Strong Convergence Theorems for a Pair of Strictly Pseudononspreading Mappings

Abstract

Let 𝐻 be a real Hilbert space. Let 𝑇₁,𝑇₂∶𝐻→𝐻 be 𝑘₁-, 𝑘₂-strictly pseudononspreading mappings; let {𝛼_𝑛} and {𝛽_𝑛} be two real sequences in (0,1). For given 𝑥₀∈𝐻, the sequence {𝑥_𝑛} is generated iteratively by 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑇_𝑤1(𝛼_𝑛𝛾𝑓(𝑥_𝑛)+(𝐼−𝜇𝛼_𝑛𝐵)𝑇_𝑤2𝑥_𝑛), ∀𝑛∈𝑁, where 𝑇_𝑤𝑖=(1−𝑤_𝑖)𝐼+𝑤_𝑖𝑇_𝑖 with 𝑖=1,2 and 𝐵∶𝐻→𝐻 is strongly monotone and Lipschitzian. Under some mild conditions on parameters {𝛼_𝑛} and {𝛽_𝑛}, we prove that the sequence {𝑥_𝑛} converges strongly to the set 𝐹_𝑖𝑥(𝑇₁)∩𝐹_𝑖𝑥(𝑇₂) of fixed points of a pair of strictly pseudononspreading mappings 𝑇₁ and 𝑇₂.

1. Introduction

Let 𝐻 be a real Hilbert space whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, and let 𝐶 be a nonempty, closed, and convex subset of 𝐻, respectively. Recall the following definitions.

Definition 1. Let 𝐴∶𝐶→𝐻 be a nonlinear mapping. (1)𝐴is said to be monotone if 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶. (1)(2)𝐴 is said to be strongly monotone if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (2) For such a case, 𝐴 is said to be 𝛼-strongly-monotone.(3)𝐴 is said to be inverse strongly if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶. (3) For such a case, 𝐴 is said to be 𝛼-inverse-strongly monotone.
The classical variational inequality which is denoted by VI(𝐴,𝐶) is to find 𝑥∈𝐶 such that ⟨𝐴𝑥,𝑦−𝑥⟩≥0,∀𝑦∈𝐶. (4) The variational inequality has been extensively studied in the literature; see, for example, [1, 2] and the reference therein. Recall that 𝑇 is a nonexpansive mapping of 𝐶 into itself; that is, 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (5) The set of fixed points of   𝑇 is the set 𝐹_𝑖𝑥(𝑇)={𝑥∈𝐻,𝑇𝑥=𝑥}.
In 2011, Osilike and Isiogugu [3] introduced a new class of mappings, the so-called 𝑘-strictly pseudononspreading; that is, a mapping 𝑇∶𝐷(𝑇)⊆𝐻→𝐻 is 𝑘-strictly pseudononspreading if there exists 𝑘∈[0,1) such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘𝑥−𝑇𝑥−𝑦−𝑇𝑦²+2𝑥−𝑇𝑥,𝑦−𝑇𝑦, (6) for all 𝑥,𝑦∈𝐷(𝑇). They showed that the class of nonspreading mappings is properly contained in the class of strictly pseudononspreading mappings.
The iteration procedure of Mann’s type for approximating fixed points of a nonexpansive mapping 𝑇 is the following: 𝑥₁∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛∈𝑁, (7) where {𝛼_𝑛} is a sequence in [0,1]; see [4]. For two nonexpansive mappings 𝑆 and 𝑇, Takahashi and Tamura [5] considered the following iteration procedure: 𝑥₁∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑥_𝑛,∀𝑛∈𝑁, (8) where {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1].
In 2010, Tian [6] introduced the following general viscosity iterative scheme for finding an element of set of solutions to the fixed point of nonexpansive mapping in Hilbert space. Define sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇𝑥_𝑛,𝑛≥0, (9) where 𝐵 is 𝑘-Lipschitzian and 𝜂-strongly monotone operator and 𝑇 is a nonexpansive mapping on 𝐻; then he proved that if the sequence {𝛼_𝑛} satisfies appropriate conditions, the sequence {𝑥_𝑛} generated by (9) converges strongly to the unique solution 𝑥^∗∈𝐶 of the variational inequality 𝛾𝑓−𝜇𝐵𝑥^∗,𝑥−𝑥^∗≤0,∀𝑥∈𝐶, (10) where 𝐶=Fix(𝑇).
In this paper, motivated by Takahashi and Tamura [5] and Tian [6], we introduce the following iteration scheme for finding a common point of the set 𝐹_𝑖𝑥(𝑇₁)⋂𝐹_𝑖𝑥(𝑇₂) of fixed points of a pair of strictly pseudononspreading mappings 𝑇₁ and 𝑇₂: 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇_𝜔1×𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇_𝜔2𝑥_𝑛,∀𝑛∈𝑁, (11) where 𝑇_𝜔𝑖=(1−𝜔_𝑖)𝐼+𝜔_𝑖𝑇_𝑖 with 𝑖=1,2 and 𝐵∶𝐻→𝐻  is  𝜂-strongly monotone and 𝑘-Lipschitzian on 𝐻 with 𝑘>0, 𝜂>0. Under suitable conditions, we prove a strong convergence theorem, which is different from the results of general viscosity iterative scheme in [6].

