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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 417234, 13 pages
http://dx.doi.org/10.1155/2012/417234
Research Article

Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

Department of Mathematics, Institute of Applied Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

Received 14 October 2011; Accepted 11 December 2011

Academic Editor: RudongΒ Chen

Copyright Β© 2012 Dan Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 {π‘₯𝑛} generated by the iterative method π‘₯0∈𝐻,π‘₯𝑛+1=𝛼𝑛𝛾𝑓(π‘₯𝑛)+𝛽𝑛π‘₯𝑛+((1βˆ’π›½π‘›)πΌβˆ’π›Όπ‘›π΄)(1/𝑠𝑛)βˆ«π‘ π‘›0𝑇(𝑠)π‘₯𝑛𝑑𝑠,𝑛β‰₯0 converges strongly to a common fixed point π‘₯βˆ—βˆˆπΉ(𝑆), where 𝐹(𝑆) denotes the common fixed point of the nonexpansive semigroup. The point π‘₯βˆ— solves the variational inequality ⟨(π›Ύπ‘“βˆ’π΄)π‘₯βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰€0 for all π‘₯∈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 π‘₯∈𝐷(𝑇) is a fixed point of 𝑇 provided 𝑇π‘₯=π‘₯. Denote by 𝐹(𝑇) the set of fixed points of 𝑇; that is, 𝐹(𝑇)={π‘₯∈𝐷(𝑇)βˆΆπ‘‡π‘₯=π‘₯}. Recall that 𝑇 is said to be nonexpansive if‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπ·(𝐴).(1.1)

Recall that a family 𝑆={𝑇(𝑠)βˆ£π‘ β‰₯0} of mappings from 𝐻 into itself is called a one-parameter nonexpansive semigroup if it satisfies the following conditions:(i)𝑇(0)π‘₯=π‘₯, forallπ‘₯∈𝐻; (ii)𝑇(𝑠+𝑑)π‘₯=𝑇(𝑠)𝑇(𝑑)π‘₯, forall𝑠,𝑑β‰₯0 and forallπ‘₯∈𝐻;(iii)‖𝑇(𝑠)π‘₯βˆ’π‘‡(𝑠)𝑦‖≀‖π‘₯βˆ’π‘¦β€–, forall𝑠β‰₯0 and forallπ‘₯,π‘¦βˆˆπ»;(iv)for all π‘₯∈𝐢, 𝑠↦𝑇(𝑠)π‘₯ 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 𝐢 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 π‘‘βˆˆ(0,1) and define a contraction π‘‡π‘‘βˆΆπΆβ†’πΆ by 𝑇𝑑π‘₯=𝑑𝑒+(1βˆ’π‘‘)𝑇π‘₯,π‘₯∈𝐢,(1.2) where π‘’βˆˆπΆ is a fixed point. Banach’s contraction mapping principle guarantees that 𝑇𝑑 has a unique fixed point π‘₯𝑑 in 𝐢. If 𝑇 enjoys a nonempty fixed point set, Browder [2] proved the following well-known strong convergence theorem.

Theorem B. Let 𝐢 be a bounded closed convex subset of a Hilbert space 𝐻 and let 𝑇 be a nonexpansive mapping on 𝐢. Fix π‘’βˆˆπΆ and define π‘§π‘‘βˆˆπΆπ‘Žπ‘ π‘§π‘‘=𝑑𝑒+(1βˆ’π‘‘)𝑇𝑧𝑑 for π‘‘βˆˆ(0,1). Then as 𝑑→0, {𝑧𝑑} converges strongly to a element of 𝐹(𝑇) nearest to 𝑒.

As motivated by Theorem B, Halpern [4] considered the following explicit iteration: π‘₯0∈𝐢,π‘₯𝑛+1=𝛼𝑛𝑒+1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯𝑛,𝑛β‰₯0,(1.3) and proved the following theorem.

Theorem H. Let 𝐢 be a bounded closed convex subset of a Hilbert space 𝐻 and let 𝑇 be a nonexpansive mapping on 𝐢. Define a real sequence {𝛼𝑛} in [0,1] by 𝛼𝑛=π‘›βˆ’πœƒ,0<πœƒ<1. Define a sequence {π‘₯𝑛} by (1.3). Then {π‘₯𝑛} converges strongly to the element of 𝐹(𝑇) nearest to 𝑒.

In 1977, Lions [5] improved the result of Halpern [4], still in Hilbert spaces, by proving the strong convergence of {π‘₯𝑛} to a fixed point of 𝑇 where the real sequence {𝛼𝑛} satisfies the following conditions:(C1)limπ‘›β†’βˆžπ›Όπ‘›=0; (C2)βˆ‘βˆžπ‘›=1𝛼n=∞; (C3)limπ‘›β†’βˆž(𝛼𝑛+1βˆ’π›Όπ‘›)/𝛼2𝑛+1=0.

It was observed that both Halpern’s and Lions’s conditions on the real sequence {𝛼𝑛} excluded the canonical choice 𝛼𝑛=1/(𝑛+1). This was overcome in 1992 by Wittmann [6], who proved, still in Hilbert spaces, the strong convergence of {π‘₯𝑛} to a fixed point of 𝑇 if {𝛼𝑛} satisfies the following conditions:(C1)limπ‘›β†’βˆžπ›Όπ‘›=0; (C2)βˆ‘βˆžπ‘›=1𝛼𝑛=∞; (C3)βˆ‘βˆžπ‘›=1|𝛼𝑛+1βˆ’π›Όπ‘›|<∞.

