Abstract
Using the fixed point method, we prove the generalized Hyers-Ulam stabilityof the following additive-cubic-quartic functional equation:
in fuzzy Banach spaces.
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Introduction
Katsaras [1] defined a fuzzy norm on a vector space to construct a fuzzy vectortopological structure on the space. Some mathematicians have defined fuzzy norms asa vector space from various points of view (see [2–4]). In particular, Bag and Samanta [5], following Cheng and Mordeson [6], gave an idea of fuzzy norm in such a manner that the corresponding fuzzymetric is of Karmosil and Michalek type [7]. They established a decomposition theorem of a fuzzy norm into a familyof crisp norms and investigated some properties of fuzzy normed spaces [8].
Definition 1
Let X be a real vector space. A function is called a fuzzy norm on X if for all x,y ∈ X and all (Bag and Samanta [5]): (N 1) N(x, t) = 0for t≤0;(N 2) x = 0 if and only ifN(x, t) = 1 for allt > 0;(N 3) if c ≠ 0;(N 4)N(x + y,c + t) ≥ min{N(x,s), N(y, t)};(N 5)N(x, .) is a non-decreasing function of and ;(N 6) for x ≠ 0,N(x, .) is continuous on .
Example 1
Let (X, ∥.∥) be a normed linear space and α,β > 0. Then
is a fuzzy norm on X.
Definition 2
Let (X, N) be a fuzzy normed vector space. A sequence{x n } in X is said to be convergent or converges if there exists anx ∈ X such that for all t > 0. In this case,x is called the limit of the sequence {x n } in X, and we denote it by (Bag and Samanta [5]).
Definition 3
Let (X, N) be a fuzzy normed vector space. A sequence{x n } in X is called Cauchy if for each ϵ > 0 and eacht > 0 there exists an such that for all n ≥ n0 and all p > 0, we haveN(xn + p − x n t) > 1 − ϵ (Bag andSamanta [5]).
It is well known that every convergent sequence in a fuzzy normed vector space isCauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to becomplete, and the fuzzy normed vector space is called a fuzzy Banach space.
We say that a mapping f : X → Y betweenfuzzy normed vector spaces X and Y is continuous at a pointx ∈ X if for each sequence {x n } converging to x0∈X, then the sequence {f(x n )} converges to f(x0). If f : X → Y iscontinuous at each x ∈ X, then f :X → Y is said to be continuous on X(see [8]).
Definition 4
Let X be a set. A function d :X × X → [0,∞] is called a generalized metric on X if dsatisfies the following conditions:
-
(1)
d(x, y) = 0 if and only if x = y for all x, y ∈ X;
-
(2)
d(x, y) = d(y,x) for all x, y ∈ X;
-
(3)
d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.
Theorem 1
Let (X, d) be a complete generalized metric space andJ : X → X be a strictlycontractive mapping with Lipschitz constant L < 1 [9, 10]. Then, for all x ∈ X, either
for all nonnegative integers n or there exists a positive integern0 such that
-
(1)
d(J n x,J n + 1 x) < ∞ for all n 0 ≥ n 0;
-
(2)
the sequence {J n x} converges to a fixed point y ∗ of J;
-
(3)
y ∗ is the unique fixed point of J in the set ;
-
(4)
for all y ∈ Y.
The stability problem of functional equations originated from a question of Ulam [11] concerning the stability of group homomorphisms. Hyers [12] gave a first affirmative partial answer to the question of Ulam forBanach spaces. Hyers’ theorem was generalized by Themistocles M Rassias [13] for linear mappings by considering an unbounded Cauchy difference.
The functional equationf(x + y) + f(x − y) = 2f(x) + 2f(y)is called a quadratic functional equation. In particular, every solution ofthe quadratic functional equation is said to be a quadratic mapping. TheHyers-Ulam stability of the quadratic functional equation was proved by Skof [14] for mappings f : X → Y,where X is a normed space and Y is a Banach space. Cholewa [15] noticed that the theorem of Skof is still true if the relevant domainX is replaced by an Abelian group. Czerwik [16] proved the Hyers-Ulam stability of the quadratic functional equation.
