Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

Abstract

Let 𝐻 be a real Hilbert space. Consider on 𝐻 a nonexpansive semigroup 𝑆={𝑇(𝑠)∶0≤𝑠<∞} with a common fixed point, a contraction 𝑓 with the coefficient 0<𝛼<1, and a strongly positive linear bounded self-adjoint operator 𝐴 with the coefficient 𝛾>  0. Let 0<𝛾<𝛾/𝛼. It is proved that the sequence {𝑥_𝑛} generated by the iterative method 𝑥₀∈𝐻,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓(𝑥_𝑛)+𝛽_𝑛𝑥_𝑛+((1−𝛽_𝑛)𝐼−𝛼_𝑛𝐴)(1/𝑠_𝑛)∫^𝑠𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,𝑛≥0 converges strongly to a common fixed point 𝑥^∗∈𝐹(𝑆), where 𝐹(𝑆) denotes the common fixed point of the nonexpansive semigroup. The point 𝑥^∗ solves the variational inequality ⟨(𝛾𝑓−𝐴)𝑥^∗,𝑥−𝑥^∗⟩≤0 for all 𝑥∈𝐹(𝑆).

1. Introduction and Preliminaries

Let 𝐻 be a real Hilbert space and 𝑇 be a nonlinear mapping with the domain 𝐷(𝑇). A point 𝑥∈𝐷(𝑇) is a fixed point of 𝑇 provided 𝑇𝑥=𝑥. Denote by 𝐹(𝑇) the set of fixed points of 𝑇; that is, 𝐹(𝑇)={𝑥∈𝐷(𝑇)∶𝑇𝑥=𝑥}. Recall that 𝑇 is said to be nonexpansive if‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐷(𝐴).(1.1)

Recall that a family 𝑆={𝑇(𝑠)∣𝑠≥0} of mappings from 𝐻 into itself is called a one-parameter nonexpansive semigroup if it satisfies the following conditions:(i)𝑇(0)𝑥=𝑥, forall𝑥∈𝐻; (ii)𝑇(𝑠+𝑡)𝑥=𝑇(𝑠)𝑇(𝑡)𝑥, forall𝑠,𝑡≥0 and forall𝑥∈𝐻;(iii)‖𝑇(𝑠)𝑥−𝑇(𝑠)𝑦‖≤‖𝑥−𝑦‖, forall𝑠≥0 and forall𝑥,𝑦∈𝐻;(iv)for all 𝑥∈𝐶, 𝑠↦𝑇(𝑠)𝑥 is continuous.

We denote by 𝐹(𝑆) the set of common fixed points of 𝑆, that is, 𝐹(𝑆)=⋂_0≤𝑠<∞𝐹(𝑇(𝑠)). It is known that 𝐹(𝑆) is closed and convex; see [1]. Let 𝐶 be a nonempty closed and convex subset of 𝐻. One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping; see [2, 3]. More precisely, take 𝑡∈(0,1) and define a contraction 𝑇_𝑡∶𝐶→𝐶 by 𝑇_𝑡𝑥=𝑡𝑢+(1−𝑡)𝑇𝑥,𝑥∈𝐶,(1.2) where 𝑢∈𝐶 is a fixed point. Banach’s contraction mapping principle guarantees that 𝑇_𝑡 has a unique fixed point 𝑥_𝑡 in 𝐶. If 𝑇 enjoys a nonempty fixed point set, Browder [2] proved the following well-known strong convergence theorem.

Theorem B. Let 𝐶 be a bounded closed convex subset of a Hilbert space 𝐻 and let 𝑇 be a nonexpansive mapping on 𝐶. Fix 𝑢∈𝐶 and define 𝑧_𝑡∈𝐶𝑎𝑠𝑧_𝑡=𝑡𝑢+(1−𝑡)𝑇𝑧_𝑡 for 𝑡∈(0,1). Then as 𝑡→0, {𝑧_𝑡} converges strongly to a element of 𝐹(𝑇) nearest to 𝑢.

As motivated by Theorem B, Halpern [4] considered the following explicit iteration: 𝑥₀∈𝐶,𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0,(1.3) and proved the following theorem.

Theorem H. Let 𝐶 be a bounded closed convex subset of a Hilbert space 𝐻 and let 𝑇 be a nonexpansive mapping on 𝐶. Define a real sequence {𝛼_𝑛} in [0,1] by 𝛼_𝑛=𝑛^−𝜃,0<𝜃<1. Define a sequence {𝑥_𝑛} by (1.3). Then {𝑥_𝑛} converges strongly to the element of 𝐹(𝑇) nearest to 𝑢.

In 1977, Lions [5] improved the result of Halpern [4], still in Hilbert spaces, by proving the strong convergence of {𝑥_𝑛} to a fixed point of 𝑇 where the real sequence {𝛼_𝑛} satisfies the following conditions:(C1)lim_𝑛→∞𝛼_𝑛=0; (C2)∑^∞𝑛=1𝛼_n=∞; (C3)lim_𝑛→∞(𝛼_𝑛+1−𝛼_𝑛)/𝛼^2𝑛+1=0.