Recall that a mapping π‘“βˆΆπ»β†’π» is an 𝛼-contraction if there exists a constant π›Όβˆˆ(0,1) such that‖𝑓(π‘₯)βˆ’π‘“(𝑦)‖≀𝛼‖π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπ».(1.4)

Recall that an operator 𝐴 is strongly positive on 𝐻 if there exists a constant 𝛾>0 such that⟨𝐴π‘₯,π‘₯⟩β‰₯𝛾‖π‘₯β€–2,βˆ€π‘₯∈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: minπ‘₯∈𝐹(𝑇)12⟨𝐴π‘₯,π‘₯βŸ©βˆ’βŸ¨π‘₯,π‘βŸ©,(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 {π‘₯𝑛} defined by the iterative method below, with the initial guess π‘₯0∈H chosen arbitrarily, π‘₯𝑛+1=ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘₯𝑛+𝛼𝑛𝑏,𝑛β‰₯0,(1.7) strongly converges to the unique solution of the minimization problem (1.6) provided that the sequence {𝛼𝑛} satisfies certain conditions.

Recently, Marino and Xu [9] studied the following continuous scheme: π‘₯𝑑π‘₯=𝑑𝛾𝑓𝑑+(πΌβˆ’π‘‘π΄)𝑇π‘₯𝑑,(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 {π‘₯𝑑} strongly converges to a fixed point π‘₯ of 𝑇. Also in [9], they introduced a general explicit iterative scheme by the viscosity approximation method:π‘₯π‘›βˆˆπ»,π‘₯𝑛+1=𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘₯𝑛,𝑛β‰₯0(1.9) and proved that the sequence {π‘₯𝑛} generated by (1.9) converges strongly to a unique solution of the variational inequality⟨(π΄βˆ’π›Ύπ‘“)π‘₯βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐹(𝑇),(1.10)

which is the optimality condition for the minimization problem minπ‘₯∈𝐹(𝑇)12⟨𝐴π‘₯,π‘₯βŸ©βˆ’β„Ž(π‘₯),(1.11)

where β„Ž is a potential function for 𝛾𝑓 (i.e., β„Žβ€²(π‘₯)=𝛾𝑓(π‘₯) for π‘₯∈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 𝑇𝑑 defined as follows: 𝑇𝑑1π‘₯=𝑑𝛾𝑓(π‘₯)+(πΌβˆ’π‘‘π΄)πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑𝑠,(1.12) where 𝛾>0 is a constant, 𝑓 is an 𝛼-contraction, 𝐴 is a bounded linear strongly positive self-adjoint operator and {πœ†π‘‘} is a positive real divergent net. If 𝛾𝛼<𝛾 for each 0<𝑑<β€–π΄β€–βˆ’1, one can see that 𝑇𝑑 is a (1βˆ’π‘‘(π›Ύβˆ’π›Ύπ›Ό))-contraction. So, by Banach’s contraction mapping principle, there exists an unique solution π‘₯𝑑 of the fixed point equationπ‘₯𝑑π‘₯=𝑑𝛾𝑓𝑑1+(πΌβˆ’π‘‘π΄)πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑𝑑𝑠.(1.13) We show that the sequence {π‘₯𝑑} generated by above continuous scheme strongly converges to a common fixed point π‘₯βˆ—βˆˆπΉ(𝑆), which is the unique point in 𝐹(𝑆) solving the variational inequality ⟨(π›Ύπ‘“βˆ’π΄)π‘₯βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰€0 for all π‘₯∈𝐹(𝑆). Furthermore, we also study the following explicit iterative scheme: π‘₯0∈𝐻,π‘₯𝑛+1=𝛼𝑛π‘₯𝛾𝑓𝑛+𝛽𝑛π‘₯𝑛+ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’π›Όπ‘›π΄ξ€Έ1π‘ π‘›ξ€œπ‘ π‘›0𝑇(𝑠)π‘₯𝑛𝑑𝑠,𝑛β‰₯0.(1.14) We prove that the sequence {π‘₯𝑛} generated by (1.14) converges strongly to the same π‘₯βˆ—.

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(𝑑)∢0≀𝑑<∞} be a nonexpansive semigroup on 𝐷. Then, for any 0β‰€β„Ž<∞, limπ‘‘β†’βˆžsupπ‘₯βˆˆπ·β€–β€–β€–1π‘‘ξ€œπ‘‘01𝑇(𝑠)π‘₯π‘‘π‘ βˆ’π‘‡(β„Ž)π‘‘ξ€œπ‘‘0‖‖‖𝑇(𝑠)π‘₯𝑑𝑠=0.(1.15)

Lemma 1.2 (see [17]). Let 𝐻 be a Hilbert space, 𝐢 a closed convex subset of 𝐻, and π‘‡βˆΆπΆβ†’πΆ a nonexpansive mapping with 𝐹(𝑇)β‰ βˆ…. Then πΌβˆ’π‘‡ is demiclosed, that is, if {π‘₯𝑛} is a sequence in 𝐢 weakly converging to π‘₯ and if {(πΌβˆ’π‘‡)π‘₯𝑛} strongly converges to 𝑦, then (πΌβˆ’π‘‡)π‘₯=𝑦.

Lemma 1.3 (see [18]). Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝑃𝐢 be the metric projection from 𝐻 onto 𝐢( i.e., for π‘₯∈𝐻,𝑃𝐢π‘₯ is the only point in 𝐢 such that β€–π‘₯βˆ’π‘ƒπΆπ‘₯β€–=inf{β€–π‘₯βˆ’π‘§β€–βˆΆπ‘§βˆˆπΆ}). Given π‘₯∈𝐻 and π‘§βˆˆπΆ. Then 𝑧=𝑃𝐢π‘₯ if and only if there holds the relations ⟨π‘₯βˆ’π‘§,π‘¦βˆ’π‘§βŸ©β‰€0,βˆ€π‘¦βˆˆπΆ.(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, ξ€·βŸ¨π‘₯βˆ’π‘¦,(π΄βˆ’π›Ύπ‘“)π‘₯βˆ’(π΄βˆ’π›Ύπ‘“)π‘¦βŸ©β‰₯ξ€Έπ›Ύβˆ’π›Ύπ›Όβ€–π‘₯βˆ’π‘¦β€–2,π‘₯,π‘¦βˆˆπ».(1.17) That is, π΄βˆ’π›Ύπ‘“ is strongly monotone with coefficient π›Ύβˆ’π›Όπ›Ύ.

