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On stability of a cubic functional equation in intuitionistic fuzzy normed spaces
Chaos, Solitons and Fractals 42 (2009) 2997–3005
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On stability of a cubic functional equation in intuitionistic fuzzy normed spaces q M. Mursaleen *, S.A. Mohiuddine Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
a r t i c l e
i n f o
Article history: Accepted 3 April 2009
a b s t r a c t In this paper, we determine some stability results concerning the cubic functional equation
f ð2x þ yÞ þ f ð2x yÞ ¼ 2f ðx þ yÞ þ 2f ðx yÞ þ 12f ðxÞ in intuitionistic fuzzy normed spaces (IFNS). We define the intuitionistic fuzzy continuity of the cubic mappings and prove that the existence of a solution for any approximately cubic mapping implies the completeness of IFNS. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries Fuzzy set theory is a powerful hand set for modelling uncertainty and vagueness in various problems arising in the field of science and engineering. It has also very useful applications in various fields, e.g. population dynamics [2], chaos control [10,11], computer programming [12], nonlinear dynamical systems [13], nonlinear operators [25], statistical convergence [24], etc. The fuzzy topology proves to be a very useful tool to deal with such situations where the use of classical theories breaks down. The most fascinating application of fuzzy topology in quantum particle physics arises in string and ð1Þ -theory of El-Naschie (cf. [3–9,18,19]). There are many situations where the norm of a vector is not possible to find and the concept of intuitionistic fuzzy norm [26,29,30] seems to be more suitable in such cases, that is, we can deal with such situations by modelling the inexactness through the intuitionistic fuzzy norm. Stability problem of a functional equation was first posed by Ulam [31] which was answered by Hyers [14] and then generalized by Aoki [1] and Rassias [27] for additive mappings and linear mappings, respectively. Since then several stability problems for various functional equations have been investigated in [15,28]; and various fuzzy stability results concerning Cauchy, Jensen and quadratic functional equations were discussed in [20,21,23]. The stability problem for the cubic functional equation was proved by Jun and Kim [16] for mappings f : X ! Y, where X is a real normed space and Y is a Banach space. Later on, in [17,22] the problem of stability of some cubic equation were discussed. In this paper, we determine some stability results concerning the cubic functional equation
f ð2x þ yÞ þ f ð2x yÞ ¼ 2f ðx þ yÞ þ 2f ðx yÞ þ 12f ðxÞ
q Research of the first author was supported by the Department of Science and Technology, New Delhi, under Grant No. SR/S4/MS:505/07; and research of the second author was supported by the Department of Atomic Energy, Government of India under the NBHM-Post Doctoral Fellowship Programme Number 40/10/2008-R&D II/892. * Corresponding author. E-mail addresses: [email protected] (M. Mursaleen), [email protected] (S.A. Mohiuddine).
0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.04.041
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M. Mursaleen, S.A. Mohiuddine / Chaos, Solitons and Fractals 42 (2009) 2997–3005
in intuitionistic fuzzy normed spaces. We define the intuitionistic fuzzy continuity of the cubic mappings and prove that the existence of a solution for any approximately cubic mapping implies the completeness of intuitionistic fuzzy normed spaces (IFNS). In this section we recall some notations and basic definitions used in this paper. Definition 1.1. A binary operation : ½0; 1 ½0; 1 ! ½0; 1 is said to be a continuous t-norm if it satisfies the following conditions: ðaÞ is associative and commutative, ðbÞ is continuous, ðcÞ a 1 ¼ a for all a 2 ½0; 1, ðdÞ a b 6 c d whenever a 6 c and b 6 d for each a; b; c; d 2 ½0; 1. Definition 1.2. A binary operation } : ½0; 1 ½0; 1 ! ½0; 1 is said to be a continuous t-conorm if it satisfies the following conditions: 0 0 (a0 ) } is associative and commutative, (b ) } is continuous, (c0 ) a}0 ¼ a for all a 2 ½0; 1, (d ) a}b 6 c}d whenever a 6 c and b 6 d for each a; b; c; d 2 ½0; 1. Using the notions of continuous t-norm and t-conorm, Saadati and Park [29] have recently introduced the concept of intuitionistic fuzzy normed space as follows: Definition 1.3. The five-tuple ðX; l; m; ; }Þ is said to be an intuitionistic fuzzy normed spaces (for short, IFNS) if X is a vector space, is a continuous t-norm, } is a continuous t-conorm, and l; m are fuzzy sets on X ð0; 1Þ satisfying the following conditions. For every x; y 2 X and s; t > 0 (i) lðx; tÞ þ mðx; tÞ 6 1, (ii) lðx; tÞ > 0, (iii) lðx; tÞ ¼ 1 if and only if x ¼ 0, (iv) lðax; tÞ ¼ lðx; jat jÞ for each a–0, (v) lðx; tÞ lðy; sÞ 6 lðx þ y; t þ sÞ, (vi) lðx; Þ : ð0; 1Þ ! ½0; 1 is continuous, (vii) limt!1 lðx; tÞ ¼ 1 and limt!0 lðx; tÞ ¼ 0, (viii) mðx; tÞ < 1, (ix) mðx; tÞ ¼ 0 if and only if x ¼ 0, (x) mðax; tÞ ¼ mðx; jat jÞ for each a–0, (xi) lðx; tÞ}lðy; sÞ P mðx þ y; t þ sÞ, (xii) mðx; Þ : ð0; 1Þ ! ½0; 1 is continuous, (xiii) limt!1 mðx; tÞ ¼ 0 and limt!0 mðx; tÞ ¼ 1. In this case ðl; mÞ is called an intuitionistic fuzzy norm. Example 1.1. Let ðX; k kÞ be a normed space, a b ¼ ab and a}b ¼ minfa þ b; 1g for all a; b 2 ½0; 1. For all x 2 X and every t > 0, consider
(
lðx; tÞ ¼
t tþkxk
if t > 0
0
if t 6 0;
(
and
mðx; tÞ ¼
kxk tþkxk
if t > 0
0
if t 6 0:
Then ðX; l; m; ; }Þ is an IFNS. The concepts of convergence and Cauchy sequences in an intuitionistic fuzzy normed space are studied in [29]. Let ðX; l; m; ; }Þ be an IFNS. Then, a sequence x ¼ ðxk Þ is said to be intuitionistic fuzzy conv ergent to L 2 X if IF lim lðxk L; tÞ ¼ 1 and lim mðxk L; tÞ ¼ 0 for all t > 0. In this case we write xk ! L as k ! 1. Let ðX; l; m; ; }Þ be an IFNS. Then, x ¼ ðxk Þ is said to be intuitionistic fuzzy Cauchy sequence if lim lðxkþp xk ; tÞ ¼ 1 and lim mðxkþp xk ; tÞ ¼ 0 for all t > 0 and p ¼ 1; 2; . . . Let ðX; l; m; ; }Þ be an IFNS. Then ðX; l; m; ; }Þ is said to be complete if every intuitionistic fuzzy Cauchy sequence in ðX; l; m; ; }Þ is intuitionistic fuzzy convergent in ðX; l; m; ; }Þ. 2. Intuitionistic fuzzy stability The functional equation
f ð2x þ yÞ þ f ð2x yÞ ¼ 2f ðx þ yÞ þ 2f ðx yÞ þ 12f ðxÞ
ð2:1:0Þ 3
is called the cubic functional equation, since the function f ðxÞ ¼ cx is its solution. Every solution of the cubic functional equation is said to be a cubic mapping. We begin with a generalized Hyers–Ulam–Rassias type theorem in IFNS for the cubic functional equation. Theorem 2.1. Let X be a linear space and let ðZ; l0 ; m0 Þ be an IFNS. Let u : X X ! Z be a function such that for some 0 < a < 8
l0 ðuð2x; 0Þ; tÞ P l0 ðauðx; 0Þ; tÞ and m0 ðuð2x; 0Þ; tÞ 6 m0 ðauðx; 0Þ; tÞ; n
n
n
n
n
ð2:1:1Þ
n
and limn!1 l ðuð2 x; 2 yÞ; 8 tÞ ¼ 1 and limn!1 m ðuð2 x; 2 yÞ; 8 tÞ ¼ 0 for all x; y in X and t > 0. Let ðY; l; mÞ be an intuitionistic fuzzy Banach space and let f : X ! Y be a u-approximately cubic mapping in the sense that 0
0
lðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ P l0 ðuðx; yÞ; tÞ and mðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ 6 m0 ðuðx; yÞ; tÞ
ð2:1:2Þ
for all t > 0 and all x; y 2 X. Then there exists a unique cubic mapping C : X ! Y such that
lðCðxÞ f ðxÞ; tÞ P l0 ðuðx; 0Þ; ð8 aÞtÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðuðx; 0Þ; ð8 aÞtÞ for all x 2 X and all t > 0.
