AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 817392 10.1155/2013/817392 817392 Research Article The Problem of Image Recovery by the Metric Projections in Banach Spaces Kimura Yasunori 1 Nakajo Kazuhide 2 Janno Jaan 1 Department of Information Science Toho University Miyama Funabashi Chiba 274-8510 Japan toho-u.ac.jp 2 Sundai Preparatory School Kanda-Surugadai Chiyoda-ku Tokyo 101-8313 Japan 2013 14 2 2013 2013 24 09 2012 28 12 2012 2013 Copyright © 2013 Yasunori Kimura and Kazuhide Nakajo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the problem of image recovery by the metric projections in a real Banach space. For a countable family of nonempty closed convex subsets, we generate an iterative sequence converging weakly to a point in the intersection of these subsets. Our convergence theorems extend the results proved by Bregman and Crombez.

1. Introduction

Let C1,C2,,Cr be nonempty closed convex subsets of a real Hilbert space H such that i=1rCi. Then, the problem of image recovery may be stated as follows: the original unknown image z is known a priori to belong to the intersection of {Ci}i=1r; given only the metric projections PCi of H onto Ci for i=1,2,,r, recover z by an iterative scheme. Such a problem is connected with the convex feasibility problem and has been investigated by a large number of researchers.

Bregman  considered a sequence {xn} generated by cyclic projections, that is, x0=xH,x1=PC1x,x2=PC2x1,x3=PC3x2,,xr=PCrxr-1,xr+1=PC1xr,xr+2=PC2xr+1,. It was proved that {xn} converges weakly to an element of i=1rCi for an arbitrary initial point xH.

Crombez  proposed a sequence {yn} generated by y0=yH, yn+1=α0yn+i=1rαi(yn+λi(PCiyn-yn)) for n=0,1,2,, where 0<αi<1 for all i=0,1,2,,r with i=0rαi=1 and 0<λi<2 for every i=1,2,,r. It was proved that {yn} converges weakly to an element of i=1rCi for an arbitrary initial point yH.

Later, Kitahara and Takahashi  and Takahashi and Tamura  dealt with the problem of image recovery by convex combinations of nonexpansive retractions in a uniformly convex Banach space E. Alber  took up the problem of image recovery by the products of generalized projections in a uniformly convex and uniformly smooth Banach space E whose duality mapping is weakly sequentially continuous (see also [6, 7]).

On the other hand, using the hybrid projection method proposed by Haugazeau  (see also  and references therein) and the shrinking projection method proposed by Takahashi et al.  (see also ), Nakajo et al.  and Kimura et al.  considered this problem by the metric projections and proved convergence of the iterative sequence to a common point of countable nonempty closed convex subsets in a uniformly convex and smooth Banach space E and in a strictly convex, smooth, and reflexive Banach space E having the Kadec-Klee property, respectively. Kohsaka and Takahashi  took up this problem by the generalized projections and obtained the strong convergence to a common point of a countable family of nonempty closed convex subsets in a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable (see also [17, 18]). Although these results guarantee the strong convergence, they need to use metric or generalized projections onto different subsets for each step, which are not given in advance.

In this paper, we consider this problem by the metric projections, which are one of the most familiar projections to deal with. The advantage of our results is that we use projections onto the given family of subsets only, to generate the iterative scheme. Our convergence theorems extend the results of [1, 2] to a Banach space E, and they deduce the weak convergence to a common point of a countable family of nonempty closed convex subsets of E.

There are a number of results dealing with the image recovery problem from the aspects of engineering using nonlinear functional analysis (see, e.g., ). Comparing with these researches, we may say that our approach is more abstract and theoretical; we adopt a general Banach space with several conditions for an underlying space, and therefore, the technique of the proofs can be applied to various mathematical results related to nonlinear problems defined on Banach spaces.