It was observed that both Halpern’s and Lions’s conditions on the real sequence {𝛼_𝑛} excluded the canonical choice 𝛼_𝑛=1/(𝑛+1). This was overcome in 1992 by Wittmann [6], who proved, still in Hilbert spaces, the strong convergence of {𝑥_𝑛} to a fixed point of 𝑇 if {𝛼_𝑛} satisfies the following conditions:(C1)lim_𝑛→∞𝛼_𝑛=0; (C2)∑^∞𝑛=1𝛼_𝑛=∞; (C3)∑^∞𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞.

Recall that a mapping 𝑓∶𝐻→𝐻 is an 𝛼-contraction if there exists a constant 𝛼∈(0,1) such that‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐻.(1.4)

Recall that an operator 𝐴 is strongly positive on 𝐻 if there exists a constant 𝛾>0 such that⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻.(1.5)

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [7–13] and the references therein. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping 𝑇 on a real Hilbert space 𝐻: min_{𝑥∈𝐹(𝑇)}12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩,(1.6) where 𝐴 is a linear bounded operator on 𝐻 and 𝑏 is a given point in 𝐻. In [11], it is proved that the sequence {𝑥_𝑛} defined by the iterative method below, with the initial guess 𝑥₀∈H chosen arbitrarily, 𝑥_𝑛+1=𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛼_𝑛𝑏,𝑛≥0,(1.7) strongly converges to the unique solution of the minimization problem (1.6) provided that the sequence {𝛼_𝑛} satisfies certain conditions.

Recently, Marino and Xu [9] studied the following continuous scheme: 𝑥_𝑡=𝑡𝛾𝑓𝑥_𝑡+(𝐼−𝑡𝐴)𝑇𝑥_𝑡,(1.8) where 𝑓 is an 𝛼-contraction on a real Hilbert space 𝐻, 𝐴 is a bounded linear strongly positive operator and 𝛾>0 is a constant. They showed that {𝑥_𝑡} strongly converges to a fixed point 𝑥 of 𝑇. Also in [9], they introduced a general explicit iterative scheme by the viscosity approximation method:𝑥_𝑛∈𝐻,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛,𝑛≥0(1.9) and proved that the sequence {𝑥_𝑛} generated by (1.9) converges strongly to a unique solution of the variational inequality⟨(𝐴−𝛾𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐹(𝑇),(1.10)

which is the optimality condition for the minimization problem min_{𝑥∈𝐹(𝑇)}12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥),(1.11)

where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^′(𝑥)=𝛾𝑓(𝑥) for 𝑥∈𝐻).

In this paper, motivated by Li et al. [8], Marino and Xu [9], Plubtieng and Punpaeng [14], Shioji and Takahashi [15], and Shimizu and Takahashi [16], we consider the mapping 𝑇_𝑡 defined as follows: 𝑇_𝑡𝑥=𝑡𝛾𝑓(𝑥)+(𝐼−𝑡𝐴)1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥𝑑𝑠,(1.12) where 𝛾>0 is a constant, 𝑓 is an 𝛼-contraction, 𝐴 is a bounded linear strongly positive self-adjoint operator and {𝜆_𝑡} is a positive real divergent net. If 𝛾𝛼<𝛾 for each 0<𝑡<‖𝐴‖⁻¹, one can see that 𝑇_𝑡 is a (1−𝑡(𝛾−𝛾𝛼))-contraction. So, by Banach’s contraction mapping principle, there exists an unique solution 𝑥_𝑡 of the fixed point equation𝑥_𝑡=𝑡𝛾𝑓𝑥_𝑡+(𝐼−𝑡𝐴)1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠.(1.13) We show that the sequence {𝑥_𝑡} generated by above continuous scheme strongly converges to a common fixed point 𝑥^∗∈𝐹(𝑆), which is the unique point in 𝐹(𝑆) solving the variational inequality ⟨(𝛾𝑓−𝐴)𝑥^∗,𝑥−𝑥^∗⟩≤0 for all 𝑥∈𝐹(𝑆). Furthermore, we also study the following explicit iterative scheme: 𝑥₀∈𝐻,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴1𝑠_𝑛^𝑠𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,𝑛≥0.(1.14) We prove that the sequence {𝑥_𝑛} generated by (1.14) converges strongly to the same 𝑥^∗.

The results presented in this paper improve and extend the corresponding results announced by Marino and Xu [9], Plubtieng and Punpaeng [14], Shioji and Takahashi [15], and Shimizu and Takahashi [16].

In order to prove our main result, we need the following lemmas.

