Research | Open | Published:
Strong convergence of a relaxed three-step iterative algorithm for countable families of pseudocontractions
Fixed Point Theory and Applicationsvolume 2013, Article number: 217 (2013)
An up-to-date method for the approximation of common fixed points of countable families of nonlinear operators is introduced, by which a relaxed three-step iterative algorithm is developed for the class of pseudocontractive mappings, and a strong convergence theorem is established in the framework of Hilbert spaces. Since there is no need to impose uniformity assumption on the involved Lipschitzian and closed mappings, the results improve and extend those announced by Cheng et al. (Fixed Point Theory Appl. 2013:100, 2013) and other authors with the related interest.
MSC:47H05, 47H09, 47H10.
Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product and the corresponding norm . A mapping is said to be nonexpansive if
A mapping is called pseudocontractive or a pseudocontraction if
Note that inequality (1.2) can be equivalently written as
where I denotes the identity operator. A mapping A with the domain and the range in H is called monotone if the inequality
holds for any and for all .
Not only from its being an important generalization of nonexpansive mappings, but also from the firm connection with the important class of nonlinear monotone mappings stems the interest in the class of pseudocontractions. We observe that A is monotone if and only if is pseudocontractive, and hence a zero of A, that is, , is just a fixed point of T. It is well known (see, e.g., ) that if A is monotone, then the solutions to the equation correspond to the equilibrium points of some evolution systems. Considerable efforts have then been devoted to developing iterative techniques for approximating fixed points of pseudocontractive mappings (see, for example, [2–4] and the references contained therein).
In 2013, Cheng et al.  constructed the following three-step iteration method and obtain the convergence theorem for a countable family of Lipschitz pseudocontractive mappings in Hilbert space H. For the iteration format,
they proved that the sequence generated from above converges strongly to a common fixed point of . But it is worth mentioning that the involved mappings were assumed to be uniformly closed and uniformly Lipschitz pseudocontractive, which are obviously two quite strong conditions for countable families of nonlinear operators. Recall that a countable family of mapping is called uniformly Lipschitz with Lipschitz constant , if there exists an such that
A countable family of mapping is called uniformly closed if as , and imply .
Inspired and motivated by the studies mentioned above, in this paper, we introduced an up-to-date method for the approximation of common fixed points of countable families of nonlinear operators, by which a relaxed three-step iterative algorithm is developed for the class of pseudocontractive mappings, and a strong convergence theorem is established in the framework of Hilbert spaces. No compactness assumption is imposed either on the involved mappings or on the set C. The results are more applicable than those of other authors with the related interest.
In the sequel, we shall need the following definitions. Let H be a real Hilbert space. The function , defined by
The function ϕ has also the following property
In what follows, we shall make use of the following lemmas.
Lemma 2.1 
Let H be a Hilbert space. Then for all and for such that the following equality holds
Lemma 2.2 
Let , , and be the sequences of nonnegative real numbers satisfying
If and , then exists.
Lemma 2.3 
The unique solutions to the positive integer equation
where denotes the maximal integer that is not larger than x.
3 Main results
Recall that an operator T on a Hilbert space is closed if and as , then .
Theorem 3.1 Let H be a real Hilbert space, and let C be a closed convex nonempty subset of H. Let be a sequence of closed and Lipschitz pseudocontractive mappings with Lipschitzian constants for each and the interior of . Starting from an arbitrary , define by
where satisfying the following conditions: (i) and (ii) for each ; is the solutions to the positive integer equation: (, ), that is, for each , there exists a unique such that
Then converges strongly to an .
Proof Let . Using the similar argument presented in the proof of [, Theorem 3.1], we have from (3.1) and Lemma 2.1,
In addition, from (3.1), we also have
Substituting (3.4) and (3.5) into (3.3), we obtain that
it then follows from (3.7) and (3.8) that
Substituting (3.6) and (3.9) into (3.2), we obtain that
which, together with condition (i), that is, and , yields that
where . Noting that, in the light of condition (ii), , we have
So, by Lemma 2.2, we conclude that exists.
Furthermore, from (2.3), we also have that
This implies that
Moreover, since the interior of F is nonempty, there exists a and such that whenever . Thus, from (3.12) and (3.13), we obtain that
Then from (3.13) and (3.14), we obtain that
Since h with is arbitrary, we have
So, if , then we have that
Since converges, it then follows from (3.15) that is a Cauchy sequence, and hence there exists an such that
Next, set for each . For example, by Lemma 2.3 and the definition of , we have and . Note that , whenever for each . We have, from (3.11),
Since and are subsequences of , the existence of implies that
Note that, from (3.16), as . It immediately follows from (3.18) and the closedness of that for each , and hence . This completes the proof. □
We now give an example, to which the results of Cheng et al.  cannot be applied.
Example 3.2 Let and . Let be a sequence of nonlinear mappings defined by
It is clear that , and hence the interior of the common fixed points is nonempty. We show that is a countable family of pseudocontractive mappings. If and , then
Noting that , we have
The rest is trivial, and it is easy to show that each is Lipschitz and closed. However, is not uniformly closed. In fact, for any as , we have
while is obviously not a member of F.
Remark 3.3 By using a specific way of choosing the indexes of the involved mappings and parameters, we propose an up-to-date iterative approach to approximating common fixed points of countable families of pseudocontractive mappings. The results extend previous results, announced by the authors with the related research interest.
Zeidler E: Nonlinear Functional Analysis and Its Applications, Part II: Monotone Operators. Springer, Berlin; 1985.
Chidume CE, Moore C: The solution by iteration of nonlinear equations in uniformly smooth Banach spaces. J. Math. Anal. Appl. 1997, 215: 132–146. 10.1006/jmaa.1997.5628
Liu Q: The convergence theorems of the sequence of Ishikawa iterates for hemi-con tractive mappings. J. Math. Anal. Appl. 1990, 148: 55–62. 10.1016/0022-247X(90)90027-D
Zhang S: On the convergence problems of Ishikawa and Mann iteration processes with error for ϕ -pseudocontractive type mappings. Appl. Math. Mech. 2000, 21: 1–10.
Cheng Q, Su Y, Zhang J: Convergence theorems of a three-step iteration method for pseudocontractive mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 100 10.1186/1687-1812-2013-100
Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. Lecture Notes in Pure and Appl. Math. 178. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996:15–50.
Kamimura S, Takahashi W: Strong convergence of proximal-type algorithm in a Banach space. SIAM J. Optim. 2002, 13: 938–945. 10.1137/S105262340139611X
Reich S: A weak convergence theorem for the alternating method with Bergman distance. Lecture Notes in Pure and Appl. Math. 178. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996:313–318.
Zegeye H, Shahzad N: Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 2011, 62: 4007–4014. 10.1016/j.camwa.2011.09.018
Osilike MO, Aniagbosor SC, Akuchu BG: Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces. Panam. Math. J. 2002, 12: 77–88.
Deng WQ, Bai P: An implicit iteration process for common fixed points of two infinite families of asymptotically nonexpansive mappings in Banach spaces. J. Appl. Math. 2013., 2013: Article ID 602582
The author is very grateful to the referees for their useful suggestions, by which the contents of this article has been improved. This work is supported by the National Natural Science Foundation of China (Grant No. 11061037).
The author declares that they have no competing interests.