Approximate Multi-Jensen, Multi-Euler-Lagrange Additive and Quadratic Mappings in 𝑛-Banach Spaces

Abstract

We prove the generalized Hyers-Ulam stability of multi-Jensen, multi-Euler-Lagrange additive, and quadratic mappings in 𝑛-Banach spaces, using the socalled direct method. The corollaries from our main results correct some outcomes from Park (2011).

1. Introduction and Preliminaries

In 2005, Prager and Schwaiger (see [1] and also [2]) introduced the notion of multi-Jensen functions with the connection with generalized polynomials and obtained their general form. In 2008, (see [3]) they also proved the Hyers-Ulam stability of multi-Jensen equation, whereas Ciepliński (see [4, 5]) showed its generalized stability: in the spirit of Bourgin (see [6]) and Găvruţa (see [7]), and in the spirit of Aoki (see [8]) and Rassias (see [9]). Recently, some further results on the stability of multi-Jensen mappings were obtained in [10–14]. We refer the reader to [15–19] for more information on different aspects of stability of functional equations.

In this paper, we deal with the generalized Hyers-Ulam stability of multi-Jensen, multi-Euler-Lagrange additive, and quadratic mappings in 𝑛-Banach spaces. The corollaries from our main results correct some outcomes from [20]. The results of Sections 2 and 4 generalize those from [12].

The concept of 2-normed spaces was initially developed by Gähler [21, 22] in the middle of the 1960s, while that of 𝑛-normed spaces can be found in [23, 24]. Since then, many others have studied this concept and obtained various results (see [23, 25–27]).

Throughout this paper, ℕ stands for the set of all positive integers and ℝ represents the set of all real numbers. Moreover, we fix two positive integers 𝑘 and 𝑛.

We recall some basic facts concerning 𝑛-normed spaces.

Definition 1. Let 𝑛∈ℕ and let 𝑋 be a real linear space with dim𝑋≥𝑛, and let ‖⋅,…,⋅‖∶𝑋^𝑛→ℝ be a function satisfying the following properties:(N1) ‖𝑥₁,…,𝑥_𝑛‖=0 if and only if 𝑥₁,…,𝑥_𝑛 are linearly dependent,(N2)‖𝑥₁,…,𝑥_𝑛‖ is invariant under permutation,(N3)‖𝛼𝑥₁,…,𝑥_𝑛‖=|𝛼|‖𝑥₁,…,𝑥_𝑛‖,(N4)‖𝑥+𝑦,𝑥₂,…,𝑥_𝑛‖≤‖𝑥,𝑥₂,…,𝑥_𝑛‖+‖𝑦,𝑥₂,…,𝑥_𝑛‖for all 𝛼∈ℝ and 𝑥,𝑦,𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑋. Then the function ‖⋅,…,⋅‖ is called an 𝑛-norm on 𝑋, and the pair (𝑋,‖⋅,…,⋅‖) is called an 𝑛-normed space.

A sequence {𝑥_𝑗}_𝑗∈ℕ in an 𝑛-normed space 𝑋 is said to a converge to some 𝑥∈𝑋 in the 𝑛-norm if lim_𝑗→∞𝑥_𝑗−𝑥,𝑦₂,…,𝑦_𝑛=0, (1) for every 𝑦₂,…,𝑦_𝑛∈𝑋. Every convergent sequence has exactly one limit. If 𝑥 is the limit of the sequence {𝑥_𝑗}_𝑗∈ℕ, then we write lim_𝑗→∞𝑥_𝑗=𝑥. For any convergent sequences {𝑥_𝑗}_𝑗∈ℕ and {𝑦_𝑗}_𝑗∈ℕ of elements of 𝑋, the sequence {𝑥_𝑗+𝑦_𝑗}_𝑗∈ℕ is convergent and lim_𝑗→∞𝑥_𝑗+𝑦_𝑗=lim_𝑗→∞𝑥_𝑗+lim_𝑗→∞𝑦_𝑗. (2) If, moreover, {𝛼_𝑗}_𝑗∈ℕ is a convergent sequence of real numbers, then the sequence {𝛼_𝑗⋅𝑥_𝑗}_𝑗∈ℕ is also convergent and lim_𝑗→∞𝛼_𝑗⋅𝑥_𝑗=lim_𝑗→∞𝛼_𝑗⋅lim_𝑗→∞𝑥_𝑗. (3) A sequence {𝑥_𝑗}_𝑗∈ℕ in an 𝑛-normed space 𝑋 is said to be a Cauchy sequence with respect to the 𝑛-norm if lim_{𝑗,𝑙→∞}𝑥_𝑗−𝑥_𝑙,𝑦₂,…,𝑦_𝑛=0, (4) for every 𝑦₂,…,𝑦_𝑛∈𝑋. A linear 𝑛-normed space in which every Cauchy sequence is convergent is called an 𝑛-Banach space.

Example 2. For 𝑥₁,…,𝑥_𝑛∈ℝ^𝑛, the Euclidean 𝑛-norm ‖𝑥₁,…,𝑥_𝑛‖_𝐸 is defined by 𝑥₁,…,𝑥_𝑛𝐸=det𝑥_𝑖𝑗=abs𝑥₁₁⋯𝑥_1𝑛𝑥⋮⋱⋮_𝑛1⋯𝑥_𝑛𝑛, (5) where 𝑥_𝑖=(𝑥_𝑖1,…,𝑥_𝑖𝑛)∈ℝ^𝑛 for each 𝑖=1,…,𝑛.

Example 3. The standard 𝑛-norm on 𝑋, a real inner product space of dimension dim 𝑋≥𝑛, is as follows: 𝑥₁,…,𝑥_𝑛𝑆=𝑥₁,𝑥₁⋯𝑥₁,𝑥_𝑛𝑥⋮⋱⋮_𝑛,𝑥₁⋯𝑥_𝑛,𝑥_𝑛^1/2, (6) where ⟨⋅,⋅⟩ denotes the inner product on 𝑋. If 𝑋=ℝ^𝑛, then this 𝑛-norm is exactly the same as the Euclidean 𝑛-norm ‖𝑥₁,…,𝑥_𝑛‖_𝐸 mentioned earlier. For 𝑛=1, this 𝑛-norm is the usual norm ‖𝑥₁‖=⟨𝑥₁,𝑥₁⟩^1/2.

In what follows, we will also use the following lemma from [19].

Lemma 4. Let 𝑋 be an 𝑛-normed space. Then,(1)for 𝑥_𝑖∈𝑋(𝑖=1,…,𝑛) and 𝛾, a real number, 𝑥₁,…,𝑥_𝑖,…,𝑥_𝑗,…,𝑥_𝑛=𝑥₁,…,𝑥_𝑖,…,𝑥_𝑗+𝛾𝑥_𝑖,…,𝑥_𝑛, (7) for all 1≤𝑖≠𝑗≤𝑛,(2)|‖𝑥,𝑦₂,…,𝑦_𝑛‖−‖𝑦,𝑦₂,…,𝑦_𝑛‖|≤‖𝑥−𝑦,𝑦₂,…,𝑦_𝑛‖ for all 𝑥,𝑦,𝑦₂,…,𝑦_𝑛∈𝑋,(3)if ‖𝑥,𝑦₂,…,𝑦_𝑛‖=0 for all 𝑦₂,…,𝑦_𝑛∈𝑋, then 𝑥=0,(4)for a convergent sequence {𝑥_𝑗} in 𝑋, lim_𝑗→∞𝑥_𝑗,𝑦₂,…,𝑦_𝑛=lim_𝑗→∞𝑥_𝑗,𝑦₂,…,𝑦_𝑛, (8) for all 𝑦₂,…,𝑦_𝑛∈𝑋.