Lemma 1.1 (see [16]). Let 𝐷 be a nonempty bounded closed convex subset of a Hilbert space 𝐻 and let 𝑆={𝑇(𝑡)∶0≤𝑡<∞} be a nonexpansive semigroup on 𝐷. Then, for any 0≤ℎ<∞, lim_𝑡→∞sup_𝑥∈𝐷‖‖‖1𝑡^𝑡0𝑇(𝑠)𝑥𝑑𝑠−𝑇(ℎ)1𝑡^𝑡0𝑇(𝑠)𝑥𝑑𝑠‖‖‖=0.(1.15)

Lemma 1.2 (see [17]). Let 𝐻 be a Hilbert space, 𝐶 a closed convex subset of 𝐻, and 𝑇∶𝐶→𝐶 a nonexpansive mapping with 𝐹(𝑇)≠∅. Then 𝐼−𝑇 is demiclosed, that is, if {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} strongly converges to 𝑦, then (𝐼−𝑇)𝑥=𝑦.

Lemma 1.3 (see [18]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝑃_𝐶 be the metric projection from 𝐻 onto 𝐶( i.e., for 𝑥∈𝐻,𝑃_𝐶𝑥 is the only point in 𝐶 such that ‖𝑥−𝑃_𝐶𝑥‖=inf{‖𝑥−𝑧‖∶𝑧∈𝐶}). Given 𝑥∈𝐻 and 𝑧∈𝐶. Then 𝑧=𝑃_𝐶𝑥 if and only if there holds the relations ⟨𝑥−𝑧,𝑦−𝑧⟩≤0,∀𝑦∈𝐶.(1.16)

Lemma 1.4. Let 𝐻 be a Hilbert space, 𝑓 a 𝛼-contraction, and 𝐴 a strongly positive linear bounded self-adjoint operator with the coefficient 𝛾>0. Then, for 0<𝛾<𝛾/𝛼, ⟨𝑥−𝑦,(𝐴−𝛾𝑓)𝑥−(𝐴−𝛾𝑓)𝑦⟩≥𝛾−𝛾𝛼‖𝑥−𝑦‖²,𝑥,𝑦∈𝐻.(1.17) That is, 𝐴−𝛾𝑓 is strongly monotone with coefficient 𝛾−𝛼𝛾.

Proof. From the definition of strongly positive linear bounded operator, we have ⟨𝑥−𝑦,𝐴(𝑥−𝑦)⟩≥𝛾‖𝑥−𝑦‖².(1.18) On the other hand, it is easy to see ⟨𝑥−𝑦,𝛾𝑓𝑥−𝛾𝑓𝑦⟩≤𝛾𝛼‖𝑥−𝑦‖².(1.19) Therefore, we have ⟨𝑥−𝑦,(𝐴−𝛾𝑓)𝑥−(𝐴−𝛾𝑓)𝑦⟩=≥⟨𝑥−𝑦,𝐴(𝑥−𝑦)⟩−⟨𝑥−𝑦,𝛾𝑓𝑥−𝛾𝑓𝑦⟩𝛾−𝛾𝛼‖𝑥−𝑦‖²(1.20) for all 𝑥,𝑦∈𝐻. This completes the proof.

Remark 1.5. Taking 𝛾=1 and 𝐴=𝐼, the identity mapping, we have the following inequality: ⟨𝑥−𝑦,(𝐼−𝑓)𝑥−(𝐼−𝑓)𝑦⟩≥(1−𝛼)‖𝑥−𝑦‖²,𝑥,𝑦∈𝐻.(1.21) Furthermore, if 𝑓 is a nonexpansive mapping in Remark 1.5, we have ⟨𝑥−𝑦,(𝐼−𝑓)𝑥−(𝐼−𝑓)𝑦⟩≥0,𝑥,𝑦∈𝐻.(1.22)

Lemma 1.6 (see [9]). Assume 𝐴 is a strongly positive linear bounded self-adjoint operator on a Hilbert space 𝐻 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹. Then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

Lemma 1.7 (see [12]). Let {𝛼_𝑛} be a sequence of nonnegative real numbers satisfying the following condition: 𝛼_𝑛+1≤1−𝛾_𝑛𝛼_𝑛+𝛾_𝑛𝜎_𝑛,∀𝑛≥0,(1.23) where {𝛾_𝑛} is a sequence in (0,1) and {𝜎_𝑛} is a sequence of real numbers such that(i)lim_𝑛→∞𝛾_𝑛=0 and ∑^∞𝑛=0𝛾_𝑛=∞,(ii)either limsup_𝑛→∞𝜎_𝑛≤0 or ∑^∞𝑛=0|𝛾_𝑛𝜎_𝑛|<∞. Then {𝛼_𝑛}^∞𝑛=0 converges to zero.

