Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space

Abstract

Let {𝑇_𝑖}^𝑁𝑖=1 be 𝑁 strictly pseudononspreading mappings defined on closed convex subset 𝐶 of a real Hilbert space 𝐻. Consider the problem of finding a common fixed point of these mappings and introduce cyclic algorithms based on general viscosity iteration method for solving this problem. We will prove the strong convergence of these cyclic algorithm. Moreover, the common fixed point is the solution of the variational inequality ⟨(𝛾𝑓−𝜇𝐵)𝑥^∗,𝑣−𝑥^∗⟩≤0,∀𝑣∈⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖).

1. Introduction

Throughout this paper, we always assume that 𝐶 is a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝐵∶𝐶→𝐻 be a nonlinear mapping. Recall the following definitions.

Definition 1.1. 𝐵 is said to be(i)monotone if ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥0,∀𝑥,𝑦∈𝐶,(1.1)(ii)strongly monotone if there exists a constant 𝛼>0 such that ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥𝛼‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐶,(1.2) for such a case, 𝐵 is said to be 𝛼-strongly-monotone, (iii)inverse-strongly monotone if there exists a constant 𝛼>0 such that ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥𝛼‖𝐵𝑥−𝐵𝑦‖²,∀𝑥,𝑦∈𝐶,(1.3) for such a case, 𝐵 is said to be 𝛼-inverse-strongly monotone, (iv)𝑘-Lipschitz continuous if there exists a constant 𝑘≥0 such that ‖𝐵𝑥−𝐵𝑦‖≤𝑘‖𝑥−𝑦‖∀𝑥,𝑦∈𝐶.(1.4)

Remark 1.2. Let 𝐹=𝜇𝐵−𝛾𝑓, where 𝐵 is a 𝑘-Lipschitz and 𝜂-strongly monotone operator on 𝐻 with 𝑘>0 and 𝑓 is a Lipschitz mapping on 𝐻 with coefficient 𝐿>0, 0<𝛾≤𝜇𝜂/𝐿. It is a simple matter to see that the operator 𝐹 is (𝜇𝜂−𝛾𝐿)-strongly monotone over  𝐻,  that is: ⟨𝐹𝑥−𝐹𝑦,𝑥−𝑦⟩≥(𝜇𝜂−𝛾𝐿)‖𝑥−𝑦‖²,∀(𝑥,𝑦)∈𝐻×𝐻.(1.5)
Following the terminology of Browder-Petryshyn [1], we say that a mapping 𝑇∶𝐷(𝑇)⊆𝐻→𝐻 is(1)𝑘-strict pseudocontraction if there exists 𝑘∈[0,1) such that ‖𝑇𝑥−𝑇𝑦‖²≤‖𝑥−𝑦‖²+𝑘‖𝑥−𝑇𝑥−(𝑦−𝑇𝑦)‖²,∀𝑥,𝑦∈𝐷(𝑇),(1.6)(2)𝑘-strictly pseudononspreading if there exists 𝑘∈[0,1) such that ‖𝑇𝑥−𝑇𝑦‖²≤‖𝑥−𝑦‖²+𝑘‖𝑥−𝑇𝑥−(𝑦−𝑇𝑦)‖²+2⟨𝑥−𝑇𝑥,𝑦−𝑇𝑦⟩,(1.7) for all 𝑥,𝑦∈𝐷(𝑇),(3)nonspreading in [2] if ‖𝑇𝑥−𝑇𝑦‖²≤‖𝑇𝑥−𝑦‖²+‖𝑇𝑦−𝑥‖²,∀𝑥,𝑦∈𝐶.(1.8) It is shown in [3] that (1.8) is equivalent to ‖𝑇𝑥−𝑇𝑦‖²≤‖𝑥−𝑦‖²+2⟨𝑥−𝑇𝑥,𝑦−𝑇𝑦⟩,∀𝑥,𝑦∈𝐶.(1.9)

Clearly every nonspreading mapping is 0-strictly pseudononspreading. Iterative methods for strictly pseudononspreading mapping have been extensively investigated; see [2, 4–6].

Let 𝐶 be a closed convex subset of 𝐻, and let {𝑇_𝑖}^𝑁𝑖=1 be 𝑛  𝑘_𝑖-strictly pseudocontractive mappings on 𝐶 such that ⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖)≠∅. Let 𝑥₁∈𝐶 and {𝛼_𝑛}^∞𝑛=1 be a sequence in (0,1). In [7], Acedo and Xu introduced an explicit iteration scheme called the followting cyclic algorithm for iterative approximation of common fixed points of {𝑇_𝑖}^𝑁𝑖=1 in Hilbert spaces. They define the sequence {𝑥_𝑛} cyclically by 𝑥𝑥₁₂=𝛼₀𝑥₀+1−𝛼₀𝑇₀𝑥₀;𝑥_𝑁=𝛼₁𝑥₁+⋮1−𝛼₁𝑇₁𝑥₁;=𝑥𝛼_{𝑁−1𝑁+1}𝑥_𝑁−1+1−𝛼_𝑁−1𝑇_𝑁−1𝑥_𝑁−1;=𝛼_𝑁𝑥_𝑁+⋮.1−𝛼_𝑁𝑇₀𝑥_𝑁;((11..1101)) In a more compact form, they rewrite 𝑥_𝑛+1 as 𝑥_𝑁+1=𝛼_𝑁𝑥_𝑁+1−𝛼_𝑁𝑇_𝑁𝑥_𝑛,(1.12) where 𝑇_𝑁=𝑇_𝑖, with 𝑖=𝑛 (mod 𝑁), 0≤𝑖≤𝑁−1. Using the cyclic algorithm (1.12), Acedo and Xu [7] show that this cyclic algorithm (1.12) is weakly convergent if the sequence {𝛼_𝑛} of parameters is appropriately chosen.

