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

Abstract

Let 𝐸 be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let 𝔍={𝑇(𝑡)∶𝑡≥0} be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of 𝐸, with functions 𝑢,𝑣∶[0,∞)→[0,∞). Let 𝐹∶=𝐹(𝔍)=∩_𝑡≥0𝐹(𝑇(𝑡))≠∅ and 𝑓∶𝐾→𝐾 be a weakly contractive map. For some positive real numbers 𝜆 and 𝛿 satisfying 𝛿+𝜆>1, let 𝐺∶𝐸→𝐸 be a 𝛿-strongly accretive and 𝜆-strictly pseudocontractive map. Let {𝑡_𝑛} be an increasing sequence in [0,∞) with lim_𝑛→∞𝑡_𝑛=∞, and let {𝛼_𝑛} and {𝛽_𝑛} 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 ⟨(𝐺−𝛾𝑓)𝑝,𝑗(𝑝−𝑥)⟩≤0, for all 𝑥∈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 2^𝐸∗ (𝐸^∗ is the dual space of 𝐸), and it is defined by 𝐽(𝑥)=𝑓∈𝐸^∗∶⟨𝑥,𝑓⟩=‖𝑥‖²=‖𝑓‖².(1.1)

A mapping 𝑇∶𝐸→𝐸 is said to be contractive if ‖𝑇𝑥−𝑇𝑦‖≤𝛼‖𝑥−𝑦‖, for 𝑥,𝑦∈𝐸, and some constant 𝛼∈[0,1). It is said to be weakly contractive if there exists a nondecreasing function 𝜓∶[0,∞)→[0,∞) satisfying 𝜓(𝑡)=0 if and only if 𝑡=0 and ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖−𝜓(‖𝑥−𝑦‖), for all 𝑥,𝑦∈𝐸. It is known that the class of weakly contractive maps contain properly the class of contractive ones, see [1, 2]. A mapping 𝑇∶𝐸→𝐸 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐸 and asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 as 𝑛→∞ and ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐸. We denote by 𝐹(𝑇)={𝑥∈𝐾∶𝑇𝑥=𝑥} the set of fixed points of a map 𝑇.

A mapping 𝑇∶𝐸→𝐸 is said to be total asymptotically nonexpansive (see [3]) if there exist nonnegative real sequences {𝑢_𝑛} and {𝑣_𝑛}, with 𝑢_𝑛→0 and 𝑣_𝑛→0 as 𝑛→∞ and strictly increasing and continuous functions 𝜓∶ℝ⁺→ℝ⁺ with 𝜓(0)=0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤‖𝑥−𝑦‖+𝑢_𝑛𝜓(‖𝑥−𝑦‖)+𝑣_𝑛,∀𝑥,𝑦∈𝐾.(1.2)