2. Main Results

Lemma 2.1. Let 𝐻 a real Hilbert space and 𝑆={𝑇(𝑠)∶0≤𝑠<∞} a nonexpansive semigroup on 𝐻 such that 𝐹(𝑆)≠∅. Let {𝜆_𝑡}_0<𝑡<1 be a continuous net of positive real numbers such that lim_𝑡→0𝜆_𝑡=∞. Let 𝑓∶𝐻→𝐻 be an 𝛼-contraction, 𝐴 a strongly positive linear bounded self-adjoint operator of 𝐻 into itself with coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Let {𝑥_𝑡} be a sequence defined by (1.13). Then (i){𝑥_𝑡} is bounded for all 𝑡∈(0,‖𝐴‖⁻¹);(ii)lim_𝑡→0‖𝑇(𝜏)𝑥_𝑡−𝑥_𝑡‖=0 for all 0≤𝜏<∞;(iii)𝑥_𝑡 defines a continuous curve from (0,‖𝐴‖⁻¹) into 𝐻.

Proof. (i) Taking 𝑝∈𝐹(𝑆), we have ‖‖𝑥_𝑡−𝑝‖‖≤‖‖‖𝑡𝛾𝑓𝑥_𝑡+(𝐼−𝑡𝐴)1𝜆_𝑡^𝜆𝑡₀≤𝑡‖‖𝛾𝑓𝑥_𝑡𝑇(𝑠)𝑥_𝑡𝑑𝑠−𝑝‖‖‖≤𝑡‖‖𝛾𝑓𝑥_𝑡−𝐴𝑝‖‖+1−𝑡𝛾𝜆1_𝑡^𝜆𝑡₀‖‖𝑇(𝑠)𝑥_𝑡−𝑝‖‖𝑑𝑠≤𝑡𝛾‖‖𝑓𝑥_𝑡−𝐴𝑝‖‖+1−𝑡𝛾‖‖𝑥_𝑡−𝑝‖‖≤1−𝑡−𝑓(𝑝)‖‖+𝑡‖𝛾𝑓(𝑝)−𝐴𝑝‖+1−𝑡𝛾‖‖𝑥_𝑡−𝑝‖‖𝛾−𝛾𝛼‖‖𝑥_𝑡−𝑝‖‖+𝑡‖𝛾𝑓(𝑝)−𝐴𝑝‖.(2.1) It follows that ‖‖𝑥_𝑡−𝑝‖‖≤1𝛾−𝛼𝛾‖𝛾𝑓(𝑝)−𝐴𝑝‖.(2.2) This implies that {𝑥_𝑡} is not only bounded, but also that {𝑥_𝑡} is contained in 𝐵(𝑝,1/(𝛾−𝛾𝛼)‖𝛾𝑓(𝑝)−𝐴𝑝‖) of center 𝑝 and radius 1/(𝛾−𝛾𝛼)‖𝛾𝑓(𝑝)−𝐴𝑝‖, for all fixed 𝑝∈𝐹(𝑆). Moreover for 𝑝∈𝐹(𝑆) and 𝑡∈(0,‖𝐴‖⁻¹), ‖‖‖1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠−𝑝‖‖‖=‖‖‖1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡≤‖‖𝑥−𝑇(𝑠)𝑝𝑑𝑠_𝑡‖‖‖≤1−𝑝‖‖𝛾−𝛾𝛼‖𝛾𝑓(𝑝)−𝐴𝑝‖.(2.3)
(ii) Observe that ‖‖𝑇(𝜏)𝑥_𝑡−𝑥_𝑡‖‖≤‖‖‖𝑇(𝜏)𝑥_𝑡−𝑇(𝜏)1𝜆_𝑡^𝜆𝑡₀+‖‖‖𝑇(𝜏)1𝑇(𝑠)𝑥_𝑡𝑑𝑠‖‖‖𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠−1𝜆_𝑡^𝜆𝑡₀+‖‖‖1𝑇(𝑠)𝑥_𝑡𝑑𝑠‖‖‖𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠−𝑥_𝑡≤2‖‖‖𝑥_𝑡‖‖‖−1𝜆_𝑡^𝜆𝑡₀+‖‖‖𝑇(𝜏)1𝑇(𝑠)𝑥_𝑡𝑑𝑠