Ulam-Hyers Stability of a 2-Variable AC-Mixed Type Functional Equation in Felbin ’ s Type Spaces : Fixed Point Method

In this paper, the authors obtain the generalized Ulam Hyers stability of a 2 variable AC mixed type functional equation f(2x+ y, 2z + w) − f(2x− y, 2z − w) = 4[f(x+ y, z + w) − f(x− y, z − w)] − 6f(y,w) in Felbin’s type spaces using fixed point method. Mathematics Subject Classification: 39B52, 32B72, 32B82

The general solution and Ulam stability of mixed type additive and cubic functional equation of the form 3f (x + y + z) + f (−x + y + z) + f (x − y + z) + f (x + y − z) (1.1) introduced by J.M. Rassias [29].The stability of generalized mixed type functional equation of the form for fixed integers k with k = 0, ±1 in quasi -Banach spaces was investigated by M. Eshaghi Gordji and H. Khodaie [10].The mixed type functional equation (1.2) is having the property additive, quadratic and cubic.
The solution and stability of a n− dimensional additive functional equation where a are integers, a ≥ 1 with fixed point Alternative was investigated by K. Ravi, M. Arunkumar [34].Also, Y.S. Jung, I.S. Chang [17] discussed the Hyers -Ulam -Rassias stability for the cubic functional equation and its Hyers -Ulam -Rassias stability with the help of alternative fixed point idea was dealt by H.Y. Chu and D.S. Kang [5].Recently, F. Moradlou et al., [25] proved the stability of Cauchy functional equation in Felbin's type spaces using fixed point approach.J.H. Bae and W.G. Park [6] proved the general solution of the 2-variable quadratic functional equation and investigated the generalized Hyers-Ulam-Rassias stability of (1.7).The above functional equation has solution Very recently, M. Arunkumar etal., [3] first time introduced and investigated the solution and generalized Ulam-Hyers stability of a 2 -variable AC -mixed type functional equation in Banach space via direct and fixed point approach.
The solution of the AC functional equation (1.9) is given in the following lemmas.
Lemma 1.1.[3] If f : U 2 → V be a mapping satisfying (1.9) and let g : U 2 → V be a mapping given by for all x ∈ U such that g is additive.
Lemma 1.2.[3] If f : U 2 → V be a mapping satisfying (1.9) and let h : U 2 → V be a mapping given by V be a mapping satisfying (1.9) and let g, h : U 2 → V be a mapping defined in (1.12) and (1.14) then for all x ∈ U .
In this paper, the authors established the generalized Ulam-Hyers stability using fixed point method in Felbin's type spaces is discussed in Section 3.

Fuzzy real number
In this section, we give some preliminaries in the theory of fuzzy real numbers.Furthermore, we give some definition which help to investigate the stability in Felbin's type normed linear spaces.
In [12] Grantner takes the fuzzy real number as a decreasing mapping from the real line to the unit interval or lattice in general.Lowen [23] applies the fuzzy real numbers as non-decreasing, left continuous mapping from the real line to the unit interval so that its supremum over R is 1.Also fuzzy arithmetic operations on L−fuzzy real line were studied by Rodabaugh [39], where he showed that the binary addition is the only extension of addition to R((L)).
Hoehle [14] especially emphasized the role of fuzzy real numbers as modeling a fuzzy threshold softening the notion of Dedekind cut.In this paper a fuzzy real number is taken as a fuzzy normal and convex mapping from the real line to the unit interval.The concept of the fuzzy metric space has been studied by Kaleva [19,20] by using fuzzy number as a fuzzy set on the real axis.Kaleva also has recently showed that a fuzzy metric space can be embedded in a complete fuzzy metric space [21].
In [8], Felbin introduced the concept of fuzzy normed linear space (FNLS); Xiao and Zhu [44] studied its linear topological structures and some basic properties of a fuzzy normed linear space.It is known that theories of classical normed space and Menger probabilistic normed spaces are special cases of fuzzy normed linear spaces.
Let η be a fuzzy subset on R, i.e., a mapping η : R → [0, 1] associating with each real number t its grade of membership ηt.Definition 2.1.[8] A fuzzy subset η on R is called a fuzzy real number, whose α−level set is denoted by [η] if it satisfies two axioms: (N 1) There exists The set of all fuzzy real numbers denoted by F (R).If η ∈ F (R) and η(t) = 0 whenever t < 0, then η is called a non-negative fuzzy real number and F * (R) denotes the set of all non-negative fuzzy real numbers.
The number 0 stands for the fuzzy real number as: Clearly, 0 ∈ F * (R).Also the set of all real numbers can be embedded in Definition 2.2.[8] Fuzzy arithmetic operations ⊕, , ⊗, on F (R) × F (R) can be defined as: The additive and multiplicative identities in F (R) are 0 and 1, respectively.Let η be defined as 0 η.It is clear that η δ = η ⊕ ( δ).Definition 2.3.[8] For k ∈ R 0, fuzzy scalar multiplication k η is defined as (k η)(t) = η(t/k) and 0 η is defined to be 0. [44] Let X be a real linear space, L and R (respectively, left norm and right norm) be symmetric and non-decreasing mappings from

