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On the Ulam stability of mixed type QA mappings in IFN-spaces

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Abstract

We give Ulam-type stability results concerning the quadratic-additive functional equation in intuitionistic fuzzy normed spaces.

1 Introduction

In 1940, Ulam [1] proposed the following stability problem: ‘When is it true that a function which satisfies some functional equation approximately must be close to one satisfying the equation exactly?’. Hyers [2] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Aoki [3] presented a generalization of Hyers results by considering additive mappings, and later on Rassias [4] did for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias has significantly influenced the development of what we now call the Hyers-Ulam-Rassias stability of functional equations. Various extensions, generalizations and applications of the stability problems have been given by several authors so far; see, for example, [524] and references therein.

The notion of intuitionistic fuzzy set introduced by Atanassov [25] has been used extensively in many areas of mathematics and sciences. Using the idea of intuitionistic fuzzy set, Saadati and Park [26] presented the notion of intuitionistic fuzzy normed space which is a generalization of the concept of a fuzzy metric space due to Bag and Samanta [27]. The authors of [2834] defined and studied some summability problems in the setting of an intuitionistic fuzzy normed space.

In the recent past, several Hyers-Ulam stability results concerning the various functional equations were determined in [3546], respectively, in the fuzzy and intuitionistic fuzzy normed spaces. Quite recently, Alotaibi and Mohiuddine [47] established the stability of a cubic functional equation in random 2-normed spaces, while the notion of random 2-normed spaces was introduced by Goleţ [48] and further studied in [4951].

The Hyers-Ulam stability problems of quadratic-additive functional equation

f(x+y+z)+f(x)+f(y)+f(z)=f(x+y)+f(y+z)+f(x+z)

under the approximately even (or odd) condition were established by Jung [52] and the solution of the above functional equation where the range is a field of characteristic 0 was determined by Kannappan [53]. In this paper we determine the stability results concerning the above functional equation in the setting of intuitionistic fuzzy normed spaces. This work indeed presents a relationship between two various disciplines: the theory of fuzzy spaces and the theory of functional equations.

2 Definitions and preliminaries

We shall assume throughout this paper that the symbol denotes the set of all natural numbers.

A binary operation :[0,1]×[0,1][0,1] is said to be a continuous t-norm if it satisfies the following conditions:

(a) is associative and commutative, (b) is continuous, (c) a1=a for all a[0,1], (d) abcd whenever ac and bd for each a,b,c,d[0,1].

A binary operation :[0,1]×[0,1][0,1] is said to be a continuous t-conorm if it satisfies the following conditions:

(a′) is associative and commutative, (b′) is continuous, (c′) a0=a for all a[0,1], (d′) abcd whenever ac and bd for each a,b,c,d[0,1].

The five-tuple (X,μ,ν,,) is said to be intuitionistic fuzzy normed spaces (for short, IFN-spaces) [26] if X is a vector space, is a continuous t-norm, is a continuous t-conorm, and μ, ν are fuzzy sets on X×(0,) satisfying the following conditions. For every x,yX and s,t>0,

  1. (i)

    μ(x,t)+ν(x,t)1,

  2. (ii)

    μ(x,t)>0,

  3. (iii)

    μ(x,t)=1 if and only if x=0,

  4. (iv)

    μ(αx,t)=μ(x, t | α | ) for each α0,

  5. (v)

    μ(x,t)μ(y,s)μ(x+y,t+s),

  6. (vi)

    μ(x,):(0,)[0,1] is continuous,

  7. (vii)

    lim t μ(x,t)=1 and lim t 0 μ(x,t)=0,

  8. (viii)

    ν(x,t)<1,

  9. (ix)

    ν(x,t)=0 if and only if x=0,

  10. (x)

    ν(αx,t)=ν(x, t | α | ) for each α0,

  11. (xi)

    ν(x,t)ν(y,s)ν(x+y,t+s),

  12. (xii)

    ν(x,):(0,)[0,1] is continuous,

  13. (xiii)

    lim t ν(x,t)=0 and lim t 0 ν(x,t)=1.

In this case (μ,ν) is called an intuitionistic fuzzy norm. For simplicity in notation, we denote the intuitionistic fuzzy normed spaces by (X,μ,ν) instead of (X,μ,ν,,). For example, let (X,) be a normed space, and let ab=ab and ab=min{a+b,1} for all a,b[0,1]. For all xX and every t>0, consider

μ(x,t):= t t + x andν(x,t):= x t + x .

Then (X,μ,ν) is an intuitionistic fuzzy normed space.

The notions of convergence and Cauchy sequence in the setting of IFN-spaces were introduced by Saadati and Park [26] and further studied by Mursaleen and Mohiuddine [30].

Let (X,μ,ν) be an intuitionistic fuzzy normed space. Then the sequence x=( x k ) is said to be:

  1. (i)

    Convergent to LX with respect to the intuitionistic fuzzy norm (μ,ν) if, for every ϵ>0 and t>0, there exists k 0 N such that μ( x k L,t)>1ϵ and ν( x k L,t)<ϵ for all k k 0 . In this case, we write (μ,ν)-lim x k =L or x k ( μ , ν ) L as k.

  2. (ii)

    Cauchy sequence with respect to the intuitionistic fuzzy norm (μ,ν) if, for every ϵ>0 and t>0, there exists k 0 N such that μ( x k x ,t)>1ϵ and ν( x k x ,t)<ϵ for all k, k 0 . An IFN-space (X,μ,ν) is said to be complete if every Cauchy sequence in (X,μ,ν) is convergent in the IFN-space. In this case, (X,μ,ν) is called an intuitionistic fuzzy Banach space.

3 Stability of a quadratic-additive functional equation in the IFN-space

We shall assume the following abbreviation throughout this paper:

Df(x,y,z)=f(x+y+z)f(x+y)f(y+z)f(x+z)+f(x)+f(y)+f(z).

