On nonlinear stability in various random normed

Rassias et al. Journal of Inequalities and Applications 2011, 2011:62 http://www.journalofinequalitiesandapplications.com/content/2011/1/62

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On nonlinear stability in various random normed spaces John Michael Rassias1, Reza Saadati2*, Ghadir Sadeghi3 and J Vahidi4 * Correspondence: RSAADATI@EML. CC 2 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran Full list of author information is available at the end of the article

Abstract In this article, we prove the nonlinear stability of the quartic functional equation

y + f (x + 2y) 16f (x + 4y) + f (4x − y) =306 9f x + 3 + 136f (x − y) − 1394f (x + y) + 425f (y) − 1530f (x)

in the setting of random normed spaces Furthermore, the interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean space, the theory of fixed point theory, the theory of intuitionistic spaces and the theory of functional equations are also presented in the article. Keywords: generalized Hyers-Ulam stability, quartic functional equation, random normed space, intuitionistic random normed space

1. Introduction The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [2]. Subsequently, this 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 article of Rassias [4] has provided a lot of influence in the development of what we now call generalized Ulam-Hyers stability of functional equations. We refer the interested readers for more information on such problems to the article [5-17]. Recently, Alsina [18], Chang, et al. [19], Mirmostafaee et al. [20], [21], Miheţ and Radu [22], Miheţ et al. [23], [24], [25], [26], Baktash et al. [27], Eshaghi et al. [28], Saadati et al. [29], [30] investigated the stability in the settings of fuzzy, probabilistic, and random normed spaces. In this article, we study the stability of the following functional equation y + f (x + 2y) 16f (x + 4y) + f (4x − y) =306 9f x + 3 (1:1) + 136f (x − y) − 1394f (x + y) + 425f (y) − 1530f (x)

in the various random normed spaces via different methods. Since ax4 is a solution of above functional equation, we say it quartic functional equation.

© 2011 Rassias et al; licensee Springer. 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.


Page 2 of 17

2. Preliminaries In this section, we recall some definitions and results which will be used later on in the article. A triangular norm (shorter 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 Î [0, 1] the following four axioms satisfied: (i) T(a, b) = T(b, a) (commutativity); (ii) T(a, (T(b, c))) = T(T(a, b), c) (associativity); (iii) T(a, 1) = a (boundary condition); (iv) T(a, b) ≤ T(a, c) whenever b ≤ c (monotonicity). Basic examples are the Lukasiewicz t-norm TL, TL(a, b) = max (a + b - 1, 0) ∀a, b Î [0, 1] and the t-norms TP, TM, TD, where TP (a, b) := ab, TM (a, b) := min {a, b}, min(a, b), if max(a, b) = 1; TD (a, b) := 0, otherwise. If T is a t-norm then x(n) is defined for every x Î [0, 1] and n Î N ∪ {0} by 1, if n = T 0 and T(x(n−1) , x), if n ≥ 1. A t-norm T is said to be of Hadžić-type (we denote by T is equicontinuous at x = 1 (cf. [31]). T ∈ H) if the family (x(n) T )n∈N Other important triangular norms are (see [32]): SW SW = T and -the Sugeno-Weber family {TλSW }λ∈[−1,∞] is defined by T−1 = TD, T∞ P x + y − 1 + λxy TλSW (x, y) = max 0, 1+λ

if l Î (-1, ∞). -the Domby family {TλD }λ∈[0,∞], defined by TD, if l = 0, TM , if l = ∞ and TλD (x, y) =

1 λ 1/λ

1−y λ 1 + (( 1−x x ) +( y ) )

if l Î (0, ∞). -the Aczel-Alsina family {TλAA }λ∈[0,∞], defined by TD, if l = 0, TM , if l = ∞ and TλAA (x, y) = e−(| log x|

λ

1/λ

+| log y|λ )

if l Î (0, ∞). A t-norm T can be extended (by associativity) in a unique way to an n-array operation taking for (x1, ..., xn) Î [0, 1]n the value T (x1, ..., xn) defined by T0i=1 xi = 1, Tni=1 xi = T(Tn−1 i=1 xi , xn ) = T(x1 , . . . , xn ).

T can also be extended to a countable operation taking for any sequence (xn)nÎN in [0, 1] the value n T∞ i=1 xi = lim Ti=1 xi . n→∞

(2:1)


The limit on the right side of (2.1) exists since the sequence {Tni=1 xi }n∈N is nonincreasing and bounded from below. Proposition 2.1. [32] (i) For T ≥ TL the following implication holds: lim T∞ i=1 xn+i = 1 ⇔

n→∞

∞

(1 − xn ) < ∞.

n=1

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

=1

for every sequence {xn}nÎN 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→∞

∞

(1 − xn )α < ∞.

n=1

(iv) If T ∈ {TλSW }λ∈[−1,∞), then lim T∞ i=1 xn+i = 1 ⇔

n→∞

∞

(1 − xn ) < ∞.

