Finding Minimum Norm Fixed Point of Nonexpansive Mappings and Applications

Abstract

We construct two new methods for finding the minimum norm fixed point of nonexpansive mappings in Hilbert spaces. Some applications are also included.

1. Introduction

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Recall that a mapping 𝑇∶𝐶→𝐶 is nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.1)

Iterative algorithms for finding fixed point of nonexpansive mappings are very interesting topic due to the fact that many nonlinear problems can be reformulated as fixed point equations of nonexpansive mappings. Related works can be found in [1–32].

On the other hand, we notice that it is quite often to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. In an abstract way, we may formulate such problems as finding a point 𝑥^† with the property𝑥^†∈𝐶,‖‖𝑥^†‖‖=min_𝑥∈𝐶‖𝑥‖,(1.2) where 𝐶 is a nonempty closed convex subset of a real Hilbert space 𝐻. In other words, 𝑥^† is the (nearest point or metric) projection of the origin onto 𝐶,𝑥^†=𝑃_𝐶(0),(1.3) where 𝑃_𝐶 is the metric (or nearest point) projection from 𝐻 onto 𝐶.

A typical example is the least-squares solution to the constrained linear inverse problem𝐴𝑥𝑥∈=𝐶𝑏,,(1.4) where 𝐴 is a bounded linear operator from 𝐻 to another real Hilbert space 𝐻₁ and 𝑏 is a given point in 𝐻₁. The least-squares solution to (1.4) is the least-norm minimizer of the minimization problemmin_𝑥∈𝐶‖𝐴𝑥−𝑏‖².(1.5)

Let 𝑆_𝑏 denote the (closed convex) solution set of (1.4) (or equivalently (1.5)). It is known that 𝑆_𝑏 is nonempty if and only if 𝑃_𝐴(𝐶)(𝑏)∈𝐴(𝐶). In this case, 𝑆_𝑏 has a unique element with minimum norm (equivalently, (1.4) has a unique least-squares solution); that is, there exists a unique point 𝑥^†∈𝑆_𝑏 satisfying‖‖𝑥^†‖‖=min‖𝑥‖∶𝑥∈𝑆_𝑏.(1.6) The so-called 𝐶-constrained pseudoinverse of 𝐴 is then defined as the operator 𝐴^†𝐶 with domain and values given by 𝐷𝐴^†𝐶=𝑏∈𝐻∶𝑃^𝐴(𝐶)(𝑏)∈𝐴(𝐶);𝐴^†𝐶(𝑏)=𝑥^†,𝑏∈𝐷𝐴^†𝐶,(1.7)

where 𝑥^†∈𝑆_𝑏 is the unique solution to (1.6).

Note that the optimality condition for the minimization (1.5) is the variational inequality (VI)𝑥̂∈𝐶,⟨𝐴^∗(𝐴𝑥̂−𝑏),𝑥−𝑥̂⟩≥0,𝑥∈𝐶,(1.8) where 𝐴^∗ is the adjoint of 𝐴.

If 𝑏∈𝐷(𝐴^†𝐶), then (1.5) is consistent and its solution set 𝑆_𝑏 coincides with the solution set of VI (1.8). On the other hand, VI (1.8) can be rewritten as𝑥̂∈𝐶,𝑥̂−𝜆𝐴^∗(𝐴𝑥̂)−𝑏−𝑥̂,𝑥−𝑥̂≤0,𝑥∈𝐶,(1.9) where 𝜆>0 is any positive scalar. In the terminology of projections, (1.10) is equivalent to the fixed point equation𝑥̂=𝑃_𝐶𝑥̂−𝜆𝐴^∗(𝐴𝑥̂−𝑏).(1.10) It is not hard to find that for 0<𝜆<2/‖𝐴‖², the mapping 𝑥↦𝑃_𝐶(𝑥−𝜆𝐴^∗(𝐴𝑥−𝑏)) is nonexpansive. Therefore, finding the least-squares solution of the constrained linear inverse problem (1.6) is equivalent to finding the minimum-norm fixed point of the nonexpansive mapping 𝑥↦𝑃_𝐶(𝑥−𝜆𝐴^∗(𝐴𝑥−𝑏)).