2. Preliminaries

We need some facts and tools in Hilbert space 𝐻 which are listed as in the following lemmas.

Definition 2. 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 3. 𝑇∶𝐻→𝐻 is called demicontractive on 𝐻, if there exists a constant 𝛼<1 such that 𝑇𝑥−𝑞²≤𝑥−𝑞²+𝛼‖𝑥−𝑇𝑥‖²,∀𝑥,𝑞∈𝐻×𝐹_𝑖𝑥(𝑇). (12)

Remark 4. Every 𝑘-strictly pseudononspreading mapping with a nonempty fixed point set 𝐹_𝑖𝑥(𝑇) is demicontractive (see [7, 8]).

Remark 5 (see [9]). 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 ≥1𝑥−𝑇𝑥,𝑥−𝑞2(1−𝛼)‖𝑥−𝑇𝑥‖²,∀𝑥,𝑞∈𝐻×𝐹_𝑖𝑥(𝑇). (13)(A2) 𝐹_𝑖𝑥(𝑇)=𝐹_𝑖𝑥(𝑇_𝜔) if 𝜔≠0.

Remark 6. According to 𝐼−𝑇_𝜔=𝜔(𝐼−𝑇) with 𝑇 being a 𝑘-strictly pseudononspreading mapping, we obtain 𝑥−𝑇_𝜔≥𝜔()𝑥,𝑥−𝑞1−𝑘2‖𝑥−𝑇𝑥‖²,∀𝑥,𝑞∈𝐻×𝐹_𝑖𝑥(𝑇). (14)

Proposition 7 (see [3]). 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 8 (see [3]). 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 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 10 (see [10]). 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_𝜔∶𝐶→𝐶 by 𝑇_𝜔𝑥=(1−𝜔)𝑥+𝜔𝑇𝑥,∀𝑥∈𝐶. (15) Then 𝐹_𝑖𝑥(𝑇_𝜔)=𝐹_𝑖𝑥(𝑇).

Lemma 11 (see [11]). Assume {𝛼_𝑛} is a sequence of nonnegative real numbers such that 𝛼_𝑛+1≤1−𝛾_𝑛𝛼_𝑛+𝛿_𝑛,𝑛≥0, (16) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in ℝ such that (i)∑^∞_𝑛=1𝛾_𝑛=∞, (ii)limsup_𝑛→∞(𝛿_𝑛/𝛾_𝑛)=0 or ∑^∞_𝑛=1|𝛿_𝑛|<∞. Then lim_𝑛→∞𝛼_𝑛=0.

