A Fixed Point Approach to the Stability of Pexider Quadratic Functional Equation with Involution

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Volume 2010, Article ID 839639,18pages doi:10.1155/2010/839639

Research Article

A Fixed Point Approach to the Stability of Pexider

Quadratic Functional Equation with Involution

M. M. Pourpasha,

1

_{J. M. Rassias,}

2

_{R. Saadati,}

3

and S. M. Vaezpour

3

1_{Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU), Tehran, Iran} 2_{Section of Mathematics and Informatics, Pedagogical Department, National and Kapodistrian University}

of Athens, 4, Agamemnonos St., Aghia Paraskevi, Athens 15342, Greece

3_{Department of Mathematics, Amirkabir University of Technology, Hafez Avenue, P. O. Box 15914,}

Tehran, Iran

Correspondence should be addressed to

R. Saadati,[email protected] S. M. Vaezpour,[email protected]

Received 10 May 2010; Revised 13 July 2010; Accepted 31 July 2010

Academic Editor: S. Reich

Copyrightq2010 M. M. Pourpasha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We apply the fixed point method to investigate the Hyers-Ulam stability of the Pexider functional equationfxy gxσy hx ky, for allx, y∈ E, whereEis a normed space and

σ:E → Eis an involution.

1. Introduction and Preliminary

A basic question in the theory of functional equations is as follows. “When is it true that a function, which approximately satisfies a functional equation must be close to an exact solution of the equation?” The first stability problem concerning group homomorphisms was raised by Ulam1in 1940 and aﬃrmatively answered by Hyers in 2. Subsequently, the result of Hyers was generalized by Aoki 3for additive mappings and by Rassias4 for linear mappings by considering an unbounded Cauchy diﬀerence. The paper of Rassias has provided a lot of influence in the development of what we now call Hyers-Ulam-Rassias stability of functional equations. For more information, see 5–7. Specially, Maligranda 8 and Moszner9provided a very interesting discussion on the definition of functional equations’ stability.

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A Hyers-Ulam stability theorem for the quadratic functional equation

fxyfx−y2fx 2fy 1.1

was proved by Skof15for the functionf :E1 → E2, whereE1is a normed space andE2is

a Banach space. Cholewa16 noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an abelian group. Czerwik17 proved the generalized

Hyers-Ulam stability of the quadratic functional equation 1.1. Recently, Brzde¸k 18, Jung19, and Jung and Sahoo20investigated the Hyers-Ulam-Rassias stability of1.1. Furthermore they proved the Hyers-Ulam-Rassias stability of the functional equation of Pexider type

f1

xyf2

x−yf3x f4

y. 1.2

The stability problem of several functional equations has been extensively investigated by a number of authors, and there are many interesting results concerning this problemsee 4,21–37.

Let E1 and E2 be real vector spaces. If an additive functionσ : E1 → E1 satisfies

σxy σx σyandσσx xfor allx, y∈E1, thenσis called an involution ofE1,

see21,37. For a given involutionσ:E1 → E1, the functional equation

fxyfxσy2fx 2fy, ∀x, y∈E 1.3

is called the quadratic functional equation with involution. According to37, Corollary 8, a functionf : E1 → E2is a solution of1.3if and only if there exists an additive function

A:E1 → E2, and a biadditive symmetric functionB:E1×E1 → E2such thatAσx Ax,

Bσx, y −Bσx, yandfx Bx, y Axfor allx∈E1.

Indeed, if we setσx I in1.3, whereI : E1 → E1denotes the identity function,

then1.3reduces to the additive functional equation

fxyfx fy, ∀x, y∈E. 1.4

On the other hand, ifσx −Iin1.3, then1.3is transformed into the quadratic functional equation

fxyfx−y2fx 2fy, ∀x, y∈E. 1.5

Recently, Bouikhalene et al. have proved the Hyers-Ulam-Rassias stability of the quadratic functional equation with involution1.2, see21.

In this paper, we will apply the fixed point method to prove the Hyers-Ulam-Rassias stability of the functional equation1.3in the Pexider type

fxygxσy2hx 2ky. 1.6

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LetXbe a set. A functiond:X×X → 0,∞is called a generalized metric onXif and only ifdsatisfies the following

1dx, y 0, if and only ifxy

2dx, y dy, x, for allx, y∈X;

3dx, z≤dx, y dy, z, for allx, y, z∈X.