Motivated by the above least-squares solution to constrained linear inverse problems, we will study the general case of finding the minimum-norm fixed point of a nonexpansive mapping 𝑇∶𝐶→𝐶:𝑥^†∈Fix(𝑇),‖𝑥^†‖=min{‖𝑥‖∶𝑥∈Fix(𝑇)},(1.11) where Fix(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥} denotes the set of fixed points of 𝑇 (throughout we always assume that Fix(𝑇)≠∅).

We next briefly review two historic approaches which relate to the minimum-norm fixed point problem (1.11).

Browder [1] introduced an implicit scheme as follows. Fix a 𝑢∈𝐶, and for each 𝑡∈(0,1), let 𝑥_𝑡 be the unique fixed point in 𝐶 of the contraction 𝑇_𝑡 which maps 𝐶 into 𝐶:𝑇_𝑡𝑥=𝑡𝑢+(1−𝑡)𝑇𝑥,𝑥∈𝐶.(1.12) Browder proved that𝑠−lim_𝑡↓0𝑥_𝑡=𝑃_Fix(𝑇)𝑢.(1.13) That is, the strong limit of {𝑥_𝑡} as 𝑡→0⁺ is the fixed point of 𝑇 which is nearest from Fix(𝑇) to 𝑢.

Halpern [4], on the other hand, introduced an explicit scheme. Again fix a 𝑢∈𝐶. Then with a sequence {𝑡_𝑛} in (0,1) and an arbitrary initial guess 𝑥₀∈𝐶, we can define a sequence {𝑥_𝑛} through the recursive formula𝑥_𝑛+1=𝑡_𝑛𝑢+1−𝑡_𝑛𝑇𝑥_𝑛,𝑛≥0.(1.14) It is now known that this sequence {𝑥_𝑛} converges in norm to the same limit 𝑃_Fix(𝑇)𝑢 as Browder's implicit scheme (1.12) if the sequence {𝑡_𝑛} satisfies, assumptions (𝐴₁), (𝐴₂), and (𝐴₃) as follows: (𝐴₁)lim_𝑛→∞𝑡_𝑛=0, (𝐴₂)∑^∞𝑛=1𝑡_𝑛=∞, (𝐴₃)either ∑^∞𝑛=1|𝑡_𝑛+1−𝑡_𝑛|=∞ or lim_𝑛→∞(𝑡_𝑛/𝑡_𝑛+1)=1.

Some more progress on the investigation of the implicit and explicit schemes (1.12) and (1.14) can be found in [33–42]. We notice that the above two 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 method works to find the minimum-norm element 𝑥^†. The reason is simple: if 0∉𝐶, we cannot take 𝑢=0 either in (1.12) or (1.14) since the contraction 𝑥↦(1−𝑡)𝑇𝑥 is no longer a self-mapping of 𝐶 (hence may fail to have a fixed point), or (1−𝑡_𝑛)𝑇𝑥_𝑛 may not belong to 𝐶, and consequently, 𝑥_𝑛+1 may be undefined. In order to overcome the difficulties caused by possible exclusion of the origin from 𝐶, we introduce the following two remedies.

For Browder's method, we consider the contraction 𝑥↦(1−𝛽)𝑃_𝐶[(1−𝑡)𝑥]+𝛽𝑇𝑥 for some 𝛽∈(0,1). Since this contraction clearly maps 𝐶 into 𝐶, it has a unique fixed point which is still denoted by 𝑥_𝑡, that is, 𝑥_𝑡=(1−𝛽)𝑃_𝐶[(1−𝑡)𝑥_𝑡]+𝛽𝑇𝑥_𝑡. For Halpern's method, we consider the following iterative algorithm 𝑥_𝑛+1=(1−𝛽)𝑃_𝐶[(1−𝑡_𝑛)𝑥_𝑛]+𝛽𝑇𝑥_𝑛,𝑛≥0. It is easily seen that the net {𝑥_𝑡} and the sequence {𝑥_𝑛} are well defined (i.e., 𝑥_𝑡∈𝐶 and 𝑥_𝑛∈𝐶).

The purpose of this paper is to prove that the above both implicit and explicit methods converge strongly to the minimum-norm fixed point 𝑥^† of the nonexpansive mapping 𝑇. Some applications are also included.

