ANNALI DELL'UNIVERSITA' DI FERRARA

, Volume 60, Issue 2, pp 307–319

Stability of mixed additive-quadratic Jensen type functional equation in non-Archimedean \(\ell \)-fuzzy normed spaces

Authors

    • Department of MathematicsUrmia University
  • Ali Ebadian
    • Department of MathematicsUrmia University
  • Rasoul Aghalary
    • Department of MathematicsUrmia University
Article

DOI: 10.1007/s11565-013-0182-z

Cite this article as:
Abolfathi, M.A., Ebadian, A. & Aghalary, R. Ann Univ Ferrara (2014) 60: 307. doi:10.1007/s11565-013-0182-z
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Abstract

In this papers we prove the generalized Hyers–Ulam–Rassias stability of the following mixed additive-quadratic Jensen functional equation
$$\begin{aligned} 2f\left( \frac{x+y}{2}\right) +f\left( \frac{x-y}{2}\right) +f\left( \frac{y-x}{2}\right) =f(x)+f(y) \end{aligned}$$
in non- Archimedean \(\ell \)-fuzzy normed spaces.

Keywords

Generalized Hyers–Ulam–Rassias stabilityAdditive equationQuadratic equation\(\ell \)-fuzzy metric and normedNon-Archimedean \(\ell \)-fuzzy normed spaces

Mathematics Subject Classification (2000)

39B8239B52

1 Introduction

The stability problem for functional equations is related to a question of Ulam [23] in 1940 concerning the stability of group homomorphisms: Let \(G_{1}\) be a group and \(G_{2}\) a group with a metric \(d\). Given \(\varepsilon >0\), does there exist a \(\delta >0\) such that if function \(h:G_{1}\rightarrow G_{2}\) satisfies, \(d(h(xy)-h(x)h(y))<\delta \) for all \( x,y\in G_{1} \), then there is a homomorphism \(H:G_{1}\rightarrow G_{2}\) with \(d(h(x)-H(x))<\varepsilon \) for all \( x\in G_{1}\)?

Hyers [10] gave a partial affirmative answer to the quation of Ulam in context of Banach spaces. Subsequently, the result of Hyers was generalized by Aoki [1] for additive mappings. In 1978, Rassias [17] extended the theorem of Hyers by considering the unbounded cauchy difference inequality
$$\begin{aligned} \Vert f(x+y)-f(x)-f(y)\Vert \leqslant \varepsilon (\Vert x\Vert ^p+\Vert y\Vert ^p)\quad (\varepsilon \ge 0,\, p\in [0,1)). \end{aligned}$$
In 1990, Rassias [18] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for \(p\ge 1\). In 1991, Gajda [7] following the same approach as in Rassias [17] gave an affirmative solution to this question for \(p>1\). It was proved by Gajda [7], as well as by Rassias’ and Semrl [19] that one can prove Rassias’ type theorem when \(p=1\). The paper of Rassias [17] has provided a lot of influence in the development of what we now call Hyers–Ulam–Rassias stability of functional equations. For more information on such problems, w refer the interested readers to [3, 11, 12]. Several mathematicians following the spirit of the approach in the paper of Rassias [17] for the unbounded Cauchy difference obtained various results. For example in 1982, Rassias [16] obtained an analogous stability theorem in which he replace the factor \(\Vert x\Vert ^p+\Vert y\Vert ^p\) by \(\Vert x\Vert ^p.\Vert y\Vert ^q\) for \(p, q\in \mathbb R \) with \(p+q\ne 1\). In 1994, G\(\check{a}\)vruţa [8] provided a further generalization of Rassias’ Theorem in which he replaced the bound \(\varepsilon (\Vert x\Vert ^p+\Vert y\Vert ^p)\) by a general control function \(\varphi (x,y)\). Since then the stability problems of various functional equations and mappings, such as the Cauchy equation, the Jensen equation, the quadratic equation, the cubic equation, homomorphisms, derivations and their Pexiderized versions with more general domains and ranges have been investigated by a number of authors [3, 11, 12, 20]. The theory of fuzzy sets was introduced by Zadeh in 1965 [25]. After the pioneering work of Zadeh, there has been a great effort to obtain fuzzy analogues of classical theories and this branch finds a wide range of application in the field of science and engineering. Katsaras [13] introduced an idea of fuzzy norm on linear space in 1984, in the same year Wu and Fang [24]introduced a notion of fuzzy normed spaces to give a generalization of the Kolmogoroff normalized theorem for fuzzy topological linear spaces. The pioneering work of Zadeh provided some influence to several mathematicians to study fuzzy analogues of classical theories connected with functional equations in the framework of mathematical analysis. In 1897, Hensel [9] introduced a field with a valuation in which does not have the Archimedean property. The theory of non-Archimedean spaces has gained the interest of physicists for their research, in particular the problems that emerge in quantum physics. In 2007 Moslehian and Rassias [14] proved the generalized Hyers–Ulam stability of Cauchy and quadratic functional equations in non-Archimedean normed spaces. After their resulats some papers on the stability of other equations is such spaces have been published. in [2, 6, 15].

