About this Journal Submit a Manuscript Table of Contents 
Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 417234, 13 pages
doi:10.1155/2012/417234
Research Article

Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

Dan Zhang,Β Xiaolong Qin,Β and Feng Gu

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 = 𝛼 𝑛 𝛾 𝑓 ( π‘₯ 𝑛 ) + 𝛽 𝑛 π‘₯ 𝑛 + ( ( 1 βˆ’ 𝛽 𝑛 ) 𝐼 βˆ’ 𝛼 𝑛 𝐴 ) ( 1 / 𝑠 𝑛 ) ∫ 𝑠 𝑛 0 𝑇 ( 𝑠 ) π‘₯ 𝑛 𝑑 𝑠 , 𝑛 β‰₯ 0 converges strongly to a common fixed point π‘₯ βˆ— ∈ 𝐹 ( 𝑆 ) , where 𝐹 ( 𝑆 ) denotes the common fixed point of the nonexpansive semigroup. The point π‘₯ βˆ— solves the variational inequality ⟨ ( 𝛾 𝑓 βˆ’ 𝐴 ) π‘₯ βˆ— , π‘₯ βˆ’ π‘₯ βˆ— ⟩ ≀ 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 𝑇 π‘₯ = π‘₯ . Denote by 𝐹 ( 𝑇 ) the set of fixed points of 𝑇 ; that is, 𝐹 ( 𝑇 ) = { π‘₯ ∈ 𝐷 ( 𝑇 ) ∢ 𝑇 π‘₯ = π‘₯ } . Recall that 𝑇 is said to be nonexpansive if β€– 𝑇 π‘₯ βˆ’ 𝑇 𝑦 β€– ≀ β€– π‘₯ βˆ’ 𝑦 β€– , βˆ€ π‘₯ , 𝑦 ∈ 𝐷 ( 𝐴 ) . ( 1 . 1 )

Recall that a family 𝑆 = { 𝑇 ( 𝑠 ) ∣ 𝑠 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‘₯ , f o r a l l π‘₯ ∈ 𝐻 ; (ii) 𝑇 ( 𝑠 + 𝑑 ) π‘₯ = 𝑇 ( 𝑠 ) 𝑇 ( 𝑑 ) π‘₯ , f o r a l l 𝑠 , 𝑑 β‰₯ 0 and f o r a l l π‘₯ ∈ 𝐻 ;(iii) β€– 𝑇 ( 𝑠 ) π‘₯ βˆ’ 𝑇 ( 𝑠 ) 𝑦 β€– ≀ β€– π‘₯ βˆ’ 𝑦 β€– , f o r a l l 𝑠 β‰₯ 0 and f o r a l l π‘₯ , 𝑦 ∈ 𝐻 ;(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, β‹‚ 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 𝑑 ∈ ( 0 , 1 ) and define a contraction 𝑇 𝑑 ∢ 𝐢 β†’ 𝐢 by 𝑇 𝑑 π‘₯ = 𝑑 𝑒 + ( 1 βˆ’ 𝑑 ) 𝑇 π‘₯ , π‘₯ ∈ 𝐢 , ( 1 . 2 ) where 𝑒 ∈ 𝐢 is a fixed point. Banach’s contraction mapping principle guarantees that 𝑇 𝑑 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 𝑒 ∈ 𝐢 and define 𝑧 𝑑 ∈ 𝐢 π‘Ž 𝑠 𝑧 𝑑 = 𝑑 𝑒 + ( 1 βˆ’ 𝑑 ) 𝑇 𝑧 𝑑 for 𝑑 ∈ ( 0 , 1 ) . Then as 𝑑 β†’ 0 , { 𝑧 𝑑 } 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 = 𝛼 𝑛 ξ€· 𝑒 + 1 βˆ’ 𝛼 𝑛 ξ€Έ 𝑇 π‘₯ 𝑛 , 𝑛 β‰₯ 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 { 𝛼 𝑛 } in [ 0 , 1 ] by 𝛼 𝑛 = 𝑛 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 { 𝛼 𝑛 } satisfies the following conditions:(C1) l i m 𝑛 β†’ ∞ 𝛼 𝑛 = 0 ; (C2) βˆ‘ ∞ 𝑛 = 1 𝛼 n = ∞ ; (C3) l i m 𝑛 β†’ ∞ ( 𝛼 𝑛 + 1 βˆ’ 𝛼 𝑛 ) / 𝛼 2 𝑛 + 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 { π‘₯ 𝑛 } to a fixed point of 𝑇 if { 𝛼 𝑛 } satisfies the following conditions:(C1) l i m 𝑛 β†’ ∞ 𝛼 𝑛 = 0 ; (C2) βˆ‘ ∞ 𝑛 = 1 𝛼 𝑛 = ∞ ; (C3) βˆ‘ ∞ 𝑛 = 1 | 𝛼 𝑛 + 1 βˆ’ 𝛼 𝑛 | < ∞ .

Recall that a mapping 𝑓 ∢ 𝐻 β†’ 𝐻 is an 𝛼 -contraction if there exists a constant 𝛼 ∈ ( 0 , 1 ) such that β€– 𝑓 ( π‘₯ ) βˆ’ 𝑓 ( 𝑦 ) β€– ≀ 𝛼 β€– π‘₯ βˆ’ 𝑦 β€– , βˆ€ π‘₯ , 𝑦 ∈ 𝐻 . ( 1 . 4 )

Recall that an operator 𝐴 is strongly positive on 𝐻 if there exists a constant 𝛾 > 0 such that ⟨ 𝐴 π‘₯ , π‘₯ ⟩ β‰₯ 𝛾 β€– 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 : m i n π‘₯ ∈ 𝐹 ( 𝑇 ) 1 2 ⟨ 𝐴 π‘₯ , π‘₯ ⟩ βˆ’ ⟨ π‘₯ , 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 { π‘₯ 𝑛 } defined by the iterative method below, with the initial guess π‘₯ 0 ∈ H chosen arbitrarily, π‘₯ 𝑛 + 1 = ξ€· 𝐼 βˆ’ 𝛼 𝑛 𝐴 ξ€Έ 𝑇 π‘₯ 𝑛 + 𝛼 𝑛 𝑏 , 𝑛 β‰₯ 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: π‘₯ 𝑑 ξ€· π‘₯ = 𝑑 𝛾 𝑓 𝑑 ξ€Έ + ( 𝐼 βˆ’ 𝑑 𝐴 ) 𝑇 π‘₯ 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‘₯ 𝑛 ∈ 𝐻 , π‘₯ 𝑛 + 1 = 𝛼 𝑛 ξ€· π‘₯ 𝛾 𝑓 𝑛 ξ€Έ + ξ€· 𝐼 βˆ’ 𝛼 𝑛 𝐴 ξ€Έ 𝑇 π‘₯ 𝑛 , 𝑛 β‰₯ 0 ( 1 . 9 ) and proved that the sequence { π‘₯ 𝑛 } generated by (1.9) converges strongly to a unique solution of the variational inequality ⟨ ( 𝐴 βˆ’ 𝛾 𝑓 ) π‘₯ βˆ— , π‘₯ βˆ’ π‘₯ βˆ— ⟩ β‰₯ 0 , βˆ€ π‘₯ ∈ 𝐹 ( 𝑇 ) , ( 1 . 1 0 )

