Abstract
In this paper we approximate the quartic functional equations in Lipschitz spaces.
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1 Introduction
Let \({G}\) be an Abelian group and \({E}\) a vector space. Let \(S({E})\) be a family of subsets of \({E}\). We say that \(S({E})\) is linearly invariant if it is closed under the addition and scalar multiplication defined as usual sense and translation invariant, i.e, \(x+A\in S({E})\), for every \(A\in S({E})\) and every \(x\in {E}\) (see [1]). It is easy to verify that \(S({E})\) contains all singleton subsets of \({E}\). In particular, \(CB({E})\) the family of all closed balls with center at zero is a linearly invariant family in a normed vector space \({E}\). By \(B(G\times G,S({E}))\) we denote the family of all functions \(f:G\times G\longrightarrow {E}\) such that \(\mathtt{Im }f\subset A\) for some \(A\in S({E})\), where \(G\times G\) is the Cartesian product of \(G\) with itself. Obviously, this family is a vector space and contains all constant functions.
We say that \(B(G\times G,S({E}))\) admits a left invariant mean (briefly LIM), if the family \(S({E})\) is linearly invariant and there exists a linear operator \(M:B(G\times G,S({E}))\longrightarrow {E}\) such that
-
(i)
if \(\mathtt{Im }f\subset A\) for some \(A\in S({E})\), then \(M[f]\in A\),
-
(ii)
if \(f\in B(G\times G,S({E}))\) and \((a,b)\in G\times G\), then \(M[f^{a,b}]=M[f]\),
where \(f^{a,b}(x,y)=f(x+a,y+b)\). Following [2, 3] and for 2-variable functions let \(\mathbf{d }:(G\times G)\times (G\times G)\longrightarrow S({E})\) be a set-valued function such that
for all \((a,b),(x,y),(w,z)\in G\times G\). A function \(f:G\times G\longrightarrow {E}\) is said to be \(\mathbf{d }\)-Lipschitz if \(f(x,y)-f(w,z)\in \mathbf{d }((x,y),(w,z))\) for all \((x,y),(w,z)\in G\times G\).
Let \((G\times G,d)\) be a metric group and \({E}\) a normed space. A function \(m_{f}:{\mathbb {R}}^+\longrightarrow {\mathbb {R}}^+\) is a module of continuity of \(f:G\times G\longrightarrow {E}\) if \(d((x,y),(w,z))\le \delta \) implies \(||f(x,y)-f(w,z)||\le m_{f}(\delta )\) for every \(\delta >0\) and every \((x,y),(w,z)\in G\times G\). A function \(f:G\times G\longrightarrow {E}\) is called Lipschitz function if it satisfies the condition
for every \((x,y),(w,z)\in G\times G\). The smallest constant \(L\) with this property is denoted by \(\mathtt{lip }(f)\).
We define \(Lip(G\times G,{E})\) to be the Lipschitz space consisting of all bounded Lipschitz functions with the norm
The study of stability problems for functional equations is related to a question of Ulam [4] concerning the stability of group homomorphisms, which was affirmatively answered for Banach spaces by Hyers [5] (see for example [6, 7] and references therein).
In Lipschitz spaces the stability type problems for some functional equations were studied by Czerwik and Dlutek [8] and Tabor [3, 9]. Czerwik and Dlutek [8] established the stability of the quadratic functional equation
and the author of the present paper [10] proved the stability of the cubic functional equation in Lipschitz spaces. The stability problem for the following quartic functional equation
first was considered by Rassias [11] for mappings from a real normed space into a Banach space. Najati [12] proved the generalized Hyers–Ulam stability for the above quartic functional equation for functions from a linear space into a Banach space. Bae [13] obtained the general solution and the stability of the following 2-variable quadratic functional equation
in complete normed spaces. In this paper, we verify the stability of the quartic functional equation in the Lipschitz norms.
2 Approximation with \(\mathbf{d }\)-Lipschitz approach
For a given function \(f:G\times G\longrightarrow {E}\) we define its quadratic difference as follows
for all \((x,y),(z,w)\in G\times G\). By \(\Delta (G)\) we denote the diagonal set on \(G\), i.e.,
Theorem 2.1
Let \({G}\) be an Abelian group, and let \({E}\) be a vector space. Assume that the family \(B(G\times G,S({E}))\) admits LIM. If \(f:G\times G\longrightarrow {E}\) is a function and \(Qf(t,s,\cdot ,\cdot ):G\times G\longrightarrow {E}\) is \(\mathbf{d }\)-Lipschitz for every \((t,s)\in G\times G\), then there exists a quartic function \(Q\) such that \(f_{|_{\Delta (G)}}-Q\) is \(\frac{1}{2}\,\mathbf d \)-Lipschitz.
Proof
For every \((a,b)\in G\times G\) we define \(F_{a,b}:G\times G\longrightarrow {E}\) by
We prove that \(F_{a,b}\in B(G\times G,S({E}))\). We have for \((x,y),(a,b)\in G\times G\),
Set \(A:=\frac{1}{2}\mathbf{d }((0,0),(a,b))+f(a,b)+f(0,0)\). It is clear that \(A\in S({E})\). In view of our assumptions it follows that \(\mathtt{Im }F_{a,b}\subset A\) and so we obtain the result. The fact that the family \(B(G\times G,S({E}))\) admits LIM ensures there exists a linear operator \(M:B(G\times G,S({E}))\longrightarrow {E}\) such that
-
(i)
\(M[F_{a,b}]\in A\) for some \(A\in S({E})\),
-
(ii)
if for \((z,w)\in G\times G\), \(F_{a,b}^{z,w}:G\times G\longrightarrow {E}\) is defined by \(F_{a,b}^{z,w}(t,s):=F_{a,b}(t+z,s+w)\) for every \((t,s)\in G\times G\), then \(F_{a,b}^{z,w}\in B(G\times G,S({E}))\) and \(M[F_{a,b}]=M[F_{a,b}^{z,w}]\).
