# Fuzzy Stability of a Quadratic-Additive Functional Equation

Authors: Sun Sook Jin; Yang Hi Lee
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We investigate a fuzzy version of stability for the functional equationf(x+y+z+w)+2f(x)+2f(y)+2f(z)+2f(w)-f(x+y)-f(x+z)-f(x+w)-f(y+z)-f(y+w)-f(z+w)=0.


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## Body

## 1. Introduction

A classical question in the theory of functional equations is “when is it true that a mapping, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?”  Such a problem, calleda stability problem of the functional equation,  was formulated by   Ulam [1] in 1940. In the next year,   Hyers [2] gave a partial solution of Ulam's problem for the case of approximate additive mappings. Subsequently, his result was generalized by   Aoki [3] for additive mappings, and by   Rassias [4] for linear mappings, to considering the stability problem with the unbounded Cauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians, see [5–16].In 1984,   Katsaras [17] defined a fuzzy norm on a linear space to construct a fuzzy structure on the space. Since then, some mathematicians have introduced several types of fuzzy norm in different points of view. In particular,   Bag and  Samanta [18], following Cheng and Mordeson [19], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of the Kramosil and Michálek type [20]. In 2008,   Mirmostafaee and   Moslehian [21] introduced for the first time the notion of fuzzy Hyers-Ulam-Rassias stability. They obtained a fuzzy version of stability for the Cauchy functional equation(1.1)f(x+y)-f(x)-f(y)=0
whose solution is called an additive mapping. In the same year, they [22] proved a fuzzy version of stability for the quadratic functional equation(1.2)f(x+y)+f(x-y)-2f(x)-2f(y)=0
whose solution is called a quadratic mapping. Now we consider the quadratic-additive functional equation(1.3)f(x+y+z+w)+2f(x)+2f(y)+2f(z)+2f(w)-f(x+y)-f(x+z)-f(x+w)-f(y+z)-f(y+w)-f(z+w)=0
whose solution is called a quadratic-additive mapping. In [23], Chang et al. obtained a stability of the quadratic-additive functional equation by taking and composing an additive mapping A and a quadratic mapping Q to prove the existence of a quadratic-additive mapping F which is close to the given mapping f. In their processing, A is approximate to the odd part f((x)-f(-x))/2 of f and Q is close to the even part (f(x)+f(-x))/2 of it, respectively.In this paper, we get a general stability result of the quadratic-additive functional equation in the fuzzy normed linear space. To do it, we introduce a Cauchy sequence{Jnf(x)} starting from a given mapping f, which converges to the desired mapping F in the fuzzy sense. As we mentioned before, in previous studies of stability problem of (1.3), Chang et al. attempted to get stability theorems by handling the odd and even part of f, respectively. According to our proposal in this paper, we can take the desired approximate solution F at once.


## 2. Fuzzy Stability of the Quadratic-Additive Functional Equation

We use the definition of a fuzzy normed space given in [18] to exhibit a reasonable fuzzy version of stability for the quadratic-additive functional equation in the fuzzy normed linear space.Definition 2.1 (see [18]).
LetX be a real linear space. A mapping N:X×ℝ→[0,1] (the so-called fuzzy subset) is said to be a fuzzy norm onX if for all x,y∈X and all s,t∈ℝ,  (N1)
N(x,c)=0 for c≤0,(N2)
x=0 if and only if N(x,c)=1 for all c>0,(N3)
N(cx,t)=N(x,t/|c|) if c≠0,(N4)
N(x+y,s+t)≥min⁡{N(x,s),N(y,t)},(N5)
N(x,·) is a nondecreasing mapping on ℝ and lim⁡t→∞N(x,t)=1.The pair(X,N) is called a fuzzy normed linear space. Let (X,N) be a fuzzy normed linear space. Let {xn} be a sequence in X. Then {xn} is said to be convergent if there exists x∈X such that lim⁡n→∞N(xn-x,t)=1 for all t>0. In this case, x is called the limit of the sequence{xn},  and we denote it by N-lim⁡n→∞xn=x. A sequence {xn} in X is called Cauchy if for each ɛ>0 and each t>0 there exists n0 such that for all n≥n0 and all p>0 we have N(xn+p-xn,t)>1-ɛ. It is known that every convergent sequence in a fuzzy normed space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space.Let(X,N) be a fuzzy normed space and (Y,N′) a fuzzy Banach space. For a given mapping f:X→Y, we use the abbreviation(2.1)Df(x,y,z,w)∶=f(x+y+z+w)+2f(x)+2f(y)+2f(z)+2f(w)-f(x+y)-f(x+z)-f(x+w)-f(y+z)-f(y+w)-f(z+w)
for all x,y,z,w∈X. For given q>0, the mapping f is called a fuzzyq-almost quadratic-additive mapping  if(2.2)N′(Df(x,y,z,w),s+t+u+v)≥min⁡{N(x,sq),N(y,tq),N(z,uq),N(w,vq)}
for all x,y,z,w∈X and s,t,u,v∈(0,∞). Now we get the general stability result in the fuzzy normed linear setting.Theorem 2.2.