For an extensive theory of fixed point and other nonlinear methods, the reader is referred to the book of Hyers et al.44and C˘adariu and Radu45.

Theorem 1.1. LetX, dbe a generalized complete metric space. Assume thatJ:X → Xis a strictly contractive operator with the Lipschitz constant0< L <1. If there exists a nonnegative integerksuch thatdJk1_{x, J}k_x_<_∞_{for some}_x_∈_X_{, then the following are true:}

athe sequence{Jn_x_}_{converges to a fixed point}_x∗_of_J bx∗is the unique fixed point ofJin

X∗y∈X:dJk_{x, x}_<_∞_; _1.7

cify∈X∗, then

dy, x∗≤ 1

1−Ld

Jy, y. 1.8

2. Main Results

In this section, we prove the Hyers-Ulam-Rassias stability of the quadratic functional equation with involution1.6by applying the fixed point method.

Theorem 2.1. LetE1 be a commutative semigroup (with the divisibility by2), and letE2be a real Banach space. Suppose that a functionϕ:E1×E1 → 0,∞is given and there exists a constantL,

0< L <1, such that

ϕ2x,2y≤2Lϕx, y,

ϕxσx, yσy≤2Lϕx, y, 2.1

for allx, y∈E1. Furthermore, letf, g, h, k:E1 → E2be even functions satisfying the inequality

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for allx, y ∈E1, whereσ : E1 → E1 is an involution ofE1andf0 g0 h0 k0 0. Then there exists a unique solutionT :E1 → E2of2.2such that

₂_f_x₋_T_x_≤ 1 41−LM

_{x, x} _M_x,₀_,

₂_g_x₋_Tx≤ 1

41−LM

_{x, x} _M_x,₀ 1

2

ϕx, x ϕx,−x,

hx−Tx ≤ 1

41−LM

_{x, x}_,

kx−Tx ≤ 1

41−LM

_{x, x} 1

4

ϕ0, x ϕx,0,

2.3

for allx∈E1,where

Mx, yϕx, yϕ0, yϕy,0ϕx

2,

x

2

ϕx

2,− _x

2

,

Mx, yMx, yMxy,0Mxσy,0,

Tx lim n→ ∞

1 22n h2

n_x₂n₋₁_h₂n−1_x₂n−1_σ_x_.

2.4

Proof. Lettingy0 in2.2, we obtain

_f_x _g_x₋₂_h_x_≤_ϕ_x,₀_. _2.5

Similarly, for everyy∈E1, we can putx0 in2.2to obtain

fygσy−2ky≤ϕ0, y. 2.6

Sinceσ:E1 → E1is an involution, we replaceybyxσx:din2.6, then we have

fd gd−2kd≤ϕ0, d. 2.7

Also, we can replaceybyxand replaceyby−xin2.2to get

_f₂_x _g_x_σ_x₋₂_h_x₋₂_k_x_≤_ϕ_{x, x}_, _2.8 _g_x₋_σ_x₋₂_h_x₋₂_k₋_x_≤_ϕ_x,₋_x_. _2.9

Sinceσ :E1 → E1is an involution, by replacingxbyxσx:din2.8and using2.1,

we have

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Also, by replacingxbyx−σx:din2.9, we have

_g₂_d₋₂_h_d₋₂_k_d_≤_ϕ_d,₋_d_. _2.11

In view of2.5and2.7, we see that

hd−kd ≤ 1

2

ϕ0, d ϕd,0, 2.12

and it follows from2.10and2.11that

f2d−g2d≤ϕd, d ϕd,−d. 2.13

By using2.2,2.12and2.13, we have

fxyfxσy−2hx−2hy

≤fxygxσy−2hx−2ky2ky−hy

fxσy−gxσy

≤ϕx, yϕ0, yϕy,0ϕ

xσy

2 ,

xσy

2

ϕ

xσy

2 ,−

xσy

2

≤ϕx, yϕ0, yϕy,0ϕx

2,

x

2

ϕx

2,− _x

2

Mx, y.