which is the optimality condition for the minimization problem m i n π‘₯ ∈ 𝐹 ( 𝑇 ) 1 2 ⟨ 𝐴 π‘₯ , π‘₯ ⟩ βˆ’ β„Ž ( π‘₯ ) , ( 1 . 1 1 )

where β„Ž is a potential function for 𝛾 𝑓 (i.e., β„Ž β€² ( π‘₯ ) = 𝛾 𝑓 ( 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 𝑇 𝑑 defined as follows: 𝑇 𝑑 1 π‘₯ = 𝑑 𝛾 𝑓 ( π‘₯ ) + ( 𝐼 βˆ’ 𝑑 𝐴 ) πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 𝑠 , ( 1 . 1 2 ) where 𝛾 > 0 is a constant, 𝑓 is an 𝛼 -contraction, 𝐴 is a bounded linear strongly positive self-adjoint operator and { πœ† 𝑑 } is a positive real divergent net. If 𝛾 𝛼 < 𝛾 for each 0 < 𝑑 < β€– 𝐴 β€– βˆ’ 1 , one can see that 𝑇 𝑑 is a ( 1 βˆ’ 𝑑 ( 𝛾 βˆ’ 𝛾 𝛼 ) ) -contraction. So, by Banach’s contraction mapping principle, there exists an unique solution π‘₯ 𝑑 of the fixed point equation π‘₯ 𝑑 ξ€· π‘₯ = 𝑑 𝛾 𝑓 𝑑 ξ€Έ 1 + ( 𝐼 βˆ’ 𝑑 𝐴 ) πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 𝑑 𝑠 . ( 1 . 1 3 ) We show that the sequence { π‘₯ 𝑑 } generated by above continuous scheme strongly converges to a common fixed point π‘₯ βˆ— ∈ 𝐹 ( 𝑆 ) , which is the unique point in 𝐹 ( 𝑆 ) solving the variational inequality ⟨ ( 𝛾 𝑓 βˆ’ 𝐴 ) π‘₯ βˆ— , π‘₯ βˆ’ π‘₯ βˆ— ⟩ ≀ 0 for all π‘₯ ∈ 𝐹 ( 𝑆 ) . Furthermore, we also study the following explicit iterative scheme: π‘₯ 0 ∈ 𝐻 , π‘₯ 𝑛 + 1 = 𝛼 𝑛 ξ€· π‘₯ 𝛾 𝑓 𝑛 ξ€Έ + 𝛽 𝑛 π‘₯ 𝑛 + ξ€· ξ€· 1 βˆ’ 𝛽 𝑛 ξ€Έ 𝐼 βˆ’ 𝛼 𝑛 𝐴 ξ€Έ 1 𝑠 𝑛 ξ€œ 𝑠 𝑛 0 𝑇 ( 𝑠 ) π‘₯ 𝑛 𝑑 𝑠 , 𝑛 β‰₯ 0 . ( 1 . 1 4 ) 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 0 ≀ β„Ž < ∞ , l i m 𝑑 β†’ ∞ s u p π‘₯ ∈ 𝐷 β€– β€– β€– 1 𝑑 ξ€œ 𝑑 0 1 𝑇 ( 𝑠 ) π‘₯ 𝑑 𝑠 βˆ’ 𝑇 ( β„Ž ) 𝑑 ξ€œ 𝑑 0 β€– β€– β€– 𝑇 ( 𝑠 ) π‘₯ 𝑑 𝑠 = 0 . ( 1 . 1 5 )

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

Lemma 1.3 (see [18]). Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝑃 𝐢 be the metric projection from 𝐻 onto 𝐢 ( i.e., for π‘₯ ∈ 𝐻 , 𝑃 𝐢 π‘₯ is the only point in 𝐢 such that β€– π‘₯ βˆ’ 𝑃 𝐢 π‘₯ β€– = i n f { β€– π‘₯ βˆ’ 𝑧 β€– ∢ 𝑧 ∈ 𝐢 } ). Given π‘₯ ∈ 𝐻 and 𝑧 ∈ 𝐢 . Then 𝑧 = 𝑃 𝐢 π‘₯ if and only if there holds the relations ⟨ π‘₯ βˆ’ 𝑧 , 𝑦 βˆ’ 𝑧 ⟩ ≀ 0 , βˆ€ 𝑦 ∈ 𝐢 . ( 1 . 1 6 )

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€· ⟨ π‘₯ βˆ’ 𝑦 , ( 𝐴 βˆ’ 𝛾 𝑓 ) π‘₯ βˆ’ ( 𝐴 βˆ’ 𝛾 𝑓 ) 𝑦 ⟩ β‰₯ ξ€Έ 𝛾 βˆ’ 𝛾 𝛼 β€– π‘₯ βˆ’ 𝑦 β€– 2 , π‘₯ , 𝑦 ∈ 𝐻 . ( 1 . 1 7 ) That is, 𝐴 βˆ’ 𝛾 𝑓 is strongly monotone with coefficient 𝛾 βˆ’ 𝛼 𝛾 .

Proof. From the definition of strongly positive linear bounded operator, we have ⟨ π‘₯ βˆ’ 𝑦 , 𝐴 ( π‘₯ βˆ’ 𝑦 ) ⟩ β‰₯ 𝛾 β€– π‘₯ βˆ’ 𝑦 β€– 2 . ( 1 . 1 8 ) On the other hand, it is easy to see ⟨ π‘₯ βˆ’ 𝑦 , 𝛾 𝑓 π‘₯ βˆ’ 𝛾 𝑓 𝑦 ⟩ ≀ 𝛾 𝛼 β€– π‘₯ βˆ’ 𝑦 β€– 2 . ( 1 . 1 9 ) Therefore, we have β‰₯ ξ€· ⟨ π‘₯ βˆ’ 𝑦 , ( 𝐴 βˆ’ 𝛾 𝑓 ) π‘₯ βˆ’ ( 𝐴 βˆ’ 𝛾 𝑓 ) 𝑦 ⟩ = ⟨ π‘₯ βˆ’ 𝑦 , 𝐴 ( π‘₯ βˆ’ 𝑦 ) ⟩ βˆ’ ⟨ π‘₯ βˆ’ 𝑦 , 𝛾 𝑓 π‘₯ βˆ’ 𝛾 𝑓 𝑦 ⟩ ξ€Έ 𝛾 βˆ’ 𝛾 𝛼 β€– π‘₯ βˆ’ 𝑦 β€– 2 ( 1 . 2 0 ) for all π‘₯ , 𝑦 ∈ 𝐻 . This completes the proof.

