Abstract

Let 𝐸 be a real Banach space satisfying local uniform Opial's condition, whose duality map is weakly sequentially continuous. Let 𝒥={𝑇(𝑡)𝑡0} be a uniformly asymptotically regular family of asymptotically nonexpansive semigroup of 𝐸 with function 𝑘[0,)[0,). Let =𝑡0𝐹(𝑇(𝑡)) and 𝑓𝐸𝐸 be weakly contractive map. Let 𝐺𝐸𝐸 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive map with 𝛿+𝜆>1. Let {𝑡𝑛} 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 {𝑥𝑛} be a sequence defined by 𝑥0𝐸, 𝑥𝑛+1=𝛽𝑛𝑥𝑛+(1𝛽𝑛)𝑦𝑛,  𝑦𝑛=(𝐼𝛼𝑛𝐺)𝑇(𝑡𝑛)𝑥𝑛+𝛼𝑛𝛾𝑓(𝑥𝑛),  𝑛0. It is proved that {𝑥𝑛} converges strongly to a common fixed point 𝑞 of the family 𝒥 which is also the unique solution of the variational inequality (𝐺𝛾𝑓)𝑞,𝑗(𝑞𝑥)0,forall𝑥.

1. Introduction

Let 𝐸 be a real Banach space and let 𝐸 be the dual space of 𝐸. A mapping 𝜑[0,)[0,) is called a gauge function if it is strictly increasing, continuous and 𝜑(0)=0. Let 𝜑 be a gauge function, a generalized duality mapping with respect to 𝜑,𝐽𝜑𝐸2𝐸 is defined by, 𝑥𝐸, 𝐽𝜑𝑥𝑥=𝐸𝑥,𝑥(=𝑥𝜑𝑥),𝑥()=𝜑𝑥,(1.1) where , denotes the duality pairing between element of 𝐸 and that of 𝐸. If 𝜑(𝑡)=𝑡, then 𝐽𝜑 is simply called the normalized duality mapping and is denoted by 𝐽. For any 𝑥𝐸, an element of 𝐽𝜑𝑥 is denoted by 𝑗𝜑(𝑥).

The modulus of convexity of 𝐸 is the function 𝛿𝐸(0,2][0,1] defined by 𝛿𝐸(𝜖)=inf1𝑥+𝑦2𝑥=𝑦=1,𝜖=𝑥𝑦,(1.2) and 𝐸 is called uniformly convex if 𝛿𝐸(𝜖)>0forall𝜖(0,2]. A Banach space 𝐸 is said to satisfy Opial’s condition [1] if, for any sequence {𝑥𝑛} in 𝐸, 𝑥𝑛𝑥 as 𝑛 implies that liminf𝑛𝑥𝑛𝑥<liminf𝑛𝑥𝑛𝑦,𝑦𝐸,𝑦𝑥.(1.3)All Hilbert spaces and 𝑙𝑝 spaces, 1𝑝< satisfy Opial’s condition. However 𝐿𝑝, 𝑝2 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 𝜑 such that 𝐽𝜑 is singled valued and (sequentially) continuous from 𝐸 with weak topology to 𝐸 with weak 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 𝑟>0 such that 1+𝑟liminf𝑛𝑥+𝑥𝑛(1.4) for each 𝑥𝐸 with 𝑥𝑐 and each sequence {𝑥𝑛} satisfying 𝑥𝑛0 as 𝑛, and liminf𝑛𝑥𝑛1.

𝐸 is said to satisfy the local uniform Opial’s condition [5] if, for any weak null sequence {𝑥𝑛} in 𝐸 with liminf𝑛𝑥𝑛1 and any 𝑐>0, there exists 𝑟>0 such that 1+𝑟liminf𝑛𝑥+𝑥𝑛(1.5) for all 𝑥𝐸 with 𝑥𝑐. Observe that uniform Opial’s condition implies local uniform Opial’s condition which in turn implies Opial’s condition.

A self-mapping 𝑇𝐸𝐸 is said to be contraction if 𝑇𝑥𝑇𝑦𝛼𝑥𝑦,forall𝑥,𝑦𝐸, where 𝛼[0,1) is a fixed constant. 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 𝑇𝑥𝑇𝑦𝑥𝑦𝜓(𝑥𝑦),forall𝑥,𝑦𝐸. 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 𝑇𝑥𝑇𝑦𝑥𝑦forall𝑥,𝑦𝐸 and asymptotically nonexpansive if there exists a sequence {𝑘𝑛}[0,) with lim𝑛𝑘𝑛=0 such that 𝑇𝑛𝑥𝑇𝑛𝑦(1+𝑘𝑛)𝑥𝑦,forall𝑥,𝑦𝐸 and 𝑛. The set of fixed point of 𝑇 is defined as 𝐹(𝑇)={𝑥𝐸𝑇𝑥=𝑥}.