One of the problems in \(\ell \)-fuzzy topology is to obtain an appropriate concept of \(\ell \)-fuzzy metric spaces and \(\ell \)-fuzzy normed spaces. Saadati and Park [22], respectively, introduced and studied a notion of intuitionistic fuzzy metric (normed) spaces and then Deschrijver et al. and Saadati generalized this concept and introduced and studied a notion of \(\ell \)-fuzzy metric spaces and \(\ell \)-fuzzy normed spaces [4, 21]. In 2011 Ebadian and et al. [5] investigated as well the stability of a mixed type cubic and quartic functional equation in non-Archimedean \(\ell \)-fuzzy normed spaces.

A triangular norm (shortly, t-norm) is a binary operation \(T:[0,1]\times [0,1]\rightarrow [0,1]\) which is commutative, associative, monotone and has \(1\) as the unit element. A t-norm \(T\) can be extended (by associativity) in a unique way to an n-ary operation taking, for all \((x_{1},\ldots ,x_{n})\in [0,1]^{n},\) the value \(T(x_{1},\ldots ,x_{n})\) defined by
$$\begin{aligned} T_{i=1}^{0}x_{i}=1,\quad \quad T_{i=1}^{n}x_{i}=T(T_{i=1}^{n-1}x_{i},x_{n})=T(x_{1},\ldots ,x_{n}). \end{aligned}$$
A t-norm \(T\) can also be extended to a countable operation taking, for any sequence \(\{x_{n}\}_{n\in \mathbb N }\) in \([0,1]\), the value
$$\begin{aligned} T_{i=1}^{\infty }x_{i}=\lim _{n\rightarrow \infty }T_{i=1}^{n}x_{i}. \end{aligned}$$

Definition 1.1

Let \(\ell =(L,\ge _{L})\) be a complete lattice and let \(U\) a nonempty set called the universe. An \(\ell \)-fuzzy set in \(U\) is defined as a mapping \(A:U\rightarrow L.\) For each in \(U\), \(A(u)\) represents the degree (in \(L\)) to which is an element of \(U.\)

Definition 1.2

The triple \((X,M,T)\) is said to be an \(\ell \)-fuzzy metric space if \(X\) is an arbitrary (non-empty) set, \(T\) is a continuous t-norm on \(L\) and \(M\) is an \(\ell \)-fuzzy set on \(X^{2}\times ]0,+\infty [\) satisfying the following condition: for all \(x,y,z\in X\) and \(t,s\in ]0,+\infty [\),
  1. (1)

    \(M(x,y,t)>_{L}0_{L};\)

     
  2. (2)

    \(M(x,y,t)=1_L\) for all \(t>0\) if and only if \(x=y;\)

     
  3. (3)

    \(M(x,y,t)=M(y,x,t);\)

     
  4. (4)

    \(T(M(x,y,t),M(y,z,s))\le _{L}M(x,z,t+s);\)

     
  5. (5)

    \(M(x,y,.):]o,+\infty [\rightarrow L\) is continuous.

     

In this case, \(M\) is called an \(\ell \)-fuzzy metric.

Definition 1.3

The triple \((V,P,T)\) is said to be an \(\ell \)-fuzzy normed space if \(V\) is a vector space, \(T\) is a continuous t-norm on \(L\) and \(P\) is an \(\ell \)-fuzzy set on \(V\times ]0,+\infty [\) which satisfying the following condition: for all \(x,y\in V\) and \(t,s\in ]0,+\infty [\),
  1. (1)

    \(P(x,t)>_{L}0_{L};\)

     
  2. (2)

    \(P(x,t)=1_{L}\) for all \(t>0\) if and only if \(x=0;\)

     
  3. (3)

    \(P(\alpha x,t)=P(x,\frac{t}{|\alpha |})\) for each \(\alpha \ne 0;\)

     
  4. (4)

    \(T(P(x,t),P(y,s))\le _{L}P(x+y,t+s);\)

     
  5. (5)

    \(P(x,.):]o,+\infty [\rightarrow L\) is continuous.

     
  6. (6)

    \(\lim _{t\rightarrow 0}P(x,t)=0_L\) and \( \lim _{t\rightarrow \infty }P(x,t)=1_L.\)

     
In this case, \(P\) is called an \(\ell \)-fuzzy norm.
  1. (1)

    A sequence \(\{x_{n}\}_{n\in \mathbb N }\) in an \(\ell \)-fuzzy normed space \((V,P,T)\) is called a Cauchy sequence if, for each \(\varepsilon \in L\setminus \{0_{L}\}\) and \(t>0,\) there exists \(n_{0}\in \mathbb N \) such that, for all \(n,m\ge n_{0},\)\(P(x_{n}-x_{m},t)>_{L}N(\varepsilon ),\) where \(N\) is a negator on \(\ell \).

     
  2. (2)

    A sequence \(\{x_{n}\}_{n\in \mathbb N }\) is said to be convergent to \(x\in V\) in the \(\ell \)-fuzzy normed space \((V,P,T)\) which is denoted by \(x_{n} \rightarrow x\) if \(P(x_{n}-x,t)\rightarrow 1_{\ell }\) where \(n\rightarrow \infty \) for all \(t>0\).

     
  3. (3)

    A \(\ell \)-fuzzy normed space \((V,P,T)\) is said be complete if and only if every Cauchy sequence in \(V\) is convergent.

