Open Access Research

Iterative algorithms for minimum-norm fixed point of non-expansive mapping in hilbert space

Yong Cai1, Yuchao Tang12* and Liwei Liu1

Author Affiliations

1 Department of Mathematics, Nanchang University, Nanchang 330031, PR China

2 Department of Mathematics, Xi'an Jiaotong University, Xi'an 710049, PR China

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Fixed Point Theory and Applications 2012, 2012:49  doi:10.1186/1687-1812-2012-49


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/49


Received:12 January 2012
Accepted:26 March 2012
Published:26 March 2012

© 2012 Cai et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of this article is to introduce two iterative algorithms for finding the least norm fixed point of nonexpansive mappings. We provide two algorithms, one implicit and another explicit, from which strong convergence theorems are obtained in Hilbert spaces. Then we apply these algorithms to solve some convex optimization problems. Furthermore, we use them to solve some split feasibility problems. The results of this article extend and improve several results presented in the literature in the recent past.

Mathematics Subject Classification (2000): 47H09; 47H10; 90C25.

Keywords:
nonexpansive mapping; minimum-norm; fixed point; convex optimization

1 Introduction

Throughout, H is a real Hilbert space with inner product 〈·,·〉 and induced norm ║·║. Let C be a nonempty closed convex subset of H. Then, a mapping T, from C into itself is said to be a nonexpansive mapping if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M1">View MathML</a>

for any x, y C. Fix(T) denotes the fixed point set of T, that is Fix(T) = {x C : Tx = x}. Iterative methods for finding fixed points of nonexpansive mappings are an important topic in the theory of nonexpansive mappings and have wide applications in a number of applied areas, such as, image reconstruction in computerized tomography [1], optics and neural networks [2], collective sensing [3], and image denoising and deblurring [4] etc. However, the Picard sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M2">View MathML</a> often fails to converge even in the weak topology. To overcome the difficulties, the Krasnoselskii-Mann iteration algorithm become prevail. This algorithm generates from an arbitrary initial guess x0 C and a sequence {xn} by the recursive formula

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M3">View MathML</a>

(1.1)

where {αn} is a sequence in (0, 1). Reich [5] proved that if X is a uniformly convex Banach space with a Fréchet differential norm and if {αn} is chosen such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M4">View MathML</a>, then the sequence {xn} defined by (1.1) converges weakly to a fixed point of T. On the other hand, Maingé [6] proposed the so-called inertial Krasnoselskii-Mann-type algorithm as follows

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M5">View MathML</a>

(1.2)

where I : H H is the identity operator, x0, x1 H, {θn} ⊂ [0, 1], {αn} ⊂ (0, 1) are relaxation factor. The proposed algorithm unifies Krasnoselskii-Mann iteration and inertial type extrapolation. He established some weak convergence theorems of the sequence {xn} generated by (1.2). It is clear that if θn= 0 for all n, then the algorithm (1.2) reduces to the Krasnoselskii-Mann iteration (1.1). The sequence {υn} is intended to speed up the convergence of algorithms. As a matter of fact, the above algorithms (1.1) and (1.2) have only weak convergence except in a finite dimensional space. To obtain strong convergence in the setting of an infinite dimensional Hilbert or Banach spaces, there exist several iterative algorithms to nonexpansive mappings (e.g., Viscosity iteration algorithm [7], Hybrid projection algorithm [8], Hybrid steepest descent algorithm [9], Halpern-type iteration algorithm [10,11], Shrinking projection algorithm [12], etc.). In general, the nonexpansive mapping may have more than one fixed point. Without loss of generality, we may assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M6">View MathML</a> (otherwise, C is additionally bounded), then Fix(T) is closed and convex (It is worth mentioning that Ferreira [13] proved that Fix(T) is closed and convex even in a strictly convex Banach space which includes Hilbert spaces as a special case). So there exists a unique x* ∈ Fix(T) satisfies:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M7">View MathML</a>