Remark 1.5. Taking 𝛾 = 1 and 𝐴 = 𝐼 , the identity mapping, we have the following inequality: ⟨ π‘₯ βˆ’ 𝑦 , ( 𝐼 βˆ’ 𝑓 ) π‘₯ βˆ’ ( 𝐼 βˆ’ 𝑓 ) 𝑦 ⟩ β‰₯ ( 1 βˆ’ 𝛼 ) β€– π‘₯ βˆ’ 𝑦 β€– 2 , π‘₯ , 𝑦 ∈ 𝐻 . ( 1 . 2 1 ) Furthermore, if 𝑓 is a nonexpansive mapping in Remark 1.5, we have ⟨ π‘₯ βˆ’ 𝑦 , ( 𝐼 βˆ’ 𝑓 ) π‘₯ βˆ’ ( 𝐼 βˆ’ 𝑓 ) 𝑦 ⟩ β‰₯ 0 , π‘₯ , 𝑦 ∈ 𝐻 . ( 1 . 2 2 )

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 ≀ β€– 𝐴 β€– βˆ’ 1 . Then β€– 𝐼 βˆ’ 𝜌 𝐴 β€– ≀ 1 βˆ’ 𝜌 𝛾 .

Lemma 1.7 (see [12]). Let { 𝛼 𝑛 } be a sequence of nonnegative real numbers satisfying the following condition: 𝛼 𝑛 + 1 ≀ ξ€· 1 βˆ’ 𝛾 𝑛 ξ€Έ 𝛼 𝑛 + 𝛾 𝑛 𝜎 𝑛 , βˆ€ 𝑛 β‰₯ 0 , ( 1 . 2 3 ) where { 𝛾 𝑛 } is a sequence in ( 0 , 1 ) and { 𝜎 𝑛 } is a sequence of real numbers such that(i) l i m 𝑛 β†’ ∞ 𝛾 𝑛 = 0 and βˆ‘ ∞ 𝑛 = 0 𝛾 𝑛 = ∞ ,(ii)either l i m s u p 𝑛 β†’ ∞ 𝜎 𝑛 ≀ 0 or βˆ‘ ∞ 𝑛 = 0 | 𝛾 𝑛 𝜎 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 ) β‰  βˆ… . Let { πœ† 𝑑 } 0 < 𝑑 < 1 be a continuous net of positive real numbers such that l i m 𝑑 β†’ 0 πœ† 𝑑 = ∞ . Let 𝑓 ∢ 𝐻 β†’ 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 { π‘₯ 𝑑 } be a sequence defined by (1.13). Then (i) { π‘₯ 𝑑 } is bounded for all 𝑑 ∈ ( 0 , β€– 𝐴 β€– βˆ’ 1 ) ;(ii) l i m 𝑑 β†’ 0 β€– 𝑇 ( 𝜏 ) π‘₯ 𝑑 βˆ’ π‘₯ 𝑑 β€– = 0 for all 0 ≀ 𝜏 < ∞ ;(iii) π‘₯ 𝑑 defines a continuous curve from ( 0 , β€– 𝐴 β€– βˆ’ 1 ) into 𝐻 .

