stability of a leibniz type additive and quadratic ... · functional equation in intuitionistic...
TRANSCRIPT
![Page 1: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/1.jpg)
Advances in Theoretical and Applied Mathematics ISSN 0973-4554 Volume 11, Number 2 (2016), pp. 145-169 © Research India Publications http://www.ripublication.com
Stability Of A Leibniz Type Additive And Quadratic Functional Equation In Intuitionistic Fuzzy Normed
Spaces
M. Arunkumar 1 , John M. Rassias 2 and S. Karthikeyan 3
1 Department of Mathematics, Government Arts College, Tiruvannamalai-606 603, Tamil Nadu, India
E-mail:[email protected] 2 Pedagogical Department E.E., Section of Mathematics and Informatics,
National and Capodistrian University of Athens, 4, Agamemnonos Str., Aghia Paraskevi, Athens 15342, Greece.
E-mail: [email protected] URL: http://users.uoa.gr/~jrassias 3 Department of Mathematics, R.M.K. Engineering College,
Kavaraipettai-601 206, Tamil Nadu, India E-mail:[email protected]
Abstract
In this paper, the authors investigate the generalized Ulam-Hyers stability of a Leibniz type additive and quadratic functional equation
⎟⎠⎞
⎜⎝⎛ −−+⎟
⎠⎞
⎜⎝⎛ −++−+−+−
32
33=)()()( zyxftzyxftzftyftxf
⎟⎠⎞
⎜⎝⎛ +−−+⎟
⎠⎞
⎜⎝⎛ −+−+
32
32 zyxfzyxf
in the setting of intuitionistic fuzzy normed spaces using direct and fixed point methods.
1. INTRODUCTION The stability problem of functional equations was first described by S.M. Ulam [43]. This topic further addressed by D.H. Hyers [23] and then generalized by T. Aoki [2], Th.M. Rassias [35] and J.M. Rassias [33] for additive and linear mappings. Further generalizations on the above stability results was have been described in references [15, 20, 21, 37]. Since then, several stability problems for various functional
![Page 2: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/2.jpg)
146 M. Arunkumar et al
equations have been investigated in referenes [1, 3-13, 16, 24, 31, 34, 36, 44]. Various fuzzy stability results concerning Cauchy, Jensen, quadratic and cubic functional equations have also been discussed in studies such that [18, 19, 26-29, 40-42]. Recently, Matina J. Rassias et. al., [32] introduced the Leibniz type additive-quadratic functional equation of the form
⎟⎠⎞
⎜⎝⎛ −−+⎟
⎠⎞
⎜⎝⎛ −++−+−+−
32
33=)()()( zyxftzyxftzftyftxf
⎟⎠⎞
⎜⎝⎛ +−−+⎟
⎠⎞
⎜⎝⎛ −+−+
32
32 zyxfzyxf (1.1)
and obtained its general solution and generalized Ulam-Hyers stability of Leibniz AQ-mixed type functional equation in quasi-beta normed space using direct and fixed point methods. The solution of the Leibniz type additive and quadratic functional equation (1.1) is given in the following lemmas. Lemma 1.1 [32] If an odd function YXf →: satisfies the functional equation (1.1) then f is additive. Lemma 1.2 [32] If an even function YXf →: satisfies the functional equation (1.1) then f is quadratic. In this paper, the authors investigate the Generalized Ulam-Hyers stability of a Leibniz type additive and quadratic functional equation (1.1), using direct and fixed point methods, in the setting of intuitionistic fuzzy normed spaces. 2. PRELIMINARIES OF INTUITIONISTIC FUZZY NORMED SPACES In this section, some preliminaries about intuitionistic fuzzy normed space is given. Lemma 2.1 [17] Consider the set ∗L and the order relation ∗≤
L defined by:
( ) ( ) [ ]{ },10,1,:,= 212
2121 ≤+∈∗ xxandxxxxL ( ) ( ) ( ) ( ) ∗
∗ ∈∀≥≤⇔≤ LyyxxyxyxyyxxL 212122112121 ,,,,,,,
Then ( )∗∗ ≤
LL , is a complete lattice.
Definition 2.2 [14] An intuitionistic fuzzy set ηζ ,A in a universal set U is an object
( ) ( )( ){ }UuuuA AA ∈ηζηζ ,=, for all ( ) [ ]0,1, ∈∈ uUu Aζ and ( ) [ ]0,1∈uAη are called the membership degree and the non-membership degree, respectively, of u in ηζ ,A . Furthermore, they satisfy
( ) ( ) 1≤+ uuAA ηζ .
![Page 3: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/3.jpg)
Stability Of A Leibniz Type Additive And Quadratic Functional Equation 147
We denote its units by ( )0,1=0 ∗L and ( )1,0=1 ∗L
. Classically, a triangular norm
[ ]0,1on=* T is defined as an increasing, commutative, associative mapping [ ] [ ]0,10,1: 2 →T satisfying ( ) xxxT =*1=1, for all [ ]0,1∈x . A triangular conorm ◊=S is defined as an increasing, commutative, associative mapping [ ] [ ]0,10,1: 2 →S
satisfying ( ) xxxS =0=0, ◊ for all [ ]0,1∈x . Using the lattice ( )∗∗ ≤
LL , , these
definitions can be directly extended. Definition 2.3 [14] A triangular norm ( −t norm) on ∗L is a mapping ( ) ∗∗ → LLT
2:
satisfying the following conditions: • ( ) ( )( )xxTL
L=,1 ∗
∗∈∀ (boundary condition);
• ( ) ( )( ) ( ) ( )( )xyTyxTLyx ,=,,2∗∈∀ (commutativity);
• ( ) ( )( ) ( )( ) ( )( )( )zyxTTzyTxTLzyx ,,=,,,,3∗∈∀ (associativity);
• ( ) ( )( ) ( ) ( )( )yxTyxTyyandxxLyyxxLLL
′′≤⇒′≤′≤∈′′∀ ∗∗∗∗ ,,,,,
4 (monotonicity).
If ( )TL
L,, ∗
∗ ≤ is an Abelian topological monoid with unit ∗L1 , then ∗L is said to be a
continuous −t norm. Definition 2.4 [14] A continuous −t norms T on ∗L is said to be continuous −t representable if −t there exist a continuous −t norm and a continuous
−t conorm◊ on [0, 1] such that, for all ( ) ( ) ∗∈ Lyyyxxx 2121 ,=,,= , ( ) ( ).,*=, 2211 yxyxyxT ◊
For example, ( ) { }( ),1min,=, 2211 bababaT +
and ( ) { } { }( )2211 ,max,,min=, bababaM
for all ( ) ( ) ∗∈ Lbbbaaa 2121 ,=,,= are continuous −t representable. Now, we define a sequence nT recursively by TT =1 and
( ) ( )( ) ( ) ( )( ) ( )( ) ( ) .2,,,,,=,, 11111 ∗+−+ ∈≥∀ LxnxxxTTxxT innnnn KK Definition 2.5 [40] A negator on ∗L is any decreasing mapping ∗∗ → LLN : satisfying
( ) ∗∗ LLN 1=0: and ( ) ∗∗ LL
N 0=1 . If ( )( ) xxNN = for all ∗∈ Lx , then N is called an
involutive negator. A negator on [ ]0,1 is a decreasing mapping [ ] [ ]0,10,1: →N satisfying ( ) 1=0,νμP and ( ) sNP 0.=1,νμ denotes the standard negator on [ ]0,1 defined by ( ) [ ].0,1,1= ∈∀− xxxNs
![Page 4: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/4.jpg)
148 M. Arunkumar et al
Definition 2.6 [40] Let μ and ν be membership and nonmembership degree of an intuitionistic fuzzy set from ( )+∞× 0,X to [ ]0,1 such that ( ) ( ) 1≤+ tt xx νμ for all
Xx ∈ and all 0>t . The triple ( )TPX ,, ,νμ is said to be an intuitionistic fuzzy normed space (briefly IFN-space) if X is a vector space, T is a continuous −t representable and νμ ,P is a mapping ( ) ∗→+∞× LX 0, satisfying the following conditions: for all
Xyx ∈, and 0>, st , • ( ) ;0=,0, ∗L
xP νμ
• ( ) 0;=ifonlyandif1=,, xtxPL∗νμ
• ( ) 0;allfor,=, ,, ≠⎟⎟⎠
⎞⎜⎜⎝
⎛α
αα νμνμ
txPtxP
• ( ) ( ) ( )( ).,,,, ,,, syPtxPTstyxPL νμνμνμ ∗≥++
In this case, νμ ,P is called an intuitionistic fuzzy norm. Here, ( ) ( ) ( )( ).,=,, tttxP xx νμνμ Example 2.7 [40] Let ( )⋅,X be a normed space. Let ( ) ( )( ),1min,,=, 22 bababaT +
for all ( ) ( ) ∗∈ Lbbbaaa 2121 ,=,,= and νμ , be membership and non-membership degree of an intuitionistic fuzzy set defined by
( ) ( ) ( )( ) .,,=,=,,+∈∀⎟
⎟⎠
⎞⎜⎜⎝
⎛
++Rt
xtx
xtttvttxP xxv μμ
Then ( )TPX ,, ,νμ is an IFN-sapce. Definition 2.8 [40] A sequence }{ nx in an IFN-space ( )TPX ,, ,νμ is called a Cauchy sequence if, for any 0>ε and 0>t , there exists Nn ∈0 such that
( ) ( )( ) ,,,,>, 0, nmnNLtxxP smn ≥∀− ∗ εενμ where sN is the standard negator. Definition 2.9 [40] The sequence }{ nx is said to be convergent to a point Xx ∈
(denoted by ),
xxP
n
νμ→ if ( ) ∗→−
Ln txxP 1,,νμ as ∞→n for every 0>t .
