Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the additive

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Vol. 19 (2018), No. 2, pp. 1107–1115 DOI: 10.18514/MMN.2018.2239

ADDITIVE.˛; ˇ/-FUNCTIONAL EQUATIONS AND LINEAR MAPPINGS

CHOONKIL PARK Received 18 February, 2017

Abstract. In this paper, we investigate the additive.˛; ˇ/-functional equation

f .x/C˛f .˛y/Cf .´/Dˇ 1f .ˇ.xCyC´// (0.1) for all complex numbers˛withj˛j D1and for a fixed nonzero complex numberˇ.

Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the additive.˛; ˇ/-functional equation (0.1) in complex Banach spaces.

2010Mathematics Subject Classification: 39B52; 39B62; 47H10

Keywords: Hyers-Ulam stability, additive.˛; ˇ/-functional equation,C-linear mapping, fixed point method, direct method

1. INTRODUCTION AND PRELIMINARIES

The stability problem of functional equations originated from a question of Ulam [19] concerning the stability of group homomorphisms.

The functional equationf .xCy/Df .x/Cf .y/is called theCauchy equation.

In particular, every solution of the Cauchy equation is said to be anadditive mapping.

Hyers [10] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Rassias [18] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [9] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’

approach. See [2,6,8,12,15,16,20] for more information on functional equations.

We recall a fundamental result in fixed point theory.

Theorem 1 ([3,7]). Let .X; d / be a complete generalized metric space and let J WX!Xbe a strictly contractive mapping with Lipschitz constant˛ < 1. Then for

This work was supported by Basic Science Research Program through the National Re- search Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF- 2017R1D1A1B04032937).

c 2018 Miskolc University Press

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each given elementx2X, either

d.Jnx; JnC1x/D 1

for all nonnegative integersnor there exists a positive integern0such that .1/ d.Jnx; JnC1x/ <1; 8nn0;

.2/the sequencefJnxgconverges to a fixed pointyofJ;

.3/ yis the unique fixed point ofJ in the setY D fy2X jd.Jn0x; y/ <1g; .4/ d.y; y/ 1 ˛1 d.y; Jy/for ally2Y.

In 1996, G. Isac and Th.M. Rassias [11] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [4,5,17]).

In Section2, we solve the additive.˛; ˇ/-functional equation (0.1) in vector spaces and prove the Hyers-Ulam stability of the additive.˛; ˇ/-functional equation (0.1) in Banach spaces by using the fixed point method.

In Section3, we prove the Hyers-Ulam stability of the additive.˛; ˇ/-functional equation (0.1) in Banach spaces by using the direct method.

Throughout this paper, assume thatX is a complex normed space and thatY is a complex Banach space. Letˇbe a fixed nonzero complex number.

2. ADDITIVE.˛; ˇ/-FUNCTIONAL EQUATION(0.1)IN COMPLEXBANACH SPACESI

We solve the additive.˛; ˇ/-functional equation (0.1) in complex vector spaces.

Lemma 1. Let X andY be complex vector spaces. If a mapping f WX !Y satisfies

f .x/C˛f .˛y/Cf .´/Dˇ 1f .ˇ.xCyC´// (2.1) for allx; y; ´2X and all˛2TWD f2Cj jj D1g, thenf WX !Y isC-linear.

Proof. Assume thatf WX !Y satisfies (2.1).

LettingxDyD´D0in (2.1), we get.2C˛/f .0/Dˇ 1f .0/for all˛2T. So f .0/D0.

Letting ˛ D1, y D x and ´D0 in (2.1), we get f .x/Cf . x/D0 and so f . x/D f .x/for allx2X.

Letting˛D1and´D x yin (2.1), we get

f .x/Cf .y/ f .xCy/Df .x/Cf .y/Cf . x y/D0 and so

f .xCy/Df .x/Cf .y/

for allx; y2X.

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Letting´D0andyD xin (2.1), we getf .x/C˛f . ˛x/D0and sof .˛x/D

˛f .x/for all x2X and all˛ 2T. By the same reasoning as in the proof of [14,

Theorem 2.1], the mappingf WX!Y isC-linear.

Using the fixed point method, we prove the Hyers-Ulam stability of the additive .˛; ˇ/-functional equation (2.1) in complex Banach spaces.

Theorem 2. Let' WX3!Œ0;1/be a function such that there exists an L < 1 with

' x

2;y 2;´

2

L

2' .x; y; ´/ (2.2)

for allx; y; ´2X. Letf WX !Y be a mapping satisfyingf .0/D0and f .x/C˛f .˛y/Cf .´/ ˇ 1f .ˇ.xCyC´//

'.x; y; ´/ (2.3) for all x; y; ´2X and all ˛ 2 T. Then there exists a unique C-linear mapping AWX!Y such that

kf .x/ A.x/k L

2.1 L/.' . x; x; 2x/C2' . x; x; 0// (2.4) for allx2X.

Proof. Let˛D1.