Proof. (i) Taking 𝑝 ∈ 𝐹 ( 𝑆 ) , we have β€– β€– π‘₯ 𝑑 β€– β€– ≀ β€– β€– β€– ξ€· π‘₯ βˆ’ 𝑝 𝑑 𝛾 𝑓 𝑑 ξ€Έ 1 + ( 𝐼 βˆ’ 𝑑 𝐴 ) πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 β€– β€– β€– β€– β€– ξ€· π‘₯ 𝑑 𝑠 βˆ’ 𝑝 ≀ 𝑑 𝛾 𝑓 𝑑 ξ€Έ β€– β€– + ξ€· βˆ’ 𝐴 𝑝 1 βˆ’ 𝑑 𝛾 ξ€Έ 1 πœ† 𝑑 ξ€œ πœ† 𝑑 0 β€– β€– 𝑇 ( 𝑠 ) π‘₯ 𝑑 β€– β€– β€– β€– ξ€· π‘₯ βˆ’ 𝑝 𝑑 𝑠 ≀ 𝑑 𝛾 𝑓 𝑑 ξ€Έ β€– β€– + ξ€· βˆ’ 𝐴 𝑝 1 βˆ’ 𝑑 𝛾 ξ€Έ β€– β€– π‘₯ 𝑑 β€– β€– β€– β€– 𝑓 ξ€· π‘₯ βˆ’ 𝑝 ≀ 𝑑 𝛾 𝑑 ξ€Έ β€– β€– ξ€· βˆ’ 𝑓 ( 𝑝 ) + 𝑑 β€– 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– + 1 βˆ’ 𝑑 𝛾 ξ€Έ β€– β€– π‘₯ 𝑑 β€– β€– ≀ ξ€Ί ξ€· βˆ’ 𝑝 1 βˆ’ 𝑑 β€– β€– π‘₯ 𝛾 βˆ’ 𝛾 𝛼 ξ€Έ ξ€» 𝑑 β€– β€– βˆ’ 𝑝 + 𝑑 β€– 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– . ( 2 . 1 ) It follows that β€– β€– π‘₯ 𝑑 β€– β€– ≀ 1 βˆ’ 𝑝 𝛾 βˆ’ 𝛼 𝛾 β€– 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– . ( 2 . 2 ) This implies that { π‘₯ 𝑑 } is not only bounded, but also that { π‘₯ 𝑑 } is contained in 𝐡 ( 𝑝 , 1 / ( 𝛾 βˆ’ 𝛾 𝛼 ) β€– 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– ) of center 𝑝 and radius 1 / ( 𝛾 βˆ’ 𝛾 𝛼 ) β€– 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– , for all fixed 𝑝 ∈ 𝐹 ( 𝑆 ) . Moreover for 𝑝 ∈ 𝐹 ( 𝑆 ) and 𝑑 ∈ ( 0 , β€– 𝐴 β€– βˆ’ 1 ) , β€– β€– β€– 1 πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 β€– β€– β€– = β€– β€– β€– 1 𝑑 𝑠 βˆ’ 𝑝 πœ† 𝑑 ξ€œ πœ† 𝑑 0 ξ€· 𝑇 ( 𝑠 ) π‘₯ 𝑑 ξ€Έ β€– β€– β€– ≀ β€– β€– π‘₯ βˆ’ 𝑇 ( 𝑠 ) 𝑝 𝑑 𝑠 𝑑 β€– β€– ≀ 1 βˆ’ 𝑝 𝛾 βˆ’ 𝛾 𝛼 β€– 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– . ( 2 . 3 )
(ii) Observe that β€– β€– 𝑇 ( 𝜏 ) π‘₯ 𝑑 βˆ’ π‘₯ 𝑑 β€– β€– ≀ β€– β€– β€– 𝑇 ( 𝜏 ) π‘₯ 𝑑 ξ‚΅ 1 βˆ’ 𝑇 ( 𝜏 ) πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 ξ‚Ά β€– β€– β€– + β€– β€– β€– ξ‚΅ 1 𝑑 𝑠 𝑇 ( 𝜏 ) πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 ξ‚Ά βˆ’ 1 𝑑 𝑠 πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 β€– β€– β€– + β€– β€– β€– 1 𝑑 𝑠 πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 𝑑 𝑠 βˆ’ π‘₯ 𝑑 β€– β€– β€– β€– β€– β€– π‘₯ ≀ 2 𝑑 βˆ’ 1 πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 β€– β€– β€– + β€– β€– β€– ξ‚΅ 1 𝑑 𝑠 𝑇 ( 𝜏 ) πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 ξ‚Ά βˆ’ 1 𝑑 𝑠 πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 β€– β€– β€– β€– β€– β€– ξ€· π‘₯ 𝑑 𝑠 = 2 𝑑 𝛾 𝑓 𝑑 ξ€Έ 1 βˆ’ 𝐴 πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 β€– β€– β€– + β€– β€– β€– 𝑇 ξ‚΅ 1 𝑑 𝑠 ( 𝜏 ) πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 ξ‚Ά βˆ’ 1 𝑑 𝑠 πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 β€– β€– β€– . 𝑑 𝑠 ( 2 . 4 ) Taking 𝐡 ( 𝑝 , 1 / ( 𝛾 βˆ’ 𝛾 𝛼 ) β€– 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– ) as 𝐷 in Lemma 1.1 and passing to l i m 𝑑 β†’ 0 in (2.4), we can obtain (ii) immediately.
(iii) Taking 𝑑 1 , 𝑑 2 ∈ ( 0 , β€– 𝐴 β€– βˆ’ 1 ) and fixing 𝑝 ∈ 𝐹 ( 𝑆 ) , we see that β€– β€– π‘₯ 𝑑 1 βˆ’ π‘₯ 𝑑 2 β€– β€– ≀ β€– β€– β€– ξ€· 𝑑 1 βˆ’ 𝑑 2 ξ€Έ ξ€· π‘₯ 𝛾 𝑓 𝑑 1 ξ€Έ + 𝑑 2 𝛾 ξ€· 𝑓 ξ€· π‘₯ 𝑑 1 ξ€Έ ξ€· π‘₯ βˆ’ 𝑓 𝑑 2 βˆ’ ξ€· 𝑑 ξ€Έ ξ€Έ 1 βˆ’ 𝑑 2 ξ€Έ 𝐴 1 πœ† 𝑑 1 ξ€œ πœ† 𝑑 1 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 1 + ξ€· 𝑑 𝑠 𝐼 βˆ’ 𝑑 2 𝐴 ξ€Έ  1 πœ† 𝑑 1 ξ€œ πœ† 𝑑 1 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 1 1 𝑑 𝑠 βˆ’ πœ† 𝑑 2 ξ€œ πœ† 𝑑 2 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 2 ξƒͺ β€– β€– β€– β€– ≀ | | 𝑑 𝑑 𝑠 1 βˆ’ 𝑑 2 | | 𝛾 β€– β€– 𝑓 ξ€· π‘₯ 𝑑 1 ξ€Έ β€– β€– + 𝑑 2 β€– β€– π‘₯ 𝛾 𝛼 𝑑 1 βˆ’ π‘₯ 𝑑 2 β€– β€– + | | 𝑑 1 βˆ’ 𝑑 2 | | β€– β€– β€– 1 β€– 𝐴 β€– πœ† 𝑑 1 ξ€œ πœ† 𝑑 1 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 1 β€– β€– β€– + ξ€· 𝑑 𝑠 1 βˆ’ 𝑑 2 𝛾 ξ€Έ β€– β€– β€– β€– 1 πœ† 𝑑 1 ξ€œ πœ† 𝑑 1 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 1 1 𝑑 𝑠 βˆ’ πœ† 𝑑 2 ξ€œ πœ† 𝑑 1 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 2 1 𝑑 𝑠 βˆ’ πœ† 𝑑 2 ξ€œ πœ† 𝑑 2 πœ† 𝑑 1 𝑇 ( 𝑠 ) π‘₯ 𝑑 2 β€– β€– β€– β€– ≀ | | 𝑑 𝑑 𝑠 1 βˆ’ t 2 | | 𝛾 β€– β€– 𝑓 ξ€· π‘₯ 𝑑 1 ξ€Έ β€– β€– + 𝑑 2 β€– β€– π‘₯ 𝛾 𝛼 𝑑 1 βˆ’ π‘₯ 𝑑 2 β€– β€– + | | 𝑑 1 βˆ’ 𝑑 2 | | β€– β€– β€– 1 β€– 𝐴 β€– πœ† 𝑑 1 ξ€œ πœ† 𝑑 1 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 1 β€– β€– β€– + ξ€· 𝑑 𝑠 1 βˆ’ 𝑑 2 𝛾 ξ€Έ  β€– β€– π‘₯ 𝑑 1 βˆ’ π‘₯ 𝑑 2 β€– β€– + | | | | 1 πœ† 𝑑 1 βˆ’ 1 πœ† 𝑑 2 | | | | β€– β€– β€– ξ€œ πœ† 𝑑 1 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 2 β€– β€– β€– + 1 𝑑 𝑠 πœ† 𝑑 2 β€– β€– β€– β€– ξ€œ πœ† 𝑑 2 πœ† 𝑑 1 𝑇 ( 𝑠 ) π‘₯ 𝑑 2 β€– β€– β€– β€– ξƒͺ . 𝑑 𝑠 ( 2 . 5 ) Thus applying (2.3), we arrive at β€– β€– π‘₯ 𝑑 1 βˆ’ π‘₯ 𝑑 2 β€– β€– ≀ | | 𝑑 1 βˆ’ 𝑑 2 | | 𝛾 β€– β€– 𝑓 ξ€· π‘₯ 𝑑 1 ξ€Έ β€– β€– + 𝑑 2 β€– β€– π‘₯ 𝛾 𝛼 𝑑 1 βˆ’ π‘₯ 𝑑 2 β€– β€– + | | 𝑑 1 βˆ’ 𝑑 2 | | ξ‚΅ 1 β€– 𝐴 β€– ξ‚Ά + ξ€· 𝛾 βˆ’ 𝛾 𝛼 β€– 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– + β€– 𝑝 β€– 1 βˆ’ 𝑑 2 𝛾 ξ€Έ  β€– β€– π‘₯ 𝑑 1 βˆ’ π‘₯ 𝑑 2 β€– β€– + 2 πœ† 𝑑 2 | | πœ† 𝑑 2 βˆ’ πœ† 𝑑 1 | | ξ‚΅ 1 ( ξ‚Ά ξƒͺ ≀ | | 𝑑 𝛾 βˆ’ 𝛾 𝛼 β€– 𝛾 𝑓 𝑝 ) βˆ’ 𝐴 𝑝 β€– + β€– 𝑝 β€– 1 βˆ’ 𝑑 2 | | ξ‚΅ 𝛾 β€– β€– 𝑓 ξ€· π‘₯ 𝑑 1 ξ€Έ β€– β€– ξ‚΅ 1 + β€– 𝐴 β€– β€– + ξ€· 𝛾 βˆ’ 𝛾 𝛼 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– + β€– 𝑝 β€– ξ‚Ά ξ‚Ά 1 βˆ’ 𝑑 2 ξ€· β€– β€– π‘₯ 𝛾 βˆ’ 𝛾 𝛼 ξ€Έ ξ€Έ 𝑑 1 βˆ’ π‘₯ 𝑑 2 β€– β€– + 2 πœ† 𝑑 2 | | πœ† 𝑑 2 βˆ’ πœ† 𝑑 1 | | ξ‚΅ 1 ξ‚Ά . 𝛾 βˆ’ 𝛾 𝛼 β€– 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– + β€– 𝑝 β€– ( 2 . 6 ) It follows that β€– β€– π‘₯ 𝑑 1 βˆ’ π‘₯ 𝑑 2 β€– β€– ≀ 𝑀 1 | | 𝑑 1 βˆ’ 𝑑 2 | | + 𝑀 2 | | πœ† 𝑑 2 βˆ’ πœ† 𝑑 1 | | , ( 2 . 7 ) where 𝑀 1 = 𝛾 ξ€· ξ€Έ β€– β€– 𝑓 ξ€· π‘₯ 𝛾 βˆ’ 𝛾 𝛼 𝑑 1 ξ€Έ β€– β€– ξ€· + β€– 𝐴 β€– β€– 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– + ξ€Έ 𝛾 βˆ’ 𝛾 𝛼 β€– 𝐴 β€– β€– 𝑝 β€– 𝑑 2 ξ€· ξ€Έ 𝛾 βˆ’ 𝛾 𝛼 2 ( 2 . 8 ) and 𝑀 2 = 2 ξ€· ξ€· β€– 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– + ξ€Έ ξ€Έ 𝛾 βˆ’ 𝛾 𝛼 β€– 𝑝 β€– πœ† 𝑑 2 𝑑 2 ξ€· ξ€Έ 𝛾 βˆ’ 𝛾 𝛼 2 . ( 2 . 9 ) This inequality, together with the continuity of the net { πœ† 𝑑 } , 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† 𝑑 } 0 < 𝑑 < 1 be a net of positive real numbers such that 𝑙 𝑖 π‘š 𝑑 β†’ 0 πœ† 𝑑 = ∞ . 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 β†’ 0 to π‘₯ βˆ— ∈ 𝐹 ( 𝑆 ) , which solves the following variational inequality: ⟨ ( 𝛾 𝑓 βˆ’ 𝐴 ) π‘₯ βˆ— , 𝑝 βˆ’ π‘₯ βˆ— ⟩ ≀ 0 , βˆ€ 𝑝 ∈ 𝐹 ( 𝑆 ) . ( 2 . 1 0 ) Equivalently, one has 𝑃 𝐹 ( 𝑆 ) ( 𝐼 βˆ’ 𝐴 + 𝛾 𝑓 ) π‘₯ βˆ— = π‘₯ βˆ— . ( 2 . 1 1 )