Letq be a positive real number with q≠1/2,1. And let f be a fuzzy q-almost quadratic-additive mapping from a fuzzy normed space (X,N) into a fuzzy Banach space (Y,N′). Then there is a unique quadratic-additive mapping F:X→Y such that
(2.3)N′(F(x)-f(x),t)≥sup⁡t′<tN(x,(|4-2p||2-2p|t′)q(2(|4-2p|+|2-2p|))q)
for all x∈X and t>0, where p=1/q.Proof.
It follows from (2.2) and (N4) that
(2.4)N′(f(0),t)=N′(Df(0,0,0,0),3t)≥N(0,(3t4)q)=1
for all t∈(0,∞). By (N2), we have f(0)=0. We will prove the theorem in three cases:  q>1, 1/2<q<1, and 0<q<1/2. Case 1.
Letq>1 and let Jnf:X→Y be a mapping defined by
(2.5)Jnf(x)=f(2nx)+f(-2nx)2⋅4n+f(2nx)-f(-2nx)2n+1
for all x∈X. Notice that J0f(x)=f(x) and
(2.6)Jjf(x)-Jj+1f(x)=Df(2jx,-2jx,2jx,-2jx)2⋅4j+1+Df(2jx,2jx,2jx,-2jx)2j+3-Df(-2jx,-2jx,-2jx,2jx)2j+3
for all x∈X and j≥0. Together with (N3), (N4), and (2.2), this equation implies that if n+m>m≥0, then
(2.7)N′(Jmf(x)-Jn+mf(x),∑j=mn+m-1(12(2p4)j+(2p2)j)tp)≥N′(∑j=mn+m-1(Jjf(x)-Jj+1f(x)),∑j=mn+m-1(1+2j+1)2jp2⋅4jtp)≥min⁡⋃j=mn+m-1{N′(Jjf(x)-Jj+1f(x),(1+2j+1)2jp2⋅4jtp)}≥min⁡⋃j=mn+m-1{min⁡{N′(Df(2jx,-2jx,2jx,-2jx)2⋅4j+1,2jptp2⋅4j),N′(Df(2jx,2jx,2jx,-2jx)2j+3,2jptp2j+1),N′(-Df(-2jx,-2jx,-2jx,2jx)2j+3,2jptp2j+1)}}≥min⁡⋃j=mn+m-1{N(2jx,2jt)}=N(x,t)
for all x∈X and t>0. Let ɛ>0 be given. Since lim⁡t→∞N(x,t)=1, there is t0>0 such that
(2.8)N(x,t0)≥1-ɛ.