In the study of Eshaghi Gordji et. al [17], they proved that the following functional equation is anadditive-cubic-quartic functional equation:
In this paper, we prove the generalized Hyers-Ulam stability of the functionalequation (Equation 1) in fuzzy Banach spaces.
The stability problems of several functional equations have been extensivelyinvestigated by a number of authors, and there are many interesting resultsconcerning this problem (see [18]–[43]).
Methods
Fuzzy stability of the functional equation (Equation 1): an odd case
In this section, using the fixed point alternative approach, we prove thegeneralized Hyers-Ulam stability of the functional equation (Equation 1) infuzzy Banach spaces: an odd case. Throughout this paper, assume that Xis a vector space and that (Y, N) is a fuzzy Banach space.
In the work of Lee et al. [32], they considered the following quartic functional equation:
It is easy to show that the functionf(x) = x4 satisfies the functional equation (Equation 2), which is called aquartic functional equation, and every solution of the quartic functionalequation is said to be a quartic mapping.
One can easily show that an even mapping f :X → Y satisfies Equation 1 if and onlyif the even mapping f : X → Y is aquartic mapping, that is,
and an odd mapping f : X → Ysatisfies Equation 1 if and only if the odd mapping f :X → Y is a additive-cubic mapping, thatis,
It was shown in Lemma 2.2 in the study of Eshaghi Gordji et. al [17] thatg(x) = f(2x) − 2f(x)andh(x) = f(2x) − 8f(x)are cubic and additive, respectively, and that .
For a given mapping f : X → Y, wedefine the following:
for all x, y ∈ X.
Theorem 2
Let φ : X2 → [ 0, ∞) be a function suchthat there exists an α < 1 with
for all x, y ∈ X. Let f: X → Y be an odd mapping, satisfying
for all x, y ∈ X and allt > 0, and then the limit
exists for each x ∈ X and defines aunique cubic mapping C : X → Ysuch that
Proof
Putting x = 0 in Equation 6, we have the following:
for all y ∈ X andt > 0.
Replacing x by 2y in Equation 6, we obtain the following:
for all y ∈ X andt > 0.
By Equations 8 and 9, we have the following:
for all y ∈ X and allt > 0. Letting andg(x) = f(2x) − 2f(x)for all x ∈ X, we get the following:
Consider the set S := {g :X → Y} and the generalized metricd in S defined by the following:
where inf ∅ = + ∞. It is easyto show that (S, d) is complete (see Lemma 2.1 of [33]).
Now, we consider a linear mapping J :S → S such that
for all x ∈ X. Let g,h ∈ S satisfy d(g,h) = ϵ and then
for all x ∈ X andt > 0. Hence,
for all x ∈ X andt > 0. Thus, d(g,h) = ϵ implies thatd(Jg,Jh) ≤ α ϵ.This means that
for all g, h ∈ S. It followsfrom Equation 11 that
Thus,
By Theorem 1, there exists a mapping C :X → Y, satisfying the following:
-
(1)
C is a fixed point of J, that is,
(12)
for all x ∈ X. The mapping C isa unique fixed point of J in the following set:Ω = {h ∈ S: d(g,h) < ∞}.
This implies that C is a unique mapping, satisfying Equation 12,such that there exists μ ∈ (0,∞), satisfying the following:
for all x ∈ X andt > 0.
-
(2)
d(J n g, C) → 0 as n → ∞. This implies the following equality:
for all x ∈ X.
-
(3)
with f ∈ Ω, which implies the following inequality:
This implies that the inequality (Equation 7) holds.