2.14

Therefore

fxyfxσy−2hx−2hy≤Mx, y. 2.15

By puttingy0 in2.15, we get

fx−hx≤Mx,0. 2.16

Hence,2.15and2.16imply that

hxyhxσy−2hx−2hy

≤fxyfxσy−2hx−2hyfxy−hxy

fxσy−hxσy

≤Mx, yMxy,0Mxσy,0Mx, y.

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Therefore

hxyhxσy−2hx−2hy≤Mx, y. 2.18

Now, we defineX to be the set of all functionsf : E1 → E2 and introduce a generalized

metric onXas follows:

dg, hinfC∈0,∞:gx−hx≤CMx, x, ∀x∈E1

. 2.19

Let{fn}be a Cauchy sequence inX, d. According to the definition of Cauchy sequences, for any given >0, there exists a positive integerNsuch that

dfm, fn

≤ 2.20

for allm, n≥N.

By considering the definition of the generalized metricd, we see that

∀ >0, ∃N∈N, ∀m, n≥N, ∀x∈E1:fmx−fnx≤Mx, x. 2.21

Ifxis any given point inE1,2.21implies that{fnx}is a Cauchy sequence inE2. Since

E2 is complete, {fnx}converges inE2 for eachx ∈ E1. Hence, we can define a function

f:E1 → E2by

fx lim

n→ ∞fnx. 2.22

We define an operatorJ:X → Xby

JLx 1

4L2x Lxσx 2.23

for allx∈E1.

First, we assert thatJ is strictly contractive onX. Giveng, h∈X, letC∈0,∞be an arbitrary constant with

dg, h≤C, 2.24

that is,

gx−hx≤CMx, x 2.25

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If we replaceybyxin2.18, then we obtain

h2x hxσx−4hx ≤Mx, x 2.26

for everyx∈E1.

It follows from2.23and2.25that

Jgx−Jhx 1

4g2x gxσx−h2x−hxσx

≤ 1

4g2x−h2x 1

4gxσx−hxσx

≤ 1 4CM

₂_x,₂_x 1

4CM

_x_σ_x_{, x}_σ_x

≤LCMx, x

2.27

for allx ∈E1, that is,dJg, Jh ≤ LC. Hence, we conclude thatdJg, Jh ≤Ldg, hfor any

g, h∈X. Therefore,Jis strictly contractive becauseLis a constant with 0< L <1.

Now, we claim thatdJh, h < ∞. If we puty xin2.26and divide both sides by 1/4, then we get

Jhx−hx 1

4h2x hxσx−hx ≤ 1

4M

_{x, x} _2.28

for allx∈E1, that is,

dJh, h≤ 1

4 <∞. 2.29

Now, byTheorem 1.1there exists a functionT :E1 → E2which is a fixed point ofJ, such that

dJn_{h, T} _→ _{0 as}_n _{→ ∞}_{. By induction, we can easily show that}

Jnhx 1

22n h2

n_x₂n₋₁_h₂n−1_x₂n−1_σ_x _2.30

for eachn ∈ N. SincedJn_{h, T} _→ _{0 as} _n _{→ ∞}_{, there exists a sequence} _{_C_n}_{such that}

Cn → 0 asn → ∞anddJnh, T≤Cnfor everyn∈N. Hence, by the definition ofd, we have

Jnhx−Tx ≤CnMx, x 2.31

for allx∈E1. Thus, for each fixedx∈E1, we have

lim n→ ∞ J

n_h_x₋_T_x ₀_.

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Therefore

Tx lim n→ ∞

1 22n h2

n_x₂n₋₁_h₂n−1_x₂n−1_σ_x _2.33

for allx∈E1. It follows from2.1,2.2, and2.33that

TxyTxσy−2Tx−2Ty

lim n→ ∞

1 22n

h2n_x₂n_y ₂n₋₁_h₂n−1_x_y₂n−1_σ_x _σ_y

h2nx2nσy 2n−1h2n−1xσy2n−1σx y

−2h2nx 22n−1h2n−1x2n−1σx

−2h2ny22n−1h2n−1y2n−1σy

≤ lim n→ ∞

1 22nh

2nx2nyh2nx2nσy−2h2nx−2h2ny

lim n→ ∞

2n₋₁ 22n

h2n−1xσx 2n−1yσy

−2h2n−1x2n−1σx−2h2n−1x2n−1σx

≤ lim n→ ∞

1 22nM

2nx,2ny lim n→ ∞

2n−1 22n M

2nxσx,2nyσy0

2.34

for allx, y∈E1, which implies thatT is a solution of1.6.