A one parameter family 𝒥={𝑇(𝑡)𝑡0} of self-mapping of 𝐸 is called nonexpansive semigroup if the following conditions are satisfied:(i)𝑇(0)𝑥=𝑥forall𝑥𝐸; (ii)𝑇(𝑡+𝑠)=𝑇(𝑡)𝑇(𝑠)forall𝑡,𝑠0; (iii)for each 𝑥𝐸, the mapping 𝑡𝑇(𝑡)𝑥 is continuous;(iv)for 𝑥,𝑦𝐸 and 𝑡0, 𝑇(𝑡)𝑥𝑇(𝑡)𝑦𝑥𝑦.

The family 𝒥 is said to be asymptotically nonexpansive semigroup if conditions (i)–(iii) are satisfied and, in addition, there exists a function 𝑘[0,)[0,) satisfying lim𝑡𝑘(𝑡)=0 and 𝑇(𝑡)𝑥𝑇(𝑡)𝑦(1+𝑘(𝑡))𝑥𝑦forall𝑥,𝑦𝐸.

The family 𝒥={𝑇(𝑡)𝑡0} is said to be asymptotically regular if lim𝑠𝑇(𝑡+𝑠)𝑥𝑇(𝑠)𝑥=0,(1.6) for all 𝑡(0,) and 𝑥𝐾. It is said to be uniformly asymptotically regular if, for any 𝑡0 and for any bounded subset 𝐶 of 𝐾, lim𝑠sup𝑥𝐶𝑇(𝑡+𝑠)𝑥𝑇(𝑠)𝑥=0.(1.7) For some positive real numbers 𝛿 and 𝜆, the mapping 𝐺𝐸𝐸 is said to be 𝛿-strongly accretive if for any 𝑥,𝑦𝐸, there exists 𝑗(𝑥𝑦)𝐽(𝑥𝑦) such that 𝐺𝑥𝐺𝑦,𝑗(𝑥𝑦)𝛿𝑥𝑦2(1.8) and it is called 𝜆-strictly pseudocontractive if 𝐺𝑥𝐺𝑦,𝑗(𝑥𝑦)𝑥𝑦2𝜆(𝐼𝐺)𝑥(𝐼𝐺)𝑦2.(1.9) Let 𝐶 be a nonempty closed convex subset of 𝐸 and 𝑇𝐸𝐸 be a map. Then, a variational inequality problem with respect to 𝐶 and 𝑇 is find 𝑥𝐶 such that𝑇𝑥,𝑗𝑦𝑥0,𝑦𝐶,𝑗𝑦𝑥𝐽𝑦𝑥.(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, [810] 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., [1117] 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: min𝑥𝐶12𝐴𝑥,𝑥𝑥,𝑏.(1.11) Here, 𝐶 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 𝐴𝐻𝐻 is said to be strongly positive if there exists a constant 𝛾>0 such that 𝐴𝑥,𝑥𝛾𝑥2,forall𝑥𝐻.(1.12) For a strongly positive bonded linear operator 𝐴 and any 𝑥,𝑦𝐻, we have 𝐴𝑥𝐴𝑦,𝑥𝑦𝛾𝑥𝑦2.(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: 𝐴𝑥𝐴𝑦,𝑥𝑦1+𝐴22𝑥𝑦212(𝐼𝐴)𝑥(𝐼𝐴)𝑦2.(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 {𝑥𝑛} defined by 𝑥0𝐻 chosen arbitrarily, 𝑥𝑛+1=𝐼𝛼𝑛𝐴𝑇𝑥𝑛+𝛼𝑛𝑏,𝑛0,(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 𝑥0𝐻, define a sequence {𝑥𝑛} recursively by 𝑥𝑛+1=1𝛼𝑛𝑇𝑥𝑛+𝛼𝑛𝑓𝑥𝑛,𝑛0,(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 {𝑥𝑛} generated by (1.16) converges strongly to the unique solution 𝑥𝐶 of the variational inequality: (𝐼𝑓)𝑥,𝑥𝑥0,𝑥𝐶.(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: 𝑥𝑛+1=𝐼𝛼𝑛𝐴𝑇𝑥𝑛+𝛼𝑛𝑥𝛾𝑓𝑛,𝑛0.(1.18) They proved that if the sequence {𝛼𝑛} of parameters satisfies appropriate conditions, then the sequence {𝑥𝑛} generated by (1.18) converges strongly to the unique solution of the variational inequality: (𝐴𝛾𝑓)𝑥,𝑥𝑥0,𝑥𝐶,(1.19) which is also the optimality condition for the minimization problem min𝑥𝐶(1/2)𝐴𝑥,𝑥𝑥, where is a potential function for 𝛾𝑓(𝑖.𝑒.,(𝑥)=𝛾𝑓(𝑥),for𝑥𝐻).

