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

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.

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:CC be a mapping. In this paper, we denote the fixed point set of T by F(T), that is, F(T):={xC:Tx=x}.

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

T n x T n y k n xy,n1,x,yC.
(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:

lim sup n sup x , y C ( T n x T n y x y ) 0.
(1.2)

Noticing that if we define

ρ n =max { 0 , sup x , y C ( T n x T n y x y ) } ,
(1.3)

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

T n x T n y xy+ ρ n ,n1,x,yC.
(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 { k n }[1,) with k n 1 as n such that

T n x T n y , x y k n x y 2 ,x,yC.
(1.5)

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

T n x T n y 2 (2 k n 1) x y 2 + x y ( T n x T n y ) 2 ,n1,x,yC.
(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 { k n }[1,) with k n 1 as n such that

lim sup n sup x , y C ( T n x T n y , x y k n x y 2 ) 0.
(1.7)

Put

τ n =max { 0 , sup x , y C ( T n x T n y , x y k n x y 2 ) } .
(1.8)

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

T n x T n y , x y k n x y 2 + τ n ,n1,x,yC.
(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 { k n },{ ν n }[0,) with k n , ν n 0 as n such that

T n x T n y , x y x y 2 + k n ϕ ( x y ) + ν n ,n1,x,yC,
(1.10)

where ϕ:[0,)[0,) is a continuous and strictly increasing function with ϕ(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 n1, x,yC,

T n x T n y 2 x y 2 +2 k n ϕ ( x y ) + x y ( T n x T n y ) 2 +2 ν n .
(1.11)

If ϕ(λ)= λ 2 , then (1.10) is reduced to

T n x T n y , x y (1+ k n ) x y 2 + ν n ,n1,x,yC.
(1.12)

Put

ν n =max { 0 , sup x , y C ( T n x T n y , x y ( 1 + k n ) x y 2 ) } .
(1.13)

If ϕ(λ)= λ 2 , 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:CC is said to be total quasi-asymptotically pseudo-contractive if F(T), and there exist sequences { μ n }[0,) and { ξ n }[0,) with μ n 0 and ξ n 0 as n such that

T n x p , x p x p 2 + μ n ϕ ( x p ) + ξ n ,n1,xC,pF(T),
(1.14)

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

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

T n x p 2 x p 2 + 2 μ n ϕ ( x p ) + x T n x 2 + 2 ξ n , n 1 , x C , p F ( T ) .
(1.15)

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

Remark 2 If ϕ(λ)= λ 2 , the (1.14) is reduced to

T n x p , x p (1+ μ n ) x p 2 + ξ n ,n1,xC,pF(T).
(1.16)

Remark 3 Put

ξ n =max { 0 , sup x , y C ( T n x p , x p ( 1 + μ n ) x p 2 ) } .
(1.17)

If ϕ(λ)= λ 2 , 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, 913]. 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:CC is said to be uniformly L-Lipschitzian if there exists some L>0 such that

T n x T n y Lxy,x,yC,n1.
(2.1)

Let C be a nonempty closed convex subset of a real Hilbert space H. For every point xH, there exists a unique nearest point in C, denoted by P C x, such that x P C xxy holds for all yC, where P C 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 let P C be the metric projection from H onto C (i.e., for xH, P C is the only point in C such that x P C x=inf{xz:zC}). Given xH and zC, then z= P C x if and only if the relation

xz,yz0,yC
(2.2)

holds.

Lemma 2.2 Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let T:CC be a uniformly L-Lipschitzian and total quasi-asymptotically pseudo-contractive mapping with F(T). Suppose there exist positive constants M and M such that ϕ(ζ) M ζ 2 for all ζ>M. Then F(T) is a closed convex subset of C.

Proof Since ϕ is an increasing function, it follows that ϕ(ζ)ϕ(M) if ζM and ϕ(ζ) M ζ 2 if ζM. In either case, we can always obtain that

ϕ(ζ)ϕ(M)+ M ζ 2 .
(2.3)

Since T is uniformly L-Lipschitzian continuous, F(T) is closed. We need to show that F(T) is convex. To this end, let p i F(T) (i=1,2), and write p=t p 1 +(1t) p 2 for t(0,1). We take α(0, 1 1 + L ), and define y α , n =(1α)p+α T n p for each nN. Then, for all zF(T), we have from (2.3) that

p T n p 2 = p T n p , p T n p = 1 α p y α , n , p T n p = 1 α p y α , n , p T n p ( y α , n T n y α , n ) + 1 α p y α , n , y α , n T n y α , n 1 + L α p y α , n 2 + 1 α p z , y α , n T n y α , n + 1 α z y α , n , y α , n T n y α , n = 1 + L α p y α , n 2 + 1 α p z , y α , n T n y α , n + 1 α z y α , n , y α , n z + z T n y α , n 1 + L α p y α , n 2 + 1 α p z , y α , n T n y α , n + 1 α { μ n [ ϕ ( M ) + M ( diam C ) 2 ] + ξ n } = α ( 1 + L ) p T n p 2 + 1 α p z , y α , n T n y α , n + 1 α { μ n [ ϕ ( M ) + M ( diam C ) 2 ] + ξ n } .

