Abstract and Applied Analysis

Volume 2013 (2013), Article ID 817392, 6 pages

http://dx.doi.org/10.1155/2013/817392

Research Article

The Problem of Image Recovery by the Metric Projections in Banach Spaces

Yasunori Kimura¹ and Kazuhide Nakajo²

¹Department of Information Science, Toho University, Miyama, Funabashi, Chiba 274-8510, Japan

²Sundai Preparatory School, Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8313, Japan

Received 24 September 2012; Accepted 28 December 2012

Academic Editor: Jaan Janno

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.

Abstract.

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 be nonempty closed convex subsets of a real Hilbert space such that . Then, the problem of image recovery may be stated as follows: the original unknown image is known a priori to belong to the intersection of ; given only the metric projections of onto for , recover 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 [1] considered a sequence generated by cyclic projections, that is, . It was proved that converges weakly to an element of for an arbitrary initial point .

Crombez [2] proposed a sequence generated by , for , where for all with and for every . It was proved that converges weakly to an element of for an arbitrary initial point .

Later, Kitahara and Takahashi [3] and Takahashi and Tamura [4] dealt with the problem of image recovery by convex combinations of nonexpansive retractions in a uniformly convex Banach space . Alber [5] took up the problem of image recovery by the products of generalized projections in a uniformly convex and uniformly smooth Banach space whose duality mapping is weakly sequentially continuous (see also [6, 7]).

On the other hand, using the hybrid projection method proposed by Haugazeau [8] (see also [9–11] and references therein) and the shrinking projection method proposed by Takahashi et al. [12] (see also [13]), Nakajo et al. [14] and Kimura et al. [15] 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 and in a strictly convex, smooth, and reflexive Banach space having the Kadec-Klee property, respectively. Kohsaka and Takahashi [16] 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 , and they deduce the weak convergence to a common point of a countable family of nonempty closed convex subsets of .

There are a number of results dealing with the image recovery problem from the aspects of engineering using nonlinear functional analysis (see, e.g., [19]). 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, a real Banach space with norm , and the dual of . For and , we denote by the value of at . We write to indicate that a sequence converges strongly to . Similarly, and will symbolize weak and weak* convergence, respectively. We define the modulus of convexity of as follows: is a function of into such that

for every . is called uniformly convex if for each . Let . is said to be -uniformly convex if there exists a constant such that for every . It is obvious that a -uniformly convex Banach space is uniformly convex. is said to be strictly convex if for all with and . We know that a uniformly convex Banach space is strictly convex and reflexive. For every , the (generalized) duality mapping of is defined by

for all . When , is called the normalized duality mapping. We have that for , for all . is said to be smooth if the limit

exists for every with . We know that the duality mapping of is single valued for each if is smooth. It is also known that if is strictly convex, then the duality mapping of is one to one in the sense that implies that for all . If is reflexive, then is surjective, and is identical to the duality mapping defined by

for every , where satisfies . For , the duality mapping of a smooth Banach space is said to be weakly sequentially continuous if implies that (see [20, 21] for details). The following is proved by Xu [22] (see also [23]).

Theorem 1 (Xu [22]). Let be a smooth Banach space and . Then, is -uniformly convex if and only if there exists a constant such that holds for every .

Remark 2. For a -uniformly convex and smooth Banach space , we have that the constant in the theorem above satisfies . Let

Then, there exists a positive real sequence such that and for all and . So, we get for every . Therefore, is the maximum of constants. In the case of Hilbert spaces, the normalized duality mapping is the identity mapping and .

Let be a smooth Banach space and . The function is defined by

for every . We have for all and for every . It is known that if is strictly convex and smooth, then, for if and only if . Indeed, suppose that . Then, since

we have . It follows that and , which implies that . Since is one to one, we have (see also [17]). We have the following result from Theorem 1.

Lemma 3. Let and be a -uniformly convex and smooth Banach space. Then, for each ,

holds, where is maximum in Remark 2.

Proof. Let . By Theorem 1, we have

where is maximum in Remark 2. Hence, we get

which is the desired result.

Let be a nonempty closed convex subset of a strictly convex and reflexive Banach space , and let . Then, there exists a unique element such that . Putting , we call the metric projection onto (see [24]). We have the following result [25, p. 196] for the metric projection.

Lemma 4. Let be a nonempty closed convex subset of a strictly convex, reflexive, and smooth Banach space , and let . Then, if and only if for all , where is the metric projection onto .

Remark 5. For , it holds that for every . Therefore, under the same assumption as Lemma 4, we have that if and only if for all .

3. Main Results

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

Theorem 6. Let be such that . Let be a family of nonempty closed convex subsets of a -uniformly convex and smooth Banach space whose duality mapping is weakly sequentially continuous. Suppose that . Let and for all and with for every , where is maximum in Remark 2. Let be a sequence generated by and

for every . If and for each , then converges weakly to a point in .