2. Approximate Multi-Jensen Mappings

First, we prove the stability of the system of equations defining multi-Jensen mappings in 𝑛-Banach spaces. For a given mapping 𝑓∶𝑉^𝑘→𝑊, we define the difference operators 𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑘+1∶=2𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖+𝑥_𝑖+12,𝑥_𝑖+2,…,𝑥_𝑘+1−𝑓𝑥₁,…,𝑥_𝑖,𝑥_𝑖+2,…,𝑥_𝑘+1−𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖+1,…,𝑥_𝑘+1,𝑥₁,…,𝑥_𝑘+1∈𝑉^𝑘+1,𝑖∈{1,…,𝑘}. (9)

Theorem 5. Let 𝑉 be a commutative group uniquely divisible by 2, and, 𝑊 be an 𝑛-Banach space. Assume also that for every 𝑖∈{1,…,𝑘}, 𝜑_𝑖∶𝑉^𝑘+1→[0,∞) is a mapping such that ^∞_𝑗=013^𝑗+1𝜑_𝑖3^𝑗𝑥₁,𝑥₂,…,𝑥_𝑘+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑖−2,3^𝑗𝑥_𝑖−1,𝑥_𝑖,…,𝑥_𝑘+1+𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,3^𝑗𝑥_𝑖,3^𝑗𝑥_𝑖+1,𝑥_𝑖+2,…,𝑥_𝑘+1+𝜑_𝑖𝑥₁,…,𝑥_𝑖+1,3^𝑗𝑥_𝑖+2,𝑥_𝑖+3,…,𝑥_𝑘+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑘,3^𝑗𝑥_𝑘+1𝑥<∞,₁,…,𝑥_𝑘+1∈𝑉^𝑘+1. (10) If 𝑓∶𝑉^𝑘→𝑊 is a function satisfying 𝑓𝑥₁,…,𝑥_𝑖−1,0,𝑥_𝑖+1,…,𝑥_𝑘𝑥=0,₁,…,𝑥_𝑖−1,𝑥_𝑖+1,…,𝑥_𝑘∈𝑉^𝑘−1,𝑖∈{1,…,𝑘},𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑘+1,𝑦₂,…,𝑦_𝑛≤𝜑_𝑖𝑥₁,…,𝑥_𝑘+1,𝑥₁,…,𝑥_𝑘+1∈𝑉^𝑘+1,𝑖∈{1,…,𝑘},𝑦₂,…,𝑦_𝑛∈𝑊, (11)(12) then for every 𝑖∈{1,…,𝑘}, there exists a multi-Jensen mapping 𝐹_𝑖∶𝑉^𝑘→𝑊 for which 𝑓𝑥₁,…,𝑥_𝑘−𝐹_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤^∞_𝑗=013^𝑗+1×𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,3^𝑗𝑥_𝑖,−3^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘+𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,−3^𝑗𝑥_𝑖,3^𝑗+1𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑥₁,…,𝑥_𝑘∈𝑉^𝑘,𝑦₂,…,𝑦_𝑛∈𝑊. (13) For every 𝑖∈{1,…,𝑘}, the function 𝐹_𝑖 is given by 𝐹_𝑖𝑥₁,…,𝑥_𝑘∶=lim_𝑗→∞13^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,3^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑥₁,…,𝑥_𝑘∈𝑉^𝑘. (14)

Proof. Fix 𝑥₁,…,𝑥_𝑘∈𝑉, 𝑦₂,…,𝑦_𝑛∈𝑊 and 𝑖∈{1,…,𝑘}. By (12) and (11), we get 𝑓𝑥₁,…,𝑥_𝑘+𝑓𝑥₁,…,𝑥_𝑖−1,−𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤𝜑_𝑖𝑥₁,…,𝑥_𝑖,−𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,2𝑓𝑥₁,…,𝑥_𝑘−𝑓𝑥₁,…,𝑥_𝑖−1,−𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−𝑓𝑥₁,…,𝑥_𝑖−1,3𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,−𝑥_𝑖,3𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘. (15) Hence, 3𝑓𝑥₁,…,𝑥_𝑘−𝑓𝑥₁,…,𝑥_𝑖−1,3𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤𝜑_𝑖𝑥₁,…,𝑥_𝑖,−𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘+𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,−𝑥_𝑖,3𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘, (16) and consequently for any nonnegative integers 𝑙 and 𝑚 such that 𝑙<𝑚, we obtain 13^𝑙𝑓𝑥₁,…,𝑥_𝑖−1,3^𝑙𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−13^𝑚𝑓𝑥₁,…,𝑥_𝑖−1,3^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤^𝑚−1_𝑗=𝑙13^𝑗+1𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,3^𝑗𝑥_𝑖,−3^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘+𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,−3^𝑗𝑥_𝑖,3^𝑗+1𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘. (17) Therefore, from (10), it follows that {(1/3^𝑗)𝑓(𝑥₁,…,𝑥_𝑖−1,3^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘)}_𝑗∈ℕ is a Cauchy sequence. Since 𝑊 is an 𝑛-Banach space, this sequence is convergent and we define 𝐹_𝑖∶𝑉^𝑘→𝑊 by (14). Putting 𝑙=0, letting 𝑚→∞ in (17), and using Lemma 4 and (10), we see that (13) holds.
Finally, fix 𝑥^′_𝑖∈𝑉, 𝑗∈ℕ, and note that according to (12), we have 13^𝑗𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑖−1,3^𝑗𝑥_𝑖,3^𝑗𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤13^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,3^𝑗𝑥_𝑖,3^𝑗𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘. (18)
Next, fix 𝑠∈{1,…,𝑘}⧵{𝑖},𝑥^′_𝑠∈𝑉, and assume that 𝑠<𝑖 (the same arguments apply to the case where 𝑠>𝑖). From (12), it follows that 13^𝑗𝐷_𝑠𝑓𝑥₁,…,𝑥_𝑠,𝑥^′_𝑠,𝑥_𝑠+1,…,𝑥_𝑖−1,3^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤13^𝑗𝜑_𝑠𝑥₁,…,𝑥_𝑠,𝑥^′_𝑠,𝑥_𝑠+1𝑥,…,_𝑖−1,3^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘. (19) Letting 𝑗→∞ in the above two inequalities and using (10) and Lemma 4, we see that the mapping 𝐹_𝑖 is multi-Jensen.

