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Fuzzy Hyers-Ulam stability of an additive functional equation

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Journal of Inequalities and Applications20112011:140

https://doi.org/10.1186/1029-242X-2011-140

  • Received: 10 October 2011
  • Accepted: 19 December 2011
  • Published:

Abstract

In this paper, using the fixed point and direct methods, we prove the Hyers-Ulam stability of the following additive functional equation

2 f x + y + z 2 = f x + f y + f z
(0.1)

in fuzzy normed spaces.

Mathematics Subject Classification (2010): 39B22; 39B52; 39B82; 46S10; 47S10; 46S40.

Keywords

  • Hyers-Ulam stability
  • additive functional equation
  • fuzzy normed space

1. Introduction

A classical question in the theory of functional equations is the following: When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation? If the problem accepts a solution, we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940. In the next year, Hyers [2] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In 1978, Rassias [3] proved a generalization of the Hyers' theorem for additive mappings.

Theorem 1.1. (Th.M. Rassias) Let f : XY be a mapping from a normed vector space X into a Banach space Y subject to the inequality
f ( x + y ) - f ( x ) - f ( y ) ε ( x p + y p )
for all x, y X, where ε and p are constants with ε > 0 and 0 ≤ p < 1. Then the limit
L x = lim n f 2 n x 2 n
exists for all x X and L : XY is the unique additive mapping which satisfies
f ( x ) + L ( x ) 2 ε 2 - 2 p x p

for all x X. Also, if for each x X, the function f(tx) is continuous in t , then L is -linear.

Furthermore, in 1994, a generalization of Rassias' theorem was obtained by Gǎvruta [4] by replacing the bound ε(||x|| p + ||y|| p ) by a general control function φ(x, y).

In 1983, a Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [5] for mappings f : XY, where X is a normed space and Y is a Banach space. In 1984, Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group and, in 2002, Czerwik [7] proved the Hyers-Ulam stability of the quadratic functional equation. The reader is referred to ([820]) and references therein for detailed information on stability of functional equations.

Katsaras [21] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view (see [22, 23]). In particular, Bag and Samanta [24], following Cheng and Mordeson [25], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Karmosil and Michalek type [26]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [27].

Definition 1.2. Let X be a real vector space. A function N : X × → [0, 1] is called a fuzzy norm on X if for all x, y X and all s, t ,

(N 1) N(x, t) = 0 for t ≤ 0;

(N 2) x = 0 if and only if N(x, t) = 1 for all t > 0;

(N 3) N ( c x , t ) = N x , t c if c ≠ 0;

(N 4) N(x + y, c + t) ≥ min{N(x, s), N(y, t)};

(N 5) N(x,.) is a non-decreasing function of and limt→∞N(x, t) = 1;

(N 6) for x ≠ 0, N(x,.) is continuous on .

The pair (X, N) is called a fuzzy normed vector space.

Example 1.3. Let (X, ||.||) be a normed linear space and α, β > 0. Then
N ( x , t ) = α t α t + β x t > 0 , x X 0 t 0 , x X

is a fuzzy norm on X.

Definition 1.4. Let (X, N) be a fuzzy normed vector space. A sequence {x n } in X is said to be convergent or converge if there exists an x X such that limt→∞N(x n - x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {x n } in X and we denote it by N - limt→∞x n = x.

Definition 1.5. Let (X, N) be a fuzzy normed vector space. A sequence {x n } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 such that for all nn0and all p > 0, we have N(xn+p- x n , t) > 1 - ε.

It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

We say that a mapping f : XY between fuzzy normed vector spaces X and Y is continuous at a point x X if for each sequence {x n } converging to x0 X, then the sequence {f(x n )} converges to f(x0). If f : XY is continuous at each x X, then f : XY is said to be continuous on X.

Definition 1.6. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies the following conditions:

(a) d(x, y) = 0 if and only if x = y for all x, y X;

(b) d(x, y) = d(y, x) for all x, y X;

(c) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z X.

