Abstract
Iterative algorithms have been extensively studied over the class of nonexpansive mappings in Hilbert spaces. Recall that nonexpansive mappings belong to quasi-nonexpansive mappings. The aim of this article is expanding the general approximation method proposed by Marino and Xu to quasi-nonexpansive mappings in Hilbert spaces.
Similar content being viewed by others
1. Introduction
Let H be a real Hilbert space with inner product 〈·, ·〉, and induced norm || · ||. A mapping T: H → H is called nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y ∈ H. The set of the fixed points of T is denoted by Fix(T): = {x ∈ H: Tx = x}.
Iterative theory and methods for nonlinear mappings and variational inequalities have recently been applied to solve convex minimization problems, zero point problems and many others; see, e.g., [1–9] and references therein.
The viscosity approximation method was first introduced by Moudafi [10]. Starting with an arbitrary initial x0 ∈ H, define a sequence {x n } generated by:
where f is a contraction with a coefficient α ∈ [0,1) on H, i.e., ||f(x) - f(y)|| ≤ α||x - y|| for all x, y ∈ H, and {ε n } is a sequence in (0,1) satisfying the following given conditions:
-
(1)
limn→∞ε n = 0;
-
(2)
;
-
(3)
.
It is proved that the sequence {x n } generated by (1.1) converges strongly to the unique solution x* ∈ C(C: = Fix(T)) of the variational inequality:
In [1], Xu proved that the sequence {x n } defined by the below process started with an arbitrary initial x0 ∈ H:
converges strongly to the unique solution of the minimization problem (1.3) provided the the sequence {α n } satisfies certain conditions:
where C is the set of fixed points set of T on H and b is a given point in H.
In [2], Marino and Xu combined the iterative method (1.2) with the viscosity approximation method (1.1) and considered the following general iterative method:
It is proved that if the sequence {α n } satisfies appropriate conditions, the sequence {x n } generated by (1.4) converges strongly to the unique solution of the variational inequality:
or equivalently , where C is the fixed point set of a nonexpansive mapping T.
In [11], Maingé considered the viscosity approximation method (1.1), and expanded the strong convergence to quasi-nonexpansive mappings in Hilbert space. Motivated by Marino and Xu [2] and Maingé [11], we consider the following iterative process:
where T ω = (1 - ω)I + ωT, and T is a quasi-nonexpansive mapping. Under some appropriate conditions on ω and {α n }, we obtain strong convergence over the class of quasi-nonexpansive mappings in Hilbert spaces. Our result is more general than Maingé's [11] conclusion, and also extends the iterative method (1.4) to quasi-nonexpansive mappings.
2. Preliminaries
Throughout this article, we write x n ⇀ x to indicate that the sequence {x n } converges weakly to x. x n → x implies that the sequence {x n } converges strongly to x. The following lemmas are useful for our article.
The following identities are valid in a Hilbert space H: for each x,y ∈ H, t ∈ [0, 1]
-
(i)
||x + y||2 ≤ ||x||2 + 2〈y, x + y〉;
-
(ii)
||(1 - t)x + ty||2 = (1 - t)||x||2 + t||y||2 - (1 - t) t||x - y||2;
-
(iii)
.
Lemma 2.1. [2] Let H be a Hilbert space H. Given x ∈ H, C is a closed convex subset of H, f : H → H is a contraction with coefficient 0 < α < 1, and A is a strongly positive linear bounded operator with coefficient . Then for ,
That is, A - γ f is strongly monotone with coefficient .
Lemma 2.2. [2] Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient and 0 < ρ ≤ ||A||-1. Then .
Lemma 2.3. [11] Let T ω : = (1 - ω)I + ωT, with T being a quasi-nonexpansive mapping on H, , and ω ∈ (0, 1]. Then the following statements are reached:
(a1) Fix(T) = Fix(T ω );
(a2) T ω is quasi-nonexpansive;
(a3) ||T ω x - q||2 ≤ ||x - q||2 - ω(1 - ω)||Tx - x||2 for all x ∈ H and q ∈ Fix(T);
(a4) for all x ∈ H and q ∈ Fix(T).
Remark 2.4. (a4) was revised by Wongchan and Saejung [12] (Proposition 2).