     
Let \((V,P,T)\) be an \(\ell \)-fuzzy normed space. If we define M(x,y,t)=P(x-y,t) for all \(x,y\in V\) and \(t\in ]0,+\infty [\), then \(M\) is an \(\ell \)-fuzzy metric on \(V\) which is called the \(\ell \)-fuzzy metric induced by the \(\ell \)-fuzzy norm \(P\).

Definition 1.4

Let \(K\) be a field. A non-Archimedean absolute value on \(K\) is a function \(|.|:K\rightarrow [o,+\infty [\) such that , for any \(a,b\in K,\)
  1. (1)

    \(|a|\ge 0\) and equality holds if and only if \(a=0,\)

     
  2. (2)

    \(|ab|\le |a||b|,\)

     
  3. (3)

    \(|a+b|\le max\{|a|,|b|\}\) (the strict triangle inequality).

     

Note that \(|n|\le 1\) for each integer \(n\). We always assume, in addition, that \(|.|\) is non-trivial, i.e., there exists an \(a_{0}\in K\) such that \(|a_{0}|\ne 0,1.\)

Definition 1.5

A non-Archimedean \(\ell \)-fuzzy normed space is a triple \((V,P,T),\) where is \(V\) is a vector space, \(T\) is a continuous t-norm on \(L\) and \(P\) is an \(\ell \)-fuzzy set on \(V\times ]0,+\infty [\) satisfying the following condition: for all \(x,y\in V\) and \(t,s\in ]0,+\infty [\),
  1. (1)

    \( P(x,t)>_{L}0_{L};\)

     
  2. (2)

    \(P(x,t)=1_{L}\) for all \(t>0\) if and only if \(x=0;\)

     
  3. (3)

    \(P(\alpha x,t)=P(x,\frac{t}{|\alpha |})\) for each \(\alpha \ne 0;\)

     
  4. (4)

    \(T(P(x,t),P(y,s))\le _{L}P(x+y,max\{t,s\});\)

     
  5. (5)

    \(P(x,.):]o,+\infty [\rightarrow L\) is continuous.

     
  6. (6)

    \(\lim _{t\rightarrow 0}P(x,t)=0_{L}\) and \( \lim _{t\rightarrow \infty }P(x,t)=1_{L}.\)

     

Example 1.6

Let \((X,\Vert .\Vert )\) be a non-Archimedean normed linear space and
$$\begin{aligned} P(x,t)=\left\{ \begin{array}{ll} 0, &{}\quad t\le \Vert x\Vert \\ 1, &{}\quad t>\Vert x\Vert \end{array} \right. \end{aligned}$$
then triple \((V,P,T),\) is a non-Archimedean \(\ell \)-fuzzy normed space in which \(\ell =[0,1],\) where \(T(a,b)=\min (a,b)\).

2 Main results

In this section we investigate the generalized Hyers–Ulam–Rassias stability of the mixed additive-quadratic Jensen type functional equation
$$\begin{aligned} 2f\left( \frac{x+y}{2}\right) +f\left( \frac{x-y}{2}\right) +f\left( \frac{y-x}{2}\right) =f(x)+f(y) \end{aligned}$$
in non-Archimedean \(\ell \)-fuzzy normed spaces.
Let \(\Psi \) be an \(\ell \)-fuzzy set on \(X\times X\times [0,\infty )\) such that \(\Psi (x,y,.)\) is nondecreasing,
$$\begin{aligned} \Psi (cx,cx,t)\ge _{L}\Psi \left( x,x,\frac{t}{|c|}\right) , \quad \quad ( x\in X ,c\ne 0) \end{aligned}$$
and
$$\begin{aligned} \lim _{t\rightarrow \infty }\Psi (x,y,t)=1_{\ell },\quad \quad \quad ( x,y\in X ,t>0). \end{aligned}$$

Theorem 2.1

Let \(K\) be a non-Archimedean field, \(X\) a vector space over \(K\) and \((Y,P,T)\) a non-Archimedean \(\ell \)-fuzzy Banach space over \(K\). Suppose that \(f:X\rightarrow Y\) is an odd mapping and \(\Psi \) is an \(\ell \)-fuzzy set on \(X\times X\times [0,\infty )\) satisfying the inequality
$$\begin{aligned} P(2f\left( \frac{x+y}{2}\right) +f\left( \frac{x-y}{2}\right) + f\left( \frac{y-x}{2}\right) -f(x)-f(y),t)\ge _{L}\Psi (x,y,t) \nonumber \\ \end{aligned}$$
(2.1)
for all \(x,y\in X\) and all \(t>0\) and \(\Psi \) be an \(\ell \)-fuzzy on \(X\times X\times [0,\infty )\). If there exists an \(\alpha \in \mathbb R \) and an positive integer \(k\) , \(k\ge 2\) with \(|2^{k}|<\alpha \), and \(|2|\ne 0\) such that
$$\begin{aligned} \Psi (2^{-k}x,2^{-k}y,t)\ge _{L}\Psi (x,y,\alpha t),\quad \quad ( x,y\in X ,t>0), \end{aligned}$$
(2.2)
and
$$\begin{aligned} \lim _{n\rightarrow \infty }T_{j=n}^{\infty }M\left( x,\frac{\alpha ^{j}t}{|2^{k}|^{j}}\right) =1_{\ell } \end{aligned}$$
then there exists a unique additive mapping \(A:X\rightarrow Y\) such that
$$\begin{aligned} P(f(x)-A(x),t)\ge _{L}T_{i=0}^{\infty }M\left( x,\frac{\alpha ^{i}t}{|2^{k}|^{i}}\right) \quad \quad ( x\in X,t> 0), \end{aligned}$$
(2.3)
where
$$\begin{aligned} M(x,t)=T(\Psi (0,x,t),\Psi (0,2^{-1}x,t),\ldots ,\Psi (0,2^{-(k-1)}x,t)) \end{aligned}$$
for all \(x\in X ,t>0.\)

