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Computers and Mathematics with Applications
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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 a b s t r a c t
Article history: The purpose of this paper is first to introduce the notation of intuitionistic random normed Received 20 January 2010 spaces, and then by virtue of this notation to study the stability of a quartic functional Received in revised form 21 July 2010 equation in the setting of these spaces under arbitrary triangle norms. Then we prove the Accepted 22 July 2010 stability of above quartic functional equation in non-Archimedean random normed spaces. Furthermore, the interdisciplinary relation among the theory of random spaces, the theory Keywords: of non-Archimedean spaces, the theory of intuitionistic spaces and the theory of functional Stability equations are also presented in the paper. Quartic functional equation Intuitionistic random normed space © 2010 Elsevier Ltd. All rights reserved. Non-Archimedean random normed spaces
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) = cx4 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
∗ 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) 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 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}, min(a, b), if max(a, b) = 1; T (a, b) := D 0, otherwise. (n) ∈ [ ] ∈ ∪ { } = (n−1) ≥ 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 , x) if n 1. A t-norm ∈ (n) = 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 } SW = SW = The Sugeno–Weber family Tλ λ∈[−1,∞] is defined by T−1 TD, T∞ TP and x + y − 1 + λxy T SW (x, y) = max 0, λ 1 + λ if λ ∈ (−1, ∞). D The Domby family {Tλ }λ∈[0,∞] is defined by TD, if λ = 0, TM , if λ = ∞ and 1 T D(x, y) = λ + 1−x λ + 1−y λ 1/λ 1 (( x ) ( y ) ) if λ ∈ (0, ∞). AA The Aczel–Alsina family {Tλ }λ∈[0,∞] is defined by TD, if λ = 0, TM , if λ = ∞ and AA −(| log x|λ+| log y|λ)1/λ Tλ (x, y) = e if λ ∈ (0, ∞). n A t-norm T can be extended (by associativity) in a unique way to an n-array operation. For any (x1, ..., xn) ∈ [0, 1] , the value T (x1, ..., xn) is defined by 0 = n = n−1 = Ti=1xi 1, Ti=1xi T (Ti=1 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 Ti=1xi lim Ti=1xi. (2.1) n→∞ { n } The limit on the right-hand side of (2.1) exists since the sequence Ti=1xi n∈N is non-increasing and bounded from below.
Proposition 2.1 ([31]). (i) For T ≥ TL, the following implication holds: ∞ ∞ = ⇐⇒ X − ∞ lim Ti=1xn+i 1 (1 xn) < . n→∞ n=1 (ii) If T is of Hadžić type, then ∞ = lim Ti=1xn+i 1 n→∞ for every sequence {xn}n∈N in [0, 1] such that limn→∞ xn = 1. AA D (iii) If T ∈ {Tλ }λ∈(0,∞) ∪ {Tλ }λ∈(0,∞), then ∞ ∞ = ⇐⇒ X − α ∞ lim Ti=1xn+i 1 (1 xn) < . n→∞ n=1 SW (iv) If T ∈ {Tλ }λ∈[−1,∞), then ∞ ∞ = ⇐⇒ X − ∞ lim Ti=1xn+i 1 (1 xn) < . n→∞ n=1 1996 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 ∈ 6= (NA-RN2) µαx(t) µx( |α| ) for all x X, t > 0, α 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
0 if t ≤ kxk, µ (t) = x 1 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
t Ψ (cx, cx, t) ≥ Ψ x, x, , ∀x ∈ X, c 6= 0. |c|
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), ∀x, y ∈ X, t > 0. (4.1)
Our main result, in this section, is the following. R. Saadati et al. / Computers and Mathematics with Applications 60 (2010) 1994–2002 1997
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 kx, 2 ky, t) ≥ Ψ (x, y, αt), ∀x ∈ X, t > 0, (4.2) and αjt ∞ = ∀ ∈ lim Tj=nM x, 1, x X, t > 0, (4.3) n→∞ |2|kj then there exists a unique quartic mapping Q : X → Y such that
i+1 ∞ α t µf (x)−Q (x)(t) ≥ T = M x, , ∀x ∈ X, t > 0, (4.4) i 1 |2|ki where − M(x, t) := T (Ψ (x, 0, t), Ψ (2x, 0, t), . . . , Ψ (2k 1x, 0, t)), ∀x ∈ X, t > 0.
