Strong Convergence of a Unified General Iteration for k-Strictly Pseudononspreading Mapping in Hilbert Spaces

Abstract

We introduce a unified general iterative method to approximate a fixed point of k-strictly pseudononspreading mapping. Under some suitable conditions, we prove that the iterative sequence generated by the proposed method converges strongly to a fixed point of a k-strictly pseudononspreading mapping with an idea of mean convergence, which also solves a class of variational inequalities as an optimality condition for a minimization problem. The results presented in this paper may be viewed as a refinement and as important generalizations of the previously known results announced by many other authors.

1. Introduction

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. Recall that a mapping 𝑇∶𝐶→𝐶 is said to be 𝑘-strict pseudocontractive if there exists a constant 𝑘∈[0,1) such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶. (1) If 𝑘=0, 𝑇 is said to be nonexpansive mapping; that is, 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2) The set of fixed points of 𝑇 is denoted by 𝐹(𝑇); that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. Recall also that a mapping 𝑇∶𝐶→𝐶 is said to be nonspreading if 2𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑦²+𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐶. (3) It is shown in the study by Iemoto and Takahashi [1] that (3) is equivalent to 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+2𝑥−𝑇𝑥,𝑦−𝑇𝑦,∀𝑥,𝑦∈𝐶. (4) Observe that every nonspreading mapping is quasinonexpansive; that is, ‖𝑇𝑥−𝑝‖≤‖𝑥−𝑝‖ for all 𝑥∈𝐶 and all 𝑝∈𝐹(𝑇). Following the terminology of Browder and Petryshyn [2], a mapping 𝑇∶𝐶→𝐶 is called 𝑘-strictly pseudononspreading if there exists a constant 𝑘∈[0,1) such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²+2𝑥−𝑇𝑥,𝑦−𝑇𝑦,∀𝑥,𝑦∈𝐶. (5) Clearly, every nonspreading mapping is 𝑘-strictly pseudononspreading, but the converse is not true. This shows that the class of 𝑘-strictly pseudononspreading mappings is more general than the class of nonspreading mappings. Moreover, we remark also that the class of 𝑘-strictly pseudononspreading mappings is independent of the class of 𝑘-strict pseudo-contractions.

Fixed point problem of nonlinear mappings recently becomes an attractive subject because of its application in solving variational inequalities and equilibrium problems arising in various fields of pure and applied sciences. Moreover, various iterative schemes and methods have been developed for finding fixed points of nonlinear mappings. It is worth mentioning that iterative methods for nonexpansive and nonspreading mappings have been extensively investigated. However, iterative methods for strict pseudo-contractions are far less developed than those for nonexpansive mappings though Browder and Petryshyn [2] initiated their work in 1967; the reason is probably that the second term appearing in the right-hand side of (1) impedes the convergence analysis for iterative algorithms used to find a fixed point of the strict pseudo-contraction. This case is aggravated by adding another inner product to the right-hand side of (5) for 𝑘-strictly pseudononspreading mapping; see, for example, [3–13] and the references therein. On the other hand, 𝑘-strictly pseudononspreading mappings have more powerful applications than nonexpansive mappings do in solving mean ergodic problems; see, for example, [14, 15]. Therefore, it is interesting to develop the effective numerical methods for approximating fixed point of 𝑘-strictly pseudononspreading mapping.

