Hybrid Algorithms of Nonexpansive Semigroups for Variational Inequalities

Abstract

Two hybrid algorithms for the variational inequalities over the common fixed points set of nonexpansive semigroups are presented. Strong convergence results of these two hybrid algorithms have been obtained in Hilbert spaces. The results improve and extend some corresponding results in the literature.

1. Introduction

Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻. Recall that a mapping 𝑇∶𝐶→𝐶 is called nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖(1.1) for every 𝑥,𝑦∈𝐶. A family 𝑆={𝑇(𝜏)∶0<𝜏<∞} of mappings from 𝐶 into itself is called a nonexpansive semigroup on 𝐶 if it satisfies the following conditions:(i)𝑇(0)𝑥=𝑥 for all 𝑥∈𝐶,(ii)𝑇(𝑠+𝑡)=𝑇(𝑠)𝑇(𝑡) for all 𝑠,𝑡≥0,(iii)‖𝑇(𝑠)𝑥−𝑇(𝑠)𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶,𝑠≥0,(iv)for all 𝑥∈𝐶,𝑠↦𝑇(𝑠)𝑥 is continuous.We denote by Fix(𝑆) the set of all common fixed points of 𝑆, that is, Fix(𝑆)=⋂_0≤𝜏<∞Fix(𝑇(𝜏)). It is known that Fix(𝑆) is closed and convex.

Approximation of fixed points of nonexpansive mappings has been considered extensively by many authors, see, for instance, [1–18]. Nonlinear ergodic theorem for nonexpansive semigroups have been researched by some authors, see, for example, [19–23]. Our main purpose in the present paper is devoted to finding the common fixed points of nonexpansive semigroups.

Let 𝐹∶𝐶→𝐶 be a nonlinear operator. The variational inequality problem is formulated as finding a point 𝑥^∗∈𝐶 such that VI(𝐹,𝐶)∶⟨𝐹𝑥^∗,𝜐−𝑥^∗⟩≥0,∀𝜐∈𝐶.(1.2) Now it is well known that VI problem is an interesting problem and it covers as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance. Several numerical methods including the projection and its variant forms have been developed for solving the variational inequalities and related problems, see [24–41].

It is clear that the VI(𝐹,𝐶) is equivalent to the fixed point equation 𝑥^∗=𝑃_𝐶𝑥^∗−𝜇𝐹𝑥^∗,(1.3) where 𝑃_𝐶 is the projection of 𝐻 onto the closed convex set 𝐶 and 𝜇>0 is an arbitrarily fixed constant. So, fixed point methods can be implemented to find a solution of the VI(𝐹,𝐶) provided 𝐹 satisfies some conditions and 𝜇>0 is chosen appropriately. The fixed point formulation (1.3) involves the projection 𝑃_𝐶, which may not be easy to compute, due to the complexity of the convex set 𝐶. In order to reduce the complexity probably caused by the projection 𝑃_𝐶, Yamada [24] (see also [42]) recently introduced a hybrid steepest-descent method for solving the VI(𝐹,𝐶).

Assume that 𝐹 is an 𝜂-strongly monotone and 𝜅-Lipschitzian mapping with 𝜅>0,𝜂>0 on 𝐶. An equally important problem is how to find an approximate solution of the VI(𝐹,𝐶) if any. A great deal of effort has been done in this problem; see [43, 44].

Take a fixed number 𝜇 such that 0<𝜇<2𝜂/𝜅². Assume that a sequence {𝜆_𝑛} of real numbers in (0,1) satisfies the following conditions:(C1)lim_𝑛→∞𝜆_𝑛=0,(C2)∑^∞𝑛=0𝜆_𝑛=∞,(C3)lim_𝑛→∞(𝜆_𝑛−𝜆_𝑛+1)/𝜆^2𝑛=0.

Starting with an arbitrary initial guess 𝑥₀∈𝐻, one can generate a sequence {𝑥_𝑛} by the following algorithm: 𝑥_𝑛+1=𝑇𝑥_𝑛−𝜆_𝑛+1𝜇𝐹𝑇𝑥_𝑛,𝑛≥0.(1.4) Yamada [24] proved that the sequence {𝑥_𝑛} generated by (1.4) converges strongly to the unique solution of the VI(𝐹,𝐶). Xu and Kim [30] proved the strong convergence of {𝑥_𝑛} to the unique solution of the VI(𝐹,𝐶) if {𝜆_𝑛} satisfies conditions (C1), (C2), and (C4): lim_𝑛→∞𝜆_𝑛/𝜆_𝑛+1=1, or equivalently, lim_𝑛→∞(𝜆_𝑛−𝜆_𝑛+1)/𝜆_𝑛+1=1. Recently, Yao et al. [25] presented the following hybrid algorithm: 𝑥_𝑛+1𝑦_𝑛=𝑥_𝑛−𝜆_𝑛𝐹𝑥_𝑛,=1−𝛼_𝑛𝑦_𝑛+𝛼_𝑛𝑊_𝑛𝑦_𝑛,𝑛≥0,(1.5) where 𝐹 is a 𝜅-Lipschitzian and 𝜂-strongly monotone operator on 𝐻 and 𝑊_𝑛 is a 𝑊-mapping. It is shown that the sequences {𝑥_𝑛} and {𝑦_𝑛} defined by (1.5) converge strongly to 𝑥^∗∈⋂^∞𝑛=1𝐹(𝑇_𝑛), which solves the following variational inequality: ⟨𝐹𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈^∞_𝑛=1𝐹𝑇_𝑛.(1.6) Very recently, Wang [26] proved that the sequence {𝑦_𝑛} generated by the iterative algorithm (1.5) converges to a common fixed point of an infinite family of nonexpansive mappings under some weaker assumptions.