2. Preliminaries

Throughout this paper, let be the set of all positive integers, the set of all real numbers, E a real Banach space with norm ·, and E* the dual of E. For xE and x*E*, we denote by x,x* the value of x* at x. We write xnx to indicate that a sequence {xn} converges strongly to x. Similarly, xnx and xn*x will symbolize weak and weak* convergence, respectively. We define the modulus δE of convexity of E as follows: δE is a function of [0,2] into [0,1] such that (1)δE(ϵ)=inf{1-x+y2:x,yE,x=1,=infkky=1,x-yϵx+y2} for every ϵ[0,2]. E is called uniformly convex if δE(ϵ)>0 for each ϵ>0. Let p>1. E is said to be p-uniformly convex if there exists a constant c>0 such that δE(ϵ)cϵp for every ϵ[0,2]. It is obvious that a p-uniformly convex Banach space is uniformly convex. E is said to be strictly convex if x+y/2<1 for all x,yE with x=y=1 and xy. We know that a uniformly convex Banach space is strictly convex and reflexive. For every p>1, the (generalized) duality mapping Jp:E2E* of E is defined by (2)Jpx={y*E*:x,y*=xp,y*=xp-1} for all xE. When p=2, J2 is called the normalized duality mapping. We have that for p,q>1, xpJqx=xqJpx for all xE. E is said to be smooth if the limit (3)limt0x+ty-xt exists for every x,yE with x=y=1. We know that the duality mapping Jp of E is single valued for each p>1 if E is smooth. It is also known that if E is strictly convex, then the duality mapping Jp of E is one to one in the sense that xy implies that JpxJpy= for all p>1. If E is reflexive, then Jp is surjective, and Jp-1 is identical to the duality mapping Jq*:E*2E defined by (4)Jq*y*={xE:x,y*=y*q,x=y*q-1} for every y*E*, where q satisfies 1/p+1/q=1. For p>1, the duality mapping Jp of a smooth Banach space E is said to be weakly sequentially continuous if xnx implies that Jpxn*Jpx (see [20, 21] for details). The following is proved by Xu  (see also ).

Theorem 1 (Xu [<xref ref-type="bibr" rid="B25">22</xref>]).

Let E be a smooth Banach space and p>1. Then, E is p-uniformly convex if and only if there exists a constant c>0 such that x+ypxp+py,Jpx+cyp holds for every x,yE.

Remark 2.

For a p-uniformly convex and smooth Banach space E, we have that the constant c in the theorem above satisfies c1. Let (5)c0=sup{c>0:x+ypxp+py,Jpx=sump+cypx,yExppy,Jpx}. Then, there exists a positive real sequence {cn} such that limncn=c0 and x+ypxp+py,Jpx+cnyp for all x,yE and n. So, we get x+ypxp+py,Jpx+c0yp for every x,yE. Therefore, c0 is the maximum of constants. In the case of Hilbert spaces, the normalized duality mapping J2 is the identity mapping and c0=1.

Let E be a smooth Banach space and p>1. The function ϕp:E×E is defined by (6)ϕp(y,x)=yp-py,Jpx+(p-1)xp for every x,yE. We have ϕp(x,y)0 for all x,yE and ϕp(z,x)+ϕp(x,y)=ϕp(z,y)+px-z,Jpx-Jpy for every x,y,zE. It is known that if E is strictly convex and smooth, then, for x,yE,ϕp(y,x)=ϕp(x,y)=0 if and only if x=y. Indeed, suppose that ϕp(y,x)=ϕp(x,y)=0. Then, since (7)0=ϕp(y,x)+ϕp(x,y)=p(xp+yp-x,Jpy-y,Jpx)p(xp+yp-xyp-1-yxp-1)=p(x-y)(xp-1-yp-1)0, we have x=y. It follows that y,Jpx=p-1(yp+(p-1)xp-ϕp(y,x))=yp and Jpx=xp-1=yp-1, which implies that Jpy=Jpx. Since Jp is one to one, we have x=y (see also ). We have the following result from Theorem 1.

Lemma 3.

Let p>1 and E be a p-uniformly convex and smooth Banach space. Then, for each x,yE, (8)ϕp(x,y)c0x-yp holds, where c0 is maximum in Remark 2.

Proof.

Let x,yE. By Theorem 1, we have (9)xpyp+px-y,Jpy+c0x-yp, where c0 is maximum in Remark 2. Hence, we get (10)ϕp(x,y)=xp-yp-px-y,Jpyc0x-yp, which is the desired result.

Let C be a nonempty closed convex subset of a strictly convex and reflexive Banach space E, and let xE. Then, there exists a unique element x0C such that x0-x=infyCy-x. Putting x0=PCx, we call PC the metric projection onto C (see ). We have the following result [25, p. 196] for the metric projection.

Lemma 4.

Let C be a nonempty closed convex subset of a strictly convex, reflexive, and smooth Banach space E, and let xE. Then, y=PCx if and only if y-z,J2(x-y)0 for all zC, where PC is the metric projection onto C.

Remark 5.

For p>1, it holds that xJpx=xp-1J2x for every xE. Therefore, under the same assumption as Lemma 4, we have that y=PCx if and only if y-z,Jp(x-y)0 for all zC.

3. Main Results

Firstly, we consider the iteration of Crombez’s type and get the following result.