ð2:1:3Þ
2999
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Proof. Put y ¼ 0 in (2.1.2). Then for all x 2 X and t > 0
l
f ð2xÞ t P l0 ðuðx; 0Þ; tÞ and f ðxÞ; 8 16
f ð2xÞ t 6 m0 ðuðx; 0Þ; tÞ: f ðxÞ; 8 16
m
ð2:1:4Þ
Replacing x by 2n x in (2.1.4) and using (2.1.1), we obtain
! f ð2n xÞ t P l0 ðuð2n x; 0Þ; tÞ P l0 ðuðx; 0Þ; t=an Þ and ; 8n 16ð8n Þ 8nþ1 ! f ð2nþ1 xÞ f ð2n xÞ t 6 m0 ðuð2n x; 0Þ; tÞ 6 m0 ðuðx; 0Þ; t=an Þ; ; 8n 16ð8n Þ 8nþ1 f ð2nþ1 xÞ
l m
for all x 2 X, t > 0 and n P 0. By replacing t by an t, we get
f ð2nþ1 xÞ
l
8nþ1
It follows from
l
f ð2n xÞ 8n
f ð2n xÞ 8n
m
f ð2n xÞ an t ; 8n 16ð8n Þ
f ð2n xÞ 8n
f ðxÞ ¼
f ðxÞ;
n1 P k¼0
f ðxÞ;
n1 P k¼0
Pn1
ak t
16ð8 Þ
ak t
P l ðuðx; 0Þ; tÞ and
f ð2kþ1 xÞ 8kþ1
n1 Q
l
6
n1 ‘
f ð2kþ1 xÞ 8kþ1
m
k¼0
for all x 2 X, t > 0 and n > 0 where have
k
m
8nþ1
f ð2n xÞ an t ; 8n 16ð8n Þ
! 6 m0 ðuðx; 0Þ; tÞ:
ð2:1:5Þ
j¼1 aj
9 k > ak t 0 > P f ð28k xÞ ; 16ð8 l ð u ðx; 0Þ; tÞ and > k = Þ
k ak t f ð28k xÞ ; 16ð8 6 m0 ðuðx; 0Þ; tÞ; k Þ
f ð2kþ1 xÞ 8kþ1
Qn
f ð2nþ1 xÞ
f ð28k xÞ and (2.1.5) that
k¼0
16ð8k Þ
0
k¼0
P
k
!
¼ a1 a2 an ,
‘n
j¼1 aj
ð2:1:6Þ
> > > ;
¼ a1 }a2 } }an . By replacing x with 2m x in (2.1.6), we
! n1 f ð2nþm xÞ f ð2m xÞ X ak t P l0 ðuð2m x; 0Þ; tÞ P l0 ðuðx; 0Þ; t=am Þ and ; kþm 8m 8nþm 16ð8Þ k¼0 ! n1 f ð2nþm xÞ f ð2m xÞ X ak t 6 m0 ðuð2m x; 0Þ; tÞ 6 m0 ðuðx; 0Þ; t=am Þ: ; kþm 8m 8nþm 16ð8Þ k¼0
l m Thus,
! X f ð2nþm xÞ f ð2m xÞ nþm1 ak t P l0 ðuðx; 0Þ; tÞ and ; k 8m 8nþm 16ð8Þ k¼m ! X f ð2nþm xÞ f ð2m xÞ nþm1 ak t 6 m0 ðuðx; 0Þ; tÞ; ; k 8m 8nþm k¼m 16ð8Þ
l m
for all x 2 X, t > 0 m > 0 and n P 0. Hence
l
f ð2nþm xÞ 8nþm
m t f ð28m xÞ ; t P l0 uðx; 0Þ; Pnþm1 k¼m
m
f ð2nþm xÞ 8nþm
f ð2m xÞ ;t 8m
6m
0
! ak 16ð8Þk
!
t
uðx; 0Þ; Pnþm1 k¼m
;
ak 16ð8Þk
9 > > > and > > =
ð2:1:7Þ
> > > > > ;
P1 ak for all x n2 X, t > 0 m P 0 and n P 0. Since 0 < a < 8 and k¼0 8 < 1, the Cauchy criterion for convergence in IFNS shows f ð2 xÞ that 8n is a Cauchy sequence in ðY; l; mÞ. Since ðY; l; mÞ is complete, this sequence converges to some point CðxÞ 2 Y. Fix x 2 X and put m ¼ 0 in (2.1.7) to obtain
l
f ð2n xÞ f ðxÞ; t 8n
0 Pl
0@
1
t
uðx; 0Þ; Pn1
ak
k¼0 16ð8Þ
0 n f ð2 xÞ t 0 A and m f ðxÞ; t 6 m @uðx; 0Þ; Pn1 n 8 k k¼0
1 ak
A;
16ð8Þk
for all t > 0 and n > 0. Thus, we obtain
0 1 n f ð2n xÞ f ð2 xÞ t lðCðxÞ f ðxÞ; tÞ P l CðxÞ n ; t=2 l f ðxÞ; t=2 P l0 @uðx; 0Þ; Pn1 k A and a 8 8n k¼0 8ð8Þk 0 1 n f ð2n xÞ f ð2 xÞ t 0@ mðCðxÞ f ðxÞ; tÞ 6 m CðxÞ n ; t=2 }m f ðxÞ; t=2 6 m uðx; 0Þ; Pn1 k A; a 8 8n k k¼0 8ð8Þ
3000
M. Mursaleen, S.A. Mohiuddine / Chaos, Solitons and Fractals 42 (2009) 2997–3005
for large n. Taking the limit as n ! 1 and using the definition of IFNS, we get
lðCðxÞ f ðxÞ; tÞ P l0 ðuðx; 0Þ; ð8 aÞtÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðuðx; 0Þ; ð8 aÞtÞ: Replace x and y by 2n x and 2n y, respectively, in (2.1.2), we have
n f ð2 ð2x þ yÞÞ f ð2n ð2x yÞÞ 2f ð2n ðx þ yÞÞ 2f ð2n ðx yÞÞ 12f ð2n xÞ þ ; t P l0 ðuð2n x; 2n yÞ; 8n tÞ n n n n n 8 8 8 8 8
l and
n f ð2 ð2x þ yÞÞ f ð2n ð2x yÞÞ 2f ð2n ðx þ yÞÞ 2f ð2n ðx yÞÞ 12f ð2n xÞ þ ; t 6 m0 ðuð2n x; 2n yÞ; 8n tÞ; n n n n n 8 8 8 8 8
m
for all x; y 2 X and all t > 0. Since
lim l0 ðuð2n x; 2n yÞ; 8n tÞ ¼ 1 and
n!1
lim m0 ðuð2n x; 2n yÞ; 8n tÞ ¼ 0;
n!1
we observe that C fulfills (2.1.0). To prove the uniqueness of the cubic function C, assume that there exists a cubic function D : X ! Y which satisfies (2.1.3). For fix x 2 X, clearly Cð2n xÞ ¼ 8n CðxÞ and Dð2n xÞ ¼ 8n DðxÞ for all n 2 N. It follows from (2.1.3) that
n Cð2n xÞ Dð2n xÞ Cð2n xÞ f ð2n xÞ t f ð2 xÞ Dð2n xÞ t l ;t P l ; ; n n n n n n 2 2 8 8 8 8 8 8 n n 8 ð8 aÞt 8 ð8 aÞt P l0 uð2n x; 0Þ; P l0 uðx; 0Þ; 2 2an
lðCðxÞ DðxÞ; tÞ ¼ l
n n aÞt aÞt . Since limn!1 8 ð8 and similarly mðCðxÞ DðxÞ; tÞ 6 m0 uðx; 0Þ; 8 ð8 ¼ 1, we get 2an 2an
8n ð8 aÞt lim l0 uðx; 0Þ; ¼ 1 and n!1 2an
8n ð8 aÞt lim m0 uðx; 0Þ; ¼ 0: n!1 2an
Therefore,
lðCðxÞ DðxÞ; tÞ ¼ 1 and mðCðxÞ DðxÞ; tÞ ¼ 0; for all t > 0. Hence CðxÞ ¼ DðxÞ.
Example 2.1. Let X be a Hilbert space and Z be a normed space. Denote by ðl; mÞ and ðl0 ; m0 Þ the intuitionistic fuzzy norms given as in Example 1.1 on X and Z, respectively. Let u : X X ! Z be defined by uðx; yÞ ¼ 8ðkxk2 þ kyk2 Þzo , where zo is a fixed unit vector in Z. Define f : X ! X by f ðxÞ ¼ kxk2 x þ kxk2 xo for some unit vector xo 2 X. Then
lðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ ¼
t t þ 8kxk2 þ 2kyk2
P
t t þ 8kxk2 þ 8kyk2
¼ l0 ðuðx; yÞ; tÞ
and
mðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ 6 m0 ðuðx; yÞ; tÞ: Also
l0 ðuð2x; 0Þ; tÞ ¼
t t þ 32kxk2
¼ l0 ð4uðx; 0Þ; tÞ and
m0 ðuð2x; 0Þ; tÞ ¼ m0 ð4uðx; 0Þ; tÞ:
Thus,
lim l0 ðuð2n x; 2n yÞ; 8n tÞ ¼ lim
n!1
n!1
8n t 8 t þ 8ð4 Þðkxk2 þ kyk2 Þ n
n
¼1
and
lim m0 ðuð2n x; 2n yÞ; 8n tÞ ¼ lim
n!1
n!1
8ð4n Þðkxk2 þ kyk2 Þ 8 t þ 8ð4n Þðkxk2 þ kyk2 Þ n
¼ 0:
Hence, conditions of Theorem 2.1 for a ¼ 4 are fulfilled. Therefore, there is a unique cubic mapping C : X ! X such that
lðCðxÞ f ðxÞ; tÞ P l0 ðuðx; 0Þ; 4tÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðuðx; 0Þ; 4tÞ: In the following theorem we consider the case a > 8. Theorem 2.2. Let X be a linear space and let ðZ; l0 ; m0 Þ be an IFNS. Let u : X X ! Z be a function such that for some a > 8
l0 ðuðx=2; 0Þ; tÞ P l0 ðuðx; 0Þ; atÞ and m0 ðuðx=2; 0Þ; tÞ 6 m0 ðuðx; 0Þ; atÞ;
3001
M. Mursaleen, S.A. Mohiuddine / Chaos, Solitons and Fractals 42 (2009) 2997–3005
and limn!1 l0 ð8n uð2n x; 2n yÞ; tÞ ¼ 1 and limn!1 m0 ð8n uð2n x; 2n yÞ; tÞ ¼ 0 for all x; y 2 X and all t > 0. Let ðY; l; mÞ be an intuitionistic fuzzy Banach space and let f : X ! Y be a u-approximately cubic mapping in the sense of (2.1.2). Then there exists a unique cubic mapping C : X ! Y such that
lðCðxÞ f ðxÞ; tÞ P l0 ðuðx; 0Þ; ða 8ÞtÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðuðx; 0Þ; ða 8ÞtÞ; for all x 2 X and all t > 0. Proof. The techniques are similar to that of Theorem 2.1. Hence we present a sketch of proof. Put y ¼ 0 in (2.1.2), we get
lð2f ð2xÞ 16f ðxÞ; tÞ P l0 ðuðx:0Þ; tÞ and mð2f ð2xÞ 16f ðxÞ; tÞ 6 m0 ðuðx:0Þ; tÞ; for all x 2 X, t > 0. Therefore,
l f ðxÞ 8f
x ; t P l0 ðuðx; 0Þ; 2atÞ and 2
m f ðxÞ 8f
x ; t 6 m0 ðuðx; 0Þ; 2atÞ; 2
for all x 2 X and t > 0. For each x 2 X; n P 0; m P 0 and t > 0, we can deduce
lð8nþm f ð2ðnþmÞ xÞ 8m f ð2m xÞ; tÞ P l0 uðx; 0Þ; Pnþmt
8k k¼mþ1 16ak
mð8nþm f ð2ðnþmÞ xÞ 8m f ð2m xÞ; tÞ 6 m0 uðx; 0Þ; Pnþmt
8k k¼mþ1 16ak
:
9 > and > > =
ð2:3:1Þ
> > > ;
Thus, ð8n f ð2n xÞÞ is a Cauchy sequence in intuitionistic fuzzy Banach space. Therefore, there is a function C : X ! Y defined by CðxÞ ¼ limn!1 8n f ð2n xÞ. (2.3.1) with m ¼ 0 implies
lðCðxÞ f ðxÞ; tÞ P l0 ðuðx; 0Þ; ða 8ÞtÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðuðx; 0Þ; ða 8ÞtÞ; for all x 2 X and all t > 0.