Proof. From the definition of strongly positive linear bounded operator, we have ⟨π‘₯βˆ’π‘¦,𝐴(π‘₯βˆ’π‘¦)⟩β‰₯𝛾‖π‘₯βˆ’π‘¦β€–2.(1.18) On the other hand, it is easy to see ⟨π‘₯βˆ’π‘¦,𝛾𝑓π‘₯βˆ’π›Ύπ‘“π‘¦βŸ©β‰€π›Ύπ›Όβ€–π‘₯βˆ’π‘¦β€–2.(1.19) Therefore, we have β‰₯ξ€·βŸ¨π‘₯βˆ’π‘¦,(π΄βˆ’π›Ύπ‘“)π‘₯βˆ’(π΄βˆ’π›Ύπ‘“)π‘¦βŸ©=⟨π‘₯βˆ’π‘¦,𝐴(π‘₯βˆ’π‘¦)βŸ©βˆ’βŸ¨π‘₯βˆ’π‘¦,𝛾𝑓π‘₯βˆ’π›Ύπ‘“π‘¦βŸ©ξ€Έπ›Ύβˆ’π›Ύπ›Όβ€–π‘₯βˆ’π‘¦β€–2(1.20) for all π‘₯,π‘¦βˆˆπ». This completes the proof.

Remark 1.5. Taking 𝛾=1 and 𝐴=𝐼, the identity mapping, we have the following inequality: ⟨π‘₯βˆ’π‘¦,(πΌβˆ’π‘“)π‘₯βˆ’(πΌβˆ’π‘“)π‘¦βŸ©β‰₯(1βˆ’π›Ό)β€–π‘₯βˆ’π‘¦β€–2,π‘₯,π‘¦βˆˆπ».(1.21) Furthermore, if 𝑓 is a nonexpansive mapping in Remark 1.5, we have ⟨π‘₯βˆ’π‘¦,(πΌβˆ’π‘“)π‘₯βˆ’(πΌβˆ’π‘“)π‘¦βŸ©β‰₯0,π‘₯,π‘¦βˆˆπ».(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<πœŒβ‰€β€–π΄β€–βˆ’1. Then β€–πΌβˆ’πœŒπ΄β€–β‰€1βˆ’πœŒπ›Ύ.

Lemma 1.7 (see [12]). Let {𝛼𝑛} be a sequence of nonnegative real numbers satisfying the following condition: 𝛼𝑛+1≀1βˆ’π›Ύπ‘›ξ€Έπ›Όπ‘›+π›Ύπ‘›πœŽπ‘›,βˆ€π‘›β‰₯0,(1.23) where {𝛾𝑛} is a sequence in (0,1) and {πœŽπ‘›} is a sequence of real numbers such that(i)limπ‘›β†’βˆžπ›Ύπ‘›=0 and βˆ‘βˆžπ‘›=0𝛾𝑛=∞,(ii)either limsupπ‘›β†’βˆžπœŽπ‘›β‰€0 or βˆ‘βˆžπ‘›=0|π›Ύπ‘›πœŽπ‘›|<∞. Then {𝛼𝑛}βˆžπ‘›=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)β‰ βˆ…. Let {πœ†π‘‘}0<𝑑<1 be a continuous net of positive real numbers such that lim𝑑→0πœ†π‘‘=∞. Let π‘“βˆΆπ»β†’π» 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 {π‘₯𝑑} be a sequence defined by (1.13). Then (i){π‘₯𝑑} is bounded for all π‘‘βˆˆ(0,β€–π΄β€–βˆ’1);(ii)lim𝑑→0‖𝑇(𝜏)π‘₯π‘‘βˆ’π‘₯𝑑‖=0 for all 0β‰€πœ<∞;(iii)π‘₯𝑑 defines a continuous curve from (0,β€–π΄β€–βˆ’1) into 𝐻.

