Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011Β (2011), Article IDΒ 684158, 14 pages
http://dx.doi.org/10.5402/2011/684158
Research Article

Common Fixed Points Approximation for Asymptotically Nonexpansive Semigroup in Banach Spaces

Department of Mathematical Sciences, Bayero University, P.M.B. 3011, Kano, Nigeria

Received 3 May 2011; Accepted 9 June 2011

Academic Editors: J. S.Β Jung and C.Β Zhu

Copyright Β© 2011 Bashir Ali. 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 Banach space satisfying local uniform Opial's condition, whose duality map is weakly sequentially continuous. Let π’₯∢={𝑇(𝑑)𝑑β‰₯0} be a uniformly asymptotically regular family of asymptotically nonexpansive semigroup of 𝐸 with function π‘˜βˆΆ[0,∞)β†’[0,∞). Let β‹‚β„±βˆΆ=𝑑β‰₯0𝐹(𝑇(𝑑))β‰ βˆ… and π‘“βˆΆπΈβ†’πΈ be weakly contractive map. Let πΊβˆΆπΈβ†’πΈ be 𝛿-strongly accretive and πœ†-strictly pseudocontractive map with 𝛿+πœ†>1. Let {𝑑𝑛} be an increasing sequence in [0,∞)and let{𝛼𝑛} and {𝛽𝑛} be sequences in [0,1] satisfying some conditions. For some positive real number 𝛾 appropriately chosen, let {π‘₯𝑛} be a sequence defined by π‘₯0∈𝐸, π‘₯𝑛+1=𝛽𝑛π‘₯𝑛+(1βˆ’π›½π‘›)𝑦𝑛,  𝑦𝑛=(πΌβˆ’π›Όπ‘›πΊ)𝑇(𝑑𝑛)π‘₯𝑛+𝛼𝑛𝛾𝑓(π‘₯𝑛),  𝑛β‰₯0. It is proved that {π‘₯𝑛} converges strongly to a common fixed point π‘ž of the family π’₯ which is also the unique solution of the variational inequality ⟨(πΊβˆ’π›Ύπ‘“)π‘ž,𝑗(π‘žβˆ’π‘₯)⟩β‰₯0,forallπ‘₯βˆˆβ„±.

1. Introduction

Let 𝐸 be a real Banach space and let πΈβˆ— be the dual space of 𝐸. A mapping πœ‘βˆΆ[0,∞)β†’[0,∞) is called a gauge function if it is strictly increasing, continuous and πœ‘(0)=0. Let πœ‘ be a gauge function, a generalized duality mapping with respect to πœ‘,π½πœ‘βˆΆπΈβ†’2πΈβˆ— is defined by, π‘₯∈𝐸, π½πœ‘ξ€½π‘₯π‘₯=βˆ—βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘₯βˆ—(⟩=β€–π‘₯β€–πœ‘β€–π‘₯β€–),β€–π‘₯βˆ—()ξ€Ύβ€–=πœ‘β€–π‘₯β€–,(1.1) where βŸ¨β‹…,β‹…βŸ© denotes the duality pairing between element of 𝐸 and that of πΈβˆ—. If πœ‘(𝑑)=𝑑, then π½πœ‘ is simply called the normalized duality mapping and is denoted by 𝐽. For any π‘₯∈𝐸, an element of π½πœ‘π‘₯ is denoted by π‘—πœ‘(π‘₯).

The modulus of convexity of 𝐸 is the function π›ΏπΈβˆΆ(0,2]β†’[0,1] defined by 𝛿𝐸(ξ‚†β€–β€–β€–πœ–)=inf1βˆ’π‘₯+𝑦2β€–β€–β€–ξ‚‡βˆΆβ€–π‘₯β€–=‖𝑦‖=1,πœ–=β€–π‘₯βˆ’π‘¦β€–,(1.2) and 𝐸 is called uniformly convex if 𝛿𝐸(πœ–)>0forallπœ–βˆˆ(0,2]. A Banach space 𝐸 is said to satisfy Opial’s condition [1] if, for any sequence {π‘₯𝑛} in 𝐸, π‘₯𝑛⇀π‘₯ as π‘›β†’βˆž implies that liminfπ‘›β†’βˆžβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘₯<liminfπ‘›β†’βˆžβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘¦,βˆ€π‘¦βˆˆπΈ,𝑦≠π‘₯.(1.3)All Hilbert spaces and 𝑙𝑝 spaces, 1≀𝑝<∞ satisfy Opial’s condition. However 𝐿𝑝, 𝑝≠2 do not satisfy this condition; see, for example, [2]. The space 𝐸 is said to have weakly (sequentially) continuous duality map if there exists a gauge function πœ‘ such that π½πœ‘ is singled valued and (sequentially) continuous from 𝐸 with weak topology to πΈβˆ— with weakβˆ— topology. It is known that every Banach space with weakly sequentially continuous duality mapping satisfies Opial’s condition (see [3]). Every 𝑙𝑝 space, (1<𝑝<∞) has a weakly sequentially continuous duality map.

The space 𝐸 is said to have uniform Opial’s condition [4] if for each 𝑐>0, there exists an π‘Ÿ>0 such that 1+π‘Ÿβ‰€liminfπ‘›β†’βˆžβ€–β€–π‘₯+π‘₯𝑛‖‖(1.4) for each π‘₯∈𝐸 with β€–π‘₯β€–β‰₯𝑐 and each sequence {π‘₯𝑛} satisfying π‘₯𝑛⇀0 as π‘›β†’βˆž, and liminfπ‘›β†’βˆžβ€–π‘₯𝑛‖β‰₯1.