That is, x* is the minimum-norm fixed point of T. In other words, x* is the metric projection of the origin into Fix(T), i.e., x* = PFix(T)0. It is an interesting thing to construct iterative sequence to find the minimum-norm fixed point of a nonexpansive mapping T, i.e., the minimum-norm solutions of x = Tx. Recently, Yao and Xu [14] and Cui and Liu [15] independently introduced two iterative methods (one implicit and one explicit) for finding the minimum-norm fixed point of nonexpansive mapping which is defined on a closed convex subset C of H. The proposed algorithms are based on the well-known Browder's iterative method [16] and Halpern's iterative method [17]. We next briefly recall the Browder's iterative method and the Halpern's iterative method. Browder [16] introduced an implicit scheme as follows. Let u C and t ∈ (0, 1), xtbe the unique fixed point in C of the contraction Ttfrom C into C:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M8">View MathML</a>

(1.3)

Browder proved that the strong limit of {xt} as t → 0+ is the fixed point of T which is nearest from Fix(T) to u, i.e., limt→0+ xt= PFix(T)u. Besides, Halpern [17] introduced an explicit scheme. Let x0 C, define a sequence {xn} by the following:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M9">View MathML</a>

(1.4)

where {αn} ⊂ (0, 1). It is known that the sequence {xn} generated by (1.4) converges in norm to the same limit PFix(T)u as Browder's implicit scheme (1.3) if the sequence {αn} satisfies the following conditions:

(C1) limn→∞ αn= 0;

(C2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M10">View MathML</a>;

(C3) either <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M11">View MathML</a>.

It is noticed that the Browder's and the Halpern's iterative methods do find the minimum-norm fixed point x* of T if 0 ∈ C. However, if 0 ∉ C, then neither Browder's nor Halpern's methods works to find the minimum-norm element x*. The reason is simple: if 0 ∉ C, we cannot take u = 0 either in (1.3) or (1.4) since the contraction Ttx = (1 - t)Tx is no longer a self-mapping of C or the point (1 - αn)Txnmay not belong to C and consequently, {xn+1} may be undefined. In order to overcome this difficulties caused by possible exclusion of the origin from C, Yao and Xu [14] and Cui and Liu [15] put forward the improvement strategy to impose the metric projection PCon the right side of the (1.3) and (1.4) when u = 0. The role of the metric projection PCis to pull the substituted sequence back to C, then the iterative sequences are well-defined.

Motivated and inspired by the above studies, the purpose of this article is to consider another way to ensure the well defined of the iterative sequence. That is, we replace the closed convex subset C by a closed convex cone C (C is said to be a closed convex cone if (i) C is closed and convex; (ii) αx C, for all α ≥ 0 and x C; (iii) C ≠ {0}). We present new strongly convergent methods for approximating minimum-norm fixed point of nonexpansive mappings. The proposed algorithms consist of two types and generated by the following. For each λ ∈ (0, 1), (i) The implicit method

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M12">View MathML</a>

(1.5)

(ii) The explicit method

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M13">View MathML</a>

(1.6)

where {αn} ⊂ (0, 1).

We prove that the sequence {xn} generated by (1.5) and (1.6) converge strongly to the element of minimal norm fixed point of nonexpansive mappings. As applications, we provide iterative processes for solving the constrained convex optimization problem. And we use them to solve some split feasibility problems which attracted great attention in recent years. Our results improve and generalize the corresponding results of Cui and Liu [15], Yao and Xu [14], and Wang and Xu [18] et al.

2 Preliminaries

Let H be a Hilbert space with inner product 〈·,·〉 and norm ║·║, and let C be a nonempty closed convex subset of H.

We use the following notions in the sequel:

(i) for weak convergence and → for strong convergence;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M14">View MathML</a> denotes the weak ω-limit set of {xn}.