Yao et al. [19] proved that the iterative scheme defined by 𝑥0𝑥𝐻,𝑛+1=𝛽𝑛𝑥𝑛+1𝛽𝑛𝑦𝑛,𝑦𝑛=1𝛼𝑛𝐴𝑇𝑥𝑛+𝛼𝑛𝑥𝛾𝑓𝑛,𝑛0,(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, 𝑥0,𝑢𝐶, 𝑥𝑛=𝛼𝑛𝑢+1𝛼𝑛𝑇𝑡𝑛𝑥𝑛,𝑛0,(1.21) to a common fixed point of a uniformly asymptotically regular family {𝑇(𝑡)𝑡0} of nonexpansive semigroup in the framework of a real Hilbert space.

Let 𝑆 be a semigroup and let 𝐵(𝑆) be the subspace of all bounded real valued functions defined on 𝑆 with supremum norm. For each 𝑠𝑆, the left translator operator 𝑙(𝑠) on 𝐵(𝑆) is defined by (𝑙(𝑠)𝑓)(𝑡)=𝑓(𝑠𝑡) for each 𝑡𝑆 and 𝑓𝐵(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 𝑙𝑠 invariant; that is, 𝑙𝑠(𝑋)𝑋 for each 𝑠𝑆. A mean 𝜇 on 𝑋 is said to be left invariant if 𝜇(𝑙𝑠𝑓)=𝜇(𝑓) for each 𝑠𝑆 and 𝑓𝑋.

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 𝒥={𝑇(𝑡)𝑡𝑆} 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 1𝑋, and the function 𝑡𝑇(𝑡)𝑥,𝑦 is an element of 𝑋 for each 𝑥,𝑦𝐻. Let 𝑓𝐸𝐸 be a contraction with constant 𝛼 and let 𝐺𝐻𝐻 be strongly positive map with constant 𝛾>0. Let {𝜇𝑛} 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) 𝛼𝑛=. Let 𝛾(0,𝛾/𝛼) and {𝑥𝑛} be a sequence generated by 𝑥0𝐻𝑥𝑛+1=𝐼𝛼𝑛𝐺𝑇𝜇𝑛𝑥𝑛+𝛼𝑛𝑥𝛾𝑓𝑛,𝑛0.(1.22) Then, {𝑥𝑛} converges strongly to a common fixed point of the family 𝒥 which is the unique solution of the variational inequality (𝐺𝛾𝑓)𝑥,𝑥𝑥0forall𝑥𝐹(𝒥). Equivalently one has 𝑃𝐹(𝒥)(𝐼𝐺+𝛾𝑓)𝑥=𝑥.

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 𝒥={𝑇(𝑡)𝑡𝑆} 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 1𝑋, and the function 𝑡𝑇(𝑡)𝑥,𝑦 is an element of 𝑋 for each 𝑥,𝑦𝐻. Let 𝑓𝐸𝐸 be a contraction and let 𝐺𝐻𝐻 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1. Let {𝜇𝑛} 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)𝛼𝑛=.Let {𝑥𝑛} be generated by 𝑥0𝐻: 𝑥𝑛+1=𝐼𝛼𝑛𝐺𝑇𝜇𝑛𝑥𝑛+𝛼𝑛𝑥𝛾𝑓𝑛,𝑛0,(1.23) where 0<𝛾<(1(1𝛿)/𝜆)/𝛼. Then, {𝑥𝑛} converges strongly to a common fixed point of the family 𝒥 which is the unique solution of the variational inequality (𝐺𝛾𝑓)𝑥,𝑥𝑥0forall𝑥𝐹(𝒥). Equivalently one has 𝑃𝐹(𝒥)(𝐼𝐺+𝛾𝑓)𝑥=𝑥.

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: 𝑥+𝑦2𝑥2+2𝑦,𝑗(𝑥+𝑦),𝑥,𝑦𝐸,𝑗(𝑥+𝑦)𝐽(𝑥+𝑦).(2.1)

Lemma 2.2 (Suzuki [22]). Let {𝑥𝑛} and {𝑦𝑛} be bounded sequences in a Banach space 𝐸 and let {𝛽𝑛} be a sequence in [0,1] with 0<liminf𝛽𝑛limsup𝛽𝑛<1. Suppose that 𝑥𝑛+1=𝛽𝑛𝑦𝑛+(1𝛽𝑛)𝑥𝑛 for all integers 𝑛1 and limsup(𝑦𝑛+1𝑦𝑛𝑥𝑛+1𝑥𝑛)0. Then, lim𝑦𝑛𝑥𝑛=0.