𝐸 is said to satisfy the local uniform Opial’s condition [5] if, for any weak null sequence {π‘₯𝑛} in 𝐸 with liminfπ‘›β†’βˆžβ€–π‘₯𝑛‖β‰₯1 and any 𝑐>0, there exists π‘Ÿ>0 such that 1+π‘Ÿβ‰€liminfπ‘›β†’βˆžβ€–β€–π‘₯+π‘₯𝑛‖‖(1.5) for all π‘₯∈𝐸 with β€–π‘₯β€–β‰₯𝑐. Observe that uniform Opial’s condition implies local uniform Opial’s condition which in turn implies Opial’s condition.

A self-mapping π‘‡βˆΆπΈβ†’πΈ is said to be contraction if ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€π›Όβ€–π‘₯βˆ’π‘¦β€–,forallπ‘₯,π‘¦βˆˆπΈ, where π›Όβˆˆ[0,1) is a fixed constant. 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€–),forallπ‘₯,π‘¦βˆˆπΈ. It is known that the class of weakly contractive maps contain properly the class of contractive ones; see [6, 7]. The map 𝑇 is said to be nonexpansive if ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–forallπ‘₯,π‘¦βˆˆπΈ and asymptotically nonexpansive if there exists a sequence {π‘˜π‘›}βŠ‚[0,∞) with limπ‘›β†’βˆžπ‘˜π‘›=0 such that ‖𝑇𝑛π‘₯βˆ’π‘‡π‘›π‘¦β€–β‰€(1+π‘˜π‘›)β€–π‘₯βˆ’π‘¦β€–,forallπ‘₯,π‘¦βˆˆπΈ and π‘›βˆˆβ„•. The set of fixed point of 𝑇 is defined as 𝐹(𝑇)∢={π‘₯βˆˆπΈβˆΆπ‘‡π‘₯=π‘₯}.

A one parameter family π’₯={𝑇(𝑑)βˆΆπ‘‘β‰₯0} of self-mapping of 𝐸 is called nonexpansive semigroup if the following conditions are satisfied:(i)𝑇(0)π‘₯=π‘₯forallπ‘₯∈𝐸; (ii)𝑇(𝑑+𝑠)=𝑇(𝑑)βˆ˜π‘‡(𝑠)forall𝑑,𝑠β‰₯0; (iii)for each π‘₯∈𝐸, the mapping 𝑑→𝑇(𝑑)π‘₯ is continuous;(iv)for π‘₯,π‘¦βˆˆπΈ and 𝑑β‰₯0, ‖𝑇(𝑑)π‘₯βˆ’π‘‡(𝑑)𝑦‖≀‖π‘₯βˆ’π‘¦β€–.

The family π’₯ is said to be asymptotically nonexpansive semigroup if conditions (i)–(iii) are satisfied and, in addition, there exists a function π‘˜βˆΆ[0,∞)β†’[0,∞) satisfying limπ‘‘β†’βˆžπ‘˜(𝑑)=0 and ‖𝑇(𝑑)π‘₯βˆ’π‘‡(𝑑)𝑦‖≀(1+π‘˜(𝑑))β€–π‘₯βˆ’π‘¦β€–forallπ‘₯,π‘¦βˆˆπΈ.

The family π’₯={𝑇(𝑑)βˆΆπ‘‘β‰₯0} is said to be asymptotically regular if limπ‘ β†’βˆžβ€–π‘‡(𝑑+𝑠)π‘₯βˆ’π‘‡(𝑠)π‘₯β€–=0,(1.6) for all π‘‘βˆˆ(0,∞) and π‘₯∈𝐾. It is said to be uniformly asymptotically regular if, for any 𝑑β‰₯0 and for any bounded subset 𝐢 of 𝐾, limπ‘ β†’βˆžsupπ‘₯βˆˆπΆβ€–π‘‡(𝑑+𝑠)π‘₯βˆ’π‘‡(𝑠)π‘₯β€–=0.(1.7) For some positive real numbers 𝛿 and πœ†, the 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(1.8) and it is called πœ†-strictly pseudocontractive if ⟨𝐺π‘₯βˆ’πΊπ‘¦,𝑗(π‘₯βˆ’π‘¦)βŸ©β‰€β€–π‘₯βˆ’π‘¦β€–2β€–βˆ’πœ†β€–(πΌβˆ’πΊ)π‘₯βˆ’(πΌβˆ’πΊ)𝑦2.(1.9) Let 𝐢 be a nonempty closed convex subset of 𝐸 and π‘‡βˆΆπΈβ†’πΈ be a map. Then, a variational inequality problem with respect to 𝐢 and 𝑇 is find 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) The problem of solving a variational inequality of the form (1.10) has been intensively studied by numerous authors due to its various applications in several physical problems, such as in operations research, economics, and engineering design; see, for example, [8–10] and the references therein. Iterative methods for approximating fixed points of nonexpansive mappings, nonexpansive semigroups, and their generalizations which solves some variational inequalities problems have been studied by a number of authors (see, e.g., [11–17] and the references therein).

