A support theorem for delta (s, t)-convex mappings

In the present paper a notion of delta (s, t)-convexity in the sense of Veselý and Zajic̆ek is studied as a natural generalization of the classical (s, t)-convexity. The main result of this paper is a support theorem for delta (s, t)-convex mappings.


Introduction and terminology
Throughout this paper (X, · ) and (Y, · ) stand for two real Banach spaces. Definition 1. Let s, t ∈ (0, 1) be fixed real numbers, and let D ⊂ X be a convex set. A function f : D → [−∞, ∞) is said to be (s, t)-convex, if x,y∈D (s, t)-affine, if x,y∈D If the inequality (1) is satisfied for t = s then we say that f is t-convex, if t = s = 1 2 then f is said to be Jensen-convex. If the Eq. (2) is satisfied for s = t, and all t ∈ [0, 1], then we say that f is affine.
In [7] Kuhn proved that every t-convex function is Jensen-convex (cf. Daróczy and Páles [1] for a simple argument). Some properties of (s, t)-convex functions are contained in [8]. In particular in [8] Kuhn remarks that f must be constant if t is rational. He also mentions that he does not know any example of a non constant (s, t)-convex functions for s = t. In a natural way a problem 938 A. Olbryś AEM of a characterization of (s, t)-convex functions, for s = t appears in this context (independently asked by Rolewicz [5]). The complete solution of this problem was given by Matkowski and Pycia in [10] (see also [5] for partial solution). To present the main result contained in [10] we need the following is a constant function.
In the sequel we will use the following remark contained in [10].(Actually in [10] the authors assume that D is an open interval but such a restriction is inessential).
A survey of results concerning (s, t)-convex functions may be found in the papers [5,8,10], in particular, the following version of the theorem of Rodé (cf. [12]), has been proved by Kuhn in [8] and by Kominek in [5].
In 1989 Veselý and Zajicek introduced an interesting generalization of functions which are representable as a difference of two convex functions. In the paper [13] the authors introduced the following definition Definition 3. Let D ⊂ X be a convex and open set. A map F : D → Y is called 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 is satisfied for all x, y ∈ D (Corollary 1.18 in [13]). In the paper [2] Ger generalized this result. He has shown that if the inequality (3) holds for all x, y ∈ D and the function is upper bounded on a set T ⊂ D whose Q-convex hull conv Q (T ) forms a second cathegory Baire subset of X then F is locally Lipschtzian, in particular, F is a delta-convex mapping controlled by f . Moreover, if Y is a separable space and the function given by the formula (4) is Christensen measurable then it provides the same effect. The inequality (3) may obviously be investigated without any regularity assumption upon F and f . Motivated by these two concepts we introduce the following definition.
holds for all x, y ∈ D. If the above inequality is satisfied for t = s, then we say that F is a delta t-convex mapping, if t = s = 1 2 then F is said to be delta Jensen-convex.
As an immediate consequence of Theorem 1 we get

Results
In the proof of our first result we use the following corollary, which is a consequence of the Hahn-Banach theorem.
The following result establishes necessary and sufficient conditions for a given map to be delta (s, t)-convex.
The following conditions are pairwise equivalent: for all x, y ∈ D. Let y * ∈ Y * , y * = 1 be arbitrary. From the above inequality and Corollary 2 it follows that or, equivalently, (iii) implies (ii). Replace y * by −y * in (iii).
Let us observe, that delta (s, t)-convex mappings provide a generalization of functions which are representable as a difference of two (s, t)-convex functions. A support theorem for delta (s, t)-convex mappings 941 Proof. Assume f : D → R is a control function for F . For all x, y ∈ D we have Put It is easy to see that both φ 1 and φ 2 are (s, t)-convex functions, moreover, which finishes the proof.
Using a well-known Daróczy and Páles representation of the mean x+y 2 we get the following Proof. Take an arbitrary y * ∈ Y * such that y * = 1. From the identity (cf. Daróczy which means that and consequently, whence, in view of the arbitrariness of y * and on account of Corollary 2 we get The proof of our lemma is finished. Observe that, on account of the above lemma the results obtained by Ger in [2] concerning delta Jensen-convex mappings are also true for delta (s, t)convex mappings. Now, we are able to prove our main result. The following theorem corresponds to a classical support theorem for (s, t)-convex functions.

Proof. Fix an arbitrary point y ∈ D. Consider the following family of pairs of maps
Observe that H = ∅, because (F, f ) ∈ H. Define an order relation on H as follows We will show that every chain has a lower bound in H. Observe that h 0 is (s, t)-convex in D. To see it take arbitrary x, z ∈ D and arbitrary c 1 , c 2 ∈ R such that By the definition of h 0 there exist ( Therefore (say (H 1 , h 1 ) (H 2 , h 2 )) we obtain by the (s, t)-convexity of h 1 and h 2

Vol. 89 (2015)
A support theorem for delta (s, t)-convex mappings 943 Tending in the above inequalities with c 1 → h 0 (x), c 2 → h 0 (z) we get the (s, t)-convexity of h 0 . Since h 0 (y) = f (y) > −∞, then by Remark 1 h 0 has finite values. There exists a sequence {(H n , h n )} n∈N ⊂ L such that Since the sequence {h n } n∈N is convergent, in particular it is a Cauchy sequence, so ε>0 n0∈N n,m≥n0 Therefore {H n } n∈N is also a Cauchy sequence and consequently it is convergent. Let H 0 be its limit. First we must check whether the definition of H 0 is correct. If a sequence {(K n , k n )} n∈N ⊂ L satisfies the condition then using the same argumentation it is easy to check that {K n } n∈N is a Cauchy sequence, and consequently converges. Let Since then, because L is a chain we have for all n ∈ N and letting n → ∞ we obtain and then (H 0 , h 0 ) (H, h). Due to the arbitrariness of (H, h) ∈ L we infer that (H 0 , h 0 ) is a lower bound of the chain L. Now, we will show that (H 0 , h 0 ) ∈ H. Indeed, since h 0 (y) = f (y) also H 0 (y) = F (y). Note that, because To see that H 0 is delta (s, t)-convex with a control function h 0 fix arbitrary points x, z ∈ D. Let x 1 := x, x 2 := z, This completes the proof.
The following Theorem states that the existence of a support mapping at an arbitrary point in fact characterizes delta (s, t)-convexity.
Proof. The sufficiency results from Theorem 4. To prove the necessity fix arbitrary x, z ∈ D. Put y := sx + (1 − s)z. By our assumption we get The proof is complete.
It follows from the proof of Theorems 4 and 5 that for delta convex mappings the following theorem holds true

Vol. 89 (2015)
A support theorem for delta (s, t)-convex mappings 947 Remarks 1. Substituting F := 0 in our theorems we obtain the well-known results concerning classical (s, t)-convexity. 2. Let us observe that there is a close relationship between the theory of delta convex mappings and the problems of Hyers-Ulam stability, developed by studying the following functional inequality