# Approximate Multi-Jensen, Multi-Euler-Lagrange Additive and Quadratic Mappings inn-Banach Spaces

Authors: Tian Zhou Xu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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


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## Body

## 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 inn-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 concerningn-normed spaces.Definition 1.
Letn∈ℕ and let X be a real linear space with dimX≥n, and let ∥·,…,·∥:Xn→ℝ be a function satisfying the following properties:(N1)
∥x1,…,xn∥=0 if and only if x1,…,xn are linearly dependent,(N2)
∥x1,…,xn∥ is invariant under permutation,(N3)
∥αx1,…,xn∥=|α|∥x1,…,xn∥,(N4)
∥x+y,x2,…,xn∥≤∥x,x2,…,xn∥+∥y,x2,…,xn∥for all α∈ℝ and x,y,x1,x2,…,xn∈X. Then the function ∥·,…,·∥ is called an n-norm on X, and the pair (X,∥·,…,·∥) is called an n-normed space.A sequence{xj}j∈ℕ in an n-normed space X is said to a converge to some x∈X in the n-norm if
(1)limj→∞∥xj-x,y2,…,yn∥=0,
for every y2,…,yn∈X. Every convergent sequence has exactly one limit. If x is the limit of the sequence {xj}j∈ℕ, then we write limj→∞xj=x. For any convergent sequences {xj}j∈ℕ and {yj}j∈ℕ of elements of X, the sequence {xj+yj}j∈ℕ is convergent and
(2)limj→∞(xj+yj)=limj→∞xj+limj→∞yj.
If, moreover, {αj}j∈ℕ is a convergent sequence of real numbers, then the sequence {αj·xj}j∈ℕ is also convergent and
(3)limj→∞(αj·xj)=limj→∞αj·limj→∞xj.
A sequence {xj}j∈ℕ in an n-normed space X is said to be a Cauchy sequence with respect to the n-norm if
(4)limj,l→∞∥xj-xl,y2,…,yn∥=0,
for every y2,…,yn∈X. A linear n-normed space in which every Cauchy sequence is convergent is called an n-Banach space.Example 2.
Forx1,…,xn∈ℝn, the Euclidean n-norm ∥x1,…,xn∥E is defined by
(5)∥x1,…,xn∥E=|det(xij)|=abs(|x11⋯x1n⋮⋱⋮xn1⋯xnn|),
where xi=(xi1,…,xin)∈ℝn for each i=1,…,n.Example 3.
The standardn-norm on X, a real inner product space of dimension dim X≥n, is as follows:
(6)∥x1,…,xn∥S=|〈x1,x1〉⋯〈x1,xn〉⋮⋱⋮〈xn,x1〉⋯〈xn,xn〉|1/2,
where 〈·,·〉 denotes the inner product on X. If X=ℝn, then this n-norm is exactly the same as the Euclidean n-norm ∥x1,…,xn∥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.
LetX be an n-normed space. Then,(1)
forxi∈X(i=1,…,n) and γ, a real number,
(7)∥x1,…,xi,…,xj,…,xn∥=∥x1,…,xi,…,xj+γxi,…,xn∥,
for all 1≤i≠j≤n,(2)
|∥x,y2,…,yn∥-∥y,y2,…,yn∥|≤∥x-y,y2,…,yn∥ for all x,y,y2,…,yn∈X,(3)
if∥x,y2,…,yn∥=0 for all y2,…,yn∈X, then x=0,(4)
for a convergent sequence{xj} in X,
(8)limj→∞∥xj,y2,…,yn∥=∥limj→∞xj,y2,…,yn∥,
for all y2,…,yn∈X.


## 2. Approximate Multi-Jensen Mappings

First, we prove the stability of the system of equations defining multi-Jensen mappings inn-Banach spaces. For a given mapping f:Vk→W, we define the difference operators
(9)Dif(x1,…,xk+1)∶=2f(x1,…,xi-1,xi+xi+12,xi+2,…,xk+1)-f(x1,…,xi,xi+2,…,xk+1)-f(x1,…,xi-1,xi+1,…,xk+1),(x1,…,xk+1)∈Vk+1,i∈{1,…,k}.Theorem 5.
LetV be a commutative group uniquely divisible by 2, and, W be an n-Banach space. Assume also that for every i∈{1,…,k}, φi:Vk+1→[0,∞) is a mapping such that
(10)∑j=0∞‍13j+1[φi(3jx1,x2,…,xk+1)+⋯+φi(x1,…,xi-2,3jxi-1,xi,…,xk+1)+φi(x1,…,xi-1,3jxi,3jxi+1,xi+2,…,xk+1)+φi(x1,…,xi+1,3jxi+2,xi+3,…,xk+1)+⋯+φi(x1,…,xk,3jxk+1)]<∞,(x1,…,xk+1)∈Vk+1.