A typical problem is to minimize a quadratic function over the set of the fixed points of some nonexpansive mapping in a real Hilbert space 𝐻: minπ‘₯∈𝐢12⟨𝐴π‘₯,π‘₯βŸ©βˆ’βŸ¨π‘₯,π‘βŸ©.(1.11) Here, 𝐢 is the fixed point set of a nonexpansive mapping 𝑇 of 𝐻,𝑏 is a point in 𝐻, and 𝐴 is some bounded, linear, and strongly positive operator on 𝐻, where a map π΄βˆΆπ»β†’π» is said to be strongly positive if there exists a constant 𝛾>0 such that ⟨𝐴π‘₯,π‘₯⟩β‰₯𝛾‖π‘₯β€–2,forallπ‘₯∈𝐻.(1.12) For a strongly positive bonded linear operator 𝐴 and any π‘₯,π‘¦βˆˆπ», we have ⟨𝐴π‘₯βˆ’π΄π‘¦,π‘₯βˆ’π‘¦βŸ©β‰₯𝛾‖π‘₯βˆ’π‘¦β€–2.(1.13) This implies that 𝐴 is 𝛾-strongly accretive (or in particular 𝛾-strongly monotone). On the other hand, by simple calculation, the following relation also holds: ξ€·βŸ¨π΄π‘₯βˆ’π΄π‘¦,π‘₯βˆ’π‘¦βŸ©β‰€1+‖𝐴‖2ξ€Έ2β€–π‘₯βˆ’π‘¦β€–2βˆ’12β€–(πΌβˆ’π΄)π‘₯βˆ’(πΌβˆ’π΄)𝑦‖2.(1.14) This implies that 𝐴/‖𝐴‖ is 1/2-strictly pseudocontractive.

Let 𝐻 be a real Hilbert space. In 2003, Xu [18] proved that the sequence {π‘₯𝑛} defined by π‘₯0∈𝐻 chosen arbitrarily, π‘₯𝑛+1=ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘₯𝑛+𝛼𝑛𝑏,𝑛β‰₯0,(1.15) converges strongly to the unique solution of the minimization problem (1.11) provided that the sequence {𝛼𝑛} satisfies certain control conditions.

In 2000, Moudafi [12] introduced the viscosity approximation method for nonexpansive mappings. Let 𝑓 be a contraction on 𝐻. Starting with an arbitrary initial point π‘₯0∈𝐻, define a sequence {π‘₯𝑛} recursively by π‘₯𝑛+1=ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯𝑛+𝛼𝑛𝑓π‘₯𝑛,𝑛β‰₯0,(1.16) where {𝛼𝑛} is a sequence in (0,1). It was proved in [12] that, under certain appropriate conditions impose on {𝛼𝑛}, the sequence {π‘₯𝑛} generated by (1.16) converges strongly to the unique solution π‘₯βˆ—βˆˆπΆ of the variational inequality: ⟨(πΌβˆ’π‘“)π‘₯βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0,π‘₯∈𝐢.(1.17) For a strongly positive linear bounded map 𝐴 on 𝐻 with coefficient 𝛾, Marino and Xu [11] combined the iterative method (1.15) with the viscosity approximation method (1.16) and studied the following general iterative method: π‘₯𝑛+1=ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘₯𝑛+𝛼𝑛π‘₯𝛾𝑓𝑛,𝑛β‰₯0.(1.18) They proved that if the sequence {𝛼𝑛} of parameters satisfies appropriate conditions, then the sequence {π‘₯𝑛} generated by (1.18) converges strongly to the unique solution of the variational inequality: ⟨(π΄βˆ’π›Ύπ‘“)π‘₯βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0,π‘₯∈𝐢,(1.19) which is also the optimality condition for the minimization problem minπ‘₯∈𝐢(1/2)⟨𝐴π‘₯,π‘₯βŸ©βˆ’β„ŽβŸ¨π‘₯⟩, where β„Ž is a potential function for 𝛾𝑓(𝑖.𝑒.,β„Žξ…ž(π‘₯)=𝛾𝑓(π‘₯),forπ‘₯∈𝐻).

Yao et al. [19] proved that the iterative scheme defined by π‘₯0π‘₯∈𝐻,𝑛+1=𝛽𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘¦π‘›,𝑦𝑛=ξ€·1βˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘₯𝑛+𝛼𝑛π‘₯𝛾𝑓𝑛,𝑛β‰₯0,(1.20) where {𝛽𝑛} and {𝛼𝑛} are sequences in [0,1] satisfying some control conditions, converges to a fixed point of a nonexpansive mapping 𝑇 which solves the variational inequality (1.19).

Acedo and Suzuki [20], 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,(1.21) 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.

Let 𝑆 be a semigroup and let 𝐡(𝑆) be 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 [14] studied the problem of approximating common fixed point of a family of nonexpansive semigroup and solution of some variational inequality problem and proved the following theorem.

Theorem 1.1 (Saeidi and Naseri [14]). Let π’₯={𝑇(𝑑)βˆΆπ‘‘βˆˆπ‘†} be a nonexpansive semigroup on a real Hilbert space 𝐻 such that 𝐹(π’₯)β‰ βˆ…. Let 𝑋 be a left invariant subspace of B(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 (i) lim𝛼𝑛=0 and (ii) βˆ‘π›Όπ‘›=∞. Let π›Ύβˆˆ(0,𝛾/𝛼) and {π‘₯𝑛} be a sequence generated by π‘₯0∈𝐻π‘₯𝑛+1=ξ€·πΌβˆ’π›Όπ‘›πΊξ€Έπ‘‡ξ€·πœ‡π‘›ξ€Έπ‘₯𝑛+𝛼𝑛π‘₯𝛾𝑓𝑛,𝑛β‰₯0.(1.22) Then, {π‘₯𝑛} converges strongly to a common fixed point of the family π’₯ which is the unique solution of the variational inequality ⟨(πΊβˆ’π›Ύπ‘“)π‘₯βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0forallπ‘₯∈𝐹(π’₯). Equivalently one has 𝑃𝐹(π’₯)(πΌβˆ’πΊ+𝛾𝑓)π‘₯βˆ—=π‘₯βˆ—.

More recently, as commented by Golkarmanesh and Naseri [21], Piri and Vaezi [13] gave a minor variation of Theorem 1.1 as follows.

