A New General Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces

We introduce a new general iterative method by using the -mapping for finding a common fixed point of a finite family of nonexpansive mappings in the framework of Hilbert spaces. A strong convergence theorem of the purposed iterative method is established under some certain control conditions. Our results improve and extend the results announced by many others.

Let be a real Hilbert space, and let be a nonempty closed convex subset of . A mapping of into itself is called nonexpansive if for all A point is called a fixed point of provided that . We denote by the set of fixed points of (i.e., ). Recall that a self-mapping is a contraction on , if there exists a constant such that for all A bounded linear operator on is called strongly positive with coefficient if there is a constant with the property

In 1953, Mann [1] introduced a well-known classical iteration to approximate a fixed point of a nonexpansive mapping. This iteration is defined as

where the initial guess is taken in arbitrarily, and the sequence is in the interval . But Mann's iteration process has only weak convergence, even in a Hilbert space setting. In general for example, Reich [2] showed that if is a uniformly convex Banach space and has a Frehet differentiable norm and if the sequence is such that , then the sequence generated by process (1.2) converges weakly to a point in . Therefore, many authors try to modify Mann's iteration process to have strong convergence.

In 2005, Kim and Xu [3] introduced the following iteration process:

They proved in a uniformly smooth Banach space that the sequence defined by (1.3) converges strongly to a fixed point of under some appropriate conditions on and .

In 2008, Yao et al. [4] alsomodified Mann's iterative scheme 1.2 to get a strong convergence theorem.

Let be a finite family of nonexpansive mappings with There are many authors introduced iterative method for finding an element of which is an optimal point for the minimization problem. For , is understood as with the mod function taking values in . Let be a fixed element of

In 2003, Xu [5] proved that the sequence generated by

converges strongly to the solution of the quadratic minimization problem

under suitable hypotheses on and under the additional hypothesis

In 1999, Atsushiba and Takahashi [6] defined the mapping as follows:

where This mapping is called the mapping generated by and .

In 2000, Takahashi and Shimoji [7] proved that if is strictly convex Banach space, then , where .

In 2007,Shang et al.[8] introduced a composite iteration scheme as follows:

where is a contraction, and is a linear bounded operator.

Note that the iterative scheme (1.8) is not well-defined, because may not lie in , so is not defined. However, if , the iterative scheme (1.8) is well-defined and Theorem [8] is obtained. In the case , we have to modify the iterative scheme (1.8) in order to make it well-defined.

In 2009, Kangtunyakarn and Suantai [9] introduced a new mapping, called -mapping, for finding a common fixed point of a finite family of nonexpansive mappings. For a finite family of nonexpansive mappings and sequence in , the mapping is defined as follows:

The mapping is called the K-mapping generated by and .

In this paper, motivated by Kim and Xu [3], Marino and Xu [10], Xu [5], Yao et al. [4], andShang et al. [8], we introduce a composite iterative scheme as follows:

where is a contraction, and is a bounded linear operator. We prove, under certain appropriate conditions on the sequences and that defined by (1.10) converges strongly to a common fixed point of the finite family of nonexpansive mappings , which solves a variational inequaility problem.

In order to prove our main results, we need the following lemmas.

Lemma 1.1.

For all there holds the inequality

Lemma 1.2 (see [11]).

Let and be bounded sequences in a Banach space , and let be a sequence in with . Suppose that

for all integer , and

Then

Lemma 1.3 (see [5]).

Assume that is a sequence of nonnegative real numbers such that   , where and is a sequence in such that

(i),

(ii) or .

Then .

Lemma 1.4 (see [10]).

Let be a strongly positive linear bounded operator on a Hilbert space with coefficient and . Then .

Lemma 1.5 (see [10]).

Let be a Hilbert space. Let be a strongly positive linear bounded operator with coefficient . Assume that . Let be a nonexpansive mapping with a fixed point of the contraction . Then converges strongly as to a fixed point of , which solves the variational inequality

Lemma 1.6 (see [1]).

Demiclosedness principle. Assume that is nonexpansive self-mapping of closed convex subset of a Hilbert space . If has a fixed point, then is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that . Here, is identity mapping of .

Lemma 1.7 (see [9]).

Let be a nonempty closed convex subset of a strictly convex Banach space. Let be a finite family of nonexpansive mappings of into itself with , and let be real numbers such that for every and Let be the -mapping of into itself generated by and . Then .

By using the same argument as in [9, Lemma ], we obtain the following lemma.

Lemma 1.8.

Let be a nonempty closed convex subset of Banach space. Let be a finite family of nonexpanxive mappings of into itself and sequences in such that Moreover, for every , let and be the K -mappings generated by and , and and , respectively. Then, for every bounded sequence , one has

Let be real Hilbert space with inner product , a nonempty closed convex subset of . Recall that the metric (nearest point) projection from a real Hilbert space to a closed convex subset of is defined as follows. Given that  , is the only point in with the property . Below Lemma 1.9 can be found in any standard functional analysis book.

Lemma 1.9.