Theorem 3.1 Let X be a linear space and (X,μ,ν) be an IFN-space. Suppose that f is an intuitionistic fuzzy q-almost quadratic-additive mapping from (X,μ,ν) to an intuitionistic fuzzy Banach space (Y, μ , ν ) such that

μ ( D f ( x , y , z ) , s + t + u ) μ ( x , s q ) μ ( y , t q ) μ ( z , u q ) and ν ( D f ( x , y , z ) , s + t + u ) ν ( x , s q ) ν ( y , t q ) ν ( z , u q ) }
(3.1)

for all x,y,zX and s,t,u>0, where q is a positive real number with q 1 2 ,1. Then there exists a unique quadratic-additive mapping T:XY such that

μ ( T ( x ) f ( x ) , t ) { sup t < t μ ( x , ( 2 2 p 3 ) q t q ) if q > 1 , sup t < t μ ( x , ( ( 4 2 p ) ( 2 2 p ) 6 ) q t q ) if 1 2 < q < 1 , sup t < t μ ( x , ( 2 p 4 3 ) q t q ) if 0 < q < 1 2 and ν ( T ( x ) f ( x ) , t ) { sup t < t ν ( x , ( 2 2 p 3 ) q t q ) if q > 1 , sup t < t ν ( x , ( ( 4 2 p ) ( 2 2 p ) 6 ) q t q ) if 1 2 < q < 1 , sup t < t ν ( x , ( 2 p 4 3 ) q t q ) if 0 < q < 1 2 , }
(3.2)

for all xX and all t>0 with t (0,t), where p=1/q.

Proof Putting x=0=y=z in (3.1), it follows that

μ ( f ( 0 ) , t ) μ ( 0 , ( t / 3 ) q ) μ ( 0 , ( t / 3 ) q ) μ ( 0 , ( t / 3 ) q ) =1

and

ν ( f ( 0 ) , t ) ν ( 0 , ( t / 3 ) q ) ν ( 0 , ( t / 3 ) q ) ν ( 0 , ( t / 3 ) q ) =0

for all t>0. Using the definition of IFN-space, we have f(0)=0. Now we are ready to prove our theorem for three cases. We consider the cases as q>1, 1 2 <q<1 and 0<q< 1 2 .

Case 1. Let q>1. Consider a mapping J n f:XY to be such that

J n f(x)= 1 2 ( 4 n ( f ( 2 n x ) + f ( 2 n x ) ) + 2 n ( f ( 2 n x ) f ( 2 n x ) ) )

for all xX. Notice that J 0 f(x)=f(x) and

J j f ( x ) J j + 1 f ( x ) = D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 + D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 + D f ( 2 j x , 2 j x , 2 j x ) 2 j + 2 D f ( 2 j x , 2 j x , 2 j x ) 2 j + 2
(3.3)

for all xX and j0. Using the definition of IFN-space and (3.1), this equation implies that if n+m>m0, then

μ ( J m f ( x ) J n + m f ( x ) , j = m n + m 1 3 2 ( 2 p 2 ) j t p ) = μ ( j = m n + m 1 ( J j f ( x ) J j + 1 f ( x ) ) , j = m n + m 1 3 2 j p 2 j + 1 t p ) j = m n + m 1 μ ( J j ( f ( x ) J j + 1 f ( x ) ) , 3 2 j p 2 j + 1 ) j = m n + m 1 { μ ( ( 2 j + 1 + 1 ) D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 , 3 ( 2 j + 1 + 1 ) 2 j p t p 2 4 j + 1 ) μ ( 1 ( 2 j + 1 ) D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 , 3 ( 2 j + 1 1 ) 2 j p t p 2 4 j + 1 ) } j = m n + m 1 μ ( 2 j x , 2 j t ) = μ ( x , t )
(3.4)

and

ν ( J m f ( x ) J n + m f ( x ) , j = m n + m 1 3 2 ( 2 p 2 ) j t p ) = ν ( j = m n + m 1 ( J j f ( x ) J j + 1 f ( x ) ) , j = m n + m 1 3 2 j p 2 j + 1 t p ) j = m n + m 1 ν ( J j ( f ( x ) J j + 1 f ( x ) ) , 3 2 j p 2 j + 1 ) j = m n + m 1 { ν ( ( 2 j + 1 + 1 ) D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 , 3 ( 2 j + 1 + 1 ) 2 j p t p 2 4 j + 1 ) ν ( 1 ( 2 j + 1 ) D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 , 3 ( 2 j + 1 1 ) 2 j p t p 2 4 j + 1 ) } j = m n + m 1 ν ( 2 j x , 2 j t ) = ν ( x , t )
(3.5)

for all xX and t>0, where j = 1 n a j = a 1 a 2 a n , j = 1 n a j = a 1 a 2 a n . Let ϵ>0 and δ>0 be given. Since lim t μ(x,t)=1 and lim t ν(x,t)=0, there exists t 0 >0 such that μ(x, t 0 )1ϵ and ν(x, t 0 )ϵ for all xX. We observe that for some t ˜ > t 0 , the series j = 0 3 2 j p 2 j + 1 t ˜ p converges for p= 1 q <1, there exists some n 0 0 such that j = m n + m 1 3 2 j p 2 j + 1 t ˜ p <δ for each m n 0 and n>0. Using (3.4) and (3.5), we have

μ ( J m f ( x ) J n + m f ( x ) , δ ) μ ( J m f ( x ) J n + m f ( x ) , j = m n + m 1 3 2 j p 2 j + 1 t ˜ p ) μ ( x , t ˜ ) μ ( x , t 0 ) 1 ϵ

and

ν ( J m f ( x ) J n + m f ( x ) , δ ) ν ( J m f ( x ) J n + m f ( x ) , j = m n + m 1 3 2 j p 2 j + 1 t ˜ p ) ν(x, t ˜ )ν(x, t 0 )ϵ

for all xX and δ>0. Hence { J n f(x)} is a Cauchy sequence in the fuzzy Banach space (Y, μ , ν ). Thus, we define a mapping T:XY such that T(x):=( μ , ν ) lim n J n f(x) for all xX. Moreover, if we put m=0 in (3.4) and (3.5), we get

μ ( f ( x ) J n f ( x ) , t ) μ ( x , t q ( j = 0 n 1 3 2 j p 2 j + 1 ) q ) and ν ( f ( x ) J n f ( x ) , t ) ν ( x , t q ( j = 0 n 1 3 2 j p 2 j + 1 ) q ) }
(3.6)

for all xX and t>0. Now we have to show that T is quadratic additive. Let x,y,zX. Then