Since Φ g (x, y) = Φ f (2x, 2y) − 2Φ f (x, y), using Equation 6, we obtain the following:
for all x, y ∈ X,t > 0 and all . Thus, by Equation 5, we have the following:
for all x, y ∈ X,t > 0 and all . Since for all x,y ∈ X and allt > 0, we deduce that N(Φ C (x, y), t) = 1 for allx, y ∈ X and allt > 0. Thus, the mapping C :X → Y, satisfying Equation 1, asdesired. This completes the proof. □
Corollary 1
Let θ ≥ 0 and let r be a real numberwith r > 1. Let X be a normed vector spacewith norm ∥ · ∥. Let f :X → Y be an odd mapping, satisfyingthe following:
for all x, y ∈ X and allt > 0, and then,
exists for each x ∈ X and defines aunique cubic mapping C : X → Ysuch that
for all x ∈ X and allt > 0.
Proof
The proof follows from Theorem 2 by taking φ(x,y) := θ(∥x∥r + ∥y∥r) for all x, y ∈ X, andthen we can choose α = 81−r and get the desired result. □
Theorem 3
Let φ : X2 → [ 0, ∞) be a function suchthat there exists an α < 1 with the following:
for all x, y ∈ X. Let f: X → Y be an odd mapping, satisfyingEquation 6, and then the limit
exists for each x ∈ X and defines aunique cubic mapping C : X → Ysuch that
Proof
Let (S, d) be the generalized metric space defined as inthe proof of Theorem 2. Consider the linear mapping J :S → S such that for all x ∈ X. Letg, h ∈ S be such thatd(g, h) = ϵ, and
for all x ∈ X andt > 0. Hence,
for all x ∈ X andt > 0. Thus, d(g,h) = ϵ implies thatd(Jg,Jh) ≤ αϵ. This means thatd(Jg,Jh) ≤ αd(g,h), for all g, h ∈ S.It follows from Equation 10 that
for all x ∈ X andt > 0. Thus, .
By Theorem 1, there exists a mapping C :X → Y, satisfying the following:
-
(1)
C is a fixed point of J, that is,
(17)
for all x ∈ X. The mapping C isa unique fixed point of J in the setΩ = {h∈S :d(g,h) < ∞}.
This implies that C is a unique mapping, satisfying Equation 17,such that there exists μ ∈ (0,∞), satisfying the following:
for all x ∈ X andt > 0.
-
(2)
d(J n g, C) → 0 as n → ∞. This implies the following equality:
for all x ∈ X.
-
(3)
with f ∈ Ω, which implies the following inequality: This implies that the inequality (Equation 16) holds.
The rest of the proof is similar to that of the proof of Theorem 2.□
Corollary 2
Let θ ≥ 0 and let r be a real numberwith 0 < r < 1. Let X be anormed vector space with norm ∥ · ∥. Letf : X → Y be an oddmapping, satisfying Equation 14, and the limit
exists for each x ∈ X and defines aunique cubic mapping C : X → Ysuch that
for all x ∈ X and allt > 0.
Proof
The proof follows from Theorem 3 by taking φ(x,y) := θ(∥x∥r + ∥y∥r) for all x, y ∈ X, andthen we can choose α = 8−r and get the desired result. □
Theorem 4
Let φ : X2 → [ 0, ∞) be a function suchthat there exists an α < 1 with the following:
for all x, y ∈ X. Let f: X → Y be an odd mapping, satisfyingEquation 6, and then the limit
exists for each x ∈ X and defines aunique additive mapping A :X → Y such that
Proof
Let (S, d) be the generalized metric space defined as inthe proof of Theorem 2.
Letting and h(x) :f(2x) − 8f(x) forall x ∈ X in Equation 10, we obtain thefollowing:
Consider the linear mapping J :S → S such that
for all x ∈ X. Let g,h ∈ S be such thatd(g, h) = ϵ, andthen
for all x ∈ X andt > 0. Hence,
for all x ∈ X andt > 0. Thus, d(g,h) = ϵ implies thatd(Jg,Jh) ≤ αϵ. This means thatd(Jg,Jh) ≤ αd(g,h), for all g, h ∈ S.It follows from Equation 20 that
for all x ∈ X andt > 0. Thus, .