Motivated and inspired by Acedo and Xu [7], we consider the following cyclic algorithm for finding a common element of the set of solutions of 𝑘_𝑖-strictly pseudononspreading mappings {𝑇_𝑖}^𝑁𝑖=1. The sequence {𝑥_𝑛}^∞𝑖=1 generated from an arbitrary 𝑥₁∈𝐻 as follows: 𝑥𝑥₁₂=𝛼₀𝛾𝑓𝑥₀+𝐼−𝜇𝛼₀𝐵𝑇_𝜔0𝑥₀;𝑥_𝑁=𝛼₁𝛾𝑓𝑥₁+⋮𝐼−𝜇𝛼₁𝐵𝑇_𝜔1𝑥₁;=𝑥𝛼_{𝑁−1𝑁+1}𝛾𝑓𝑥_𝑁−1+𝐼−𝜇𝛼_𝑁−1𝐵𝑇_𝜔𝑁−1𝑥_𝑁−1;=𝛼_𝑁𝛾𝑓𝑥_𝑁⋮+.𝐼−𝜇𝛼_𝑁𝐵𝑇_𝜔0𝑥_𝑁;((11..1134)) Indeed, the algorithm above can be rewritten as 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇_𝜔[𝑛]𝑥_𝑛,(1.15) where 𝑇_𝜔[𝑛]=(𝐼−𝜔_[𝑛])𝐼+𝜔_[𝑛]𝑇_[𝑛], 𝑇_[𝑛]=𝑇_𝑛mod𝑁; namely, 𝑇_[𝑁] is one of 𝑇₁,𝑇₂,…,𝑇_𝑁 circularly.

2. Preliminaries

Throughout this paper, we write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥. 𝑥_𝑛→𝑥 implies that {𝑥_𝑛} converges strongly to 𝑥. The following definitions and lemmas are useful for main results.

Definition 2.1. A mapping 𝑇 is said to be demiclosed, if for any sequence {𝑥_𝑛}which weakly converges to 𝑦, and if the sequence {𝑇𝑥_𝑛} strongly converges to 𝑧, then 𝑇(𝑦)=𝑧.

Definition 2.2. 𝑇∶𝐻→𝐻 is called demicontractive on 𝐻, if there exists a constant 𝛼<1 such that ‖𝑇𝑥−𝑞‖²≤‖𝑥−𝑞‖²+𝛼‖𝑥−𝑇𝑥‖²,∀(𝑥,𝑞)∈𝐻×𝐹_𝑖𝑥(𝑇).(2.1)

Remark 2.3. Every 𝑘-strictly pseudononspreading mapping with a nonempty fixed point set 𝐹_𝑖𝑥(𝑇) is demicontractive (see [8, 9]).

Remark 2.4 (see [10]). Let 𝑇 be a 𝛼-demicontractive mapping on 𝐻 with 𝐹_𝑖𝑥(𝑇)≠∅ and 𝑇_𝜔=(1−𝜔)𝐼+𝜔𝑇 for 𝜔∈(0,∞): (A1)𝑇𝛼-demicontractive is equivalent to ⟨𝑥−𝑇𝑥,𝑥−𝑞⟩≥12(1−𝛼)‖𝑥−𝑇𝑥‖²,∀(𝑥,𝑞)∈𝐻×𝐹_𝑖𝑥(𝑇),(2.2)(A2)𝐹_𝑖𝑥(𝑇)=𝐹_𝑖𝑥(𝑇_𝜔) if 𝜔≠0.

Remark 2.5. According to 𝐼−𝑇_𝜔=𝜔(𝐼−𝑇) with 𝑇 being a 𝑘-strictly pseudononspreading mapping, we obtain ⟨𝑥−𝑇_𝜔𝑥,𝑥−𝑞⟩≥𝜔(1−𝑘)2‖𝑥−𝑇𝑥‖²,∀(𝑥,𝑞)∈𝐻×𝐹_𝑖𝑥(𝑇).(2.3)

Proposition 2.6 (see [2]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a 𝑘-strictly pseudononspreading mapping. If 𝐹_𝑖𝑥(𝑇)≠∅, then it is closed and convex.

Proposition 2.7 (see [2]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a 𝑘-strictly pseudononspreading mapping. Then (𝐼−𝑇) is demiclosed at 0.

Lemma 2.8 (see [11]). Let {𝒯_𝑛} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {𝒯_𝑛𝑗}_𝑗≥0 of {𝒯_𝑛} which satisfies 𝒯_𝑛𝑗<𝒯_𝑛𝑗+1 for all 𝑗≥0. Also consider the sequence of integers {𝛿(𝑛)}_𝑛≥𝑛0 defined by 𝛿(𝑛)=max𝑘≤𝑛∣𝒯_𝑘<𝒯_𝑘+1.(2.4) Then {𝛿(𝑛)}_𝑛≥𝑛0 is a nondecreasing sequence verifying lim_𝑛→∞𝛿(𝑛)=∞,∀𝑛≥𝑛₀; it holds that 𝒯_𝛿(𝑛)<𝒯_𝛿(𝑛)+1 and we have 𝒯_𝑛<𝒯_𝛿(𝑛)+1.(2.5)

Lemma 2.9. Let 𝐻 be a real Hilbert space. The following expressions hold: (i)‖𝑡𝑥+(1−𝑡)𝑦‖²=𝑡‖𝑥‖²+(1−𝑡)‖𝑦‖²−𝑡(1−𝑡)‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐻,∀𝑡∈[0,1], (ii)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩,∀𝑥,𝑦∈𝐻.

Lemma 2.10 (see [6]). Let 𝐶 be a closed convex subset of a Hilbert space 𝐻, and let 𝑇∶𝐻→𝐻 be a 𝑘-strictly pseudononspreading mapping with a nonempty fixed point set. Let 𝑘≤𝜔<1 be fixed and define 𝑇_𝜔𝐶→𝐶 by 𝑇_𝜔(𝑥)=(1−𝜔)(𝑥)+𝜔𝑇(𝑥),∀𝑥∈𝐶.(2.6) Then 𝐹_𝑖𝑥(𝑇_𝜔)=𝐹_𝑖𝑥(𝑇).

