On the Weak Relatively Nonexpansive Mappings in Banach Spaces

(1)

Volume 2010, Article ID 189751,7pages doi:10.1155/2010/189751

Research Article

On the Weak Relatively Nonexpansive Mappings in Banach Spaces

Yongchun Xu

¹

and Yongfu Su

²

1Department of Mathematics, Hebei North University, Zhangjiakou 075000, China

2Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Correspondence should be addressed to Yongfu Su,[email protected] Received 23 March 2010; Accepted 20 May 2010

Academic Editor: Billy Rhoades

Copyrightq2010 Y. Xu and Y. Su. 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.

In recent years, the definition of weak relatively nonexpansive mapping has been presented and studied by many authors. In this paper, we give some results about weak relatively nonexpansive mappings and give two examples which are weak relatively nonexpansive mappings but not relatively nonexpansive mappings in Banach spacel²andL^p0,1 1< p <∞.

1. Introduction

LetE be a smooth Banach space, and let Cbe a nonempty closed convex subset ofE. We denote byφthe function defined by

φ x, y

x²−2 x, Jy

y² forx, y∈E. 1.1

Following Alber1, the generalized projectionΠC fromEontoCis defined by ΠCx arg min

y∈C φ y, x

, ∀x∈E. 1.2

The generalized projectionΠCfromEontoCis well defined, single value and satisfies x −y²≤φ

x, y

≤

xy² for x, y∈E. 1.3 IfEis a Hilbert space, thenφy, x y−x², andΠCis the metric projection ofEontoC.

(2)

LetCbe a closed convex subset ofE, and letT be a mapping from Cinto itself. We denote byFTthe set of fixed points ofT. A pointpinCis said to be an asymptotic fixed point ofT2–4ifCcontains a sequence{xn}which converges weakly topsuch that lim_n_{→ ∞}Txn− xn0. The set of asymptotic fixed point ofT will be denoted by bFT .

Following Matsushita and Takahashi 2, a mapping T of Cinto itself is said to be relatively nonexpansive if the following conditions are satisfied:

1FTis nonempty;

2φu, Tx≤φu, x, for allu∈FT, x∈C;

3FT FT.

The hybrid algorithms for fixed point of relatively nonexpansive mappings and applications have been studied by many authors, for example2–7

In recent years, the definition of weak relatively nonexpansive mapping has been presented and studied by many authors5–8, but they have not given the example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping. In this paper, we give an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping in Banach spacel².

A pointpinCis said to be a strong asymptotic fixed point ofT 5,6 ifCcontains a sequence{xn}which converges strongly top such that lim_n_{→ ∞}Txn −xn 0. The set of strong asymptotic fixed points ofTwill be denoted byFT . A mappingT fromCinto itself is called weak relatively nonexpansive if

1FTis nonempty;

2φu, Tx≤φu, x, for allu∈FT, x∈C;

3FT FT.

Remark 1.1. In6, the weak relatively nonexpansive mapping is also said to be relatively weak nonexpansive mapping.

Remark 1.2. In 7, the authors have given the definition of hemirelatively nonexpansive mapping as follows. A mappingTfromCinto itself is called hemirelatively nonexpansive if

1FTis nonempty;

2φu, Tx≤φu, x, for allu∈FT, x∈C.

The following conclusion is obvious.

Conclusion 1. A mapping is closed hemi-relatively nonexpansive if and only if it is weak relatively nonexpansive.

IfE is strictly convex and reflexive Banach space, and A ⊂ E×E^∗ is a continuous monotone mapping withA⁻¹0/∅, then it is proved in2thatJr : JrA⁻¹J, forr >0 is relatively nonexpansive. Moreover, if T : E → Eis relatively nonexpansive, then using the definition ofφ, one can show thatFTis closed and convex. It is obvious that relatively nonexpansive mapping is weak relatively nonexpansive mapping. In fact, for any mapping T :C → C, we haveFT⊂FT ⊂FT. Therefore, if T is relatively nonexpansive mapping, thenFT FT FT .

