Scholarly article on topic 'On the convergence of hybrid projection algorithms for total quasi-asymptotically pseudo-contractive mapping'

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Academic research paper on topic "On the convergence of hybrid projection algorithms for total quasi-asymptotically pseudo-contractive mapping"

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On the convergence of hybrid projection algorithms for total quasi-asymptotically pseudo-contractive mapping

Zi-Ming Wang1* and Yongfu Su2

"Correspondence:

wangziming@ymail.com

1 Department of Foundation,

Shandong Yingcai University, Jinan,

250104, P.R. China

Full list of author information is

available at the end of the article

Abstract

In this paper, a hybrid projection algorithm for a total quasi-asymptotically pseudo-contractive mapping is introduced in a Hilbert space. A strong convergence theorem of the proposed algorithm to a fixed point of a total quasi-asymptotically pseudo-contractive mapping is proved. Our main result extends and improves many corresponding results. MSC: 47H05; 47H09

Keywords: total quasi-asymptotically pseudo-contractive; hybrid projection algorithm; fixed point; Hilbert space

ft Spri

ringer

1 Introduction

Throughout this paper, we always assume that H is a real Hilbert space, whose inner product and norm are denoted by <•, •) and || • ||. The symbol — is denoted by a strong convergence. Let C be a nonempty closed and convex subset of H, and let T : C — C be a mapping. In this paper, we denote the fixed point set of T by F(T), that is, F(T) := {x e C: Tx = x}.

Recall that T is said to be asymptotically nonexpansive if there exists a sequence {kn} c [1, to) with kn — 1 as n — to such that

|| Tnx - Tny|| < knllx-y||, "in > 1,Vx,y e C. (1.1)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalization of the class of nonexpansive mappings.

T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

limsup sup (||Tnx - Tny|| - ||x -y||) < 0. (1.2)

n—>to x,yeC

Noticing that if we define

Pn = max J 0, sup (| |Tnx - Tny|| - ||x - y||)}, (.3)

1 x,yeC '

© 2013 Wang and Su; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.Org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

then pn — 0 as n — to. It follows that (1.2) is reduced to

\\Tnx - Tny|| < ||x - y|| + pn, ^n > 1, Vx, y e C. (1.4)

The class of mappings, which are asymptotically nonexpansive in the intermediate sense, was introduced by Bruck et al. [2] (see also [3]). It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings.

Recall that T is said to be asymptotically pseudocontractive if there exists a sequence {kn} c [1, to) with kn — 1 as n — to such that

(Tnx - Tny,x -y) < kn ||x -y||2, Vx,y e C. (.5)

It is not hard to see that (1.5) is equivalent to

|| Tnx - Tny||2 < (2kn - 1)||x -y||2 + ||x -y - (Tnx - Tny) ||2, Vn > 1,x,y e C. (1.6)

The class of an asymptotically pseudocontractive mapping was introduced by Schu [4] (see also [5]). In [6], Rhoades gave an example to show that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings, see [6] for more details. Zhou [7] showed that every uniformly Lipschitz and asymptotically pseudocontractive mapping, which is also uniformly asymptotically regular, has a fixed point.

T is said to be an asymptotically pseudocontractive mapping in the intermediate sense if there exists a sequence {kn} c [1, to) with kn — 1 as n — to such that

limsup sup ((Tnx - Tny,x -y) - kn||x -y||2) < 0. (1.7)

n—>to xyyeC

Tn = max J 0, sup ((Tnx - Tny, x - y) - kn||x - y||2)}. (1.8)

I x,yeC '

It follows that Tn — 0 as n — to. Then, (1.8) is reduced to the following:

(Tnx - Tny,x -y) < kn ||x -y||2 + Tn, Vn > 1,x,y e C. (1.9)

The class of asymptotically pseudocontractive mappings in the intermediate sense was introduced by Qin et al. [8].

