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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012, Article IDΒ 986426, 16 pages
http://dx.doi.org/10.1155/2012/986426
Research Article

Convergence Theorem for a Family of Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces

1Department of Mathematical Sciences, Bayero University, P.M.B. 3011 Kano, Nigeria
2Department of Mathematics, University of Nigeria, Nsukka, Nigeria

Received 15 March 2012; Revised 8 June 2012; Accepted 8 June 2012

Academic Editor: Ram U.Β Verma

Copyright Β© 2012 Bashir Ali and G. C. Ugwunnadi. 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 𝐸 be a real reflexive and strictly convex Banach space with a uniformly GΓ’teaux differentiable norm. Let 𝔍={𝑇(𝑑)βˆΆπ‘‘β‰₯0} be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of 𝐸, with functions 𝑒,π‘£βˆΆ[0,∞)β†’[0,∞). Let 𝐹∢=𝐹(𝔍)=βˆ©π‘‘β‰₯0𝐹(𝑇(𝑑))β‰ βˆ… and π‘“βˆΆπΎβ†’πΎ be a weakly contractive map. For some positive real numbers πœ† and 𝛿 satisfying 𝛿+πœ†>1, let πΊβˆΆπΈβ†’πΈ be a 𝛿-strongly accretive and πœ†-strictly pseudocontractive map. Let {𝑑𝑛} be an increasing sequence in [0,∞) with limπ‘›β†’βˆžπ‘‘π‘›=∞, and let {𝛼𝑛} and {𝛽𝑛} be sequences in (0,1] satisfying some conditions. Strong convergence of a viscosity iterative sequence to common fixed points of the family 𝔍 of uniformly asymptotically regular asymptotically nonexpansive semigroup, which also solves the variational inequality ⟨(πΊβˆ’π›Ύπ‘“)𝑝,𝑗(π‘βˆ’π‘₯)βŸ©β‰€0, for all π‘₯∈𝐹, is proved in a framework of a real Banach space.

1. Introduction

Let 𝐸 be a real Banach space. We denote by 𝐽 the normalized duality map from 𝐸 to 2πΈβˆ— (πΈβˆ— is the dual space of 𝐸), and it is defined by 𝐽(π‘₯)=π‘“βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘“βŸ©=β€–π‘₯β€–2=‖𝑓‖2ξ€Ύ.(1.1)

A mapping π‘‡βˆΆπΈβ†’πΈ is said to be contractive if ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€π›Όβ€–π‘₯βˆ’π‘¦β€–, for π‘₯,π‘¦βˆˆπΈ, and some constant π›Όβˆˆ[0,1). It is said to be weakly contractive if there exists a nondecreasing function πœ“βˆΆ[0,∞)β†’[0,∞) satisfying πœ“(𝑑)=0 if and only if 𝑑=0 and ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–βˆ’πœ“(β€–π‘₯βˆ’π‘¦β€–), for all π‘₯,π‘¦βˆˆπΈ. It is known that the class of weakly contractive maps contain properly the class of contractive ones, see [1, 2]. A mapping π‘‡βˆΆπΈβ†’πΈ is said to be nonexpansive if ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–, for all π‘₯,π‘¦βˆˆπΈ and asymptotically nonexpansive if there exists a sequence {π‘˜π‘›}βŠ‚[1,∞) with π‘˜π‘›β†’1 as π‘›β†’βˆž and ‖𝑇𝑛π‘₯βˆ’π‘‡π‘›π‘¦β€–β‰€π‘˜π‘›β€–π‘₯βˆ’π‘¦β€–, for all π‘₯,π‘¦βˆˆπΈ. We denote by 𝐹(𝑇)={π‘₯βˆˆπΎβˆΆπ‘‡π‘₯=π‘₯} the set of fixed points of a map 𝑇.

A mapping π‘‡βˆΆπΈβ†’πΈ is said to be total asymptotically nonexpansive (see [3]) if there exist nonnegative real sequences {𝑒𝑛} and {𝑣𝑛}, with 𝑒𝑛→0 and 𝑣𝑛→0 as π‘›β†’βˆž and strictly increasing and continuous functions πœ“βˆΆβ„+→ℝ+ with πœ“(0)=0 such that ‖𝑇𝑛π‘₯βˆ’π‘‡π‘›π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–+π‘’π‘›πœ“(β€–π‘₯βˆ’π‘¦β€–)+𝑣𝑛,βˆ€π‘₯,π‘¦βˆˆπΎ.(1.2)

