Multiple-Set Split Feasibility Problems for 𝜿-Strictly Pseudononspreading Mapping in Hilbert Spaces

Abstract

The purpose of this paper is to prove some weak and strong convergence theorems for solving the multiple-set split feasibility problems for 𝜅-strictly pseudononspreading mapping in infinite-dimensional Hilbert spaces by using the proposed iterative method. The main results presented in this paper extend and improve the corresponding results of Xu et al. (2006), of Osilike et al. (2011), and of many other authors.

1. Introduction and Preliminaries

Censor and Elfving first introduced the split feasibility problem (SFP) [1] in finite dimensional spaces for modeling inverse problems. The SFP can be used in various disciplines such as medical image reconstruction [2], image restoration, computer tomograph, and radiation therapy treatment planning [3–5]. The multiple-set split feasibility problem (MSSFP) was studied in [4–6].

In the sequel, we always assume that 𝐻₁, 𝐻₂ are two real Hilbert spaces and denote by “→” and “⇀” the strong and weak convergence, respectively.

The so-called multiple-set split feasibility problem (MSSFP) is to find 𝑥^∗∈𝐶 such that 𝐴𝑥^∗∈𝑄, where 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator, 𝑆_𝑖 and 𝑇_𝑖, 𝑖=1,2,…,𝑁 are the families of mappings, 𝑆_𝑖∶𝐻₁→𝐻₁ and 𝑇_𝑖∶𝐻₂→𝐻₂, 𝐶∶=⋂^𝑁_𝑖=1𝐹(𝑆_𝑖), and 𝑄∶=⋂^𝑁_𝑖=1𝐹(𝑇_𝑖), where 𝐹(𝑆_𝑖)={𝑥∈𝐻₁∶𝑆_𝑖𝑥=𝑥} and 𝐹(𝑇_𝑖)={𝑦∈𝐻₂∶𝑇_𝑖𝑦=𝑦} denote the sets of fixed points of 𝑆_𝑖 and 𝑇_𝑖, respectively. In the sequel, we use Γ to denote the set of solutions of the MSSFP; that is, Γ={𝑥∈𝐶∶𝐴𝑥∈𝑄}. (1)

Recently, Kohsaka and Takahashi [7, 8] introduced an important class of mappings which is called the class of nonspreading mappings.

Definition 1 (see [7, 8]). Let 𝐾 to be a nonempty closed convex subset of a Hilbert space 𝐻. A mapping 𝑇∶𝐾→𝐾 is said to be nonspreading, if 2𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑦²+𝑇𝑦−𝑥²,∀𝑥,𝑦∈𝐾. (2)

In [9], Iemoto and Takahashi proved that this definition is equivalent to the following.

Definition 2 (see [9]). Let 𝐾 be a nonempty closed convex subset of a Hilbert space 𝐻. A mapping 𝑇∶𝐾→𝐾 is said to be nonspreading, if 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+2𝑥−𝑇𝑥,𝑦−𝑇𝑦,∀𝑥,𝑦∈𝐾. (3)

Browder and Petryshyn [10] proposed the following 𝜅-strictly pseudononspreading mapping.

Definition 3 (see [10]). Let 𝐻 be a real Hilbert space. We say that a mapping 𝑇∶𝐷(𝑇)⊂𝐻→𝐻 is 𝜅-strictly pseudononspreading if there exists 𝜅∈[0,1), such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝜅𝑥−𝑇𝑥−𝑦−𝑇𝑦²+2𝑥−𝑇𝑥,𝑦−𝑇𝑦,∀𝑥,𝑦∈𝐷(𝑇). (4) Clearly every nonspreading mapping is 𝜅-strictly pseudononspreading.

Osilike and Isiogugu [11] introduced a class of nonspreading type mappings which is more general than that of the mappings studied in [12] in Hilbert spaces and proved some weak and strong convergence theorems in real Hilbert spaces. Recently, the split feasibility problem also was considered in the work by Deepho and Kumam [13, 14] and Sunthrayuth et al. [15], and some weak and strong convergence theorems are proved in real Hilbert spaces.