By Theorem 1, there exists a mapping A :X → Y, satisfying the following:
-
(1)
A is a fixed point of J, that is,
(21)
for all x ∈ X. The mapping A isa unique fixed point of J in the setΩ = {h ∈ S: d(g,h) < ∞}.
This implies that A is a unique mapping, satisfying Equation 21,such that there exists μ ∈ (0,∞), satisfying the following:
for all x ∈ X andt > 0.
-
(2)
d(J n h, A) → 0 as n → ∞. This implies the following equality:
for all x ∈ X.
-
(3)
with f ∈ Ω, which implies the following inequality: This implies that the inequality (Equation 2. □
Corollary 3
Let θ ≥ 0 and let r be a real numberwith r > 1. Let X be a normed vector spacewith norm ∥ · ∥. Let f :X → Y be an odd mapping, satisfyingEquation 14, and then
exists for each x ∈ X and defines aunique additive mapping A :X → Y such that
for all x ∈ X and allt > 0.
Proof
The proof follows from Theorem 4 by taking φ(x,y) := θ(∥x∥r + ∥y∥r) for all x, y ∈ X, andthen we can choose α = 21−r and get the desired result. □
Theorem 5
Let φ : X2 → [ 0, ∞) be a function suchthat there exists an α < 1 with the following:
for all x, y ∈ X. Let f: X → Y be an odd mapping, satisfyingEquation 6, and then the limit
exists for each x ∈ X and defines aunique additive mapping A :X → Y such that
Proof
Let (S, d) be the generalized metric space defined as inthe proof of Theorem 2. Consider the linear mapping J :S → S such that for all x ∈ X. Letg, h ∈ S be such thatd(g, h) = ϵ.
Then
for all x ∈ X andt > 0. Hence,
for all x ∈ X andt > 0. Thus, d(g,h) = ϵ implies thatd(Jg,Jh) ≤ αϵ. This means thatd(Jg,Jh) ≤ αd(g,h),for all g, h ∈ S. It followsfrom Equation 10 that
for all x ∈ X andt > 0. Thus,
By Theorem 1, there exists a mapping A :X → Y, satisfying the following:
-
(1)
A is a fixed point of J, that is,
(24)
for all x ∈ X. The mapping A isa unique fixed point of J in the setΩ = {h ∈ S: d(g,h) < ∞}.
This implies that A is a unique mapping, satisfying Equation 24,such that there exists μ ∈ (0,∞), satisfying the following:
for all x ∈ X andt > 0.
-
(2)
d(J n h,A) → 0 as n → ∞. This implies the following equality:
for all x ∈ X.
-
(3)
with f ∈ Ω, which implies the following inequality:
This implies that the inequality (Equation 23) holds. The rest of the proofis similar to that of the proof of Theorem 2. □
Corollary 4
Let θ ≥ 0 and let r be a real numberwith 0 < r < 1. Let X be anormed vector space with norm ∥ · ∥. Letf : X → Y be an oddmapping, satisfying Equation 14, and then the limit
exists for each x ∈ X and defines aunique additive mapping A :X → Y such that
for all x ∈ X and allt > 0.
Proof
The proof follows from Theorem 5 by taking φ(x,y) := θ(∥x∥r + ∥y∥r), for all x, y ∈ X, andthen we can choose α = 2−r and get the desired result. □
Fuzzy stability of the functional equation (Equation 1): an even case
Throughout this section, using the fixed point alternative approach, we prove thegeneralized Hyers-Ulam stability of the functional equation (Equation 1) infuzzy Banach spaces: an even case.