Theorem 1.2 (Piri and Vaezi [13]). Let π’₯={𝑇(𝑑)βˆΆπ‘‘βˆˆπ‘†} be a nonexpansive semigroup on a real Hilbert space 𝐻 such that 𝐹(π’₯)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 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 (i) lim𝛼𝑛=0 and (ii)βˆ‘π›Όπ‘›=∞.Let {π‘₯𝑛} be 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,(1.23) where √0<𝛾<(1βˆ’(1βˆ’π›Ώ)/πœ†)/𝛼. Then, {π‘₯𝑛} converges strongly to a common fixed point of the family π’₯ which is the unique solution of the variational inequality ⟨(πΊβˆ’π›Ύπ‘“)π‘₯βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0forallπ‘₯∈𝐹(π’₯). Equivalently one has 𝑃𝐹(π’₯)(πΌβˆ’πΊ+𝛾𝑓)π‘₯βˆ—=π‘₯βˆ—.

Motivated by these results, it is our purpose in this paper to continue the study of this problem and prove new strong convergence theorem for common fixed point of family of asymptotically nonexpansive semigroup and solution of some variational inequality problem in the framework of a real Banach space much more general than Hilbert. Our theorem, proved for more general classes of maps, is applicable in 𝑙𝑝 spaces, 1<𝑝<∞.

2. Preliminaries

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

Lemma 2.1. 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)⟩,βˆ€π‘₯,π‘¦βˆˆπΈ,𝑗(π‘₯+𝑦)∈𝐽(π‘₯+𝑦).(2.1)

Lemma 2.2 (Suzuki [22]). Let {π‘₯𝑛} and {𝑦𝑛} be bounded sequences in a 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 integers 𝑛β‰₯1 and limsup(‖𝑦𝑛+1βˆ’π‘¦π‘›β€–βˆ’β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖)≀0. Then, limβ€–π‘¦π‘›βˆ’π‘₯𝑛‖=0.

Lemma 2.3 (Kim et al. [23]). Let 𝐸 be a real Banach space satisfying the local uniform Opial’s condition and 𝐢 a nonempty weakly compact convex subset of 𝐸. If {𝑇(𝑑)βˆΆπ‘‘β‰₯0} is asymptotically nonexpansive semigroup on 𝐢, then (πΌβˆ’π‘‡(𝑑)) is demiclosed at zero.

Lemma 2.4 (Acedo and Suzuki, [20]). Let 𝐢 be a set of a separated topological vector space 𝐸. Let {𝑇(𝑑)βˆΆπ‘‘β‰₯0} be a family of mappings on 𝐢 such that 𝑇(𝑠)βˆ˜π‘‡(𝑑)=𝑇(𝑠+𝑑) for all 𝑠,π‘‘βˆˆ[0,∞). Assume that {𝑇(𝑑)βˆΆπ‘‘β‰₯0} is asymptotically regular, then ⋂𝐹(𝑇(𝑑))=𝑠β‰₯0𝐹(𝑇(𝑠)) holds for all π‘‘βˆˆ(0,∞).

Lemma 2.5 (Xu [24]). 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,(2.2) where, (i) {π›Όπ‘›βˆ‘π›Ό}βŠ‚[0,1],𝑛=∞, (ii) limsupπœŽπ‘›β‰€0, (iii) 𝛾𝑛β‰₯0andβˆ‘π›Ύπ‘›<∞. Then, π‘Žπ‘›β†’0 as π‘›β†’βˆž.

Let 𝐸 be a real Banach space and 𝛿,πœ†, and 𝜏 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, as shown in [13], (πΌβˆ’πΊ) and (πΌβˆ’πœπΊ) are strict contractions. In fact, for π‘₯,π‘¦βˆˆπΈ, β€–πœ†β€–(πΌβˆ’πΊ)π‘₯βˆ’(πΌβˆ’πΊ)𝑦2≀‖π‘₯βˆ’π‘¦β€–2βˆ’βŸ¨πΊπ‘₯βˆ’πΊπ‘¦,𝑗(π‘₯βˆ’π‘¦)βŸ©β‰€(1βˆ’π›Ώ)β€–π‘₯βˆ’π‘¦β€–2,(2.3) which implies β€–ξ‚™(πΌβˆ’πΊ)π‘₯βˆ’(πΌβˆ’πΊ)𝑦‖≀1βˆ’π›Ώπœ†β€–π‘₯βˆ’π‘¦β€–.(2.4) Also, for𝜏∈(0,1), β€–β€–ξ‚™(πΌβˆ’πœπΊ)π‘₯βˆ’(πΌβˆ’πœπΊ)𝑦‖≀‖(1βˆ’πœ)(π‘₯βˆ’π‘¦)+𝜏((πΌβˆ’πΊ)π‘₯βˆ’(πΌβˆ’πΊ)𝑦)‖≀(1βˆ’πœ)(π‘₯βˆ’π‘¦)β€–+𝜏1βˆ’π›Ώπœ†ξƒ©ξ‚™β€–π‘₯βˆ’π‘¦β€–=1βˆ’πœ1βˆ’1βˆ’π›Ώπœ†ξƒͺβ€–(π‘₯βˆ’π‘¦)β€–.(2.5)

3. Main Results

Theorem 3.1. Let 𝐸 be a real Banach space with local uniform Opial’s property whose duality mapping is sequentially continuous. Let π’₯={𝑇(𝑑)βˆΆπ‘‘β‰₯0} be uniformly asymptotically regular family of asymptotically nonexpansive semigroup of 𝐸, with function π‘˜βˆΆ[0,∞)β†’[0,∞) and β‹‚β„±βˆΆ=𝐹(π’₯)=𝑑β‰₯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‚/2,𝛿}). 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‘˜ξ€·π‘‘π‘›ξ€Έπ›Όπ‘›ξ“π›Ό=0,𝑛=∞,0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1.(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 π’₯ which solves the variational inequality: ⟨(πΊβˆ’π›Ύπ‘“)π‘ž,𝑗(π‘₯βˆ’π‘ž)⟩β‰₯0,βˆ€π‘₯βˆˆβ„±.(3.3)

