An Iterative Approximation Method for a Common Fixed Point of Two Pseudocontractive Mappings
Habtu Zegeye1
Academic Editor: G. L. Karakostas, C. Zhu, N. Shioji, G. Mantica, O. Miyagaki
Received24 Jan 2011
Accepted22 Mar 2011
Published01 Jun 2011
Abstract
We introduce an iterative process for finding an element in the common fixed point sets of two continuous pseudocontractive
mappings. As a consequence, we provide an approximation method for a common fixed point of a finite family of pseudocontractive mappings. Furthermore,
our convergence theorem is applied to a convex minimization problem. Our
theorems extend and unify most of the results that have been proved for this
class of nonlinear mappings.
1. Introduction
Let be a real Hilbert space. A mapping with domain and range in is called pseudocontractive if for each we have is called strongly pseudocontractive if there exists such that
and is said to be -strict pseudocontractive if there exists a constant such that
The operator is called Lipschitzian if there exists such that for all . If , then is called nonexpansive, and if , then is called a contraction. As a result of Kato [1], it follows from inequality (1.1) that is pseudocontractive if and only if the inequality
holds for each and for all .
Apart from being an important generalization of nonexpansive, strongly pseudocontractive and -strict pseudocontractive mappings, interest in pseudocontractive mappings stems mainly from their firm connection with the important class of nonlinear accretive operators, where a mapping with domain and range in is called accretive if the inequality
holds for every and for all . We observe that is accretive if and only if is pseudocontractive, and thus a zero of , , is a fixed point of , . It is now well known that if is accretive then the solutions of the equation correspond to the equilibrium points of some evolution systems. Consequently, considerable research efforts have been devoted to iterative methods for approximating fixed points of when is pseudocontractive (see, e.g., [2β4] and the references contained therein).
For a sequence of real numbers in and an arbitrary , let the sequence in be iteratively defined by and
where is a nonexpansive mapping of into itself. Halpern [11] was the first to study the convergence of Algorithm (1.7) in the framework of Hilbert spaces. Lions [14] and Wittmann [21] improved the result of Halpern by proving strong convergence of to a fixed point of if the real sequence satisfies certain conditions. Reich [22], Shioji and Takahashi [16], and Zegeye and Shahzad [23] extend the result of Wittmann [21] to the case of Banach space.
In 2000, Moudafi [24] introduced viscosity approximation method and proved that if is a real Hilbert space, for given , the sequence generated by the algorithm
where is a contraction mapping and satisfies certain conditions, converges strongly to a common fixed point of . Moudafi [24] generalizes Halpernβs theorems in the direction of viscosity approximations. In [25], Zegeye et al. extended Moudafi's result to the class of Lipschitz pseudocontractive mappings in Banach spaces more general than Hilbert spaces. Viscosity approximations are very important because they are applied to convex optimization, linear programming, monotone inclusions, and elliptic differential equations.
Our concern now is the following. Is it possible to construct a viscosity approximation sequence that converges strongly to a fixed point of pseudocontractive mappings more general than nonexpansive mappings?
In this paper, motivated and inspired by the work of Halpern [11], Moudafi [24], and the methods of Takahashi and Zembayashi [26], we introduce a viscosity approximation method for finding a common fixed point of two continuous pseudocontractive mappings. As a consequence, we provide an approximation method for a common fixed point of finite family of pseudocontractive mappings. This provides affirmative answer to the above concern. Furthermore, we apply our convergence theorem to the convex minimization problem. Our theorems extend and unify most of the results that have been proved for this important class of nonlinear operators.
2. Preliminaries
Let be closed and convex subset of a real Hilbert space . For every point , there exists a unique nearest point in , denoted by , such that is called the metric projection of onto . We know that is a nonexpansive mapping of onto . In connection with metric projection, we have the following lemma.
Lemma 2.1. Let be a nonempty convex subset of a Hilbert space . Let and . Then, if and only if
Lemma 2.2 (see [27]). Let be a sequence of nonnegative real numbers satisfying the following relation:
where (i)ββ, and (ii)ββ or . Then, as .
By a similar argument in [28], we have the following lemma.
Lemma 2.3. Let be a nonempty closed convex subset of a real Hilbert space H. Let be a continuous accretive mapping. Then, for and , there exists such that
Moreover, by a similar argument of the proof of Lemmas 2.8 and 2.9 of [26], we get the following lemma.
Lemma 2.4. Let be a nonempty closed convex subset of a real Hilbert space . Let be a continuous accretive mapping. For and , define a mapping as follows:
for all . Then, the following hold: (1) is single valued;(2) is firmly nonexpansive type mapping, that is, for all ,
(3);(4) is closed and convex.
3. Main Results
In the sequel, we will make use of the following lemmas.
Lemma 3.1. Let be a nonempty closed convex subset of a real Hilbert space H. Let be a continuous pseudocontractive mapping. Then, for and , there exists such that
Proof. Let and . Let , where is the identity mapping on . Then, clearly is continuous accretive mapping. Thus, by Lemma 2.3, there exists such that , for all . But this is equivalent to , for all . Hence, the lemma holds.