We observe that for some t̃>t0, the series ∑j=0∞(1+2j+1)2jp/2·4jt̃p converges for p=(1/q)<1. It guarantees that, for an arbitrary given c>0, there exists n0≥0 such that
(2.9)∑j=mn+m-1(1+2j+1)2jp2⋅4jt̃p<c
for each m≥n0 and n>0. By (N5) and (2.7), we have
(2.10)N′(Jmf(x)-Jn+mf(x),c)≥N′(Jmf(x)-Jn+mf(x),∑j=mn+m-1(1+2j+1)2jp2⋅4jt̃p)≥N(x,t̃)≥N(x,t0)≥1-ɛ
for all x∈X. Hence {Jnf(x)} is a Cauchy sequence in the fuzzy Banach space (Y,N′), and so we can define a mapping F:X→Y by
(2.11)F(x)∶=N′-lim⁡n→∞Jnf(x)
for all x∈X. Moreover, if we put m=0 in (2.7), we have
(2.12)N′(f(x)-Jnf(x),t)≥N(x,tq(∑j=0n-1((1/2)(2p/4)j+(2p/2)j))q)
for all x∈X. Next we will show that F is a desired quadratic-additive mapping. Using (N4), we have
(2.13)N′(DF(x,y,z,w),t)≥min⁡{N′(F(x+y+z+w)-Jnf(x+y+z+w),t30),N′(2F(x)-2Jnf(x),t15),N′(2F(y)-2Jnf(y),t15),N′(2F(z)-2Jnf(z),t15),N′(2F(w)-2Jnf(w),t15),N′(F(x+y)-Jnf(x+y),t30),N′(F(x+z)-Jnf(x+z),t30),N′(F(x+w)-Jnf(x+w),t30),N′(F(y+z)-Jnf(y+z),t30),N′(F(y+w)-Jnf(y+w),t30),N′(F(z+w)-Jnf(z+w),t30),N′(DJnf(x,y,z,w),t2)}
for all x,y,z,w∈X and n∈ℕ. The first eleven terms on the right-hand side of (2.13) tend to 1 as n→∞ by the definition of F and (N2), and the last term holds:
(2.14)N′(DJnf(x,y,z,w),t2)≥min⁡{N′(Df(2nx,2ny,2nz,2nw)2⋅4n,t8),N′(Df(-2nx,-2ny,-2nz,-2nw)2⋅4n,t8),N′(Df(2nx,2ny,2nz,2nw)2⋅2n,t8),N′(Df(-2nx,-2ny,-2nz,-2nw)2⋅2n,t8)}
for all x,y,z,w∈X. By (N3) and (2.2), we obtain
(2.15)N′(Df(±2nx,±2ny,±2nz,±2nw)2⋅4n,t8)=N′(Df(±2nx,±2ny,±2nz,±2nw),4nt4)≥min⁡{N(2nx,(4nt16)q),N(2ny,(4nt16)q),N(2nz,(4nt16)q),N(2nw,(4nt16)q)}≥min⁡{N(x,2(2q-1)n24qtq),N(y,2(2q-1)n24qtq),N(z,2(2q-1)n24qtq),N(w,2(2q-1)n24qtq)},N′(Df(±2nx,±2ny,±2nz,±2nw)2⋅2n,t8)≥min⁡{N(x,2(q-1)n24qtq),N(y,2(q-1)n24qtq),N(z,2(q-1)n24qtq),N(w,2(q-1)n24qtq)}
for all x,y,z,w∈X and n∈ℕ. Since q>1, together with (N5), we can deduce that the last term of (2.13) also tends to 1 as n→∞. It follows from (2.13) that
(2.16)N′(DF(x,y,z,w),t)=1
for each x,y,z,w∈X and t>0. By (N2), this means that DF(x,y,z,w)=0 for all x,y,z,w∈X.  Now we approximate the difference between f and F in a fuzzy sense. For an arbitrary fixed x∈X and t>0, choose 0<ɛ<1 and 0<t′<t. Since F is the limit of {Jnf(x)}, there is n∈ℕ such that N′(F(x)-Jnf(x),t-t′)≥1-ɛ. By (2.12), we have
(2.17)N′(F(x)-f(x),t)≥min⁡{N′(F(x)-Jnf(x),t-t′),N′(Jnf(x)-f(x),t′)}≥min⁡{1-ɛ,N(x,t′q(∑j=0n-1((1/2)(2p/4)j+(2p/2)j))q)}≥min⁡{1-ɛ,N(x,(4-2p)q(2-2p)qt′q(2(4-2p+2-2p))q)}
since ∑j=0n-1((1/2)(2p/4)j+(2p/2)j)≤2(4-2p+2-2p)/((4-2p)(2-2p)). Because 0<ɛ<1 is arbitrary, we get  inequality (2.3) in this case. Finally, to prove the uniqueness of F, let F′:X→Y be another quadratic-additive mapping satisfying (2.3). Then by (2.6), we get
(2.18)F(x)-JnF(x)=∑j=0n-1(JjF(x)-Jj+1F(x))=0,F′(x)-JnF′(x)=∑j=0n-1(JjF′(x)-Jj+1F′(x))=0
for all x∈X and n∈ℕ. Together with (N4) and (2.3), this implies that
(2.19)N′(F(x)-F′(x),t)=N′(JnF(x)-JnF′(x),t)≥min⁡{N′(JnF(x)-Jnf(x),t2),N′(Jnf(x)-JnF′(x),t2)}≥min⁡{N′((F-f)(2nx)2⋅4n,t8),N′((f-F′)(2nx)2⋅4n,t8),N′((F-f)(-2nx)2⋅4n,t8),N′((f-F′)(-2nx)2⋅4n,t8),N′((F-f)(2nx)2⋅2n,t8),N′((f-F′)(2nx)2⋅2n,t8),N′((F-f)(-2nx)2⋅2n,t8),N′((f-F′)(-2nx)2⋅2n,t8)}≥sup⁡t′<tN(x,2(q-1)n-2q(4-2p)q(2-2p)qt′q(4(3-2p))q)
for all x∈X and n∈ℕ. Observe that, for q=(1/p)>1, the last term of the above inequality tends to 1 as n→∞ by (N5). This implies that N′(F(x)-F′(x),t)=1, and so we get
(2.20)F(x)=F′(x)
for all x∈X by (N2). Case 2.
