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In 1940, Ulam [24] posed the stability problem concerning group homomorphisms. For Banach spaces, the problem was solved by Hyers [7] in the case of approximate additive mappings. And then Hyers' result was extended by Aoki [1] and Rassias [18] for additive mappings and linear mappings, respectively. In 1994, another further generalization, the so-called generalized Hyer-Ulam stability, was obtained by Gavruta [6]. Later, the stability of several functional equations has been extensively discussed by many mathematicians and there are many interesting results concerning this problem (see [2,8,9,10,19,20] and references therein); also, some stability results of different functional equations and inequalities were studied and generalized [5,11,12,15,16,17,26] in various matrix normed spaces like matrix fuzzy normed spaces, matrix paranormed spaces and matrix non-Archimedean random normed spaces.
In 2017, Wang and Xu [25] introduced the following functional equation
| $ 2k[f(x+ky)+f(kx+y)]=k(1−s+k+ks+2k2)f(x+y)+k(1−s−3k+ks+2k2)f(x−y)+2kf(kx)+2k(s+k−ks−2k2)f(x)+2(1−k−s)f(ky)+2ksf(y) $ | (1.1) |
where $ s $ is a parameter, $ k > 1 $ and $ s\neq 1-2k $. It is easy to verify that $ f(x) = ax+bx^{2}(x\in {\mathbb{R}}) $ satisfies the functional Eq (1.1), where $ a, b $ are arbitrary constants. They considered the general solution of the functional Eq (1.1), and then determined the generalized Hyers-Ulam stability of the functional Eq (1.1) in quasi-Banach spaces by applying the direct method.
The main purpose of this paper is to employ the direct and fixed point methods to establish the Hyers-Ulam stability of the functional Eq (1.1) in matrix intuitionistic fuzzy normed spaces. The paper is organized as follows: In Sections $ 1 $ and $ 2 $, we present a brief introduction and introduce related basic definitions and preliminary results, respectively. In Section $ 3 $, we prove the Hyers-Ulam stability of the functional Eq (1.1) in matrix intuitionistic fuzzy normed spaces by applying the direct method. In Section $ 4 $, we prove the Hyers-Ulam stability of the functional Eq (1.1) in matrix intuitionistic fuzzy normed spaces by applying the fixed point method. Our results may be viewed as a continuation of the previous contribution of the authors in the setting of fuzzy stability (see [14,17]).
For the sake of completeness, in this section, we present some basic definitions and preliminary results, which will be useful to investigate the Hyers-Ulam stability results in matrix intuitionistic fuzzy normed spaces. The notions of continuous $ t $-norm and continuous $ t $-conorm can be found in [14,22]. Using these, an intuitionistic fuzzy normed space (for short, IFNS) is defined as follows:
Definition 2.1. ([14,21]) The five-tuple $ (X, \mu, \nu, \ast, \diamond) $ is said to be an IFNS if $ X $ is a vector space, $ \ast $ is a continuous $ t $-norm, $ \diamond $ is a continuous $ t $-conorm, and $ \mu, \nu $ are fuzzy sets on $ X \times (0, \infty) $ satisfy the following conditions. For every $ x, y\in X $ and $ s, t > 0 $,
(i) $ \mu(x, t)+\nu(x, t)\leq 1 $;
(ii) $ \mu(x, t) > 0 $, (iii) $ \mu(x, t) = 1 $ if and only if $ x = 0 $;
(iii) $ \mu(\alpha x, t) = \mu(x, \frac{t}{|\alpha|}) $ for each $ \alpha\neq 0 $, (v) $ \mu(x, t)\ast \mu(y, s)\leq \mu(x+y, t+s) $;
(iv) $ \mu(x, \cdot): (0, \infty)\to [0, 1] $ is continuous;
(v) $ \lim_{t\to \infty}\mu(x, t) = 1 $ and $ \lim_{t\to 0}\mu(x, t) = 0 $;
(vi) $ \nu(x, t) < 1 $, (ix) $ \nu(x, t) = 0 $ if and only if $ x = 0 $;
(vii) $ \nu(\alpha x, t) = \nu(x, \frac{t}{|\alpha|}) $ for each $ \alpha\neq 0 $, (xi) $ \nu(x, t)\diamond \; \nu(y, s)\geq \nu(x+y, t+s) $;
(xiii) $ \nu(x, \cdot): (0, \infty)\to [0, 1] $ is continuous;
(ix) $ \lim_{t\to \infty}\nu(x, t) = 0 $ and $ \lim_{t\to 0}\nu(x, t) = 1 $.
In this case, $ (\mu, \nu) $ is called an intuitionistic fuzzy norm.
The following concepts of convergence and Cauchy sequences are considered in [14,21]:
Let $ (X, \mu, \nu, \ast, \diamond) $ be an IFNS. Then, a sequence $ \{x_{k}\} $ is said to be intuitionistic fuzzy convergent to $ x\in X $ if for every $ \varepsilon > 0 $ and $ t > 0 $, there exists $ k_{0}\in \mathbb{N} $ such that
| $ \mu(x_{k}-x,t) > 1-\varepsilon $ |
and
| $ \nu(x_{k}-x,t) < \varepsilon $ |
for all $ k\geq k_{0} $. In this case we write
| $ (\mu,\nu)-\lim x_{k} = x. $ |
The sequence $ \{x_{k}\} $ is said to be an intuitionistic fuzzy Cauchy sequence if for every $ \varepsilon > 0 $ and $ t > 0 $, there exists $ k_{0}\in \mathbb{N} $ such that
| $ \mu(x_{k}-x_{\ell},t) > 1-\varepsilon $ |
and
| $ \nu(x_{k}-x_{\ell},t) < \varepsilon $ |
for all $ k, \ell\geq k_{0} $. $ (X, \mu, \nu, \ast, \diamond) $ is said to be complete if every intuitionistic fuzzy Cauchy sequence in $ (X, \mu, \nu, \ast, \diamond) $ is intuitionistic fuzzy convergent in $ (X, \mu, \nu, \ast, \diamond) $.
Following [11,12], we will also use the following notations: The set of all $ m\times n $-matrices in $ X $ will be denoted by $ M_{m, n}(X) $. When $ m = n $, the matrix $ M_{m, n}(X) $ will be written as $ M_{n}(X) $. The symbols $ e_{j}\in M_{1, n}({\mathbb{C}}) $ will denote the row vector whose $ j $th component is $ 1 $ and the other components are $ 0 $. Similarly, $ E_{ij}\in M_{n}({\mathbb{C}}) $ will denote the $ n \times n $ matrix whose $ (i, j) $-component is $ 1 $ and the other components are $ 0 $. The $ n \times n $ matrix whose $ (i, j) $-component is $ x $ and the other components are $ 0 $ will be denoted by $ E_{ij}\otimes x \in M_{n}(X) $.
Let $ (X, \|\cdot\|) $ be a normed space. Note that $ (X, \{\|\cdot\|_{n}\}) $ is a matrix normed space if and only if $ (M_{n}(X), \|\cdot\|_{n}) $ is a normed space for each positive integer $ n $ and
| $ \|AxB\|_{k}\leq \|A\|\|B\|\|x\|_{n} $ |
holds for $ A\in M_{k, n} $, $ x = [x_{ij}]\in M_{n}(X) $ and $ B\in M_{n, k} $, and that $ (X, \{\|\cdot\|_{n}\}) $ is a matrix Banach space if and only if $ X $ is a Banach space and $ (X, \{\|\cdot\|_{n}\}) $ is a matrix normed space.