Theorem 1.7. ([28, 29]) Let (X, d) be a complete generalized metric space and J : XX be a strictly contractive mapping with Lipschitz constant L < 1. Then, for all x X, either d(J n x, Jn+1x) = ∞ for all nonnegative integers n or there exists a positive integer n0such that

(a) d(J n x, Jn+1x) < ∞ for all n0n0;

(b) the sequence {J n x} converges to a fixed point y* of J;

(c) y* is the unique fixed point of J in the set Y = { y X : d ( J n 0 x , y ) < } ;

(d) d ( y , y * ) d ( y , J y ) 1 - L for all y Y.

2. Fuzzy stability of the functional Eq. (0.1)

Throughout this section, using the fixed point and direct methods, we prove the Hyers-Ulam stability of functional Eq. (0.1) in fuzzy normed spaces.

2.1. Fixed point alternative approach

Throughout this subsection, using the fixed point alternative approach, we prove the Hyers-Ulam stability of functional Eq. (0.1) in fuzzy Banach spaces.

In this subsection, assume that X is a vector space and that (Y, N) is a fuzzy Banach space.

Theorem 2.1. Let φ : X3 → [0, ∞) be a function such that there exists an L < 1 with
φ ( x , y , z ) L φ ( 2 x , 2 y , 2 z ) 2
for all x, y, z X. Let f : XY be a mapping satisfying
N 2 f x + y + z 2 - f ( x ) - f ( y ) - f ( z ) , t t t + φ ( x , y , z )
(2.1)
for all x, y, z X and all t > 0. Then the limit
A ( x ) : = N - lim n 2 n f x 2 n
exists for each x X and defines a unique additive mapping A : XY such that
N ( f ( x ) - A ( x ) , t ) ( 2 - 2 L ) t ( 2 - 2 L ) t + L φ ( x , 2 x , x ) .
(2.2)
Proof. Putting y = 2x and z = x in (2.1) and replacing x by x 2 , we have
N 2 f x 2 - f ( x ) , t t t + φ x 2 , x , x 2
(2.3)
for all x X and t > 0. Consider the set
S : = { g : X Y }
and the generalized metric d in S defined by
d ( f , g ) = inf μ + : N ( g ( x ) - h ( x ) , μ t ) t t + φ ( x , 2 x , x ) , x X , t > 0 ,
where inf = +∞. It is easy to show that (S, d) is complete (see [30, Lemma 2.1]). Now, we consider a linear mapping J : SS such that
J g ( x ) : = 2 g x 2
for all x X. Let g, h S be such that d(g, h) = ε. Then
N ( g ( x ) - h ( x ) , ε t ) t t + φ ( x , 2 x , x )
for all x X and t > 0. Hence,
N ( J g ( x ) - J h ( x ) , L ε t ) = N 2 g x 2 - 2 h x 2 , L ε t = N g x 2 - h x 2 , L ε t 2 L t 2 L t 2 + φ x 2 , x , x 2 L t 2 L t 2 + L φ ( x , 2 x , x ) 2 = t t + φ ( x , 2 x , x )
for all x X and t > 0. Thus, d(g, h) = ε implies that d(Jg, Jh) ≤ . This means that
d ( J g , J h ) L d ( g , h )
for all g, h S. It follows from (2.3) that
N f ( x ) - 2 f x 2 , t t t + φ x 2 , x , x 2 t t + L φ ( x , 2 x , x ) 2 = 2 t L 2 t L + φ ( x 2 x , x ) .
(2.4)
Therefore,
N f ( x ) - 2 f x 2 , L t 2 t t + φ ( x , 2 x , x ) .
(2.5)
This means that
d ( f , J f ) L 2 .
By Theorem 1.7, there exists a mapping A : XY satisfying the following:
  1. (1)
    A is a fixed point of J, that is,
    A x 2 = A ( x ) 2
    (2.6)
     
for all x X. The mapping A is a unique fixed point of J in the set
Ω = { h S : d ( g , h ) < } .
This implies that A is a unique mapping satisfying (2.6) such that there exists μ (0, ∞) satisfying
N ( f ( x ) - A ( x ) , μ t ) t t + φ ( x , 2 x , x )
for all x X and t > 0.
  1. (2)
    d(J n f, A) → 0 as n → ∞. This implies the equality
    N - lim n 2 n f x 2 n = A ( x )
     
for all x X.
  1. (3)
    d ( f , A ) d ( f , J f ) 1 - L with f Ω, which implies the inequality
    d ( f , A ) L 2 - 2 L .
     