The purpose of this paper is to study the multiple-set split feasibility problem (MSSFP) for 𝜅-strictly pseudononspreading mappings in the framework of infinite dimensional Hilbert spaces.

In the sequel, we recall some definitions, notations, and conclusions which will be needed in proving our main results.

Definition 4 (see [3]). Let 𝐸 be a real Banach space. A mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐸 is said to be demiclosed at origin if for any sequence {𝑥_𝑛} in 𝐷(𝑇) which converges weakly to a point 𝑥^∗∈𝐷(𝑇) and ‖(𝐼−𝑇)𝑥_𝑛‖ converges strongly to 0, then 𝑇𝑥^∗=𝑥^∗.

Definition 5. A Banach space 𝐸 is said to have Opial property if, for any sequence {𝑥_𝑛} with 𝑥_𝑛⇀𝑥^∗, we have liminf_𝑛→∞𝑥_𝑛−𝑥^∗<liminf_𝑛→∞𝑥_𝑛−𝑦 (5) for all 𝑦∈𝐸 with 𝑦≠𝑥^∗.

Remark 6. It is well known that each Hilbert space possesses Opial property.

Definition 7. A mapping 𝑆∶𝐾→𝐾 is said to be semicompact, if, for any bounded sequence {𝑥_𝑛}⊂𝐾 with lim_𝑛→∞‖𝑥_𝑛−𝑆𝑥_𝑛‖=0, there exists a subsequence {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that {𝑥_𝑛𝑖} converges strongly to some point 𝑥^∗∈𝐾.

Lemma 8 (see [11]). Let H be a real Hilbert space; then the following results hold.(i)For all 𝑥, 𝑦∈𝐻 and for all 𝑡∈[0,1], 𝑡𝑥+(1−𝑡)𝑦²=𝑡‖𝑥‖²+(1−𝑡)𝑦²−𝑡(1−𝑡)𝑥−𝑦². (6)(ii)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩.(iii)If {𝑥_𝑛}^∞_𝑛=1 is a sequence in 𝐻 which converges weakly to 𝑧∈𝐻, then limsup_𝑛→∞𝑥_𝑛−𝑦²=limsup_𝑛→∞𝑥_𝑛−𝑧²+𝑧−𝑦²,∀𝑦∈𝐻. (7)

Definition 9. Let 𝐾 be a nonempty closed convex subset of a real Hilbert space 𝐻. The metric projection𝑃_𝐾∶𝐻→𝐾 is a mapping such that, for each 𝑥∈𝐻, 𝑃_𝐾𝑥 is the unique point in 𝐾 such that ‖𝑥−𝑃_𝐾𝑥‖≤‖𝑥−𝑦‖, ∀𝑦∈𝐾. It is known that, for each 𝑥∈𝐻, 𝑥−𝑃_𝐾𝑥,𝑦−𝑃_𝐾𝑥≤0,∀𝑦∈𝐾. (8)

Lemma 10 (see [11]). Let 𝐾 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐾→𝐾 be a 𝜅-strictly pseudononspreading mapping. If 𝐹(𝑇)≠∅, then it is closed and convex.

Lemma 11 (see [11]). Let 𝐾 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐾→𝐾 be a 𝜅-strictly pseudononspreading mapping. Then (𝐼−𝑇) is demiclosed at 0.

2. Main Results

Theorem 12. Let 𝐻₁, 𝐻₂, 𝐴, {𝑆_𝑖}, {𝑇_𝑖}, 𝐶, 𝑄 be the same as aforementioned. For each 𝑖=1,2,…,𝑁, let 𝑇_𝑖 be a 𝜅_𝑖-strictly pseudononspreading mapping and let 𝑆_𝑖 be a 𝜚_𝑖-strictly pseudononspreading mapping. Let {𝑥_𝑛} be the sequence generated by 𝑥₁∈𝐻₁𝑢𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝑥_𝑛+𝛾𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑆_{𝑛(mod𝑁)}𝑢_𝑛, (9) where 𝛾∈(0,(1−𝜅)/𝜆) with 𝜆 being the spectral of the operator 𝐴^∗𝐴 and 𝜅=max{𝜅₁,𝜅₂,…,𝜅_𝑁}∈(0,1), and 𝛼_𝑛 is a sequence in (0,1−𝜚] with 𝜚=max{𝜚₁,𝜚₂,…,𝜚_𝑁}∈(0,1). If Γ≠∅ (where Γ is the set of solutions of the MSSFP defined by (1)), then the sequence {𝑥_𝑛} converges weakly to a point 𝑥^∗∈Γ.