Following [23], we introduce the concept of a matrix intuitionistic fuzzy normed space as follows:
Definition 2.2. ([23]) Let $ (X, \mu, \nu, \ast, \diamond) $ be an intuitionistic fuzzy normed space, and the symbol $ \theta $ for a rectangular matrix of zero elements over $ X $. Then:
(1) $ (X, \{\mu_{n}\}, \{\nu_{n}\}, \ast, \diamond) $ is called a matrix intuitionistic fuzzy normed space (briefly, MIFNS) if for each positive integer $ n $, $ (M_{n}(X), \mu_{n}, \nu_{n}, \ast, \diamond) $ is an intuitionistic fuzzy normed space, $ \mu_{n} $ and $ \nu_{n} $ satisfy the following conditions:
(i) $ \mu_{n+m}(\theta+x, t) = \mu_{n}(x, t), \nu_{n+m}(\theta+x, t) = \nu_{n}(x, t) $ for all $ t > 0 $, $ x = [x_{ij}]\in M_{n}(X) $, $ \theta\in M_{n}(X) $;
(ii) $ \mu_{k}(AxB, t)\geq \mu_{n}(x, \frac{t}{\|A\|\cdot\|B\|}) $, $ \nu_{k}(AxB, t)\leq \nu_{n}(x, \frac{t}{\|A\|\cdot\|B\|}) $ for all $ t > 0 $, $ A\in M_{k, n}({\mathbb{R}}) $, $ x = [x_{ij}]\in M_{n}(X) $ and $ B\in M_{n, k}({\mathbb{R}}) $ with $ \|A\|\cdot\|B\|\neq 0 $.
(2) $ (X, \{\mu_{n}\}, \{\nu_{n}\}, \ast, \diamond) $ is called a matrix intuitionistic fuzzy Banach space if $ (X, \mu, \nu, \ast, \diamond) $ is an intuitionistic fuzzy Banach space and $ (X, \{\mu_{n}\}, \{\nu_{n}\}, \ast, \diamond) $ is a matrix intuitionistic fuzzy normed space.
The following Lemma 2.3 was found in [23].
Lemma 2.3. ([23]) Let $ (X, \{\mu_{n}\}, \{\nu_{n}\}, \ast, \diamond) $ be a matrix intuitionistic fuzzy normed space. Then,
(1) $ \mu_{n}({E_{kl}\otimes x}, t) = \mu(x, t) $, $ \nu_{n}({E_{kl}\otimes x}, t) = \nu(x, t) $ for all $ t > 0 $ and $ x\in X $.
(2) For all $ [x_{ij}]\in M_{n}(X) $ and $ t = \sum\limits^{n}_{i, j = 1}t_{ij} > 0 $,
| $ μ(xkl,t)≥μn([xij],t)≥min{μ(xij,tij):i,j=1,2,…,n},μ(xkl,t)≥μn([xij],t)≥min{μ(xij,tn2):i,j=1,2,…,n}, $ |
and
| $ ν(xkl,t)≤νn([xij],t)≤max{ν(xij,tij):i,j=1,2,…,n},ν(xkl,t)≤νn([xij],t)≤max{ν(xij,tn2):i,j=1,2,…,n}. $ |
(3) $ \lim\limits_{m\to\infty}x_{m} = x $ if and only if $ \lim\limits_{m\to\infty}x_{ijm} = x_{ij} $ for $ x_{m} = [x_{ijm}], x = [x_{ij}]\in M_{n}(X) $.
For explicit later use, we also recall the following Lemma 2.4 is due to Diaz and Margolis [4], which will play an important role in proving our stability results in this paper.
Lemma 2.4. (The fixed point alternative theorem [4]) Let $ (E, d) $ be a complete generalized metric space and $ J $: $ E\to E $ be a strictly contractive mapping with Lipschitz constant $ L < 1 $. Then for each fixed element $ x\in E $, either
| $ d(Jnx,Jn+1x)=∞,∀n≥0, $ |
or
| $ d(Jnx,Jn+1x)<∞,∀n≥n0, $ |
for some natural number $ n_{0} $. Moreover, if the second alternative holds then:
$ (i) $ The sequence $ \{J^{n}x\} $ is convergent to a fixed point $ y^{*} $ of $ J $.
$ (ii) \; y^{*} $ is the unique fixed point of $ J $ in the set $ E^{\ast}: = \{y\in E\mid d(J^{n_{0}}x, y) < +\infty\} $ and $ d(y, y^{*})\leq \frac{1}{1-L}d(y, Jy), \; \; \forall x, y \in E^{\ast} $.
From now on, let $ (X, \{\mu_{n}\}, \{\nu_{n}\}, \ast, \diamond) $ be a matrix intuitionistic fuzzy normed space and $ (Y, \{\mu_{n}\}, \{\nu_{n}\}, \ast, \diamond) $ be a matrix intuitionistic fuzzy Banach space. In this section, we will prove the Hyers-Ulam stability of the functional Eq (1.1) in matrix intuitionistic fuzzy normed spaces by using the direct method. For the sake of convenience, given mapping $ f $: $ X\to Y $, we define the difference operators $ Df $: $ X^{2}\to Y $ and $ Df_{n} $: $ M_{n}(X^{2})\to M_{n}(Y) $ of the functional Eq (1.1) by
| $ Df(a,b):=2k[f(a+kb)+f(ka+b)]−k(1−s+k+ks+2k2)f(a+b)−k(1−s−3k+ks+2k2)f(a−b)−2kf(ka)−2k(s+k−ks−2k2)f(a)−2(1−k−s)f(kb)−2ksf(b),Dfn([xij],[yij]):=2k[fn([xij]+k[yij])+fn(k[xij]+[yij])]−k(1−s+k+ks+2k2)fn([xij]+[yij])−k(1−s−3k+ks+2k2)fn([xij]−[yij])−2kfn(k[xij])−2k(s+k−ks−2k2)fn([xij])−2(1−k−s)fn(k[yij])−2ksfn([yij]) $ |
for all $ a, b\in X $ and all $ x = [x_{ij}], y = [y_{ij}]\in M_{n}(X) $.
We start with the following lemmas which will be used in this paper.
Lemma 3.1. ([25]) Let $ V $ and $ W $ be real vector spaces. If an odd mapping $ f $: $ V\to W $ satisfies the functional Eq (1.1), then $ f $ is additive.
Lemma 3.2. ([25]) Let $ V $ and $ W $ be real vector spaces. If an even mapping $ f $: $ V\to W $ satisfies the functional Eq (1.1), then $ f $ is quadratic.
Theorem 3.3. Let $ \varphi_{o} $: $ X^{2}\to [0, \infty) $ be a function such that for some real number $ \alpha $ with $ 0 < \alpha < k $,
| $ φo(ka,kb)=αφo(a,b) $ | (3.1) |
for all $ a, b\in X $. Suppose that an odd mapping $ f $: $ X\to Y $ satisfies the inequality
| $ {μn(Dfn([xij],[yij]),t)≥tt+∑ni,j=1φo(xij,yij),νn(Dfn([xij],[yij]),t)≤∑ni,j=1φo(xij,yij)t+∑ni,j=1φo(xij,yij) $ | (3.2) |
for all $ x = [x_{ij}], y = [y_{ij}]\in M_{n}(X) $ and all $ t > 0 $. Then there exists a unique additive mapping $ A $: $ X\to Y $ such that
| $ {μn(fn([xij])−An([xij]),t)≥(k−α)(2k+s−1)t(k−α)(2k+s−1)t+n2∑ni,j=1φo(0,xij),νn(fn([xij])−An([xij]),t)≤n2∑ni,j=1φo(0,xij)(k−α)(2k+s−1)t+n2∑ni,j=1φo(0,xij) $ | (3.3) |
for all $ x = [x_{ij}]\in M_{n}(X) $ and all $ t > 0 $.
