The stability of the quartic functional equation in various spaces

Computers and Mathematics with Applications 60 (2010) 1994–2002

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The stability of the quartic functional equation in various spaces Reza Saadati a,b , Yeol J. Cho c,∗ , Javad Vahidi b a

Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran

b

Faculty of Sciences, Shomal University, Amol, Iran

c

Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea

article

info

Article history: Received 20 January 2010 Received in revised form 21 July 2010 Accepted 22 July 2010 Keywords: Stability Quartic functional equation Intuitionistic random normed space Non-Archimedean random normed spaces

abstract The purpose of this paper is first to introduce the notation of intuitionistic random normed spaces, and then by virtue of this notation to study the stability of a quartic functional equation in the setting of these spaces under arbitrary triangle norms. Then we prove the stability of above quartic functional equation in non-Archimedean random normed spaces. Furthermore, the interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces, the theory of intuitionistic spaces and the theory of functional equations are also presented in the paper. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms, which was affirmatively answered for Banach spaces by Hyers [2]. Subsequently, the result of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper by Rassias has provided a lot of influence in the development of what we now call Hyers–Ulam–Rassias stability of functional equations. For more information on such problems, we refer interested readers to [5–9] and [10,11]. The functional equation f (2x + y) + f (2x − y) = 4f (x + y) + 4f (x − y) + 24f (x) − 6f (y)

(1.1)

is said to be a quartic functional equation since the function f (x) = cx is a solution. Every solution of a quartic functional equation is said to be a quartic mapping. The stability problem for a quartic functional equation first was considered by Rassias [12] for mappings f : X −→ Y , where X is a real normed space and Y is a Banach space. In addition, Alsina [13], Miheţ and Radu [14], Miheţ, Saadati and Vaezpour [15,16] Mirmostafaee, Mirzavaziri and Moslehian [17–19], Saadati et al., [20,21] and Shakeri [22] investigated the stability in the settings of probabilistic, random and fuzzy normed spaces. The usual uncertainty principle of Werner Heisenberg leads to a generalized uncertainty principle, which has been motivated by string theory and non-commutative geometry. In a strong quantum gravity regime, space–time points are determined in a random manner. Thus the impossibility of determining the position of articles gives space–time a fuzzy and random structure [23–27]. Because of this random structure, a position space representation of quantum mechanics breaks down and therefore a generalized normed space of a quasi-position eigenfunction is required [26]. Thus, one needs to discuss a new family of random norms. Also, the concept of topology introduced by a random norm has important applications in quantum particle physics, particularly in connection with both string and E-infinity theory; see [24,28]. One of the most 4

∗

Corresponding author. E-mail addresses: [email protected], [email protected], [email protected] (R. Saadati), [email protected] (Y.J. Cho).

0898-1221/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.07.034

R. Saadati et al. / Computers and Mathematics with Applications 60 (2010) 1994–2002

1995

important problems in topology introduced by a random norm is to obtain an appropriate concept of intuitionistic random normed space. These problems have been investigated by Chang–Rassias–Saadati [29]. 2. Preliminaries In this section, we recall some definitions and results which will be used later on in the paper. A triangular norm (briefly, a t-norm) is a binary operation on the unit interval [0, 1], i.e., a function T : [0, 1] × [0, 1] → [0, 1] such that for all a, b, c ∈ [0, 1] the following four axioms are satisfied. (i) (ii) (iii) (iv)

T (a, b) = T (b, a) (commutativity); T (a, (T (b, c ))) = T (T (a, b), c ) (associativity); T (a, 1) = a (boundary condition); T (a, b) ≤ T (a, c ) whenever b ≤ c (monotonicity).