μ ( D T ( x , y , z ) , t ) μ ( ( T J n f ) ( x + y + z ) , t 28 ) μ ( ( T J n f ) ( x ) , t 28 ) μ ( ( T J n f ) ( y ) , t 28 ) μ ( ( T J n f ) ( z ) , t 28 ) μ ( ( J n f T ) ( x + y ) , t 28 ) μ ( ( J n f T ) ( x + z ) , t 28 ) μ ( ( J n f T ) ( y + z ) , t 28 ) μ ( D J n f ( x , y , z ) , 3 t 4 )
(3.7)

and

ν ( D T ( x , y , z ) , t ) ν ( ( T J n f ) ( x + y + z ) , t 28 ) ν ( ( T J n f ) ( x ) , t 28 ) ν ( ( T J n f ) ( y ) , t 28 ) ν ( ( T J n f ) ( z ) , t 28 ) ν ( ( J n f T ) ( x + y ) , t 28 ) ν ( ( J n f T ) ( x + z ) , t 28 ) ν ( ( J n f T ) ( y + z ) , t 28 ) ν ( D J n f ( x , y , z ) , 3 t 4 )
(3.8)

for all t>0 and nN. Taking the limit as n in the inequalities (3.7) and (3.8), we can see that first seven terms on the right-hand side of (3.7) and (3.8) tend to 1 and 0, respectively, by using the definition of T. It is left to find the value of the last term on the right-hand side of (3.7) and (3.8). By using the definition of J n f(x), write

μ ( D J n f ( x , y , z ) , 3 t 4 ) μ ( D f ( 2 n x , 2 n y , 2 n z ) 2 4 n , 3 t 16 ) μ ( D f ( 2 n x , 2 n y , 2 n z ) 2 4 n , 3 t 16 ) μ ( D f ( 2 n x , 2 n y , 2 n z ) 2 2 n , 3 t 16 ) μ ( D f ( 2 n x , 2 n y , 2 n z ) 2 2 n , 3 t 16 )
(3.9)

and, similarly,

ν ( D J n f ( x , y , z ) , 3 t 4 ) ν ( D f ( 2 n x , 2 n y , 2 n z ) 2 4 n , 3 t 16 ) ν ( D f ( 2 n x , 2 n y , 2 n z ) 2 4 n , 3 t 16 ) ν ( D f ( 2 n x , 2 n y , 2 n z ) 2 2 n , 3 t 16 ) ν ( D f ( 2 n x , 2 n y , 2 n z ) 2 2 n , 3 t 16 )
(3.10)

for all x,y,zX, t>0 and nN. Also, from (3.1), we have

μ ( D f ( ± 2 n x , ± 2 n y , ± 2 n z ) 2 4 n , 3 t 16 ) = μ ( D f ( ± 2 n x , ± 2 n y , ± 2 n z ) , 3 4 n t 8 ) μ ( 2 n x , ( 4 n t 8 ) q ) μ ( 2 n y , ( 4 n t 8 ) q ) μ ( 2 n z , ( 4 n t 8 ) q ) μ ( x , 2 ( 2 q 1 ) n 3 q t q ) μ ( y , 2 ( 2 q 1 ) n 3 q t q ) μ ( z , 2 ( 2 q 1 ) n 3 q t q )
(3.11)

and

μ ( D f ( ± 2 n x , ± 2 n y , ± 2 n z ) 2 2 n , 3 t 16 ) μ ( x , 2 ( 2 q 1 ) n 3 q t q ) μ ( y , 2 ( 2 q 1 ) n 3 q t q ) μ ( z , 2 ( 2 q 1 ) n 3 q t q )
(3.12)

for all x,y,zX, t>0 and nN. Since q>1, therefore (3.9) tends to 1 as n with the help of (3.11) and (3.12). Similarly, by proceeding along the same lines as in (3.11) and (3.12), we can show that (3.10) tends to 0 as n. Thus, inequalities (3.7) and (3.8) become

μ ( D T ( x , y , z ) , t ) =1and ν ( D T ( x , y , z ) , t ) =0

for all x,y,zX and t>0. Accordingly, DT(x,y,z)=0 for all x,y,zX. Now we approximate the difference between f and T in a fuzzy sense. Choose ϵ(0,1) and 0< t <t. Since T is the intuitionistic fuzzy limit of { J n f(x)} such that

μ ( T ( x ) J n f ( x ) , t t ) 1ϵand ν ( T ( x ) J n f ( x ) , t t ) ϵ

for all xX, t>0 and nN. From (3.6), we have

μ ( T ( x ) f ( x ) , t ) μ ( T ( x ) J n f ( x ) , t t ) μ ( J n f ( x ) f ( x ) , t ) ( 1 ϵ ) μ ( x , t q ( j = 0 n 1 3 2 j p 2 j + 1 ) q ) ( 1 ϵ ) μ ( x , ( ( 2 2 p ) t 3 ) q )

and

ν ( T ( x ) f ( x ) , t ) ν ( T ( x ) J n f ( x ) , t t ) ν ( J n f ( x ) f ( x ) , t ) ( 1 ϵ ) ν ( x , ( ( 2 2 p ) t 3 ) q ) .

Since ϵ(0,1) is arbitrary, we get the inequality (3.2) in this case.

To prove the uniqueness of T, assume that T is another quadratic-additive mapping from X into Y, which satisfies the required inequality, i.e., (3.2). Then, by (3.3), for all xX and n N ,

T ( x ) J n T ( x ) = j = 0 n 1 ( J j T ( x ) J j + 1 T ( x ) ) = 0 , T ( x ) J n T ( x ) = j = 0 n 1 ( J j T ( x ) J j + 1 T ( x ) ) = 0 . }
(3.13)

