Approximation of fixed points of strongly pseudocontractive mappings in uniformly smooth Banach spaces

APPROXIMATION OF FIXED POINTS OF STRONGLY

PSEUDOCONTRACTIVE MAPPINGS IN UNIFORMLY

SMOOTH BANACH SPACES

XUE ZHIQUN

Received 3 March 2005; Revised 21 March 2006; Accepted 4 April 2006

LetEbe a real uniformly smooth Banach space, andK a nonempty closed convex sub-set ofE. Assume thatT1+T2:K→K is a continuous and strongly pseudocontractive mapping, whereT1:K→K is Lipschitz andT2:K→K has the bounded range map-ping. Then the Ishikawa iterative sequence converges strongly to the unique fixed point ofT1+T2.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

1. Introduction

Let Ebe an arbitrary real Banach space andE∗the dual space on E. The normalized duality mappingJ:E→2E∗is defined by

Jx=f ∈E∗_{:x,f = x · f = f}2_{, (1.1)}

for allx∈E, where·,·denotes the generalized duality pairing. It is well known that if Eis a uniformly smooth Banach space, thenJ is single valued such thatJ(−x)= −J(x), J(tx)=tJ(x) for allt≥0,x∈E; andJ is uniformly continuous on any bounded subset ofE. In the sequel we will denote single-valued normalized duality mapping byj. In the following we give some concepts.

LetT:D(T)→Ebe a mapping with domainD(T) and rangeR(T). A mappingT is said to be pseudocontractive if for anyx,y∈D(T) there exists j(x−y)∈J(x−y) such that

Tx−T y,j(x−y)≤ x−y2_{. (1.2)}

The mappingTis said to be strongly pseudocontractive if for anyx,y∈D(T) there exists j(x−y)∈J(x−y) such that

Tx−T y,j(x−y)≤kx−y2 _(1.3)

for some constantk∈(0, 1).

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 46561, Pages1–6

Recently, Zhou and Jia [5] proved the following result: letEbe a real Banach space with a uniformly convex dualE∗, and letK be a nonempty closed convex and bounded subset ofE. Assume thatT:K→K is a continuous and strong pseudocontraction, the Ishikawa iteration sequence{xn}∞n=1generated by (IS) converges strongly to the unique fixed point ofT. However, whenT is continuous strongly pseudocontractive mapping, one question arises naturally: ifT neither is Lipschitzian nor has the bounded range, whether or not the Ishikawa iterative sequence{xn}∞n=1generated converges strongly to the unique fixed point ofT. It is our purpose in this note to solve the above question by proving the following much more general result:Eis a real uniformly smooth Banach space, andK is a nonempty closed convex subset ofE. Assume thatT:K→K is a con-tinuous and strong pseudocontraction, and T neither is Lipschizian nor has the bounded range, then the Ishikawa iteration sequence converges strongly to the unique fixed point ofT.

Lemma 1.1 [5]. LetEbe a real Banach space, then for allx,y∈E, there exists j(x+y)∈

J(x+y) such that

x+y2_{≤ x}2_{+ 2y,j(x+y). (1.4)}

Lemma 1.2 [5]. Let{ρn}∞n=1be a nonnegative real sequence satisfying ρn+1≤

1−λnρn+σn, (1.5)

whereλn∈[0, 1],∞_n=1λn= ∞andσn=o(λn). Thenρn→0 asn→ ∞.

2. Main results

Now we prove the main results of this note, In the sequel, we always assume thatEis a real uniformly smooth Banach space.

Theorem 2.1. LetKbe a nonempty closed convex subset ofE. Assume thatT1+T2:K→K is a continuous and strongly pseudocontractive mapping, whereT1:K→Kis Lipschitz and T2:K→Khas the bounded range mapping. Let{αn}∞

n=1and{βn}∞n=1be two real sequences in [0, 1] satisfying the following conditions: (i)αn,βn→0 asn→ ∞; (ii)∞_n=1αn= ∞. Then the Ishikawa iterative sequence generated from an arbitraryx1∈Kby (IS1),

xn+1=1−αnxn+αnT1+T2yn,

yn=1−βnxn+βnT1+T2xn,

(2.1)

converges strongly to the unique fixed point ofT1+T2.