Proof

First we show, by induction on \(j\), that for each \(x\in X , t>0\) and \(j\ge 1\),
$$\begin{aligned} P(2^{j}f(2^{-j}x)-f(x),t)&\ge _{L}M_{j}(x,t):\nonumber \\&= T(\Psi (0,x,t), \Psi (0,2^{-1}x,t),\ldots ,\Psi (0,2^{-(j-1)}x,t)). \nonumber \\ \end{aligned}$$
(2.4)
Put \(x=0\) in (2.1) to obtain
$$\begin{aligned} P(2f(2^{-1}y)-f(y),t)\ge _{L}\Psi (0,y,t) \quad \quad (y\in X ,t>0). \end{aligned}$$
If we replace \(y\) in above equation by \(x\), we get
$$\begin{aligned} P(2f(2^{-1}x)-f(x),t)\ge _{L}\Psi (0,x,t) \quad \quad (x\in X ,t>0). \end{aligned}$$
This proves (2.4) for \(j=1\). Let (2.4) hold for some \(j>1\). Putting \(x=0\) and \(y=2^{-j}x\) in (2.1), we get
$$\begin{aligned} P(2f(2^{-(j+1)}x)-f(2^{-j}x),t)\ge _{L}\Psi (0,2^{-j}x,t) \quad \quad (x\in X ,t>0). \end{aligned}$$
Hence since \(|2|<1\), we have
$$\begin{aligned}&P(2^{j+1}f(2^{-(j+1)}x)-f(x),t)\\&\qquad \ge _{L}T(P(2^{j+1}f(2^{-(j+1)}x)-2^{j}f(2^{-j}x),t),P(2^{j}f(2^{-j}x)-f(x),t))\\&\qquad =T\left( P(2f(2^{-(j+1)}x)-f(2^{-j}x)),\frac{t}{|2^{j}|}\right) ,P(2^{j}f(2^{-j}x)-f(x),t))\\&\qquad \ge _{L}T(P(2f(2^{-(j+1)}x)-f(2^{-j}x),t),P(2^{j}f(2^{-j}x)-f(x),t))\\&\qquad \ge _{L}T(\Psi (0,2^{-j},t),M_{j}(x,t))=M_{j+1}(x,t),\quad \quad (x\in X,t>0). \end{aligned}$$
Thus (2.4) holds for all \(j\ge 1\). In particular
$$\begin{aligned} P(2^{k}f(2^{-k}x)-f(x),t)\ge _{L} M(x,t),\quad \quad (x\in X ,t>0). \end{aligned}$$
(2.5)
Replacing \(x\) by \(2^{-kn}x\) in (2.5) and using inequality (2.2) we obtain
$$\begin{aligned} P(2^{k}f(2^{-k(n+1)}x)\!-\!f(2^{-kn}x),t)\!\ge \!_{L} M(x,\alpha ^{n}t),\quad (x\in X ,t>0, n=0,1,2,\ldots ). \end{aligned}$$
Therefore
$$\begin{aligned}&P(2^{kn+k}f(2^{-k(n+1)}x)-2^{kn}f(2^{-kn}x),t)\nonumber \\&\qquad \ge _{L}M\left( x,\frac{\alpha ^{n}t}{|2^{k}|^{n}}\right) ,\quad (x\in X ,t>0, n=0,1,2,\ldots ). \end{aligned}$$
Hence it follows that
$$\begin{aligned}&P(2^{k(n+p)}f(2^{-k(n+p)}x)-2^{kn}f(2^{-kn}x),t)\nonumber \\&\qquad \ge _{L}T_{j=n}^{n+p-1}(P(2^{k(j+1)}f(2^{-k(j+1)}x)-2^{kj}f(2^{-kj}x),t))\\&\qquad \ge _{L}T_{j=n}^{n+p-1}M\left( x,\frac{\alpha ^{j}t}{|2^{k}|^{j}}\right) ,\quad (x\in X ,t>0, n=0,1,2,\ldots ).\nonumber \end{aligned}$$
(2.6)
Since \(\lim _{n\rightarrow \infty }T_{j=n}^{\infty }M(x,\frac{\alpha ^{j}t}{|2^{k}|^{j}})=1_{\ell }\) for all \(x\in X ,t>0\) this shows that \(\{2^{kn}f(2^{-kn}x)\}_{n\in \mathbb N }\) is a Cauchy sequence in the non-Archimedean \(\ell \)-fuzzy Banach space \((Y,P,T).\) Hence, we can define a mapping \(A:X\rightarrow Y\) such that
$$\begin{aligned} \lim _{n\rightarrow \infty }P(2^{kn}f(2^{-kn}x)-A(x),t)=1_{\ell },\quad \quad (x\in X ,t>0). \end{aligned}$$
(2.7)
For each \(n\ge 1,x\in X\) and \(t>0,\)
$$\begin{aligned}&P(f(x)-2^{kn}f(2^{-kn}x),t)\nonumber \\&\qquad =P\left( \sum _{i=0}^{n-1}2^{ki}f(2^{-ki}x)-2^{k(i+1)}f(2^{-k(i+1)}x),t\right) \nonumber \\&\qquad \ge _{L}T_{i=0}^{n-1}P(2^{ki}f(2^{-ki}x)-2^{k(i+1)}f(2^{-k(i+1)}x),t)\nonumber \\&\qquad \ge _{L}T_{i=0}^{n-1}M\left( x,\frac{\alpha ^{i}t}{|2^{k}|^{i}}\right) \end{aligned}$$
(2.8)
and so for each \(x\in X , t>0,\) and large enough \(n\), we have
$$\begin{aligned}&P(f(x)-A(x),t)\\&\qquad \ge _{L}T(P(f(x)-2^{kn}f(2^{-kn}x),t),P(2^{kn}f(2^{-kn}x)-A(x),t))\\&\qquad \ge _{L}T(T_{i=0}^{n-1}M\left( x,\frac{\alpha ^{i}t}{|2^{k}|^{i}}\right) ,P(2^{kn}f(2^{-kn}x)-A(x),t)) \end{aligned}$$
Taking the limit as \(n\rightarrow \infty \) in above, we obtain
$$\begin{aligned} P(f(x)-A(x),t)\ge _{L}T_{i=0}^{\infty }M\left( x,\frac{\alpha ^{i}t}{|2^{k}|^{i}}\right) \end{aligned}$$
which proves (2.3). Replacing \(x\), \(y\) by \(2^{-kn}\), \(2^{-kn}\)in (2.1), we get
$$\begin{aligned}&P\left( 2^{kn+1}f\left( \frac{x+y}{2^{kn+1}}\right) +2^{kn}f\left( \frac{x-y}{2^{kn+1}}\right) +2^{kn}f\left( \frac{y-x}{2^{kn+1}}\right) \right. \\&\qquad \left. \left. -2^{kn}f\left( \frac{x}{2^{kn}}\right) -2^{kn}f\left( \frac{y}{2^{kn}}\right) ,t\right) \right) \\&\qquad \ge _{L}\psi \left( 2^{-kn}x,2^{-kn}y,\frac{t}{|2^{k}|^{n}}\right) \\&\qquad \ge _{L}\psi \left( x,y,\frac{\alpha ^{n}t}{|2^{k}|^{n}}\right) \quad \quad (x\in X ,t>0). \end{aligned}$$
Since \(\lim _{n\rightarrow \infty }\Psi (x,y,\frac{\alpha ^{n}t}{|2^{k}|^{n}})=1_{\ell }.\) We infer that \(A\) is additive [2]. Now if \(A^{\prime }:X\rightarrow Y\) is another additive mapping such that \(P(f(x)-A^{\prime }(x),t)\ge _{L}T_{i=0}^{\infty }M(x,\frac{\alpha ^{i}t}{|2^{k}|^{i}})\) for each \(x\in X , t>0,\) then
$$\begin{aligned} P(A(x)-A^{\prime }(x),t)\ge _{L}T(P(A(x)-2^{kn}f(2^{-kn}x),t),P(2^{kn}f(2^{-kn}x)-A^{\prime }(x),t) \end{aligned}$$
Therefore, from (2.7) we conclude that \(A=A^{\prime }\). \(\square \)