Lemma 2.11. Assume 𝐶 is a closed convex subset of a Hilbert space 𝐻. (a)Given an integer 𝑁≥1, assume, 𝑇_𝑖∶𝐻→𝐻 is a 𝑘_𝑖-strictly pseudononspreading mapping for some 𝑘_𝑖∈[0,1), (𝑖=1,2,…,𝑁). Let {𝜆_𝑖}^𝑁𝑖=1 be a positive sequence such that ∑^𝑁𝑖=1𝜆_𝑖=1. Suppose that {𝑇_𝑖}^𝑁𝑖=1 has a common fixed point and ⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖)≠∅. Then, 𝐹_𝑖𝑥^𝑁_𝑖=1𝜆_𝑖𝑇_𝑖=^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝑖.(2.7)(b)Assuming 𝑇_𝑖∶𝐻→𝐻 is a 𝑘_𝑖-strictly pseudononspreading mapping for some 𝑘_𝑖∈[0,1), (𝑖=1,2,…,𝑁), let 𝑇_𝜔𝑖=(1−𝜔_𝑖)𝐼+𝜔_𝑖𝑇_𝑖, 1≤𝑖≤𝑁. If ⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖)≠∅, then 𝐹_𝑖𝑥𝑇_𝜔1𝑇_𝜔2⋯𝑇_𝜔𝑁=^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝜔𝑖.(2.8)

Proof. To prove (a), we can assume 𝑁=2. It suffices to prove that 𝐹_𝑖𝑥(𝐹)⊂𝐹_𝑖𝑥(𝑇₁)∩𝐹_𝑖𝑥(𝑇₂), where 𝐹=(1−𝜆)𝑇₁+𝜆𝑇₂ with 0<𝜆<1. Let 𝑥∈𝐹_𝑖𝑥(𝐹) and write 𝑉₁=𝐼−𝑇₁ and 𝑉₂=𝐼−𝑇₂.
From Lemma 2.9 and taking 𝑧∈𝐹_𝑖𝑥(𝑇₁)∩𝐹_𝑖𝑥(𝑇₂) to deduce that ‖𝑧−𝑥‖²=‖‖(1−𝜆)(𝑧−𝑇₁𝑥)+𝜆(𝑧−𝑇₂𝑥)‖‖²=(1−𝜆)‖‖𝑧−𝑇₁𝑥‖‖²+𝜆‖‖𝑧−𝑇₂𝑥‖‖²−𝜆(1−𝜆)‖‖𝑇₁𝑥−𝑇₂𝑥‖‖²≤−(1𝜆−(1𝜆−)𝜆‖)𝑧‖‖𝑇−₁𝑥‖²+𝑘‖‖𝑥−𝑇₁𝑥‖‖²+𝜆‖𝑧−𝑥‖²+𝑘‖‖𝑥−𝑇₂𝑥‖‖²𝑥−𝑇₂𝑥‖‖²=‖𝑧−𝑥‖²+𝑘(1−𝜆)‖‖𝑉₁𝑥‖‖²+𝜆‖‖𝑉₂𝑥‖‖²−𝜆(1−𝜆)‖‖𝑉¹𝑥−𝑉₂𝑥‖‖²,(2.9) it follows that 𝜆(1−𝜆)‖‖𝑉₁𝑥−𝑉₂𝑥‖‖²≤𝑘(1−𝜆)‖‖𝑉₁𝑥‖‖²+𝜆‖‖𝑉₂𝑥‖‖².(2.10) Since (1−𝜆)𝑉₁𝑥+𝜆𝑉₂𝑥=0, we obtain (1−𝜆)‖‖𝑉₁𝑥‖‖²+𝜆‖‖𝑉₂𝑥‖‖²=𝜆(1−𝜆)‖‖𝑉₁𝑥−𝑉₂𝑥‖‖².(2.11)
This together with (2.10) implies that (1−𝑘)𝜆(1−𝜆)‖‖𝑉₁𝑥−𝑉₂𝑥‖‖²≤0.(2.12) Since 0<𝜆<1 and 𝑘<1, we get ‖𝑉₁𝑥−𝑉₂𝑥‖=0 which implies 𝑇₁𝑥=𝑇₂𝑥 which in turn implies that 𝑇₁𝑥=𝑇₂𝑥=𝑥 since (1−𝜆)𝑇₁𝑥+𝜆𝑇₂𝑥=𝑥. Thus, 𝑥∈𝐹_𝑖𝑥(𝑇₁)∩𝐹_𝑖𝑥(𝑇₂).
By induction, we also claim that 𝐹_𝑖𝑥(∑^𝑁𝑖=1𝜆_𝑖𝑇_𝑖)=⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖) with {𝜆_𝑖}^𝑁𝑖=1 is a positive sequence such that ∑^𝑁𝑖=1𝜆_𝑖=1, (𝑖=1,2,…,𝑁).
To prove (b), we can assume 𝑁=2. Set 𝑇_𝜔1=(1−𝜔₁)𝐼+𝜔₁𝑇₁ and 𝑇_𝜔2=(1−𝜔₂)𝐼+𝜔₁𝑇₂, 0<𝑘_𝑖<𝜔_𝑖<1/2, 𝑖=1,2. Obviously 𝐹_𝑖𝑥𝑇_𝜔1∩𝐹_𝑖𝑥𝑇_𝜔2⊂𝐹_𝑖𝑥𝑇_𝜔1𝑇_𝜔2.(2.13) Now we prove 𝐹_𝑖𝑥𝑇_𝜔1∩𝐹_𝑖𝑥𝑇_𝜔2⊃𝐹_𝑖𝑥𝑇_𝜔1𝑇_𝜔2,(2.14) for all 𝑞∈𝐹_𝑖𝑥(𝑇_𝜔1𝑇_𝜔2) and 𝑇_𝜔1𝑇_𝜔2𝑞=𝑞. If 𝑇_𝜔2𝑞=𝑞, then 𝑇_𝜔1𝑞=𝑞; the conclusion holds. From Lemma 2.10, we can know that 𝐹_𝑖𝑥(𝑇_𝜔1)∩𝐹_𝑖𝑥(𝑇_𝜔2)=𝐹_𝑖𝑥(𝑇₁)∩𝐹_𝑖𝑥(𝑇₂)≠∅. Taking 𝑝∈𝐹_𝑖𝑥(𝑇_𝜔1)∩𝐹_𝑖𝑥(𝑇_𝜔2), then ‖𝑝−𝑞‖²=‖‖𝑝−𝑇_𝜔1𝑇_𝜔2𝑞‖‖²=‖‖𝑝−1−𝜔₁𝑇_𝜔2𝑞+𝜔₁𝑇₁𝑇_𝜔2𝑞‖‖²=‖‖1−𝜔₁𝑝−𝑇_𝜔2𝑞+𝜔₁𝑝−𝑇₁𝑇_𝜔2𝑞‖‖²=1−𝜔₁‖‖𝑝−𝑇_𝜔2𝑞‖‖²+𝜔₁‖‖𝑝−𝑇₁𝑇_𝜔2𝑞‖‖²−𝜔₁1−𝜔₁‖‖𝑇_𝜔2𝑞−𝑇₁𝑇_𝜔2𝑞‖‖²≤1−𝜔₁‖‖𝑝−𝑇_𝜔2𝑞‖‖²−𝜔₁1−𝜔₁‖‖𝑇_𝜔2𝑞−𝑇₁𝑇_𝜔2𝑞‖‖²=+1𝜔−₁𝜔‖‖₁𝑝−𝑇^𝜔₂𝑞‖‖²+𝑘₁‖‖𝑇_𝜔2𝑞−𝑇₁𝑇_𝜔2𝑞‖‖²+2𝑝−𝑇₁𝑇_𝜔2𝑝,𝑇_𝜔2𝑞−𝑇₁𝑇_𝜔2𝑞‖‖𝑝−𝑇_𝜔2𝑞‖‖²−𝜔₁1−𝜔₁‖‖𝑇_𝜔2𝑞−𝑇₁𝑇_𝜔2𝑞‖‖²≤+‖‖𝑝𝜔−¹𝑇‖‖𝑝_𝜔2−𝑇^𝜔₂𝑞‖‖²+𝑘₁‖‖𝑇_𝜔2𝑞−𝑇₁𝑇_𝜔2𝑞‖‖²𝑞‖‖²−𝜔₁1−𝜔₁−𝑘₁‖‖𝑇_𝜔2𝑞−𝑇₁𝑇_𝜔2𝑞‖‖²≤‖𝑝−𝑞‖²−𝜔₁1−𝜔₁−𝑘₁‖‖𝑇_𝜔2𝑞−𝑇₁𝑇_𝜔2𝑞‖‖².(2.15) Since 0<𝑘₁<𝜔₁<1/2, we obtain ‖‖𝑇_𝜔2𝑞−𝑇₁𝑇_𝜔2𝑞‖‖²≤0.(2.16) Namely, 𝑇_𝜔2𝑞=𝑇₁𝑇_𝜔2𝑞, that is: 𝑇_𝜔2𝑞∈𝐹_𝑖𝑥𝑇₁=𝐹_𝑖𝑥𝑇_𝜔1,𝑇_𝜔2𝑞=𝑇_𝜔1𝑇_𝜔2𝑞.(2.17)
By induction, we also claim that the Lemma 2.11(b) holds.