Remark 1.1. If 𝜓(𝜆)=𝜆, the total asymptotically nonexpansive mapping coincides with generalized asymptotically nonexpansive mapping. In addition, for all 𝑛∈ℕ, if 𝑣_𝑛=0, then generalized asymptotically nonexpansive mapping coincides with asymptotically nonexpansive mapping; if 𝑢_𝑛=0,𝑣_𝑛=max{0,𝑝_𝑛} where 𝑝_𝑛∶=sup_{𝑥,𝑦∈𝐾}(‖𝑇_𝑛𝑥−𝑇_𝑛𝑦‖−‖𝑥−𝑦‖), then generalized asymptotically nonexpansive mapping coincide with asymptotically nonexpansive mapping in the intermediate sense; if 𝑢_𝑛=0, and 𝑣_𝑛=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(𝑡)∶𝑡≥0} of self-mapping of 𝐸 such that (i)𝑇(0)𝑥=𝑥 for 𝑥∈𝐸,(ii)𝑇(𝑠+𝑡)𝑥=𝑇(𝑠)𝑇(𝑡)𝑥 for all 𝑡,𝑠≥0 and 𝑥∈𝐸, (iii)lim_𝑡→0𝑇(𝑡)𝑥=𝑥 for 𝑥∈𝐸, (iv)there exist functions 𝑢,𝑣∶[0,∞)→[0,∞) such that 𝑢(𝑡)→0,𝑣(𝑡)→0 as 𝑡→∞, and ‖𝑇(𝑡)𝑥−𝑇(𝑡)𝑦‖≤(1+𝑢(𝑡))‖𝑥−𝑦‖+𝑣(𝑡)∀𝑥,𝑦∈𝐸.(1.3)
We will denote by 𝐹 the common fixed-point set of 𝔍, that is, 𝐹∶=Fix(𝔍)={𝑥∈𝐸∶𝑇(𝑡)𝑥=𝑥,𝑡≥0}=_𝑡≥0Fix(𝑇(𝑡)).(1.4)
The family 𝔍={𝑇(𝑡)∶𝑡≥0} is said to be asymptotically regular if lim_𝑠→∞‖𝑇(𝑠+𝑡)𝑥−𝑇(𝑠)𝑥‖=0,(1.5) for all 𝑡∈[0,∞) and 𝑥∈𝐸. It is said to be uniformly asymptotically regular if, for any 𝑡≥0 and for any bounded subset 𝐶 of 𝐸, lim_𝑠→∞sup_𝑥∈𝐶‖𝑇(𝑠+𝑡)𝑥−𝑇(𝑠)𝑥‖=0.(1.6)
For some positive real numbers 𝛿 and 𝜆, a mapping 𝐺∶𝐸→𝐸 is said to be 𝛿-strongly accretive if for any 𝑥,𝑦∈𝐸, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that ⟨𝐺𝑥−𝐺𝑦,𝑗(𝑥−𝑦)⟩≥𝛿‖𝑥−𝑦‖²,(1.7) and it is called 𝜆-strictly pseudocontractive if ⟨𝐺𝑥−𝐺𝑦,𝑗(𝑥−𝑦)⟩≤‖𝑥−𝑦‖²−𝜆‖(𝐼−𝐺)𝑥−(𝐼−𝐺)𝑦‖².(1.8)
Let 𝐸 be a real Banach space, and let 𝛿,𝜆, and 𝜏 be positive real numbers satisfying 𝛿+𝜆>1 and 𝜏∈(0,1). Let 𝐺∶𝐸→𝐸 be a 𝛿-strongly accretive and 𝜆-strictly pseudocontractive, then the following holds, see [4], for 𝑥,𝑦∈𝐸∶‖(𝐼−𝐺)𝑥−(𝐼−𝐺)𝑦‖≤1−𝛿‖(𝐼−𝜏𝐺)𝑥−(𝐼−𝜏𝐺)𝑦‖≤1−𝜏1−𝜆‖𝑥−𝑦‖,1−𝛿𝜆‖𝑥−𝑦‖,(1.9) that is, (𝐼−𝐺) and (𝐼−𝜏𝐺) are contractive mappings.
Let 𝐶 be a nonempty closed-convex subset of 𝐸 and 𝑇∶𝐸→𝐸 a map. Then, a variational inequality problem with respect to 𝐶 and 𝑇 is found to be 𝑥^∗∈𝐶 such that 𝑇𝑥^∗,𝑗𝑦−𝑥^∗≥0,∀𝑦∈𝐶,𝑗𝑦−𝑥^∗∈𝐽𝑦−𝑥^∗.(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, 𝑥₀,𝑢∈𝐶, 𝑥_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛,𝑛≥0,(1.11) to a common fixed point of a uniformly asymptotically regular family {𝑇(𝑡)∶𝑡≥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 𝐵(𝑆) the subspace of all bounded real-valued functions defined on 𝑆 with supremum norm. For each 𝑠∈𝑆, the left translator operator 𝑙(𝑠) on 𝐵(𝑆) is defined by (𝑙(𝑠)𝑓)(𝑡)=𝑓(𝑠𝑡) for each 𝑡∈𝑆 and 𝑓∈𝐵(𝑆). Let 𝑋 be a subspace of 𝐵(𝑆) containing 1, and let 𝑋^∗ be its topological dual. An element 𝜇 of 𝑋^∗ is said to be a mean on 𝑋 if ‖𝜇‖=𝜇(1)=1. Let 𝑋 be 𝑙_𝑠 invariant, that is, 𝑙_𝑠(𝑋)⊂𝑋 for each 𝑠∈𝑆. A mean 𝜇 on 𝑋 is said to be left invariant if 𝜇(𝑙_𝑠𝑓)=𝜇(𝑓) for each 𝑠∈𝑆 and 𝑓∈𝑋.
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(𝑡)∶𝑡∈𝑆} be a nonexpansive semigroup in a real Hilbert space 𝐻 such that 𝐹(𝔍)≠∅. Let 𝑋 be a left invariant subspace of 𝐵(𝑆) such that 1∈𝑋, and the function 𝑡→⟨𝑇(𝑡)𝑥,𝑦⟩ is an element of 𝑋 for each 𝑥,𝑦∈𝐻. Let 𝑓∶𝐸→𝐸 be a contraction with constant 𝛼, and let 𝐺∶𝐻→𝐻 be strongly positive map with constant 𝛾>0. Let {𝜇_𝑛} be a left regular sequence of means on 𝑋, and let {𝛼_𝑛} be a sequence in (0,1) such that lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞. Let 𝛾∈(0,𝛾/𝛼), and let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐻, 𝑥_𝑛+1=𝐼−𝛼_𝑛𝐺𝑇𝜇_𝑛𝑥_𝑛+𝛼_𝑛𝛾𝑓𝑥_𝑛,𝑛≥0.(1.12)
Then, {𝑥_𝑛} converges strongly to a common fixed point of the family 𝔍 which is the unique solution of the variational inequality ⟨(𝐺−𝛾𝑓)𝑥^∗,𝑗(𝑥−𝑥^∗)⟩≥0 for all 𝑥∈𝐹(𝔍). Equivalently one has 𝑃_𝐹(𝔍)(𝐼−𝐺+𝛾𝑓)𝑥^∗=𝑥^∗.

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(𝑡)∶𝑡∈𝑆} be a nonexpansive semigroup on a real Hilbert space 𝐻 such that 𝐹(𝔍)≠∅. Let 𝑋 be a left invariant subspace of 𝐵(𝑆) such that 1∈𝑋, and the function 𝑡→⟨𝑇(𝑡)𝑥,𝑦⟩ is an element of 𝑋 for each 𝑥,𝑦∈𝐻. Let 𝑓∶𝐸→𝐸 be a contraction with constant 𝛼, and let 𝐺∶𝐻→𝐻 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1. Let {𝜇_𝑛} be a left regular sequence of means on 𝑋, and let {𝛼_𝑛} be a sequence in (0,1) such that lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞. Let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐻, 𝑥_𝑛+1=𝐼−𝛼_𝑛𝐺𝑇𝜇_𝑛𝑥_𝑛+𝛼_𝑛𝛾𝑓𝑥_𝑛,𝑛≥0,(1.13) where 0<𝛾<(1−√(1−𝛿/𝜆))/𝛼, then, {𝑥_𝑛} converges strongly to a common fixed point of the family 𝐹(𝔍) which is the unique solution of the variational inequality ⟨(𝐺−𝛾𝑓)𝑥^∗,𝑗(𝑥−𝑥^∗)⟩≥0 for all 𝑥∈𝐹(𝔍). Equivalently one has 𝑃_𝐹(𝔍)(𝐼−𝐺+𝛾𝑓)𝑥^∗=𝑥^∗.

