Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E*. Let 𝒮={T(s):0≤s<∞} be a nonexpansive semigroup on E such that Fix(𝒮):=⋂t≥0Fix(T(t))≠∅, and f is a contraction on E with coefficient 0<α<1. Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1 and γ a positive real number such that γ<1/α(1-1-δ/λ). When the sequences of real numbers {αn} and {tn} satisfy some appropriate conditions, the three iterative processes given as follows: xn+1=αnγf(xn)+(I-αnF)T(tn)xn, n≥0, yn+1=αnγf(T(tn)yn)+(I-αnF)T(tn)yn, n≥0, and zn+1=T(tn)(αnγf(zn)+(I-αnF)zn), n≥0 converge strongly to x̃, where x̃ is the unique solution in Fix(𝒮) of the variational inequality 〈(F-γf)x̃,j(x-x̃)〉≥0, x∈Fix(𝒮). Our results extend and improve corresponding ones of Li et al. (2009) Chen and He (2007), and many others.

1. Introduction

Let E be a real Banach space. A mapping T of E into itself is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for each x,y∈E. We denote by Fix(T) the set of fixed points of T. A mapping f:E→E is called α-contraction if there exists a constant 0<α<1 such that ∥f(x)-f(y)∥≤α∥x-y∥ for all x,y∈E. A family 𝒮={T(t):0≤t<∞} of mappings of E into itself is called a nonexpansive semigroup on E if it satisfies the following conditions:

T(0)x=x for all x∈E;

T(s+t)=T(s)T(t) for all s,t≥0;

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

for all x∈E, the mapping t↦T(t)x is continuous.

We denote by Fix(𝒮) the set of all common fixed points of 𝒮, that is, Fix(S)∶={x∈E:T(t)x=x,0≤t<∞}=⋂t≥0Fix(T(t)).

In [1], Shioji and Takahashi introduced the following implicit iteration in a Hilbert spacexn=αnx+(1-αn)1tn∫0tnT(s)xnds,∀n∈N,
where {αn} is a sequence in (0,1) and {tn} is a sequence of positive real numbers which diverges to ∞. Under certain restrictions on the sequence {αn}, Shioji and Takahashi [1] proved strong convergence of the sequence {xn} to a member of F(𝒮). In [2], Shimizu and Takahashi studied the strong convergence of the sequence {xn} defined byxn+1=αnx+(1-αn)1tn∫0tnT(s)xnds,∀n∈N
in a real Hilbert space where {T(t):t≥0} is a strongly continuous semigroup of nonexpansive mappings on a closed convex subset C of a Banach space E and limn→∞tn=∞. Using viscosity method, Chen and Song [3] studied the strong convergence of the following iterative method for a nonexpansive semigroup {T(t):t≥0} with Fix(𝒮)≠∅ in a Banach space:xn+1=αnf(x)+(1-αn)1tn∫0tnT(s)xnds,∀n∈N,
where f is a contraction. Note however that their iterate xn at step n is constructed through the average of the semigroup over the interval (0,t). Suzuki [4] was the first to introduce again in a Hilbert space the following implicit iteration process:xn=αnu+(1-αn)T(tn)xn,∀n∈N,
for the nonexpansive semigroup case. In 2002, Benavides et al. [5], in a uniformly smooth Banach space, showed that if 𝒮 satisfies an asymptotic regularity condition and {αn} fulfills the control conditions limn→∞αn=0, ∑n=1∞αn=∞, and limn→∞αn/αn+1=0, then both the implicit iteration process (1.5) and the explicit iteration process (1.6),xn+1=αnu+(1-αn)T(tn)xn,∀n∈N,
converge to a same point of F(𝒮). In 2005, Xu [6] studied the strong convergence of the implicit iteration process (1.2) and (1.5) in a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping. Recently, Chen and He [7] introduced the viscosity approximation process:xn+1=αnf(xn)+(1-βn)T(tn)xn,∀n∈N,
where f is a contraction and {αn} is a sequence in (0,1) and a nonexpansive semigroup {T(t):t≥0}. The strong convergence theorem of {xn} is proved in a reflexive Banach space which admits a weakly sequentially continuous duality mapping. In [8], Chen et al. introduced and studied modified Mann iteration for nonexpansive mapping in a uniformly convex Banach space.

