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, …, C_r be nonempty closed convex subsets of a real Hilbert space H such that . 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 ; given only the metric projections of H onto C_i 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 [1] considered a sequence {x_n} generated by cyclic projections, that is, . It was proved that {x_n} converges weakly to an element of for an arbitrary initial point x ∈ H.

Crombez [2] proposed a sequence {y_n} generated by y₀ = y ∈ H, for n = 0,1, 2, …, where 0 < α_i < 1 for all i = 0,1, 2, …, r with and 0 < λ_i < 2 for every i = 1,2, …, r. It was proved that {y_n} converges weakly to an element of for an arbitrary initial point y ∈ H.

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 E. Alber [5] 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 [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 E and in a strictly convex, smooth, and reflexive Banach space E 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 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., [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 ℕ be the set of all positive integers, ℝ the set of all real numbers, E a real Banach space with norm ∥·∥, and E^* the dual of E. For x ∈ E and x^* ∈ E^*, we denote by 〈x, x^*〉 the value of x^* at x. We write x_n → x to indicate that a sequence {x_n} converges strongly to x. Similarly, x_n⇀x and

will symbolize weak and weak* convergence, respectively. We define the modulus δ_E of convexity of E as follows: δ_E is a function of [0,2] into [0,1] such that

()

for every ϵ ∈ [0,2]. E is called uniformly convex if δ_E(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 δ_E(ϵ) ≥ cϵ^p 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, y ∈ E with ∥x∥ = ∥y∥ = 1 and x ≠ y. We know that a uniformly convex Banach space is strictly convex and reflexive. For every p > 1, the (generalized) duality mapping

of E is defined by

()

for all x ∈ E. When p = 2, J₂ is called the normalized duality mapping. We have that for p, q > 1, ∥x∥^pJ_qx = ∥x∥^qJ_px for all x ∈ E. E is said to be smooth if the limit

()

exists for every x, y ∈ E with ∥x∥ = ∥y∥ = 1. We know that the duality mapping J_p 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 J_p of E is one to one in the sense that x ≠ y implies that J_px∩J_py = ∅ for all p > 1. If E is reflexive, then J_p is surjective, and

is identical to the duality mapping

defined by

()

for every y^* ∈ E^*, where q ∈ ℝ satisfies 1/p + 1/q = 1. For p > 1, the duality mapping J_p of a smooth Banach space E is said to be weakly sequentially continuous if x_n⇀x implies that

(see [20, 21] for details). The following is proved by Xu [22] (see also [23]).

Theorem 1 (Xu [22]). 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+y∥^p ≥ ∥x∥^p + p〈y, J_px〉 + c∥y∥^p holds for every x, y ∈ E.

Remark 2. For a p-uniformly convex and smooth Banach space E, we have that the constant c in the theorem above satisfies c ≤ 1. Let

()

Then, there exists a positive real sequence {c_n} such that lim _n→∞c_n = c₀ and ∥x+y∥^p ≥ ∥x∥^p + p〈y, J_px〉 + c_n∥y∥^p for all x, y ∈ E and n ∈ ℕ. So, we get ∥x+y∥^p ≥ ∥x∥^p + p〈y, J_px〉 + c₀∥y∥^p for every x, y ∈ E. 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 ϕ_p : E × E → ℝ is defined by

()

for every x, y ∈ E. We have ϕ_p(x, y) ≥ 0 for all x, y ∈ E and ϕ_p(z, x) + ϕ_p(x, y) = ϕ_p(z, y) + p〈x − z, J_px − J_py〉 for every x, y, z ∈ E. It is known that if E is strictly convex and smooth, then, for x, y ∈ E, ϕ_p(y, x) = ϕ_p(x, y) = 0 if and only if x = y. Indeed, suppose that ϕ_p(y, x) = ϕ_p(x, y) = 0. Then, since

()

we have ∥x∥ = ∥y∥. It follows that 〈y, J_px〉 = p⁻¹(∥y∥^p + (p − 1)∥x∥^p − ϕ_p(y, x)) = ∥y∥^p and ∥J_px∥ = ∥x∥^p−1 = ∥y∥^p−1, which implies that J_py = J_px. Since J_p is one to one, we have x = y (see also [17]). 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, y ∈ E,

()

holds, where c₀ is maximum in Remark 2.