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(𝑡)∶𝑡≥0} be a uniformly asymptotically regular family of asymptotically nonexpansive semigroup of 𝐸 with function 𝑘∶[0,∞)→[0,∞) and 𝐹∶=𝐹(𝔍)=∩_𝑡≥0𝐹(𝑇(𝑡))≠∅. Let 𝑓∶𝐸→𝐸 be weakly contractive, and let 𝐺∶𝐸→𝐸 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1. Let 𝜂∶=(1−√(1−𝛿)/𝜆) and 𝛾∈(0,min{𝜂,𝛿/2}). Let {𝛽_𝑛} and {𝛼_𝑛} be sequences in (0,1], and let {𝑡_𝑛} be an increasing sequence in [0,∞) satisfying the following conditions: lim_𝑛→∞𝛼_𝑛=0,lim_𝑛→∞𝑘_𝑛𝛼_𝑛=0,^∞_𝑛=1𝛼_𝑛=∞,0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.(1.14)
Define a sequence {𝑥_𝑛} by 𝑥₀∈𝐸, 𝑦_𝑛𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛,=𝐼−𝛼_𝑛𝐺𝑇𝑡_𝑛𝑥_𝑛+𝛼_𝑛𝛾_𝑛𝑓𝑥_𝑛,𝑛≥0.(1.15)
Then, the sequence {𝑥_𝑛} converges strongly to a common fixed point of the family 𝔍 which solves the variational inequality ⟨(𝐺−𝛾𝑓)𝑞,𝑗(𝑥−𝑞)⟩≥0,∀𝑥∈𝐹.(1.16)

Remark 1.5. It is well known that all 𝑙^𝑝(1<𝑝<∞) spaces satisfy Opial's condition and possess a weakly sequentially continuous duality mapping. However, 𝐿^𝑝  (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 Gâteaux differentiable norm. Our theorem is applicable in 𝐿_𝑝(ℓ_𝑝) 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 𝐿^𝑝(1<𝑝<∞) spaces on one hand and for more general class of maps on the other hand.

2. Preliminaries

Let 𝑆∶={𝑥∈𝐸∶||𝑥||=1} denote the unit sphere of a real Banach space 𝐸. 𝐸 is said to have a Gâteaux differentiable norm if the limit lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.1) exists for each 𝑥,𝑦∈𝑆; 𝐸 is said to have a uniformly Gâteaux differentiable norm if for each 𝑦∈𝑆, the limit is attained uniformly for 𝑥∈𝑆. A Banach space 𝐸 is said to be strictly convex if ‖𝑥+𝑦‖/2<1 for 𝑥≠𝑦 and ‖𝑥‖=‖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{‖𝑥−𝑦‖∶𝑥,𝑦∈𝐾}. For each 𝑥∈𝐾, let 𝑟(𝑥,𝐾)∶=sup{‖𝑥−𝑦‖∶𝑦∈𝐾} and 𝑟(𝐾)∶=inf{𝑟(𝑥,𝐾)∶𝑥∈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)/𝑟(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 Gâteaux differentiable norm, then the normalized duality mapping 𝐽∶𝐸→2^𝐸∗, defined by (1.1), is singled valued and uniformly continuous from the norm topology of 𝐸 to the weak^∗ topology of 𝐸^∗ on each bounded subset of 𝐸, see, for example [30].

Definition 2.1. Let 𝜇 be a continuous linear functional on 𝑙^∞, and let (𝑎₀,𝑎₁,…)∈𝑙^∞. We write 𝜇_𝑛(𝑎_𝑛) instead of 𝜇(𝑎₀,𝑎₁,…). The function 𝜇 is called a Banach limit when 𝜇 satisfies ||𝜇||=𝜇_𝑛(1)=1 and 𝜇_𝑛(𝑎_𝑛+1)=𝜇_𝑛(𝑎_𝑛) for each (𝑎₀,𝑎₁,…)∈𝑙^∞.
For a Banach limit 𝜇, it is known that liminf_𝑛→∞𝑎_𝑛≤𝜇_𝑛(𝑎_𝑛)≤limsup_𝑛→∞𝑎_𝑛 for every 𝑎=(𝑎₀,𝑎₁,…)∈𝑙^∞. So if 𝑎=(𝑎₀,𝑎₁,…)∈𝑙^∞ and 𝑎_𝑛−𝑏_𝑛→0 as 𝑛→∞, we have 𝜇_𝑛(𝑎_𝑛)=𝜇_𝑛(𝑏_𝑛).
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: ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑗(𝑥+𝑦)⟩∀𝑥,𝑦∈𝐸,𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦).(2.2)

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

Lemma 2.3 (Suzuki [31]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a real Banach space 𝐸, and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf𝛽_𝑛≤limsup𝛽_𝑛<1. Suppose that 𝑥_𝑛+1=𝛽_𝑛𝑦_𝑛+(1−𝛽_𝑛)𝑥_𝑛 for all integer 𝑛≥1 and limsup_𝑛→∞(||𝑦_𝑛+1−𝑦_𝑛||−||𝑥_𝑛+1−𝑥_𝑛||)≤0. Then, lim_𝑛→∞||𝑦_𝑛−𝑥_𝑛||=0.

Lemma 2.4 (Shioji and Takahashi [32]). Let (𝑎₀,𝑎₁,𝑎₂,…)∈𝑙^∞ be such that 𝜇_𝑛𝑎_𝑛≤0 for all Banach limits 𝜇. If limsup_𝑛→∞(𝑎_𝑛+1−𝑎_𝑛)≤0, then limsup_𝑛→∞𝑎_𝑛≤0.

Lemma 2.5 (Xu [33]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝑎_𝑛+1≤1−𝛼_𝑛𝑎_𝑛+𝛼_𝑛𝜎_𝑛+𝛾_𝑛,𝑛≥0,(2.3) where (i){𝛼_𝑛}⊂[0,1],∑^∞𝑛=0𝛼_𝑛=∞  (ii)limsup_𝑛→∞𝜎_𝑛≤0   (iii)𝛾_𝑛≥0 and (𝑛≥0),∑^∞𝑛=0𝛾_𝑛<∞. Then, 𝑎_𝑛→0 as 𝑛→∞.