Therefore

μ ( T ( x ) T ( x ) , t ) = μ ( J n T ( x ) J n T ( x ) , t ) μ ( J n T ( x ) J n f ( x ) , t 2 ) μ ( J n f ( x ) J n T ( x ) , t 2 ) μ ( ( T f ) ( 2 n x ) 2 4 n , t 8 ) μ ( ( f T ) ( 2 n x ) 2 4 n , t 8 ) μ ( ( T f ) ( 2 n x ) 2 4 n , t 8 ) μ ( ( f T ) ( 2 n x ) 2 4 n , t 8 ) μ ( ( T f ) ( 2 n x ) 2 2 n , t 8 ) μ ( ( f T ) ( 2 n x ) 2 2 n , t 8 ) μ ( ( T f ) ( 2 n x ) 2 2 n , t 8 ) μ ( ( f T ) ( 2 n x ) 2 2 n , t 8 ) sup t < t μ ( x , 2 ( q 1 ) n 2 q ( 2 2 p 3 ) q t q )

and

ν ( T ( x ) T ( x ) , t ) = ν ( J n T ( x ) J n T ( x ) , t ) ν ( J n T ( x ) J n f ( x ) , t 2 ) ν ( J n f ( x ) J n T ( x ) , t 2 ) ν ( ( T f ) ( 2 n x ) 2 4 n , t 8 ) ν ( ( f T ) ( 2 n x ) 2 4 n , t 8 ) ν ( ( T f ) ( 2 n x ) 2 4 n , t 8 ) ν ( ( f T ) ( 2 n x ) 2 4 n , t 8 ) ν ( ( T f ) ( 2 n x ) 2 2 n , t 8 ) ν ( ( f T ) ( 2 n x ) 2 2 n , t 8 ) ν ( ( T f ) ( 2 n x ) 2 2 n , t 8 ) ν ( ( f T ) ( 2 n x ) 2 2 n , t 8 ) sup t < t ν ( x , 2 ( q 1 ) n 2 q ( 2 2 p 3 ) q t q )

for all xX, t>0 and nN. Since q=1/p>1 and taking limit as n in the last two inequalities, we get μ (T(x) T (x),t)=1 and ν (T(x) T (x),t)=0 for all xX and t>0. Hence T(x)= T (x) for all xX.

Case 2. Let 1 2 <q<1. Consider a mapping J n f:XY to be such that

J n f(x)= 1 2 ( 4 n ( f ( 2 n x ) + f ( 2 n x ) ) + 2 n ( f ( x 2 n ) f ( x 2 n ) ) )

for all xX. Then J 0 f(x)=f(x) and

J j f ( x ) J j + 1 f ( x ) = D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 + D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 2 j 1 ( D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) )

for all xX and j0. Thus, for each n+m>m0, we have

μ ( J m f ( x ) J n + m f ( x ) , j = m n + m 1 ( 3 4 ( 2 p 4 ) j + 3 2 p ( 2 2 p ) j ) t p ) j = m n + m 1 { μ ( D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 , 3 2 j p t p 2 4 j + 1 ) μ ( D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 , 3 2 j p t p 2 4 ( j + 1 ) ) μ ( 2 j 1 D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) , 3 2 j 1 t p 2 ( j + 1 ) p ) μ ( 2 j 1 D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) , 3 2 j 1 t p 2 ( j + 1 ) p ) } j = m n + m 1 { μ ( 2 j x , 2 j t ) μ ( x 2 j + 1 , t 2 j + 1 ) } = μ ( x , t ) and ν ( J m f ( x ) J n + m f ( x ) , j = m n + m 1 ( 3 4 ( 2 p 4 ) j + 3 2 p ( 2 2 p ) j ) t p ) j = m n + m 1 { ν ( D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 , 3 2 j p t p 2 4 j + 1 ) ν ( D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 , 3 2 j p t p 2 4 ( j + 1 ) ) ν ( 2 j 1 D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) , 3 2 j 1 t p 2 ( j + 1 ) p ) ν ( 2 j 1 D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) , 3 2 j 1 t p 2 ( j + 1 ) p ) } j = m n + m 1 { ν ( 2 j x , 2 j t ) ν ( x 2 j + 1 , t 2 j + 1 ) } = ν ( x , t ) ,

where ∏ and are the same as in Case 1. Proceeding along a similar argument as in Case 1, we see that { J n f(x)} is a Cauchy sequence in (Y, μ , ν ). Thus, we define T(x):=( μ , ν )- lim n J n f(x) for all xX. Putting m=0 in the last two inequalities, we get

μ ( f ( x ) J n f ( x ) , t ) μ ( x , t p ( j = 0 n 1 ( 3 4 ( 2 p 4 ) j + 3 2 p ( 2 2 p ) j ) ) q ) and ν ( f ( x ) J n f ( x ) , t ) ν ( x , t p ( j = 0 n 1 ( 3 4 ( 2 p 4 ) j + 3 2 p ( 2 2 p ) j ) ) q ) }
(3.14)

for all xX and t>0. To prove that t is a quadratic-additive function, it is enough to show that the last term on the right-hand side of (3.7) and (3.8) tends to 1 and 0, respectively, as n. Using the definition of J n f(x) and (3.1), we obtain

μ ( D J n f ( x , y , z ) , 3 t 4 ) μ ( D f ( 2 n x , 2 n y , 2 n z ) 2 4 n , 3 t 16 ) μ ( D f ( 2 n x , 2 n y , 2 n z ) 2 4 n , 3 t 16 ) μ ( 2 n 1 D f ( x 2 n , y 2 n , z 2 n ) , 3 t 16 ) μ ( 2 n 1 D f ( x 2 n , y 2 n , z 2 n ) , 3 t 16 ) μ ( x , 2 ( 2 q 1 ) n 3 q t q ) μ ( y , 2 ( 2 q 1 ) n 3 q t q ) μ ( z , 2 ( 2 q 1 ) n 3 q t q ) μ ( x , 2 ( 1 q ) n 3 q t q ) μ ( y , 2 ( 1 q ) n 3 q t q ) μ ( z , 2 ( 1 q ) n 3 q t q )
(3.15)

and

ν ( D J n f ( x , y , z ) , 3 t 4 ) ν ( D f ( 2 n x , 2 n y , 2 n z ) 2 4 n , 3 t 16 ) ν ( D f ( 2 n x , 2 n y , 2 n z ) 2 4 n , 3 t 16 ) ν ( 2 n 1 D f ( x 2 n , y 2 n , z 2 n ) , 3 t 16 ) ν ( 2 n 1 D f ( x 2 n , y 2 n , z 2 n ) , 3 t 16 ) ν ( x , 2 ( 2 q 1 ) n 3 q t q ) ν ( y , 2 ( 2 q 1 ) n 3 q t q ) ν ( z , 2 ( 2 q 1 ) n 3 q t q ) ν ( x , 2 ( 1 q ) n 3 q t q ) ν ( y , 2 ( 1 q ) n 3 q t q ) ν ( z , 2 ( 1 q ) n 3 q t q )
(3.16)

for each x,y,zX, t>0 and nN. Since 1/2<q<1 and taking the limit as n, we see that (3.15) and (3.16) tend to 1 and 0, respectively. As in Case 1, we have DT(x,y,z)=0 for all x,y,zX. Using the same argument as in Case 1, we see that (3.2) follows from (3.14). To prove the uniqueness of T, assume that T is another quadratic-additive mapping from X into Y satisfying (3.2). Using (3.2) and (3.13), we have