Basic examples are the Łukasiewicz t-norm TL , TL (a, b) = max(a + b − 1, 0) ∀a, b ∈ [0, 1] and the t-norms TP , TM , TD , where TP (a, b) := ab, TM (a, b) := min{a, b}, TD (a, b) :=



min(a, b), 0,

if max(a, b) = 1; otherwise. (n−1)

(n)

If T is a t-norm then xT is defined for every x ∈ [0, 1] and n ∈ N ∪ {0} by 1 if n = 0 and by T (xT (n)

, x) if n ≥ 1. A t-norm

T is said to be of Hadžić type (we denote this by T ∈ H ) if the family (xT )n∈N is equicontinuous at x = 1 (see [30]). Other important triangular norms are as follows (see [31]). SW The Sugeno–Weber family {TλSW }λ∈[−1,∞] is defined by T−SW 1 = TD , T∞ = TP and



Tλ (x, y) = max 0, SW

x + y − 1 + λxy



1+λ

if λ ∈ (−1, ∞). The Domby family {TλD }λ∈[0,∞] is defined by TD , if λ = 0, TM , if λ = ∞ and TλD (x, y) =

1 x λ 1 + (( 1− ) +( x

1−y λ 1/λ y

) )

if λ ∈ (0, ∞). The Aczel–Alsina family {TλAA }λ∈[0,∞] is defined by TD , if λ = 0, TM , if λ = ∞ and λ λ 1/λ TλAA (x, y) = e−(| log x| +| log y| )

if λ ∈ (0, ∞). A t-norm T can be extended (by associativity) in a unique way to an n-array operation. For any (x1 , ..., xn ) ∈ [0, 1]n , the value T (x1 , ..., xn ) is defined by T0i=1 xi = 1, Tni=1 xi = T (Tni=−11 xi , xn ) = T (x1 , ..., xn ). T can also be extended to a countable operation. For any sequence (xn )n∈N in [0, 1], the value is defined by n T∞ i=1 xi = lim Ti=1 xi .

(2.1)

n→∞

Tni=1 xi n∈N

The limit on the right-hand side of (2.1) exists since the sequence {

Proposition 2.1 ([31]). (i) For T ≥ TL , the following implication holds: lim T∞ i=1 xn+i = 1 ⇐⇒ n→∞

∞ X (1 − xn ) < ∞. n=1

(ii) If T is of Hadžić type, then lim T∞ i=1 xn+i = 1

n→∞

for every sequence {xn }n∈N in [0, 1] such that limn→∞ xn = 1. (iii) If T ∈ {TλAA }λ∈(0,∞) ∪ {TλD }λ∈(0,∞) , then lim T∞ i=1 xn+i = 1 ⇐⇒ n→∞

∞ X (1 − xn )α < ∞. n=1

(iv) If T ∈ {Tλ }λ∈[−1,∞) , then SW

lim T∞ i=1 xn+i = 1 ⇐⇒

n→∞

∞ X (1 − xn ) < ∞. n=1

}

is non-increasing and bounded from below.

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R. Saadati et al. / Computers and Mathematics with Applications 60 (2010) 1994–2002

3. Non-Archimedean random normed spaces In 1897, Hensel [32] introduced a field with a valuation norm which does not have the Archimedean property. Definition 3.1. Let K be a field. A non-Archimedean absolute value on K is a function | · | : K → R such that, for any a, b ∈ K , we have (i) |a| ≥ 0 and equality holds if and only if a = 0, (ii) |ab| = |a||b|, (iii) |a + b| ≤ max{|a|, |b|} (the strict triangle inequality). Note that |n| ≤ 1 for each integer n. We always assume in addition that | · | is non-trivial, i.e., that there is an a0 ∈ K such that |a0 | 6= 0, 1. Definition 3.2. A non-Archimedean random normed space (briefly, a non-Archimedean RN-space) is a triple (X , µ, T ), where X is a linear space over a non-Archimedean field K , T is a continuous t-norm, and µ is a mapping from X into D+ , such that the following conditions hold: (NA-RN1) µx (t ) = ε0 (t ) for all t > 0 if and only if x = 0; t (NA-RN2) µα x (t ) = µx ( |α| ) for all x ∈ X , t > 0, α 6= 0;

(NA-RN3) µx+y (max{t , s}) ≥ T (µx (t ), µy (s)) for all x, y, z ∈ X and t , s ≥ 0. It is easy to see that if (NA-RN3) holds then we have (RN3) µx+y (t + s) ≥ T (µx (t ), µy (s)). As a classical example, if (X , k.k) is a non-Archimedean normed linear space, then the triple (X , N , min), where