Theorem 6
Let φ : X2 → [ 0, ∞) be a function suchthat there exists an α < 1 with the following:
for all x, y ∈ X. Let f: X → Y be an even mapping, satisfyingthe following:
for all x, y ∈ X and allt > 0, and then the limit
exists for each x ∈ X and defines aunique quartic mapping Q :X → Y such that
Proof
Putting x = 0 in Equation 26, we have the following:
for all y ∈ X andt > 0.
Substituting x = y in Equation 26, we obtainthe following:
for all y ∈ X andt > 0.
By Equations 28 and 29, we have the following:
for all y ∈ X and allt > 0. By replacing for all x ∈ X, we getthe following:
Consider the set S := {g :X → Y}, and the generalized metricd in S defined by
where inf ∅ = + ∞. It is easyto show that (S, d∗) is complete (see Lemma 2.1 in [33]).
Now, we consider a linear mapping J :S → S such that for all x ∈ X. Letg, h ∈ S satisfy d∗(g, h) = ϵ,and then
for all x ∈ X andt > 0. Hence,
for all x ∈ X andt > 0. Thus, d∗(g, h) = ϵimplies that d∗(Jg,Jh) ≤ αϵ. This means thatd∗(Jg,Jh) ≤ α d∗(g, h), for all g,h ∈ S. It follows from Equation 31that
Thus, By Theorem 1, there exists a mapping Q :X → Y, satisfying the following:
-
(1)
Q is a fixed point of J, that is,
(32)
for all x ∈ X. The mapping Q isa unique fixed point of J in the following set:Ω = {h ∈ S: d∗(g,h) < ∞}. This implies thatQ is a unique mapping, satisfying Equation 32, such that thereexists μ ∈ (0, ∞), satisfyingthe following:
for all x ∈ X andt > 0.
-
(2)
d ∗(J n f, Q) → 0 as n → ∞. This implies the following equality: for all x ∈ X.
-
(3)
with f ∈ Ω, which implies the following inequality: This implies that the inequality (Equation 27) holds.
On the other hand, by Equation 26, we obtain the following:
for all x, y ∈ X,t > 0 and all . Thus,
for all x, y ∈ X,t > 0 and all . Since for all x,y ∈ X and allt > 0, we deduce that N(Φ Q (x, y), t) = 1 for allx, y ∈ X and allt > 0. Thus, the mappingQ:X → Y, satisfyingEquation 1, as desired. This completes the proof. □
Corollary 5
Let θ ≥ 0 and let r be a real numberwith r > 1. Let X be a normed vector spacewith norm ∥ · ∥. Let f :X → Y be an even mapping, satisfyingEquation 14, and then the limit
exists for each x ∈ X and defines aunique quartic mapping Q:X→Y such that
for all x ∈ X and allt > 0.
Proof
The proof follows from Theorem 6 by taking φ(x,y) := θ(∥x∥r + ∥y∥r), for all x, y ∈ X, andthen we can choose α = 16−r and get the desired result. □
Theorem 7
Let φ : X2 → [ 0, ∞) be a function suchthat there exists an α < 1 with the following:
for all x, y ∈ X. Let f: X → Y be an even mapping, satisfyingEquation 26, and then the limit
exists for each x ∈ X and defines aunique quartic mapping Q :X → Y such that
Proof
Let (S, d∗) be the generalized metric space defined as in the proofof Theorem 6. Consider the linear mapping J :S → S such that
for all x ∈ X. Let g,h ∈ S be such that d∗(g, h) = ϵ,and then
for all x ∈ X andt > 0. Hence,
for all x ∈ X andt > 0. Thus, d∗(g, h) = ϵimplies that d∗(Jg,Jh) ≤ αϵ. This means thatd∗(Jg,Jh) ≤ α d∗(g, h) for all g,h ∈ S. It follows from Equation 30that
for all x ∈ X andt > 0. Thus, .
By Theorem 1, there exists a mapping Q :X → Y satisfying the following:
-
(1)
Q is a fixed point of J, that is,
(35)
for all x ∈ X. The mapping Q isa unique fixed point of J in the setΩ = {h ∈ S: d∗(g,h) < ∞}.