Remark 1.1. If πœ“(πœ†)=πœ†, the total asymptotically nonexpansive mapping coincides with generalized asymptotically nonexpansive mapping. In addition, for all π‘›βˆˆβ„•, if 𝑣𝑛=0, then generalized asymptotically nonexpansive mapping coincides with asymptotically nonexpansive mapping; if 𝑒𝑛=0,𝑣𝑛=max{0,𝑝𝑛} where π‘π‘›βˆΆ=supπ‘₯,π‘¦βˆˆπΎ(‖𝑇𝑛π‘₯βˆ’π‘‡π‘›π‘¦β€–βˆ’β€–π‘₯βˆ’π‘¦β€–), then generalized asymptotically nonexpansive mapping coincide with asymptotically nonexpansive mapping in the intermediate sense; if 𝑒𝑛=0, and 𝑣𝑛=0 then we obtain from (1.2) the class of nonexpansive mapping.
A one-parameter family of generalized asymptotically nonexpansive semigroup is a family 𝔍={𝑇(𝑑)βˆΆπ‘‘β‰₯0} of self-mapping of 𝐸 such that (i)𝑇(0)π‘₯=π‘₯ for π‘₯∈𝐸,(ii)𝑇(𝑠+𝑑)π‘₯=𝑇(𝑠)𝑇(𝑑)π‘₯ for all 𝑑,𝑠β‰₯0 and π‘₯∈𝐸, (iii)lim𝑑→0𝑇(𝑑)π‘₯=π‘₯ for π‘₯∈𝐸, (iv)there exist functions 𝑒,π‘£βˆΆ[0,∞)β†’[0,∞) such that 𝑒(𝑑)β†’0,𝑣(𝑑)β†’0 as π‘‘β†’βˆž, and ‖𝑇(𝑑)π‘₯βˆ’π‘‡(𝑑)𝑦‖≀(1+𝑒(𝑑))β€–π‘₯βˆ’π‘¦β€–+𝑣(𝑑)βˆ€π‘₯,π‘¦βˆˆπΈ.(1.3)
We will denote by 𝐹 the common fixed-point set of 𝔍, that is, ξ™πΉβˆΆ=Fix(𝔍)={π‘₯βˆˆπΈβˆΆπ‘‡(𝑑)π‘₯=π‘₯,𝑑β‰₯0}=𝑑β‰₯0Fix(𝑇(𝑑)).(1.4)
The family 𝔍={𝑇(𝑑)βˆΆπ‘‘β‰₯0} is said to be asymptotically regular if limπ‘ β†’βˆžβ€–π‘‡(𝑠+𝑑)π‘₯βˆ’π‘‡(𝑠)π‘₯β€–=0,(1.5) for all π‘‘βˆˆ[0,∞) and π‘₯∈𝐸. It is said to be uniformly asymptotically regular if, for any 𝑑β‰₯0 and for any bounded subset 𝐢 of 𝐸, limπ‘ β†’βˆžsupπ‘₯βˆˆπΆβ€–π‘‡(𝑠+𝑑)π‘₯βˆ’π‘‡(𝑠)π‘₯β€–=0.(1.6)
For some positive real numbers 𝛿 and πœ†, a mapping πΊβˆΆπΈβ†’πΈ is said to be 𝛿-strongly accretive if for any π‘₯,π‘¦βˆˆπΈ, there exists 𝑗(π‘₯βˆ’π‘¦)∈𝐽(π‘₯βˆ’π‘¦) such that ⟨𝐺π‘₯βˆ’πΊπ‘¦,𝑗(π‘₯βˆ’π‘¦)⟩β‰₯𝛿‖π‘₯βˆ’π‘¦β€–2,(1.7) and it is called πœ†-strictly pseudocontractive if ⟨𝐺π‘₯βˆ’πΊπ‘¦,𝑗(π‘₯βˆ’π‘¦)βŸ©β‰€β€–π‘₯βˆ’π‘¦β€–2βˆ’πœ†β€–(πΌβˆ’πΊ)π‘₯βˆ’(πΌβˆ’πΊ)𝑦‖2.(1.8)
Let 𝐸 be a real Banach space, and let 𝛿,πœ†, and 𝜏 be positive real numbers satisfying 𝛿+πœ†>1 and 𝜏∈(0,1). Let πΊβˆΆπΈβ†’πΈ be a 𝛿-strongly accretive and πœ†-strictly pseudocontractive, then the following holds, see [4], for π‘₯,π‘¦βˆˆπΈβˆΆβ€–β€–β‰€ξƒ©ξ‚™(πΌβˆ’πΊ)π‘₯βˆ’(πΌβˆ’πΊ)𝑦1βˆ’π›Ώπœ†ξƒͺ‖‖π‘₯βˆ’π‘¦β€–,(πΌβˆ’πœπΊ)π‘₯βˆ’(πΌβˆ’πœπΊ)𝑦‖≀1βˆ’πœ1βˆ’1βˆ’π›Ώπœ†ξƒͺβ€–π‘₯βˆ’π‘¦β€–,(1.9) that is, (πΌβˆ’πΊ) and (πΌβˆ’πœπΊ) are contractive mappings.
Let 𝐢 be a nonempty closed-convex subset of 𝐸 and π‘‡βˆΆπΈβ†’πΈ a map. Then, a variational inequality problem with respect to 𝐢 and 𝑇 is found to be π‘₯βˆ—βˆˆπΆ such that 𝑇π‘₯βˆ—ξ€·,π‘—π‘¦βˆ’π‘₯βˆ—ξ€·ξ€Έξ¬β‰₯0,βˆ€π‘¦βˆˆπΆ,π‘—π‘¦βˆ’π‘₯βˆ—ξ€Έξ€·βˆˆπ½π‘¦βˆ’π‘₯βˆ—ξ€Έ.(1.10)
Recently, convergence theorems for fixed points of nonexpansive mappings, common fixed points of family of nonexpansive mappings, nonexpansive semigroup, and their generalisation have been studied by numerous authors (see, e.g., [5–21]).
Acedo and Suzuki [22], recently, proved the strong convergence of the Browder's implicit scheme, π‘₯0,π‘’βˆˆπΆ, π‘₯𝑛=𝛼𝑛𝑒+1βˆ’π›Όπ‘›ξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯𝑛,𝑛β‰₯0,(1.11) to a common fixed point of a uniformly asymptotically regular family {𝑇(𝑑)βˆΆπ‘‘β‰₯0} of nonexpansive semigroup in the framework of a real Hilbert space.
Li et al. [23] proved strong convergence theorems for implicit viscosity schemes for common fixed points of family of generalized asymptotically nonexpansive semigroups in Banach spaces.
Let 𝑆 be a semigroup and 𝐡(𝑆) the subspace of all bounded real-valued functions defined on 𝑆 with supremum norm. For each π‘ βˆˆπ‘†, the left translator operator 𝑙(𝑠) on 𝐡(𝑆) is defined by (𝑙(𝑠)𝑓)(𝑑)=𝑓(𝑠𝑑) for each π‘‘βˆˆπ‘† and π‘“βˆˆπ΅(𝑆). Let 𝑋 be a subspace of 𝐡(𝑆) containing 1, and let π‘‹βˆ— be its topological dual. An element πœ‡ of π‘‹βˆ— is said to be a mean on 𝑋 if β€–πœ‡β€–=πœ‡(1)=1. Let 𝑋 be 𝑙𝑠 invariant, that is, 𝑙𝑠(𝑋)βŠ‚π‘‹ for each π‘ βˆˆπ‘†. A mean πœ‡ on 𝑋 is said to be left invariant if πœ‡(𝑙𝑠𝑓)=πœ‡(𝑓) for each π‘ βˆˆπ‘† and π‘“βˆˆπ‘‹.
Recently, Saeidi and Naseri [24] studied the problem of approximating common fixed point of a family of nonexpansive semigroup and solution of some variational inequality problem in a real Hilbert space. They proved the following theorem.

Theorem 1.2 (Saeidi and Naseri [24]). Let 𝔍={𝑇(𝑑)βˆΆπ‘‘βˆˆπ‘†} be a nonexpansive semigroup in a real Hilbert space 𝐻 such that 𝐹(𝔍)β‰ βˆ…. Let 𝑋 be a left invariant subspace of 𝐡(𝑆) such that 1βˆˆπ‘‹, and the function π‘‘β†’βŸ¨π‘‡(𝑑)π‘₯,π‘¦βŸ© is an element of 𝑋 for each π‘₯,π‘¦βˆˆπ». Let π‘“βˆΆπΈβ†’πΈ be a contraction with constant 𝛼, and let πΊβˆΆπ»β†’π» be strongly positive map with constant 𝛾>0. Let {πœ‡π‘›} be a left regular sequence of means on 𝑋, and let {𝛼𝑛} be a sequence in (0,1) such that limπ‘›β†’βˆžπ›Όπ‘›=0 and βˆ‘βˆžπ‘›=1𝛼𝑛=∞. Let π›Ύβˆˆ(0,𝛾/𝛼), and let {π‘₯𝑛} be a sequence generated by π‘₯0∈𝐻, π‘₯𝑛+1=ξ€·πΌβˆ’π›Όπ‘›πΊξ€Έπ‘‡ξ€·πœ‡π‘›ξ€Έπ‘₯𝑛+𝛼𝑛π‘₯𝛾𝑓𝑛,𝑛β‰₯0.(1.12)
Then, {π‘₯𝑛} converges strongly to a common fixed point of the family 𝔍 which is the unique solution of the variational inequality ⟨(πΊβˆ’π›Ύπ‘“)π‘₯βˆ—,𝑗(π‘₯βˆ’π‘₯βˆ—)⟩β‰₯0 for all π‘₯∈𝐹(𝔍). Equivalently one has 𝑃𝐹(𝔍)(πΌβˆ’πΊ+𝛾𝑓)π‘₯βˆ—=π‘₯βˆ—.

More recently, as commented by Golkarmanesh and Naseri [25], Piri and Vaezi [4] gave a minor variation of Theorem 1.2 as follows.

Theorem 1.3 (Piri and Vaezi [4]). Let 𝔍={𝑇(𝑑)βˆΆπ‘‘βˆˆπ‘†} be a nonexpansive semigroup on a real Hilbert space 𝐻 such that 𝐹(𝔍)β‰ βˆ…. Let 𝑋 be a left invariant subspace of 𝐡(𝑆) such that 1βˆˆπ‘‹, and the function π‘‘β†’βŸ¨π‘‡(𝑑)π‘₯,π‘¦βŸ© is an element of 𝑋 for each π‘₯,π‘¦βˆˆπ». Let π‘“βˆΆπΈβ†’πΈ be a contraction with constant 𝛼, and let πΊβˆΆπ»β†’π» be 𝛿-strongly accretive and πœ†-strictly pseudocontractive with 𝛿+πœ†>1. Let {πœ‡π‘›} be a left regular sequence of means on 𝑋, and let {𝛼𝑛} be a sequence in (0,1) such that limπ‘›β†’βˆžπ›Όπ‘›=0 and βˆ‘βˆžπ‘›=1𝛼𝑛=∞. Let {π‘₯𝑛} be a sequence generated by π‘₯0∈𝐻, π‘₯𝑛+1=ξ€·πΌβˆ’π›Όπ‘›πΊξ€Έπ‘‡ξ€·πœ‡π‘›ξ€Έπ‘₯𝑛+𝛼𝑛π‘₯𝛾𝑓𝑛,𝑛β‰₯0,(1.13) where √0<𝛾<(1βˆ’(1βˆ’π›Ώ/πœ†))/𝛼, then, {π‘₯𝑛} converges strongly to a common fixed point of the family 𝐹(𝔍) which is the unique solution of the variational inequality ⟨(πΊβˆ’π›Ύπ‘“)π‘₯βˆ—,𝑗(π‘₯βˆ’π‘₯βˆ—)⟩β‰₯0 for all π‘₯∈𝐹(𝔍). Equivalently one has 𝑃𝐹(𝔍)(πΌβˆ’πΊ+𝛾𝑓)π‘₯βˆ—=π‘₯βˆ—.