Lemma 12 (see [12]). 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 that the sequence of integers {𝛿(𝑛)}_𝑛≥𝑛0 is defined by 𝛿(𝑛)=max𝑘≤𝑛∣𝒯_𝑘<𝒯_𝑘+1. (17) 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 one has 𝒯_𝑛<𝒯_𝛿(𝑛)+1. (18)

Lemma 13. 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. (19)

3. Main Results

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑇₁,𝑇₂∶𝐶→𝐶 be 𝑘₁-, 𝑘₂-strictly pseudononspreading mappings with nonempty fixed point set Ω=𝐹_𝑖𝑥(𝑇₁)⋂𝐹_𝑖𝑥(𝑇₂). Let 𝑓 be a 𝐿-Lipschitz mapping on 𝐻 with coefficient 𝐿>0. Assume that the set Ω is nonempty. Since Ω is closed and convex, the nearest point projection from 𝐶 onto Ω is well defined. Recall that 𝐵∶𝐻→𝐻 is 𝜂-strongly monotone and 𝑘-Lipschitzian on 𝐻 with 𝑘>0, 𝜂>0.

Lemma 14 (see [13]). Let 𝐻 be a real Hilbert space and let 𝐵 be a 𝑘-Lipschitzian and 𝜂-strongly monotone operator with 𝑘>0, 𝜂>0. Let 0<𝜇<(2𝜂/𝑘²) and 𝜏=𝜇(𝜂−(𝜂𝑘²/2)). Then for 𝑡∈min{1,(1/𝜏)}, 𝐼−𝑡𝜇𝐵 is in contradiction with constant 1−𝑡𝜏.

Lemma 15. Let 𝑇 be a 𝑘-strictly pseudononspreading mapping on 𝐻, and 𝑇_𝜔𝑥=(1−𝜔)𝑥+𝜔𝑇𝑥 with 0<𝑘<𝜔<1/2. Then 𝑇_𝜔𝑥−𝑇_𝜔𝑦²≤𝑥−𝑦²+2𝜔𝑥−𝑇_𝜔𝑥,𝑦−𝑇_𝜔𝑦,∀𝑥,𝑦∈𝐶. (20)

Proof. For 𝑥,𝑦∈𝐻, we have 𝑇_𝜔𝑥−𝑇_𝜔𝑦²=(1−𝜔)𝑥−𝑦²+𝜔𝑇𝑥−𝑇𝑦²−𝜔(1−𝜔)𝑥−𝑇𝑥−𝑦−𝑇𝑦²≤(1−𝜔)𝑥−𝑦²+𝜔𝑥−𝑦²+𝑘𝑥−𝑇𝑥−𝑦−𝑇𝑦²(+2𝑥−𝑇𝑥,𝑦−𝑇𝑦−𝜔1−𝜔)𝑥−𝑇𝑥−𝑦−𝑇𝑦²=𝑥−𝑦²(+2𝜔𝑥−𝑇𝑥,𝑦−𝑇𝑦−𝜔1−𝜔)(1−𝑘)𝑥−𝑇𝑥−𝑦−𝑇𝑦²≤𝑥−𝑦²=+2𝜔𝑥−𝑇𝑥,𝑦−𝑇𝑦𝑥−𝑦²+2𝜔𝑥−𝑇_𝜔𝑥,𝑦−𝑇_𝜔𝑦. (21)

Remark 16. If 𝑦=𝑝∈𝐹_𝑖𝑥(𝑇) and from Lemma 15, we can easily claim that ‖𝑇_𝜔𝑥−𝑝‖≤‖𝑥−𝑝‖, ∀𝑥∈𝐶.