Proof. We start by showing that the solution of the variational inequality (3.3) in β„± is unique. Assume π‘ž,π‘βˆˆβ„± 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 relations, 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)⟩=βŸ¨πΊπ‘βˆ’πΊπ‘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‘žβˆ’π‘β€–β‰₯π›Ώβ€–π‘βˆ’π‘žβ€–2+π›Ύπœ“(β€–π‘βˆ’π‘žβ€–)β€–π‘žβˆ’π‘β€–βˆ’π›Ύβ€–π‘žβˆ’π‘β€–2=(π›Ώβˆ’π›Ύ)β€–π‘žβˆ’π‘β€–2)+π›Ύπœ“(β€–π‘βˆ’π‘žβ€–β€–π‘žβˆ’π‘β€–.(3.6) Since 𝛿>𝛾, we obtain that 𝑝=π‘ž and so the solution is unique in β„±.
Now, let π‘žβˆˆβ„±, since (1βˆ’π›Όπ‘›π›Ύ)(π‘˜(𝑑𝑛)/𝛼𝑛)β†’0 as π‘›β†’βˆž,there exists 𝑛0βˆˆβ„• such that (1βˆ’π›Όπ‘›π›Ύ)(π‘˜(𝑑𝑛)/𝛼𝑛)<(πœ‚βˆ’π›Ύ)/2,forall𝑛β‰₯𝑛0. Hence, for 𝑛β‰₯𝑛0, we have the following: β€–β€–π‘¦π‘›β€–β€–ξ€·βˆ’π‘ž=β€–πΌβˆ’π›Όπ‘›πΊξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯π‘›βˆ’ξ€·πΌβˆ’π›Όπ‘›πΊξ€Έπ‘ž+𝛼𝑛π‘₯π›Ύπ‘“π‘›ξ€Έβˆ’π›Όπ‘›π›Ύπ‘“(π‘ž)+𝛼𝑛𝛾𝑓(π‘ž)βˆ’π›Όπ‘›πΊβ‰€ξ€·(π‘ž)β€–1βˆ’π›Όπ‘›πœ‚ξ€·π‘‘ξ€Έξ€·1+π‘˜π‘›β€–β€–π‘₯ξ€Έξ€Έπ‘›β€–β€–βˆ’π‘ž+𝛼𝑛𝛾‖‖𝑓π‘₯π‘›ξ€Έβ€–β€–βˆ’π‘“(π‘ž)+𝛼𝑛(β‰€ξ€Ίβ€–π›Ύπ‘“π‘ž)βˆ’πΊ(π‘ž)β€–1βˆ’π›Όπ‘›ξ€·(πœ‚βˆ’π›Ύ)+1βˆ’π›Όπ‘›πœ‚ξ€Έπ‘˜ξ€·π‘‘π‘›β€–β€–π‘₯ξ€Έξ€»π‘›β€–β€–βˆ’π‘ž+𝛼𝑛‖𝛾𝑓(π‘ž)βˆ’πΊ(π‘ž)β€–,(3.7) so that β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘žβ‰€π›½π‘›β€–β€–π‘₯𝑛‖‖+ξ€·βˆ’π‘ž1βˆ’π›½π‘›ξ€Έβ€–β€–π‘¦π‘›β€–β€–β‰€ξ€Ίπ›½βˆ’π‘žπ‘›+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‘₯ξ€Έξ€»ξ€»π‘›β€–β€–βˆ’π‘ž+𝛼𝑛1βˆ’π›½π‘›ξ€Έβ€–=‖𝛾𝑓(π‘ž)βˆ’πΊ(π‘ž)1βˆ’π›Όπ‘›ξ€·1βˆ’π›½π‘›ξ€Έξƒ©ξ€·(πœ‚βˆ’π›Ύ)βˆ’1βˆ’π›Όπ‘›π›Ύξ€Έπ‘˜ξ€·π‘‘π‘›ξ€Έπ›Όπ‘›β€–β€–π‘₯ξƒͺξƒ­π‘›β€–β€–βˆ’π‘ž+𝛼𝑛1βˆ’π›½π‘›ξ€Έβ€–β‰€ξƒ¬β€–π›Ύπ‘“(π‘ž)βˆ’πΊ(π‘ž)1βˆ’π›Όπ‘›ξ€·1βˆ’π›½π‘›ξ€Έξƒ©ξ€·(πœ‚βˆ’π›Ύ)βˆ’1βˆ’π›Όπ‘›π›Ύξ€Έπ‘˜ξ€·π‘‘π‘›ξ€Έπ›Όπ‘›β€–β€–π‘₯ξƒͺξƒ­π‘›β€–β€–βˆ’π‘ž+𝛼𝑛1βˆ’π›½π‘›ξ€Έξƒ©ξ€·(πœ‚βˆ’π›Ύ)βˆ’1βˆ’π›Όπ‘›π›Ύξ€Έπ‘˜ξ€·π‘‘π‘›ξ€Έπ›Όπ‘›ξƒͺ2‖𝛾𝑓(π‘ž)βˆ’πΊ(π‘ž)β€–ξ‚»β€–β€–π‘₯(πœ‚βˆ’π›Ύ)≀max𝑛‖‖,βˆ’π‘ž2‖𝛾𝑓(π‘ž)βˆ’πΊ(π‘ž)β€–ξ‚Ό.πœ‚βˆ’π›Ύ(3.8) By induction, we have β€–β€–π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘žβ‰€max𝑛0β€–β€–,βˆ’π‘ž2‖𝛾𝑓(π‘ž)βˆ’πΊ(π‘ž)β€–ξ‚Όπœ‚βˆ’π›Ύ.(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+π‘˜π‘›+1β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+||π›Όπ‘›βˆ’π›Όπ‘›+1||‖‖𝑑𝐺𝑇𝑛+1ξ€Έπ‘₯𝑛‖‖+ξ€·1βˆ’π›Όπ‘›πœ‚ξ€Έβ€–β€–π‘‡ξ€·π‘‘π‘›+1ξ€Έπ‘₯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‖‖=ξ€·1βˆ’π›Όπ‘›+1πœ‚ξ€·π‘‘ξ€Έξ€·1+π‘˜π‘›+1β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+||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+π‘˜π‘›+1β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+||π›Όπ‘›βˆ’π›Όπ‘›+1||‖‖𝑑𝐺𝑇𝑛+1ξ€Έπ‘₯𝑛‖‖+ξ€·1βˆ’π›Όπ‘›πœ‚ξ€Έsupξ€½π‘₯π‘§βˆˆπ‘›ξ€Ύ,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+π‘˜π‘›+1ξ€»β€–β€–π‘₯ξ€Έξ€Έβˆ’1𝑛+1βˆ’π‘₯𝑛‖‖+||π›Όπ‘›βˆ’π›Όπ‘›+1||‖‖𝑑𝐺𝑇𝑛+1ξ€Έπ‘₯𝑛‖‖+ξ€·1βˆ’π›Όπ‘›πœ‚ξ€Έsupξ€½π‘₯π‘§βˆˆπ‘›ξ€Ύ,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 