‖‖‖𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠−1𝜆_𝑡^𝜆𝑡₀=2𝑡‖‖‖𝛾𝑓𝑥_𝑡𝑇(𝑠)𝑥_𝑡𝑑𝑠‖‖‖+‖‖‖𝑇(𝜏)1−𝐴𝜆1_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠‖‖‖𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠−1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠‖‖‖.(2.4) Taking 𝐵(𝑝,1/(𝛾−𝛾𝛼)‖𝛾𝑓(𝑝)−𝐴𝑝‖) as 𝐷 in Lemma 1.1 and passing to lim_𝑡→0 in (2.4), we can obtain (ii) immediately.
(iii) Taking 𝑡₁,𝑡₂∈(0,‖𝐴‖⁻¹) and fixing 𝑝∈𝐹(𝑆), we see that ‖‖𝑥_𝑡1≤−𝑥‖‖‖_𝑡2‖‖𝑡₁−𝑡₂𝛾𝑓𝑥_𝑡1+𝑡₂𝛾𝑓𝑥_𝑡1−𝑓𝑥_𝑡2−𝑡₁−𝑡₂𝐴1𝜆_𝑡1^𝜆𝑡1₀+𝐼−𝑡₂𝑇(𝑠)𝑥_𝑡1𝑑𝑠𝐴1𝜆_𝑡1^𝜆𝑡1₀𝑇(𝑠)𝑥_𝑡1𝑑𝑠−1𝜆_𝑡2^𝜆𝑡2₀≤||𝑡𝑇(𝑠)𝑥_𝑡2𝑑𝑠₁‖‖‖‖−𝑡₂||𝛾‖‖𝑓𝑥_𝑡1‖‖+𝑡₂𝛾𝛼‖‖𝑥_𝑡1−𝑥_𝑡2‖‖+||𝑡₁−𝑡₂||‖𝐴‖‖‖‖1𝜆_𝑡1^𝜆𝑡1₀+1−𝑡₂𝑇(𝑠)𝑥_𝑡1𝑑𝑠‖‖‖𝛾‖‖‖‖1𝜆_𝑡1^𝜆𝑡1₀𝑇(𝑠)𝑥_𝑡1𝑑𝑠−1𝜆_𝑡2^𝜆𝑡1₀𝑇(𝑠)𝑥_𝑡2𝑑𝑠−1𝜆_𝑡2^𝜆𝑡2_𝜆𝑡1≤||𝑡𝑇(𝑠)𝑥_𝑡2𝑑𝑠₁‖‖‖‖−t₂||𝛾‖‖𝑓𝑥_𝑡1‖‖+𝑡₂𝛾𝛼‖‖𝑥_𝑡1−𝑥_𝑡2‖‖+||𝑡₁−𝑡₂||‖𝐴‖‖‖‖1𝜆_𝑡1^𝜆𝑡1₀+1−𝑡₂𝑇(𝑠)𝑥_𝑡1𝑑𝑠‖‖‖𝛾‖‖𝑥_𝑡1−𝑥_𝑡2‖‖+||||1𝜆_𝑡1−1𝜆_𝑡2||||‖‖‖^𝜆_𝑡10𝑇(𝑠)𝑥_𝑡2𝑑𝑠‖‖‖+1𝜆_𝑡2‖‖‖‖^𝜆𝑡2_𝜆𝑡1𝑇(𝑠)𝑥_𝑡2𝑑𝑠‖‖‖‖.(2.5) Thus applying (2.3), we arrive at ‖‖𝑥_𝑡1≤−𝑥||𝑡_𝑡21‖‖−𝑡₂||𝛾‖‖𝑓𝑥_𝑡1‖‖+𝑡₂𝛾𝛼‖‖𝑥_𝑡1−𝑥_𝑡2‖‖+||𝑡₁−𝑡₂||‖𝐴‖1+1−𝑡₂𝛾−𝛾𝛼‖𝛾𝑓(𝑝)−𝐴𝑝‖+‖𝑝‖𝛾‖‖𝑥_𝑡1−𝑥_𝑡2‖‖+2𝜆_𝑡2||𝜆_𝑡2−𝜆_𝑡1||1≤||𝑡𝛾−𝛾𝛼‖𝛾𝑓(𝑝)−𝐴𝑝‖+‖𝑝‖₁−𝑡₂||𝛾‖‖𝑓𝑥_𝑡1‖‖+‖𝐴‖1+1−𝑡₂𝛾−𝛾𝛼‖𝛾𝑓(𝑝)−𝐴𝑝‖+‖𝑝‖𝛾−𝛾𝛼‖‖𝑥_𝑡1−𝑥_𝑡2‖‖+𝜆2_𝑡2||𝜆_𝑡2−𝜆_𝑡1||1𝛾−𝛾𝛼‖𝛾𝑓(𝑝)−𝐴𝑝‖+‖𝑝‖.(2.6) It follows that ‖‖𝑥_𝑡1−𝑥_𝑡2‖‖≤𝑀₁||𝑡₁−𝑡₂||+𝑀₂||𝜆_𝑡2−𝜆_𝑡1||,(2.7) where 𝑀₁=𝛾𝛾−𝛾𝛼‖‖𝑓𝑥_𝑡1‖‖+‖𝐴‖‖𝛾𝑓(𝑝)−𝐴𝑝‖+𝛾−𝛾𝛼‖𝐴‖‖𝑝‖𝑡₂𝛾−𝛾𝛼²(2.8) and 𝑀₂=2‖𝛾𝑓(𝑝)−𝐴𝑝‖+𝛾−𝛾𝛼‖𝑝‖𝜆_𝑡2𝑡₂𝛾−𝛾𝛼².(2.9) This inequality, together with the continuity of the net {𝜆_𝑡}, gives the continuity of the curve {𝑥_𝑡}.