Proof. The existence of a fixed point follows from Deimling [4]. Letqbe a fixed point of T1+T2. SinceT1+T2is strongly pseudocontractive, thus for allx,y∈K,

T1+T2x−T1+T2y,J(x−y)≤kx−y2_{, (2.2)}

(2.1), we have

_{yn−q ≤}

1−βn xn−q +βn T1xn−T1q + T2xn−T2q ≤1−βn xn−q +βnL xn−q +M

≤1−βn+βnL xn−q +βnM.

(2.3)

SetAn= J((xn+1−q)/(1+xn−q))−J((yn−q)/(1+xn−q)),Dn= J((yn−q)/(1 +

xn−q))−J((xn−q)/(1 +xn−q)), thenAn→0,Dn→0 asn→ ∞. Indeed{(yn−

q)/(1 +xn−q)}, {(xn−q)/(1 +xn−q)}, and {(xn+1−q)/(1 +xn−q)} are all

bounded, using that J is uniformly continuous on bounded subset, hence An→0 as n→ ∞andDn→0 asn→ ∞. ApplyingLemma 1.1, we obtain

_xn+1−q 2

= 1−αnxn−q+αnT yn−Tq 2

≤1−αn2 xn−q 2+ 2αnT yn−Tq,Jxn+1−q

≤1−αn2 xn−q 2+ 2αnT yn−Tq,Jyn−q

+ 2αnT yn−Tq,Jxn+1−q

−Jyn−q

≤1−αn2 xn−q 2+ 2αnk yn−q 2

+ 2αn

T yn−Tq,J

xn+1−q 1 + xn−q

−J

yn−q

1 + xn−q

×1 + xn−q

≤1−αn2 xn−q 2+ 2αnk yn−q 2

+ 2αnAnL yn−q +M1 + xn−q ).

(2.4)

Again usingLemma 1.1, we obtain

_yn−q 2

= 1−βnxn−q+βnTxn−Tq 2

≤1−βn2 xn−q 2+2βnTxn−Tq,Jyn−q

≤1−βn2 xn−q 2+ 2βnTxn−Tq,Jyn−q−Jxn−q

+ 2βnTxn−Tq,Jxn−q

≤1−βn2 xn−q 2+2βn

Txn−Tq,J

_yn−q

1 + xn−q

−J_{1 +}x n−_x q

n−q

×1+ xn−q +2kβn xn−q 2

≤1−βn2+2kβn xn−q 2

≤1−βn2+ 2kβn xn−q 2

+ 2βnL xn−q +MDn1 + xn−q

≤1−βn2+2kβn xn−q 2+2βn(L+M)1+ xn−q 2

≤1−βn2+2kβn xn−q 2+4βn(L+M)1 + xn−q 2

≤1−βn2+2kβn+4βn(L+M) xn−q 2+4βn(L+M).

(2.5)

Furthermore, we have the following estimates to a part of (2.4):

2αnAnL yn−q +M1 + xn−q

≤2αnLAn1−βn+βnL xn−q LMAnβn+MAn1 + xn−q

≤2LAnαn1−βn+βnL xn−q 2+ 2MαnAnLβn+ 1

+ 2LAnαn1−βn+βnL+ 2MAnαnLβn+ 1 xn−q

≤(2L+M)Anαn1 +βnL xn−q 2+ (3L+M)Anαn1 +Lβn

≤En xn−q 2+En,

(2.6)

whereEn=(3L+M)Anαn(1 +Lβn). Substituting (2.5) and (2.6) in (2.4), we have

_xn+1−q 2

≤1−αn2+ 2αnk1−βn2+ 2kβn+ 4βn(L+M)+En xn−q 2

+ 8kαnβn(L+M) +En=1−2(1−k)αn+Fn xn−q 2+Gn,

(2.7)

where Fn =α2n−4kαnβn+ 2kαnβ2n+ 4k2αnβn+ 8kαnβn(L+M) + 2kαnEn, Gn=8k(L

+M)αnβn+En, thenFn=o(αn),Gn=o(αn). Hence, we may choose a large positive inte-gerNsuch that for alln≥N,