If f:Vk→W is a function satisfying
(11)f(x1,…,xi-1,0,xi+1,…,xk)=0,(x1,…,xi-1,xi+1,…,xk)∈Vk-1,i∈{1,…,k},(12)∥Dif(x1,…,xk+1),y2,…,yn∥≤φi(x1,…,xk+1),(x1,…,xk+1)∈Vk+1,i∈{1,…,k},y2,…,yn∈W,
then for every i∈{1,…,k}, there exists a multi-Jensen mapping Fi:Vk→W for which
(13)∥f(x1,…,xk)-Fi(x1,…,xk),y2,…,yn∥≤∑j=0∞‍13j+1×[φi(x1,…,xi-1,3jxi,-3jxi,xi+1,…,xk)+φi(x1,…,xi-1,-3jxi,3j+1xi,xi+1,…,xk)],(x1,…,xk)∈Vk,y2,…,yn∈W.
For every i∈{1,…,k}, the function Fi is given by
(14)Fi(x1,…,xk):=limj→∞13jf(x1,…,xi-1,3jxi,xi+1,…,xk),(x1,…,xk)∈Vk.Proof.
Fixx1,…,xk∈V, y2,…,yn∈W and i∈{1,…,k}. By (12) and (11), we get
(15)∥f(x1,…,xk)+f(x1,…,xi-1,-xi,xi+1,…,xk),y2,…,yn∥≤φi(x1,…,xi,-xi,xi+1,…,xk),∥2f(x1,…,xk)-f(x1,…,xi-1,-xi,xi+1,…,xk)-f(x1,…,xi-1,3xi,xi+1,…,xk),y2,…,yn∥≤φi(x1,…,xi-1,-xi,3xi,xi+1,…,xk).
Hence,
(16)∥3f(x1,…,xk)-f(x1,…,xi-1,3xi,xi+1,…,xk),y2,…,yn∥≤φi(x1,…,xi,-xi,xi+1,…,xk)+φi(x1,…,xi-1,-xi,3xi,xi+1,…,xk),
and consequently for any nonnegative integers l and m such that l<m, we obtain
(17)∥13lf(x1,…,xi-1,3lxi,xi+1,…,xk)-13mf(x1,…,xi-1,3mxi,xi+1,…,xk),y2,…,yn∥≤∑j=lm-1‍13j+1[φi(x1,…,xi-1,3jxi,-3jxi,xi+1,…,xk)+φi(x1,…,xi-1,-3jxi,3j+1xi,xi+1,…,xk)].
Therefore, from (10), it follows that {(1/3j)f(x1,…,xi-1,3jxi,xi+1,…,xk)}j∈ℕ is a Cauchy sequence. Since W is an n-Banach space, this sequence is convergent and we define Fi:Vk→W by (14). Putting l=0, letting m→∞ in (17), and using Lemma 4 and (10), we see that (13) holds.
Finally, fixxi′∈V, j∈ℕ, and note that according to (12), we have
(18)∥13jDif(x1,…,xi-1,3jxi,3jxi′,xi+1,…,xk),y2,…,yn∥≤13jφi(x1,…,xi-1,3jxi,3jxi′,xi+1,…,xk).
Next, fixs∈{1,…,k}∖{i},xs′∈V, and assume that s<i (the same arguments apply to the case where s>i). From (12), it follows that
(19)∥13jDsf(x1,…,xs,xs′,xs+1,…,xi-1,13j3jxi,xi+1,…,xk),y2,…,yn∥≤13jφs(x1,…,xs,xs′,xs+1,…,xs′xi-1,3jxi,xi+1,…,xk).
Letting j→∞ in the above two inequalities and using (10) and Lemma 4, we see that the mapping Fi is multi-Jensen.Theorem 6.
LetV be a real linear space and, W be an n-Banach space. Assume also that for every i∈{1,…,k}, φi:Vk+1→[0,∞) is a mapping such that
(20)∑j=0∞‍3j[φi(x13j,x2,…,xk+1)+⋯+φi(x1,…,xi-2,xi-13j,xi,…,xk+1)+φi(x1,…,xi-1,xi3j,xi+13j,xi+2,…,xk+1)+φi(x1,…,xi+1,xi+23j,xi+3,…,xk+1)+⋯+φi(x1,…,xk,xk+13j)]<∞,(x1,…,xk+1)∈Vk+1.
If f:Vk→W is a function satisfying conditions  (11) and (12), then for every i∈{1,…,k} there exists a multi-Jensen mapping Fi:Vk→W for which
(21)∥f(x1,…,xk)-Fi(x1,…,xk),y2,…,yn∥≤∑j=0∞‍3j[φi(x1,…,xi-1,xi3j+1,-xi3j+1,xi+1,…,xk)+φi(x1,…,xi-1,-xi3j+1,xi3j,xi+1,…,xk)],(x1,…,xk)∈Vk,y2,…,yn∈W.