In 2006, Marino and Xu [10] introduced a general iterative method and proved that, for a given 𝑥₀∈𝐻, the sequence {𝑥_𝑛} generated by 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐵𝑇𝑥_𝑛,∀𝑛≥1, (6) where 𝑇 is a self-nonexpansive mapping on 𝐻, 𝑓 is a contraction of 𝐻 into itself, {𝛼_𝑛}⊆(0,1) satisfies certain conditions, and 𝐵 is a strongly positive bounded linear operator on 𝐻, converges strongly to 𝑥^∗∈𝐹(𝑇), which is the unique solution of the following variational inequality: 𝐵−𝛾𝑓𝑥^∗,𝑥^∗−𝑤≤0,∀𝑤∈𝐹(𝑇), (7) and is also the optimality condition of problem min_𝑥∈𝐶(1/2)⟨𝐵𝑥,𝑥⟩−ℎ(𝑥), where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^′(𝑥)=𝛾𝑓(𝑥),∀𝑥∈𝐻). Thereafter, the general iterative method is used to find a common element of the set of fixed point problems and the set of solutions of variational inequalities and equilibrium problems (see, e.g., [11–13]).

Recently, Kurokawa and Takahashi [14] obtained a weak mean ergodic theorem for nonspreading mappings in Hilbert spaces. Furthermore, they proved a strong convergence theorem using an idea of mean convergence. In 2011, Osilike and Isiogugu [15] introduced a more general 𝑘-strictly pseudononspreading mapping and considered the following iterative schemes: 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑧_𝑛,𝑧_𝑛=1𝑛^𝑛−1_𝑘=0𝑇^𝑘_𝛽𝑥_𝑛,𝑛≥1, (8) where auxiliary mapping 𝑇_𝛽=𝛽𝐼+(1−𝛽)𝑇. They proved that the sequences {𝑥_𝑛} and {𝑧_𝑛} converge strongly to 𝑃_𝐹(𝑇)𝑢, which is the metric projection of 𝐻 onto 𝐹(𝑇). Moreover, they considered the following Halpern type iterative scheme: 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇_𝛽𝑥_𝑛,𝑛≥1. (9) They also proved that {𝑥_𝑛} generated by (9) converges strongly to 𝑞∈𝐹(𝑇) under some suitable conditions and hence resolved in the affirmative the open problem raised by Kurokawa and Takahashi [14] in their final remark for the case where the mapping 𝑇 is averaged.

In 2013, Kangtunyakarn [16] further studied variational inequalities and fixed point problem of 𝑘-strictly pseudononspreading mapping 𝑇 by modifying the auxiliary mapping with projection technique. To be more precise, he introduced the following iterative scheme: 𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑃_𝐶𝐼−𝜆_𝑛(𝐼−𝑇)𝑥_𝑛+𝛾_𝑛𝑆𝑥_𝑛,𝑛≥1, (10) where 𝛼_𝑛,𝛽_𝑛,𝛾_𝑛∈(0,1) such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1 and 𝛽_𝑛∈[𝑐,𝑑]⊂(0,1) and 𝑆 is a nonexpansive mapping generated by a finite family of defining operators, whose fixed point problems are equivalent to variational inequalities. Moreover, under some suitable conditions, he proved that the sequence {𝑥_𝑛} converges strongly to 𝑃_Ω𝑢, where Ω is the intersection of the set of fixed point problems and the set of solutions for variational inequalities.

Inspired and motivated by research going on in this area, we introduce a modified general iterative method for 𝑘-strictly pseudononspreading mapping, which is defined in the following way: 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐵𝑇_𝜆𝑛𝑥_𝑛,𝑛≥1, (11) where 𝑇_𝜆𝑛=𝑃_𝐶[𝐼−𝜆_𝑛(𝐼−𝑇)] with 𝜆_𝑛∈(0,1) and sequences {𝛼_𝑛} and {𝛽_𝑛} in [0,1]. Note that, if 𝛽_𝑛=0, scheme (11) reduces to general iterative method (6), which is mainly due to Marino and Xu [10]. If 𝛽_𝑛=0, 𝛾=1, and 𝐵=𝐼, scheme (11) reduces to viscosity approximate method introduced by Moudafi [17] and developed by Inchan [18], which also extends the Halpern type results of [19, 20] with an idea of mean convergence for 𝑘-strictly pseudononspreading mapping.