Lemma 2.3 (Kim et al. [23]). Let 𝐸 be a real Banach space satisfying the local uniform Opial’s condition and 𝐶 a nonempty weakly compact convex subset of 𝐸. If {𝑇(𝑡)𝑡0} is asymptotically nonexpansive semigroup on 𝐶, then (𝐼𝑇(𝑡)) is demiclosed at zero.

Lemma 2.4 (Acedo and Suzuki, [20]). Let 𝐶 be a set of a separated topological vector space 𝐸. Let {𝑇(𝑡)𝑡0} be a family of mappings on 𝐶 such that 𝑇(𝑠)𝑇(𝑡)=𝑇(𝑠+𝑡) for all 𝑠,𝑡[0,). Assume that {𝑇(𝑡)𝑡0} is asymptotically regular, then 𝐹(𝑇(𝑡))=𝑠0𝐹(𝑇(𝑠)) holds for all 𝑡(0,).

Lemma 2.5 (Xu [24]). Let {𝑎𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝑎𝑛+11𝛼𝑛𝑎𝑛+𝛼𝑛𝜎𝑛+𝛾𝑛,𝑛0,(2.2) where, (i) {𝛼𝑛𝛼}[0,1],𝑛=, (ii) limsup𝜎𝑛0, (iii) 𝛾𝑛0and𝛾𝑛<. Then, 𝑎𝑛0 as 𝑛.

Let 𝐸 be a real Banach space and 𝛿,𝜆, and 𝜏 positive real numbers satisfying 𝛿+𝜆>1 and 𝜏(0,1). Let 𝐺𝐸𝐸 be a 𝛿-strongly accretive and 𝜆-strictly pseudocontractive then, as shown in [13], (𝐼𝐺) and (𝐼𝜏𝐺) are strict contractions. In fact, for 𝑥,𝑦𝐸, 𝜆(𝐼𝐺)𝑥(𝐼𝐺)𝑦2𝑥𝑦2𝐺𝑥𝐺𝑦,𝑗(𝑥𝑦)(1𝛿)𝑥𝑦2,(2.3) which implies (𝐼𝐺)𝑥(𝐼𝐺)𝑦1𝛿𝜆𝑥𝑦.(2.4) Also, for𝜏(0,1), (𝐼𝜏𝐺)𝑥(𝐼𝜏𝐺)𝑦(1𝜏)(𝑥𝑦)+𝜏((𝐼𝐺)𝑥(𝐼𝐺)𝑦)(1𝜏)(𝑥𝑦)+𝜏1𝛿𝜆𝑥𝑦=1𝜏11𝛿𝜆(𝑥𝑦).(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 𝒥={𝑇(𝑡)𝑡0} be uniformly asymptotically regular family of asymptotically nonexpansive semigroup of 𝐸, with function 𝑘[0,)[0,) and =𝐹(𝒥)=𝑡0𝐹(𝑇(𝑡)). Let 𝑓𝐸𝐸 be weakly contractive and let 𝐺𝐸𝐸 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1. Let 𝜂=(1(1𝛿)/𝜆) and 𝛾(0,min{𝜂/2,𝛿}). Let {𝛽𝑛} and {𝛼𝑛} be sequences in [0,1] and let {𝑡𝑛} be an increasing sequence in [0,) satisfying the following conditions: lim𝑛𝛼𝑛=0,lim𝑛𝑘𝑡𝑛𝛼𝑛𝛼=0,𝑛=,0<liminf𝑛𝛽𝑛limsup𝑛𝛽𝑛<1.(3.1) Define a sequence {𝑥𝑛} by 𝑥0𝐸, 𝑥𝑛+1=𝛽𝑛𝑥𝑛+1𝛽𝑛𝑦𝑛,𝑦𝑛=𝐼𝛼𝑛𝐺𝑇𝑡𝑛𝑥𝑛+𝛼𝑛𝑥𝛾𝑓𝑛,𝑛0.(3.2) Then, the sequence {𝑥𝑛} converges strongly to a common fixed point of the family 𝒥 which solves the variational inequality: (𝐺𝛾𝑓)𝑞,𝑗(𝑥𝑞)0,𝑥.(3.3)