Proof. Let x, y ∈ E. By Theorem 1, we have

()

where c₀ is maximum in Remark 2. Hence, we get

()

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 x₀ ∈ C such that ∥x₀ − x∥ = inf _y∈C∥y − x∥. Putting x₀ = P_Cx, we call P_C the metric projection onto C (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_Cx if and only if 〈y − z, J₂(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_px = ∥x∥^p−1J₂x for every x ∈ E. Therefore, under the same assumption as Lemma 4, we have that y = P_Cx if and only if 〈y − z, J_p(x − y)〉 ≥ 0 for all z ∈ C.

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 {C_n} _n∈ℕ be a family of nonempty closed convex subsets of a p-uniformly convex and smooth Banach space E whose duality mapping J_p is weakly sequentially continuous. Suppose that ⋂_n∈ℕC_n ≠ ∅. Let λ_n,k ∈ ]0, (1 + 1/(p − 1)) ^p−1c₀[ and α_n,k ∈ [0,1] for all n ∈ ℕ and k = 1,2, …, n with for every n ∈ ℕ, where c₀ is maximum in Remark 2. Let {x_n} be a sequence generated by x₁ = x ∈ E and

()

for every n ∈ ℕ. If 0 < liminf _n→∞λ_n,k ≤ limsup _n→∞λ_n,k < (1 + 1/(p − 1)) ^p−1c₀ and liminf _n→∞α_n,k > 0 for each k ∈ ℕ, then {x_n} converges weakly to a point in

Proof. Let for n ∈ ℕ and k = 1,2, …, n. Then, for z ∈ ⋂_n∈ℕC_n, we obtain

()

for all n ∈ ℕ and k = 1,2, …, n. Using Remark 5 with that z ∈ C_k, we get

()

for every n ∈ ℕ and k = 1,2 … , n. Thus, by Lemma 3 we have

()

for each n ∈ ℕ and k = 1,2, …, n. Since it holds that

()

for s, t ≥ 0, p, q > 1 with 1/p + 1/q = 1, and β > 0, we have

()

for every k ∈ ℕ, β_k > 0 and n ≥ k. Therefore, it follows that

()

for every n ∈ ℕ, k = 1,2, …, n, and β_k > 0. Since

()

for every n ∈ ℕ, we have

()

for all n ∈ ℕ and β₁, β₂, …, β_n > 0. Since λ_n,k ∈ ]0, (1 + 1/(p − 1)) ^p−1c₀[, α_n,k ∈ [0,1] for all n ∈ ℕ and k = 1,2, …, n,

()

for each k ∈ ℕ, we can choose β_k > 0 for every k ∈ ℕ such that α_n,k(λ_n,k/β_k − c₀) ≤ 0,

for all n ≥ k and

()

for each k ∈ ℕ. Hence, there exists lim _n→∞ϕ_p(z, x_n) for every z ∈ ⋂_n∈ℕC_n and

()

for all k ∈ ℕ. It follows from Lemma 3 that {x_n} is bounded. Let

and

be subsequences of {x_n} such that

and

. Then, we get

which implies that u₁ ∈ C_k for every k ∈ ℕ. In the same way, we also have u₂ ∈ C_k for every k ∈ ℕ. Let lim _n→∞ϕ_p(u₁, x_n) = μ₁ and lim _n→∞ϕ_p(u₂, x_n) = μ₂. Since

()

and J_p is weakly sequentially continuous, we have

()

Similarly, we obtain μ₂ − μ₁ = −ϕ_p(u₁, u₂). So, we get ϕ_p(u₁, u₂) + ϕ_p(u₂, u₁) = 0, that is, u₁ = u₂. Therefore, {x_n} converges weakly to a point in ⋂_n∈ℕC_n.