On the other hand, iterative approximation methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [9–11] and the references therein. Let H be a real Hilbert space, whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let A be a strongly positive bounded linear operator on H; that is, there is a constant γ¯>0 with property〈Ax,x〉≥γ¯‖x‖2∀x∈H.
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:minx∈C12〈Ax,x〉-〈x,b〉,
where C is the fixed point set of a nonexpansive mapping T on H and b is a given point in H. In 2003, Xu [10] proved that the sequence {xn} defined by the iterative method below, with the initial guess x0∈H chosen arbitrarily,xn+1=(I-αnA)Txn+αnu,n≥0,
converges strongly to the unique solution of the minimization problem (1.9) provided the sequence {αn} satisfies certain conditions. Using the viscosity approximation method, Moudafi [12] introduced the following iterative process for nonexpansive mappings (see [13] for further developments in both Hilbert and Banach spaces). Let f be a contraction on H. Starting with an arbitrary initial x0∈H, define a sequence {xn} recursively byxn+1=(1-αn)Txn+αnf(xn),n≥0,
where {αn} is a sequence in (0,1). It is proved [12, 13] that, under certain appropriate conditions imposed on {αn}, the sequence {xn} generated by (1.11) strongly converges to the unique solution x* in C of the variational inequality〈(I-f)x*,x-x*〉≥0,x∈H.
Recently, Marino and Xu [14] mixed the iterative method (1.10) and the viscosity approximation method (1.11) and considered the following general iterative method:xn+1=(I-αnA)Txn+αnγf(xn),n≥0,
where A is a strongly positive bounded linear operator on H. They proved that if the sequence {αn} of parameters satisfies the certain conditions, then the sequence {xn} generated by (1.13) converges strongly to the unique solution x* in H of the variational inequality〈(A-γf)x*,x-x*〉≥0,x∈H
which is 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) for x∈H).

Very recently, Li et al. [15] introduced the following iterative procedures for the approximation of common fixed points of a one-parameter nonexpansive semigroup on a Hilbert space H:x0=x∈H,xn+1=(I-αnA)1tn∫0tnT(s)xnds+αnγf(xn),n≥0,
where A is a strongly positive bounded linear operator on H.

Let δ and λ be two positive real numbers such that δ,λ<1. Recall that a mapping F with domain D(F) and range R(F) in E is called δ-strongly accretive if, for each x,y∈D(F), there exists j(x-y)∈J(x-y) such that〈Fx-Fy,j(x-y)〉≥δ‖x-y‖2,
where J is the normalized duality mapping from E into the dual space E*. Recall also that a mapping F is called λ-strictly pseudocontractive if, for each x,y∈D(F), there exists j(x-y)∈J(x-y) such that〈Fx-Fy,j(x-y)〉≤‖x-y‖2-λ‖(x-y)-(Fx-Fy)‖2.
It is easy to see that (1.17) can be rewritten as〈(I-F)x-(I-F)y,j(x-y)〉≥λ‖(I-F)x-(I-F)y‖2,
see [16].

In this paper, motivated by the above results, we introduce and study the strong convergence theorems of the general iterative scheme {xn} defined by (1.19) in the framework of a reflexive Banach space E which admits a weakly sequentially continuous duality mapping:x0=x∈E,xn+1=αnγf(xn)+(I-αnF)T(tn)xn,n≥0,
where F is δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1, f is a contraction on E with coefficient 0<α<1, γ is a positive real number such that γ<(1/α)(1-(1-δ)/λ), and 𝒮={T(t):0≤t<∞} is a nonexpansive semigroup on E. The strong convergence theorems are proved under some appropriate control conditions on parameters {αn} and {tn}. Furthermore, by using these results, we obtain strong convergence theorems of the following new general iterative schemes {yn} and {zn} defined byy0=y∈E,yn+1=αnγf(T(tn)yn)+(I-αnF)T(tn)yn,n≥0,z0=z∈E,zn+1=T(tn)(αnγf(zn)+(I-αnF)zn),n≥0.
The results presented in this paper extend and improve the main results in Li et al. [15], Chen and He [7], and many others.

2. Preliminaries

Throughout this paper, it is assumed that E is a real Banach space with norm ∥·∥ and let J denote the normalized duality mapping from E into E* given by J(x)={f∈E*:〈x,f〉=‖x‖2=‖f‖2}
for each x∈E, where E* denotes the dual space of E,〈·,·〉 denotes the generalized duality pairing, and ℕ denotes the set of all positive integers. In the sequel, we will denote the single-valued duality mapping by j, and consider F(T)={x∈C:Tx=x}. When {xn} is a sequence in E, then xn→x (resp., xn⇀x, xn⇀*x ) will denote strong (resp., weak, weak*) convergence of the sequence {xn} to x. In a Banach space E, the following result (the subdifferential inequality) is well known [17, Theorem 4.2.1]: for all x,y∈E, for all j(x+y)∈J(x+y), for all j(x)∈J(x),‖x‖2+2〈y,j(x)〉≤‖x+y‖2≤‖x‖2+〈y,j(x+y)〉.
A real Banach space E is said to be strictly convex if ∥x+y∥/2<1 for all x,y∈E with ∥x∥=∥y∥=1 and x≠y. It is said to be uniformly convex if, for all ϵ∈[0,2], there exits δϵ>0 such that ‖x‖=‖y‖=1with‖x-y‖≥ϵimplies‖x+y‖2<1-δϵ.
The following results are well known and can be founded in [17]:

a uniformly convex Banach space E is reflexive and strictly convex [17, Theorems 4.2.1 and 4.1.6],

if E is a strictly convex Banach space and T:E→E is a nonexpansive mapping, then fixed point set F(T) of T is a closed convex subset of E [17, Theorem 4.5.3].

