y)≤ε, x, y ∈ S, where ε>0, then there exists a unique additive map A : S → Y such that F (x) − A(x)≤ε, x ∈ S. Proof. Observe that by our assumption F is a delta-subadditive mapping with a control function f(x):=ε, x ∈ S, and it is a delta-superadditive with a control function g(x):=−ε, x ∈ S. Since sup[f(x) − g(x)] = 2ε<∞, x∈S and (F (x), −ε) (F (x),ε),x∈ S, then by Theorem 1 there exist unique additive mappings A : S → Y and a : S → R such that F (x) − A(x)≤ε − a(x), and F (x) − A(x)≤a(x)+ε, which means that F (x) − A(x)≤ε, and ﬁnishes the proof of our theorem.

5. Support Theorem In this section (X, +) is assumed to be a uniquely 2-divisible abelian group. It means that the mapping ω : X → X, ω(x)=2x, x ∈ X is bijective. Then −1 1 −1 both ω and ω are authomorphism of (X, +), and we write 2 x for ω (x). In the sequel we will use an additive notation + for a group operation and 0 stands for a neutral element of X. We consider the following problem: Whether for a given delta-subadditive map F : X → Y with a control function f : X → R and a given point y ∈ X there exist additive functions Ay : X → Y, ay : X → R such that a map

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Ay =(Ay,ay):X → Y support F =(F, f)aty in the sense of an order generated by a Lorenz cone. It means that

F (x) − Ay(x)≤f(x) − ay(x),x∈ X (8) and Ay(y)=f(y),ay(y)=f(y). (9) In the proof of our main result of this section we apply the following theorem, which is a particular case of Theorem 4 proved in [18] (Actually this theorem has been proved in the case when X is a real Banach space but its proof in our case runs without any essential changes). Theorem 2. Assume that (X, +) is a uniquely divisible by 2 abelian group, and (Y,·) is a real Banach space. If F : X → Y is a delta Jensen-convex map with a control function f : X → R that is x + z x + z F (x)+F (z) − 2F ≤ f(x)+f(z) − 2f ,x,z∈ X, (10) 2 2 then for an arbitrary point y ∈ X there exist aﬃne maps By : X → Y and by : X → R such that

F (x) − By(x)≤f(x) − by(x),x∈ X, (11) moreover, By(y)=f(y),by(y)=f(y). (12) Main result of this section reads as follows. Theorem 3. Let (X, +) be a uniquely divisible by 2 abelian group and let (Y,·) be a real Banach space. Let F : X → Y be a delta-subadditive map with a control function f : X → R, and let y ∈ Y be arbitrary. Then there exist additive maps Ay : X → Y and ay : X → R such that (8)and(9) hold if and only if f is N-homogeneous at y.

Proof. Assume that (F, f) ∈ Ds(X), moreover, f(yn)=nf(y),n∈ N. Observe, that N-homogeneity of F at y follows from the condition (i) from Corollary 1. Let deﬁne the map G : X → Y by formula G(x)=(G(x),g(x)) := (−F (−x), −f(−x)),x∈ X.

Obviously, −G ∈ Ds(X), moreover, using a delta-subadditivity of F and condition (ii) from Corollary 1 we obtain F (x) − G(x) = F (x)+F (−x) − F (0) + F (0) ≤F (x)+F (−x) − F (0) + F (0) ≤ f(x)+f(−x) − f(0) + f(0) = f(x)+f(−x)=f(x) − g(x),

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which means that G(x) F (x),x∈ X.

On account of Lemma 2 there exist N-homogeneous maps F 0, G0 : X → Y such that F 0, −G0 ∈ Ds(X)and

F (x) F 0(x) G0(x) G(x),x∈ X. Since f and g are N-homogeneous at y then by condition (e) of Lemma 2 we know that

F (y)=F0(y), and G(y)=G0(y).

Observe that F0 is a delta Jensen-convex map with a control function f0. Indeed, for any x, z ∈ X by 2-homogeneity of F0 and f0 we get x + z F0(x)+F0(z) − 2F0 = F0(x)+F0(z) − F0(x + z) 2 ≤ f0(x)+f0(z) − f0(x + z) x + z = f0(x)+f0(z) − 2f0 . 2 Now, we are able to apply the Theorem 2. By virtue of this theorem there are aﬃne maps By : X → Y and by : X → R such that

F0(x) − By(x)≤f0(x) − by(x),x∈ X, (13) moreover, By(y)=F0(y),by(y)=f0(y). (14)