Proof. The uniqueness of the solution of the variational inequality (2.10) is a consequence of the strong monotonicity of 𝐴 βˆ’ 𝛾 𝑓 (Lemma 1.4) and it was proved in [9]. Next, we will use π‘₯ βˆ— ∈ 𝐹 ( 𝑆 ) to denote the unique solution of (2.10). To prove that π‘₯ 𝑑 β†’ π‘₯ βˆ— ( 𝑑 β†’ 0 ) , we write, for a given 𝑝 ∈ 𝐹 ( 𝑆 ) , π‘₯ 𝑑 ξ€· ξ€· π‘₯ βˆ’ 𝑝 = 𝑑 𝛾 𝑓 𝑑 ξ€Έ ξ€Έ ξ‚΅ 1 βˆ’ 𝐴 𝑝 + ( 𝐼 βˆ’ 𝑑 𝐴 ) πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 ξ‚Ά 𝑑 𝑠 βˆ’ 𝑝 . ( 2 . 1 2 ) Using π‘₯ 𝑑 βˆ’ 𝑝 to make inner product, we obtain that β€– β€– π‘₯ 𝑑 β€– β€– βˆ’ 𝑝 2 =  ξ‚΅ 1 ( 𝐼 βˆ’ 𝑑 𝐴 ) πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 ξ‚Ά 𝑑 𝑠 βˆ’ 𝑝 , π‘₯ 𝑑 ξƒ’  ξ€· π‘₯ βˆ’ 𝑝 + 𝑑 𝛾 𝑓 𝑑 ξ€Έ βˆ’ 𝐴 𝑝 , π‘₯ 𝑑  ≀ ξ€· βˆ’ 𝑝 1 βˆ’ 𝑑 𝛾 ξ€Έ β€– β€– π‘₯ 𝑑 β€– β€– βˆ’ 𝑝 2  ξ€· π‘₯ + 𝑑 𝛾 𝑓 𝑑 ξ€Έ βˆ’ 𝐴 𝑝 , π‘₯ 𝑑  . βˆ’ 𝑝 ( 2 . 1 3 ) It follows that β€– β€– π‘₯ 𝑑 β€– β€– βˆ’ 𝑝 2 ≀ 1 𝛾 ξ€· 𝛾  𝑓 ξ€· π‘₯ 𝑑 ξ€Έ βˆ’ 𝑓 ( 𝑝 ) , π‘₯ t  βˆ’ 𝑝 + ⟨ 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 , π‘₯ 𝑑 ξ€Έ ≀ βˆ’ 𝑝 ⟩ 𝛾 𝛼 𝛾 β€– β€– π‘₯ 𝑑 β€– β€– βˆ’ 𝑝 2 + 1 𝛾 ⟨ 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 , π‘₯ 𝑑 βˆ’ 𝑝 ⟩ , ( 2 . 1 4 ) which yields that β€– β€– π‘₯ 𝑑 β€– β€– βˆ’ 𝑝 2 ≀ 1 𝛾 βˆ’ 𝛼 𝛾 ⟨ 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 , π‘₯ 𝑑 βˆ’ 𝑝 ⟩ . ( 2 . 1 5 ) Since 𝐻 is a Hilbert space and { π‘₯ 𝑑 } is bounded as 𝑑 β†’ 0 , we have that if { 𝑑 𝑛 } is a sequence in ( 0 , 1 ) such that 𝑑 𝑛 β†’ 0 and π‘₯ 𝑑 𝑛 ⇀ π‘₯ . By (2.15), we see π‘₯ 𝑑 𝑛 β†’ 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 ( 𝐴 βˆ’ 𝛾 𝑓 ) π‘₯ 𝑑 1 = βˆ’ 𝑑 ξ‚Έ π‘₯ ( 𝐼 βˆ’ 𝑑 𝐴 ) 𝑑 βˆ’ 1 πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 ξ‚Ή 𝑑 𝑠 . ( 2 . 1 6 ) For 𝑝 ∈ 𝐹 ( 𝑆 ) , it follows from (1.22) that ⟨ ( 𝐴 βˆ’ 𝛾 𝑓 ) π‘₯ 𝑑 , π‘₯ 𝑑 1 βˆ’ 𝑝 ⟩ = βˆ’ 𝑑  ξ‚Έ π‘₯ ( 𝐼 βˆ’ 𝑑 𝐴 ) 𝑑 βˆ’ 1 πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 ξ‚Ή 𝑑 𝑠 , π‘₯ 𝑑 ξƒ’ 1 βˆ’ 𝑝 = βˆ’ 𝑑  1 πœ† 𝑑 ξ€œ πœ† 𝑑 0 ξ€Ί ( 𝐼 βˆ’ 𝑇 ( 𝑠 ) ) π‘₯ 𝑑 ξ€» βˆ’ ( 𝐼 βˆ’ 𝑇 ( 𝑠 ) ) 𝑝 𝑑 𝑠 , π‘₯ 𝑑 ξƒ’ +  𝐴 1 βˆ’ 𝑝 πœ† 𝑑 ξ€œ πœ† 𝑑 0 ( 𝐼 βˆ’ 𝑇 ( 𝑠 ) ) π‘₯ 𝑑 𝑑 𝑠 , π‘₯ 𝑑 ξƒ’ 1 βˆ’ 𝑝 = βˆ’ 𝑑 πœ† 𝑑 ξ€œ πœ† 𝑑 0 ⟨ ( 𝐼 βˆ’ 𝑇 ( 𝑠 ) ) π‘₯ 𝑑 βˆ’ ( 𝐼 βˆ’ 𝑇 ( 𝑠 ) ) 𝑝 , π‘₯ 𝑑 +  𝐴 1 βˆ’ 𝑝 ⟩ 𝑑 𝑠 πœ† 𝑑 ξ€œ πœ† 𝑑 0 ( 𝐼 βˆ’ 𝑇 ( 𝑠 ) ) π‘₯ 𝑑 𝑑 𝑠 , π‘₯ 𝑑 ξƒ’ ≀  𝐴 1 βˆ’ 𝑝 πœ† 𝑑 ξ€œ πœ† 𝑑 0 ( 𝐼 βˆ’ 𝑇 ( 𝑠 ) ) π‘₯ 𝑑 𝑑 𝑠 , π‘₯ 𝑑 ξƒ’ =  𝐴 ξ‚΅ ξ€· π‘₯ βˆ’ 𝑝 𝑑 𝛾 𝑓 𝑑 ξ€Έ 1 βˆ’ 𝑑 𝐴 πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 ξ‚Ά 𝑑 𝑠 , π‘₯ 𝑑 ξƒ’  𝐴 ξ‚΅ ξ€· π‘₯ βˆ’ 𝑝 = 𝑑 𝛾 𝑓 𝑑 ξ€Έ 1 βˆ’ 𝐴 πœ† 𝑑 ξ€œ πœ† 𝑑 0 𝑇 ( 𝑠 ) π‘₯ 𝑑 ξ‚Ά 𝑑 𝑠 , π‘₯ 𝑑 ξƒ’ . βˆ’ 𝑝 ( 2 . 1 7 ) Passing to l i m 𝑑 β†’ 0 , since { π‘₯ 𝑑 } is a bounded sequence, we obtain  ( 𝐴 βˆ’ 𝛾 𝑓 ) π‘₯ ,  π‘₯ βˆ’ 𝑝 ≀ 0 , ( 2 . 1 8 ) 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 β†’ 0 ) equals π‘₯ βˆ— . Therefore, π‘₯ 𝑑 β†’ π‘₯ βˆ— as 𝑑 β†’ 0 . The variational inequality (2.10) can be rewritten as  ξ€Ί ( 𝐼 βˆ’ 𝐴 + 𝛾 𝑓 ) π‘₯ βˆ— ξ€» βˆ’ π‘₯ βˆ— , π‘₯ βˆ—  βˆ’ 𝑝 , 𝑝 ∈ 𝐹 ( 𝑆 ) . ( 2 . 1 9 ) This, by Lemma 1.3, is equivalent to 𝑃 𝐹 ( 𝑆 ) ( 𝐼 βˆ’ 𝐴 + 𝛾 𝑓 ) π‘₯ βˆ— = π‘₯ βˆ— . ( 2 . 