Lemma 3.2. Let be a nonempty closed convex subset of a real Hilbert space . Let be continuous pseudocontractive mapping. For and , define a mapping as follows:
for all . Then, the following hold: (1) is single valued;(2) is firmly nonexpansive type mapping, that is, for all ,
(3);(4) is closed and convex.
Proof. We note that , for all , is equivalent to , for all , where is continuous accretive mapping and the identity mapping on . Moreover, as is self-map, we have that . Thus, by Lemma 2.4, the conclusions of (1)β(4) hold.
Let be a nonempty closed convex subset of a real Hilbert space . Let , for , be continuous pseudocontractive mappings. Then, in what follows, are defined as follows. For and , define
Now, we prove our main convergence theorem.
Theorem 3.3. Let be a nonempty closed convex subset of a real Hilbert space . Let , for , be continuous pseudocontractive mappings such that . Let be a contraction of into itself, and let be a sequence generated by and
where and such that , , , , and . Then, the sequence converges strongly to , where .
Proof. Let . Then, is a contraction of into . In fact, we have that
for all , where is contraction constant of . So is a contraction of into itself. Since is closed subset of , there exists a unique element of such that . Let , and let , where . Then, we have from Lemma 3.2 that
Moreover, from (3.5) and (3.7), we get that
By induction, we get that
Therefore, is bounded. Consequently, we get that , , , and are bounded. Next, we show that . But from (3.5) we have that
where . Moreover, since and , we get that
Putting in (3.11) and in (3.12), we get that
Adding (3.13) and (3.14), we have
which implies that
Now, using the fact that is pseudocontractive, we get that
and hence
Without loss of generality, let us assume that there exists a real number such that for all . Then, we have
and hence from (3.19) we obtain that
where . Furthermore, from (3.10) and (3.20), we have that
Now, using conditions of , and Lemma 2.2, we have that
Consequently, from (3.20) and (3.22), we obtain that
Similarly, taking and and following the method used for , we get that . Furthermore, since , we have that
Thus, since , we obtain that
Moreover, for , using Lemma 3.2, we get that
and hence
Therefore, from (3.5), the convexity of , (3.7) and (3.27) we get that
and hence
So we have as . This implies with (3.25) that as . Next, we show that
where . To show this inequality, we choose a subsequence of such that
Since is bounded, there exists a subsequence of and such that . Without loss of generality, we may assume that . Since and is convex and closed, we get that . Moreover, since as , we have that . Now, we show that . Note that, from the definition of , we have
Put for all and . Consequently, we get that . From (3.32) and pseudocontractivity of , it follows that
Then, since , as , we obtain that as . Thus, as , it follows that
and hence
Letting and using the fact that is continuous, we obtain that
Now, let . Then, we obtain that , and hence . Furthermore, the fact that and imply that , following the method used for , we obtain that , and hence . Therefore, since , by Lemma 2.1, we have
Now, we show that as . From , we have that
This implies that,
where , , for . But note that , , and . Therefore, by Lemma 2.2, we conclude that converges to , where . This completes the proof.
If, in Theorem 3.3, is a constant mapping, then we get . In fact, we have the following corollary.
Corollary 3.4. Let be a nonempty closed convex subset of a real Hilbert space H. Let , for , be continuous pseudocontractive mappings such that . Let be a sequence generated by and
where and such that , , , , and . Then, the sequence converges strongly to , where .
If, in Theorem 3.3, we have that , identity mapping on , then we obtain the following corollary.
Corollary 3.5. Let be a nonempty closed convex subset of a real Hilbert space H. Let be continuous pseudocontractive mapping such that . Let be a contraction of into itself, and let be a sequence generated by and
where and such that , , , , and . Then, the sequence converges strongly to , where .
Let be a real Hilbert space. Let , for , be accretive mappings. Let , for all , , for all . Then we have the following convergence theorem for a zero of two accretive mappings.
Theorem 3.6. Let be a real Hilbert space. Let , for , be continuous accretive mappings such that . Let be a contraction of into itself, and let be a sequence generated by and
where and such that , , , , and . Then, the sequence converges strongly to , where .
Proof. Let , for . Then, we get that , for , are continuous pseudocontractive mappings with . Thus, the conclusion follows from Theorem 3.3.
The proof of the following theorem can be easily obtained from the method of proof of Theorem 3.3.
Theorem 3.7. Let be a nonempty closed convex subset of a real Hilbert space H. Let , for , be continuous pseudocontractive mappings such that . Let be a contraction of into itself, and let be a sequence generated by and
where , for all , for , and and such that , , , , and . Then, the sequence converges strongly to , where .
Remark 4.2. Our theorems extend and unify most of the results that have been proved for this important class of nonlinear operators. In particular, Theorem 3.3 extends Theorem 2.2 of Moudafi [24] and Theorem 4.1 of Iiduka and Takahashi [12] in the sense that our convergence is for the more general class of continuous pseudocontractive mappings. Moreover, this provides affirmative answer to the concern raised.
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