Proof. First, we show by induction on j that, for all x ∈ X, t > 0 and j ≥ 1,
j−1 µf (2jx)−16jf (x)(t) ≥ Mj(x, t) := T (Ψ (x, 0, t), . . . , Ψ (2 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 2jx in (4.1), we get j µf (2j+1x)−16f (2jx)(t) ≥ Ψ (2 x, 0, t), ∀x ∈ X, t > 0. Since |16| ≤ 1, µf (2j+1x)−16j+1f (x)(t) ≥ T µf (2j+1x)−16f (2jx)(t), µ16f (2jx)−16j+1f (x)(t) t = T µ j+1 − j (t), µ j − j f (2 x) 16f (2 x) f (2 x) 16 f (x) |16| ≥ T µf (2j+1x)−16f (2jx)(t), µf (2jx)−16jf (x) (t) j ≥ T (Ψ (2 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 (2kx)−16kf (x)(t) ≥ M(x, t), ∀x ∈ X, t > 0. (4.6) Replacing x by 2−(kn+k)x in (4.6) and using inequality (4.2), we obtain x µ (t) ≥ M , t f x −16kf x kn+k 2kn 2kn+k 2 + ≥ M(x, αn 1t), ∀x ∈ X, t > 0, n ≥ 0. (4.7) Then it follows that αn+1 µ (t) ≥ M x, t (24k)nf x −(24k)n+1f x |(24k)n| (2k)n (2k)n+1 αn+1 ≥ M x, t , ∀x ∈ X, t > 0, n ≥ 0, (4.8) |(2k)n| and so ! n+p ≥ µ (t) Tj=n µ (t) (24k)nf x −(24k)n+pf x (24k)jf x −(24k)j+pf x (2k)n (2k)n+p (2k)j (2k)j+p j+1 n+p α ≥ T = M x, t , ∀x ∈ X, t > 0, n ≥ 0. j n |(2k)j| 1998 R. Saadati et al. / Computers and Mathematics with Applications 60 (2010) 1994–2002
j+1 n o ∞ α = ∈ 4k n x Since limn→∞ Tj=nM x, | k j| t 1 (x X, t > 0), (2 ) f k n is a Cauchy sequence in the non-Archimedean (2 ) (2 ) n∈N random Banach space (Y , µ, T ). Hence we can define a mapping Q : X → Y such that
lim µ (t) = 1, ∀x ∈ X, t > 0. (4.9) n→∞ (24k)nf x −Q (x) (2k)n Next, for each n ≥ 1, x ∈ X and t > 0,
= µ (t) µ − (t) f (x)−(24k)nf x Pn 1(24k)if x −(24k)i+1f x (2k)n i=0 (2k)i (2k)i+1 ! − ≥ n 1 Ti=0 µ (t) (24k)if x −(24k)i+1f x (2k)i (2k)i+1 i+1 n−1 α t ≥ T = M x, i 0 |2k|i and so !
µf (x)−Q (x)(t) ≥ T µ (t), µ (t) f (x)−(24k)nf x (24k)nf x −Q (x) (2k)n (2k)n
i+1 ! − α t ≥ n 1 T Ti=0 M x, , µ (t) . (4.10) |2k|i (24k)nf x −Q (x) (2k)n Taking the limit as n → ∞ in (4.10), we obtain i+1 ∞ α t µf (x)−Q (x)(t) ≥ T = M x, , ∀x ∈ X, t > 0. i 1 |2k|i 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)nf (2−kn(2x+y))+(24k)nf (2−kn(2x−y))−4(24k)nf (2−kn(x+y)) 4k n −kn 4k n −kn 4k n −kn (t) n→∞ −4(2 ) f (2 (x−y))−24(2 ) f (2 x)+6(2 ) f (2 (y))
= 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−knx, 2−kny in (4.1) and (4.2), we get
µ k n −kn k n −kn k n −kn (t) (16 ) f (2 (2x+y))+(16 ) f (2 (2x−y))−4(16 ) f (2 (x+y))−4(16k)nf (2−kn(x−y))−24(16k)nf (2−knx)+6(16k)nf (2−kny) − − t ≥ Ψ 2 knx, 2 kny, |2k|n αnt ≥ Ψ x, y, , ∀x, y ∈ X, t > 0. |2k|n
αnt Since lim →∞ Ψ x, y, = 1, we infer that Q is a quartic mapping. n |2k|n 0 If Q : 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 (t), µ (t) . Q (x)−(24k)nf x (24k)nf x −Q 0(x) (2k)n (2k)n 0 Therefore, from (4.9), we conclude that Q = Q . 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 kx, 2 ky, t) ≥ Ψ (x, y, αt), ∀x ∈ X, t > 0, then there exists a unique quartic mapping Q : X → Y such that i+1 ∞ α t µf (x)−Q (x)(t) ≥ T = M x, , ∀x ∈ X, t > 0, i 1 |2|ki 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 1x, 0, t)), ∀x ∈ X, t > 0.