This implies that the inequality (2.2) holds. Furthermore, since
N 2 A x + y + z 2 - A ( x ) - A ( y ) - A ( z ) , t N - lim n 2 n + 1 f x + y + z 2 n + 1 - 2 n f x 2 n - 2 n f y 2 n - 2 n f z 2 n , t lim n t 2 n t 2 n + L n φ ( x , y , z ) 2 n 1

for all x, y, z X, t > 0. So N A x + y + z 2 - A x - A y - A z , t = 1 for all x, y, z X and all t > 0. Thus the mapping A : XY is additive, as desired.    □

Corollary 2.2. Let θ ≥ 0 and let p be a real number with p > 1. Let X be a normed vector space with norm ||.||. Let f : XY be a mapping satisfying
N 2 f x + y + z 2 - f ( x ) - f ( y ) - f ( z ) , t t t + θ ( x p + y p + z p )
for all x, y, z X and all t > 0. Then the limit
A ( x ) : = N - lim n 2 n f x 2 n
exists for each x X and defines a unique additive mapping A : XY such that
N ( f ( x ) - A ( x ) , t ) ( 2 p - 1 ) t ( 2 p - 1 ) t + ( 2 r - 1 + 1 ) θ x p

for all x X.

Proof. The proof follows from Theorem 2.1 by taking φ(x, y, z): = θ(||x|| p + ||y|| p + ||z|| p ) for all x, y, z X. Then we can choose L = 2-pand we get the desired result.    □

Theorem 2.3. Let φ : X3 → [0, ∞) be a function such that there exists an L < 1 with
φ ( 2 x , 2 y , 2 z ) 2 L φ ( x , y , z )
for all x, y, z X. Let f : XY be a mapping satisfying (2.1). Then
A ( x ) : = N - lim n f ( 2 n x ) 2 n
exists for each x X and defines a unique additive mapping A : XY such that
N ( f ( x ) - A ( x ) , t ) ( 2 - 2 L ) t ( 2 - 2 L ) t + φ ( x , 2 x , x )
(2.7)

for all x X and all t > 0.

Proof. Let (S, d) be the generalized metric space defined as in the proof of Theorem 2.1.

Consider the linear mapping J : SS such that
J g ( x ) : = g ( 2 x ) 2
for all x X. Let g, h S be such that d(g, h) = ε. Then
N ( g ( x ) - h ( x ) , ε t ) t t + φ ( x , 2 x , x )
for all x X and t > 0. Hence,
N ( J g ( x ) - J h ( x ) , L ε t ) = N g ( 2 x ) 2 - h ( 2 x ) 2 , L ε t = N g ( 2 x ) - h ( 2 x ) , 2 L ε t 2 L t 2 L t + φ ( 2 x , , 4 x , 2 x ) 2 L t 2 L t + 2 L φ ( x , 2 x , x ) = t t + φ ( x , 2 x , x )
for all x X and t > 0. Thus, d(g, h) = ε implies that d(Jg, Jh) ≤ . This means that
d ( J g , J h ) L d ( g , h )
for all g, h S. It follows from (2.3) that
N f ( 2 x ) 2 - f ( x ) , t 2 t t + φ ( x , 2 x , x ) .
Therefore,
d ( f , J f ) 1 2 .
By Theorem 1.7, there exists a mapping A : XY satisfying the following:
  1. (1)
    A is a fixed point of J, that is,
    2 A ( x ) = A ( 2 x )
    (2.8)
     
for all x X. The mapping A is a unique fixed point of J in the set
Ω = { h S : d ( g , h ) < } .
This implies that A is a unique mapping satisfying (2.8) such that there exists μ (0, ∞) satisfying
N ( f ( x ) - A ( x ) , μ t ) t t + φ ( x , 2 x , x )
for all x X and t > 0.
  1. (2)
    d(J n f, A) → 0 as n → ∞. This implies the equality
    N - lim n f ( 2 n x ) 2 n
     
for all x X.
  1. (3)
    d ( f , A ) d ( f , J f ) 1 - L with f Ω which implies the inequality
    d ( f , A ) 1 2 - 2 L .
     