If a Banach space E admits a sequentially continuous duality mapping J from weak topology to weak star topology, then from Lemma 1 of [18], it follows that the duality mapping J is single-valued and also E is smooth. In this case, duality mapping J is also said to be weakly sequentially continuous, that is, for each {xn}⊂E with xn⇀x, then J(xn)⇀*J(x) (see [18, 19]).

In the sequel, we will denote the single-valued duality mapping by j. A Banach space E is said to satisfy Opial's condition if, for any sequence {xn} in E, xn⇀x as n→∞ implieslimsupn→∞‖xn-x‖<limsupn→∞‖xn-y‖∀y∈Ewithx≠y.
By Theorem 1 of [18], we know that if E admits a weakly sequentially continuous duality mapping, then E satisfies Opial’s condition and E is smooth; for the details, see [18].

Now, we present the concept of uniformly asymptotically regular semigroup (also see [20, 21]). Let C be a nonempty closed convex subset of a Banach space E, 𝒮={T(t):0≤t<∞} a continuous operator semigroup on C. Then, 𝒮 is said to be uniformly asymptotically regular (in short, u.a.r.) on C if, for all h≥0 and any bounded subset D of C,limt→∞supx∈D‖T(h)(T(t)x)-T(t)x‖=0.
The nonexpansive semigroup {σt:t>0} defined by the following lemma is an example of u.a.r. operator semigroup. Other examples of u.a.r. operator semigroup can be found in [20, Examples 17 and 18].

Lemma 2.1 (see [<xref ref-type="bibr" rid="B3">3</xref>, Lemma 2.7]).

Let C be a nonempty closed convex subset of a uniformly convex Banach space E, D a bounded closed convex subset of C, and 𝒮={T(s):0≤s<∞} a nonexpansive semigroup on C such that F(𝒮)≠∅. For each h>0, set σt(x)=(1/t)∫0tT(s)xds, then
limt→∞supx∈D‖σt(x)-T(h)σt(x)‖=0.

Example 2.2.

The set {σt:t>0} defined by Lemma 2.1 is u.a.r. nonexpansive semigroup. In fact, it is obvious that {σt:t>0} is a nonexpansive semigroup. For each h>0, we have
‖σt(x)-σhσt(x)‖=‖σt(x)-1h∫0hT(s)σt(x)ds‖=‖1h∫0h(σt(x)-T(s)σt(x))ds‖≤1h∫0h‖σt(x)-T(s)σt(x)‖ds.
Applying Lemma 2.1, we have
limt→∞supx∈D‖σt(x)-σhσt(x)‖≤1h∫0hlimt→∞supx∈D‖σt(x)-T(s)σt(x)‖ds=0.

Let C be a nonempty closed and convex subset of a Banach space E and D a nonempty subset of C. A mapping Q:C→D is said to be sunny if
Q(Qx+t(x-Qx))=Qx,
whenever Qx+t(x-Qx)∈C for x∈C and t=0. A mapping Q:C→D is called a retraction if Qx=x for all x∈D. Furthermore, Q is a sunny nonexpansive retraction from C onto D if Q is a retraction from C onto D which is also sunny and nonexpansive. A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D. The following lemma concerns the sunny nonexpansive retraction.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B22">22</xref>, <xref ref-type="bibr" rid="B23">23</xref>]).

Let C be a closed convex subset of a smooth Banach space E. Let D be a nonempty subset of C and Q:C→D be a retraction. Then, Q is sunny and nonexpansive if and only if
〈u-Qu,j(y-Qu)〉≤0
for all u∈C and y∈D.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B24">24</xref>, Lemma 2.3]).

Let {an} be a sequence of nonnegative real numbers satisfying the property
an+1≤(1-tn)an+tncn+bn,∀n≥0,
where {tn},{bn},and{cn} satisfy the restrictions

∑n=1∞tn=∞;

∑n=1∞bn<∞;

limsupn→∞cn≤0.

Then, limn→∞an=0.

The following lemma will be frequently used throughout the paper and can be found in [25].

Lemma 2.5 (see [<xref ref-type="bibr" rid="B25">25</xref>, Lemma 2.7]).

Let E be a real smooth Banach space and F:E→E a mapping.

If F is δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1, then I-F is contractive with constant (1-δ)/λ.

If F is δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1, then, for any fixed number τ∈(0,1), I-τF is contractive with constant 1-τ(1-(1-δ)/λ).

3. Main Results

Now, we are in a position to state and prove our main results.

Theorem 3.1.

Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮={T(t):0≤t<∞} be a u.a.r. nonexpansive semigroup on E such that Fix(𝒮)≠∅. Suppose that the real sequences {αn}⊂[0,1], {tn}⊂(0,∞) satisfy the conditions
limn→∞αn=0,∑n=0∞αn=∞,limn→∞tn=∞.
Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1, f:E→E a contraction mapping with coefficient α∈(0,1), and γ a positive real number such that γ<(1/α)(1-(1-δ)/λ). Then, the sequence {xn} defined by (1.19) converges strongly to x̃, where x̃ is the unique solution in Fix(𝒮) of the variational inequality
〈(F-γf)x̃,j(x-x̃)〉≥0,x∈Fix(S)
or equivalently x̃=QFix(𝒮)(I-F+γf)x̃, where QFix(𝒮) is the sunny nonexpansive retraction of E onto Fix(𝒮).

Proof.

Note that Fix(𝒮) is a nonempty closed convex set. We first show that {xn} is bounded. Let q∈Fix(𝒮). Thus, by Lemma 2.5, we have
‖xn+1-q‖=‖αnγf(xn)+(I-αnF)T(tn)xn-(I-αnF)q-αnFq‖≤αn‖γf(xn)-Fq‖+‖I-αnF‖‖T(tn)xn-q‖≤αnγ‖f(xn)-f(q)‖+αn‖γf(q)-Fq‖+‖I-αnF‖‖xn-q‖≤αnαγ‖xn-q‖+αn‖γf(q)-Fq‖+(1-αn(1-1-δλ))‖xn-q‖=(1-αn(1-1-δλ-αγ))‖xn-q‖+αn(1-1-δλ-αγ)‖γf(q)-Fq‖1-(1-δ)/λ-αγ≤max{‖xn-q‖,11-(1-δ)/λ-αγ‖γf(q)-Fq‖},∀n≥0.
By induction, we get
‖xn-q‖≤max{‖x0-q‖,11-(1-δ)/λ-αγ‖γf(q)-Fq‖},n≥0.
This implies that {xn} is bounded and, hence, so are {f(xn)} and {FT(tn)xn}. This implies that
limn→∞‖xn+1-T(tn)xn‖=limn→∞αn‖γf(xn)-FT(tn)xn‖=0.
Since {T(t)} is a u.a.r. nonexpansive semigroup and limn→∞tn=∞, we have, for all h>0,
limn→∞‖T(h)(T(tn)xn)-T(tn)xn‖≤limn→∞supx∈{xn}‖T(h)(T(tn)x)-T(tn)x‖=0.
Hence, for all h>0,
‖xn+1-T(h)xn+1‖≤‖xn+1-T(tn)xn‖+‖T(tn)xn-T(h)T(tn)xn‖+‖T(h)T(tn)xn-T(h)xn+1‖≤2‖xn+1-T(tn)xn‖+‖T(tn)xn-T(h)T(tn)xn‖⟶0.
That is, for all h>0,
limn→∞‖xn-T(h)xn‖=0.
Let Φ=QFix(𝒮). Then, Φ(I-F-γf) is a contraction on E. In fact, from Lemma 2.5(i), we have
‖Φ(I-F-γf)x-Φ(I-F-γf)y‖≤‖(I-F-γf)x-(I-F-γf)y‖≤‖(I-F)x-(I-F)y‖+γ‖f(x)-f(y)‖≤1-δλ‖x-y‖+αγ‖x-y‖=(1-δλ+αγ)‖x-y‖,∀x,y∈E.
Therefore, Φ(I-F-γf) is a contraction on E due to ((1-δ)/λ+αγ)∈(0,1). Thus, by Banach contraction principle, QFix(𝒮)(I-F-γf) has a unique fixed point x̃. Then, using Lemma 2.3, x̃ is the unique solution in Fix(S) of the variational inequality (3.2). Next, we show that