Our purpose is not only to modify the general iterative method (6) and projection method (10) to the case of a 𝑘-strictly pseudononspreading mapping, but also to establish a new strong convergence theorem with an idea of mean convergence for a 𝑘-strictly pseudononspreading mapping, which also solves a class of variational inequalities as an optimality condition for a minimization problem. Our main results presented in this paper improve and extend the corresponding results of [10, 14–17] and many others.

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of real Hilbert 𝐻 space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. For every point 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶, such that 𝑥−𝑃_𝐶𝑥≤𝑥−𝑦,∀𝑦∈𝐶. (12) Then 𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. It is well known that 𝑃_𝐶 is a nonexpansive mapping and the following inequality holds: 𝑥−𝑢,𝑢−𝑦≥0,∀𝑦∈𝐶, (13) if and only if 𝑢=𝑃_𝐶𝑥 for given 𝑥∈𝐻 and 𝑢∈𝐶.

Let 𝐴 be a mapping from 𝐶 into 𝐻. The normal variational inequality problem is to find a point 𝑢∈𝐶 such that ⟨𝐴𝑢,𝑣−𝑢⟩≥0,∀𝑣∈𝐶. (14) The set of all solutions of the variational inequality is denoted by 𝑉𝐼(𝐶,𝐴). Note that 𝑢∈𝑉𝐼(𝐶,𝐴) if and only if 𝑢=𝑃_𝐶(𝐼−𝜆𝐴)𝑢 for some 𝜆>0.

Recall that an operator 𝐵 is strongly positive if there exists a constant 𝛾>0 with the property ⟨𝐵𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻. (15) Recall also that an operator 𝑓∶𝐶→𝐶 is a contraction, if there exists a constant 𝜌∈(0,1) such that 𝑓(𝑥)−𝑓𝑦≤𝜌𝑥−𝑦,∀𝑥,𝑦∈𝐶. (16)

In order to prove our main results, we need the following lemmas and propositions.

Lemma 1. Let 𝐻 be a real Hilbert space. There hold the following well-known results: (i)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,(𝑥+𝑦)⟩,  ∀𝑥,𝑦∈𝐻;(ii)‖𝑡𝑥+(1−𝑡)𝑦‖²=𝑡‖𝑥‖²+(1−𝑡)‖𝑦‖²−𝑡(1−𝑡)‖𝑥−𝑦‖²,  𝑡∈[0,1],  ∀𝑥,𝑦∈𝐻.

Lemma 2 (see [6]). Let {𝑥_𝑛} and {𝑧_𝑛} be bounded sequences in Banach space 𝐸 and let {𝛽_𝑛} be a sequence in [0,1] such that 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑧_𝑛 and limsup_𝑛→∞𝑧_𝑛+1−𝑧_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0,∀𝑛≥0. (17) Then lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0.

Lemma 3 (see [10]). Let 𝐵 be a strongly positive linear bounded operator on a Hilbert space 𝐻 with a coefficient 𝛾>0 and 0<𝜚<‖𝐵‖⁻¹. Then ‖𝐼−𝜚𝐵‖≤1−𝜚𝛾.

Lemma 4 (see [10]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. Assume that 𝑓∶𝐶→𝐶 is a contraction with a coefficient 𝜌∈(0,1) and 𝐵 is a strongly positive linear bounded operator with a coefficient 𝛾>0. Then, for 0<𝛾<𝛾/𝜌, 𝑥−𝑦,𝐵−𝛾𝑓𝑥−𝐵−𝛾𝑓𝑦≥𝛾−𝛾𝜌𝑥−𝑦²,∀𝑥,𝑦∈𝐻. (18) That is, 𝐵−𝛾𝑓 is strongly monotone with coefficient 𝛾−𝛾𝜌.

Lemma 5 (see [15]). 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 zero.

Lemma 6 (see [15]). 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(𝑇)≠∅, then it is closed and convex.

Lemma 7 (see [16]). 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 with 𝐹(𝑇)≠∅. Then 𝐹(𝑇)=𝑉𝐼(𝐶,(𝐼−𝑇)).