ByTheorem 1.1cand2.29, we obtain

dh, T≤ 1

1−Ldh, Jh<

1

41−L, 2.35

that is,2.3is true for allx ∈ E1. Assume thatT1 : E1 → E2 is another solution of2.2

satisfying2.3. We know thatT1 is a fixed point ofJ. In view of2.3and the definition of

d, we can conclude that2.35is true withT1in place ofT. Due toTheorem 1.1b, we get

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By2.16,2.28, we obtain

_f_x₋_T_x_≤_f_x₋_h_x _h_x₋_T_x

≤Mx,0 1 41−LM

_{x, x}_. 2.36

Also by2.12,2.28, we obtain

kx−Tx ≤ kx−hx hx−Tx

≤ 1 4

ϕ0, x ϕx,0 1 41−LM

_{x, x}_, 2.37

and by2.13,2.28, we obtain

gx−Tx≤fx−gxfx−Tx

≤ 1 2

ϕx, x ϕx,−xMx,0 1 41−LM

_{x, x}_. 2.38

In the following, we will investigate some special cases ofTheorem 2.1.

Remark 2.2. LetE1be a commutative semigroupwith the divisibility by 2, and letE2be a a

real Banach space. Suppose that a functionϕ :E1×E1 → 0,∞is given and there exists a

constantL, 0< L <1, such that

ϕx, y≤ L

8ϕ

2x,2y,

ϕxσx, yσy≤ L

4ϕ

2x,2y,

2.39

for all x, y ∈ E1. Furthermore, letf, g, h, k : E1 → E2 be even functions satisfying the

inequality

_f

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for allx, y∈E1, whereσ:E1 → E1is an involution ofE1andf0 g0 h0 k0 0.

Then there exists a unique solutionT :E1 → E2of2.40such that

2fx−Tx≤ L

41−LM

_{x, x} _M_x,₀_,

2gx−Tx≤ L

41−LM

_{x, x} _M_x,₀ 1

2

ϕx, x ϕx,−x,

hx−Tx ≤ L

41−LM

_{x, x}_,

kx−Tx ≤ L

41−LM

_{x, x} 1

4

ϕ0, x ϕx,0

2.41

for allx∈E1,where

Mx, yϕx, yϕ0, yϕy,0ϕx

2,

x

2

ϕx

2,− _x

2

,

Mx, yMx, yMxy,0Mxσy,0,

Tx lim n→ ∞

22n

hx

2n

1 2n −1

h

x

2n1

σx

2n1

.

2.42

Remark 2.3. LetE1 andE2 be real Banach spaces. Let the hypotheses ofTheorem 2.1hold. If

we put

φx, yδ, δ >0 2.43

for allx, y∈E1, then, there exists a unique solutionT :E1 → E2such that

₂_f_x₋_T_x_≤ 25 2 δ, ₂_g_x₋_T_x_≤ 27

2 δ,

hx−Tx ≤ 15

2 δ,

Kx−Tx ≤8δ,

2.44

for allx∈E1,where

Tx lim n→ ∞

1 22n h2

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Also, if we putφx, y x p y pfor 0 ≤ p < 1 and > 0, then there exists a unique solutionT :E1 → E2such that

2fx−Tx≤ 1

22−2p

103 −1 p

2p−1 2−2 p

x p

13 −1 p

2p

x p,

₂_g_x₋_T_x_≤ 1 22−2p

103 −1 p

2p−1 2−2

p _x p

5 2

3 −1p 2p

−1p 2

x p,

hx−Tx ≤ 1

22−2p

103 −1 p

2p−1 2−2 p

x p,

kx−Tx ≤ 1

22−2p

103 −1 p

2p−1 2−2 p

x p1

2 x p_.

2.46

Similarly, let, p, q≥0 be real numbers such thatpq <1. If we putφx, y x p y q

see22,28, then there exists a unique solutionT :E1 → E2such that

2fx−Tx≤ 1

4−2pq1

5 1

2pq−1

x pq

2pq

x pq,

₂_g_x₋_T_x_≤ 1 4−2pq1

5 1

2pq−1

x pq

1 2pq 1

x pq,

hx−Tx ≤ 1

4−2pq1

5 1

2pq−1

x pq,

kx−Tx ≤ 1

4−2pq1

5 1

2pq−1

x pq.

