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

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 𝐶₁,𝐶₂,…,𝐶_𝑟 be nonempty closed convex subsets of a real Hilbert space 𝐻 such that ⋂^𝑟_𝑖=1𝐶_𝑖≠∅. 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 {𝐶_𝑖}^𝑟_𝑖=1; given only the metric projections 𝑃_𝐶𝑖 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, 𝑥₀=𝑥∈𝐻,𝑥₁=𝑃_𝐶1𝑥,𝑥₂=𝑃_𝐶2𝑥₁,𝑥₃=𝑃_𝐶3𝑥₂,…,𝑥_𝑟=𝑃_𝐶𝑟𝑥_𝑟−1,𝑥_𝑟+1=𝑃_𝐶1𝑥_𝑟,𝑥_𝑟+2=𝑃_𝐶2𝑥_𝑟+1,…. It was proved that {𝑥_𝑛} converges weakly to an element of ⋂^𝑟_𝑖=1𝐶_𝑖 for an arbitrary initial point 𝑥∈𝐻.

Crombez [2] proposed a sequence {𝑦_𝑛} generated by 𝑦₀=𝑦∈𝐻, 𝑦_𝑛+1=𝛼₀𝑦_𝑛+∑^𝑟_𝑖=1𝛼_𝑖(𝑦_𝑛+𝜆_𝑖(𝑃_𝐶𝑖𝑦_𝑛−𝑦_𝑛)) for 𝑛=0,1,2,…, where 0<𝛼_𝑖<1 for all 𝑖=0,1,2,…,𝑟 with ∑^𝑟_𝑖=0𝛼_𝑖=1 and 0<𝜆_𝑖<2 for every 𝑖=1,2,…,𝑟. It was proved that {𝑦_𝑛} converges weakly to an element of ⋂^𝑟_𝑖=1𝐶_𝑖 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 𝑥⇀_𝑛^∗𝑥 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 𝛿_𝐸(𝜖)=inf1−𝑥+𝑦2‖𝑥‖𝑦∶𝑥,𝑦∈𝐸,=1,=1,𝑥−𝑦≥𝜖 (1) 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 𝐽_𝑝∶𝐸→2^𝐸∗ of 𝐸 is defined by 𝐽_𝑝𝑥=𝑦^∗∈𝐸^∗∶𝑥,𝑦^∗=‖𝑥‖^𝑝,𝑦^∗=‖𝑥‖^𝑝−1 (2) 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 lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (3) 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 𝐽_𝑝⁻¹ is identical to the duality mapping 𝐽^∗_𝑞∶𝐸^∗→2^𝐸 defined by 𝐽^∗_𝑞𝑦^∗=𝑥∈𝐸∶𝑥,𝑦^∗=𝑦^∗𝑞,‖𝑥‖=𝑦^∗𝑞−1 (4) 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 𝐽_𝑝𝑥⇀_𝑛^∗𝐽_𝑝𝑥 (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 𝑐₀=sup𝑐>0∶𝑥+𝑦^𝑝≥‖𝑥‖^𝑝+𝑝𝑦,𝐽_𝑝𝑥+𝑐𝑦^𝑝∀𝑥,𝑦∈𝐸. (5) 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 𝜙_𝑝𝑦,𝑥=𝑦^𝑝−𝑝𝑦,𝐽_𝑝𝑥+𝑝−1‖𝑥‖^𝑝 (6) 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 0=𝜙_𝑝𝑦,𝑥+𝜙_𝑝‖𝑥‖𝑥,𝑦=𝑝^𝑝+𝑦^𝑝−𝑥,𝐽_𝑝𝑦−𝑦,𝐽_𝑝𝑥≥𝑝‖𝑥‖^𝑝+𝑦^𝑝−‖𝑥‖𝑦^𝑝−1−𝑦‖𝑥‖^𝑝−1=𝑝‖𝑥‖−𝑦‖𝑥‖^𝑝−1−𝑦^𝑝−1≥0, (7) 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, 𝜙_𝑝𝑥,𝑦≥𝑐₀𝑥−𝑦^𝑝 (8) holds, where 𝑐₀ is maximum in Remark 2.

