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R E S E A R C H

Open Access

Strong convergence of approximated

iterations for asymptotically

pseudocontractive mappings

Yonghong Yao

1

, Mihai Postolache

2

and Shin Min Kang

3*

*Correspondence:

smkang@gnu.ac.kr

3Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, Korea Full list of author information is available at the end of the article

Abstract

The asymptotically nonexpansive mappings have been introduced by Goebel and Kirk in 1972. Since then, a large number of authors have studied the weak and strong convergence problems of the iterative algorithms for such a class of mappings. It is well known that the asymptotically nonexpansive mappings is a proper subclass of the class of asymptotically pseudocontractive mappings. In the present paper, we devote our study to the iterative algorithms for finding the fixed points of asymptotically pseudocontractive mappings in Hilbert spaces. We suggest an iterative algorithm and prove that it converges strongly to the fixed points of asymptotically pseudocontractive mappings.

MSC: 47J25; 47H09; 65J15

Keywords: asymptotically pseudocontractive mappings; iterative algorithms; fixed point; Hilbert space

1 Introduction

LetHbe a real Hilbert space with inner product·,·and norm · , respectively. LetC be a nonempty, closed, and convex subset ofH. LetT:CCbe a nonlinear mapping. We useF(T) to denote the fixed point set ofT.

Recall thatTis said to beL-Lipschitzianif there existsL>  such that

TxTyLxy

for all x,yC. In this case, ifL< , then we callT anL-contraction. IfL= , we callT nonexpansive.Tis said to beasymptotically nonexpansiveif there exists a sequence{kn} ⊂ [,∞) withlimn→∞kn=  such that

TnxTnyknxy (.)

for allx,yCand alln≥.

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [] in . They proved that, ifCis a nonempty bounded, closed, and convex subset of a uniformly convex Banach spaceE, then every asymptotically nonexpansive self-mapping TofChas a fixed point. Further, the setF(T) of fixed points ofTis closed and convex.

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Since then, a large number of authors have studied the following algorithms for the iter-ative approximation of fixed points of asymptotically nonexpansive mappings (see,e.g., [–] and the references therein).

(A) The modified Mann iterative algorithm. For arbitraryx∈C, the modified Mann iteration generates a sequence{xn}by

xn+= ( –αn)xn+αnTnxn, ∀n≥. (.)

(B) The modified Ishikawa iterative algorithm. For arbitraryx∈C, the modified Ishikawa iteration generates a sequence{xn}by

⎧ ⎨ ⎩

yn= ( –βn)xn+βnTnxn,

xn+= ( –αn)xn+αnTnyn, ∀n. (.)

(C) TheCQalgorithm. For arbitraryxC, theCQalgorithm generates a sequence {xn}by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

yn=αnxn+ ( –αn)Tnxn,

Cn={zC:ynz≤ xnz+θn}, Qn={z∈C:xn–z,xxn ≥}, xn+=PCnQn(x), ∀n≥.

(.)

An important class of asymptotically pseudocontractive mappings generalizing the class of asymptotically nonexpansive mapping has been introduced and studied by Schu in ; see [].

Recall thatT:CCis called anasymptotically pseudocontractive mappingif there ex-ists a sequence{kn} ⊂[,∞) withlimn→∞kn=  for which the following inequality holds:

TnxTny,xyknxy (.)

for allx,yCand alln≥. It is clear that (.) is equivalent to

TnxTnyknxy+ xTnx yTny(.)

for allx,yCand alln≥.

Recall also thatTis calleduniformly L-Lipschitzianif there existsL>  such that

TnxTnyLxy

for allx,yCand alln≥.

Now, we know that the class of asymptotically nonexpansive mappings is a proper sub-class of the sub-class of asymptotically pseudocontractive mappings. If we define a mapping T: [, ]→[, ] by the formulaTx= ( –x), then we can verify thatTis asymptotically

pseudocontractive but it is not asymptotically nonexpansive.