n=1

Definition 2.2. [33] A random normed space (briefly, RN-space) is a triple (X, μ, T), where X is a vector space, T is a continuous t-norm, and μ is a mapping from X into D+ such that, the following conditions hold: (RN1) μx(t) = ε0(t) for all t >0 if and only if x = 0; t for all x Î X, a ≠ 0; (RN2) μαx (t) = μx |α| (RN3) μx+y(t + s) ≥ T (μx(t), μy(s)) for all x, y, z Î X and t, s ≥ 0. Definition 2.3. Let (X, μ, T) be an RN-space. (1) A sequence {xn} in X is said to be convergent to x in X if, for every ε >0 and l >0, there exists a positive integer N such that μxn −x (ε) > 1 − λ whenever n ≥ N. (2) A sequence {xn} in X is called Cauchy if, for every ε >0 and l >0, there exists a positive integer N such that μxn −xm (ε) > 1 − λ whenever n ≥ m ≥ N. (3) An RN-space (X, μ, T) is said to be complete if every Cauchy sequence in X is convergent to a point in X. Theorem 2.4. [34]If (X, μ, T) is an RN-space and {xn} is a sequence such that xn ® x, then limn→∞ μxn (t) = μx (t)almost everywhere.

3. Non-Archimedean random normed space By a non-Archimedean field we mean a field K equipped with a function (valuation) | · | from K into [0, ∞] such that |r| = 0 if and only if r = 0, |rs| = |r| |s|, and |r + s| ≤ max{| r|, |s|} for all r, s ∈ K. Clearly |1| = | -1| = 1 and |n| ≤ 1 for all n Î N. By the trivial valuation we mean the mapping | · | taking everything but 0 into 1 and |0| = 0. Let X be a vector space over a field K with a non-Archimedean non-trivial valuation | · |.

Page 3 of 17


Page 4 of 17

A function || · || : X → [0, ∞] is called a non-Archimedean norm if it satisfies the following conditions: (i) ||x|| = 0 if and only if x = 0; (ii) for any r ∈ K, x ∈ X , ||rx|| = ||r|||x||; (iii) the strong triangle inequality (ultrametric); namely, ||x + y|| ≤ max{||x||, ||y||}

(x, y ∈ X ).

Then (X , || · ||) is called a non-Archimedean normed space. Due to the fact that ||xn − xm || ≤ max{||xj+1 − xj || : m ≤ j ≤ n − 1}

(n > m),

a sequence {x n } is Cauchy if and only if {x n+1 - x n } converges to zero in a nonArchimedean normed space. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent. In 1897, Hensel [35] discovered the p-adic numbers as a number theoretical analogue of power series in complex analysis. Fix a prime number p. For any non-zero rational number x, there exists a unique integer nx Î ℤ such that x = ab pnx, where a and b are integers not divisible by p. Then |x|p := p−nx defines a non-Archimedean norm on Q. The completion of Q with respect to the metric d(x, y) = |x - y| p is denoted by Qp, which is called the p-adic number field. Throughout the article, we assume that X is a vector space and Y is a complete nonArchimedean normed space. Definition 3.1. A non-Archimedean random normed space (briefly, 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 for all x ∈ X , t >0, a ≠ 0; (NA-RN2) μαx (t) = μx |α| (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 so is (RN3) μx+y(t + s) ≥ T (μx(t), μy(s)). As a classical example, if (X , ||.||) is a non-Archimedean normed linear space, then the triple (X , μ, TM ), where 0 t ≤ ||x|| μx (t) = 1 t > ||x|| is a non-Archimedean RN-space. Example 3.2. Let (X , ||.||) be is a non-Archimedean normed linear space. Define μx (t) =

t , t + ||x||

∀x ∈ X

t > 0.

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


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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 Cauchy if for each ε >0 and each 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. Remark 3.4. [36] Let (X , μ, TM ) be a non-Archimedean RN-space, then μxn+p −xn (t) ≥ min{μxn+j+1 −xn+j (t) : j = 0, 1, 2, . . . , p − 1}

So, the sequence {xn} is Cauchy if for each ε >0 and t >0 there exists n0 such that for all n ≥ n0 we have μxn+1 −xn (t) > 1 − ε.

4. Generalized Ulam-Hyers stability for a quartic functional equation in nonArchimedean RN-spaces Let K be a non-Archimedean field, X a vector space over K and let (Y , μ, T) be a nonArchimedean random Banach space over K. We investigate the stability of the quartic functional equation y + f (x + 2y) 16f (x + 4y) + f (4x − y) = 306 9f x + 3 + 136f (x − y) − 1394f (x + y) + 425f (y) − 1530f (x),

where f is a mapping from X to Y and f(0) = 0. Next, we define a random approximately quartic mapping. Let Ψ be a distribution function on X × X × [0, ∞] such that Ψ (x, y, ·) is symmetric, nondecreasing and t Ψ (cx, cx, t) ≥ Ψ x, x, (x ∈ X , c = 0). |c| Definition 4.1. A mapping f : X → Y is said to be Ψ-approximately quartic if

μ

(t) y +f (x+2y) −136f (x−y)+1394f (x+y)−425f (y)+1530f (x) 3 (x, y ∈ X , t > 0).