Proof. The proof is divided into four steps.
(I) We first prove that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for any 𝑝∈Γ.
Since 𝑝∈Γ, 𝑝∈𝐶∶=⋂^𝑁_𝑖=1𝐹(𝑆_𝑖) and 𝐴𝑝∈𝑄∶=⋂^𝑁_𝑖=1𝐹(𝑇_𝑖). It follows from (9) that 𝑥_𝑛+1−𝑝²=𝑢_𝑛−𝑝+𝛼_𝑛𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑢_𝑛²=𝑢_𝑛−𝑝²+2𝛼_𝑛𝑢_𝑛−𝑝,𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑢_𝑛+𝛼²_𝑛𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑢_𝑛². (10)
Because 𝑆_𝑖 is 𝜚_𝑖-strictly pseudononspreading, for each 𝑣∈𝐻₁, we have 𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑣²≤𝑢_𝑛−𝑣²+𝜚𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑢_𝑛−(𝑣−𝑆_{𝑛(mod𝑁)}𝑣)²+2𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑢_𝑛,𝑣−𝑆_{𝑛(mod𝑁)}𝑣. (11)
Taking 𝑣=𝑝, we have 𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑝²≤𝑢_𝑛−𝑝²+𝜚𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑢_𝑛². (12)
This implies that 𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑝²=𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑢_𝑛+𝑢_𝑛−𝑝²=𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑢_𝑛²+2𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑢_𝑛,𝑢_𝑛+𝑢−𝑝_𝑛−𝑝²≤𝑢_𝑛−𝑝²+𝜚𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑢_𝑛². (13)
Thus it yields that 2𝛼_𝑛𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑢_𝑛,𝑢_𝑛−𝑝≤𝛼_𝑛𝜚−1𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑢_𝑛². (14)
Substituting (14) into (10) and simplifying, we have 𝑥_𝑛+1−𝑝²≤𝑢_𝑛−𝑝²+𝛼_𝑛𝜚−1𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑢_𝑛²+𝛼²_𝑛𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑢_𝑛²=𝑢_𝑛−𝑝²−𝛼_𝑛1−𝜚−𝛼_𝑛𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑢_𝑛². (15)
On the other hand, 𝑢_𝑛−𝑝²=𝑥_𝑛−𝑝+𝛾𝐴^∗(𝑇_{𝑛(mod𝑁)}−𝐼)𝐴𝑥_𝑛²=𝑥_𝑛−𝑝²+2𝛾𝑥_𝑛−𝑝,𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛+𝛾²𝐴^∗(𝑇_{𝑛(mod𝑁)}−𝐼)𝐴𝑥_𝑛²=𝑥_𝑛−𝑝²+2𝛾𝑥_𝑛−𝑝,𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛+𝛾²𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛,𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛=𝑥_𝑛−𝑝²+2𝛾𝑥_𝑛−𝑝,𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛+𝛾²𝐴𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛,𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛≤𝑥_𝑛−𝑝²+2𝛾𝑥_𝑛−𝑝,𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛+𝛾²‖𝐴‖²𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛². (16)
Since 𝑇_𝑖 is 𝜅_𝑖-strictly pseudononspreading and noting that 𝐴𝑝∈⋂^𝑁_𝑖=1𝐹(𝑇_𝑖), we have 𝑇_{𝑛(mod𝑁)}𝐴𝑥_𝑛−𝐴𝑝²=𝑇_{𝑛(mod𝑁)}𝐴𝑥_𝑛−𝐴𝑥_𝑛+𝐴𝑥_𝑛−𝐴𝑝²=𝑇_{𝑛(mod𝑁)}𝐴𝑥_𝑛−𝐴𝑥_𝑛²+𝐴𝑥_𝑛−𝐴𝑝²+2𝑇_{𝑛(mod𝑁)}𝐴𝑥_𝑛−𝐴𝑥_𝑛,𝐴𝑥_𝑛≤−𝐴𝑝𝐴𝑥_𝑛−𝐴𝑝²+𝜅𝑇_{𝑛(mod𝑁)}𝐴𝑥_𝑛−𝐴𝑥_𝑛². (17)
This leads to 𝑇_{𝑛(mod𝑁)}𝐴𝑥_𝑛−𝐴𝑥_𝑛,𝐴𝑥_𝑛≤−𝐴𝑝𝜅−12𝑇_{𝑛(mod𝑁)}𝐴𝑥_𝑛−𝐴𝑥_𝑛². (18)
By (18), we have 𝑇_{𝑛(mod𝑁)}𝐴𝑥_𝑛−𝐴𝑥_𝑛,𝑇_{𝑛(mod𝑁)}𝐴𝑥_𝑛=𝑇−𝐴𝑝_{𝑛(mod𝑁)}𝐴𝑥_𝑛−𝐴𝑥_𝑛,𝑇_{𝑛(mod𝑁)}𝐴𝑥_𝑛+𝐴𝑥_𝑛−𝐴𝑥_𝑛=𝑇−𝐴𝑝_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛²+𝑇_{𝑛(mod𝑁)}𝐴𝑥_𝑛−𝐴𝑥_𝑛,𝐴𝑥_𝑛≤−𝐴𝑝(𝑇_{𝑛(mod𝑁)}−𝐼)𝐴𝑥_𝑛²+𝜅−12(𝑇_{𝑛(mod𝑁)}−𝐼)𝐴𝑥_𝑛²=𝜅+12(𝑇_{𝑛(mod𝑁)}−𝐼)𝐴𝑥_𝑛². (19)
It follows from (19) that 2𝛾𝑥_𝑛−𝑝,𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛=2𝛾𝐴𝑥_𝑛−𝑝,𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛=2𝛾𝐴𝑥_𝑛−𝑝+𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛−𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛,𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛=2𝛾𝑇_{𝑛(mod𝑁)}𝐴𝑥_𝑛−𝐴𝑝,𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛−2𝛾𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛²≤𝛾(1+𝜅)−2𝛾𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛²=𝛾(𝜅−1)(𝑇_{𝑛(mod𝑁)}−𝐼)𝐴𝑥_𝑛². (20)