Very recently, Ali [26] continued the study of the problem in [4, 24] and proved a strong convergence theorem in a Banach space setting much more general than Hilbert space. He actually proved the following theorem.

Theorem 1.4 (Ali [26]). Let 𝐸 be a real Banach space with local uniform Opial's property whose duality mapping is sequentially continuous. Let 𝔍={𝑇(𝑑)βˆΆπ‘‘β‰₯0} be a uniformly asymptotically regular family of asymptotically nonexpansive semigroup of 𝐸 with function π‘˜βˆΆ[0,∞)β†’[0,∞) and 𝐹∢=𝐹(𝔍)=βˆ©π‘‘β‰₯0𝐹(𝑇(𝑑))β‰ βˆ…. Let π‘“βˆΆπΈβ†’πΈ be weakly contractive, and let πΊβˆΆπΈβ†’πΈ be 𝛿-strongly accretive and πœ†-strictly pseudocontractive with 𝛿+πœ†>1. Let βˆšπœ‚βˆΆ=(1βˆ’(1βˆ’π›Ώ)/πœ†) and π›Ύβˆˆ(0,min{πœ‚,𝛿/2}). Let {𝛽𝑛} and {𝛼𝑛} be sequences in (0,1], and let {𝑑𝑛} be an increasing sequence in [0,∞) satisfying the following conditions: limπ‘›β†’βˆžπ›Όπ‘›=0,limπ‘›β†’βˆžπ‘˜π‘›π›Όπ‘›=0,βˆžξ“π‘›=1𝛼𝑛=∞,0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1.(1.14)
Define a sequence {π‘₯𝑛} by π‘₯0∈𝐸, π‘₯𝑛+1=𝛽𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘¦π‘›,𝑦𝑛=ξ€·πΌβˆ’π›Όπ‘›πΊξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯𝑛+𝛼𝑛𝛾𝑛𝑓π‘₯𝑛,𝑛β‰₯0.(1.15)
Then, the sequence {π‘₯𝑛} converges strongly to a common fixed point of the family 𝔍 which solves the variational inequality ⟨(πΊβˆ’π›Ύπ‘“)π‘ž,𝑗(π‘₯βˆ’π‘ž)⟩β‰₯0,βˆ€π‘₯∈𝐹.(1.16)

Remark 1.5. It is well known that all 𝑙𝑝(1<𝑝<∞) spaces satisfy Opial's condition and possess a weakly sequentially continuous duality mapping. However, 𝐿𝑝  (1<𝑝<∞) spaces and consequently all Sobolev spaces do not satisfy either of the properties.
It is our purpose in this paper to prove a strong convergence theorem for approximating common fixed points of family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup in a real reflexive and strictly convex Banach space 𝐸 with a uniformly GΓ’teaux differentiable norm. Our theorem is applicable in 𝐿𝑝(ℓ𝑝) spaces, 1<𝑝<∞ (and consequently in sobolev spaces). Our theorem extends and improves some recent important results. For instance, our theorem presents a convergence of an explicit scheme that extends Theorem 1.4 to a more general setting of Banach spaces that includes 𝐿𝑝(1<𝑝<∞) spaces on one hand and for more general class of maps on the other hand.

2. Preliminaries

Let π‘†βˆΆ={π‘₯∈𝐸∢||π‘₯||=1} denote the unit sphere of a real Banach space 𝐸. 𝐸 is said to have a GΓ’teaux differentiable norm if the limit lim𝑑→0β€–π‘₯+π‘‘π‘¦β€–βˆ’β€–π‘₯‖𝑑(2.1) exists for each π‘₯,π‘¦βˆˆπ‘†; 𝐸 is said to have a uniformly GΓ’teaux differentiable norm if for each π‘¦βˆˆπ‘†, the limit is attained uniformly for π‘₯βˆˆπ‘†. A Banach space 𝐸 is said to be strictly convex if β€–π‘₯+𝑦‖/2<1 for π‘₯≠𝑦 and β€–π‘₯β€–=‖𝑦‖=1.

Let 𝐾 be a nonempty, closed, convex, and bounded subset of a real Banach space 𝐸, and let the diameter of 𝐾 be defined by 𝑑(𝐾)∢=sup{β€–π‘₯βˆ’π‘¦β€–βˆΆπ‘₯,π‘¦βˆˆπΎ}. For each π‘₯∈𝐾, let π‘Ÿ(π‘₯,𝐾)∢=sup{β€–π‘₯βˆ’π‘¦β€–βˆΆπ‘¦βˆˆπΎ} and π‘Ÿ(𝐾)∢=inf{π‘Ÿ(π‘₯,𝐾)∢π‘₯∈𝐾} denote the Chebyshev radius of 𝐾 relative to itself. The normal structure coefficient 𝑁(𝐸) of 𝐸 (introduced in 1980 by Bynum [27], see also Lim [28] and the references contained therein) is defined by 𝑁(𝐸)∢=inf{(𝑑(𝐾)/π‘Ÿ(𝐾)): 𝐾 is a closed convex and bounded subset of 𝐸 with 𝑑(𝐾)>0}. A space 𝐸 such that 𝑁(𝐸)>1 is said to have uniform normal structures . It is known that every space with a uniform normal structure is reflexive, and that all uniformly convex and uniformly smooth Banach spaces have uniform normal structure (see, e.g., [29]).

Let 𝐸 be a real Banach space with uniformly GΓ’teaux differentiable norm, then the normalized duality mapping π½βˆΆπΈβ†’2πΈβˆ—, defined by (1.1), is singled valued and uniformly continuous from the norm topology of 𝐸 to the weakβˆ— topology of πΈβˆ— on each bounded subset of 𝐸, see, for example [30].

Definition 2.1. Let πœ‡ be a continuous linear functional on π‘™βˆž, and let (π‘Ž0,π‘Ž1,…)βˆˆπ‘™βˆž. We write πœ‡π‘›(π‘Žπ‘›) instead of πœ‡(π‘Ž0,π‘Ž1,…). The function πœ‡ is called a Banach limit when πœ‡ satisfies ||πœ‡||=πœ‡π‘›(1)=1 and πœ‡π‘›(π‘Žπ‘›+1)=πœ‡π‘›(π‘Žπ‘›) for each (π‘Ž0,π‘Ž1,…)βˆˆπ‘™βˆž.
For a Banach limit πœ‡, it is known that liminfπ‘›β†’βˆžπ‘Žπ‘›β‰€πœ‡π‘›(π‘Žπ‘›)≀limsupπ‘›β†’βˆžπ‘Žπ‘› for every π‘Ž=(π‘Ž0,π‘Ž1,…)βˆˆπ‘™βˆž. So if π‘Ž=(π‘Ž0,π‘Ž1,…)βˆˆπ‘™βˆž and π‘Žπ‘›βˆ’π‘π‘›β†’0 as π‘›β†’βˆž, we have πœ‡π‘›(π‘Žπ‘›)=πœ‡π‘›(𝑏𝑛).
We will make use of the following well-known result.

Lemma 2.2. Let 𝐸 be a real-normed linear space. Then, the following inequality holds: β€–π‘₯+𝑦‖2≀‖π‘₯β€–2+2βŸ¨π‘¦,𝑗(π‘₯+𝑦)βŸ©βˆ€π‘₯,π‘¦βˆˆπΈ,𝑗(π‘₯+𝑦)∈𝐽(π‘₯+𝑦).(2.2)

In the sequel, we shall also make use of the following lemmas.

Lemma 2.3 (Suzuki [31]). Let {π‘₯𝑛} and {𝑦𝑛} be bounded sequences in a real Banach space 𝐸, and let {𝛽𝑛} be a sequence in [0,1] with 0<liminf𝛽𝑛≀limsup𝛽𝑛<1. Suppose that π‘₯𝑛+1=𝛽𝑛𝑦𝑛+(1βˆ’π›½π‘›)π‘₯𝑛 for all integer 𝑛β‰₯1 and limsupπ‘›β†’βˆž(||𝑦𝑛+1βˆ’π‘¦π‘›||βˆ’||π‘₯𝑛+1βˆ’π‘₯𝑛||)≀0. Then, limπ‘›β†’βˆž||π‘¦π‘›βˆ’π‘₯𝑛||=0.