Proof. We start by showing that the solution of the variational inequality (3.3) in is unique. Assume 𝑞,𝑝 are solutions of the variational inequality (3.3), then (𝐺𝛾𝑓)𝑝,𝑗(𝑞𝑝)0,(𝐺𝛾𝑓)𝑞,𝑗(𝑝𝑞)0.(3.4) Adding these two relations, we get (𝐺𝛾𝑓)𝑝(𝐺𝛾𝑓)𝑞,𝑗(𝑝𝑞)0.(3.5) Therefore, 0(𝐺𝛾𝑓)𝑝(𝐺𝛾𝑓)𝑞,𝑗(𝑝𝑞)=𝐺𝑝𝐺𝑞,𝑗(𝑝𝑞)𝛾𝑓(𝑝)𝑓(𝑞),𝑗(𝑝𝑞)𝛿𝑝𝑞2𝛾𝑓(𝑝)𝑓(𝑞)𝑞𝑝𝛿𝑝𝑞2+𝛾𝜓(𝑝𝑞)𝑞𝑝𝛾𝑞𝑝2=(𝛿𝛾)𝑞𝑝2)+𝛾𝜓(𝑝𝑞𝑞𝑝.(3.6) Since 𝛿>𝛾, we obtain that 𝑝=𝑞 and so the solution is unique in .
Now, let 𝑞, since (1𝛼𝑛𝛾)(𝑘(𝑡𝑛)/𝛼𝑛)0 as 𝑛,there exists 𝑛0 such that (1𝛼𝑛𝛾)(𝑘(𝑡𝑛)/𝛼𝑛)<(𝜂𝛾)/2,forall𝑛𝑛0. Hence, for 𝑛𝑛0, we have the following: 𝑦𝑛𝑞=𝐼𝛼𝑛𝐺𝑇𝑡𝑛𝑥𝑛𝐼𝛼𝑛𝐺𝑞+𝛼𝑛𝑥𝛾𝑓𝑛𝛼𝑛𝛾𝑓(𝑞)+𝛼𝑛𝛾𝑓(𝑞)𝛼𝑛𝐺(𝑞)1𝛼𝑛𝜂𝑡1+𝑘𝑛𝑥𝑛𝑞+𝛼𝑛𝛾𝑓𝑥𝑛𝑓(𝑞)+𝛼𝑛(𝛾𝑓𝑞)𝐺(𝑞)1𝛼𝑛(𝜂𝛾)+1𝛼𝑛𝜂𝑘𝑡𝑛𝑥𝑛𝑞+𝛼𝑛𝛾𝑓(𝑞)𝐺(𝑞),(3.7) so that 𝑥𝑛+1𝑞𝛽𝑛𝑥𝑛+𝑞1𝛽𝑛𝑦𝑛𝛽𝑞𝑛+1𝛽𝑛1𝛼𝑛(𝜂𝛾)+1𝛼𝑛𝜂𝑘𝑡𝑛𝑥𝑛𝑞+𝛼𝑛1𝛽𝑛=𝛾𝑓(𝑞)𝐺(𝑞)1𝛼𝑛1𝛽𝑛(𝜂𝛾)1𝛼𝑛𝛾𝑘𝑡𝑛𝛼𝑛𝑥𝑛𝑞+𝛼𝑛1𝛽𝑛𝛾𝑓(𝑞)𝐺(𝑞)1𝛼𝑛1𝛽𝑛(𝜂𝛾)1𝛼𝑛𝛾𝑘𝑡𝑛𝛼𝑛𝑥𝑛𝑞+𝛼𝑛1𝛽𝑛(𝜂𝛾)1𝛼𝑛𝛾𝑘𝑡𝑛𝛼𝑛2𝛾𝑓(𝑞)𝐺(𝑞)𝑥(𝜂𝛾)max𝑛,𝑞2𝛾𝑓(𝑞)𝐺(𝑞).𝜂𝛾(3.8) By induction, we have 𝑥𝑛𝑥𝑞max𝑛0,𝑞2𝛾𝑓(𝑞)𝐺(𝑞)𝜂𝛾.(3.9) Thus, {𝑥𝑛} is bounded and so are {𝑇(𝑡𝑛)𝑥𝑛},{𝐺𝑇(𝑡𝑛)𝑥𝑛},{𝑦𝑛}, and {𝑓(𝑥𝑛)}. Observe that 𝑦𝑛+1𝑦𝑛=𝐼𝛼𝑛+1𝐺𝑇𝑡𝑛+1𝑥𝑛+1𝐼𝛼𝑛+1𝐺𝑇𝑡𝑛+1𝑥𝑛+𝐼𝛼𝑛+1𝐺𝑇𝑡𝑛+1𝑥𝑛𝐼𝛼𝑛𝐺𝑇𝑡𝑛+1𝑥𝑛+𝐼𝛼𝑛𝐺𝑇𝑡𝑛+1𝑥𝑛𝐼𝛼𝑛𝐺𝑇𝑡𝑛𝑥𝑛+𝛼𝑛+1𝑥𝛾𝑓𝑛+1𝛼𝑛+1𝑥𝛾𝑓𝑛+𝛼𝑛+1𝑥𝛾𝑓𝑛𝛼𝑛𝑥𝛾𝑓𝑛,(3.10) so that 𝑦𝑛+1𝑦𝑛1𝛼𝑛+1𝜂𝑡1+𝑘𝑛+1𝑥𝑛+1𝑥𝑛+||𝛼𝑛𝛼𝑛+1||𝑡𝐺𝑇𝑛+1𝑥𝑛+1𝛼𝑛𝜂𝑇𝑡𝑛+1𝑥𝑛𝑡𝑇𝑛𝑥𝑛+𝛼𝑛+1𝛾𝑓𝑥𝑛+1𝑥𝑓𝑛+||𝛼𝑛+1𝛼𝑛||𝛾𝑓𝑥𝑛=1𝛼𝑛+1𝜂𝑡1+𝑘𝑛+1𝑥𝑛+1𝑥𝑛+||𝛼𝑛𝛼𝑛+1||𝑡𝐺𝑇𝑛+1𝑥𝑛+1𝛼𝑛𝜂𝑇𝑡𝑛+1𝑡𝑛+𝑡𝑛𝑥𝑛𝑡𝑇𝑛𝑥𝑛+𝛼𝑛+1𝛾𝑓𝑥𝑛+1𝑥𝑓𝑛+||𝛼𝑛+1𝛼𝑛||𝛾𝑓𝑥𝑛1𝛼𝑛+1𝜂𝑡1+𝑘𝑛+1𝑥𝑛+1𝑥𝑛+||𝛼𝑛𝛼𝑛+1||𝑡𝐺𝑇𝑛+1𝑥𝑛+1𝛼𝑛𝜂sup𝑥𝑧𝑛,𝑠+𝑇𝑠+𝑡𝑛𝑡𝑧𝑇𝑛𝑧+𝛼𝑛+1𝛾𝑓𝑥𝑛+1𝑥𝑓𝑛+||𝛼𝑛+1𝛼𝑛||𝛾𝑓𝑥𝑛.