Proof. Let for and . Then, for , we obtain

for all and . Using Remark 5 with that , we get

for every and . Thus, by Lemma 3 we have

for each and . Since it holds that

for , with , and , we have

for every , and . Therefore, it follows that

for every , , and . Since

for every , we have

for all and . Since , for all and ,

for each , we can choose for every such that , for all and

for each . Hence, there exists for every and

for all . It follows from Lemma 3 that is bounded. Let and be subsequences of such that and . Then, we get which implies that for every . In the same way, we also have for every . Let and . Since

and is weakly sequentially continuous, we have

Similarly, we obtain . So, we get , that is, . Therefore, converges weakly to a point in .

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

Theorem 7. Let be such that . Let be a countable set and a family of nonempty closed convex subsets of a -uniformly convex and smooth Banach space whose duality mapping is weakly sequentially continuous. Suppose that . Let for all , where is maximum in Remark 2, and let be a sequence generated by and

for every , where the index mapping satisfies that, for every , there exists such that for each . If , then, converges weakly to a point in .

Proof. Let . As in the proof of Theorem 6, we have

for all and . Since for all and , we can find that such that

Then, there exists for every and

So, we have that is bounded from Lemma 3. Let be a subsequence of such that . For fixed , there exists a strictly increasing sequence such that and for every . It follows that

for all which implies that . Since , for every . So, we get . As in the proof of Theorem 6, using that is weakly sequentially continuous, we get that converges weakly to a point in .

Suppose that the index set is a finite set . For the cyclic iteration, the index mapping is defined by for each . Clearly it satisfies the assumption in Theorem 7. In the case where the index set is countably infinite, that is, , one of the simplest examples of can be defined as follows:

Then, the assumption in Theorem 7 is satisfied by letting for each .

4. Deduced Results

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

Theorem 8. Let be a family of nonempty closed convex subsets of such that . Let and for all and with for every . Let be a sequence generated by and

for every . If it holds that and for each , then, converges weakly to a point in .

Next, we have the following theorem which extends the result of [1] by Theorem 7.

Theorem 9. Let be a countable set and a family of nonempty closed convex subsets of such that . Let for all , and let be a sequence generated by and

for every , where the index mapping satisfies that, for every , there exists such that for each . If , then, converges weakly to a point in .

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.

References

L. M. Bregman, “The method of successive projection for finding a common point of convexsets,” Soviet Mathematics, vol. 6, pp. 688–692, 1965.

G. Crombez, “Image recovery by convex combinations of projections,” Journal of Mathematical Analysis and Applications, vol. 155, no. 2, pp. 413–419, 1991.

S. Kitahara and W. Takahashi, “Image recovery by convex combinations of sunny nonexpansive retractions,” Topological Methods in Nonlinear Analysis, vol. 2, no. 2, pp. 333–342, 1993.

W. Takahashi and T. Tamura, “Limit theorems of operators by convex combinations of nonexpansive retractions in Banach spaces,” Journal of Approximation Theory, vol. 91, no. 3, pp. 386–397, 1997.

Y. I. Alber, “Metric and generalized projections operators in Banach spaces: properties andapplications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartasatos, Ed., vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.

Y. I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations in Banach spaces,” Panamerican Mathematical Journal, vol. 4, no. 2, pp. 39–54, 1994.

S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.

Y. Haugazeau, Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes, [Thése], These, Universit é de Paris, Paris, France, 1968.

H. H. Bauschke and P. L. Combettes, “A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces,” Mathematics of Operations Research, vol. 26, no. 2, pp. 248–264, 2001.

K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003.

S. Ohsawa and W. Takahashi, “Strong convergence theorems for resolvents of maximal monotone operators in Banach spaces,” Archiv der Mathematik, vol. 81, no. 4, pp. 439–445, 2003.

W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276–286, 2008.

Y. Kimura and W. Takahashi, “On a hybrid method for a family of relatively nonexpansive mappings in a Banach space,” Journal of Mathematical Analysis and Applications, vol. 357, no. 2, pp. 356–363, 2009.

K. Nakajo, K. Shimoji, and W. Takahashi, “Strong convergence theorems by the hybrid method for families of mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 812–818, 2009.

Y. Kimura, K. Nakajo, and W. Takahashi, “Strongly convergent iterative schemes for a sequence of nonlinear mappings,” Journal of Nonlinear and Convex Analysis, vol. 9, no. 3, pp. 407–416, 2008.

F. Kohsaka and W. Takahashi, “Iterative scheme for finding a common point of infinitely many convex sets in a Banach space,” Journal of Nonlinear and Convex Analysis, vol. 5, no. 3, pp. 407–414, 2004.

S.-y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.

S.-y. Matsushita, K. Nakajo, and W. Takahashi, “Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 73, no. 6, pp. 1466–1480, 2010.

X. Liu and L. Huang, “Total bounded variation-based Poissonian images recovery by split Bregman iteration,” Mathematical Methods in the Applied Sciences, vol. 35, no. 5, pp. 520–529, 2012.

I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.

J. Diestel, Geometry of Banach Spaces, Selected Topics, vol. 485 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.

H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1127–1138, 1991.

B. Beauzamy, Introduction to Banach Spaces and Their Geometry, vol. 68 of North-Holland Mathematics Studies, North-Holland Publishing, Amsterdam, The Netherlands, 2nd edition, 1985.

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984.

W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, 2000.