Lemma 2.12. Let 𝐾 be a closed convex subset of a real Hilbert space 𝐻, given 𝑥∈𝐻 and 𝑦∈𝐾. Then 𝑦=𝑃_𝐾𝑥 if and only if there holds the inequality ⟨𝑥−𝑦,𝑦−𝑧⟩≥0,∀𝑧∈𝐾.(2.18)

3. Cyclic Algorithm

In this section, we are concerned with the problem of finding a point 𝑝 such that 𝑝∈^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝜔𝑖=^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝑖,𝑁≥1,(3.1) where 𝑇_𝜔𝑖=(1−𝜔_𝑖)𝐼+𝜔_𝑖𝑇_𝑖, {𝜔_𝑖}^𝑁𝑖=1∈(0,1/2] and {𝑇_𝑖}^𝑁𝑖=1 are 𝑘_𝑖-strictly pseudononspreading mappings with 𝑘_𝑖∈[0,𝜔_𝑖), (𝑖=1,2,…,𝑁), defined on a closed convex subset 𝐶 in Hilbert space 𝐻. Here 𝐹_𝑖𝑥(𝑇_𝜔𝑖)={𝑞∈𝐶∶𝑇_𝜔𝑖𝑞=𝑞} is the set of fixed points of 𝑇_𝑖, 1≤𝑖≤𝑁.

Let 𝐻 be a real Hilbert space, and let 𝐵∶𝐻→𝐻 be 𝜂-strongly monotone and 𝜌-Lipschitzian on 𝐻 with 𝜌>0, 𝜂>0. Let 0<𝜇<2𝜂/𝜌², 0<𝛾<𝜇(𝜂−(𝜇𝜌²/2))/𝐿=𝜏/𝐿. Let 𝑁 be a positive integer, and let 𝑇_𝑖∶𝐻→𝐻 be a 𝑘_𝑖-strictly pseudononspreading mapping for some 𝑘_𝑖∈[0,1), (𝑖=1,2,…,𝑁), such that ⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖)≠∅. We consider the problem of finding 𝑝∈⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖) such that ⟨(𝛾𝑓−𝜇𝐵)𝑝,𝑣−𝑝⟩≤0,∀𝑣∈^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝑖.(3.2)

Since ⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖) is a nonempty closed convex subset of 𝐻, VI (3.2) has a unique solution. The variational inequality has been extensively studied in literature; see, for example, [12–16].

Remark 3.1. Let 𝐻 be a real Hilbert space. Let 𝐵 be a 𝜌-Lipschitzian and 𝜂-strongly monotone operator on 𝐻 with 𝜌>0,𝜂>0. Leting 0<𝜇<2𝜂/𝜌² and leting 𝑆=(𝐼−𝑡𝜇𝐵) and 𝜇(𝜂−(𝜇𝜌²/2))=𝜏, then for ∀𝑥,𝑦∈𝐻 and 𝑡∈(0,min{1,1/𝜏}), 𝑆 is a contraction.