Proof. When $ n = 1 $, (3.2) is equivalent to
| $ μ(Df(a,b),t)≥tt+φo(a,b)andν(Df(a,b),t)≤φo(a,b)t+φo(a,b) $ | (3.4) |
for all $ a, b\in X $ and all $ t > 0 $. Putting $ a = 0 $ in (3.4), we have
| $ {μ(2(2k+s−1)f(kb)−2(2k+s−1)kf(b),t)≥tt+φo(0,b),ν(2(2k+s−1)f(kb)−2(2k+s−1)kf(b),t)≤φo(0,b)t+φo(0,b) $ | (3.5) |
for all $ b\in X $ and all $ t > 0 $. Replacing $ a $ by $ k^{p}a $ in (3.5) and using (3.1), we get
| $ {μ(f(kp+1a)kp+1−f(kpa)kp,t2k(2k+s−1)kp)≥tt+αpφo(0,a),ν(f(kp+1a)kp+1−f(kpa)kp,t2k(2k+s−1)kp)≤αpφo(0,a)t+αpφo(0,a) $ | (3.6) |
for all $ a\in X $ and all $ t > 0 $. It follows from (3.6) that
| $ {μ(f(kp+1a)kp+1−f(kpa)kp,αpt2k(2k+s−1)kp)≥tt+φo(0,a),ν(f(kp+1a)kp+1−f(kpa)kp,αpt2k(2k+s−1)kp)≤φo(0,a)t+φo(0,a) $ | (3.7) |
for all $ a\in X $ and all $ t > 0 $. It follows from
| $ \frac{f(k^{p}a)}{k^{p}}-f(a) = \sum\limits^{p-1}_{\ell = 0}(\frac{f(k^{\ell+1}a)}{k^{\ell+1}}-\frac{f(k^{\ell}a)}{k^{\ell}}) $ |
and (3.7) that
| $ {μ(f(kpa)kp−f(a),∑p−1ℓ=0αℓt2k(2k+s−1)kℓ)≥∏p−1ℓ=0μ(f(kℓ+1a)kℓ+1−f(kℓa)kℓ,αℓt2k(2k+s−1)kℓ)≥tt+φo(0,a),ν(f(kpa)kp−f(a),∑p−1ℓ=0αℓt2k(2k+s−1)kℓ)≤∐p−1ℓ=0ν(f(kℓ+1a)kℓ+1−f(kℓa)kℓ,αℓt2k(2k+s−1)kℓ)≤φo(0,a)t+φo(0,a) $ | (3.8) |
for all $ a\in X $ and all $ t > 0 $, where
| $ \prod\limits^{p}_{j = 0}a_{j} = a_{1} \ast a_{2}\ast\cdots\ast a_{p},\ \ \ \coprod\limits^{p}_{j = 0}a_{j} = a_{1} \diamond a_{2}\diamond\cdots\diamond a_{p}. $ |
By replacing $ a $ with $ k^{q}a $ in (3.8), we have
| $ {μ(f(kp+qa)kp+q−f(kqa)kq,∑p−1ℓ=0αℓt2k(2k+s−1)kℓ+q)≥tt+αqφo(0,a),ν(f(kp+qa)kp+q−f(kqa)kq,∑p−1ℓ=0αℓt2k(2k+s−1)kℓ+q)≤αqφo(0,a)t+αqφo(0,a) $ | (3.9) |
for all $ a\in X $, $ t > 0 $, $ p > 0 $ and $ q > 0 $. Thus
| $ {μ(f(kp+qa)kp+q−f(kqa)kq,∑p+q−1ℓ=qαℓt2k(2k+s−1)kℓ)≥tt+φo(0,a),ν(f(kp+qa)kp+q−f(kqa)kq,∑p+q−1ℓ=qαℓt2k(2k+s−1)kℓ)≤φo(0,a)t+φo(0,a) $ | (3.10) |
for all $ a\in X $, $ t > 0 $, $ p > 0 $ and $ q > 0 $. Hence
| $ {μ(f(kp+qa)kp+q−f(kqa)kq,t)≥tt+∑p+q−1ℓ=qαℓ2k(2k+s−1)kℓφo(0,a),ν(f(kp+qa)kp+q−f(kqa)kq,t)≤∑p+q−1ℓ=qαℓ2k(2k+s−1)kℓφo(0,a)t+∑p+q−1ℓ=qαℓ2k(2k+s−1)kℓφo(0,a) $ | (3.11) |
for all $ a\in X $, $ t > 0 $, $ p > 0 $ and $ q > 0 $. Since $ 0 < \alpha < k $ and
| $ \sum\limits^{\infty}_{\ell = 0} \frac{\alpha^{\ell}}{2k(2k+s-1)k^{\ell}} < \infty, $ |
the Cauchy criterion for convergence in IFNS shows that $ \{\frac{f(k^{p}a)}{k^{p}}\} $ is a Cauchy sequence in $ (Y, \mu, \nu, \ast, \diamond) $. Since $ (Y, \mu, \nu, \ast, \diamond) $ is an intuitionistic fuzzy Banach space, this sequence converges to some point $ A(a)\in Y $. So one can define the mapping $ A $: $ X\to Y $ such that
| $ A(a): = (\mu,\nu)-\lim\limits_{p\to\infty}\frac{f(k^{p}a)}{k^{p}}. $ |
Moreover, if we put $ q = 0 $ in (3.11), we get
| $ {μ(f(kpa)kp−f(a),t)≥tt+∑p−1ℓ=0αℓ2k(2k+s−1)kℓφo(0,a),ν(f(kpa)kp−f(a),t)≤∑p−1ℓ=0αℓ2k(2k+s−1)kℓφo(0,a)t+∑p−1ℓ=0αℓ2k(2k+s−1)kℓφo(0,a) $ | (3.12) |
for all $ a\in X $, $ t > 0 $ and $ p > 0 $. Thus, we obtain
| $ {μ(f(a)−A(a),t)≥μ(f(a)−f(kpa)kp,t2)∗μ(f(kpa)kp−A(a),t2)≥tt+∑p−1ℓ=0αℓk(2k+s−1)kℓφo(0,a),ν(f(a)−A(a),t)≤ν(f(a)−f(kpa)kp,t2)∗ν(f(kpa)kp−A(a),t2)≤∑p−1ℓ=0αℓk(2k+s−1)kℓφo(0,a)t+∑p−1ℓ=0αℓk(2k+s−1)kℓφo(0,a) $ | (3.13) |
for every $ a\in X $, $ t > 0 $ and large $ p $. Taking the limit as $ p\to\infty $ and using the definition of IFNS, we get
| $ {μ(f(a)−A(a),t)≥(k−α)(2k+s−1)t(k−α)(2k+s−1)t+φo(0,a),ν(f(a)−A(a),t)≤φo(0,a)(k−α)(2k+s−1)t+φo(0,a). $ | (3.14) |
Replacing $ a $ and $ b $ by $ k^{p}a $ and $ k^{p}b $ in (3.4), respectively, and using (3.1), we obtain
| $ μ(1kpDf(kpa,kpb),t)≥tt+(αk)pφo(a,b)andν(1kpDf(kpa,kpb),t)≤(αk)pφo(a,b)t+(αk)pφo(a,b) $ | (3.15) |
for all $ a, b\in X $ and all $ t > 0 $. Letting $ p\to \infty $ in (3.15), we obtain
| $ μ(DA(a,b),t)=1andν(DA(a,b),t)=0 $ | (3.16) |
for all $ a, b\in X $ and all $ t > 0 $. This means that $ A $ satisfies the functional Eq (1.1). Since $ f $: $ X\to Y $ is an odd mapping, and using the definition $ A $, we have $ A(-a) = -A(a) $ for all $ a\in X $. Thus by Lemma 3.1, the mapping $ A $: $ X\to Y $ is additive. To prove the uniqueness of $ A $, let $ A^{\prime} $: $ X\to Y $ be another additive mapping satisfying (3.14). Let $ n = 1 $. Then we have
| $ {μ(A(a)−A′(a),t)=μ(A(kpa)kp−A′(kpa)kp,t)≥μ(A(kpa)kp−f(kpa)kp,t2)∗μ(f(kpa)kp−A′(kpa)kp,t2)≥(k−α)(2k+s−1)t(k−α)(2k+s−1)t+2(αk)pφo(0,a),ν(A(a)−A′(a),t)=ν(A(kpa)kp−A′(kpa)kp,t)≤ν(A(kpa)kp−f(kpa)kp,t2)⋄ν(f(kpa)kp−A′(kpa)kp,t2)≤2(αk)pφo(0,a)(k−α)(2k+s−1)t+2(αk)pφo(0,a) $ | (3.17) |
for all $ a\in X $, $ t > 0 $ and $ p > 0 $. Letting $ p\to \infty $ in (3.17), we get
| $ μ(A(a)−A′(a),t)=1andν(A(a)−A′(a),t)=0 $ |
for all $ a\in X $ and $ t > 0 $. Hence we get $ A(a) = A^{\prime}(a) $ for all $ a\in X $. Thus the mapping $ A $: $ X\to Y $ is a unique additive mapping.