Theorem 2.2

Let \(K\) be a non-Archimedean field, \(X\) a vector space over \(K\) and \((Y,P,T)\) a non-Archimedean \(\ell \)-fuzzy Banach space over \(K\). Suppose that \(f:X\rightarrow Y\) is an even mapping and \(\Psi \) is an \(\ell \)-fuzzy set on \(X\times X\times [0,\infty )\) satisfying the inequality
$$\begin{aligned} P\left( 2f\left( \frac{x+y}{2}\right) +f\left( \frac{x-y}{2}\right) + f\left( \frac{y-x}{2}\right) -f(x)-f(y),t\right) \ge _{L}\Psi (x,y,t) \nonumber \\ \end{aligned}$$
(2.9)
for all \(x,y\in X\) and all \(t>0\) and \(\Psi \) be an \(\ell \)-fuzzy on \(X\times X\times [0,\infty )\). If there exists an \(\alpha \in \mathbb R \) and an positive integer \(k\) , \(k\ge 2\) with \(|2^{k}|<\alpha \), and \(|2|\ne 0\) such that
$$\begin{aligned} \Psi (2^{-k}x,2^{-k}y,t)\ge _{L}\Psi (x,y,\alpha t),\quad \quad ( x,y\in X ,c\ne 0), \end{aligned}$$
(2.10)
and
$$\begin{aligned} \lim _{n\rightarrow \infty }T_{j=n}^{\infty }M\left( x,\frac{\alpha ^{j}t}{|2^{k}|^{j}}\right) =1_{\ell } \end{aligned}$$
then there exists a unique quadratic mapping \(Q:X\rightarrow Y\) such that
$$\begin{aligned} P(f(x)-Q(x),t)\ge _{L}T_{i=0}^{\infty }M\left( x,\frac{\alpha ^{i}t}{|2^{k}|^{i}}\right) \quad \quad ( x\in X,t> 0), \end{aligned}$$
(2.11)
where
$$\begin{aligned} M(x,t)=T(\Psi (0,x,t),\Psi (0,2^{-1}x,t),\ldots ,\Psi (0,2^{-(k-1)}x,t)) \end{aligned}$$
for all \(x\in X ,t>0.\)