It is known (see for instance [15]) that the maps By and by have the form

By(x)=Ay(x)+C, by(x)=ay(x)+c, x ∈ X, (15)

where Ay : X → Y,ay : X → R are additive maps and C ∈ X and c ∈ R are constants. We need to show that C =0andc =0.By(13), (14)and N-homogeneity of Ay,ay,F0 and f0 for all x ∈ X we obtain n n n n n n F0(2 x)−Ay(2 x)−C=2 F0(x) − 2 Ay(x) − C≤2 f0(x) − 2 ay(x) − c. The above inequality is equivalent to the following one C c F0(x) − Ay(x) − ≤ f0(x) − ay(x) − ,x∈ X, 2n 2n therefore,

F0(x) − Ay(x)≤f0(x) − ay(x),x∈ X. By (14) and (15)wehave

C = F0(y) − Ay(y), and c = f0(y) − ay(y), whence, C = F0(y) − Ay(y)≤f0(y) − ay(y)=c. (16)

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On the other hand, using again the N-homogeneity of Ay,ay,F0 and f0 for all x ∈ X we obtain 1 1 1 1 F0(x) − Ay(x) − C ≤ f0(x) − ay(x) − c, 2n 2n 2n 2n or, equivalently, n n F0(x) − Ay(x) − 2 C≤f0(x) − ay(x) − 2 c. In particular, n 2 c ≤ f0(x) − ay(x),x∈ X, n ∈ N, hence c ≤ 0 and this together with (16) implies that c =0andC =0. To end the proof of suﬃciency it is enough to observe that

F (x) − Ay(x)≤F (x) − F0(x) + F0(x) − Ay(x) ≤ f(x) − f0(x)+f0(x) − ay(x)=f(x) − ay(x),x∈ X. Conversely, observe that the N-homogeneity at y ∈ X is necessary for existence of additive map (Ay,ay) supporting (F, f)aty. Indeed, if (F, f) ∈ Ds(X) and additive map (Ay,ay) supports (F, f)aty, then for any n ∈ N we have

F (ny) − nF (y)≤F (ny) − Ay(ny) + nAy(y) − nF (y) = F (ny) − Ay(ny)≤f(ny) − ay(ny) = f(ny) − nay(y) ≤ nf(y) − nay(y)=0.

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[5] Buskes, G.: The Hahn–Banach Theorem surveyed. Dissertationes Math. (Rozprawy Mat.), 327, 49 (1993) [6] Cabello Sánchez, F., Castillo, J.M.F.: Banach space techniques underpinning a theory of nearly additive mappings. Dissertationes Math. (Rozprawy Mat.) 404, 4 (2002) [7] Chaljub-Simon, A., Volkmann, P.: Bemerkungen zu einem Satz von Rodé. Arch. Math. (Basel) 57, 180–188 (1991) [8] Fuchssteiner, B., Lusky, W.: Convex Cones, vol. 56. North-Holland Mathematics Studies, Amsterdam (1981) [9] Gajda, Z., Kominek, Z.: On separations theorems for subadditive and superadditive functionals. Stud. Math. 100, 25–38 (1991) [10] Ger, R.: On Functional Inequalities Stemming from Stability Questions, General Inequalities 6, ISNM, vol. 103, pp. 227–240. Birkhäuser Verlag, Basel (1992) [11] Ger, R.: Stability aspects of delta-convexity. In: Rassias, ThM, Tabor, J. (eds.) Stability of Hyers–Ulam Type, pp. 99–109. Hardonic Press, Palm Harbor (1994) [12] Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27, 222–224 (1941) [13] Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Sev- eral Variables. Birkhäuser, Boston (1998) [14] Kaufman, R.: Interpolation of additive functionals. Stud. Math. 27, 269–272 (1966) [15] Kominek, Z.: Convex Functions in Linear Spaces, Prace Naukowe Uniwersytetu Sla´ ˛skiego w Katowicach nr 1087, Katowice (1989) [16] Köning, H.: On the abstract Hahn–Banach theorem due to Rodé. Aequ. Math. 34, 89–95 (1987) [17] Kranz, P.: Additive functionals on abelian semigroup. Comment. Math. Pr. Mat. 16, 239–246 (1972) [18] Olbry´s, A.: A support theorem for delta(s, t)-convex mappings. Aequ. Math. 89, 937–948 (2015) [19] Rodé, G.: Eine abstrakte version des Satzen von Hahn–Banach. Arch. Mat. (Basel) 31, 474–481 (1978) [20] Veselý, L., Zaji˘cek, L.: Delta-convex mappings between Banach spaces and ap- plications. Dissertationes Math. Polish Scientific Publishers (289), Warszawa (1989) [21] Tabor, J.: Remark 18, Report of Meeting, the 22nd Internat. In: Symposium on Functional Equations. Aequationes Math., vol. 29, pp. 96 (1985)

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Andrzej Olbry´s Institute of Mathematics University of Silesia Bankowa 14 40-007 Katowice Poland e-mail: [email protected]

Received: September 7, 2016. Accepted: November 19, 2016.

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