3. Intuitionistic fuzzy continuity Recently, the intuitionistic fuzzy continuity is discussed in [25]. In this section, we establish some interesting results of continuous approximately cubic mappings. Definition 3.1. Let f : R ! X be a function, where R is endowed with the Euclidean topology and X is an intuitionistic fuzzy normed space equipped with intuitionistic fuzzy norm ðl; mÞ. Then, f is called intuitionistic fuzzy continuous at a point so 2 R if for all > 0 and all 0 < a < 1 there exists d > 0 such that for each s with 0 < js so j < d
lðf ðsxÞ f ðso xÞ; Þ P a and mðf ðsxÞ f ðso xÞ; Þ 6 1 a: Theorem 3.2. Let X be a normed space and ðZ; l0 ; m0 Þ be an IFNS. Let ðY; l; mÞ be an intuitionistic fuzzy Banach space and f : X ! Y be a ðp; qÞ-approximately cubic mapping in the sense that for some p; q and some zo 2 Z
lðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ P l0 ððkxkp þ kykq Þzo ; tÞ and mðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ 6 m0 ððkxkp þ kykq Þzo ; tÞ; for all x; y 2 X and all t > 0. If p; q < 3, there exists a unique cubic mapping C : X ! Y such that
lðCðxÞ f ðxÞ; tÞ P l0 ðkxkp zo ; ð8 2p ÞtÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðkxkp zo ; ð8 2p ÞtÞ;
ð3:2:1Þ n
for all x 2 X and all t > 0. Furthermore, if for some x 2 X and all n 2 N, the mapping g : R ! Y defined by gðsÞ ¼ f ð2 sxÞ is intuitionistic fuzzy continuous. Then the mapping s#CðsxÞ from R to Y is intuitionistic fuzzy continuous; in this case, CðrxÞ ¼ r3 CðxÞ for all r 2 R. Proof. If we define u : X X ! Z by uðx; yÞ ¼ ðkxkp þ kykq Þzo . Existence and uniqueness of the cubic mapping C satisfying (3.2.1) are deduced from Theorem 2.1 (see Example 2.2). Note that for each x 2 X; t 2 R and n 2 N, we have
n
l CðxÞ f ð28n xÞ ; t
m CðxÞ
f ð2n xÞ ;t 8n
9 n > f ð28n xÞ ; t ¼ lðCð2n xÞ f ð2n xÞ; 8n tÞ > > > = n p p p np n p 8 ð82 Þt 0 0 P l ð2 kxk zo ; 8 ð8 2 ÞtÞ ¼ l kxk zo ; 2np and > > > n p > Þt ; : 6 m0 kxkp zo ; 8 ð82 np 2 ¼
l
Cð2n xÞ 8n
ð3:2:2Þ
3002
M. Mursaleen, S.A. Mohiuddine / Chaos, Solitons and Fractals 42 (2009) 2997–3005
Fix x 2 X and so 2 R. Given
> 0 and 0 < a < 1. From (3.2.2) follows that
f ð2n sxÞ 8n ð8 2p Þt 8n ð8 2p Þt p 0 P and l CðsxÞ n ; t P l0 kxkp zo ; l kxk z ; o 8 jsjp 2np ð1 þ jso jÞp 2np f ð2n sxÞ 8n ð8 2p Þt 8n ð8 2p Þt 6 m0 kxkp zo ; ; m CðsxÞ ; t 6 m0 kxkp zo ; n p np 8 jsj 2 ð1 þ jso jÞp 2np n
p
8 ð82 Þt for all j s so j< 1 and s 2 R. Since limn!1 ð1þjs p np ¼ 1, there exists no 2 N such that o jÞ 2
l CðsxÞ
f ð2no sxÞ P a and ; 3 8no
m CðsxÞ
f ð2no sxÞ 61a ; 3 8no
for all j s so j< 1 and s 2 R. By the intuitionistic fuzzy continuity of the mapping t ! f ð2no txÞ, there exists d < 1 such that for each s with 0 < js so j < d, we have
no f ð2 sxÞ f ð2no so xÞ ; P a and no no 3 8 8
l
m
no f ð2 sxÞ f ð2no so xÞ ; 6 1 a: no no 3 8 8
It follows that
lðCðsxÞ Cðso xÞ; Þ P l CðsxÞ
no f ð2no sxÞ f ð2 sxÞ f ð2no so xÞ f ð2no so xÞ Pa ; l ; l Cðs xÞ ; o 3 3 3 8no 8no 8no 8no
and mðCðsxÞ Cðso xÞ; Þ 6 1 a, for each s with 0 0. Then, for 0 < a < 1 there exists d > 0 such that
lðCðrxÞ Cðro xÞ; t=3Þ P a and mðCðrxÞ Cðro xÞ; t=3Þ 6 1 a for each r 2 R and 0 < jr ro j < d. Choose a rational number r with 0 < jr r o j < d and jr 3 r 3o j < 1 a. Then
lðCðro xÞ r3o CðxÞ; tÞ P lðCðro xÞ CðrxÞ; t=3Þ lðCðrxÞ r3 CðxÞ; t=3Þ lðr3 CðxÞ r3o CðxÞ; t=3Þ P a 1 lðCðxÞ; t=3ð1 aÞÞ and mðCðro xÞ r 3o CðxÞ; tÞ 6 ð1 aÞ}0}mðCðxÞ; t=3ð1 aÞÞ: Letting a ! 1 and using definition of IFNS, we get
lðCðro xÞ r3o CðxÞ; tÞ ¼ 1 and mðCðro xÞ r3o CðxÞ; tÞ ¼ 0: Hence, we conclude that Cðr o xÞ ¼ r3o CðxÞ.
In the following theorem we proof a result similar to Theorem 3.2, for the case p; q > 3. Theorem 3.3. Let X be a normed space and ðZ; l0 ; m0 Þ be an IFNS. Let ðY; l; mÞ be an intuitionistic fuzzy Banach space and f : X ! Y be a ðp; qÞ-approximately cubic mapping in the sense that for some p; q and some zo 2 Z
lðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ P l0 ððkxkp þ kykq Þzo ; tÞ and mðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ 6 m0 ððkxkp þ kykq Þzo ; tÞ; for all x; y 2 X and all t > 0. If p; q > 3, there exists a unique cubic mapping C : X ! Y such that
lðCðxÞ f ðxÞ; tÞ P l0 ðkxkp zo ; ð2p 8ÞtÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðkxkp zo ; ð2p 8ÞtÞ;
ð3:3:1Þ n
for all x 2 X and all t > 0. Furthermore, if for some x 2 X and all n 2 N, the mapping g : R ! Y defined by gðsÞ ¼ f ð2 sxÞ is intuitionistic fuzzy continuous. Then the mapping s#CðsxÞ from R to Y is intuitionistic fuzzy continuous; in this case, CðrxÞ ¼ r 3 CðxÞ for all r 2 R. Proof. If we define u : X X ! Z by uðx; yÞ ¼ ðkxkp þ kykq Þzo . Then
l0 ðuðx=2; 0Þ; tÞ ¼ l0 ðkxkp zo ; 2p tÞ and m0 ðuðx=2; 0Þ; tÞ ¼ m0 ðkxkp zo ; 2p tÞ; for all x 2 X and t > 0. Since p > 3, we have a ¼ 2p > 23 ¼ 8. By Theorem 2.2, there exists a unique cubic mapping C which satisfies (3.3.1). Rest of the proof can be done on the same lines as in Theorem 3.2.