Lemma 2.4 (Shioji and Takahashi [32]). Let (π‘Ž0,π‘Ž1,π‘Ž2,…)βˆˆπ‘™βˆž be such that πœ‡π‘›π‘Žπ‘›β‰€0 for all Banach limits πœ‡. If limsupπ‘›β†’βˆž(π‘Žπ‘›+1βˆ’π‘Žπ‘›)≀0, then limsupπ‘›β†’βˆžπ‘Žπ‘›β‰€0.

Lemma 2.5 (Xu [33]). Let {π‘Žπ‘›} be a sequence of nonnegative real numbers satisfying the following relation: π‘Žπ‘›+1≀1βˆ’π›Όπ‘›ξ€Έπ‘Žπ‘›+π›Όπ‘›πœŽπ‘›+𝛾𝑛,𝑛β‰₯0,(2.3) where (i){π›Όπ‘›βˆ‘}βŠ‚[0,1],βˆžπ‘›=0𝛼𝑛=βˆžβ€‰β€‰(ii)limsupπ‘›β†’βˆžπœŽπ‘›β‰€0   (iii)𝛾𝑛β‰₯0 and βˆ‘(𝑛β‰₯0),βˆžπ‘›=0𝛾𝑛<∞. Then, π‘Žπ‘›β†’0 as π‘›β†’βˆž.

3. Main Results

Theorem 3.1. Let 𝐸 be a real reflexive and strictly convex Banach space with a uniformly GΓ’teaux differentiable norm, and let 𝔍={𝑇(𝑑)βˆΆπ‘‘β‰₯0} be uniformly asymptotically regular family of generalized asymptotically nonexpansive semigroup of 𝐸, with functions 𝑒,π‘£βˆΆ[0,∞)β†’[0,∞) and 𝐹∢=𝐹(𝔍)=βˆ©π‘‘β‰₯0𝐹(𝑇(𝑑))β‰ βˆ…. Let π‘“βˆΆπΈβ†’πΈ be weakly contractive, and let πΊβˆΆπΈβ†’πΈ be 𝛿-strongly accretive and πœ†-strictly pseudocontractive with 𝛿+πœ†>1. Let βˆšπœ‚βˆΆ=(1βˆ’(1βˆ’π›Ώ)/πœ†) and π›Ύβˆˆ(0,min{𝛿,πœ‚/2}). Let {𝛽𝑛} and {𝛼𝑛} be sequences in (0,1] and {𝑑𝑛} an increasing sequence in [0,∞) satisfying the following conditions: limπ‘›β†’βˆžπ›Όπ‘›=0,limπ‘›β†’βˆžπ‘’ξ€·π‘‘π‘›ξ€Έπ›Όπ‘›=0,limπ‘›β†’βˆžπ‘£ξ€·π‘‘π‘›ξ€Έπ›Όπ‘›=0,βˆžξ“π‘›=1𝛼𝑛=∞,0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1,limπ‘›β†’βˆžπ‘‘π‘›=∞.(3.1)
Define a sequence {π‘₯𝑛} by π‘₯0∈𝐸, π‘₯𝑛+1=𝛽𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘¦π‘›,𝑦𝑛=ξ€·πΌβˆ’π›Όπ‘›πΊξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯𝑛+𝛼𝑛π‘₯𝛾𝑓𝑛,𝑛β‰₯0.(3.2)
Then, the sequence {π‘₯𝑛} converges strongly to a common fixed point of the family 𝔍 which solves the variational inequality ⟨(πΊβˆ’π›Ύπ‘“)π‘ž,𝑗(π‘₯βˆ’π‘ž)⟩β‰₯0,βˆ€π‘₯∈𝐹.(3.3)