For every i∈{1,…,k}, the function Fi is given by
(22)Fi(x1,…,xk)∶=limj→∞3jf(x1,…,xi-1,xi3j,xi+1,…,xk),(x1,…,xk)∈Vk.Proof.
Fixx1,…,xk∈V, y2,…,yn∈W, j∈ℕ∪{0} and i∈{1,…,k}. By (12) and (11), we get
(23)∥3j+1f(x1,…,xi-1,xi3j+1,xi+1,…,xk)-3jf(x1,…,xi-1,xi3j,xi+1,…,xk),y2,…,yn∥≤3j[φi(x1,…,xi-1,xi3j+1,-xi3j+1,xi+1,…,xk)+φi(x1,…,xi-1,-xi3j+1,xi3j,xi+1,…,xk)],
and consequently for any non-negative integers l and m such that l<m, we obtain
(24)∥3lf(x1,…,xi-1,xi3l,xi+1,…,xk)-3mf(x1,…,xi-1,xi3m,xi+1,…,xk),y2,…,yn∥≤∑j=lm-1‍3j[φi(x1,…,xi-1,xi3j+1,-xi3j+1,xi+1,…,xk)+φi(x1,…,xi-1,-xi3j+1,xi3j,xi+1,…,xk)].
Therefore, from (20), it follows that {3jf(x1,…,xi-1,xi/3j,xi+1,…,xk)}j∈ℕ is a Cauchy sequence. Since W is an n-Banach space, this sequence is convergent and we define Fi:Vk→W by (22). Putting l=0, letting m→∞ in (24), and using Lemma 4 and (20), we see that (21) holds.
Finally, fixxi′∈V, and note that according to (12), we have
(25)∥3jDif(x1,…,xi-1,xi3j,xi′3j,xi+1,…,xk),y2,…,yn∥≤3jφi(x1,…,xi-1,xi3j,xi′3j,xi+1,…,xk).
Next, fix s∈{1,…,k}∖{i},xs′∈V, and assume that s<i (the same arguments apply to the case where s>i). From (12), it follows that
(26)∥3jDsf(x1,…,xs,xs′,xs+1,…,xi-1,xi3j,xi+1,…,xk),xi3jy2,…,yn∥≤3jφs(x1,…,xs,xs′,xs+1,…,xi-1,xi3j,xi+1,…,xk).
Letting j→∞ in the previous two inequalities and using (20) and Lemma 4, we see that the mapping Fi is multi-Jensen.As applications of Theorems5 and 6 we get the following corollaries.Corollary 7.
LetV 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:Vk→W is a function satisfying  (11) and
(27)∥Dif(x1,…,xk+1),y2,…,yn∥≤θ[∥x1∥r⋯∥xi-1∥r(∥xi∥r+∥xi+1∥r)∥xk+11∥r×∥xk+11∥r∥xi+2∥r⋯∥xk+1∥r],(x1,…,xk+1)∈Vk+1,i∈{1,…,k},y2,…,yn∈W,
then for every i∈{1,…,k} there exists a multi-Jensen mapping Fi:Vk→W for which
(28)∥f(x1,…,xk)-Fi(x1,…,xk),y2,…,yn∥≤θ∥x1∥r⋯∥xk∥r(3+3r)|3-3r|,
for all x1,…,xk∈V, y2,…,yn∈W.Corollary 8.
LetV 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:Vk→W is a function satisfying  (11) and
(29)∥Dif(x1,…,xk+1),y2,…,yn∥≤θ∥x1∥r⋯∥xi-1∥r(∥xi∥p∥xi+1∥q)×∥xi+2∥r⋯∥xk+1∥r,(x1,…,xk+1)∈Vk+1,i∈{1,…,k},y2,…,yn∈W,
then for every i∈{1,…,k}, there exists a multi-Jensen mapping Fi:Vk→W for which
(30)∥f(x1,…,xk)-Fi(x1,…,xk),y2,…,yn∥≤θ∥x1∥r⋯∥xi-1∥r∥xi∥p+q∥xi+1∥r⋯∥xk∥r(1+3q)|3-3p+q|,
for all x1,…,xk∈V, y2,…,yn∈W.From Corollary8, we obtain the following corollary which corrects Theorems 3.1 and 3.2 from [20].Corollary 9.
LetV 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
(31)∥2f(x1+x22)-f(x1)-f(x2),y∥≤θ∥x1∥p∥x2∥q,x1,x2∈V,y∈W,
then there exists a Jensen mapping F:V→W for which
(32)∥f(x)-F(x),y∥≤θ∥x∥p+q(1+3q)|3-3p+q|,x∈V,y∈W.