Definition 2.10 [40] An IFN-space ( )TPX ,, ,νμ is said to be complete if every Cauchy sequence in X is convergent to a point Xx ∈ . Throughout this paper, we assume that X is a linear space, ( )TPZ ,', ,νμ is an IFN-space and ( )TPY ,', ,νμ is a complete IFN-space.
![Page 5: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/5.jpg)
Stability Of A Leibniz Type Additive And Quadratic Functional Equation 149
3. STABILITY RESULTS: DIRECT METHOD In this section, the authors present the generalized Ulam-Hyers stability of the Leibniz additive-quadratic functional equation (1.1) in intuitionistic fuzzy normed spaces. Now we use the following notation for a given mapping YXDf →: such that
⎟⎠⎞
⎜⎝⎛ −++−−+−+− tzyxftzftyftxftzyxDf
33)()()(=),,,(
⎟⎠⎞
⎜⎝⎛ +−−−⎟
⎠⎞
⎜⎝⎛ −+−−⎟
⎠⎞
⎜⎝⎛ −−−
32
32
32 zyxfzyxfzyxf
for all .,,, Xtzyx ∈ Theorem 3.1 Let 1}{1,−∈τ . Let ZXXXX →×××:σ be a function such that for
some 1<2
<0τ
⎟⎠⎞
⎜⎝⎛ a ,
( )( ) ( )( )rxxxxaPrxxxxPL
,,,, ',,2,2,22' ,, σσ τνμ
ττττνμ ∗≥ (3.1)
for all Xx ∈ and all 0>r and ( )( ) ∗
∞→ L
nnnnn
nrtzyxP 1=,2,2,2,22'lim ,
τττττνμ σ (3.2)
for all Xtzyx ∈,,, and all 0>r . Let YXfo →: be an odd function satisfies the inequality
( ) ( )( )rtzyxPrtzyxDfPLo ,,,,'),,,,( ,, σνμνμ ∗≥ (3.3)
for all Xtzyx ∈,,, and all 0>r . Then the limit
0> , ,1,2
)(2)(, rnasrxfxAP Ln
no ∞→→⎟⎟
⎠
⎞⎜⎜⎝
⎛− ∗
νμ (3.4)
exists for all Xx ∈ and the mapping YXA →: is a unique additive mapping satisfying (1.1) and
( ) ( )( ) ( )( )raxxPrxAxfPLo |2|,,0,0,2', ,, −≥− ∗ σνμνμ (3.5)
for all Xx ∈ and all 0>r . Proof. Let 1=τ . Since of is an odd function, replacing ),,,( tzyx by ,0,0),(2 xx in (3.3), we get
( ) ( )( )rxxPrxfxfPLoo ,,0,0,2'),(2)(2 ,, σνμνμ ∗≥− (3.6)
for all Xx ∈ and all 0>r . Using 3)(IFN in (3.6), we obtain
( )( )rxxPrxfxfPLo
o ,,0,0,2'2
),(2
)(2,, σνμνμ ∗≥⎟
⎠⎞
⎜⎝⎛ − (3.7)
for all Xx ∈ and all 0>r . Replacing x by xn2 in (3.7), we have
( )( )rxxPrxfxfP nnL
no
no ,,0,0,22'
2),(2
2)(2 1
,
1
,+
∗
+
≥⎟⎟⎠
⎞⎜⎜⎝
⎛− σνμνμ (3.8)
for all Xx ∈ and all 0>r . Using (1), 3)(IFN in (3.8), we arrive
![Page 6: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/6.jpg)
150 M. Arunkumar et al
( ) ⎟⎠⎞
⎜⎝⎛≥⎟⎟
⎠
⎞⎜⎜⎝
⎛− ∗
+
nLn
o
no
arxxPrxfxfP ,,0,0,2'
2),(2
2)(2
,
1
, σνμνμ (3.9)
for all Xx ∈ and all 0>r . It is easy to verify from (3.9), that
( ) ⎟⎠⎞
⎜⎝⎛≥⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅
− ∗+
+
nLnn
no
n
no
arxxPrxfxfP ,,0,0,2'
22,
2)(2
2)(2
,1
1
, σνμνμ (3.10)
holds for all Xx ∈ and all 0>r . Replacing r by ran in (3.10), we get
( )( )rxxPraxfxfPLn
n
n
no
n
no ,,0,0,2'
22 ,
2)(2
2)(2
,1
1
, σνμνμ ∗+
+
≥⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
− (3.11)
for all Xx ∈ and all 0>r . It is easy to see that
i
io
i
io
n
ion
no xfxfxfxf
2)(2
2)(2=)(
2)(2
1)(
11
0=−− +
+−
∑ (3.12)
for all Xx ∈ . From equations (3.11) and (3.12), we have
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
−≥⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
− ∑∑∑−
+
+−−
∗
−
i
in
ii
io
i
io
n
i
niLi
in
ion
no raxfxfPTraxfxfP
22 ,
2)(2
2)(2'
22 ),(
2)(2 1
0=1)(
11
0=,
10=
1
0=, νμνμ
( )( )( )rxxPT niL
,,0,0,2' ,1
0= σνμ−
∗≥
( )( )rxxPL
,,0,0,2' , σνμ∗≥ (3.13)
for all Xx ∈ and all 0>r . Replacing x by xm2 in (3.13) and using (3.1), 3)(IFN , we obtain
⎟⎠⎞
⎜⎝⎛≥⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅
− ∗+
−
+
+
∑ mLmi
in
im
mo
mn
mno
arxxPraxfxfP ,0,0),,(2'
22 ,
2)(2
2)(2
,
1
0=, σνμνμ (3.14)
for all Xx ∈ and all 0>r and all 0, ≥nm . Replacing r by ram in (3.14), we get
( )rxxPraxfxfPLmi
min
im
mo
mn
mno ,0,0),,(2'
22 ,
2)(2
2)(2
,
1
0=, σνμνμ ∗+
+−
+
+
≥⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
− ∑ (3.15)
for all Xx ∈ and all 0>r and all 0, ≥nm . It follows from (3.15), that
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⋅
≥⎟⎟⎠
⎞⎜⎜⎝
⎛−
∑−+∗+
+
i
imn
mi
Lm
mo
mn
mno
arxxPrxfxfP
22
,0,0),,(2',2
)(22
)(21
=
,, σνμνμ (3.16)
holds for all Xx ∈ and all 0>r and all 0, ≥nm . Since 2<<0 a and ∞⎟⎠⎞
⎜⎝⎛∑ <
20=
in
i
a .