16f (x+4y)+f (4x−y)−306 9f x+

≥ Ψ (x, y, t)

(4:1)

In this section, we assume that 4 ≠ 0 in K (i.e., characteristic of K is not 4). Our main result, in this section, is the following: Theorem 4.2. Let Kbe a non-Archimedean field, X a vector space over Kand let (Y , μ, T)be a non-Archimedean random Banach space over K. Let f : X → Y be a Ψapproximately quartic mapping. If for some a Î ℝ, a >0, and some integer k, k >3 with |4k| 0)

(4:2)

and αj t lim T∞ M x, =1 n→∞ j=n |4|kj

(x ∈ X ,

t > 0),

(4:3)


Page 6 of 17

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 |4|ki

(4:4)

for all x Î X and t >0, where M(x, t) := T(Ψ (x, 0, t), Ψ (4x, 0, t), · · · , Ψ (4k−1 x, 0, t))

(x ∈ X ,

t > 0).

Proof. First, we show by induction on j that for each x ∈ X , t >0 and j ≥ 1, μf (4j x)−256j f (x) (t) ≥ Mj (x, t) := T(Ψ (x, 0, t), · · · , Ψ (4j−1 x, 0, t)).

(4:5)

Putting y = 0 in (4.1), we obtain (x ∈ X ,

μf (4x)−256f (x) (t) ≥ Ψ (x, 0, t)

t > 0).

This proves (4.5) for j = 1. Assume that (4.5) holds for some j ≥ 1. Replacing y by 0 and x by 4jx in (4.1), we get μf (4j+1 x)−256f (4j x) (t) ≥ Ψ (4j x, 0, t)

(x ∈ X ,

t > 0).

Since |256| ≤ 1,

μf (4j+1 x)−256j+1 f (x) (t) ≥ T μf (4j+1 x)−256f (4j x) (t), μ256f (4j x)−256j+1 f (x) (t) t = T μf (4j+1 x)−256f (4j x) (t), μf (4j x)−256j f (x) |256| ≥ T μf (4j+1 x)−256f (4j x) (t), μf (4j x)−256j f (x) (t) ≥ T(Ψ (4j x, 0, t), Mj (x, t)) = Mj+1 (x, t)

for all x ∈ X . Thus (4.5) holds for all j ≥ 1. In particular μf (4k x)−256k f (x) (t) ≥ M(x, t)

(x ∈ X ,

t > 0).

(4:6)

Replacing x by 4-(kn+k)x in (4.6) and using inequality (4.2), we obtain x μf x −256k f x (t) ≥ M kn+k , t 4 4kn 4kn+k ≥ M(x, α n+1 t) (x ∈ X , t > 0, n = 0, 1, 2, . . .). Then (t) μ n x x 4k n+1 (44k ) f f n+1 k n −(4 ) k (4 ) (4 )

≥ M x,

α n+1 |(44k )n |

t

(x ∈ X ,

t > 0,

Hence, μ

n

(44k ) f

(t) x x 4k n+p f k n −(4 ) k n+p (4 ) (4 )

⎞

⎛

n+p ⎜ ≥ Tj=n ⎝μ

j

(44k ) f

(t)⎟ ⎠ x x 4k j+p −(4 ) f j j+p k k (4 ) (4 )

j+1 n+p ≥ Tj=n M x, α4k j t |(4 ) | j+1 n+p (x ∈ X , ≥ Tj=n M x, α k j t |(4 ) |

t > 0,

n = 0, 1, 2, . . .).

n = 0, 1, 2, . . .).

(4:7)


j+1 ∞ M x, α k j t = 1 Since limn→∞ Tj=n |(4 ) |

Page 7 of 17

(x ∈ X ,

t > 0),

n

(44k ) f

x n (4k )

n∈N

, is a

Cauchy sequence in the non-Archimedean random Banach space (Y , μ, T). Hence, we can define a mapping Q : X → Y such that lim μ

n→∞

n

(44k ) f

(t) x k n −Q(x) (4 )

(x ∈ X,

=1

t > 0).

(4:8)

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

n

f (x)−(44k ) f

x n (4k )

(t)

= μ

n−1 i=0

i

(44k ) f

⎛ n−1 ⎜ ≥ Ti=0 ⎝μ

(t) x x 4k i+1 −(4 ) f i i+1 (4k ) (4k )

⎞

(t)⎟ ⎠ x x 4k i+1 (4 ) f f i −(4 ) i+1 (4k ) (4k )

4k i

α i+1 t n−1 . ≥ Ti=0 M x, 4k |4 |i

Therefore,

μf (x)−Q(x)(t) ≥ T μ

n

f (x)−(44k ) f

n−1 ≥ T Ti=0 M x,

x n (4k )

α i+1 t |44k |i

(t), μ

n

(44k ) f

(t) x n −Q(x) (4k )

,μ

n

(44k ) f

(t) x k n −Q(x) (4 )

.