Theorem 2.2. Let 𝐻 be a real Hilbert space 𝐻 and 𝑆={𝑇(𝑠)∶0≤𝑠<∞} a nonexpansive semigroup such that 𝐹(𝑆)≠∅. Let {𝜆_𝑡}_0<𝑡<1 be a net of positive real numbers such that 𝑙𝑖𝑚_𝑡→0𝜆_𝑡=∞. Let 𝑓 be an 𝛼-contraction and let 𝐴 be a strongly positive linear bounded self-adjoint operator on 𝐻 with the coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Then sequence {𝑥_𝑡} defined by (1.13) strongly converges as 𝑡→0 to 𝑥^∗∈𝐹(𝑆), which solves the following variational inequality: ⟨(𝛾𝑓−𝐴)𝑥^∗,𝑝−𝑥^∗⟩≤0,∀𝑝∈𝐹(𝑆).(2.10) Equivalently, one has 𝑃_𝐹(𝑆)(𝐼−𝐴+𝛾𝑓)𝑥^∗=𝑥^∗.(2.11)

Proof. The uniqueness of the solution of the variational inequality (2.10) is a consequence of the strong monotonicity of 𝐴−𝛾𝑓 (Lemma 1.4) and it was proved in [9]. Next, we will use 𝑥^∗∈𝐹(𝑆) to denote the unique solution of (2.10). To prove that 𝑥_𝑡→𝑥^∗(𝑡→0), we write, for a given 𝑝∈𝐹(𝑆), 𝑥_𝑡−𝑝=𝑡𝛾𝑓𝑥_𝑡−𝐴𝑝+(𝐼−𝑡𝐴)1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠−𝑝.(2.12) Using 𝑥_𝑡−𝑝 to make inner product, we obtain that ‖‖𝑥_𝑡−𝑝‖‖²=(𝐼−𝑡𝐴)1𝜆_𝑡^𝜆𝑡₀≤1−𝑡𝑇(𝑠)𝑥_𝑡𝑑𝑠−𝑝,𝑥_𝑡−𝑝+𝑡𝛾𝑓𝑥_𝑡−𝐴𝑝,𝑥_𝑡−𝑝𝛾‖‖𝑥_𝑡−𝑝‖‖²+𝑡𝛾𝑓𝑥_𝑡−𝐴𝑝,𝑥_𝑡−𝑝.(2.13) It follows that ‖‖𝑥_𝑡−𝑝‖‖²≤1𝛾𝛾𝑓𝑥^𝑡≤𝛾𝛼−𝑓(𝑝),𝑥_t−𝑝+⟨𝛾𝑓(𝑝)−𝐴𝑝,𝑥_𝑡−𝑝⟩𝛾‖‖𝑥^𝑡−𝑝‖‖²+1𝛾⟨𝛾𝑓(𝑝)−𝐴𝑝,𝑥^𝑡−𝑝⟩,(2.14) which yields that ‖‖𝑥_𝑡−𝑝‖‖²≤1𝛾−𝛼𝛾⟨𝛾𝑓(𝑝)−𝐴𝑝,𝑥^𝑡−𝑝⟩.(2.15) Since 𝐻 is a Hilbert space and {𝑥_𝑡} is bounded as 𝑡→0, we have that if {𝑡_𝑛} is a sequence in (0,1) such that 𝑡_𝑛→0 and 𝑥_𝑡𝑛⇀𝑥. By (2.15), we see 𝑥_𝑡𝑛→𝑥. Moreover, by (ii) of Lemma 2.1 we have 𝑥∈𝐹(𝑆). We next prove that 𝑥 solves the variational inequality (2.10). From (1.13), we arrive at (𝐴−𝛾𝑓)𝑥_𝑡=−1𝑡(𝐼−𝑡𝐴)𝑥^𝑡−1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠.(2.16) For 𝑝∈𝐹(𝑆), it follows from (1.22) that ⟨(𝐴−𝛾𝑓)𝑥_𝑡,𝑥_𝑡−𝑝⟩=−1𝑡(𝐼−𝑡𝐴)𝑥^𝑡−1𝜆_𝑡^𝜆𝑡₀=−1𝑇(𝑠)𝑥_𝑡𝑑𝑠,𝑥_𝑡−𝑝𝑡𝜆1_𝑡^𝜆𝑡₀(𝐼−𝑇(𝑠))𝑥_𝑡+𝐴1−(𝐼−𝑇(𝑠))𝑝𝑑𝑠,𝑥_𝑡−𝑝𝜆_𝑡^𝜆𝑡₀=−1(𝐼−𝑇(𝑠))𝑥_𝑡𝑑𝑠,𝑥_𝑡−𝑝𝑡𝜆_𝑡^𝜆𝑡₀+𝐴1⟨(𝐼−𝑇(𝑠))𝑥_𝑡−(𝐼−𝑇(𝑠))𝑝,𝑥_𝑡−𝑝⟩𝑑𝑠𝜆_𝑡^𝜆𝑡₀≤𝐴1(𝐼−𝑇(𝑠))𝑥_𝑡𝑑𝑠,𝑥_𝑡−𝑝𝜆_𝑡^𝜆𝑡₀=𝐴𝑡𝛾𝑓(𝐼𝑥_𝑡−𝑇(𝑠))𝑥_𝑡𝑑𝑠,𝑥_𝑡−𝑝=𝑡𝐴𝛾𝑓𝑥_𝑡−𝑡𝐴𝜆1_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠,𝑥_𝑡−𝑝−𝐴𝜆1_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠,𝑥_𝑡−𝑝.(2.17) Passing to lim_𝑡→0, since {𝑥_𝑡} is a bounded sequence, we obtain (𝐴−𝛾𝑓)𝑥,𝑥−𝑝≤0,(2.18) that is, 𝑥 satisfies the variational inequality (2.10). By the uniqueness it follows 𝑥=𝑥^∗. In a summary, we have shown that each cluster point of {𝑥_𝑡} (as 𝑡→0) equals 𝑥^∗. Therefore, 𝑥_𝑡→𝑥^∗ as 𝑡→0. The variational inequality (2.10) can be rewritten as (𝐼−𝐴+𝛾𝑓)𝑥^∗−𝑥^∗,𝑥^∗−𝑝,𝑝∈𝐹(𝑆).(2.19) This, by Lemma 1.3, is equivalent to 𝑃_𝐹(𝑆)(𝐼−𝐴+𝛾𝑓)𝑥^∗=𝑥^∗.(2.20) This completes the proof.