Thus ⎭⎬⎫
⎩⎨⎧
n
no xf
2)(2 is a Cauchy sequence in ( )TPY ,, ,νμ . Since ( )TPY ,, ,νμ is a complete
IFN-space this sequence convergent to some point ( ) YxA ∈ . So, one can define the mapping YXA →: by
![Page 7: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/7.jpg)
Stability Of A Leibniz Type Additive And Quadratic Functional Equation 151
0> , ,1,2
)(2)(, rnasrxfxAP Ln
no ∞→→⎟⎟
⎠
⎞⎜⎜⎝
⎛− ∗
νμ (3.17)
for all .Xx ∈ Letting 0=m in (3.16), we get
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⋅
≥⎟⎟⎠
⎞⎜⎜⎝
⎛−
∑−∗
i
in
i
Lon
no
arxxPrxfxfP
22
,0,0),,(2'),(2
)(21
0=
,, σνμνμ (3.18)
for all Xx ∈ and all 0>r . Now for every 0>δ and from (3.18), we have
( )δνμ +− rxfxAP o ),()(,( ) ( ) ( )
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−≥ ∗ rxfxfPxfxAPT n
no
on
no
L,
22',,
22)(' ,, νμνμ δ
( ) ( ) ( )⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⋅
⎟⎟⎠
⎞⎜⎜⎝
⎛−≥
∑−∗
i
in
i
n
no
L arxxPxfxAPT
22
,,0,0,2',,22' 1
0=
,, σδ νμνμ (3.19)
for all Xx ∈ and all 0>r . Taking the limit as ∞→n in (3.19), we get ( ) ( )( )( )raxxPTrxfxAP
LLo )(2,,0,0,2',1),()( ,, −≥+− ∗∗ σδ νμνμ
( )( )raxxPL
)(2,,0,0,2' , −≥ ∗ σνμ (3.20)
for all Xx ∈ and all 0>r and 0>δ . Since δ is arbitrary, by taking 0→δ in (3.20), we obtain
( ) ( )( ) ( ) ( )( )arxxPrxfxAPLo −≥− ∗ 2,,0,0,2', ,, σνμνμ (3.21)
for all Xx ∈ and all 0>r . To prove A satisfies (1.1), replacing ),,,( tzyx by ),2,2,2(2 tzyx nnnn in (3.3), we get
( )rtzyxPrtzyxDfP nnnnn
L
nnnnon ),2,2,2,2(2'),,2,2,2(2
21
,, σνμνμ ∗≥⎟⎠⎞
⎜⎝⎛ (3.22)
for all Xtzyx ∈,,, and all 0>r . Now,
⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −++−−+−+− tzyxAtzAtyAtxAP
33)()()(,νμ
⎟⎟⎠
⎞⎟⎠⎞
⎜⎝⎛ +−−−⎟
⎠⎞
⎜⎝⎛ −+−−⎟
⎠⎞
⎜⎝⎛ −−− rzyxAzyxAzyxA ,
32
32
32
,8
)),((221)(',
8)),((2
21)(' ,,
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ −−−⎟
⎠⎞
⎜⎝⎛ −−−≥ ∗
rtyftyAPrtxftxAPT non
nonL νμνμ
,8
)),((221)(' , ⎟
⎠⎞
⎜⎝⎛ −−− rtzftzAP n
onνμ
,8
,3
223
33' , ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −+++⎟
⎠⎞
⎜⎝⎛ −++− rtzyxftzyxAP n
onνμ
![Page 8: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/8.jpg)
152 M. Arunkumar et al
,8
,3
223
33' , ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −+++⎟
⎠⎞
⎜⎝⎛ −++− rtzyxftzyxAP n
onνμ
,8
,3
2221
32' , ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −−+⎟
⎠⎞
⎜⎝⎛ −−− rzyxfzyxAP n
onνμ
,8
,322
21
32' , ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −+−+⎟
⎠⎞
⎜⎝⎛ −+−− rzyxfzyxAP n
onνμ
⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −++−−+−+− tzyxftzftyftxfP n
onn
onn
onn
on 32
23))((2
21))((2
21))((2
21' ,νμ
⎪⎭
⎪⎬⎫⎟⎟⎠
⎞⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −+−−⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −−−
8322
21
322
21 rzyxfzyxf n
onn
on (3.23)
for all Xtzyx ∈,,, and all 0>r . Letting ∞→n in (3.23) and using (3.22),(3.2), we arrive
⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −++−−+−+− tzyxAtzAtyAtxAP
33)()()(,νμ
⎟⎟⎠
⎞⎟⎠⎞
⎜⎝⎛ +−−−⎟
⎠⎞
⎜⎝⎛ −+−−⎟
⎠⎞
⎜⎝⎛ −−− rzyxAzyxAzyxA ,
32
32
32
( )( )rtzyxPT nnnnn
LLLLLLLL),2,2,2,2(2',,1,1,1,1,1,11 , σνμ∗∗∗∗∗∗∗∗≥
( )rtzyxP nnnnn
L),2,2,2,2(2' , σνμ∗≥ (3.24)
for all Xtzyx ∈,,, and all 0>r . Letting ∞→n in (3.24) and using (3.2), 2)(IFN , we arrive
⎟⎠⎞
⎜⎝⎛ −−+⎟
⎠⎞
⎜⎝⎛ −++−+−+−
32
33=)()()( zyxAtzyxAtzAtyAtxA
⎟⎠⎞
⎜⎝⎛ +−−+⎟
⎠⎞
⎜⎝⎛ −+−+
32
32 zyxAzyxA
for all Xtzyx ∈,,, . Hence A satisfies the functional equation (1.1). In order to prove )(xA is unique, let )(xA′ be another additive functional equation satisfying (1.1) and
(3.5). Hence,
⎟⎟⎠
⎞⎜⎜⎝
⎛ ′−′− rxAxAPrxAxAP n
n
n
n
,2
)(22
)(2=)),()(( ,, νμνμ
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ ′−⎟⎟⎠
⎞⎜⎜⎝
⎛−≥ ∗ 2
.2),(22
)(2',2.2,
2)(2)(2' ,,
nn
n
no
n
n
non
L
rxAxfPrxfxAPT νμνμ
⎟⎟⎠
⎞⎜⎜⎝
⎛ −≥⎟⎟⎠
⎞⎜⎜⎝
⎛ −≥ ∗∗ n
n
L
nnn
L aarxxParxxP
2)(22 ,0,0),,(2'
2)(22 ,0,0),,22 (2' ,, σσ νμνμ
![Page 9: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/9.jpg)
Stability Of A Leibniz Type Additive And Quadratic Functional Equation 153
for all Xx ∈ and all 0>r . Since ,=2
)(22 lim ∞−
∞→ n
n
n aar we obtain
.1=2
)(22 ,0,0),,(2'lim , ∗∞→ ⎟⎟
⎠
⎞⎜⎜⎝
⎛ −Ln
n
n aarxxP σνμ
Thus ∗′−
LrxAxAP 1=)),()((,νμ
for all Xx ∈ and all 0>r , hence )(=)( xAxA ′ . Therefore )(xA is unique. For 1= −τ , we can prove a similar stability result. This completes the proof of the theorem. The following corollary is an immediate consequence of Theorem 3.1, regarding the stability of(1.1) Corollary 3.2 Suppose that an odd function YXfo →: satisfies the inequality
( )rtzyxfDP o ),,,,( ,νμ
( )( )( )
( ){ }( )⎪⎩
⎪⎨
⎧
+++++++≥ ∗
,,|||||||||||||||||||| |||| |||| ||||',,||||||||||||||||'
,,'
4444,
,
,
rtzyxtzyxPrtzyxP
rP
ssssssss
ssssL
λλλ
νμ
νμ
νμ
(3.25)
for all Xtzyx ∈,,, and all 0>r , where s,λ are constants with 0>λ . Then there exists a unique additive mapping YXA →: such that
( )( )( )( )⎪
⎪
⎩
⎪⎪
⎨
⎧
≠−+
≠−+≥− ∗
;41,|22|,||||)2(1'
1;,|22|,||||)2(1',,'
),()(44
,
,
,
,
srxP
srxPrP
rxAxfPsss
sss
Lo
λ
λλ
νμ
νμ
νμ
νμ (3.26)
for all Xx ∈ and all 0>r . Proof. Replacing
( )( ){ }⎪
⎩
⎪⎨
⎧
+++++++
,|||||||||||||||||||| |||| |||| ||||,||||||||||||||||
,=),,,(
4444 ssssssss
ssss
tzyxtzyxtzyxtzyx
λλλ
σ
then the corollary follows from Theorem 3.1, if we define
⎪⎩
⎪⎨
⎧
.2,2
1,=
4s
sa
Example 3.3 Let X be a normed space and νμ ,P and νμ ,'P be an intuitionistic fuzzy norms on X and Ρ defined by
![Page 10: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/10.jpg)
154 M. Arunkumar et al
( )⎪⎩
⎪⎨⎧
∈≤
∈+
.0,0,
,0,>=,,
Xxr
Xxrxr
rrxP νμ
( )⎪⎩
⎪⎨⎧
∈≤
∈+
.0,0,
,0,>=,' ,
Ρ
Ρ
xr
xrxr
rrxP νμ
Let ( ) ( )∞→∞ 0,0,:α be a function such that ( ) ( )lal αα <2 for all 0>l and 2<<0 a . Define
( ) ( ) ( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −++−−+−+− tzyxtztytxtzyx3
3=,,, ααααβ
⎟⎟⎠
⎞⎜⎜⎝
⎛ +−−+⎟⎟⎠
⎞⎜⎜⎝
⎛ −+−−⎟⎟⎠
⎞⎜⎜⎝
⎛ −−−3
232
32 zyxzyxzyx ααα
for all Xtzyx ∈,,, . Let Xx ∈0 be a unit vector and define XXf →: by ( ) ( ) 0= xxxxf α+ . Now for any Xtzyx ∈,,, and 0>r , we have
( ) ( ) 0, ,,,
=),,,,(xtzyxr
rrtzyxDfP o ⋅+ βνμ
( ) |,,, tzyxrr
L βΠ+≥ ∗
( )( ).,,,,'= , rtzyxP βνμ For any Xtzyx ∈,,, and 0>r , we have
( )( ) ( )tzyxrrrtzyxP
,2,2,22=,,2,2,22' , β
βνμ +
( )tzyxarr
L ,,,β+≥ ∗
( )( ).,,,,'= , rtzyxaP βνμ Hence the inequalities (3.1) and (3.3) are satisfied. Using Theorem 3.1, there exists a unique additive mapping YXA →: such that
( ) ( )( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛
−≥− ∗ r
axxPrxfxAP
Lo ,2
,0,0,2', ,,β
νμνμ
Xx ∈ and 0>r . The proof of the following Theorem and Corollary is similar tracing to that of Theorem 3.1 and Corollary 3.2, when ef is even. Hence the details of the proof is omitted.