Define the function \(K:G\times G\longrightarrow {E}\) by \(K(x,y):=M[F_{x,y}]\) for \((x,y)\in G\times G\). We know that \(B(G\times G,S({E}))\) contains constant functions. By using property (i) of \(M\) it is easy to verify that if \(f:G\times G\longrightarrow {E}\) is constant, i.e., \(f(x,y)=c\) for \((x,y)\in G\times G\), where \(c\in {E}\), then \(M[f]=c\). We now show that \(f-K\) is \(\frac{1}{2}\,\mathbf d \)-Lipschitz. Let for any \((x,y)\in G\times G\) the constant function \(R_{x,y}:G\times G\longrightarrow {E}\) be the function \(R_{x,y}(z,w):=f(x,y)\) for all \((z,w)\in G\times G\). We have
for all \((x,y),(z,w)\in G\times G\). On the other hand
for all \((x,y),(z,w)\in G\times G\). From this we deduce that
In view of property (i) of \(M\) we conclude that
for all \((x,y),(z,w)\in G\times G\). This shows that
for all \((x,y),(z,w)\in G\times G\), i.e., \(f-K\) is a \(\frac{1}{2}\,\mathbf d \)-Lipschitz function. We now have
Furthermore, applying property (ii) of \(M\), one gets
for \((z,w)\in G\times G\). Consequently, we have
On the other hand we have
This shows that \(K\) is 2-variable quadratic. Define \(Q: G\longrightarrow E\) by \(Q(x):=K(x,x)\). We have
The function \(f-K\) is \(\frac{1}{2}\,\mathbf d \)-Lipschitz and so is \(f_{|_{\Delta (G)}}-Q\). The following equality entails that \(Q\) is quartic.
\(\square \)
Remark 2.2
Assuming the hypotheses of Theorem 2.1 and \(\mathtt{Im }Qf\subset A\) for some \(A\in S({E})\), we then obtain \(\mathtt{Im }(f_{|_{\Delta (G)}}-Q)\subset \frac{1}{2}A\). In fact,
and so \(\frac{1}{2}Qf(x,y,\cdot ,\cdot )\in B(G\times G,S({E}))\) for all \((x,y)\in G\times G\). Thus, property (i) of \(M\) implies
for all \((x,y)\in G\times G\). Therefore, \(\mathtt{Im }(f_{|_{\Delta (G)}}-Q)\subset \mathtt{Im }(f-K)\subset \frac{1}{2}A\).
3 Approximation with Lipschitz norm
Consider an Abelian group \((G\times G,+)\) with a metric \(d\) invariant under translation, i.e., satisfying the condition
for all \((a,b),(x,y),(w,z)\in G\times G\). We say that a metric \(D\) on \(G\times G\times G\times G\) is a product metric if it is an invariant metric and the following condition holds
Theorem 3.1
Let \((G\times G,+,d,D)\) be a product metric, and let \({E}\) be a normed space such that \(B(G\times G,CB({E}))\) admits LIM. Assume that \(f:G\times G\longrightarrow E\) be a function. If \(Qf\in Lip(G\times G\times G\times G, E)\), then there exists a quartic function \(Q\) such that
Proof
Assume that \(m_{{Qf}}: {\mathbb {R}}^{+}\rightarrow {\mathbb {R}}^{+}\) is the module of continuity of \(Qf:G\times G\times G\times G\rightarrow E\) with the product metric \(D\) on \(G\times G\times G\times G\). It is immediate that
for all \((t,s),(x,y),(w,z)\in G\times G\). Define the set-valued function \(\mathbf{d }:{G}\times {G}\longrightarrow CB({E})\) by
where \(B(0,1)\) is the closed unit ball with center at zero. We now conclude that \(Qf(t,s,\cdot ,\cdot )\) is d-Lipschitz and so Theorem 2.1 implies there exists a quartic function \(Q\) such that \(f_{|_{\Delta (G)}}-Q\) is \(\frac{1}{2}\,\mathbf d \)-Lipschitz. Hence,
which shows that \(m_{{f_{|_{\Delta (G)}}-Q}}=\frac{1}{2}m_{{Qf}}\). Moreover, \(||Qf||_{\sup }<\infty \) and clearly \(\mathtt{Im }Qf\subset ||Qf||_{\sup }B(0,1)\). Using Remark 2.2 we get
We may also prove that \(m_{Qf}=\mathtt{lip }(Qf)\) and so \(\mathtt{lip }(f_{|_{\Delta (G)}}-Q)\le \frac{1}{2}\mathtt{lip }(Qf)\). Applying the inequality (3.1) we get
\(\square \)
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Nikoufar, I. Quartic functional equations in Lipschitz spaces. Rend. Circ. Mat. Palermo 64, 171–176 (2015). https://doi.org/10.1007/s12215-014-0187-1
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DOI: https://doi.org/10.1007/s12215-014-0187-1