Equations in Non-Archimedean Intuitionistic Fuzzy Normed Spaces

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Volume 2012, Article ID 234727,16pages doi:10.1155/2012/234727

Research Article

Stability of Various Functional

Equations in Non-Archimedean Intuitionistic Fuzzy Normed Spaces

Syed Abdul Mohiuddine, Abdullah Alotaibi, and Mustafa Obaid

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Syed Abdul Mohiuddine,[email protected] Received 18 May 2012; Revised 25 October 2012; Accepted 9 November 2012

Academic Editor: Seenith Sivasundaram

Copyrightq2012 Syed Abdul Mohiuddine et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We define and study the concept of non-Archimedean intuitionistic fuzzy normed space by using the idea of t-norm and t-conorm. Furthermore, by using the non-Archimedean intuitionistic fuzzy normed space, we investigate the stability of various functional equations. That is, we determine some stability results concerning the Cauchy, Jensen and its Pexiderized functional equations in the framework of non-Archimedean IFN spaces.

1. Introduction

The study of stability problem of functional equations originated from a question of Ulam1 concerning the stability of group homomorphisms.

LetG,∗be a group and letG,◦, dbe a metric group with the metricd·,·. Given >0, does there exist aδ> 0 such that if a mappingh:G → Gsatisfies the inequality dhx∗y, hx◦hy< δfor allx, y∈G, then there exists a homomorphismH:G → G withdhx, Hx< for allx∈G?

If the answer is affirmative, we would say that the equation of homomorphism Hx∗y Hx◦Hyis stable. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Hyers2gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers theorem was generalized by Aoki 3for additive mappings and by Rassias 4 for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias has significantly influenced the development of what we now call the Hyers-Ulam- Rassias stability of functional equations. Since then several stability problems for various functional equations have been investigated in5–20. Quite recently, the stability problem

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for Pexiderized quadratic functional equation, Jensen functional equation, cubic functional equation, functional equations associated with inner product spaces, and additive functional equation was considered in21–26, respectively, in the intuitionistic fuzzy normed spaces;

while the idea of intuitionistic fuzzy normed space was introduced in27and further studied in28–34to deal with some summability problems. Quite recently, Alotaibi and Mohiuddine 35established the stability of a cubic functional equation in random 2-normed spaces, while the notion of random 2-normed spaces was introduced by Golet¸36and further studied in 37–39.

By modifying the definition of intuitionistic fuzzy normed space27, in this paper, we introduce the notion of non-Archimedean intuitionistic fuzzy normed space and also estab- lish Hyers-Ulam-Rassias-type stability results concerning the Cauchy, Pexiderized Cauchy, Jensen, and Pexiderized Jension functional equations in this new setup. This work indeed pre- sents a relationship between four various disciplines: the theory of fuzzy spaces, the theory of non-Archimedean spaces, the theory of Hyers-Ulam-Rassias stability, and the theory of functional equations.

2. Non-Archimedean Intuitionistic Fuzzy Normed Space

In this section, we introduce the concept of non-Archimedean intuitionistic fuzzy normed space and further define the notions of convergence and Cauchy sequences in this new framework. We will assume throughout this paper that the symbolsN,R,C, andQwill denote the set of all natural, real, complex, and rational numbers, respectively.

A valuation is a map|·|from a fieldKinto0,∞such that 0 is the unique element hav- ing the 0 valuation, |k1k₂| |k1||k2|, and the triangle inequality holds, that is, |k1 k₂| ≤

|k1| |k2|, for allk1, k2 ∈K. We say that a fieldKis valued ifKcarries a valuation. The usual absolute values ofRandCare examples of valuations.

Let us consider a valuation which satisfies stronger condition than the triangle inequality. If the triangle inequality is replaced by|k1 k2| ≤max{|k1|,|k2|}, for allk1, k2∈Kthen, a map|·|is called non-Archimedean or ultrametric valuation, and field is called a non-Archimedean field. Clearly|1|| −1|1 and|n| ≤1, for alln∈N. A trivial example of a non-Archimedean valuation is the map| · |taking everything but 0 into 1 and|0|0.

LetXbe a vector space over a fieldKwith a non-Archimedean valuation| · |. A non- Archimedean normed space is a pairX, · , where · :X → 0,∞is such that

i x 0 if and only ifx0, ii αx |α| x forα∈K, and

iiithe strong triangle inequality, x y ≤max{ x , y }, forx, y∈X.