This implies that the inequality (2.7) holds.

The rest of the proof is similar to that of the proof of Theorem 2.1.    □

Corollary 2.4. Let θ ≥ 0 and let p be a real number with 0 < p < 1 3 . Let X be a normed vector space with norm || . ||. Let f : XY be a mapping satisfying
N 2 f x + y + z 2 - f ( x ) - f ( y ) - f ( z ) , t t t + θ ( x p . y p . z p )
for all x, y, z X and all t > 0. Then
A ( x ) : = N - lim n f ( 2 n x ) 2 n
exists for each x X and defines a unique additive mapping A : XY such that
N ( f ( x ) - A ( x ) , t ) ( 2 3 p - 1 ) t ( 2 3 p - 1 ) t + 2 3 p - 1 θ x 3 p .

for all x X.

Proof. The proof follows from Theorem 2.3 by taking φ(x, y, z): = θ(||x|| p · ||y|| p · ||z|| p ) for all x, y, z X. Then we can choose L = 2-3pand we get the desired result.    □

2.2. Direct method. In this subsection, using direct method, we prove the Hyers-Ulam stability of the functional Eq. (0.1) in fuzzy Banach spaces.

Throughout this subsection, we assume that X is a linear space, (Y, N) is a fuzzy Banach space and (Z, N') is a fuzzy normed spaces. Moreover, we assume that N(x,.) is a left continuous function on .

Theorem 2.5. Assume that a mapping f : XY satisfies the inequality
N 2 f x + y + z 2 - f ( x ) - f ( y ) - f ( z ) , t N ( φ ( x , y , z ) , t )
(2.9)
for all x, y, z X, t > 0 and φ : X3Z is a mapping for which there is a constant r satisfying 0 < r < 1 2 and
N ( φ ( x , y , z ) , t ) N φ ( 2 x , 2 y , 2 z ) , t r
(2.10)
for all x, y, z X and all t > 0. Then there exist a unique additive mapping A : XY satisfying (0.1) and the inequality
N ( f ( x ) - A ( x ) , t ) N φ ( x , 2 x , x ) , ( 1 - 2 r ) t r
(2.11)

for all x X and all t > 0.

