Let E be a real Banach space satisfying local uniform Opial's condition,
whose duality map is weakly 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,∞). Let ℱ:=⋂t≥0F(T(t))≠∅ and f:E→E be weakly contractive map. Let G:E→E 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 x0∈E, xn+1=βnxn+(1−βn)yn, yn=(I−αnG)T(tn)xn+αnγf(xn), n≥0. 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 〈(G−γf)q,j(q−x)〉≥0, for all x∈ℱ.

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,Jφ:E→2E* is defined by, x∈E,
Jφx={x*∈E*:〈x,x*〉=‖x‖φ(‖x‖),‖x*‖=φ(‖x‖)},
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 x∈E, 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)=inf{1-‖x+y2‖:‖x‖=‖y‖=1,ϵ=‖x-y‖},
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, xn⇀x as n→∞ implies that
liminfn→∞‖xn-x‖<liminfn→∞‖xn-y‖,∀y∈E,y≠x.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 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+r≤liminfn→∞‖x+xn‖
for each x∈E with ∥x∥≥c and each sequence {xn} satisfying xn⇀0 as n→∞, and liminfn→∞∥xn∥≥1.

E is said to satisfy the local uniform Opial’s condition [5] if, for any weak null sequence {xn} in E with liminfn→∞∥xn∥≥1 and any c>0, there exists r>0 such that
1+r≤liminfn→∞‖x+xn‖
for all x∈E with ∥x∥≥c. Observe that uniform Opial’s condition implies local uniform Opial’s condition which in turn implies Opial’s condition.

A self-mapping T:E→E is said to be contraction if ∥Tx-Ty∥≤α∥x-y∥,forallx,y∈E, 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 ∥Tx-Ty∥≤∥x-y∥-ψ(∥x-y∥),forallx,y∈E. 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 ∥Tx-Ty∥≤∥x-y∥forallx,y∈E and asymptotically nonexpansive if there exists a sequence {kn}⊂[0,∞) with limn→∞kn=0 such that ∥Tnx-Tny∥≤(1+kn)∥x-y∥,forallx,y∈E and n∈ℕ. The set of fixed point of T is defined as F(T):={x∈E:Tx=x}.

A one parameter family 𝒥={T(t):t≥0} of self-mapping of E is called nonexpansive semigroup if the following conditions are satisfied:

T(0)x=xforallx∈E;

T(t+s)=T(t)∘T(s)forallt,s≥0;

for each x∈E, the mapping t→T(t)x is continuous;

for x,y∈E and t≥0, ∥T(t)x-T(t)y∥≤∥x-y∥.

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 limt→∞k(t)=0 and ∥T(t)x-T(t)y∥≤(1+k(t))∥x-y∥forallx,y∈E.

The family 𝒥={T(t):t≥0} is said to be asymptotically regular if
lims→∞‖T(t+s)x-T(s)x‖=0,
for all t∈(0,∞) and x∈K. It is said to be uniformly asymptotically regular if, for any t≥0 and for any bounded subset C of K,
lims→∞supx∈C‖T(t+s)x-T(s)x‖=0.
For some positive real numbers δ and λ, the 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
and it is called λ-strictly pseudocontractive if
〈Gx-Gy,j(x-y)〉≤‖x-y‖2-λ‖(I-G)x-(I-G)y‖2.
Let C be a nonempty closed convex subset of E and T:E→E be a map. Then, a variational inequality problem with respect to C and T is find x*∈C such that〈Tx*,j(y-x*)〉≥0,∀y∈C,j(y-x*)∈J(y-x*).
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:
minx∈C12〈Ax,x〉-〈x,b〉.
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 A:H→H is said to be strongly positive if there exists a constant γ¯>0 such that
〈Ax,x〉≥γ¯‖x‖2,forallx∈H.
For a strongly positive bonded linear operator A and any x,y∈H, we have
〈Ax-Ay,x-y〉≥γ¯‖x-y‖2.
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:
〈Ax-Ay,x-y〉≤(1+‖A‖2)2‖x-y‖2-12‖(I-A)x-(I-A)y‖2.
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 x0∈H chosen arbitrarily,
xn+1=(I-αnA)Txn+αnb,n≥0,
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 x0∈H, define a sequence {xn} recursively by
xn+1=(1-αn)Txn+αnf(xn),n≥0,
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 x*∈C of the variational inequality:
〈(I-f)x*,x-x*〉≥0,x∈C.
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=(I-αnA)Txn+αnγf(xn),n≥0.
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:
〈(A-γf)x*,x-x*〉≥0,x∈C,
which is also the optimality condition for the minimization problem minx∈C(1/2)〈Ax,x〉-h〈x〉, where h is a potential function for γf(i.e.,h′(x)=γf(x),forx∈H).