## 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, letV 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 mappingf:Vk→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:
(33)f(x1,…,xi-1,axi+bxi′,xi+1,…,xk)+f(x1,…,xi-1,bxi+axi′,xi+1,…,xk)=(a+b)[f(x1,…,xk)(xi′)+f(x1,…,xi-1,xi′,xi+1,…,xk)],
for all i∈{1,…,k} and all x1,…,xi-1,xi,xi′,xi+1,…,xk∈V. If a=b=1, then the multi-Euler-Lagrange additive mapping is multiadditive (see [28]). For a given mapping f:Vk→W, we define the difference operators
(34)D~if(x1,…,xk+1):=f(x1,…,xi-1,axi+bxi+1,xi+2,…,xk+1)+f(x1,…,xi-1,bxi+axi+1,xi+2,…,xk+1)-(a+b)[f(x1,…,xi,xi+2,…,xk+1)+f(x1,…,xi-1,xi+1,…,xk+1)],(x1,…,xk+1)∈Vk+1,i∈{1,…,k}.Theorem 10.
Assume that for everyi∈{1,…,k}, φi:Vk+1→[0,∞) is a mapping such that
(35)∑j=0∞‍1|λ|j[φi(λjx1,…,xk+1)+⋯+φi(x1,…,xi-2,λjxi-1,xi,…,xk+1)+φi(x1,…,xi-1,λjxi,λjxi+1,xi+2,…,xk+1)+φi(x1,…,xi+1,λjxi+2,xi+3,…,xk+1)+⋯+φi(x1,…,xk,λjxk+1)]<∞,(x1,…,xk+1)∈Vk+1.
If f:Vk→W is a function satisfying
(36)∥D~if(x1,…,xk+1),y2,…,yn∥≤φi(x1,…,xk+1),(x1,…,xk+1)∈Vk+1,i∈{1,…,k},y2,…,yn∈W,
then for every i∈{1,…,k}, there exists a unique multi-Euler-Lagrange additive mapping Ai:Vk→W for which
(37)∥f(x1,…,xk)-Ai(x1,…,xk),y2,…,yn∥≤12∑j=0∞‍1|λ|j+1×φi(x1,…,xi-1,λjxi,λjxi,xi+1,…,xk),(x1,…,xk)∈Vk,y2,…,yn∈W.
For every i∈{1,…,k}, the function Ai is given by
(38)Ai(x1,…,xk):=limj→∞1λjf(x1,…,xi-1,λjxi,xi+1,…,xk),(x1,…,xk)∈Vk.Proof.
Fixx1,…,xk∈V, y2,…,yn∈W, j∈ℕ∪{0} and i∈{1,…,k}. By (36), we get
(39)∥f(x1,…,xk)-1λf(x1,…,xi-1,λxi,xi+1,…,xk),1λy2,…,yn∥≤1|2λ|φi(x1,…,xi,xi,xi+1,…,xk),
whence
(40)∥1λjf(x1,…,xi-1,λjxi,xi+1,…,xk)-1λj+1f(x1,…,xi-1,λj+1xi,xi+1,…,xk),y2,…,yn∥≤12|λ|j+1φi(x1,…,xi-1,λjxi,λjxi,xi+1,…,xk).
For any nonnegative integers l and m with l<m, using (40) we get
(41)∥1λlf(x1,…,xi-1,λlxi,xi+1,…,xk)-1λmf(x1,…,xi-1,λmxi,xi+1,…,xk),y2,…,yn∥≤∑j=lm-1‍12|λ|j+1×φi(x1,…,xi-1,λjxi,λjxi,xi+1,…,xk),
which tends to zero as l tends to infinity. Therefore, from (35) it follows that {(1/λj)f(x1,…,xi-1,λjxi,xi+1,…,xk)}j∈ℕ is a Cauchy sequence in n-Banach space W and it thus converges. Hence, we can define Ai:Vk→W by
(42)Ai(x1,…,xk):=limj→∞1λjf(x1,…,xi-1,λjxi,xi+1,…,xk).
Putting l=0, letting m→∞ in (41), and using (35), we see that (37) holds.
Now, fix alsoxi′∈V, and from (36), we have
(43)∥1λjD~if(x1,…,xi-1,λjxi,λjxi′,xi+1,…,xk),1λjy2,…,yn∥≤1|λ|jφi(x1,…,xi-1,λjxi,λjxi′,xi+1,…,xk).
Next, fixs∈{1,…,k}∖{i}, xs′∈V, and assume that s<i (the same arguments apply to the case where s>i). From (36) it follows that
(44)∥1λjD~sf(x1,…,xs,xs′,xs+1,…,xi-1,1λjλjxi,xi+1,…,xk),y2,…,yn∥≤1|λ|jφs(x1,…,xs,xs′,xs+1,…,xi-1,λjxi,xi+1,…,xk).