Proof. We start by showing that solution of the variational inequality (3.3) in 𝐹 is at most one. Assume that π‘ž,π‘βˆˆπΉ are solutions of the variational inequality (3.3), then ⟨(πΊβˆ’π›Ύπ‘“)𝑝,𝑗(π‘žβˆ’π‘)⟩β‰₯0,⟨(πΊβˆ’π›Ύπ‘“)π‘ž,𝑗(π‘βˆ’π‘ž)⟩β‰₯0.(3.4)
Adding these two inequalities, we get ⟨(πΊβˆ’π›Ύπ‘“)π‘βˆ’(πΊβˆ’π›Ύπ‘“)π‘ž,𝑗(π‘βˆ’π‘ž)βŸ©β‰€0.(3.5)
Therefore, 0β‰₯⟨(πΊβˆ’π›Ύπ‘“)π‘βˆ’(πΊβˆ’π›Ύπ‘“)π‘ž,𝑗(π‘βˆ’π‘ž)⟩=⟨𝐺(𝑝)βˆ’πΊ(π‘ž),𝑗(π‘βˆ’π‘ž)βŸ©βˆ’π›ΎβŸ¨π‘“(𝑝)βˆ’π‘“(π‘ž),𝑗(π‘βˆ’π‘ž)⟩β‰₯π›Ώβ€–π‘βˆ’π‘žβ€–2βˆ’π›Ύβ€–π‘“(𝑝)βˆ’π‘“(π‘ž)β€–β€–π‘βˆ’π‘žβ€–β‰₯π›Ώβ€–π‘βˆ’π‘žβ€–2+π›Ύπœ“(β€–π‘βˆ’π‘žβ€–)β€–π‘βˆ’π‘žβ€–βˆ’π›Ύβ€–π‘βˆ’π‘žβ€–2=(π›Ώβˆ’π›Ύ)β€–π‘βˆ’π‘žβ€–2)+π›Ύπœ“(β€–π‘βˆ’π‘žβ€–β€–π‘βˆ’π‘žβ€–.(3.6) Since 𝛿>𝛾, we obtain that 𝑝=π‘ž, and so the solution is unique in 𝐹.
Now, let π‘βˆˆπΉ, since (1βˆ’π›Όπ‘›πœ‚)(𝑒(𝑑𝑛)/𝛼𝑛)β†’0 and (1βˆ’π›Όπ‘›πœ‚)(𝑣(𝑑𝑛)/𝛼𝑛)β†’0 as π‘›β†’βˆž, then there exists 𝑛0βˆˆβ„• such that (1βˆ’π›Όπ‘›πœ‚)(𝑒(𝑑𝑛)/𝛼𝑛)<(πœ‚βˆ’π›Ύ)/2 and (1βˆ’π›Όπ‘›πœ‚)(𝑣(𝑑𝑛)/𝛼𝑛)<(πœ‚βˆ’π›Ύ)/2 for all 𝑛β‰₯𝑛0. Hence, for 𝑛β‰₯𝑛0, we have the following: β€–β€–π‘¦π‘›β€–β€–β‰€β€–β€–ξ€·βˆ’π‘πΌβˆ’π›Όπ‘›πΊπ‘‡ξ€·π‘‘ξ€Έξ€·π‘›ξ€Έπ‘₯π‘›ξ€Έβ€–β€–βˆ’π‘+𝛼𝑛‖‖π‘₯π›Ύπ‘“π‘›ξ€Έβ€–β€–β‰€ξ€·βˆ’πΊ(𝑝)1βˆ’π›Όπ‘›πœ‚ξ€·π‘‘ξ€Έξ€Ίξ€·1+𝑒𝑛‖‖π‘₯ξ€Έξ€Έπ‘›β€–β€–ξ€·π‘‘βˆ’π‘+𝑣𝑛+𝛼𝑛𝛾‖‖𝑓π‘₯π‘›ξ€Έβ€–β€–βˆ’π‘“(𝑝)+𝛼𝑛(≀‖𝛾𝑓𝑝)βˆ’πΊ(𝑝)β€–1βˆ’π›Όπ‘›ξ€·(πœ‚βˆ’π›Ύ)+1βˆ’π›Όπ‘›πœ‚ξ€Έπ‘’ξ€·π‘‘π‘›β€–β€–π‘₯𝑛‖‖+ξ€·βˆ’π‘1βˆ’π›Όπ‘›πœ‚ξ€Έπ‘£ξ€·π‘‘π‘›ξ€Έ+𝛼𝑛‖𝛾𝑓(𝑝)βˆ’πΊ(𝑝)β€–,(3.7)
so that β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘β‰€π›½π‘›β€–β€–π‘₯𝑛‖‖+ξ€·βˆ’π‘1βˆ’π›½π‘›ξ€Έβ€–β€–π‘¦π‘›β€–β€–β‰€ξ€Ίπ›½βˆ’π‘π‘›+ξ€·1βˆ’π›½π‘›ξ€Έξ€Ί1βˆ’π›Όπ‘›(ξ€·πœ‚βˆ’π›Ύ)+1βˆ’π›Όπ‘›πœ‚ξ€Έπ‘’ξ€·π‘‘π‘›β€–β€–π‘₯𝑛‖‖+ξ€·βˆ’π‘1βˆ’π›Όπ‘›πœ‚ξ€Έξ€·1βˆ’π›½π‘›ξ€Έπ‘£ξ€·π‘‘π‘›ξ€Έ+𝛼𝑛1βˆ’π›½π‘›ξ€Έβ€–β‰€ξƒ¬β€–π›Ύπ‘“(𝑝)βˆ’πΊ(𝑝)1βˆ’π›Όπ‘›ξ€·1βˆ’π›½π‘›ξ€Έξƒ©ξ€·(πœ‚βˆ’π›Ύ)βˆ’1βˆ’π›Όπ‘›πœ‚ξ€Έπ‘’ξ€·π‘‘π‘›ξ€Έπ›Όπ‘›β€–β€–π‘₯ξƒͺξƒ­π‘›β€–β€–βˆ’π‘+𝛼𝑛1βˆ’π›½π‘›ξ€Έξƒ¬ξ€·β€–π›Ύπ‘“(𝑝)βˆ’πΊ(𝑝)β€–+1βˆ’π›Όπ‘›πœ‚ξ€Έπ‘£ξ€·π‘‘π‘›ξ€Έπ›Όπ‘›ξƒ­β‰€ξƒ¬1βˆ’π›Όπ‘›ξ€·1βˆ’π›½π‘›ξ€Έξƒ©ξ€·(πœ‚βˆ’π›Ύ)βˆ’1βˆ’π›Όπ‘›πœ‚ξ€Έπ‘’ξ€·π‘‘π‘›ξ€Έπ›Όπ‘›β€–β€–π‘₯ξƒͺξƒ­π‘›β€–β€–βˆ’π‘+𝛼𝑛1βˆ’π›½π‘›ξ€Έξƒ©ξ€·(πœ‚βˆ’π›Ύ)βˆ’1βˆ’π›Όπ‘›πœ‚ξ€Έπ‘’ξ€·π‘‘π‘›ξ€Έπ›Όπ‘›ξƒͺΓ—2‖𝛾𝑓(𝑝)βˆ’πΊ(𝑝)β€–+1βˆ’π›Όπ‘›πœ‚π‘£ξ€·π‘‘ξ€Έξ€·π‘›ξ€Έ/𝛼𝑛‖‖π‘₯πœ‚βˆ’π›Ύβ‰€max𝑛‖‖,βˆ’π‘2‖𝛾𝑓(𝑝)βˆ’πΊ(𝑝)β€–ξ‚Ό.πœ‚βˆ’π›Ύ+1(3.8)
By induction, we have β€–β€–π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘β‰€max𝑛0β€–β€–,βˆ’π‘2‖𝛾𝑓(𝑝)βˆ’πΊ(𝑝)β€–ξ‚Όπœ‚βˆ’π›Ύ+1,βˆ€π‘›β‰₯0.(3.9)
Thus, {π‘₯𝑛} is bounded and so are {𝑇(𝑑𝑛)π‘₯𝑛},{𝐺𝑇(𝑑𝑛)π‘₯𝑛},{𝑦𝑛}, and {𝑓(π‘₯𝑛)}.
Observe that 𝑦𝑛+1βˆ’π‘¦π‘›=ξ€·ξ€·πΌβˆ’π›Όπ‘›+1𝐺𝑇𝑑𝑛+1ξ€Έπ‘₯𝑛+1βˆ’ξ€·πΌβˆ’π›Όπ‘›+1𝐺𝑇𝑑𝑛+1ξ€Έπ‘₯𝑛+ξ€·ξ€·πΌβˆ’π›Όπ‘›+1𝐺𝑇𝑑𝑛+1ξ€Έπ‘₯π‘›βˆ’ξ€·πΌβˆ’π›Όπ‘›πΊξ€Έπ‘‡ξ€·π‘‘π‘›+1ξ€Έπ‘₯𝑛+ξ€·ξ€·πΌβˆ’π›Όπ‘›πΊξ€Έπ‘‡ξ€·π‘‘π‘›+1ξ€Έπ‘₯π‘›βˆ’ξ€·πΌβˆ’π›Όπ‘›πΊξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯𝑛+𝛼𝑛+1ξ€·π‘₯𝛾𝑓𝑛+1ξ€Έβˆ’π›Όπ‘›+1ξ€·π‘₯𝛾𝑓𝑛+𝛼𝑛+1ξ€·π‘₯π›Ύπ‘“π‘›ξ€Έβˆ’π›Όπ‘›ξ€·π‘₯𝛾𝑓𝑛,ξ€Έξ€Έ(3.10) so that ‖‖𝑦𝑛+1βˆ’π‘¦π‘›β€–β€–β‰€ξ€·1βˆ’π›Όπ‘›+1πœ‚ξ€·π‘‘ξ€Έξ€·1+𝑒𝑛+1β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+ξ€·1βˆ’π›Όπ‘›+1πœ‚ξ€Έπ‘£ξ€·π‘‘π‘›+1ξ€Έ+||π›Όπ‘›βˆ’π›Όπ‘›+1||‖‖𝑑𝐺𝑇𝑛+1ξ€Έπ‘₯𝑛‖‖+ξ€·1βˆ’π›Όπ‘›πœ‚ξ€Έβ€–β€–π‘‡π‘‘ξ€·ξ€·π‘›+1βˆ’π‘‘π‘›ξ€Έ+𝑑𝑛π‘₯π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘₯𝑛‖‖+𝛼𝑛+1𝛾‖‖𝑓π‘₯𝑛+1ξ€Έξ€·π‘₯βˆ’π‘“π‘›ξ€Έβ€–β€–+||𝛼𝑛+1βˆ’π›Όπ‘›||𝛾‖‖𝑓π‘₯𝑛‖‖≀1βˆ’π›Όπ‘›+1πœ‚ξ€·π‘‘ξ€Έξ€·1+𝑒𝑛+1β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+ξ€·1βˆ’π›Όπ‘›+1πœ‚ξ€Έπ‘£ξ€·π‘‘π‘›+1ξ€Έ+||π›Όπ‘›βˆ’π›Όπ‘›+1||‖‖𝑑𝐺𝑇𝑛+1ξ€Έπ‘₯𝑛‖‖+ξ€·1βˆ’π›Όπ‘›πœ‚ξ€Έsupπ‘§βˆˆ{π‘₯𝑛},π‘ βˆˆβ„+‖‖𝑇𝑠+π‘‘π‘›ξ€Έξ€·π‘‘π‘§βˆ’π‘‡π‘›ξ€Έπ‘§β€–β€–+𝛼𝑛+1𝛾‖‖𝑓π‘₯𝑛+1ξ€Έξ€·π‘₯βˆ’π‘“π‘›ξ€Έβ€–β€–+||𝛼𝑛+1βˆ’π›Όπ‘›||𝛾‖‖𝑓π‘₯𝑛‖‖.(3.11)
From this, we obtain that ‖‖𝑦𝑛+1βˆ’π‘¦π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀1βˆ’π›Όπ‘›+1πœ‚ξ€·π‘‘ξ€Έξ€·1+𝑒𝑛+1ξ€»β€–β€–π‘₯ξ€Έξ€Έβˆ’1𝑛+1βˆ’π‘₯𝑛‖‖+ξ€·1βˆ’π›Όπ‘›+1πœ‚ξ€Έπ‘£ξ€·π‘‘π‘›+1ξ€Έ+||π›Όπ‘›βˆ’π›Όπ‘›+1||‖‖𝑑𝐺𝑇𝑛+1ξ€Έπ‘₯𝑛‖‖+ξ€·1βˆ’π›Όπ‘›πœ‚ξ€Έsupπ‘§βˆˆ{π‘₯𝑛},π‘ βˆˆβ„+‖‖𝑇𝑠+π‘‘π‘›ξ€Έξ€·π‘‘π‘§βˆ’π‘‡π‘›ξ€Έπ‘§β€–β€–+𝛼𝑛+1𝛾‖‖𝑓π‘₯𝑛+1ξ€Έξ€·π‘₯βˆ’π‘“π‘›ξ€Έβ€–β€–+||𝛼𝑛+1βˆ’π›Όπ‘›||𝛾‖‖𝑓π‘₯𝑛‖‖,(3.12) which implies that limsupπ‘›β†’βˆžξ€·β€–β€–π‘¦π‘›+1βˆ’π‘¦π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀0,(3.13) and by Lemma 2.3, limπ‘›β†’βˆžβ€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖=0.(3.14)
