stability of a leibniz type additive and quadratic ... · functional equation in intuitionistic...

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

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 .

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


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.


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


( ) ⎟⎠⎞

⎜⎝⎛≥⎟⎟

⎠

⎞⎜⎜⎝

⎛− ∗

+

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


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


,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


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


( )⎪⎩

⎪⎨⎧

∈≤

∈+

.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.


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


( )⎪⎩

⎪⎨⎧

∈≤

∈+

.0,0,

,0,>=,' ,

Ρ

Ρ

xr

xrxr

rrxP νμ


( ) ( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝


3=,,, ααααβ

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−−+⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−3

232

32 zyxzyxzyx ααα


( ) ( ) 0, ,,,

=),,,,(xtzyxr

rrtzyxDfP e ⋅+ βνμ

( ) |,,, tzyxrr

L βΠ+≥ ∗



,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)


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


( ) ( ) ( ){ }|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


( ) ( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝


3=,,, ααααβ

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−−+⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−−⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−3

232

32 zyxzyxzyx ααα



( ) ( ) 0, ,,,

=),,,,(xtzyxr

rrtzyxDfP⋅+ βνμ

( ) |,,, tzyxrr

L βΠ+≥ ∗



,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


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)


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.


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


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



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


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


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

,=)( ⎟⎠⎞


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


( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≠⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

≠⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

⎟⎠⎞

⎜⎝⎛

≥− ∗

;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

,=)( ⎟⎠⎞


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

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−

≥−−

∗


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)

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