4. Intuitionistic fuzzy completeness In this section, we obtain intuitionistic fuzzy completeness through the existence of some solution of a fuzzy stability problem for cubic functional equation.
M. Mursaleen, S.A. Mohiuddine / Chaos, Solitons and Fractals 42 (2009) 2997–3005
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Definition 4.1. Let ðX; l; mÞ be an IFNS. A mapping f : N [ f0g ! X is said to be approximately cubic if for each a 2 ð0; 1Þ there exists some na 2 N such that
lðf ð2n þ mÞ þ f ð2n mÞ 2f ðn þ mÞ 2f ðn mÞ 12f ðnÞ; 1Þ P a and mðf ð2n þ mÞ þ f ð2n mÞ 2f ðn þ mÞ 2f ðn mÞ 12f ðnÞ; 1Þ 6 1 a; for all n P 2m P na . By a conditional cubic mapping we mean a mapping f : N [ f0g ! X such that (2.1.0) holds whenever x P 2y. It can be easily verified that for each conditional cubic mapping f : N [ f0g ! X, we have f ð2n Þ ¼ 23n f ð1Þ. Theorem 4.2. Let ðX; l; mÞ be an IFNS such that for each approximately cubic mapping f : N [ f0g ! X, there exists a conditional cubic mapping C : N [ f0g ! X such that
lim lðCðnÞ f ðnÞ; 1Þ ¼ 1 and
lim mðCðnÞ f ðnÞ; 1Þ ¼ 0:
n!1
n!1
Then ðX; l; mÞ is an intuitionistic fuzzy Banach space. Proof. Let ðxn Þ be a Cauchy sequence in an IFNS. By induction on k, we can find a strictly increasing sequence ðnk Þ of natural numbers such that
l x n xm ;
!
1
1 P1 k
ð10kÞ3
and
m x n xm ;
!
1
6
ð10kÞ3
1 ; k 3
for each n; m P na . Let yk ¼ xnk and define f : N [ f0g ! X by f ðkÞ ¼ k yk . Let a 2 ð0; 1Þ and find some no 2 N such that 1 n1o > a. Since
f ð2k þ jÞ þ f ð2k jÞ 2f ðk þ jÞ 2f ðk jÞ 12f ðkÞ 3
3
3
3
2
¼ 2k ðy2kþj ykþj Þ þ 2k ðy2kþj ykj Þ þ 4k ðy2kþj yk Þ þ 8k ðy2kj yk Þ þ 12k jðy2kþj y2kj Þ 2
2
2
3
3
þ 6k jðykj ykþj Þ þ 6kj ðy2kþj ykþj Þ þ 6kj ðy2kj ykj Þ þ j ðy2kþj y2kj Þ þ 2j ðykj ykþj Þ for each k P 2j, we have
lðf ð2k þ jÞ þ f ð2k jÞ 2f ðk þ jÞ 2f ðk jÞ 12f ðkÞ; 1Þ
1 1 1 1 l y y ; l y y ; l y y ; P l y2kþj ykþj ; 2kþj kj 2kþj k 2kj k 3 3 3 3 20k 20k 40k 80k ! ! ! l y2kþj y2kj ; l y2kj ykj ;
1
l ykj ykþj ;
2
120k j ! 1 2
60kj
l y2kþj y2kj ;
1
l y2kþj ykþj ;
2
60k j ! 1
10j
3
l
1
60kj ! 1 : ykj ykþj ; 3 20j
2
Then for j > no
l y2kþj ykþj ;
1 20k
P l y2kþj ykþj ;
3
!
1
P l y2kþj ykþj ;
20ðk þ jÞ3
!
1 103 ðk þ jÞ3
P a:
Similarly
m y2kþj ykþj ;
1 20k
3
6 m y2kþj ykþj ;
!
1 20ðk þ jÞ3
Since k j P 2k and k j P j, we have
l y2kþj ykj ; l ykj ykþj ;
3
20k
2
60k j 1 2
60kj ! 1
20j
3
P l y2kþj ykj ;
!
1
l y2kj ykj ; l ykj ykþj ;
1
P l ykj ykþj ; !
1
!
103 ðk jÞ3 ! 1
103 ðk jÞ3
P l y2kj ykj ; P l y2kj ykj ;
6 m y2kþj ykþj ;
1 103 ðk jÞ3 ! 1
103 ðk jÞ3
P a; P a;
! P a; P a:
1 103 ðk þ jÞ3
! 6 1 a:
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M. Mursaleen, S.A. Mohiuddine / Chaos, Solitons and Fractals 42 (2009) 2997–3005
Clearly
and
l
1 1 1 l y y ; a ; l y y ; l y y ; P P P P a; 2kþj k 2kj k 2kj k 3 3 3 3 40k 103 k 80k 103 k ! ! 1 1 P l y2kþj ykþj ; 3 P a: y2kþj ykþj ; 2 10 ðk þ jÞ3 60kj 1
l y2kþj yk ;
Similarly
m y2kþj yk ;
1 3
40k
6 1 a;
m y2kj yk ;
1 80k
3
6 1 a;
and
m y2kþj ykþj ;
!
1 60kj
2
6 1 a:
The inequalities 2k j P j and 2k j > k imply
l y2kþj y2kj ; l
!
1
!
1
1
!
P l y2kþj y2kj ; 3 P a; P l y2kþj y2kj ; 2 120ð2k jÞ3 10 ð2k jÞ3 120k j ! ! ! 1 1 1 P P P a; y2kþj y2kj ; l y y ; l y y ; 2kþj 2kj 2kþj 2kj 3 10ð2k jÞ3 103 ð2k jÞ3 10j
and similarly
m y2kþj y2kj ;
!
1 2
120k j
6 1 a; m y2kþj y2kj ;
!
1 10j
3
6 1 a:
Therefore,
lðf ð2k þ jÞ þ f ð2k jÞ 2f ðk þ jÞ 2f ðk jÞ 12f ðkÞ; 1Þ P a and
mðf ð2k þ jÞ þ f ð2k jÞ 2f ðk þ jÞ 2f ðk jÞ 12f ðkÞ; 1Þ 6 1 a: This shows that f is approximately cubic type mapping. By our assumption, there exists a conditional cubic mapping C : N [ f0g ! X such that
lim lðCðkÞ f ðkÞ; 1Þ ¼ 1 and
k!1
lim mðCðkÞ f ðkÞ; 1Þ ¼ 0:
k!1
In particular,
lim lðCð2k Þ f ð2k Þ; 1Þ ¼ 1 and
k!1
lim mðCð2k Þ f ð2k Þ; 1Þ ¼ 0:
k!1
This means that
1 lim l Cð1Þ y2k ; 3k ¼ 1 and k!1 2
1 lim m Cð1Þ y2k ; 3k ¼ 0: k!1 2
Hence the subsequence ðy2k Þ converges to y ¼ Cð1Þ. Therefore, the Cauchy sequence ðxn Þ is also converges to y.
5. Conclusion We linked here two different disciplines, namely, the fuzzy spaces and functional equations. We established Hyers– Ulam–Rassias stability of a cubic functional equation
f ð2x þ yÞ þ f ð2x yÞ ¼ 2f ðx þ yÞ þ 2f ðx yÞ þ 12f ðxÞ in the set of intuitionistic fuzzy normed spaces. We also studied the intuitionistic fuzzy continuity and completeness through the existence of a certain solution of a fuzzy stability problem for approximately cubic functional equation. References [1] [2] [3] [4] [5] [6] [7]
Aoki T. On the stability of the linear transformation in Banach spaces. J Math Soc Jpn 1950;2:64–6. Barros LC, Bassanezi RC, Tonelli PA. Fuzzy modelling in population dynamics. Ecol Model 2000;128:27–33. El Naschie MS. On the uncertainty of Cantorian geometry and two-slit experiment. Chaos, Solitons and Fractals 1998;9:517–29. El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons and Fractals 2004;19:209–36. El Naschie MS. Fuzzy dodecahedron topology and E-infinity spacetime as a model for quantum physics. Chaos, Solitons and Fractals 2006;30:1025–33. El Naschie MS. Holographic dimensional reduction: center manifold theorem and E-infinity. Chaos, Solitons and Fractals 2006;29:816–22. El Naschie MS. A review of applications and results of E-infinity theory. Int J Nonlinear Sci Numer Simul 2007;8:11–20.