µx (t ) =



0 1

if t ≤ kxk, if t > kxk,

is a non-Archimedean RN-space. Definition 3.3. Let (X , µ, T ) be a non-Archimedean RN-space. Let {xn } be a sequence in X . Then {xn } is said to be convergent if there exists x ∈ X such that lim µxn −x (t ) = 1

n→∞

for all t > 0. In that case, x is called the limit of the sequence {xn }. A sequence {xn } in X is called a Cauchy sequence if, for each ε > 0 and t > 0, there exists n0 such that, for all n ≥ n0 and all p > 0, we have µxn+p −xn (t ) > 1 − ε . If each Cauchy sequence is convergent, then the random norm is said to be complete and the non-Archimedean RN-space is called a non-Archimedean random Banach space. 4. Hyers–Ulam–Rassias stability for a quartic equation in non-Archimedean RN-space Let K be a non-Archimedean field, X be a vector space over K and (Y , µ, T ) be a non-Archimedean random Banach space over K . We investigate the stability of the quartic equation (1.1), where f is a mapping from X to Y and f (0) = 0. It is known that a function f satisfies the above functional equation if and only if it is quartic. Next, we define a random approximately quartic mapping. Let Ψ be a distribution function on X × X × [0, ∞) such that Ψ (x, y, ·) is nondecreasing and

Ψ (cx, cx, t ) ≥ Ψ



x, x,

t

|c |



,

∀x ∈ X , c 6= 0.

Definition 4.1. A mapping f : X → Y is said to be Ψ -approximately quartic if

µf (2x+y)+f (2x−y)−4f (x+y)−4f (x−y)−24f (x)+6f (y) (t ) ≥ Ψ (x, y, t ), Our main result, in this section, is the following.

∀x, y ∈ X , t > 0.

(4.1)

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Theorem 4.2. Let K be a non-Archimedean field, X be a vector space over K and (Y , µ, T ) be a non-Archimedean random Banach space over K . Let f : X → Y be a Ψ -approximately quartic function. If, for some α ∈ R, α > 0, and some integer k, k ≥ 3 with |2k | < α ,

Ψ (2−k x, 2−k y, t ) ≥ Ψ (x, y, α t ),

∀x ∈ X , t > 0,

(4.2)

and lim T∞ j =n M n→∞



x,

αj t |2|kj



= 1,

∀x ∈ X , t > 0,

(4.3)

then there exists a unique quartic mapping Q : X → Y such that

µf (x)−Q (x) (t ) ≥

T∞ i =1 M



α i +1 t x, |2|ki



,

∀x ∈ X , t > 0,

(4.4)

where M (x, t ) := T (Ψ (x, 0, t ), Ψ (2x, 0, t ), . . . , Ψ (2k−1 x, 0, t )),

∀x ∈ X , t > 0.

Proof. First, we show by induction on j that, for all x ∈ X , t > 0 and j ≥ 1,

µf (2j x)−16j f (x) (t ) ≥ Mj (x, t ) := T (Ψ (x, 0, t ), . . . , Ψ (2j−1 x, 0, t )).

(4.5)

Put y = 0 in (4.1) to obtain

µ2f (2x)−32f (x) (t ) ≥ Ψ (x, 0, t ), ∀x ∈ X , t > 0, µf (2x)−16f (x) (t ) ≥ Ψ (x, 0, 2t ) ≥ Ψ (x, 0, t ), ∀x ∈ X , t > 0. This proves (4.5) for j = 1. Let (4.5) hold for some j > 1. Replacing y by 0 and x by 2j x in (4.1), we get

µf (2j+1 x)−16f (2j x) (t ) ≥ Ψ (2j x, 0, t ),

∀x ∈ X , t > 0.

Since |16| ≤ 1,


µf (2j+1 x)−16j+1 f (x) (t ) ≥ T µf (2j+1 x)−16f (2j x) (t ), µ16f (2j x)−16j+1 f (x) (t )    t = T µf (2j+1 x)−16f (2j x) (t ), µf (2j x)−16j f (x) |16|
≥ T µf (2j+1 x)−16f (2j x) (t ), µf (2j x)−16j f (x) (t ) ≥ T (Ψ (2j x, 0, t ), Mj (x, t )) = Mj+1 (x, t ), ∀x ∈ X , t > 0. Thus (4.5) holds for all j ≥ 1. In particular,

µf (2k x)−16k f (x) (t ) ≥ M (x, t ), −(kn+k)

Replacing x by 2

µf 

x 2kn



−16k f



∀x ∈ X , t > 0.

