Open Access

On the stability of a cubic functional equation in random 2-normed spaces

Advances in Difference Equations20122012:39

https://doi.org/10.1186/1687-1847-2012-39

Received: 16 December 2011

Accepted: 29 March 2012

Published: 29 March 2012

Abstract

In this article, we propose to determine some stability results for the functional equation of cubic in random 2-normed spaces which seems to be a quite new and interesting idea. Also, we define the notion of continuity, approximately and conditional cubic mapping in random 2-normed spaces and prove some interesting results.

Keywords

distribution function t-norm triangle function random 2-normed space cubic functional equation Hyers-Ulam-Rassias stability

1 Introduction and preliminaries

In 1940, Ulam [1] proposed the following question concerning the stability of group homomorphisms:

Let G1 be a group and let G2 be a metric group with the metric d(., .). Given ϵ > 0, does there exist a δ > 0 such that if a function h : G1G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y G1, then there exists a homomorphism H : G1G1 with d(h(x), H(x)) < ϵ for all x G1?

In next year, Hyers [2] answers the problem of Ulam under the assumption that the groups are Banach spaces and then generalized by Aoki [3] and Rassias [4] for additive mappings and linear mappings, respectively. Since then several stability problems for various functional equations have been investigated in [512].

The stability problem for the cubic functional equation was proved by Jun and Kim [5] for mappings f: XY, where X is a real normed space and Y is a Banach space. Later on, the problem of stability of cubic functional equation were discussed by many mathematician.

An interesting and important generalization of the notion of a metric space was introduced by Menger [13] under the name of statistical metric space, which is now called a probabilistic metric space. An important family of probabilistic metric spaces is that of probabilistic normed spaces. The theory of probabilistic normed spaces is important as a generalization of deterministic results of linear normed spaces. The theory of probabilistic normed spaces was initiated and developed in [14, 15] and further it was extended to random 2-normed spaces by Goleţ [16] using the concept of 2-norm of Gahler [17]. For more details of probabilistic and random/fuzzy 2-normed space, we refer to [1822] and references therein.

In this article, we establish Hyers-Ulam stability concerning the cubic functional equations in random 2-normed spaces which is quite a new and interesting idea to study with.

In this section, we recall some notations and basic definitions used in this article.

A distribution function is an element of Δ+, where Δ+ = {f : → [0, 1]; f is left-continuous, nondecreasing, f(0) = 0 and f(+∞) = 1} and the subset D+ Δ+ is the set D+ = {f Δ+; l-f(+∞) = 1}. Here l-f(+∞) denotes the left limit of the function f at the point x. The space Δ+ is partially ordered by the usual point-wise ordering of functions, i.e., fg if and only if f(x) ≤ g(x) for all x . For any a , H a is a distribution function defined by
H a ( x ) = 0 if x a ; 1 if x > a .

The set Δ, as well as its subsets, can be partially ordered by the usual pointwise order: in this order, H0 is the maximal element in Δ+.

A triangle function is a binary operation on Δ+, namely a function τ : Δ+ × Δ+ → Δ+ that is associative, commutative nondecreasing and which has ε0 as unit, that is, for all f, g, h Δ+, we have:
  1. (i)

    τ(τ(f, g), h) = τ(f, τ(g, h)),

     
  2. (ii)

    τ(f, g) = τ(g, f),

     
  3. (iii)

    τ(f, g) = τ(g, f) whenever fg,

     
  4. (iv)

    τ(f, H 0) = f.

     

A t-norm is a continuous mapping * : [0, 1] × [0, 1] → [0, 1] such that ([0, 1], *) is abelian monoid with unit one and c * da * b if ca and db for all a, b, c, d [0, 1].

The concept of 2-normed space was first introduced in [17] and further studied in [2325].

Let X is a linear space of a dimension d, where 2 ≤ d < ∞. A 2-normed on X is a function ., . : X × X satisfying the following conditions, for every x, y X (i) x, y = 0 if and only if x and y are linearly dependent; (ii) x, y = y, x; (iii) αx, y = |α|x, y, α ; (iv) x + y, zx, z + y, z. In this case (X, ., . ) is called a 2-norm space.