Let be a closed convex subset of a real Hilbert space . Given that and then

(i) if and only if the inequality for all ,

(ii) is nonexpansive,

(iii) for all ,

(iv) for all and .

In this section, we prove strong convergence of the sequences defined by the iteration scheme (1.10).

Theorem 2.1.

Let be a Hilbert space, a closed convex nonempty subset of . Let be a strongly positive linear bounded operator with coefficient , and let Let be a finite family of nonexpansive mappings of into itself, and let be defined by (1.9). Assume that and . Let , given that and are sequences in , and suppose that the following conditions are satisfied:

(C1)

(C2)

(C3)

(C4) and , where ;

(C5)

(C6).

If is the composite process defined by (1.10), then converges strongly to , which also solves the following variational inequality:

Proof.

First, we observe that is bounded. Indeed, take a point , and notice that

Since , we may assume that for all . By Lemma 1.4, we have for all .

It follows that

By simple inductions, we have

Therefore is bounded, so are and . Since is nonexpansive and , we also have

By using the inequalities (2.6) and (2.11) of [9, Lemma ], we can conclude that

where .

By (2.5) and (2.6), we have

where , . Since , , and , for all , by Lemma 1.3, we obtain . It follows that

Since and , are bounded, we have as . Since

it implies that as .

On the other hand, we have

which implies that .

From condition and as , we obtain

By (C4), we have for all . Let be the -mapping generated by and . Next, we show that

where with being the fixed point of the contraction . Thus, solves the fixed point equation . By Lemma 1.5 and Lemma 1.7, we have and for all . It follows by (2.11) and Lemma 1.8 that Thus, we have . It follows from Lemma 1.1 that for ,

where

It follows that

Letting in (2.15) and (2.14), we get

where is a constant such that for all and . Taking in (2.16), we have

On the other hand, one has

It follows that

Therefore, from   (2.17) and , we have

Hence (2.12) holds. Finally, we prove that  . By using (2.2) and together with the Schwarz inequality, we have

Since , , and are bounded, we can take a constant such that

for all . It then follows that

where . By , we get . By applying Lemma 1.3 to (2.23), we can conclude that . This completes the proof.

If and in Theorem 2.1, we obtain the following result.

Corollary 2.2.

Let be a Hilbert space, a closed convex nonempty subset of , and let . Let be a finite family of nonexpansive mappings of into itself, and let be defined by (1.9). Assume that . Let , given that and are sequences in , and suppose that the following conditions are satisfied:

(C1)

(C2)

(C3)

(C4) and , where ;

(C5)

(C6).

If is the composite process defined by

then converges strongly to , which also solves the following variational inequality:

If , , , and is a constant in Theorem 2.1, we get the results of Kim and Xu [3].

Corollary 2.3.

Let be a Hilbert space, a closed convex nonempty subset of , and let . Let be a nonexpansive mapping of into itself. . Let , given that and are sequences in , and suppose that the following conditions are satisfied:

(C1)

(C2)

(C3)

(C4)

(C5).

If is the composite process defined by

then converges strongly to , which also solves the following variational inequality:

The authors would like to thank the referees for valuable suggestions on the paper and thank the Center of Excellence in Mathematics, the Thailand Research Fund, and the Graduate School of Chiang Mai University for financial support.

1. Mann, WR: Mean value methods in iteration. Proceedings of the American Mathematical Society. 4, 506–510 (1953). Publisher Full Text

2. Reich, S: Weak convergence theorems for nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications. 67(2), 274–276 (1979). Publisher Full Text

3. Kim, T-H, Xu, H-K: Strong convergence of modified Mann iterations. Nonlinear Analysis: Theory, Methods & Applications. 61(1-2), 51–60 (2005). PubMed Abstract | Publisher Full Text

4. Yao, Y, Chen, R, Yao, J-C: Strong convergence and certain control conditions for modified Mann iteration. Nonlinear Analysis: Theory, Methods & Applications. 68(6), 1687–1693 (2008). PubMed Abstract | Publisher Full Text

5. Xu, HK: An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications. 116(3), 659–678 (2003). Publisher Full Text

6. Atsushiba, S, Takahashi, W: Strong convergence theorems for a finite family of nonexpansive mappings and applications. Indian Journal of Mathematics. 41(3), 435–453 (1999)

7. Takahashi, W, Shimoji, K: Convergence theorems for nonexpansive mappings and feasibility problems. Mathematical and Computer Modelling. 32(11–13), 1463–1471 (2000)

8. Shang, M, Su, Y, Qin, X: Strong convergence theorems for a finite family of nonexpansive mappings. Fixed Point Theory and Applications. 2007, (2007)

9. Kangtunyakarn, A, Suantai, S: A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings. Nonlinear Analysis: Theory, Methods & Applications. 71(10), 4448–4460 (2009). PubMed Abstract | Publisher Full Text

10. Marino, G, Xu, H-K: A general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications. 318(1), 43–52 (2006). Publisher Full Text

11. Suzuki, T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications. 305(1), 227–239 (2005). Publisher Full Text