Remark 2.3. Theorem 2.2 which include the corresponding results of Shioji and Takahashi [15] as a special case is reduced to Theorem 3.1 of Plubtieng and Punpaeng [14] when 𝐴=𝐼, the identity mapping and 𝛾=1.

Theorem 2.4. Let 𝐻 be a real Hilbert space 𝐻 and 𝑆={𝑇(𝑠)∶0≤𝑠<∞} a nonexpansive semigroup such that 𝐹(𝑆)≠∅. Let {𝑠_𝑛} be a positive real divergent sequence and let {𝛼_𝑛} and {𝛽_𝑛} be sequences in (0,1) satisfying the following conditions lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛽_𝑛=0 and ∑^∞𝑛=0𝛼_𝑛=∞. Let 𝑓 be an 𝛼-contraction and let 𝐴 be a strongly positive linear bounded self-adjoint operator with the coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Then sequence {𝑥_𝑛} defined by (1.14) strongly converges to 𝑥^∗∈𝐹(𝑆), which solves the variational inequality (2.10).

Proof. We divide the proof into three parts.Step 1. Show the sequence {𝑥_𝑛} is bounded.
Noticing that lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛽_𝑛=0, we may assume, with no loss of generality, that 𝛼_𝑛/(1−𝛽_𝑛)<‖𝐴‖⁻¹ for all 𝑛≥0. From Lemma 1.6, we know that ‖(1−𝛽_𝑛)𝐼−𝛼_𝑛𝐴‖≤(1−𝛽_𝑛−𝛼_𝑛𝛾). Picking 𝑝∈𝐹(𝑆), we have ‖‖𝑥=_𝑛+1−‖‖‖𝛼𝑝_𝑛‖‖≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑠1_𝑛^𝑠𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠−𝑝‖‖‖‖‖𝛾𝑓𝑥_𝑛−𝐴𝑝‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖+1−𝛽_𝑛−𝛼_𝑛𝛾‖‖‖1𝑠_𝑛^𝑠𝑛₀≤𝛼_𝑛𝑇(𝑠)𝑥_𝑛𝑑𝑠−𝑝‖‖‖≤1𝛾−‖‖𝑓𝛼_𝑛𝑥_𝑛−𝑓(𝑝)‖‖+𝛼_𝑛‖𝛾𝑓(𝑝)−𝐴𝑝‖+𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖+1−𝛽_𝑛−𝛼_𝑛𝛾‖‖𝑥_𝑛−𝑝‖‖𝛾−𝛾𝛼‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖𝛾𝑓(𝑝)−𝐴𝑝‖.(2.21) By simple inductions, we see that‖‖𝑥_𝑛−𝑝‖‖≤max‖‖𝑥₀−𝑝‖‖,‖𝐴𝑝−𝛾𝑓(𝑝)‖𝛾−𝛾𝛼,(2.22) which yields that the sequence {𝑥_𝑛} is bounded.Step 2. Show that limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑥^∗,𝑦_𝑛−𝑥^∗⟩≤0,(2.23) where 𝑥^∗ is obtained in Theorem 2.2 and 𝑦_𝑛=(1/𝑠_𝑛)∫^𝑠𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠.
Putting 𝑧₀=𝑃_𝐹(𝑆)𝑥₀, from (2.22) we see that the closed ball 𝑀 of center 𝑧₀ and radius max{‖𝑧₀−𝑝‖,‖𝐴𝑧₀−𝛾𝑓(𝑧₀)‖/(𝛾−𝛾𝛼)} is 𝑇(𝑠)-invariant for each 𝑠∈[0,∞) and contain {𝑥_𝑛}. Therefore, we assume, without loss of generality, 𝑆={𝑇(𝑠)∶0≤𝑠<∞} is a nonexpansive semigroup on 𝑀. It follows from Lemma 1.1 that lim_𝑛→∞‖‖𝑦_𝑛−𝑇(ℎ)𝑦_𝑛‖‖=0(2.24) for all 0≤ℎ<∞. Taking a suitable subsequence {𝑦_𝑛𝑖} of {𝑦_𝑛}, we see that limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑥^∗,𝑦_𝑛−𝑥^∗⟩=lim_𝑖→∞(𝛾𝑓−𝐴)𝑥^∗,𝑦_𝑛𝑖−𝑥^∗.(2.25) Since the sequence {𝑦_𝑛} is also bounded, we may assume that 𝑦_𝑛𝑖⇀𝑥. From the demiclosedness principle, we have 𝑥∈𝐹(𝑆). Therefore, we have limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑥^∗,𝑦_𝑛−𝑥^∗⟩=(𝛾𝑓−𝐴)𝑥^∗,𝑥−𝑥^∗≤0.(2.26) On the other hand, we have ‖‖𝑥_𝑛+1−𝑦_𝑛‖‖≤𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥_𝑛‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑦_𝑛‖‖.(2.27) From the assumption lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛽_𝑛=0, we see that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑦_𝑛‖‖=0,(2.28) which combines with (2.26) gives that limsup_𝑛→∞(𝛾𝑓−𝐴)𝑥^∗,𝑥_𝑛+1−𝑥^∗≤0.(2.29)Step 3. Show 𝑥_𝑛→𝑥^∗ as 𝑛→∞.