Recall that the orthogonal projection PCx of x onto C is defined by the following

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M15">View MathML</a>

The orthogonal projection has the following well-known properties. For a given x H,

(i) 〈x - PCx, z - PCx〉 ≤ 0, for all z C;

(ii) ║PCx - PCy2 ≤ 〈PCx - PCy, x - y〉, for all x, y H.

We shall make use of the following results.

Lemma 2.1. (Demiclosedness principle of nonexpansive mapping) Let T : C C a non-expansive mapping with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M6">View MathML</a>. If xn ⇀ x and (I - T)xn→ 0, then x = Tx.

Lemma 2.2. (see, [19]) Let {xn} and {yn} be bounded sequences in a Banach space E and let {βn} be a sequence in [0, 1] with 0 < lim inf βn ≤ lim sup βn < 1. Suppose xn+1 = βnyn + (1 - βn)xn for all n ≥ 0 and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M16">View MathML</a>

Then limn→∞ yn- xn║ = 0.

Lemma 2.3. (see, [20]) Let {an} be an nonnegative real sequences satisfying the following inequality:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M17">View MathML</a>

where {γn} ⊂ (0, 1) such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M18">View MathML</a>, and lim supn→∞ δn≤ 0. Then limn→∞ an= 0.

3 Main results

First, we prove the following strong convergence theorem by using the implicit method (1.5) for finding the minimum-norm fixed point of a nonexpansive mapping T.

Theorem 3.1. Let C be a closed convex cone of a real Hilbert space H. Let T : C C be a nonexpansive with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M6">View MathML</a>. For each t ∈ (0, 1), let xt be the unique fixed point in C of the contraction Tλ : = (1 - t)(λT + (1 - λ)I), where λ ∈ (0, 1) is a constant. Then xt converges strongly to the minimum-norm fixed point of T as t → 0+.

Proof. Take p ∈ Fix(T), from (1.5), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M19">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M20">View MathML</a>

Hence, {xt} is bounded and so is {Txt}. Next, we prove that ║xt- Txt║→ 0 as t → 0+. In fact, from (1.5), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M21">View MathML</a>

that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M22">View MathML</a>

(3.1)

Next we show that {xt} is relatively norm-compact as t → 0+. Since {xt} is bounded, there exists a null sequence {tn} ⊂ (0, 1) such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M23">View MathML</a>. By Lemma 2.1 and (3.1), then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M24">View MathML</a>.

For any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M25">View MathML</a>, we deduce that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M26">View MathML</a>

It turns out that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M27">View MathML</a>

(3.2)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M24">View MathML</a>, we may substitute <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M28">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M29">View MathML</a> and tnfor t in (3.2) to obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M30">View MathML</a>.

Hence, {xt} is indeed relatively compact (as t → 0+) in the norm topology.

Observe that (3.2) is equivalent to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M31">View MathML</a>

Hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M32">View MathML</a>

This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M33">View MathML</a>

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M34">View MathML</a>, where x* is the minimum-norm fixed point of T, and we conclude that xtx* as t → 0+. This completes the proof. □

Now, we are in the position to prove the strong convergence of the explicit method (1.6). Our proofs of this theorem closely follows proofs given in [11] for some related results.

Theorem 3.2. Let C be a closed convex cone of a real Hilbert space H. Let T : C → C be a nonexpansive mapping and Fix(T) is nonempty. Assume that the sequence {αn} ⊂ (0, 1) satisfies the following conditions:

(i) limn→∞ αn= 0;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M10">View MathML</a>.

Then the sequence {xn} generated by the algorithm (1.6) strongly converges to a fixed point of T which is of minimal norm.