Motivated and inspired by the above works, in this paper, we introduce two hybrid algorithms for finding a common fixed point of a nonexpansive semigroup {𝑇(𝜏)}_𝜏≥0 in Hilbert spaces. We prove that the presented algorithms converge strongly to a common fixed point 𝑥^∗ of {𝑇(𝜏)}_𝜏≥0. Such common fixed point 𝑥^∗ is the unique solution of some variational inequality in Hilbert spaces.

2. Preliminaries

In this section, we will collect some basic concepts and several lemmas that will be used in the next section.

Suppose that 𝐻 is a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. For the sequence {𝑥_𝑛} in 𝐻, we write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥. 𝑥_𝑛→𝑥 means that {𝑥_𝑛} converges strongly to 𝑥. We denote by 𝜔_𝑤(𝑥_𝑛) the weak 𝜔-limit set of {𝑥_𝑛}, that is 𝜔_𝑤𝑥_𝑛=𝑥∈𝐻∶𝑥_𝑛𝑖⇀𝑥forsomesubsequence𝑥_𝑛𝑖of𝑥_𝑛.(2.1) Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. A mapping 𝐹∶𝐶→𝐶 is called 𝜅-Lipschitzian if there exists a positive constant 𝜅 such that ‖𝐹𝑥−𝐹𝑦‖≤𝜅‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(2.2)𝐹 is said to be 𝜂-strongly monotone if there exists a positive constant 𝜂 such that ⟨𝐹𝑥−𝐹𝑦,𝑥−𝑦⟩≥𝜂‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐶.(2.3) The following equalities are well known: ‖𝜆𝑥+(1−𝜆)‖𝑦𝑥‖−₂𝑦‖²=‖𝑥‖²−2⟨𝑥,𝑦⟩+‖𝑦‖²,=𝜆‖𝑥‖²+(1−𝜆)‖𝑦‖²−𝜆(1−𝜆)‖𝑥−𝑦‖²(2.4) for all 𝑥,𝑦∈𝐻 and 𝜆∈[0,1] (see [45]).

In the sequel, we will make use of the following well-known lemmas.

Lemma 2.1 (see [46]). Let 𝐶 be a nonempty bounded closed convex subset of 𝐻 and let 𝑆={𝑇(𝑠)∣0≤𝑠<∞} be a nonexpansive semigroup on 𝐶. Then, for any ℎ≥0, lim_𝑡→∞sup_𝑥∈𝐶‖‖‖1𝑡^𝑡0𝑇(𝑠)𝑥𝑑𝑠−𝑇(ℎ)1𝑡^𝑡0𝑇(𝑠)𝑥𝑑𝑠‖‖‖=0.(2.5)

Lemma 2.2 (see [47]). Assume that 𝑇∶𝐻→𝐻 is a nonexpansive mapping. If 𝑇 has a fixed point, then 𝐼−𝑇 is demiclosed. That is, whenever {𝑥_𝑛} is a sequence in 𝐻 weakly converging to some 𝑥∈𝐻 and the sequence {(𝐼−𝑇)𝑥_𝑛} strongly converges to some 𝑦, it follows that (𝐼−𝑇)𝑥=𝑦. Here, 𝐼 is the identity operator of 𝐻.

Lemma 2.3 (see [27]). Let {𝛾_𝑛} be a real sequence satisfying 0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1. Assume that {𝑥_𝑛} and {𝑧_𝑛} are bounded sequences in Banach space 𝐸, which satisfy the following condition: 𝑥_𝑛+1=(1−𝛾_𝑛)𝑥_𝑛+𝛾_𝑛𝑧_𝑛. If limsup_𝑛→∞(‖𝑧_𝑛+1−𝑧_𝑛‖)−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0, then lim_𝑛→∞‖𝑥_𝑛−𝑧_𝑛‖=0.

Lemma 2.4 (see [48]). Let 𝐹 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator on a Hilbert space 𝐻 with 0<𝜂≤𝜅 and 0<𝑡<𝜂/𝜅². Then, 𝑆=(𝐼−𝑡𝐹)∶𝐻→𝐻 is a contraction with contraction coefficient 𝜏_𝑡=√1−𝑡(2𝜂−𝑡𝜅²).

Lemma 2.5 (see [49]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying 𝑎_𝑛+1≤1−𝜆_𝑛𝑎_𝑛+𝜆_𝑛𝛿_𝑛+𝛾_𝑛,𝑛≥0,(2.6) where {𝜆_𝑛} and {𝛾_𝑛} satisfy the following conditions:(i)𝜆_𝑛⊂[0,1] and ∑^∞𝑛=0𝜆_𝑛=∞,(ii)limsup_𝑛→∞𝛿_𝑛≤0or∑^∞𝑛=0𝜆_𝑛𝛿_𝑛<∞, (iii)𝛾_𝑛≥0(𝑛≥0),∑^∞𝑛=0𝛾_𝑛<∞.
Then, lim_𝑛→∞𝑎_𝑛=0.

3. Main Results

In this section we will show our main results.