![Page 11: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/11.jpg)
Stability Of A Leibniz Type Additive And Quadratic Functional Equation 155
Theorem 3.4 Let 1}{1,−∈τ . Let ZXXXX →×××:σ be a function such that for
some 1<4
<0τ
⎟⎠⎞
⎜⎝⎛ a ,
( )( ) ( )( )rxxxxaPrxxxxPL
,,,, ',,2,2,22' ,, σσ τνμ
ττττνμ ∗≥ (3.27)
for all Xx ∈ and all 0>r and ( )( ) ∗
∞→ L
nnnnn
nrtzyxP 1=,4,2,2,22'lim ,
τττττνμ σ (3.28)
for all Xtzyx ∈,,, and all 0>r . If YXf →: is an even function that satisfies the inequality
( ) ( )( )rtzyxPrtzyxDfPLe ,,,,'),,,,( ,, σνμνμ ∗≥ (3.29)
for all Xtzyx ∈,,, and all 0>r . Then the limit
0> , ,1,4
)(2)(, rnasrxfxQP Ln
ne ∞→→⎟⎟
⎠
⎞⎜⎜⎝
⎛− ∗
νμ (3.30)
exists for all Xx ∈ and the mapping YXQ →: is a unique quadratic mapping such that
( ) ( )( ) ( )( )raxxPrxQxfPLe |4|,,0,0,2', ,, −≥− ∗ σνμνμ (3.31)
for all Xx ∈ and all 0>r . Corollary 3.5 Suppose that an even function YXfe →: satisfies the inequality
( )rtzyxfDP e ),,,,( ,νμ
( )( )( )
( ){ }( )⎪⎩
⎪⎨
⎧
+++++++≥ ∗
,,|||||||||||||||||||| |||| |||| ||||',,||||||||||||||||'
,,'
4444,
,
,
rtzyxtzyxPrtzyxP
rP
ssssssss
ssss
L
λλλ
νμ
νμ
νμ
(3.32)
for all Xtzyx ∈,,, and all 0>r , where s,λ are constants with 0>λ . Then there exists a unique quadratic mapping YXQ →: such that
( )( )( )( )⎪
⎪
⎩
⎪⎪
⎨
⎧
≠−+
≠−+≥− ∗
;21,|24|,||||1)(23'
2;,|24|,||||1)(2',,3'
),()(444
,
,
,
,
srxP
srxPrP
rxQxfPsss
sss
Le
λ
λλ
νμ
νμ
νμ
νμ (3.33)
for all Xx ∈ and all 0>r . Example 3.6 Let X be a normed space and νμ ,P and νμ ,'P be an intuitionistic fuzzy norms on X and Ρ defined by
( )⎪⎩
⎪⎨⎧
∈≤
∈+
.0,0,
,0,>=,,
Xxr
Xxrxr
rrxP νμ
![Page 12: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/12.jpg)
156 M. Arunkumar et al
( )⎪⎩
⎪⎨⎧
∈≤
∈+
.0,0,
,0,>=,' ,
Ρ
Ρ
xr
xrxr
rrxP νμ
Let ( ) ( )∞→∞ 0,0,:α be a function such that ( ) ( )lal αα <2 for all 0>l and 4<<0 a . Define
( ) ( ) ( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −++−−+−+− tzyxtztytxtzyx3
3=,,, ααααβ
⎟⎟⎠
⎞⎜⎜⎝
⎛ +−−+⎟⎟⎠
⎞⎜⎜⎝
⎛ −+−−⎟⎟⎠
⎞⎜⎜⎝
⎛ −−−3
232
32 zyxzyxzyx ααα
for all Xtzyx ∈,,, . Let Xx ∈0 be a unit vector and define XXf →: by ( ) ( ) 0= xxxxf α+ . Now for any Xtzyx ∈,,, and 0>r , we have
( ) ( ) 0, ,,,
=),,,,(xtzyxr
rrtzyxDfP e ⋅+ βνμ
( ) |,,, tzyxrr
L βΠ+≥ ∗
( )( ).,,,,'= , rtzyxP βνμ For any Xtzyx ∈,,, and 0>r , we have
( )( ) ( )tzyxrrrtzyxP
,2,2,22=,,2,2,22' , β
βνμ +
( )tzyxarr
L ,,,β+≥ ∗
( )( ).,,,,'= , rtzyxaP βνμ Hence the inequalities (3.27) and (3.29) are satisfied. Using Theorem 3.4, there exists a unique quadratic mapping YXQ →: such that
( ) ( )( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛
−≥− ∗ r
axxPrxfxQP
Le ,4
,0,0,2', ,,β
νμνμ
for all Xx ∈ and 0>r . Theorem 3.7 Let 1= ±τ be fixed and let ZX →4:σ be a mapping such that for d
is defined as 1<2
<0τ
⎟⎠⎞
⎜⎝⎛ a and satisfies (1),(2),(27) and (28). Suppose that a function
YXf →: satisfies the inequality ( ) ( ) 0.>,,,,),,,,('),,,,( ,, rXtzyxrtzyxPrtzyxfDP
L∈∀≥ ∗ σνμνμ (3.34)
Then there exists a unique additive mapping YXA →: and unique quadratic mapping YXQ →: satisfying (1.1) and
( ) ( )rxxPrxQxAxfPL
,0,0),,(2),()()( 3,, σνμνμ ∗≥−− (3.35)
![Page 13: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/13.jpg)
Stability Of A Leibniz Type Additive And Quadratic Functional Equation 157
where ( ) ( ) ( ){ }|4|,0,0),2,(2,|2|,0,0),2,(2=,0,0),,(2 2
,1,
3, arxxParxxPTrxxP −− σσσ νμνμνμ (3.36)
for all Xx ∈ and all 0>r . Proof. Clearly
.|2||4| a≤≤
Let 2
)()(=)( xfxfxf ooa
−− for all Xx ∈ . Then 0=(0)af and )(=)( xfxf aa −− for
all Xx ∈ . Hence ( ) ( ) ( ){ }rtzyxfDPrtzyxfDPTrtzyxfDP ooLa ),,,,( ',),,,,( '),,,,( ,,, −−−−≥ ∗ νμνμνμ
( ) ( ){ }rtzyxPrtzyxPTL
),,,,(',),,,,(' ,, −−−−≥ ∗ σσ νμνμ (3.37)
for all Xtzyx ∈,,, and all 0>r . By Theorem 3.1 there exists a unique additive mapping YXA →: such that
( ) ( )|2|,0,0),,(2),()( 1,, arxxPrxAxfP
Lo −≥− ∗ σνμνμ (3.38)
for all Xx ∈ and all 0>r , where ( ) ( ){ }rtzyxPrtzyxPTrtzyxP ),,,,(',),,,,('=)),,,,(( ,,
1, −−−−σσσ νμνμνμ (3.39)
for all Xtzyx ∈,,, and all 0>r .