By letting n ® ∞, we obtain α i+1 t ∞ μf (x)−Q(x) (t) ≥ Ti=1 M x, k . |4 |i This proves (4.4). As T is continuous, from a well-known result in probabilistic metric space (see e.g., [[34], Chapter 12]), it follows that lim μ

n→∞

y n ))+(4k ) f (4−kn (x+2y)) 3 −136(4k )n f (4−kn (x−y))+1394(4k )n f (4−kn (x+y))−425(4k )n f (4−kn y)+1530(4k )n f (4−kn x) (t) n

n

n

(4k ) ·16f (4−kn (x+4y))+(4k ) f (4−kn (4x−y))−306 (4k ) ·9f (4−kn (x+

=μ

16Q(x+4y)+Q(4x−y)−306 9Q x+

y 3

+Q(x+2y) −136Q(x−y)+1394Q(x+y)−425Q(y)+1530Q(x)

(t)

for almost all t >0. On the other hand, replacing x, y by 4-knx, 4-kny, respectively, in (4.1) and using (NARN2) and (4.2), we get μ

y n ))+(4k ) f (4−kn (x+2y)) 3 −136(4k )n f (4−kn (x−y))+1394(4k )n f (4−kn (x+y))−425(4k )n f (4−kn y)+1530(4k )n f (4−kn x) (t) αn t t −kn −kn ≥ Ψ x, y, k ≥ Ψ 4 x, 4 y, k |4 |n |4 |n n for all x, y ∈ X and all t >0. Since limn→∞ Ψ x, y, |4αk |tn = 1, we infer that Q is a n

n

n

(4k ) ·16f (4−kn (x+4y))+(4k ) f (4−kn (4x−y))−306 (4k ) ·9f (4−kn (x+

quartic mapping.


Page 8 of 17

If Q : X → Y is another quartic mapping such that μ Q’(x)-f(x) (t) ≥ M(x, t) for all x ∈ X and t >0, then for each n Î N, x ∈ X and t >0, μQ(x)−Q (x) (t) ≥ T μ

n

Q(x)−(44k ) f

x n (4k )

(t), μ

n

(44k ) f

(t), t) x k n −Q (x) (4 )

.

Thanks to (4.8), we conclude that Q = Q’. □ Corollary 4.3. Let Kbe a non-Archimedean field, X a vector space over Kand let (Y , μ, T)be a non-Archimedean random Banach space over Kunder a t-norm T ∈ H. Let f : X → Y be a Ψ-approximately quartic mapping. If, for some a Î ℝ, a >0, and some integer k, k >3, with |4k| 0),

then there exists a unique quartic mapping Q : X → Y such that α i+1 t ∞ μf (x)−Q(x) (t) ≥ Ti=1 M x, |4|ki for all x ∈ X and all t >0, where M(x, t) := T(Ψ (x, 0, t), Ψ (4x, 0, t), · · · , Ψ (4k−1 x, 0, t))

(x ∈ X ,

t > 0).

Proof. Since αj t lim M x, kj = 1 n→∞ |4|

(x ∈ X ,

t > 0)

and T is of Hadžić type, from Proposition 2.1, it follows that αj t ∞ lim T M x, kj = 1 (x ∈ X , t > 0). n→∞ j=n |4| Now we can apply Theorem 4.2 to obtain the result. □ Example 4.4. Let (X , μ, TM ) non-Archimedean random normed space in which μx (t) =

t , t + ||x||

∀x ∈ X ,

t > 0,

and (Y , μ, TM ) a complete non-Archimedean random normed space (see Example 3.2). Define Ψ (x, y, t) =

t . 1+t

It is easy to see that (4.2) holds for a = 1. Also, since M(x, t) =

t , 1+t

we have αj t t ∞ m lim TM,j=n M x, kj = lim lim TM,j=n M x, kj n→∞ n→∞ m→∞ |4| |4| t = lim lim n→∞ m→∞ t + |4k |n = 1, ∀x ∈ X , t > 0.


Page 9 of 17

Let f : X → Y be a Ψ-approximately quartic mapping. Thus all the conditions of Theorem 4.2 hold and so there exists a unique quartic mapping Q : X → Y such that μf (x)−Q(x) (t) ≥

t t + |4k |

.