Theorem 3.1. Let 𝐻 be a real Hilbert space. Let 𝑆={𝑇(𝜏)∣0≤𝜏<∞}∶𝐻→𝐻 be a nonexpansive semigroup such that Fix(𝑆)≠∅. Let 𝐹 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator on 𝐻 with 0<𝜂≤𝜅. Let {𝛾_𝑡}_0<𝑡<1 be a continuous net of positive real numbers such that lim_𝑡→0+𝛾_𝑡=+∞. Putting 𝜏_𝑡=√1−𝑡(2𝜂−𝑡𝜅²), for each 𝑡∈(0,𝜂/𝜅²), let the net {𝑥_𝑡} be defined by the following implicit scheme: 𝑥_𝑡=1𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)(𝐼−𝑡𝐹)𝑥_𝑡𝑑𝜏.(3.1) Then, as 𝑡→0⁺, the net {𝑥_𝑡} converges strongly to a fixed point 𝑥^∗ of 𝑆, which is the unique solution of the following variational inequality: ⟨𝐹𝑥^∗,𝑥^∗−𝑢⟩≤0,∀𝑢∈Fix(𝑆).(3.2)

Proof. First, we note that the net {𝑥_𝑡} defined by (3.1) is well defined. We define a mapping 𝑃_𝑡𝑥∶=1𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)[(𝐼−𝑡𝐹)𝑥]𝑑𝜏,𝑡∈0,𝜂𝜅²,𝑥∈𝐻.(3.3) It follows that ‖‖𝑃_𝑡𝑥−𝑃_𝑡𝑦‖‖≤1𝛾_𝑡^𝛾𝑡₀≤‖(𝐼−‖𝑡𝑇𝐹()𝜏𝑥)[−(𝐼(𝐼−−𝑡𝐹𝑡𝐹)𝑥)]𝑦−‖.𝑇(𝜏)[(𝐼−𝑡𝐹)𝑦]‖𝑑𝜏(3.4) Obviously, 𝑃_𝑡 is a contraction. Indeed, from Lemma 2.4, we have ‖‖𝑃_𝑡𝑥−𝑃_𝑡𝑦‖‖≤‖(𝐼−𝑡𝐹)𝑥−(𝐼−𝑡𝐹)𝑦‖≤‖𝑥−𝑦‖,(3.5) for all 𝑥,𝑦∈𝐶. So it has a unique fixed point. Therefore, the net {𝑥_𝑡} defined by (3.1) is well defined.
We prove that {𝑥_𝑡} is bounded. Taking 𝑢∈Fix(𝑆) and using Lemma 2.4, we have ‖‖𝑥_𝑡−𝑢‖‖=‖‖‖1𝛾_𝑡^𝛾𝑡₀=‖‖‖1𝑇(𝜏)(𝐼−𝑡𝐹)𝑥_𝑡𝑑𝜏−𝑢‖‖‖𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)(𝐼−𝑡𝐹)𝑥_𝑡𝑑𝜏−1𝛾_𝑡^𝛾𝑡₀≤1𝑇(𝜏)𝑢𝑑𝜏‖‖‖𝛾_𝑡^𝛾𝑡₀≤‖‖(𝐼−‖‖𝑡𝑇𝐹()𝜏𝑥)_𝑡(𝐼−𝑡𝐹)𝑥_𝑡𝑑𝜏−𝑇(𝜏)𝑢‖‖𝑑𝜏≤‖‖(𝐼−𝑡𝐹)𝑥_𝑡−𝑢‖‖≤𝜏_𝑡−(𝐼−𝑡𝐹)𝑢−𝑡𝐹𝑢‖‖‖‖𝑥_𝑡−𝑢‖‖+𝑡‖𝐹𝑢‖.(3.6) It follows that ‖‖𝑥_𝑡−𝑢‖‖≤𝑡1−𝜏_𝑡‖𝐹𝑢‖.(3.7) Observe that lim_𝑡→0+𝑡1−𝜏_𝑡=1𝜂.(3.8) Thus, (3.7) and (3.8) imply that the net {𝑥_𝑡} is bounded for small enough 𝑡. Without loss of generality, we may assume that the net {𝑥_𝑡} is bounded for all 𝑡∈(0,𝜂/𝜅²). Consequently, we deduce that {𝐹𝑥_𝑡} is also bounded.
On the other hand, from (3.1), we have ‖‖𝑥_𝑡−𝑇(𝜏)𝑥_𝑡‖‖≤‖‖‖𝑇(𝜏)𝑥_𝑡−𝑇(𝜏)1𝛾_𝑡^𝛾𝑡₀+‖‖‖1𝑇(𝜏)𝑥_𝑡𝑑𝜏‖‖‖𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)𝑥_𝑡𝑑𝜏−𝑥_𝑡‖‖‖+‖‖‖𝑇(𝜏)1𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)𝑥_𝑡𝑑𝜏−1𝛾_𝑡^𝛾𝑡₀≤2‖‖‖1𝑇(𝜏)𝑥_𝑡𝑑𝜏‖‖‖𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)𝑥_𝑡𝑑𝜏−𝑥_𝑡+‖‖‖𝑇(𝜏)1‖‖‖𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)𝑥_𝑡𝑑𝜏−1𝛾_𝑡^𝛾𝑡₀=2‖‖‖1𝑇(𝜏)𝑥_𝑡𝑑𝜏‖‖‖𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