Theorem 6. Let 𝑉 be a real linear space and, 𝑊 be an 𝑛-Banach space. Assume also that for every 𝑖∈{1,…,𝑘}, 𝜑_𝑖∶𝑉^𝑘+1→[0,∞) is a mapping such that ^∞_𝑗=03^𝑗𝜑_𝑖x₁3^𝑗,𝑥₂,…,𝑥_𝑘+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑖−2,𝑥_𝑖−13^𝑗,𝑥_𝑖,…,𝑥_𝑘+1+𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖3^𝑗,𝑥_𝑖+13^𝑗,𝑥_𝑖+2,…,𝑥_𝑘+1+𝜑_𝑖𝑥₁,…,𝑥_𝑖+1,𝑥_𝑖+23^𝑗,𝑥_𝑖+3,…,𝑥_𝑘+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑘,𝑥_𝑘+13^𝑗𝑥<∞,₁,…,𝑥_𝑘+1∈𝑉^𝑘+1. (20) If 𝑓∶𝑉^𝑘→𝑊 is a function satisfying conditions (11) and (12), then for every 𝑖∈{1,…,𝑘} there exists a multi-Jensen mapping 𝐹_𝑖∶𝑉^𝑘→𝑊 for which 𝑓𝑥₁,…,𝑥_𝑘−𝐹_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤^∞_𝑗=03^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖3^𝑗+1𝑥,−^𝑖3^𝑗+1,𝑥_𝑖+1,…,𝑥_𝑘+𝜑_𝑖𝑥₁,…,𝑥_𝑖−1𝑥,−^𝑖3^𝑗+1,𝑥_𝑖3^𝑗,𝑥_𝑖+1,…,𝑥_𝑘,𝑥₁,…,𝑥_𝑘∈𝑉^𝑘,𝑦₂,…,𝑦_𝑛∈𝑊. (21) For every 𝑖∈{1,…,𝑘}, the function 𝐹_𝑖 is given by 𝐹_𝑖𝑥₁,…,𝑥_𝑘∶=lim_𝑗→∞3^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖3^𝑗,𝑥_𝑖+1,…,𝑥_𝑘,𝑥₁,…,𝑥_𝑘∈𝑉^𝑘. (22)

Proof. Fix 𝑥₁,…,𝑥_𝑘∈𝑉, 𝑦₂,…,𝑦_𝑛∈𝑊, 𝑗∈ℕ∪{0} and 𝑖∈{1,…,𝑘}. By (12) and (11), we get 3^𝑗+1𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖3^𝑗+1,𝑥_𝑖+1,…,𝑥_𝑘−3^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖3^𝑗,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤3^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖3^𝑗+1𝑥,−^𝑖3^𝑗+1,𝑥_𝑖+1,…,𝑥_𝑘+𝜑_𝑖𝑥₁,…,𝑥_𝑖−1𝑥,−^𝑖3^𝑗+1,𝑥_𝑖3^𝑗,𝑥_𝑖+1,…,𝑥_𝑘, (23) and consequently for any non-negative integers 𝑙 and 𝑚 such that 𝑙<𝑚, we obtain 3^𝑙𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖3^𝑙,𝑥_𝑖+1,…,𝑥_𝑘−3^𝑚𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖3^𝑚,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤^𝑚−1_𝑗=𝑙3^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖3^𝑗+1𝑥,−^𝑖3^𝑗+1,𝑥_𝑖+1,…,𝑥_𝑘+𝜑_𝑖𝑥₁,…,𝑥_𝑖−1𝑥,−^𝑖3^𝑗+1,𝑥_𝑖3^𝑗,𝑥_𝑖+1,…,𝑥_𝑘. (24) Therefore, from (20), it follows that {3^𝑗𝑓(𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖/3^𝑗,𝑥_𝑖+1,…,𝑥_𝑘)}_𝑗∈ℕ is a Cauchy sequence. Since 𝑊 is an 𝑛-Banach space, this sequence is convergent and we define 𝐹_𝑖∶𝑉^𝑘→𝑊 by (22). Putting 𝑙=0, letting 𝑚→∞ in (24), and using Lemma 4 and (20), we see that (21) holds.
Finally, fix 𝑥^′_𝑖∈𝑉, and note that according to (12), we have 3^𝑗𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖3^𝑗,𝑥^′_𝑖3^𝑗,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤3^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖3^𝑗,𝑥^′_𝑖3^𝑗,𝑥_𝑖+1,…,𝑥_𝑘. (25) Next, fix 𝑠∈{1,…,𝑘}⧵{𝑖},𝑥^′_𝑠∈𝑉, and assume that 𝑠<𝑖 (the same arguments apply to the case where 𝑠>𝑖). From (12), it follows that 3^𝑗𝐷_𝑠𝑓𝑥₁,…,𝑥_𝑠,𝑥^′_𝑠,𝑥_𝑠+1,…,𝑥_𝑖−1,𝑥_𝑖3^𝑗,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤3^𝑗𝜑_𝑠𝑥₁,…,𝑥_𝑠,𝑥^′_𝑠,𝑥_𝑠+1,…,𝑥_𝑖−1,𝑥_𝑖3^𝑗,𝑥_𝑖+1,…,𝑥_𝑘. (26) Letting 𝑗→∞ in the previous two inequalities and using (20) and Lemma 4, we see that the mapping 𝐹_𝑖 is multi-Jensen.

As applications of Theorems 5 and 6 we get the following corollaries.

Corollary 7. Let 𝑉 be a real normed linear space and, 𝑊 be an 𝑛-Banach space. Assume also that 𝜃∈[0,∞) and 𝑟∈(0,∞) are such that 𝑟≠1. If 𝑓∶𝑉^𝑘→𝑊 is a function satisfying (11) and 𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑘+1,𝑦₂,…,𝑦_𝑛≤𝜃𝑥₁^𝑟⋯𝑥_𝑖−1^𝑟𝑥_𝑖^𝑟+𝑥_𝑖+1^𝑟×𝑥_𝑖+2^𝑟⋯𝑥_𝑘+1^𝑟,𝑥₁,…,𝑥_𝑘+1∈𝑉^𝑘+1{},𝑦,𝑖∈1,…,𝑘₂,…,𝑦_𝑛∈𝑊, (27) then for every 𝑖∈{1,…,𝑘} there exists a multi-Jensen mapping 𝐹_𝑖∶𝑉^𝑘→𝑊 for which 𝑓𝑥₁,…,𝑥_𝑘−𝐹_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤𝜃𝑥₁^𝑟⋯𝑥_𝑘^𝑟3+3^𝑟|3−3^𝑟|, (28) for all 𝑥₁,…,𝑥_𝑘∈𝑉, 𝑦₂,…,𝑦_𝑛∈𝑊.

Corollary 8. Let 𝑉 be a real normed linear space and let 𝑊 be an 𝑛-Banach space. Assume also that 𝜃∈[0,∞) and 𝑟,𝑝,𝑞∈(0,∞) are such that 𝑟,𝑝+𝑞∈(0,1) or 𝑟,𝑝+𝑞∈(1,∞). If 𝑓∶𝑉^𝑘→𝑊 is a function satisfying (11) and 𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑘+1,𝑦₂,…,𝑦_𝑛≤𝜃𝑥₁^𝑟⋯𝑥_𝑖−1^𝑟𝑥_𝑖^𝑝𝑥_𝑖+1^𝑞×𝑥_𝑖+2^𝑟⋯𝑥_𝑘+1^𝑟,𝑥₁,…,𝑥_𝑘+1∈𝑉^𝑘+1{},𝑦,𝑖∈1,…,𝑘₂,…,𝑦_𝑛∈𝑊, (29) then for every 𝑖∈{1,…,𝑘}, there exists a multi-Jensen mapping 𝐹_𝑖∶𝑉^𝑘→𝑊 for which 𝑓𝑥₁,…,𝑥_𝑘−𝐹_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤𝜃𝑥₁^𝑟⋯𝑥_𝑖−1^𝑟𝑥_𝑖^𝑝+𝑞𝑥_𝑖+1^𝑟⋯𝑥_𝑘^𝑟1+3^𝑞3−3^𝑝+𝑞, (30) for all 𝑥₁,…,𝑥_𝑘∈𝑉, 𝑦₂,…,𝑦_𝑛∈𝑊.