In40, Hensel discovered thep-adic numbers as a number theoretical analogue of power series in complex analysis. The most interesting example of non-Archimedean spaces isp-adic numbers.

Example 2.1. Letpbe a prime number. For any nonzero rational numberap^rm/nsuch that mandnare coprime to the prime numberp, define thep-adic absolute value|a|pp^−r. Then

| · |is a non-Archimedean norm onQ. The completion ofQwith respect to| · |is denoted by Qpand is called thep-adic number field.

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A binary operation ∗ : 0,1×0,1 → 0,1is said to be a continuous t-norm if it satisfies the following conditions.

a∗is associative and commutative,b∗is continuous,ca∗1afor alla∈0,1, andda∗b≤c∗dwhenevera≤candb≤dfor eacha, b, c, d∈0,1.

A binary operation♦ : 0,1×0,1 → 0,1is said to be a continuous t-conorm if it satisfies the following conditions.

a’♦is associative and commutative,b’♦is continuous,c’a♦0afor alla∈0,1, andd’a♦b≤c♦dwhenevera≤candb≤dfor eacha, b, c, d∈0,1.

Definition 2.2. The five-tupleX,E,F,∗,♦is said to be an non-Archimedean intuitionistic fuzzy normed space for short, non-Archimedean IFN space if X is a vector space over a non- Archimedean field K, ∗ is a continuous t-norm, ♦ is a continuous t-conorm, and E,F are functions fromX ×Rto 0,1satisfying the following conditions. For every x, y ∈ X and s, t ∈ K i Ex, t Fx, t ≤ 1, ii Ex, t > 0, iiiEx, t 1 if and only if x 0, iv Eαx, t Ex, t/|α| for each α /0, v Ex, t ∗ Ey, s ≤ Ex y,max{t, s}, vi Ex,· : 0,∞ → 0,1is continuous,viilim_{t→ ∞}Ex, t 1 and lim_t_→0Ex, t 0,viii Fx, t< 1,ixFx, t 0 if and only ifx 0,x Fαx, t Fx, t/|α|for eachα /0,xi Fx, t♦Fy, s ≥ Fx y,max{t, s},xiiFx,· :0,∞ → 0,1is continuous, andxiii lim_{t→ ∞}Fx, t 0 and lim_t→₀Fx, t 1.

In this caseE,Fis called a non-Archimedean intuitionistic fuzzy norm.

Example 2.3. LetX, · be a non-Archimedean normed space,a∗babanda♦b min {a b,1}for alla, b∈0,1. For allx∈X, everyt >0 andk1,2, consider the following:

Ekx, t

⎧⎨

⎩ t

t k x if t >0,

0 if t≤0;

Fkx, t

⎧⎨

⎩ k x

t k x if t >0,

1 if t≤0.

2.1

ThenX,Ek,Fk,∗,♦is a non-Archimedean intuitionistic fuzzy normed space.

Definition 2.4. Let X,E,F,∗,♦ be a non-Archimedean intuitionistic fuzzy normed space.

Then, a sequences snis said to be

iconvergent inX,E,F,∗,♦or simplyE,F-convergent toξ∈Xif for every >0 and t >0, there existsn₀∈Nsuch thatEsn−ξ, t>1−andFsn−ξ, t< for alln≥n₀. In this case we writeE,F-limns_nξandξis called theE,F-limit ofs sn. iiCauchy in X,E,F,∗,♦ or simply E,F-Cauchy if for every > 0 and t > 0,

there exists n₀ ∈ N such that Esn −s_m, t > 1− and Fsn −s_m, t < for all n, m ≥ n₀. A non-Archimedean IFN-space X,E,F,∗,♦ is said to be complete if every E,F-Cauchy is E,F-convergent. In this case X,E,F,∗,♦ is called non- Archimedean intuitionistic fuzzy Banach space.

3. Stability of Cauchy Functional Equation

In this section, we determine stability result concerning the Cauchy functional equationfx y fx fyin non-Archimedean intuitionistic fuzzy normed space.