2.47

Let , p, q ≥ 0 be real numbers such that p q < 1. Another control function is

x p y q x pq y pq see35. Then, there exists a unique solutionT : E1 → E2

such that

2fx−Tx≤ 1

4−2pq1

15 9 2pq−1

x pq

1 3

2pq−1

x pq,

₂_g_x₋_T_x_≤ 1 4−2pq1

15 9 2pq−1

x pq

4 3

2pq−1

x pq,

hx−Tx ≤ 1

4−2pq1

15 9 2pq−1

x pq,

kx−Tx ≤ 1

4−2pq1

15 9 2pq−1

x pq1

2 x pq

2.48

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Theorem 2.4. LetE1 be a commutative semigroup (with the divisibility by2) and letE2 be a real Banach space. Suppose that a functionϕ:E1×E1 → 0,∞is given and there exists a constantL,

0< L <1, such that

ϕ2x,2y≤Lϕx, y,

ϕxσx, yσy≤Lϕx, y, 2.49

for allx, y∈E1. Furthermore, letf, g, h, k:E1 → E2be odd functions satisfying the inequality

fxygxσy−hx−ky≤ϕx, y, 2.50

for allx, y ∈ E1, whereσ : E1 → E1is an involution ofE1. Then, there exists a unique solution

T :E1 → E2of 2.50such that

_f_x₋_T_x_≤ 1 41−LM

_{x, x} 3

2

ϕx,0 ϕ0, x,

gx−Tx≤ 1

21−LM

_{x, x} ₂_ϕ_x,₀ _ϕ₀_{, x}_,

hx−Tx ≤ 1

21−LM

_{x, x}_,

kx−Tx ≤ 1

21−LM

_{x, x} 1

2

ϕx,0 ϕ0, x,

2.51

for allx∈E1,where

Mx, yϕx, yϕxy,0ϕ0, xy 3

2

ϕx,0 ϕ0, x

ϕx,−x 1

2

ϕy,0ϕ0, yϕy,−y,

Mx, yMx, yMx, σy,

Tx lim n→ ∞

1 2n h2

n_x₂n₋₁_h₂n−1_x₂n−1_σ_x_.

2.52

Proof. As in theTheorem 2.1, if we puty0,x0and replaceybyx,y x, andy−x

in2.50separately, then we obtain

fx gx−2hx≤ϕx,0, 2.53 _f

ygσy−2ky≤ϕ0, y, 2.54 f2x gxσx−2hx−2kx≤ϕx, x, 2.55 gx−σx−2hx 2kx≤ϕx,−x, 2.56

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We replaceybyx−σx:din2.54. Sinceσ:E1 → E1is an involution, then

_f_d₋_g_d₋₂_k_d_≤_ϕ₀_{, d}_. _2.57

Also, replaceybyxσx:din2.54, then

fd gd−2kd≤ϕ0, d. 2.58

Also, we replacexbyx−σx:din2.55, then

_f₂_d₋₂_h_d₋₂_k_d_≤_ϕ_{d, d}_. _2.59

We replacexbyx−σx:din2.56, then

g2d−2hd 2kd≤ϕd,−d. 2.60

Due to2.53and2.57, we have

₂_f_d₋₂_h_d₋₂_k_d_≤_ϕ_d,₀ _ϕ₀_{, d}_, _2.61 ₂_g_d₋₂_h_d ₂_k_d_≤_ϕ_d,₀ _ϕ₀_{, d}_. _2.62

Combining2.54with2.61yields

h2d k2d−2hd−2kd ≤ 1

2

ϕd,0 ϕ0, dϕd, d. 2.63

Due to2.60and2.62, we have

h2d−k2d−2hd 2kd ≤ 1

2

ϕd,0 ϕ0, dϕd,−d. 2.64

Now, it follows from2.63and2.64that

h2d−2hd ≤ 1

2

ϕd,0 ϕ0, dϕd,−d,

k2d−2kd ≤ 1

2

ϕd,0 ϕ0, dϕd,−d.