Proof. Consider ‖𝑆𝑥−𝑆𝑦‖²=‖(𝐼−𝑡𝜇𝐵)𝑥−(𝐼−𝑡𝜇𝐵)𝑦‖²==⟨‖𝑥(𝐼−−𝑦𝑡‖𝜇₂𝐵)𝑥−(𝐼−𝑡𝜇𝐵)𝑦,(𝐼−𝑡𝜇𝐵)𝑥−(𝐼−𝑡𝜇𝐵)𝑦⟩≤‖𝑥−𝑦‖₂+𝑡²𝜇²‖𝐵𝑥−𝐵𝑦‖²−2𝑡𝜇⟨𝑥−𝑦,𝐵𝑥−𝐵𝑦⟩+𝑡²𝜇²𝜌²‖𝑥−𝑦‖²−2𝑡𝜇𝜂‖𝑥−𝑦‖²≤1−2𝑡𝜇𝜂−𝜇𝜌²2‖𝑥−𝑦‖²=(1−2𝑡𝜏)‖𝑥−𝑦‖²≤(1−𝑡𝜏)²‖𝑥−𝑦‖².(3.3) It follows that ‖𝑆𝑥−𝑆𝑦‖≤(1−𝑡𝜏)‖𝑥−𝑦‖.(3.4) So 𝑆 is a contraction.

Next, we consider the cyclic algorithm (1.15), respectively, for solving the variational inequality over the set of the common fixed points of finite strictly pseudononspreading mappings.

Lemma 3.2. Assume that {𝑥_𝑛} is defined by (1.15); if 𝑝 is solution of (3.2) with 𝑇∶𝐻→𝐻 being strictly pseudononspreading mapping and demiclosed and {𝑦_𝑛}⊂𝐻 is a bounded sequence such that ‖𝑇𝑦_𝑛−𝑦_𝑛‖→0, then liminf_𝑛→∞⟨(𝛾𝑓−𝜇𝐵)𝑝,𝑦_𝑛−𝑝⟩≤0.(3.5)

Proof. By ‖𝑇𝑦_𝑛−𝑦_𝑛‖→0 and 𝑇∶𝐻→𝐻 demi-closed, we know that any weak cluster point of {𝑦_𝑛} belongs to ⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖). Furthermore, we can also obtain that there exists 𝑦⁻ and a subsequence {𝑦_𝑛𝑗} of {𝑦_𝑛} such that 𝑦_𝑛𝑗⇀𝑦⁻ as 𝑗→∞ (hence 𝑦⁻∈⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖)) and liminf_𝑛→∞⟨(𝛾𝑓−𝜇𝐵)𝑝,𝑦_𝑛−𝑝⟩=lim_𝑗→∞(𝛾𝑓−𝜇𝐵)𝑝,𝑦^𝑛_𝑗−𝑝.(3.6) From (3.2), we can derive that liminf_𝑛→∞⟨(𝛾𝑓−𝜇𝐵)𝑝,𝑦_𝑛−𝑝⟩=(𝛾𝑓−𝜇𝐵)𝑝,𝑦⁻−𝑝≤0.(3.7) It is the desired result. In addition, the variational inequality (3.7) can be written as (𝐼−𝜇𝐵+𝛾𝑓)𝑝−𝑝,⁻𝑦−𝑝≤0,𝑦⁻∈^𝑁_𝑖=1𝐹_𝑖𝑥𝑇_𝑖.(3.8) So, by Lemma 2.12, it is equivalent to the fixed point equation 𝑃_⋂^𝑁𝑖=1_{𝐹𝑖𝑥(𝑇𝑖)}(𝐼−𝜇𝐵+𝛾𝑓)𝑝=𝑝.(3.9)

Theorem 3.3. Let 𝐶 be a nonempty closed convex subset of 𝐻 and for 1≤𝑖≤𝑁. Let 𝑇_𝑖∶𝐻→𝐻 be 𝑘_𝑖-strictly pseudononspreading mappings for some 𝑘_𝑖∈[0,𝜔_𝑖), 𝜔_𝑖∈(0,1/2), (𝑖=1,2,…,𝑁), and 𝑘=max{𝑘_𝑖∶1≤𝑖≤𝑁}. Let 𝑓 be 𝐿-Lipschitz mapping on 𝐻 with coefficient 𝐿>0, and let 𝐵∶𝐻→𝐻 be 𝜂-strongly monotone and 𝜌-Lipschitzian on 𝐻 with 𝜌>0, 𝜂>0. Let {𝛼_𝑛} being a sequence in (0,min{1,1/𝜏}) satisfying the following conditions: (c1)lim_𝑛→∞𝛼_𝑛=0, (c2)∑^∞𝑛=0𝛼_𝑛=∞.
Given 𝑥₀∈𝐶, let {𝑥_𝑛}^∞𝑛=1 be the sequence generated by the cyclic algorithm (1.15). Then {𝑥_𝑛} converges strongly to the unique element 𝑝 in ⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖) verifying 𝑝=𝑃_⋂^𝑁𝑖=1_{𝐹𝑖𝑥(𝑇𝑖)}(𝐼−𝜇𝐵+𝛾𝑓)𝑝,(3.10) which equivalently solves the variational inequality problem (3.2).