Also, let 2
)()(=)( xfxfxf eeq
−+ for all Xx ∈ . Then 0=(0)qf and )(=)( xfxf qq −
for all Xx ∈ . Hence ( ) ( )rtzyxfDtzyxfDPrtzyxfDP eeq ),2,,,( ),,,( =),,,,( ,, −−−−−νμνμ
( ) ( ){ }rtzyxPrtzyxPTL
),,,,(',),,,,(' ,, −−−−≥ ∗ σσ νμνμ (3.40)
for all Xtzyx ∈,,, and all 0>r . By Theorem 3.4, there exists a unique quadratic mapping YXQ →: such that
( ) ( )|4|,0,0),,(2),()( 2,, arxxPrxQxfP
Le −≥− ∗ σνμνμ (3.41)
for all Xx ∈ and all 0>r , where ( ) ( ){ }rtzyxPrtzyxPTrtzyxP ),,,,(',),,,,('=)),,,,(( ,,
2, −−−−σσσ νμνμνμ (3.42)
for all Xtzyx ∈,,, and all 0>r . Define )()(=)( xfxfxf qa + (3.43)
for all Xx ∈ . From (3.35),(3.38) and (3.39), we arrive ( ) ( )rxQxAxfxfPrxQxAxfP qa ),()()()(=),()()( ,, −−+−− νμνμ
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ −⎟
⎠⎞
⎜⎝⎛ −≥ ∗ 2
),()(,2
),()( ,,rxQxfPrxAxfPT qaL νμνμ
( ) ( ){ }|4|,0,0),2,(2,|2|,0,0),2,(2 2,
1, arxxParxxPT
L−−≥ ∗ σσ νμνμ
( ),,0,0),,(2= 3, rxxP σνμ
where
![Page 14: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/14.jpg)
158 M. Arunkumar et al
( ) ( ) ( ){ }|4|,0,0),2,(2,|2|,0,0),2,(2=,0,0),,(2 2,
1,
3, arxxParxxPTrxxP −− σσσ νμνμνμ (3.44)
for all Xx ∈ and all 0>r . Thus, the theorem is proved. The following corollary is the immediate consequence of corollaries 3.2, 3.5 and Theorem 3.7 concerning the stability for the functional equation (1.1). Corollary 3.8 Suppose that a function YXf →: satisfies the inequality
( )rtzyxfDP ),,,,( ,νμ
( )( )( )
( ){ }( )⎪⎩
⎪⎨
⎧
+++++++≥ ∗
,,|||||||||||||||||||| |||| |||| ||||',,||||||||||||||||'
,,'
4444,
,
,
rvuyxtzyxPrtzyxP
rP
ssssssss
ssss
L
λλλ
νμ
νμ
νμ
(3.45)
for all Xtzyx ∈,,, and all 0>r , where s,λ are constants with 0>λ . Then there exists a unique additive mapping YXA →: and a unique quadratic mapping
YXQ →: such that
( ) ( )( )⎪
⎪
⎩
⎪⎪
⎨
⎧
≠+
≠+
⎟⎠⎞
⎜⎝⎛
≥−− ∗
;21,
41,,||||1)(2
2;1,,,||||1)(2
,43,
),()()(443
,
3,
3,
,
srxP
srxP
rP
rxQxAxfPss
ss
L
λ
λ
λ
νμ
νμ
νμ
νμ (3.46)
for all Xx ∈ and all 0>r . Example 3.9 Let X be a normed space and νμ ,P and νμ ,'P be an intuitionistic fuzzy norms on X and Ρ defined by
( )⎪⎩
⎪⎨⎧
∈≤
∈+
.0,0,
,0,>=,,
Xxr
Xxrxr
rrxP νμ
( )⎪⎩
⎪⎨⎧
∈≤
∈+
.0,0,
,0,>=,' ,
Ρ
Ρ
xr
xrxr
rrxP νμ
Let ( ) ( )∞→∞ 0,0,:α be a function such that ( ) ( )lal αα <2 for all 0>l and 2<<0 a . Define
( ) ( ) ( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −++−−+−+− tzyxtztytxtzyx3
3=,,, ααααβ
⎟⎟⎠
⎞⎜⎜⎝
⎛ +−−+⎟⎟⎠
⎞⎜⎜⎝
⎛ −+−−⎟⎟⎠
⎞⎜⎜⎝
⎛ −−−3
232
32 zyxzyxzyx ααα
for all Xtzyx ∈,,, . Let Xx ∈0 be a unit vector and define XXf →: by ( ) ( ) 0= xxxxf α+ . Now for any Xtzyx ∈,,, and 0>r , we have
![Page 15: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/15.jpg)
Stability Of A Leibniz Type Additive And Quadratic Functional Equation 159
( ) ( ) 0, ,,,
=),,,,(xtzyxr
rrtzyxDfP⋅+ βνμ
( ) |,,, tzyxrr
L βΠ+≥ ∗
( )( ).,,,,'= , rtzyxP βνμ For any Xtzyx ∈,,, and 0>r , we have
( )( ) ( )tzyxrrrtzyxP
,2,2,22=,,2,2,22' , β
βνμ +
( )tzyxarr
L ,,,β+≥ ∗
( )( ).,,,,'= , rtzyxaP βνμ Hence the inequalities (3.1),(3.27) and (3.34) are satisfied. Using Theorem 3.7, there exists a unique additive mapping YXA →: and a unique quadratic mapping
YXQ →: such that ( ) ( ) ( )( ) ( )( )rxxPrxQxAxfP
L,,0,0,2, 3
,, βνμνμ ∗≥−−
for all Xx ∈ and 0>r . 4. STABILITY RESULTS: FIXED POINT METHOD In this section, we discuss the generalized Ulam-Hyers stability of the functional equation (1.1) in intuitionistic fuzzy normed space using fixed point method. First, we will recall the fundamental results in fixed point theory. Theorem 4.1 [25](The alternative of fixed point) Suppose that for a complete generalized metric space ),( dX and a strictly contractive mapping XXT →: with Lipschitz constant L . Then, for each given element ,Xx ∈ either
0, =),(1)( 1 ≥∀∞+ nxTxTdB nn or
2)(B there exists a natural number 0n such that: )(i ∞+ <),( 1xTxTd nn for all 0nn ≥ ; )(ii The sequence )( xT n is convergent to a fixed point ∗y of T
)(iii ∗y is the unique fixed point of T in the set };<),(:{= 0 ∞∈ yxTdXyY n
)(iv ),( 1
1),( TyydL
yyd−
≤∗ for all .Yy ∈
In order to prove the fixed point stability result, we define a constant iχ such that:
⎪⎩
⎪⎨⎧
1,=21
0,=2= iif
iifiχ
![Page 16: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/16.jpg)
160 M. Arunkumar et al
and Ω is the set such that { }.0=(0),: | = gYXgg →Ω
Theorem 4.2 Let YXfo →: be an odd mapping for which there exist a function
ZX →4:σ with the condition ( )( ) 0>,,,,,1=,,,,'lim , rXtzyxrtzyxP
L
ni
ni
ni
ni
ni
n∈∀∗
∞→χχχχχσνμ (4.1)
and satisfying the functional inequality ( ) ( ) 0.>,,,,,),,,,('),,,,( ,, rXtzyxrtzyxPrtzyxfDP
Lo ∈∀≥ ∗ σνμνμ (4.2)
If there exists 0>)(= iLL such that the function
,,0,02
,=)( ⎟⎠⎞
⎜⎝⎛→ xxxx σρ
has the property
( ) 0.>,,),('=,)(' ,, rXxrxPrxLPi
i ∈∀⎟⎟⎠
⎞⎜⎜⎝
⎛ρ
χχρ
νμνμ (4.3)
Then there exists a unique additive function YXA →: satisfying the functional equation (1.1) and
( ) 0.>,,1
),('),()(1
,, rXxrL
LxPrxAxfPi
Lo ∈∀⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥−−
∗ ρνμνμ (4.4)
Proof. Let d be a general metric on ,Ω such that
( ) ( ){ }.,),('),()(|)(0,=),( ,, XxKrxPrxhxgPKinfhgdL
∈≥−∞∈ ∗ ρνμνμ
It is easy to see that ),( dΩ is complete. Define Ω→Ω:T by ),(1=)( xgxTg ii
χχ
for
all .Xx ∈ For Ω∈hg, , we have ( ) ( )KrxPrxhxgPKhgd
L),('),()(),( ,, ρνμνμ ∗≥−⇒≤