(3)

2. Results for Weak Relatively Nonexpansive Mappings in Banach Space

Theorem 2.1. LetEbe a smooth Banach space andCa nonempty closed convex and balanced subset ofE. Let{xn}be a sequence inCsuch that{xn}converges weakly tox0/0 andxn−xm ≥r >0 for alln /m. Define a mappingT:C → Cas follows:

Tx

⎧⎨

⎩ n

n1xn ifxxn ∃n≥1,

−x ifx /xn ∀n≥1. 2.1

Then the following conclusions hold:

1T is a weak relatively nonexpansive mapping but not relatively nonexpansive mapping;

2T is not continuous;

3T is not pseudo-contractive;

4if{xn} ⊂intC, thenT is also not monotone (accretive), where intCis the interior ofC.

Proof. 1It is obvious thatT has a unique fixed point 0, that is,FT {0}. Firstly, we show thatx0is an asymptotic fixed point ofT. In fact since{xn}converges weakly tox0,

Txn−xn n

n1xn−xn

1

n1xn −→0 2.2

asn → ∞, so, x0 is an asymptotic fixed point ofT. Secondly, we show thatT has a unique strong asymptotic fixed point 0, so that, FT FT. In fact, for any strong convergent sequence,{zn} ⊂Csuch thatzn → z0andzn−Tzn → 0 asn → ∞, from the conditions of Theorem 2.1, there exists suﬃciently large nature number N such that zn/xm, for any n, m > N. ThenTzn−znforn > N, it follows fromzn−Tzn → 0 that 2zn → 0,and hence zn → z00. Observe that

φ0, Tx Tx²≤ x²φ0, x, ∀x∈C. 2.3

Then T is a weak relatively nonexpansive mapping. On the other hand, since x0 is an asymptotic fixed point of T but not fixed point, hence T is not a relatively nonexpansive mapping.

2For anyxn/0, we can take 0≤λm → 0 such thatλmxn∈{xn}^∞_n1, then we have

xn−λmxn −→0, m−→ ∞, Txn−Tλmxn

n

n1xnλmxn

n

n1 λm

xn ≥

n n1

xn>0, 2.4

thenT is not continuous.

(4)

3Sincexn−xm ≥ r > 0 for alln /m, without loss of generality, we assume that xn/0 for alln≥1. In this case, we can take 1≥δn → 1 such thatδnxn∈{xi}^∞_i1for alln≥1.

Therefore we have

Txn−Tδnxn, Jxn−δnxn n

n1xnδnxn, Jxn−δnxn

n

n1 δn

xn, J1−δnxn

n n1 δn

1

1−δn1−δnxn, J1−δnxn

n n1 δn

1

1−δn1−δnxn²

n n1 δn

1

1−δnxn−δnxn².

2.5

Sincen/n1δn1/1−δn → ∞asn → ∞, we know thatTis not pseudo-contractive.

4In the same as2, we can take 1 ≤ δn → 1 such thatδnxn∈{xi}^∞_i1 for alln ≥ 1.

Therefore we have

Txn−Tδnxn, Jxn−δnxn n

n1xnδnxn, Jxn−δnxn

n

n1 δn

xn, J1−δnxn

n n1 δn

1

1−δn1−δnxn, J1−δnxn

n n1 δn

1

1−δn1−δnxn²

n n1 δn

1

1−δnxn−δnxn².

2.6

Sincen/n1 δn1/1 −δn → −∞as n → ∞, we know that T is not monotone accretive.

3. An Example in Banach Space l

²

In this section, we will give an example which is a weak relatively nonexpansive mapping but not a relatively nonexpansive mapping.

(5)

Example 3.1. LetEl², where

l²

ξ ξ1, ξ2, ξ3, . . . , ξn, . . .: ∞ n1

|xn|²<∞

,

ξ _∞

n1

|ξn|² 1/2

, ∀ξ∈l², ξ, η

^∞

n1

ξnηn, ∀ξ ξ1, ξ2, ξ3, . . . , ξn, . . ., η

η1, η2, η3, . . . , ηn, . . . .

∈l².

3.1

It is well known thatl²is a Hilbert space, so thatl²^∗l². Let{xn} ⊂Ebe a sequence defined by

x0 1,0,0,0, . . ., x1 1,1,0,0, . . ., x2 1,0,1,0,0, . . ., x3 1,0,0,1,0,0, . . .,

...

xn ξn,1, ξn,2, ξn,3, . . . , ξn,k, . . .,

3.2

where

ξn,k

⎧⎨

⎩

1 ifk1, n1,

0 ifk /1, k /n1, 3.3

for alln≥1. Define a mappingT :E → Eas follows:

Tx

⎧⎨

⎩ n

n1xn ifxxn ∃n≥1,

−x ifx /xn∀n≥1. 3.4

Conclusion 1. {xn}converges weakly tox0.