Proof. It follows from (2.10) that
N φ x 2 j , y 2 j , z 2 j , t N φ ( x , y , z ) , t r j .
(2.12)
So
N φ x 2 j , y 2 j , z 2 j , r j t N ( φ ( x , y , z ) , t )
for all x, y, z X and all t > 0. Substituting y = 2x and z = x in (2.9), we obtain
N ( f ( 2 x ) - 2 f ( x ) , t ) N ( φ ( x , 2 x , x ) , t )
(2.13)
So
N f ( x ) - 2 f x 2 , t N φ x 2 , x , x 2 , t
(2.14)
for all x X and all t > 0. Replacing x by x 2 j in (2.14), we have
N 2 j + 1 f x 2 j + 1 - 2 j f x 2 j , 2 j t N φ x 2 j + 1 , x 2 j , x 2 j + 1 , t N φ ( x , 2 x , x ) , t r j + 1
(2.15)
for all x X, all t > 0 and any integer j ≥ 0. So
N f ( x ) - 2 n f x 2 n , j = 0 n - 1 2 j r j + 1 t = N j = 0 n - 1 2 j + 1 f x 2 j + 1 - 2 j f x 2 j , j = 0 n - 1 2 j r j + 1 t min 0 j n - 1 N 2 j + 1 f x 2 j + 1 - 2 j f x 2 j , 2 j r j + 1 t N ( φ ( x , 2 x , x ) , t ) .
(2.16)
Replacing x by x 2 p in the above inequality, we find that
N 2 n + p f x 2 n + p - 2 p f x 2 p , j = 0 n - 1 2 j r j + 1 t N φ x 2 p , 2 x 2 p , x 2 p , t N φ ( x , 2 x , x ) , t r p
for all x X, t > 0 and all integers n ≥ 0, p ≥ 0. So
N 2 n + p f x 2 n + p - 2 p f x 2 p , j = 0 n - 1 2 j + p r j + p + 1 t N ( φ ( x , 2 x , x ) , t )
for all x X, t > 0 and all integers n > 0, p ≥ 0. Hence, one obtains
N 2 n + p f x 2 n + p - 2 p f x 2 p , t N φ ( x , 2 x , x ) , t j = 0 n - 1 2 j + p r j + p + 1
(2.17)
for all x X, t > 0 and all integers n > 0, p ≥ 0. Since the series j = 0 2 j r j is convergent, by taking the limit p → ∞ in the last inequality, we know that a sequence 2 n f x 2 n is a Cauchy sequence in the fuzzy Banach space (Y, N) and so it converges in Y. Therefore, a mapping A: XY defined by
A ( x ) : = N - lim n 2 n f x 2 n
is well defined for all x X. It means that
lim n N A ( x ) - 2 n f x 2 n , t = 1
(2.18)
for all x X and all t > 0. In addition, it follows from (2.17) that
N 2 n f x 2 n - f ( x ) , t N φ ( x , 2 x , x ) , t j = 0 n - 1 2 j r j + 1
for all x X and all t > 0. So
N ( f ( x ) - A ( x ) , t ) min N f ( x ) - 2 n f x 2 n , ( 1 - ε ) t , N A ( x ) - 2 n f x 2 n , ε t N φ ( x , 2 x , x ) , t j = 0 n - 1 2 j r j + 1 N φ ( x , 2 x , x ) , ( 1 - 2 r ) ε t r
for sufficiently large n and for all x X, t > 0 and N with 0 < N < 1. Since N is arbitrary and N' is left continuous, we obtain
N ( f ( x ) - A ( x ) , t ) N φ ( x , 2 x , x ) , ( 1 - 2 r ) t r
for all x X and t > 0. It follows from (2.9) that
N 2 n + 1 f x + y + z 2 n + 1 - 2 n f x 2 n - 2 n f y 2 n - 2 n f z 2 n , t N φ x 2 n , y 2 n , z 2 n , t 2 n N φ ( x , y , z ) , t 2 n r n
for all x, y, z X, t > 0 and all n . Since
lim n N φ ( x , y , z ) , t 2 n r n = 1
and so
N 2 n + 1 f x + y + z 2 n + 1 - 2 n f x 2 n - 2 n f y 2 n - 2 n f z 2 n , t 1
for all x, y, z X and all t > 0. Therefore, we obtain in view of (2.18)
N 2 A x + y + z 2 - A ( x ) - A ( y ) - A ( z ) , t min N ( A x + y + z 2 - A ( x ) - A ( y ) - A ( z ) - 2 n + 1 f x + y + z 2 n + 1 - 2 n f x 2 n - 2 n f y 2 n - 2 n f z 2 n , t 2 , N 2 n + 1 f x + y + z 2 n + 1 - 2 n f x 2 n - 2 n f y 2 n - 2 n f z 2 n , t 2 = N 2 n + 1 f x + y + z 2 n + 1 - 2 n f x 2 n - 2 n f y 2 n - 2 n f z 2 n , t 2 N φ ( x , y , z ) , t 2 n + 1 r n 1 as  n
which implies
2 A x + y + z 2 = A ( x ) + A ( y ) + A ( z )

for all x, y, z X. Thus, A: XY is a mapping satisfying the Eq. (0.1) and the inequality (2.11).