limsupn→∞〈γf(x̃)-Fx̃,j(xn-x̃)〉≤0.
Indeed, we can take a subsequence {xnk} of {xn} such that
limsupn→∞〈γf(x̃)-Fx̃,j(xn-x̃)〉=limk→∞〈γf(x̃)-Fx̃,j(xnk-x̃)〉.
We may assume that xnk⇀p∈E as k→∞, since a Banach space E has a weakly sequentially continuous duality mapping J satisfying Opial's condition [13]. We will prove that p∈Fix(𝒮). Suppose the contrary, p∉Fix(𝒮), that is, T(h0)p≠p for some h0>0. It follows from (3.8) and Opial’s condition that
liminfk→∞‖xnk-p‖<liminfk→∞‖xnk-T(h0)p‖≤liminfk→∞{‖xnk-T(h0)xnk‖+‖T(h0)xnk-T(h0)p‖}≤liminfk→∞{‖xnk-T(h0)xnk‖+‖xnk-p‖}=liminfk→∞‖xnk-p‖.
This is a contradiction, which shows that p∈F(T(h)) for all h>0, that is, p∈Fix(𝒮). In view of the variational inequality (3.2) and the assumption that duality mapping J is weakly sequentially continuous, we conclude
limsupn→∞〈γf(x̃)-Fx̃,j(xn-x̃)〉=limk→∞〈γf(x̃)-Fx̃,j(xnk-x̃)〉≤〈γf(x̃)-Fx̃,j(p-x̃)〉≤0.
Finally, we will show that xn→x̃. For each n≥0, we have
‖xn+1-x̃‖2=‖αnγf(xn)+(I-αnF)T(tn)xn-(I-αnF)x̃-αnFx̃‖2≤‖αnγf(xn)-αnFx̃+(I-αnF)T(tn)xn-(I-αnF)x̃‖2=‖(I-αnF)T(tn)xn-(I-αnF)x̃‖2+2αn〈γf(xn)-Fx̃,j(xn+1-x̃)〉≤(1-αn(1-1-δλ))2‖xn-x̃‖2+2αn〈γf(xn)-γf(x̃),j(xn+1-x̃)〉+2αn〈γf(x̃)-Fx̃,j(xn+1-x̃)〉.
On the other hand,
〈γf(xn)-γf(x̃),j(xn+1-x̃)〉≤γα‖xn-x̃‖‖xn+1-x̃‖≤γα‖xn-x̃‖[(1-αn(1-1-δλ))2‖xn-x̃‖2+2αn|〈γf(xn)-Fx̃,j(xn+1-x̃)〉|]≤γα(1-αn(1-1-δλ))‖xn-x̃‖2+γα‖xn-x̃‖2|〈γf(xn)-Fx̃,j(xn+1-x̃)〉|αn≤γα(1-αn(1-1-δλ))‖xn-x̃‖2+αnM0,
where M0 is a constant satisfying M0≥γα∥xn-x̃∥2|〈γf(xn)-Fx̃,j(xn+1-x̃)〉|. Substituting (3.15) in (3.14), we obtain
‖xn+1-x̃‖2≤(1-αn(1-1-δλ))2‖xn-x̃‖2+2αnγα(1-αn(1-1-δλ))×‖xn-x̃‖2+2αnαnM0+2αn〈γf(x̃)-Fx̃,j(xn+1-x̃)〉=(1-2αn(1-1-δλ)+αn2(1-1-δλ)2)‖xn-x̃‖2+2αnγα(1-αn(1-1-δλ))‖xn-x̃‖2+2αnαnM0+2αn〈γf(x̃)-Fx̃,j(xn+1-x̃)〉=(1-2αn[(1-1-δλ)-αγ+αnγα(1-1-δλ)])‖xn-x̃‖2+αn[αn(1-1-δλ)2‖xn-x̃‖2+2M0αn+2〈γf(x̃)-Fx̃,j(xn+1-x̃)〉]=(1-αnγn)‖xn-x̃‖2+αnγnβnγn,
where
γn=2[(1-1-δλ)-αγ+αnγα(1-1-δλ)],βn=[αn(1-1-δλ)2‖xn-x̃‖2+2M0αn+2〈γf(x̃)-Fx̃,j(xn+1-x̃)〉].
It is easily seen that ∑n=1∞αnγn=∞. Since {xn} is bounded and limn→∞αn=0, by (3.46), we obtain limsupn→∞βn/γn≤0, applying Lemma 2.4 to (3.16) to conclude xn→x̃ as n→∞. This completes the proof.