Lemma 8 (see [21]). Assume {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝛿_𝑛,𝑛≥0, (19) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that (i)∑^∞_𝑛=1𝛾_𝑛=∞;(ii)limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞_𝑛=1|𝛾_𝑛𝛿_𝑛|<∞.Then lim_𝑛→∞𝑎_𝑛=0.

3. Main Results

Theorem 9. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and let 𝑇∶𝐶→𝐶 be a 𝑘-strictly pseudononspreading mapping such that 𝐹(𝑇)≠∅. Let 𝑓∶𝐶→𝐶 be a contraction with a coefficient 𝜌∈(0,1) and let 𝐵 be a strongly positive bounded linear operator with 𝛾>0. For a given point 𝑥₀∈𝐶 and 0<𝛾<𝛾/𝜌, assume that 𝛼_𝑛,𝛽_𝑛,𝜆_𝑛∈[0,1] satisfying the following conditions: (i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;(iii)𝜆_𝑛∈(0,1−𝑘) and lim_𝑛→∞𝜆_𝑛=0.Then the sequence {𝑥_𝑛} generated by (11) converges strongly to 𝑞∈𝐹(𝑇), which is the unique solution of the following variational inequality: 𝐵−𝛾𝑓𝑞,𝑞−𝑤≤0,∀𝑤∈𝐹(𝑇). (20)