This implies that

α [ 1 α ( 1 + L ) ] p T n p 2 p z , y α , n T n y α , n + μ n [ ϕ ( M ) + M ( diam C ) 2 ] + ξ n .
(2.4)

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

α [ 1 α ( 1 + L ) ] p T n p 2 μ n [ ϕ ( M ) + M ( diam C ) 2 ] + ξ n .
(2.5)

Letting n in (2.5), we obtain T n pp. Since T is continuous, we have T n + 1 pTp as n, 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:CC be a uniformly L-Lipschitzian and total quasi-asymptotically pseudo-contractive mapping with F(T). Suppose that there exist positive constants M and M such that ϕ(ζ) M ζ 2 for all ζ>M. Let { x n } be a sequence generated by the following iterative scheme:

{ x 1 C chosen arbitrarily , C 1 = C , Q 1 = C , y n = ( 1 α n ) x n + α n T n x n , C n + 1 = { z C n : α n [ 1 α n ( 1 + L ) ] x n T n x n 2 x n z , y n T n y n + θ n } , Q n + 1 = { z Q n : x n z , x 1 x n 0 } , x n + 1 = P C n + 1 Q n + 1 x 1 , n 1 ,
(3.1)

where θ n = μ n [ϕ(M)+ M ( diam C ) 2 ]+ ξ n , { α n } is a sequence in [a,b] with a,b(0, 1 1 + L ). Then the sequence { x n } converges strongly to a point P F ( T ) x 1 , where P F ( T ) is the projection from C onto F(T).

Proof We split the proof into seven steps.

Step 1. Show that P F ( T ) x 1 is well defined for every x 1 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), P F ( T ) x 1 is well defined for every x 1 C.

Step 2. Show that C n and Q n are closed and convex for all n1.

From the definitions of C n and Q n , it is obvious that C n and Q n are closed and convex for all n1. We omit the details.

Step 3. Show that F(T) C n Q n for all n1.

To this end, we first prove that F(T) C n for all n1. This can be proved by induction on n. It is obvious that F(T) C 1 =C. Assume that F(T) C n for some nN. Then, using the uniform L-Lipschitzian continuity of T, the total quasi-asymptotic pseudo-contractiveness of T and (2.3), we have for any wF(T) that

x n T n x n 2 = x n T n x n , x n T n x n = 1 α n x n y n , x n T n x n = 1 α n x n y n , x n T n x n ( y n T n y n ) + 1 α n x n y n , y n T n y n = 1 α n x n y n , x n T n x n ( y n T n y n ) + 1 α n x n w + w y n , y n T n y n 1 + L α n x n y n 2 + 1 α n x n w , y n T n y n + 1 α n w y n , y n T n y n = 1 + L α n x n y n 2 + 1 α n x n w , y n T n y n + 1 α n w y n , y n w + w T n y n = 1 + L α n x n y n 2 + 1 α n x n w , y n T n y n 1 α n w y n 2 + 1 α n w y n , w T n y n 1 + L α n x n y n 2 + 1 α n x n w , y n T n y n + 1 α n { μ n [ ϕ ( M ) + M ( diam C ) 2 ] + ξ n } = ( 1 + L ) α n x n T n x n 2 + 1 α n x n w , y n T n y n + 1 α n { μ n [ ϕ ( M ) + M ( diam C ) 2 ] + ξ n } ,

which implies that

α n [ 1 α n ( 1 + L ) ] x n T n x n 2 x n w , y n T n y n + μ n [ ϕ ( M ) + M ( diam C ) 2 ] + ξ n ,

which shows that w C n + 1 . By the mathematical induction principle, F(T) C n for all n1.

Next, we prove F(T) Q n for all n1. By induction, for n=1, we have F(T)C= Q 1 . Assume that F(T) Q n for some nN. Since x n is the projection of x 1 onto C n Q n , by Lemma 2.1, we have

x n z, x 1 x n 0,z C n Q n .
(3.2)

Since F(T) C n Q n , we easily see that

x n w, x 1 x n 0,wF(T),
(3.3)

which implies that F(T) Q n + 1 . This proves that F(T) C n Q n for all n1.