In order to approximate the fixed point of asymptotically pseudocontractive mappings, the following two results are interesting.

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Theorem . Let H be a Hilbert space, C be a nonempty closed bounded and convex subset of H.Let T be a completely continuous,uniformly L-Lipschitzian and asymptoti-cally pseudocontractive self-mapping of C with{kn} ⊂[,∞)andn=(qn– ) <∞,where qn= (kn– )for all n≥.Let{αn} ⊂[, ]and{βn} ⊂[, ]be two sequences satisfying  <αnβnb<L–(

 +L– )for all n.Then the sequence{xn}generated by the modified Ishikawa iteration(.)converges strongly to some fixed point of T.

Another one is due to Chidume and Zegeye [] who introduced the following algorithm in .

Let a sequence{xn}be generated fromx∈Cby

xn+=λnθnx+ ( –λnλnθn)xn+λnTnxn, ∀n≥, (.)

where the sequences{λn}and{θn}satisfy (i) ∞n=λnθn=∞andλn( +θn)≤; (ii) λn

θn →,θn→and

(θn– θn –)

λnθn →;

(iii) kn–kn–

λnθn →;

(iv) kn–

θn →.

They gave the strong convergence analysis for the above algorithm (.) with some fur-ther assumptions on the mappingT in Banach spaces.

Remark . Note that there are some additional assumptions imposed on the underlying spaceCand the mappingTin the above two results. In (.), the parameter control is also restricted.

Inspired by the results above, the main purpose of this article is to construct an iterative method for finding the fixed points of asymptotically pseudocontractive mappings. We construct an algorithm which is based on the algorithms (.) and (.). Under some mild conditions, we prove that the suggested algorithm converges strongly to the fixed point of asymptotically pseudocontractive mappingT.

2 Preliminaries

It is well known that in a real Hilbert spaceH, the following inequality and equality hold:

x+y≤ x+ y,x+y, ∀x,yH (.)

and

tx+ ( –t)y=tx+ ( –t)yt( –t)xy(.)

for allx,yHandt∈[, ].

Lemma .([]) Let C be a nonempty bounded and closed convex subset of a real Hilbert space H.Let T:CC be a uniformly L-Lipschtzian and asymptotically pseudocontrac-tion.Then IT is demiclosed at zero.

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all k≥.For every nN,define an integer sequence{τ(n)}as

τ(n) =max{i≤n:rni<rni+}.

Thenτ(n)→ ∞as n→ ∞,and for all nN

max{rτ(n),rn} ≤rτ(n)+.

Lemma .([]) Assume that{an}is a sequence of nonnegative real numbers such that

an+≤( –δn)an+ξn,

where{δn}is a sequence in(, )and{ξn}is a sequence such that (i) ∞n=δn=∞;

(ii) lim supn→∞ξn

δn≤or

n=|ξn|<∞. Thenlimn→∞an= .

3 Main results

Now we introduce the following iterative algorithm for asymptotically pseudocontractive mappings.

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let T : CCbe a uniformlyL-Lipschitzian asymptotically pseudocontractive mapping satis-fying∞n=(kn– ) <∞. Letf :CCbe aρ-contractive mapping. Let{αn},{βn}, and{γn}

be three real number sequences in [, ].

Algorithm . Forx∈C, define the sequence{xn}by

⎧ ⎨ ⎩

yn= ( –γn)xn+γnTnxn,

xn+=αnf(xn) + ( –αnβn)xn+βnTnyn, ∀n≥.

(.)

Next, we prove our main result as follows.

Theorem . Suppose that F(T)=∅.Assume the sequences{αn},{βn},and{γn}satisfy the following conditions:

(i) limn→∞αn= and

n=αn=∞; (ii) αn+βnγnand <lim infn→∞βn; (iii)  <aγnb<√ 

(+kn)+L++kn for alln≥.