Proof. Take a 𝑝∈⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖). Let 𝑇_𝜔𝑥=(1−𝜔)𝑥+𝜔𝑇𝑥 and 0<𝑘<𝜔<1/2. Then ∀𝑥,𝑦∈𝐶, we have ‖𝑇_𝜔𝑥−𝑇_𝜔𝑦‖²=𝜔‖𝑥−𝑦‖²+1−𝜔‖𝑇𝑥−𝑇𝑦‖²−𝜔1−𝜔‖𝑥−𝑇𝑥−(𝑦−𝑇𝑦)‖²≤−𝜔𝜔‖𝑥−1−𝑦‖𝜔²+‖𝑥1−−𝑇𝜔𝑥−(‖𝑦𝑥−−𝑇𝑦𝑦‖)‖²+₂𝑘‖𝑥−𝑇𝑥−(𝑦−𝑇𝑦)‖²+2⟨𝑥−𝑇𝑥,𝑦−𝑇𝑦⟩=‖𝑥−𝑦‖²+21−𝜔⟨𝑥−𝑇𝑥,𝑦−𝑇𝑦⟩−1−𝜔𝜔−𝑘‖𝑥−𝑇𝑥−(𝑦−𝑇𝑦)‖²≤=‖‖𝑥𝑥−−𝑦𝑦‖‖²₂+21−𝜔⟨𝑥−𝑇𝑥,𝑦−𝑇𝑦⟩+21−𝜔𝜔²𝑥−𝑇_𝜔𝑥,𝑦−𝑇_𝜔𝑦.(3.11)
From 𝑝∈𝐹_𝑖𝑥(𝑇) and (3.11), we also have ‖‖𝑇_𝜔𝑥_𝑛−𝑝‖‖≤‖‖𝑥_𝑛−𝑝‖‖.(3.12) Using (1.15) and (3.12), we obtain ‖‖𝑥_𝑛+1−𝑝‖‖=≤𝛼‖‖𝛼_𝑛𝑛𝛾𝑓𝑥_𝑛−𝑓(𝑝)+𝛼_𝑛(𝛾𝑓(𝑝)−𝑝)+𝐼−𝜇𝛼_𝑛𝐵𝑇_𝜔[𝑛]𝑥_𝑛−𝑝‖‖𝛾‖‖𝑓𝑥_𝑛−𝑓(𝑝)‖‖+𝛼_𝑛‖𝛾𝑓(𝑝)−𝑝‖+1−𝛼_𝑛𝜏‖‖𝑥_𝑛−𝑝‖‖,(3.13) which combined with (3.12) and ‖𝑓(𝑥_𝑛)−𝑓(𝑝)‖≤𝐿‖𝑥_𝑛−𝑝‖ amounts to ‖‖𝑥_𝑛+1−𝑝‖‖≤1−𝛼_𝑛(𝜏−𝛾𝐿)‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖𝛾𝑓(𝑝)−𝑝‖.(3.14) Putting 𝑀₁=max{‖𝑥₀−𝑝‖,‖𝛾𝑓(𝑝)−𝑝‖}, we clearly obtain ‖𝑥_𝑛−𝑝‖≤𝑀₁. Hence {𝑥_𝑛} is bounded. We can also prove that the sequences {𝑓(𝑥_𝑛)} and {𝑇_𝜔[𝑛]𝑥_𝑛} are all bounded.
From (1.15) we obtain that 𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝜇𝐵𝑥_𝑛−𝛾𝑓𝑥_𝑛=𝐼−𝜇𝛼_𝑛𝐵𝑇_𝜔[𝑛]𝑥_𝑛−𝑥_𝑛,(3.15) hence =1−𝛼_𝑛𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝜇𝐵𝑥_𝑛−𝛾𝑓𝑥_𝑛,𝑥_𝑛−𝑝=1−𝛼_𝑛𝑇_𝜔[𝑛]𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑝+𝛼_𝑛(𝐼−𝜇𝐵)𝑇^𝜔_[𝑛]𝑥_𝑛−𝑥_𝑛,𝑥^𝑛−𝑝𝑇_𝜔[𝑛]𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑝+𝛼_𝑛‖‖(𝐼−𝜇𝐵)𝑇_𝜔[𝑛]𝑥_𝑛−𝑥_𝑛=1−𝛼_𝑛‖‖‖‖𝑥^𝑛−𝑝‖‖=1−𝛼_𝑛𝑇_𝜔[𝑛]𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑝+𝛼_𝑛(1−𝜏)‖‖𝑇_𝜔[𝑛]𝑥_𝑛−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖𝑇_𝜔[𝑛]𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑝+𝜔_[𝑛]𝛼_𝑛(1−𝜏)‖‖𝑇_[𝑛]𝑥_𝑛−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖.(3.16) Moreover, by 𝑝∈⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖) and using Remark 2.5, we obtain 𝑥^𝑛−𝑇_𝜔[𝑛]𝑥_𝑛,𝑥_𝑛−𝑝≥21𝜔^[𝑛]1−𝑘_[𝑛]‖‖𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛‖‖²,(3.17) which combined with (3.16) entails 𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛(𝜇𝐵−𝛾𝑓)𝑥_𝑛,𝑥_𝑛−𝑝≤−12𝜔^[𝑛]1−𝑘_[𝑛]1−𝛼_𝑛‖‖𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛‖‖²+𝜔_[𝑛]𝛼_𝑛(1−𝜏)‖‖𝑇_[𝑛]𝑥_𝑛−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖,(3.18) or equivalently ≤−𝛼_𝑛−⟨𝑥_𝑛−𝑥_𝑛+1,𝑥_𝑛−𝑝⟩⟨(𝜇𝐵−𝛾𝑓)𝑥_𝑛,𝑥_𝑛−𝑝⟩−12𝜔^[𝑛]1−𝑘_[𝑛]1−𝛼_𝑛‖‖𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛‖‖²+𝜔_[𝑛]𝛼_𝑛(1−𝜏)‖‖𝑇_[𝑛]𝑥_𝑛−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖.(3.19) Furthermore, using the following classical equality ⟨𝑢,𝑣⟩=12‖𝑢‖²−12‖𝑢−𝑣‖²+12‖𝑣‖²,∀𝑢,𝑣∈𝐶,(3.20) and setting 𝒯_𝑛=1/2‖𝑥_𝑛−𝑝‖², we have 𝑥_𝑛−𝑥_𝑛+1,𝑥_𝑛−𝑝=𝒯_𝑛−𝒯_𝑛+1+12‖‖𝑥^𝑛−𝑥_𝑛+1‖‖².(3.21) So (3.19) can be equivalently rewritten as 𝒯_𝑛+1−𝒯_𝑛−12‖‖𝑥^𝑛−𝑥_𝑛+1‖‖²≤−−1𝛼_𝑛⟨(𝜇𝐵−𝛾𝑓)𝑥_𝑛,𝑥_𝑛−𝑝⟩2𝜔^[𝑛]1−𝑘_[𝑛]1−𝛼_𝑛‖‖𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛‖‖²+𝜔_[𝑛]𝛼_𝑛(1−𝜏)‖‖𝑇_[𝑛]𝑥_𝑛−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖.(3.22) Now using (3.15) again, we have ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖²=‖‖𝛼_𝑛(𝛾𝑓(𝑥_𝑛)−𝜇𝐵𝑥_𝑛)+(𝐼−𝜇𝛼_𝑛𝐵)(𝑇_𝜔[𝑛]𝑥_𝑛−𝑥_𝑛)‖‖².(3.23) Since 𝐵∶𝐻→𝐻 is 𝜂-strongly monotone and 𝑘-Lipschitzian on 𝐻, hence it is a classical matter to see that ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖²≤2𝛼^2𝑛‖‖𝛾𝑓(𝑥_𝑛)−𝜇𝐵𝑥_𝑛‖‖²+21−𝛼_𝑛𝜏²‖‖𝑇_𝜔[𝑛]𝑥_𝑛−𝑥_𝑛‖‖²,(3.24) which by ‖𝑇_𝜔[𝑛]𝑥_𝑛−𝑥_𝑛‖=𝜔_[𝑛]‖𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛‖ and (1−𝛼_𝑛𝜏)²≤(1−𝛼_𝑛𝜏) yields 12‖‖𝑥^𝑛+1−𝑥_𝑛‖‖²≤𝛼^2𝑛‖‖𝛾𝑓(𝑥_𝑛)−𝜇𝐵𝑥_𝑛‖‖²+1−𝛼_𝑛𝜏𝜔^2[𝑛]‖‖𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛‖‖².(3.25) Then from (3.22) and (3.25), we have 𝒯_𝑛+1−𝒯_𝑛+𝜔_[𝑛]121−𝑘^[𝑛]1−𝛼_𝑛−𝜔_[𝑛]1−𝛼_𝑛𝜏‖‖𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛‖‖²≤𝛼_𝑛𝛼^𝑛‖‖𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥_𝑛‖‖²−⟨(𝜇𝐵−𝛾𝑓)𝑥_𝑛,𝑥_𝑛−𝑝⟩+𝜔_[𝑛](1−𝜏)‖‖𝑇_[𝑛]𝑥_𝑛−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖.(3.26) The rest of the proof will be divided into two parts.