Let(1/2)<q<1 and let Jnf:X→Y be a mapping defined by
(2.21)Jnf(x)=12(4-n(f(2nx)+f(-2nx))+2n(f(x2n)-f(-x2n)))
for all x∈X. Then we also have J0f(x)=f(x) and
(2.22)Jjf(x)-Jj+1f(x)=Df(2jx,-2jx,2jx,-2jx)2⋅4j+1-2j-2Df(x2j+1,x2j+1,x2j+1,-x2j+1)+2j-2Df(-x2j+1,-x2j+1,-x2j+1,x2j+1)
for all x∈X and j≥0. If n+m>m≥0, then we have
(2.23)N′(Jmf(x)-Jn+mf(x),∑j=mn+m-1(12(2p4)j+(22p)j+1)tp)≥min⁡⋃j=mn+m-1{min⁡{N′(Df(2jx,-2jx,2jx,-2jx)2⋅4j+1,2jptp2⋅4j),N′(-2j-2Df(x2j+1,x2j+1,x2j+1,-x2j+1),2jtp2(j+1)p),N′(2j-2Df(-x2j+1,-x2j+1,-x2j+1,x2j+1),2jtp2(j+1)p)(Df(2jx,-2jx,2jx,-2jx)2⋅4j+1,2jptp2⋅4j)}}≥min⁡⋃j=mn+m-1{N(2jx,2jt),N(x2j+1,t2j+1)}=N(x,t)
for all x∈X and t>0. In a similar argument following (2.7) of the previous case, we can define the limit F(x):=N′-lim⁡n→∞Jnf(x) of the Cauchy sequence {Jnf(x)} in the Banach fuzzy space Y. Moreover, putting m=0 in the above inequality, we have
(2.24)N′(f(x)-Jnf(x),t)≥N(x,tq(∑j=0n-1((1/2)(2p/4)j+(2/2p)j+1))q)
for each x∈X and t>0. To prove that F is a quadratic-additive mapping, we have enough to show that the last term of (2.13) in Case 1 tends to 1 as n→∞. By (N3) and (2.2), we get
(2.25)N′(DJnf(x,y,z,w),t2)≥min⁡{N′(Df(2nx,2ny,2nz,2nw)2⋅4n,t8),N′(Df(-2nx,-2ny,-2nz,-2nw)2⋅4n,t8),N′(2n-1Df(x2n,y2n,z2n,w2n),t8),N′(2n-1Df(-x2n,-y2n,-z2n,-w2n),t8)(Df(2nx,2ny,2nz,2nw)2⋅4n,t8)}≥min⁡{N(x,2(2q-1)n-4qtq),N(y,2(2q-1)n-4qtq),N(z,2(2q-1)n-4qtq),N(w,2(2q-1)n-4qtq),N(x,2(1-q)n-4qtq),N(y,2(1-q)n-4qtq),N(z,2(1-q)n-4qtq),N(w,2(1-q)n-4qtq)}
for all x,y,z,w∈X and t>0. Observe that all the terms on the right-hand side of the above inequality tend to 1 as n→∞, since 1/2<q<1. Hence, with arguments similar to those of (2.13)–(2.15), we can say that DF(x,y,z,w)=0 for all x,y,z,w∈X. Recall  that, in Case 1,   inequality (2.3) follows from (2.12). By the same reasoning, we get (2.3) from (2.24) in this case.