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 2.2, limπ‘›β†’βˆžβ€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖=0.(3.14) Thus, β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖=(1βˆ’π›½π‘›)β€–π‘¦π‘›βˆ’π‘₯𝑛‖→0 as π‘›β†’βˆž.
Next, we show that limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘‡(𝑑)π‘₯𝑛‖=0 and limπ‘›β†’βˆžβ€–π‘¦π‘›βˆ’π‘‡(𝑑)𝑦𝑛‖=0,forall𝑑β‰₯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‖‖,(3.15) 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‖‖.(3.16) From 𝛼𝑛→0 as π‘›β†’βˆž, we obtain limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘‡(𝑑𝑛)π‘₯𝑛‖=0.
Now, for any 𝑑β‰₯0, we have ‖‖𝑇(𝑑)π‘₯π‘›βˆ’π‘₯𝑛‖‖≀‖‖𝑇(𝑑)π‘₯π‘›ξ€·π‘‘βˆ’π‘‡(𝑑)𝑇𝑛π‘₯𝑛‖‖+‖‖𝑇𝑑(𝑑)𝑇𝑛π‘₯π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘₯𝑛‖‖+‖‖𝑇𝑑𝑛π‘₯π‘›βˆ’π‘₯𝑛‖‖≀‖‖𝑇𝑑+𝑑𝑛π‘₯π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘₯𝑛‖‖‖‖𝑇𝑑+(2+π‘˜(𝑑))𝑛π‘₯π‘›βˆ’π‘₯𝑛‖‖≀supξ€½π‘₯π‘§βˆˆπ‘›ξ€Ύ,π‘ βˆˆβ„+‖‖𝑇𝑠+π‘‘π‘›ξ€Έξ€·π‘‘π‘§βˆ’π‘‡π‘›ξ€Έπ‘§β€–β€–+‖‖𝑇𝑑(2+π‘˜(𝑑))𝑛π‘₯π‘›βˆ’π‘₯𝑛‖‖.(3.17) Using this and the uniform asymptotic regularity of π’₯, we get limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘‡(𝑑)π‘₯𝑛‖‖=0,βˆ€π‘‘β‰₯0.(3.18) We also have β€–β€–π‘¦π‘›βˆ’π‘‡(𝑑)π‘¦π‘›β€–β€–β‰€β€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖+β€–β€–π‘₯π‘›βˆ’π‘‡(𝑑)π‘₯𝑛‖‖+‖‖𝑇(𝑑)π‘₯π‘›βˆ’π‘‡(𝑑)𝑦𝑛‖‖‖‖𝑦≀(2+π‘˜(𝑑))π‘›βˆ’π‘₯𝑛‖‖+β€–β€–π‘₯π‘›βˆ’π‘‡(𝑑)π‘₯𝑛‖‖.(3.19) This implies that limπ‘›β†’βˆžβ€–β€–π‘¦π‘›βˆ’π‘‡(𝑑)𝑦𝑛‖‖=0,βˆ€π‘‘β‰₯0.(3.20) Let {𝑦𝑛𝑗} be a subsequence of {𝑦𝑛} such that limsupπ‘›β†’βˆžξ«ξ€·π‘¦π›Ύπ‘“(π‘ž)βˆ’πΊ(π‘ž),π‘—π‘›βˆ’π‘žξ€Έξ¬=limπ‘—β†’βˆžξ‚¬ξ‚€π‘¦π›Ύπ‘“(π‘ž)βˆ’πΊ(π‘ž),π‘—π‘›π‘—βˆ’π‘žξ‚ξ‚­,(3.21) and assume without loss of generality that π‘¦π‘›π‘—β‡€π‘§βˆˆπΈ. By Lemma 2.3, (πΌβˆ’π‘‡(𝑑)) is demiclosed at zero, so π‘§βˆˆπΉ(𝑇(𝑑)) and, by Lemma 2.4, π‘§βˆˆβ„±.