Theorem 17. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝑇₁,𝑇₂∶𝐶→𝐶 be 𝑘₁-, 𝑘₂-strictly pseudononspreading mappings and demiclosed. Let 𝑓 be 𝐿-Lipschitz mapping on 𝐻 with coefficient 𝐿>0 and let 𝐵∶𝐻→𝐻 be 𝜂-strongly monotone and 𝑘-Lipschitzian on 𝐻 with 𝑘>0, 𝜂>0, 0<𝑘₁, and 𝑘₂<𝜔<1/2. Let the sequence of {𝛼_𝑛} and {𝛽_𝑛} satisfy the following conditions: (c1)lim_𝑛→∞𝛼_𝑛=0,(c2)∑^∞_𝑛=0𝛼_𝑛=∞ and lim_𝑛→∞𝛼_𝑛+1/𝛼_𝑛=1,(c3) 0<𝑐≤𝛽_𝑛≤𝑑<1. Let 0<𝜇<2𝜂/𝑘², 0<𝛾<𝜇(𝜂−(𝜇𝑘²/2))/𝐿=𝜏/𝐿, and let the following sequence {𝑥_𝑛}^∞_𝑛=0 be a sequence in 𝐶 generated from an arbitrary 𝑥₁∈𝐶 by 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛×𝑇_𝜔1𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇_𝜔2𝑥_𝑛, (22) where 𝑇_𝜔𝑖=(1−𝜔_𝑖)𝐼+𝜔_𝑖𝑇_𝑖 with 0<𝑘_𝑖<𝜔_𝑖<1, 𝑖=1,2. Then {𝑥_𝑛} converges strongly to the unique element 𝑥^∗ in Ω=𝐹_𝑖𝑥(𝑇₁)⋂𝐹_𝑖𝑥(𝑇₂) verifying 𝑥^∗=𝑃_Ω⋅𝑓𝑥^∗, (23) which equivalently solves the following variational inequality problem: 𝑥^∗∈Ω,𝛾𝑓−𝜇𝐵𝑥^∗,𝑣−𝑥^∗≤0,∀𝑣∈Ω. (24)