Letting j→∞ in the above two inequalities and using (35) and Lemma 4, we see that the mapping Ai is multi-Euler-Lagrange additive.
Now, let us finally assume thatAi′:Vk→W is another multi-Euler-Lagrange additive mapping satisfying (37). Then we have
(45)∥Ai(x1,…,xk)-Ai′(x1,…,xk),y2,…,yn∥=limm→∞1|λ|m×∥Ai(x1,…,xi-1,λmxi,xi+1,…,xk)Ai′-Ai′(x1,…,xi-1,λmxi,xi+1,…,xk),y2,…,yn∥≤limm→∞1|λ|m×[∥Ai(x1,…,xi-1,λmxi,xi+1,…,xk)λ22k-f(x1,…,xi-1,λmxi,xi+1,…,xk),y2,…,yn∥+∥f(x1,…,xi-1,λmxi,xi+1,…,xk)λ22k-Ai(x1,…,xi-1,λmxi,xi+1,…,xk),y2,…,yn∥]≤limm→∞1|λ|×∑j=0∞‍1|λ|m+j×φi(x1,…,xi-1,λm+jxi,λm+jxi,xi+1,…,xk)=0,
and therefore Ai=Ai′.Theorem 11.
Assume that for everyi∈{1,…,k}, φi:Vk+1→[0,∞) is a mapping such that
(46)∑j=0∞‍|λ|j[φi(x1λj,x2,…,xk+1)+⋯+φi(x1,…,xi-2,xi-1λj,xi,…,xk+1)+φi(x1,…,xi-1,xiλj,xi+1λj,xi+2,…,xk+1)+φi(x1,…,xi+1,xi+2λj,xi+3,…,xk+1)+⋯+φi(x1,…,xk,xk+1λj)]<∞,(x1,…,xk+1)∈Vk+1.
If f:Vk→W is a function satisfying  (36), then for every i∈{1,…,k}, there exists a unique multi-Euler-Lagrange additive mapping Ai:Vk→W for which
(47)∥f(x1,…,xk)-Ai(x1,…,xk),y2,…,yn∥≤12∑j=1∞‍|λ|j-1×φi(x1,…,xi-1,xiλj,xiλj,xi+1,…,xk),(x1,…,xk)∈Vk,y2,…,yn∈W.
For every i∈{1,…,k}, the function Ai is given by
(48)Ai(x1,…,xk):=limj→∞λjf(x1,…,xi-1,xiλj,xi+1,…,xk),(x1,…,xk)∈Vk.Proof.
Fixx1,…,xk∈V, y2,…,yn∈W, j∈ℕ∪{0} and i∈{1,…,k}. By (36) we get
(49)∥f(x1,…,xk)-λf(x1,…,xi-1,xiλ,xi+1,…,xk),(xiλ)y2,…,yn∥≤12φi(x1,…,xi-1,xiλ,xiλ,xi+1,…,xk).
For any non-negative integers l and m with 0≤l<m, using (49), we get
(50)∥λlf(x1,…,xi-1,xiλl,xi+1,…,xk)-λmf(x1,…,xi-1,xiλm,xi+1,…,xk),y2,…,yn∥≤∑j=lm-1‍|λ|j2×φi(x1,…,xi-1,xiλj+1,xiλj+1,xi+1,…,xk),
which tends to zero as l tends to infinity. Therefore from (46), it follows that {λjf(x1,…,xi-1,xi/λj,xi+1,…,xk)}j∈ℕ is a Cauchy sequence in n-Banach space W and it thus converges. Hence, we can define Ai:Vk→W by
(51)Ai(x1,…,xk):=limj→∞λjf(x1,…,xi-1,xiλj,xi+1,…,xk).
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 Theorems10 and 11, we get the following corollaries.Corollary 12.
LetV 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:Vk→W is a function satisfying
(52)∥D~if(x1,…,xk+1),y2,…,yn∥≤θ[∥x1∥r⋯∥xi-1∥r(∥xi∥r+∥xi+1∥r)×∥xi+2∥r⋯∥xk+1∥r],(x1,…,xk+1)∈Vk+1,i∈{1,…,k},y2,…,yn∈W,
then for every i∈{1,…,k} there exists a unique multi-Euler-Lagrange additive mapping Ai:Vk→W for which
(53)∥f(x1,…,xk)-Ai(x1,…,xk),y2,…,yn∥≤θ∥x1∥r⋯∥xk∥r||λ|-|λ|r|,
for all x1,…,xk∈V, y2,…,yn∈W.Corollary 13.