Then the sequence {xn}defined by(.)converges strongly to u=PF(T)f(u),which is the unique solution of the variational inequality(I–f)x∗,xx∗ ≥for all xF(T). Proof From (.), we have

xn+u

=αnf(xn) + ( –αnβn)xn+βnTnynu ≤αn f(xn) –u

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=αn f(xn) –u

+ ( –αn)

 –αnβn  –αn

(xn–u) + βn  –αn

Tnynu

≤( –αn)( –αnβn)(xn–u)  –αn

+βn(T nynu)  –αn

+αnf(xn) –u. (.)

Using the equality (.), we get

( –αnβn)(xnu)  –αn

+βn(T nynu)  –αn

= –αnβn  –αn xn

u+ βn  –αn

Tnynu

βn( –αnβn) ( –αn)xnT

nyn. (.)

Picking upy=uin (.) we deduce

Tnxu≤knxu+xTnx (.)

for allxC.

From (.), (.), and (.), we obtain

Tnynu

knynu+ynTnyn

=kn( –γn)xn+γnTnxnu+( –γn)xn+γnTnxnTnyn

=kn( –γn)(xnu) +γn Tnxnu

+( –γn) xnTnyn

+γn TnxnTnyn

=kn

( –γn)xnu+γnTnxnu

γn( –γn)xnTnxn

+ ( –γn)xnTnyn+γnTnxnTnyn–γn( –γn)xnTnxn ≤kn( –γn)xnu+γn knxnu+xnTnxn

γn( –γn)xnTnxn+ ( –γn)xnTnyn

+γnTnxnTnyn

γn( –γn)xnTnxn. (.)

By (.), we have

xnyn=γnxnTnxn. (.)

Noting thatT is uniformlyL-Lipschitzian, from (.) and (.), we deduce

Tnynu

kn( –γn)xnu+γn knxnu+xnTnxn

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+γnLxn–yn–γn( –γn)xnTnxn

=kn( –γn)xnu+γn knxnu+xnTnxn

γn( –γn)xnTnxn+ ( –γn)xnTnyn

+γnLxnTnxn–γn( –γn)xnTnxn

= + (knγn+ )(kn– )

xnu+ ( –γn)xnTnyn

γn  –γnknγnγnLxnTnxn

. (.)

By condition (iii), we know thatγnb<√  (+kn)+L+kn+

for alln. Then we deduce that  –γnknγnγnL>  for alln≥.

Therefore, from (.), we derive

Tnynu≤ + (knγn+ )(kn– )

xnu

+ ( –γn)xnTnyn

. (.)

Note thatβn≤ –αnand substituting (.) to (.), we obtain

( –αnβn)(xnu)  –αn

+βn(T ny

nu)  –αn

βn  –αn

 + (knγn+ )(kn– )

xn–u+ ( –γn)xnTnyn

+ –αnβn  –αn xn

u–βn( –αnβn) ( –αn)xnT

nyn

=

 + βn  –αn

(knγn+ )(kn– )

xn–u+βn(αn+βnγn) ( –αn)xnT

nyn

 + βn  –αn

(knγn+ )(kn– )

xn–u

≤ + (knγn+ )(kn– )

xnu.

Thus,

( –αnβn)(xnu)  –αn

+βn(T nynu)  –αn

≤ + (knγn+ )(kn– )xnu ≤ + (knγn+ )(kn– )

xn–u. (.)

Sincekn→, without loss of generality, we assume thatkn≤ for alln≥. It follows from (.) and (.) that

nxn+u

αnf(xn) –u+ ( –αn)

 + (knγn+ )(kn– )

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+ ( –αn) + (knγn+ )(kn– )

xn–uαnρxn–u+αnf(u) –u

+ ( –αn)

 + (knγn+ )(kn– )

xnuαnf(u) –u+

 – ( –ρ)αn

xnu

+ (knγn+ )(kn– )xn–u

≤( –ρ)αn

f(u) –u  –ρ +

 – ( –ρ)αn

xnu+ (kn– )xnu.

An induction induces

xn+–u ≤ + (kn– )max

xn–u,f(u) –u  –ρ

n

j=

 + (kj– )max

x–u,f(u) –u  –ρ

.