Proof. First, we show that sequences {𝑥_𝑛} and {𝑇𝑥_𝑛} are bounded. Indeed, from the property of 𝑘-strictly pseudononspreading mapping defined on 𝑇 and 𝑝∈𝐹(𝑇), we have 𝑇𝑥_𝑛−𝑇𝑝²=(𝑥_𝑛−𝑝)−[(𝐼−𝑇)𝑥_𝑛−(𝐼−𝑇)𝑝]²=𝑥_𝑛−𝑝²−2𝑥_𝑛−𝑝,(𝐼−𝑇)𝑥_𝑛+(𝐼−𝑇)𝑥_𝑛²≤𝑥_𝑛−𝑝²+𝑘(𝐼−𝑇)𝑥_𝑛−(𝐼−𝑇)𝑝²+2(𝐼−𝑇)𝑥_𝑛,()𝑝=𝑥𝐼−𝑇_𝑛−𝑝²+𝑘(𝐼−𝑇)𝑥_𝑛², (21) which implies that (1−𝑘)(𝐼−𝑇)𝑥_𝑛²≤2𝑥_𝑛−𝑝,(𝐼−𝑇)𝑥_𝑛. (22) From 𝑇_𝜆𝑛=𝑃_𝐶[𝐼−𝜆_𝑛(𝐼−𝑇)] and (22), we obtain 𝑇_𝜆𝑛𝑥_𝑛−𝑝²≤(𝑥_𝑛−𝑝)−𝜆_𝑛[(𝐼−𝑇)𝑥_𝑛−(𝐼−𝑇)𝑝]²=𝑥_𝑛−𝑝²−2𝜆_𝑛𝑥_𝑛−𝑝,(𝐼−𝑇)𝑥_𝑛+𝜆²_𝑛(𝐼−𝑇)𝑥_𝑛²≤𝑥_𝑛−𝑝²−𝜆_𝑛(1−𝑘)(𝐼−𝑇)𝑥_𝑛²+𝜆²_𝑛(𝐼−𝑇)𝑥_𝑛²=𝑥_𝑛−𝑝²−𝜆_𝑛(1−𝑘)−𝜆_𝑛(𝐼−𝑇)𝑥_𝑛²≤𝑥_𝑛−𝑝². (23) By (i) and Lemma 3, we have that (1−𝛽_𝑛)𝐼−𝛼_𝑛𝐵 is positive and ‖(1−𝛽_𝑛)𝐼−𝛼_𝑛𝐵‖≤1−𝛽_𝑛−𝛼_𝑛𝛾 for all 𝑛≥1 (see, i.e., [8]). It follows from (11) and (23) that 𝑥_𝑛+1=𝛼−𝑝_𝑛𝛾𝑓𝑥_𝑛−𝐵𝑝+𝛽_𝑛𝑥_𝑛+−𝑝1−𝛽_𝑛𝐼−𝛼_𝑛𝐵𝑇_𝜆𝑛𝑥_𝑛−𝑝≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐵𝑝+𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛−𝛼_𝑛𝛾𝑇_𝜆𝑛𝑥_𝑛−𝑝≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑝+𝛼_𝑛𝛾𝑓𝑝−𝐵𝑝+𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛−𝛼_𝑛𝛾𝑥_𝑛≤−𝑝1−𝛼_𝑛𝛾−𝛾𝜌𝑥_𝑛−𝑝+𝛼_𝑛𝛾𝑓𝑝−𝐵𝑝. (24) By induction, we have that 𝑥_𝑛−𝑝≤max𝑥₀,1−𝑝𝑝.𝛾−𝛾𝜌𝛾𝑓−𝐵𝑝 (25) Therefore, {𝑥_𝑛} is bounded and so is {𝑇_𝜆𝑛𝑥_𝑛}. On the other hand, we estimate 𝑇𝑥_𝑛−𝑝²≤𝑥_𝑛−𝑝²+𝑘(𝐼−𝑇)𝑥_𝑛−(𝐼−𝑇)𝑝²+2𝑥_𝑛−𝑇𝑥_𝑛=𝑥,𝑝−𝑇𝑝_𝑛−𝑝²+𝑘𝑥_𝑛−𝑝−𝑇𝑥_𝑛−𝑝²=𝑥_𝑛−𝑝²+𝑘𝑥_𝑛−𝑝²−2⟨𝑥_𝑛−𝑝,𝑇𝑥_𝑛+−𝑝⟩𝑇𝑥_𝑛−𝑝², (26) which implies that (1−𝑘)𝑇𝑥_𝑛−𝑝²≤(1+𝑘)𝑥_𝑛−𝑝²+2𝑘𝑥_𝑛−𝑝𝑇𝑥_𝑛−𝑝. (27) From (27), we can obtain 0≥(1−𝑘)𝑇𝑥_𝑛−𝑝²−(1+𝑘)𝑥_𝑛−𝑝²−2𝑘𝑥_𝑛−𝑝𝑇𝑥_𝑛=()−𝑝1−𝑘𝑇𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝𝑇𝑥_𝑛−()𝑥−𝑝1+𝑘_𝑛−𝑝²+𝑥_𝑛−𝑝𝑇𝑥_𝑛=(−𝑝1−𝑘)𝑇𝑥_𝑛−𝑝𝑇𝑥_𝑛−𝑝+𝑥_𝑛−(𝑥−𝑝1+𝑘)_𝑛−𝑝𝑥_𝑛−𝑝+𝑇𝑥_𝑛−𝑝. (28) It follows that 𝑇𝑥_𝑛−𝑝≤1+𝑘𝑥1−𝑘^𝑛−𝑝. (29) Combining (25) and (29), we conclude that {𝑇𝑥_𝑛} is bounded.