(4.6)

x in (4.6) and using inequality (4.2), we obtain

x 2kn+k

 (t )

≥M

 x

,t kn+k



2

≥ M (x, α n+1 t ),

∀x ∈ X , t > 0, n ≥ 0.

(4.7)

Then it follows that

µ

(24k )n f



x



(2k )n

−(24k )n+1 f



x

 (t )

(2k )n+1

 α n +1 ≥ M x, 4k n t |(2 ) |   α n+1 ≥ M x, k n t , |(2 ) | 

∀x ∈ X , t > 0, n ≥ 0,

and so

! µ

(24k )n f



x

(2k )n



−(24k )n+p f



x

(2k )n+p

 (t )

≥

n +p T j =n

µ

(24k )j f



x

(2k )j



−(24k )j+p f

  α j +1 ≥ Tjn=+np M x, k j t , |(2 ) |



x

 (t )

(2k )j+p

∀x ∈ X , t > 0, n ≥ 0.

(4.8)

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n



o

α x 4k n Since limn→∞ T∞ is a Cauchy sequence in the non-Archimedean j=n M x, |(2k )j | t = 1 (x ∈ X , t > 0), (2 ) f (2k )n n∈N random Banach space (Y , µ, T ). Hence we can define a mapping Q : X → Y such that

lim µ

n→∞

(24k )n f



x (2k )n



j+1

−Q (x)

(t ) = 1,

∀x ∈ X , t > 0.

(4.9)

Next, for each n ≥ 1, x ∈ X and t > 0,

µ

f (x)−(24k )n f



 (t )

x

= µPn−1 i=0

(2k )n

(24k )i f





x

(2k )i

−(24k )i+1 f



 (t )

x

(2k )i+1

! Tni=−01

≥

µ

(24k )i f



≥ Tni=−01 M x,





x

(2k )i

α i+1 t |2k |i



−(24k )i+1 f

 (t )

x

(2k )i+1



and so

! µf (x)−Q (x) (t ) ≥ T µ

≥T

f (x)−(24k )n f

Tni=−01 M





x

(2k )n

α i +1 t x, |2k |i

 (t ), µ  (24k )n f





x

(2k )n

−Q (x)

(t ) !

,µ

(24k )n f





x

(2k )n

−Q (x)

(t ) .

(4.10)

Taking the limit as n → ∞ in (4.10), we obtain

µf (x)−Q (x) (t ) ≥

T∞ i=1 M

α i+1 t x, |2k |i





,

∀x ∈ X , t > 0.

This proves (4.4). As T is continuous, from a well-known result in probabilistic metric space (see, e.g., [33], Chapter 12) it follows that lim µ(24k )n f (2−kn (2x+y))+(24k )n f (2−kn (2x−y))−4(24k )n f (2−kn (x+y)) −4(24k )n f (2−kn (x−y))−24(24k )n f (2−kn x)+6(24k )n f (2−kn (y)) (t )

n→∞

= µQ (2x+y)+Q (2x−y)−4Q (x+y)−4Q (x−y)−24Q (x)+6Q (y) (t ),

∀ x, y ∈ X , t > 0 .

On the other hand, replacing x, y by 2−kn x, 2−kn y in (4.1) and (4.2), we get

µ(16k )n f (2−kn (2x+y))+(16k )n f (2−kn (2x−y))−4(16k )n f (2−kn (x+y)) −4(16k )n f (2−kn (x−y))−24(16k )n f (2−kn x)+6(16k )n f (2−kn y) (t )   t −kn −kn ≥ Ψ 2 x, 2 y, k n |2 |   n α t ≥ Ψ x, y, k n , ∀x, y ∈ X , t > 0. |2 |   n Since limn→∞ Ψ x, y, |α2k |tn = 1, we infer that Q is a quartic mapping. If Q 0 : X → Y is another quartic mapping such that µQ 0 (x)−f (x) (t ) ≥ M (x, t ) for all x ∈ X and t > 0, then, for each n ∈ N, x ∈ X and t > 0, ! µQ (x)−Q 0 (x) (t ) ≥ T µ

Q (x)−(24k )n f



x

(2k )n

 (t ), µ  (24k )n f

x



(2k )n

−Q 0 (x)

(t ) .