( )rKxPrxhxgP iiLi
i
i
i χχρχχ
χχ
νμνμ ),(',)()(,, ∗≥⎟⎟
⎠
⎞⎜⎜⎝
⎛−⇒
( ) ( )KLrxPrxThxTgPL
),('),()( ,, ρνμνμ ∗≥−⇒
( ) KLxThxTgd ≤⇒ )(),( (4.5) ( ) ),(, hgLdThTgd ≤⇒ (4.6)
for all ., Ω∈hg Therefore T is strictly contractive mapping on Ω with Lipschitz constant .L Replacing ),,,( tzyx by ,0,0),(2 xx in (4.2) and using oddness, we get
( ) ( ).,0,0),,(2'),(2)(2 ,, rxxPrxfxfPLoo σνμνμ ∗≥− (4.7)
for all 0.>, rXx ∈ Using (IFN2) in (4.7), we arrive at
( )rxxPrxfxfPLo
o ,0,0),2,(2'),(2
)(2,, σνμνμ ∗≥⎟
⎠⎞
⎜⎝⎛ − (4.8)
![Page 17: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/17.jpg)
Stability Of A Leibniz Type Additive And Quadratic Functional Equation 161
for all 0>, rXx ∈ . With the help of (4.3), when 0=i , it follows from (4.8), that
( ) 0>,,),('),()(,
0
0, rXxLrxPrxfxfP
Loo ∈∀≥⎟⎟
⎠
⎞⎜⎜⎝
⎛− ∗ ρ
μμ
νμνμ
.=),( 01−≤⇒ LLfTfd oo (9)
Replacing x by 2x in (4.7), we obtain
⎟⎠⎞
⎜⎝⎛≥⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛− ∗ rxxPrxfxfP
Loo ,0,0),2
,(',2
2)( ,, σνμνμ (10)
for all 0>, rXx ∈ . With the help of (4.3), when 1=i , it follows from (4.10) that,
( ) 0>,,),(',2
2)( ,, rXxrxPrxfxfPLoo ∈∀≥⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛− ∗ ρνμνμ
.==1),( 10 ioo LLTffd −≤⇒ (4.11)
One can conclude from (4.9) and (4.11) that ∞≤ − <),( 1 i
oo LTffd Now, using fixed point alternative in both cases, it follows that there exists a fixed point A of T in Ω such that
( ) 0.>,1),(lim , rXxrxAxfPLn
i
nio
n∈∀→⎟⎟
⎠
⎞⎜⎜⎝
⎛− ∗
∞→ χχ
νμ (4.12)
Replacing ),,,( tzyx by ),,,( tzyx ni
ni
ni
ni χχχχ in (4.2), we get
( )rtzyxPrtzyxDfP ni
ni
ni
ni
niL
ni
ni
ni
nion
i
χχχχχσχχχχχ νμνμ ),,,,('),,,,(1
,, ∗≥⎟⎟⎠
⎞⎜⎜⎝
⎛ (4.13)
for all Xtzyx ∈,,, and all 0>r . By following the same procedure as in Theorem 3.1, we see that the function
YXA →: is additive and satisfies the functional equation (1.1). By fixed point alternative, since A is unique fixed point of T in the set
{ },<),(|= ∞Ω∈Δ Afdf oo A is a uniqe function such that
( ) ( )KrxPrxAxfPLo ),('),()( ,, ρνμνμ ∗≥− (4.14)
for all Xx ∈ , 0>r and 0.>K Again, using the fixed point alternative, we get
),(1
1),( ooo TffdL
Afd−
≤
LLAfd
i
o −≤⇒
−
1),(
1
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥−⇒−
∗ rL
LxPrxAxfPi
Lo 1),('),()(
1
,, ρνμνμ (4.15)
for all Xx ∈ and all 0>r . This completes the proof of the theorem.
![Page 18: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/18.jpg)
162 M. Arunkumar et al
From Theorem 4.2, we obtain the following corollary concerning the stability for the functional equation (1.1). Corollary 4.3 Suppose that an odd function YXfo →: satisfies the inequality
( )rtzyxfDP o ),,,,( ,νμ
( )( )( )
( ){ }( )⎪⎩
⎪⎨
⎧
+++++++≥ ∗
,,|||||||||||||||||||| |||| |||| ||||',,||||||||||||||||'
,,'
4444,
,
,
rtzyxtzyxPrtzyxP
rP
ssssssss
ssss
L
λλλ
νμ
νμ
νμ
(4.16)
for all 0>r and all Xtzyx ∈,,, , where s,λ are constants with 0>λ . Then there exists a unique aditive mapping YXA →: such that
( )
( )
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
≠⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−+
≠⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−+≥− ∗
;41,
222,||||
21)(2'
1;,22
2,||||2
1)(2'
,|2|,'
),()(
44
4
4
,
,
,
,
srxP
srxP
rP
rxAxfP
ss
s
s
ss
s
s
Lo
λ
λ
λ
νμ
νμ
νμ
νμ (4.17)
for all Xx ∈ and all 0>r . Proof. Set
{ }[ ]{ }⎪
⎩
⎪⎨
⎧
+++++++
,||||||||||||||||||||||||||||||||,||||||||||||||||
,=),,,(
4444 ssssssss
ssss
tzyxtzyxtzyxtzyx
λλλ
σ
for all Xtzyx ∈,,, . Then, ( )rtzyxP n
ini
ni
ni
ni χχχχχσνμ ),,,,(' ,
( ){ }( )
[ ]( )⎪⎩
⎪⎨
⎧
+++++++
,},|||||||||||||||||||| |||| |||| ||{||',,||||||||||||||||'
,,'=
44444,
,
,
rtzyxtzyxPrtzyxP
rP
ni
ssssssssnsi
ni
ssssnsi
ni
χλχχλχ
χλ
νμ
νμ
νμ
( ){ }( ){ }( )⎪
⎩
⎪⎨
⎧
+++++++
−
−
,)(,|||| |||| |||| ||||||||||||||||||||',)(,||||||||||||||||'
,,'=
414444,
1,
,
rtzyxtzyxPrtzyxP
rP
nsi
ssssssss
nsi
ssss
ni
χλχλ
χλ
νμ
νμ
νμ
⎪⎪⎩
⎪⎪⎨
⎧
∞→→∞→→∞→→
∗
∗
∗
. 1, 1, 1
=nasnasnas
L
L
L
![Page 19: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/19.jpg)
Stability Of A Leibniz Type Additive And Quadratic Functional Equation 163
Thus, (4.1) is holds. But, ⎟⎠⎞
⎜⎝⎛ ,0,0
2,=)( xxx σρ and has the property
( ) 0.>,,),('),(1' ,, rXxrxPrxLPLi
i
∈∀≥⎟⎟⎠
⎞⎜⎜⎝
⎛∗ ρχρ
χ νμνμ
Hence
( )
( )
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ +
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ +⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
.,2
21||||'
,,2
21||||'
,,'
=,,,0,02
,'=),('
4
44
,
,
,
,,
rxP
rxP
rP
rxxPrxP
s
ss
s
ss
λ
λ
λ
σρ
νμ
νμ
νμ
νμνμ
Now,
( )( )( )⎪
⎩
⎪⎨
⎧
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ +
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ +
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
−
−
.),(',),('
,,'=
.,2
21||||'
,,2
21||||'
,,'
=),(1'41
,
1,
,
4
44
,
,
,
,
rxPrxP
rP
rxP
rxP
rP
rxPs
i
si
i
s
ss
ii
s
ss
ii
i
ii χρ
χρχλ
χχλ
χχλχλ
χρχ
νμ
νμ
νμ
νμ
νμ
νμ
νμ
Hence, inequality (4.3) holds for the following cases. Now from (4.4), we prove the following cases. Case:1 12=L for 0=s if 0=i
( ) ( ).2)(,'=1
),(' ),()( ,
01
,, rPrL
LxPrxAxfPLo −⎟⎟
⎠
⎞⎜⎜⎝
⎛−
≥−−
∗ λρ νμνμνμ
Case:2 12= −L for 0=s if 1=i
( ) ( ).,2'=1
),(' ),()( ,
11
,, rPrL
LxPrxAxfPLo λρ νμνμνμ ⎟⎟
⎠
⎞⎜⎜⎝
⎛−
≥−−
∗
Case:3 sL −12= for 1>s if 0=i
( ) .22
2,||||2
1)(2'=1
),(' ),()( ,
01