LetV 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:Vk→W is a function satisfying
(54)∥D~if(x1,…,xk+1),y2,…,yn∥≤θ∥x1∥r⋯∥xi-1∥r(∥xi∥p∥xi+1∥q)×∥xi+2∥r⋯∥xk+1∥r,(x1,…,xk+1)∈Vk+1,i∈{1,…,k},y2,…,yn∈W,
then for every i∈{1,…,k}, there exists a unique multi-Euler-Lagrange additive mapping Ai:Vk→W for which
(55)∥f(x1,…,xk)-Ai(x1,…,xk),y2,…,yn∥≤θ∥x1∥r⋯∥xi-1∥r∥xi∥p+q∥xi+1∥r⋯∥xk∥r2||λ|-|λ|p+q|,
for all x1,…,xk∈V, y2,…,yn∈W.From Corollary13 we obtain the following corollary which corrects Theorems 2.1 and 2.2 from [20].Corollary 14.
LetV 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
(56)∥f(x1+x2)-f(x1)-f(x2),y∥≤θ∥x1∥p∥x2∥q,x1,x2∈V,y∈W,
then there exists a unique additive mapping A:V→W for which
(57)∥f(x)-A(x),y∥≤θ∥x∥p+q|2-2p+q|,x∈V,y∈W.


## 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, letV 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 mappingf:Vk→W is called a multi-Euler-Lagrange quadratic mapping, if it satisfies the Euler-Lagrange quadratic equations in each of their k arguments:
(58)f(x1,…,xi-1,axi+bxi′,xi+1,…,xk)+f(x1,…,xi-1,bxi-axi′,xi+1,…,xk)=(a2+b2)×[f(x1,…,xk)+f(x1,…,xi-1,xi′,xi+1,…,xk)],
for all i∈{1,…,k} and all x1,…,xi-1,xi,xi′,xi+1,…,xk∈V.Ifa=b=1, then the multi-Euler-Lagrange quadratic mapping is multiquadratic (see [30]). Letting xi=xi′=0 in (58), we get f(x1,…,xi-1,0,xi+1,…,xk)=0. Putting xi′=0 in (58), we have
(59)f(x1,…,xi-1,axi,xi+1,…,xk)+f(x1,…,xi-1,bxi,xi+1,…,xk)=λf(x1,…,xk).
Replacing xi by axi and xi′ by bxi in (58), respectively, we obtain
(60)f(x1,…,xi-1,λxi,xi+1,…,xk)=λ[f(x1,…,xi-1,axi,xi+1,…,xk)+f(x1,…,xi-1,bxi,xi+1,…,xk)].
From (59) and (60), one gets
(61)f(x1,…,xi-1,λxi,xi+1,…,xk)=λ2f(x1,…,xk),
for all i∈{1,…,k} and all x1,…,xk∈V.For a given mappingf:Vk→W, we define the difference operators
(62)D^if(x1,…,xk+1):=f(x1,…,xi-1,axi+bxi+1,xi+2,…,xk+1)+f(x1,…,xi-1,bxi-axi+1,xi+2,…,xk+1)-(a2+b2)×[f(x1,…,xi,xi+2,…,xk+1)+f(x1,…,xi-1,xi+1,…,xk)],(x1,…,xk+1)∈Vk+1,i∈{1,…,k}.Theorem 15.
Assume that for everyi∈{1,…,k}, φi:Vk+1→[0,∞) is a mapping such that
(63)∑j=0∞‍1λ2j×[φi(λjx1,x2,…,xk+1)+⋯+φi(x1,…,xi-2,λjxi-1,xi,…,xk+1)+φi(x1,…,xi-1,λjxi,λjxi+1,xi+2,…,xk+1)+φi(x1,…,xi+1,λjxi+2,xi+3,…,xk+1)+⋯+φi(x1,…,xk,λjxk+1)]<∞,(x1,…,xk+1)∈Vk+1.
If f:Vk→W is a function satisfying condition  (11) and
(64)∥D^if(x1,…,xk+1),y2,…,yn∥≤φi(x1,…,xk+1),(x1,…,xk+1)∈Vk+1,i∈{1,…,k},y2,…,yn∈W,
then for every i∈{1,…,k}, there exists a unique multi-Euler-Lagrange quadratic mapping Qi:Vk→W for which
(65)∥f(x1,…,xk)-Qi(x1,…,xk),y2,…,yn∥≤∑j=0∞[1λ2j+1φi(x1,…,xi-1,λjxi,0,xi+1,…,xk)‍+1λ2j+2φi(x1,…,xi-1,aλjxi,1λ2j+1bλjxi,xi+1,…,xk)],(x1,…,xk)∈Vk,y2,…,yn∈W.
For every i∈{1,…,k}, the function Qi is given by
(66)Qi(x1,…,xk):=limj→∞1λ2j×f(x1,…,xi-1,λjxi,xi+1,…,xk),(x1,…,xk)∈Vk.Proof.