Example 1.1. Take X = 2 being equipped with the 2-norm x, y = the area of the parallelogram spanned by the vectors x and y, which may be given explicitly by the formula
x , y = x 1 y 2 - x 2 y 1 , where x = ( x 1 , x 2 ) , y = ( y 1 , y 2 ) .

Recently, Goleţ [16] introduced the notion of random 2-normed space and further studied by Mursaleen [26].

Let X be a linear space of a dimension greater than one, τ is a triangle function, and : X × X Δ + . Then is called a probabilistic 2-norm on X and ( X , , τ ) a probabilistic 2-normed space if the following conditions are satisfied:
  1. (i)

    x , y ( t ) = H 0 ( t ) if x and y are linearly dependent, where x , y ( t ) denotes the value of x , y at t ,

     
  2. (ii)

    x , y H 0 if x and y are linearly independent,

     
  3. (iii)

    x , y = y , x for every x, y in X,

     
  4. (iv)

    α x , y ( t ) = x , y ( t α ) for every t > 0, α ≠ 0 and x, y X,

     
  5. (v)

    x + y , z τ ( x , z , y , z ) whenever x, y, z X.

     

If (v) is replaced by

(v') x + y , z ( t 1 + t 2 ) x , z ( t 1 ) * y , z ( t 2 ) , for all x, y, z X and t 1 , t 2 0 + , then triple ( X , , * ) is called a random 2-normed space (for short, RTN-space).

Example 1.2. Let (X, ., .) be a 2-normed space with x, z = x1z2 - x2z1, x = (x1, x2), z = (z1, z2) and a * b = ab for a, b [0, 1]. For all x X, t > 0 and nonzero z X, consider
x , z ( t ) = t t + x , z if t > 0 0 if t 0 ;

Then ( X , , * ) is a random 2-normed space.

Remark 1.3. Note that every 2-normed space (X, ., .) can be made a random 2-normed space in a natural way, by setting x , y ( t ) = H 0 ( t - x , y ) , for every x, y X, t > 0 and a * b = min{a, b}, a, b [0, 1].

2 Stability of cubic functional equation

In the present section, we define the notion of convergence, Cauchy sequence and completeness in RTN-space and determine some stability results of the cubic functional equation in RTN-space.

The functional equation
f ( 2 x + y ) + f ( 2 x - y ) = 2 f ( x + y ) + 2 f ( x - y ) + 12 f ( x )
(1)

is called the cubic functional equation, since the function f(x) = cx3 is its solution. Every solution of the cubic functional equation is said to be a cubic mapping.

We shall assume throughout this article that X and Y are linear spaces; ( X , , * ) and ( Z , , * ) are random 2-normed spaces; and ( Y , , * ) is a random 2-Banach space.

Let φ be a function from X × X to Z. A mapping f : XY is said to be φ-approximately cubic function if
E x , y , z ( t ) φ ( x , y ) , z ( t ) ,
(2)
for all x, y X, t > 0 and nonzero z X, where
E x , y = f ( 2 x + y ) + f ( 2 x - y ) - 2 f ( x + y ) - 2 f ( x - y ) - 12 f ( x ) .

We define:

We say that a sequence x = (x k ) is convergent in ( X , , * ) or simply -convergent to if for every ϵ > 0 and θ (0, 1) there exists k0 such that x k - , z ( ε ) > 1 - θ whenever kk0 and nonzero z X. In this case we write - lim k x k = and is called the -limit of x = (x k ).

A sequence x = (x k ) is said to be Cauchy sequence in ( X , , * ) or simply -Cauchy if for every ϵ > 0, θ > 0 and nonzero z X there exists a number N = N(ϵ, z) such that lim x n - x m , z ( ε ) > 1 - θ for all n, mN. RTN-space ( X , , * ) is said to be complete if every -Cauchy is -convergent. In this case ( X , , * ) is called random 2-Banach space.