From Corollary 8, we obtain the following corollary which corrects Theorems 3.1 and 3.2 from [20].

Corollary 9. Let 𝑉 be a real normed linear space and 𝑊 be an 𝑛-Banach space. Assume also that 𝜃∈[0,∞) and 𝑝,𝑞∈(0,∞) are such that 𝑝+𝑞≠1. If 𝑓∶𝑉→𝑊 is a function satisfying 𝑓(0)=0 and 𝑥2𝑓¹+𝑥₂2−𝑓𝑥₁−𝑓𝑥₂𝑥,𝑦≤𝜃₁^𝑝𝑥₂^𝑞,𝑥₁,𝑥₂∈𝑉,𝑦∈𝑊, (31) then there exists a Jensen mapping 𝐹∶𝑉→𝑊 for which 𝑓(𝑥)(𝑥)≤𝜃‖𝑥‖−𝐹,𝑦_𝑝+𝑞1+3^𝑞3−3^𝑝+𝑞,𝑥∈𝑉,𝑦∈𝑊. (32)

3. Approximate Multi-Euler-Lagrange Additive Mappings

In this section, we prove the stability of the system of equations defining multi-Euler-Lagrange additive mappings.

Throughout this section, let 𝑉 be a real linear space and let 𝑊 be an 𝑛-Banach space, and 𝑎,𝑏∈ℝ⧵{0} are fixed with 𝜆∶=𝑎+𝑏≠0,±1.

A mapping 𝑓∶𝑉^𝑘→𝑊 is called a multiEuler-Lagrange additive mapping as follows if it satisfies the Euler-Lagrange additive equations in each of their 𝑘 arguments as follows: 𝑓𝑥₁,…,𝑥_𝑖−1,𝑎𝑥_𝑖+𝑏𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘+𝑓𝑥₁,…,𝑥_𝑖−1,𝑏𝑥_𝑖+𝑎𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘=(𝑎+𝑏)𝑓𝑥₁,…,𝑥_𝑘+𝑓𝑥₁,…,𝑥_𝑖−1,𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘, (33) for all 𝑖∈{1,…,𝑘} and all 𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖,𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘∈𝑉. If 𝑎=𝑏=1, then the multi-Euler-Lagrange additive mapping is multiadditive (see [28]). For a given mapping 𝑓∶𝑉^𝑘→𝑊, we define the difference operators 𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑘+1∶=𝑓𝑥₁,…,𝑥_𝑖−1,𝑎𝑥_𝑖+𝑏𝑥_𝑖+1,𝑥_𝑖+2,…,𝑥_𝑘+1+𝑓𝑥₁,…,𝑥_𝑖−1,𝑏𝑥_𝑖+𝑎𝑥_𝑖+1,𝑥_𝑖+2,…,𝑥_𝑘+1−(𝑎+𝑏)𝑓𝑥₁,…,𝑥_𝑖,𝑥_𝑖+2,…,𝑥_𝑘+1+𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖+1,…,𝑥_𝑘+1,𝑥₁,…,𝑥_𝑘+1∈𝑉^k+1,𝑖∈{1,…,𝑘}. (34)

Theorem 10. Assume that for every 𝑖∈{1,…,𝑘}, 𝜑_𝑖∶𝑉^𝑘+1→[0,∞) is a mapping such that ^∞_𝑗=01|𝜆|^𝑗𝜑_𝑖𝜆^𝑗𝑥₁,…,𝑥_𝑘+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑖−2,𝜆^𝑗𝑥_𝑖−1,𝑥_𝑖,…,𝑥_𝑘+1+𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝜆^𝑗𝑥_𝑖+1,𝑥_𝑖+2,…,𝑥_𝑘+1+𝜑_𝑖𝑥₁,…,𝑥_𝑖+1,𝜆^𝑗𝑥_𝑖+2,𝑥_𝑖+3,…,𝑥_𝑘+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑘,𝜆^𝑗𝑥_𝑘+1𝑥<∞,₁,…,𝑥_𝑘+1∈𝑉^𝑘+1. (35) If 𝑓∶𝑉^𝑘→𝑊 is a function satisfying 𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑘+1,𝑦₂,…,𝑦_𝑛≤𝜑_𝑖𝑥₁,…,𝑥_𝑘+1,𝑥₁,…,𝑥_𝑘+1∈𝑉^𝑘+1,𝑖∈{1,…,𝑘},𝑦₂,…,𝑦_𝑛∈𝑊, (36) then for every 𝑖∈{1,…,𝑘}, there exists a unique multi-Euler-Lagrange additive mapping 𝐴_𝑖∶𝑉^𝑘→𝑊 for which 𝑓𝑥₁,…,𝑥_𝑘−𝐴_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤12^∞_𝑗=01|𝜆|^𝑗+1×𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑥₁,…,𝑥_𝑘∈𝑉^𝑘,𝑦₂,…,𝑦_𝑛∈𝑊. (37) For every 𝑖∈{1,…,𝑘}, the function 𝐴_𝑖 is given by 𝐴_𝑖𝑥₁,…,𝑥_𝑘∶=lim_𝑗→∞1𝜆^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑥₁,…,𝑥_𝑘∈𝑉^𝑘. (38)