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[8] El Naschie MS. On zero-dimensional points curvature in the dynamics of Cantorian-fractal spacetime setting and high energy particle physics. Chaos, Solitons and Fractals 2009;41:2725–32. [9] El Naschie MS. Arguments for the compactness and multiple connectivity of our cosmic spacetime. Chaos, Solitons and Fractals 2009;41:2787–9. [10] Feng G, Chen G. Adaptative control of discrete-time chaotic systems: a fuzzy control approach. Chaos, Solitons and Fractals 2005;253:459–67. [11] Fradkov AL, Evans RJ. Control of chaos: methods and applications in engineering. Chaos, Solitons and Fractals 2005;29:33–56. [12] Giles R. A computer program for fuzzy reasoning. Fuzzy Sets Syst 1980;4:221–34. [13] Hong L, Sun JQ. Bifurcations of fuzzy nonlinear dynamical systems. Commun Nonlinear Sci Numer Simul 2006;1:1–12. [14] Hyers DH. On the stability of the linear functional equation. Proc Natl Acad Sci USA 1941;27:222–4. [15] Hyers DH, Isac G, Rassias TM. Stability of functional equations in several variables. Basel: Birkhäuser; 1998. [16] Jun KW, Kim HM. The generalized Hyers–Ulam–Rassias stability of a cubic functional equation. J Math Anal Appl 2002;274:867–78. [17] Jun KW, Kim HM, Chang IS. On the Hyers–Ulam stability of an Euler–Lagrange type cubic functional equation. J Comput Anal Appl 2005;7:21–33. [18] Marek-Crnjac L. A short history of fractal-Cantorian space-time. Chaos, Solitons and Fractals 2009;41:2697–705. [19] Marek-Crnjac L. A Feynman path integral-like method for deriving the four dimensionality of spacetime from first principles. Chaos, Solitons and Fractals 2009;41:2471–3. [20] Mirmostafaee AK, Mirzavaziri M, Moslehian MS. Fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst 2008;159:730–8. [21] Mirmostafaee AK, Moslehian MS. Fuzzy versions of Hyers–Ulam–Rassias theorem. Fuzzy Sets Syst 2008;159:720–9. [22] Mirmostafaee AK, Moslehian MS. Fuzzy approximately cubic mappings. Inf Sci 2008;178:3791–8. [23] Mirmostafaee AK, Moslehian MS. Fuzzy almost quadratic functions. Results Math. doi:10.1007/s00025-007-0278-9. [24] Mursaleen M, Mohiuddine SA. Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos, Solitons and Fractals 2009;41:2414–21. [25] Mursaleen M, Mohiuddine SA. Nonlinear operators between intuitionistic fuzzy normed spaces and Fréchet differentiation. Chaos, Solitons and Fractals 2009;42:1010–5. [26] Park JH. Intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 2004;22:1039–46. [27] Rassias TM. On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978;72:297–300. [28] Rassias TM. On the stability of functional equations and a problem of Ulam. Acta Appl Math 2000;62:23–130. [29] Saadati R, Park JH. On the intuitionistic fuzzy topological spaces. Chaos, Solitons and Fractals 2006;27:331–44. [30] Saadati R. A note on ‘‘Some results on the IF-normed spaces”. Chaos, Solitons and Fractals 2009;41:206–13. [31] Ulam SM. Problems in modern mathematics. Science ed. New York: John Wiley & Sons; 1940 [chapter VI, Some questions in analysis: Section 1, Stability].
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
On stability of a cubic functional equation in intuitionistic fuzzy normed spaces q M. Mursaleen *, S.A. Mohiuddine Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
a r t i c l e
i n f o
Article history: Accepted 3 April 2009
a b s t r a c t In this paper, we determine some stability results concerning the cubic functional equation
f ð2x þ yÞ þ f ð2x yÞ ¼ 2f ðx þ yÞ þ 2f ðx yÞ þ 12f ðxÞ in intuitionistic fuzzy normed spaces (IFNS). We define the intuitionistic fuzzy continuity of the cubic mappings and prove that the existence of a solution for any approximately cubic mapping implies the completeness of IFNS. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries Fuzzy set theory is a powerful hand set for modelling uncertainty and vagueness in various problems arising in the field of science and engineering. It has also very useful applications in various fields, e.g. population dynamics [2], chaos control [10,11], computer programming [12], nonlinear dynamical systems [13], nonlinear operators [25], statistical convergence [24], etc. The fuzzy topology proves to be a very useful tool to deal with such situations where the use of classical theories breaks down. The most fascinating application of fuzzy topology in quantum particle physics arises in string and ð1Þ -theory of El-Naschie (cf. [3–9,18,19]). There are many situations where the norm of a vector is not possible to find and the concept of intuitionistic fuzzy norm [26,29,30] seems to be more suitable in such cases, that is, we can deal with such situations by modelling the inexactness through the intuitionistic fuzzy norm. Stability problem of a functional equation was first posed by Ulam [31] which was answered by Hyers [14] and then generalized by Aoki [1] and Rassias [27] for additive mappings and linear mappings, respectively. Since then several stability problems for various functional equations have been investigated in [15,28]; and various fuzzy stability results concerning Cauchy, Jensen and quadratic functional equations were discussed in [20,21,23]. The stability problem for the cubic functional equation was proved by Jun and Kim [16] for mappings f : X ! Y, where X is a real normed space and Y is a Banach space. Later on, in [17,22] the problem of stability of some cubic equation were discussed. In this paper, we determine some stability results concerning the cubic functional equation
f ð2x þ yÞ þ f ð2x yÞ ¼ 2f ðx þ yÞ þ 2f ðx yÞ þ 12f ðxÞ
q Research of the first author was supported by the Department of Science and Technology, New Delhi, under Grant No. SR/S4/MS:505/07; and research of the second author was supported by the Department of Atomic Energy, Government of India under the NBHM-Post Doctoral Fellowship Programme Number 40/10/2008-R&D II/892. * Corresponding author. E-mail addresses: [email protected] (M. Mursaleen), [email protected] (S.A. Mohiuddine).
0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.04.041
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in intuitionistic fuzzy normed spaces. We define the intuitionistic fuzzy continuity of the cubic mappings and prove that the existence of a solution for any approximately cubic mapping implies the completeness of intuitionistic fuzzy normed spaces (IFNS). In this section we recall some notations and basic definitions used in this paper. Definition 1.1. A binary operation : ½0; 1 ½0; 1 ! ½0; 1 is said to be a continuous t-norm if it satisfies the following conditions: ðaÞ is associative and commutative, ðbÞ is continuous, ðcÞ a 1 ¼ a for all a 2 ½0; 1, ðdÞ a b 6 c d whenever a 6 c and b 6 d for each a; b; c; d 2 ½0; 1. Definition 1.2. A binary operation } : ½0; 1 ½0; 1 ! ½0; 1 is said to be a continuous t-conorm if it satisfies the following conditions: 0 0 (a0 ) } is associative and commutative, (b ) } is continuous, (c0 ) a}0 ¼ a for all a 2 ½0; 1, (d ) a}b 6 c}d whenever a 6 c and b 6 d for each a; b; c; d 2 ½0; 1. Using the notions of continuous t-norm and t-conorm, Saadati and Park [29] have recently introduced the concept of intuitionistic fuzzy normed space as follows: Definition 1.3. The five-tuple ðX; l; m; ; }Þ is said to be an intuitionistic fuzzy normed spaces (for short, IFNS) if X is a vector space, is a continuous t-norm, } is a continuous t-conorm, and l; m are fuzzy sets on X ð0; 1Þ satisfying the following conditions. For every x; y 2 X and s; t > 0 (i) lðx; tÞ þ mðx; tÞ 6 1, (ii) lðx; tÞ > 0, (iii) lðx; tÞ ¼ 1 if and only if x ¼ 0, (iv) lðax; tÞ ¼ lðx; jat jÞ for each a–0, (v) lðx; tÞ lðy; sÞ 6 lðx þ y; t þ sÞ, (vi) lðx; Þ : ð0; 1Þ ! ½0; 1 is continuous, (vii) limt!1 lðx; tÞ ¼ 1 and limt!0 lðx; tÞ ¼ 0, (viii) mðx; tÞ < 1, (ix) mðx; tÞ ¼ 0 if and only if x ¼ 0, (x) mðax; tÞ ¼ mðx; jat jÞ for each a–0, (xi) lðx; tÞ}lðy; sÞ P mðx þ y; t þ sÞ, (xii) mðx; Þ : ð0; 1Þ ! ½0; 1 is continuous, (xiii) limt!1 mðx; tÞ ¼ 0 and limt!0 mðx; tÞ ¼ 1. In this case ðl; mÞ is called an intuitionistic fuzzy norm. Example 1.1. Let ðX; k kÞ be a normed space, a b ¼ ab and a}b ¼ minfa þ b; 1g for all a; b 2 ½0; 1. For all x 2 X and every t > 0, consider
(
lðx; tÞ ¼
t tþkxk
if t > 0
0
if t 6 0;
(
and
mðx; tÞ ¼
kxk tþkxk
if t > 0
0
if t 6 0:
Then ðX; l; m; ; }Þ is an IFNS. The concepts of convergence and Cauchy sequences in an intuitionistic fuzzy normed space are studied in [29]. Let ðX; l; m; ; }Þ be an IFNS. Then, a sequence x ¼ ðxk Þ is said to be intuitionistic fuzzy conv ergent to L 2 X if IF lim lðxk L; tÞ ¼ 1 and lim mðxk L; tÞ ¼ 0 for all t > 0. In this case we write xk ! L as k ! 1. Let ðX; l; m; ; }Þ be an IFNS. Then, x ¼ ðxk Þ is said to be intuitionistic fuzzy Cauchy sequence if lim lðxkþp xk ; tÞ ¼ 1 and lim mðxkþp xk ; tÞ ¼ 0 for all t > 0 and p ¼ 1; 2; . . . Let ðX; l; m; ; }Þ be an IFNS. Then ðX; l; m; ; }Þ is said to be complete if every intuitionistic fuzzy Cauchy sequence in ðX; l; m; ; }Þ is intuitionistic fuzzy convergent in ðX; l; m; ; }Þ. 2. Intuitionistic fuzzy stability The functional equation
f ð2x þ yÞ þ f ð2x yÞ ¼ 2f ðx þ yÞ þ 2f ðx yÞ þ 12f ðxÞ
ð2:1:0Þ 3
is called the cubic functional equation, since the function f ðxÞ ¼ cx is its solution. Every solution of the cubic functional equation is said to be a cubic mapping. We begin with a generalized Hyers–Ulam–Rassias type theorem in IFNS for the cubic functional equation. Theorem 2.1. Let X be a linear space and let ðZ; l0 ; m0 Þ be an IFNS. Let u : X X ! Z be a function such that for some 0 < a < 8
l0 ðuð2x; 0Þ; tÞ P l0 ðauðx; 0Þ; tÞ and m0 ðuð2x; 0Þ; tÞ 6 m0 ðauðx; 0Þ; tÞ; n
n
n
n
n
ð2:1:1Þ
n
and limn!1 l ðuð2 x; 2 yÞ; 8 tÞ ¼ 1 and limn!1 m ðuð2 x; 2 yÞ; 8 tÞ ¼ 0 for all x; y in X and t > 0. Let ðY; l; mÞ be an intuitionistic fuzzy Banach space and let f : X ! Y be a u-approximately cubic mapping in the sense that 0
0
lðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ P l0 ðuðx; yÞ; tÞ and mðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ 6 m0 ðuðx; yÞ; tÞ
ð2:1:2Þ
for all t > 0 and all x; y 2 X. Then there exists a unique cubic mapping C : X ! Y such that
lðCðxÞ f ðxÞ; tÞ P l0 ðuðx; 0Þ; ð8 aÞtÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðuðx; 0Þ; ð8 aÞtÞ for all x 2 X and all t > 0.