Case  1. Suppose that there exists 𝑛₀ such that {𝒯_𝑛}_𝑛≥𝑛0 is nonincreasing. In this situation, {𝒯_𝑛} is then convergent because it is also nonnegative (hence it is bounded from below), so that lim_𝑛→∞(𝒯_𝑛+1−𝒯_𝑛)=0; hence, in light of (3.26) together with lim_𝑛→∞𝛼_𝑛=0 and the boundedness of {𝑥_𝑛}, we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑇_[𝑛]𝑥_𝑛‖‖=0.(3.27) It also follows from (3.26) that 𝒯_𝑛−𝒯_𝑛+1≥𝛼_𝑛+−𝜔𝛼^𝑛_[𝑛]‖‖𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥_𝑛‖‖²+⟨(𝜇𝐵−𝛾𝑓)𝑥_𝑛,𝑥_𝑛−𝑝⟩(1−𝜏)‖‖𝑇_[𝑛]𝑥_𝑛−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖.(3.28) Then, by ∑^∞𝑛=0𝛼_𝑛=∞, we obviously deduce that liminf_𝑛→∞𝛼_𝑛−+𝛼𝜔_𝑛‖‖𝛾_[𝑛]𝑓𝑥_𝑛−𝜇𝐵𝑥_𝑛‖‖²+⟨(𝜇𝐵−𝛾𝑓)𝑥_𝑛,𝑥_𝑛−𝑝⟩(1−𝜏)‖‖𝑇_[𝑛]𝑥_𝑛−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖≤0,(3.29) or equivalently (as 𝛼_𝑛‖𝛾𝑓(𝑥_𝑛)−𝜇𝐵𝑥_𝑛‖²→0) liminf_𝑛→∞⟨(𝜇𝐵−𝛾𝑓)𝑥_𝑛,𝑥_𝑛−𝑝⟩≤0.(3.30) Moreover, by Remark 1.2, we have 2(𝜇𝜂−𝛾𝐿)𝒯_𝑛+⟨(𝜇𝐵−𝛾𝑓)𝑝,𝑥_𝑛−𝑝⟩≤⟨(𝜇𝐵−𝛾𝑓)𝑥_𝑛,𝑥_𝑛−𝑝⟩,(3.31) which by (3.30) entails liminf_𝑛→∞⟨2(𝜇𝜂−𝛾𝐿)𝒯_𝑛+(𝜇𝐵−𝛾𝑓)𝑝,𝑥_𝑛−𝑝⟩≤0,(3.32) hence, recalling that lim_𝑛→∞𝒯_𝑛 exists, we equivalently obtain 2(𝜇𝜂−𝛾𝐿)lim_𝑛→∞𝒯_𝑛+liminf_𝑛→∞⟨(𝜇𝐵−𝛾𝑓)𝑝,𝑥_𝑛−𝑝⟩≤0,(3.33) namely 2(𝜇𝜂−𝛾𝐿)lim_𝑛→∞𝒯_𝑛≤−liminf_𝑛→∞⟨(𝜇𝐵−𝛾𝑓)𝑝,𝑥_𝑛−𝑝⟩.(3.34) From (3.27) and Lemma 3.2, we obtain liminf_𝑛→∞⟨(𝜇𝐵−𝛾𝑓)𝑝,𝑥_𝑛−𝑝⟩≥0,(3.35) which yields lim_𝑛→∞𝒯_𝑛=0, so that {𝑥_𝑛} converges strongly to 𝑝.
Case  2. Suppose there exists a subsequence {𝒯_𝑛𝑘}_𝑘≥0 of {𝒯_𝑛}_𝑛≥0 such that 𝒯_𝑛𝑘≤𝒯_𝑛𝑘+1 for all 𝑘≥0. In this situation, we consider the sequence of indices {𝛿(𝑛)} as defined in Lemma 2.8. It follows that 𝒯_𝛿(𝑛+1)−𝒯_𝛿(𝑛)>0, which by (3.26) amounts to 𝜔_[𝑛]121−𝑘^[𝑛]1−𝛼_𝛿(𝑛)−𝜔_[𝑛]1−𝛼_𝛿(𝑛)𝜏‖‖𝑥_𝛿(𝑛)−𝑇_[𝑛]𝑥_𝛿(𝑛)‖‖²<𝛼_𝛿(𝑛)+𝛼𝜔_𝛿(𝑛)‖‖_[𝑛]𝛾𝑓𝑥_𝛿(𝑛)−𝜇𝐵𝑥_𝛿(𝑛)‖‖²−(𝜇𝐵−𝛾𝑓)𝑥_𝛿(𝑛),𝑥_𝛿(𝑛)−𝑝(1−𝜏)‖‖𝑇_[𝑛]𝑥_𝛿(𝑛)−𝑥_𝛿(𝑛)‖‖‖‖𝑥_𝛿(𝑛)−𝑝‖‖.(3.36) By the boundedness of {𝑥_𝑛} and lim_𝑛→∞𝛼_𝑛=0, we immediately obtain lim_𝑛→∞‖‖𝑥_𝛿(𝑛)−𝑇_[𝑛]𝑥_𝛿(𝑛)‖‖=0.(3.37) Using (1.15), we have ‖𝑥_𝛿(𝑛)+1−𝑥_𝛿(𝑛)‖≤≤𝛼𝛼_{𝛿(𝑛)𝛿(𝑛)}‖‖𝛾𝑓𝑥_𝛿(𝑛)−𝜇𝐵𝑥_𝛿(𝑛)‖‖+1−𝛼_𝛿(𝑛)𝜏‖‖𝑇_𝜔[𝑛]𝑥_𝛿(𝑛)−𝑥_𝛿(𝑛)‖‖‖‖𝛾𝑓𝑥_𝛿(𝑛)−𝜇𝐵𝑥_𝛿(𝑛)‖‖+𝜔_[𝑛]1−𝛼_𝛿(𝑛)𝜏‖‖𝑇_[𝑛]𝑥_𝛿(𝑛)−𝑥_𝛿(𝑛)‖‖,(3.38) which together with (3.37) and lim_𝑛→∞𝛼_𝑛=0 yields lim_𝑛→∞‖‖𝑥_𝛿(𝑛)+1−𝑥_𝛿(𝑛)‖‖=0.(3.39)
Now by (3.36) we clearly have 𝛼_𝛿(𝑛)‖‖𝛾𝑓(𝑥_𝛿(𝑛))−𝜇𝐵≥𝑥_𝛿(𝑛)(‖‖𝜇²𝐵+−𝜔𝛾𝑓_[𝑛])(1𝑥−_𝛿(𝑛)𝜏)‖‖𝑇_[𝑛]𝑥_𝛿(𝑛)−𝑥_𝛿(𝑛)‖‖‖‖𝑥_𝛿(𝑛)−𝑝‖‖,𝑥_𝛿(𝑛)−𝑝,(3.40) which in the light of (3.31) yields ≤𝛼_𝛿(𝑛)2(𝜇𝜂−𝛾𝐿)𝒯_𝛿(𝑛)+(𝜇𝐵−𝛾𝑓)𝑝,𝑥_𝛿(𝑛)−𝑝‖‖𝛾𝑓(𝑥_𝛿(𝑛))−𝜇𝐵𝑥_𝛿(𝑛)‖‖²+𝜔_[𝑛](1−𝜏)‖‖𝑇_[𝑛]𝑥_𝛿(𝑛)−𝑥_[𝑛]‖‖‖‖𝑥_𝛿(𝑛)−𝑝‖‖,(3.41) hence (as lim_𝑛→∞𝛼_𝛿(𝑛)‖𝛾𝑓(𝑥_𝛿(𝑛))−𝜇𝐵𝑥_𝛿(𝑛)‖²=0 and (3.37)) it follows that 2(𝜇𝜂−𝛾𝐿)limsup_𝑛→∞𝒯_𝛿(𝑛)≤−liminf_𝑛→∞(𝜇𝐵−𝛾𝑓)𝑝,𝑥_𝛿(𝑛)−𝑝.(3.42) From (3.37) and Lemma 3.2, we obtain lim_𝑛→∞⟨(𝜇𝐵−𝛾𝑓)𝑝,𝑥_𝛿(𝑛)−𝑝⟩≥0,(3.43) which by (3.42) yields limsup_𝑛→∞𝒯_𝛿(𝑛)=0, so that lim_𝑛→∞𝒯_𝛿(𝑛)=0. Combining (3.39), we have lim_𝑛→∞𝒯_𝛿(𝑛)+1=0. Then, recalling that 𝒯_𝑛<𝒯_𝛿(𝑛)+1 (by Lemma 2.8), we get lim_𝑛→∞𝒯_𝑛=0, so that 𝑥_𝑛→𝑝 strongly.