3. Main Results

Theorem 3.1. Let 𝐸 be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm, and let 𝔍={𝑇(𝑡)∶𝑡≥0} be uniformly asymptotically regular family of generalized asymptotically nonexpansive semigroup of 𝐸, with functions 𝑢,𝑣∶[0,∞)→[0,∞) and 𝐹∶=𝐹(𝔍)=∩_𝑡≥0𝐹(𝑇(𝑡))≠∅. Let 𝑓∶𝐸→𝐸 be weakly contractive, and let 𝐺∶𝐸→𝐸 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1. Let 𝜂∶=(1−√(1−𝛿)/𝜆) and 𝛾∈(0,min{𝛿,𝜂/2}). Let {𝛽_𝑛} and {𝛼_𝑛} be sequences in (0,1] and {𝑡_𝑛} an increasing sequence in [0,∞) satisfying the following conditions: lim_𝑛→∞𝛼_𝑛=0,lim_𝑛→∞𝑢𝑡_𝑛𝛼_𝑛=0,lim_𝑛→∞𝑣𝑡_𝑛𝛼_𝑛=0,^∞_𝑛=10<liminf_𝑛→∞𝛼_𝑛=∞,𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1,lim_𝑛→∞𝑡_𝑛=∞.(3.1)
Define a sequence {𝑥_𝑛} by 𝑥₀∈𝐸, 𝑦_𝑛𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛,=𝐼−𝛼_𝑛𝐺𝑇𝑡_𝑛𝑥_𝑛+𝛼_𝑛𝛾𝑓𝑥_𝑛,𝑛≥0.(3.2)
Then, the sequence {𝑥_𝑛} converges strongly to a common fixed point of the family 𝔍 which solves the variational inequality ⟨(𝐺−𝛾𝑓)𝑞,𝑗(𝑥−𝑞)⟩≥0,∀𝑥∈𝐹.(3.3)