Next, we will show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. To do this, define a sequence {𝑧_𝑛} by 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛,𝑛≥1. (30) Observe that 𝑧_𝑛+1−𝑧_𝑛=𝑥_𝑛+2−𝛽_𝑛+1𝑥_𝑛+11−𝛽_𝑛+1−𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛=𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1+1−𝛽_𝑛+1𝐼−𝛼_𝑛+1𝐵𝑤_𝑛+11−𝛽_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐵𝑤_𝑛1−𝛽_𝑛=𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐵𝑤_𝑛+1+𝑤_𝑛+1−𝑤_𝑛−𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐵𝑤_𝑛, (31) where 𝑤_𝑛=𝑇_𝜆𝑛𝑥_𝑛, and 𝑤_𝑛+1−𝑤_𝑛≤𝐼−𝜆_𝑛+1(𝐼−𝑇)𝑥_𝑛+1−𝐼−𝜆_𝑛(𝐼−𝑇)𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛−𝜆_𝑛+1(𝐼−𝑇)𝑥_𝑛+1+𝜆_𝑛(𝐼−𝑇)𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛+1(𝐼−𝑇)𝑥_𝑛+1−(𝐼−𝑇)𝑥_𝑛+𝜆_𝑛+1−𝜆_𝑛(𝐼−𝑇)𝑥_𝑛. (32) From (31) and (32), we obtain 𝑧_𝑛+1−𝑧_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐵𝑤_𝑛+1+𝑤_𝑛+1−𝑤_𝑛+𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐵𝑤_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐵𝑤_𝑛+1+𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐵𝑤_𝑛+𝜆_𝑛+1(𝐼−𝑇)𝑥_𝑛+1−(𝐼−𝑇)𝑥_𝑛+𝜆_𝑛+1−𝜆_𝑛(𝐼−𝑇)𝑥_𝑛. (33) It follows from conditions (i)–(iii) and Lemma 2 that lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0. (34) From (30) and (34) and condition (ii), we have lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛=0. (35) Moreover, note that 𝑤_𝑛=𝑇_𝜆𝑛𝑥_𝑛 and 𝑥_𝑛−𝑤_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑤_𝑛=𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐵𝑤_𝑛−𝑤_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐵𝑤_𝑛+𝛽_𝑛𝑥_𝑛−𝑤_𝑛, (36) which implies that 𝑥_𝑛−𝑤_𝑛≤11−𝛽_𝑛𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐵𝑤_𝑛. (37) Combining conditions (i)-(ii) and (35), we obtain lim_𝑛→∞𝑥_𝑛−𝑤_𝑛=lim_𝑛→∞𝑥_𝑛−𝑇_𝜆𝑛𝑥_𝑛=0. (38) That is, lim_𝑛→∞𝑥_𝑛−𝑃_𝐶𝐼−𝜆_𝑛(𝐼−𝑇)𝑥_𝑛=0. (39)