2 0 ) 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 { 𝑠 𝑛 } be a positive real divergent sequence and let { 𝛼 𝑛 } and { 𝛽 𝑛 } be sequences in ( 0 , 1 ) satisfying the following conditions l i m 𝑛 β†’ ∞ 𝛼 𝑛 = l i m 𝑛 β†’ ∞ 𝛽 𝑛 = 0 and βˆ‘ ∞ 𝑛 = 0 𝛼 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 { π‘₯ 𝑛 } defined by (1.14) strongly converges to π‘₯ βˆ— ∈ 𝐹 ( 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 l i m 𝑛 β†’ ∞ 𝛼 𝑛 = l i m 𝑛 β†’ ∞ 𝛽 𝑛 = 0 , we may assume, with no loss of generality, that 𝛼 𝑛 / ( 1 βˆ’ 𝛽 𝑛 ) < β€– 𝐴 β€– βˆ’ 1 for all 𝑛 β‰₯ 0 . From Lemma 1.6, we know that β€– ( 1 βˆ’ 𝛽 𝑛 ) 𝐼 βˆ’ 𝛼 𝑛 𝐴 β€– ≀ ( 1 βˆ’ 𝛽 𝑛 βˆ’ 𝛼 𝑛 𝛾 ) . Picking 𝑝 ∈ 𝐹 ( 𝑆 ) , we have β€– β€– π‘₯ 𝑛 + 1 β€– β€– = β€– β€– β€– 𝛼 βˆ’ 𝑝 𝑛 ξ€· ξ€· π‘₯ 𝛾 𝑓 𝑛 ξ€Έ ξ€Έ βˆ’ 𝐴 𝑝 + 𝛽 𝑛 ξ€· π‘₯ 𝑛 ξ€Έ + βˆ’ 𝑝 ξ€· ξ€· 1 βˆ’ 𝛽 𝑛 ξ€Έ 𝐼 βˆ’ 𝛼 𝑛 𝐴 ξ€Έ ξ‚΅ 1 𝑠 𝑛 ξ€œ 𝑠 𝑛 0 𝑇 ( 𝑠 ) π‘₯ 𝑛 ξ‚Ά β€– β€– β€– 𝑑 𝑠 βˆ’ 𝑝 ≀ 𝛼 𝑛 β€– β€– ξ€· π‘₯ 𝛾 𝑓 𝑛 ξ€Έ β€– β€– βˆ’ 𝐴 𝑝 + 𝛽 𝑛 β€– β€– π‘₯ 𝑛 β€– β€– + ξ€· βˆ’ 𝑝 1 βˆ’ 𝛽 𝑛 βˆ’ 𝛼 𝑛 𝛾 ξ€Έ β€– β€– β€– 1 𝑠 𝑛 ξ€œ 𝑠 𝑛 0 𝑇 ( 𝑠 ) π‘₯ 𝑛 β€– β€– β€– 𝑑 𝑠 βˆ’ 𝑝 ≀ 𝛼 𝑛 𝛾 β€– β€– 𝑓 ξ€· π‘₯ 𝑛 ξ€Έ β€– β€– βˆ’ 𝑓 ( 𝑝 ) + 𝛼 𝑛 β€– 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– + 𝛽 𝑛 β€– β€– π‘₯ 𝑛 β€– β€– + ξ€· βˆ’ 𝑝 1 βˆ’ 𝛽 𝑛 βˆ’ 𝛼 𝑛 𝛾 ξ€Έ β€– β€– π‘₯ 𝑛 β€– β€– ≀ ξ€Ί βˆ’ 𝑝 1 βˆ’ 𝛼 𝑛 ξ€· β€– β€– π‘₯ 𝛾 βˆ’ 𝛾 𝛼 ξ€Έ ξ€» 𝑛 β€– β€– βˆ’ 𝑝 + 𝛼 𝑛 β€– 𝛾 𝑓 ( 𝑝 ) βˆ’ 𝐴 𝑝 β€– . ( 2 . 2 1 ) By simple inductions, we see that β€– β€– π‘₯ 𝑛 β€– β€– ξ‚» β€– β€– π‘₯ βˆ’ 𝑝 ≀ m a x 0 β€– β€– , ( βˆ’ 𝑝 β€– 𝐴 𝑝 βˆ’ 𝛾 𝑓 𝑝 ) β€– ξ‚Ό 𝛾 βˆ’ 𝛾 𝛼 , ( 2 . 2 2 ) which yields that the sequence { π‘₯ 𝑛 } is bounded.Step 2. Show that l i m s u p 𝑛 β†’ ∞ ⟨ ( 𝛾 𝑓 βˆ’ 𝐴 ) π‘₯ βˆ— , 𝑦 𝑛 βˆ’ π‘₯ βˆ— ⟩ ≀ 0 , ( 2 . 2 3 ) where π‘₯ βˆ— is obtained in Theorem 2.2 and 𝑦 𝑛 = ( 1 / 𝑠 𝑛 ) ∫ 𝑠 𝑛 0 𝑇 ( 𝑠 ) π‘₯ 𝑛 𝑑 𝑠 .
Putting 𝑧 0 = 𝑃 𝐹 ( 𝑆 ) π‘₯ 0 , from (2.22) we see that the closed ball 𝑀 of center 𝑧 0 and radius m a x { β€– 𝑧 0 βˆ’ 𝑝 β€– , β€– 𝐴 𝑧 0 βˆ’ 𝛾 𝑓 ( 𝑧 0 ) β€– / ( 𝛾 βˆ’ 𝛾 𝛼 ) } is 𝑇 ( 𝑠 ) -invariant for each 𝑠 ∈ [ 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 l i m 𝑛 β†’ ∞ β€– β€– 𝑦 𝑛 βˆ’ 𝑇 ( β„Ž ) 𝑦 𝑛 β€– β€– = 0 ( 2 . 2 4 ) for all 0 ≀ β„Ž < ∞ . Taking a suitable subsequence { 𝑦 𝑛 𝑖 } of { 𝑦 𝑛 } , we see that l i m s u p 𝑛 β†’ ∞ ⟨ ( 𝛾 𝑓 βˆ’ 𝐴 ) π‘₯ βˆ— , 𝑦 𝑛 βˆ’ π‘₯ βˆ— ⟩ = l i m 𝑖 β†’ ∞  ( 𝛾 𝑓 βˆ’ 𝐴 ) π‘₯ βˆ— , 𝑦 𝑛 𝑖 βˆ’ π‘₯ βˆ—  . ( 2 . 2 5 ) Since the sequence { 𝑦 𝑛 } is also bounded, we may assume that 𝑦 𝑛 𝑖 ⇀ π‘₯ . From the demiclosedness principle, we have π‘₯ ∈ 𝐹 ( 𝑆 ) . Therefore, we have l i m s u p 𝑛 β†’ ∞ ⟨ ( 𝛾 𝑓 βˆ’ 𝐴 ) π‘₯ βˆ— , 𝑦 𝑛 βˆ’ π‘₯ βˆ—  ⟩ = ( 𝛾 𝑓 βˆ’ 𝐴 ) π‘₯ βˆ— , π‘₯ βˆ’ π‘₯ βˆ—  ≀ 0 . ( 2 . 2 6 ) On the other hand, we have β€– β€– π‘₯ 𝑛 + 1 βˆ’ 𝑦 𝑛 β€– β€– ≀ 𝛼 𝑛 β€– β€– ξ€· π‘₯ 𝛾 𝑓 𝑛 ξ€Έ βˆ’ 𝐴 π‘₯ 𝑛 β€– β€– + 𝛽 𝑛 β€– β€– π‘₯ 𝑛 βˆ’ 𝑦 𝑛 β€– β€– . ( 2 . 2 7 ) From the assumption l i m 𝑛 β†’ ∞ 𝛼 𝑛 = l i m 𝑛 β†’ ∞ 𝛽 𝑛 = 0 , we see that l i m 𝑛 β†’ ∞ β€– β€– π‘₯ 𝑛 + 1 βˆ’ 𝑦 𝑛 β€– β€– = 0 , ( 2 . 2 8 ) which combines with (2.26) gives that l i m s u p 𝑛 β†’ ∞  ( 𝛾 𝑓 βˆ’ 𝐴 ) π‘₯ βˆ— , π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ—  ≀ 0 . ( 2 . 2 9 ) Step 3. Show π‘₯ 𝑛 β†’ π‘₯ βˆ— as 𝑛 β†’ ∞ .