Therefore, from (4.9), we conclude that Q = Q 0 . This completes the proof.



Corollary 4.3. Let K be a non-Archimedean field, X be a vector space over K and (Y , µ, T ) be a non-Archimedean random Banach space over K under a t-norm T ∈ H . Let f : X → Y be a Ψ -approximately quartic function. If, for some α ∈ R, α > 0, and some integer k, k ≥ 2 with |2k | < α ,

Ψ (2−k x, 2−k y, t ) ≥ Ψ (x, y, α t ),

∀x ∈ X , t > 0,

then there exists a unique quartic mapping Q : X → Y such that

  α i+1 t µf (x)−Q (x) (t ) ≥ T∞ M x , , i=1 |2|ki

∀x ∈ X , t > 0,

R. Saadati et al. / Computers and Mathematics with Applications 60 (2010) 1994–2002

1999

where M (x, t ) := T (Ψ (x, 0, t ), Ψ (2x, 0, t ), . . . , Ψ (2k−1 x, 0, t )),

∀x ∈ X , t > 0.

Proof. Since

αj t lim M x, n→∞ |2|kj 



= 1,

∀x ∈ X , t > 0,

and T is of Hadžić type, from Proposition 2.1 it follows that lim T∞ j =n M n→∞



αj t x, |2|kj



= 1,

Now we can apply Theorem 4.2.

∀x ∈ X , t > 0. 

5. Intuitionistic random normed spaces In this section, we shall adopt the usual terminology, notations and conventions of the theory of random normed spaces and intuitionistic random normed spaces, as in [29,34–36,30,37,14,8,38,39,33,40]. Definition 5.1. A measure distribution function is a function µ : R → [0, 1] which is left continuous on R, non-decreasing and inft ∈R µ(t ) = 0, supt ∈R µ(t ) = 1. We will denote by D the family of all measure distribution functions and by H a special element of D defined by H (t ) =



0 1

if t ≤ 0, if t > 0.

If X is a nonempty set, then µ : X −→ D is called a random measure on X and µ(x) is denoted by µx . Definition 5.2. A non-measure distribution function is a function ν : R → [0, 1] which is right continuous on R, nonincreasing and inft ∈R ν(t ) = 0, supt ∈R ν(t ) = 1. We will denote by B the family of all non-measure distribution functions and by G a special element of B defined by G(t ) =



1 0

if t ≤ 0, if t > 0.

If X is a nonempty set, then ν : X −→ B is called a random non-measure on X and ν(x) is denoted by νx . Lemma 5.3 ([41,42,39]). Consider the set L∗ and the operation ≤L∗ defined by L∗ = {(x1 , x2 ) : (x1 , x2 ) ∈ [0, 1]2 and x1 + x2 ≤ 1},

(x1 , x2 ) ≤L∗ (y1 , y2 ) ⇐⇒ x1 ≤ y1 , x2 ≥ y2 ,

∀(x1 , x2 ), (y1 , y2 ) ∈ L∗ .

Then (L∗ , ≤L∗ ) is a complete lattice. In Section 2, we presented the classical definition of a t-norm, which can be straightforwardly extended to any lattice

(L∗ , ≤L∗ ). Define first units by 0L∗ = (0, 1) and 1L∗ = (1, 0).