Now to prove the uniqueness ofF, let F′ be another quadratic-additive mapping satisfying (2.3). Then, together with (N4), (2.3), and (2.18), we have
(2.26)N′(F(x)-F′(x),t)=N′(JnF(x)-JnF′(x),t)≥min⁡{N′(JnF(x)-Jnf(x),t2),N′(Jnf(x)-JnF′(x),t2)}≥min⁡{N′((F-f)(2nx)2⋅4n,t8),N′((f-F′)(2nx)2⋅4n,t8),N′((F-f)(-2nx)2⋅4n,t8),N′((f-F′)(-2nx)2⋅4n,t8),N′(2n-1((F-f)(x2n)),t8),N′(2n-1((f-F′)(x2n)),t8),N′(2n-1((F-f)(-x2n)),t8),N′(2n-1((f-F′)(-x2n)),t8)((F-f)(2nx)2⋅4n,t8)}≥min⁡{sup⁡t′<tN(x,2(2q-1)n-4q(4-2p)q(2p-2)qt′q),sup⁡t′<tN(x,2(1-q)n-4q(4-2p)q(2p-2)qt′q)}
for all x∈X and n∈ℕ. Since lim⁡n→∞2(2q-1)n-4q=lim⁡n→∞2(1-q)n-4q=∞ in this case, both terms on the right-hand side of the above inequality tend to 1 as n→∞ by (N5). This implies that N′(F(x)-F′(x),t)=1 and so F(x)=F′(x) for all x∈X by (N2).Case 3.
Finally, we take0<q<(1/2) and define Jnf:X→Y by
(2.27)Jnf(x)=12(4n(f(2-nx)+f(-2-nx))+2n(f(x2n)-f(-x2n)))
for all x∈X. Then we have J0f(x)=f(x) and
(2.28)Jjf(x)-Jj+1f(x)=-4j2Df(x2j+1,-x2j+1,x2j+1,-x2j+1)-2j-2Df(x2j+1,x2j+1,x2j+1,-x2j+1)+2j-2Df(-x2j+1,-x2j+1,-x2j+1,x2j+1)
which implies that if n+m>m≥0 then
(2.29)N′(Jmf(x)-Jn+mf(x),22p∑j=mn+m-1((42p)j+(22p)j)tp)≥min⁡⋃j=mn+m-1{min⁡{N′(-4j2Df(x2j+1,-x2j+1,x2j+1,-x2j+1),2⋅4jtp2(j+1)p),N′(-2j-2Df(x2j+1,x2j+1,x2j+1,-x2j+1),2jtp2(j+1)p),N′(2j-2Df(-x2j+1,-x2j+1,-x2j+1,x2j+1),2jtp2(j+1)p)}}≥min⁡⋃j=mn+m-1{N(x2j+1,t2j+1)}=N(x,t)
for all x∈X and t>0. Similar to the previous cases, it leads us to define the mapping F:X→Y by F(x):=N′-lim⁡n→∞Jnf(x). Putting m=0 in the above inequality, we have
(2.30)N′(f(x)-Jnf(x),t)≥N(x,tq(∑j=0n-1((1/2)(4/2p)j+1+(2/2p)j+1))q)
for all x∈X and t>0. Notice that
(2.31)N′(DJnf(x,y,z,w),t2)≥min⁡{N′(4n2Df(x2n,y2n,z2n,w2n),t8),N′(4n2Df(-x2n,-y2n,-z2n,-w2n),t8),N′(2n-1Df(x2n,y2n,z2n,w2n),t8),N′(2n-1Df(-x2n,-y2n,-z2n,-w2n),t8)(4n2Df(x2n,y2n,z2n,w2n),t8)}≥min⁡{N(x,2(1-2q)n-4qtq),N(y,2(1-2q)n-4qtq),N(z,2(1-2q)n-4qtq),N(w,2(1-2q)n-4qtq),N(x,2(1-q)n-4qtq),N(y,2(1-q)n-4qtq),N(z,2(1-q)n-4qtq),N(w,2(1-q)n-4qtq)}
for each x,y,z,w∈X and t>0. Since 0<q<(1/2), all terms on the right-hand side tend to 1 as n→∞, which implies that the last term of (2.13) tends to 1 as n→∞. Therefore, we can say that DF≡0. Moreover, using arguments similar to those of (2.13)–(2.15) in Case 1, we get  inequality (2.3) from (2.30) in this case. To prove the uniqueness of F, let F′:X→Y be another quadratic-additive mapping satisfying (2.3). Then by (2.18), we get
(2.32)N′(F(x)-F′(x),t)≥min⁡{N′(JnF(x)-Jnf(x),t2),N′(Jnf(x)-JnF′(x),t2)}≥min⁡{N′(4n2((F-f)(x2n)),t8),N′(4n2((f-F′)(x2n)),t8),N′(4n2((F-f)(-x2n)),t8),N′(4n2((f-F′)(-x2n)),t8),N′(2n-1((F-f)(x2n)),t8),N′(2n-1((f-F′)(x2n)),t8),N′(2n-1((F-f)(-x2n)),t8),N′(2n-1((f-F′)(-x2n)),t8)(4n2((F-f)(x2n)),t8)}≥sup⁡t′<tN(x,2(1-2q)n-4q(2p-4)(2p-2)t′q(2p-3)q)
for all x∈X and n∈ℕ. Observe that, for 0<q<(1/2), the last term tends to 1 as n→∞ by (N5). This implies that N′(F(x)-F′(x),t)=1 and F(x)=F′(x) for all x∈X by (N2).Remark 2.3.