Proof. Fix 𝑥₁,…,𝑥_𝑘∈𝑉, 𝑦₂,…,𝑦_𝑛∈𝑊, 𝑗∈ℕ∪{0} and 𝑖∈{1,…,𝑘}. By (36), we get 𝑓𝑥₁,…,𝑥_𝑘−1𝜆𝑓𝑥₁,…,𝑥_𝑖−1,𝜆𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤1||𝜑2𝜆^𝑖𝑥₁,…,𝑥_𝑖,𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘, (39) whence 1𝜆^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−1𝜆^𝑗+1𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗+1𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤12|𝜆|^𝑗+1𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘. (40) For any nonnegative integers 𝑙 and 𝑚 with 𝑙<𝑚, using (40) we get 1𝜆^𝑙𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑙𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−1𝜆^𝑚𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤^𝑚−1_𝑗=𝑙12|𝜆|^𝑗+1×𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘, (41) which tends to zero as 𝑙 tends to infinity. Therefore, from (35) it follows that {(1/𝜆^𝑗)𝑓(𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘)}_𝑗∈ℕ is a Cauchy sequence in 𝑛-Banach space 𝑊 and it thus converges. Hence, we can define 𝐴_𝑖∶𝑉^𝑘→𝑊 by 𝐴_𝑖𝑥₁,…,𝑥_𝑘∶=lim_𝑗→∞1𝜆^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘. (42) Putting 𝑙=0, letting 𝑚→∞ in (41), and using (35), we see that (37) holds.
Now, fix also 𝑥^′_𝑖∈𝑉, and from (36), we have 1𝜆^𝑗𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝜆^𝑗𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤1|𝜆|^𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝜆^𝑗𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘. (43)
Next, fix 𝑠∈{1,…,𝑘}⧵{𝑖}, 𝑥^′_𝑠∈𝑉, and assume that 𝑠<𝑖 (the same arguments apply to the case where 𝑠>𝑖). From (36) it follows that 1𝜆^𝑗𝐷_𝑠𝑓𝑥₁,…,𝑥_𝑠,𝑥^′_𝑠,𝑥_𝑠+1,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤1|𝜆|^𝑗𝜑_𝑠𝑥₁,…,𝑥_𝑠,𝑥^′_𝑠,𝑥_𝑠+1,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘. (44) Letting 𝑗→∞ in the above two inequalities and using (35) and Lemma 4, we see that the mapping 𝐴_𝑖 is multi-Euler-Lagrange additive.
Now, let us finally assume that 𝐴^′_𝑖∶𝑉^𝑘→𝑊 is another multi-Euler-Lagrange additive mapping satisfying (37). Then we have 𝐴_𝑖𝑥₁,…,𝑥_𝑘−𝐴^′_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛=lim_𝑚→∞1|𝜆|^𝑚×𝐴_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−𝐴^′_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤lim_𝑚→∞1|𝜆|^𝑚×𝐴_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛+𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−𝐴_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤lim_𝑚→∞1|𝜆|×^∞_𝑗=01|𝜆|^𝑚+𝑗×𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚+𝑗𝑥_𝑖,𝜆^𝑚+𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘=0, (45) and therefore 𝐴_𝑖=𝐴^′_𝑖.

Theorem 11. Assume that for every 𝑖∈{1,…,𝑘}, 𝜑_𝑖∶𝑉^𝑘+1→[0,∞) is a mapping such that ^∞_𝑗=0|𝜆|^𝑗𝜑_𝑖𝑥₁𝜆^𝑗,𝑥₂,…,𝑥_𝑘+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑖−2,𝑥_𝑖−1𝜆^𝑗,𝑥_𝑖,…,𝑥_𝑘+1+𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑗,𝑥_𝑖+1𝜆^𝑗,𝑥_𝑖+2,…,𝑥_𝑘+1+𝜑_𝑖𝑥₁,…,𝑥_𝑖+1,𝑥_𝑖+2𝜆^𝑗,𝑥_𝑖+3,…,𝑥_𝑘+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑘,𝑥_𝑘+1𝜆^𝑗𝑥<∞,₁,…,𝑥_𝑘+1∈𝑉^𝑘+1. (46) If 𝑓∶𝑉^𝑘→𝑊 is a function satisfying (36), then for every 𝑖∈{1,…,𝑘}, there exists a unique multi-Euler-Lagrange additive mapping 𝐴_𝑖∶𝑉^𝑘→𝑊 for which 𝑓𝑥₁,…,𝑥_𝑘−𝐴_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤12^∞_𝑗=1|𝜆|^𝑗−1×𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑗,𝑥_𝑖𝜆^𝑗,𝑥_𝑖+1,…,𝑥_𝑘,𝑥₁,…,𝑥_𝑘∈𝑉^𝑘,𝑦₂,…,𝑦_𝑛∈𝑊. (47) For every 𝑖∈{1,…,𝑘}, the function 𝐴_𝑖 is given by 𝐴_𝑖𝑥₁,…,𝑥_𝑘∶=lim_𝑗→∞𝜆^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑗,𝑥_𝑖+1,…,𝑥_𝑘,𝑥₁,…,𝑥_𝑘∈𝑉^𝑘. (48)

Proof. Fix 𝑥₁,…,𝑥_𝑘∈𝑉, 𝑦₂,…,𝑦_𝑛∈𝑊, 𝑗∈ℕ∪{0} and 𝑖∈{1,…,𝑘}. By (36) we get 𝑓𝑥₁,…,𝑥_𝑘−𝜆𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤12𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆,𝑥_𝑖𝜆,𝑥_𝑖+1,…,𝑥_𝑘. (49) For any non-negative integers 𝑙 and 𝑚 with 0≤𝑙<𝑚, using (49), we get 𝜆^𝑙𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑙,𝑥_𝑖+1,…,𝑥_𝑘−𝜆^𝑚𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑚,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤^𝑚−1_𝑗=𝑙|𝜆|^𝑗2×𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑗+1,𝑥_𝑖𝜆^𝑗+1,𝑥_𝑖+1,…,𝑥_𝑘, (50) which tends to zero as 𝑙 tends to infinity. Therefore from (46), it follows that {𝜆^𝑗𝑓(𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖/𝜆^𝑗,𝑥_𝑖+1,…,𝑥_𝑘)}_𝑗∈ℕ is a Cauchy sequence in 𝑛-Banach space 𝑊 and it thus converges. Hence, we can define 𝐴_𝑖∶𝑉^𝑘→𝑊 by 𝐴_𝑖𝑥₁,…,𝑥_𝑘∶=lim_𝑗→∞𝜆^𝑗𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑗,𝑥_𝑖+1,…,𝑥_𝑘. (51) Putting 𝑙=0, letting 𝑚→∞ in (50), and using (46), we see that (47) holds. The further part of the proof is similar to the proof of Theorem 10.

As applications of Theorems 10 and 11, we get the following corollaries.

Corollary 12. Let 𝑉 be a real normed linear space and, 𝑊 be an 𝑛-Banach space. Assume also that 𝜃∈[0,∞) and 𝑟∈(0,∞) are such that 𝑟≠1. If 𝑓∶𝑉^𝑘→𝑊 is a function satisfying 𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑘+1,𝑦₂,…,𝑦_𝑛≤𝜃𝑥₁^𝑟⋯𝑥_𝑖−1^𝑟𝑥_𝑖^𝑟+𝑥_𝑖+1^𝑟×𝑥_𝑖+2^𝑟⋯𝑥_𝑘+1^𝑟,𝑥₁,…,𝑥_𝑘+1∈𝑉^𝑘+1{},𝑦,𝑖∈1,…,𝑘₂,…,𝑦_𝑛∈𝑊, (52) then for every 𝑖∈{1,…,𝑘} there exists a unique multi-Euler-Lagrange additive mapping 𝐴_𝑖∶𝑉^𝑘→𝑊 for which 𝑓𝑥₁,…,𝑥_𝑘−𝐴_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤𝜃𝑥₁^𝑟⋯𝑥_𝑘^𝑟|𝜆|−|𝜆|^𝑟, (53) for all 𝑥₁,…,𝑥_𝑘∈V, 𝑦₂,…,𝑦_𝑛∈𝑊.