Lemma 2.5. [13] Let {Γ n } be a sequence of real numbers that does not decrease at infinity, in the sense that there exist a subsequence of {Γ n } which satisfies for all j ≥ 0. Also consider the sequence of integers defined by
Then is a nondecreasing sequence verifying limn→∞τ(n) = ∞ and for all n ≥ n0, it holds that Γτ(n)< Γτ(n)+1and we have
Recall the metric projection P K form a Hilbert space H to a closed convex subset K of H is defined: for each x ∈ H, there exists a unique element P K x ∈ K such that
Lemma 2.6. Let K be a closed convex subset of H. Given x ∈ H, and z ∈ K, z = P K x, if and only if there holds the inequality:
Lemma 2.7. If x* is the solution of the variational inequality (1.5) with demi-closedness of T and {y n } ∈ H is a bounded sequence such that ||Ty n - y n || → 0, then
Proof. We assume that there exists a subsequence of {y n } such that . From the given conditions ||Ty n - y n || → 0 and T: H → H demi-closed, we have that any weak cluster point of {y n } belongs to the fixed point set Fix(T). Hence, we conclude that , and also have that
Recalling the (1.5), we immediately obtain
This completes the proof.
3. Main results
Let H be a real Hilbert space, let A be a bounded linear operator on H, and let T be a quasi-nonexpansive mapping on H, and f is a contraction with coefficient α; that is ||f (x) - f(y)|| ≤ α||x - y|| for all x, y ∈ H. Assume the set Fix(T) of fixed points of T is nonempty and we note that Fix(T) is closed and convex (see [14] for more general results).
Throughout this article, we assume that A is strongly positive; that is, there exist a constant such that , for all x ∈ H. Let .
Theorem 3.1. Starting with an arbitrary chosen x0 ∈ H, let the sequence {x n } be generated by
where the sequence {α n } ⊂ (0,1) satisfies limn→∞α n = 0, and . Also , T ω : = (1 - ω)I + ωT with two conditions on T:
(C1) ||Tx - q|| ≤ ||x - q|| for any x ∈ H, and q ∈ Fix(T); this means that T is a quasi-nonexpansive mapping;
(C2) T is demiclosed on H; that is: if {y k } ∈ H, y k ⇀ z, and (I - T)y k → 0, then z ∈ Fix(T).
Then {x n } converges strongly to the x* ∈ Fix(T) which is the unique solution of the VIP:
Remark 3.2. Equivalently, from the VIP (3.2), we have
Proof. First we show that {x n } is bounded.
Take any p ∈ Fix(T), from Lemma 2.3 (a3), we have
By induction
Hence {x n } is bounded, so are the {f(x n )} and {A(x n )}.
Let x* = PFix(T)o(I - A + γf)x* From (3.1), we have
Since x* ∈ Fix(T), from (a4), and together with (3.5), we obtain
it follows from the previous inequality that
From (iii), we obviously have
Set , and combine with (3.6), it follows that
Now, we calculate ||xn+1- x n ||.
From the given condition: T ω : = (1 - ω)I + ωT, it is easy to deduce that ||T ω x n - x n || = ω||x n - Txn||. Thus, it follows from (3.5) that
Then from (3.8) and (3.9), we have
Finally, we prove x n → x*. To this end, we consider two cases.
Case 1: Suppose that there exists n0 such that is nonincreasing, it is equal to Γ n+1≤ Γ n for all n ≥ n0. It follows that limn→∞Γ n exists, so we conclude that
It follows from (3.10), (3.11) and the fact that limn→∞α n = 0, we have limn→∞||x n -Tx n || = 0. Again, from (3.10), we have
Then, by , we conclude that
Since {f(x n )} and {x n } are both bounded, as well as α n → 0, and limn→∞||x n - Tx n || = 0, it follows from (3.13) that
From Lemma 2.1, it is obvious that
Thus, from (3.14), (3.15) and the fact that limn→∞Γ n exists, we immediately obtain
or equivalently
Finally, by Lemma 2.7, we have
so we conclude that limn→∞Γ n = 0, which equivalently means that {x n } converges strongly to x*.