Proof. Let 𝑥,𝑦∈𝐸. By Theorem 1, we have ‖𝑥‖^𝑝≥𝑦^𝑝+𝑝𝑥−𝑦,𝐽_𝑝𝑦+𝑐₀𝑥−𝑦^𝑝, (9) where 𝑐₀ is maximum in Remark 2. Hence, we get 𝜙_𝑝𝑥,𝑦=‖𝑥‖^𝑝−𝑦^𝑝−𝑝𝑥−𝑦,𝐽_𝑝𝑦≥𝑐₀𝑥−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 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 ∑^𝑛_𝑘=1𝛼_𝑛,𝑘=1 for every 𝑛∈ℕ, where 𝑐₀ is maximum in Remark 2. Let {𝑥_𝑛} be a sequence generated by 𝑥₁=𝑥∈𝐸 and 𝑥_𝑛+1=𝐽^∗_𝑞^𝑛_𝑘=1𝛼_𝑛,𝑘𝐽_𝑝𝑥_𝑛−𝜆_𝑛,𝑘𝐽_𝑝𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛 (11) 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 ⋂^∞_𝑛=1𝐶_𝑛.

Proof. Let 𝑦_𝑛,𝑘=𝐽^∗_𝑞(𝐽_𝑝𝑥_𝑛−𝜆_𝑛,𝑘𝐽_𝑝(𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛)) for 𝑛∈ℕ and 𝑘=1,2,…,𝑛. Then, for 𝑧∈⋂_𝑛∈ℕ𝐶_𝑛, we obtain 𝜙_𝑝𝑧,𝑦_𝑛,𝑘−𝜙_𝑝𝑧,𝑥_𝑛=−𝜙_𝑝𝑦_𝑛,𝑘,𝑥_𝑛+𝑝𝑦_𝑛,𝑘−𝑧,𝐽_𝑝𝑦_𝑛,𝑘−𝐽_𝑝𝑥_𝑛=−𝜙_𝑝𝑦_𝑛,𝑘,𝑥_𝑛−𝑝𝜆_𝑛,𝑘𝑦_𝑛,𝑘−𝑧,𝐽_𝑝𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛=−𝜙_𝑝𝑦_𝑛,𝑘,𝑥_𝑛−𝑝𝜆_𝑛,𝑘𝑦_𝑛,𝑘−𝑥_𝑛,𝐽_𝑝𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛−𝑝𝜆_𝑛,𝑘𝑥_𝑛−𝑧,𝐽_𝑝𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛 (12) for all 𝑛∈ℕ and 𝑘=1,2,…,𝑛. Using Remark 5 with that 𝑧∈𝐶_𝑘, we get 𝑥_𝑛−𝑧,𝐽_𝑝𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛=𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛,𝐽_𝑝𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛+𝑃_𝐶𝑘𝑥_𝑛−𝑧,𝐽_𝑝𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛≥𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛^𝑝 (13) for every 𝑛∈ℕ and 𝑘=1,2…,𝑛. Thus, by Lemma 3 we have 𝜙_𝑝𝑧,𝑦_𝑛,𝑘−𝜙_𝑝𝑧,𝑥_𝑛≤−𝑐₀𝑦_𝑛,𝑘−𝑥_𝑛^𝑝−𝑝𝜆_𝑛,𝑘𝑦_𝑛,𝑘−𝑥_𝑛,𝐽_𝑝𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛−𝑝𝜆_𝑛,𝑘𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛^𝑝≤−𝑐₀𝑦_𝑛,𝑘−𝑥_𝑛^𝑝+𝑝𝜆_𝑛,𝑘𝑦_𝑛,𝑘−𝑥_𝑛𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛^𝑝−1−𝑝𝜆_𝑛,𝑘𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛^𝑝 (14) for each 𝑛∈ℕ and 𝑘=1,2,…,𝑛. Since it holds that 1𝑠𝑡≤𝛽𝑠^𝑝𝑝+𝛽^𝑞−1𝑡^𝑞𝑞 (15) for 𝑠,𝑡≥0, 𝑝,𝑞>1 with 1/𝑝+1/𝑞=1, and 𝛽>0, we have 𝑦_𝑛,𝑘−𝑥_𝑛𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛^𝑝−1≤1𝛽_𝑘𝑝𝑦_𝑛,𝑘−𝑥_𝑛^𝑝+𝛽_𝑘^1/(𝑝−1)𝑝−1𝑝𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛^𝑝 (16) for every 𝑘∈ℕ, 𝛽_𝑘>0 and 𝑛≥𝑘. Therefore, it follows that 𝜙_𝑝𝑧,𝑦_𝑛,𝑘−𝜙_𝑝𝑧,𝑥_𝑛≤𝜆_𝑛,𝑘𝛽_𝑘−𝑐₀𝑦_𝑛,𝑘−𝑥_𝑛^𝑝+𝜆_𝑛,𝑘𝑝−1𝛽_𝑘^1/(𝑝−1)−𝑝𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛^𝑝 (17) for every 𝑛∈ℕ, 𝑘=1,2,…,𝑛, and 𝛽_𝑘>0. Since 𝜙_𝑝𝑧,𝑥_𝑛+1=‖𝑧‖^𝑝−𝑝𝑧,^𝑛_𝑘=1𝛼_𝑛,𝑘𝐽_𝑝𝑦_𝑛,𝑘+𝑝−1^𝑛_𝑘=1𝛼_𝑛,𝑘𝐽_𝑝𝑦_𝑛,𝑘^{𝑝/(𝑝−1)}≤‖𝑧‖^𝑝−𝑝^𝑛_𝑘=1𝛼_𝑛,𝑘𝑧,𝐽_𝑝𝑦_𝑛,𝑘+𝑝−1^𝑛_𝑘=1𝛼_𝑛,𝑘𝑦_𝑛,𝑘^𝑝=^𝑛_𝑘=1𝛼_𝑛,𝑘𝜙_𝑝𝑧,𝑦_𝑛,𝑘 (18) for every 𝑛∈ℕ, we have 𝜙_𝑝𝑧,𝑥_𝑛+1−𝜙_𝑝𝑧,𝑥_𝑛≤^𝑛_𝑘=1𝛼_𝑛,𝑘𝜆_𝑛,𝑘𝛽_𝑘−𝑐₀𝑦_𝑛,𝑘−𝑥_𝑛^𝑝+^𝑛_𝑘=1𝛼_𝑛,𝑘𝜆_𝑛,𝑘𝑝−1𝛽_𝑘^1/(𝑝−1)−𝑝𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛^𝑝 (19) 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, 0<liminf_𝑛→∞𝜆_𝑛,𝑘≤limsup_𝑛→∞𝜆_𝑛,𝑘<11+𝑝−1^𝑝−1𝑐₀,liminf_𝑛→∞𝛼_𝑛,𝑘>0 (20) 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, 𝛼_𝑛,𝑘𝜆_𝑛,𝑘((𝑝−1)𝛽_𝑘^1/(𝑝−1)−𝑝)≤0 for all 𝑛≥𝑘 and limsup_𝑛→∞𝛼_𝑛,𝑘𝜆_𝑛,𝑘𝛽_𝑘−𝑐₀<0,limsup_𝑛→∞𝛼_𝑛,𝑘𝜆_𝑛,𝑘𝑝−1𝛽_𝑘^1/(𝑝−1)−𝑝<0 (21) for each 𝑘∈ℕ. Hence, there exists lim_𝑛→∞𝜙_𝑝(𝑧,𝑥_𝑛) for every 𝑧∈⋂_𝑛∈ℕ𝐶_𝑛 and lim_𝑛→∞𝑦_𝑛,𝑘−𝑥_𝑛=lim_𝑛→∞𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛=0 (22) for all 𝑘∈ℕ. It follows from Lemma 3 that {𝑥_𝑛} is bounded. Let {𝑥_𝑛𝑖} and {𝑥_𝑚𝑗} be subsequences of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑢₁ and 𝑥_𝑚𝑗⇀𝑢₂. Then, we get ‖𝑥_𝑛𝑖−𝑃_𝐶𝑘𝑥_𝑛𝑖‖→0 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 𝜇₁−𝜇₂=lim_𝑖→∞𝜙_𝑝𝑢₁,𝑥_𝑛𝑖−𝜙_𝑝𝑢₂,𝑥_𝑛𝑖=𝑢₁^𝑝−𝑢₂^𝑝+𝑝lim_𝑖→∞𝑢₂−𝑢₁,𝐽_𝑝𝑥_𝑛𝑖 (23) and 𝐽_𝑝 is weakly sequentially continuous, we have 𝜇₁−𝜇₂=𝑢₁^𝑝−𝑢₂^𝑝+𝑝𝑢₂−𝑢₁,𝐽_𝑝𝑢₁=−𝜙_𝑝𝑢₂,𝑢₁. (24) 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 𝑥_𝑛+1=𝐽^∗_𝑞𝐽_𝑝𝑥_𝑛−𝜆_𝑛𝐽_𝑝𝑥_𝑛−𝑃_{𝐶𝑖(𝑛)}𝑥_𝑛 (25) 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 𝜙_𝑝𝑧,𝑥_𝑛+1−𝜙_𝑝𝑧,𝑥_𝑛≤𝜆_𝑛𝛽−𝑐₀𝑥_𝑛+1−𝑥_𝑛^𝑝+𝜆_𝑛𝑝−1𝛽^1/(𝑝−1)−𝑝𝑥_𝑛−𝑃_{𝐶𝑖(𝑛)}𝑥_𝑛^𝑝 (26) 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 limsup_𝑛→∞𝜆_𝑛𝛽−𝑐₀<0,limsup_𝑛→∞𝜆_𝑛𝑝−1𝛽^1/(𝑝−1)−𝑝<0. (27) Then, there exists lim_𝑛→∞𝜙_𝑝(𝑧,𝑥_𝑛) for every 𝑧∈⋂_𝑖∈𝐼𝐶_𝑖 and lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞𝑥_𝑛−𝑃_{𝐶𝑖(𝑛)}𝑥_𝑛=0. (28) So, we have that {𝑥_𝑛} is bounded from Lemma 3. Let {𝑥_𝑛𝑘} be a subsequence of {𝑥_𝑛} such that 𝑥_𝑛𝑘⇀𝑢. 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 𝑥_𝑚𝑘−𝑥_𝑛𝑘≤^{𝑛𝑘+𝑀𝑗−1}_{𝑙=𝑛𝑘}𝑥_𝑙+1−𝑥_𝑙 (29) for all 𝑘∈ℕ which implies that 𝑥_𝑚𝑘⇀𝑢. Since lim_𝑘→∞‖𝑥_𝑚𝑘−𝑃_𝐶𝑗𝑥_𝑚𝑘‖=0, 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)=𝑗mod𝑁 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: 𝑖(𝑛)=1(),2(()),3(()),𝑘𝑛=2𝑚−1forsome𝑚∈ℕ𝑛=22𝑚−1forsome𝑚∈ℕ𝑛=42𝑚−1forsome𝑚∈ℕ…,𝑛=2^𝑘−1(2𝑚−1)forsome𝑚∈ℕ,…. (30) 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 ∑^𝑛_𝑘=1𝛼_𝑛,𝑘=1 for every 𝑛∈ℕ. Let {𝑥_𝑛} be a sequence generated by 𝑥₁=𝑥∈𝐻 and 𝑥_𝑛+1=^𝑛_𝑘=1𝛼_𝑛,𝑘𝑥_𝑛−𝜆_𝑛,𝑘𝑥_𝑛−𝑃_𝐶𝑘𝑥_𝑛 (31) 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 ⋂^∞_𝑛=1𝐶_𝑛.

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 𝑥_𝑛+1=𝑥_𝑛−𝜆_𝑛𝑥_𝑛−𝑃_{𝐶𝑖(𝑛)}𝑥_𝑛 (32) 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.