This implies that Q is a unique mapping, satisfying Equation 35,such that there exists μ ∈ (0,∞), satisfying the following:
for all x ∈ X andt > 0.
-
(2)
d ∗(J n f, Q) → 0 as n → ∞. This implies the following equality: for all x ∈ X.
-
(3)
with f ∈ Ω, which implies the following inequality: This implies that the inequality (Equation 2. □
Corollary 6
Let θ ≥ 0 and let r be a real numberwith 0 < r < 1. Let X be anormed vector space with norm ∥·∥. Letf:X→Y be an even mapping, satisfyingEquation 14, and then the limit
exists for each x ∈ X and defines aunique quartic mapping Q :X → Y such that
for all x ∈ X and allt > 0.
Proof
The proof follows from Theorem 7 by taking the following:φ(x, y) := θ(∥x∥r + ∥y∥r), for all x, y ∈ X, andthen we can choose α = 16r−1 and get the desired result. □
Results and discussion
We linked here three different disciplines, namely fuzzy Banach spaces, functionalequations, and fixed point theory. We established the Hyers-Ulam-Rassias stabilityof functional Equation 1 in fuzzy Banach spaces by fixed point method.
Conclusions
Throughout this paper, using the fixed point method, we proved the Hyers-Ulam-Rassiasstability of a mixed type ACQ functional equation in fuzzy Banach spaces.
References
Katsaras AK: Fuzzy topological vector spaces. Fuzzy Sets and Syst 1984, 12: 143–154.
Felbin C: Finite-dimensional fuzzy normed linear space. Fuzzy Sets and Syst 1992, 48: 239–248.
Krishna SV, Sarma KKM: Separation of fuzzy normed linear spaces. Fuzzy Sets and Syst 1994, 63: 207–217.
Park C: Fuzzy stability of a functional equation associated with inner productspaces. Fuzzy Sets and Syst 2009, 160: 1632–1642.
Bag T, Samanta SK: Finite dimensional fuzzy normed linear spaces. J Fuzzy Mathematics 2003, 11: 687–705.
Cheng SC, Mordeson JN: Fuzzy linear operators and fuzzy normed linear spaces. Bull. Calcutta Math. Soc 1994, 86: 429–436.
Karmosil I, Michalek J: Fuzzy metric and statistical metric spaces. Kybernetica 1975, 11: 326–334.
Bag T, Samanta SK: Fuzzy bounded linear operators. Fuzzy Sets and Syst 2005, 151: 513–547.
Cădariu L, Radu V: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math 2003, 4(1):4.
Diaz J, Margolis B: A fixed point theorem of the alternative for contractions on a generalizedcomplete metric space. Bull. Amer. Math. Soc 1968, 74: 305–309.
Ulam SM: Problems in Modern Mathematics. John Wiley and Sons, New York; 1964.
Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224.
Rassias ThM: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc 1978, 72: 297–300.
Skof F: Local properties and approximation of operators. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113–129.
Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984, 27: 76–86.
Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hambourg 1992, 62: 239–248.
Eshaghi-Gordji M, Kaboli-Gharetapeh S, Park C, Zolfaghri S: Stability of an additive-cubic-quartic functional equation. Advances in Difference Equations 2009, 2009: 395693.
Agarwal RP, Cho YJ, Saadati R, Wang S: Nonlinear L-fuzzy stability of cubic functional equations. J. Inequal. Appl 2012, 2012: 77.
Azadi Kenary H: Non-Archimedean stability of Cauchy-Jensen type functional equation. Int. J. Nonlinear Anal. Appl 2010, 1(2):1–10.
Azadi Kenary H: Stability of a Pexiderial functional equation in random normed spaces. Rend. Del Circolo Math. Di Palermo 2011, 60(1):59–68.