Case 2: Assume that there exists a subsequence of {Γ n }n≥0such that for all j ∈ ℕ. In this case, it follows from Lemma 2.5 that there exists a subsequence {Γτ(n)} of {Γ n } such that Γτ(n)+1> Γτ(n), and {τ(n)} is defined as in Lemma 2.5.
Invoking the (3.10) again, it follows that
Recalling the fact that Γτ(n)+1> Γτ(n), we have
From the preceding results, we get the boundedness of {x n } and α n → 0, which obviously lead to
Hence, combining (3.19) with (3.20), we immediately deduce that
Again, (3.15) and (3.21) yield
Recall that limn→∞α τ(n)= 0 and (3.20), we immediately have
By Lemma 2.7, we have
Consider (3.23) again, we conclude that
which means that limn→∞Γτ(n)= 0. By Lemma 2.5, it follows that Γ n ≤ Γτ(n), thus, we get limn→∞Γ n = 0, which is equivalent to x n → x*.
Corollary 3.3. [11] Let the sequence {x n } be generated by
where the sequence {α n } ⊂ (0,1) satisfies limn→∞α n = 0, and . Also , and T ω : = (1 - ω)I + ωT with two conditions on T:
(C1) ||Tx - q|| ≤ ||x - q|| for any x ∈ H, and q ∈ Fix(T); this means that T is a quasi-nonexpansive mapping;
(C2) T is demiclosed on H; that is: if{yk} ∈ H, y k ⇀ z, and (I - T)y k → 0, z ∈ Fix(T).
Then {x n } converges strongly to the x* ∈ Fix(T) which is the unique solution of the VIP (3.27):
References
Xu HK: An iterative approach to quadratic optimizaton. J Optim Theory Appl 2003, 116: 659–678. 10.1023/A:1023073621589
Marino G, Xu HK: An general iterative method for nonexpansive mapping in Hilbert space. J Math Anal Appl 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028
Ansari, QH (Eds): Topics in Nonlinear Analysis and Optimization. World Education, Delhi; 2011.
Khan AR, Yao JC: Convergence of composite iterative schemes for zeros ofm-accretive operators in Banach spaces. In Nonlinear Anal Theory Methods Appl Edited by: Ceng LC, Ansari QH. 2009, 70: 1830–1840. 10.1016/j.na.2008.02.083
Khan AR, Yao JC: Viscosity Approximation Methods for Strongly Positive and Monotone Operators. In Fixed Point Theory Edited by: Ceng LC, Ansari QH. 2009, 10: 35–71.
Yao Y, Shahzad N: New methods with perturbations for nonexpansive mappings in Hilbert spaces. Fixed Point Theory and Applications 2011, 2011: 79. 10.1186/1687-1812-2011-79
Yao Y, Shahzad N: Strong convergence of a proximal point algorithm with general errors. Optim Lett
Yao Y, Liou YC, Chen CP: Algorithms construction for nonexpansive mappings and inverse-strongly monotone mappings. Taiwanese J Math 2011, 15: 1979–1998.
Yao Y, Liou YC, Chen CP: A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem. Math Comput Model 2012, 55(3–4):1506–1515. 10.1016/j.mcm.2011.10.041
Moudafi A: Viscosity approximation methods for fixed-points problems. J Math Anal Appl 2000, 241: 46–55. 10.1006/jmaa.1999.6615
Maingé PE: The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces. Com-put Math Appl 2009, 59(1):74–79.
Wongchan K, Saejung S: On the strong convergence of viscosity approximation process of quasi-nonexpansive mappings in Hilbert spaces. J Abstr Appl Anal 2011., 2011: Article ID 385843, 9
Maingé PE: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal 2008, 16(7–8):899–912. 10.1007/s11228-008-0102-z
Itoh S, Takahashi W: The common fixed point theory of singlevalued mappings and multivalued mappings. Pacific J Math 2008, 79(2):493–508.
Acknowledgements
M. Tian was supported by the Fundamental Research Funds for the Central Universities (No. ZXH2011C002).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Tian, M., Jin, X. A general iterative method for quasi-nonexpansive mappings in Hilbert space. J Inequal Appl 2012, 38 (2012). https://doi.org/10.1186/1029-242X-2012-38
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2012-38