To prove the uniqueness, assume that there is another mapping L : XY which satisfies the inequality (2.11). Since L ( x ) = 2 n L ( x 2 n ) for all x X, we have
N ( A ( x ) - L ( x ) , t ) = 2 n A x 2 n - 2 n L x 2 n , t min N 2 n A x 2 n - 2 n f x 2 n , t 2 , N 2 n f x 2 n - 2 n L x 2 n , t 2 N φ x 2 n , 2 x 2 n , x 2 n , ( 1 - 2 r ) t r 2 n + 1 N φ ( x , 2 x , x ) , ( 1 - 2 r ) t r n + 1 2 n + 1 1 as  n

for all t > 0. Therefore, A(x) = L(x) for all x X, this completes the proof.    □

Corollary 2.6. Let X be a normed spaces and (, N') a fuzzy Banach space. Assume that there exist real numbers θ ≥ 0 and 0 < p < 1 such that a mapping f : XY satisfies the following inequality
N 2 f x + y + z 2 - f ( x ) - f ( y ) - f ( z ) , t N ( θ ( x p + y p + z p ) , t )
for all x, y, z X and t > 0. Then there is a unique additive mapping A : XY satisfying (0.1) and the inequality
N ( f ( x ) - A ( x ) , t ) N θ x p , 2 t 2 r + 2

Proof. Let φ(x, y, z): = θ(||x|| p + ||y|| p + ||z|| p ) and r = 1 4 . Applying Theorem 2.5, we get the desired result.    □

Theorem 2.7. Assume that a mapping f : XY satisfies (2.9) and φ : X2Z is a mapping for which there is a constant r satisfying 0 < |r| < 2 and
N ( φ ( 2 x , 2 y , 2 z ) , r t ) N ( φ ( x , y , z ) , t )
(2.19)
for all x, y, z X and all t > 0. Then there is a unique additive mapping A : XY satisfying (0.1) and the following inequality
N ( f ( x ) - A ( x ) , t ) N ( φ ( x , 2 x , x ) , ( 2 - r ) t ) .
(2.20)

for all x X and all t > 0.

Proof. It follows from (2.13) that
N f ( 2 x ) 2 - f ( x ) , t 2 N ( φ ( x , 2 x , x ) , t )
(2.21)
for all x X and all t > 0. Replacing x by 2 n x in (2.21), we obtain
N f ( 2 n + 1 x ) 2 n + 1 - f ( 2 n x ) 2 n , t 2 n + 1 N ( φ ( 2 n x , 2 n + 1 x , 2 n x ) , t ) N φ ( x , 2 x , x ) , t r n .
So
N f ( 2 n + 1 x ) 2 n + 1 - f ( 2 n x ) 2 n , r n t 2 n + 1 N ( φ ( x , 2 x , x ) , t )
(2.22)
for all x X and all t > 0. Proceeding as in the proof of Theorem 2.5, we obtain that
N f ( x ) - f ( 2 n x ) 2 n , j = 0 n - 1 r j t 2 j + 1 N ( φ ( x , 2 x , x ) , t )
for all x X, all t > 0 and all integers n > 0. So
N f ( x ) - f ( 2 n x ) 2 n , t N φ ( x , 2 x , x ) , t j = 0 n - 1 r j 2 j + 1 N ( φ ( x , 2 x , x ) , ( 2 - r ) t ) .

The rest of the proof is similar to the proof of Theorem 2.5.    □

Corollary 2.8. Let X be a normed spaces and (, N') a fuzzy Banach space. Assume that there exist real numbers θ ≥ 0 and 0 < p < 1 3 such that a mapping f : XY satisfies the following inequality
N 2 f x + y + z 2 - f ( x ) - f ( y ) - f ( z ) , t N ( θ ( x p y p z p ) , t )
for all x, y, z X and t > 0. Then there is a unique additive mapping A : XY satisfying (0.1) and the inequality
N ( f ( x ) - A ( x ) , t ) N θ x p , t 2 r + 2

Proof. Let φ ( x , y , z ) : = θ ( x p 1 y p 2 z p 3 ) and |r| = 1. Applying Theorem 2.7, we get the desired result.    □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, College of Sciences, Yasouj University, 75914-353 Yasouj, Iran
(2)
Department of Mathematics, Firoozabad Branch, Islamic Azad University, Firoozabad, Iran
(3)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea
(4)
Department of Mathematics, Daejin University, Kyeonggi, 487-711, Korea

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© Kenary et al; licensee Springer. 2011

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.

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