ð2:1:3Þ
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Proof. Put y ¼ 0 in (2.1.2). Then for all x 2 X and t > 0
l
f ð2xÞ t P l0 ðuðx; 0Þ; tÞ and f ðxÞ; 8 16
f ð2xÞ t 6 m0 ðuðx; 0Þ; tÞ: f ðxÞ; 8 16
m
ð2:1:4Þ
Replacing x by 2n x in (2.1.4) and using (2.1.1), we obtain
! f ð2n xÞ t P l0 ðuð2n x; 0Þ; tÞ P l0 ðuðx; 0Þ; t=an Þ and ; 8n 16ð8n Þ 8nþ1 ! f ð2nþ1 xÞ f ð2n xÞ t 6 m0 ðuð2n x; 0Þ; tÞ 6 m0 ðuðx; 0Þ; t=an Þ; ; 8n 16ð8n Þ 8nþ1 f ð2nþ1 xÞ
l m
for all x 2 X, t > 0 and n P 0. By replacing t by an t, we get
f ð2nþ1 xÞ
l
8nþ1
It follows from
l
f ð2n xÞ 8n
f ð2n xÞ 8n
m
f ð2n xÞ an t ; 8n 16ð8n Þ
f ð2n xÞ 8n
f ðxÞ ¼
f ðxÞ;
n1 P k¼0
f ðxÞ;
n1 P k¼0
Pn1
ak t
16ð8 Þ
ak t
P l ðuðx; 0Þ; tÞ and
f ð2kþ1 xÞ 8kþ1
n1 Q
l
6
n1 ‘
f ð2kþ1 xÞ 8kþ1
m
k¼0
for all x 2 X, t > 0 and n > 0 where have
k
m
8nþ1
f ð2n xÞ an t ; 8n 16ð8n Þ
! 6 m0 ðuðx; 0Þ; tÞ:
ð2:1:5Þ
j¼1 aj
9 k > ak t 0 > P f ð28k xÞ ; 16ð8 l ð u ðx; 0Þ; tÞ and > k = Þ
k ak t f ð28k xÞ ; 16ð8 6 m0 ðuðx; 0Þ; tÞ; k Þ
f ð2kþ1 xÞ 8kþ1
Qn
f ð2nþ1 xÞ
f ð28k xÞ and (2.1.5) that
k¼0
16ð8k Þ
0
k¼0
P
k
!
¼ a1 a2 an ,
‘n
j¼1 aj
ð2:1:6Þ
> > > ;
¼ a1 }a2 } }an . By replacing x with 2m x in (2.1.6), we
! n1 f ð2nþm xÞ f ð2m xÞ X ak t P l0 ðuð2m x; 0Þ; tÞ P l0 ðuðx; 0Þ; t=am Þ and ; kþm 8m 8nþm 16ð8Þ k¼0 ! n1 f ð2nþm xÞ f ð2m xÞ X ak t 6 m0 ðuð2m x; 0Þ; tÞ 6 m0 ðuðx; 0Þ; t=am Þ: ; kþm 8m 8nþm 16ð8Þ k¼0
l m Thus,
! X f ð2nþm xÞ f ð2m xÞ nþm1 ak t P l0 ðuðx; 0Þ; tÞ and ; k 8m 8nþm 16ð8Þ k¼m ! X f ð2nþm xÞ f ð2m xÞ nþm1 ak t 6 m0 ðuðx; 0Þ; tÞ; ; k 8m 8nþm k¼m 16ð8Þ
l m
for all x 2 X, t > 0 m > 0 and n P 0. Hence
l
f ð2nþm xÞ 8nþm
m t f ð28m xÞ ; t P l0 uðx; 0Þ; Pnþm1 k¼m
m
f ð2nþm xÞ 8nþm
f ð2m xÞ ;t 8m
6m
0
! ak 16ð8Þk
!
t
uðx; 0Þ; Pnþm1 k¼m
;
ak 16ð8Þk
9 > > > and > > =
ð2:1:7Þ
> > > > > ;
P1 ak for all x n2 X, t > 0 m P 0 and n P 0. Since 0 < a < 8 and k¼0 8 < 1, the Cauchy criterion for convergence in IFNS shows f ð2 xÞ that 8n is a Cauchy sequence in ðY; l; mÞ. Since ðY; l; mÞ is complete, this sequence converges to some point CðxÞ 2 Y. Fix x 2 X and put m ¼ 0 in (2.1.7) to obtain
l
f ð2n xÞ f ðxÞ; t 8n
0 Pl
0@
1
t
uðx; 0Þ; Pn1
ak
k¼0 16ð8Þ
0 n f ð2 xÞ t 0 A and m f ðxÞ; t 6 m @uðx; 0Þ; Pn1 n 8 k k¼0
1 ak
A;
16ð8Þk
for all t > 0 and n > 0. Thus, we obtain
0 1 n f ð2n xÞ f ð2 xÞ t lðCðxÞ f ðxÞ; tÞ P l CðxÞ n ; t=2 l f ðxÞ; t=2 P l0 @uðx; 0Þ; Pn1 k A and a 8 8n k¼0 8ð8Þk 0 1 n f ð2n xÞ f ð2 xÞ t 0@ mðCðxÞ f ðxÞ; tÞ 6 m CðxÞ n ; t=2 }m f ðxÞ; t=2 6 m uðx; 0Þ; Pn1 k A; a 8 8n k k¼0 8ð8Þ
3000
M. Mursaleen, S.A. Mohiuddine / Chaos, Solitons and Fractals 42 (2009) 2997–3005
for large n. Taking the limit as n ! 1 and using the definition of IFNS, we get
lðCðxÞ f ðxÞ; tÞ P l0 ðuðx; 0Þ; ð8 aÞtÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðuðx; 0Þ; ð8 aÞtÞ: Replace x and y by 2n x and 2n y, respectively, in (2.1.2), we have
n f ð2 ð2x þ yÞÞ f ð2n ð2x yÞÞ 2f ð2n ðx þ yÞÞ 2f ð2n ðx yÞÞ 12f ð2n xÞ þ ; t P l0 ðuð2n x; 2n yÞ; 8n tÞ n n n n n 8 8 8 8 8
l and
n f ð2 ð2x þ yÞÞ f ð2n ð2x yÞÞ 2f ð2n ðx þ yÞÞ 2f ð2n ðx yÞÞ 12f ð2n xÞ þ ; t 6 m0 ðuð2n x; 2n yÞ; 8n tÞ; n n n n n 8 8 8 8 8
m
for all x; y 2 X and all t > 0. Since
lim l0 ðuð2n x; 2n yÞ; 8n tÞ ¼ 1 and
n!1
lim m0 ðuð2n x; 2n yÞ; 8n tÞ ¼ 0;
n!1
we observe that C fulfills (2.1.0). To prove the uniqueness of the cubic function C, assume that there exists a cubic function D : X ! Y which satisfies (2.1.3). For fix x 2 X, clearly Cð2n xÞ ¼ 8n CðxÞ and Dð2n xÞ ¼ 8n DðxÞ for all n 2 N. It follows from (2.1.3) that
n Cð2n xÞ Dð2n xÞ Cð2n xÞ f ð2n xÞ t f ð2 xÞ Dð2n xÞ t l ;t P l ; ; n n n n n n 2 2 8 8 8 8 8 8 n n 8 ð8 aÞt 8 ð8 aÞt P l0 uð2n x; 0Þ; P l0 uðx; 0Þ; 2 2an
lðCðxÞ DðxÞ; tÞ ¼ l
n n aÞt aÞt . Since limn!1 8 ð8 and similarly mðCðxÞ DðxÞ; tÞ 6 m0 uðx; 0Þ; 8 ð8 ¼ 1, we get 2an 2an
8n ð8 aÞt lim l0 uðx; 0Þ; ¼ 1 and n!1 2an
8n ð8 aÞt lim m0 uðx; 0Þ; ¼ 0: n!1 2an
Therefore,
lðCðxÞ DðxÞ; tÞ ¼ 1 and mðCðxÞ DðxÞ; tÞ ¼ 0; for all t > 0. Hence CðxÞ ¼ DðxÞ.