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Theorem 3.1. LetXbe a linear space over a non-Archimedean fieldKand letZ,E,Fbe a non- Archimedean IFN space. Suppose thatϕ :X×X → Zis a function such that for someα > 0 and some positive integerkwith|k|< α

E ϕ

k⁻¹x, k⁻¹y , t

≥ E ϕ

x, y , αt

, F

ϕ

k⁻¹x, k⁻¹y , t

≤ F ϕ

x, y , αt

,

3.1

for allx, y∈Xandt >0. LetY,E,Fbe a non-Archimedean intuitionistic fuzzy Banach space over Kand letf :X → Y be aϕ-approximately Cauchy mapping in the sense that

E f

x y

−fx−f y

, t

≥ E ϕ

x, y , t

, F

f x y

−fx−f y

, t

≤ F ϕ

x, y , t

, 3.2

for allx, y∈Xandt >0. Then there exists a unique additive mappingC:X → Y such that

E

fx−Cx, t

≥ Mx, αt, F

fx−Cx, t

≤ Nx, αt, 3.3

for allx∈Xandt >0, where

Mx, t E

ϕx, x, t

∗ E

ϕx,2x, t

∗ · · · ∗ E

ϕx,k−1x, t , Nx, t F

ϕx, x, t

♦F

ϕx,2x, t

♦ · · · ♦F

ϕx,k−1x, t

. 3.4

Proof. By induction onjwe will show that for eachx∈X, t >0 andj≥2

E f

jx

−jfx, t

≥ Mjx, t E

ϕx, x, t

∗ · · · ∗ E ϕ

x, j−1

x , t

, F

f jx

−jfx, t

≤ Njx, t F

ϕx, x, t

♦ · · · ♦F ϕ

x, j−1

x , t

. 3.5

Puttingxyin3.2, we obtain

E

f2x−2fx, t

≥ E

ϕx, x, t

, F

f2x−2fx, t

≤ F

ϕx, x, t

, 3.6

for allx∈Xandt >0. This proves3.5forj 2. Let3.5hold for somej >2. Replacingy byjxin3.2, we get

E f

j 1 x

−fx−f jx

, t

≥ E ϕ

x, jx , t

, F

f j 1

x

−fx−f jx

, t

≤ F ϕ

x, jx , t

, 3.7

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for eachx∈Xandt >0. Thus E

f j 1

x

− j 1

fx, t E

f j 1

x

−fx−f jx

f jx

−jfx, t

≥ E f

j 1 x

−fx−f jx

, t

∗ E f

jx

−jfx, t

≥ E ϕ

x, jx , t

∗ Mjx, t Mj 1x, t, F

f j 1

x

− j 1

fx, t F

f

j 1 x

−fx−f jx

f jx

−jfx, t

≤ F

f

j 1 x

−fx−f jx

, t

♦F f

jx

−jfx, t

≤ F ϕ

x, jx , t

♦Njx, t N_{j 1}x, t,

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for eachx∈Xandt >0. Hence3.5holds for allj≥2. In particular E

fkx−kfx, t

≥ Mx, t, F

fkx−kfx, t

≤ Nx, t. 3.9

Replacingxbyk⁻ⁿ⁻¹xin3.9and using3.1, we get E

f k⁻ⁿx

−kf

k^{−n 1}x , t

≥ M

x, α^{n 1}t , F

f k⁻ⁿx

−kf

k^{−n 1}x , t

≤ N

x, α^{n 1}t ,

3.10

for allx∈X,t >0 andn0,1,2, . . .. Therefore

E kⁿf

k⁻ⁿx

−k^{n 1}f

k^{−n 1}x , t

≥ M x,α^{n 1}t

|k|ⁿ

,

F kⁿf

k⁻ⁿx

−k^{n 1}f

k^{−n 1}x , t

≤ N x,α^{n 1}t

|k|ⁿ

,

3.11

for allx∈X,t >0 andn0,1,2, . . .. Since

mlim→ ∞M x,α^{m 1}t

|k|^m

1, lim

m→ ∞N x,α^{m 1}t

|k|^m

0, 3.12

so 3.11 shows thatkⁿfk⁻ⁿx is a Cauchy sequence in non-Archimedean intuitionistic fuzzy Banach spaceY,E,F. Therefore, we can define a mapping C : X → Y by Cx E,F−limn→ ∞kⁿfk⁻ⁿx. Hence