Fn<1−₂kαn. (2.8)

Thus the above inequality (2.7) yields

_xn+1−q 2

≤

1−3(1−k)

2 αn xn−q 2

+Gn. (2.9)

ByLemma 1.2we see that asxn−q →0 asn→ ∞. The proof of theorem is completed.

Remark 2.2. Concrete the following example: let E=(−∞, +∞), K=[0, +∞), where

by

T2x=

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

−

1−(x−1)2

3 , ifx∈[0, 1],

−1

3, ifx∈(1, +∞).

(2.10)

ThenT1is Lipschitz,T2has the bounded range, andT1+T2is strongly pseudocontractive mapping. ButT1+T2neither is Lipschitzian nor has a bounded range.

Remark 2.3. Theorem 2.1contains a good number of the known results as its special cases. In particular, if the mappingT considered here satisfies one of the following as-sumptions: (i)T:K→Kis a Lipschitzian; (ii)Thas the bounded range, thenTsatisfied the conditions ofTheorem 2.1.

Remark 2.4. In [1], Bogin proved thatTis strongly pseudocontractive if and only if (I−

T) is strongly accretive, whereI denotes the identity operator. It is well known that ifT is continuous and strongly accretive, thenTis surjective, so that, for any given f ∈E, the equationTx=f has unique solution.

Theorem 2.5. Assume thatT=T1+T2:E→Eis a continuous strongly accretive operator, whereT1:E→Eis Lipschitz, T2:E→Ehas the bounded range operator. For any given

f ∈E, defineS:E→EbySx= f −Tx+xfor allx∈E. Let{αn}∞

n=1 and{βn}∞n=1be two real sequences [0, 1] in satisfying the conditions: (i)αn,βn→0 asn→ ∞; (ii)∞_n=1αn= ∞. Then the Ishikawa iterative sequence generated from an arbitraryx1∈Eby (IS2),

xn+1=

1−αnxn+αnSyn, yn=1−βnxn+βnSxn,

(2.11)

converges strongly to the unique solution of the equationTx=f.

Proof. By virtue ofRemark 2.3, the equationTx= f has unique solution. SetS1x=x− T1x,S2x= −T2x,x∈E. ThenS1 is Lipschitz,S2has the bounded range operator, and Sx=S1x+S2x+ f. HenceS is a continuous and strongly pseudocontractive mapping.

We obtain directly the conclusion fromTheorem 2.1.

Acknowledgments

The author is highly grateful to the referee for valuable comments and good suggestions for improving the note. This project is supported by the National Science Foundation of China and Shijiazhuang Railway College Sciences Foundation.

References

[1] J. Bogin, On strict pseudo-contractions and a fixed point theorem, Technion Preprint MT-29, Haifa, 1974.

[3] C. E. Chidume and M. O. Osilike, Ishikawa iteration process for nonlinear Lipschitz strongly

accre-tive mappings, Journal of Mathematical Analysis and Applications 192 (1995), no. 3, 727–741.

[4] K. Deimling, Zeros of accretive operators, Manuscripta Mathematica 13 (1974), no. 4, 365–374. [5] H. Zhou and Y. Jia, Approximation of fixed points of strongly pseudocontractive maps without

Lip-schitz assumption, Proceedings of the American Mathematical Society 125 (1997), no. 6, 1705–

1709.

Xue Zhiqun: Department of Mathematics, Shijiazhuang Railway College, Shijiazhuang 050043, China