(3.11) From this, we obtain that 𝑦𝑛+1𝑦𝑛𝑥𝑛+1𝑥𝑛1𝛼𝑛+1𝜂𝑡1+𝑘𝑛+1𝑥1𝑛+1𝑥𝑛+||𝛼𝑛𝛼𝑛+1||𝑡𝐺𝑇𝑛+1𝑥𝑛+1𝛼𝑛𝜂sup𝑥𝑧𝑛,𝑠+𝑇𝑠+𝑡𝑛𝑡𝑧𝑇𝑛𝑧+𝛼𝑛+1𝛾𝑓𝑥𝑛+1𝑥𝑓𝑛+||𝛼𝑛+1𝛼𝑛||𝛾𝑓𝑥𝑛,(3.12) which implies limsup𝑛𝑦𝑛+1𝑦𝑛𝑥𝑛+1𝑥𝑛0(3.13) and by Lemma 2.2, lim𝑛𝑦𝑛𝑥𝑛=0.(3.14) Thus, 𝑥𝑛+1𝑥𝑛=(1𝛽𝑛)𝑦𝑛𝑥𝑛0 as 𝑛.
Next, we show that lim𝑛𝑥𝑛𝑇(𝑡)𝑥𝑛=0 and lim𝑛𝑦𝑛𝑇(𝑡)𝑦𝑛=0,forall𝑡0.
Since 𝑥𝑛𝑡𝑇𝑛𝑥𝑛𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑡𝑇𝑛𝑥𝑛𝑥𝑛𝑥𝑛+1+𝛽𝑛𝑥𝑛𝑡𝑇𝑛𝑥𝑛+1𝛽𝑛𝑦𝑛𝑡𝑇𝑛𝑥𝑛,(3.15) we have 1𝛽𝑛𝑥𝑛𝑡𝑇𝑛𝑥𝑛𝑥𝑛𝑥𝑛+1+1𝛽𝑛𝑦𝑛𝑡𝑇𝑛𝑥𝑛=𝑥𝑛𝑥𝑛+1+𝛼𝑛1𝛽𝑛𝑥𝛾𝑓𝑛𝑡𝐺𝑇𝑛𝑥𝑛.(3.16) From 𝛼𝑛0 as 𝑛, we obtain lim𝑛𝑥𝑛𝑇(𝑡𝑛)𝑥𝑛=0.
Now, for any 𝑡0, we have 𝑇(𝑡)𝑥𝑛𝑥𝑛𝑇(𝑡)𝑥𝑛𝑡𝑇(𝑡)𝑇𝑛𝑥𝑛+𝑇𝑡(𝑡)𝑇𝑛𝑥𝑛𝑡𝑇𝑛𝑥𝑛+𝑇𝑡𝑛𝑥𝑛𝑥𝑛𝑇𝑡+𝑡𝑛𝑥𝑛𝑡𝑇𝑛𝑥𝑛𝑇𝑡+(2+𝑘(𝑡))𝑛𝑥𝑛𝑥𝑛sup𝑥𝑧𝑛,𝑠+𝑇𝑠+𝑡𝑛𝑡𝑧𝑇𝑛𝑧+𝑇𝑡(2+𝑘(𝑡))𝑛𝑥𝑛𝑥𝑛.(3.17) Using this and the uniform asymptotic regularity of 𝒥, we get lim𝑛𝑥𝑛𝑇(𝑡)𝑥𝑛=0,𝑡0.(3.18) We also have 𝑦𝑛𝑇(𝑡)𝑦𝑛𝑦𝑛𝑥𝑛+𝑥𝑛𝑇(𝑡)𝑥𝑛+𝑇(𝑡)𝑥𝑛𝑇(𝑡)𝑦𝑛𝑦(2+𝑘(𝑡))𝑛𝑥𝑛+𝑥𝑛𝑇(𝑡)𝑥𝑛.(3.19) This implies that lim𝑛𝑦𝑛𝑇(𝑡)𝑦𝑛=0,𝑡0.(3.20) Let {𝑦𝑛𝑗} be a subsequence of {𝑦𝑛} such that limsup𝑛𝑦𝛾𝑓(𝑞)𝐺(𝑞),𝑗𝑛𝑞=lim𝑗𝑦𝛾𝑓(𝑞)𝐺(𝑞),𝑗𝑛𝑗𝑞,(3.21) and assume without loss of generality that 𝑦𝑛𝑗𝑧𝐸. By Lemma 2.3, (𝐼𝑇(𝑡)) is demiclosed at zero, so 𝑧𝐹(𝑇(𝑡)) and, by Lemma 2.4, 𝑧.
