On the stability of a cubic functional equation in

Alotaibi and Mohiuddine Advances in Difference Equations 2012, 2012:39 http://www.advancesindifferenceequations.com/content/2012/1/39

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On the stability of a cubic functional equation in random 2-normed spaces Abdullah Alotaibi and Syed Abdul Mohiuddine* * Correspondence: [email protected] Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

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 > 0, does there exist a δ > 0 such that if a function h : G1 ® G2 satisfies the inequality d(h (xy), h(x)h(y)) 0, a ≠ 0 and x, y Î X,

(v) Fx+y,z ≥ τ (Fx,z , Fy,z ) whenever x, y, z Î X. If (v) is replaced by (v’) Fx+y,z (t1 + t2 ) ≥ Fx,z (t1 ) ∗ Fy,z(t2 ) , for all x, y, z Î X and t1 , t2 ∈ R+0 , then triple (X, F , ∗) 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 Î [0, 1]. For all x Î X, t > 0 and nonzero z Î X, consider t if t > 0 Fx,z (t) = t+x,z 0 if t ≤ 0; Then (X, F , ∗) 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 Fx,y (t) = H0 (t − x, y) , for every x, y Î X, t > 0 and a * b = min{a, b}, a, b Î [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 (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (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, F , ∗) and (Z, F , ∗) are random 2-normed spaces; and (Y, F , ∗) is a random 2-Banach space.

Let be a function from X × X to Z. A mapping f : X ® Y is said to be -approximately cubic function if FEx,y ,z (t) ≥ Fϕ(x,y),z (t),

(2)

for all x, y Î X, t > 0 and nonzero z Î X, where Ex,y = f (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x).

We define: We say that a sequence x = (xk) is convergent in (X, F , ∗) or simply ℱ-convergent to ℓ if for every > 0 and θ Î (0, 1) there exists k 0 Î N such that Fxk −,z (ε) > 1 − θ lim xk = and ℓ is whenever k ≥ k0 and nonzero z Î X. In this case we write F − k→∞

called the ℱ-limit of x = (xk). A sequence x = (xk) is said to be Cauchy sequence in (X, F , ∗) or simply ℱ-Cauchy if for every > 0, θ > 0 and nonzero z Î X there exists a number N = N(, z) such that lim Fxn −xm ,z (ε) > 1 − θ for all n, m ≥ N. RTN-space (X, F , ∗) is said to be complete if


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every ℱ-Cauchy is ℱ-convergent. In this case (X, F , ∗) is called random 2-Banach space. Theorem 2.1. Suppose that a function : X × X ® Z satisfies (2x, 2y) = a(x, y) for all x, y Î X and a ≠ 0. Let f : X ® Y be a -approximately cubic function. If for some 0 0 and nonzero z Î X. Then there and n→∞

exists a unique cubic mapping C : X ® Y such that FC(x)−f (x),z(t) ≥ Fϕ(x,0),z ((8 − α)t),

(4)

for all x Î X, t > 0 and nonzero z Î X. Proof. For convenience, let us fix y = 0 in (2). Then for all x Î X, t > 0 and nonzero zÎX t F f (2x) (t). ≥ Fϕ(x,0),z (5) 8 −f (x),z 16 Replacing x by 2nx in (5) and using (3), we obtain t n F f (2n+1 x) f (2n x) ≥ Fϕ(2 n x,0),z (t) ≥ Fϕ(x,0),z (t/α ), n) 16(8 − ,z n n+1 8 8 for all x Î X, t > 0 and nonzero z Î X; and for all n ≥ 0. By replacing t by ant, we get αn t F f (2n+1 x) f (2n x) (t). ≥ Fϕ(x,0),z (6) n) 16(8 − ,z n n+1 8 8 n=1 f (2k+1 x) f (2k x) f (2n x) and (6) that It follows from − f (x) = − 8n 8k+1 8k k=0 F f (2n x) 8n

n−1 −f (x),z

k=0

αk t 16(8k )

≥

n−1

F

k=0

for all x Î X, t > 0 and n > 0 where

f (2k+1 x) − 8k+1

n

j=1 aj

f (2k x) 8k

,z

αk t 16(8k )

≥ F ϕ(x,0),z (t),

= a1 ∗ a2 ∗ · · · ∗ an . By replacing x with

m

2 x in (7), we have

F f (2n+m x) 8n+m

n−1 −

f (2m x) 8m ,z

k=0

αk t 16(8)

m ≥ Fϕ(2 m x,0),z (t) ≥ Fϕ(x,0),z (t/α ).