Fixx1,…,xk∈V, y2,…,yn∈W, j∈ℕ∪{0} and i∈{1,…,k}. By (64), we get
(67)∥f(x1,…,xi-1,axi,xi+1,…,xk)+f(x1,…,xi-1,bxi,xi+1,…,xk)-λf(x1,…,xk),y2,…,yn∥≤φi(x1,…,xi,0,xi+1,…,xk),∥f(x1,…,xi-1,λxi,xi+1,…,xk)-λf(x1,…,xi-1,axi,xi+1,…,xk)-λf(x1,…,xi-1,bxi,xi+1,…,xk),y2,…,yn∥≤φi(x1,…,xi-1,axi,bxi,xi+1,…,xk).
From (67), we obtain
(68)∥1λ2f(x1,…,xi-1,λxi,xi+1,…,xk)1λ2-f(x1,…,xk),y2,…,yn∥≤1λφi(x1,…,xi,0,xi+1,…,xk)+1λ2×φi(x1,…,xi-1,axi,bxi,xi+1,…,xk),
and consequently for any non-negative integers l and m such that l<m, we get
(69)∥1λ2lf(x1,…,xi-1,λlxi,xi+1,…,xk)-1λ2m1λ2l×f(x1,…,xi-1,λmxi,xi+1,…,xk),y2,…,yn∥≤∑j=lm-1‍∥1λ2(j+1)f(x1,…,xi-1,λj+1xi,xi+1,…,xk)-1λ2jf(x1,…,xi-1,λjxi,xi+1,…,xk),1λ2(j+1)y2,…,yn∥≤∑j=lm-1‍[1λ2j+1φi(x1,…,xi-1,λjxi,0,xi+1,…,xk)+1λ2j+21λ2j+1×φi(x1,…,xi-1,aλjxi,bλjxi,xi+1,…,xk)].
Therefore from (63), it follows that {(1/λ2j)f(x1,…,xi-1,λjxi,xi+1,…,xk)}j∈ℕ is a Cauchy sequence. Since W is an n-Banach space, this sequence is convergent and we define Qi:Vk→W by (66). Putting l=0, letting m→∞ in (69) and using Lemma 4 and (63), we see that (65) holds.
Now, fix alsoxi′∈V and note that according to (64), we have
(70)∥1λ2jD^if(x1,…,xi-1,λjxi,λjxi′,xi′xi+1,…,xk),1λ2jy2,…,yn∥≤1λ2jφi(x1,…,xi-1,λjxi,λjxi′,xi′xi+1,…,xk).
Next, fixs∈{1,…,k}∖{i}, xs′∈V, and assume that s<i (the same arguments apply to the case where s>i). From (64), it follows that
(71)∥1λ2jD^sf(x1,…,xs,xs′,xs+1,…,xi-1,λj81λ2jλj8λjxi,xi+1,…,xk),y2,…,yn∥≤1λ2jφs(x1,…,xs,xs′,xs+1,…,xi-1,λjλjxi,xi+1,…,xk).
Letting j→∞ in the above two inequalities and using (63), and Lemma 4 we see that the mapping Qi is multi-Euler-Lagrange quadratic.
Now, let us finally assume thatQi′:Vk→W is another multi-Euler-Lagrange quadratic mapping satisfying (65) and note that according to (61) and using Lemma 4, and (63) we have
(72)∥Qi(x1,…,xk)-Qi′(x1,…,xk),y2,…,yn∥=limm→∞1λ2m∥Qi(x1,…,xi-1,λmxi,xi+1,…,xk)-Qi′(x1,…,xi-1,λmxi,xi+1,…,xk),y2,…,yn∥≤limm→∞1λ2m[∥Qi(x1,…,xi-1,λmxi,xi+1,…,xk)-f(x1,…,xi-1,λmxi,xi+1,…,xk),y2,…,yn∥+∥f(x1,…,xi-1,λmxi,xi+1,…,xk)-Qi(x1,…,xi-1,λmxi,xi+1,…,xk),(xi-1,λmxi,)y2,…,yn∥]≤limm→∞∑j=0∞‍[1λ2(m+j)+1×φi(x1,…,xi-1,λm+jxi,0,xi+1,…,xk)+1λ2(m+j)+2×φi(x1,…,xi-1,aλm+jxi,1λ2(m+j)+1bλm+jxi,xi+1,…,xk)]=0.
Therefore, by Lemma 4, we can conclude that Qi=Qi′.Similar to Theorem15, one can get the following.Theorem 16.
Assume that for everyi∈{1,…,k}, φi:Vk+1→[0,∞) is a mapping such that
(73)∑j=0∞‍λ2j[φi(x1λj,x2,…,xk+1)+⋯+φi(x1,…,xi-2,xi-1λj,xi,…,xk+1)+φi(x1,…,xi-1,xiλj,xi+1λj,xi+2,…,xk+1)+φi(x1,…,xi+1,xi+2λj,xi+3,…,xk+1)+⋯+φi(x1,…,xk,xk+1λj)]<∞,(x1,…,xk+1)∈Vk+1.
