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

and Applied Analysis 3 Lemma 3. Let p > 1 and E be a p-uniformly convex and smooth Banach space. Then, for each x, y ∈ E, φ p (x, y) ≥ c 0 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 p (8) holds, where c 0 is maximum in Remark 2. Proof. Let x, y ∈ E. By Theorem 1, we have ‖x‖ p ≥ 󵄩󵄩󵄩󵄩y 󵄩󵄩󵄩󵄩 p + p⟨x − y, J p y⟩ + c 0 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 p , (9) where c 0 is maximum in Remark 2. Hence, we get φ p (x, y) = ‖x‖ p − 󵄩󵄩󵄩󵄩y 󵄩󵄩󵄩󵄩 p − p⟨x − y, J p y⟩ ≥ c 0 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 p , (10) which is the desired result. Let C be a nonempty closed convex subset of a strictly convex and reflexive Banach space E, and let x ∈ E. Then, there exists a unique element x 0 ∈ C such that ‖x 0 − x‖ = inf y∈C ‖y − x‖. Putting x 0 = P C x, we call P C the metric projection ontoC (see [24]).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 x ∈ E. Then, y = P C x if and only if ⟨y − z, J 2 (x − y)⟩ ≥ 0 for all z ∈ C, where P C is the metric projection onto C. Remark 5. For p > 1, it holds that ‖x‖J p x = ‖x‖ p−1 J 2 x for every x ∈ E. Therefore, under the same assumption as Lemma 4, we have that y = P C x if and only if ⟨y − z, J p (x − y)⟩ ≥ 0 for all z ∈ C.


Introduction
Let 1 , 2 , . . . , be nonempty closed convex subsets of a real Hilbert space such that ⋂ =1 ̸ = 0. 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 { } =1 ; given only the metric projections of onto for = 1, 2, . . . , , 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.
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][10][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.

Preliminaries
Throughout this paper, let N be the set of all positive integers, R 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 for every ∈ [0, 2]. is called uniformly convex if ( ) > 0 for each > 0. Let > 1. is said to be -uniformly convex if there exists a constant > 0 such that ( ) ≥ for every ∈ [0, 2]. It is obvious that a -uniformly convex Banach space is uniformly convex. is said to be strictly convex if ‖ + ‖/2 < 1 for all , ∈ with ‖ ‖ = ‖ ‖ = 1 and ̸ = .
We know that a uniformly convex Banach space is strictly convex and reflexive. For every > 1, the (generalized) duality mapping : → 2 * of is defined by for all ∈ . When = 2, 2 is called the normalized duality mapping. We have that for , > 1, ‖ ‖ = ‖ ‖ for all ∈ . is said to be smooth if the limit exists for every , ∈ with ‖ ‖ = ‖ ‖ = 1. We know that the duality mapping of is single valued for each > 1 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 ∩ = 0 for all > 1. If is reflexive, then is surjective, and −1 is identical to the duality mapping for every * ∈ * , where ∈ R satisfies 1/ + 1/ = 1. For > 1, 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]).
Let be a smooth Banach space and > 1.
Since is one to one, we have = (see also [17]). We have the following result from Theorem 1. Proof. Let , ∈ . By Theorem 1, we have where 0 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 0 ∈ such that ‖ 0 − ‖ = inf ∈ ‖ − ‖. Putting 0 = , we call the metric projection onto (see [24]). We have the following result [25, p. 196] for the metric projection.

Main Results
Firstly, we consider the iteration of Crombez's type and get the following result.
Using the idea of [9, p. 256], we also have the following result by the iteration of Bregman's type.

Theorem 7.
Let , > 1 be such that 1/ + 1/ = 1. 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.
Proof. Let ∈ ⋂ ∈ . As in the proof of Theorem 6, we have for all ∈ N and > 0.
Suppose that the index set is a finite set {0, 1, 2, . . . , − 1}. For the cyclic iteration, the index mapping is defined by ( ) = mod for each ∈ . Clearly it satisfies the assumption in Theorem 7. In the case where the index set is countably infinite, that is, = N, one of the simplest examples of : N → N can be defined as follows: Then, the assumption in Theorem 7 is satisfied by letting = 2 for each ∈ = N.

Deduced Results
Since a real Hilbert space is 2-uniformly convex and the maximum 0 in Remark 2 is equal to 1, we get the following results. At first, we have the following theorem which generalizes the results of [2] by Theorem 6. for every ∈ N, where the index mapping : N → satisfies that, for every ∈ , there exists ∈ N such that ∈ { ( ), . . . , ( + −1)} for each ∈ N. If 0 < lim inf → ∞ ≤ lim sup → ∞ < 2, then, { } converges weakly to a point in ⋂ ∈ .