By using (15), (16), (19), and (20), we have 𝑥_𝑛+1−𝑝²≤𝑥_𝑛−𝑝²+𝛾²‖𝐴‖²𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛²+𝛾(𝜅−1)𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛²−𝛼_𝑛1−𝜎−𝛼_𝑛𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑢_𝑛²≤𝑥_𝑛−𝑝²−𝛾1−𝜅−𝛾‖𝐴‖²𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛²−𝛼_𝑛1−𝜚−𝛼_𝑛𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑢_𝑛²≤𝑥_𝑛−𝑝². (21)
This shows that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists.
(II) We now prove that lim_𝑛→∞‖𝑢_𝑛−𝑝‖ exists.
In fact, by (21), we have 𝛾1−𝜅−𝛾‖𝐴‖²𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛²+𝛼_𝑛1−𝜚−𝛼_𝑛𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑢_𝑛²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝². (22)
This implies that lim_𝑛→∞𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛=0,lim_𝑛→∞𝑢_𝑛−𝑆_{𝑛(mod𝑁)}𝑢_𝑛=0. (23)(24)
By virtue of (16), (23), and (24), it follows that lim_𝑛→∞‖𝑢_𝑛−𝑝‖ exists and lim_𝑛→∞‖𝑥_𝑛−𝑝‖=lim_𝑛→∞‖𝑢_𝑛−𝑝‖.
(III) Now, we prove that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑢_𝑛+1−𝑢_𝑛‖=0.
In fact, it follows from (9) that 𝑥_𝑛+1−𝑥_𝑛=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛾𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝛾𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝛾𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑢_𝑛+𝛼_𝑛𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝛾𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑢_𝑛+𝛼_𝑛𝛾𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛=𝛾𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆_{𝑛(mod𝑁)}𝑢_𝑛−𝑢_𝑛. (25)
This together with (23) and (24) leads to lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
Similarly, it follows from (9), (23), and (25) that 𝑢_𝑛+1−𝑢_𝑛=𝑥_𝑛+1+𝛾𝐴^∗𝑇_{𝑛+1(mod𝑁)}−𝐼𝐴𝑥_𝑛+1−𝑥_𝑛+𝛾𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛾𝐴^∗𝑇_{𝑛+1(mod𝑁)}−𝐼𝐴𝑥_𝑛+1+𝛾𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛⟶0(as𝑛→∞). (26)
(IV) Finally, we prove that 𝑥_𝑛⇀𝑥^∗ and 𝑢_𝑛⇀𝑥^∗, which is a solution of the MSSFP.
In fact, since {𝑢_𝑛} is bounded, there exists a subsequence {𝑢_𝑛𝑖}⊂{𝑢_𝑛} such that {𝑢_𝑛𝑖}⇀𝑥^∗∈𝐻₁. Hence, for any positive integer 𝑗=1,2,…,𝑁, there exists a subsequence {𝑛_𝑖(𝑗)}⊂{𝑛_𝑖} with 𝑛_𝑖(𝑗)(mod𝑁)=𝑗 such that {𝑢_{𝑛𝑖(𝑗)}}⇀𝑥^∗. Again, by (24) we know that ‖𝑢_{𝑖𝑁+𝑗}−𝑆_𝑗𝑢_{𝑖𝑁+𝑗}‖→0, as 𝑛→∞; therefore, we have that 𝑢_{𝑛𝑖(𝑗)}−𝑆_𝑗𝑢_{𝑛𝑖(𝑗)}⟶0,as𝑛_𝑖(𝑗)⟶∞. (27)
Since 𝑆_𝑗 is demiclosed at zero, it follows that 𝑥∈𝐹(𝑆_𝑗). By the arbitrariness of 𝑗=1,2,…,𝑁, we have 𝑥^∗∈𝐶∶=^𝑁_𝑖=1𝐹𝑆_𝑖. (28)
Moreover, from (9) and (24), we have 𝑥_𝑛𝑖=𝑢_𝑛𝑖−𝛾𝐴^∗(𝑇_{𝑛𝑖(mod𝑁)}−𝐼)𝐴𝑥_𝑛𝑖⇀𝑥^∗. Since 𝐴 is a bounded linear operator, it follows that 𝐴𝑥_𝑛𝑖⇀𝐴𝑥^∗. For any positive integer 𝑘=1,2,…,𝑁, there exists a subsequence {𝑥_{𝑛𝑖(𝑘)}}⊂{𝑥_𝑛𝑖} with 𝑛_𝑖(𝑘)(mod𝑁)=𝑘 such that 𝐴𝑥_{𝑛𝑖(𝑘)}⇀𝐴𝑥^∗ and ‖𝐴𝑥_{𝑛𝑖(𝑘)}−𝑇_𝑘𝐴𝑥_{𝑛𝑖(𝑘)}‖→0. Since 𝑇_𝑘 is demiclosed at zero, we have 𝐴𝑥^∗∈𝐹(𝑇_𝑘). By the arbitrariness of 𝑘∈{1,2,…,𝑁}, it follows that 𝐴𝑥^∗∈𝑄∶=⋂^𝑁_𝑘=1𝐹(𝑇_𝑘). This together with 𝑥^∗∈𝐶 shows that 𝑥^∗∈Γ; that is, 𝑥^∗ is a solution to the MSSFP.
Now, we prove that 𝑥_𝑛⇀𝑥^∗ and 𝑢_𝑛⇀𝑥^∗.
Suppose on the contrary that there exists another subsequence {𝑢_𝑛𝑙}⊂{𝑢_𝑛} such that {𝑢_𝑛𝑙}⇀𝑦^∗∈Γ with 𝑦^∗≠𝑥^∗. Consequently, by virtue of the existence of lim_𝑛→∞‖𝑥_𝑛−𝑝‖ and the Opial property of Hilbert space, we have liminf_{𝑛𝑖→∞}𝑢_𝑛𝑖−𝑥^∗<liminf_{𝑛𝑖→∞}𝑢_𝑛𝑖−𝑦^∗=liminf_𝑛→∞𝑢_𝑛−𝑦^∗=liminf_{𝑛𝑗→∞}𝑢_𝑛𝑗−𝑦^∗<liminf_{𝑛𝑗→∞}𝑢_𝑛𝑗−𝑥^∗=liminf_𝑛→∞𝑢_𝑛−𝑥^∗=liminf_{𝑛𝑖→∞}𝑢_𝑛𝑖−𝑥^∗. (29)
This is a contradiction. Therefore 𝑢_𝑛⇀𝑥^∗. By (9) and (24), we have 𝑥_𝑛=𝑢_𝑛−𝛾𝐴^∗𝑇^𝑛_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛⇀𝑥^∗. (30)
Therefore, the conclusion follows.
This completes the proof of Theorem 12.