Definition 5.4 ([35,42,39]). A triangular norm (t-norm) on L∗ is a mapping T : (L∗ )2 −→ L∗ satisfying the following conditions: (a) (∀x ∈ L∗ )(T (x, 1L∗ ) = x) (boundary condition); (b) (∀(x, y) ∈ (L∗ )2 )(T (x, y) = T (y, x)) (commutativity); (c) (∀(x, y, z ) ∈ (L∗ )3 )(T (x, T (y, z )) = T (T (x, y), z )) (associativity); (d) (∀(x, x0 , y, y0 ) ∈ (L∗ )4 )(x ≤L∗ x0 and y ≤L∗ y0 H⇒ T (x, y) ≤L∗ T (x0 , y0 )) (monotonicity). If (L∗ , ≤L∗ , T ) is an Abelian topological monoid with unit 1L∗ , then T is said to be a continuous t-norm. Definition 5.5 ([35,42,39]). A continuous t-norm T on L∗ is said to be continuous t-representable if there exist a continuous t-norm ∗ and a continuous t-conorm  on [0, 1] such that, for all x = (x1 , x2 ), y = (y1 , y2 ) ∈ L∗ ,

T (x, y) = (x1 ∗ y1 , x2  y2 ).

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R. Saadati et al. / Computers and Mathematics with Applications 60 (2010) 1994–2002

For example,

T (a, b) = (a1 b1 , min{a2 + b2 , 1}) and M(a, b) = (min{a1 , b1 }, max{a2 , b2 }) for all a = (a1 , a2 ), b = (b1 , b2 ) ∈ L∗ are continuous t-representable. Now, we define a sequence T n recursively by T 1 = T and

T n (x(1) , . . . , x(n+1) ) = T (T n−1 (x(1) , . . . , x(n) ), x(n+1) ),

∀n ≥ 2, x(i) ∈ L∗ .

Definition 5.6 ([35,39]). A negator on L∗ is any decreasing mapping N : L∗ −→ L∗ satisfying N (0L∗ ) = 1L∗ and N (1L∗ ) = 0L∗ . If N (N (x)) = x for all x ∈ L∗ , then N is called an involutive negator. A negator on [0, 1] is a decreasing mapping N : [0, 1] −→ [0, 1] satisfying N (0) = 1 and N (1) = 0. Ns denotes the standard negator on [0, 1] defined by Ns (x) = 1 − x,

∀x ∈ [0, 1].

Definition 5.7 ([29]). Let µ and ν be measure and non-measure distribution functions from X × (0, +∞) to [0, 1] such that µx (t ) + νx (t ) ≤ 1 for all x ∈ X and t > 0. The triple (X , Pµ,ν , T ) is said to be an intuitionistic random normed space (briefly, an IRN-space) if X is a vector space, T is a continuous t-representable and Pµ,ν is a mapping X × (0, +∞) → L∗ satisfying the following conditions. For all x, y ∈ X and t , s > 0: (a) Pµ,ν (x, 0) = 0L∗ ; (b) Pµ,ν (x, t ) = 1L∗ if and only if x = 0; t (c) Pµ,ν (α x, t ) = Pµ,ν (x, |α| ) for all α 6= 0; (d) Pµ,ν (x + y, t + s) ≥L∗ T (Pµ,ν (x, t ), Pµ,ν (y, s)). In this case, Pµ,ν is called an intuitionistic random norm. Here,

Pµ,ν (x, t ) = (µx (t ), νx (t )). Example 5.8. Let (X , k · k) be a normed space. Let T (a, b) = (a1 b1 , min(a2 + b2 , 1)) for all a = (a1 , a2 ), b = (b1 , b2 ) ∈ L∗ and µ, ν be measure and non-measure distribution functions defined by

Pµ,ν (x, t ) = (µx (t ), νx (t )) =



 kxk , , t + k xk t + k xk t

∀t ∈ R+ .

Then (X , Pµ,ν , T ) is an IRN-space. Definition 5.9. (1) A sequence {xn } in an IRN-space (X , Pµ,ν , T ) is called a Cauchy sequence if, for any ε > 0 and t > 0, there exists n0 ∈ N such that