Thus, β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘¦π‘›βˆ’π‘₯π‘›β€–β€–βŸΆ0asπ‘›βŸΆβˆž.(3.15)
Next, we show that limπ‘›β†’βˆžβ€–π‘¦π‘›βˆ’π‘‡(𝑑)𝑦𝑛‖=0, for all 𝑑β‰₯0.
Since β€–β€–π‘₯π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘₯𝑛‖‖≀‖‖π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+β€–β€–π‘₯𝑛+1ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘₯𝑛‖‖≀‖‖π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+𝛽𝑛‖‖π‘₯π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘₯𝑛‖‖+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘¦π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘₯𝑛‖‖,(3.16) we have ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘₯𝑛‖‖≀‖‖π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘¦π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘₯𝑛‖‖=β€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+𝛼𝑛1βˆ’π›½π‘›ξ€Έβ€–β€–ξ€·π‘₯π›Ύπ‘“π‘›ξ€Έξ€·π‘‘βˆ’πΊπ‘‡π‘›ξ€Έπ‘₯𝑛‖‖.(3.17)
From 𝛼𝑛→0 as π‘›β†’βˆž and (3.15), we obtain limπ‘›β†’βˆžβ€–β€–π‘₯π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘₯𝑛‖‖=0.(3.18)
Also, β€–β€–π‘¦π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘¦π‘›β€–β€–β‰€β€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖+β€–β€–π‘₯π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘₯𝑛‖‖+‖‖𝑇𝑑𝑛π‘₯π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘¦π‘›β€–β€–β‰€ξ€·ξ€·π‘‘2+π‘’π‘›β€–β€–π‘¦ξ€Έξ€Έπ‘›βˆ’π‘₯𝑛‖‖𝑑+𝑣𝑛+β€–β€–π‘₯π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘₯π‘›β€–β€–βŸΆ0asπ‘›βŸΆβˆž.(3.19)
Since limπ‘›β†’βˆžπ‘‘π‘›=∞ and {𝑇(𝑑)βˆΆπ‘‘β‰₯0} is uniformly asymptotically regular, limπ‘›β†’βˆžβ€–β€–π‘‡ξ€·π‘‘(𝑑)𝑇𝑛π‘₯π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘₯𝑛‖‖≀limπ‘›β†’βˆžsupπ‘₯βˆˆπΆβ€–β€–π‘‡ξ€·π‘‘(𝑑)𝑇𝑛𝑑π‘₯βˆ’π‘‡π‘›ξ€Έπ‘₯β€–β€–=0,limπ‘›β†’βˆžβ€–β€–ξ€·π‘‘π‘‡(𝑑)π‘‡π‘›ξ€Έπ‘¦π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘¦π‘›β€–β€–β‰€limπ‘›β†’βˆžsupπ‘¦βˆˆπΆβ€–β€–ξ€·π‘‘π‘‡(𝑑)π‘‡π‘›ξ€Έξ€·π‘‘π‘¦βˆ’π‘‡π‘›ξ€Έπ‘¦β€–β€–=0,(3.20) where 𝐢 is any bounded subset of 𝐸 containing {π‘₯𝑛}. Since {𝑇(𝑑)} is continuous, we get that β€–β€–π‘¦π‘›βˆ’π‘‡(𝑑)π‘¦π‘›β€–β€–β‰€β€–β€–π‘¦π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘¦π‘›β€–β€–+β€–β€–π‘‡ξ€·π‘‘π‘›ξ€Έπ‘¦π‘›ξ€·π‘‡ξ€·π‘‘βˆ’π‘‡(𝑑)𝑛𝑦𝑛‖‖+‖‖𝑇𝑑𝑇(𝑑)π‘›ξ€Έπ‘¦π‘›ξ€Έβˆ’π‘‡(𝑑)𝑦𝑛‖‖.(3.21)
This implies that limπ‘›β†’βˆžβ€–β€–π‘¦π‘›βˆ’π‘‡(𝑑)𝑦𝑛‖‖=0,βˆ€π‘‘β‰₯0.(3.22)
Next, we show that limsupπ‘›β†’βˆžξ«ξ€·π‘¦(π›Ύπ‘“βˆ’πΊ)𝑝,π‘—π‘›βˆ’π‘ξ€Έξ¬β‰€0.(3.23)
Define a map πœ™βˆΆπΈβ†’β„ by πœ™(𝑦)∢=πœ‡π‘›β€–β€–π‘¦π‘›β€–β€–βˆ’π‘¦2,βˆ€π‘¦βˆˆπΈ.(3.24)
Then, πœ™(𝑦)β†’βˆž as β€–π‘¦β€–β†’βˆž, πœ™ is continuous and convex, so as 𝐸 is reflexive, there exists π‘žβˆˆπΈ such that πœ™(π‘ž)=minπ‘’βˆˆπΈπœ™(𝑒). Hence, the set πΎβˆ—ξ‚»βˆΆ=π‘¦βˆˆπΈβˆΆπœ™(𝑦)=minπ‘’βˆˆπΈξ‚Όπœ™(𝑒)β‰ βˆ….(3.25)
Since limπ‘›β†’βˆžβ€–π‘¦π‘›βˆ’π‘‡(𝑑)𝑦𝑛‖=0,limπ‘‘β†’βˆžπ‘’(𝑑)=0,limπ‘‘β†’βˆžπ‘£(𝑑)=0, and πœ™ is continuous for all π‘§βˆˆπΎβˆ—, we have πœ™ξ‚΅limπ‘‘β†’βˆžξ‚Άπ‘‡(𝑑)𝑧=limπ‘‘β†’βˆžπœ™(𝑇(𝑑)𝑧)=limπ‘‘β†’βˆžπœ‡π‘›β€–β€–π‘¦π‘›β€–β€–βˆ’π‘‡(𝑑)𝑧2≀limπ‘‘β†’βˆžπœ‡π‘›ξ€·β€–β€–π‘¦(1+𝑒(𝑑))π‘›β€–β€–ξ€Έβˆ’π‘§+(𝑣(𝑑))2=πœ‡π‘›β€–β€–π‘¦π‘›β€–β€–βˆ’π‘§2=πœ™(𝑧).(3.26)
Hence, limπ‘‘β†’βˆžπ‘‡(𝑑)π‘§βˆˆπΎβˆ—.
Let π‘βˆˆπΉ. Since πΎβˆ— is a closed-convex set, there exists a unique π‘žβˆˆπΎβˆ— such that β€–π‘βˆ’π‘žβ€–=minπ‘₯βˆˆπΎβˆ—β€–π‘βˆ’π‘₯β€–.(3.27)
Since 𝑝=limπ‘‘β†’βˆžπ‘‡(𝑑)𝑝 and limπ‘‘β†’βˆžπ‘‡(𝑑)π‘žβˆˆπΎβˆ—, β€–β€–β€–π‘βˆ’limπ‘‘β†’βˆžβ€–β€–β€–=‖‖‖𝑇(𝑑)π‘žlimπ‘‘β†’βˆžπ‘‡(𝑑)π‘βˆ’limπ‘‘β†’βˆžβ€–β€–β€–π‘‡(𝑑)π‘ž=limπ‘‘β†’βˆžβ€–β€–π‘‡(𝑑)π‘βˆ’π‘‡(𝑑)π‘žβ‰€limπ‘‘β†’βˆžβ€–β‰€((1+𝑒(𝑑))π‘βˆ’π‘žβ€–+𝑣(𝑑))β€–π‘βˆ’π‘žβ€–.(3.28)
Therefore, limπ‘‘β†’βˆžπ‘‡(𝑑)π‘ž=π‘ž. Since 𝑇(𝑠+β„Ž)π‘₯=𝑇(𝑠)𝑇(β„Ž)π‘₯ for all π‘₯∈𝐸 and 𝑠β‰₯0, we have π‘ž=limπ‘‘β†’βˆžπ‘‡(𝑑)π‘ž=limπ‘‘β†’βˆžπ‘‡(𝑠+𝑑)π‘ž=limπ‘‘β†’βˆžπ‘‡(𝑠)𝑇(𝑑)π‘ž=𝑇(𝑠)limπ‘‘β†’βˆžπ‘‡(𝑑)π‘ž=𝑇(𝑠)π‘ž.(3.29)
Therefore, π‘žβˆˆπΉ and so πΎβˆ—βˆ©πΉβ‰ βˆ….
Let π‘βˆˆπΎβˆ—βˆ©πΉ(𝑇) and 𝜏∈(0,1). Then, it follows that πœ™(𝑝)β‰€πœ™(π‘βˆ’πœ(πΊβˆ’π›Ύπ‘“)𝑝), and using Lemma 2.2, we obtain that β€–β€–π‘¦π‘›β€–β€–βˆ’π‘+𝜏(πΊβˆ’π›Ύπ‘“)𝑝2β‰€β€–β€–π‘¦π‘›β€–β€–βˆ’π‘2𝑦+2𝜏(πΊβˆ’π›Ύπ‘“)𝑝,𝑗𝑛,βˆ’π‘+𝜏(πΊβˆ’π›Ύπ‘“)𝑝(3.30) which implies that πœ‡π‘›ξ«ξ€·π‘¦(π›Ύπ‘“βˆ’πΊ)𝑝,π‘—π‘›βˆ’π‘+𝜏(πΊβˆ’π›Ύπ‘“)𝑝≀0.(3.31)