Theorem 13. Let 𝐻₁, 𝐻₂, 𝐴, {𝑆_𝑖}, {𝑇_𝑖}, 𝐶, 𝑄 be the same as in Theorem 12. For each 𝑖=1,2,…,𝑁, let 𝑇_𝑖 be a 𝜅_𝑖-strictly pseudononspreading mapping, and 𝑆_𝑖 be a 𝜚_𝑖-strictly pseudononspreading mapping. Let {𝑥_𝑛} be the sequence generated by 𝑥₁∈𝐻₁𝑢𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝑥_𝑛+𝛾𝐴^∗𝑇_{𝑛(mod𝑁)}−𝐼𝐴𝑥_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑆_{𝑛(mod𝑁)}𝑢_𝑛, (31) where 𝛾∈(0,(1−𝜅)/𝜆) with 𝜆 being the spectral of the operator 𝐴^∗𝐴 and 𝜅=max{𝜅₁,𝜅₂,…,𝜅_𝑁}∈(0,1), and {𝛼_𝑛} is a sequence in (0,1−𝜚] with 𝜚=max{𝜚₁,𝜚₂,…,𝜚_𝑁}∈(0,1). If Γ≠∅ and if there exists a positive integer 𝑗 such that 𝑆_𝑗 is semicompact, then the sequence {𝑥_𝑛} converges strongly to a point 𝑥^∗∈Γ.