By Lemma 2.3 and (3.14), we get
| $ {μn(fn([xij])−An([xij]),t)≥min{μ(f(xij)−A(xij),tn2):i,j=1,…,n} ≥(k−α)(2k+s−1)t(k−α)(2k+s−1)t+n2∑ni,j=1φo(0,xij),νn(fn([xij])−An([xij]),t)≤max{ν(f(xij)−A(xij),tn2):i,j=1,…,n} ≤n2∑ni,j=1φo(0,xij)(k−α)(2k+s−1)t+n2∑ni,j=1φo(0,xij) $ |
for all $ x = [x_{ij}]\in M_{n}(X) $ and all $ t > 0 $. Thus $ A $: $ X\to Y $ is a unique additive mapping satisfying (3.3), as desired. This completes the proof of the theorem.
Theorem 3.4. Let $ \varphi_{e} $: $ X^{2}\to [0, \infty) $ be a function such that for some real number $ \alpha $ with $ 0 < \alpha < k^{2} $,
| $ φe(ka,kb)=αφe(a,b) $ | (3.18) |
for all $ a, b\in X $. Suppose that an even mapping $ f $: $ X\to Y $ with $ f(0) = 0 $ satisfies the inequality
| $ {μn(Dfn([xij],[yij]),t)≥tt+∑ni,j=1φe(xij,yij),νn(Dfn([xij],[yij]),t)≤∑ni,j=1φe(xij,yij)t+∑ni,j=1φe(xij,yij) $ | (3.19) |
for all $ x = [x_{ij}], \ y = [y_{ij}]\in M_{n}(X) $ and all $ t > 0 $. Then there exists a unique quadratic mapping $ Q $: $ X\to Y $ such that
| $ {μn(fn([xij])−Qn([xij]),t)≥(k2−α)(2k+s−1)t(k2−α)(2k+s−1)t+n2∑ni,j=1φe(0,xij),νn(fn([xij])−Qn([xij]),t)≤n2∑ni,j=1φe(0,xij)(k2−α)(2k+s−1)t+n2∑ni,j=1φe(0,xij) $ | (3.20) |
for all $ x = [x_{ij}]\in M_{n}(X) $ and all $ t > 0 $.
Proof. When $ n = 1 $, (3.19) is equivalent to
| $ μ(Df(a,b),t)≥tt+φe(a,b)andν(Df(a,b),t)≤φe(a,b)t+φe(a,b) $ | (3.21) |
for all $ a, b\in X $ and all $ t > 0 $. Letting $ a = 0 $ in (3.21), we obtain
| $ {μ(2(2k+s−1)f(kb)−2(2k+s−1)k2f(b),t)≥tt+φe(0,b),ν(2(2k+s−1)f(kb)−2(2k+s−1)k2f(b),t)≤φe(0,b)t+φe(0,b) $ | (3.22) |
for all $ b\in X $ and all $ t > 0 $. Replacing $ a $ by $ k^{p}a $ in (3.22) and using (3.18), we get
| $ {μ(f(kp+1a)k2(p+1)−f(kpa)k2p,t2k2(2k+s−1)k2p)≥tt+αpφe(0,a),ν(f(kp+1a)k2(p+1)−f(kpa)k2p,t2k2(2k+s−1)k2p)≤αpφe(0,a)t+αpφe(0,a) $ | (3.23) |
for all $ a\in X $ and all $ t > 0 $. It follows from (3.23) that
| $ {μ(f(kp+1a)k2(p+1)−f(kpa)k2p,αpt2k2(2k+s−1)k2p)≥tt+φe(0,a),ν(f(kp+1a)k2(p+1)−f(kpa)k2p,αpt2k2(2k+s−1)k2p)≤φe(0,a)t+φe(0,a) $ | (3.24) |
for all $ a\in X $ and all $ t > 0 $. It follows from
| $ \frac{f(k^{p}a)}{k^{2p}}-f(a) = \sum\limits^{p-1}_{\ell = 0}(\frac{f(k^{\ell+1}a)}{k^{2(\ell+1)}}-\frac{f(k^{\ell}a)}{k^{2\ell}}) $ |
and (3.24) that
| $ {μ(f(kpa)k2p−f(a),∑p−1ℓ=0αℓt2k2(2k+s−1)k2ℓ)≥∏p−1ℓ=0μ(f(kℓ+1a)k2(ℓ+1)−f(kℓa)k2ℓ,αℓt2k2(2k+s−1)k2ℓ)≥tt+φe(0,a),ν(f(kpa)k2p−f(a),∑p−1ℓ=0αℓt2k2(2k+s−1)k2ℓ)≤∐p−1ℓ=0ν(f(kℓ+1a)k2(ℓ+1)−f(kℓa)k2ℓ,αℓt2k2(2k+s−1)k2ℓ)≤φe(0,a)t+φe(0,a) $ | (3.25) |
for all $ a\in X $ and all $ t > 0 $, where
| $ \prod\limits^{p}_{j = 0}a_{j} = a_{1} \ast a_{2}\ast\cdots\ast a_{p},\ \ \ \coprod\limits^{p}_{j = 0}a_{j} = a_{1} \diamond a_{2}\diamond\cdots\diamond a_{p}. $ |
By replacing $ a $ with $ k^{q}a $ in (3.25), we have
| $ {μ(f(kp+qa)k2(p+q)−f(kqa)k2q,∑p−1ℓ=0αℓt2k2(2k+s−1)k2(ℓ+q))≥tt+αqφe(0,a),ν(f(kp+qa)k2(p+q)−f(kqa)k2q,∑p−1ℓ=0αℓt2k2(2k+s−1)k2(ℓ+q))≤αqφe(0,a)t+αqφe(0,a) $ | (3.26) |
for all $ a\in X $, $ t > 0 $, $ p > 0 $ and $ q > 0 $. Thus
| $ {μ(f(kp+qa)k2(p+q)−f(kqa)k2q,∑p+q−1ℓ=qαℓt2k2(2k+s−1)k2ℓ)≥tt+φe(0,a),ν(f(kp+qa)k2(p+q)−f(kqa)k2q,∑p+q−1ℓ=qαℓt2k2(2k+s−1)k2ℓ)≤φe(0,a)t+φe(0,a) $ | (3.27) |
for all $ a\in X $, $ t > 0 $, $ p > 0 $ and $ q > 0 $. Hence
| $ {μ(f(kp+qa)k2(p+q)−f(kqa)k2q,t)≥tt+∑p+q−1ℓ=qαℓ2k2(2k+s−1)k2ℓφe(0,a),ν(f(kp+qa)k2(p+q)−f(kqa)k2q,t)≤∑p+q−1ℓ=qαℓ2k2(2k+s−1)k2ℓφe(0,a)t+∑p+q−1ℓ=qαℓ2k2(2k+s−1)k2ℓφe(0,a) $ | (3.28) |
for all $ a\in X $, $ t > 0 $, $ p > 0 $ and $ q > 0 $. Since $ 0 < \alpha < k^{2} $ and
| $ \sum\limits^{\infty}_{\ell = 0} \frac{\alpha^{\ell}}{2k^{2}(2k+s-1)k^{2\ell}} < \infty, $ |
the Cauchy criterion for convergence in IFNS shows that $ \{\frac{f(k^{p}a)}{k^{2p}}\} $ is a Cauchy sequence in $ (Y, \mu, \nu, \ast, \diamond) $. Since $ (Y, \mu, \nu, \ast, \diamond) $ is an intuitionistic fuzzy Banach space, this sequence converges to some point $ Q(a)\in Y $. So one can define the mapping $ Q $: $ X\to Y $ such that