Proof. First we prove that the sequence {xn} is bounded. Let p ∈ Fix(T). By (1.6), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M35">View MathML</a>

By induction,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M36">View MathML</a>

for all n ≥ 0. Then {xn} is bounded. Therefore, {Txn} is also bounded.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M37">View MathML</a>, then the iterative sequence (1.6) is equivalent to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M38">View MathML</a>

(3.3)

Observe that limn→∞(αn+ (1 - αn)λ) = λ, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M39">View MathML</a>

Thus, {yn} is bounded. Consequently, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M40">View MathML</a>

From the fact that {xn} and {Txn} are bounded sequences and limn→∞ αn= 0, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M41">View MathML</a>

With the help of Lemma 2.2, we obtain that limn→∞ ║yn- xn║ = 0. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M42">View MathML</a>

(3.4)

On the other hand,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M43">View MathML</a>

From the above inequality, and (3.4), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M44">View MathML</a>

Next, we prove that lim supn→∞ x* - xn, x*〉 ≤ 0. To achieve this, we take a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M45">View MathML</a> of {xn} such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M46">View MathML</a>

Since {xn} is bounded, without loss of generality, we may assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M47">View MathML</a>. Consequently,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M48">View MathML</a>

Notice that limn→∞ xn- Txn║ = 0. By the demiclosedness principle of nonexpansive mapping T, we have x' ∈ Fix(T). Since x* = PFix(T)0. It follows from the properties of Projection operator that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M49">View MathML</a>

(3.5)

By (1.6), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M50">View MathML</a>

(3.6)

Observe that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M51">View MathML</a>

(3.7)

Therefore, by (3.6) and (3.7), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M52">View MathML</a>

(3.8)

By the condition of (ii) and the inequality (3.5), we can apply Lemma 2.3 to (3.8) and conclude that {xn} converges strongly to x* as n → ∞, that is, the minimum-norm fixed point of T. This completes the proof. □

Remark 3.1. (i) If the closed convex cone C in Theorems 3.1 and 3.2 are replaced by closed convex C with 0 ∈ C, then Theorems 3.1 and 3.2 are still true because of the iterative sequence (1.5) and (1.6) are well-defined now.

(ii) Theorem 3.2 also improve the [[14], Theorem 3.2] and [[15], Theorem 3.3], in which the restrictions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M53">View MathML</a> are removed.

4 Some applications

From now on, we apply the proposed methods for approximating the minimum-norm solution of convex function and to split feasibility problems. Let's recall that the standard constrained convex optimization problem as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M54">View MathML</a>

(4.1)

where f : C R is a convex, Fréchet differentiable function, C is closed convex subset of H.

It is known that the above optimization problem is equivalent to the following variational inequality:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M55">View MathML</a>

(4.2)

where ∇f : H H is the gradient of f.

It is well-known that the optimality condition (4.2) is equivalent to the following fixed point problem:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M56">View MathML</a>

where PCis the metric projection onto C and μ > 0 is a positive constant. Based on the fixed point problem, we deduce the projected gradient method.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M57">View MathML</a>

(4.3)

Using Theorems 3.1 and 3.2, we immediately obtain the following result.

Theorem 4.1. Suppose that the solution set of (4.1) is nonempty. Let the objective function f be convex, fréchet differentiable and its gradient f is Lipschitz continuous with Lipschitz constant L. In addition, if 0 ∈ C or C is closed convex cone. Let μ ∈ (0, 2/L),

(i) For each t ∈ (0, 1), let xt be the unique solution of the fixed point equation

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M58">View MathML</a>

(4.4)

Then {xt} converges in norm as t → 0+ to the minimum-norm solution of the minimization (4.1)

(ii) Define a sequence {xn} by the following

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M59">View MathML</a>

where λ ∈ (0, 1) and the sequence {αn} ⊂ (0, 1) satisfies conditions in Theorem 3.2. Then the sequence {xn} converges strongly to the minimum-norm solution of the minimization (4.1).