Consider a mappingf:X→Y satisfying (2.2) for all x,y,z,w∈X and a real number q<0. Take any t>0. If we choose a real number s with 0<4s<t, then
(2.33)N′(Df(x,y,z,w),t)≥N′(Df(x,y,z,w),4s)≥min⁡{N(x,sq),N(y,sq),N(z,sq),N(w,sq)}
for all x,y,z,w∈X. Since q<0, we have lim⁡s→0+sq=∞. This implies that
(2.34)lim⁡s→0+N(x,sq)=lim⁡s→0+N(y,sq)=lim⁡z→0+N(x,sq)=lim⁡w→0+N(w,sq)=1,
and so
(2.35)N′(Df(x,y,z,w),t)=1
for all x,y,z,w∈X and t>0. By (N2), we are allowed to get Df(x,y,z,w)=0 for all x,y,z,w∈X. In other words, f is itself a quadratic-additive mapping if f is a fuzzy q-almost quadratic-additive mapping for the case q<0.Corollary 2.4.
Letf be an even mapping satisfying all of the conditions of Theorem 2.2. Then there is a unique quadratic mapping F:X→Y such that
(2.36)N′(F(x)-f(x),t)≥sup⁡t′<tN(x,(|4-2p|t′2)q)
for all x∈X and t>0, where p=1/q.Proof.
LetJnf be defined as in Theorem 2.2. Since f is an even mapping, we obtain
(2.37)Jnf(x)={f(2nx)+f(-2nx)2⋅4nifq>12,12(4n(f(2-nx)+f(-2-nx)))if0<q<12
for all x∈X. Notice that J0f(x)=f(x) and
(2.38)Jjf(x)-Jj+1f(x)={Df(2jx,-2jx,2jx,-2jx)2⋅4j+1ifq>12,-4j2Df(x2j+1,-x2j+1,x2j+1,-x2j+1)if0<q<12
for all x∈X and j∈ℕ∪{0}. From these, using the similar method in Theorem 2.2, we obtain the quadratic-additive mapping F satisfying (2.36). Notice that F(x):=N′-lim⁡n→∞Jnf(x) for all x∈X, F is even, F(2x)=4F(x), and DF(x,y,z,w)=0 for all x,y,z,w∈X. Hence, we get
(2.39)F(x+y)+F(x-y)-2F(x)-2F(y)=-DF(x,y,x,-x)=0
for all x,y∈X. This means that F is a quadratic mapping.Corollary 2.5.
Letf be an odd mapping satisfying all of the conditions of Theorem 2.2. Then there is a unique additive mapping F:X→Y such that
(2.40)N′(F(x)-f(x),t)≥sup⁡t′<tN(x,(|2-2p|t′2)q)
for all x∈X and t>0, where p=1/q.Proof.