μ ( T ( x ) T ( x ) , t ) = μ ( J n T ( x ) J n T ( x ) , t ) μ ( J n T ( x ) J n f ( x ) , t 2 ) μ ( J n f ( x ) J n T ( x ) , t 2 ) μ ( ( T f ) ( 2 n x ) 2 4 n , t 8 ) μ ( ( f T ) ( 2 n x ) 2 4 n , t 8 ) μ ( ( T f ) ( 2 n x ) 2 4 n , t 8 ) μ ( ( f T ) ( 2 n x ) 2 4 n , t 8 ) μ ( 2 n 1 ( ( T f ) ( x 2 n ) ) , t 8 ) μ ( 2 n 1 ( ( f T ) ( x 2 n ) ) , t 8 ) μ ( 2 n 1 ( ( T f ) ( x 2 n ) ) , t 8 ) μ ( 2 n 1 ( ( f T ) ( x 2 n ) ) , t 8 ) sup t < t μ ( x , 2 ( 2 q 1 ) n 2 q ( ( 4 2 p ) ( 2 p 2 ) 6 ) q t q ) sup t < t μ ( x , 2 2 ( 1 q ) n 2 q ( ( 4 2 p ) ( 2 p 2 ) 6 ) q t q )
(3.17)

and

ν ( T ( x ) T ( x ) , t ) ν ( J n T ( x ) J n f ( x ) , t 2 ) ν ( J n f ( x ) J n T ( x ) , t 2 ) ν ( ( T f ) ( 2 n x ) 2 4 n , t 8 ) ν ( ( f T ) ( 2 n x ) 2 4 n , t 8 ) ν ( ( T f ) ( 2 n x ) 2 4 n , t 8 ) ν ( ( f T ) ( 2 n x ) 2 4 n , t 8 ) ν ( 2 n 1 ( ( T f ) ( x 2 n ) ) , t 8 ) ν ( 2 n 1 ( ( f T ) ( x 2 n ) ) , t 8 ) ν ( 2 n 1 ( ( T f ) ( x 2 n ) ) , t 8 ) ν ( 2 n 1 ( ( f T ) ( x 2 n ) ) , t 8 ) sup t < t μ ( x , 2 ( 2 q 1 ) n 2 q ( ( 4 2 p ) ( 2 p 2 ) 6 ) q t q ) sup t < t μ ( x , 2 2 ( 1 q ) n 2 q ( ( 4 2 p ) ( 2 p 2 ) 6 ) q t q )
(3.18)

for all xX, t>0 and nN. Letting n in (3.17) and (3.18), and using the fact that lim n 2 ( 2 q 1 ) n 2 q = lim n 2 ( 1 q ) n 2 q = together with the definition of IFN-space, we get μ (T(x) T (x),t)=1 and ν (T(x) T (x),t)=0 for all xX and t>0. Hence T(x)= T (x) for all xX.

Case 3. Let 0<q< 1 2 . Define a mapping J n f:XY by

J n f(x)= 1 2 ( 4 n ( f ( 2 n x ) + f ( 2 n x ) ) + 2 n ( f ( x 2 n ) f ( x 2 n ) ) )

for all xX. In this case, J 0 f(x)=f(x) and

J j f ( x ) J j + 1 f ( x ) = 4 j 2 ( D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) + D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) ) 2 j 1 ( D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) )

for all xX and j0. Thus, for each n+m>m0, we have

μ ( J m f ( x ) J n + m f ( x ) j = m n + m 1 3 2 p ( 4 2 p ) j t p ) j = m n + m 1 { μ ( ( 4 j + 2 j ) D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) 2 , 3 ( 4 j + 2 j ) t p 2 2 ( j + 1 ) p ) μ ( ( 4 j 2 j ) D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) 2 , 3 ( 4 j 2 j ) t p 2 2 ( j + 1 ) p ) } j = m n + m 1 μ ( x 2 j + 1 , t 2 j + 1 ) = μ ( x , t ) and ν ( J m f ( x ) J n + m f ( x ) j = m n + m 1 3 2 p ( 4 2 p ) j t p ) j = m n + m 1 { ν ( ( 4 j + 2 j ) D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) 2 , 3 ( 4 j + 2 j ) t p 2 2 ( j + 1 ) p ) ν ( ( 4 j 2 j ) D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) 2 , 3 ( 4 j 2 j ) t p 2 2 ( j + 1 ) p ) } j = m n + m 1 ν ( x 2 j + 1 , t 2 j + 1 ) = ν ( x , t )

for all xX and t>0. Proceeding along a similar argument as in the previous cases, we see that { J n f(x)} is a Cauchy sequence in (Y, μ , ν ). Thus, we define T(x):=( μ , ν ) lim n J n f(x) for all xX. Putting m=0 in the last two inequalities, we get