Replacingxby xand lettingyD xand´D2xin (2.3), we get

k2f . x/Cf .2x/k '. x; x; 2x/ (2.5) for allx2X.

Replacingxby xand lettingyDxand´D0in (2.3), we get

kf . x/Cf .x/k '. x; x; 0/ (2.6) for allx2X.

It follows from (2.5) and (2.6) that

kf .2x/ 2f .x/k '. x; x; 2x/C2'. x; x; 0/ (2.7) for allx2X.

Consider the set

SWD fhWX!Y; h.0/D0g and introduce the generalized metric onS:

d.g; h/

Dinff2RCW kg.x/ h.x/k .' . x; x; 2x/C2' . x; x; 0//; 8x2Xg; where, as usual, infD C1. It is easy to show that.S; d /is complete (see [13]).

Now we consider the linear mappingJ WS!S such that J g.x/WD2gx

2

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for allx2X.

Letg; h2S be given such thatd.g; h/D". Then

kg.x/ h.x/k ".' . x; x; 2x/C2' . x; x; 0//

for allx2X. Hence

kJ g.x/ J h.x/k D 2gx

2

2hx 2

2"

' x 2; x

2; x

C2' x 2;x

2; 0 2"L

2.' . x; x; 2x/C2' . x; x; 0//

L".' . x; x; 2x/C2' . x; x; 0//

for allx2X. Sod.g; h/D"implies thatd.J g; J h/L". This means that d.J g; J h/Ld.g; h/

for allg; h2S.

It follows from (2.7) that

f .x/ 2f x 2

' x 2; x

2; x

C2' x 2;x

2; 0 L

2.' . x; x; 2x/C2' . x; x; 0//

for allx2X. Sod.f; Jf / L2.

By Theorem1, there exists a mappingAWX !Y satisfying the following:

(1)Ais a fixed point ofJ, i.e.,

A .x/D2Ax 2

(2.8) for allx2X. The mappingAis a unique fixed point ofJ in the set

M D fg2SWd.f; g/ <1g:

This implies thatAis a unique mapping satisfying (2.8) such that there exists a2 .0;1/satisfying

kf .x/ A.x/k .' . x; x; 2x/C2' . x; x; 0//

for allx2X;

(2)d.Jlf; A/!0asl! 1. This implies the equality

llim!12nf x 2n

DA.x/

for allx2X;

(3)d.f; A/ 1 L1 d.f; Jf /, which implies kf .x/ A.x/k L

2.1 L/.' . x; x; 2x/C2' . x; x; 0//

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for allx2X.

It follows from (2.2) and (2.3) that

A.x/C˛A.˛y/CA.´/ ˇ 1A .ˇ.xCyC´//

D lim

n!12n

f x 2n

C˛f ˛y 2n

Cf ´ 2n

ˇ 1f

ˇ

xCyC´ 2n

lim

n!12n' x 2n; y

2n; ´ 2n

D0 for allx; y; ´2X and all˛2T. So

A.x/C˛A.˛y/CA.´/ ˇ 1A .ˇ.xCyC´//D0

for all x; y; ´2X and all ˛ 2T. By Lemma 1, the mapping AW X !Y is C-

linear.

Corollary 1. Letr > 1andbe nonnegative real numbers, and letf WX !Y be a mapping satisfying

f .x/C˛f .˛y/Cf .´/ ˇ 1f .ˇ.xCyC´//

.kxkrCkykrCk´kr/ (2.9) for all x; y; ´2X and all ˛ 2 T. Then there exists a unique C-linear mapping AWX!Y such that

kf .x/ A.x/k 2rC6 2r 2kxkr for allx2X.

Proof. The proof follows from Theorem2by taking'.x; y; ´/D.kxkrCkykrC k´kr/for allx; y; ´2X. Then we can chooseLD21 r and we get the desired res-

ult.

Theorem 3. Let' WX3!Œ0;1/be a function such that there exists an L < 1 with

' .x; y; ´/2L'x 2;y

2;´ 2

for allx; y; ´2X. Letf WX!Y be a mapping satisfyingf .0/D0and(2.3). Then there exists a uniqueC-linear mappingAWX!Y such that

kf .x/ A.x/k 1

2.1 L/.' . x; x; 2x/C2' . x; x; 0//

for allx2X.

Proof. It follows from (2.7) that

f .x/ 1 2f .2x/

1

2.' . x; x; 2x/C2' . x; x; 0//

for allx2X.

Let.S; d /be the generalized metric space defined in the proof of Theorem2.

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Now we consider the linear mappingJ WS!S such that J g.x/WD1

2g .2x/

for allx2X.

The rest of the proof is similar to the proof of Theorem2.

Corollary 2. Letr < 1and be positive real numbers, and letf WX !Y be a mapping satisfying(2.9). Then there exists a uniqueC-linear mappingAWX !Y such that

kf .x/ A.x/k 6C2r 2 2rkxkr for allx2X.