nlim→ ∞E kⁿf

k⁻ⁿx

−Cx, t

1, lim

n→ ∞F kⁿf

k⁻ⁿx

−Cx, t

0. 3.13

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For eachn≥1,x∈Xandt >0

E

fx−kⁿf k⁻ⁿx

, t

E ⁿ⁻¹

i0

kⁱf k⁻ⁱx

−k^{i 1}f

k^{−i 1}x , t

≥ⁿ⁻¹

i0

E kⁱf

k⁻ⁱx

−k^{i 1}f

k^{−i 1}x , t Mx, αt,

F

fx−kⁿf k⁻ⁿx

, t

≤ⁿ⁻¹

i0

F kⁱf

k⁻ⁱx

−k^{i 1}f

k^{−i 1}x , t Nx, αt,

3.14

where_n

j1a_ja₁∗a₂∗ · · · ∗a_nand_n

j1a_ja₁♦a2♦ · · · ♦an. It follows from3.13and3.14 that

E

fx−Cx, t

≥ E

fx−kⁿf k⁻ⁿx

, t

∗ E kⁿf

k⁻ⁿx

−Cx, t

≥ Mx, αt, F

fx−Cx, t

≤ F

fx−kⁿf k⁻ⁿx

, t

♦F kⁿf

k⁻ⁿx

−Cx, t

≤ Nx, αt, 3.15

for eachx ∈X, t > 0 and for suﬃciently largen; that is,3.3holds. Also, from3.1,3.2, and3.13, we have

E C

x y

−Cx−C y

, t

≥ E C

x y

−kⁿf k⁻ⁿ

x y

, t

∗ E kⁿf

k⁻ⁿx

−Cx, t

∗ E kⁿf

k⁻ⁿy

−C y

, t

∗ E kⁿf

k⁻ⁿ

x y

−kⁿf k⁻ⁿx

−kⁿf k⁻ⁿy

, t

≥ E

ϕ

k⁻ⁿx, k⁻ⁿy , t

|k|ⁿ

≥ E

ϕ x, y

,αⁿt

|k|ⁿ

, F

C x y

−Cx−C y

, t

≤ F C

x y

−kⁿf k⁻ⁿ

x y

, t

♦F kⁿf

k⁻ⁿx

−Cx, t

♦F kⁿf

k⁻ⁿy

−C y

, t

♦F kⁿf

k⁻ⁿ

x y

−kⁿf k⁻ⁿx

−kⁿf k⁻ⁿy

, t

≤ F

ϕ

k⁻ⁿx, k⁻ⁿy , t

|k|ⁿ

≥ F

ϕ x, y

,αⁿt

|k|ⁿ

,

3.16

for allx, y∈X,t >0 and for largen. Since

nlim→ ∞E

ϕ x, y

,αⁿt

|k|ⁿ

1, lim

n→ ∞F

ϕ x, y

,αⁿt

|k|ⁿ

0, 3.17

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which shows thatCis additive. Now ifC:X → Y is another additive mapping such that

E

Cx−fx, t

≥ Mx, t, F

Cx−fx, t

≥ Nx, t, 3.18

for allx∈Xandt >0. Then, for allx∈X, t >0 andn∈N, we have E

Cx−Cx, t

≥ E

Cx−kⁿf k⁻ⁿx

, t

∗ E kⁿf

k⁻ⁿx

−Cx, t

≥ E

C k⁻ⁿx

−f k⁻ⁿx

, t

|k|ⁿ

∗ E

f k⁻ⁿx

−C k⁻ⁿx

, t

|k|ⁿ

≥ M

k⁻ⁿx, αt

|k|ⁿ

≥ M x,α^{n 1}t

|k|ⁿ

, F

Cx−Cx, t

≤ F

Cx−kⁿf k⁻ⁿx

, t

♦F kⁿf

k⁻ⁿx

−Cx, t

≤ F

C k⁻ⁿx

−f k⁻ⁿx

, t

|k|ⁿ

♦F

f k⁻ⁿx

−C k⁻ⁿx

, t

|k|ⁿ

≥ N

k⁻ⁿx, αt

|k|ⁿ

≤ N x,α^{n 1}t

|k|ⁿ

.

3.19

Therefore

nlim→ ∞M x,α^{n 1}t

|k|ⁿ

1, lim

n→ ∞N x,α^{n 1}t

|k|ⁿ

0. 3.20

HenceCx Cxfor allx∈X.