Proof. Since ∇f is Lipschitz continuous with Lipschitz constant L, then the PC(I - μf) is nonexpansive mapping (see [[21], Sect. 4). Replace the mapping T in (1.5) and (1.6) with PC(I - μf). Therefore, the conclusion of Theorem 4.1 follows from Theorems 3.1 and 3.2 immediately. □

Next, we give an application of Theorem 4.1 to the split feasibility problem (say SFP, for short) which was introduced by Censor and Elfving [22].

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M60">View MathML</a>

(4.5)

where C and Q are nonempty closed convex subset of Hilbert space H1 and H2, respectively. A : H1 H2 is a bounded linear operator.

It is clear that x* is a solution to the split feasibility problem (4.5) if and only if x* ∈ C and Ax* - PQAx* = 0. We define the proximity function f by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M61">View MathML</a>

and consider the convex optimization problem

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M62">View MathML</a>

(4.6)

Then, x* solves the split feasibility problem (4.5) if and only if x* solves the minimization (4.6) with the minimize equal to 0. Byrne [21] introduced the so-called CQ algorithm to solve the (SFP).

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M63">View MathML</a>

(4.7)

where 0 < μ < 2/ρ(A*A) and where PCdenotes the projection onto C and ρ(A*A) is the spectral radius of the self-adjoint operator A*A. He obtained that the sequence {xn} generated by (4.7) converges weakly to a solution of the (SFP).

In order to obtain strong convergence iterative sequence to solve the (SFP), Xu [23] investigated the following algorithm:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M64">View MathML</a>

(4.8)

where 0 < μ < 2/ρ(A*A). He showed that when the sequence {αn} satisfies the conditions (C1)-(C3), then {xn} converges strongly to the projection of u onto the solution set of the (SFP). In particular, if u = 0 in the algorithm (4.8), then the corresponding algorithms converges strongly to the minimal norm solution of the (SFP). Lately, Wang and Xu [18] introduced a modification of CQ algorithm (4.7) with strong convergence by introducing an approximating curve for the (SFP) in infinite dimensional Hilbert space, and obtained the minimum-norm solution of the (SFP) as the strong limit of the approximating curve. The sequence {xn} is generated by the iterative algorithm

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M65">View MathML</a>

(4.9)

where {αn} ⊂ (0, 1) such that (C1)-(C3).

Applying Theorem 4.1, we obtain the following result which improve the corresponding results of Xu [23] and Wang and Xu [18].

Theorem 4.2. Assume that the split feasibility problem (4.5) is consistent. In addition, if 0 ∈ C or C is closed convex cone. Let the sequence {xn} be generated by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M66">View MathML</a>

(4.10)

where the sequence {αn} ⊂ (0, 1) satisfies the conditions: (i) lim n→∞, αn= 0, (ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M67">View MathML</a>, λ ∈ (0, 1) and μ ∈ (0, 2/ρ(A*A)), where ρ(A*A) denotes the spectral radius of the self-adjoint operator A*A. Then the sequence {xn} converges strongly to the minimum-norm solution of the split feasibility problem (4.5).

Proof. By the definition of the proximity function f, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M68">View MathML</a>

and ∇f is Lipschitz continuous (Lemma 8.1 of [21]), i.e.,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M69">View MathML</a>

where L = ρ(A*A). Then the iterative scheme (4.10) is equivalent to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M70">View MathML</a>

Due to Theorem 4.1, we have the conclusion immediately. □

Remark 4.1. Theorem 4.2 extends the corresponding results of Wang and Xu [18] and Xu [23] by discarding the assumption "<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/49/mathml/M71">View MathML</a> or limn → ∞ (αn/αn+1) = 1".

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments and helpful suggestions, which greatly improved the original manuscript of this article. The authors are also deeply grateful to Prof. Hongkun Xu (Editor) for managing the review process. This study was supported partly by the National Natural Science Foundations of China (11101204, 11102078), the Natural Science Foundations of Jiangxi Province (2009GZS0021, CA201107114) and the Youth Science Funds of The Education Department of Jiangxi Province (GJJ12141).

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