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. Let 𝐶 be a nonempty closed convex subset of 𝐻. Recall that the nearest point (or metric) projection from 𝐻 onto 𝐶 is defined as follows: for each point 𝑥∈𝐻, 𝑃_𝐶𝑥 is the unique point in 𝐶 with the property ‖‖𝑥−𝑃_𝐶𝑥‖‖≤‖𝑥−𝑦‖,𝑦∈𝐶.(2.1)

Note that 𝑃_𝐶 is characterized by the inequality 𝑃_𝐶𝑥∈𝐶,⟨𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥⟩≤0,𝑦∈𝐶.(2.2)

Consequently, 𝑃_𝐶 is nonexpansive.

Below is the so-called demiclosedness principle for nonexpansive mappings.

Lemma 2.1 (cf. [7]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with fixed points. If (𝑥_𝑛) is a sequence in 𝐶 such that 𝑥_𝑛→𝑥^∗ weakly and 𝑥_𝑛−𝑇𝑥_𝑛→𝑦 strongly, then (𝐼−𝑇)𝑥^∗=𝑦.

Finally we state the following elementary result on convergence of real sequences.

Lemma 2.2 (see [19]). Let {𝑎_𝑛}^∞𝑛=0 be a sequence of nonnegative real numbers satisfying 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝜎_𝑛,𝑛≥0,(2.3)where {𝛾_𝑛}^∞𝑛=0⊂(0,1) and {𝜎_𝑛}^∞𝑛=0 are satisfied that (i)∑^∞𝑛=0𝛾_𝑛=∞; (ii)either limsup_𝑛→∞𝜎_𝑛≤0 or ∑^∞𝑛=0|𝛾_𝑛𝜎_𝑛|<∞. Then {𝑎_𝑛}^∞𝑛=0 converges to 0.

We use the following notation: (i)Fix(𝑇) stands for the set of fixed points of 𝑇; (ii)𝑥_𝑛⇀𝑥 stands for the weak convergence of (𝑥_𝑛) to 𝑥; (iii)𝑥_𝑛→𝑥 stands for the strong convergence of (𝑥_𝑛) to 𝑥.

3. Main Results

The aim of this section is to introduce some methods for finding the minimum-norm fixed point of a nonexpansive mapping 𝑇. First, we prove the following theorem by using an implicit method.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝑇∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑇)≠∅. For 𝛽∈(0,1) and each 𝑡∈(0,1), let 𝑥_𝑡 be defined as the unique solution of fixed point equation 𝑥_𝑡=𝛽𝑇𝑥_𝑡+(1−𝛽)𝑃_𝐶(1−𝑡)𝑥_𝑡,𝑡∈(0,1).(3.1) Then the net {𝑥_𝑡} converges in norm, as 𝑡→0⁺, to the minimum-norm fixed point of 𝑇.