Corollary 3.2. LetXbe a linear space over non-Archimedean fieldKand letY, · be a non-Archi- medean normed space. Suppose that a functionϕ:X×X → R satisfies

ϕ

k⁻¹x, k⁻¹y

≤α⁻¹ϕ x, y

, 3.21

for allx, y∈X, whereα >0 andkis an integer with|k|< α. If a mapf:X → Ysatisfies f

x y

−fx−f

y≤ϕ x, y

, 3.22

for allx, y∈X, then there exists a unique additive mappingC:X → Ysatisfies fx−Cx≤ 1

αmax

ϕx, x∗ϕx,2x∗ · · · ∗ϕx,k−1x

. 3.23

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Proof. Consider the non-Archimedean intuitionistic fuzzy norm

E y, t

⎧⎪

⎨

⎪⎩ t

t y ift >0,

0 ift≤0;

Fx, t

⎧⎪

⎨

⎪⎩ y

t y if t >0,

1 if t≤0,

3.24

onY. LetZRand let the functionE,F:R×R → 0,1be defined by

Ez, t

⎧⎨

⎩ t

t |z| if t >0, 0 if t≤0;

Fz, t

⎧⎨

⎩

|z|

t |z| ift >0, 1 ift≤0.

3.25

ThenE,Fis a non-Archimedean intuitionistic fuzzy norm onR. The result follows from the fact that3.21,3.22, and3.23are equivalent to3.1,3.2, and3.3, respectively.

Example 3.3. LetXbe a linear space over non-Archimedean fieldKand letY, · be a non- Archimedean normed space. Suppose that a functionf:X → Y satisfies

f x y

−fx−f

y≤ x ^p y^p, 3.26

for allx, y ∈Xandp ∈0,1. Suppose that there exists an integerksuch that|k|<1. Since p < 1, by applyingCorollary 3.2forϕx, y x ^p y ^p, we observe that3.21holds for α |k|^p. Inequality3.23assures the existence of a unique additive mappingC : X → Y such that

fx−Cx≤ 1 k−1^p

|k|^p x ^p, 3.27

for allx∈X.

4. Stability of Pexiderized Cauchy Functional Equation

The functional equationfx y gx hyis said to be Pexiderized Cauchy, wheref, g, andh are mappings between linear spaces. In the case f g h, it is called Cauchy functional equation.

Theorem 4.1. LetX be a linear space over a non-Archimedean fieldKand let Y,E,Fbe a non- Archimedean intuitionistic fuzzy Banach space. Suppose thatf,g, andhare mappings fromXtoY withf0 g0 h0 0. Suppose thatϕis a function fromX×X to a non-Archimedean IFN spaceZ,E,Fsuch that

E f

x y

−gx−h y

, t

≥ E ϕ

x, y , t

, F

f x y

−gx−h y

, t

≤ F ϕ

x, y , t

, 4.1

for allx, y∈Xandt >0. If E

ϕ

k⁻¹x, k⁻¹y , t

≥ E ϕ

x, y , αt

, F

ϕ

k⁻¹x, k⁻¹y , t

≤ F ϕ

x, y , αt

, 4.2

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for some positive real numberα >0 and some positive integerkwith|k|< α, then there exists a unique additive mappingC:X → Y such that

E

fx−Cx, t

≥ Mx, αt, F

fx−Cx, t

≤ Nx, αt, 4.3

E

gx−Cx, t

≥ Mx,min{1, α}t, F

gx−Cx, t

≤ Nx,min{1, α}t, 4.4 Ehx−Cx, t≥ Mx,min{1, α}t, Fhx−Cx, t≤ Nx,min{1, α}t, 4.5

for allx∈Xandt >0, where Mx, t E

ϕx, x, t

∗ · · · ∗ E

ϕx,k−1x, t

∗ E

ϕ0, x, t

∗ · · · ∗ E

ϕ0,k−1x, t

∗ E

ϕx,0, t

∗ · · · ∗ E

ϕk−1x,0, t , Nx, t F

ϕx, x, t

♦ · · · ♦F

ϕx,k−1x, t

♦F

ϕ0, x, t

♦ · · · ♦F

ϕ0,k−1x, t

♦F

ϕx,0, t

♦ · · · ♦F

ϕk−1x,0, t .