Recall that T is said to be total asymptotically pseudocontractive if there exist sequences {kn}, {vn} c [0, to) with kn, vn — 0 as n —to such that

(Tnx - Tny,x -y) < ||x -y||2 + kn$(Hx -y||) + vn, Vn > 1,x,y e C, (1.10)

where $ : [0, to) — [0, to) is a continuous and strictly increasing function with ^(0) = 0. The class of a total asymptotically pseudocontractive mapping was introduced by Qin [9].

It is easy to see that (1.10) is equivalent to the following: for all n > 1, x, y e C,

|| Tnx - Tny||2 < ||x -y||2 + 2kn$(||x -y||) + ||x -y - (Tnx - Tny)||2 + 2vn. (1.11) If $(k) = k2, then (1.10) is reduced to

(Tnx - Tny,x-y) < (1 + kn)||x -y||2 + vn, Vn > 1,x,y e C. (1.12)

vn = max!0, sup ((Tnx - Tny,x -y) -(1 + kn)||x -y||2)!. (1.13)

^ x,yeC '

If $(k) = k2, then the class of total asymptotically pseudocontractive mappings is reduced to the class of asymptotically pseudocontractive mappings in the intermediate sense. In this paper, we introduce and study the following mapping.

Definition 1.1 A mapping T : C — C is said to be total quasi-asymptotically pseudocontractive if F(T) = 0, and there exist sequences {^n} c [0, to) and {fn} c [0, to) with ¡xn — 0 and fn — 0 as n — to such that

(Tnx -p,x -p) < ||x -p||2 + iin$(Hx -p||) + fn, Vn > 1,x e C,p e F(T), (1.14)

where $ : [0, to) — [0, to) is a continuous and strictly increasing function with $(0) = 0.

It is easy to see that (1.14) is equivalent to the following:

||Tnx-p||2 < ||x -p||2 + 2/Xn$(||x -p||)

+ ||x- Tnx\2 + 2fn, Vn > 1,x e C,p e F(T). (1.15)

Remark 1 It is clear that every total asymptotically pseudo-contractive mapping with F(T) = 0 is total quasi-asymptotically pseudo-contractive, but the converse maybe not true.

Remark 2 If $(k) = k2, the (1.14) is reduced to

(Tnx -p,x -p) < (1 + ^n)|x -p||2 + fn, Vn > 1,x e C,p e F(T). (1.16)

Remark 3 Put

fn = max J 0, sup ((Tnx - p, x - p) -(1 + ||x - p||2)}. (1.17)

1 x,yeC '

If $(k) = k2, then the class of total quasi-asymptotically pseudo-contractive mappings is reduced to the class of quasi-asymptotically pseudo-contractive mappings in the intermediate sense.

Recently, the iterative approximation of fixed points for asymptotically pseudo-contractive mappings, total asymptotically pseudo-contractive mappings in Hilbert, or Banach spaces has been studied extensively by many authors, see, for example, [7, 9-13]. In this paper, we shall consider and study a total quasi-asymptotically pseudo-contractive mapping as a generalization of (total) asymptotically pseudo-contractive mappings. Furthermore, we shall introduce an iterative algorithm for finding a fixed point of a total quasi-asymptotically pseudo-contractive mapping.

2 Preliminaries

A mapping T : C ^ C is said to be uniformly L-Lipschitzian if there exists some L > 0 such that

\\Tnx - Tny|| < L||x - y||, Vx, y e C, n > 1. (2.1)

Let C be a nonempty closed convex subset of a real Hilbert space H. For every point x e H, there exists a unique nearest point in C, denotedby PCx, suchthat ||x - PCx|| < ||x -y|| holds for all y e C, where PC is said to be the metric projection of H onto C. In order to prove our main results, we also need the following lemmas.

Lemma 2.1 [14] Let C be a nonempty closed convex subset of a real Hilbert space H and letPC be the metric projectionfrom H onto C (i.e., forx e H, PC is the only point in C such that ||x -PCx|| = inf{||x -z||: z e C}). Given x e Handze C, then z = PCx ifandonlyifthe relation

<x - z, y - z) < 0, Vy e C (2.2)

holds.

