The Hybrid Projection Methods for Pseudocontractive, Nonexpansive Semigroup, and Monotone Mapping

Abstract

We modify the three-step iterative schemes to prove the strong convergence theorems by using the hybrid projection methods for finding a common element of the set of solutions of fixed points for a pseudocontractive mapping and a nonexpansive semigroup mapping and the set of solutions of a variational inequality problem for a monotone mapping in a Hilbert space under some appropriate control conditions. Our theorems extend and unify most of the results that have been proved for this class of nonlinear mappings.

1. Introduction

Let be a nonempty closed convex subset of a real Hilbert space with inner product and norm . Recall that a mapping is said to be a -strict pseudocontraction if there exists such that where denotes the identity operator on . When , is said to be nonexpansive if And when , is said to be pseudocontraction if Clearly, the class of -strict pseudocontraction falls into the one between classes of nonexpansive mappings and pseudocontraction mapping. We denote the set of fixed points of by .

A mapping of into is called monotone if The classical variational inequality is used for finding such that The set of solutions of variational inequality problems is denoted by . See, for example, [1–7] and the references therein.

In 1953, Mann [8] introduced the iteration as follows: where the initial guess element is arbitrary and is a real sequence in . The Mann iteration has been extensively investigated for nonexpansive mappings. In an infinite-dimensional Hilbert space, the Mann iteration can conclude only weak convergence [9]. Attempts to modify the Mann iteration method (6) so that strong convergence is guaranteed have recently been made. In 1974, The Ishikawa’s iteration process is defined by Ishikawa [10] as the following: where the initial guess is taken in arbitrarily and the sequences . This is called Ishikawa Iteration. This has been studied in strong convergence theorem for lipschitzian pseudocontractive mapping in Hilbert spaces. Several years later, inspired by the idea of one and two step iterative scheme, Noor [11, 12] introduced a three-step iterative scheme and studied the approximate solution of variational inclusion in Hilbert spaces by using the techniques of updating the solution and the auxiliary principle. It has been shown in [13] by Goebel and Kirk that the three-step iterative scheme gives better numerical results than the two-step and one-step approximate iterations.

A family of mappings of into itself is called a nonexpansive semigroup on if it satisfies the following conditions:(i) for all ;(ii) for all ;(iii) for all and ;(iv)for all is continuous.

We denote by the set of all common fixed points of ; that is, . It is known that is closed and convex. In the sense of nonexpansive semigroup mapping, we also see [14–24].

In 2003, Nakajo and Takahashi [25] proposed the following modification of Mann iteration method for a nonexpansive mapping from into itself in a Hilbert space where denotes the metric projection from a Hilbert space onto a close convex subset of and proves that the sequence converges strongly to . A projection onto intersection of two halfspaces is computed by solving a linear system of two equations with two unknowns. In 2008, Takahashi et al. [26] proved the following strong convergence theorem by the new hybrid method in a Hilbert space. They assume , , , and defined the sequence by (8) where where . Then converges strongly to .

Recently, Zegeye and Shahzad [27] defined the mappings as follows: for all and , where is a continuous pseudocontractive mapping and is a continuous monotone mapping. In the following year, Tang [28] introduced a viscosity iterative process, which converges strongly to a common element of the set of fixed points of a pseudocontractive mapping and the set of solutions of a monotone mapping as the following: where and are defined by (10) and (11), respectively.

In this paper, we modify the three-step iterative schemes to prove the strong convergence theorems by using the hybrid projection methods for finding a common element of the set of solutions of fixed points for a pseudocontractive mapping and a nonexpansive semigroup mapping and the set of solutions of a variational inequality problem for a monotone mapping in a Hilbert space under some appropriate control conditions. The results that are presented in this paper extend and improve the corresponding ones announced by Nakajo and Takahashi [25], Takahashi et al. [26], Zegeye and Shahzad [27], Tang [28], and many authors.

2. Preliminaries

Let be a real Hilbert space with norm and inner product and let be a closed convex subset of . Then for all and .

Recall that, the metric projection    from a Hilbert space    to a closed convex subset    of    is defined as the following: given is the only point in with the property is characterized as follows: for all .

Hilbert space satisfies the Kadec-Klee property [28, 29]; that is, for any sequence . and together imply .

A normed space is said to satisfy Opial’s condition [30], if for each sequence in which converges weakly to a point , we have

Lemma 1 (see [27]). Let be a nonempty closed convex subset of a Hilbert space . Let be a continuous pseudocontractive mapping and let be a continuous monotone mapping and define mappings and as follows: Then, the following hold: (1) and are single-valued;(2) and are firmly nonexpansive; that is, for any , ,  ;(3), ;(4) and are closed and convex.

3. Main Results

3.1. The Hybrid Method

Theorem 2. Let be a nonempty bounded closed and convex subset of a real Hilbert space . Let be a continuous pseudocontractive mapping and let be a continuous monotone mapping. Let be a nonexpansive semigroup on such that , , and . Let , , and be the sequences in for some , such that and suppose . The mappings and are defined by (10) and (11). Let be sequences generated by and Then the sequence converges strongly to .