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 {C_j} _j∈I a family of nonempty closed convex subsets of a p-uniformly convex and smooth Banach space E whose duality mapping J_p is weakly sequentially continuous. Suppose that ⋂_j∈IC_j ≠ ∅. Let λ_n ∈ ]0, (1 + 1/(p − 1)) ^p−1c₀[ for all n ∈ ℕ, where c₀ is maximum in Remark 2, and let {x_n} be a sequence generated by x₁ = x ∈ E and

()

for every n ∈ ℕ, where the index mapping i : ℕ → I satisfies that, for every j ∈ I, there exists M_j ∈ ℕ such that j ∈ {i(n), …, i(n + M_j − 1)} for each n ∈ ℕ. If 0 < liminf _n→∞λ_n ≤ limsup _n→∞λ_n < (1 + 1/(p − 1)) ^p−1c₀, then, {x_n} converges weakly to a point in ⋂_j∈IC_j.

Proof. Let z ∈ ⋂_j∈IC_j. As in the proof of Theorem 6, we have

()

for all n ∈ ℕ and β > 0. Since λ_n ∈ ]0, (1 + 1/(p − 1)) ^p−1c₀[ for all n ∈ ℕ and 0 < liminf _n→∞λ_n ≤ limsup _n→∞λ_n < (1 + 1/(p − 1)) ^p−1c₀, we can find that β > 0 such that

()

Then, there exists lim _n→∞ϕ_p(z, x_n) for every z ∈ ⋂_i∈IC_i and

()

So, we have that {x_n} is bounded from Lemma 3. Let

be a subsequence of {x_n} such that

. For fixed j ∈ I, there exists a strictly increasing sequence {m_k} ⊂ ℕ such that n_k ≤ m_k ≤ n_k + M_j − 1 and i(m_k) = j for every k ∈ ℕ. It follows that

()

for all k ∈ ℕ which implies that

. Since

, u ∈ C_j for every j ∈ I. So, we get u ∈ ⋂_j∈IC_j. As in the proof of Theorem 6, using that J_p is weakly sequentially continuous, we get that {x_n} converges weakly to a point in ⋂_j∈IC_j.

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) = j mod N for each j ∈ I. Clearly it satisfies the assumption in Theorem 7. In the case where the index set I is countably infinite, that is, I = ℕ, one of the simplest examples of i : ℕ → ℕ can be defined as follows:

()

Then, the assumption in Theorem 7 is satisfied by letting M_j = 2^j for each j ∈ I = ℕ.

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 [2] by Theorem 6.

Theorem 8. Let {C_n} _n∈ℕ be a family of nonempty closed convex subsets of H such that ⋂_n∈ℕC_n ≠ ∅. Let λ_n,k ∈ ]0,2[ and α_n,k ∈ [0,1] for all n ∈ ℕ and k = 1,2, …, n with for every n ∈ ℕ. Let {x_n} be a sequence generated by x₁ = x ∈ H and

()

for every n ∈ ℕ. If it holds that 0 < liminf _n→∞λ_n,k ≤ limsup _n→∞λ_n,k < 2 and liminf _n→∞α_n,k > 0 for each k ∈ ℕ, then, {x_n} converges weakly to a point in

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

Theorem 9. Let I be a countable set and {C_j} _j∈I a family of nonempty closed convex subsets of H such that ⋂_j∈IC_j ≠ ∅. Let λ_n ∈ ]0,2[ for all n ∈ ℕ, and let {x_n} be a sequence generated by x₁ = x ∈ H and

()

for every n ∈ ℕ, where the index mapping i : ℕ → I satisfies that, for every j ∈ I, there exists M_j ∈ ℕ such that j ∈ {i(n), …, i(n + M_j − 1)} for each n ∈ ℕ. If 0 < liminf _n→∞λ_n ≤ limsup _n→∞λ_n < 2, then, {x_n} converges weakly to a point in ⋂_j∈IC_j.

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

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Deduced Results

Acknowledgment

References

References

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

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Deduced Results

Acknowledgment

References

References

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