4.6

Proof. Puty0 in4.1. Then, for allx∈Xandt >0 E

fx−gx, t

≥ E

ϕx,0, t

, F

fx−gx, t

≤ F

ϕx,0, t

. 4.7

Forx0,4.1becomes E

f y

−h y

, t

≥ E ϕ

0, y , t

, F

f y

−h y

, t

≤ F ϕ

0, y , t

, 4.8

for ally∈Xandt >0. Combining4.1,4.7, and4.8, we obtain E

f x y

−fx−f y

, t

≥ E ϕ

x, y , t

∗ E

ϕx,0, t

∗ E ϕ

0, y , t

, F

f x y

−fx−f y

, t

≤ F ϕ

x, y , t

♦F

ϕx,0, t

♦F ϕ

0, y , t

, 4.9

for each x, y ∈ X and t > 0. Replacing Eϕx, y, t and Fϕx, y, t by Eϕx, y, t ∗ Eϕx,0, t ∗ Eϕ0, y, t and Fϕx, y, t♦Fϕx,0, t♦Fϕ0, y, t, respectively, in Theorem 3.1, we can find that there exists a unique additive mappingC:X → Ythat satisfies 4.3. From4.3and4.7, we see that

E

gx−Tx, t

≥ E

gx−fx, t

∗ E

fx−Tx, t

≥ Mx, t, F

gx−Tx, t

≤ F

gx−fx, t

♦F

fx−Tx, t

≤ Nx, t, 4.10

for allx, y∈Xandt >0, which proves4.4. Similarly, we can prove4.5.

Corollary 4.2. LetXbe a linear space over a non-Archimedean fieldKand letZ,E,Fbe a non- Archimedean IFN space. Let Y,E,F be a non-Archimedean intuitionistic fuzzy Banach space.

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Suppose thatf,gandhare functions fromXtoY such thatf0 g0 h0 0, and there is an integerkwith|k|<1 and satisfies

E f

x y

−gx−h y

, t

≥ E

x ^ry^sz_◦, t , F

f x y

−gx−h y

, t

≤ F

x ^ry^sz_◦, t ,

4.11

for allx, y∈X, t >0 and for some fixedz_◦∈Zandr, s≥0 withr s <1. Then there exists a unique additive mappingT :X → Y such that

E

fx−Tx, t

≥ E

k−1^s x ^{r s}z_◦,|k|^{r s}t , F

fx−Tx, t

≤ F

k−1^s x ^{r s}z_◦,|k|^{r s}t , E

gx−Tx, t

≥ E

k−1^s x ^{r s}z_◦,|k|^{r s}t , F

gx−Tx, t

≤ F

k−1^s x ^{r s}z_◦,|k|^{r s}t , Ehx−Tx, t≥ E

k−1^s x ^{r s}z_◦,|k|^{r s}t , Fhx−Tx, t≤ F

k−1^s x ^{r s}z_◦,|k|^{r s}t ,

4.12

for allx∈Xandt >0.

Proof. Let the functionϕ:X×X → Zbe defined byϕx, y x ^r y ^sz₀for allx, y∈Xand z_◦is a fixed unit vector inZ. Then4.1holds. Since

E ϕ

k⁻¹x, k⁻¹y , t

Ek⁻¹x^rk⁻¹y^s z_◦, t

E

x ^ry^s

z_◦,|k|^{r s}t , F

ϕ

k⁻¹x, k⁻¹y , t

F

x ^ry^s

z_◦,|k|^{r s}t ,

4.13

for eachx, y ∈ X andt > 0. Ifα |k|^{r s}andr s < 1, thenα > |k|holds. It follows from Theorem 4.1that there exists a unique additive mappingC : X → Y such that4.3–4.5 hold.

5. Stability of Jensen Functional Equation

The stability problem for the Jensen functional equation was first proved by Kominek13 and since then several generalizations and applications of this notion have been investigated by various authors, namely, Jung12, Mohiuddine23, Parnami and Vasudeva41, and many others. The Jensen functional equation is 2fx y/2 fx fy, wheref is a mapping between linear spaces. It is easy to see that a mappingf :X → Y between linear spaces withf0 0 satisfies the Jensen equation if and only if it is additivecf.41.