This implies that the sequence{xn}is bounded by the condition∞n=(kn– ) <∞. From (.) and (.), we have

xn+–u

=( –αn)(xnu) –βn xnTnyn

+αn f(xn) –u ≤( –αn)(xnu) –βn xnTnyn

 + αn

f(xn) –u,xn+u

=( –αn)(xnu)– βn( –αn)xnTnyn,xnu

+βnxnTnyn+ αn

f(xn) –u,xn+u. (.)

From (.), we deduce

xnTnyn,xnu

γnxnTnyn+γn  –γnknγnγnLxnTnxn

– (knγn+ )(kn– )xn–u ≥γn  –γnknγnγnLxnTnxn

+γnxnTnyn

. (.)

By condition (ii), we have ≥γnαn+βnαnγn+βnfor alln≥. Hence, by (.) and (.), we get

xn+–u

≤( –αn)xnu–βn( –αn)γnxnTnyn

+βnxnTnyn

βn( –αn)γn  –γnknγnγnLxnTnxn

+ αn

f(xn) –u,xn+u ≤( –αn)xnu+ αn

f(xn) –u,xn+u

βn( –αn)γn  –γnknγnγnLxnTnxn

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It follows that

xn+u–xnu+βn( –αn)γn  –γnknγnγnLxnTnxn

αn

f(xn) –u,xn+u

xnu

.

Since{xn}and{f(xn)}are bounded, there existsM>  such thatsupn{f(xn) –u,xn+u–xn–u} ≤M. So,

βn( –αn)γn  –γnknγnγnLxnTnxn

+xn+–u–xn–u≤αnM. (.)

Next, we consider two possible cases.

Case . Assume there exists some integerm>  such that{xnu}is decreasing for allnm.

In this case, we know thatlimn→∞xn–uexists. From (.), we deduce

βn( –αn)γn  –γnknγnγnLxnTnxn

≤ xn–u–xn+–u+Mαn. (.)

By conditions (ii) and (iii), we havelim infn→∞βn( –αn)γn( –γnknγnγnL) > . Thus, from (.), we get

lim

n→∞xnT

nxn= . (.)

It follows from (.) and (.) that

lim

n→∞xnyn= . (.)

SinceT is uniformlyL-Lipschitzian, we haveTny

nTnxn ≤Lxnyn. This together with (.) implies that

lim

n→∞T

nynTnxn= . (.)

Note that

xnTnynxnTnxn+TnxnTnyn. (.)

Combining (.), (.), and (.), we have

lim

n→∞xnT

nyn= . (.)

From (.), we deduce

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Therefore,

lim

n→∞xn+–xn= . (.)

SinceTis uniformlyL-Lipschitzian, we derive

xn–Txn

xnTnxn+TnxnTxnxnTnxn+LTn–xnxnxnTnxn+LTn–xnTn–xn–

+LTn–xn–xn–+Lxn–xnxnTnxn+ L+L

xnxn–+LTn–xn–xn–. (.)

By (.), (.), and (.), we have immediately

lim

n→∞xn–Txn= . (.)

Since{xn}is bounded, there exists a subsequence{xnk}of{xn}satisfying

xnk→ ˜xC

and

lim sup

n→∞

f(u) –u,xnu

= lim

k→∞

f(u) –u,xnku

.

Thus, we use the demiclosed principle ofT(Lemma .) and (.) to deduce

˜ xF(T).

So,

lim sup

n→∞

f(u) –u,xnu= lim

k→∞

f(u) –u,xnku

=f(u) –u,x˜–u ≤.

Returning to (.) to obtain

xn+–u≤( –αn)xnu+ αn

f(xn) –u,xn+u

= ( –αn)xnu+ αn

f(xn) –f(u),xn+u

+ αn

f(u) –u,xn+u

≤( –αn)xnu+ αnρxn–uxn+u

+ αn

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≤( –αn)xnu+αnρ xn–u+xn+–u

+ αn

f(u) –u,xn+u.