Theorem 2.1. Suppose that a function φ : X × XZ satisfies φ(2x, 2y) = αφ(x, y) for all x, y X and α ≠ 0. Let f : XY be a φ-approximately cubic function. If for some 0 < α < 8,
φ ( 2 x , 2 y ) , z ( t ) α φ ( x , y ) , z ( t ) ,
(3)
and lim n φ ( 2 n x , 2 n y ) , z ( 8 n t ) = 1 for all x, y X, t > 0 and nonzero z X. Then there exists a unique cubic mapping C : XY such that
C ( x ) - f ( x ) , z ( t ) φ ( x , 0 ) , z ( ( 8 - α ) t ) ,
(4)

for all x X, t > 0 and nonzero z X.

Proof. For convenience, let us fix y = 0 in (2). Then for all x X, t > 0 and nonzero z X
f ( 2 x ) 8 - f ( x ) , z t 16 φ ( x , 0 ) , z ( t ) .
(5)
Replacing x by 2 n x in (5) and using (3), we obtain
f ( 2 n + 1 x ) 8 n + 1 - f ( 2 n x ) 8 n , z t 16 ( 8 n ) φ ( 2 n x , 0 ) , z ( t ) φ ( x , 0 ) , z ( t / α n ) ,
for all x X, t > 0 and nonzero z X; and for all n ≥ 0. By replacing t by α n t, we get
f ( 2 n + 1 x ) 8 n + 1 - f ( 2 n x ) 8 n , z α n t 16 ( 8 n ) φ ( x , 0 ) , z ( t ) .
(6)
It follows from f ( 2 n x ) 8 n - f ( x ) = k = 0 n = 1 f ( 2 k + 1 x ) 8 k + 1 - f ( 2 k x ) 8 k and (6) that
f ( 2 n x ) 8 n - f ( x ) , z k = 0 n - 1 α k t 16 ( 8 k ) k = 0 n - 1 f ( 2 k + 1 x ) 8 k + 1 - f ( 2 k x ) 8 k , z α k t 16 ( 8 k ) φ ( x , 0 ) , z ( t ) ,
(7)
for all x X, t > 0 and n > 0 where j = 1 n a j = a 1 * a 2 * * a n . By replacing x with 2 m x in (7), we have
f ( 2 n + m x ) 8 n + m - f ( 2 m x ) 8 m , z k = 0 n - 1 α k t 16 ( 8 ) k + m φ ( 2 m x , 0 ) , z ( t ) φ ( x , 0 ) , z ( t / α m ) .
Thus
f ( 2 n + m x ) 8 n + m - f ( 2 m x ) 8 m , z k = m n + m - 1 α k t 16 ( 8 ) k φ ( x , 0 ) , z ( t ) ,
for all x X, t > 0, m > 0, n ≥ 0 and nonzero z X. Hence
f ( 2 n + m x ) 8 n + m - f ( 2 m x ) 8 m , z ( t ) φ ( x , 0 ) , z t k = m n + m - 1 α k 16 ( 8 ) k ,
(8)
for all x X, t > 0 m ≥ 0, n ≥ 0 and nonzero z X. Since 0 < α < 8 and k = 0 ( α 8 ) k < , the Cauchy criterion for convergence shows that ( f ( 2 n x ) 8 n ) is a Cauchy sequence in ( Y , , * ) . Since ( Y , , * ) is complete, this sequence converges to some point C(x) Y. Fix x X and put m = 0 in (8) to obtain
f ( 2 n x ) 8 n - f ( x ) , z ( t ) φ ( x , 0 ) , z t k = 0 n - 1 α k 16 ( 8 ) k ,
for all t > 0, n > 0 and nonzero z X. Thus we obtain
C ( x ) - f ( x ) , z ( t ) C ( x ) - f ( 2 n x ) 8 n , z ( t / 2 ) * f ( 2 n x ) 8 n - f ( x ) , z ( t / 2 ) φ ( x , 0 ) , z t k = 0 n - 1 a k 8 ( 8 ) k ,
for large n. Taking the limit as n → ∞ and using the definition of RTN-space, we get
C ( x ) - f ( x ) , z ( t ) φ ( x , 0 ) , z ( ( 8 - α ) t ) .
Replace x and y by 2 n x and 2 n y, respectively, in (2), we have
E 2 n x , 2 n y 8 n , z ( t ) φ ( 2 n x , 2 n y ) , z ( 8 n t ) ,
for all x, y X, t > 0 and nonzero z X. Since
lim n φ ( 2 n x , 2 n y ) , z ( 8 n t ) = 1 ,
we observe that C fulfills (1). To Prove the uniqueness of the cubic function C, assume that there exists a cubic function D : XY which satisfies (4). For fix x X, clearly C(2 n x) = 8 n C(x) and D(2 n x) = 8 n D(x) for all n . It follows from (4) that
C ( x ) - D ( x ) , z ( t ) = C ( 2 n x ) 8 n - D ( 2 n x ) 8 n , z ( t ) C ( 2 n x ) 8 n - f ( 2 n x ) 8 n , z t 2 * f ( 2 n x ) 8 n - D ( 2 n x ) 8 n , z t 2 φ ( 2 n x , 0 ) , z 8 n ( 8 - α ) t 2 φ ( x , 0 ) , z 8 n ( 8 - α ) t 2 α n .
Therefore
φ ( x , 0 ) , z 8 n ( 8 - α ) t 2 α n = 1 .

