On Stability and Hyperstability of an Equation Characterizing Multi-Cauchy–Jensen Mappings

Recently, functions of several variables satisfying, with respect to each variable, some functional equation (among them Cauchy’s, Jensen’s, quadratic and other ones) have been studied. We give a new characterization of multi-Cauchy–Jensen mappings, which states that a function fulfilling some equation on a restricted domain is multi-Cauchy–Jensen. Next, using a fixed point theorem, it is proved that a function which approximately satisfies (on restricted domain) the equation characterizing such functions is close (in some sense) to the solution of the equation. This result is a tool for obtaining a generalized Hyers–Ulam stability or hyperstability of this equation for particular control functions, which is presented in several examples.


Introduction
It is well-known that among functional equations the Cauchy equation f (x + y) = f (x) + f (y) (1) and the Jensen equation (which is closely connected with the notion of convex function) play a prominent role. A lot of information about them and their applications can be found for instance in [29,30]. The first positive answer to celebrated Ulam's question concerning the problem of stability of functional equations was given by Hyers in the case of Eq. (1) in Banach spaces (see [23]). The history and recent results concerning the notion of Hyers-Ulam stability can be found in many papers (see e.g. [12,13,16,21,28,29] and references included there). The multi-Cauchy-Jensen mappings mentioned in the title are functions of several variables satisfying Cauchy's functional equation in each of some chosen variables and Jensen's functional equation in each of the remaining ones. Namely, if it holds for k and l variables, respectively, such a function is called k-Cauchy and l-Jensen (see [15]). Without loss of generality it can be assumed that such functions satisfy (1) for the first few variables, and (2) for the next ones.
Let us note that for k = n the above definition leads to the so-called multi-additive mappings (some basic facts on such mappings can be found for instance in [30], where their application to the representation of polynomial functions is also presented); for k = 0 we obtain the notion of multi-Jensen function (which was introduced in 2005 by Prager and Schwaiger (see [33]) in the connection with generalized polynomials), and an 1-Cauchy and 1-Jensen mapping is just a Cauchy-Jensen mapping defined by Park and Bae [32].
In this paper, we give a new characterization of multi-Cauchy-Jensen mappings, which states that a function fulfilling some equation on a restricted domain is multi-Cauchy-Jensen on the whole space. Next it is proved that a function which approximately satisfies (on restricted domain) the equation characterizing such functions is close (in some sense) to the solution of the equation. This result is a tool for obtaining a generalized Hyers-Ulam stability or hyperstability of this equation for particular control functions, which is presented in several examples. Our results are significant counterparts of some classical outcomes from [1,11,22,23,35] and recent results from [2][3][4]10,[17][18][19][20][24][25][26][27]31,32,34,36].
In the proof of our stability result (Theorem 6) we use the fixed point method, which was used for the investigation of the Hyers-Ulam stability of functional equations for the first time by Baker [5]. For more information about this method we refer the reader to recent survey papers [13,21].
Let us recall that an abelian semigroup G is called uniquely divisible by 2 provided for every x ∈ G there exists a unique y ∈ G (which is denoted in the sequel by x 2 or 1 2 x) such that x = y + y. The symbol my denotes (m − 1)y + y for m ∈ N, m ≥ 2. We will denote by G 0 the set G \ {0}, where 0 is an identity element. G is said to be torsion free, if the identity element is the only one of finite order. In this case in particular, 2x, 3x For a nonempty set X and l, m ∈ N we identify m-tuple Moreover, we assume that V and W are linear spaces over the rationals, In [9], a characterization of multi-Cauchy-Jensen mappings was proved. Using our notations we can rephrase it in the following form.

Theorem 1.
Assume that G is a semigroup uniquely divisible by 2 and with an identity element, and W is a linear space over the rationals. Then a function f : G n → W is k-Cauchy and n − k-Jensen if and only if for any x = (x 1 , x 2 ), y = (y 1 , y 2 ) ∈ G k × G n−k we have Substituting k = n we have a characterization of multi-additive mappings. A counterpart of this theorem, for mappings defined on linear space over rationals which satisfy (1) on a restricted domain, was proved in [8]. The theorem is still true if we assume that the domain of f is a group satisfying some additional assumptions, which will be proven with the aid of the following characterization of multi-Jensen mappings.

Theorem 2.
Assume that G is a torsion free group uniquely divisible by 2, G 0 = ∅. A function f : G k → W satisfies the equation for all x, y ∈ G k 0 , if and only if f is a multi-Jensen mapping. Proof. First observe that by Theorem 1, every multi-Jensen mapping of k variables satisfies (4) on G k . The proof of the converse theorem is by induction on k. It is true for k = 1. Indeed, in this case (4) means that for x = 0 and and it suffices to prove the above equality for x ∈ G and y = 0 which is equivalent that for every x = 0 Obviously, the above equality holds for x = 0. If x = 0, applying (5) for pairs of nonzero elements 2x, x and next 2x, −x we get and 2f Adding the above equalities and applying (5) for elements 3x 2 , x 2 and next for and the proof of the base case is complete. Now assume that every function of k variables satisfying (4) is multi- for all x, y ∈ G k+1 Similarly, for Vol. 73 (2018) On Stability and Hyperstability Page 5 of 18 55 Fix z ∈ G 0 and define the function g z (x) := f (x, 2z) for x ∈ G k . Applying (7) and (9) we have for all x, y ∈ G k which with the inductive assumption implies that g z is a multi-Jensen function.
Similarly, a function g 0 : G k → W given by the formula g 0 (x) = f (x, 0) is multi-Jensen, since according to (7) and (8) It suffices to show that f is a Jensen function with respect to the last variable. Fix Thus u = (u 1 , . . . , u k ), w = (w 1 , . . . , w k ) ∈ G k 0 and x = u+w 2 . We will show that h x fulfills (4) on G 0 . To this end take y, z ∈ G 0 , then (u, y), (w, z) ∈ G k+1 0 and the functions g y , g z are multi-Jensen. Therefore From what has already been proved in the base step, we conclude that h x is Jensen, and finally induction completes the proof.
We are thus led to the following new proof of a refinement of a characterization for multi-additive mappings given in [8].