Proof. Let 𝑥^∗∈Ω. According to Remark 16, we have 𝑇_𝜔𝑥_𝑛−𝑥^∗≤𝑥_𝑛−𝑥^∗. (25) Let 𝑦_𝑛=𝛼_𝑛𝛾𝑓(𝑥_𝑛)+(𝐼−𝜇𝛼_𝑛𝐵)𝑇_𝜔2𝑥_𝑛. From (22), we obtain 𝑥_𝑛+1−𝑥^∗=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇_𝜔1𝑦_𝑛−𝑥^∗≤𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝑇_𝜔1𝑦_𝑛−𝑥^∗≤𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝑦_𝑛−𝑥^∗, (26) And, from Lemma 14, we also obtain that 𝑦_𝑛−𝑥^∗=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑥^∗+𝐼−𝜇𝛼_𝑛𝐵𝑇_𝜔2𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝜇𝐵𝑥^∗≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑥^∗+1−𝛼_𝑛𝜏𝑇_𝜔2𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝜇𝐵𝑥^∗≤𝛼_𝑛𝛾𝐿𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝜏𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝜇𝐵𝑥^∗. (27) Together with (26) and (27), we obtain 𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛1−𝛽_𝑛𝜏−𝛾𝐿𝑥_𝑛−𝑥^∗+𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥^∗−𝜇𝐵𝑥^∗. (28) Putting 𝑀₁=max{‖𝑥₀−𝑥^∗‖,‖𝛾𝑓(𝑥^∗)−𝑥^∗‖}, we clearly obtain ‖𝑥_𝑛−𝑥^∗‖≤𝑀₁. By induction, we can deduce that {𝑥_𝑛} is bounded and the sequences {𝑓(𝑥_𝑛)} and {𝑇_𝜔𝑖𝑥_𝑛},  𝑖=1,2, are also bounded.
Next, we show the following estimation: 𝒯_𝑛+1≤1−𝛼_𝑛𝛿𝒯_𝑛−𝛿‖𝑥_𝑛−𝑇₂𝑥_𝑛‖²+‖𝑇_𝜔1𝑦_𝑛−𝑥_𝑛‖²+2𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥^∗,𝑦_𝑛−𝑥^∗, (29) where 𝒯_𝑛=‖𝑥_𝑛−𝑥^∗‖² and 𝛿>0 is chosen so that 𝛿≤min𝜔₂1−𝛼_𝑛𝜏1−𝛽_𝑛1−𝜔₂1−𝑘₂,𝛽_𝑛1−𝛽_𝑛. (30) From Lemma 15, we obtain 𝑇_𝜔2𝑥_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝜔₂1−𝜔₂1−𝑘₂𝑥_𝑛−𝑇₂𝑥_𝑛². (31) By (25) and (31), we get 𝑦_𝑛−𝑥^∗2=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥^∗+𝐼−𝜇𝛼_𝑛𝐵𝑇_𝜔2𝑥_𝑛−𝑥^∗2≤1−𝛼_𝑛𝜏𝑇_𝜔2𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥^∗,𝑦_𝑛−𝑥^∗≤1−𝛼_𝑛𝜏𝑥_𝑛−𝑥^∗2−𝜔₂1−𝛼_𝑛𝜏1−𝜔₂1−𝑘₂𝑥_𝑛−𝑇₂𝑥_𝑛²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥^∗,𝑦_𝑛−𝑥^∗. (32) Consequently, 𝑥_𝑛+1−𝑥^∗2=𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝑇_𝜔1𝑦_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑇_𝜔1𝑦_𝑛−𝑥_𝑛²≤𝛽_𝑛𝑥_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑇_𝜔1𝑦_𝑛−𝑥_𝑛²+1−𝛽_𝑛𝑦_𝑛−𝑥^∗2≤𝛽_𝑛𝑥_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑇_𝜔1𝑦_𝑛−𝑥_𝑛²+1−𝛽_𝑛1−𝛼_𝑛𝜏𝑥_𝑛−𝑥^∗2−𝜔₂1−𝛼_𝑛𝜏1−𝛽_𝑛1−𝜔₂1−𝑘₂×𝑥_𝑛−𝑇₂𝑥_𝑛²+2𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥^∗,𝑦_𝑛−𝑥^∗≤1−𝛼_𝑛𝜏1−𝛽_𝑛𝑥_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑇_𝜔1𝑦_𝑛−𝑥_𝑛²−𝜔₂1−𝛼_𝑛𝜏1−𝛽_𝑛1−𝜔₂1−𝑘₂×𝑥_𝑛−𝑇₂𝑥_𝑛²+2𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥^∗,𝑦_𝑛−𝑥^∗, (33) and the desired inequality (29) follows.
Finally, we show lim_𝑛→∞𝒯_𝑛=0 by considering two possible cases.