Azadi Kenary H, Shafaat Kh, Shafei M, Takbiri G: Hyers-Ulam-Rassias stability of the Appollonius quadratic mapping inRN-spaces. J. Nonlinear Sci. Appl 2011, 4(1):110–119.
Eshaghi-Gordji M, Abbaszadeh S, Park C: J. Inequal. Appl. 2009, 153084.
Eshaghi Gordji M, Khodaei H, Rassias JM: Fixed point methods for the stability of general quadratic functionalequation. Fixed Point Theory 2011, 12(1):71–82.
Eshaghi Gordji M, Park C, Savadkouhi MB: The stability of quartic functional equation with the fixed pointalternative. Fixed Point Theory 2010, 11(2):265–272.
Eshaghi Gordji M: Stability of a functional equation deriving from quartic and additivefunctions. Bull. Korean Math. Soc 2010, 47(3):491–502.
Eshaghi Gordji M, Ebadian A, Zolfaghari S: Stability of a functional equation deriving from cubic and quarticfunctions. Abstract and Appl. Anal 2008, 2008: 801904.
Eshaghi Gordji M, Khodaei H: Solution and stability of generalized mixed type cubic, quadratic andadditive functional equation in quasi–Banach spaces. Nonlinear Analysis–TMA 2009, 71: 5629–5643.
Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximatelyadditive mappings. J. Math. Anal. Appl 1994, 184: 431–436.
Jun K, Cho Y: Stability of generalized Jensen quadratic functional equations. J. Chungcheong Math. Soc 2007, 29: 515–523.
Jung S: Hyers-Ulam-Rassias Stability of Functional Equations in MathematicalAnalysis. Hadronic Press, Palm Harbor; 2001.
Khodaei H, Eshaghi Gordji M, Kim SS, Cho YJ: Approximation of radical functional equations related to quadratic andquartic mappings. J. Math. Anal. Appl 2012, 397: 284–297.
Lee SH, Im SM, Hwang IS: Quartic functional equations. J. Math. Anal. Appl 2005, 307(2):387–394.
Mihet D, Radu V: On the stability of the additive Cauchy functional equation in random normedspaces. J. Math. Anal. Appl 2008, 343: 567–572.
Park C: On the stability of the linear mapping in Banach modules. J. Math. Anal. Appl 2002, 275: 711–720.
Park C: Modefied Trif’s functional equations in Banach modules over aC∗-algebra and approximate algebra homomorphism. J. Math. Anal. Appl 2003, 278: 93–108.
Park C, Moradlou F: Stability of homomorphisms and derivations on C∗-ternary rings. Taiwanese J. Math 2009, 13: 1985–1999.
Rassias ThM: On the stability of the quadratic functional equation and it’sapplication. Studia Univ. Babes-Bolyai 1998, XLIII: 89–124.
Rassias ThM: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl 2000, 251: 264–284.
Rassias ThM, Šemrl P: On the Hyers-Ulam stability of linear mappings. J. Math. Anal. Appl. 1993, 173: 325–338.
Saadati R, Park C: Non-Archimedean $\mathcal {L}$ℒ-fuzzy normed spaces andstability of functional equations. Journal Computers & Mathematics with Applications 2010, 60: 2488–2496.
Saadati R, Park JH: On the intuitionistic fuzzy topological spaces. Chaos, Solitons and Fractals 2006, 27: 331–44.
Saadati R, Vaezpour M, Cho YJ: J. Ineq. Appl. 2009, 214530.
Saadati R, Zohdi MM, Vaezpour SM: Nonlinear L-random stability of an ACQ functional equation. J. Ineq. Appl 2011, 2011: 194394.
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Kenary, H.A. Nonlinear fuzzy approximation of a mixed type ACQ functional equation via fixedpoint alternative. Math Sci 6, 54 (2012). https://doi.org/10.1186/2251-7456-6-54
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DOI: https://doi.org/10.1186/2251-7456-6-54