Proof. (i) Taking π‘βˆˆπΉ(𝑆), we have β€–β€–π‘₯𝑑‖‖≀‖‖‖π‘₯βˆ’π‘π‘‘π›Ύπ‘“π‘‘ξ€Έ1+(πΌβˆ’π‘‘π΄)πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑‖‖‖‖‖π‘₯π‘‘π‘ βˆ’π‘β‰€π‘‘π›Ύπ‘“π‘‘ξ€Έβ€–β€–+ξ€·βˆ’π΄π‘1βˆ’π‘‘π›Ύξ€Έ1πœ†π‘‘ξ€œπœ†π‘‘0‖‖𝑇(𝑠)π‘₯𝑑‖‖‖‖π‘₯βˆ’π‘π‘‘π‘ β‰€π‘‘π›Ύπ‘“π‘‘ξ€Έβ€–β€–+ξ€·βˆ’π΄π‘1βˆ’π‘‘π›Ύξ€Έβ€–β€–π‘₯𝑑‖‖‖‖𝑓π‘₯βˆ’π‘β‰€π‘‘π›Ύπ‘‘ξ€Έβ€–β€–ξ€·βˆ’π‘“(𝑝)+𝑑‖𝛾𝑓(𝑝)βˆ’π΄π‘β€–+1βˆ’π‘‘π›Ύξ€Έβ€–β€–π‘₯π‘‘β€–β€–β‰€ξ€Ίξ€·βˆ’π‘1βˆ’π‘‘β€–β€–π‘₯π›Ύβˆ’π›Ύπ›Όξ€Έξ€»π‘‘β€–β€–βˆ’π‘+𝑑‖𝛾𝑓(𝑝)βˆ’π΄π‘β€–.(2.1) It follows that β€–β€–π‘₯𝑑‖‖≀1βˆ’π‘π›Ύβˆ’π›Όπ›Ύβ€–π›Ύπ‘“(𝑝)βˆ’π΄π‘β€–.(2.2) This implies that {π‘₯𝑑} is not only bounded, but also that {π‘₯𝑑} is contained in 𝐡(𝑝,1/(π›Ύβˆ’π›Ύπ›Ό)‖𝛾𝑓(𝑝)βˆ’π΄π‘β€–) of center 𝑝 and radius 1/(π›Ύβˆ’π›Ύπ›Ό)‖𝛾𝑓(𝑝)βˆ’π΄π‘β€–, for all fixed π‘βˆˆπΉ(𝑆). Moreover for π‘βˆˆπΉ(𝑆) and π‘‘βˆˆ(0,β€–π΄β€–βˆ’1), β€–β€–β€–1πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑‖‖‖=β€–β€–β€–1π‘‘π‘ βˆ’π‘πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑‖‖‖≀‖‖π‘₯βˆ’π‘‡(𝑠)𝑝𝑑𝑠𝑑‖‖≀1βˆ’π‘π›Ύβˆ’π›Ύπ›Όβ€–π›Ύπ‘“(𝑝)βˆ’π΄π‘β€–.(2.3)
(ii) Observe that ‖‖𝑇(𝜏)π‘₯π‘‘βˆ’π‘₯𝑑‖‖≀‖‖‖𝑇(𝜏)π‘₯𝑑1βˆ’π‘‡(𝜏)πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑‖‖‖+β€–β€–β€–ξ‚΅1𝑑𝑠𝑇(𝜏)πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯π‘‘ξ‚Άβˆ’1π‘‘π‘ πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑‖‖‖+β€–β€–β€–1π‘‘π‘ πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯π‘‘π‘‘π‘ βˆ’π‘₯𝑑‖‖‖‖‖‖π‘₯≀2π‘‘βˆ’1πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑‖‖‖+β€–β€–β€–ξ‚΅1𝑑𝑠𝑇(𝜏)πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯π‘‘ξ‚Άβˆ’1π‘‘π‘ πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑‖‖‖‖‖‖π‘₯𝑑𝑠=2𝑑𝛾𝑓𝑑1βˆ’π΄πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑‖‖‖+‖‖‖𝑇1𝑑𝑠(𝜏)πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯π‘‘ξ‚Άβˆ’1π‘‘π‘ πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑‖‖‖.𝑑𝑠(2.4) Taking 𝐡(𝑝,1/(π›Ύβˆ’π›Ύπ›Ό)‖𝛾𝑓(𝑝)βˆ’π΄π‘β€–) as 𝐷 in Lemma 1.1 and passing to lim𝑑→0 in (2.4), we can obtain (ii) immediately.