Case 1. {𝒯_𝑛} is nonincreasing sequence with some 𝑛≥𝑛₀. In this case, {𝒯_𝑛} is then convergent because it is also nonnegative (hence it is bounded form blow). From (29), we have 𝛿𝑥_𝑛−𝑇₂𝑥_𝑛²+𝑇_𝜔1𝑦_𝑛−𝑥_𝑛²≤𝛼_𝑛𝑀+𝒯_𝑛−𝒯_𝑛+1, (34) where 0<𝑀=2⟨𝛾𝑓(𝑥_𝑛)−𝜇𝐵𝑥^∗,𝑦_𝑛−𝑥^∗⟩. Consequently, both ‖𝑥_𝑛−𝑇₂𝑥_𝑛‖ and ‖𝑇_𝜔1𝑦_𝑛−𝑥_𝑛‖ converge to zero. From (22) and (25), lim_𝑛→∞𝛼_𝑛=0 and the sequence {𝑥_𝑛} is bounded, we obtain 𝑦_𝑛−𝑥_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇_𝜔2𝑥_𝑛−𝑥_𝑛≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥_𝑛+1−𝛼_𝑛𝜏𝑇_𝜔2𝑥_𝑛−𝑥_𝑛≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥_𝑛+1−𝛼_𝑛𝜏𝜔₂𝑇₂𝑥_𝑛−𝑥_𝑛⟶0, (35) which implies 𝑥_𝑛−𝑇_𝜔1𝑥_𝑛²≤𝑥_𝑛−𝑇_𝜔1𝑦_𝑛²+𝑇_𝜔1𝑦_𝑛−𝑇_𝜔1𝑥_𝑛²≤𝑥_𝑛−𝑇_𝜔1𝑦_𝑛²+𝑥_𝑛−𝑦_𝑛²+2𝜔₁𝑥_𝑛−𝑇_𝜔1𝑥_𝑛,𝑦_𝑛−𝑇_𝜔1𝑦_𝑛≤𝑥_𝑛−𝑇_𝜔1𝑦_𝑛²+𝑥_𝑛−𝑦_𝑛²+2𝜔₁𝑥_𝑛−𝑇_𝜔1𝑥_𝑛,𝑦_𝑛−𝑥_𝑛+2𝜔₁𝑥_𝑛−𝑇_𝜔1𝑥_𝑛,𝑥_𝑛−𝑇_𝜔1𝑦_𝑛. (36) From lim_𝑛→∞‖𝑇_𝜔1𝑦_𝑛−𝑥_𝑛‖=0, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0, and 𝑥_𝑛−𝑇_𝜔1𝑥_𝑛=𝜔₁(𝑥_𝑛−𝑇₁𝑥_𝑛), we obtain lim_𝑛→∞𝑥_𝑛−𝑇₁𝑥_𝑛=lim_𝑛→∞𝑥_𝑛−𝑇_𝜔1𝑥_𝑛=0. (37) Using the demiclosedness principle (Proposition 8) and (35), we know that 𝜔_𝑤(𝑥_𝑛)∈Ω; hence limsup_𝑛→∞𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥^∗,𝑦_𝑛−𝑥^∗=limsup_𝑛→∞𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥^∗,𝑥_𝑛−𝑥^∗=max_{𝑤∈𝜔𝑤(𝑥𝑛)}𝛾𝑓(𝑤)−𝜇𝐵𝑥^∗,𝑤−𝑥^∗≤0. (38) It then follows from (29) that 𝒯_𝑛+1≤1−𝛼_𝑛𝛿𝒯_𝑛+2𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥^∗,𝑦_𝑛−𝑥^∗. (39) According to Lemma 11, we obtain lim_𝑛→∞𝒯_𝑛=0.

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 12. It follows that 𝒯_𝛿(𝑛+1)−𝒯_𝛿(𝑛)>0, which by (29) amounts to 𝛿𝑥_𝛿(𝑛)−𝑇₂𝑥_𝛿(𝑛)²+𝑇_𝜔1𝑦_𝛿(𝑛)−𝑥_𝛿(𝑛)²≤𝛼_𝛿(𝑛)𝑀. (40) In a similar way to Case 1, we obtain limsup_𝑛→∞𝛾𝑓𝑥_𝛿(𝑛)−𝜇𝐵𝑥^∗,𝑦_𝛿(𝑛)−𝑥^∗≤0. (41) From Lemma 12 and (29) 𝒯_𝛿(𝑛)≤21−𝛽_𝛿(𝑛)𝛾𝑓𝑥_𝛿(𝑛)−𝜇𝐵𝑥^∗,𝑦_𝛿(𝑛)−𝑥^∗, (42) for all 𝑛>𝑛₀. Taking limsup in this inequality, we obtain lim_𝑛→∞𝒯_𝛿(𝑛)=0. Moreover, it follows from (22) that 𝒯²_𝛿(𝑛)+1=𝑥_𝛿(𝑛)−𝑥^∗−𝑥_𝛿(𝑛)−𝑥_𝛿(𝑛)+1²≤𝒯²_𝛿(𝑛)+𝑥_𝛿(𝑛)−𝑥_𝛿(𝑛)+1²≤𝒯²_𝛿(𝑛)+𝑥_𝛿(𝑛)−𝑇_𝜔1𝑦_𝛿(𝑛)², (43) which together with (40) implies lim_𝑛→∞𝒯_𝛿(𝑛)+1=0. Consequently, from Lemma 12, we obtain lim_𝑛→∞𝒯_𝑛=0.
In addition, from (38), we obtain limsup_𝑛→∞𝛾𝑓𝑥_𝑛−𝜇𝐵𝑥^∗,𝑥_𝑛−𝑥^∗=limsup_𝑛→∞𝛾𝑓𝑥_𝑛−𝛾𝑓𝑥^∗,𝑥_𝑛−𝑥^∗+limsup_𝑛→∞𝛾𝑓𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛−𝑥^∗=𝛾𝐿lim_𝑛→∞𝒯_𝑛+limsup_𝑛→∞𝛾𝑓𝑥^∗−𝜇𝐵𝑥^∗,𝑥_𝑛−𝑥^∗=𝛾𝑓𝑥^∗−𝜇𝐵𝑥^∗,𝑦−𝑥^∗,𝑦∈Ω; (44) then variational inequality (44) can be written as 𝐼−𝜇𝐵+𝛾𝑓𝑥^∗−𝑥^∗,𝑦−𝑥^∗≥0,𝑦∈Ω. (45) So, by Lemma 13, it is equivalent to the fixed point equation 𝑃_{𝐹𝑖𝑥(𝑇)}𝐼−𝜇𝐵+𝛾𝑓𝑥^∗=𝑥^∗. (46)