Using Theorem 3.1, we obtain the following two strong convergence theorems of new iterative approximation methods for a nonexpansive semigroup {T(t):0≤t<∞}.

Corollary 3.2.

Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮={T(t):0≤t<∞} be a u.a.r. nonexpansive semigroup on E such that Fix(𝒮)≠∅. Suppose that the real sequences {αn}⊂[0,1], {tn}⊂(0,∞) satisfy the conditions
limn→∞αn=0,∑n=0∞αn=∞,limn→∞tn=∞.
Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1, f:E→E a contraction mapping with coefficient α∈(0,1), and γ a positive real number such that γ<(1/α)(1-(1-δ)/λ). Then, the sequence {yn} defined by (1.20) converges strongly to x̃, where x̃ is the unique solution in Fix(𝒮) of the variational inequality
〈(F-γf)x̃,j(x-x̃)〉≥0,x∈Fix(S)
or equivalently x̃=QFix(𝒮)(I-F+γf)x̃, where QFix(𝒮) is the sunny nonexpansive retraction of E onto Fix(𝒮).

Proof.

Let {xn} be the sequence given by x0=y0 and
xn+1=αnγf(xn)+(I-αnF)T(tn)xn,∀n≥0.
Form Theorem 3.1, xn→x̃. We claim that yn→x̃. Indeed, we estimate
‖xn+1-yn+1‖≤αnγ‖f(T(tn)yn)-f(xn)‖+‖I-αnF‖‖T(tn)xn-T(tn)yn‖≤αnγα‖T(tn)yn-xn‖+(1-αn(1-1-δλ))‖xn-yn‖≤αnγα‖T(tn)yn-T(tn)x̃‖+αnγα‖T(tn)x̃-xn‖+(1-αn(1-1-δλ))‖xn-yn‖≤αnγα‖yn-x̃‖+αnγα‖x̃-xn‖+(1-αn(1-1-δλ))‖xn-yn‖≤αnγα‖yn-xn‖+αnγα‖xn-x̃‖+αnγα‖x̃-xn‖+(1-αn(1-1-δλ))‖xn-yn‖=(1-αn(1-1-δλ-γα))‖xn-yn‖+αn(1-1-δλ-γα)2αγ(1-(1-δ)/λ-γα)‖x̃-xn‖.
It follows from ∑n=1∞αn=∞, limn→∞∥xn-x̃∥=0, and Lemma 2.4 that ∥xn-yn∥→0. Consequently, yn→x̃ as required.

Corollary 3.3.

Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮={T(t):0≤t<∞} be a u.a.r. nonexpansive semigroup on E such that Fix(𝒮)≠∅. Suppose that the real sequences {αn}⊂[0,1], {tn}⊂(0,∞) satisfy the conditions
limn→∞αn=0,∑n=0∞αn=∞,limn→∞tn=∞.
Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1, f:E→E a contraction mapping with coefficient α∈(0,1), and γ a positive real number such that γ<(1/α)(1-(1-δ)/λ). Then, the sequence {zn} defined by (1.21) converges strongly to x̃, where x̃ is the unique solution in Fix(𝒮) of the variational inequality
〈(F-γf)x̃,j(x-x̃)〉≥0,x∈Fix(S)
or equivalently x̃=QFix(𝒮)(I-F+γf)x̃, where QFix(𝒮) is the sunny nonexpansive retraction of E onto Fix(𝒮).

Proof.

Define the sequences {yn} and {βn} by
yn=αnγf(zn)+(I-αnF)zn,βn=αn+1∀n∈N.
Taking p∈Fix(𝒮), we have
‖zn+1-p‖=‖T(tn)yn-T(tn)p‖≤‖yn-p‖=‖αnγf(zn)+(I-αnF)zn-(I-αnF)p-αnFp‖≤(1-αn(1-1-δλ))‖zn-p‖+αn‖γf(zn)-F(p)‖=(1-αn(1-1-δλ))‖zn-p‖+αn(1-1-δλ)‖γf(zn)-F(p)‖(1-(1-δ)/λ).
It follows from induction that
‖zn+1-p‖≤max{‖z0-p‖,‖γf(z0)-F(p)‖1-(1-δ)/λ},n≥0.
Thus, both {zn} and {yn} are bounded. We observe that
yn+1=αn+1γf(zn+1)+(I-αn+1F)zn+1=βnγf(T(tn)yn)+(I-βnF)T(tn)yn.
Thus, Corollary 3.2 implies that {yn} converges strongly to some point x̃. In this case, we also have
‖zn-x̃‖≤‖zn-yn‖+‖yn-x̃‖=αn‖γf(zn)-Fzn‖+‖yn-x̃‖⟶0.
Hence, the sequence {zn} converges strongly to some point x̃. This complete the proof.

Using Theorem 3.1, Lemma 2.1, and Example 2.2, we have the following result.

Corollary 3.4.

Let E be a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮={T(t):0≤t<∞} be a nonexpansive semigroup on E such that Fix(𝒮)≠∅. Suppose that the real sequences {αn}⊂[0,1], {tn}⊂(0,∞) satisfy the conditions
limn→∞αn=0,∑n=0∞αn=∞,limn→∞tn=∞.
Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1, f:E→E a contraction mapping with coefficient α∈(0,1), and γ a positive real number such that γ<(1/α)(1-(1-δ)/λ). Then, the sequence {xn} defined by
x0=x∈E,xn+1=αnγf(xn)+(I-αnF)1tn∫0tnT(t)xnds,n≥0
converges strongly to x̃, where x̃ is the unique solution in Fix(𝒮) of the variational inequality
〈(F-γf)x̃,j(x-x̃)〉≥0,x∈Fix(S)
or equivalently x̃=QFix(𝒮)((I-F+γf)x̃), where QFix(𝒮) is the sunny nonexpansive retraction of E onto Fix(𝒮).

Corollary 3.5.