5. Fixed point method for random stability of the quartic functional equation In this section, we apply a fixed point method for achieving random stability of the quartic functional equation. The notion of generalized metric space has been introduced by Luxemburg [37], by allowing the value +∞ for the distance mapping. The following lemma (Luxemburg-Jung theorem) will be used in the proof of Theorem 5.3. Lemma 5.1. [38]. Let (X, d) be a complete generalized metric space and let A : X ® X be a strict contraction with the Lipschitz constant k such that d(x0, A(x0)) < +∞ for some x0 Î X. Then A has a unique fixed point in the set Y := {y Î X, d(x0, y) < ∞} and the sequence (An(x))nÎN converges to the fixed point x* for every x Î Y. Moreover, d(x0, δ A(x0)) ≤ δ implies d(x∗ , x0 ) ≤ 1−k . Let X be a linear space, (Y, ν, TM ) a complete RN-space and let G be a mapping from X × R into [0, 1], such that G(x, .) Î D+ for all x. Consider the set E := {g : X ® Y, g(0) = 0} and the mapping dG defined on E × E by dG (g, h) = inf{u ∈ R+ , νg(x)−h(x) (ut) ≥ G(x, t) for all x ∈ X and t > 0}

where, as usual, inf ∅ = +∞. The following lemma can be proved as in [22]: Lemma 5.2. cf. [22,39]dG is a complete generalized metric on E. Theorem 5.3. Let X be a real linear space, t f a mapping from X into a complete RNspace (Y, μ , TM ) with f(0) = 0 and let F : X2 ® D+ be a mapping with the property ∃α ∈ (0, 256) : Φ4x,4y (αt) ≥ Φx,y (t), ∀x, y ∈ X,

∀t > 0.

(5:1)

If

μ

16f (x+4y)+f (4x−y)−306 9f x+

y 3

+f (x+2y) −136f (x−y)+1394f (x+y)−425f (y)+1530f (x)

(t)

(5:2)

≥ Φx,y (t), ∀x, y ∈ X,

then there exists a unique quartic mapping g : X ® Y such that μg(x)−f (x)(t) ≥ Φx,0 (Mt) , ∀x ∈ X,

∀t > 0,

where M = (256 − α).

Moreover, f (4n x) . n→∞ 44n

g(x) = lim

Proof. By setting y = 0 in (5.2), we obtain μf (4x)−256f (x) (t) ≥ Φx,0 (t)

(5:3)


Page 10 of 17

for all x Î X, whence μ

(t) 1 256 f (4x)−f (x)

=μ

(t) 1 256 (f (4x)−256f (x))

= μf (4x)−256f (x) (256t) ≥ Φx,0 (256t) ,

∀x ∈ X,

∀t > 0.

Let G(x, t) := Φx,0 (256t) .

Consider the set E := {g : X → Y, g(0) = 0}

and the mapping dG defined on E × E by dG (g, h) = inf{u ∈ R+ , μg(x)−h(x) (ut) ≥ G(x, t) for all x ∈ X and t > 0}.

By Lemma 5.2, (E, dG) is a complete generalized metric space. Now, let us consider the linear mapping J : E ® E, Jg(x) :=

1 g(4x). 256

We show that J is a strictly contractive self-mapping of E with the Lipschitz constant k = a/256. Indeed, let g, h Î E be mappings such that dG(g, h) 0,

whence μJg(x)−Jh(x) (

α α ( εt) = μ 1 εt) (g(4x)−h(4x)) 256 256 256 = μg(4x)−h(4x) (αεt) ≥ G(4x, αt)

(x ∈ X,

t > 0).

α εt) ≥ G(x, t), that is, Since G(4x, at) ≥ G(x, t), μJg(x)−Jh(x) ( 256

dG (g, h) < ε ⇒ dG (Jg, Jh) ≤

α ε. 256

This means that dG (Jg, Jh) ≤

α dG (g, h) 256

for all g, h in E. Next, from μ

(t) 1 f (x)− 256 f (4x)

≥ G(x, t)

it follows that dG(f, Jf ) ≤ 1. Using the Luxemburg-Jung theorem, we deduce the existence of a fixed point of J, that is, the existence of a mapping g : X ® Y such that g(4x) = 256g(x) for all x Î X.


Since, for any x Î X and t >0,

t dG (u, v) < ε ⇒ μu(x)−v(x) (t) ≥ G x, ε

Page 11 of 17

from dG(Jnf, g) ® 0, it follows that limn→∞

, f (4n x) 44n

= g(x) for any x Î X. 1 1 Also, dG (f , g) ≤ 1−L d(f , Jf ) implies the inequality dG (f , g) ≤ α from which it 1 − 256 256 t) ≥ G(x, t) for all t >0 and all x Î X. This means immediately follows νg(x)−f (x) ( 256−α

that

256 − α μg(x)−f (x)(t) ≥ G x, t , 256

∀x ∈ X,

∀t > 0.

It follows that μg(x)−f (x)(t) ≥ Φx,0 ((256 − α)t)

∀x ∈ X,

∀t > 0.