Step 4. Show that lim n x n x 1 exists.

In view of (3.1) and Lemma 2.1, we have x n = P Q n x 1 and x n + 1 Q n , which implies

x n x 1 x n + 1 x 1 ,n1.

On the other hand, since F(T) Q n , we also have

x n x 1 w x 1 ,wF(T),n1.

Therefore, lim n x n x 1 exists and { x n } is bounded.

Step 5. Show that { x n } is a Cauchy sequence.

Noticing the construction of C n , one has C m C n and x m = P C m x 1 C n for any positive integer m>n. From (3.2), we have

x n x n + m , x 1 x n 0.

It follows that

x n x n + m 2 = x n x 1 + x 1 x n + m 2 = x n x 1 2 + x 1 x n + m 2 2 x 1 x n , x 1 x n + m = x n x 1 2 + x 1 x n + m 2 2 x 1 x n , x 1 x n + x n x n + m x 1 x n + m 2 x n x 1 2 x 1 x n , x n x n + m x 1 x n + m 2 x n x 1 2 .
(3.4)

Letting n in (3.4), one has lim n x n x n + m =0, mn. Hence, { x n } is a Cauchy sequence. Since H is a Hilbert space and C is closed and convex, one can assume that x n qC as n.

Step 6. Show that lim n x n T x n =0.

It follows from x n + 1 C n and (3.1) that

α n [ 1 α n ( 1 + L ) ] x n T n x n 2 x n x n + 1 , y n T n y n + θ n x n x n + 1 y n T n y n + θ n .
(3.5)

Since { y n } is bounded, { T n y n } is bounded, lim n x n + 1 x n =0 and α n (a,b), we have from (3.5) that

lim n x n T n x n =0.

On the other hand, we notice that

x n T x n x n x n + 1 + x n + 1 T n + 1 x n + 1 + T n + 1 x n + 1 T n + 1 x n + T n + 1 x n T x n ( 1 + L ) x n x n + 1 + x n + 1 T n + 1 x n + 1 + L T n x n x n .

From lim n x n + 1 x n =0 and lim n x n T n x n =0, we have

lim n x n T x n =0.

It follows that T x n q as n. Since T is continuous, one has that q is a fixed point of T; that is, qF(T).

Step 7. Finally, we prove q= P F ( T ) x 1 .

By taking the limit in (3.3), we have

qw, x 1 q0,wF(T),

which implies that q= P F ( T ) x 1 by using Lemma 2.1. This completes the proof. □

Since every total asymptotically pseudo-contractive mapping with F(T) 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:CC be a uniformly L-Lipschitzian and total asymptotically pseudo-contractive mapping with F(T). Suppose there exist positive constants M and M such that ϕ(ζ) M ζ 2 for all ζ>M. Let { x n } be a sequence generated by the following iterative scheme:

{ x 1 C chosen arbitrarily , C 1 = C , Q 1 = C , y n = ( 1 α n ) x n + α n T n x n , C n + 1 = { z C n : α n [ 1 α n ( 1 + L ) ] x n T n x n 2 x n z , y n T n y n + θ n } , Q n + 1 = { z Q n : x n z , x 1 x n 0 } , x n + 1 = P C n + 1 Q n + 1 x 1 , n 1 ,

where θ n = μ n [ϕ(M)+ M ( diam C ) 2 ]+ ξ n , { α n } is a sequence in [a,b] with a,b(0, 1 1 + L ). Then the sequence { x n } converges strongly to a point P F ( T ) x 1 , where P F ( 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 3.1 improves the corresponding results in Zhou [7], Qin et al. [9], Chang [10] and Qin et al. [12].

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Acknowledgements

The authors are grateful to the referees for their helpful and useful comments. The first author is supported by the Project of Shandong Province Higher Educational Science and Technology Program (grant No. J13LI51) and the Natural Science Foundation of Shandong Yingcai University (grant No. 12YCZDZR03). The second author is supported by the National Natural Science Foundation of China under grant (11071279) and the Project of Shandong Province Higher Educational Science and Technology Program (grant No. J13LI51).

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Wang, ZM., Su, Y. On the convergence of hybrid projection algorithms for total quasi-asymptotically pseudo-contractive mapping. J Inequal Appl 2013, 375 (2013). https://doi.org/10.1186/1029-242X-2013-375

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Keywords

  • total quasi-asymptotically pseudo-contractive
  • hybrid projection algorithm
  • fixed point
  • Hilbert space