Yao et al. [19] proved that the iterative scheme defined by
x0∈H,xn+1=βnxn+(1-βn)yn,yn=(1-αnA)Txn+αnγf(xn),n≥0,
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,u∈C,
xn=αnu+(1-αn)T(tn)xn,n≥0,
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.

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 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 ls invariant; that is, ls(X)⊂X for each s∈S. A mean μ on X is said to be left invariant if μ(lsf)=μ(f) for each s∈S and f∈X.

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 [<xref ref-type="bibr" rid="B17">14</xref>]).

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 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 x0∈Hxn+1=(I-αnG)T(μn)xn+αnγf(xn),n≥0.
Then, {xn} converges strongly to a common fixed point of the family 𝒥 which is the unique solution of the variational inequality 〈(G-γf)x*,x-x*〉≥0forallx∈F(𝒥). Equivalently one has PF(𝒥)(I-G+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 [<xref ref-type="bibr" rid="B14">13</xref>]).

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 and let G:H→H 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 x0∈H:
xn+1=(I-αnG)T(μn)xn+αnγf(xn),n≥0,
where 0<γ<(1-(1-δ)/λ)/α. Then, {xn} converges strongly to a common fixed point of the family 𝒥 which is the unique solution of the variational inequality 〈(G-γf)x*,x-x*〉≥0forallx∈F(𝒥). Equivalently one has PF(𝒥)(I-G+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+y‖2≤‖x‖2+2〈y,j(x+y)〉,∀x,y∈E,j(x+y)∈J(x+y).

Let {xn} 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 xn+1=βnyn+(1-βn)xn for all integers n≥1 and limsup(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, lim∥yn-xn∥=0.

Lemma 2.3 (Kim et al. [<xref ref-type="bibr" rid="B7">23</xref>]).

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):t≥0} is asymptotically nonexpansive semigroup on C, then (I-T(t)) is demiclosed at zero.

Lemma 2.4 (Acedo and Suzuki, [<xref ref-type="bibr" rid="B1">20</xref>]).

Let C be a set of a separated topological vector space E. Let {T(t):t≥0} 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):t≥0} is asymptotically regular, then F(T(t))=⋂s≥0F(T(s)) holds for all t∈(0,∞).

Let {an} be a sequence of nonnegative real numbers satisfying the following relation:
an+1≤(1-αn)an+αnσn+γn,n≥0,
where, (i) {αn}⊂[0,1],∑αn=∞, (ii) limsupσn≤0, (iii) γn≥0and∑γn<∞. Then, an→0 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 G:E→E be a δ-strongly accretive and λ-strictly pseudocontractive then, as shown in [13], (I-G) and (I-τG) are strict contractions. In fact, for x,y∈E,
λ‖(I-G)x-(I-G)y‖2≤‖x-y‖2-〈Gx-Gy,j(x-y)〉≤(1-δ)‖x-y‖2,
which implies
‖(I-G)x-(I-G)y‖≤1-δλ‖x-y‖.
Also, forτ∈(0,1),
‖(I-τG)x-(I-τG)y‖≤‖(1-τ)(x-y)+τ((I-G)x-(I-G)y)‖≤(1-τ)‖(x-y)‖+τ1-δλ‖x-y‖=1-τ(1-1-δλ)‖(x-y)‖.

3. Main ResultsTheorem 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):t≥0} be 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/2,δ}). Let {βn} and {αn} be sequences in [0,1] and let {tn} be an increasing sequence in [0,∞) satisfying the following conditions:
limn→∞αn=0,limn→∞k(tn)αn=0,∑αn=∞,0<liminfn→∞βn≤limsupn→∞βn<1.
Define a sequence {xn} by x0∈E,
xn+1=βnxn+(1-βn)yn,yn=(I-αnG)T(tn)xn+αnγf(xn),n≥0.
Then, the sequence {xn} 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.

Proof.

We start by showing that the solution of the variational inequality (3.3) in ℱ is unique. Assume 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.
Adding these two relations, we get
〈(G-γf)p-(G-γf)q,j(p-q)〉≤0.
Therefore,
0≥〈(G-γf)p-(G-γf)q,j(p-q)〉=〈Gp-Gq,j(p-q)〉-γ〈f(p)-f(q),j(p-q)〉≥δ‖p-q‖2-γ‖f(p)-f(q)‖‖q-p‖≥δ‖p-q‖2+γψ(‖p-q‖)‖q-p‖-γ‖q-p‖2=(δ-γ)‖q-p‖2+γψ(‖p-q‖)‖q-p‖.
Since δ>γ, we obtain that p=q and so the solution is unique in ℱ.