Note that‖‖𝑥_𝑛+1−𝑥^∗‖‖²==𝛼_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛾𝑓1−𝑥𝛽_𝑛_𝑛−𝐴𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝛼_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛽_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥^∗‖‖+‖‖1−𝛽_𝑛𝐼−𝛼_𝑛𝐴‖‖‖‖𝑦_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥^∗‖‖+𝛽𝛼_𝑛𝛾‖‖𝑥_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥^∗‖‖+𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝑥_𝑛+1−𝑥^∗=1−‖‖𝛼𝑥_𝑛𝑛−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥^∗‖‖+1−𝛽_𝑛−𝛼_𝑛𝛾‖‖𝑥_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥^∗‖‖≤1−𝛼_𝑛𝛾−𝛾𝛼‖‖𝑥_𝑛−𝑥^∗‖‖‖‖𝑥_𝑛+1−𝑥^∗‖‖+𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝑥_𝑛+1−𝑥^∗𝛾−𝛾𝛼2‖‖𝑥^𝑛≤1−𝛼_𝑛−𝑥^∗‖‖²+‖‖𝑥_𝑛+1−𝑥^∗‖‖²+𝛼^𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝑥_𝑛+1−𝑥^∗.𝛾−𝛾𝛼2‖‖𝑥^𝑛−𝑥^∗‖‖²+12‖‖𝑥^𝑛+1−𝑥^∗‖‖²+𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝑥_𝑛+1−𝑥^∗.(2.30) It follows that ‖‖𝑥_𝑛+1−𝑥^∗‖‖²≤1−𝛼_𝑛𝛾−𝛾𝛼‖‖𝑥_𝑛−𝑥^∗‖‖²+2𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝑥_𝑛+1−𝑥^∗.(2.31)By using Lemma 1.7, we can obtain the desired conclusion easily.

Remark 2.5. If 𝛾=1 and 𝐴=𝐼, the identity mapping, then Theorem 2.4 is reduced to Theorem 3.3 of Plubtieng and Punpaeng [14].
If the sequence {𝛽_𝑛}≡0, then Theorem 2.4 is reduced to the following.

Corollary 2.6. Let 𝐻 be a real Hilbert space 𝐻 and 𝑆={𝑇(𝑠)∶0≤𝑠<∞} a nonexpansive semigroup such that 𝐹(𝑆)≠∅. Let {𝑠_𝑛} be a positive real divergent sequence and let {𝛼_𝑛} be a sequence in (0,1) satisfying the following conditions lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=0𝛼_𝑛=∞. Let 𝑓 be a 𝛼-contraction and let 𝐴 be a strongly positive linear bounded self-adjoint operator with the coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Let {𝑥_𝑛} be a sequence generated by the following manner: 𝑥₀∈𝐻,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴1𝑠_𝑛^𝑠𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,𝑛≥0.(2.32) Then the sequence {𝑥_𝑛} defined by above iterative algorithm converges strongly to 𝑥^∗∈𝐹(𝑆), which solves the variational inequality (2.10).

Remark 2.7. Corollary 2.6 includes Theorem 2 of Shioji and Takahashi [15] as a special case.

Remark 2.8. Theorem 2.2 and Corollary 2.6 improve Theorem 3.2 and Theorem 3.4 of Marino and Xu [9] from a single nonexpansive mapping to a nonexpansive semigroup, respectively.

Acknowledgment

The present studies were supported by the National Natural Science Foundation of China (11071169), (11126334) and the Natural Science Foundation of Zhejiang Province (Y6110287).