Proof. We start by showing that solution of the variational inequality (3.3) in 𝐹 is at most one. Assume that 𝑞,𝑝∈𝐹 are solutions of the variational inequality (3.3), then ⟨(𝐺−𝛾𝑓)𝑝,𝑗(𝑞−𝑝)⟩≥0,⟨(𝐺−𝛾𝑓)𝑞,𝑗(𝑝−𝑞)⟩≥0.(3.4)
Adding these two inequalities, we get ⟨(𝐺−𝛾𝑓)𝑝−(𝐺−𝛾𝑓)𝑞,𝑗(𝑝−𝑞)⟩≤0.(3.5)
Therefore, 0≥=≥⟨𝛿⟨𝐺‖(𝑝𝐺(𝑝−−)𝑞−𝛾‖𝑓𝐺₂)(𝑝𝑞−),(𝑗𝐺(𝑝−−𝛾𝑞𝑓))⟩𝑞−,𝑗𝛾(𝑝⟨𝑓−(𝑞𝑝))⟩−𝑓(𝑞),𝑗(𝑝−𝑞)⟩≥𝛿‖𝑝−𝑞‖₂−𝛾‖𝑓(𝑝)−𝑓(𝑞)‖‖𝑝−𝑞‖+𝛾𝜓(‖𝑝−𝑞‖)‖𝑝−𝑞‖−𝛾‖𝑝−𝑞‖²=(𝛿−𝛾)‖𝑝−𝑞‖²+𝛾𝜓(‖𝑝−𝑞‖)‖𝑝−𝑞‖.(3.6) Since 𝛿>𝛾, we obtain that 𝑝=𝑞, and so the solution is unique in 𝐹.
Now, let 𝑝∈𝐹, since (1−𝛼_𝑛𝜂)(𝑢(𝑡_𝑛)/𝛼_𝑛)→0 and (1−𝛼_𝑛𝜂)(𝑣(𝑡_𝑛)/𝛼_𝑛)→0 as 𝑛→∞, then there exists 𝑛₀∈ℕ such that (1−𝛼_𝑛𝜂)(𝑢(𝑡_𝑛)/𝛼_𝑛)<(𝜂−𝛾)/2 and (1−𝛼_𝑛𝜂)(𝑣(𝑡_𝑛)/𝛼_𝑛)<(𝜂−𝛾)/2 for all 𝑛≥𝑛₀. Hence, for 𝑛≥𝑛₀, we have the following: ‖‖𝑦_𝑛−𝑝‖‖≤≤‖‖1𝐼−−𝛼_𝑛𝛼_𝑛𝐺𝑇𝑡_𝑛𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐺(𝑝)‖‖≤1−𝛼_𝑛𝜂1+𝑢𝑡_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝑣𝑡_𝑛+𝛼_𝑛𝛾‖‖𝑓𝑥_𝑛−𝑓(𝑝)‖‖+𝛼_𝑛‖𝛾𝑓(𝑝)−𝐺(𝑝)‖(𝜂−𝛾)+1−𝛼_𝑛𝜂𝑢𝑡_𝑛‖‖𝑥_𝑛−𝑝‖‖+1−𝛼_𝑛𝜂𝑣𝑡_𝑛+𝛼_𝑛‖𝛾𝑓(𝑝)−𝐺(𝑝)‖,(3.7)
so that ‖‖𝑥_𝑛+1−𝑝‖‖≤≤𝛽_𝑛𝛽‖‖𝑥_𝑛−𝑝‖‖+1−𝛽_𝑛‖‖𝑦_𝑛−𝑝_𝑛‖‖+1+−𝛼1_𝑛−𝛽_𝑛1−𝛼_𝑛(𝜂−𝛾)+1−𝛼_𝑛𝜂𝑢𝑡_𝑛‖‖𝑥_𝑛−𝑝‖‖≤1−𝛼_𝑛𝜂1−𝛽_𝑛𝑣𝑡_𝑛+𝛼_𝑛1−𝛽_𝑛‖𝛾𝑓(𝑝)−𝐺(𝑝)‖1−𝛽_𝑛(𝜂−𝛾)−1−𝛼_𝑛𝜂𝑢𝑡^𝑛𝛼_𝑛‖‖𝑥^𝑛+𝛼_𝑛−𝑝‖‖1−𝛽_𝑛‖𝛾𝑓(𝑝)−𝐺(𝑝)‖+1−𝛼_𝑛𝜂𝑣𝑡^𝑛≤1−𝛼_𝑛𝛼_𝑛1−𝛽_𝑛(𝜂−𝛾)−1−𝛼_𝑛𝜂𝑢𝑡^𝑛𝛼_𝑛‖‖𝑥^𝑛+𝛼_𝑛−𝑝‖‖1−𝛽_𝑛(𝜂−𝛾)−1−𝛼_𝑛𝜂𝑢𝑡^𝑛×2‖𝛾𝑓(𝑝)−𝐺(𝑝)‖+1−𝛼_𝑛𝛼_𝑛𝜂𝑣𝑡_𝑛/𝛼_𝑛≤max‖‖𝑥_𝑛𝜂−𝛾−𝑝‖‖,2‖𝛾𝑓(𝑝)−𝐺(𝑝)‖𝜂−𝛾+1.(3.8)
By induction, we have ‖‖𝑥_𝑛−𝑝‖‖≤max‖‖𝑥_𝑛0−𝑝‖‖,2‖𝛾𝑓(𝑝)−𝐺(𝑝)‖𝜂−𝛾+1,∀𝑛≥0.(3.9)
Thus, {𝑥_𝑛} is bounded and so are {𝑇(𝑡_𝑛)𝑥_𝑛},{𝐺𝑇(𝑡_𝑛)𝑥_𝑛},{𝑦_𝑛}, and {𝑓(𝑥_𝑛)}.
Observe that 𝑦_𝑛+1−𝑦_𝑛=+𝐼𝐼−−𝛼𝛼_𝑛+1𝐺_𝑛+1𝑇𝑡_𝑛+1𝑥_𝑛+1−𝐼−𝛼_𝑛+1𝐺𝑇𝑡_𝑛+1𝑥_𝑛+𝐼−𝛼_𝑛𝐺𝑇𝑡_𝑛+1𝑥_𝑛−𝐼−𝛼_𝑛𝐺𝑇𝑡_𝑛+1𝑥_𝑛+𝛼_𝑛+1𝐺𝑇𝑡_𝑛+1𝑥_𝑛−𝐼−𝛼_𝑛𝐺𝑇𝑡_𝑛𝑥_𝑛𝛾𝑓𝑥_𝑛+1−𝛼_𝑛+1𝛾𝑓𝑥_𝑛+𝛼_𝑛+1𝛾𝑓𝑥_𝑛−𝛼_𝑛𝛾𝑓𝑥_𝑛,(3.10) so that ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖≤+1||𝛼−_𝑛𝛼_𝑛+1𝜂1+𝑢𝑡_𝑛+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+1−𝛼_𝑛+1𝜂𝑣𝑡_𝑛+1+𝛼−_𝑛+1𝛼_𝑛+1||‖‖𝐺𝑇𝑡_𝑛+1𝑥_𝑛‖‖+1−𝛼_𝑛𝜂‖‖𝑇𝑡_𝑛+1−𝑡_𝑛+𝑡_𝑛𝑥_𝑛−𝑇𝑡_𝑛𝑥_𝑛‖‖≤1−𝛼𝛾‖‖_𝑛+1𝑓𝑥_𝑛+1−𝑓𝑥_𝑛‖‖+||𝛼_𝑛+1−𝛼_𝑛||𝛾‖‖𝑓𝑥_𝑛‖‖+||𝛼_𝑛𝜂1+𝑢𝑡_𝑛+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+1−𝛼_𝑛+1𝜂𝑣𝑡_𝑛+1+1−−𝛼𝛼_𝑛+1𝑛||‖‖𝐺𝑇𝑡_𝑛+1𝑥_𝑛‖‖+𝛼_𝑛+1𝜂sup_{𝑧∈{𝑥𝑛},𝑠∈ℝ+}‖‖𝑇𝑠+𝑡_𝑛𝑧−𝑇𝑡_𝑛𝑧‖‖𝛾‖‖𝑓𝑥_𝑛+1−𝑓𝑥_𝑛‖‖+||𝛼_𝑛+1−𝛼_𝑛||𝛾‖‖𝑓𝑥_𝑛‖‖.(3.11)