Thus C ( x ) - D ( x ) , z ( t ) = 1 for all x X, t > 0 and nonzero z X. Hence C(x) = D(x).

Example 2.2. Let X be a Hilbert space and Z be a normed space. By and , we denote the random 2-norms given as in Example 1.1 on X and Z, respectively. Let φ : X × XZ be defined by φ(x, y) = 8(x2 + y2)zο, where zο is a fixed unit vector in Z. Define f : XX by f(x) = x2x + x2xο for some unit vector xο X. Then
E x , y , z ( t ) = t t + 8 x , z 2 + 2 y , z 2 t t + 8 x , z 2 + 8 y , z 2 = φ ( x , y ) , z ( t ) .
Also
φ ( 2 x , 0 ) , z ( t ) = t t + 32 x , z 2 = 4 φ ( x , 0 ) , z ( t ) .
Thus,
lim n φ ( 2 n x , 2 n y ) , z ( 8 n t ) = lim n 8 n t 8 n t + 8 ( 4 n ) ( x , z 2 + y , z 2 ) = 1 .

Hence, conditions of Theorem 2.1 for α = 4 are fulfilled. Therefore, there is a unique cubic mapping C : XX such that C ( x ) - f ( x ) , z ( t ) φ ( x , 0 ) , z ( 4 t ) for all x X, t > 0 and nonzero z X.

By a modification in the proof of Theorem 2.1, one can easily prove the following:

Theorem 2.3. Suppose that a function φ : X × XZ satisfies φ ( x / 2 , y / 2 ) = 1 α φ ( x , y ) for all x, y X and α ≠ 0. Let f : XY be a φ-approximately cubic function. If for some α > 8
φ ( x / 2 , y / 2 ) , z ( t ) φ ( x , y ) , z ( α t )
and lim n 8 n φ ( 2 - n x , 2 - n y ) , z ( t ) = 1 for all x, y X, t > 0 and nonzero z X. Then there exists a unique cubic mapping C : XY such that
C ( x ) - f ( x ) , z ( t ) φ ( x , 0 ) , z ( ( α - 8 ) t ) ,

for all x X, t > 0 and nonzero z X.

3 Continuity in random 2-normed spaces

In this section, we establish some interesting results of continuous approximately cubic mappings.

Let f : X be a function, where is endowed with the Euclidean topology and X is an random 2-normed space equipped with random 2-norm . Then, f is said to be random 2-continuous or simply -continuous at a point sο if for all ϵ > 0 and all 0 < α < 1 there exists δ > 0 such that
f ( s x ) - f ( s x ) , z ( ε ) α ,

for each s with 0 < |s - sο| < δ and nonzero z X.

A mapping f : XY is said to be (p, q)-approximately cubic function if, for some p, q and some zο Z,
E x , y , z ( t ) ( x p + y q ) z , z ( t ) ,

for all x, y X, t > 0 and nonzero z X.