Corollary 13. Let 𝑉 be a real normed linear space and let 𝑊 be an 𝑛-Banach space. Assume also that 𝜃∈[0,∞) and 𝑟,𝑝,𝑞∈(0,∞) are such that 𝑟,𝑝+𝑞∈(0,1) or 𝑟,𝑝+𝑞∈(1,∞). If 𝑓∶𝑉^𝑘→𝑊 is a function satisfying 𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑘+1,𝑦₂,…,𝑦_𝑛≤𝜃𝑥₁^𝑟⋯𝑥_𝑖−1^𝑟𝑥_𝑖^𝑝𝑥_𝑖+1^𝑞×𝑥_𝑖+2^𝑟⋯𝑥_𝑘+1^𝑟,𝑥₁,…,𝑥_𝑘+1∈𝑉^𝑘+1,𝑖∈{1,…,𝑘},𝑦₂,…,𝑦_𝑛∈𝑊, (54) then for every 𝑖∈{1,…,𝑘}, there exists a unique multi-Euler-Lagrange additive mapping 𝐴_𝑖∶𝑉^𝑘→𝑊 for which 𝑓𝑥₁,…,𝑥_𝑘−𝐴_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤𝜃𝑥₁^𝑟⋯𝑥_𝑖−1^𝑟𝑥_𝑖^𝑝+𝑞𝑥_𝑖+1^𝑟⋯𝑥_𝑘^𝑟2|𝜆|−|𝜆|^𝑝+𝑞, (55) for all 𝑥₁,…,𝑥_𝑘∈𝑉, 𝑦₂,…,𝑦_𝑛∈𝑊.

From Corollary 13 we obtain the following corollary which corrects Theorems 2.1 and 2.2 from [20].

Corollary 14. Let 𝑉 be a real normed linear space and 𝑊 be an 2-Banach space. Assume also that 𝜃∈[0,∞) and 𝑝,𝑞∈(0,∞) are such that 𝑝+𝑞≠1. If 𝑓∶𝑉→𝑊 is a function satisfying 𝑓𝑥₁+𝑥₂−𝑓𝑥₁−𝑓𝑥₂𝑥,𝑦≤𝜃₁^𝑝𝑥₂^𝑞,𝑥₁,𝑥₂∈𝑉,𝑦∈𝑊, (56) then there exists a unique additive mapping 𝐴∶𝑉→𝑊 for which 𝑓(𝑥)(𝑥)≤𝜃‖𝑥‖−𝐴,𝑦^𝑝+𝑞2−2^𝑝+𝑞,𝑥∈𝑉,𝑦∈𝑊. (57)

4. Approximate Multi-Euler-Lagrange Quadratic Mappings

In this section, we prove the stability of the system of equations defining multi-Euler-Lagrange quadratic mappings.

Throughout this section, let 𝑉 be a real linear space and let 𝑊 be an 𝑛-Banach space, and 𝑎,𝑏∈ℝ⧵{0} are fixed with 𝜆∶=𝑎²+𝑏²≠1.

Rassias [29] introduced the notion of a generalized Euler-Lagrange-type quadratic mapping, and investigated its generalized stability.

A mapping 𝑓∶𝑉^𝑘→𝑊 is called a multi-Euler-Lagrange quadratic mapping, if it satisfies the Euler-Lagrange quadratic equations in each of their 𝑘 arguments: 𝑓𝑥₁,…,𝑥_𝑖−1,𝑎𝑥_𝑖+𝑏𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘+𝑓𝑥₁,…,𝑥_𝑖−1,𝑏𝑥_𝑖−𝑎𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘=𝑎²+𝑏²×𝑓𝑥₁,…,𝑥_𝑘+𝑓𝑥₁,…,𝑥_𝑖−1,𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘, (58) for all 𝑖∈{1,…,𝑘} and all 𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖,𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘∈𝑉.

If 𝑎=𝑏=1, then the multi-Euler-Lagrange quadratic mapping is multiquadratic (see [30]). Letting 𝑥_𝑖=𝑥^′_𝑖=0 in (58), we get 𝑓(𝑥₁,…,𝑥_𝑖−1,0,𝑥_𝑖+1,…,𝑥_𝑘)=0. Putting 𝑥^′_𝑖=0 in (58), we have 𝑓𝑥₁,…,𝑥_𝑖−1,𝑎𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘+𝑓𝑥₁,…,𝑥_𝑖−1,𝑏𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘=𝜆𝑓𝑥₁,…,𝑥_𝑘. (59) Replacing 𝑥_𝑖 by 𝑎𝑥_𝑖 and 𝑥^′_𝑖 by 𝑏𝑥_𝑖 in (58), respectively, we obtain 𝑓𝑥₁,…,𝑥_𝑖−1,𝜆𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘=𝜆𝑓𝑥₁,…,𝑥_𝑖−1,𝑎𝑥_𝑖,x_𝑖+1,…,𝑥_𝑘+𝑓𝑥₁,…,𝑥_𝑖−1,𝑏𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘. (60) From (59) and (60), one gets 𝑓𝑥₁,…,𝑥_𝑖−1,𝜆𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘=𝜆²𝑓𝑥₁,…,𝑥_𝑘, (61) for all 𝑖∈{1,…,𝑘} and all 𝑥₁,…,𝑥_𝑘∈𝑉.

For a given mapping 𝑓∶𝑉^𝑘→𝑊, we define the difference operators 𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑘+1∶=𝑓𝑥₁,…,𝑥_𝑖−1,𝑎𝑥_𝑖+𝑏𝑥_𝑖+1,𝑥_𝑖+2,…,𝑥_𝑘+1+𝑓𝑥₁,…,𝑥_𝑖−1,𝑏𝑥_𝑖−𝑎𝑥_𝑖+1,𝑥_𝑖+2,…,𝑥_𝑘+1−𝑎²+𝑏²×𝑓𝑥₁,…,𝑥_𝑖,𝑥_𝑖+2,…,𝑥_𝑘+1+𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖+1,…,𝑥_𝑘,𝑥₁,…,𝑥_𝑘+1∈𝑉^𝑘+1,𝑖∈{1,…,𝑘}. (62)

Theorem 15. Assume that for every 𝑖∈{1,…,𝑘}, 𝜑_𝑖∶𝑉^𝑘+1→[0,∞) is a mapping such that ^∞_𝑗=01𝜆^2𝑗×𝜑_𝑖𝜆^𝑗𝑥₁,𝑥₂,…,𝑥_𝑘+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑖−2,𝜆^𝑗𝑥_𝑖−1,𝑥_𝑖,…,𝑥_𝑘+1+𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝜆^𝑗𝑥_𝑖+1,𝑥_𝑖+2,…,𝑥_𝑘+1+𝜑_𝑖𝑥₁,…,𝑥_𝑖+1,𝜆^𝑗𝑥_𝑖+2,𝑥_𝑖+3,…,𝑥_𝑘+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑘,𝜆^𝑗𝑥_𝑘+1𝑥<∞,₁,…,𝑥_𝑘+1∈𝑉^𝑘+1. (63) If 𝑓∶𝑉^𝑘→𝑊 is a function satisfying condition (11) and 𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑘+1,𝑦₂,…,𝑦_𝑛≤𝜑_𝑖𝑥₁,…,𝑥_𝑘+1,𝑥₁,…,𝑥_𝑘+1∈𝑉^𝑘+1{},𝑦,𝑖∈1,…,𝑘₂,…,𝑦_𝑛∈𝑊, (64) then for every 𝑖∈{1,…,𝑘}, there exists a unique multi-Euler-Lagrange quadratic mapping 𝑄_𝑖∶𝑉^𝑘→𝑊 for which 𝑓𝑥₁,…,𝑥_𝑘−𝑄_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤^∞_𝑗=01𝜆^2𝑗+1𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,0,𝑥_𝑖+1,…,𝑥_𝑘+1𝜆^2𝑗+2𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑎𝜆^𝑗𝑥_𝑖,𝑏𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑥₁,…,𝑥_𝑘∈𝑉^𝑘,𝑦₂,…,𝑦_𝑛∈𝑊. (65) For every 𝑖∈{1,…,𝑘}, the function 𝑄_𝑖 is given by 𝑄_𝑖𝑥₁,…,𝑥_𝑘∶=lim_𝑗→∞1𝜆^2𝑗×𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑥₁,…,𝑥_𝑘∈𝑉^𝑘. (66)