Proof

First we show, by induction on \(j\), that for each \(x\in X , t>0\) and \(j\ge 1\),
$$\begin{aligned}&P(4^{j}f(2^{-j}x)-f(x),t)\ge _{L}M_{j}(x,t):\nonumber \\&\qquad = T(\Psi (0,x,t),\Psi (0,2^{-1}x,t),\ldots ,\Psi (0,2^{-(j-1)}x,t)). \end{aligned}$$
(2.12)
Put \(x=0\) in (2.9) to obtain
$$\begin{aligned} P(4f(2^{-1}y)-f(y),t)\ge _{L}\Psi (0,y,t) \quad \quad (y\in X ,t>0). \end{aligned}$$
If we replace \(y\) in above equation by \(x\), we get
$$\begin{aligned} P(4f(2^{-1}x)-f(x),t)\ge _{L}\Psi (0,x,t) \quad \quad (x\in X ,t>0). \end{aligned}$$
This proves (2.12) for \(j=1\). Let (2.12) hold for some \(j>1\). Putting \(x=0\) and \(y=2^{j}x\) in (2.9), we get
$$\begin{aligned} P(4f(2^{-(j+1)}x)-f(2^{-j}x),t)\ge _{L}\Psi (0,2^{-j}x,t) \quad \quad (x\in X ,t>0). \end{aligned}$$
Hence since \(|2|<1\), we have
$$\begin{aligned}&P(4^{j+1}f(2^{-(j+1)}x)-f(x),t)\\&\qquad \ge _{L}T(P(4^{j+1}f(2^{-(j+1)}x)-4^{j}f(2^{-j}x),t),P(4^{j}f(2^{-j}x)-f(x),t))\\&\qquad =T\left( P\left( 4f(2^{-(j+1)}x)-f(2^{-j}x)),\frac{t}{|4^{j}|}\right) ,P(4^{j}f(2^{-j}x)-f(x),t)\right) \\&\qquad \ge _{L}T(P(4f(2^{-(j+1)}x)-f(2^{-j}x),t),P(4^{j}f(2^{-j}x)-f(x),t))\\&\qquad \ge _{L}T(\Psi (0,2^{-j},t),M_{j}(x,t))=M_{j+1}(x,t),\quad \quad (x\in X,t>0). \end{aligned}$$
Thus (2.12) holds for all \(j\ge 1\). In particular
$$\begin{aligned} P(4^{k}f(2^{-k}x)-f(x),t)\ge _{L} M(x,t),\quad \quad (x\in X ,t>0). \end{aligned}$$
(2.13)
Replacing \(x\) by \(2^{-kn}x\) in (2.13) and using inequality (2.10) we obtain
$$\begin{aligned} P(4^{k}f(2^{-k(n+1)}x)\!-\!f(2^{-kn}x),t)\!\ge \!_{L} M(x,\alpha ^{n}t),\quad (x\in X ,t>0, n=0,1,2,\ldots ). \end{aligned}$$
Therefore
$$\begin{aligned}&P(4^{kn+k}f(2^{-k(n+1)}x)-4^{kn}f(2^{-kn}x),t)\\&\qquad \ge _{L}M\left( x,\frac{\alpha ^{n}t}{|4^{k}|^{n}}\right) \\&\qquad \ge _{L}M\left( x,\frac{\alpha ^{n}t}{|2^{k}|^{n}}\right) ,\quad (x\in X ,t>0, n=0,1,2,\ldots ). \end{aligned}$$
Hence it follows that
$$\begin{aligned}&P(4^{k(n+p)}f(2^{-k(n+p)}x)-4^{kn}f(2^{-kn}x),t)\nonumber \\&\qquad \ge _{L}T_{j=n}^{n+p-1}(P(4^{k(j+1)}f(2^{-k(j+1)}x) -4^{kj}f(2^{-kj}x),t))\nonumber \\&\qquad \ge _{L}T_{j=n}^{n+p-1}M\left( x,\frac{\alpha ^{j}t}{|2^{k}|^{j}}\right) ,\quad (x\in X ,t>0, n=0,1,2,\ldots ). \end{aligned}$$
(2.14)