Theorem 3.2. Let X be a normed space and let f : XY be a (p, q)-approximately cubic function. If p, q < 3, there exists a unique cubic mapping C : XY such that
C ( x ) - f ( x ) , z ( t ) x p z , z ( ( 8 - 2 p ) t ) ,
(9)

for all x X, t > 0 and nonzero z X. Furthermore, if for some x X and all n , the mapping g : Y defined by g(s) = f(2 n sx) is -continuous. Then the mapping s C(sx) from to Y is -continuous; in this case, C(rx) = r3C(x) for all r .

Proof. Suppose that a function φ : X × XZ satisfies φ(x, y) = (x p +y q )zο. Existence and uniqueness of the cubic mapping C satisfying (9) are deduced from Theorem 2.1. Note that for each x X, t and n , we have
C ( x ) - f ( 2 n x ) 8 n , z ( t ) = C ( 2 n x ) - f ( 2 n x ) , z ( 8 n t ) 2 n p x p z , z ( 8 n ( 8 - 2 p ) t ) = x p z , z 8 n ( 8 - 2 p ) t 2 n p .
(10)
Fix x X and sο . Given ϵ > 0 and 0 < α < 1. From (10) follows that
C ( s x ) - f ( 2 n s x ) 8 n , z ( t ) x p z , z 8 n ( 8 - 2 p ) t s p 2 n p x p z , z 8 n ( 8 - 2 p ) t ( 1 + s ) p 2 n p ,
for all |s - sο| < 1 and s . Since lim n 8 n ( 8 - 2 p ) t ( 1 + s ) p 2 n p = , there exists nο such that
C ( s x ) - f ( 2 n s x ) 8 n , z ε 3 α ,
for all |s - sο| < 1 and s . By the -continuity of the mapping t f ( 2 n t x ) , there exists δ < 1 such that for each s with 0 < |s - sο| < δ, we have
f ( 2 n s x ) 8 n - f ( 2 n s x ) 8 n , z ε 3 α .
It follows that
C ( s x ) - C ( s x ) , z ( ε ) C ( s x ) - f ( 2 n s x ) 8 n , z ε 3 * f ( 2 n s x ) 8 n - f ( 2 n s x ) 8 n , z ε 3 * C ( s x ) - f ( 2 n s x ) 8 n , z ε 3 α ,

for each s with 0 < |s - sο| < δ. Hence, the mapping s C(sx) is -continuous.

Now, we use the -continuity of s C(sx) to establish that C ( r x ) = r 3 C ( x ) for all rο . For each r, is a dense subset of , we have C(rx) = r3C(x). Fix rο and t > 0. Then, for 0 < α < 1 there exists δ > 0 such that
C ( r x ) - C ( r x ) , z ( t / 3 ) α ,
for each r and 0 < |r - rο| < δ. Choose a rational number r with 0 < |r - rο| < δ and r 3 - r 3 < 1 - α . Then
C ( r x ) - r 3 C ( x ) , z ( t ) C ( r x ) - C ( r x ) , z ( t / 3 ) * C ( r x ) - r 3 C ( x ) , z ( t / 3 ) * r 3 C ( x ) - r 3 C ( x ) , z ( t / 3 ) α * 1 * C ( x ) , z ( t / 3 ( 1 - α ) ) .

Thus C ( r x ) - r 3 C ( x ) , z ( t ) = 1 . Hence, we conclude that C ( r x ) = r 3 C ( x ) .

Remark 3.2. We can also prove Theorem 3.1 for the case when p, q > 3. In this case, there exists a unique cubic mapping C : XY such that C ( x ) - f ( x ) , z ( t ) x p z , z ( ( 2 p - 8 ) t ) for all x X, t > 0 and nonzero z X.

4 Approximately and conditional cubic mapping in random 2-normed spaces

In this section, we obtain completeness in RTN-space through the existence of some solution of a stability problem for cubic functional equation.

A mapping f : {0} → X is said to be approximately cubic if for each α (0, 1) there exists some n α such that E ( n , m ) , z ( 1 ) α , for all n ≥ 2mn α and nonzero z X.