Taking 𝑘_𝑖=0, we know that 𝑘_𝑖-strictly pseudononspreading mapping is nonspreading mapping and 𝑖=𝑛 (mod 𝑁), 0≤𝑖≤𝑁−1. According to the proof Theorem 3.3, we obtain the following corollary.

Corollary 3.4. Let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝑇_𝑖∶𝐶→𝐶 be nonspreading mappings and 𝜔_𝑖∈(0,1/2), (𝑖=1,2,…,𝑁). Let 𝑓 be 𝐿-Lipschitz mapping on 𝐻 with coefficient 𝐿>0 and let 𝐵∶𝐻→𝐻 be 𝜂-strongly monotone and 𝜌-Lipschitzian on 𝐻 with 𝜌>0, 𝜂>0. Let {𝛼_𝑛} be a sequence in (0,min{1,1/𝜏}) satisfying the following conditions: (c1)lim_𝑛→∞𝛼_𝑛=0, (c2)∑^∞𝑛=0𝛼_𝑛=∞.
Given 𝑥₀∈𝐶, let {𝑥_𝑛}^∞𝑛=1 be the sequence generated by the cyclic algorithm (1.15). Then {𝑥_𝑛} converges strongly to the unique element 𝑝 in ⋂^𝑁𝑖=1𝐹_𝑖𝑥(𝑇_𝑖) verifying 𝑝=𝑃_⋂^𝑁𝑖=1_{𝐹𝑖𝑥(𝑇𝑖)}(𝐼−𝜇𝐵+𝛾𝑓)𝑝,(3.44) which equivalently solves the variational inequality problem (3.2).

Acknowledgment

This work is supported in part by China Postdoctoral Science Foundation (Grant no. 20100470783).