Lemma 2.2 Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let T : C ^ C be a uniformly L-Lipschitzian and total quasi-asymptotically pseudo-contractive mapping with F (T) = 0. Suppose there exist positive constants M and M* such that ) < M*Z2 for all Z > M. Then F(T) is a closed convex subset ofC.

Proof Since $ is an increasing function, it follows that $(Z) < $(M) if Z < M and $(Z) < M*Z2 if Z > M. In either case, we can always obtain that

$(Z) < $(M)+M*Z2. (.3)

Since T is uniformly L-Lipschitzian continuous, F(T) is closed. We need to show that F(T) is convex. To this end, letpi e F(T) (i = 1,2), and writep = tp1 + (1- t)p2 for t e (0,1). We take a e (0, ), and defineyan = (1-a)p + aTnp for each n e N. Then, forallz e F(T), we have from (2.3) that

||p - Tnp||2 = p - Tnp, p - Tnp)

= -ip - ya,n,p - T"p) a

= - ip - ya,n,p - Tnp - (ya,n - Tnya,n))

+ 1 (p - ya,n, ya,n - T"ya,n) a * 1

1+ L 2 1 , n ,

-|p -ya,n| + "[p - z,ya,n - T ya,n)

+ 1 (z -ya,n,ya,n - Tnya,n) a

1 + L 2 1 / n \

-|p -ya,n|| + "(p - z,ya,n - Tya,n)

+ 1 (z - ya,n, ya,n - Z + Z - Tnya,n) a

1 + L ||p -ya,n ||2 + 1 (p - Z,ya,n - Tnya,n)

+ a^n [$ (M) + (diam C)^ + fn}

= a(1 + L) |p - Tnp | 2 + a(p - Z,ya,n - Tnya,n) + 1 {^n [$ (M) + M* (diam C)2 ] + fn}.

This implies that

a[1 - a(1 + L)] ||p - Tnp||2 < (p - z,ya,n - Tnya,,

+ /Xn [$(M) + M*(diam C)^ + fn. (2.4)

Now, we take z = pi (i = 1,2) in (2.4), multiplying t and (1 -1) on the both sides of the above inequality (2.4), respectively, and adding up, and we can get

a[1-a(1+ L)]||p - Tnp||2 < /n[$(M) + M*(diamC)2] + fn. (2.5)

Letting n —to in (2.5), we obtain Tnp — p. Since T is continuous, we have Tn+1p — Tp as n — to, therefore, p = Tp. This proves that F(T) is a closed convex subset of C. □

3 Main results

In this section, we shall give our main results of this paper.

Theorem 3.1 Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let T : C — C be a uniformly L-Lipschitzian and total quasi-asymptotically pseudo-contractive mapping with F (T) = 0. Suppose that there exist positive constants M and M* such that $(Z) < M*Z2 for all Z > M. Let {xn} be a sequence generated by thefol-lowing iterative scheme:

xi e C chosen arbitrarily, C1 = C, Q1 = C,

yn = (1 - an)xn + anT xn, , ,

Cn+1 = {z e Cn : an [1 - an(1 + L)] ||xn - Tnxn ||2 < {xn - z,yn - Tnyn) + 0n}, Qn+1 = {z e Qn : {xn - z, x1 - xn) > 0},

xn+1 = ^Cn+1HQn+1 x1, Vn > 1,

where dn = /n[$(M) + M*(diamC)2] + fn, {an} is a sequence in [a,b] with a,b e (0, ^^). Then the sequence {xn} converges strongly to a point Pf(t)x1, where Pf(T) is the projection from C onto F(T).

Proof We split the proof into seven steps. Step 1. Show that Pf(T)x1 is well defined for every xi e C.