Proposition 3. Assume that G is a torsion free group uniquely divisible by 2,
for all x, y ∈ G k 0 , if and only if f is a multi-additive mapping.
Proof. First observe that by Theorem 1, every multi-additive mapping of k variables satisfies (10) on G k . Now assume that (10) is fulfilled for x, y ∈ G k 0 . According to Theorem 1, it suffices to show that it holds on G k . We begin by proving that f is 2-homogeneous of degree k, namely Indeed, if x ∈ G k 0 we conclude from (10) that For any x = (x 1 , . . . , x k ) ∈ G k \G k 0 , fix v ∈ G 0 and define Then 2x = y + z with y = (y 1 , . . . , y k ), z = (z 1 , . . . , z k ) ∈ G k 0 , and Since (10) holds on G k 0 and f is 2-homogeneous of degree k, for x, y ∈ G k 0 we obtain Therefore f is k-Jensen and satisfies (4) on G k , by Theorems 2 and 1. Finally, applying (4) and (11), for x, y ∈ G k we see that which completes the proof.
We are now in a position to show the second characterization of multi-Cauchy-Jensen mappings.
Since for any z ∈ G n−k 0 a mapping g z : G k −→ W given by satisfies the Eq. (10) for all x, y ∈ G k 0 , Lemma 3 shows that the function g z is multi-additive, which means that for z ∈ G n−k 0 , x, y ∈ G k . On the other hand, setting y = 0 := (0, . . . , 0) ∈ G k in (13) we have for any x ∈ G k and z, w ∈ G n−k . Therefore using (3) for all x ∈ G k and z, w ∈ G n−k 0 we have Thus for any x ∈ G k the function h x : G n−k −→ W given by . Lemma 2 shows that the function h x is multi-Jensen, which means (15) holds for all x ∈ G k and z, w ∈ G n−k , and finishes the proof that f is a multi-Cauchy-Jensen mapping.

Stability of Multi-Cauchy-Jensen Mappings on Restricted Domain
In this section we prove stability of Eq. (3) on restricted domain. This result generalizes Theorem 3.2 from [9]. The proof is based on a fixed point result that can be derived from [14] (Theorem 1). To present it we need the following three hypothesis: (H1) E is a nonempty set, Y is a Banach space, f 1 , . . . , f k : E → E and L 1 , . . . , L k : E → R + are given.
For the convenience of the reader, we recall the above mentioned fixed point theorem.
Then there exists a unique fixed point ψ of T with In the sequel, we assume that W is a Banach space, k ≥ 1 and D, E are the nonempty subsets of V such that E ⊂ D and Theorem 6. Let f : D n → W and θ : E 2n → R + be mappings satisfying the inequality for x, y ∈ E n . Assume also that there is an s ∈ {−1, 1} such that for x 1 , y 1 ∈ E k , x 2 , y 2 ∈ E n−k , and the set E fulfils a condition Then there exists a unique function F : E n → W satisfying Eq. (3) for all x = (x 1 , x 2 ), y = (y 1 , y 2 ) ∈ E k × E n−k and such that Proof. If s = 1 putting in (16) x = y = (x 1 , x 2 ) ∈ E k ×E n−k and then dividing by 2 n we get If s = −1 putting in (16) x = y = 1 2 x 1 , x 2 ∈ E k × E n−k and then dividing by 2 n−k we have Then (21) Then it is easily seen that Λ has the form described in (H3) with j = 1 and It is easy to check that for x = (x 1 , Hence and from (17), according to Theorem 5 there exists a unique solu- such that (20) holds. Moreover, One can now show, by induction, that for every l ∈ N 0 , x = (x 1 , x 2 ), y = (y 1 , y 2 ) ∈ E k × E n−k . Letting l → ∞ in (23) and using (18) we obtain which means the function F satisfies Eq. (3) for all x = (x 1 , x 2 ), y = (y 1 , y 2 ) ∈ E k × E n−k . Now, we assume that F : E n → W is another function satisfying the Eq. (3) for all x = (x 1 , x 2 ), y = (y 1 , y 2 ) ∈ E k × E n−k and the inequality (20). Then using (3) and (20), we have for whence letting l → ∞ and using (17) we obtain F (x) = F (x) for x ∈ E n , which finishes the proof.

Stability and Hyperstability Results
Applying the above Theorem 6 for specific functions θ and D = V, E = V 0 yields the following stability results. These results are significant generalizations of some outcomes from [8,27].
From (25) we get that d < 0 or d > 0. Then there exists s ∈ {1, −1} such that Using Theorem 6, because we obtain that there exists a unique function F * : V n 0 → W satisfying Eq. (3) for all x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ V n 0 and such that otherwise . Finally, using Theorem 4 we get that F is k-Cauchy and n−k-Jensen mapping.
Using the above corollary we can obtain the following hyperstability result.
Proof. According to Corollary 7, there exists a unique k-Cauchy and n − k- Let t ∈ {1, . . . , n} be such that p t + q t < 0. Then at least one of p t , q t must be negative. Without loss of generality we can assume that p t < 0.
Then by (26) and (27) We now turn to the case t > k and apply similar arguments with sequences defined as follows