1,3 Department of Mathematics, Sacred Heart College, Tirupattur - 635 601,
TamilNadu, India. 2 Departamento de Ciencias Exatas e Naturais, Academia Militar,
2720-113 Amadora, Portugal.
Abstract
The aim of present paper is to obtain some results for the stability of viginti unus functional equation in matrix non-archimedean fuzzy normed spaces and matrix normed spaces by using the fixed point method.
1. INTRODUCTION
In 1940, an interesting talk presented by S. M. Ulam [20] triggered the study of stability problems for various functional equations. He raised a question concerning the stability of homomorphism. In the following year 1941, D. H. Hyers [8] was able to give a partial solution to Ulam’s question. The result of Hyers was generalized by Aoki [4] for additive mappings. In 1978, Th. M. Rassias [15] succeeded in extending the result of Hyers theorem by weakening the condition for the Cauchy difference. In 1982 J. M. Rassias [16] solved the Ulam problem for different mappings and for many Euler-Lagrange type quadratic mappings by involving a product of different powers of norms. In 1994, a generalization of the Rassias theorem was obtained by Gavruta [7] by
replacing the unbounded Cauchy difference by a general control function. A further generalization of the Hyers-Ulam stability for a large class of mapping was obtained by Isac and Th. M. Rassias [9]. They also presented some applications in non-linear analysis, especially in fixed point theory. This terminology may also be applied to the cases of other functional equations [1, 2, 6, 10, 17]. Mihet and Radu [12] investigated the stability in the settings of fuzzy, probabilistic and random normed spaces. The cocept of non-Archimedean fuzzy normed spaces has been introduced by Mirmostafafe and Moslehian [14]. Quite recently, the new results on stability of functional equations in non-Archimedean fuzzy normed spaces [3, 13, 18, 19].
In this paper, we introduce the following new functional equation
where 00000000005109094217=21! in matrix normed spaces by using the fixed point method. The above functional equation is said to be viginti unus functional equation since the function 21=)( cxxf is its solution.
In this paper, we study the general solution of the functional equation (1) and we also investigate the Ulam-Hyers stability of the functional equation (1) in matrix non-Archimedean fuzzy normed spaces by using fixed point approach.
2. PRELIMINARIES
In this section, we firstly restate the usual terminology, notations and conventions of the theory of non-Archimedean fuzzy normed space and we introduce the new concept of matrix non-Archimedean fuzzy normed spaces.
Definition 1. [3] Let X be a linear space over a non-Archimedean field K . A function [0,1]: RXN is said to be a non-Archimedean fuzzy norm on X if for all
Xyx , and all Rt .
THE STABILITY OF VIGINTI UNUS FUNCTIONAL EQUATION IN VARIOUS SPACES 5737
(N1) 0=),( txN for 0t ;
(N2) 1=),(0= txNx for all 0>t ;
(N3) ),( tcxN = ),(c
txN if 0c ;
(NA4) ,( yxN max )ts min ),(),,( tyNsxN ;
(NA5) 1=),(lim txNt
;
The pair ),( NX is called a non-Archimedean fuzzy normed space. Clearly, if 4)(NA holds then so is
(N4) ),( tsyxN min ),(),,( tyNsxN
A classical vector space over a complex or real field satisfying 1)(N and 5)(N is called fuzzy normed space. It is easy to see that 4)(NA is equivalent to the following condition
(NA4 ' ) ),( tyxN min );,(),(,),( RtXyxtyNtxN .
We will use the following notations:
)(XM n is the set of all nn -matrices in X ;
)(1, Cnj Me is that jth component is 1 and the other components are zero ;
)(Cnij ME is that (i,j)-component is 1 and the other components are zero;
)(XMxE nij is that (i,j)-component is x and the other components are
zero. For )(),( XMyXMx kn ,
yx =
y
x
00
.
Note that ).,(n
X is a matrix normed space if and only if ).),((nn XM is a normed
space for each positive integer n and nk
xBAAxB holds for )(, CnkMA ,
)(, CknMB and )()(= XMxx nij , and that ).,(n
X is a matrix Banach space if
5738 R. Murali, Sandra Pinelas and V. Vithya
and only if X is a Banach space and ).,(n
X is a matrix normed space. A matrix
normed space ).,(n
X is called an L -matrix normed space if
knkn
yxmaxyx ,=
holds for all )(XMx n and all ).(XMy k Let FE,
be vector space. For a given mapping FEh : and a given positive integer n , define )()(: FMEMh nnn by, )]([=])([ ijijn xhxh for all ).(][ EMx nij
Definition 2. Let ),( NX be a non-Archimedean fuzzy normed space.
(M1) ),( nNX is called a matrix non-Archimedean fuzzy normed space if for each
positive integer n , )),(( nn NXM is a non-archimedean fuzzy normed space and
BA
txNtAxBN nk .,),( for all 0>t , )(, RnkMA , )(, RknMB and
)(][= XMxx nij with 0. BA .
(M2) ),( nNX is called a complete matrix non-Archimedean fuzzy normed space if
),( NX is a non-Archimedean fuzzy Banach space and ),( nNX is a matrix
non-archimedean fuzzy normed space.