Remark 18. For a nonspreading mapping 𝑇, we have 𝜚=0 in Theorem 17 to obtain the following corollary.

Corollary 19. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝑇₁,𝑇₂∶𝐻→𝐻 be nonspreading mappings and demiclosed. Let 𝑓 be 𝐿-Lipschitz mapping on 𝐻 with coefficient 𝐿>0 and let 𝐵∶𝐻→𝐻 be 𝜂-strongly monotone and 𝑘-Lipschitzian on 𝐻 with 𝑘>0, 𝜂>0. Let the sequence of {𝛼_𝑛} and {𝛽_𝑛} satisfy the following conditions: (c1)lim_𝑛→∞𝛼_𝑛=0,(c2)∑^∞_𝑛=0𝛼_𝑛=∞ and lim_𝑛→∞𝛼_𝑛+1/𝛼_𝑛=1,(c3)0<𝑐≤𝛽_𝑛≤𝑑<1. Let the following sequence {𝑥_𝑛}^∞_𝑛=0 be a sequence in 𝐶 generated from an arbitrary 𝑥₁∈𝐶 by 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇_𝜔1𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇_𝜔2𝑥_𝑛, (47) where 𝑇_𝜔𝑖=(1−𝜔_𝑖)𝐼+𝜔_𝑖𝑇_𝑖, 𝜔_𝑖∈(0,1), 𝑖=1,2. Then {𝑥_𝑛} converges strongly to the unique element 𝑥^∗ in Ω=𝐹_𝑖𝑥(𝑇₁)⋂𝐹_𝑖𝑥(𝑇₂) verifying 𝑥^∗=𝑃_Ω⋅𝑓𝑥^∗, (48) which equivalently solves the following variational inequality problem: 𝑥^∗∈Ω𝛾𝑓−𝜇𝐵𝑥^∗,𝑣−𝑥^∗≤0,∀𝑣∈Ω. (49)

Remark 20. If 𝐵=𝐼 and 𝜇=1 in Corollary 19, we obtain the following corollary.