Since the duality map of 𝐸 is weakly sequentially continuous, we obtain limsupπ‘›β†’βˆžξ«ξ€·π‘¦π›Ύπ‘“(π‘ž)βˆ’πΊ(π‘ž),π‘—π‘›βˆ’π‘žξ€Έξ¬=limπ‘—β†’βˆžξ‚¬ξ‚€π‘¦π›Ύπ‘“(π‘ž)βˆ’πΊ(π‘ž),π‘—π‘›π‘—βˆ’π‘žξ‚ξ‚­=⟨(π›Ύπ‘“βˆ’πΊ)π‘ž,𝑗(π‘§βˆ’π‘ž)βŸ©β‰€0.(3.22) We now conclude by showing that π‘₯π‘›β†’π‘ž as π‘›β†’βˆž. Since limπ‘›β†’βˆž(π‘˜(𝑑𝑛)/𝛼𝑛)=0, if we denote by πœŽπ‘› the value 2π‘˜(𝑑𝑛)+π‘˜(𝑑𝑛)2, then we clearly have that limπ‘›β†’βˆž(πœŽπ‘›/𝛼𝑛)=0. Let 𝑁0βˆˆβ„• be large enough such that (1βˆ’π›Όπ‘›πœ‚)(πœŽπ‘›/𝛼𝑛)<(πœ‚βˆ’2𝛿)/2,forall𝑛β‰₯𝑁0, and let 𝑀 be a positive real number such that β€–π‘₯π‘›βˆ’π‘žβ€–β‰€π‘€forall𝑛β‰₯0. Then, using the recursion formula (3.2) and for 𝑛β‰₯𝑁0, we have β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘ž2≀𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘ž2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘¦π‘›β€–β€–βˆ’π‘ž2=𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘ž2+ξ€·1βˆ’π›½π‘›ξ€ΈΓ—β€–β€–π›Όπ‘›ξ€·ξ€·π‘₯𝛾𝑓𝑛+ξ€·βˆ’πΊ(π‘ž)πΌβˆ’π›Όπ‘›πΊξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯π‘›βˆ’ξ€·πΌβˆ’π›Όπ‘›πΊξ€Έβ€–β€–(π‘ž)2≀𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘ž2+ξ€·1βˆ’π›½π‘›ξ€Έξ€·1βˆ’π›Όπ‘›πœ‚ξ€Έ2‖‖𝑇𝑑𝑛π‘₯π‘›β€–β€–βˆ’π‘ž2+2𝛼𝑛1βˆ’π›½π‘›ξ€·π‘₯ξ€Έξ«π›Ύπ‘“π‘›ξ€Έξ€·π‘¦βˆ’πΊπ‘ž,π‘—π‘›βˆ’π‘žξ€Έξ¬β‰€π›½π‘›β€–β€–π‘₯π‘›β€–β€–βˆ’π‘ž2+ξ€·1βˆ’π›½π‘›ξ€Έξ€·1βˆ’π›Όπ‘›πœ‚ξ€·π‘‘ξ€Έξ€·1+π‘˜π‘›ξ€Έξ€Έ2β€–β€–π‘₯π‘›β€–β€–βˆ’π‘ž2+2𝛼𝑛1βˆ’π›½π‘›ξ€·π‘₯ξ€Έξ«π›Ύπ‘“π‘›ξ€Έξ€·π‘¦βˆ’π›Ύπ‘“(π‘ž)+𝛾𝑓(π‘ž)βˆ’πΊπ‘ž,π‘—π‘›βˆ’π‘žξ€Έξ¬β‰€π›½π‘›β€–β€–π‘₯π‘›β€–β€–βˆ’π‘ž2+ξ€·1βˆ’π›½π‘›ξ€Έξ€·1βˆ’π›Όπ‘›πœ‚ξ€Έξ€·1+πœŽπ‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘ž2+2𝛼𝑛1βˆ’π›½π‘›ξ€·π‘¦ξ€Έξ«π›Ύπ‘“(π‘ž)βˆ’πΊπ‘ž,π‘—π‘›βˆ’π‘žξ€Έξ¬βˆ’2𝛼𝑛1βˆ’π›½π‘›ξ€Έπ›Ύβ€–β€–π‘¦π‘›β€–β€–πœ“ξ€·β€–β€–π‘₯βˆ’π‘žπ‘›β€–β€–ξ€Έβˆ’π‘ž+2𝛼𝑛1βˆ’π›½π‘›ξ€Έπ›Ύβ€–β€–π‘₯π‘›β€–β€–β€–β€–ξ€·π‘¦βˆ’π‘žπ‘›βˆ’π‘₯𝑛+ξ€·π‘₯π‘›ξ€Έβ€–β€–β‰€ξ€Ίπ›½βˆ’π‘žπ‘›+ξ€·1βˆ’π›½π‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€·πœ‚+1βˆ’π›Όπ‘›πœ‚ξ€ΈπœŽπ‘›β€–β€–π‘₯ξ€Έξ€»π‘›β€–β€–βˆ’π‘ž2+2𝛼𝑛1βˆ’π›½π‘›ξ€·π‘¦ξ€Έξ€Ίξ«π›Ύπ‘“(π‘ž)βˆ’πΊπ‘ž,π‘—π‘›β€–β€–π‘¦βˆ’π‘žξ€Έξ¬+π›Ύπ‘›βˆ’π‘₯𝑛‖‖‖‖π‘₯π‘›β€–β€–ξ€»βˆ’π‘ž+2𝛼𝑛1βˆ’π›½π‘›ξ€Έπ›Ύβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘ž2≀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‘ž2+2𝛼𝑛1βˆ’π›½π‘›ξ€·π‘¦ξ€Έξ€Ίξ«π›Ύπ‘“(π‘ž)βˆ’πΊπ‘ž,π‘—π‘›β€–β€–π‘¦βˆ’π‘žξ€Έξ¬+π›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‘ž2+𝛼𝑛1βˆ’π›½π‘›ξ€Έξ‚΅ξ€·(πœ‚βˆ’2𝛾)βˆ’1βˆ’π›Όπ‘›πœ‚ξ€ΈπœŽπ‘›π›Όπ‘›ξ‚ΆΓ—2𝑦𝛾𝑓(π‘ž)βˆ’πΊπ‘ž,π‘—π‘›β€–β€–π‘¦βˆ’π‘žξ€Έξ¬+π›Ύπ‘›βˆ’π‘₯𝑛‖‖𝑀(πœ‚βˆ’2𝛾)βˆ’1βˆ’π›Όπ‘›πœ‚πœŽξ€Έξ€·π‘›/𝛼𝑛.ξ€Έξ€Έ(3.23) Observe that βˆ‘π›Όπ‘›(1βˆ’π›½π‘›)[(πœ‚βˆ’2𝛾)βˆ’(1βˆ’π›Όπ‘›πœ‚)(πœŽπ‘›/𝛼𝑛)]=∞ and 2𝑦limsup𝛾𝑓(π‘ž)βˆ’πΊπ‘ž,π‘—π‘›β€–β€–π‘¦βˆ’π‘žξ€Έξ¬+π›Ύπ‘›βˆ’π‘₯𝑛‖‖𝑀(πœ‚βˆ’2𝛾)βˆ’1βˆ’π›Όπ‘›πœ‚πœŽξ€Έξ€·π‘›/𝛼𝑛ξƒͺ≀0.(3.24) Applying Lemma 2.5, we obtain β€–π‘₯π‘›βˆ’π‘žβ€–β†’0 as π‘›β†’βˆž. This completes the proof.