μ ( f ( x ) J n f ( x ) , t ) μ ( x , t q ( j = 0 n 1 3 2 p ( 4 2 p ) j ) q ) and ν ( f ( x ) J n f ( x ) , t ) ν ( x , t q ( j = 0 n 1 3 2 p ( 4 2 p ) j ) q ) }
(3.19)

for all xX and t>0. Write

μ ( D J n f ( x , y , z ) , 3 t 4 ) μ ( 4 n 2 D f ( x 2 n , y 2 n , z 2 n ) , 3 t 16 ) μ ( 4 n 2 D f ( x 2 n , y 2 n , z 2 n ) , 3 t 16 ) μ ( 2 n 1 D f ( x 2 n , y 2 n , z 2 n ) , 3 t 16 ) μ ( 2 n 1 D f ( x 2 n , y 2 n , z 2 n ) , 3 t 16 ) μ ( x , 2 ( 1 2 q ) n 3 q t q ) μ ( y , 2 ( 1 2 q ) n 3 q t q ) μ ( z , 2 ( 1 2 q ) n 3 q t q ) μ ( x , 2 ( 1 q ) n 3 q t q ) μ ( y , 2 ( 1 q ) n 3 q t q ) μ ( z , 2 ( 1 q ) n 3 q t q )
(3.20)

and

ν ( D J n f ( x , y , z ) , 3 t 4 ) ν ( 4 n 2 D f ( x 2 n , y 2 n , z 2 n ) , 3 t 16 ) ν ( 4 n 2 D f ( x 2 n , y 2 n , z 2 n ) , 3 t 16 ) ν ( 2 n 1 D f ( x 2 n , y 2 n , z 2 n ) , 3 t 16 ) ν ( 2 n 1 D f ( x 2 n , y 2 n , z 2 n ) , 3 t 16 ) ν ( x , 2 ( 1 2 q ) n 3 q t q ) ν ( y , 2 ( 1 2 q ) n 3 q t q ) ν ( z , 2 ( 1 2 q ) n 3 q t q ) ν ( x , 2 ( 1 q ) n 3 q t q ) ν ( y , 2 ( 1 q ) n 3 q t q ) ν ( z , 2 ( 1 q ) n 3 q t q )
(3.21)

for all x,y,zX, t>0 and nN. Since 1/2<q<1 and taking the limit as n, we see that (3.20) and (3.21) tend to 1 and 0, respectively. As in the previous cases, we have that DT(x,y,z)=0 for all x,y,zX. By the same argument as in previous cases, we can see that (3.2) follows from (3.19). To prove the uniqueness of T, assume that T is another quadratic-additive mapping from X into Y satisfying (3.2). From (3.2) and (3.13), for all xX and t>0, write

μ ( T ( x ) T ( x ) , t ) = ν ( J n T ( x ) J n T ( x ) , t ) μ ( J n T ( x ) J n f ( x ) , t 2 ) μ ( J n f ( x ) J n T ( x ) , t 2 ) μ ( 4 n 2 ( ( T f ) ( x 2 n ) ) , t 8 ) μ ( 4 n 2 ( ( f T ) ( x 2 n ) ) , t 8 ) μ ( 4 n 2 ( ( T f ) ( x 2 n ) ) , t 8 ) μ ( 4 n 2 ( ( f T ) ( x 2 n ) ) , t 8 ) μ ( 2 n 1 ( ( T f ) ( x 2 n ) ) , t 8 ) μ ( 2 n 1 ( ( f T ) ( x 2 n ) ) , t 8 ) μ ( 2 n 1 ( ( T f ) ( x 2 n ) ) , t 8 ) μ ( 2 n 1 ( ( f T ) ( x 2 n ) ) , t 8 ) sup t < t μ ( x , 2 ( 1 2 q ) n 2 q ( 2 p 4 3 ) q t q )

and, similarly,

ν ( T ( x ) T ( x ) , t ) ν ( 4 n 2 ( ( T f ) ( x 2 n ) ) , t 8 ) ν ( 4 n 2 ( ( f T ) ( x 2 n ) ) , t 8 ) ν ( 4 n 2 ( ( T f ) ( x 2 n ) ) , t 8 ) ν ( 4 n 2 ( ( f T ) ( x 2 n ) ) , t 8 ) ν ( 2 n 1 ( ( T f ) ( x 2 n ) ) , t 8 ) ν ( 2 n 1 ( ( f T ) ( x 2 n ) ) , t 8 ) ν ( 2 n 1 ( ( T f ) ( x 2 n ) ) , t 8 ) ν ( 2 n 1 ( ( f T ) ( x 2 n ) ) , t 8 ) sup t < t ν ( x , 2 ( 1 2 q ) n 2 q ( 2 p 4 3 ) q t q )

for nN. Letting n in (3.17) and (3.18), and using the fact that lim n 2 ( 2 q 1 ) n 2 q = lim n 2 ( 1 q ) n 2 q = together with the definition of IFN-space, we get μ (T(x) T (x),t)=1 and ν (T(x) T (x),t)=0 for all xX and t>0. Hence T(x)= T (x) for all xX. □

Remark 3.2 Let (X,μ,ν) be an IFN-space and (X,μ,ν) be an intuitionistic fuzzy Banach space (Y, μ , ν ). Let f:XY be a mapping satisfying (3.1) with a real number q<0 and for all t>0. If we choose a real number α with 0<3α<t, then

μ ( D f ( x , y , z ) , t ) μ ( D f ( x , y , z ) , 3 α ) μ ( x , α q ) μ ( y , α q ) μ ( z , α q ) and ν ( D f ( x , y , z ) , t ) ν ( D f ( x , y , z ) , 3 α ) ν ( x , α q ) ν ( y , α q ) ν ( z , α q )

for all x,y,zX, t>0 and q<0. Since q<0, we have lim α 0 + α q =. This implies that

lim α 0 + μ ( x , α q ) = 1 = lim α 0 + μ ( y , α q ) = lim α 0 + μ ( z , α q ) and lim α 0 + ν ( x , α q ) = 0 = lim α 0 + ν ( y , α q ) = lim α 0 + ν ( z , α q ) .

Thus, we have μ (Df(x,y,z),t)=1 and ν (Df(x,y,z),t)=0 for all x,y,zX and t>0. Hence Df(x,y,z)=0 for all x,y,zX. In other words, if f is an intuitionistic fuzzy q-almost quadratic-additive mapping for the case q<0, then f is itself a quadratic-additive mapping.

Corollary 3.3 Suppose that f is an even mapping satisfying the conditions of Theorem 3.1. Then there exists a unique quadratic mapping T:XY such that

μ ( T ( x ) f ( x ) , t ) sup t < t μ ( x , ( | 4 2 p | t 3 ) q ) and ν ( T ( x ) f ( x ) , t ) sup t < t ν ( x , ( | 4 2 p | t 3 ) q ) }
(3.22)

for all xX and t>0, where p=1/q.