)𝑥_𝑡𝑑𝜏−1𝛾_𝑡^𝛾𝑡₀+‖‖‖𝑇(𝜏)1𝑇(𝜏)[(𝐼−𝑡𝐹)]𝑥_𝑡𝑑𝜏‖‖‖𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)𝑥_𝑡𝑑𝜏−1𝛾_𝑡^𝛾𝑡₀≤21𝑇(𝜏)𝑥_𝑡𝑑𝜏‖‖‖𝛾_𝑡^𝛾𝑡₀+‖‖‖𝑇(‖‖𝜏𝑇)(𝜏1)𝑥_𝑡−𝑇(𝜏)(𝐼−𝑡𝐹)𝑥_𝑡‖‖𝑑𝜏𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)𝑥_𝑡𝑑𝜏−1𝛾_𝑡^𝛾𝑡₀≤2𝑡‖‖𝐹𝑥_𝑡𝑇(𝜏)𝑥_𝑡𝑑𝜏‖‖‖‖‖+‖‖‖𝑇(𝜏)1𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)𝑥_𝑡𝑑𝜏−1𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)𝑥_𝑡𝑑𝜏‖‖‖.(3.9) This together with Lemma 2.1 implies that lim_𝑡→0+‖‖𝑥_𝑡−𝑇(𝜏)𝑥_𝑡‖‖=0.(3.10) Let {𝑡_𝑛}⊂(0,1) be a sequence such that 𝑡_𝑛→0 as 𝑛→∞. Put 𝑥_𝑛∶=𝑥_𝑡𝑛. Since {𝑥_𝑛} is bounded, without loss of generality, we may assume that {𝑥_𝑛} converges weakly to a point 𝑥̃∈𝐶. Noticing (3.10), we can use Lemma 2.2 to get 𝑥̃∈Fix(𝑆).
Again, from (3.1), we have ‖‖𝑥_𝑡−𝑢‖‖²=‖‖‖1𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)(𝐼−𝑡𝐹)𝑥_𝑡𝑑𝜏−𝑢‖‖‖²≤‖‖‖1𝛾_𝑡^𝛾𝑡₀𝑇(𝜏)(𝐼−𝑡𝐹)𝑥_𝑡−𝑇(𝜏)𝑢𝑑𝜏‖‖‖²≤‖‖(𝐼−𝑡𝐹)𝑥_𝑡−(𝐼−𝑡𝐹)𝑢−𝑡𝐹𝑢‖‖²≤≤𝜏𝜏^2𝑡_𝑡‖‖𝑥_𝑡−𝑢‖‖²+𝑡²‖𝐹𝑢‖²+2𝑡⟨(𝐼−𝑡𝐹)𝑢−(𝐼−𝑡𝐹)𝑥_𝑡,𝐹𝑢⟩≤𝜏_𝑡‖‖𝑥_𝑡−𝑢‖‖²+𝑡²‖𝐹𝑢‖²+2𝑡⟨𝑢−𝑥_𝑡,𝐹𝑢⟩+2𝑡²⟨𝐹𝑥_𝑡−𝐹𝑢,𝐹𝑢⟩‖‖𝑥_𝑡−𝑢‖‖²+𝑡²‖𝐹𝑢‖²+2𝑡⟨𝑢−𝑥_𝑡,𝐹𝑢⟩+2𝜅𝑡²‖‖𝑥_𝑡−𝑢‖‖‖𝐹𝑢‖.(3.11) Therefore, ‖‖𝑥_𝑡−𝑢‖‖²≤𝑡²1−𝜏_𝑡‖𝐹𝑢‖²+2𝑡1−𝜏_𝑡⟨𝑢−𝑥_𝑡,𝐹𝑢⟩+2𝑡²𝜅1−𝜏_𝑡‖‖𝑥_𝑡−𝑢‖‖‖𝐹𝑢‖.(3.12) It follows that ‖‖𝑥_𝑛−𝑥̃‖‖²≤𝑡^2𝑛1−𝜏_𝑡𝑛‖𝐹𝑥̃‖²+2𝑡^𝑛1−𝜏_𝑡𝑛⟨𝑥̃−𝑥_𝑛,𝐹𝑥̃⟩+2𝑡^2𝑛𝜅1−𝜏_𝑡𝑛‖‖𝑥_𝑛−𝑥̃‖‖‖𝐹𝑥̃‖.(3.13) Thus, 𝑥_𝑛⇀𝑥̃ implies that 𝑥_𝑛→𝑥̃.
Again, from (3.12), we obtain ‖‖𝑥_𝑛−𝑢‖‖²≤𝑡^2𝑛1−𝜏_𝑡𝑛‖𝐹𝑢‖²+2𝑡^𝑛1−𝜏_𝑡𝑛⟨𝑢−𝑥_𝑛,𝐹𝑢⟩+2𝑡^2𝑛𝜅1−𝜏_𝑡𝑛‖‖𝑥_𝑛−𝑢‖‖‖𝐹𝑢‖.(3.14) It is clear that lim_𝑛→∞(𝑡^2𝑛/(1−𝜏_𝑡𝑛))=0, lim_𝑛→∞(2𝑡_𝑛/(1−𝜏_𝑡𝑛))=2/𝜂, and lim_𝑛→∞(2𝑡^2𝑛𝜅/(1−𝜏_𝑡𝑛))=0. We deduce immediately from (3.14) that ⟨𝐹𝑢,𝑥̃−𝑢⟩≤0,(3.15) which is equivalent to its dual variational inequality ⟨𝐹𝑥̃,𝑥̃−𝑢⟩≤0.(3.16) That is, 𝑥̃∈Fix(𝑆) is a solution of the variational inequality (3.2).
Suppose that 𝑥^∗∈Fix(𝑆) and 𝑥̃∈Fix(𝑆) both are solutions to the variational inequality (3.2); then ⟨⟨𝐹𝐹𝑥𝑥^∗̃,,𝑥𝑥̃^∗−−𝑥𝑥_∗̃⟩≤0,⟩≤0.(3.17) Adding up (3.17) and the last inequality yields ⟨𝐹𝑥^∗−𝐹𝑥̃,𝑥^∗−𝑥̃⟩≤0.(3.18)
The strong monotonicity of 𝐹 implies that 𝑥^∗=𝑥̃ and the uniqueness is proved. Later, we use 𝑥^∗∈Fix(𝑆) to denote the unique solution of (3.2).
Therefore, 𝑥̃=𝑥^∗ by uniqueness. In a nutshell, we have shown that each cluster point of {𝑥_𝑡}(𝑡→0) equals 𝑥^∗. Hence 𝑥_𝑡→𝑥^∗ as 𝑡→0. This completes the proof.