(iii) Taking 𝑑1,𝑑2∈(0,β€–π΄β€–βˆ’1) and fixing π‘βˆˆπΉ(𝑆), we see that β€–β€–π‘₯𝑑1βˆ’π‘₯𝑑2‖‖≀‖‖‖𝑑1βˆ’π‘‘2ξ€Έξ€·π‘₯𝛾𝑓𝑑1ξ€Έ+𝑑2𝛾𝑓π‘₯𝑑1ξ€Έξ€·π‘₯βˆ’π‘“π‘‘2βˆ’ξ€·π‘‘ξ€Έξ€Έ1βˆ’π‘‘2𝐴1πœ†π‘‘1ξ€œπœ†π‘‘10𝑇(𝑠)π‘₯𝑑1+ξ€·π‘‘π‘ πΌβˆ’π‘‘2𝐴1πœ†π‘‘1ξ€œπœ†π‘‘10𝑇(𝑠)π‘₯𝑑11π‘‘π‘ βˆ’πœ†π‘‘2ξ€œπœ†π‘‘20𝑇(𝑠)π‘₯𝑑2ξƒͺ‖‖‖‖≀||𝑑𝑑𝑠1βˆ’π‘‘2||𝛾‖‖𝑓π‘₯𝑑1ξ€Έβ€–β€–+𝑑2β€–β€–π‘₯𝛾𝛼𝑑1βˆ’π‘₯𝑑2β€–β€–+||𝑑1βˆ’π‘‘2||β€–β€–β€–1β€–π΄β€–πœ†π‘‘1ξ€œπœ†π‘‘10𝑇(𝑠)π‘₯𝑑1β€–β€–β€–+𝑑𝑠1βˆ’π‘‘2𝛾‖‖‖‖1πœ†π‘‘1ξ€œπœ†π‘‘10𝑇(𝑠)π‘₯𝑑11π‘‘π‘ βˆ’πœ†π‘‘2ξ€œπœ†π‘‘10𝑇(𝑠)π‘₯𝑑21π‘‘π‘ βˆ’πœ†π‘‘2ξ€œπœ†π‘‘2πœ†π‘‘1𝑇(𝑠)π‘₯𝑑2‖‖‖‖≀||𝑑𝑑𝑠1βˆ’t2||𝛾‖‖𝑓π‘₯𝑑1ξ€Έβ€–β€–+𝑑2β€–β€–π‘₯𝛾𝛼𝑑1βˆ’π‘₯𝑑2β€–β€–+||𝑑1βˆ’π‘‘2||β€–β€–β€–1β€–π΄β€–πœ†π‘‘1ξ€œπœ†π‘‘10𝑇(𝑠)π‘₯𝑑1β€–β€–β€–+𝑑𝑠1βˆ’π‘‘2𝛾‖‖π‘₯𝑑1βˆ’π‘₯𝑑2β€–β€–+||||1πœ†π‘‘1βˆ’1πœ†π‘‘2||||β€–β€–β€–ξ€œπœ†π‘‘10𝑇(𝑠)π‘₯𝑑2β€–β€–β€–+1π‘‘π‘ πœ†π‘‘2β€–β€–β€–β€–ξ€œπœ†π‘‘2πœ†π‘‘1𝑇(𝑠)π‘₯𝑑2β€–β€–β€–β€–ξƒͺ.𝑑𝑠(2.5) Thus applying (2.3), we arrive at β€–β€–π‘₯𝑑1βˆ’π‘₯𝑑2‖‖≀||𝑑1βˆ’π‘‘2||𝛾‖‖𝑓π‘₯𝑑1ξ€Έβ€–β€–+𝑑2β€–β€–π‘₯𝛾𝛼𝑑1βˆ’π‘₯𝑑2β€–β€–+||𝑑1βˆ’π‘‘2||ξ‚΅1‖𝐴‖+ξ€·π›Ύβˆ’π›Ύπ›Όβ€–π›Ύπ‘“(𝑝)βˆ’π΄π‘β€–+‖𝑝‖1βˆ’π‘‘2𝛾‖‖π‘₯𝑑1βˆ’π‘₯𝑑2β€–β€–+2πœ†π‘‘2||πœ†π‘‘2βˆ’πœ†π‘‘1||ξ‚΅1(ξ‚Άξƒͺ≀||π‘‘π›Ύβˆ’π›Ύπ›Όβ€–π›Ύπ‘“π‘)βˆ’π΄π‘β€–+‖𝑝‖1βˆ’π‘‘2||𝛾‖‖𝑓π‘₯𝑑1ξ€Έβ€–β€–ξ‚΅1+‖𝐴‖‖+ξ€·π›Ύβˆ’π›Ύπ›Όπ›Ύπ‘“(𝑝)βˆ’π΄π‘β€–+‖𝑝‖1βˆ’π‘‘2ξ€·β€–β€–π‘₯π›Ύβˆ’π›Ύπ›Όξ€Έξ€Έπ‘‘1βˆ’π‘₯𝑑2β€–β€–+2πœ†π‘‘2||πœ†π‘‘2βˆ’πœ†π‘‘1||ξ‚΅1ξ‚Ά.π›Ύβˆ’π›Ύπ›Όβ€–π›Ύπ‘“(𝑝)βˆ’π΄π‘β€–+‖𝑝‖(2.6) It follows that β€–β€–π‘₯𝑑1βˆ’π‘₯𝑑2‖‖≀𝑀1||𝑑1βˆ’π‘‘2||+𝑀2||πœ†π‘‘2βˆ’πœ†π‘‘1||,(2.7) where 𝑀1=𝛾‖‖𝑓π‘₯π›Ύβˆ’π›Ύπ›Όπ‘‘1ξ€Έβ€–β€–ξ€·+‖𝐴‖‖𝛾𝑓(𝑝)βˆ’π΄π‘β€–+ξ€Έπ›Ύβˆ’π›Ύπ›Όβ€–π΄β€–β€–π‘β€–π‘‘2ξ€·ξ€Έπ›Ύβˆ’π›Ύπ›Ό2(2.8) and 𝑀2=2‖𝛾𝑓(𝑝)βˆ’π΄π‘β€–+ξ€Έξ€Έπ›Ύβˆ’π›Ύπ›Όβ€–π‘β€–πœ†π‘‘2𝑑2ξ€·ξ€Έπ›Ύβˆ’π›Ύπ›Ό2.(2.9) This inequality, together with the continuity of the net {πœ†π‘‘}, gives the continuity of the curve {π‘₯𝑑}.