Proof. Fix 𝑥₁,…,𝑥_𝑘∈𝑉, 𝑦₂,…,𝑦_𝑛∈𝑊, 𝑗∈ℕ∪{0} and 𝑖∈{1,…,𝑘}. By (64), we get 𝑓𝑥₁,…,𝑥_𝑖−1,𝑎𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘+𝑓𝑥₁,…,𝑥_𝑖−1,𝑏𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−𝜆𝑓𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤𝜑_𝑖𝑥₁,…,𝑥_𝑖,0,𝑥_𝑖+1,…,𝑥_𝑘,𝑓𝑥₁,…,𝑥_𝑖−1,𝜆𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−𝜆𝑓𝑥₁,…,𝑥_𝑖−1,𝑎𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−𝜆𝑓𝑥₁,…,𝑥_𝑖−1,𝑏𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑎𝑥_𝑖,𝑏𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘. (67)
From (67), we obtain 1𝜆²𝑓𝑥₁,…,𝑥_𝑖−1,𝜆𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−𝑓𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤1𝜆𝜑_𝑖𝑥₁,…,𝑥_𝑖,0,𝑥_𝑖+1,…,𝑥_𝑘+1𝜆²×𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑎𝑥_𝑖,𝑏𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘, (68) and consequently for any non-negative integers 𝑙 and 𝑚 such that 𝑙<𝑚, we get 1𝜆^2𝑙𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑙𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−1𝜆^2𝑚×𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤^𝑚−1_𝑗=𝑙1𝜆^2(𝑗+1)𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗+1𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−1𝜆^2𝑗𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤^𝑚−1_𝑗=𝑙1𝜆^2𝑗+1𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,0,𝑥_𝑖+1,…,𝑥_𝑘+1𝜆^2𝑗+2×𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑎𝜆^𝑗𝑥_𝑖,𝑏𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘. (69) Therefore from (63), it follows that {(1/𝜆^2𝑗)𝑓(𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘)}_𝑗∈ℕ is a Cauchy sequence. Since 𝑊 is an 𝑛-Banach space, this sequence is convergent and we define 𝑄_𝑖∶𝑉^𝑘→𝑊 by (66). Putting 𝑙=0, letting 𝑚→∞ in (69) and using Lemma 4 and (63), we see that (65) holds.
Now, fix also 𝑥^′_𝑖∈𝑉 and note that according to (64), we have 1𝜆^2𝑗𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝜆^𝑗𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤1𝜆^2𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝜆^𝑗𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑘. (70)
Next, fix 𝑠∈{1,…,𝑘}⧵{𝑖}, 𝑥^′_𝑠∈𝑉, and assume that 𝑠<𝑖 (the same arguments apply to the case where 𝑠>𝑖). From (64), it follows that 1𝜆^2𝑗𝐷_𝑠𝑓𝑥₁,…,𝑥_𝑠,𝑥^′_𝑠,𝑥_𝑠+1,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤1𝜆^2𝑗𝜑_𝑠𝑥₁,…,𝑥_𝑠,𝑥^′_𝑠,𝑥_𝑠+1,…,𝑥_𝑖−1,𝜆^𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘. (71) Letting 𝑗→∞ in the above two inequalities and using (63), and Lemma 4 we see that the mapping 𝑄_𝑖 is multi-Euler-Lagrange quadratic.
Now, let us finally assume that 𝑄^′_𝑖∶𝑉^𝑘→𝑊 is another multi-Euler-Lagrange quadratic mapping satisfying (65) and note that according to (61) and using Lemma 4, and (63) we have 𝑄_𝑖𝑥₁,…,𝑥_𝑘−𝑄^′_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛=lim_𝑚→∞1𝜆^2𝑚𝑄_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−𝑄^′_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤lim_𝑚→∞1𝜆^2𝑚𝑄_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛+𝑓𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘−𝑄_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤lim^∞_{𝑚→∞𝑗=0}1𝜆^{2(𝑚+𝑗)+1}×𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝜆^𝑚+𝑗𝑥_𝑖,0,𝑥_𝑖+1,…,𝑥_𝑘+1𝜆^{2(𝑚+𝑗)+2}×𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑎𝜆^𝑚+𝑗𝑥_𝑖,𝑏𝜆^𝑚+𝑗𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑘=0. (72) Therefore, by Lemma 4, we can conclude that 𝑄_𝑖=𝑄^′_𝑖.

Similar to Theorem 15, one can get the following.

Theorem 16. Assume that for every 𝑖∈{1,…,𝑘}, 𝜑_𝑖∶𝑉^𝑘+1→[0,∞) is a mapping such that ^∞_𝑗=0𝜆^2𝑗𝜑_𝑖𝑥₁𝜆^𝑗,𝑥₂,…,𝑥_𝑘+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑖−2,𝑥_𝑖−1𝜆^𝑗,𝑥_𝑖,…,𝑥_𝑘+1+𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑗,𝑥_𝑖+1𝜆^𝑗,𝑥_𝑖+2,…,𝑥_𝑘+1+𝜑_𝑖𝑥₁,…,𝑥_𝑖+1,𝑥_𝑖+2𝜆^𝑗,𝑥_𝑖+3,…,𝑥_𝑘+1+⋯+𝜑_𝑖𝑥₁,…,𝑥_𝑘,𝑥_𝑘+1𝜆^𝑗𝑥<∞,₁,…,𝑥_𝑘+1∈𝑉^𝑘+1. (73)
If 𝑓∶𝑉^𝑘→𝑊 is a function satisfying condition (11) and 𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑘+1,𝑦₂,…,𝑦_𝑛≤𝜑_𝑖𝑥₁,…,𝑥_𝑘+1,𝑥₁,…,𝑥_𝑘+1∈𝑉^𝑘+1{},𝑦,𝑖∈1,…,𝑘₂,…,𝑦_𝑛∈𝑊, (74) then for every 𝑖∈{1,…,𝑘}, there exists a unique multi-Euler-Lagrange quadratic mapping 𝑄_𝑖∶𝑉^𝑘→𝑊 for which 𝑓𝑥₁,…,𝑥_𝑘−𝑄_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,y_𝑛≤^∞_𝑗=0𝜆^2𝑗+1×𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑗+1,0,𝑥_𝑖+1,…,𝑥_𝑘+𝜆^2𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑎𝑥_𝑖𝜆^𝑗+1,𝑏𝑥_𝑖𝜆^𝑗+1,𝑥_𝑖+1,…,𝑥_𝑘,𝑥₁,…,𝑥_𝑘∈𝑉^𝑘,𝑦₂,…,𝑦_𝑛∈𝑊. (75) For every 𝑖∈{1,…,𝑘} the function 𝑄_𝑖 is given by 𝑄_𝑖𝑥₁,…,𝑥_𝑘∶=lim_𝑗→∞𝜆^2𝑗𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑗,𝑥_𝑖+1,…,𝑥_𝑘,𝑥₁,…,𝑥_𝑘∈𝑉^𝑘. (76)