Moreover, πœ‡π‘›ξ«ξ€·π‘¦(π›Ύπ‘“βˆ’πΊ)𝑝,π‘—π‘›βˆ’π‘ξ€Έξ¬=πœ‡π‘›ξ«ξ€·π‘¦(π›Ύπ‘“βˆ’πΊ)𝑝,π‘—π‘›ξ€Έξ€·π‘¦βˆ’π‘βˆ’π‘—π‘›βˆ’π‘+𝜏(πΊβˆ’π›Ύπ‘“)𝑝+πœ‡π‘›ξ«(ξ€·π‘¦π›Ύπ‘“βˆ’πΊ)𝑝,π‘—π‘›βˆ’π‘+𝜏(πΊβˆ’π›Ύπ‘“)π‘ξ€Έξ¬β‰€πœ‡π‘›ξ«ξ€·π‘¦(π›Ύπ‘“βˆ’πΊ)𝑝,π‘—π‘›ξ€Έξ€·π‘¦βˆ’π‘βˆ’π‘—π‘›.βˆ’π‘+𝜏(πΊβˆ’π›Ύπ‘“)𝑝(3.32)
Since 𝑗 is norm-to-weak* uniformly continuous on bounded subsets of 𝐸, we have that πœ‡π‘›ξ«ξ€·π‘¦(π›Ύπ‘“βˆ’πΊ)𝑝,π‘—π‘›βˆ’π‘ξ€Έξ¬β‰€0.(3.33)
Observe that from (3.14) and (3.15), we have limπ‘›β†’βˆžβ€–β€–π‘¦π‘›+1βˆ’π‘¦π‘›β€–β€–=0.(3.34)
This implies that limsupπ‘›β†’βˆžξ€ΊβŸ¨ξ€·π‘¦(π›Ύπ‘“βˆ’πΊ)𝑝,π‘—π‘›ξ€Έξ€·π‘¦βˆ’π‘βŸ©βˆ’βŸ¨(π›Ύπ‘“βˆ’πΊ)𝑝,𝑗𝑛+1ξ€ΈβŸ©ξ€»βˆ’π‘β‰€0,(3.35) and so we obtain by Lemma 2.4 that limsupπ‘›β†’βˆžξ«ξ€·π‘¦(π›Ύπ‘“βˆ’πΊ)𝑝,π‘—π‘›βˆ’π‘ξ€Έξ¬β‰€0.(3.36)
Finally, we show that π‘₯𝑛→𝑝 as π‘›β†’βˆž. Since limπ‘›β†’βˆž(𝑒(𝑑𝑛)/𝛼𝑛)=0, if we denote by 𝜎(𝑑𝑛) the value 2𝑒(𝑑𝑛)+𝑒(𝑑𝑛)2, then we clearly have limπ‘›β†’βˆž(𝜎(𝑑𝑛)/𝛼𝑛)=0. Let 𝑁0βˆˆβ„• be large enough such that (1βˆ’π›Όπ‘›πœ‚)(𝜎(𝑑𝑛)/𝛼𝑛)<(πœ‚βˆ’2𝛾)/2, for all 𝑛β‰₯𝑁0, and let 𝑀 be a positive real number such that ||π‘₯π‘›βˆ’π‘||≀𝑀 for all 𝑛β‰₯0. Then, using the recursion formula (3.2) and for 𝑛β‰₯𝑁0, we have β€–β€–π‘¦π‘›β€–β€–βˆ’π‘2=‖‖𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·βˆ’πΊ(𝑝)πΌβˆ’π›Όπ‘›πΊπ‘‡ξ€·π‘‘ξ€Έξ€·π‘›ξ€Έπ‘₯π‘›ξ€Έβ€–β€–βˆ’π‘2≀1βˆ’π›Όπ‘›πœ‚ξ€Έβ€–β€–π‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯π‘›β€–β€–βˆ’π‘2+2𝛼𝑛π‘₯π›Ύπ‘“π‘›ξ€Έξ€·π‘¦βˆ’πΊ(𝑝),π‘—π‘›β‰€ξ€·βˆ’π‘ξ€Έξ¬1βˆ’π›Όπ‘›πœ‚ξ€·π‘‘ξ€Έξ€Ίξ€·1+𝑒𝑛‖‖π‘₯ξ€Έξ€Έπ‘›β€–β€–ξ€·π‘‘βˆ’π‘+𝑣𝑛2+2𝛼𝑛π‘₯π›Ύπ‘“π‘›ξ€Έξ€·π‘¦βˆ’π›Ύπ‘“(𝑝)+𝛾𝑓(𝑝)βˆ’πΊ(𝑝),π‘—π‘›β‰€ξ€·βˆ’π‘ξ€Έξ¬1βˆ’π›Όπ‘›πœ‚ξ€Έξ‚ƒξ€·ξ€·π‘‘1+𝑒𝑛2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘2𝑑+21+𝑒𝑛𝑣𝑑𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘2𝑑+𝑣𝑛2ξ‚„+2𝛼𝑛𝑦𝛾𝑓(𝑝)βˆ’πΊ(𝑝),π‘—π‘›βˆ’π‘ξ€Έξ¬βˆ’2π›Όπ‘›π›Ύβ€–β€–π‘¦π‘›β€–β€–πœ“ξ€·β€–β€–π‘₯βˆ’π‘π‘›β€–β€–ξ€Έβˆ’π‘+2π›Όπ‘›π›Ύβ€–β€–ξ€·π‘¦π‘›βˆ’π‘₯𝑛+ξ€·π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘π‘›β€–β€–β‰€βˆ’π‘ξ€Ίξ€·1βˆ’π›Όπ‘›πœ‚ξ€·π‘‘ξ€Έξ€·1+πœŽπ‘›ξ€Έξ€Έ+2𝛼𝑛𝛾‖‖π‘₯π‘›β€–β€–βˆ’π‘2+𝛼𝑛2𝑦𝛾𝑓(𝑝)βˆ’πΊ(𝑝),π‘—π‘›ξ€·βˆ’π‘ξ€Έξ¬+21βˆ’π›Όπ‘›πœ‚ξ€·π‘‘ξ€Έξ€·1+𝑒𝑛𝑣𝑑𝑛𝛼𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘2+ξ€·1βˆ’π›Όπ‘›πœ‚ξ€Έπ‘£ξ€·π‘‘π‘›ξ€Έ2𝛼𝑛‖‖𝑦+2π›Ύπ‘›βˆ’π‘₯𝑛‖‖‖‖π‘₯𝑛‖‖=ξ‚Έβˆ’π‘1βˆ’π›Όπ‘›ξ‚΅ξ€·(πœ‚βˆ’2𝛾)βˆ’1βˆ’π›Όπ‘›πœ‚ξ€ΈπœŽπ‘›π›Όπ‘›β€–β€–π‘₯ξ‚Άξ‚Ήπ‘›β€–β€–βˆ’π‘2+𝛼𝑛2𝑦𝛾𝑓(𝑝)βˆ’πΊ(𝑝),π‘—π‘›ξ€·βˆ’π‘ξ€Έξ¬+21βˆ’π›Όπ‘›πœ‚ξ€·π‘‘ξ€Έξ€·1+𝑒𝑛𝑣𝑑𝑛𝛼𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘2+ξ€·1βˆ’π›Όπ‘›πœ‚ξ€Έπ‘£ξ€·π‘‘π‘›ξ€Έ2𝛼𝑛‖‖𝑦+2π›Ύπ‘›βˆ’π‘₯𝑛‖‖‖‖π‘₯𝑛‖‖,βˆ’π‘(3.37)
so that β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘2≀𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘¦π‘›β€–β€–βˆ’π‘2≀𝛽𝑛+ξ€·1βˆ’π›½π‘›ξ€Έξ‚Έ1βˆ’π›Όπ‘›ξ‚΅ξ€·(πœ‚βˆ’2𝛾)βˆ’1βˆ’π›Όπ‘›πœ‚ξ€ΈπœŽπ‘›π›Όπ‘›β€–β€–π‘₯ξ‚Άξ‚Ήξ‚Άπ‘›β€–β€–βˆ’π‘2+𝛼𝑛1βˆ’π›½π‘›ξ€Έξƒ¬2𝑦𝛾𝑓(𝑝)βˆ’πΊ(𝑝),π‘—π‘›ξ€·βˆ’π‘ξ€Έξ¬+21βˆ’π›Όπ‘›πœ‚ξ€·π‘‘ξ€Έξ€·1+𝑒𝑛×𝑣𝑑𝑛𝛼𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘2+ξ€·1βˆ’π›Όπ‘›πœ‚ξ€Έπ‘£ξ€·π‘‘π‘›ξ€Έ2𝛼𝑛‖‖𝑦+2π›Ύπ‘›βˆ’π‘₯𝑛‖‖‖‖π‘₯π‘›β€–β€–ξƒ­β‰€ξ‚Έβˆ’π‘1βˆ’π›Όπ‘›ξ€·1βˆ’π›½π‘›ξ€Έξ‚΅(ξ€·πœ‚βˆ’2𝛾)βˆ’1βˆ’π›Όπ‘›πœ‚ξ€ΈπœŽπ‘›π›Όπ‘›β€–β€–π‘₯ξ‚Άξ‚Ήπ‘›β€–β€–βˆ’π‘2+𝛼𝑛1βˆ’π›½π‘›ξ€Έξƒ¬2𝑦𝛾𝑓(𝑝)βˆ’πΊ(𝑝),π‘—π‘›ξ€·βˆ’π‘ξ€Έξ¬+21βˆ’π›Όπ‘›πœ‚ξ€·π‘‘ξ€Έξ€·1+𝑒𝑛𝑣𝑑𝑛𝛼𝑛𝑀2+ξ€·1βˆ’π›Όπ‘›πœ‚ξ€Έπ‘£ξ€·π‘‘π‘›ξ€Έ2𝛼𝑛‖‖𝑦+2π›Ύπ‘›βˆ’π‘₯𝑛‖‖𝑀=ξ‚Έ1βˆ’π›Όπ‘›ξ€·1βˆ’π›½π‘›ξ€Έξ‚΅ξ€·(πœ‚βˆ’2𝛾)βˆ’1βˆ’π›Όπ‘›πœ‚ξ€ΈπœŽπ‘›π›Όπ‘›β€–β€–π‘₯ξ‚Άξ‚Ήπ‘›β€–β€–βˆ’π‘2+𝛼𝑛1βˆ’π›½π‘›ξ€Έξ‚΅ξ€·(πœ‚βˆ’2𝛾)βˆ’1βˆ’π›Όπ‘›πœ‚ξ€ΈπœŽπ‘›π›Όπ‘›ξ‚ΆΓ—ξƒ¬2𝑦𝛾𝑓(𝑝)βˆ’πΊ(𝑝),π‘—π‘›ξ€·βˆ’π‘ξ€Έξ¬+21βˆ’π›Όπ‘›πœ‚ξ€·π‘‘ξ€Έξ€·1+𝑒𝑛𝑣𝑑𝑛𝛼𝑛ξƒͺ𝑀2+π’œπ‘›ξƒ­ξ‚΅(ξ€·πœ‚βˆ’2𝛾)βˆ’1βˆ’π›Όπ‘›πœ‚ξ€Έξ‚΅πœŽπ‘›π›Όπ‘›,ξ‚Άξ‚Ά(3.38) where π’œπ‘› denotes (1βˆ’π›Όπ‘›πœ‚)(𝑣(𝑑𝑛)2/𝛼𝑛)+2π›Ύβ€–π‘¦π‘›βˆ’π‘₯𝑛‖𝑀.
Observe  that βˆ‘βˆžπ‘›=1𝛼𝑛(1βˆ’π›½π‘›)((πœ‚βˆ’2𝛾)βˆ’(1βˆ’π›Όπ‘›πœ‚)(πœŽπ‘›/𝛼𝑛))=∞ and limsupπ‘›β†’βˆžξƒ©2𝑦𝛾𝑓(𝑝)βˆ’πΊ(𝑝),π‘—π‘›ξ€·βˆ’π‘ξ€Έξ¬+21βˆ’π›Όπ‘›πœ‚ξ€·π‘‘ξ€Έξ€·1+𝑒𝑛𝑣𝑑𝑛/𝛼𝑛𝑀2+π’œπ‘›ξ€·ξ€·(πœ‚βˆ’2𝛾)βˆ’1βˆ’π›Όπ‘›πœ‚πœŽξ€Έξ€·π‘›/𝛼𝑛ξƒͺ≀0.(3.39)
Applying Lemma 2.5, we obtain β€–π‘₯π‘›βˆ’π‘β€–β†’0 as π‘›β†’βˆž. This completes the proof.