STABILITY RESULTS: FIXED POINT METHOD
In this section, we apply a fixed point method for achieving stability of the 2-variable AC functional equation (1.9).Now, we present the following theorem due to B. Margolis and J.B. Diaz [24] for fixed point Theory.
Theorem 3.1.[24] Suppose that for a complete generalized metric space (Ω, δ) and a strictly contractive mapping T : Ω → Ω with Lipschitz constant L.Then, for each given x ∈ Ω , either y, T y) for all y ∈ Δ.Using the above theorem, we now obtain the generalized Ulam -Hyers stability of (1.9).Through out this section let U be a normed space and V be a Banach space.Define a mapping for all x, y, z, w ∈ U .Theorem 3.2.Let F : U 2 → V be a mapping for which there exist a function ϕ : where for all x, y, z, w ∈ U and α ∈ (0, 1].If there exists L = L(i) < 1 such that the function has the property Then there exists a unique 2-variable additive mapping A : U 2 → V satisfying the functional equation (1.9) and for all x ∈ U , where ψ(x) + α is defined in (3.9), for all x ∈ U .Proof.Consider the set Ω = {p/p : U 2 → V, p(0, 0) = 0} and introduce the generalized metric on Ω, for all x ∈ U .Now p, q ∈ Ω, This implies d(T p, T q) ≺ Ld(p, q), for all p, q ∈ Ω .i.e., T is a strictly contractive mapping on Ω with Lipschitz constant L. Letting (x, y, z, w) by (x, x, x, x) in (3.2), we obtain for all x ∈ U .Replacing (x, y, z, w) by (x, 2x, x, 2x) in (3.2), we get for all x ∈ U .Now, from (3.5) and (3.6), we have for all x ∈ U .From (3.7), we arrive where for all x ∈ U .It is easy to see from (3.8) that for all x ∈ U .Using (1.12) in (3.10), we obtain for all x ∈ U .From (3.11), we arrive for all x ∈ U .Using (3.3) for the case i = 0 it reduces to Again replacing x = x 2 in (3.12), we get, Using (3.3) for the case i = 1 it reduces to In both cases, we arrive d(g, T g)≺L 1−i .
Therefore (A1) holds.By (A2), it follows that there exists a fixed point A of T in Ω such that for all x ∈ U .
To prove A : U 2 → V is additive.Replacing (x, y, z, w) by (μ n i x, μ n i y, μ n i z, μ n i ) in (3.2) and dividing by μ n i , it follows from (3.1) that

A(x, y, z, w)
for all x, y, z, w ∈ U i.e., A satisfies the functional equation (1.9).By (A3), A is the unique fixed point of T in the set Δ = {A ∈ Ω : d(f, A) < ∞}, A is the unique function such that for all x ∈ U and K > 0. Finally by (A4), we obtain this completes the proof of the theorem.
The following Corollary is an immediate consequence of Theorem 3.2 concerning the stability of (1.9).Corollary 3.3.Let F : U 2 → V be a mapping and there exits real numbers λ and s such that for all x, y, z, w ∈ U , then there exists a unique 2-variable additive function A : for all x ∈ U.
In similar manner we can prove the following cases L = 2 4s−1 for s < 1  4 if i = 0 and L = 1 2 4s−1 for s > 1  4 if i = 1 for conditions (ii) and (iii) respectively.Hence the proof is complete.Theorem 3.4.Let F : U 2 → V be a mapping for which there exist a function ϕ : U 4 → F * (R) with the condition where μ i = 2 if i = 0 and μ 1 = 1 2 if i = 1 such that the functional inequality F (x, y, z, w) + α ≺ ϕ(x, y, z, w) + α (3.18) for all x, y, z, w ∈ U .If there exists L = L(i) < 1 such that the function Then there exists a unique 2-variable cubic mapping C : U 2 → V satisfying the functional equation (1.9) and for all x ∈ U .The mapping ψ(x) + α is defined in (3.9) for all x ∈ U .
In both cases, we arrive d(h, T h)≺L 1−i .Therefore (A1) holds.The rest of the proof is similar to that of Theorem 3.2.
The following Corollary is an immediate consequence of Theorem 3.4 concerning the stability of (1.9).Corollary 3.5.Let F : U 2 → V be a mapping and there exits real numbers λ and s such that for all x, y, z, w ∈ U , then there exists a unique 2-variable cubic function C : for all x ∈ U.
Proof.The proof of the corollary is similar tracing as that of Corollary 3.3.Now, we are ready to prove the main fixed point stability results.
Theorem 3.6.Let F : U 2 → V be a mapping for which there exist a function ϕ : U 4 → F * (R) with the conditions (3.1) and (3.17)where  α is defined in (3.9) for all x ∈ U .
The following Corollary is an immediate consequence of Theorem 3.6, using Corollaries 3.3 and 3.5 concerning the stability of (1.9).Corollary 3.7.Let F : U 2 → V be a mapping and there exits real numbers λ and s such that