Theorem 6.

Let p,q>1 be such that 1/p+1/q=1. Let {Cn}n be a family of nonempty closed convex subsets of a p-uniformly convex and smooth Banach space E whose duality mapping Jp is weakly sequentially continuous. Suppose that nCn. Let λn,k]0,(1+1/(p-1))p-1c0[ and αn,k[0,1] for all n and k=1,2,,n with k=1nαn,k=1 for every n, where c0 is maximum in Remark 2. Let {xn} be a sequence generated by x1=xE and (11)xn+1=Jq*(k=1nαn,k(Jpxn-λn,kJp(xn-PCkxn))) for every n. If 0<liminfnλn,klimsupnλn,k<(1+1/(p-1))p-1c0 and liminfnαn,k>0 for each k, then {xn} converges weakly to a point in n=1Cn.

Proof.

Let yn,k=Jq*(Jpxn-λn,kJp(xn-PCkxn)) for n and k=1,2,,n. Then, for znCn, we obtain (12)ϕp(z,yn,k)-ϕp(z,xn)=-ϕp(yn,k,xn)+pyn,k-z,Jpyn,k-Jpxn=-ϕp(yn,k,xn)-pλn,kyn,k-z,Jp(xn-PCkxn)=-ϕp(yn,k,xn)-pλn,kyn,k-xn,Jp(xn-PCkxn)-pλn,kxn-z,Jp(xn-PCkxn) for all n and k=1,2,,n. Using Remark 5 with that zCk, we get (13)xn-z,Jp(xn-PCkxn)=xn-PCkxn,Jp(xn-PCkxn)+PCkxn-z,Jp(xn-PCkxn)xn-PCkxnp for every n and k=1,2,n. Thus, by Lemma 3 we have (14)ϕp(z,yn,k)-ϕp(z,xn)-c0yn,k-xnp-pλn,kyn,k-xn,Jp(xn-PCkxn)-pλn,kxn-PCkxnp-c0yn,k-xnp+pλn,kyn,k-xnxn-PCkxnp-1-pλn,kxn-PCkxnp for each n and k=1,2,,n. Since it holds that (15)st1βspp+βq-1tqq for s,t0, p,q>1 with 1/p+1/q=1, and β>0, we have (16)yn,k-xnxn-PCkxnp-11βkpyn,k-xnp+βk1/(p-1)p-1pxn-PCkxnp for every k, βk>0 and nk. Therefore, it follows that (17)ϕp(z,yn,k)-ϕp(z,xn)(λn,kβk-c0)yn,k-xnp+λn,k((p-1)βk1/(p-1)-p)xn-PCkxnp for every n, k=1,2,,n, and βk>0. Since (18)ϕp(z,xn+1)=zp-pz,k=1nαn,kJpyn,k+(p-1)k=1nαn,kJpyn,kp/(p-1)zp-pk=1nαn,kz,Jpyn,k+(p-1)k=1nαn,kyn,kp=k=1nαn,kϕp(z,yn,k) for every n, we have (19)ϕp(z,xn+1)-ϕp(z,xn)k=1nαn,k(λn,kβk-c0)yn,k-xnp+k=1nαn,kλn,k((p-1)βk1/(p-1)-p)xn-PCkxnp for all n and β1,β2,,βn>0. Since λn,k]0,(1+1/(p-1))p-1c0[, αn,k[0,1] for all n and k=1,2,,n, (20)0<liminfnλn,klimsupnλn,k<(1+1(p-1))p-1c0,liminfnαn,k>0 for each k, we can choose βk>0 for every k such that αn,k(λn,k/βk-c0)0, αn,kλn,k((p-1)βk1/(p-1)-p)0 for all nk and (21)limsupnαn,k(λn,kβk-c0)<0,limsupnαn,kλn,k((p-1)βk1/(p-1)-p)<0 for each k. Hence, there exists limnϕp(z,xn) for every znCn and (22)limnyn,k-xn=limnxn-PCkxn=0 for all k. It follows from Lemma 3 that {xn} is bounded. Let {xni} and {xmj} be subsequences of {xn} such that xniu1 and xmju2. Then, we get xni-PCkxni0 which implies that u1Ck for every k. In the same way, we also have u2Ck for every k. Let limnϕp(u1,xn)=μ1 and limnϕp(u2,xn)=μ2. Since (23)μ1-μ2=limi(ϕp(u1,xni)-ϕp(u2,xni))=u1p-u2p+plimiu2-u1,Jpxni and Jp is weakly sequentially continuous, we have (24)μ1-μ2=u1p-u2p+pu2-u1,Jpu1=-ϕp(u2,u1). Similarly, we obtain μ2-μ1=-ϕp(u1,u2). So, we get ϕp(u1,u2)+ϕp(u2,u1)=0, that is, u1=u2. Therefore, {xn} converges weakly to a point in nCn.