Next we introduce an explicit algorithm for finding a solution of the variational inequality (3.2).

Algorithm 3.2. For given 𝑥₀∈𝐶 arbitrarily, define a sequence {𝑥_𝑛} iteratively by 𝑥_𝑛+1𝑦_𝑛=𝑥_𝑛−𝜆_𝑛𝐹𝑥_𝑛,=1−𝛼_𝑛𝑦_𝑛+𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝜏)𝑦_𝑛𝑑𝜏,𝑛≥0,(3.19) where {𝜆_𝑛} and {𝑡_𝑛} are sequences in (0,∞) and {𝛼_𝑛} is a sequence in [0,1].

Theorem 3.3 3.3. Let 𝐻 be a real Hilbert space. Let 𝐹 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator on 𝐻 with 0<𝜂≤𝜅. Let 𝑆={𝑇(𝜏)∣0≤𝜏<∞}∶𝐻→𝐻 be a nonexpansive semigroup with Fix(𝑆)≠∅. Assume that(i)limsup_𝑛→∞𝜆_𝑛<𝜂/𝜅² and ∑^∞𝑛=1𝜆_𝑛=∞,(ii)lim_𝑛→∞𝑡_𝑛=∞ and lim_𝑛→∞(𝑡_𝑛+1/𝑡_𝑛)=1,(iii)0<𝛾≤liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1, for some 𝛾∈(0,1).
Then, the sequences {𝑥_𝑛} and {𝑦_𝑛} generated by (3.19) converge strongly to 𝑥^∗∈Fix(𝑆) if and only if 𝜆_𝑛𝐹(𝑥_𝑛)→0, where 𝑥^∗ solves the variational inequality (3.2).