LetJnf be defined as in Theorem 2.2. Since f is an odd mapping, we obtain
(2.41)Jnf(x)={f(2nx)+f(-2nx)2n+1ifq>1,2n-1(f(2-nx)+f(-2-nx))if0<q<1
for all x∈X. Notice that J0f(x)=f(x) and
(2.42)Jjf(x)-Jj+1f(x)={Df(2jx,2jx,2jx,-2jx)2j+3-Df(-2jx,-2jx,-2jx,2jx)2j+3ifq>1,2j-2(Df(-x2j+1,-x2j+1,-x2j+1,x2j+1)-Df(x2j+1,x2j+1,x2j+1,-x2j+1))if0<q<1
for all x∈X and j∈ℕ∪{0}. From these, using the similar method in Theorem 2.2, we obtain the quadratic-additive mapping F satisfying (2.40). Notice that F(x):=N′-lim⁡n→∞Jnf(x) for all x∈X,F is odd, F(2x)=2F(x), and DF(x,y,z,w)=0 for all x,y,z,w∈X. Hence, we get
(2.43)F(x+y)-F(x)-F(y)=-DF(x-y2,-x-y2,x-y2,-x+y2)=0
for all x,y∈X. This means that F is an additive mapping.We can use Theorem2.2 to get a classical result in the framework of normed spaces. Let (X,∥·∥) be a normed linear space. Then we can define a fuzzy norm NX on X by (2.44)NX(x,t)={0ift≤‖x‖,1ift>‖x‖,
where x∈X and t∈ℝ, see [22]. Suppose that f:X→Y is a mapping into a Banach space (Y,|∥·∥|) such that (2.45)|‖Df(x,y,z,w)‖|≤‖x‖p+‖y‖p+‖z‖p+‖w‖p
for all x,y,z,w∈X, where p>0 and p≠1,2. Let NY be a fuzzy norm on Y. Then we get (2.46)NY(Df(x,y,z,w),s+t+u+v)={0ifs+t+u+v≤|‖Df(x,y,z,w)‖|,1ifs+t+u+v>|‖Df(x,y,z,w)‖|
for all x,y,z,w∈X and s,t,u,v∈ℝ. Consider the case (2.47)NY(Df(x,y,z,w),s+t+u+v)=0.This implies that(2.48)‖x‖p+‖y‖p+‖z‖p+‖w‖p≥|‖Df(x,y,z,w)‖|≥s+t+u+v,
and so ∥x∥p≥s, ∥y∥p≥t, ∥z∥p≥u, or ∥w∥p≥v in this case. Hence, for q=1/p, we have (2.49)min⁡{NX(x,sq),NX(y,tq),NX(z,uq),NX(w,vq)}=0
for all x,y,z,w∈X and s,t,u,v>0. Therefore, in every case, the inequality (2.50)NY(Df(x,y,z,w),s+t+u+v)≥min⁡{NX(x,sq),NX(y,tq),NX(z,uq),NX(w,vq)}
holds. It means that f is a fuzzy q-almost quadratic-additive mapping, and by Theorem 2.2, we get the following stability result.Corollary 2.6.
Let(X,∥·∥) be a normed linear space and let (Y,|∥·∥|) be a Banach space. If f:X→Y satisfies
(2.51)|‖Df(x,y,z,w)‖|≤‖x‖p+‖y‖p+‖z‖p+‖w‖p
for all x,y,z,w∈X, where p>0 and p≠1,2, then there is a unique quadratic-additive mapping F:X→Y such that
(2.52)|‖F(x)-f(x)‖|≤(2|2-2p|+2|4-2p|)‖x‖p
for all x∈X.Lemma 2.7.
IfF:X→Y is a mapping such that
(2.53)Df(x,y,z,w)=0
for all x,y,z,w∈X∖{0} with F(0)=0, then
(2.54)DF(x,y,z,w)=0
for all x,y,z,w∈X.Proof.
We easily know that(2.55)DF(x,y,z,0)=DF(2x,y,z,-x)-DF(2x,y,-x,-x)+DF(2x,y,x,-x)-DF(2x,z,-x,-x)+DF(2x,z,x,-x)-2DF(2x,x,x,-x)+DF(x,-x,-x,-x)=0
for all x,y,z∈X∖{0}. Using a similar method, we have DF(x,y,0,w)=0, DF(x,0,z,w)=0, DF(0,y,z,w)=0, DF(x,y,0,0)=0, DF(x,0,z,0)=0, DF(x,0,0,w)=0, DF(0,y,z,0)=0, DF(0,0,z,w)=0, DF(0,y,0,w)=0, DF(x,0,0,0)=0, DF(0,y,0,0)=0, DF(0,0,z,0)=0, DF(0,0,0,w)=0, and DF(0,0,0,0)=0 for all x,y,z,w∈X∖{0}, as we desired.



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*Source: 504802-2011-10-20.xml*