| $ Q(a): = (\mu,\nu)-\lim\limits_{p\to\infty}\frac{f(k^{p}a)}{k^{2p}}. $ |
Moreover, if we put $ q = 0 $ in (3.28), we get
| $ {μ(f(kpa)k2p−f(a),t)≥tt+∑p−1ℓ=0αℓ2k2(2k+s−1)k2ℓφe(0,a),ν(f(kpa)k2p−f(a),t)≤∑p−1ℓ=0αℓ2k2(2k+s−1)k2ℓφe(0,a)t+∑p−1ℓ=0αℓ2k2(2k+s−1)k2ℓφe(0,a) $ | (3.29) |
for all $ a\in X $, $ t > 0 $ and $ p > 0 $. Thus, we obtain
| $ {μ(f(a)−Q(a),t)≥μ(f(a)−f(kpa)k2p,t2)∗μ(f(kpa)k2p−Q(a),t2)≥tt+∑p−1ℓ=0αℓk2(2k+s−1)k2ℓφe(0,a),ν(f(a)−Q(a),t)≤ν(f(a)−f(kpa)k2p,t2)∗ν(f(kpa)k2p−Q(a),t2)≤∑p−1ℓ=0αℓk2(2k+s−1)k2ℓφe(0,a)t+∑p−1ℓ=0αℓk2(2k+s−1)k2ℓφe(0,a) $ | (3.30) |
for every $ a\in X $, $ t > 0 $ and large $ p $. Taking the limit as $ p\to\infty $ and using the definition of IFNS, we get
| $ {μ(f(a)−Q(a),t)≥(k2−α)(2k+s−1)t(k2−α)(2k+s−1)t+φe(0,a),ν(f(a)−Q(a),t)≤φe(0,a)(k2−α)(2k+s−1)t+φe(0,a). $ | (3.31) |
Replacing $ a $ and $ b $ by $ k^{p}a $ and $ k^{p}b $ in (3.21), respectively, and using (3.18), we obtain
| $ μ(1k2pDf(kpa,kpb),t)≥tt+(αk2)pφe(a,b),ν(1k2pDf(kpa,kpb),t)≤(αk2)pφe(a,b)t+(αk2)pφe(a,b) $ | (3.32) |
for all $ a, b\in X $ and all $ t > 0 $. Letting $ p\to \infty $ in (3.32), we obtain
| $ μ(DQ(a,b),t)=1andν(DQ(a,b),t)=0 $ | (3.33) |
for all $ a, b\in X $ and all $ t > 0 $. This means that $ Q $ satisfies the functional Eq (1.1). Since $ f $: $ X\to Y $ is an even mapping, and using the definition $ Q $, we have $ Q(-a) = -Q(a) $ for all $ a\in X $. Thus by Lemma 3.2, the mapping $ Q $: $ X\to Y $ is quadratic. To prove the uniqueness of $ Q $, let $ Q^{\prime} $: $ X\to Y $ be another quadratic mapping satisfying (3.31). Let $ n = 1 $. Then we have
| $ {μ(Q(a)−Q′(a),t)=μ(Q(kpa)k2p−Q′(kpa)k2p,t) ≥μ(Q(kpa)k2p−f(kpa)k2p,t2)∗μ(f(kpa)k2p−Q′(kpa)k2p,t2) ≥(k2−α)(2k+s−1)t(k2−α)(2k+s−1)t+2(αk2)pφe(0,a),ν(Q(a)−Q′(a),t)=ν(Q(kpa)k2p−Q′(kpa)k2p,t) ≤ν(Q(kpa)k2p−f(kpa)k2p,t2)⋄ν(f(kpa)kp−Q′(kpa)k2p,t2) ≤2(αk2)pφe(0,a)(k2−α)(2k+s−1)t+2(αk2)pφe(0,a) $ | (3.34) |
for all $ a\in X $, $ t > 0 $ and $ p > 0 $. Letting $ p\to \infty $ in (3.34), we get
| $ μ(Q(a)−Q′(a),t)=1andν(Q(a)−Q′(a),t)=0 $ |
for all $ a\in X $ and $ t > 0 $. Hence we get $ Q(a) = Q^{\prime}(a) $ for all $ a\in X $. Thus the mapping $ Q $: $ X\to Y $ is a unique quadratic mapping.
By Lemma 2.3 and (3.31), we get
| $ {μn(fn([xij])−Qn([xij]),t)≥min{μ(f(xij)−Q(xij),tn2):i,j=1,…,n}≥(k2−α)(2k+s−1)t(k2−α)(2k+s−1)t+n2∑ni,j=1φe(0,xij),νn(fn([xij])−Qn([xij]),t)≤max{ν(f(xij)−Q(xij),tn2):i,j=1,…,n}≤n2∑ni,j=1φe(0,xij)(k2−α)(2k+s−1)t+n2∑ni,j=1φe(0,xij) $ |
for all $ x = [x_{ij}]\in M_{n}(X) $ and all $ t > 0 $. Thus $ Q $: $ X\to Y $ is a unique quadratic mapping satisfying (3.20), as desired. This completes the proof of the theorem.
Theorem 3.5. Let $ \varphi $: $ X^{2}\to [0, \infty) $ be a function such that for some real number $ \alpha $ with $ 0 < \alpha < k $,
| $ φ(ka,kb)=αφ(a,b) $ | (3.35) |
for all $ a, b\in X $. Suppose that a mapping $ f $: $ X\to Y $ with $ f(0) = 0 $ satisfies the inequality
| $ {μn(Dfn([xij],[yij]),t)≥tt+∑ni,j=1φ(xij,yij),νn(Dfn([xij],[yij]),t)≤∑ni,j=1φ(xij,yij)t+∑ni,j=1φ(xij,yij) $ | (3.36) |
for all $ x = [x_{ij}], y = [y_{ij}]\in M_{n}(X) $ and all $ t > 0 $. Then there exist a unique quadratic mapping $ Q $: $ X\to Y $ and a unique additive mapping $ A $: $ X\to Y $ such that
| $ {μn(fn([xij])−Qn([xij])−An([xij]),t)≥(k−α)(2k+s−1)t(k−α)(2k+s−1)t+2n2∑ni,j=1˜φ(0,xij),νn(fn([xij])−Qn([xij])−An([xij]),t)≤2n2∑ni,j=1˜φ(0,xij)(k−α)(2k+s−1)t+2n2∑ni,j=1˜φ(0,xij) $ | (3.37) |
for all $ x = [x_{ij}]\in M_{n}(X) $ and all $ t > 0 $, $ \widetilde{\varphi}(a, b) = \varphi(a, b)+\varphi(-a, -b) $ for all $ a, b\in X $.