Proof. The necessity is obvious. We only need to prove the sufficiency. Suppose that 𝜆_𝑛𝐹(𝑥_𝑛)→0. First, we show that 𝑥_𝑛 is bounded. In fact, letting 𝑢∈Fix(𝑆), we have ‖‖𝑥_𝑛+1−𝑢‖‖=‖‖‖1−𝛼_𝑛𝑦_𝑛+𝛼^𝑛𝑡_𝑛^𝑡𝑛₀=‖‖‖1−𝛼_𝑛𝑇(𝜏)𝑦_𝑛𝑑𝜏−𝑢‖‖‖≤1−𝛼_𝑛𝑦_𝑛−𝑢+𝛼_𝑛𝑡1_𝑛^𝑡𝑛₀𝑇(𝜏)𝑦_𝑛𝑑𝜏−𝑢‖‖‖‖‖𝑦_𝑛−𝑢‖‖+𝛼_𝑛‖‖‖1𝑡_𝑛^𝑡𝑛₀≤1−𝛼_𝑛𝑇(𝜏)𝑦_𝑛𝑑𝜏−𝑢‖‖‖‖‖𝑦_𝑛−𝑢‖‖+𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀≤1−𝛼_𝑛‖‖𝑇(𝜏)𝑦_𝑛−𝑇(𝜏)𝑢‖‖𝑑𝜏=‖‖𝑦‖‖𝑦_𝑛−𝑢‖‖+𝛼_𝑛‖‖𝑦_𝑛−𝑢_𝑛‖‖−𝑢‖‖.(3.20) From condition (i), without loss of generality, we can assume that 𝜆_𝑛≤𝑎<𝜂/𝜅² for all 𝑛. By (3.19) and Lemma 2.4, we have ‖‖𝑦_𝑛−𝑢‖‖==‖‖‖‖𝑥_𝑛𝐼−−𝜆𝜆_𝑛𝑛𝐹𝑥_𝑛−𝑢‖‖≤𝜏_𝜆𝑛𝐹𝑥_𝑛−𝐼−𝜆_𝑛𝐹𝑢−𝜆_𝑛𝐹𝑢‖‖‖‖𝑥_𝑛−𝑢‖‖+𝜆_𝑛‖𝐹𝑢‖,(3.21) where 𝜏_𝜆𝑛=√1−𝜆_𝑛(2𝜂−𝜆_𝑛𝜅²)∈(0,1).
Then, from (3.20) and (3.21), we obtain ‖‖𝑥_𝑛+1−𝑢‖‖=≤𝜏1_𝜆𝑛−‖‖𝑥_𝑛1−−𝑢𝜏‖‖_𝜆𝑛+𝜆_𝑛‖𝐹𝑢‖≤max‖‖𝑥_𝑛‖‖𝑥_𝑛−𝑢‖‖+1−𝜏_𝜆𝑛1−𝜆^𝑛𝜏_𝜆𝑛‖𝐹𝑢‖−𝑢‖‖,𝜆^𝑛‖𝐹𝑢‖1−𝜏_𝜆𝑛.(3.22) Since lim_𝑛→∞(𝜆_𝑛/(1−𝜏_𝜆𝑛))=1/𝜂, we have by induction ‖‖𝑥_𝑛+1−𝑢‖‖≤max‖‖𝑥₀−𝑢‖‖,𝑀₁‖𝐹𝑢‖,(3.23) where 𝑀₁=sup_𝑛{𝜆_𝑛/(1−𝜏_𝜆𝑛)}<∞. Hence, {𝑥_𝑛} is bounded. We also obtain that {𝑦_𝑛},{𝑇(𝜏)𝑦_𝑛}, and {𝐹𝑥_𝑛} are all bounded.
Define 𝑥_𝑛+1=(1−𝛼_𝑛)𝑥_𝑛+𝛼_𝑛𝑧_𝑛 for all 𝑛. Observe that ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖=‖‖‖‖𝑥^𝑛+2−1−𝛼_𝑛+1𝑥_𝑛+1𝛼_𝑛+1−𝑥^𝑛+1−1−𝛼_𝑛𝑥_𝑛𝛼_𝑛=‖‖‖‖1−𝛼_𝑛+1‖‖‖‖𝑦_𝑛+1+𝛼_𝑛+11/𝑡_𝑛+1∫^𝑡𝑛+1₀𝑇(𝜏)𝑦_𝑛+1𝑑𝜏−1−𝛼_𝑛+1𝑥_𝑛+1𝛼_𝑛+1−1−𝛼^𝑛𝑦_𝑛+𝛼_𝑛1/𝑡_𝑛∫^𝑡𝑛₀𝑇(𝜏)𝑦_𝑛𝑑𝜏−1−𝛼_𝑛𝑥_𝑛𝛼_𝑛=‖‖‖‖𝛼_𝑛+1‖‖‖‖1/𝑡_𝑛+1∫^𝑡𝑛+1₀𝑇(𝜏)𝑦_𝑛+1𝑑𝜏−1−𝛼_𝑛+1𝜆_𝑛+1𝐹𝑥_𝑛+1𝛼_𝑛+1−𝛼^𝑛1/𝑡_𝑛∫^𝑡𝑛₀𝑇(𝜏)𝑦_𝑛𝑑𝜏−1−𝛼_𝑛𝜆_𝑛𝐹𝑥_𝑛𝛼_𝑛≤1−𝛼_𝑛+1‖‖‖‖𝛼_𝑛+1‖‖𝜆_𝑛+1𝐹𝑥_𝑛+1‖‖+1−𝛼^𝑛𝛼_𝑛+‖‖‖1‖‖𝜆_𝑛𝐹𝑥_𝑛‖‖𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝜏)𝑦_𝑛+1𝑑𝜏−1𝑡_𝑛^𝑡𝑛₀𝑇(𝜏)𝑦_𝑛𝑑𝜏‖‖‖.(3.24) Next, we estimate ‖(1/𝑡_𝑛+1)∫^𝑡𝑛+1₀𝑇(𝜏)𝑦_𝑛+1𝑑𝜏−(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝜏)𝑦_𝑛𝑑𝜏‖. As a matter of fact, we have ‖‖‖1𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝜏)𝑦_𝑛+1𝑑𝜏−1𝑡_𝑛^𝑡𝑛₀≤‖‖‖1𝑇(𝜏)𝑦_𝑛𝑑𝜏‖‖‖𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝜏)𝑦_𝑛+1𝑑𝜏−1𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝜏)𝑦_𝑛𝑑𝜏‖‖‖+‖‖‖1𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝜏)𝑦_𝑛𝑑𝜏−1𝑡_𝑛^𝑡𝑛₀≤‖‖𝑦𝑇(𝜏)𝑦_𝑛𝑑𝜏_𝑛+1‖‖‖−𝑦_𝑛‖‖+‖‖‖1𝑡_𝑛+1^𝑡𝑛+1₀𝑇(𝜏)𝑦_𝑛𝑑