By Lemma 2.2, we know that F(T) is a closed and convex subset of C. Therefore, in view of the assumption of F(T) = 0, Pf(T)x1 is well defined for every x1 e C. Step 2. Show that Cn and Qn are closed and convex for all n > 1. From the definitions of Cn and Qn, it is obvious that Cn and Qn are closed and convex for all n > 1. We omit the details. Step 3. Show that F(T) c Cn n Qn for all n > 1.

To this end, we first prove that F(T) c Cn for all n > 1. This can be proved by induction on n. It is obvious that F(T) c C1 = C. Assume that F(T) c Cn for some n e N. Then, using the uniform L-Lipschitzian continuity of T, the total quasi-asymptotic pseudo-contractiveness of T and (2.3), we have for any w e F(T) that

xn - Tnxn 2 = xn - Tnxn, xn - Tnxn

= \xn -yn,xn - T xn)

= — ixn - yn, xn - Tnxn - (yn - Tnyn)) + — ixn - yn,yn - Tny„)

= (xn - yn, xn - T xn - (yn - T yn)) an

+ — ixn - w + w - yn, yn - Tnyn)

||xn - yn ||2 + — (xn - w,yn - Tnyn)

+ — (w - yn, yn - Tnyn)

||xn - yn|2 + (xn - w, yn - Tnyn)

+ — (w - yn,yn - w + w - Tnyn) an

1+L,, 1,2 1 / ™

-||xn - yn|2 + — (xn - w, yn - Tnyn/

- — ||w - yn |2 + — (w - yn, w - Tnyn)

^ 1+L,, || 2 1 / rrn

< -||xn - yn |2 + — (xn - w,yn - Tnyn

+ — {/¿n [4>(M) + M*(diam C)^ + £n}

n 2 1 n = (1 + L)a J xn - TnxJ + — (xn - w, yn - Tnyn)

+ — n [4> (M)+ M* (diam C)2}+ £n},

which implies that

n[l -an(l + i)]|\xn - Tnxn\2 < (xn - w,yn - Tnyn) + fin\_4>(M) +M*(diamC)2] +

which shows that w e Cn+l. By the mathematical induction principle, F(T) c Cn for all n > l.

Next, we prove F(T) c Qn for all n > l. By induction, for n = l, we have F(T) c C = Ql. Assume that F(T) c Qn for some n e N. Since xn is the projection of xi onto Cn n Qn, by Lemma 2.l, we have

<xn - z,xl -xn) >0, Vz e Cn n Qn. (.2)

Since F(T) c Cn n Qn, we easily see that

<xn - w,xl - xn) > 0, Vw e F(T), (3.3)

which implies that F(T) c Qn+l. This proves that F(T) c Cn n Qn for all n > l. Step 4. Show that limn^TO ||xn - xly exists.

In view of (3.l) and Lemma 2.l, we have xn = PQnxl and xn+l e Qn, which implies

l|xn - xly<|xn+l- xl|, Vn > l.

On the other hand, since F(T) c Qn, we also have

l|xn -xl| < ||w -xl|, Vw e F(T),Vn > l.

Therefore, limn^TO ||xn - xl| exists and {xn} is bounded. Step 5. Show that {xn} is a Cauchy sequence.

Noticing the construction of Cn, one has Cm c Cn and xm = PCmxl e Cn for any positive integer m > n. From (3.2), we have

<xn xn+m, ^^l xn) >> 0.

It follows that

||xn - xn+m y = ||xn - xl + xl - xn+m ^

— 11xn xl|| + ||xl xn+m | 2 <xl xn, xl xn+m)

— 11xn xl| + ||xl xn+m | 2 <xl xn, xl xn + xn xn+m)

< ||xl - xn+m || - ||xn - xl|| - <xl- xn, xn xn+m)

< ||xl - xn+m|| - ||xn - xl^ . ((4)

Letting n ^to in (3.4), one has limn^TO ||xn -xn+m || = 0, Vm > n. Hence, {xn} is a Cauchy sequence. Since H is a Hilbert space and C is closed and convex, one can assume that xn ^ q e C as n ^to. Step 6. Show that limn^ to ||xn Txn || = 0.