Now, let q∈ℱ, since (1-αnγ)(k(tn)/αn)→0 as n→∞,there exists n0∈ℕ such that (1-αnγ)(k(tn)/αn)<(η-γ)/2,foralln≥n0. Hence, for n≥n0, we have the following:
‖yn-q‖=∥(I-αnG)T(tn)xn-(I-αnG)q+αnγf(xn)-αnγf(q)+αnγf(q)-αnG(q)∥≤(1-αnη)(1+k(tn))‖xn-q‖+αnγ‖f(xn)-f(q)‖+αn‖γf(q)-G(q)‖≤[1-αn(η-γ)+(1-αnη)k(tn)]‖xn-q‖+αn‖γf(q)-G(q)‖,
so that
‖xn+1-q‖≤βn‖xn-q‖+(1-βn)‖yn-q‖≤[βn+(1-βn)[1-αn(η-γ)+(1-αnη)k(tn)]]‖xn-q‖+αn(1-βn)‖γf(q)-G(q)‖=[1-αn(1-βn)((η-γ)-(1-αnγ)k(tn)αn)]‖xn-q‖+αn(1-βn)‖γf(q)-G(q)‖≤[1-αn(1-βn)((η-γ)-(1-αnγ)k(tn)αn)]‖xn-q‖+αn(1-βn)((η-γ)-(1-αnγ)k(tn)αn)2‖γf(q)-G(q)‖(η-γ)≤max{‖xn-q‖,2‖γf(q)-G(q)‖η-γ}.
By induction, we have
‖xn-q‖≤max{‖xn0-q‖,2‖γf(q)-G(q)‖η-γ}.
Thus, {xn} is bounded and so are {T(tn)xn},{GT(tn)xn},{yn}, and {f(xn)}. Observe that
yn+1-yn=((I-αn+1G)T(tn+1)xn+1-(I-αn+1G)T(tn+1)xn)+((I-αn+1G)T(tn+1)xn-(I-αnG)T(tn+1)xn)+((I-αnG)T(tn+1)xn-(I-αnG)T(tn)xn)+(αn+1γf(xn+1)-αn+1γf(xn))+(αn+1γf(xn)-αnγf(xn)),
so that
‖yn+1-yn‖≤(1-αn+1η)(1+k(tn+1))‖xn+1-xn‖+|αn-αn+1|‖GT(tn+1)xn‖+(1-αnη)‖T(tn+1)xn-T(tn)xn‖+αn+1γ‖f(xn+1)-f(xn)‖+|αn+1-αn|γ‖f(xn)‖=(1-αn+1η)(1+k(tn+1))‖xn+1-xn‖+|αn-αn+1|‖GT(tn+1)xn‖+(1-αnη)‖T((tn+1-tn)+tn)xn-T(tn)xn‖+αn+1γ‖f(xn+1)-f(xn)‖+|αn+1-αn|γ‖f(xn)‖≤(1-αn+1η)(1+k(tn+1))‖xn+1-xn‖+|αn-αn+1|‖GT(tn+1)xn‖+(1-αnη)supz∈{xn},s∈R+‖T(s+tn)z-T(tn)z‖+αn+1γ‖f(xn+1)-f(xn)‖+|αn+1-αn|γ‖f(xn)‖.
From this, we obtain that
‖yn+1-yn‖-‖xn+1-xn‖≤[(1-αn+1η)(1+k(tn+1))-1]‖xn+1-xn‖+|αn-αn+1|‖GT(tn+1)xn‖+(1-αnη)supz∈{xn},s∈R+‖T(s+tn)z-T(tn)z‖+αn+1γ‖f(xn+1)-f(xn)‖+|αn+1-αn|γ‖f(xn)‖,
which implies
limsupn→∞(‖yn+1-yn‖-‖xn+1-xn‖)≤0
and by Lemma 2.2,
limn→∞‖yn-xn‖=0.
Thus, ∥xn+1-xn∥=(1-βn)∥yn-xn∥→0 as n→∞.

Next, we show that limn→∞∥xn-T(t)xn∥=0 and limn→∞∥yn-T(t)yn∥=0,forallt≥0.

Since
‖xn-T(tn)xn‖≤‖xn-xn+1‖+‖xn+1-T(tn)xn‖≤‖xn-xn+1‖+βn‖xn-T(tn)xn‖+(1-βn)‖yn-T(tn)xn‖,
we have
(1-βn)‖xn-T(tn)xn‖≤‖xn-xn+1‖+(1-βn)‖yn-T(tn)xn‖=‖xn-xn+1‖+αn(1-βn)‖γf(xn)-GT(tn)xn‖.
From αn→0 as n→∞, we obtain limn→∞∥xn-T(tn)xn∥=0.