The uniqueness of g follows from the fact that g is the unique fixed point of J with the property: there is C Î (0, ∞) such that μg(x)-f(x)(Ct) ≥ G(x, t) for all x Î X and all t >0, as desired. □

6. Intuitionistic random normed spaces Recently, the notation of intuitionistic random normed space introduced by Chang et al. [19]. In this section, we shall adopt the usual terminology, notations, and conventions of the theory of intuitionistic random normed spaces as in [22], [31], [33], [34], [40], [41], [42]. Definition 6.1. A measure distribution function is a function μ : R ® [0, 1] which is left continuous, non-decreasing on R, inftÎR μ(t) = 0 and suptÎR 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, if t ≤ 0, 1, if t > 0.

If X is a nonempty set, then μ : X ® D is called a probabilistic measure on X and μ (x) is denoted by μx. Definition 6.2. A non-measure distribution function is a function ν : R ® [0, 1] which is right continuous, non-increasing on R, inftÎR ν(t) = 0 and suptÎR 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 1, if t ≤ 0, G(t) = 0, if t > 0. If X is a nonempty set, then ν : X ® B is called a probabilistic non-measure on X and ν (x) is denoted by νx. Lemma 6.3. [43], [44]Consider the set L* and 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∗ .


Page 12 of 17

Then (L∗ , ≤L∗ )is a complete lattice. We denote its units by 0L∗ = (0, 1) and 1L∗ = (1, 0). In Section 2, we presented classical t-norm. Using the lattice (L∗ , ≤L∗ ), these definitions can be straightforwardly extended. Definition 6.4. [44] 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, x , y, y ) ∈ (L∗ )4 )(x≤L∗ x and y≤L∗ y ⇒ T (x, y)≤L∗ T (x , y )) (monotonicity). If (L∗ , ≤L∗ , T ) is an Abelian topological monoid with unit 1L∗, then T is said to be a continuous t-norm. Definition 6.5. [44] 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) Î L*,

T (x, y) = (x1 ∗ y1 , x2 ♦y2 ). For example,

T (a, b) = (a1 b1 , min{a2 + b2 , 1}) and M(a, b) = (min{a1 , b1 }, max{a2 , b2 })

are continuous t-representable for all a = (a1, a2), b = (b1, b2) Î L*. 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 6.6. A negator on L* is any decreasing mapping N : L∗ → L∗ satisfying N (1L∗ ) = 0L∗and N (1L∗ ) = 0L∗. If N (N (x)) = x for all x Î L*, then N is called an involutive negator. A negator on [0, 1] is a decreasing function 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 6.7. Let μ and ν be measure and non-measure distribution functions from X × (0, +∞) to [0, 1] such that μx (t) + ν x (t) ≤ 1 for all x Î X and t >0. The triple (X, Pμ,ν , T ) is said to be an intuitionistic random normed space (briefly IRN-space) if X is a vector space, T is continuous t-representable and Pμ,ν is a mapping X × (0, +∞) ® L* satisfying the following conditions: for all x, y Î X and t, s >0, (a) Pμ,ν (x, 0) = 0L∗; (b) Pμ,ν (x, t) = 1L∗ if and only if x = 0; t ) for all a ≠ 0; (c) Pμ,ν (αx, t) = Pμ,ν (x, |α| (d) Pμ,ν (x + y, t + s)≥L∗ T (Pμ,ν (x, t), Pμ,ν (y, s)).


Page 13 of 17

In this case, Pμ,ν is called an intuitionistic random norm. Here,

Pμ,ν (x, t) = (μx (t), νx (t)). Example 6.8. Let (X, || · ||) be a normed space. Let T (a, b) = (a1 b1 , min(a2 + b2 , 1)) for all a = (a1, a2), b = (b1, b2) Î L* and let μ, ν be measure and non-measure distribution functions defined by t ||x|| Pμ,ν (x, t) = (μx (t), νx (t)) = , ∀t ∈ R+ . , t + ||x|| t + ||x|| Then (X, Pμ,ν , T ) is an IRN-space. Definition 6.9. (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 an n0 Î N such that

Pμ,ν (xn − xm , t) >L∗ (Ns (ε), ε),

∀n, m ≥ n0 ,

where Ns is the standard negator. μ,ν (2) The sequence {xn} is said to be convergent to a point x Î X (denoted byxn P→ 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 Î X.

7. Stability results in intuitionistic random normed spaces In this section, we prove the generalized Ulam-Hyers stability of the quartic functional equation in intuitionistic random normed spaces. Theorem 7.1. Let X be a linear space and let (X, Pμ,ν , T )be a complete IRN-space. Let f : X ® Y be a mapping with f(0) = 0 for which there are ξ, ζ : X2 ® D+, where 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: y + f (x + 2y)] Pμ,ν (16f (x + 4y) + f (4x − y) − 306[9f x + 3 −136f (x − y) + 1394f (x + y) − 425f (y) + 1530f (x), t) ≥L∗ Qξ ,ζ (x, y, t).

(7:1)

If ∞ Ti=1 (Qξ ,ζ (4n+i−1 x, 0, 44n+3i+3 t)) = 1L∗

(7:2)

and lim Qξ ,ζ (4n x, 4n y, 44n t) = 1L∗

n→∞

(7:3)

for all x, y Î X and all t >0, then there exists a unique quartic mapping Q : X ® Y such that ∞ Pμ,ν (f (x) − Q(x), t)≥L∗ Ti=1 (Qξ ,ζ (4i−1 x, 0, 43i+3 t)).