Proof. First observe that, for each 𝑡∈(0,1), 𝑥_𝑡 is well defined. Indeed, we define a mapping 𝑆_𝑡∶𝐶→𝐶 by 𝑆_𝑡𝑥=𝛽𝑇𝑥+(1−𝛽)𝑃_𝐶[(1−𝑡)𝑥],𝑥∈𝐶.(3.2) For 𝑥,𝑦∈𝐶, we have ‖‖𝑆_𝑡𝑥−𝑆_𝑡𝑦‖‖=≤𝛽‖‖𝛽‖𝑇(𝑇𝑥𝑥−−𝑇𝑇𝑦𝑦‖)++((11−−𝛽𝛽))‖‖𝑃𝑃_𝐶𝐶[(1−𝑡)𝑥]−𝑃_𝐶[(1−𝑡)𝑦]‖‖≤[1−(1−𝛽)𝑡]‖𝑥−𝑦‖,[(1−𝑡)𝑥]−𝑃_𝐶[(1−𝑡)𝑦]‖‖(3.3) which implies that 𝑆_𝑡 is a self-contraction of 𝐶. Hence 𝑆_𝑡 has a unique fixed point 𝑥_𝑡∈𝐶 which is the unique solution of the fixed point equation (3.1).
Next we prove that {𝑥_𝑡} is bounded. Take 𝑢∈Fix(𝑇). From (3.1), we have ‖‖𝑥_𝑡−𝑢‖‖≤=‖‖𝛽𝛽‖‖𝑇𝑇𝑥𝑥_𝑡𝑡+(1−𝛽)𝑃_𝐶(1−𝑡)𝑥_𝑡−𝑢‖‖≤𝛽‖‖𝑥_𝑡−𝑢‖‖+(1−𝛽)‖‖𝑃_𝐶(1−𝑡)𝑥_𝑡−𝑢‖‖≤𝛽‖‖𝑥_𝑡−𝑢‖‖+(1−𝛽)‖‖(1−𝑡)𝑥_𝑡−𝑢‖‖−𝑢‖‖+(1−𝛽)(1−𝑡)‖‖𝑥_𝑡−𝑢‖‖+𝑡‖𝑢‖,(3.4) that is, ‖‖𝑥_𝑡−𝑢‖‖≤‖𝑢‖.(3.5) Hence, {𝑥_𝑡} is bounded and so is {𝑇𝑥_𝑡}.
From (3.1), we have ‖‖𝑥_𝑡−𝑇𝑥_𝑡‖‖≤≤((11−−𝛽𝛽))‖‖‖‖𝑥𝑃_𝐶𝑡(1−𝑡)𝑥_𝑡−𝑃_𝐶𝑇𝑥_𝑡‖‖≤(1−𝛽)‖‖𝑥_𝑡−𝑇𝑥_𝑡−𝑡𝑥_𝑡‖‖−𝑇𝑥_𝑡‖‖+(1−𝛽)𝑡‖‖𝑥_𝑡‖‖,(3.6)that is, ‖‖𝑥_𝑡−𝑇𝑥_𝑡‖‖≤1−𝛽𝛽𝑡‖‖𝑥^𝑡‖‖⟶0as𝑡⟶0⁺.(3.7) Next we show that {𝑥_𝑡} is relatively norm-compact as 𝑡→0⁺. Let {𝑡_𝑛}⊂(0,1) be a sequence such that 𝑡_𝑛→0⁺ as 𝑛→∞. Put 𝑥_𝑛∶=𝑥_𝑡𝑛. From (3.7), we have ‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖⟶0.(3.8) Again from (3.1), we get ‖‖𝑥_𝑡−𝑢‖‖²≤𝛽‖‖𝑇𝑥_𝑡−𝑢‖‖²+(1−𝛽)‖‖𝑃_𝐶(1−𝑡)𝑥_𝑡−𝑢‖‖²≤𝛽‖‖𝑥_𝑡−𝑢‖‖²+(1−𝛽)‖‖𝑥_𝑡−𝑢−𝑡𝑥_𝑡‖‖²=𝛽‖‖𝑥_𝑡−𝑢‖‖²+(1−𝛽)‖‖𝑥_𝑡−𝑢‖‖²−2𝑡⟨𝑥_𝑡−𝑢,𝑥_𝑡−𝑢⟩−2𝑡⟨𝑢,𝑥_𝑡−𝑢⟩+𝑡²‖‖𝑥_𝑡‖‖².(3.9)It turns out that ‖‖𝑥_𝑡−𝑢‖‖²≤⟨𝑢,𝑢−𝑥_𝑡⟩+𝑡𝑀,(3.10) where 𝑀>0 is some constant such that sup{(1/2)‖𝑥_𝑡‖²∶𝑡∈(0,1)}≤𝑀. In particular, we get from (3.10) ‖‖𝑥_𝑛−𝑢‖‖²≤⟨𝑢,𝑢−𝑥_𝑛⟩+𝑡_𝑛𝑀,𝑢∈Fix(𝑇).(3.11) Since {𝑥_𝑛} is bounded, without loss of generality, we may assume that {𝑥_𝑛} converges weakly to a point 𝑥^∗∈𝐶. Noticing (3.8) we can use Lemma 2.1 to get 𝑥^∗∈Fix(𝑇). Therefore we can substitute 𝑥^∗ for 𝑢 in (3.11) to get ‖‖𝑥_𝑛−𝑥^∗‖‖²≤⟨𝑥^∗,𝑥^∗−𝑥_𝑛⟩+𝑡_𝑛𝑀.(3.12) However, 𝑥_𝑛⇀𝑥^∗. This together with (3.12) guarantees that 𝑥_𝑛→𝑥^∗. The net {𝑥_𝑡} is therefore relatively compact, as 𝑡→0⁺, in the norm topology.
Now we return to (3.11) and take the limit as 𝑛→∞ to get ‖𝑥^∗−𝑢‖²≤⟨𝑢,𝑢−𝑥^∗⟩,𝑢∈Fix(𝑇).(3.13)This is equivalent to 0≤⟨𝑥^∗,𝑢−𝑥^∗⟩,𝑢∈Fix(𝑇).(3.14)Therefore, 𝑥^∗=𝑃_Fix(𝑇)0. This is sufficient to conclude that the entire net {𝑥_𝑡} converges in norm to 𝑥^∗ and 𝑥^∗ is the minimum-norm fixed point of 𝑇. This completes the proof.