𝜏−1𝑡_𝑛^𝑡𝑛₀≤‖‖𝑦𝑇(𝜏)𝑦_𝑛𝑑𝜏_𝑛+1‖‖‖−𝑦_𝑛‖‖+||||1𝑡_𝑛+1−1𝑡_𝑛||||‖‖‖^𝑡_𝑛0𝑇(𝜏)𝑦_𝑛𝑑𝜏‖‖‖+1𝑡_𝑛+1‖‖‖^𝑡𝑛+1_𝑡𝑛≤‖‖𝑥𝑇(𝜏)𝑦_𝑛𝑑𝜏_𝑛+1‖‖‖−𝜆_𝑛+1𝐹𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛𝐹𝑥_𝑛‖‖+||||𝑡^𝑛𝑡_𝑛+1−1||||𝑀₂≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝜆_𝑛+1𝐹𝑥_𝑛+1‖‖+‖‖𝜆_𝑛𝐹𝑥_𝑛‖‖+𝑀₂||||𝑡_𝑛𝑡_𝑛+1−1||||,(3.25) where 𝑀₂=sup_𝑛{2‖𝑇(𝜏)𝑦_𝑛‖}<∞. From (3.24) and (3.25), we have ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤1−𝛾𝛾‖‖𝜆^𝑛+1𝐹𝑥_𝑛+1‖‖+1−𝛾𝛾‖‖𝜆^𝑛+‖‖𝜆_𝑛𝐹𝑥_𝑛‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝜆_𝑛+1𝐹𝑥_𝑛+1‖‖𝐹𝑥_𝑛‖‖+𝑀₂||||𝑡_𝑛𝑡_𝑛+1≤1−1||||𝛾‖‖𝜆^𝑛+1𝐹𝑥_𝑛+1‖‖+1𝛾‖‖𝜆^𝑛𝐹𝑥_𝑛‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+𝑀₂||||𝑡_𝑛𝑡_𝑛+1−1||||.(3.26) Namely, ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤1𝛾‖‖𝜆^𝑛+1𝐹𝑥_𝑛+1‖‖+1𝛾‖‖𝜆^𝑛𝐹𝑥_𝑛‖‖+𝑀₂||||𝑡_𝑛𝑡_𝑛+1−1||||.(3.27) Since 𝜆_𝑛𝐹(𝑥_𝑛)→0 and (𝑡_𝑛/𝑡_𝑛+1)−1→0, we get limsup_𝑛→∞‖‖𝑧_𝑛+1−𝑧_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(3.28) Consequently, by Lemma 2.3, we deduce lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0. Therefore, lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞𝛼_𝑛‖‖𝑧_𝑛−𝑥_𝑛‖‖=0.(3.29)
Next, we claim that lim_𝑛→∞‖𝑥_𝑛−𝑇(𝜏)𝑥_𝑛‖=0. Observe that ‖‖𝑇(𝜏)𝑥_𝑛−𝑥_𝑛‖‖≤‖‖‖𝑇(𝜏)𝑥_𝑛−𝑇(𝜏)1𝑡_𝑛^𝑡𝑛₀𝑇(𝜏)𝑥_𝑛𝑑𝜏‖‖‖+‖‖‖1𝑡_𝑛^𝑡𝑛₀𝑇(𝜏)𝑥_𝑛𝑑𝜏−𝑥_𝑛+‖‖‖𝑇(𝜏)1‖‖‖𝑡_𝑛^𝑡𝑛₀𝑇(𝜏)𝑥_𝑛𝑑𝜏−1𝑡_𝑛^𝑡𝑛₀≤2‖‖‖1𝑇(𝜏)𝑥_𝑛𝑑𝜏‖‖‖𝑡_𝑛^𝑡𝑛₀𝑇(𝜏)𝑥_𝑛𝑑𝜏−𝑥_𝑛+‖‖‖𝑇(𝜏)1‖‖‖𝑡_𝑛^𝑡𝑛₀𝑇(𝜏)𝑥_𝑛𝑑𝜏−1𝑡_𝑛^𝑡𝑛₀𝑇(𝜏)𝑥_𝑛𝑑𝜏‖‖‖.(3.30) Note that ‖‖‖𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝜏)𝑥_𝑛𝑑𝜏‖‖‖≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖‖𝑥_𝑛+1−1𝑡_𝑛^𝑡𝑛₀≤‖‖𝑥𝑇(𝜏)𝑥_𝑛𝑑𝜏_𝑛‖‖‖−𝑥_𝑛+1‖‖+1−𝛼_𝑛‖‖‖𝑦_𝑛−1𝑡_𝑛^𝑡𝑛₀+𝛼_𝑛𝑇(𝜏)𝑥_𝑛𝑑𝜏‖‖‖‖‖‖1𝑡_𝑛^𝑡𝑛₀𝑇(𝜏)𝑥_𝑛𝑑𝜏−1𝑡_𝑛^𝑡𝑛₀≤‖‖𝑥𝑇(𝜏)𝑦_𝑛𝑑𝜏_𝑛‖‖‖+1−−𝑥𝛼_𝑛+1𝑛‖‖+1−𝛼_𝑛‖‖𝑦_𝑛−𝑥_𝑛‖‖≤‖‖𝑥_𝑛‖‖‖𝑥_𝑛−𝑡1_𝑛^𝑡𝑛₀𝑇(𝜏)𝑥_𝑛𝑑𝜏‖‖‖+𝛼_𝑛‖‖𝑦_𝑛−𝑥_𝑛‖‖+1−−𝑥𝛼_𝑛+1𝑛‖‖+‖‖𝑦_𝑛−𝑥_𝑛‖‖‖‖‖𝑥_𝑛−𝑡1_𝑛^𝑡𝑛₀𝑇(𝜏)𝑥_𝑛𝑑𝜏‖‖‖.(3.31) It follows that ‖‖‖𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝜏)𝑥_𝑛𝑑𝜏‖‖‖≤1𝛼_𝑛‖‖𝑥_𝑛=1−𝑥_𝑛+1‖‖+‖‖𝑦_𝑛−𝑥_𝑛‖‖𝛼_𝑛‖‖𝑥_𝑛⟶0.−𝑥_𝑛+1‖‖+‖‖𝜆_𝑛𝐹𝑥_𝑛‖‖(3.32) By Lemma 2.1, (3.30), and (3.32), we derive lim_𝑛→∞‖‖𝑇(𝜏)𝑥_𝑛−𝑥_𝑛‖‖=0.(3.33)