Proof. Putting y = 0 in (7.1), we have f (4x) Pμ,ν − f (x), t ≥L∗ Qξ ,ζ (x, 0, 44 t). 256

(7:4)

(7:5)


Page 14 of 17

Therefore, it follows that

Pμ,ν (

f (4k+1 x) f (4k x) t − , )≥L∗ Qξ ,ζ (4k x, 0, 44 t), 44(k+1) 44k 44k

(7:6)

which implies that

Pμ,ν (

f (4k+1 x) f (4k x) − , t)≥L∗ Qξ ,ζ (4k x, 0, 44(k+1) t), 44(k+1) 44k

(7:7)

that is,

Pμ,ν

f (4k+1 x) f (4k x) t − , 44k 4k+1 44(k+1)

≥L∗ Qξ ,ζ (4k x, 0, 44(k+1) t)

(7:8)

for all k Î N and all t >0. As 1 > 1/4 + ... + 1/4n, from the triangle inequality, it follows n−1 f (4n x) f (4k+1 x) f (4k x) 1 n−1 Pμ,ν − f (x), t ≥L∗ Tk=0 Pμ,ν − , t 256n 44(k+1) 44k 4k+1 (7:9) k=0 n ≥L∗ Ti=1 (Qξ ,ζ (4i−1 x, 0, 43i+3 t)). n

m (4 x) In order to prove convergence of the sequence { f256 n }, replacing x with 4 x in (7.9),

we get that for m, n >0

Pμ,ν (

f (4n+m x) 256(n+m)

−

f (4m x) n , t)≥L∗ Ti=1 (Qξ ,ζ (4i+m−1 x, 0, 43i+4m+3 t)). 256m

(7:10)

Since the right-hand side of the inequality tends 1 L* as m tends to infinity, the n

x) sequence { f (4 is a Cauchy sequence. So we may define Q(x) = limn→∞ 44n }

f (4n x) 44n

for all

x Î X. Now, we show that Q is a quartic mapping. Replacing x, y with 4nx and 4ny, respectively, in (7.1), we obtain y n n f (4n (x + 4y)) f (4n (4x − y)) 306[9f (4 (x + 3 )) + f (4 (x + 2y)) + − 256n 256n 256n 136f (4n (x − y)) 1394f (4n (x + y)) 425f (4n (y)) 1530f (4n (x)) (7:11) − + − + , t) n n n n 256 256 256 256 ≥L∗ Qξ ,ζ (4n x, 4n y, 44n t).

Pμ,ν (

Taking the limit as n ® ∞, we find that Q satisfies (1.1) for all x, y Î X. Taking the limit as n ® ∞ in (7.9), we obtain (7.4). To prove the uniqueness of the quartic mapping Q subject to (7.4), let us assume that there exists another quartic mapping Q’ which satisfies (7.4). Obviously, we have x Î X and all n Î N. Hence it follows from (7.4) that

Pμ,ν (Q(x) − Q (x), t) ≥L∗ Pμ,ν (Q(4n x) − Q (4n x), 44n t) ≥L∗ T (Pμ,ν (Q(4n x) − f (4n x), 44n−1 t), Pμ,ν (f (4n x) − Q (4n x), 44n−1 t)) ∞ ∞ ≥L∗ T (Ti=1 (Qξ ,ζ (4n+i−1 x, 0, 44n+3i+3 t)), Ti=1 (Qξ ,ζ (4n+i−1 x, 0, 44n+3i+3 t))


for all x Î X. By letting n ® ∞ in (7.4), we prove the uniqueness of Q. This completes the proof of the uniqueness, as desired. □ Corollary 7.2. Let (X, P μ ,ν , T )be an IRN-space and let (Y, Pμ,ν , T )be a complete IRN-space. Let f : X ® Y be a mapping such that y + f (x + 2y) Pμ,ν (16f (x + 4y) + f (4x − y) − 306 9f x + 3 −136f (x − y) + 1394f (x + y) − 425f (y) + 1530f (x), t) ≥L∗ P μ ,ν (x + y, t)

for all t >0 in which lim T ∞ (Pμ ,ν (x, 44n+3i+3 t)) n→∞ i=1

= 1L∗

for all x, y Î X. Then there exists a unique quartic mapping Q : X ® Y such that ∞ Pμ,ν (f (x) − Q(x), t)≥L∗ Ti=1 (Pμ ,ν (x, 43i+3 t)).