Let H be a real Hilbert space. Let 𝒮={T(t):0≤t<∞} be a nonexpansive semigroup on H such that Fix(𝒮)≠∅. Suppose that the real sequences {αn}⊂[0,1], {tn}⊂(0,∞) satisfy the conditions
limn→∞αn=0,∑n=0∞αn=∞,limn→∞tn=∞.
Let f:E→E be a contraction mapping with coefficient α∈(0,1) and A a strongly positive bounded linear operator with coefficient γ¯>1/2 and 0<γ<(1-2-2γ¯)/α. Then, the sequence {xn} defined by
x0=x∈E,xn+1=αnγf(xn)+(I-αnA)1tn∫0tnT(t)xnds,n≥0
converges strongly to x̃, where x̃ is the unique solution in Fix(𝒮) of the variational inequality
〈(A-γf)x̃,j(x-x̃)〉≥0,x∈Fix(S)
or equivalently x̃=QFix(𝒮)((I-A+γf)x̃), where QFix(𝒮) is the sunny nonexpansive retraction of E onto Fix(𝒮).

Proof.

Since A is a strongly positive bounded linear operator with coefficient γ¯, we have
〈Ax-Ay,x-y〉≥γ¯‖x-y‖2.
Therefore, A is γ¯-strongly accretive. On the other hand,
‖(I-A)x-(I-A)y‖2=〈(x-y)-(Ax-Ay),(x-y)-(Ax-Ay)〉=〈x-y,x-y〉-2〈Ax-Ay,x-y〉+〈Ax-Ay,Ax-Ay〉=‖x-y‖2-2〈Ax-Ay,x-y〉+‖Ax-Ay‖2≤‖x-y‖2-2〈Ax-Ay,x-y〉+‖A‖2‖x-y‖2.
Since A is strongly positive if and only if (1/∥A∥)A is strongly positive, we may assume, without loss of generality, that ∥A∥=1, so that
〈Ax-Ay,x-y〉≤‖x-y‖2-12‖(I-A)x-(I-A)y‖2=‖x-y‖2-12‖(x-y)-(Ax-Ay)‖2.
Hence, A is 1⁄2-strongly pseudocontractive. Applying Corollary 3.4, we conclude the result.

Theorem 3.6.

Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮={T(t):0<t<∞} be a u.a.r. nonexpansive semigroup on E such that Fix(𝒮)≠∅. Let {αn} and {tn} be sequences of real number satisfying
0<αn<1,∑n=0∞αn=∞,tn>0,limn→∞αn=limn→∞αntn=0.
Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1, f:E→E a contraction mapping with coefficient α∈(0,1), and γ a positive real number such that γ<(1/α)(1-(1-δ)/λ). Then, the sequence {xn} defined by
x0=x∈E,xn+1=αnγf(xn)+(I-αnF)T(tn)xn,n≥0
converges strongly to x̃, where x̃ is the unique solution in Fix(𝒮) of the variational inequality
〈(F-γf)x̃,j(x-x̃)〉≥0,x∈Fix(S)
or equivalently x̃=QFix(𝒮)(I-F+γf)x̃, where QFix(𝒮) is the sunny nonexpansive retraction of E onto Fix(𝒮).

Proof.

By the same argument as in the proof of Theorem 3.1, we can obtain that {xn}, {f(xn)}, and {FT(tn)xn} are bounded and QFix(𝒮)(I-F-γf) is a contraction on E. Thus, by Banach contraction principle, QFix(𝒮)(I-F-γf) has a unique fixed point x̃. Then, using Lemma 2.3, x̃ is the unique solution in Fix(𝒮) of the variational inequality (3.40). Next, we show that
limsupn→∞〈γf(x̃)-Fx̃,j(xn-x̃)〉≤0.
Indeed, we can take a subsequence {xnk} of {xn} such that
limsupn→∞〈γf(x̃)-Fx̃,j(xn-x̃)〉=limk→∞〈γf(x̃)-Fx̃,j(xnk-x̃)〉.
We may assume that xnk⇀p∈E as k→∞. Now, we show that p∈Fix(𝒮). Put
xk=xnk,αk=αnksk=tnk∀k∈N.
Fix t>0, then we have
‖xk-T(t)p‖=∑i=0[t/si]-1‖T((i+1)sk)xk-T(isk)xk‖+‖T([tsk]sk)xk-T([tsk]sk)p‖+‖T([tsk]sk)p-T(t)p‖≤[tsk]‖T(sk)xk-xk+1‖+‖xk+1-p‖+‖T(t-[tsk]sk)p-p‖≤[tsk]αk‖FT(sk)xk-f(xk)‖+‖xk+1-p‖+‖T(t-[tsk]sk)p-p‖≤(tαksk)‖FT(sk)xk-f(xk)‖+‖xk+1-p‖+max{‖T(s)p-p‖:0≤s≤sk}.
Thus, for all k∈ℕ, we obtain
limsupk→∞‖xk-T(t)p‖≤limsupk→∞‖xk+1-p‖=limsupk→∞‖xk-p‖.
Since Banach space E has a weakly sequentially continuous duality mapping satisfying Opial's condition [13], we can conclude that T(t)p=p for all t>0, that is, p∈Fix(𝒮). In view of the variational inequality (3.2) and the assumption that duality mapping J is weakly sequentially continuous, we conclude
limsupn→∞〈γf(x̃)-Fx̃,j(xn-x̃)〉=limk→∞〈γf(x̃)-Fx̃,j(xnk-x̃)〉≤〈γf(x̃)-Fx̃,J(p-x̃)〉≤0.
By the same argument as in the proof of Theorem 3.1, we conclude that xn→x̃ as n→∞. This completes the proof.