From this, we obtain that ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤+11−−𝛼𝛼_𝑛+1𝑛+1𝜂1+𝑢𝑡_𝑛+1−1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+1−𝛼_𝑛𝜂𝑣𝑡_𝑛+1+||𝛼_𝑛−𝛼_𝑛+1||‖‖𝐺𝑇𝑡_𝑛+1𝑥_𝑛‖‖+𝛼_𝑛+1𝜂sup_{𝑧∈{𝑥𝑛},𝑠∈ℝ+}‖‖𝑇𝑠+𝑡_𝑛𝑧−𝑇𝑡_𝑛𝑧‖‖𝛾‖‖𝑓𝑥_𝑛+1−𝑓𝑥_𝑛‖‖+||𝛼_𝑛+1−𝛼_𝑛||𝛾‖‖𝑓𝑥_𝑛‖‖,(3.12) which implies that limsup_𝑛→∞‖‖𝑦_𝑛+1−𝑦_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0,(3.13) and by Lemma 2.3, lim_𝑛→∞‖‖𝑦_𝑛−𝑥_𝑛‖‖=0.(3.14)
Thus, ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=1−𝛽_𝑛‖‖𝑦_𝑛−𝑥_𝑛‖‖⟶0as𝑛⟶∞.(3.15)
Next, we show that lim_𝑛→∞‖𝑦_𝑛−𝑇(𝑡)𝑦_𝑛‖=0, for all 𝑡≥0.
Since ‖‖𝑥_𝑛−𝑇𝑡_𝑛𝑥_𝑛‖‖≤≤‖‖‖‖𝑥𝑥_𝑛𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑇𝑡_𝑛𝑥_𝑛‖‖−𝑥_𝑛+1‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑇𝑡_𝑛𝑥_𝑛‖‖+1−𝛽_𝑛‖‖𝑦_𝑛−𝑇𝑡_𝑛𝑥_𝑛‖‖,(3.16) we have 1−𝛽_𝑛‖‖𝑥_𝑛−𝑇𝑡_𝑛𝑥_𝑛‖‖=≤‖‖‖‖𝑥𝑥_𝑛𝑛−𝑥_𝑛+1‖‖+1−𝛽_𝑛‖‖𝑦_𝑛−𝑇𝑡_𝑛𝑥_𝑛‖‖−𝑥_𝑛+1‖‖+𝛼_𝑛1−𝛽_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐺𝑇𝑡_𝑛𝑥_𝑛‖‖.(3.17)
From 𝛼_𝑛→0 as 𝑛→∞ and (3.15), we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑇𝑡_𝑛𝑥_𝑛‖‖=0.(3.18)
Also, ‖‖𝑦_𝑛−𝑇𝑡_𝑛𝑦_𝑛‖‖≤≤‖‖2𝑦_𝑛+−𝑢𝑥_𝑛𝑡‖‖_𝑛+‖‖𝑥_𝑛−𝑇𝑡_𝑛𝑥_𝑛‖‖+‖‖𝑇𝑡_𝑛𝑥_𝑛−𝑇𝑡_𝑛𝑦_𝑛‖‖‖‖𝑦_𝑛−𝑥_𝑛‖‖+𝑣𝑡_𝑛+‖‖𝑥_𝑛−𝑇𝑡_𝑛𝑥_𝑛‖‖⟶0as𝑛⟶∞.(3.19)
Since lim_𝑛→∞𝑡_𝑛=∞ and {𝑇(𝑡)∶𝑡≥0} is uniformly asymptotically regular, lliimm_{𝑛→∞𝑛→∞}‖‖𝑇(𝑡)𝑇𝑡_𝑛𝑥_𝑛−𝑇𝑡_𝑛𝑥_𝑛‖‖≤lim_𝑛→∞sup_𝑥∈𝐶‖‖𝑇(𝑡)𝑇𝑡_𝑛𝑥−𝑇𝑡_𝑛𝑥‖‖=0,‖‖𝑇(𝑡)𝑇𝑡_𝑛𝑦_𝑛−𝑇𝑡_𝑛𝑦_𝑛‖‖≤lim_𝑛→∞sup_𝑦∈𝐶‖‖𝑇(𝑡)𝑇𝑡_𝑛𝑦−𝑇𝑡_𝑛𝑦‖‖=0,(3.20) where 𝐶 is any bounded subset of 𝐸 containing {𝑥_𝑛}. Since {𝑇(𝑡)} is continuous, we get that ‖‖𝑦_𝑛−𝑇(𝑡)𝑦_𝑛‖‖≤+‖‖‖‖𝑦_𝑛𝑇−(𝑡𝑇)𝑇𝑡_𝑛𝑡_𝑛𝑦_𝑛‖‖+‖‖𝑇𝑡_𝑛𝑦_𝑛−𝑇(𝑡)𝑇𝑡_𝑛𝑦_𝑛‖‖𝑦_𝑛−𝑇(𝑡)𝑦_𝑛‖‖.(3.21)
This implies that lim_𝑛→∞‖‖𝑦_𝑛−𝑇(𝑡)𝑦_𝑛‖‖=0,∀𝑡≥0.(3.22)
Next, we show that limsup_𝑛→∞(𝛾𝑓−𝐺)𝑝,𝑗𝑦_𝑛−𝑝≤0.(3.23)
Define a map 𝜙∶𝐸→ℝ by 𝜙(𝑦)∶=𝜇_𝑛‖‖𝑦_𝑛−𝑦‖‖²,∀𝑦∈𝐸.(3.24)
Then, 𝜙(𝑦)→∞ as ‖𝑦‖→∞, 𝜙 is continuous and convex, so as 𝐸 is reflexive, there exists 𝑞∈𝐸 such that 𝜙(𝑞)=min_𝑢∈𝐸𝜙(𝑢). Hence, the set 𝐾^∗∶=𝑦∈𝐸∶𝜙(𝑦)=min_𝑢∈𝐸𝜙(𝑢)≠∅.(3.25)