It follows from xn+i e Cn and (3.1) that

an [1 - an (1 + L)] | |xn - Tnxn\2 < (xn - xn+1, yn - Tnyn) + On

<\\xn - xn+i III |yn - Tny^| + On. (3.5)

Since {yn} is bounded, {Tnyn} is bounded, limn—TO \\xn+1 -xn\\ = 0 and an e (a,b), we have from (3.5) that

lim ||xn - Tnxn || = 0.

On the other hand, we notice that

\\xn - Txn\\ < \\xn - xn+1 \ + ||xn+1 - T xn+11|

Wrrn+1 _ rrn+1 || II rrn+1 _ f || + T xn+1 T x^ T T xn Txn

< (1+ L)\\x n xn+1 \\ + ||xn+1 T xn+11| + L || T xn xn || . From limn—TO \\xn+1 - xn \\ = 0 and limn—TO \\xn - Tnxn \\ = 0, we have lim \\xn - Txn \\ = 0.

It follows that Txn — q as n — to. Since T is continuous, one has that q is a fixed point of T; that is, q e F(T). Step 7. Finally, we prove q = Pf(T)x1. By taking the limit in (3.3), we have

{q - w,x1 - q) > 0, Vw e F(T),

which implies that q = Pf(T)x1 by using Lemma 2.1. This completes the proof. □

Since every total asymptotically pseudo-contractive mapping with F(T) = 0 is total quasi-asymptotically pseudo-contractive, we immediately obtain the following corollary:

Corollary 3.2 Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let T : C — C be a uniformly L-Lipschitzian and total asymptotically pseudo-contractive mapping with F (T) = 0. Suppose there exist positive constants M and M* such that ) < M*Z2 for all Z > M. Let {xn} be a sequence generated by the following iterative scheme:

x1 e C chosen arbitrarily, C1 = C, Q1 = C, yn — (1 an)xn + an T' xn,

Cn+1 = {z e Cn : an [1 - an(1 + L)] \\xn - Tnxn \\2 < {xn - z,yn - Tnyn) + On}, Qn+1 = {z e Qn : {xn - z, x1 - xn) > 0}, xn+1 = PCn+1nQn+1 x1, Vn > 1,

where dn = fin[$(M) + M*(diamC)2] + fn, {an} is a sequence in [a,b] with a,b e (0, ). Then the sequence {xn} converges strongly to a point Pf(t)x1, where Pf(T) is the projection from C onto F(T).

Remark 3.3 Since the class of the total quasi-asymptotically pseudo-contractive mappings includes the class of asymptotically pseudocontractive mappings, the class of asymptotically pseudocontractive mappings in the intermediate sense, the class of the total asymptotically pseudo-contractive mappings, the class of quasi-asymptotically pseudo-contractive mappings in the intermediate sense as special cases, Theorem 3.l improves the corresponding results in Zhou [7], Qin etal. [9], Chang [10] and Qin etal. [12].

Competing interests

The authors declare that they have no competing interests. Authors' contributions

Allauthors read and approved the finalmanuscript Author details

1 Department of Foundation, Shandong Yingcai University, Jinan, 250104, P.R. China. 2 Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, P.R. China.

Acknowledgements

The authors are gratefulto the referees for their helpfuland usefulcomments. The first author is supported by the Project of Shandong Province Higher EducationalScience and Technology Program (grant No. J13LI51) and the NaturalScience Foundation of Shandong Yingcai University (grant No. 12YCZDZR03). The second author is supported by the National NaturalScience Foundation of China under grant (11071279) and the Project of Shandong Province Higher Educational Science and Technology Program (grant No. J13LI51).

Received: 25 April 2013 Accepted: 25 July 2013 Published: 9 August 2013

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doi:10.1186/1029-242X-2013-375

Cite this article as: Wang and Su: On the convergence of hybrid projection algorithms for total quasi-asymptotically pseudo-contractive mapping. Journal of Inequalities and Applications 2013 2013:375.