Proof. Fix 𝑥₁,…,𝑥_𝑘∈𝑉, 𝑦₂,…,𝑦_𝑛∈𝑊, 𝑗∈ℕ∪{0} and 𝑖∈{1,…,𝑘}. By (74), we obtain 𝜆²𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆,𝑥_𝑖+1,…,𝑥_𝑘−𝑓𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤𝜆𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆,0,𝑥_𝑖+1,…,𝑥_𝑘,+𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑎𝑥_𝑖𝜆,𝑏𝑥_𝑖𝜆,𝑥_𝑖+1,…,𝑥_𝑘, (77) and consequently for any non-negative integers 𝑙 and 𝑚 such that 𝑙<𝑚, we get 𝜆^2𝑙𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑙,𝑥_𝑖+1,…,𝑥_𝑘−𝜆^2𝑚𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑚,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤^𝑚−1_𝑗=𝑙𝜆^2(𝑗+1)𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑗+1,𝑥_𝑖+1,…,𝑥_𝑘−𝜆^2𝑗𝑓𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑗,𝑥_𝑖+1,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤^𝑚−1_𝑗=𝑙𝜆^2𝑗+1𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖𝜆^𝑗+1,0,𝑥_𝑖+1,…,𝑥_𝑘+𝜆^2𝑗𝜑_𝑖𝑥₁,…,𝑥_𝑖−1,𝑎𝑥_𝑖𝜆^𝑗+1,𝑏𝑥_𝑖𝜆^𝑗+1,𝑥_𝑖+1,…,𝑥_𝑘. (78) Therefore, from (73), it follows that {𝜆^2𝑗𝑓(𝑥₁,…,𝑥_𝑖−1,𝑥_𝑖/𝜆^𝑗,𝑥_𝑖+1,…,𝑥_𝑘)}_𝑗∈ℕ is a Cauchy sequence. Since 𝑊 is an 𝑛-Banach space, this sequence is convergent and we define 𝑄_𝑖∶𝑉^𝑘→𝑊 by (76). Putting 𝑙=0, letting 𝑚→∞ in (78), and using Lemma 4 and (73) we see that (75) holds. The further part of the proof is similar to the proof of Theorem 15.

As applications of Theorems 15 and 16, we get the following corollaries.

Corollary 17. Let 𝑉 be a real normed linear space and, 𝑊 be an 𝑛-Banach space. Assume also that 𝜃∈[0,∞) and 𝑟∈(0,∞) are such that 𝑟≠1. If 𝑓∶𝑉^𝑘→𝑊 is a function satisfying 𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑘+1,𝑦₂,…,𝑦_𝑛≤𝜃𝑥₁^𝑟⋯𝑥_𝑖−1^𝑟𝑥_𝑖^𝑟+𝑥_𝑖+1^𝑟×𝑥_𝑖+2^𝑟⋯𝑥_𝑘+1^𝑟,𝑥₁,…,𝑥_𝑘+1∈𝑉^𝑘+1{},𝑦,𝑖∈1,…,𝑘₂,…,𝑦_𝑛∈𝑊, (79) then for every 𝑖∈{1,…,𝑘}, there exists a unique multi-Euler-Lagrange quadratic mapping 𝑄_𝑖∶𝑉^𝑘→𝑊 for which 𝑓𝑥₁,…,𝑥_𝑘−𝑄_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤𝜃𝑥₁^𝑟⋯𝑥_𝑘^𝑟𝜆+|𝑎|^𝑟+|𝑏|^𝑟𝜆²−𝜆^𝑟, (80) for all 𝑥₁,…,𝑥_𝑘∈𝑉, 𝑦₂,…,𝑦_𝑛∈𝑊.

Corollary 18. Let 𝑉 be a real normed linear space and let 𝑊 be an 𝑛-Banach space. Assume also that 𝜃∈[0,∞) and 𝑟,𝑝,𝑞∈(0,∞) are such that 𝑟,𝑝+𝑞∈(0,2) or 𝑟,𝑝+𝑞∈(2,∞). If 𝑓∶𝑉^𝑘→𝑊 is a function satisfying 𝐷_𝑖𝑓𝑥₁,…,𝑥_𝑘+1,𝑦₂,…,𝑦_𝑛≤𝜃𝑥₁^𝑟⋯𝑥_𝑖−1^𝑟𝑥_𝑖^𝑝𝑥_𝑖+1^𝑞×𝑥_𝑖+2^𝑟⋯𝑥_𝑘+1^𝑟,𝑥₁,…,𝑥_𝑘+1∈𝑉^𝑘+1{},𝑦,𝑖∈1,…,𝑘₂,…,𝑦_𝑛∈𝑊, (81) then for every 𝑖∈{1,…,𝑘}, there exists a unique multi-Euler-Lagrange quadratic mapping 𝑄_𝑖∶𝑉^𝑘→𝑊 for which 𝑓𝑥₁,…,𝑥_𝑘−𝑄_𝑖𝑥₁,…,𝑥_𝑘,𝑦₂,…,𝑦_𝑛≤|𝑎|^𝑝|𝑏|^𝑞𝜃𝑥₁^𝑟⋯𝑥_𝑖−1^𝑟𝑥_𝑖^p+𝑞𝑥_𝑖+1^𝑟⋯𝑥_𝑘^𝑟𝜆²−𝜆^𝑝+𝑞, (82) for all 𝑥₁,…,𝑥_𝑘∈𝑉, 𝑦₂,…,𝑦_𝑛∈𝑊.

For 𝑎=𝑏=1, Corollary 18 yields the following corollary which corrects Theorems 4.1 and 4.2 from [20].

Corollary 19. Let 𝑉 be a real normed linear space and let 𝑊 be an 2-Banach space. Assume also that 𝜃∈[0,∞) and 𝑝,𝑞∈(0,∞) are such that 𝑝+𝑞≠2. If 𝑓∶𝑉→𝑊 is a function satisfying 𝑓𝑥₁+𝑥₂−𝑓𝑥₁−𝑓𝑥₂𝑥,𝑦≤𝜃₁^𝑝𝑥₂^𝑞,𝑥₁,𝑥₂∈𝑉,𝑦∈𝑊, (83) then there exists a unique quadratic mapping 𝑄∶𝑉→𝑊 for which 𝑓(𝑥)(𝑥)≤𝜃‖𝑥‖−𝑄,𝑦^𝑝+𝑞4−2^𝑝+𝑞,𝑥∈𝑉,𝑦∈𝑊. (84)

Acknowledgments

This project was supported by the National Natural Science Foundation of China (NNSFC) (Grant no. 11171022). The author would like to thank Professor Krzysztof Ciepliński and anonymous referees for their valuable comments and suggestions.