Since every Banach space whose duality map is weakly sequentially continuous satisfies Opial’s condition (see [3]) and every uniformly convex Banach space satisfying Opial’s condition also satisfies local uniform Opial’s condition (see [5]), we have the following theorem.

Theorem 3.2. Let 𝐸 be a real uniformly convex Banach space with weakly sequentially continuous duality mapping. Let π’₯={𝑇(𝑑)βˆΆπ‘‘β‰₯0},𝑓,𝐺,{𝛼𝑛},{𝛽𝑛},{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 π’₯ which solves the variational inequality (3.3).

The following corollaries follow from Theorem 3.1

Corollary 3.3. Let 𝐸=𝐻 be a real Hilbert space. Let π’₯={𝑇(𝑑)βˆΆπ‘‘β‰₯0},𝑓,𝐺,{𝛼𝑛},{𝛽𝑛},{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 π’₯ which solves the variational inequality ⟨(πΊβˆ’π›Ύπ‘“)π‘ž,π‘₯βˆ’π‘žβŸ©β‰₯0,π‘“π‘œπ‘Ÿπ‘Žπ‘™π‘™π‘₯βˆˆβ„±.

Corollary 3.4. Let 𝐸,𝑓,𝐺,{𝛼𝑛},{𝛽𝑛}, and {𝑑𝑛} be as in Theorem 3.1. Let π’₯={𝑇(𝑑)βˆΆπ‘‘β‰₯0} be a family of nonexpansive semigroup of 𝐸 with β„±βˆΆ=𝐹(π’₯)β‰ βˆ…, and let {π‘₯𝑛} be define by (3.2). Then, {π‘₯𝑛} converges strongly to a common fixed point of the family π’₯ which solves the variational inequality (3.3).

Corollary 3.5. Let 𝐸=𝑙𝑝 space, 1<𝑝<∞. Let π’₯={𝑇(𝑑)βˆΆπ‘‘β‰₯0},𝑓,𝐺,{𝛼𝑛},{𝛽𝑛},{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 π’₯ which solves the variational inequality (3.3).

Corollary 3.6. Let H be a real Hilbert space. Let π’₯={𝑇(𝑑)βˆΆπ‘‘β‰₯0} be uniformly asymptotically regular family of asymptotically nonexpansive semigroup of 𝐻, with function π‘˜βˆΆ[0,∞)β†’[0,∞) and β‹‚β„±βˆΆ=𝐹(π’₯)=𝑑β‰₯0𝐹(𝑇(𝑑))β‰ βˆ…. Let 𝑓,{𝛼𝑛},{𝛽𝑛}, and {𝑑𝑛} be as in Theorem 3.1. Let πΊβˆΆπΈβ†’πΈ be a strongly positive, bounded, and linear operator on 𝐻 with coefficient π›Ώβˆˆ(1/2,1) and ‖𝐺‖=1. For a fixed real number βˆšπ›Ύβˆˆ(0,1βˆ’2(1βˆ’π›Ώ)), let {π‘₯𝑛} be generated by (3.2). Then, the sequence {π‘₯𝑛} converges strongly to a common fixed point of the family π’₯ which solves the variational inequality ⟨(πΊβˆ’π›Ύπ‘“)π‘ž,π‘₯βˆ’π‘žβŸ©β‰₯0,π‘“π‘œπ‘Ÿπ‘Žπ‘™π‘™π‘₯βˆˆβ„±.

Acknowledgments

This work was conducted when the author was visiting the Abdus Salam International Center for Theoretical Physics Trieste Italy as an Associate. The author would like to thank the centre for hospitality and financial support.

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