Next, we introduce an explicit algorithm for finding the minimum norm fixed point of nonexpansive mappings.

Theorem 3.2. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix(𝑇)≠∅. For given 𝑥₀∈𝐶, define a sequence {𝑥_𝑛} iteratively by 𝑥_𝑛+1=𝛽𝑇𝑥_𝑛+(1−𝛽)𝑃_𝐶1−𝛼_𝑛𝑥_𝑛,𝑛≥0,(3.15) where 𝛽∈(0,1) and 𝛼_𝑛∈(0,1) satisfying the following conditions: (C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=0𝛼_𝑛=∞; (C2)lim_𝑛→∞(𝛼_𝑛/𝛼_𝑛−1)=1. Then the sequence {𝑥_𝑛} converges strongly to the minimum-norm fixed point of 𝑇.

Proof. First we prove that the sequence {𝑥_𝑛} is bounded. Pick 𝑝∈Fix(𝑇). Then, we have ‖‖𝑥_𝑛+1−𝑝‖‖=≤𝛽‖‖𝛽‖‖𝑇𝑇𝑥𝑥_𝑛𝑛−𝑝+(1−𝛽)𝑃_𝐶1−𝛼_𝑛𝑥_𝑛−𝑝‖‖≤𝛽‖‖𝑥_𝑛−𝑝‖‖+(1−𝛽)‖‖𝑃_𝐶1−𝛼_𝑛𝑥_𝑛−𝑝‖‖≤1−(−1−𝑝‖‖𝛽)+𝛼(_𝑛1−𝛽)‖‖1−𝛼_𝑛𝑥_𝑛−𝑝−𝛼_𝑛𝑝‖‖≤max‖‖𝑥_𝑛‖‖𝑥_𝑛−𝑝‖‖+(1−𝛽)𝛼_𝑛‖𝑝‖−𝑝‖‖,‖𝑝‖.(3.16)By induction, ‖‖𝑥_𝑛+1−𝑝‖‖≤max‖‖𝑥₀−𝑝‖‖,‖𝑝‖.(3.17) Next, we estimate ‖𝑥_𝑛+1−𝑥_𝑛‖. From (3.15), we have ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤=‖‖𝛽𝛽‖‖𝑇𝑇𝑥𝑥_𝑛𝑛−𝑇𝑥_𝑛−1+(1−𝛽)𝑃_𝐶1−𝛼_𝑛𝑥_𝑛−𝑃_𝐶1−𝛼_𝑛−1𝑥_𝑛−1‖‖≤𝛽‖‖𝑥_𝑛−𝑇𝑥_𝑛−1‖‖+(1−𝛽)‖‖𝑃_𝐶1−𝛼_𝑛𝑥_𝑛−𝑃_𝐶1−𝛼_𝑛−1𝑥_𝑛−1‖‖≤1−(−1−𝑥𝛽_𝑛−1)‖‖𝛼_𝑛+(1−𝛽)‖‖1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−1−𝛼_𝑛𝑥_𝑛−1‖‖‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+(1−𝛽)𝛼_𝑛||𝛼_𝑛−𝛼_𝑛−1||𝛼_𝑛‖‖𝑥_𝑛−1‖‖.(3.18)This together with Lemma 2.2 implies that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=0.(3.19)Note that ‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖≤≤‖‖‖‖𝑥𝑥_𝑛𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑇𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+(1−𝛽)‖‖𝑃_𝐶1−𝛼_𝑛𝑥_𝑛−𝑃_𝐶𝑇𝑥_𝑛‖‖−𝑥_𝑛+1‖‖+(1−𝛽)‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖+(1−𝛽)𝛼_𝑛‖‖𝑥_𝑛‖‖.(3.20)Thus, ‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖≤1𝛽‖‖𝑥^𝑛−𝑥_𝑛+1‖‖+(1−𝛽)𝛼_𝑛‖‖𝑥_𝑛‖‖⟶0.(3.21) We next show that limsup_𝑛→∞⟨𝑥̃,𝑥̃−𝑥_𝑛⟩≤0,(3.22) where 𝑥̃=𝑃_Fix(𝑇)0, the minimum norm fixed point of 𝑇. To see this, we can take a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} satisfying the properties limsup_𝑛→∞⟨𝑥𝑥̃,𝑥̃−𝑥_𝑛⟩=lim_𝑘→∞𝑥̃,𝑥̃−𝑥_𝑛𝑘,(3.23)_𝑛𝑘⇀𝑥^∗as𝑘⟶∞.(3.24)Now since 𝑥^∗∈Fix(𝑇) (this is a consequence of Lemma 2.2 and (3.21)), we get by combining (3.22) and (3.23) limsup_𝑛→∞⟨𝑥̃,𝑥̃−𝑥_𝑛⟩=⟨𝑥̃,𝑥̃−𝑥^∗⟩≤0.(3.25)Finally, we show that 𝑥_𝑛→𝑥̃. As a matter of fact, we have ‖‖𝑥_𝑛+1−𝑥̃‖‖²≤𝛽‖‖𝑇𝑥_𝑛−𝑥̃‖‖²+(1−𝛽)‖‖𝑃_𝐶1−𝛼_𝑛𝑥_𝑛−𝑥̃‖‖²≤𝛽‖‖𝑥_𝑛−𝑥̃‖‖²+(1−𝛽)‖‖𝑥_𝑛−𝑥̃−𝛼_𝑛𝑥_𝑛‖‖²==𝛽1‖‖𝑥−^𝑛2−(1𝑥̃‖‖−²𝛽+)𝛼(1_𝑛−𝛽)1−2𝛼_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²−2𝛼_𝑛⟨𝑥̃,𝑥_𝑛−𝑥̃⟩+𝛼^2𝑛‖𝑥̃‖²‖‖𝑥_𝑛−𝑥̃‖‖²+2(1−𝛽)𝛼_𝑛⟨𝑥̃,𝑥̃−𝑥^𝑛⟩+𝛼^𝑛‖𝑥̃‖²=1−𝛿_𝑛2‖‖𝑥_𝑛−𝑥̃‖‖²+𝛿_𝑛𝜃_𝑛.(3.26) By (C1) and (3.22), it is easily found that lim_𝑛→∞𝛿_𝑛=0 and limsup_𝑛→∞𝜃_𝑛≤0. We can therefore apply Lemma 2.2 to (3.26) and conclude that 𝑥_𝑛+1→𝑥̃ as 𝑛→∞. This completes the proof.