Now, we give an example to illustrate the main result of Theorem 7.1 as follows. Example 7.3. Let (X, ||.||) be a Banach algebra, (X, Pμ,ν , M) an IRN-space in which t ||x|| Pμ,ν (x, t) = , t + ||x|| t + ||x|| and let (Y, Pμ,ν , M) be a complete IRN-space for all x Î X. Define f : X ® X by f (x) = x4 + x0, where x0 is a unit vector in X. A straightforward computation shows that y + f (x + 2y) Pμ,ν (16f (x + 4y) + f (4x − y) − 306 9f x + 3 −136f (x − y) + 1394f (x + y) − 425f (y) + 1530f (x), t) ≥L∗ Pμ,ν (x + y, t),

∀t > 0.

Also n+i−1 3n+2i+4 lim M∞ x, 44n+3i+3 t)) = lim lim Mm t)) i=1 (Pμ,ν (4 i=1 (Pμ,ν (x, 4

n→∞

n→∞ m→∞

= lim lim Pμ,ν (x, 43n+6 t) n→∞ m→∞

= lim Pμ,ν (x, 43n+6 t) n→∞

= 1L∗ .

Therefore, all the conditions of 7.1 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, 46 t).

Author details 1 Section of Mathematics and Informatics, Pedagogical Department, National and Capodistrian University of Athens, 4, Agamemnonos St., Aghia Paraskevi, Athens 15342, Greece 2Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran 3Faculty of Mathematics and Computer Sciences, Sabzevar Tarbiat Moallem University, Sabzevar, Iran 4Department of Mathematics, Iust, Behshar, Iran Authors’ contributions All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

Page 15 of 17


Competing interests The authors declare that they have no competing interests. Received: 15 February 2011 Accepted: 18 September 2011 Published: 18 September 2011 References 1. Ulam, SM: Problems in Modern Mathematics. In Science Editions, vol. Chapter VI,Wiley, New York (1964) 2. Hyers, DH: On the stability of the linear functional equation. Proc Natl 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 Jpn. 2, 64–66 (1950). doi:10.2969/ jmsj/00210064 4. Rassias, ThM: 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 5. Baak, C, Moslehian, MS: On the stability of J*-homomorphisms. Nonlinear Anal TMA. 63, 42–48 (2005). doi:10.1016/j. na.2005.04.004 6. Chudziak, J, Tabor, J: Generalized Pexider equation on a restricted domain. J Math Psychol. 52, 389–392 (2008). doi:10.1016/j.jmp.2008.04.002 7. 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Mirmostafaee, M, Mirzavaziri, M, Moslehian, MS: Fuzzy stability of the Jensen functional equation. Fuzzy Set Syst. 159, 730–738 (2008). doi:10.1016/j.fss.2007.07.011 21. Mirzavaziri, M, Moslehian, MS: A fixed point approach to stability of a quadratic equation. Bull Braz Math Soc. 37, 361–376 (2006). doi:10.1007/s00574-006-0016-z 22. Miheţ, D, Radu, V: On the stability of the additive Cauchy functional equation in random normed spaces. J Math Anal Appl. 343, 567–572 (2008) 23. Miheţ, D: The probabilistic stability for a functional equation in a single variable. Acta Math Hungar. 123, 249–256 (2009). doi:10.1007/s10474-008-8101-y 24. Miheţ, D: The fixed point method for fuzzy stability of the Jensen functional equation. Fuzzy Set Syst. 160, 1663–1667 (2009). doi:10.1016/j.fss.2008.06.014 25. Miheţ, D, Saadati, R, Vaezpour, SM: The stability of the quartic functional equation in random normed spaces. Acta Appl Math. 110, 797–803 (2010). doi:10.1007/s10440-009-9476-7 26. 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38. Jung, C: On generalized complete metric spaces. Bull Am Math Soc. 75, 113–116 (1969). doi:10.1090/S0002-9904-196912165-8 39. Miheţ, D: The stability of the additive Cauchy functional equation in non-Archimedean fuzzy normed spaces. Fuzzy Set Syst. 161, 2206–2212 (2010). doi:10.1016/j.fss.2010.02.010 40. Chang, SS, Cho, Y, Kang, Y: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science Publishers Inc., New York (2001) 41. Kutukcu, S, Tuna, A, Yakut, AT: Generalized contraction mapping principle in intuitionistic Menger spaces and application to differential equations. Appl Math Mech. 28, 799–809 (2007). doi:10.1007/s10483-007-0610-z 42. Saadati, R, Park, J: On the intuitionistic fuzzy topological spaces. Chaos Soliton Fract. 27, 331–344 (2006). doi:10.1016/j. chaos.2005.03.019 43. Atanassov, KT: Intuitionistic fuzzy sets. Fuzzy Set Syst. 20, 87–96 (1986). doi:10.1016/S0165-0114(86)80034-3 44. Deschrijver, G, Kerre, EE: On the relationship between some extensions of fuzzy set theory. Fuzzy Set Syst. 23, 227–235 (2003) doi:10.1186/1029-242X-2011-62 Cite this article as: Rassias et al.: On nonlinear stability in various random normed spaces. Journal of Inequalities and Applications 2011 2011:62.

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On nonlinear stability in various random normed

On nonlinear stability in various random normed

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