Example 2.1. Let X be a Hilbert space and Z be a normed space. Denote by ðl; mÞ and ðl0 ; m0 Þ the intuitionistic fuzzy norms given as in Example 1.1 on X and Z, respectively. Let u : X X ! Z be defined by uðx; yÞ ¼ 8ðkxk2 þ kyk2 Þzo , where zo is a fixed unit vector in Z. Define f : X ! X by f ðxÞ ¼ kxk2 x þ kxk2 xo for some unit vector xo 2 X. Then
lðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ ¼
t t þ 8kxk2 þ 2kyk2
P
t t þ 8kxk2 þ 8kyk2
¼ l0 ðuðx; yÞ; tÞ
and
mðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ 6 m0 ðuðx; yÞ; tÞ: Also
l0 ðuð2x; 0Þ; tÞ ¼
t t þ 32kxk2
¼ l0 ð4uðx; 0Þ; tÞ and
m0 ðuð2x; 0Þ; tÞ ¼ m0 ð4uðx; 0Þ; tÞ:
Thus,
lim l0 ðuð2n x; 2n yÞ; 8n tÞ ¼ lim
n!1
n!1
8n t 8 t þ 8ð4 Þðkxk2 þ kyk2 Þ n
n
¼1
and
lim m0 ðuð2n x; 2n yÞ; 8n tÞ ¼ lim
n!1
n!1
8ð4n Þðkxk2 þ kyk2 Þ 8 t þ 8ð4n Þðkxk2 þ kyk2 Þ n
¼ 0:
Hence, conditions of Theorem 2.1 for a ¼ 4 are fulfilled. Therefore, there is a unique cubic mapping C : X ! X such that
lðCðxÞ f ðxÞ; tÞ P l0 ðuðx; 0Þ; 4tÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðuðx; 0Þ; 4tÞ: In the following theorem we consider the case a > 8. Theorem 2.2. Let X be a linear space and let ðZ; l0 ; m0 Þ be an IFNS. Let u : X X ! Z be a function such that for some a > 8
l0 ðuðx=2; 0Þ; tÞ P l0 ðuðx; 0Þ; atÞ and m0 ðuðx=2; 0Þ; tÞ 6 m0 ðuðx; 0Þ; atÞ;
3001
M. Mursaleen, S.A. Mohiuddine / Chaos, Solitons and Fractals 42 (2009) 2997–3005
and limn!1 l0 ð8n uð2n x; 2n yÞ; tÞ ¼ 1 and limn!1 m0 ð8n uð2n x; 2n yÞ; tÞ ¼ 0 for all x; y 2 X and all t > 0. Let ðY; l; mÞ be an intuitionistic fuzzy Banach space and let f : X ! Y be a u-approximately cubic mapping in the sense of (2.1.2). Then there exists a unique cubic mapping C : X ! Y such that
lðCðxÞ f ðxÞ; tÞ P l0 ðuðx; 0Þ; ða 8ÞtÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðuðx; 0Þ; ða 8ÞtÞ; for all x 2 X and all t > 0. Proof. The techniques are similar to that of Theorem 2.1. Hence we present a sketch of proof. Put y ¼ 0 in (2.1.2), we get
lð2f ð2xÞ 16f ðxÞ; tÞ P l0 ðuðx:0Þ; tÞ and mð2f ð2xÞ 16f ðxÞ; tÞ 6 m0 ðuðx:0Þ; tÞ; for all x 2 X, t > 0. Therefore,
l f ðxÞ 8f
x ; t P l0 ðuðx; 0Þ; 2atÞ and 2
m f ðxÞ 8f
x ; t 6 m0 ðuðx; 0Þ; 2atÞ; 2
for all x 2 X and t > 0. For each x 2 X; n P 0; m P 0 and t > 0, we can deduce
lð8nþm f ð2ðnþmÞ xÞ 8m f ð2m xÞ; tÞ P l0 uðx; 0Þ; Pnþmt
8k k¼mþ1 16ak
mð8nþm f ð2ðnþmÞ xÞ 8m f ð2m xÞ; tÞ 6 m0 uðx; 0Þ; Pnþmt
8k k¼mþ1 16ak
:
9 > and > > =
ð2:3:1Þ
> > > ;
Thus, ð8n f ð2n xÞÞ is a Cauchy sequence in intuitionistic fuzzy Banach space. Therefore, there is a function C : X ! Y defined by CðxÞ ¼ limn!1 8n f ð2n xÞ. (2.3.1) with m ¼ 0 implies
lðCðxÞ f ðxÞ; tÞ P l0 ðuðx; 0Þ; ða 8ÞtÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðuðx; 0Þ; ða 8ÞtÞ; for all x 2 X and all t > 0.
3. Intuitionistic fuzzy continuity Recently, the intuitionistic fuzzy continuity is discussed in [25]. In this section, we establish some interesting results of continuous approximately cubic mappings. Definition 3.1. Let f : R ! X be a function, where R is endowed with the Euclidean topology and X is an intuitionistic fuzzy normed space equipped with intuitionistic fuzzy norm ðl; mÞ. Then, f is called intuitionistic fuzzy continuous at a point so 2 R if for all > 0 and all 0 < a < 1 there exists d > 0 such that for each s with 0 < js so j < d
lðf ðsxÞ f ðso xÞ; Þ P a and mðf ðsxÞ f ðso xÞ; Þ 6 1 a: Theorem 3.2. Let X be a normed space and ðZ; l0 ; m0 Þ be an IFNS. Let ðY; l; mÞ be an intuitionistic fuzzy Banach space and f : X ! Y be a ðp; qÞ-approximately cubic mapping in the sense that for some p; q and some zo 2 Z
lðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ P l0 ððkxkp þ kykq Þzo ; tÞ and mðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ 6 m0 ððkxkp þ kykq Þzo ; tÞ; for all x; y 2 X and all t > 0. If p; q < 3, there exists a unique cubic mapping C : X ! Y such that
lðCðxÞ f ðxÞ; tÞ P l0 ðkxkp zo ; ð8 2p ÞtÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðkxkp zo ; ð8 2p ÞtÞ;
ð3:2:1Þ n
for all x 2 X and all t > 0. Furthermore, if for some x 2 X and all n 2 N, the mapping g : R ! Y defined by gðsÞ ¼ f ð2 sxÞ is intuitionistic fuzzy continuous. Then the mapping s#CðsxÞ from R to Y is intuitionistic fuzzy continuous; in this case, CðrxÞ ¼ r3 CðxÞ for all r 2 R. Proof. If we define u : X X ! Z by uðx; yÞ ¼ ðkxkp þ kykq Þzo . Existence and uniqueness of the cubic mapping C satisfying (3.2.1) are deduced from Theorem 2.1 (see Example 2.2). Note that for each x 2 X; t 2 R and n 2 N, we have
n
l CðxÞ f ð28n xÞ ; t
m CðxÞ
f ð2n xÞ ;t 8n
9 n > f ð28n xÞ ; t ¼ lðCð2n xÞ f ð2n xÞ; 8n tÞ > > > = n p p p np n p 8 ð82 Þt 0 0 P l ð2 kxk zo ; 8 ð8 2 ÞtÞ ¼ l kxk zo ; 2np and > > > n p > Þt ; : 6 m0 kxkp zo ; 8 ð82 np 2 ¼
l
Cð2n xÞ 8n
ð3:2:2Þ
3002
M. Mursaleen, S.A. Mohiuddine / Chaos, Solitons and Fractals 42 (2009) 2997–3005
Fix x 2 X and so 2 R. Given
> 0 and 0 < a < 1. From (3.2.2) follows that
f ð2n sxÞ 8n ð8 2p Þt 8n ð8 2p Þt p 0 P and l CðsxÞ n ; t P l0 kxkp zo ; l kxk z ; o 8 jsjp 2np ð1 þ jso jÞp 2np f ð2n sxÞ 8n ð8 2p Þt 8n ð8 2p Þt 6 m0 kxkp zo ; ; m CðsxÞ ; t 6 m0 kxkp zo ; n p np 8 jsj 2 ð1 þ jso jÞp 2np n
p
8 ð82 Þt for all j s so j< 1 and s 2 R. Since limn!1 ð1þjs p np ¼ 1, there exists no 2 N such that o jÞ 2
l CðsxÞ
f ð2no sxÞ P a and ; 3 8no
m CðsxÞ
f ð2no sxÞ 61a ; 3 8no
for all j s so j< 1 and s 2 R. By the intuitionistic fuzzy continuity of the mapping t ! f ð2no txÞ, there exists d < 1 such that for each s with 0 < js so j < d, we have
no f ð2 sxÞ f ð2no so xÞ ; P a and no no 3 8 8
l
m
no f ð2 sxÞ f ð2no so xÞ ; 6 1 a: no no 3 8 8
It follows that
lðCðsxÞ Cðso xÞ; Þ P l CðsxÞ
no f ð2no sxÞ f ð2 sxÞ f ð2no so xÞ f ð2no so xÞ Pa ; l ; l Cðs xÞ ; o 3 3 3 8no 8no 8no 8no
and mðCðsxÞ Cðso xÞ; Þ 6 1 a, for each s with 0
lðCðrxÞ Cðro xÞ; t=3Þ P a and mðCðrxÞ Cðro xÞ; t=3Þ 6 1 a for each r 2 R and 0 < jr ro j < d. Choose a rational number r with 0 < jr r o j < d and jr 3 r 3o j < 1 a. Then
lðCðro xÞ r3o CðxÞ; tÞ P lðCðro xÞ CðrxÞ; t=3Þ lðCðrxÞ r3 CðxÞ; t=3Þ lðr3 CðxÞ r3o CðxÞ; t=3Þ P a 1 lðCðxÞ; t=3ð1 aÞÞ and mðCðro xÞ r 3o CðxÞ; tÞ 6 ð1 aÞ}0}mðCðxÞ; t=3ð1 aÞÞ: Letting a ! 1 and using definition of IFNS, we get
lðCðro xÞ r3o CðxÞ; tÞ ¼ 1 and mðCðro xÞ r3o CðxÞ; tÞ ¼ 0: Hence, we conclude that Cðr o xÞ ¼ r3o CðxÞ.