By a conditional cubic mapping, we mean a mapping f : {0} → X such that (1) holds whenever x ≥ 2y.

It can be easily verified that for each conditional cubic mapping f : {0} → X, we have f(2 n ) = 23nf(1).

Theorem 4.1. Let ( X , , * ) be a RTN-space such that for each approximately cubic mapping f : {0} → X, there exists a conditional cubic mapping C : {0} → X, such that
lim n C ( n ) - f ( n ) , z ( 1 ) = 1 ,

for nonzero z X. Then ( X , , * ) is a random 2-Banach space.

Proof. Let (x n ) be a Cauchy sequence in a RTN-space. By induction on k, we can find a strictly increasing sequence (n k ) of natural numbers such that
x n - x m , z 1 ( 10 k ) 3 1 - 1 k ,
for each n, mn k and nonzero z X. Let y k = x n k and define f : {0} → X by f(k) = k3y k . Let α (0, 1). and find some nο such that 1 - 1 n > α . One can easily verified that
E k , j , z ( 1 ) y 2 k + j - y k + j , z 1 20 k 3 * y 2 k + j - y k - j , z 1 20 k 3 * y 2 k + j - y k , z 1 40 k 3 * y 2 k - j - y k , z 1 80 k 3 * y 2 k + j - y 2 k - j , z 1 120 k 2 j * y k - j - y k + j , z 1 60 k 2 j * y 2 k + j - y k + j , z 1 60 k j 2 * y 2 k - j - y k - j , z 1 60 k j 2 * y 2 k + j - y 2 k - j , z 1 10 j 3 * y k - j - y k + j , z 1 20 j 3 ,
for each k ≥ 2j, and nonzero z X. Then
y 2 k + j - y k + j , z 1 20 k 3 y 2 k + j - y k + j , z 1 20 ( k + j ) 3 y 2 k + j - y k + j , z 1 10 3 ( k + j ) 3 α ,
for j > nο and nonzero z X. Since k - j k 2 and k - jj, we have
y 2 k + j - y k - j , z 1 20 k 3 y 2 k + j - y k - j , z 1 10 3 ( k - j ) 3 α , y k - j - y k + j , z 1 60 k 2 j y k - j - y k + j , z 1 10 3 ( k - j ) 3 α , y 2 k - j - y k - j , z 1 60 k j 2 y 2 k - j - y k - j , z 1 10 3 ( k - j ) 3 α , y k - j - y k + j , z 1 20 j 3 y 2 k - j - y k - j , z 1 10 3 ( k - j ) 3 α .
Clearly
y 2 k + j - y k , z 1 40 k 3 y 2 k + j - y k , z 1 10 3 k 3 α , y 2 k - j - y k , z 1 80 k 3 y 2 k - j - y k , z 1 10 3 k 3 α , and y 2 k + j - y k + j , z 1 60 k j 2 y 2 k + j - y k + j , z 1 10 3 ( k + j ) 3 α .
The inequalities 2k - jj and 2k - j > k imply
y 2 k + j - y 2 k - j , z 1 120 k 2 j y 2 k + j - y 2 k - j , z 1 120 ( 2 k - j ) 3 y 2 k + j - y 2 k - j , z 1 10 3 ( 2 k - j ) 3 α y 2 k + j - y 2 k - j , z 1 10 j 3 y 2 k + j - y 2 k - j , z 1 10 ( 2 k - j ) 3 y 2 k + j - y 2 k - j , z 1 10 3 ( 2 k - j ) 3 α .
Therefore E k , j , z ( 1 ) α . This shows that f is approximately cubic type mapping. By our assumption, there exists a conditional cubic mapping C : {0} → X, such that lim k C ( k ) - f ( k ) , z ( 1 ) = 1 . In particular, lim k C ( 2 k ) - f ( 2 k ) , z ( 1 ) = 1 . This means that
lim k C ( 1 ) - y 2 k , z 1 2 3 k = 1

Hence the subsequence ( y 2 k ) converges to y = C(1). Therefore, the Cauchy sequence (x n ) also converges to y.

Declarations

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University

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© Alotaibi and Mohiuddine; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.