Theorem 5.1. LetXbe a linear space over a non-Archimedean fieldKand letZ,E,Fbe a non- Archimedean IFN space. Suppose thatϕ :X×X → Zis a function such that for someα > 0 and

(11)

some positive integer k with |k| < αsatisfies 3.1. Suppose thatY,E,F is a non-Archimedean intuitionistic fuzzy Banach space. If a mapf:X → Y satisfies

E

2f x y

2

−fx−f y

, t

≥ E ϕ

x, y , t

, F

2f x y

2

−fx−f y

, t

≤ F ϕ

x, y , t

,

5.1

for allx, y∈Xandt >0, then there exists a unique additive mappingC:X → Y such that

E

fx−f0−Cx, t

≥ Mx, αt, F

fx−f0−Cx, t

≤ Nx, αt, 5.2

Mx, t E

ϕx, x, t

∗ E

ϕx,2x, t

∗ · · · ∗ E

ϕx,k−1x, t

∗ E

ϕ2x,0, t

∗ E

ϕ3x,0, t

∗ · · · ∗ E

ϕkx,0, t

, 5.3

Nx, t F

ϕx, x, t

♦F

ϕx,2x, t

♦ · · · ♦F

ϕx,k−1x, t

♦F

ϕ2x,0, t

♦F

ϕ3x,0, t

♦ · · · ♦F

ϕkx,0, t

. 5.4

Proof. Suppose thatgx fx−f0for allx∈X. Then

E

2g x y

2

−gx−g y

, t

≥ E ϕ

x, y , t

, F

2g

x y 2

−gx−g y

, t

≤ F ϕ

x, y , t

,

5.5

for allx, y ∈X andt >0. Replacingxbyx yandyby 0 in5.5, then, for allx, y ∈Xand t >0, we have

E

2g x y

2

−g x y

, t

≥ E ϕ

x y,0 , t

, F

2g

x y 2

−g x y

, t

≤ F ϕ

x y,0 , t

.

5.6

From5.5and5.6, we conclude that E

g x y

−gx−g y

, t

≥ E ϕ

x, y , t

∗ E ϕ

x y,0 , t

, F

g x y

−gx−g y

, t

≤ F ϕ

x, y , t

♦F ϕ

x y,0 , t

, 5.7

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for allx, y∈X andt > 0. Proceeding the same lines as in the proof ofTheorem 3.1, one can show that there exists a unique additive mappingC:X → Y such that

E

fx−f0−Cx, t E

gx−Tx, αt

≥ Mx, t, F

fx−f0−Cx, t F

gx−Tx, αt

≤ Nx, t, 5.8

for allx∈Xandt >0.

6. Stability of Pexiderized Jensen Functional Equation

The functional equation 2fx y/2 gx hyis said to be Pexiderized Jensen, where f,g, and hare mappings between linear spaces. In the case f g h, it is called Jensen functional equation.

Theorem 6.1. LetX be a linear space over a non-Archimedean fieldKand let Y,E,Fbe a non- Archimedean intuitionistic fuzzy Banach space. Suppose thatf,g, andhare mappings from X to Y withf0 g0 h0 0. LetZ,E,Fbe non-Archimedean IFN space. Suppose thatϕ : X×X → Zis a function such that for someα >0, and some positive integerkwith|k|< αsatisfies 3.1and inequality

E

2f x y

2

−gx−h y

, t

≥ E ϕ

x, y , t

, F

2f

x y 2

−gx−h y

, t

≤ F ϕ

x, y , t

,

6.1

for allx, y∈Xandt >0. Then there exists a unique additive mappingC:X → Y such that

E

fx−Cx, t

≥ Mx, αt, F

fx−Cx, t

≤ Nx, αt, 6.2

E

gx−Cx, t

≥ M x

2,αt 2

∗ E

ϕx,0, t , F

gx−Cx, t

≤ N x

2,αt 2

♦F

ϕx,0, t ,

6.3

Ehx−Cx, t≥ M x

2,αt 2

∗ E

ϕ0, x, t , Fhx−Cx, t≤ N

x 2,αt

2

♦F

ϕ0, x, t ,

6.4

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Mx, t ^k−1

m1

E

ϕx, mx,|2|t

∗ E

ϕmx, mx,|2|t

∗^k

m0

E

ϕmx,0,|2|t

∗ E

ϕ0, mx,|2|t ,

Nx, t ^k−1

m1

E

ϕx, mx,|2|t

♦E

ϕmx, mx,|2|t

♦^k

m0

E

ϕmx,0,|2|t

♦E

ϕ0, mx,|2|t .