4. Applications

We consider the following minimization problemmin_𝑥∈𝐶𝜑(𝑥),(4.1) where 𝐶 is a closed convex subset of a real Hilbert space 𝐻 and 𝜑∶𝐶→ℝ is a continuously Fréchet differentiable convex function. Denote by 𝑆 the solution set of (4.1); that is, 𝑆=𝑧∈𝐶∶𝜑(𝑧)=min_𝑥∈𝐶𝜑(𝑥).(4.2)

Assume 𝑆=∅. It is known that a point 𝑧∈𝐶 is a solution of (4.1) if and only if the following optimality condition holds:𝑧∈𝐶,⟨∇𝜑(𝑧),𝑥−𝑧⟩≥0,𝑥∈𝐶.(4.3) (Here ∇𝜑(𝑥) denotes the gradient of 𝜑 at 𝑥∈𝐶.) It is also known that the optimality condition (4.3) is equivalent to the following fixed point problem, 𝑧=𝑇_𝛾𝑧,𝑇_𝛾=𝑃_𝐶(𝐼−𝛾∇𝜑),(4.4)

where 𝛾>0 is any positive number. Note that the solution set 𝑆 of (4.1) coincides with the set of fixed points of 𝑇_𝛾 (for any 𝛾>0).

If the gradient ∇𝜑 is L-Lipschitzian continuous on 𝐶, then it is not hard to see that the mapping 𝑇_𝛾 is nonexpansive if 0<𝛾<2/𝐿.