Proof. Since αjt lim M x, = 1, ∀x ∈ X, t > 0, n→∞ |2|kj and T is of Hadžić type, from Proposition 2.1 it follows that
αjt ∞ = ∀ ∈ lim Tj=nM x, 1, x X, t > 0. n→∞ |2|kj
Now we can apply Theorem 4.2.
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
0 if t ≤ 0, H(t) = 1 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, non- = = increasing 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
1 if t ≤ 0, G(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 ∗ 2 L = {(x1, x2) : (x1, x2) ∈ [0, 1] 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); 0 0 ∗ 4 0 0 0 0 (d)(∀(x, x , y, y ) ∈ (L ) )(x ≤L∗ x and y ≤L∗ y H⇒ 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 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). 2000 R. Saadati et al. / Computers and Mathematics with Applications 60 (2010) 1994–2002
For example,
T (a, b) = (a1b1, 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 6= (c) Pµ,ν (αx, t) Pµ,ν (x, |α| ) for all α 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) = (a1b1, 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 t kxk + Pµ,ν (x, t) = (µx(t), νx(t)) = , , ∀t ∈ R . t + kxk t + kxk
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 2 + such that there are ξ, ζ : X −→ 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
∞ + − + + n i 1 4n 3i 1 = ∗ Ti=1(Qξ,ζ (2 x, 0, 2 t)) 1L (6.2) and
n n 4n lim Qξ,ζ (2 x, 2 y, 2 t) = 1L∗ , ∀x, y ∈ X, t > 0 (6.3) n→∞ 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
∞ i−1 3i+1 − ≥ ∗ ∀ ∈ Pµ,ν (f (x) Q (x), t) L Ti=1(Qξ,ζ (2 x, 0, 2 t)), x X, t > 0. (6.4)
Proof. See [15]. 0 : −→ Corollary 6.2. Let (X, Pµ0,ν0 , T ) be an IRN-space and (Y , Pµ,ν , T ) be a complete IRN-space. If f X Y is a mapping such that 0 + + − − + − − − + ≥ ∗ + ∀ ∈ Pµ,ν (f (2x y) f (2x y) 4f (x y) 4f (x y) 24f (x) 6f (y), t) L Pµ0,ν0 (x y, t), x, y X, t > 0, and ∞ 0 + + 3n 2i 2 = ∗ ∀ ∈ lim Ti=1(P 0 0 (x, 2 t)) 1L , x X, t > 0, n−→∞ µ ,ν then there exists a unique quartic mapping Q : X −→ Y such that
∞ 0 2i+2 − ≥ ∗ ∀ ∈ Pµ,ν (f (x) Q (x), t) L Ti=1(Pµ0,ν0 (x, 2 t)), x X, t > 0.
= 0 + Proof. Put Qξ,ζ (x, y, t) Pµ0,ν0 (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 t kxk Pµ,ν (x, t) = , , ∀x ∈ X, t > 0, t + kxk t + kxk
4 and (Y , Pµ,ν , M) be a complete IRN-space for all x ∈ X. Define a mapping f : X −→ Y by f (x) = x + 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 = m 3n+2i+2 lim Mi=1(Pµ,ν (x, 2 t)) lim lim Mi=1(Pµ,ν (x, 2 t)) n−→∞ n−→∞ m−→∞ 2n+4 = lim lim Pµ,ν (x, 2 t) n−→∞ m−→∞ 2n+4 = lim Pµ,ν (x, 2 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
4 Pµ,ν (f (x) − Q (x), t) ≥L∗ Pµ,ν (x, 2 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
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