It follows that

xn+u≤

 – ( –ρ)αn

xnu

+ αn  –αnρ

f(u) –u,xn+u. (.)

In Lemma ., we takean=xn+–u,δ

n= ( –ρ)αn, andξn=–ααnnρf(u) –u,xn+u. We can easily check that∞n=δn=∞andlim supn→∞ξδnn≤. Thus, we deduce thatxnu.

Case . Assume there exists an integernsuch thatxn–u ≤ xn+–u. At this case,

we setωn={xn–u}. Then we haveωn≤ωn+. Define an integer sequence{τn}for all

nnas follows:

τ(n) =max{l∈N|n≤ln,ωlωl+}.

It is clear thatτ(n) is a non-decreasing sequence satisfying

lim

n→∞τ(n) =∞ and

ωτ(n)≤ωτ(n)+

for allnn. From (.), we get

lim

n→∞(n)–Txτ(n)= .

This implies thatωw(xτ(n))⊂F(T). Thus, we obtain

lim sup

n→∞

f(u) –u,xτ(n)u≤. (.)

Sinceωτ(n)≤ωτ(n)+, we have from (.) that

ωτ(n)ωτ(n)+≤ – ( –ρ)ατ(n)

ωτ(n)+ ατ(n)  –ατ(n)ρ

f(u) –u,xτ(n)+–u

.

It follows that

ωτ(n)≤ 

( –ατ(n)ρ)( –ρ)

f(u) –u,xτ(n)+–u

. (.)

Combining (.) and (.), we have

lim sup

n→∞

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and hence

lim

n→∞ωτ(n)= . (.)

From (.), we obtain

xτ(n)+–u≤ – ( –ρ)ατ(n)

xτ(n)–u

+ ατ(n)  –ατ(n)ρ

f(u) –u,xτ(n)+–u

.

It follows that

lim sup

n→∞ ωτ(n)+≤lim supn→∞ ωτ(n).

This together with (.) imply that

lim

n→∞ωτ(n)+= .

Applying Lemma . to get

≤ωn≤max{ωτ(n),ωτ(n)+}.

Therefore,ωn→. That is,xnu. The proof is completed.

Since the class of asymptotically nonexpansive mappings is a proper subclass of the class of asymptotically pseudocontractive mappings and asymptotically nonexpansive mapping TisL-Lipschitzian withL=supnkn. Thus, from Theorem ., we get the following corol-lary.

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H.Let T:CC be an asymptotically nonexpansive mapping satisfyingn=(kn– ) <∞. Sup-pose that F(T)=∅.Let f :CC be aρ-contractive mapping.Let{αn},{βn},and{γn}be three real number sequences in[, ].Assume the sequences{αn},{βn},and{γn}satisfy the following conditions:

(i) limn→∞αn= and

n=αn=∞; (ii) αn+βnγnand <lim infn→∞βn; (iii)  <aγnb<√ 

(+L)+L++Lfor alln≥,whereL=supnkn.

Then the sequence {xn}defined by(.)converges strongly to u=PF(T)f(u),which is the unique solution of the variational inequality(I–f)x∗,xx∗ ≥for all xF(T).

Remark . Our Theorem . does not impose any boundedness or compactness as-sumption on the spaceCor the mappingT. The parameter control conditions (i)-(iii) are mild.

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4 Conclusion

This work contains our dedicated study to develop and improve iterative algorithms for finding the fixed points of asymptotically pseudocontractive mappings in Hilbert spaces. We introduced our iterative algorithm for this class of problems, and we have proven its strong convergence. This study is motivated by relevant applications for solving classes of real-world problems, which give rise to mathematical models in the sphere of nonlinear analysis.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Author details

1Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China.2Faculty of Applied Sciences,

University Politehnica of Bucharest, Splaiul Independentei 313, Bucharest, 060042, Romania.3Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, Korea.

Received: 19 February 2014 Accepted: 21 April 2014 Published:02 May 2014

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