Since the duality map of 𝐸 is weakly sequentially continuous, we obtain limsup𝑛𝑦𝛾𝑓(𝑞)𝐺(𝑞),𝑗𝑛𝑞=lim𝑗𝑦𝛾𝑓(𝑞)𝐺(𝑞),𝑗𝑛𝑗𝑞=(𝛾𝑓𝐺)𝑞,𝑗(𝑧𝑞)0.(3.22) We now conclude by showing that 𝑥𝑛𝑞 as 𝑛. Since lim𝑛(𝑘(𝑡𝑛)/𝛼𝑛)=0, if we denote by 𝜎𝑛 the value 2𝑘(𝑡𝑛)+𝑘(𝑡𝑛)2, then we clearly have that lim𝑛(𝜎𝑛/𝛼𝑛)=0. Let 𝑁0 be large enough such that (1𝛼𝑛𝜂)(𝜎𝑛/𝛼𝑛)<(𝜂2𝛿)/2,forall𝑛𝑁0, and let 𝑀 be a positive real number such that 𝑥𝑛𝑞𝑀forall𝑛0. Then, using the recursion formula (3.2) and for 𝑛𝑁0, we have 𝑥𝑛+1𝑞2𝛽𝑛𝑥𝑛𝑞2+1𝛽𝑛𝑦𝑛𝑞2=𝛽𝑛𝑥𝑛𝑞2+1𝛽𝑛×𝛼𝑛𝑥𝛾𝑓𝑛+𝐺(𝑞)𝐼𝛼𝑛𝐺𝑇𝑡𝑛𝑥𝑛𝐼𝛼𝑛𝐺(𝑞)2𝛽𝑛𝑥𝑛𝑞2+1𝛽𝑛1𝛼𝑛𝜂2𝑇𝑡𝑛𝑥𝑛𝑞2+2𝛼𝑛1𝛽𝑛𝑥𝛾𝑓𝑛𝑦𝐺𝑞,𝑗𝑛𝑞𝛽𝑛𝑥𝑛𝑞2+1𝛽𝑛1𝛼𝑛𝜂𝑡1+𝑘𝑛2𝑥𝑛𝑞2+2𝛼𝑛1𝛽𝑛𝑥𝛾𝑓𝑛𝑦𝛾𝑓(𝑞)+𝛾𝑓(𝑞)𝐺𝑞,𝑗𝑛𝑞𝛽𝑛𝑥𝑛𝑞2+1𝛽𝑛1𝛼𝑛𝜂1+𝜎𝑛𝑥𝑛𝑞2+2𝛼𝑛1𝛽𝑛𝑦𝛾𝑓(𝑞)𝐺𝑞,𝑗𝑛𝑞2𝛼𝑛1𝛽𝑛𝛾𝑦𝑛𝜓𝑥𝑞𝑛𝑞+2𝛼𝑛1𝛽𝑛𝛾𝑥𝑛𝑦𝑞𝑛𝑥𝑛+𝑥𝑛𝛽𝑞𝑛+1𝛽𝑛1𝛼𝑛𝜂+1𝛼𝑛𝜂𝜎𝑛𝑥𝑛𝑞2+2𝛼𝑛1𝛽𝑛𝑦𝛾𝑓(𝑞)𝐺𝑞,𝑗𝑛𝑦𝑞+𝛾𝑛𝑥𝑛𝑥𝑛𝑞+2𝛼𝑛1𝛽𝑛𝛾𝑥𝑛𝑞21𝛼𝑛1𝛽𝑛(𝜂2𝛾)1𝛼𝑛𝜂𝜎𝑛𝛼𝑛𝑥𝑛𝑞2+2𝛼𝑛1𝛽𝑛𝑦𝛾𝑓(𝑞)𝐺𝑞,𝑗𝑛𝑦𝑞+𝛾𝑛𝑥𝑛𝑀=1𝛼𝑛1𝛽𝑛(𝜂2𝛾)1𝛼𝑛𝜂𝜎𝑛𝛼𝑛𝑥𝑛𝑞2+𝛼𝑛1𝛽𝑛(𝜂2𝛾)1𝛼𝑛𝜂𝜎𝑛𝛼𝑛×2𝑦𝛾𝑓(𝑞)𝐺𝑞,𝑗𝑛𝑦𝑞+𝛾𝑛𝑥𝑛𝑀(𝜂2𝛾)1𝛼𝑛𝜂𝜎𝑛/𝛼𝑛.(3.23) Observe that 𝛼𝑛(1𝛽𝑛)[(𝜂2𝛾)(1𝛼𝑛𝜂)(𝜎𝑛/𝛼𝑛)]= and 2𝑦limsup𝛾𝑓(𝑞)𝐺𝑞,𝑗𝑛𝑦𝑞+𝛾𝑛𝑥𝑛𝑀(𝜂2𝛾)1𝛼𝑛𝜂𝜎𝑛/𝛼𝑛0.(3.24) Applying Lemma 2.5, we obtain 𝑥𝑛𝑞0 as 𝑛. 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 𝒥={𝑇(𝑡)𝑡0},𝑓,𝐺,{𝛼𝑛},{𝛽𝑛},{𝑡𝑛}, and {𝑥𝑛} be as in Theorem 3.1. Then, the sequence {𝑥𝑛} converges strongly to a common fixed point of the family 𝒥 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 𝒥={𝑇(𝑡)𝑡0},𝑓,𝐺,{𝛼𝑛},{𝛽𝑛},{𝑡𝑛}, and {𝑥𝑛} be as in Theorem 3.1, then the sequence {𝑥𝑛} converges strongly to a common fixed point of the family 𝒥 which solves the variational inequality (𝐺𝛾𝑓)𝑞,𝑥𝑞0,𝑓𝑜𝑟𝑎𝑙𝑙𝑥.