,, ⎟⎟⎠
⎞⎜⎜⎝
⎛−
+⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥−−
∗ rxPrL
LxPrxAxfP ss
s
s
Lo λρ νμνμνμ
Case:4 12= −sL for 1<s if 1=i
( ) .22
2,||||2
1)(2'=1
),(' ),()( ,
11
,, ⎟⎟⎠
⎞⎜⎜⎝
⎛−
+⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥−−
∗ rxPrL
LxPrxAxfP ss
s
s
Lo λρ νμνμνμ
![Page 20: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/20.jpg)
164 M. Arunkumar et al
Case:5 sL 412= − for 41>s if 0=i
( ) .22
2,||||2
1)(2'=1
),(' ),()( 44
4
4
,
01
,, ⎟⎟⎠
⎞⎜⎜⎝
⎛−
+⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥−−
∗ rxPrL
LxPrxAxfP ss
s
s
Lo λρ νμνμνμ
Case:6 142= −sL for 41<s if 1=i
( ) .22
2,||||2
1)(2'=1
),(' ),()( 44
4
4
,
11
,, ⎟⎟⎠
⎞⎜⎜⎝
⎛−
+⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥−−
∗ rxPrL
LxPrxAxfP ss
s
s
Lo λρ νμνμνμ
Thus, the proof is complete. The proof of the following Theorem and Corollary is similar tracing to that of Theorem 4.2 and corollary 4.3, when f is even. Hence we omit the proof. Theorem 4.4 Let YXfe →: be an even mapping for which there exist a function
ZX →4:σ with the condition ( )( ) 0>,,,,,1=,,,,'lim 2
, rXtzyxrtzyxPL
ni
ni
ni
ni
ni
n∈∀∗
∞→χχχχχσνμ (4.18)
and satisfying the functional inequality ( ) ( ) 0.>,,,,,),,,,('),,,,( ,, rXtzyxrtzyxPrtzyxfDP
Le ∈∀≥ ∗ σνμνμ (4.19)
If there exists )(= iLL such that the function
,,0,02
,=)( ⎟⎠⎞
⎜⎝⎛→ xxxx σρ
has the property
( ) 0.>,,),('=),(1' ,2, rXxrxPrxLP ii
∈∀⎟⎟⎠
⎞⎜⎜⎝
⎛ρχρ
χ νμνμ (4.20)
Then there exists a unique quadratic function YXQ →: satisfying the functional equation (1.1) and
( ) 0.>,1
),('),()(1
,, rXxrL
LxPrxQxfPi
Le ∈∀⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥−−
∗ ρνμνμ (4.21)
Corollary 4.5 Suppose that an even function YXfe →: satisfies the inequality
( )rtzyxfDP e ),,,,( ,νμ
( )( )( )
( ){ }( )⎪⎩
⎪⎨
⎧
+++++++≥
.,|||||||||||||||||||| |||| |||| ||||',,||||||||||||||||'
,,'
4444,
,
,
rtzyxtzyxPrtzyxP
rP
ssssssss
ssss
λλλ
νμ
νμ
νμ
(4.22)
for all tzyx ,,, and all Xr ∈0> , where s,λ are constants with 0>λ . Then there exists a unique quadratic mapping YXQ →: such that
![Page 21: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/21.jpg)
Stability Of A Leibniz Type Additive And Quadratic Functional Equation 165
( )
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
≠⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−+
≠⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−+
⎟⎠⎞
⎜⎝⎛
≥− ∗
;21,
424,||||
21)(2'
2;,42
4,||||2
1)(2'
,|34|,'
),()(
44
4
4
,
,
,
,
srxP
srxP
rP
rxQxfP
ss
s
s
ss
s
s
Le
λ
λ
λ
νμ
νμ
νμ
νμ (4.23)
for all Xx ∈ and all 0>r . Theorem 4.7 Let YXf →: be a mapping for which there exist a function
ZX →4:σ with the condition (4.1) and (4.18) satisfying the functional inequality ( ) ( ) 0.>,,,,,),,,,('),,,,( ,, rXtzyxrtzyxPrtzyxfDP
L∈∀≥ ∗ σνμνμ (4.24)
If there exists )(= iLL such that the function
,,0,02
,=)( ⎟⎠⎞
⎜⎝⎛→ xxxx σρ
has the properties (4.3) and (4.20) for all .Xx ∈ Then there exists a unique additive function YXA →: and a unique quadratic function YXQ →: satisfying the functional equation (1.1) and
( ) ( ) 0.>,,),(),()()( 3,, rXxrxPrxQxAxfP
L∈∀≥−− ∗ ρνμνμ (4.25)
Proof. From Theorem 4.2 in (3.37), there exists a unique additive mapping
YXA →: such that
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥−−
∗ rL
LxPrxAxfPi
La 1),(),()(
11,, ρνμνμ (4.26)
for all Xx ∈ and all 0>r . Using Theorem 4.4, in (3.37) there exists a unique quadratic mapping YXQ →: such that
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥−−
∗ rL
LxPrxQxfPi
Lq 1),(),()(
12,, ρνμνμ (4.27)
for all Xx ∈ and all 0>r . Define )()(=)( xfxfxf aq + (4.28)
for all xx ∈ . From (4.25),(4.26) and (4.27), we get ( ) ( )rxQxAxfxfPrxQxAxfP aq ),()()()(=),()()( ,, −−+−− νμνμ
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ −⎟
⎠⎞
⎜⎝⎛ −≥ ∗ 2
),()(',2
),()(' ,,rxQxfPrxAxfPT qaL νμνμ
( ),),(=1
),(,1
),( 3,
12,
11, rxPr
LLxPr
LLxPT
ii
Lρρρ νμνμνμ
⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
≥−−
∗
![Page 22: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/22.jpg)
166 M. Arunkumar et al
where
( )⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−−
rL
LxPrL
LxPTrxPii
1),(,
1),(=),(
12,
11,
3, ρρρ νμνμνμ (4.29)
for all Xx ∈ and all 0>r . The theorem is thus proved. The following corollary is the immediate consequence of Corollaries 4.3, 4.5 and Theorem 4.6 concerning the stability for the functional equation (1.1) using fixed point method. Corollary 4.8 Suppose that a function YXf →: satisfies the inequality
( )rtzyxfDP ),,,,( ,νμ
( )( )( )
( ){ }( )⎪⎩
⎪⎨
⎧
+++++++≥ ∗
,,|||||||||||||||||||| |||| |||| ||||',,||||||||||||||||'
,,'
4444,
,
,
rtzyxtzyxPrtzyxP
rP
ssssssss
ssss
L
λλλ
νμ
νμ
νμ
(4.30)
for all Xtzyx ∈,,, and all 0>r , where s,λ are constants with 0>λ . Then there exists a unique additive mapping YXA →: and a unique quadratic mapping
YXQ →: such that
( )( )
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≠⎟⎟⎠
⎞⎜⎜⎝
⎛ +
≠⎟⎟⎠
⎞⎜⎜⎝
⎛ +≥−− ∗
;21,
41,,||||
21)(2
1,2;,,||||2
1)(2,,
),()()(
44
43,
3,
3,
,
srxP
srxP
rP
rxQxAxfP
ss
s
ss
s
L
λ
λ
λ
νμ
νμ
νμ
νμ (4.31)
for all Xx ∈ and all 0>r . REFERENCES [1] J. Aczel and J. Dhombres, Functional Equations in Several Variables,
Cambridge Univ, Press, 1989. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J.