6.5

Proof. Putyxin6.1. Then, for allx∈Xandt >0

E

2fx−gx−hx, t

≥ E

ϕx, x, t , F

2fx−gx−hx, t

≤ F

ϕx, x, t

. 6.6

Replacingxbyyin6.1, we get

E 2f

y

−g y

−h y

, t

≥ E ϕ

y, y , t

, F

2f y

−g y

−h y

, t

≤ F ϕ

y, y , t

, 6.7

for ally∈Xandt >0. Again replacingxbyyas well asybyxin6.1, we get

E

2f x y

2

−g y

−hx, t

≥ E ϕ

y, x , t

, F

2f

x y 2

−g y

−hx, t

≤ F ϕ

y, x , t

,

6.8

for allx, y∈Xandt >0. It follows from6.1and6.6–6.8that

E

4f x y

2

−2fx−2f y

, t

≥ E

ϕx, x, t

∗ E ϕ

x, y , t

∗ E ϕ

y, y , t

∗ E ϕ

y, x , t

, F

4f

x y 2

−2fx−2f y

, t

≤ F

ϕx, x, t

♦F ϕ

x, y , t

♦F ϕ

y, y , t

♦F ϕ

y, x , t

.

6.9

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Thus, for allx, y∈Xandt >0,

E

2f x y

2

−fx−f y

, t

≥ E

ϕx, x,|2|t

∗ E ϕ

x, y ,|2|t

∗ E ϕ

y, y ,|2|t

∗ E ϕ

y, x ,|2|t

, F

2f

x y 2

−fx−f y

, t

≤ F

ϕx, x,|2|t

♦F ϕ

x, y ,|2|t

♦F ϕ

y, y ,|2|t

♦F ϕ

y, x ,|2|t

.

6.10

Proceeding the same argument used inTheorem 5.1shows that there exists a unique additive mappingC:X → Y such that6.2holds. Therefore

E 2fx

2

−Cx, t

≥ M x

2,αt 2

, F

2fx 2

−Cx, t

≤ N x

2,αt 2

, 6.11

for allx∈Xandt >0. Puty0 in6.1, we get

E 2fx

2

−gx, t

≥ E

ϕx,0, t

, F

2fx 2

−gx, t

≤ F

ϕx,0, t

, 6.12

for allx ∈ X andt > 0. It follows from6.11and6.12that6.3holds. Similarly we can show that6.4holds.

Corollary 6.2. LetXbe a non-Archimedean normed space. Suppose thatf, g, h:X → Y such that f0 g0 h0 0, and there is an integerkwith|k|<1 and satisfies

2f x y

2

−gx−h

y≤, 6.13

for allx, y∈X. Then there exists a unique additive mappingC:X → Ysuch that

fx−Cx≤, gx−Cx≤, hx−Cx ≤, 6.14

for allx∈X.

Proof. Let the functionE,F:Y×R → 0,1be defined by

Ex, t

⎧⎨

⎩ t

t x if t >0, 0 if t≤0;

Fx, t

⎧⎨

⎩ x

t x ift >0, 1 ift≤0,

6.15

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onY. It is easy to see thatY,E,Fis a non-Archimedean intuitionistic fuzzy Banach space.

Consider the non-Archimedean intuitionistic fuzzy norm

Ez, t

⎧⎨

⎩ t

t |z| if t >0, 0 if t≤0;

Fz, t

⎧⎨

⎩

|z|

t |z| ift >0, 1 ift≤0.

6.16

ThenE,Fis a non-Archimedean intuitionistic fuzzy norm onR. It is easy to see that4.1 holds forϕx, y and α 1 satisfies 3.1. Therefore the condition of Theorem 6.1is fulfilled. Hence there exists a unique additive mappingC:X → Ysuch that6.14holds.

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Equations in Non-Archimedean Intuitionistic Fuzzy Normed Spaces

Research Article

Stability of Various Functional

Equations in Non-Archimedean Intuitionistic Fuzzy Normed Spaces

Syed Abdul Mohiuddine, Abdullah Alotaibi, and Mustafa Obaid

1. Introduction

2. Non-Archimedean Intuitionistic Fuzzy Normed Space

3. Stability of Cauchy Functional Equation

4. Stability of Pexiderized Cauchy Functional Equation

5. Stability of Jensen Functional Equation

6. Stability of Pexiderized Jensen Functional Equation

References

Submit your manuscripts at http://www.hindawi.com

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Complex Analysis

Optimization

Combinatorics

Function Spaces

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