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By2.50,2.61,2.62, and2.65, we have

hxykxyhxσy−kxσy−h2x−k2y

≤fxygxσy−2hx−2ky

hxykxy−fxy

hxσy−kxσy−gxσy

h2x−2hx k2y−2ky

≤ϕx, yϕxy,0ϕ0, xy3

2

ϕx,0 ϕ0, x

ϕx,−x 1

2

ϕy,0ϕ0, yϕy,−yMx, y

2.66

for allx, y∈E1. If we replaceyin2.66byσy, we get

_h

xσykxσyhxy−kxy−h2x−k2σy≤Mx, σy.

2.67

By2.66and2.67, we get

2hxy2hxσy−h2x−h2σy≤Mx, yMx, σy. 2.68

If we replaceyin2.66byσyand combine2.53with2.58, we get ₂_h

xy2hxσy−h2x−h2y≤Mx, yMx, σy. 2.69

By2.65and2.69, we get

2hxy2hxσy−2hx−2hy≤Mx, yMx, σy

Mx, y. 2.70

Therefore

_h

xyhxσy−hx−hy≤Mx, y. 2.71

We defineXto be the set of all functionsf :E1 → E2and introduce a generalized metric on

Xas follows:

dg, hinfC∈0,∞:gx−hx≤CMx, x, ∀x∈E1

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Also, we define an operatorJ :X → Xby

JLx 1

2L2x Lxσx 2.73

for allx∈E1.

Then, there exists a function T : E1 → E2 which is a fixed point of J, such that

dJn_{h, T} _→ _{0 as}_n _{→ ∞}_{. By induction, we can show that}

Jnhx 1

2n h2

n_x₂n₋₁_h₂n−1_x₂n−1_σ_x _2.74

for eachn ∈ N. SincedJn_{h, T} _→ _{0 as} _n _{→ ∞}_{, there exists a sequence} _{_C_n}_{such that}

Cn → 0 asn → ∞anddJnh, T≤Cnfor everyn∈N. Hence,

Jnhx−Tx ≤CnMx, x 2.75

for allx∈E1. Thus, for each fixedx∈E1, we have

lim n→ ∞ J

n_h_x₋_T_x ₀_.

2.76

Then

Tx lim n→ ∞

1 2n h2

n_x₂n₋₁_h₂n−1_x₂n−1_σ_x _2.77

for allx∈E1. Hence,T is a solution of2.51.

Remark 2.5. LetE1andE2be real Banach spaces. Let the hypotheses ofTheorem 2.4hold. If we

replace the control functionφx, ybyδ >0, then, there exists a unique solutionT :E1 → E2

such that

_f_x₋_T_x_≤₁₂_δ, gx−Tx≤21δ,

hx−Tx ≤18δ,

kx−Tx ≤19δ,

2.78

for allx∈E1,where

Tx lim n→ ∞

1 2n h2

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If we putφx, y x p y pin which >0 andpis a nonnegative number less than 1, then, there exists a unique solutionT :E1 → E2such that

fx−Tx≤ 2

p−1

2p₋₁

122−1p x p3 x p,

gx−Tx≤ 2

p

2p₋₁

122−1p x p3 x p,

hx−Tx ≤ 2

p

2p₋₁

122−1p x p,

kx−Tx ≤ 2

p

2p₋₁

122−1p x p x p.

2.80

Let >0 andp, q >2. If we putφx, y x p y q, then, there exists a unique solution

T :E1 → E2such that

fx−Tx≤ 3

22pq2₋₁ x pq, gx−Tx≤ 3

2pq2₋₁ x pq,

hx−Tx ≤ 3

2pq2₋₁ x pq,

kx−Tx ≤ 3

2pq2₋₁ x pq_.

2.81

Finally, if we putφx, y x p y q x pq y pq, then, there exists a unique solution

T :E1 → E2such that

_f_x₋_T_x_≤ 9

22pq2₋₁ x pq, _g_x₋_T_x_≤ 9

2pq2₋₁ x pq,

hx−Tx ≤ 9

2pq2₋₁ x pq,

kx−Tx ≤ 9

2pq2₋₁ x pq

2.82

for allx∈E1.

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Acknowledgments

The authors would like to thank the referee and the associate editor Professor S. Reich for giving useful suggestions and comments for the improvement of this paper.

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