Math. Soc. Japan, 2 (1950), 64-66. [3] M. Arunkumar, G. Ganapathy, S. Murthy, S. Karthikeyan, Stability of the
Generalized Arun-additive functional equation in Intuitionistic fuzzy normed spaces, International Journal Mathematical Sciences and Engineering Applications, Vol.4, No. V, December 2010, 135-146.
[4] M. Arunkumar, S. Karthikeyan, Solution and Stability Of n -Dimensional Mixed Type Additive and Quadratic Functional Equation, Far East Journal of Applied Mathematics, 54 1 (2011) 47-64.
[5] M. Arunkumar, S. Karthikeyan, Solution and Stability of n-Dimensional Quadratic Functional Equation: Direct and Fixed Point Methods,
![Page 23: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/23.jpg)
Stability Of A Leibniz Type Additive And Quadratic Functional Equation 167
International Journal of Advanced Mathematical Sciences, Vol. 2 (1), pp, 21-33, 2014.
[6] M. Arunkumar, John M. Rassias, On the generalized Ulam-Hyers stability of an AQ-Mixed type functional equation with counter examples, Far East Journal of Applied Mathematics, Volume 71, No. 2, (2012), 279-305.
[7] M. Arunkumar, Solution and stability of modified additive and quadratic functional equation in generalized 2-normed spaces, International Journal Mathematical Sciences and Engineering Applications, Vol. 7 No. I (January, 2013), 383-391.
[8] M. Arunkumar, Generalized Ulam-Hyers stability of derivations of a AQ-functional equation, "Cubo A Mathematical Journal" dedicated to Professor Gaston M. N’Guérékata on the occasion of his 60th Birthday Vol.15, No 1, (March 2013), 159-169.
[9] M. Arunkumar, P. Agilan, Additive Quadratic functional equation are Stable in Banach space: A Fixed Point Approach, International Journal of pure and Applied Mathematics, Vol. 86, No.6, (2013), 951-963.
[10] M. Arunkumar, P. Agilan, Additive Quadratic Functional Equation are Stable in Banach space: A Direct Method, Far East Journal of Applied Mathematics, Vol. 80, No. 1, (2013), 105-121.
[11] M. Arunkumar, G.Shobana, S. Hemalatha, Ulam-Hyers, Ulam-TRassias, Ulam-GRassias,Ulam-JRassias Stabilities of A Additive-Quadratic Mixed Type Functional Equation In Banach Spaces, International Conference on Mathematics and Computing, ICMCE (Accepted).
[12] M. Arunkumar, P. Agilan, C. Devi Shyamala Mary, Permanence of A Generalized AQ Functional Equation In Quasi-Beta Normed Spaces, International Conference on Mathematics and Computing, ICMCE (Accepted).
[13] M. Arunkumar, Perturbation of n Dimensional AQ-mixed type Functional Equation via Banach Spaces and Banach Algebra : Hyers Direct and Alternative Fixed Point Methods, International Journal of Advanced Mathematical Sciences, 2 (1) (2014), 34-56.
[14] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.
[15] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951) 223-237.
[16] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002.
[17] G. Deschrijver, E.E. Kerre On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems 23 (2003), 227-235.
[18] M. Eshaghi Gordji, Stability of an Additive-Quadratic Functional Equation of Two Variables in F-Spaces, J. Nonlinear Sci. Appl. 2 (2009), no. 4, 251-259
[19] M. Eshaghi Gordji, N.Ghobadipour, J. M. Rassias, Fuzzy Stability of Additive-Quadratic Functional Equations, arxiv:0903.0842v1 [math.fa]. 2009
![Page 24: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/24.jpg)
168 M. Arunkumar et al
[20] Z. Gajda and R.Ger, Subadditive multifunctions and Hyers-Ulam stability, in General Inequalites 5, Internat Schrifenreiche Number. Math.Vol. 80, Birkhauser \ Basel, 1987.
[21] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.
[22] S.B. Hosseini, D. O’Regan, R. Saadati, Some results on intuitionistic fuzzy spaces, Iranian J. Fuzzy Syst, 4 (2007) 53-64.
[23] D.H. Hyers, On the stability of the linear functional equation, Proc.Nat. Acad.Sci.,U.S.A.,27 (1941) 222-224.
[24] D.H. Hyers, G. Isac, Th.M. Rassias, Stability of functional equations in several variables, Birkhauser, Basel, 1998.
[25] B. Margoils, J.B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull.Amer. Math. Soc. 126 74 (1968), 305-309.
[26] A.K. Mirmostafaee, M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), no. 6, 720-729.
[27] A.K. Mirmostafaee, M. Mirzavaziri, M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), no. 6, 730-738.
[28] A.K. Mirmostafaee, M.S. Moslehian, Fuzzy almost quadratic functions, Results Math. doi:10.1007/s00025-007-0278-9.
[29] M. Mursaleen, S.A. Mohiuddine, On stability of a cubic functional equation in intuitionistic fuzzy normed spaces, Chaos Solitons Fractals 42 (2009), no. 5, 2997-3005.
[30] J.H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals, 22 (2004), 1039-1046.
[31] C. Park, Orthogonal Stability of an Additive-Quadratic Functional Equation, Fixed Point Theory and Applications, doi:10.1186/1687-1812-2011-66
[32] Matina J. Rassias, M. Arunkumar, S. Ramamoorthi, Stability of the Leibniz additive-quadratic functional equation in Quasi-Beta normed space: Direct and fixed point methods, Journal Of Concrete And Applicable Mathematics (JCAAM), Vol. 14 No. 1-2, (2014), 22–46.
[33] J.M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, 46, (1982) 126-130.
[34] J.M. Rassias, M.J. Rassias, On the Ulam stability of Jensen and Jensen type mappings on the restricted domains, J. Math. Anal. Appl., 281 (2003), 516-524.
[35] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc.Amer.Math. Soc., 72 (1978), 297-300.
[36] Th.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Acedamic Publishers, Dordrecht, Bostan London, 2003.
[37] K. Ravi, M. Arunkumar, J.M. Rassias, On the Ulam stability for the orthogonally generalEuler-Lagrange type functional equation, International Journal of Mathematical Sciences, Autumn 2008 Vol.3, No. 08, 36-47.
![Page 25: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/25.jpg)
Stability Of A Leibniz Type Additive And Quadratic Functional Equation 169
[38] R. Saadati, J.H. Park, On the intuitionistic fuzzy topological spaces, Chaos, Solitons and Fractals 27 (2006), 331-344.
[39] R. Saadati, J.H. Park, Intuitionstic fuzzy Euclidean normed spaces, Commun. Math. Anal., 1 (2006), 85-90.
[40] S. Shakeri, Intuitionstic fuzzy stability of Jensen type mapping, J. Nonlinear Sci. Appli. Vol.2 No. 2 (2009), 105-112.
[41] Sun Sook Jin, Yang Hi Lee, A Fixed Point Approach to The Stability of the Cauchy Additive and Quadratic Type Functional Equation, Journal of Applied Mathematics 16 pages, doi:10.1155/2011/817079
[42] Sun Sook Jin, Yang Hi Lee, Fuzzy Stability of a Quadratic-Additive Functional Equation, International Journal of Mathematics and Mathematical Sciences 6 pages, doi:10.1155/2011/504802
[43] S.M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, NewYork, 1964.
[44] G. Zamani Eskandani, Hamid Vaezi, Y. N. Dehghan, Stability of a Mixed Additive and Quadratic Functional Equation in Non-Archimedean Banach Modules, Taiwanese Journal of Mathematics, vol. 14, no. 4, (2010), 1309-1324.
[45] Ding Xuan Zhou, On a conjecture of Z. Ditzian, J. Approx. Theory 69 (1992), 167-172.
![Page 26: Stability Of A Leibniz Type Additive And Quadratic ... · Functional Equation In Intuitionistic Fuzzy Normed Spaces M. Arunkumar1, John M. Rassias2 and S. Karthikeyan3 1Department](https://reader034.vdocuments.net/reader034/viewer/2022043000/5f771a73884ef770715e4dca/html5/thumbnails/26.jpg)
170 M. Arunkumar et al