Theorem 2.2. Let 𝐻 be a real Hilbert space 𝐻 and 𝑆={𝑇(𝑠)∢0≀𝑠<∞} a nonexpansive semigroup such that 𝐹(𝑆)β‰ βˆ…. Let {πœ†π‘‘}0<𝑑<1 be a net of positive real numbers such that π‘™π‘–π‘šπ‘‘β†’0πœ†π‘‘=∞. 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 {π‘₯𝑑} defined by (1.13) strongly converges as 𝑑→0 to π‘₯βˆ—βˆˆπΉ(𝑆), which solves the following variational inequality: ⟨(π›Ύπ‘“βˆ’π΄)π‘₯βˆ—,π‘βˆ’π‘₯βˆ—βŸ©β‰€0,βˆ€π‘βˆˆπΉ(𝑆).(2.10) Equivalently, one has 𝑃𝐹(𝑆)(πΌβˆ’π΄+𝛾𝑓)π‘₯βˆ—=π‘₯βˆ—.(2.11)

Proof. The uniqueness of the solution of the variational inequality (2.10) is a consequence of the strong monotonicity of π΄βˆ’π›Ύπ‘“ (Lemma 1.4) and it was proved in [9]. Next, we will use π‘₯βˆ—βˆˆπΉ(𝑆) to denote the unique solution of (2.10). To prove that π‘₯𝑑→π‘₯βˆ—(𝑑→0), we write, for a given π‘βˆˆπΉ(𝑆), π‘₯𝑑π‘₯βˆ’π‘=𝑑𝛾𝑓𝑑1βˆ’π΄π‘+(πΌβˆ’π‘‘π΄)πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯π‘‘ξ‚Άπ‘‘π‘ βˆ’π‘.(2.12) Using π‘₯π‘‘βˆ’π‘ to make inner product, we obtain that β€–β€–π‘₯π‘‘β€–β€–βˆ’π‘2=1(πΌβˆ’π‘‘π΄)πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯π‘‘ξ‚Άπ‘‘π‘ βˆ’π‘,π‘₯𝑑π‘₯βˆ’π‘+π‘‘π›Ύπ‘“π‘‘ξ€Έβˆ’π΄π‘,π‘₯π‘‘ξ¬β‰€ξ€·βˆ’π‘1βˆ’π‘‘π›Ύξ€Έβ€–β€–π‘₯π‘‘β€–β€–βˆ’π‘2π‘₯+π‘‘π›Ύπ‘“π‘‘ξ€Έβˆ’π΄π‘,π‘₯𝑑.βˆ’π‘(2.13) It follows that β€–β€–π‘₯π‘‘β€–β€–βˆ’π‘2≀1𝛾𝛾𝑓π‘₯π‘‘ξ€Έβˆ’π‘“(𝑝),π‘₯tξ¬βˆ’π‘+βŸ¨π›Ύπ‘“(𝑝)βˆ’π΄π‘,π‘₯π‘‘ξ€Έβ‰€βˆ’π‘βŸ©π›Ύπ›Όπ›Ύβ€–β€–π‘₯π‘‘β€–β€–βˆ’π‘2+1π›ΎβŸ¨π›Ύπ‘“(𝑝)βˆ’π΄π‘,π‘₯π‘‘βˆ’π‘βŸ©,(2.14) which yields that β€–β€–π‘₯π‘‘β€–β€–βˆ’π‘2≀1π›Ύβˆ’π›Όπ›ΎβŸ¨π›Ύπ‘“(𝑝)βˆ’π΄π‘,π‘₯π‘‘βˆ’π‘βŸ©.(2.15) Since 𝐻 is a Hilbert space and {π‘₯𝑑} is bounded as 𝑑→0, we have that if {𝑑𝑛} is a sequence in (0,1) such that 𝑑𝑛→0 and π‘₯𝑑𝑛⇀π‘₯. By (2.15), we see π‘₯𝑑𝑛→π‘₯. Moreover, by (ii) of Lemma 2.1 we have π‘₯∈𝐹(𝑆). We next prove that π‘₯ solves the variational inequality (2.10). From (1.13), we arrive at (π΄βˆ’π›Ύπ‘“)π‘₯𝑑1=βˆ’π‘‘ξ‚Έπ‘₯(πΌβˆ’π‘‘π΄)π‘‘βˆ’1πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑𝑑𝑠.(2.16) For π‘βˆˆπΉ(𝑆), it follows from (1.22) that ⟨(π΄βˆ’π›Ύπ‘“)π‘₯𝑑,π‘₯𝑑1βˆ’π‘βŸ©=βˆ’π‘‘ξƒ‘ξ‚Έπ‘₯(πΌβˆ’π‘‘π΄)π‘‘βˆ’1πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑𝑑𝑠,π‘₯𝑑1βˆ’π‘=βˆ’π‘‘ξƒ‘1πœ†π‘‘ξ€œπœ†π‘‘0ξ€Ί(πΌβˆ’π‘‡(𝑠))π‘₯π‘‘ξ€»βˆ’(πΌβˆ’π‘‡(𝑠))𝑝𝑑𝑠,π‘₯𝑑+𝐴1βˆ’π‘πœ†π‘‘ξ€œπœ†π‘‘0(πΌβˆ’π‘‡(𝑠))π‘₯𝑑𝑑𝑠,π‘₯𝑑1βˆ’π‘=βˆ’π‘‘πœ†π‘‘ξ€œπœ†π‘‘0⟨(πΌβˆ’π‘‡(𝑠))π‘₯π‘‘βˆ’(πΌβˆ’π‘‡(𝑠))𝑝,π‘₯𝑑+𝐴1βˆ’π‘βŸ©π‘‘π‘ πœ†π‘‘ξ€œπœ†π‘‘0(πΌβˆ’π‘‡(𝑠))π‘₯𝑑𝑑𝑠,π‘₯𝑑≀𝐴1βˆ’π‘πœ†π‘‘ξ€œπœ†π‘‘0(πΌβˆ’π‘‡(𝑠))π‘₯𝑑𝑑𝑠,π‘₯𝑑=𝐴π‘₯βˆ’π‘π‘‘π›Ύπ‘“π‘‘ξ€Έ1βˆ’π‘‘π΄πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑𝑑𝑠,π‘₯𝑑𝐴π‘₯βˆ’π‘=𝑑𝛾𝑓𝑑1βˆ’π΄πœ†π‘‘ξ€œπœ†π‘‘0𝑇(𝑠)π‘₯𝑑𝑑𝑠,π‘₯𝑑.βˆ’π‘(2.17) Passing to lim𝑑→0, since {π‘₯𝑑} is a bounded sequence, we obtain (π΄βˆ’π›Ύπ‘“)π‘₯,π‘₯βˆ’π‘β‰€0,(2.18) that is, π‘₯ satisfies the variational inequality (2.10). By the uniqueness it follows π‘₯=π‘₯βˆ—. In a summary, we have shown that each cluster point of {π‘₯𝑑} (as 𝑑→0) equals π‘₯βˆ—. Therefore, π‘₯𝑑→π‘₯βˆ— as 𝑑→0. The variational inequality (2.10) can be rewritten as (πΌβˆ’π΄+𝛾𝑓)π‘₯βˆ—ξ€»βˆ’π‘₯βˆ—,π‘₯βˆ—ξ¬βˆ’π‘,π‘βˆˆπΉ(𝑆).(2.19) This, by Lemma 1.3, is equivalent to 𝑃𝐹(𝑆)(πΌβˆ’π΄+𝛾𝑓)π‘₯βˆ—=π‘₯βˆ—.(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)β‰ βˆ…. Let {𝑠𝑛} be a positive real divergent sequence and let {𝛼𝑛} and {𝛽𝑛} be sequences in (0,1) satisfying the following conditions limπ‘›β†’βˆžπ›Όπ‘›=limπ‘›β†’βˆžπ›½π‘›=0 and βˆ‘βˆžπ‘›=0𝛼𝑛=∞. 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 {π‘₯𝑛} defined by (1.14) strongly converges to π‘₯βˆ—βˆˆπΉ(𝑆), which solves the variational inequality (2.10).