In the following theorem we proof a result similar to Theorem 3.2, for the case p; q > 3. Theorem 3.3. Let X be a normed space and ðZ; l0 ; m0 Þ be an IFNS. Let ðY; l; mÞ be an intuitionistic fuzzy Banach space and f : X ! Y be a ðp; qÞ-approximately cubic mapping in the sense that for some p; q and some zo 2 Z
lðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ P l0 ððkxkp þ kykq Þzo ; tÞ and mðf ð2x þ yÞ þ f ð2x yÞ 2f ðx þ yÞ 2f ðx yÞ 12f ðxÞ; tÞ 6 m0 ððkxkp þ kykq Þzo ; tÞ; for all x; y 2 X and all t > 0. If p; q > 3, there exists a unique cubic mapping C : X ! Y such that
lðCðxÞ f ðxÞ; tÞ P l0 ðkxkp zo ; ð2p 8ÞtÞ and mðCðxÞ f ðxÞ; tÞ 6 m0 ðkxkp zo ; ð2p 8ÞtÞ;
ð3:3:1Þ n
for all x 2 X and all t > 0. Furthermore, if for some x 2 X and all n 2 N, the mapping g : R ! Y defined by gðsÞ ¼ f ð2 sxÞ is intuitionistic fuzzy continuous. Then the mapping s#CðsxÞ from R to Y is intuitionistic fuzzy continuous; in this case, CðrxÞ ¼ r 3 CðxÞ for all r 2 R. Proof. If we define u : X X ! Z by uðx; yÞ ¼ ðkxkp þ kykq Þzo . Then
l0 ðuðx=2; 0Þ; tÞ ¼ l0 ðkxkp zo ; 2p tÞ and m0 ðuðx=2; 0Þ; tÞ ¼ m0 ðkxkp zo ; 2p tÞ; for all x 2 X and t > 0. Since p > 3, we have a ¼ 2p > 23 ¼ 8. By Theorem 2.2, there exists a unique cubic mapping C which satisfies (3.3.1). Rest of the proof can be done on the same lines as in Theorem 3.2.
4. Intuitionistic fuzzy completeness In this section, we obtain intuitionistic fuzzy completeness through the existence of some solution of a fuzzy stability problem for cubic functional equation.
M. Mursaleen, S.A. Mohiuddine / Chaos, Solitons and Fractals 42 (2009) 2997–3005
3003
Definition 4.1. Let ðX; l; mÞ be an IFNS. A mapping f : N [ f0g ! X is said to be approximately cubic if for each a 2 ð0; 1Þ there exists some na 2 N such that
lðf ð2n þ mÞ þ f ð2n mÞ 2f ðn þ mÞ 2f ðn mÞ 12f ðnÞ; 1Þ P a and mðf ð2n þ mÞ þ f ð2n mÞ 2f ðn þ mÞ 2f ðn mÞ 12f ðnÞ; 1Þ 6 1 a; for all n P 2m P na . By a conditional cubic mapping we mean a mapping f : N [ f0g ! X such that (2.1.0) holds whenever x P 2y. It can be easily verified that for each conditional cubic mapping f : N [ f0g ! X, we have f ð2n Þ ¼ 23n f ð1Þ. Theorem 4.2. Let ðX; l; mÞ be an IFNS such that for each approximately cubic mapping f : N [ f0g ! X, there exists a conditional cubic mapping C : N [ f0g ! X such that
lim lðCðnÞ f ðnÞ; 1Þ ¼ 1 and
lim mðCðnÞ f ðnÞ; 1Þ ¼ 0:
n!1
n!1
Then ðX; l; mÞ is an intuitionistic fuzzy Banach space. Proof. Let ðxn Þ be a Cauchy sequence in an IFNS. By induction on k, we can find a strictly increasing sequence ðnk Þ of natural numbers such that
l x n xm ;
!
1
1 P1 k
ð10kÞ3
and
m x n xm ;
!
1
6
ð10kÞ3
1 ; k 3
for each n; m P na . Let yk ¼ xnk and define f : N [ f0g ! X by f ðkÞ ¼ k yk . Let a 2 ð0; 1Þ and find some no 2 N such that 1 n1o > a. Since
f ð2k þ jÞ þ f ð2k jÞ 2f ðk þ jÞ 2f ðk jÞ 12f ðkÞ 3
3
3
3
2
¼ 2k ðy2kþj ykþj Þ þ 2k ðy2kþj ykj Þ þ 4k ðy2kþj yk Þ þ 8k ðy2kj yk Þ þ 12k jðy2kþj y2kj Þ 2
2
2
3
3
þ 6k jðykj ykþj Þ þ 6kj ðy2kþj ykþj Þ þ 6kj ðy2kj ykj Þ þ j ðy2kþj y2kj Þ þ 2j ðykj ykþj Þ for each k P 2j, we have
lðf ð2k þ jÞ þ f ð2k jÞ 2f ðk þ jÞ 2f ðk jÞ 12f ðkÞ; 1Þ
1 1 1 1 l y y ; l y y ; l y y ; P l y2kþj ykþj ; 2kþj kj 2kþj k 2kj k 3 3 3 3 20k 20k 40k 80k ! ! ! l y2kþj y2kj ; l y2kj ykj ;
1
l ykj ykþj ;
2
120k j ! 1 2
60kj
l y2kþj y2kj ;
1
l y2kþj ykþj ;
2
60k j ! 1
10j
3
l
1
60kj ! 1 : ykj ykþj ; 3 20j
2
Then for j > no
l y2kþj ykþj ;
1 20k
P l y2kþj ykþj ;
3
!
1
P l y2kþj ykþj ;
20ðk þ jÞ3
!
1 103 ðk þ jÞ3
P a:
Similarly
m y2kþj ykþj ;
1 20k
3
6 m y2kþj ykþj ;
!
1 20ðk þ jÞ3
Since k j P 2k and k j P j, we have
l y2kþj ykj ; l ykj ykþj ;
3
20k
2
60k j 1 2
60kj ! 1
20j
3
P l y2kþj ykj ;
!
1
l y2kj ykj ; l ykj ykþj ;
1
P l ykj ykþj ; !
1
!
103 ðk jÞ3 ! 1
103 ðk jÞ3
P l y2kj ykj ; P l y2kj ykj ;
6 m y2kþj ykþj ;
1 103 ðk jÞ3 ! 1
103 ðk jÞ3
P a; P a;
! P a; P a:
1 103 ðk þ jÞ3
! 6 1 a:
3004
M. Mursaleen, S.A. Mohiuddine / Chaos, Solitons and Fractals 42 (2009) 2997–3005
Clearly
and
l
1 1 1 l y y ; a ; l y y ; l y y ; P P P P a; 2kþj k 2kj k 2kj k 3 3 3 3 40k 103 k 80k 103 k ! ! 1 1 P l y2kþj ykþj ; 3 P a: y2kþj ykþj ; 2 10 ðk þ jÞ3 60kj 1
l y2kþj yk ;
Similarly
m y2kþj yk ;
1 3
40k
6 1 a;
m y2kj yk ;
1 80k
3
6 1 a;
and
m y2kþj ykþj ;
!
1 60kj
2
6 1 a:
The inequalities 2k j P j and 2k j > k imply
l y2kþj y2kj ; l
!
1
!
1
1
!
P l y2kþj y2kj ; 3 P a; P l y2kþj y2kj ; 2 120ð2k jÞ3 10 ð2k jÞ3 120k j ! ! ! 1 1 1 P P P a; y2kþj y2kj ; l y y ; l y y ; 2kþj 2kj 2kþj 2kj 3 10ð2k jÞ3 103 ð2k jÞ3 10j
and similarly
m y2kþj y2kj ;
!
1 2
120k j
6 1 a; m y2kþj y2kj ;
!
1 10j
3
6 1 a:
Therefore,
lðf ð2k þ jÞ þ f ð2k jÞ 2f ðk þ jÞ 2f ðk jÞ 12f ðkÞ; 1Þ P a and
mðf ð2k þ jÞ þ f ð2k jÞ 2f ðk þ jÞ 2f ðk jÞ 12f ðkÞ; 1Þ 6 1 a: This shows that f is approximately cubic type mapping. By our assumption, there exists a conditional cubic mapping C : N [ f0g ! X such that
lim lðCðkÞ f ðkÞ; 1Þ ¼ 1 and
k!1
lim mðCðkÞ f ðkÞ; 1Þ ¼ 0:
k!1
In particular,
lim lðCð2k Þ f ð2k Þ; 1Þ ¼ 1 and
k!1
lim mðCð2k Þ f ð2k Þ; 1Þ ¼ 0:
k!1
This means that
1 lim l Cð1Þ y2k ; 3k ¼ 1 and k!1 2
1 lim m Cð1Þ y2k ; 3k ¼ 0: k!1 2
Hence the subsequence ðy2k Þ converges to y ¼ Cð1Þ. Therefore, the Cauchy sequence ðxn Þ is also converges to y.
5. Conclusion We linked here two different disciplines, namely, the fuzzy spaces and functional equations. We established Hyers– Ulam–Rassias stability of a cubic functional equation
f ð2x þ yÞ þ f ð2x yÞ ¼ 2f ðx þ yÞ þ 2f ðx yÞ þ 12f ðxÞ in the set of intuitionistic fuzzy normed spaces. We also studied the intuitionistic fuzzy continuity and completeness through the existence of a certain solution of a fuzzy stability problem for approximately cubic functional equation. References [1] [2] [3] [4] [5] [6] [7]
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