Pµ,ν (xn − xm , t ) >L∗ (Ns (ε), ε),

∀n, m ≥ n0 ,

where Ns is the standard negator. Pµ,ν

(2) The sequence {xn } is said to be convergent to a point x ∈ X (denoted by xn −→ x) if Pµ,ν (xn − x, t ) −→ 1L∗ as n −→ ∞ for every t > 0. (3) An IRN-space (X , Pµ,ν , T ) is said to be complete if every Cauchy sequence in X is convergent to a point x ∈ X . 6. The stability result in intuitionistic random normed spaces Theorem 6.1. Let X be a linear space and (Y , Pµ,ν , T ) be a complete IRN-space. Let f : X −→ Y be a mapping with f (0) = 0 such that there are ξ , ζ : X 2 −→ D+ (ξ (x, y) is denoted by ξx,y and ζ (x, y) is denoted by ζx,y , further, (ξx,y (t ), ζx,y (t )) is denoted by Qξ ,ζ (x, y, t )) with the property

Pµ,ν (f (2x + y) + f (2x − y) − 4f (x + y) − 4f (x − y) − 24f (x) + 6f (y), t ) ≥L∗ Qξ ,ζ (x, y, t ),

∀x, y ∈ X , t > 0.

(6.1)

If

Ti=∞1 (Qξ ,ζ (2n+i−1 x, 0, 24n+3i+1 t )) = 1L∗

(6.2)

and lim Qξ ,ζ (2n x, 2n y, 24n t ) = 1L∗ ,

n→∞

∀x, y ∈ X , t > 0

(6.3)

R. Saadati et al. / Computers and Mathematics with Applications 60 (2010) 1994–2002

2001

then there exists a unique quartic mapping Q : X −→ Y such that

Pµ,ν (f (x) − Q (x), t ) ≥L∗ Ti=∞1 (Qξ ,ζ (2i−1 x, 0, 23i+1 t )), Proof. See [15].

∀x ∈ X , t > 0.

(6.4)



Corollary 6.2. Let (X , Pµ0 0 ,ν 0 , T ) be an IRN-space and (Y , Pµ,ν , T ) be a complete IRN-space. If f : X −→ Y is a mapping such that

Pµ,ν (f (2x + y) + f (2x − y) − 4f (x + y) − 4f (x − y) − 24f (x) + 6f (y), t ) ≥L∗ Pµ0 0 ,ν 0 (x + y, t ),

∀ x, y ∈ X , t > 0 ,

and ∞ 0 3n+2i+2 lim Ti= t )) = 1L∗ , 1 (Pµ0 ,ν 0 (x, 2

n−→∞

∀x ∈ X , t > 0,

then there exists a unique quartic mapping Q : X −→ Y such that

Pµ,ν (f (x) − Q (x), t ) ≥L∗ Ti=∞1 (Pµ0 0 ,ν 0 (x, 22i+2 t )),

∀x ∈ X , t > 0 .

Proof. Put Qξ ,ζ (x, y, t ) = Pµ0 0 ,ν 0 (x + y, t ); then the corollary follows immediately from Theorem 6.1.



Example 6.3. Let (X , k.k) be a Banach algebra space, (X , Pµ,ν , M) be an IRN-space in which

Pµ,ν (x, t ) =



 kxk , t + k xk t + k xk t

,

∀x ∈ X , t > 0,

and (Y , Pµ,ν , M) be a complete IRN-space for all x ∈ X . Define a mapping f : X −→ Y by f (x) = x4 + x0 , where x0 is a unit vector in X . A straightforward computation shows that

Pµ,ν (f (2x + y) + f (2x − y) − 4f (x + y) − 4f (x − y) − 24f (x) + 6f (y), t ) ≥L∗ Pµ,ν (x + y, t ),

∀x, y ∈ X , t > 0.

Also, 3n+2i+2 3n+2i+2 lim M∞ t )) = lim lim Mm t )) i=1 (Pµ,ν (x, 2 i=1 (Pµ,ν (x, 2 n−→∞ m−→∞

n−→∞

= lim

lim Pµ,ν (x, 22n+4 t )

n−→∞ m−→∞

= lim Pµ,ν (x, 22n+4 t ) n−→∞

= 1L∗ ,

∀x ∈ X , t > 0.

Therefore, all the conditions of Theorem 6.1 hold,and so there exists a unique quartic mapping Q : X −→ Y such that

Pµ,ν (f (x) − Q (x), t ) ≥L∗ Pµ,ν (x, 24 t ),

∀x ∈ X , t > 0.

Acknowledgement The second author’s work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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