Proof. Consider that On the other hand, we get that is closed and is closed and convex for all . From (19), is equivalent to for . Thus, we have is convex for all . Therefore, is closed and convex for all . Let , we have It follows that Therefore, for all . Thus, we have for all .
Next, we use mathematical induction. Start with , we have and then . Assume that is given and for some . There exists a unique such that , then we get for . From , we have . Therefore, . Thus is well defined and for all .
Since is a nonempty closed convex subset of , there exists a unique such that . From and the metric projection property, we have It follows that for all and . This implied is bounded. So, , , , , , and are bounded.
Next, we show that Since , , and , as same as the prove of (22), we get for all . Thus, is nondecreasing. By is bounded and nondecreasing, there exists the limit of . Since , we have Since exists, therefore (23) holds.
Next, we show that Since , we have Let , , and ; it follows from Lemma 1, we get Hence It follows that Consequently, we have that Equation (27) implies that On the other hand, from (18) and (32), we also have It follows from (32)-(33) we get Since and from (27) and (34), we get It follows that Therefore (26) holds.
Next, we show that . First, we show that is the unique solution in . Since is bounded, we choose subsequence of and assume that . Suppose that ; that is, . From Opial’s condition and (26), we have This is a contradiction. Thus, we obtain .
Since is bounded, there exists a subsequence of such that , . Without loss of generality, we may assume that . From the setting and (10), we have For and , let . Since and , we have and Since is a monotone and , we obtain By the continuity of , if then . Therefore, .
On the other hand, since , from (11) we have For and , let . Since and , we have and It follows from as , we get and hence . By the continuity of , if then . Let , we have ; therefore, . Consequently, we conclude that .
Finally, we show that , where . Since and , we get If  , it follows from (43), and the lower semicontinuity of the norm that Thus, we obtain that . Using the Kadec-Klee property of , we obtain that Since is an arbitrary weakly convergent subsequence of , we can conclude that converges strongly to , where . This completes the proof.

Corollary 3. Let be a nonempty bounded closed and convex subset of a real Hilbert space . Let be a continuous pseudocontractive mapping and let be a continuous monotone mapping. Let be a nonexpansive semigroup on such that , , and . Let such that and suppose . The mappings and are defined by (10) and (11). Let be a sequences generated by and Then the sequence converges strongly to .

Proof. Putting in Theorem 2, we can obtain the result.

3.2. The Shrinking Projection Method

Theorem 4. Let be a nonempty bounded closed and convex subset of a real Hilbert space . Let be a continuous pseudocontractive mapping and let be a continuous monotone mapping. Let be a nonexpansive semigroup on such that , , and . Let , , and be the sequences in for some , such that and suppose . The mappings and are defined by (10) and (11). Let be sequences generated by and Then the sequence converges strongly to .

Proof. First, we show that . By induction, it is obvious that . Suppose that for some , so we have such that . Then, we get . Therefore for all .
On the other hand, we show that is closed and convex for all . By mathematical induction, it is obvious that is closed and convex. Suppose that is closed and convex for some . For , we have that is equivalent to . Thus, we have is closed and convex for all . Therefore, is closed and convex for all . This implies that is well defined.
Next, we show that is bounded. From the metric projection property and (47), we have and for all and . Consider It follows that for all and . This implied is bounded. So, , , , , , and are bounded.
Next, we show that From the metric projection property and (47), we have , , and . As same as the prove of (48), we get for all . Thus, is nondecreasing. Since is bounded and nondecreasing, there exists the limit of . Similar to the proved of (25), we get (49) is hold.
Since sequence is bounded, we can choose subsequence of and assume that . Similar to the proof of Theorem 2, we also have .
Finally, we show that , where . Since and , we have It follows from (50), if and the lower semicontinuity of the norm thus, we obtain that . Using the Kadec-Klee property of , we obtain that Since is an arbitrary weakly convergent subsequence of , we can conclude that converges strongly to , where .

Corollary 5. Let be a nonempty bounded closed and convex subset of a real Hilbert space . Let be a continuous pseudocontractive mapping and let be a continuous monotone mapping. Let be a nonexpansive semigroup on such that , , and . Let such that and suppose . The mappings and are defined by (10) and (11). Let be a sequences generated by and Then the sequence converges strongly to .

Proof. Putting in Theorem 4, we can obtain the result.

Remark 6. According to nonexpansive semigroup mapping, it will be interesting if we replace the semigroup by an additive positive real numbers of a commutative semigroup or a left amenable semigroup or a left reversible semigroup with using an asymptotically invariant sequence. See [31, 32].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Commission on Higher Education, the Thailand Research Fund, and the Rajamangala University of Technology Lanna Tak (Grant no. MRG5580233).