Proof Since f is an even mapping, we get

J n f(x)={ f ( 2 n x ) + f ( 2 n x ) 2 4 n if  q > 1 2 , 1 2 ( 4 n ( f ( 2 n x ) + f ( 2 n x ) ) ) if  0 < q < 1 2 ,

for all xX, where J n f is defined as in Theorem 3.1. In this case, J 0 f(x)=f(x). For all xX and jN{0}, we have

J j f(x) J j + 1 f(x)={ D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 + D f ( 2 j x , 2 j x , 2 j x ) 2 4 j + 1 if  q > 1 2 , 4 j 2 ( D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) + D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) ) if  0 < q < 1 2 .

Proceeding along the same lines as in Theorem 3.1, we obtain that T is a quadratic-additive function satisfying (3.22). Notice that T(x):=( μ , ν ) lim n J n f(x), T is even and DT(x,y,z)=0 for all x,y,zX. Hence, we get

T(x+y)+T(xy)2T(x)2T(y)=DT(x,y,x)=0

for all x,yX. It follows that T is a quadratic mapping. □

Corollary 3.4 Suppose that f is an even mapping satisfying the conditions of Theorem 3.1. Then there exists a unique additive mapping T:XY such that

μ ( T ( x ) f ( x ) , t ) sup t < t μ ( x , ( | 2 2 p | t 3 ) q ) and ν ( T ( x ) f ( x ) , t ) sup t < t ν ( x , ( | 2 2 p | t 3 ) q ) }
(3.23)

for all xX and t>0, where p=1/q.

Proof Since f is an odd mapping, we get

J n f(x)={ f ( 2 n x ) + f ( 2 n x ) 2 n + 1 if  q > 1 , 2 n 1 ( f ( 2 n x ) + f ( 2 n x ) ) if  0 < q < 1 ,

for all xX, where J n f is defined as in Theorem 3.1. Here J 0 f(x)=f(x). For all xX and jN{0}, we have

J j f(x) J j + 1 f(x)={ D f ( 2 j x , 2 j x , 2 j x ) 2 j + 2 D f ( 2 j x , 2 j x , 2 j x ) 2 j + 2 if  q > 1 , 2 j 1 ( D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) D f ( x 2 j + 1 , x 2 j + 1 , x 2 j + 1 ) ) if  0 < q < 1 .

Proceeding along the same lines as in Theorem 3.1, we obtain that T is a quadratic-additive function satisfying (3.23). Here T(x):=( μ , ν ) lim n J n f(x), T is odd and DT(x,y,z)=0 for all x,y,zX. Hence, we obtain

T(x+y)T(x)T(y)=Df ( x y 2 , x + y 2 , x + y 2 ) =0

for all x,yX. It follows that T is an additive mapping. □

References

  1. 1.

    Ulam SM: A Collection of the Mathematical Problems. Interscience, New York; 1960.

  2. 2.

    Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222-224. 10.1073/pnas.27.4.222

  3. 3.

    Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 64-66. 10.2969/jmsj/00210064

  4. 4.

    Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297-300. 10.1090/S0002-9939-1978-0507327-1

  5. 5.

    Rassias TM: On the stability of functional equations and a problem of Ulam. Acta Appl. Math. 2000, 62: 123-130.

  6. 6.

    Agarwal RP, Xu B, Zhang W: Stability of functional equations in single variable. J. Math. Anal. Appl. 2003, 288: 852-869. 10.1016/j.jmaa.2003.09.032

  7. 7.

    Gajda Z: On stability of additive mappings. Int. J. Math. Math. Sci. 1991, 14: 431-434. 10.1155/S016117129100056X

  8. 8.

    Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431-436. 10.1006/jmaa.1994.1211

  9. 9.

    Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.

  10. 10.

    Isac G, Rassias TM: On the Hyers-Ulam stability of ψ -additive mappings. J. Approx. Theory 1993, 72: 131-137. 10.1006/jath.1993.1010

  11. 11.

    Najati A, Park C: On the stability of an n -dimensional functional equation originating from quadratic forms. Taiwan. J. Math. 2008, 12: 1609-1624.

  12. 12.

    Lu G, Park C: Additive functional inequalities in Banach spaces. J. Inequal. Appl. 2012., 2012: Article ID 294

  13. 13.

    Rassias JM, Kim H-M: Generalized Hyers-Ulam stability for general additive functional equations in quasi- β -normed spaces. J. Math. Anal. Appl. 2009, 356: 302-309. 10.1016/j.jmaa.2009.03.005

  14. 14.

    Rassias JM: Solution of a problem of Ulam. J. Approx. Theory 1989, 57: 268-273. 10.1016/0021-9045(89)90041-5

  15. 15.

    Rassias JM: On the Ulam stability of mixed type mappings on restricted domains. J. Math. Anal. Appl. 2002, 276: 747-762. 10.1016/S0022-247X(02)00439-0

  16. 16.

    Dadipour F, Moslehian MS, Rassias JM, Takahasi S-E: Characterization of a generalized triangle inequality in normed spaces. Nonlinear Anal. 2012, 75: 735-741. 10.1016/j.na.2011.09.004

  17. 17.

    Eskandani GZ, Rassias JM, Gavruta P: Generalized Hyers-Ulam stability for a general cubic functional equation in quasi- β -normed spaces. Asian-Eur. J. Math. 2011, 4: 413-425. 10.1142/S1793557111000332

  18. 18.

    Faziev V, Sahoo PK: On the stability of Jensen’s functional equation on groups. Proc. Indian Acad. Sci. Math. Sci. 2007, 117: 31-48. 10.1007/s12044-007-0003-3

  19. 19.

    Gordji ME, Khodaei H, Rassias JM: Fixed point methods for the stability of general quadratic functional equation. Fixed Point Theory 2011, 12: 71-82.

  20. 20.

    Jun KW, Kim HM: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 2002, 274: 867-878. 10.1016/S0022-247X(02)00415-8

  21. 21.

    Jung SM: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor; 2001.

  22. 22.