k+m

Thus

F f (2n+m x) 8n+m

n+m−1 −

f (2m x) 8m ,z

k=m

αk t 16(8)k

(7)

(t), ≥ Fϕ(x,0),z


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for all x Î X, t > 0, m > 0, n ≥ 0 and nonzero z Î X. Hence ⎛ ⎞ t ⎝ ⎠, F f (2n+m x) f (2m x) (t) ≥ Fϕ(x,0),z n+m−1 α k − ,z 8n+m

8m

k=m

(8)

16(8)k

for all x Î X, t > 0 m ≥ 0, n ≥ 0 and nonzero z Î X. Since 0 0, n > 0 and nonzero z Î X. Thus we obtain ⎛

FC(x)−f (x),z(t) ≥ F

f (2n x) (t/2) C(x)− 8n ,z

∗ F f (2n x) 8n

−f (x),z

(t/2) ≥

⎞

⎝ t k Fϕ(x,0),z n−1 a k=0 8(8)k

⎠,

for large n. Taking the limit as n ® ∞ and using the definition of RTN-space, we get FC(x)−f (x),z(t) ≥ Fϕ(x,0),z ((8 − α)t).

Replace x and y by 2nx and 2ny, respectively, in (2), we have n F E2 nx,2 ny (t) ≥ Fϕ(2 n x,2n y),z (8 t), 8n

,z

for all x, y Î X, t > 0 and nonzero z Î X. Since n lim Fϕ(2 n x,2n y),z (8 t) = 1,

n→∞

we observe that C fulfills (1). To Prove the uniqueness of the cubic function C, assume that there exists a cubic function D : X ® Y which satisfies (4). For fix x Î X, clearly C(2nx) = 8nC(x) and D(2nx) = 8nD(x) for all n Î N. It follows from (4) that t

FC(x)−D(x),z(t) = F C(2n x) ≥

∗ F f (2n x) D(2n x) D(2n x) (t) ≥ F C(2n x) f (2n x) − 8n ,z 2 n 8n − 8n ,z 8n − 8n ,z n 8 n 8 (8 − α)t 8 (8 − α)t Fϕ(2n x,0),z ≥ Fϕ(x,0),z . n 2

t 2

2α

Therefore Fϕ(x,0),z

8n (8 − α)t 2α n

= 1.

Thus FC(x)−D(x),z(t) = 1 for all x Î X, t > 0 and nonzero z Î X. Hence C(x) = D(x). Example 2.2. Let X be a Hilbert space and Z be a normed space. By ℱ and F , we denote the random 2-norms given as in Example 1.1 on X and Z, respectively. Let : X × X ® Z be defined by (x, y) = 8(∥x∥2 + ∥y∥2)zο, where zο is a fixed unit vector in Z. Define f : X ® X by f(x) = ∥x∥2x + ∥x∥2xο for some unit vector xο Î X. Then


FEx,y ,z (t) =

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t

t 2 ≥ 2 = Fϕ(x,y),z(t). 2 t + 8x, z + 2 y, z t + 8x, z + 8 y, z 2

Also Fϕ(2x,0),z (t) =

t = F4ϕ(x,0),z (t). t + 32x, z2

Thus, n lim Fϕ(2 n x,2n y),z (8 t) = lim

n→∞

n→∞

8n t

2 = 1. 8n t + 8(4n )(x, z2 + y, z )

Hence, conditions of Theorem 2.1 for a = 4 are fulfilled. Therefore, there is a unique cubic mapping C : X ® X such that FC(x)−f (x),z(t) ≥ Fϕ(x,0),z (4t) for all x Î X, t > 0 and nonzero z Î X. By a modification in the proof of Theorem 2.1, one can easily prove the following: Theorem 2.3. Suppose that a function : X × X ® Z satisfies ϕ(x/2, y/2) = α1 ϕ(x, y) for all x, y Î X and a ≠ 0. Let f : X ® Y be a -approximately cubic function. If for some a > 8 Fϕ(x/2,y/2),z (t) ≥ Fϕ(x,y),z (αt)

lim F8 n ϕ(2−n x,2−n y),z (t) = 1 for all x, y Î X, t > 0 and nonzero z Î X. Then there and n→∞

exists a unique cubic mapping C : X ® Y such that FC(x)−f (x),z(t) ≥ Fϕ(x,0),z ((α − 8)t),