Since lim_𝑛→∞‖𝑦_𝑛−𝑇(𝑡)𝑦_𝑛‖=0,lim_𝑡→∞𝑢(𝑡)=0,lim_𝑡→∞𝑣(𝑡)=0, and 𝜙 is continuous for all 𝑧∈𝐾^∗, we have 𝜙lim_𝑡→∞𝑇(𝑡)𝑧=lim_𝑡→∞𝜙(𝑇(𝑡)𝑧)=lim_𝑡→∞𝜇_𝑛‖‖𝑦_𝑛−𝑇(𝑡)𝑧‖‖²≤lim_𝑡→∞𝜇_𝑛(1+𝑢(𝑡))‖‖𝑦_𝑛−𝑧‖‖+(𝑣(𝑡))²=𝜇_𝑛‖‖𝑦_𝑛−𝑧‖‖²=𝜙(𝑧).(3.26)
Hence, lim_𝑡→∞𝑇(𝑡)𝑧∈𝐾^∗.
Let 𝑝∈𝐹. Since 𝐾^∗ is a closed-convex set, there exists a unique 𝑞∈𝐾^∗ such that ‖𝑝−𝑞‖=min_{𝑥∈𝐾∗}‖𝑝−𝑥‖.(3.27)
Since 𝑝=lim_𝑡→∞𝑇(𝑡)𝑝 and lim_𝑡→∞𝑇(𝑡)𝑞∈𝐾^∗, ‖‖‖𝑝−lim_𝑡→∞𝑇(𝑡)𝑞‖‖‖==‖‖‖lilmi_𝑡→∞m_𝑡→∞𝑇(𝑡)𝑝−lim_𝑡→∞𝑇(𝑡)𝑞‖‖‖≤lim_𝑡→∞‖𝑇(𝑡)𝑝−𝑇(𝑡)𝑞‖≤‖𝑝−((𝑞1‖+.𝑢(𝑡))‖𝑝−𝑞‖+𝑣(𝑡))(3.28)
Therefore, lim_𝑡→∞𝑇(𝑡)𝑞=𝑞. Since 𝑇(𝑠+ℎ)𝑥=𝑇(𝑠)𝑇(ℎ)𝑥 for all 𝑥∈𝐸 and 𝑠≥0, we have 𝑞==𝑇lim(_𝑡→∞𝑠)𝑇li(m𝑡_𝑡→∞)𝑞=lim_𝑡→∞𝑇(𝑠+𝑡)𝑞=lim_𝑡→∞𝑇(𝑠)𝑇(𝑡)𝑞𝑇(𝑡)𝑞=𝑇(𝑠)𝑞.(3.29)
Therefore, 𝑞∈𝐹 and so 𝐾^∗∩𝐹≠∅.
Let 𝑝∈𝐾^∗∩𝐹(𝑇) and 𝜏∈(0,1). Then, it follows that 𝜙(𝑝)≤𝜙(𝑝−𝜏(𝐺−𝛾𝑓)𝑝), and using Lemma 2.2, we obtain that ‖‖𝑦_𝑛−𝑝+𝜏(𝐺−𝛾𝑓)𝑝‖‖²≤‖‖𝑦_𝑛−𝑝‖‖²+2𝜏(𝐺−𝛾𝑓)𝑝,𝑗𝑦_𝑛−𝑝+𝜏(𝐺−𝛾𝑓)𝑝,(3.30) which implies that 𝜇_𝑛(𝛾𝑓−𝐺)𝑝,𝑗𝑦_𝑛−𝑝+𝜏(𝐺−𝛾𝑓)𝑝≤0.(3.31)
Moreover, 𝜇_𝑛(𝛾𝑓−𝐺)𝑝,𝑗𝑦_𝑛−𝑝=+𝜇_𝑛𝜇_𝑛(𝛾𝑓−𝐺)𝑝,𝑗𝑦_𝑛−𝑝−𝑗𝑦_𝑛−𝑝+𝜏(𝐺−𝛾𝑓)𝑝≤𝜇_𝑛(𝛾𝑓−𝐺)𝑝,𝑗𝑦_𝑛−𝑝+𝜏(𝐺−𝛾𝑓)𝑝(𝛾𝑓−𝐺)𝑝,𝑗𝑦_𝑛−𝑝−𝑗𝑦_𝑛−𝑝+𝜏(𝐺−𝛾𝑓)𝑝.(3.32)
Since 𝑗 is norm-to-weak* uniformly continuous on bounded subsets of 𝐸, we have that 𝜇_𝑛(𝛾𝑓−𝐺)𝑝,𝑗𝑦_𝑛−𝑝≤0.(3.33)
Observe that from (3.14) and (3.15), we have lim_𝑛→∞‖‖𝑦_𝑛+1−𝑦_𝑛‖‖=0.(3.34)
This implies that limsup_𝑛→∞⟨(𝛾𝑓−𝐺)𝑝,𝑗𝑦_𝑛−𝑝⟩−⟨(𝛾𝑓−𝐺)𝑝,𝑗𝑦_𝑛+1−𝑝⟩≤0,(3.35) and so we obtain by Lemma 2.4 that limsup_𝑛→∞(𝛾𝑓−𝐺)𝑝,𝑗𝑦_𝑛−𝑝≤0.(3.36)
Finally, we show that 𝑥_𝑛→𝑝 as 𝑛→∞. Since lim_𝑛→∞(𝑢(𝑡_𝑛)/𝛼_𝑛)=0, if we denote by 𝜎(𝑡_𝑛) the value 2𝑢(𝑡_𝑛)+𝑢(𝑡_𝑛)², then we clearly have lim_𝑛→∞(𝜎(𝑡_𝑛)/𝛼_𝑛)=0. Let 𝑁₀∈ℕ be large enough such that (1−𝛼_𝑛𝜂)(𝜎(𝑡_𝑛)/𝛼_𝑛)<(𝜂−2𝛾)/2, for all 𝑛≥𝑁₀, and let 𝑀 be a positive real number such that ||𝑥_𝑛−𝑝||≤𝑀 for all 𝑛≥0. Then, using the recursion formula (3.2) and for 𝑛≥𝑁₀, we have ‖‖𝑦_𝑛−𝑝‖‖²=‖‖𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐺(𝑝)+𝐼−𝛼_𝑛𝐺𝑇𝑡_𝑛𝑥_𝑛−𝑝‖‖²≤≤11−−𝛼𝛼_𝑛𝑛𝜂‖‖𝑇𝑡_𝑛𝑥_𝑛−𝑝‖‖²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐺(𝑝),𝑗𝑦_𝑛−𝑝𝜂1+𝑢𝑡_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝑣𝑡_𝑛²≤+12𝛼−_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑝)+𝛾𝑓(𝑝)−𝐺(𝑝),𝑗𝑦_𝑛−𝑝+2𝛼_𝑛𝜂1+𝑢𝑡_𝑛²‖‖𝑥_𝑛−𝑝‖‖²+21+𝑢𝑡_𝑛𝑣𝑡_𝑛‖‖𝑥_𝑛−𝑝‖‖²+𝑣𝑡_𝑛²+2𝛼_𝑛𝛾𝑓(𝑝)−𝐺(𝑝),𝑗𝑦_𝑛−𝑝−2𝛼_𝑛