Corollary 21. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝑇₁,𝑇₂∶𝐻→𝐻 be nonspreading mappings and demiclosed. Let 𝑓 be 𝐿-Lipschitz mapping on 𝐻 with coefficient 𝐿>0. Let the sequence of {𝛼_𝑛} and {𝛽_𝑛} satisfy the following conditions: (c1)lim_𝑛→∞𝛼_𝑛=0,(c2)∑^∞_𝑛=0𝛼_𝑛=∞ and lim_𝑛→∞𝛼_𝑛+1/𝛼_𝑛=1,(c3)0<𝑐≤𝛽_𝑛≤𝑑<1. Let the following sequence {𝑥_𝑛}^∞_𝑛=0 be a sequence in 𝐶 generated from an arbitrary 𝑥₁∈𝐶 by 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇_𝜔1𝛼_𝑛𝛾𝑓𝑥_𝑛+1−𝛼_𝑛𝑇_𝜔2𝑥_𝑛, (50) where 𝑇_𝜔𝑖=(1−𝜔_𝑖)𝐼+𝜔_𝑖𝑇_𝑖, 𝜔_𝑖∈(0,1), 𝑖=1,2. Then {𝑥_𝑛} converges strongly to the unique element 𝑥^∗ in Ω=𝐹_𝑖𝑥(𝑇₁)⋂𝐹_𝑖𝑥(𝑇₂) verifying 𝑥^∗=𝑃_Ω⋅𝑓𝑥^∗, (51) which equivalently solves the following variational inequality problem: 𝑥^∗∈Ω,𝛾𝑓−𝐼𝑥^∗,𝑣−𝑥^∗≤0,∀𝑣∈Ω. (52)

4. Application

In this section, we constructed a numerical example to illustrate that our main results are well defined. The following example is introduced by Deng et al. [14]. They prove that 𝑇 is (1/3)-strictly pseudononspreading.

Example 22. Let 𝑋=𝑙² with the norm ‖⋅‖ defined by ‖𝑥‖=^∞_𝑖=1𝑥²_𝑖,∀𝑥=𝑥₁,𝑥₂,…,𝑥_𝑛,…∈𝑋, (53)𝐶={𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛,…)∣𝑥_𝑖∈𝑅¹,𝑖=1,2,…}, and let 𝐶 be an orthogonal subspace of 𝑋 (i.e.,  ∀𝑥,𝑦∈𝐶, we have ⟨𝑥,𝑦⟩=0). Then it is obvious that 𝐶 is a nonempty closed convex subset of 𝑋. Now, for any 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛,…)∈𝐶, define a mapping 𝑇∶𝐶→𝐶 as follows: 𝑥𝑇𝑥=¹,𝑥₂,…,𝑥_𝑛,…,∏^∞_𝑖=1𝑥_𝑖<0,−2𝑥₁,−2𝑥₂,…,−2𝑥_𝑛,…,∏^∞_𝑖=1𝑥_𝑖≥0. (54)

From the definition of this example, let 𝑥=(𝑥₁,𝑥₂)∈𝑅² and 𝑓(𝑥)=8(sin(𝑥₁),cos(𝑥₂)) be a Lipschitz mapping on 𝑅². Assume that 𝑇₁(𝑥)=(0,4𝑥₁) and 𝑇₂(𝑥)=(4𝑥₂,0) be  (1/3)-strictly pseudononspreading mappings. Let 𝛼_𝑛=𝛽_𝑛=1/5, 𝛾=1/8, 𝜇=𝜔₁=𝜔₂=1, and 𝐵=𝐼. According to (22), we can obtain the following algorithm: 𝑥_𝑛+1=15𝑥_𝑛+4𝑇200¹𝑓𝑥_𝑛+16𝑇25¹𝑇₂𝑥_𝑛. (55) From Theorem 17, we can easily know that algorithm (55) converges to the unique point in 𝑅². Let 𝑥₀=(5,4) and |𝑥_𝑛+1−𝑥_𝑛|≤10⁻⁵, and let 𝑥^∗ be the fixed point of the algorithm (55). Using the software of MATLAB, we obtain 𝑥^∗=(0,0).

Acknowledgments

This work is supported in part by National Natural Science Foundation of China (71272148), the Ph.D. Programs Foundation of Ministry of Education of China (20120032110039), and China Postdoctoral Science Foundation (Grant no. 20100470783).