Proof . For anyf ζ1, ζ2, ζ3, . . . , ζk, . . .∈l² l²^∗, we have

fxn−x0 f, xn−x0^∞

k2

ζkξn,kζn1−→0, 3.5

asn → ∞. That is,{xn}converges weakly tox0. The following conclusion is obvious.

(6)

Conclusion 2. xn−xm√

2 for anyn /m.

It follows fromTheorem 2.1and the above two conclusions thatT is a weak relatively nonexpansive mapping but not relatively nonexpansive mapping. We have also the following conclusions: 1 T is not continuous; 2 T is not pseudo-contractive; 3 T is also not monotoneaccretive.

4. An Example in Banach Space L

^p

0 , 1 1 < p < ∞

LetEL^p0,1 1< p <∞, and

xn1− 1

2ⁿ, n1,2,3, . . . .. 4.1

Define a sequence of functions inL^p0,1by the following expression:

fnx

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 2

xn1−xn if xn≤x < xn1xn

2 ,

−2

xn1−xn if xn1xn

2 ≤x < xn1,

0 otherwise

4.2

for alln≥1. Firstly, we can see, for anyx∈0,1, that _x

0

fntdt−→0 _x

0

f0tdt, 4.3

wheref0x≡0. It is wellknown that the above relation4.3is equivalent to{fnx}which converges weakly tof0xin uniformly smooth Banach spaceL^p0,1 1< p <∞. On the other hand, for anyn /m, we have

fn−fm 1

0

fnx−fmx^pdx 1/p

_x_n1

xn

fnx−fmx^pdx _x_m1

xm

fnx−fmx^pdx _1/p

_x_n1

xn

fnx^pdx _x_m1

xm

fmx^pdx 1/p

2 xn1−xn

p

xn1−xn

2 xm1−xm

p

xm1−xm 1/p

2^p

xn1−xn^p−1 2^p xm1−xm^p−1

_1/p

≥2^p2^p^1/p>0.

4.4

(7)

Let

unx fnx 1, ∀n≥1. 4.5

It is obvious thatunconverges weakly tou0x≡1 and

un−umfn−fm≥2^p2^p^1/p >0, ∀n≥1. 4.6 Define a mappingT :E → Eas follows:

Tx

⎧⎨

⎩ n

n1un ifxun ∃n≥1,

−x ifx /un ∀n≥1. 4.7

Since4.6 holds, by usingTheorem 2.1, we know thatT : L^p0,1 → L^p0,1 is a weak relatively nonexpansive mapping but not relatively nonexpansive mapping. We have also the following conclusions:1T is not continuous;2T is not pseudo-contractive;3T is also not monotoneaccretive.

Acknowledgments

This project is supported by the Zhangjiakou City Technology Research and Development Projects Foundation0811024B-5, Hebei Education Department Research Projects Founda- tion2009103, and Hebei North University Research Projects Foundation2009008.

References

1 Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.

2 S. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.

3 W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis. Theory, Methods &

Applications, vol. 70, no. 1, pp. 45–57, 2009.

4 X. Qin and Y. Su, “Strong convergence theorems for relatively nonexpansive mappings in a Banach space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 6, pp. 1958–1965, 2007.

5 Y. Su, J. Gao, and H. Zhou, “Monotone CQ algorithm of fixed points for weak relatively nonexpansive mappings and applications,” Journal of Mathematical Research and Exposition, vol. 28, no. 4, pp. 957–967, 2008.

6 H. Zegeye and N. Shahzad, “Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 7, pp.

2707–2716, 2009.

7 Y. Su, D. Wang, and M. Shang, “Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings,” Fixed Point Theory and Applications, Article ID 284613, 8 pages, 2008.

8 Y. Su, Z. Wang, and H. Xu, “Strong convergence theorems for a common fixed point of two hemi- relatively nonexpansive mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 11, pp. 5616–5628, 2009.