Since \(\lim _{n\rightarrow \infty }T_{j=n}^{\infty }M(x,\frac{\alpha ^{j}t}{|2^{k}|^{j}})=1_{\ell }\) for all \(x\in X ,t>0\) this shows that \(\{4^{kn}f(2^{-kn}x)\}_{n\in \mathbb N }\) is a Cauchy sequence in the non-Archimedean \(\ell \)-fuzzy Banach space \((Y,P,T).\) Hence, we can define a mapping \(Q:X\rightarrow Y\) such that
$$\begin{aligned} \lim _{n\rightarrow \infty }P(2^{2kn}f(2^{-kn}x)-Q(x),t)=1_{\ell },\quad \quad (x\in X ,t>0). \end{aligned}$$
(2.15)
For each \(n\ge 1,x\in X\) and \(t>0,\) we have
$$\begin{aligned}&P(f(x)-2^{2kn}f(2^{-kn}x),t)\nonumber \\&\qquad =P\left( \sum _{i=0}^{n-1}2^{2ki}f(2^{-ki}x)-2^{2k(i+1)}f(2^{-k(i+1)}x),t\right) \nonumber \\&\qquad \ge _{L}T_{i=0}^{n-1}P(2^{2ki}f(2^{-ki}x)-2^{2k(i+1)}f(2^{-k(i+1)}x),t)\nonumber \\&\qquad \ge _{L}T_{i=0}^{n-1}M\left( x,\frac{\alpha ^{i}t}{|2^{k}|^{i}}\right) , \end{aligned}$$
(2.16)
and so for each \(x\in X , t>0,\) and large enough \(n\)
$$\begin{aligned}&P(f(x)-Q(x),t)\\&\qquad \ge _{L}T(P(f(x)-2^{2kn}f(2^{-kn}x),t),P(2^{2kn}f(2^{-kn}x)-Q(x),t))\\&\qquad \ge _{L}T\left( T_{i=0}^{n-1}M\left( x,\frac{\alpha ^{i}t}{|2^{k}|^{i}}\right) ,P(2^{kn}f(2^{-kn}x)-Q(x),t)\right) \end{aligned}$$
Taking the limit as \(n\rightarrow \infty \) in above, we obtain
$$\begin{aligned} P(f(x)-Q(x),t)\ge _{L}T_{i=0}^{\infty }M\left( x,\frac{\alpha ^{i}t}{|2^{k}|^{i}}\right) \end{aligned}$$
which proves (2.11). Replacing\(x\), \(y\) by \(2^{-kn}x\), \(2^{-kn}y\) in (2.9), we get
$$\begin{aligned}&P\left( 4^{kn+1}f\left( \frac{x+y}{2^{kn+1}}\right) \right. \\&\qquad \left. \left. +4^{kn}f\left( \frac{x-y}{2^{kn+1}}\right) +4^{kn}f\left( \frac{y-x}{2^{kn+1}}\right) -4^{kn}f\left( \frac{x}{2^{kn}}\right) -4^{kn}f\left( \frac{y}{2^{kn}}\right) ,t\right) \right) \\&\quad \ge _{L}\psi \left( 2^{-kn}x,2^{-kn}y,\frac{t}{|4^{k}|^{n}}\right) \\&\quad \ge _{L}\psi \left( x,y,\frac{\alpha ^{n}t}{|2^{k}|^{n}}\right) \quad \quad (x\in X ,t>0). \end{aligned}$$
Since \(\lim _{n\rightarrow \infty }\Psi (x,y,\frac{\alpha ^{n}t}{|2^{k}|^{n}})=1_{\ell }.\) We infer that \(Q\) is quadratic [2]. Now if \(Q^{\prime }:X\rightarrow Y\) is another additive mapping such that \(P(f(x)-Q^{\prime }(x),t)\ge _{L}T_{i=0}^{\infty }M(x,\frac{\alpha ^{i}t}{|2^{k}|^{i}})\) for each \(x\in X , t>0,\) then
$$\begin{aligned} P(Q(x)-Q^{\prime }(x),t)\ge _{L}T(P(A(x)-4^{kn}f(2^{-kn}x),t), P(4^{kn}f(2^{-kn}x)-Q^{\prime }(x),t) \end{aligned}$$
Therefore, from (2.13) we conclude that \(Q=Q^{\prime }\). This completes the proof. \(\square \)