Now, for any t≥0, we have
‖T(t)xn-xn‖≤‖T(t)xn-T(t)T(tn)xn‖+‖T(t)T(tn)xn-T(tn)xn‖+‖T(tn)xn-xn‖≤‖T(t+tn)xn-T(tn)xn‖+(2+k(t))‖T(tn)xn-xn‖≤supz∈{xn},s∈R+‖T(s+tn)z-T(tn)z‖+(2+k(t))‖T(tn)xn-xn‖.
Using this and the uniform asymptotic regularity of 𝒥, we get
limn→∞‖xn-T(t)xn‖=0,∀t≥0.
We also have
‖yn-T(t)yn‖≤‖yn-xn‖+‖xn-T(t)xn‖+‖T(t)xn-T(t)yn‖≤(2+k(t))‖yn-xn‖+‖xn-T(t)xn‖.
This implies that
limn→∞‖yn-T(t)yn‖=0,∀t≥0.
Let {ynj} be a subsequence of {yn} such that
limsupn→∞〈γf(q)-G(q),j(yn-q)〉=limj→∞〈γf(q)-G(q),j(ynj-q)〉,
and assume without loss of generality that ynj⇀z∈E. By Lemma 2.3, (I-T(t)) is demiclosed at zero, so z∈F(T(t)) and, by Lemma 2.4, z∈ℱ.

Since the duality map of E is weakly sequentially continuous, we obtain
limsupn→∞〈γf(q)-G(q),j(yn-q)〉=limj→∞〈γf(q)-G(q),j(ynj-q)〉=〈(γf-G)q,j(z-q)〉≤0.
We now conclude by showing that xn→q 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∈ℕ be large enough such that (1-αnη)(σn/αn)<(η-2δ)/2,foralln≥N0, and let M be a positive real number such that ∥xn-q∥≤Mforalln≥0. Then, using the recursion formula (3.2) and for n≥N0, we have
‖xn+1-q‖2≤βn‖xn-q‖2+(1-βn)‖yn-q‖2=βn‖xn-q‖2+(1-βn)×‖αn(γf(xn)-G(q))+(I-αnG)T(tn)xn-(I-αnG)(q)αn(γf(xn)-G(q))‖2≤βn‖xn-q‖2+(1-βn)(1-αnη)2‖T(tn)xn-q‖2+2αn(1-βn)〈γf(xn)-Gq,j(yn-q)〉≤βn‖xn-q‖2+(1-βn)(1-αnη)(1+k(tn))2‖xn-q‖2+2αn(1-βn)〈γf(xn)-γf(q)+γf(q)-Gq,j(yn-q)〉≤βn‖xn-q‖2+(1-βn)(1-αnη)(1+σn)‖xn-q‖2+2αn(1-βn)〈γf(q)-Gq,j(yn-q)〉-2αn(1-βn)γ‖yn-q‖ψ(‖xn-q‖)+2αn(1-βn)γ‖xn-q‖‖(yn-xn)+(xn-q)‖≤[βn+(1-βn)(1-αnη+(1-αnη)σn)]‖xn-q‖2+2αn(1-βn)[〈γf(q)-Gq,j(yn-q)〉+γ‖yn-xn‖‖xn-q‖]+2αn(1-βn)γ‖xn-q‖2≤[1-αn(1-βn)((η-2γ)-(1-αnη)σnαn)]‖xn-q‖2+2αn(1-βn)[〈γf(q)-Gq,j(yn-q)〉+γ‖yn-xn‖M]=[1-αn(1-βn)((η-2γ)-(1-αnη)σnαn)]‖xn-q‖2+αn(1-βn)((η-2γ)-(1-αnη)σnαn)×2[〈γf(q)-Gq,j(yn-q)〉+γ‖yn-xn‖M]((η-2γ)-(1-αnη)(σn/αn)).
Observe that ∑αn(1-βn)[(η-2γ)-(1-αnη)(σn/αn)]=∞ and
limsup(2[〈γf(q)-Gq,j(yn-q)〉+γ‖yn-xn‖M]((η-2γ)-(1-αnη)(σn/αn)))≤0.
Applying Lemma 2.5, we obtain ∥xn-q∥→0 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):t≥0},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):t≥0},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 〈(G-γf)q,x-q〉≥0,forallx∈ℱ.

Corollary 3.4.

Let E,f,G,{αn},{βn}, and {tn} be as in Theorem 3.1. Let 𝒥={T(t):t≥0} be a family of nonexpansive semigroup of E with ℱ:=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):t≥0},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):t≥0} be uniformly asymptotically regular family of asymptotically nonexpansive semigroup of H, with function k:[0,∞)→[0,∞) and ℱ:=F(𝒥)=⋂t≥0F(T(t))≠∅. Let f,{αn},{βn}, and {tn} be as in Theorem 3.1. Let G:E→E 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∈(0,1-2(1-δ)), 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 〈(G-γf)q,x-q〉≥0,forallx∈ℱ.

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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