Proof. We divide the proof into three parts.Step 1. Show the sequence {π‘₯𝑛} is bounded.
Noticing that limπ‘›β†’βˆžπ›Όπ‘›=limπ‘›β†’βˆžπ›½π‘›=0, we may assume, with no loss of generality, that 𝛼𝑛/(1βˆ’π›½π‘›)<β€–π΄β€–βˆ’1 for all 𝑛β‰₯0. From Lemma 1.6, we know that β€–(1βˆ’π›½π‘›)πΌβˆ’π›Όπ‘›π΄β€–β‰€(1βˆ’π›½π‘›βˆ’π›Όπ‘›π›Ύ). Picking π‘βˆˆπΉ(𝑆), we have β€–β€–π‘₯𝑛+1β€–β€–=β€–β€–β€–π›Όβˆ’π‘π‘›ξ€·ξ€·π‘₯π›Ύπ‘“π‘›ξ€Έξ€Έβˆ’π΄π‘+𝛽𝑛π‘₯𝑛+βˆ’π‘ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’π›Όπ‘›π΄ξ€Έξ‚΅1π‘ π‘›ξ€œπ‘ π‘›0𝑇(𝑠)π‘₯π‘›ξ‚Άβ€–β€–β€–π‘‘π‘ βˆ’π‘β‰€π›Όπ‘›β€–β€–ξ€·π‘₯π›Ύπ‘“π‘›ξ€Έβ€–β€–βˆ’π΄π‘+𝛽𝑛‖‖π‘₯𝑛‖‖+ξ€·βˆ’π‘1βˆ’π›½π‘›βˆ’π›Όπ‘›π›Ύξ€Έβ€–β€–β€–1π‘ π‘›ξ€œπ‘ π‘›0𝑇(𝑠)π‘₯π‘›β€–β€–β€–π‘‘π‘ βˆ’π‘β‰€π›Όπ‘›π›Ύβ€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβ€–β€–βˆ’π‘“(𝑝)+𝛼𝑛‖𝛾𝑓(𝑝)βˆ’π΄π‘β€–+𝛽𝑛‖‖π‘₯𝑛‖‖+ξ€·βˆ’π‘1βˆ’π›½π‘›βˆ’π›Όπ‘›π›Ύξ€Έβ€–β€–π‘₯π‘›β€–β€–β‰€ξ€Ίβˆ’π‘1βˆ’π›Όπ‘›ξ€·β€–β€–π‘₯π›Ύβˆ’π›Ύπ›Όξ€Έξ€»π‘›β€–β€–βˆ’π‘+𝛼𝑛‖𝛾𝑓(𝑝)βˆ’π΄π‘β€–.(2.21) By simple inductions, we see thatβ€–β€–π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘β‰€max0β€–β€–,(βˆ’π‘β€–π΄π‘βˆ’π›Ύπ‘“π‘)β€–ξ‚Όπ›Ύβˆ’π›Ύπ›Ό,(2.22) which yields that the sequence {π‘₯𝑛} is bounded.
Step 2. Show that limsupπ‘›β†’βˆžβŸ¨(π›Ύπ‘“βˆ’π΄)π‘₯βˆ—,π‘¦π‘›βˆ’π‘₯βˆ—βŸ©β‰€0,(2.23) where π‘₯βˆ— is obtained in Theorem 2.2 and 𝑦𝑛=(1/𝑠𝑛)βˆ«π‘ π‘›0𝑇(𝑠)π‘₯𝑛𝑑𝑠.
Putting 𝑧0=𝑃𝐹(𝑆)π‘₯0, from (2.22) we see that the closed ball 𝑀 of center 𝑧0 and radius max{‖𝑧0βˆ’π‘β€–,‖𝐴𝑧0βˆ’π›Ύπ‘“(𝑧0)β€–/(π›Ύβˆ’π›Ύπ›Ό)} is 𝑇(𝑠)-invariant for each π‘ βˆˆ[0,∞) and contain {π‘₯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 limπ‘›β†’βˆžβ€–β€–π‘¦π‘›βˆ’π‘‡(β„Ž)𝑦𝑛‖‖=0(2.24) for all 0β‰€β„Ž<∞. Taking a suitable subsequence {𝑦𝑛𝑖} of {𝑦𝑛}, we see that limsupπ‘›β†’βˆžβŸ¨(π›Ύπ‘“βˆ’π΄)π‘₯βˆ—,π‘¦π‘›βˆ’π‘₯βˆ—βŸ©=limπ‘–β†’βˆžξ«(π›Ύπ‘“βˆ’π΄)π‘₯βˆ—,π‘¦π‘›π‘–βˆ’π‘₯βˆ—ξ¬.(2.25) Since the sequence {𝑦𝑛} is also bounded, we may assume that 𝑦𝑛𝑖⇀π‘₯. From the demiclosedness principle, we have π‘₯∈𝐹(𝑆). Therefore, we have limsupπ‘›β†’βˆžβŸ¨(π›Ύπ‘“βˆ’π΄)π‘₯βˆ—,π‘¦π‘›βˆ’π‘₯βˆ—ξ«βŸ©=(π›Ύπ‘“βˆ’π΄)π‘₯βˆ—,π‘₯βˆ’π‘₯βˆ—ξ¬β‰€0.(2.26) On the other hand, we have β€–β€–π‘₯𝑛+1βˆ’π‘¦π‘›β€–β€–β‰€π›Όπ‘›β€–β€–ξ€·π‘₯π›Ύπ‘“π‘›ξ€Έβˆ’π΄π‘₯𝑛‖‖+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘¦π‘›β€–β€–.(2.27) From the assumption limπ‘›β†’βˆžπ›Όπ‘›=limπ‘›β†’βˆžπ›½π‘›=0, we see that limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘¦π‘›β€–β€–=0,(2.28) which combines with (2.26) gives that limsupπ‘›β†’βˆžξ«(π›Ύπ‘“βˆ’π΄)π‘₯βˆ—,π‘₯𝑛+1βˆ’π‘₯βˆ—ξ¬β‰€0.(2.29)