Next, we will prove that limsup_𝑛→∞⟨𝛾𝑓(𝑞)−𝐵𝑞,𝑥_𝑛−𝑞⟩≤0, where 𝑞=𝑃_𝐹(𝑇)(𝐼−𝐵+𝛾𝑓)𝑞. To show this inequality, take a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that limsup_𝑛→∞𝛾𝑓𝑞−𝐵𝑞,𝑥_𝑛−𝑞=lim_𝑗→∞𝛾𝑓𝑞−𝐵𝑞,𝑥_𝑛𝑗−𝑞. (40) Without loss of generality, we may assume that {𝑥_𝑛𝑗} converges weakly to 𝑤; that is, 𝑥_𝑛𝑗⇀𝑤 as 𝑗→∞, where 𝑤∈𝐶. We will show that 𝑤∈𝐹(𝑇). From Lemmas 5 and 7, we have 𝐹(𝑇)=𝐹(𝑇_𝜆𝑛𝑗)=𝐹(𝑃_𝐶[𝐼−𝜆_𝑛𝑗(𝐼−𝑇)]). Assume that 𝑤≠𝑃_𝐶[𝐼−𝜆_𝑛𝑗(𝐼−𝑇)]𝑤. By condition (iii), (38), and Opial's property, we obtain liminf_𝑗→∞𝑥_𝑛𝑗−𝑤<liminf_𝑗→∞𝑥_𝑛𝑗−𝑃_𝐶𝐼−𝜆_𝑛𝑗(𝐼−𝑇)𝑤≤liminf_𝑗→∞𝑥_𝑛𝑗−𝑇_𝜆𝑛𝑗𝑥_𝑛𝑗+𝑃_𝐶𝐼−𝜆_𝑛𝑗(𝐼−𝑇)𝑥_𝑛𝑗−𝑃_𝐶𝐼−𝜆_𝑛𝑗(𝐼−𝑇)𝑤≤liminf_𝑗→∞𝑥_𝑛𝑗−𝑇_𝜆𝑛𝑗𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑤+𝜆_𝑛𝑗(𝐼−𝑇)𝑥_𝑛𝑗−(𝐼−𝑇)𝑤≤liminf_𝑗→∞𝑥_𝑛𝑗−𝑤. (41) This is a contradiction. Then 𝑤∈𝐹(𝑇). Since 𝑥_𝑛𝑗⇀𝑤 as 𝑗→∞, we have limsup_𝑛→∞𝛾𝑓𝑞−𝐵𝑞,𝑥_𝑛−𝑞=lim_𝑗→∞𝛾𝑓𝑞−𝐵𝑞,𝑥_𝑛𝑗=𝑞−𝑞𝛾𝑓−𝐵𝑞,𝑤−𝑞≤0. (42) On the other hand, we will show the uniqueness of a solution of the variational inequality 𝐵−𝛾𝑓𝑥,𝑥−𝑤≤0,𝑤∈𝐹(𝑇). (43) Suppose 𝑞∈𝐹(𝑇) and 𝑞∈𝐹(𝑇) both are solutions to (43); then 𝐵−𝛾𝑓𝑞,𝑞−𝑞≤0,𝐵−𝛾𝑓𝑞,𝑞−𝑞≤0. (44) Adding up (44), we get 𝐵−𝛾𝑓𝑞−𝐵−𝛾𝑓𝑞,𝑞−𝑞≤0. (45) From Lemma 4, the strong monotonicity of 𝐵−𝛾𝑓, we obtain 𝑞=𝑞 and the uniqueness is proved.
Finally, we show that {𝑥_𝑛} converges strongly to 𝑞 as 𝑛→∞. From (11), (23), and Lemma 1, we have (note that 𝑤_𝑛=𝑇_𝜆𝑛𝑥_𝑛) 𝑥_𝑛+1−𝑞²=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐵𝑤_𝑛−𝑞,𝑥_𝑛+1−𝑞=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐵𝑞,𝑥_𝑛+1+−𝑞1−𝛽_𝑛𝐼−𝛼_𝑛𝐵𝑤_𝑛−𝑞,𝑥_𝑛+1−𝑞+𝛽_𝑛𝑥_𝑛−𝑞,𝑥_𝑛+1−𝑞≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑞,𝑥_𝑛+1−𝑞+𝛼_𝑛𝛾𝑓𝑞−𝐵𝑞,𝑥_𝑛+1−𝑞+𝛽_𝑛𝑥_𝑛−𝑞𝑥_𝑛+1+−𝑞1−𝛽_𝑛−𝛼_𝑛𝛾𝑤_𝑛−𝑞𝑥_𝑛+1−𝑞≤𝛼_𝑛𝛾𝜌𝑥_𝑛−𝑞𝑥_𝑛+1−𝑞+𝛼_𝑛𝛾𝑓𝑞−𝐵𝑞,𝑥_𝑛+1+−𝑞1−𝛼_𝑛𝛾𝑥_𝑛−𝑞𝑥_𝑛+1=−𝑞1−𝛾−𝛾𝜌𝛼_𝑛𝑥_𝑛−𝑞𝑥_𝑛+1−𝑞+𝛼_𝑛𝛾𝑓𝑞−𝐵𝑞,𝑥_𝑛+1≤−𝑞1−𝛾−𝛾𝜌𝛼_𝑛2𝑥_𝑛−𝑞²+𝑥_𝑛+1−𝑞²+𝛼_𝑛𝛾𝑓𝑞−𝐵𝑞,𝑥_𝑛+1≤−𝑞1−𝛾−𝛾𝜌𝛼_𝑛2𝑥_𝑛−𝑞²+12𝑥_𝑛+1−𝑞²+𝛼_𝑛𝛾𝑓𝑞−𝐵𝑞,𝑥_𝑛+1−𝑞. (46) It follows that 𝑥_𝑛+1−𝑞²≤1−𝛾−𝛾𝜌𝛼_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝛾𝑓𝑞−𝐵𝑞,𝑥_𝑛+1−𝑞. (47) Together with 0<𝛾<𝛾/𝜌, condition (i), and (42), we can arrive at the desired conclusion lim_𝑛→∞‖𝑥_𝑛−𝑞‖=0 by Lemma 8. This completes the proof.