Proof. When $ n = 1 $, (3.36) is equivalent to
| $ μ(Df(a,b),t)≥tt+φ(a,b)andν(Df(a,b),t)≤φ(a,b)t+φ(a,b) $ | (3.38) |
for all $ a, b\in X $ and all $ t > 0 $. Let
| $ f_{e}(a) = \frac{f(a)+f(-a)}{2} $ |
for all all $ a\in X $. Then $ f_{e}(0) = 0, f_{e}(-a) = f_{e}(a) $. And we have
| $ {μ(Dfe(a,b),t)=μ(12Df(a,b)+12Df(−a,−b),t)=μ(Df(a,b)+Df(−a,−b),2t)≥μ(Df(a,b),t)∗μ(Df(−a,−b),t)≥min{μ(Df(a,b),t),μ(Df(−a,−b),t)}≥tt+˜φ(a,b),ν(Dfe(a,b),t)=ν(12Df(a,b)+12Df(−a,−b),t)=ν(Df(a,b)+Df(−a,−b),2t)≤ν(Df(a,b),t)⋄ν(Df(−a,−b),t)≤max{ν(Df(a,b),t),ν(Df(−a,−b),t)}≤˜φ(a,b)t+˜φ(a,b) $ | (3.39) |
for all $ a\in X $ and all $ t > 0 $. Let
| $ f_{o}(a) = \frac{f(a)-f(-a)}{2} $ |
for all all $ a\in X $. Then $ f_{0}(0) = 0, f_{o}(-a) = -f_{o}(a) $. And we obtain
| $ {μ(Dfo(a,b),t)=μ(12Df(a,b)−12Df(−a,−b),t)=μ(Df(a,b)−Df(−a,−b),2t)≥μ(Df(a,b),t)∗μ(Df(−a,−b),t)=min{μ(Df(a,b),t),μ(Df(−a,−b),t)}≥tt+˜φ(a,b),ν(Dfo(a,b),t)=ν(12Df(a,b)−12Df(−a,−b),t)=ν(Df(a,b)−Df(−a,−b),2t)≤ν(Df(a,b),t)⋄ν(Df(−a,−b),t)=max{ν(Df(a,b),t),ν(Df(−a,−b),t)}≤˜φ(a,b)t+˜φ(a,b) $ | (3.40) |
for all $ a\in X $ and all $ t > 0 $. It follows that the definition of $ \widetilde{\varphi} $ that $ \widetilde{\varphi}(ka, kb) = \alpha\widetilde{\varphi}(a, b) $ for all $ a, b\in X $. It is easy to check that the condition of Theorems 3.3 and 3.4 are satisfying. Then applying the proofs of Theorems 3.3 and 3.4, we know that there exists a unique quadratic mapping $ Q $: $ X\to Y $ and a unique additive mapping $ A $: $ X\to Y $ satisfying
| $ {μ(fe(a)−Q(a),t)≥(k2−α)(2k+s−1)t(k2−α)(2k+s−1)t+˜φ(0,a),ν(fe(a)−Q(a),t)≤˜φ(0,a)(k2−α)(2k+s−1)t+˜φ(0,a) $ | (3.41) |
and
| $ {μ(fo(a)−A(a),t)≥(k−α)(2k+s−1)t(k−α)(2k+s−1)t+˜φ(0,a),ν(fo(a)−A(a),t)≤˜φ(0,a)(k−α)(2k+s−1)t+˜φ(0,a) $ | (3.42) |
for all $ a\in X $ and all $ t > 0 $. Therefore
| $ {μ(f(a)−Q(a)−A(a),t)=μ(fe(a)−Q(a)+fo(a)−A(a),t)≥μ(fe(a)−Q(a),t2)∗μ(fo(a)−A(a),t2)=min{μ(fe(a)−Q(a),t2),μ(fo(a)−A(a),t2)}≥min{(k2−α)(2k+s−1)t(k2−α)(2k+s−1)t+2˜φ(0,a),(k−α)(2k+s−1)t(k−α)(2k+s−1)t+2˜φ(0,a)}=(k−α)(2k+s−1)t(k−α)(2k+s−1)t+2˜φ(0,a),ν(f(a)−Q(a)−A(a),t)=ν(fe(a)−Q(a)+fo(a)−A(a),t)≤ν(fe(a)−Q(a),t2)⋄ν(fo(a)−A(a),t2)=max{ν(fe(a)−Q(a),t2),ν(fo(a)−A(a),t2)}≤max{2˜φ(0,a)(k2−α)(2k+s−1)t+2˜φ(0,a),2˜φ(0,a)(k−α)(2k+s−1)t+2˜φ(0,a)}=2˜φ(0,a)(k−α)(2k+s−1)t+2˜φ(0,a). $ | (3.43) |
By Lemma 2.3 and (3.43), we have
| $ {μn(fn([xij])−Qn([xij])−An([xij]),t)≥min{μ(f(xij)−Q(xij)−A(xij),tn2):i,j=1,…,n}≥(k−α)(2k+s−1)t(k−α)(2k+s−1)t+2n2∑ni,j=1˜φ(0,xij),νn(fn([xij])−Qn([xij])−An([xij]),t)≤max{ν(f(xij)−Q(xij)−A(xij),tn2):i,j=1,…,n}≤2n2∑ni,j=1˜φ(0,xij)(k−α)(2k+s−1)t+2n2∑ni,j=1˜φ(0,xij) $ |
for all $ x = [x_{ij}]\in M_{n}(X) $ and all $ t > 0 $. Thus $ Q $: $ X\to Y $ is a unique quadratic mapping and a unique additive mapping $ A $: $ X\to Y $ satisfying (3.37), as desired. This completes the proof of the theorem.
Corollary 3.6. Let $ r, \theta $ be positive real numbers with $ r < 1 $. Suppose that a mapping $ f $: $ X\to Y $ with $ f(0) = 0 $ satisfies the inequality
| $ {μn(Dfn([xij],[yij]),t)≥tt+∑ni,j=1θ(‖xij‖r+‖yij‖r),νn(Dfn([xij],[yij]),t)≤∑ni,j=1θ(‖xij‖r+‖yij‖r)t+∑ni,j=1θ(‖xij‖r+‖yij‖r) $ | (3.44) |
for all $ x = [x_{ij}], y = [y_{ij}]\in M_{n}(X) $ and all $ t > 0 $. Then there exist a unique quadratic mapping $ Q $: $ X\to Y $ and a unique additive mapping $ A $: $ X\to Y $ such that
| $ {μn(fn([xij])−Qn([xij])−An([xij]),t)≥(k−kr)(2k+s−1)t(k−kr)(2k+s−1)t+4n2∑ni,j=1θ‖xij‖r,νn(fn([xij])−Qn([xij])−An([xij]),t)≤4n2∑ni,j=1θ‖xij‖r(k−kr)(2k+s−1)t+4n2∑ni,j=1θ‖xij‖r $ | (3.45) |
for all $ x = [x_{ij}]\in M_{n}(X) $ and all $ t > 0 $.
Proof. The proof follows from Theorem 3.5 by taking $ \varphi(a, b) = \theta(\|a\|^{r}+\|b\|^{r}) $ for all $ a, b\in X $, we obtain the desired result.
In this section, we will prove the Hyers-Ulam stability of the functional Eq (1.1) in matrix intuitionistic fuzzy normed spaces by applying the fixed point method.
Theorem 4.1. Let $ \varphi_{o} $: $ X^{2}\to [0, \infty) $ be a function such that for some real number $ \rho $ with $ 0 < \rho < 1 $ and
| $ φo(a,b)=ρkφo(ka,kb) $ | (4.1) |
for all $ a, b\in X $. Suppose that an odd mapping $ f $: $ X\to Y $ satisfies the inequality
| $ {μn(Dfn([xij],[yij]),t)≥tt+∑ni,j=1φo(xij,yij),νn(Dfn([xij],[yij]),t)≤∑ni,j=1φo(xij,yij)t+∑ni,j=1φo(xij,yij) $ | (4.2) |
for all $ x = [x_{ij}], y = [y_{ij}]\in M_{n}(X) $ and all $ t > 0 $. Then there exists a unique additive mapping $ A $: $ X\to Y $ such that
| $ {μn(fn([xij])−An([xij]),t)≥2k(2k+s−1)(1−ρ)t2k(2k+s−1)(1−ρ)t+ρn2∑ni,j=1φo(0,xij),νn(fn([xij])−An([xij]),t)≤ρn2∑ni,j=1φo(0,xij)2k(2k+s−1)(1−ρ)t+ρn2∑ni,j=1φo(0,xij) $ | (4.3) |
for all $ x = [x_{ij}]\in M_{n}(X) $ and all $ t > 0 $.
Proof. When $ n = 1 $, similar to the proof of Theorem 3.3, we have
| $ {μ(2(2k+s−1)f(ka)−2(2k+s−1)kf(a),t)≥tt+φo(0,a),ν(2(2k+s−1)f(ka)−2(2k+s−1)kf(a),t)≤φo(0,a)t+φo(0,a) $ | (4.4) |
for all $ a\in X $ and all $ t > 0 $.