    Ravi K, Arunkumar M, Rassias JM: Ulam stability for the orthogonally general Euler-Lagrange type functional equation. Int. J. Math. Stat. 2008, 3(A08):36-46.

  23. 23.

    Saadati R, Park C: Non-Archimedean -fuzzy normed spaces and stability of functional equations. Comput. Math. Appl. 2010, 60: 2488-2496. 10.1016/j.camwa.2010.08.055

  24. 24.

    Xu TZ, Rassias MJ, Xu WX, Rassias JM: A fixed point approach to the intuitionistic fuzzy stability of quintic and sextic functional equations. Iranian J. Fuzzy Sys. 2012, 9: 21-40.

  25. 25.

    Atanassov, K: Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, June 1983 (Deposed in Central Science-Technical Library of Bulg. Academy of Science, 1697/84) (in Bulgarian)

  26. 26.

    Saadati R, Park JH: On the intuitionistic fuzzy topological spaces. Chaos Solitons Fractals 2006, 27: 331-344. 10.1016/j.chaos.2005.03.019

  27. 27.

    Bag T, Samanta SK: Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 2003, 11(3):687-705.

  28. 28.

    Mohiuddine SA, Danish Lohani QM: On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos Solitons Fractals 2009, 42: 1731-1737. 10.1016/j.chaos.2009.03.086

  29. 29.

    Mursaleen M, Karakaya V, Mohiuddine SA: Schauder basis, separability, and approximation property in intuitionistic fuzzy normed space. Abstr. Appl. Anal. 2010., 2010: Article ID 131868

  30. 30.

    Mursaleen M, Mohiuddine SA: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2009, 41: 2414-2421. 10.1016/j.chaos.2008.09.018

  31. 31.

    Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2009, 233(2):142-149. 10.1016/j.cam.2009.07.005

  32. 32.

    Mursaleen M, Mohiuddine SA: Nonlinear operators between intuitionistic fuzzy normed spaces and Fréchet differentiation. Chaos Solitons Fractals 2009, 42: 1010-1015. 10.1016/j.chaos.2009.02.041

  33. 33.

    Mursaleen M, Mohiuddine SA, Edely OHH: On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces. Comput. Math. Appl. 2010, 59: 603-611. 10.1016/j.camwa.2009.11.002

  34. 34.

    Yilmaz Y: On some basic properties of differentiation in intuitionistic fuzzy normed spaces. Math. Comput. Model. 2010, 52: 448-458. 10.1016/j.mcm.2010.03.026

  35. 35.

    Jin SS, Lee Y-H: Fuzzy stability of a mixed type functional equation. J. Inequal. Appl. 2011., 2011: Article ID 70

  36. 36.

    Mohiuddine SA, Alotaibi A: Fuzzy stability of a cubic functional equation via fixed point technique. Adv. Differ. Equ. 2012., 2012: Article ID 48

  37. 37.

    Mohiuddine SA, Alotaibi A, Obaid M: Stability of various functional equations in non-Archimedean intuitionistic fuzzy normed spaces. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 234727

  38. 38.

    Mohiuddine SA, Alghamdi MA: Stability of functional equation obtained through a fixed-point alternative in intuitionistic fuzzy normed spaces. Adv. Differ. Equ. 2012., 2012: Article ID 141

  39. 39.

    Mohiuddine SA, Şevli H: Stability of pexiderized quadratic functional equation in intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2011, 235: 2137-2146. 10.1016/j.cam.2010.10.010

  40. 40.

    Mohiuddine SA, Cancan M, Şevli H: Intuitionistic fuzzy stability of a Jensen functional equation via fixed point technique. Math. Comput. Model. 2011, 54: 2403-2409. 10.1016/j.mcm.2011.05.049

  41. 41.

    Mohiuddine SA: Stability of Jensen functional equation in intuitionistic fuzzy normed space. Chaos Solitons Fractals 2009, 42: 2989-2996. 10.1016/j.chaos.2009.04.040

  42. 42.

    Mursaleen M, Mohiuddine SA: On stability of a cubic functional equation in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2009, 42: 2997-3005. 10.1016/j.chaos.2009.04.041

  43. 43.

    Wang Z, Rassias TM: Intuitionistic fuzzy stability of functional equations associated with inner product spaces. Abstr. Appl. Anal. 2011., 2011: Article ID 456182

  44. 44.

    Xu TZ, Rassias JM, Xu WX: Intuitionistic fuzzy stability of a general mixed additive-cubic equation. J. Math. Phys. 2010., 51: Article ID 063519

  45. 45.

    Xu TZ, Rassias JM, Xu WX: Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces. J. Math. Phys. 2010., 51: Article ID 093508

  46. 46.

    Xu TZ, Rassias JM: Stability of general multi-Euler-Lagrange quadratic functional equations in non-Archimedean fuzzy normed spaces. Adv. Differ. Equ. 2012., 2012: Article ID 119

  47. 47.

    Alotaibi A, Mohiuddine SA: On the stability of a cubic functional equation in random 2-normed spaces. Adv. Differ. Equ. 2012., 2012: Article ID 39

  48. 48.

    Goleţ I: On probabilistic 2-normed spaces. Novi Sad J. Math. 2005, 35(1):95-102.

  49. 49.

    Mohiuddine SA, Aiyub M: Lacunary statistical convergence in random 2-normed spaces. Appl. Math. Inform. Sci. 2012, 6(3):581-585.

  50. 50.

    Mursaleen M: On statistical convergence in random 2-normed spaces. Acta Sci. Math. 2010, 76: 101-109.

  51. 51.

    Mohiuddine SA, Alotaibi A, Alsulami SM: Ideal convergence of double sequences in random 2-normed spaces. Adv. Differ. Equ. 2012., 2012: Article ID 149

  52. 52.

    Jung S-M: On the Hyers-Ulam stability of the functional equations that have the quadratic property. J. Math. Anal. Appl. 1998, 222: 126-137. 10.1006/jmaa.1998.5916

  53. 53.

    Kannappan P: Quadratic functional equation and inner product spaces. Results Math. 1995, 27: 368-372. 10.1007/BF03322841

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Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (405/130/1433). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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Keywords

  • t-norm
  • t-conorm
  • quadratic-additive functional equation
  • intuitionistic fuzzy normed space
  • Hyers-Ulam stability