Corollary 3.4. Let 𝐸,𝑓,𝐺,{𝛼𝑛},{𝛽𝑛}, and {𝑡𝑛} be as in Theorem 3.1. Let 𝒥={𝑇(𝑡)𝑡0} be a family of nonexpansive semigroup of 𝐸 with =𝐹(𝒥), and let {𝑥𝑛} be define by (3.2). Then, {𝑥𝑛} converges strongly to a common fixed point of the family 𝒥 which solves the variational inequality (3.3).

Corollary 3.5. Let 𝐸=𝑙𝑝 space, 1<𝑝<. Let 𝒥={𝑇(𝑡)𝑡0},𝑓,𝐺,{𝛼𝑛},{𝛽𝑛},{𝑡𝑛}, and {𝑥𝑛} be as in Theorem 3.1, then the sequence {𝑥𝑛} converges strongly to a common fixed point of the family 𝒥 which solves the variational inequality (3.3).

Corollary 3.6. Let H be a real Hilbert space. Let 𝒥={𝑇(𝑡)𝑡0} be uniformly asymptotically regular family of asymptotically nonexpansive semigroup of 𝐻, with function 𝑘[0,)[0,) and =𝐹(𝒥)=𝑡0𝐹(𝑇(𝑡)). Let 𝑓,{𝛼𝑛},{𝛽𝑛}, and {𝑡𝑛} be as in Theorem 3.1. Let 𝐺𝐸𝐸 be a strongly positive, bounded, and linear operator on 𝐻 with coefficient 𝛿(1/2,1) and 𝐺=1. For a fixed real number 𝛾(0,12(1𝛿)), let {𝑥𝑛} be generated by (3.2). Then, the sequence {𝑥𝑛} converges strongly to a common fixed point of the family 𝒥 which solves the variational inequality (𝐺𝛾𝑓)𝑞,𝑥𝑞0,𝑓𝑜𝑟𝑎𝑙𝑙𝑥.

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.