Let $ S_{1} = \left\{g_{1}:X\to Y\right\} $, and introduce a generalized metric $ d_{1} $ on $ S_{1} $ as follows:
| $ d1(g1,h1):=inf{λ∈R+|{μ(g1(a)−h1(a),λt)≥tt+φo(0,a),ν(g1(a)−h1(a),λt)≤φo(0,a)t+φo(0,a),∀a∈X,∀t>0}. $ |
It is easy to prove that $ (S_{1}, d_{1}) $ is a complete generalized metric space ([3,13]). Now, we define the mapping $ {\mathcal {J}}_{1} $: $ S_{1}\to S_{1} $ by
| $ J1g1(a):=kg1(ak),for allg1∈S1anda∈X. $ | (4.5) |
Let $ g_{1}, h_{1}\in S_{1} $ and let $ \lambda\in {\mathbb{R_{+}}} $ be an arbitrary constant with $ d_{1}(g_{1}, h_{1})\leq \lambda $. From the definition of $ d_{1} $, we get
| $ {μ(g1(a)−h1(a),λt)≥tt+φo(0,a),ν(g1(a)−h1(a),λt)≤φo(0,a)t+φo(0,a) $ |
for all $ a\in X $ and $ t > 0 $. Therefore, using (4.1), we get
| $ {μ(J1g1(a)−J1h1(a),λρt)=μ(kg1(ak)−kh1(ak),λρt)=μ(g1(ak)−h1(ak),λρtk)≥ρktρkt+ρkφo(0,a)=tt+φo(0,a),ν(J1g1(a)−J1h1(a),λρt)=ν(kg1(ak)−kh1(ak),λρt)=ν(g1(ak)−h1(ak),λρtk)≤ρkφo(0,a)ρkt+ρkφo(0,a)=φo(0,a)t+φo(0,a) $ | (4.6) |
for some $ \rho < 1 $, all $ a\in X $ and all $ t > 0 $. Hence, it holds that $ d_{1}({\mathcal {J}}_{1}g_{1}, {\mathcal {J}}_{1}h_{1})\leq \lambda \rho $, that is, $ d_{1}({\mathcal {J}}_{1}g_{1}, {\mathcal {J}}_{1}h_{1})\leq \rho d_{1}(g_{1}, h_{1}) $ for all $ g_{1}, h_{1}\in S_{1} $.
Furthermore, by (4.1) and (4.4), we obtain the inequality
| $ d(f,J1f)≤ρ2k(2k+s−1). $ |
It follows from Lemma 2.4 that the sequence $ {\mathcal {J}}^{p}_{1}f $ converges to a fixed point $ A $ of $ {\mathcal {J}}_{1} $, that is, for all $ a\in X $ and all $ t > 0 $,
| $ A:X→Y,A(a):=(μ,ν)−limp→∞kpf(akp) $ | (4.7) |
and
| $ A(ka)=kA(a). $ | (4.8) |
Meanwhile, $ A $ is the unique fixed point of $ {\mathcal {J}}_{1} $ in the set
| $ S^{\ast}_{1} = \{g_{1}\in S_{1}: d_{1}(f,g_{1}) < \infty\}. $ |
Thus, there exists a $ \lambda\in \mathfrak{}{\mathbb{R_{+}}} $ such that
| $ {μ(f(a)−A(a),λt)≥tt+φo(0,a),ν(f(a)−A(a),λt)≤φo(0,a)t+φo(0,a) $ |
for all $ a\in X $ and all $ t > 0 $. Also,
| $ d1(f,A)≤11−ρd(f,J1f)≤ρ2k(1−ρ)(2k+s−1). $ |
This means that the following inequality
| $ {μ(f(a)−A(a),t)≥2k(2k+s−1)(1−ρ)t2k(2k+s−1)(1−ρ)t+ρφo(0,a),ν(f(a)−A(a),t)≤ρφo(0,a)2k(2k+s−1)(1−ρ)t+ρφo(0,a) $ | (4.9) |
holds for all $ a\in X $ and all $ t > 0 $. It follows from (3.4) and (4.1) that
| $ μ(kpDf(akp,bkp),t)≥tt+ρpφo(a,b),ν(kpDf(akp,bkp),t)≤ρpφo(a,b)t+ρpφo(a,b) $ | (4.10) |
for all $ a, b\in X $ and all $ t > 0 $. Letting $ p\to \infty $ in (4.10), we obtain
| $ μ(DA(a,b),t)=1andν(DA(a,b),t)=0 $ | (4.11) |
for all $ a, b\in X $ and all $ t > 0 $. This means that $ A $ satisfies the functional Eq (1.1). Since $ f $: $ X\to Y $ is an odd mapping, and using the definition $ A $, we have $ A(-a) = -A(a) $ for all $ a\in X $. Thus by Lemma 3.1, the mapping $ A $: $ X\to Y $ is additive.
By Lemma 2.3 and (4.9), we get
| $ {μn(fn([xij])−An([xij]),t)≥min{μ(f(xij)−A(xij),tn2):i,j=1,⋯,n}≥2k(2k+s−1)(1−ρ)t2k(2k+s−1)(1−ρ)t+ρn2∑ni,j=1φo(0,xij),νn(fn([xij])−An([xij]),t)≤max{ν(f(xij)−A(xij),tn2):i,j=1,…,n}≤ρn2∑ni,j=1φo(0,xij)2k(2k+s−1)(1−ρ)t+ρn2∑ni,j=1φo(0,xij) $ |
for all $ x = [x_{ij}]\in M_{n}(X) $ and all $ t > 0 $. Thus $ A $: $ X\to Y $ is a unique additive mapping satisfying (4.3), as desired. This completes the proof of the theorem.
Theorem 4.2. Let $ \varphi_{e} $: $ X^{2}\to [0, \infty) $ be a function such that for some real number $ \rho $ with $ 0 < \rho < 1 $ and
| $ φe(a,b)=ρk2φe(ka,kb) $ | (4.12) |
for all $ a, b\in X $. Suppose that an even mapping $ f $: $ X\to Y $ satisfies the inequality
| $ {μn(Dfn([xij],[yij]),t)≥tt+∑ni,j=1φe(xij,yij),νn(Dfn([xij],[yij]),t)≤∑ni,j=1φe(xij,yij)t+∑ni,j=1φe(xij,yij) $ | (4.13) |
for all $ x = [x_{ij}], y = [y_{ij}]\in M_{n}(X) $ and all $ t > 0 $. Then there exists a unique quadratic mapping $ Q $: $ X\to Y $ such that
| $ {μn(fn([xij])−Qn([xij]),t)≥2k2(2k+s−1)(1−ρ)t2k2(2k+s−1)(1−ρ)t+ρn2∑ni,j=1φe(0,xij),νn(fn([xij])−Qn([xij]),t)≤ρn2∑ni,j=1φe(0,xij)2k2(2k+s−1)(1−ρ)t+ρn2∑ni,j=1φe(0,xij) $ | (4.14) |
for all $ x = [x_{ij}]\in M_{n}(X) $ and all $ t > 0 $.
Proof. When $ n = 1 $, similar to the proof of Theorem 3.4, we obtain
| $ {μ(2(2k+s−1)f(ka)−2(2k+s−1)k2f(a),t)≥tt+φe(0,a),ν(2(2k+s−1)f(ka)−2(2k+s−1)k2f(a),t)≤φe(0,a)t+φe(0,a) $ | (4.15) |
for all $ a\in X $ and all $ t > 0 $.