Proof. Without loss of generality, we can assume that 𝑆₁ is semicompact. It follows from (27) that 𝑢_𝑛𝑖(1)−𝑆₁𝑢_𝑛𝑖(1)⟶0,𝑛_𝑖(1)⟶∞. (32)
Therefore, there exists a subsequence of {𝑢_𝑛𝑖(1)} (for the sake of convenience, we still denote it by {𝑢_𝑛𝑖(1)}), such that 𝑢_𝑛𝑖(1)→𝑢^∗∈𝐻₁. Since 𝑢_𝑛𝑖(1)⇀𝑥^∗,  𝑥^∗=𝑢^∗, and so 𝑢_𝑛𝑖(1)→𝑥^∗∈Γ. By virtue of the existence of lim_𝑛→∞‖𝑥_𝑛−𝑝‖, we know that lim_𝑛→∞𝑢_𝑛−𝑥^∗=0,lim_𝑛→∞𝑥_𝑛−𝑥^∗=0. (33)
That is, {𝑢_𝑛} and {𝑥_𝑛} both converge strongly to the point 𝑥^∗∈Γ. This completes the proof of Theorem 13.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final paper.

Acknowledgments

This work was supported by the Natural Science Foundation of Yibin University (no. 2011B07) and the Scientific Research Fund Project of Sichuan Provincial Education Department (no. 12ZB345) and the Fund of Science and Technology Department of Guizhou Province ([2013] 2223).