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

3 Continuity in random 2-normed spaces In this section, we establish some interesting results of continuous approximately cubic mappings. Let f : ℝ ® X be a function, where ℝ is endowed with the Euclidean topology and X is an random 2-normed space equipped with random 2-norm ℱ. Then, f is said to be random 2-continuous or simply ℱ-continuous at a point sο Î ℝ if for all > 0 and all 0 0 such that Ff (sx)−f (s◦ x),z (ε) ≥ α, for each s with 0 < |s - sο| 0 and nonzero z Î X. Theorem 3.2. Let X be a normed space and let f : X ® Y be a (p, q)-approximately cubic function. If p, q < 3, there exists a unique cubic mapping C : X ® Y such that


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p FC(x)−f (x),z(t) ≥ Fx p z◦ ,z ((8 − 2 )t),

(9)

for all x Î X, t > 0 and nonzero z Î X. Furthermore, if for some x Î X and all n Î N, the mapping g : ℝ ® Y defined by g(s) = f(2nsx) is ℱ-continuous. Then the mapping s ↦ C(sx) from ℝ to Y is ℱ-continuous; in this case, C(rx) = r3C(x) for all r Î ℝ. Proof. Suppose that a function : X × X ® Z satisfies (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 Î X, t Î ℝ and n Î N, we have F

f (2n x) (t) C(x)− 8n ,z

= FC(2n x)−f (2n x),z (8n t) ≥ F2 np xp z◦ ,z (8n (8−2p )t) = Fx p z◦ ,z

8n (8 − 2p )t . 2np

(10)

Fix x Î X and sο Î ℝ. Given > 0 and 0 nο and nonzero z Î X. Since k − j ≥

Clearly

Fy2k+j −yk ,z

1 40k3

1 1 1 ≥ α, F ≥ F ≥ α, y −y ,z y −y ,z 2k−j k 2k−j k 80k3 103 k3 103 k3 1 1 and Fy2k+j −yk+j ,z ≥ Fy2k+j −yk+j ,z ≥ α. 3 60kj2 10 (k + j)3 ≥ Fy2k+j −yk ,z

The inequalities 2k - j ≥ j and 2k - j >k imply

1 1 1 ≥ ≥α F F ≥ y −y ,z y −y ,z 2k+j 2k−j 2k+j 2k−j 120k2 j 120(2k − j)3 103 (2k − j)3 1 1 1 Fy2k+j −y2k−j ,z ≥ Fy2k+j −y2k−j ,z ≥ Fy2k+j −y2k−j ,z ≥ α. 10j3 10(2k − j)3 103 (2k − j)3

Fy2k+j −y2k−j ,z

Therefore FEk,j ,z (1) ≥ α . This shows that f is approximately cubic type mapping. By our assumption, there exists a conditional cubic mapping C : N ∪ {0} ® X, such that lim FC(k)−f (k),z(1) = 1 . In particular, lim FC(2k )−f (2k ),z (1) = 1 . This means that k→∞

k→∞

lim FC(1)−y2 k,z

k→∞

1 23k

=1

Hence the subsequence (y2k ) converges to y = C(1). Therefore, the Cauchy sequence (xn) also converges to y. Acknowledgements The authors would like to thank the anonymous reviewers for their valuable comments. Authors’ contributions Both the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript. Competing interests The authors declare that they have no competing interests. Received: 16 December 2011 Accepted: 29 March 2012 Published: 29 March 2012 References 1. Ulam, SM: A Collection of the Mathematical Problems. Interscience Publ., New York (1960) 2. Hyers, DH: On the stability of the linear functional equation. Proc Nat Acad Sci USA. 27, 222–224 (1941). doi:10.1073/ pnas.27.4.222 3. Aoki, T: On the stability of the linear transformation in Banach spaces. J Math Soc Japan. 2, 64–66 (1950). doi:10.2969/ jmsj/00210064 4. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc. 72, 297–300 (1978). doi:10.1090/S0002-9939-1978-0507327-1

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doi:10.1186/1687-1847-2012-39 Cite this article as: Alotaibi and Mohiuddine: On the stability of a cubic functional equation in random 2-normed spaces. Advances in Difference Equations 2012 2012:39.

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