Theorem 2.3

Let \(K\) be a non-Archimedean field, \(X\) a vector space over \(K\) and \((Y,P,T)\) a non-Archimedean \(\ell \)-fuzzy Banach space over \(K\). Suppose that \(f:X\rightarrow Y\) is an even mapping and \(\Psi \) is an \(\ell \)-fuzzy set on \(X\times X\times [0,\infty )\) satisfying the inequality
$$\begin{aligned} P\left( 2f\left( \frac{x+y}{2}\right) +f\left( \frac{x-y}{2}\right) + f\left( \frac{y-x}{2}\right) -f(x)-f(y),t\right) \ge _{L}\Psi (x,y,t) \nonumber \\ \end{aligned}$$
(2.17)
for all \(x,y\in X\) and all \(t>0\) and \(\Psi \) be an \(\ell \)-fuzzy on \(X\times X\times [0,\infty )\). If there exists an \(\alpha \in \mathbb R \) and an positive integer \(k\) , \(k\ge 2\) with \(|2^{k}|<\alpha \), and \(|2|\ne 0\) such that
$$\begin{aligned} \Psi (2^{-k}x,2^{-k}y,t)\ge _{L}\Psi (x,y,\alpha t),\quad \quad \forall x,y\in X ,c\ne 0, \end{aligned}$$
(2.18)
and
$$\begin{aligned} \lim _{n\rightarrow \infty }T_{j=n}^{\infty }\left( T\left( M\left( x,\frac{\alpha ^{j}t}{|2^{k}|^{j}}\right) ,M\left( -x,\frac{\alpha ^{j}t}{|2^{k}|^{j}}\right) \right) \right) =1_{\ell } \end{aligned}$$
then there exists a unique additive mapping \(A:X\rightarrow Y\) and a unique quadratic mapping \(Q:X\rightarrow Y\) such that
$$\begin{aligned}&P(f(x)-A(x)-Q(x),t)\nonumber \\&\qquad \ge _{L}T_{i=0}^{\infty }\left( T\left( M\left( x,\frac{\alpha ^{i}t}{|2^{k}|^{i}}\right) ,M\left( -x,\frac{\alpha ^{i}t}{|2^{k}|^{i}}\right) \right) \right) \quad \quad \forall x\in X,t> 0, \nonumber \\ \end{aligned}$$
(2.19)
where
$$\begin{aligned} M(x,t)=T(\Psi (0,x,t),\Psi (0,2^{-1}x,t),\ldots ,\Psi (0,2^{-(k-1)}x,t)) \end{aligned}$$
for all \(x\in X ,t>0.\)

Proof

Let \(f_{o}(x)=\frac{1}{2}(f(x)-f(-x))\) for all \(x\in X\). Then \(f_{o}\) is an odd mapping, such that
$$\begin{aligned}&P\left( 2f_{o}\left( \frac{x+y}{2}\right) +f_{o}\left( \frac{x-y}{2}\right) +f_{o}\left( \frac{y-x}{2}\right) -f_{o}(x)-f_{o}(y),t\right) \\&\quad \ge _{L}T\left( P\left( 2f\left( \frac{x+y}{2}\right) +f\left( \frac{x-y}{2}\right) +f\left( \frac{y-x}{2}\right) -f(x)-f(y),t\right. \right) \\&\quad \ge _{L}P\left( 2f\left( \frac{-x-y}{2}\right) +f\left( \frac{-x+y}{2}\right) \right. \\&\qquad \left. +f\left( \frac{-y+x}{2}\right) -f(-x)-f(-y),t\right) \quad (x,y\in X ,t>0)\\&\quad \ge _{L}T\left( \Psi (x,y,t),\Psi (-x,-y,t)\right) \\ \end{aligned}$$
By Theorem 2.1 it follows that there exists a unique additive function \(A:X\rightarrow Y\) satisfying
$$\begin{aligned}&P(f_{o}(x)-A(x),t)\nonumber \\&\qquad \ge _{L}T_{i=0}^{\infty }\left( T\left( M\left( x,\frac{\alpha ^{}t}{|2^{k}|^{i}}\right) ,M\left( -x,\frac{\alpha ^{i}t}{|2^{k}|^{i}}\right) \right) \right) \quad (x,y\in X ,t>0). \nonumber \\ \end{aligned}$$
(2.20)
Let \(f_{e}(x)=\frac{1}{2}(f(x)-f(-x))\) for all \(x\in X\). Then \(f_{e}\) is an even mapping, such that
$$\begin{aligned}&P\left( 2f_{e}\left( \frac{x+y}{2}\right) +f_{e}\left( \frac{x-y}{2}\right) +f_{e}\left( \frac{y-x}{2}\right) -f_{e}(x)-f_{e}(y),t\right) \\&\quad \ge _{L}T\left( P\left( 2f\left( \frac{x+y}{2}\right) +f\left( \frac{x-y}{2}\right) +f\left( \frac{y-x}{2}\right) -f(x)-f(y),t\right) \right. \\&\quad \ge _{L}P\left( 2f\left( \frac{-x-y}{2}\right) +f\left( \frac{-x+y}{2}\right) \right. \\&\qquad \left. +f\left( \frac{-y+x}{2}\right) -f(-x)-f(-y),t\right) \quad (x,y\in X ,t>0)\\&\quad \ge _{L}T(\Psi (x,y,t),\Psi (-x,-y,t))\\ \end{aligned}$$
By Theorem 2.2 it follows that there exists a unique quadratic function \(Q:X\rightarrow Y\) satisfying
$$\begin{aligned}&P(f_{e}(x)-Q(x),t)\nonumber \\&\qquad \ge _{L}T_{i=0}^{\infty }\left( T\left( M\left( x,\frac{\alpha ^{i}t}{|2^{k}|^{i}}\right) ,M\left( -x,\frac{\alpha ^{i}t}{|2^{k}|^{i}}\right) \right) \right) \quad (x\in X ,t>0).\nonumber \\ \end{aligned}$$
(2.21)
Therefore by (2.20) and (2.21) we have (2.17). \(\square \)

Acknowledgments

The authors would like to thank the referee for his comments and suggestions on the manuscript.

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© Università degli Studi di Ferrara 2013