Next, we show that limsup_𝑛→∞⟨𝐹𝑥^∗,𝑥^∗−𝑥_𝑛⟩≤0, where 𝑥^∗=lim_𝑛→∞𝑥_𝑡𝑛 and 𝑥_𝑡𝑛 is defined by 𝑥_𝑡𝑛=(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝜏)[(𝐼−𝑡_𝑛𝐹)𝑥_𝑡𝑛]𝑑𝜏. Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} that converges weakly to 𝜔. It is clear that 𝑇(𝜏)𝑥_𝑛𝑘⇀𝜔. From Lemma 2.2, we have 𝜔∈Fix(𝑆). Hence, by Theorem 3.1, we have limsup_𝑛→∞⟨𝐹𝑥^∗,𝑥^∗−𝑥_𝑛⟩=lim_𝑘→∞𝐹𝑥^∗,𝑥^∗−𝑥_𝑛𝑘=⟨𝐹𝑥^∗,𝑥^∗−𝜔⟩≤0.(3.34)
Finally, we prove that {𝑥_𝑛} converges strongly to 𝑥^∗∈Fix(𝑆). From (3.19), we have ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1−𝛼_𝑛‖‖𝑦_𝑛−𝑥^∗‖‖²+𝛼_𝑛‖‖‖1𝑡_𝑛^𝑡𝑛₀𝑇(𝜏)𝑦_𝑛𝑑𝜏−𝑥^∗‖‖‖²≤1−𝛼_𝑛‖‖𝑦_𝑛−𝑥^∗‖‖²+𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀≤‖‖𝑦‖‖𝑇(𝜏)𝑦_𝑛−𝑇(𝜏)𝑥^∗‖‖²𝑑𝜏_𝑛−𝑥^∗‖‖²=‖‖𝑥_𝑛−𝜆_𝑛𝐹𝑥_𝑛−𝑥^∗‖‖²=‖‖𝐼−𝜆_𝑛𝐹𝑥_𝑛−𝐼−𝜆_𝑛𝐹𝑥^∗−𝜆_𝑛𝐹𝑥^∗‖‖²≤𝜏^2𝜆_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+𝜆^2𝑛‖‖𝐹𝑥^∗‖‖²≤𝜏+_𝜆𝑛2𝜆_𝑛𝐼−𝜆_𝑛𝐹𝑥^∗−𝐼−𝜆_𝑛𝐹𝑥_𝑛,𝐹𝑥^∗+2‖‖𝜆𝑥_𝑛𝑛−𝑥^∗‖‖²+𝜆^2𝑛‖‖𝐹𝑥^∗‖‖²+2𝜆_𝑛⟨𝑥^∗−𝑥_𝑛,𝐹𝑥^∗⟩⟨𝜆_𝑛𝐹𝑥_𝑛,𝐹𝑥^∗⟩−2𝜆^2𝑛‖𝐹𝑥^∗‖²≤+12−𝜆_𝑛1−𝜏_𝜆𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝜆_𝑛⟨𝑥^∗−𝑥_𝑛,𝐹𝑥^∗⟩‖‖𝜆_𝑛𝐹𝑥_𝑛‖‖‖𝐹𝑥^∗‖−𝜆^2𝑛‖𝐹𝑥^∗‖²≤1−1−𝜏_𝜆𝑛‖‖𝑥_𝑛−𝑥^∗‖‖²+1−𝜏_𝜆𝑛2𝜆^𝑛1−𝜏_𝜆𝑛⟨𝑥^∗−𝑥_𝑛,𝐹𝑥^∗⟩+2𝜆^𝑛‖𝐹𝑥^∗‖1−𝜏_𝜆𝑛=1−𝛿_𝑛‖‖𝜆_𝑛𝐹𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑥^∗‖‖²+𝛿_𝑛𝜎_𝑛,(3.35) where 𝛿_𝑛=1−𝜏_𝜆𝑛 and 𝜎_𝑛=(2𝜆_𝑛/1−𝜏_𝜆𝑛)⟨𝑥^∗−𝑥_𝑛,𝐹𝑥^∗⟩+(2𝜆_𝑛‖𝐹𝑥^∗‖/(1−𝜏_𝜆𝑛))‖𝜆_𝑛𝐹𝑥_𝑛‖. Obviously, we can see that ∑^∞𝑛=1𝛿_𝑛=∞ and limsup_𝑛→∞𝜎_𝑛≤0. Hence, all conditions of Lemma 2.5 are satisfied. Therefore, we immediately deduce that the sequence {𝑥_𝑛} converges strongly to 𝑥^∗∈Fix(𝑆).
Observe that ‖‖𝑦_𝑛−𝑥^∗‖‖≤‖‖𝑦_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑥^∗‖‖≤‖‖𝜆_𝑛𝐹𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑥^∗‖‖⟶0(𝑛⟶∞).(3.36)
Consequently, it is clear that {𝑦_𝑛} converges strongly to 𝑥^∗∈Fix(𝑆). From 𝑥^∗=lim_𝑡→0𝑥_𝑡 and Theorem 3.1, we get that 𝑥^∗ is the unique solution of the variational inequality ⟨𝐹𝑥^∗,𝑥^∗−𝑢⟩≤0,∀𝑢∈Fix(𝑆).(3.37) This completes the proof.

Acknowledgments

Y. Yao was supported in part by NSFC 11071279 and NSFC 71161001-G0105. Y.-C. Liou was partially supported by NSC 100-2221-E-230-012. R. Chen was supported in part by NSFC 11071279.