3. GENERAL SOLUTION OF VIGINTI UNUS FUNCTIONAL EQUATION (1)
In this section, we study the general solution of viginti unus functional equation (1). For this, let us consider A and B be real vector spaces.
Theorem 3. If a mapping BA :f satisfies the functional equation (1) for all
,, Ayx then )(2=)(2 21 xfxf for all Ax .
Proof. Letting 0== yx in (1), one gets 0=(0)f . Replacing 0=x , xy = and xx = , xy = in (1) and adding the two resulting equations, we get )(=)( xfxf .
Hence, f is an odd mapping. Replacing 0=x , xy 2= and xx 11= , xy = in (1) and subtracting the two resulting equations, we get
)(1720349)(185796)(191330)(20230)(2121 xfxfxfxfxf
THE STABILITY OF VIGINTI UNUS FUNCTIONAL EQUATION IN VARIOUS SPACES 5739
for all .Ax Multiplying (23) by 646646 , and then subtracting (22) from the
resulting equation, we can obtain that )(2=)(2 21 xfxf for all Ax . This completes
the proof.
4. ULAM-HYERS STABILITY OF VIGINTI UNUS FUNCTIONAL
EQUATION (1) IN MATRIX NON-ARCHIMEDEAN FUZZY NORMED
SPACES
In this section, we will investigate the Ulam-Hyers stability for the functional equation (1) in matrix non-Archimedean fuzzy normed spaces by using the fixed point method.
Throughout this section, we assume that K be a non-Archimedean field, X is a vector space over K and ( , )nY N is an complete matrix non-Archimedean fuzzy
normed space over K , and ),( NZ is (an Archimedean or a non-Archimedean fuzzy) normed space and minimum is denoted by T. For a mapping YXf : , define
YXf 2:G and )()(: 2 YMXMf nnn G by,
)7(5985)8(1330)9(210)10(21)11(=),( bafbafbafbafbafbaf G
THE STABILITY OF VIGINTI UNUS FUNCTIONAL EQUATION IN VARIOUS SPACES 5755
0=))((221!))10((2))9((221))8((2210 yyxyxyx nnnn
From the definition of f and (54), we obtain that
))9((2210))10((221))11((221),( 210=
yxyxyxyxDf nnn
nn
))6((220349))7((25985))8((21330 yxyxyx nnn
))3((2203490))4((2116280))5((254264 yxyxyx nnn
))((2352716))((2352716))2((2293930 xyxyx nnn
))3((2116280))2((2203490))((2293930 yxyxyx nnn
))6((25985))5((220349))4((254264 yxyxyx nnn
))9((221))8((2210))7((21330 yxyxyx nnn
))((221!))10((2 yyx nn
(2097151)2)00000000005109094217(2097152)(=
20)7000000000(510909421
2121= knkn
).((2097152)2097151
0)7000000000(510909421 21212 yx
Therefore f satisfies (52) for all Ryx, Now, We find that the viginti unus
functional equation (1) is not stable for 21=t in corollary 1. Suppose on the contrary
that there exists a viginti unus mapping RRVU : and a constant 0> satisfying
(53). Then there exists a constant Rc such that 21=)( cxxUV for any Rx . Thus
we obtain the following inequality.
21)()( xcxf (55)
Let Nm with cm > . If )2
1(0, 1m
x , then (0,1)2 xn for all
10,1,2,...,= mn .
5756 R. Murali, Sandra Pinelas and V. Vithya
For this x , we get 212121
211
0=21
0=)(>=
2)(2
2)(2=)( xcxm
xxxf
n
nm
nn
n
n
which contradicts (55). Therefore the viginti unus functional equation (1) is not stable for 21.=p
The following examples illustrates the fact that functional equation (1) is not stable for
21== wvp (when 221=,
221= wv ) in corollary 2.
Example 7. Let RR: be a function defined by (50) and define a function
RR:f by (51). Then f satisfies the inequality
).((2097152)2097151
0)7000000000(510909421),( 221
221
2 yxyxDf (56)
for all Ryx, . Then there do not exists a viginti unus mapping RRVU : and a
constant 0> such that (53).
Proof. The proof is analogous to the proof of Example 6.
The following examples illustrates the fact that functional equation (1) is not stable for
21== wvp (when 221=,
221= wv ) in corollary 3.
Example 8. Let RR: be a function defined by (50) and define a function
RR:f by (51). Then f satisfies the inequality
))(.((2097152)2097151
0)7000000000(510909421),( 2121221
221
2 yxyxyxDf (57)
for all Ryx, . Then there do not exists a viginti unus mapping RRVU : and a
constant 0> such that (53).
Proof. The proof is analogous to the proof of Example 6.
THE STABILITY OF VIGINTI UNUS FUNCTIONAL EQUATION IN VARIOUS SPACES 5757
CONCLUSION
In this investigation, we identified a general solution of a viginti unus functional equation (1) and established the Ulam -Hyers stability of the functional equation (1) in matrix non-Archimedean fuzzy normed spaces and matrix normed spaces by using the fixed point method. Also we illustrate the examples for non-stability.
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