Step 3. Show π‘₯𝑛→π‘₯βˆ— as π‘›β†’βˆž.
Note thatβ€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2=𝛼𝑛π‘₯π›Ύπ‘“π‘›ξ€Έβˆ’π΄π‘₯βˆ—ξ€Έ+𝛽𝑛π‘₯π‘›βˆ’π‘₯βˆ—ξ€Έ+ξ€·ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’π›Όπ‘›π΄π‘¦ξ€Έξ€·π‘›βˆ’π‘₯βˆ—ξ€Έ,π‘₯𝑛+1βˆ’π‘₯βˆ—ξ¬=𝛼𝑛π‘₯π›Ύπ‘“π‘›ξ€Έβˆ’π΄π‘₯βˆ—,π‘₯𝑛+1βˆ’π‘₯βˆ—ξ¬+𝛽𝑛π‘₯π‘›βˆ’π‘₯βˆ—,π‘₯𝑛+1βˆ’π‘₯βˆ—ξ¬+1βˆ’π›½π‘›ξ€ΈπΌβˆ’π›Όπ‘›π΄π‘¦ξ€Έξ€·π‘›βˆ’π‘₯βˆ—ξ€Έ,π‘₯𝑛+1βˆ’π‘₯βˆ—ξ¬β‰€π›Όπ‘›ξ€·π›Ύξ«π‘“ξ€·π‘₯𝑛π‘₯βˆ’π‘“βˆ—ξ€Έ,π‘₯𝑛+1βˆ’π‘₯βˆ—ξ¬+π‘₯π›Ύπ‘“βˆ—ξ€Έβˆ’π΄π‘₯βˆ—,π‘₯𝑛+1βˆ’π‘₯βˆ—ξ¬ξ€Έ+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–+β€–β€–ξ€·1βˆ’π›½π‘›ξ€ΈπΌβˆ’π›Όπ‘›π΄β€–β€–β€–β€–π‘¦π‘›βˆ’π‘₯βˆ—β€–β€–β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–β‰€π›Όπ‘›β€–β€–π‘₯π›Όπ›Ύπ‘›βˆ’π‘₯βˆ—β€–β€–β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛π‘₯π›Ύπ‘“βˆ—ξ€Έβˆ’π΄π‘₯βˆ—,π‘₯𝑛+1βˆ’π‘₯βˆ—ξ¬+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–+ξ€·1βˆ’π›½π‘›βˆ’π›Όπ‘›π›Ύξ€Έβ€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–=ξ€Ί1βˆ’π›Όπ‘›ξ€·β€–β€–π‘₯π›Ύβˆ’π›Ύπ›Όξ€Έξ€»π‘›βˆ’π‘₯βˆ—β€–β€–β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛π‘₯π›Ύπ‘“βˆ—ξ€Έβˆ’π΄π‘₯βˆ—,π‘₯𝑛+1βˆ’π‘₯βˆ—ξ¬β‰€1βˆ’π›Όπ‘›ξ€·ξ€Έπ›Ύβˆ’π›Ύπ›Ό2ξ‚€β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2+𝛼𝑛π‘₯π›Ύπ‘“βˆ—ξ€Έβˆ’π΄π‘₯βˆ—,π‘₯𝑛+1βˆ’π‘₯βˆ—ξ¬.≀1βˆ’π›Όπ‘›ξ€·ξ€Έπ›Ύβˆ’π›Ύπ›Ό2β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+12β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2+𝛼𝑛π‘₯π›Ύπ‘“βˆ—ξ€Έβˆ’π΄π‘₯βˆ—,π‘₯𝑛+1βˆ’π‘₯βˆ—ξ¬.(2.30) It follows that β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2≀1βˆ’π›Όπ‘›ξ€·β€–β€–π‘₯π›Ύβˆ’π›Ύπ›Όξ€Έξ€»π‘›βˆ’π‘₯βˆ—β€–β€–2+2𝛼𝑛π‘₯π›Ύπ‘“βˆ—ξ€Έβˆ’π΄π‘₯βˆ—,π‘₯𝑛+1βˆ’π‘₯βˆ—ξ¬.(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 {𝛽𝑛}≑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)β‰ βˆ…. Let {𝑠𝑛} be a positive real divergent sequence and let {𝛼𝑛} be a sequence in (0,1) satisfying the following conditions limπ‘›β†’βˆžπ›Όπ‘›=0 and βˆ‘βˆžπ‘›=0𝛼𝑛=∞. 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 {π‘₯𝑛} be a sequence generated by the following manner: π‘₯0∈𝐻,π‘₯𝑛+1=𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έ1π‘ π‘›ξ€œπ‘ π‘›0𝑇(𝑠)π‘₯𝑛𝑑𝑠,𝑛β‰₯0.(2.32) Then the sequence {π‘₯𝑛} defined by above iterative algorithm converges strongly to π‘₯βˆ—βˆˆπΉ(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).

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