The following corollaries follow from Theorem 3.1.

Corollary 3.2. Let 𝐸 be a real uniformly convex and uniformly smooth Banach space, 𝔍={𝑇(𝑑)βˆΆπ‘‘β‰₯0}, and let 𝐹,𝑓,𝐺,𝛿,πœ†,πœ‚,𝛾,{𝛽𝑛},{𝛼𝑛},{𝑑𝑛} and {π‘₯𝑛} be as in Theorem 3.1. Then, the sequence {π‘₯𝑛} converges strongly to a common fixed point of the family 𝔍 which solves the variational inequality (3.3).

Corollary 3.3. Let 𝐸=𝐻 be a real Hilbert space, and let 𝔍={𝑇(𝑑)βˆΆπ‘‘β‰₯0},𝐹,𝑓,𝐺,𝛿,πœ†,πœ‚,𝛾,{𝛽𝑛},{𝛼𝑛},{𝑑𝑛} and {π‘₯𝑛} be as in Theorem 3.1. Then, the sequence {π‘₯𝑛} converges strongly to a common fixed point of the family 𝔍 which solves the variational inequality ⟨(πΊβˆ’π›Ύπ‘“)π‘ž,π‘₯βˆ’π‘žβŸ©β‰₯0,βˆ€π‘₯∈𝐹.(3.40)

Corollary 3.4. Let 𝔍={𝑇(𝑑)βˆΆπ‘‘β‰₯0} be a family of nonexpansive semigroup of a real reflexive and strictly convex Banach space with a uniformly GΓ’teaux differentiable norm 𝐸, and let 𝐹,𝑓,𝐺,𝛿,πœ†,πœ‚,𝛾,{𝛽𝑛},{𝛼𝑛},{𝑑𝑛}, and {π‘₯𝑛} be as in Theorem 3.1. Then, the sequence {π‘₯𝑛} converges strongly to a common fixed point of the family 𝔍 which solves the variational inequality (3.3).

Acknowledgment

The authors thank the anonymous referees for useful comments and observations, that helped in improving the presentation of this paper.

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