Let $ S_{2}: = \left\{g_{2}:X\to Y\right\} $, and introduce a generalized metric $ d_{2} $ on $ S_{2} $ as follows:
| $ d2(g2,h2):=inf{λ∈R+|{μ(g2(a)−h2(a),λt)≥tt+φe(0,a),ν(g2(a)−h2(a),λt)≤φe(0,a)t+φe(0,a),∀a∈X,∀t>0}. $ |
It is easy to prove that $ (S_{2}, d_{2}) $ is a complete generalized metric space ([3,13]). Now, we define the mapping $ {\mathcal {J}}_{2} $: $ S_{2}\to S_{2} $ by
| $ J2g2(a):=k2g2(ak),for allg2∈S2anda∈X. $ | (4.16) |
Let $ g_{2}, h_{2}\in S_{2} $ and let $ \lambda\in {\mathbb{R_{+}}} $ be an arbitrary constant with $ d_{2}(g_{2}, h_{2})\leq \lambda $. From the definition of $ d_{2} $, we get
| $ {μ(g2(a)−h2(a),λt)≥tt+φe(0,a),ν(g2(a)−h2(a),λt)≤φe(0,a)t+φe(0,a) $ |
for all $ a\in X $ and $ t > 0 $. Therefore, using (4.12), we get
| $ {μ(J2g2(a)−J2h2(a),λρt)=μ(k2g2(ak)−k2h2(ak),λρt)=μ(g2(ak)−h2(ak),λρtk2)≥ρk2tρk2t+ρk2φe(0,a)=tt+φe(0,a),ν(J2g2(a)−J2h2(a),λρt)=ν(k2g2(ak)−k2h2(ak),λρt)=ν(g2(ak)−h2(ak),λρtk2)≤ρk2φe(0,a)ρk2t+ρk2φe(0,a)=φe(0,a)t+φe(0,a) $ | (4.17) |
for some $ \rho < 1 $, all $ a\in X $ and all $ t > 0 $. Hence, it holds that $ d_{2}({\mathcal {J}}_{2}g_{2}, {\mathcal {J}}_{2}h_{2})\leq \lambda \rho $, that is, $ d_{2}({\mathcal {J}}_{2}g_{2}, {\mathcal {J}}_{2}h_{2})\leq \rho d_{2}(g_{2}, h_{2}) $ for all $ g_{2}, h_{2}\in S_{2} $.
Furthermore, by (4.12) and (4.15), we obtain the inequality
| $ d(f,J2f)≤ρ2k2(2k+s−1). $ |
It follows from Lemma 2.4 that the sequence $ {\mathcal {J}}^{p}_{2}f $ converges to a fixed point $ Q $ of $ {\mathcal {J}}_{2} $, that is, for all $ a\in X $ and all $ t > 0 $,
| $ Q:X→Y,Q(a):=(μ,ν)−limp→∞k2pf(akp) $ | (4.18) |
and
| $ Q(ka)=k2Q(a). $ | (4.19) |
Meanwhile, $ Q $ is the unique fixed point of $ {\mathcal {J}}_{2} $ in the set
| $ S^{\ast}_{2} = \{g_{2}\in S_{2}: d_{2}(f,g_{2}) < \infty\}. $ |
Thus there exists a $ \lambda\in \mathfrak{}{\mathbb{R_{+}}} $ such that
| $ {μ(f(a)−Q(a),λt)≥tt+φe(0,a),ν(f(a)−Q(a),λt)≤φe(0,a)t+φe(0,a) $ |
for all $ a\in X $ and all $ t > 0 $. Also,
| $ d2(f,Q)≤11−ρd(f,J2f)≤ρ2k2(1−ρ)(2k+s−1). $ |
This means that the following inequality
| $ {μ(f(a)−Q(a),t)≥2k2(2k+s−1)(1−ρ)t2k2(2k+s−1)(1−ρ)t+ρφe(0,a),ν(f(a)−Q(a),t)≤ρφe(0,a)2k2(2k+s−1)(1−ρ)t+ρφe(0,a) $ | (4.20) |
holds for all $ a\in X $ and all $ t > 0 $. The rest of the proof is similar to the proof of Theorem 4.1. This completes the proof of the theorem.
Theorem 4.3. Let $ \varphi $: $ X^{2}\to [0, \infty) $ be a function such that for some real number $ \rho $ with $ 0 < \rho < k $,
| $ φ(a,b)=ρk2φ(ka,kb) $ | (4.21) |
for all $ a, b\in X $. Suppose that a mapping $ f $: $ X\to Y $ with $ f(0) = 0 $ satisfies the inequality
| $ {μn(Dfn([xij],[yij]),t)≥tt+∑ni,j=1φ(xij,yij),νn(Dfn([xij],[yij]),t)≤∑ni,j=1φ(xij,yij)t+∑ni,j=1φ(xij,yij) $ | (4.22) |
for all $ x = [x_{ij}], y = [y_{ij}]\in M_{n}(X) $ and all $ t > 0 $. Then there exist a unique quadratic mapping $ Q $: $ X\to Y $ and a unique additive mapping $ A $: $ X\to Y $ such that
| $ {μn(fn([xij])−Qn([xij])−An([xij]),t)≥k(2k+s−1)(1−ρ)tk(2k+s−1)(1−ρ)t+ρn2∑ni,j=1˜φ(0,xij),νn(fn([xij])−Qn([xij])−An([xij]),t)≤ρn2∑ni,j=1˜φ(0,xij)k(2k+s−1)(1−ρ)t+ρn2∑ni,j=1˜φ(0,xij) $ | (4.23) |
for all $ x = [x_{ij}]\in M_{n}(X) $ and all $ t > 0 $, $ \widetilde{\varphi}(a, b) = \varphi(a, b)+\varphi(-a, -b) $ for all $ a, b\in X $.
Proof. The proof follows from Theorems 4.1 and 4.2, and a method similar to Theorem 3.5. This completes the proof of the theorem.
Corollary 4.4. Let $ r, \theta $ be positive real numbers with $ r > 2 $. Suppose that a mapping $ f $: $ X\to Y $ with $ f(0) = 0 $ satisfies the inequality
| $ {μn(Dfn([xij],[yij]),t)≥tt+∑ni,j=1θ(‖xij‖r+‖yij‖r),νn(Dfn([xij],[yij]),t)≤∑ni,j=1θ(‖xij‖r+‖yij‖r)t+∑ni,j=1θ(‖xij‖r+‖yij‖r) $ | (4.24) |
for all $ x = [x_{ij}], y = [y_{ij}]\in M_{n}(X) $ and all $ t > 0 $. Then there exist a unique quadratic mapping $ Q $: $ X\to Y $ and a unique additive mapping $ A $: $ X\to Y $ such that
| $ {μn(fn([xij])−Qn([xij])−An([xij]),t)≥(2k+s−1)(kr−k2)t(2k+s−1)(kr−k2)t+2kn2∑ni,j=1θ‖xij‖r,νn(fn([xij])−Qn([xij])−An([xij]),t)≤2kn2∑ni,j=1θ‖xij‖r(2k+s−1)(kr−k2)t+2kn2∑ni,j=1θ‖xij‖r $ | (4.25) |
for all $ x = [x_{ij}]\in M_{n}(X) $ and all $ t > 0 $.
Proof. Taking $ \varphi(a, b) = \theta(\|a\|^{r}+\|b\|^{r}) $ for all $ a, b\in X $ and $ \rho = k^{2-r} $ in Theorem 4.3, we get the desired result.
We use the direct and fixed point methods to investigate the Hyers-Ulam stability of the functional Eq (1.1) in the framework of matrix intuitionistic fuzzy normed spaces. We therefore provide a link two various discipline: matrix intuitionistic fuzzy normed spaces and functional equations. We generalized the Hyers-Ulam stability results of the functional Eq (1.1) from quasi-Banach spaces to matrix intuitionistic fuzzy normed spaces. These circumstances can be applied to other significant functional equations.
The author declare he has not used Artificial Intelligence (AI) tools in the creation of this article.
The author is grateful to the referees for their helpful comments and suggestions that help to improve the quality of the manuscript.
The author declares no conflict of interest in this paper.
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Chris Cadonic, Benedict C. Albensi. Oscillations and NMDA Receptors: Their Interplay Create Memories[J]. AIMS Neuroscience, 2014, 1(1): 52-64. doi: 10.3934/Neuroscience.2014.1.52
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