Theorem 10. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and let 𝑇∶𝐶→𝐶 be a 𝑘-strictly pseudononspreading mapping such that 𝐹(𝑇)≠∅. Let 𝑓∶𝐶→𝐶 be a contraction with a coefficient 𝜌∈(0,1). Let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐶 in the following manner: 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛼_𝑛−𝛽_𝑛𝑇_𝜆𝑛𝑥_𝑛,𝑛≥1, (48) where {𝛼_𝑛}, {𝛽_𝑛}, and {𝜆_𝑛} are sequences in (0,1) satisfying the following conditions: (i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;(iii)𝜆_𝑛∈(0,1−𝑘) and lim_𝑛→∞𝜆_𝑛=0.Then the sequence {𝑥_𝑛} converges strongly to 𝑞∈𝐹(𝑇), which is the unique solution of the following variational inequality: 𝐼−𝑓𝑞,𝑞−𝑤≤0,∀𝑤∈𝐹(𝑇). (49)

Proof. Putting 𝐵=𝐼 and 𝛾=1, general iterative scheme (11) reduces to viscosity iteration (48). The desired conclusion follows immediately from Theorem 9. This completes the proof.

Theorem 11. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and let 𝑇∶𝐶→𝐶 be a nonspreading mapping (or quasinonexpansive) such that 𝐹(𝑇)≠∅. Let 𝑓∶𝐶→𝐶 be a contraction with a coefficient 𝜌∈(0,1) and let 𝐵 be a strongly positive bounded linear operator with 𝛾>0 and 0<𝛾<𝛾/𝜌. Let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐶 in the following manner: 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐵𝑇_𝜆𝑛𝑥_𝑛,𝑛≥1, (50) where {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) satisfying the following conditions: (i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;(iii)𝜆_𝑛∈(0,1) and lim_𝑛→∞𝜆_𝑛=0.Then the sequence {𝑥_𝑛} converges strongly to 𝑞∈𝐹(𝑇), which is the unique solution of the following variational inequality: 𝐵−𝛾𝑓𝑞,𝑞−𝑤≤0,∀𝑤∈𝐹(𝑇). (51)

Proof. Clearly, every nonspreading mapping 𝑇 is 0-strictly pseudononspreading, which is also quasinonexpansive. Therefore, the desired conclusion follows immediately from Theorem 9. This completes the proof.

Remark 12. Theorems 9 and 10 extend the Halpern type methods of [14, 15] and viscosity methods of Moudafi [17] to more general unified general iterative methods for 𝑘-strictly pseudononspreading mapping, which also solves a class of variational inequalities related to an optimality problem.

Remark 13. Theorems 9 and 10 improve and extend the main results of Kangtunyakarn [16] for 𝑘-strictly pseudononspreading mapping in different directions.

Remark 14. The auxiliary mapping 𝑇_𝛽 of Osilike and Isiogugu [15] is generalized to the averaged mapping 𝑇_𝜆𝑛 presented in scheme (11) with variable coefficient and projection operator based on the equivalence between variational inequality and fixed point problem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Science Foundation of China (11001287), Natural Science Foundation Project of Chongqing (CSTC 2012jjA00039), and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ130712).