Proof. The proof follows from Theorem3by taking'.x; y; ´/D.kxkrCkykrC k´kr/for all x; y; ´2X. Then we can chooseLD2r 1 and we get desired res-

ult.

3. ADDITIVE.˛; ˇ/-FUNCTIONAL EQUATION(0.1)IN COMPLEXBANACH SPACESII

In this section, using the direct method, we prove the Hyers-Ulam stability of the additive.˛; ˇ/-functional equation (2.1) in complex Banach spaces.

Theorem 4. Let'WX3!Œ0;1/be a function and letf WX !Y be a mapping satisfyingf .0/D0and

.x; y; ´/WD

1

X

jD1

2j'x 2j; y

2j; ´ 2j

<1;

f .x/C˛f .˛y/Cf .´/ ˇ 1f .ˇ.xCyC´//

'.x; y; ´/ (3.1) for all x; y; ´2X and all ˛ 2 T. Then there exists a unique C-linear mapping AWX!Y such that

kf .x/ A.x/k 1

2. . x; x; 2x/C2 . x; x; 0// (3.2) for allx2X.

Proof. Let˛D1.

It follows from (2.7) that

f .x/ 2f x 2

' x 2; x

2; x

C2' x 2;x

2; 0 for allx2X. Hence

2lf x

2l

2mf x 2m

m 1

X

jDl

2jf x 2j

2jC1f x 2jC1

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m 1

X

jDl

2j' x

2jC1; x 2jC1; x

2j

C2jC1' x 2jC1; x

2jC1; 0

(3.3) for all nonnegative integersmandl withm > l and allx2X. It follows from (3.3) that the sequencef2kf .2xk/gis Cauchy for allx2X. SinceY is a Banach space, the sequencef2kf .2xk/gconverges. So one can define the mappingAWX!Y by

A.x/WD lim

k!12kf x

2k

for allx2X. Moreover, lettinglD0and passing the limitm! 1in (3.3), we get (3.2).

Now, letT WX !Y be another additive mapping satisfying (3.2). Then we have kA.x/ T .x/k D

2qAx 2q

2qTx 2q

2qAx 2q

2qf x 2q

C

2qT x 2q

2qf x 2q

2q

x 2q; x

2q;2x 2q

C2qC1 x 2q; x

2q; 0

;

which tends to zero asq! 1for allx2X. So we can conclude thatA.x/DT .x/

for allx2X. This proves the uniqueness ofA.

The rest of the proof is similar to the proof of Theorem2.

Corollary 3. Letr > 1andbe nonnegative real numbers, and letf WX!Y be a mapping satisfying(2.9). Then there exists a uniqueC-linear mappingAWX !Y such that

kf .x/ A.x/k 2rC6 2r 2kxkr for allx2X.

Proof. The proof follows from Theorem4by taking'.x; y; ´/D.kxkrCkykrC

k´kr/for allx; y; ´2X.

Theorem 5. Let'WX3!Œ0;1/be a function and letf WX !Y be a mapping satisfyingf .0/D0,(3.1)and

.x; y; ´/WD

1

X

jD0

1

2j'.2jx; 2jy; 2j´/ <1

for allx; y; ´2X. Then there exists a uniqueC-linear mapping AWX !Y such that

kf .x/ A.x/k 1

2. . x; x; 2x/C2 . x; x; 0// (3.4) for allx2X.

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Proof. It follows from (2.7) that

f .x/ 1 2f .2x/

.' . x; x; 2x/C2' . x; x; 0//

for allx2X. Hence

1

2lf .2lx/ 1

2mf .2mx/

m 1

X

jDl

1 2jf

2jx 1 2jC1f

2jC1x

m 1

X

jDl

1

2jC1'. 2jx; 2jx; 2jC1x/C 1

2j'. 2jx; 2jx; 0/

(3.5) for all nonnegative integers m and l with m > l and all x 2X. It follows from (3.5) that the sequencef21nf .2nx/gis a Cauchy sequence for all x2X. SinceY is complete, the sequence f21nf .2nx/gconverges. So one can define the mapping AWX!Y by

A.x/WD lim

n!1

1

2nf .2nx/

for allx2X. Moreover, lettinglD0and passing the limitm! 1in (3.5), we get (3.4).

The rest of the proof is similar to the proofs of Theorems2and4.

Corollary 4. Letr < 1and be positive real numbers, and letf WX !Y be a mapping satisfying(2.9). Then there exists a uniqueC-linear mappingAWX !Y such that

kf .x/ A.x/k 6C2r 2 2rkxkr for allx2X.

Proof. The proof follows from Theorem5by taking'.x; y; ´/D.kxkrCkykrC

k´kr/for allx; y; ´2X.

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Author’s address

Choonkil Park

Hanyang University, Department of Department, Research Institute for Natural Sciences, Seoul 04763, Republic of Korea

E-mail address:baak@hanyang.ac.kr

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