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

Theorem 4.1. Assume 𝜑 is continuously (Fréchet) differentiable and convex and its gradient ∇𝜑 is L-Lipschitzian. Assume the solution set 𝑆 of the minimization (4.1) is nonempty. Fix 𝛾 such that 0<𝛾<2/𝐿.
(i)For each 𝑡∈(0,1), let 𝑥_𝑡 be the unique solution of the fixed point equation 𝑥_𝑡=𝛽𝑃_𝐶(𝐼−𝛾∇𝜑)𝑥_𝑡+(1−𝛽)𝑃_𝐶(1−𝑡)𝑥_𝑡.(4.5) Then {𝑥_𝑡} converges in norm as 𝑡→0⁺ to the minimum-norm solution of the minimization (4.1).(ii)Define a sequence {𝑥_𝑛} via the recursive algorithm 𝑥_𝑛+1=𝛽𝑃_𝐶(𝐼−𝛾∇𝜑)𝑥_𝑛+(1−𝛽)𝑃_𝐶1−𝛼_𝑛𝑥_𝑛,(4.6) where the sequence {𝛼_𝑛} satisfies conditions (𝐶1)-(𝐶2) in Theorem 3.2. Then {𝑥_𝑛} converges in norm to the minimum-norm solution of the minimization (4.1).

We next turn to consider a convexly constrained linear inverse problem𝐴𝑥𝑥∈=𝐾𝑏,,(4.7) where 𝐴 is a bounded linear operator with nonclosed range from a real Hilbert space 𝐻₁ to another real Hilbert space 𝐻₂ and 𝑏∈𝐻₂ is given.

Problem (4.7) models many applied problems arising from image reconstructions, learning theory, and so on.

Due to some reasons (errors, noises, etc.), (4.7) is often illposed and inconsistent; thus regularization and least-squares are taken into consideration; that is, we look for a solution to the minimization problemmin_𝑥∈𝐾12‖𝐴𝑥−𝑏‖².(4.8) Let 𝑆_𝑏 denote the solution set of (4.8). It is always closed convex (but possibly empty). It is known that 𝑆_𝑏 is nonempty if and only if 𝑃_𝐴(𝐾)(𝑏)∈𝐴(𝐾). In this case, 𝑆_𝑏 has a unique element with minimum norm; that is, there exists a unique point 𝑥^†∈𝑆_𝑏 satisfying‖‖𝑥^†‖‖=min‖𝑥‖∶𝑥∈𝑆_𝑏.(4.9) The K-constrained pseudoinverse of 𝐴, 𝐴^†𝐾, is defined as 𝐷𝐴^†𝐾=𝑏∈𝐻²∶𝑃_𝐴(𝐾)(𝑏)∈𝐴(𝐾),𝐴^†𝐾(𝑏)=𝑥^†,𝑏∈𝐷𝐴^†𝐾,(4.10)

where 𝑥^†∈𝑆_𝑏 is the unique solution to (4.9).

Set 𝜑(𝑥)=12‖𝐴𝑥−𝑏‖².(4.11)

Then 𝜑(𝑥) is quadratic with gradient ∇𝜑(𝑥)=𝐴^∗(𝐴𝑥−𝑏),𝑥∈𝐻₁,(4.12)

where 𝐴^∗ is the adjoint of 𝐴. Clearly ∇𝜑 is Lipschitzian with constant 𝐿=‖𝐴^∗𝐴‖=‖𝐴‖². Therefore, applying Theorem 4.1, we obtain the following result.

Theorem 4.2. Let 𝑏∈𝐷(𝐴^†𝐾). Fix 𝛾 such that 0<𝛾<2/‖𝐴‖².
(i)For each 𝑡∈(0,1), let 𝑥_𝑡 be the unique solution of the fixed point equation 𝑥_𝑡=𝛽𝑃_𝐾𝑥_𝑡−𝛾𝐴^∗𝐴𝑥_𝑡−𝑏+(1−𝛽)𝑃_𝐾(1−𝑡)𝑥_𝑡.(4.13)Then {𝑥_𝑡} converges in norm as 𝑡→0⁺ to 𝐴^†𝐾(𝑏).(ii)Define a sequence {𝑥_𝑛} via the recursive algorithm 𝑥_𝑛+1=𝛽𝑃_𝐾𝑥_𝑛−𝛾𝐴^∗𝐴𝑥_𝑛−𝑏+(1−𝛽)𝑃_𝐾1−𝛼_𝑛𝑥_𝑛,(4.14)where the sequence {𝛼_𝑛} satisfies conditions (𝐶1)-(𝐶2) in Theorem 3.2. Then {𝑥_𝑛} converges in norm to 𝐴^†𝐾(𝑏).

Acknowledgments

The authors are very grateful to the referees for their comments and suggestions which improved the presentation of this paper. Y. -C. Liou was supported in part by NSC 99-2221-E-230-006. Y. Yao was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin and NSFC 11071279.