Note that β€– β€– π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ— β€– β€– 2 =  𝛼 𝑛 ξ€· ξ€· π‘₯ 𝛾 𝑓 𝑛 ξ€Έ βˆ’ 𝐴 π‘₯ βˆ— ξ€Έ + 𝛽 𝑛 ξ€· π‘₯ 𝑛 βˆ’ π‘₯ βˆ— ξ€Έ + ξ€· ξ€· 1 βˆ’ 𝛽 𝑛 ξ€Έ 𝐼 βˆ’ 𝛼 𝑛 𝐴 𝑦 ξ€Έ ξ€· 𝑛 βˆ’ π‘₯ βˆ— ξ€Έ , π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ—  = 𝛼 𝑛  ξ€· π‘₯ 𝛾 𝑓 𝑛 ξ€Έ βˆ’ 𝐴 π‘₯ βˆ— , π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ—  + 𝛽 𝑛  π‘₯ 𝑛 βˆ’ π‘₯ βˆ— , π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ—  +  ξ€· ξ€· 1 βˆ’ 𝛽 𝑛 ξ€Έ 𝐼 βˆ’ 𝛼 𝑛 𝐴 𝑦 ξ€Έ ξ€· 𝑛 βˆ’ π‘₯ βˆ— ξ€Έ , π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ—  ≀ 𝛼 𝑛 ξ€· 𝛾  𝑓 ξ€· π‘₯ 𝑛 ξ€Έ ξ€· π‘₯ βˆ’ 𝑓 βˆ— ξ€Έ , π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ—  +  ξ€· π‘₯ 𝛾 𝑓 βˆ— ξ€Έ βˆ’ 𝐴 π‘₯ βˆ— , π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ—  ξ€Έ + 𝛽 𝑛 β€– β€– π‘₯ 𝑛 βˆ’ π‘₯ βˆ— β€– β€– β€– β€– π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ— β€– β€– + β€– β€– ξ€· 1 βˆ’ 𝛽 𝑛 ξ€Έ 𝐼 βˆ’ 𝛼 𝑛 𝐴 β€– β€– β€– β€– 𝑦 𝑛 βˆ’ π‘₯ βˆ— β€– β€– β€– β€– π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ— β€– β€– ≀ 𝛼 𝑛 β€– β€– π‘₯ 𝛼 𝛾 𝑛 βˆ’ π‘₯ βˆ— β€– β€– β€– β€– π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ— β€– β€– + 𝛼 𝑛  ξ€· π‘₯ 𝛾 𝑓 βˆ— ξ€Έ βˆ’ 𝐴 π‘₯ βˆ— , π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ—  + 𝛽 𝑛 β€– β€– π‘₯ 𝑛 βˆ’ π‘₯ βˆ— β€– β€– β€– β€– π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ— β€– β€– + ξ€· 1 βˆ’ 𝛽 𝑛 βˆ’ 𝛼 𝑛 𝛾 ξ€Έ β€– β€– π‘₯ 𝑛 βˆ’ π‘₯ βˆ— β€– β€– β€– β€– π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ— β€– β€– = ξ€Ί 1 βˆ’ 𝛼 𝑛 ξ€· β€– β€– π‘₯ 𝛾 βˆ’ 𝛾 𝛼 ξ€Έ ξ€» 𝑛 βˆ’ π‘₯ βˆ— β€– β€– β€– β€– π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ— β€– β€– + 𝛼 𝑛  ξ€· π‘₯ 𝛾 𝑓 βˆ— ξ€Έ βˆ’ 𝐴 π‘₯ βˆ— , π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ—  ≀ 1 βˆ’ 𝛼 𝑛 ξ€· ξ€Έ 𝛾 βˆ’ 𝛾 𝛼 2 ξ‚€ β€– β€– π‘₯ 𝑛 βˆ’ π‘₯ βˆ— β€– β€– 2 + β€– β€– π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ— β€– β€– 2  + 𝛼 𝑛  ξ€· π‘₯ 𝛾 𝑓 βˆ— ξ€Έ βˆ’ 𝐴 π‘₯ βˆ— , π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ—  . ≀ 1 βˆ’ 𝛼 𝑛 ξ€· ξ€Έ 𝛾 βˆ’ 𝛾 𝛼 2 β€– β€– π‘₯ 𝑛 βˆ’ π‘₯ βˆ— β€– β€– 2 + 1 2 β€– β€– π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ— β€– β€– 2 + 𝛼 𝑛  ξ€· π‘₯ 𝛾 𝑓 βˆ— ξ€Έ βˆ’ 𝐴 π‘₯ βˆ— , π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ—  . ( 2 . 3 0 ) It follows that β€– β€– π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ— β€– β€– 2 ≀ ξ€Ί 1 βˆ’ 𝛼 𝑛 ξ€· β€– β€– π‘₯ 𝛾 βˆ’ 𝛾 𝛼 ξ€Έ ξ€» 𝑛 βˆ’ π‘₯ βˆ— β€– β€– 2 + 2 𝛼 𝑛  ξ€· π‘₯ 𝛾 𝑓 βˆ— ξ€Έ βˆ’ 𝐴 π‘₯ βˆ— , π‘₯ 𝑛 + 1 βˆ’ π‘₯ βˆ—  . ( 2 . 3 1 ) 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 ) β‰  βˆ… . Let { 𝑠 𝑛 } be a positive real divergent sequence and let { 𝛼 𝑛 } be a sequence in ( 0 , 1 ) satisfying the following conditions l i m 𝑛 β†’ ∞ 𝛼 𝑛 = 0 and βˆ‘ ∞ 𝑛 = 0 𝛼 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 { π‘₯ 𝑛 } be a sequence generated by the following manner: π‘₯ 0 ∈ 𝐻 , π‘₯ 𝑛 + 1 = 𝛼 𝑛 ξ€· π‘₯ 𝛾 𝑓 𝑛 ξ€Έ + ξ€· 𝐼 βˆ’ 𝛼 𝑛 𝐴 ξ€Έ 1 𝑠 𝑛 ξ€œ 𝑠 𝑛 0 𝑇 ( 𝑠 ) π‘₯ 𝑛 𝑑 𝑠 , 𝑛 β‰₯ 0 . ( 2 . 3 2 ) Then the sequence { π‘₯ 𝑛 } defined by above iterative algorithm converges strongly to π‘₯ βˆ— ∈ 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).