Using the idea of [9, p. 256], we also have the following result by the iteration of Bregman’s type.

Theorem 7.

Let p,q>1 be such that 1/p+1/q=1. Let I be a countable set and {Cj}jI a family of nonempty closed convex subsets of a p-uniformly convex and smooth Banach space E whose duality mapping Jp is weakly sequentially continuous. Suppose that jICj. Let λn]0,(1+1/(p-1))p-1c0[ for all n, where c0 is maximum in Remark 2, and let {xn} be a sequence generated by x1=xE and (25)xn+1=Jq*(Jpxn-λnJp(xn-PCi(n)xn)) for every n, where the index mapping i:I satisfies that, for every jI, there exists Mj such that j{i(n),,i(n+Mj-1)} for each n. If 0<liminfnλnlimsupnλn<(1+1/(p-1))p-1c0, then, {xn} converges weakly to a point in jICj.

Proof.

Let zjICj. As in the proof of Theorem 6, we have (26)ϕp(z,xn+1)-ϕp(z,xn)(λnβ-c0)xn+1-xnp+λn((p-1)β1/(p-1)-p)xn-PCi(n)xnp for all n and β>0. Since λn]0,(1+1/(p-1))p-1c0[ for all n and 0<liminfnλnlimsupnλn<(1+1/(p-1))p-1c0, we can find that β>0 such that (27)limsupn(λnβ-c0)<0,limsupnλn((p-1)β1/(p-1)-p)<0. Then, there exists limnϕp(z,xn) for every ziICi and (28)limnxn+1-xn=limnxn-PCi(n)xn=0. So, we have that {xn} is bounded from Lemma 3. Let {xnk} be a subsequence of {xn} such that xnku. For fixed jI, there exists a strictly increasing sequence {mk} such that nkmknk+Mj-1 and i(mk)=j for every k. It follows that (29)xmk-xnkl=nknk+Mj-1xl+1-xl for all k which implies that xmku. Since limkxmk-PCjxmk=0, uCj for every jI. So, we get ujICj. As in the proof of Theorem 6, using that Jp is weakly sequentially continuous, we get that {xn} converges weakly to a point in jICj.

Suppose that the index set I is a finite set {0,1,2,,N-1}. For the cyclic iteration, the index mapping i is defined by i(j)=jmodN for each jI. Clearly it satisfies the assumption in Theorem 7. In the case where the index set I is countably infinite, that is, I=, one of the simplest examples of i: can be defined as follows: (30)i(n)={1(n=2m-1forsomem),2(n=2(2m-1)forsomem),3(n=4(2m-1)forsomem),,k(n=2k-1(2m-1)forsomem),. Then, the assumption in Theorem 7 is satisfied by letting Mj=2j for each jI=.

4. Deduced Results

Since a real Hilbert space H is 2-uniformly convex and the maximum c0 in Remark 2 is equal to 1, we get the following results. At first, we have the following theorem which generalizes the results of  by Theorem 6.

Theorem 8.

Let {Cn}n be a family of nonempty closed convex subsets of H such that nCn. Let λn,k]0,2[ and αn,k[0,1] for all n and k=1,2,,n with k=1nαn,k=1 for every n. Let {xn} be a sequence generated by x1=xH and (31)xn+1=k=1nαn,k(xn-λn,k(xn-PCkxn)) for every n. If it holds that 0<liminfnλn,klimsupnλn,k<2 and liminfnαn,k>0 for each k, then, {xn} converges weakly to a point in n=1Cn.

Next, we have the following theorem which extends the result of  by Theorem 7.

Theorem 9.

Let I be a countable set and {Cj}jI a family of nonempty closed convex subsets of H such that jICj. Let λn]0,2[ for all n, and let {xn} be a sequence generated by x1=xH and (32)xn+1=xn-λn(xn-PCi(n)xn) for every n, where the index mapping i:I satisfies that, for every jI, there exists Mj such that j{i(n),,i(n+Mj-1)} for each n. If 0<liminfnλnlimsupnλn<2, then, {xn} converges weakly to a point in jICj.

Acknowledgment

The first author was supported by the Grant-in-Aid for Scientific Research no. 22540175 from the Japan Society for the Promotion of Science.

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