Iff:Vk→W is a function satisfying condition  (11) and
(74)∥D^if(x1,…,xk+1),y2,…,yn∥≤φi(x1,…,xk+1),(x1,…,xk+1)∈Vk+1,i∈{1,…,k},y2,…,yn∈W,
then for every i∈{1,…,k}, there exists a unique multi-Euler-Lagrange quadratic mapping Qi:Vk→W for which
(75)∥f(x1,…,xk)-Qi(x1,…,xk),y2,…,yn∥≤∑j=0∞[λ2j+1(bxiλj+1)‍×φi(x1,…,xi-1,xiλj+1,0,xi+1,…,xk)+λ2jφi(x1,…,xi-1,axiλj+1,bxiλj+1,xi+1,…,xk)],(x1,…,xk)∈Vk,y2,…,yn∈W.
For every i∈{1,…,k} the function Qi is given by
(76)Qi(x1,…,xk):=limj→∞λ2jf(x1,…,xi-1,xiλj,xi+1,…,xk),(x1,…,xk)∈Vk.Proof.
Fixx1,…,xk∈V, y2,…,yn∈W, j∈ℕ∪{0} and i∈{1,…,k}. By (74), we obtain
(77)∥λ2f(x1,…,xi-1,xiλ,xi+1,…,xk)xiλ-f(x1,…,xk),y2,…,yn∥≤λφi(x1,…,xi-1,xiλ,0,xi+1,…,xk),+φi(x1,…,xi-1,axiλ,bxiλ,xi+1,…,xk),
and consequently for any non-negative integers l and m such that l<m, we get
(78)∥λ2lf(x1,…,xi-1,xiλl,xi+1,…,xk)-λ2mf(x1,…,xi-1,xiλm,xi+1,…,xk),y2,…,yn∥≤∑j=lm-1‍∥λ2(j+1)f(x1,…,xi-1,xiλj+1,xi+1,…,xk)-λ2jf(x1,…,xi-1,xiλj,xi+1,…,xk),y2,…,yn∥≤∑j=lm-1‍[λ2j+1φi(x1,…,xi-1,xiλj+1,0,xi+1,…,xk)bxiλj+1+λ2jφi(x1,…,xi-1,axiλj+1,bxiλj+1,xi+1,…,xk)].
Therefore, from (73), it follows that {λ2jf(x1,…,xi-1,xi/λj,xi+1,…,xk)}j∈ℕ is a Cauchy sequence. Since W is an n-Banach space, this sequence is convergent and we define Qi:Vk→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 Theorems15 and 16, we get the following corollaries.Corollary 17.
LetV 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:Vk→W is a function satisfying
(79)∥D^if(x1,…,xk+1),y2,…,yn∥≤θ[∥x1∥r⋯∥xi-1∥r(∥xi∥r+∥xi+1∥r)×∥xi+2∥r⋯∥xk+1∥r],(x1,…,xk+1)∈Vk+1,i∈{1,…,k},y2,…,yn∈W,
then for every i∈{1,…,k}, there exists a unique multi-Euler-Lagrange quadratic mapping Qi:Vk→W for which
(80)∥f(x1,…,xk)-Qi(x1,…,xk),y2,…,yn∥≤θ∥x1∥r⋯∥xk∥r(λ+|a|r+|b|r)|λ2-λr|,
for all x1,…,xk∈V, y2,…,yn∈W.Corollary 18.
LetV 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:Vk→W is a function satisfying
(81)∥D^if(x1,…,xk+1),y2,…,yn∥≤θ∥x1∥r⋯∥xi-1∥r(∥xi∥p∥xi+1∥q)×∥xi+2∥r⋯∥xk+1∥r,(x1,…,xk+1)∈Vk+1,i∈{1,…,k},y2,…,yn∈W,
then for every i∈{1,…,k}, there exists a unique multi-Euler-Lagrange quadratic mapping Qi:Vk→W for which
(82)∥f(x1,…,xk)-Qi(x1,…,xk),y2,…,yn∥≤|a|p|b|qθ∥x1∥r⋯∥xi-1∥r∥xi∥p+q∥xi+1∥r⋯∥xk∥r|λ2-λp+q|,
for all x1,…,xk∈V, y2,…,yn∈W.Fora=b=1, Corollary 18 yields the following corollary which corrects Theorems 4.1 and 4.2 from [20].Corollary 19.
LetV 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
(83)∥f(x1+x2)-f(x1)-f(x2),y∥≤θ∥x1∥p∥x2∥q,x1,x2∈V,y∈W,
then there exists a unique quadratic mapping Q:V→W for which
(84)∥f(x)-Q(x),y∥≤θ∥x∥p+q|4-2p+q|,x∈V,y∈W.



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*Source: 648709-2013-09-14.xml*