Using Theorem 3.6 and the method as in the proof of Corollary 3.7, we have the following result.

Corollary 3.7.

Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮={T(t):0<t<∞} be a u.a.r. nonexpansive semigroup on E such that Fix(𝒮)≠∅. Let {αn} and {tn} be sequences of real number satisfying
0<αn<1,∑n=0∞αn=∞,tn>0,limn→∞αn=limn→∞αntn=0.
Let F be a δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1, f:E→E a contraction mapping with coefficient α∈(0,1), and γ is a positive real number such that γ<1/α(1-(1-δ)/λ). Then, the sequence {yn} defined by
y0=y∈E,yn+1=αnγf(T(tn)yn)+(I-αnF)T(tn)yn,n≥0
converges strongly to x̃, where x̃ is the unique solution in Fix(𝒮) of the variational inequality
〈(F-γf)x̃,j(x-x̃)〉≥0,x∈Fix(S)
or equivalently x̃=QFix(𝒮)(I-F+γf)x̃, where QFix(𝒮) is the sunny nonexpansive retraction of E onto Fix(𝒮).

Using Theorem 3.6 and the method as in the proof of Corollary 3.8, we have the following result.

Corollary 3.8.

Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮={T(t):0<t<∞} be a u.a.r. nonexpansive semigroup on E such that Fix(𝒮)≠∅. Let {αn} and {tn} be sequences of real number satisfying
0<αn<1,∑n=0∞αn=∞,tn>0,limn→∞αn=limn→∞αntn=0.
Let F be a δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1, f:E→E a contraction mapping with coefficient α∈(0,1), and γ is a positive real number such that γ<(1/α)(1-(1-δ)/λ). Then, the sequence {zn} defined by
z0=z∈E,zn+1=T(tn)(αnγf(zn)+(I-αnF)zn),n≥0
converges strongly to x̃, where x̃ is the unique solution in Fix(𝒮) of the variational inequality
〈(F-γf)x̃,j(x-x̃)〉≥0,x∈Fix(S)
or equivalently x̃=QFix(𝒮)(I-F+γf)x̃, where QFix(𝒮) is the sunny nonexpansive retraction of E onto Fix(𝒮).

Using Theorem 3.6, Lemma 2.1, and Example 2.2, we have the following result.

Corollary 3.9.

Let E be a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮={T(t):0<t<∞} be a nonexpansive semigroup on E such that Fix(𝒮)≠∅. Let {αn} and {tn} be sequences of real numbers satisfying
0<αn<1,∑n=0∞αn=∞,tn>0,limn→∞αn=limn→∞αntn=0.
Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1, f:E→E a contraction mapping with coefficient α∈(0,1), and γ a positive real number such that γ<(1/α)(1-(1-δ)/λ). Then, the sequence {xn} defined by
x0=x∈E,xn+1=αnγf(xn)+(I-αnF)1tn∫0tnT(t)xnds,n≥0
converges strongly to x̃, where x̃ is the unique solution in Fix(𝒮) of the variational inequality
〈(F-γf)x̃,j(x-x̃)〉≥0,x∈Fix(S)
or equivalently x̃=QFix(𝒮)(I-F+γf)x̃, where QFix(𝒮) is the sunny nonexpansive retraction of E onto Fix(𝒮).

Corollary 3.10.

Let H be a real Hilbert space. Let 𝒮={T(t):0≤t<∞} be a nonexpansive semigroup on H such that Fix(𝒮)≠∅. Suppose that the real sequences {αn}⊂[0,1], {tn}⊂(0,∞) satisfy the conditions
0<αn<1,∑n=0∞αn=∞,tn>0,limn→∞αn=limn→∞αntn=0.
Let f:E→E be a contraction mapping with coefficient α∈(0,1) and A a strongly positive bounded linear operator with coefficient γ¯>1/2 and 0<γ<(1-2-2γ¯)/α. Then, the sequence {xn} defined by
x0=x∈E,xn+1=αnγf(xn)+(I-αnA)1tn∫0tnT(t)xnds,n≥0
converges strongly to x̃, where x̃ is the unique solution in Fix(𝒮) of the variational inequality
〈(A-γf)x̃,j(x-x̃)〉≥0,x∈Fix(S)
or equivalently x̃=QFix(𝒮)((I-A+γf)x̃), where QFix(𝒮) is the sunny nonexpansive retraction of E onto Fix(𝒮).

Acknowledgment

The project was supported by the “Centre of Excellence in Mathematics” under the Commission on Higher Education, Ministry of Education, Thailand.

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