𝛾‖‖𝑦_𝑛−𝑝‖‖𝜓‖‖𝑥_𝑛−𝑝‖‖≤1−𝛾𝛼‖‖_𝑛𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑝‖‖‖‖𝑥_𝑛−𝑝‖‖𝜂1+𝜎𝑡_𝑛+2𝛼_𝑛𝛾‖‖𝑥_𝑛−𝑝‖‖²+𝛼_𝑛2𝛾𝑓(𝑝)−𝐺(𝑝),𝑗𝑦^𝑛−𝑝+21−𝛼_𝑛𝜂1+𝑢𝑡_𝑛𝑣𝑡^𝑛𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖²+1−𝛼_𝑛𝜂𝑣𝑡^𝑛²𝛼_𝑛=1−𝛼_𝑛+2𝛾‖‖𝑦_𝑛−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖(𝜂−2𝛾)−1−𝛼^𝑛𝜂𝜎^𝑛𝛼_𝑛‖‖𝑥^𝑛−𝑝‖‖²+𝛼_𝑛2𝛾𝑓(𝑝)−𝐺(𝑝),𝑗𝑦^𝑛−𝑝+21−𝛼_𝑛𝜂1+𝑢𝑡_𝑛𝑣𝑡^𝑛𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖²+1−𝛼_𝑛𝜂𝑣𝑡^𝑛²𝛼_𝑛+2𝛾‖‖𝑦_𝑛−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖,(3.37)
so that ‖‖𝑥_𝑛+1−𝑝‖‖²≤𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖²+1−𝛽_𝑛‖‖𝑦_𝑛−𝑝‖‖²≤𝛽_𝑛+1−𝛽_𝑛1−𝛼_𝑛(𝜂−2𝛾)−1−𝛼^𝑛𝜂𝜎^𝑛𝛼_𝑛‖‖𝑥^𝑛−𝑝‖‖²+𝛼_𝑛1−𝛽_𝑛×2𝑣𝑡𝛾𝑓(𝑝)−𝐺(𝑝),𝑗𝑦_𝑛−𝑝+21−𝛼_𝑛𝜂1+𝑢𝑡_𝑛_𝑛𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖²+1−𝛼_𝑛𝜂𝑣𝑡^𝑛²𝛼_𝑛≤1−𝛼_𝑛+2𝛾‖‖𝑦_𝑛−𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖1−𝛽_𝑛(𝜂−2𝛾)−1−𝛼_𝑛𝜂𝜎𝛼^𝑛_𝑛‖‖𝑥^𝑛−𝑝‖‖²+𝛼_𝑛1−𝛽_𝑛2𝛾𝑓(𝑝)−𝐺(𝑝),𝑗𝑦_𝑛−𝑝+21−𝛼_𝑛𝜂1+𝑢𝑡_𝑛𝑣𝑡^𝑛𝛼_𝑛𝑀²+1−𝛼_𝑛𝜂𝑣𝑡^𝑛²𝛼_𝑛=1−𝛼_𝑛+2𝛾‖‖𝑦_𝑛−𝑥_𝑛‖‖𝑀1−𝛽_𝑛(𝜂−2𝛾)−1−𝛼_𝑛𝜂𝜎𝛼^𝑛_𝑛‖‖𝑥^𝑛−𝑝‖‖²+𝛼_𝑛1−𝛽_𝑛(𝜂−2𝛾)−1−𝛼_𝑛𝜂𝜎^𝑛×2𝛾𝑓(𝑝)−𝐺(𝑝),𝑗𝑦_𝑛𝛼_𝑛−𝑝+21−𝛼_𝑛𝜂1+𝑢𝑡_𝑛𝑣𝑡^𝑛𝛼_𝑛𝑀²+𝒜_𝑛(𝜂−2𝛾)−1−𝛼^𝑛𝜂𝜎^𝑛𝛼_𝑛,(3.38) where 𝒜_𝑛 denotes (1−𝛼_𝑛𝜂)(𝑣(𝑡_𝑛)²/𝛼_𝑛)+2𝛾‖𝑦_𝑛−𝑥_𝑛‖𝑀.
Observe  that ∑^∞𝑛=1𝛼_𝑛(1−𝛽_𝑛)((𝜂−2𝛾)−(1−𝛼_𝑛𝜂)(𝜎_𝑛/𝛼_𝑛))=∞ and limsup_𝑛→∞2𝛾𝑓(𝑝)−𝐺(𝑝),𝑗𝑦^𝑛−𝑝+21−𝛼_𝑛𝜂1+𝑢𝑡_𝑛𝑣𝑡_𝑛/𝛼_𝑛𝑀²+𝒜_𝑛(𝜂−2𝛾)−1−𝛼_𝑛𝜂𝜎_𝑛/𝛼_𝑛≤0.(3.39)
Applying Lemma 2.5, we obtain ‖𝑥_𝑛−𝑝‖→0 as 𝑛→∞. 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(𝑡)∶𝑡≥0}, and let 𝐹,𝑓,𝐺,𝛿,𝜆,𝜂,𝛾,{𝛽_𝑛},{𝛼_𝑛},{𝑡_𝑛} and {𝑥_𝑛} be as in Theorem 3.1. Then, the sequence {𝑥_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(𝑡)∶𝑡≥0},𝐹,𝑓,𝐺,𝛿,𝜆,𝜂,𝛾,{𝛽_𝑛},{𝛼_𝑛},{𝑡_𝑛} and {𝑥_𝑛} be as in Theorem 3.1. Then, the sequence {𝑥_𝑛} converges strongly to a common fixed point of the family 𝔍 which solves the variational inequality ⟨(𝐺−𝛾𝑓)𝑞,𝑥−𝑞⟩≥0,∀𝑥∈𝐹.(3.40)

Corollary 3.4. Let 𝔍={𝑇(𝑡)∶𝑡≥0} be a family of nonexpansive semigroup of a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm 𝐸, and let 𝐹,𝑓,𝐺,𝛿,𝜆,𝜂,𝛾,{𝛽_𝑛},{𝛼_𝑛},{𝑡_𝑛}, and {𝑥_𝑛} be as in Theorem 3.1. Then, the sequence {𝑥_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.