References

  1. F. E. Browder, β€œNonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, pp. 1041–1044, 1965. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  2. F. E. Browder, β€œConvergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, pp. 82–90, 1967.
  3. S. Reich, β€œStrong convergence theorems for resolvents of accretive operators in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287–292, 1980. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  4. B. Halpern, β€œFixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957–961, 1967. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  5. P.-L. Lions, β€œApproximation de points fixes de contractions,” Comptes Rendus de l'Académie des Sciences, vol. 284, no. 21, pp. A1357–A1359, 1977. View at Zentralblatt MATH
  6. R. Wittmann, β€œApproximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol. 58, no. 5, pp. 486–491, 1992. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  7. F. Deutsch and I. Yamada, β€œMinimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 33–56, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  8. S. Li, L. Li, and Y. Su, β€œGeneral iterative methods for a one-parameter nonexpansive semigroup in Hilbert space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 9, pp. 3065–3071, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  9. G. Marino and H.-K. Xu, β€œA general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  10. G. Marino, V. Colao, X. Qin, and S. M. Kang, β€œStrong convergence of the modified Mann iterative method for strict pseudo-contractions,” Computers & Mathematics with Applications, vol. 57, no. 3, pp. 455–465, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. H. K. Xu, β€œAn iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  12. H.-K. Xu, β€œIterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  13. I. Yamada, N. Ogura, Y. Yamashita, and K. Sakaniwa, β€œQuadratic optimization of fixed points of nonexpansive mappings in Hilbert space,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 165–190, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  14. S. Plubtieng and R. Punpaeng, β€œFixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces,” Mathematical and Computer Modelling, vol. 48, no. 1-2, pp. 279–286, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  15. N. Shioji and W. Takahashi, β€œStrong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 34, no. 1, pp. 87–99, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  16. T. Shimizu and W. Takahashi, β€œStrong convergence to common fixed points of families of nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 71–83, 1997. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  17. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28, Cambridge University Press, Cambridge, UK, 1990. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  18. S.-S. Chang, Y. J. Cho, and H. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach Spaces, Nova Science Publishers, Huntington, NY, USA, 2002.