Strong convergence theorems of quasi-ϕ-asymptotically nonexpansive semi-groups in Banach spaces

Abstract

The purpose of this article is to modify the Halpern-Mann-type iteration algorithm for quasi-ϕ-asymptotically nonexpansive semi-groups to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the article improve and extend the corresponding recent results announced by many authors.

2000 AMS Subject Classification: 47J05; 47H09; 49J25.

1. Introduction

Throughout this article, we assume that E is a real Banach space with the dual E*, C is a nonempty closed convex subset of E and J : E → 2^E*is the normalized duality mapping defined by

$$J\left(x\right)=\left\{f^{*}\in E^{*}:⟨x,f^{*}⟩=\left|\right|x|{|}^2=|\left|f^{*}\right|{|}^2\right\},x\in E.$$

Let T : C → E be a nonlinear mapping, we denote by F(T) the set of fixed points of T.

Recalled that a mapping T : C → C is said to be nonexpansive , if

$$\left|\right|Tx-Ty\left|\right|\le \left|\right|x-y\left|\right|,\forall x,y\in C.$$

T : C → C is said to be quasi-nonexpansive, if F(T) ≠ ∅ and

$$\left|\right|Tx-p\left|\right|\le \left|\right|x-p\left|\right|,\forall x\in C,p\in F\left(T\right).$$

T : C → C is said to be asymptotically nonexpansive, if there exists a sequence {k _n }⊂ [1, ∞) with k _n → 1 such that

$$\left|\right|T^{n}x-T^{n}y\left|\right|\le k_n\left|\right|x-y\left|\right|,\forall x,y\in C,n\ge 1.$$

T : C → C is said to be quasi-asymptotically nonexpansive, if F(T) ≠ ∅ and there exists a sequence $\left\{k_n\right\}\subset [1,\mathrm{\infty }]$ with k _n → 1 such that

$$\left|\right|T^{n}x-p\left|\right|\le k_n\left|\right|x-p\left|\right|,\forall x\in C,p\in F\left(T\right),n\ge 1.$$

One parameter family $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$ of mappings from C into C is said to be nonexpansive semi-group, if the following conditions are satisfied:

• (i) T(0)x = x for all x ∈ C;

• (ii) T(s + t) = T(s)T(t) ∀s, t ≥ 0;

• (iii) for each x ∈ C, the mapping t ↦ T(t)x is continuous;

(iv)$\left|\right|T\left(t\right)x-T\left(t\right)y\left|\right|\le \left|\right|x-y\left|\right|,\forall x,y\in C.$

We use $F\left(\mathcal{J}\right)$ to denote the common fixed point set of the nonexpansive semi-group $\mathcal{J}$, i.e., $F\left(\mathcal{J}\right):={\bigcap }_{t\ge 0}F\left(T\left(t\right)\right).$

One parameter family $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$ of mappings from C into C is said to be quasi-nonexpansive semi-group, if $F\left(\mathcal{J}\right)\ne \varnothing ,$ and the above conditions (i)-(iii) and the following condition (v) are satisfied:

• (v) ||T(t)x - p|| ≤ ||x - p||, ∀x∈C, $p\in F\left(\mathcal{J}\right),$ t ≥ 0.

One parameter family $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$ of mappings from C into C is said to be asymptotically nonexpansive semi-group, if there exists a sequence {k _n } ⊂ [1, ∞) with k _n → 1 such that the above conditions (i)-(iii) and the following condition (vi) are satisfied:

• (vi) ||Tⁿ (t)x - Tⁿ (t)y|| ≤ k _n ||x - y||, ∀x, y ∈ C, n ≥ 1, t ≥ 0.

One parameter family $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$ of mappings from C into C is said to be quasi asymptotically nonexpansive semi-group, if $F\left(\mathcal{J}\right)\ne \mathrm{\varnothing },$ and there exists a sequence {k _n } ⊂ [1, ∞) with k _n → 1 such that the above conditions (i)-(iii) and the following condition (vii) are satisfied:

• (vii) ||Tⁿ (t)x - p|| ≤ k _n ||x - p||, ∀x ∈ C, $p\in F\left(\mathcal{J}\right),$ t ≥ 0, n ≥ 1.

As well known, the construction of fixed points of nonexpansive mappings (asymptotically nonexpansive mappings), and of common fixed points of nonexpansive semi-groups (asymptotically nonexpansive semi-groups) is an important problem in the theory of nonexpansive mappings and its applications, in particular, in image recovery, convex feasibility problem, and signal processing problem (see, for example [1–4]).

Iterative approximation of fixed point for nonexpansive mappings, asymptotically nonexpansive mappings, nonexpansive semi-groups, and asymptotically nonexpansive semi-groups in Hilbert or Banach spaces has been studied extensively by many authors (see, for example, [5–30] and the references therein).

The purpose of this article is to introduce the concept of quasi-ϕ-asymptotically nonexpansive semi-groups and to modify the Halpern and Mann-type iteration algorithm [14, 15] for quasi-ϕ-asymptotically nonexpansive semi-groups and to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the article improve and extend the corresponding results of Suzuki [5], Xu [6], Chang et al. [7], Zhang [8], Chang et al. [9], Cho et al. [11], Thong [12], Buong [13], Mann [14], Halpern [15], Qin et al. [16], Nakajo and Takahashi [19], Kang et al. [23], Chang et al. [24], and others.

2.Preliminaries

In the sequel, we assume that E is a smooth, strictly convex and reflexive Banach space and C is a nonempty closed convex subset of E. In what follows, we always use ϕ : $E\times E\to {\mathcal{R}}^{+}$ to denote the Lyapunov functional defined by

$$\varphi \left(x,y\right)=\left|\right|x|{|}^2+2⟨x,Jy⟩+|\left|y\right|{|}^2,\forall x,y\in E.$$

(2.1)

It is obvious from the definition of ϕ that

$${\left(\left|\right|x\left|\right|-\left|\right|y\left|\right|\right)}^2\le \varphi \left(x,y\right)\le {\left(\left|\right|x\left|\right|+\left|\right|y\left|\right|\right)}^2,\forall x,y\in E.$$

(2.2)

and

$$\varphi (x,J^{-1}\left(\lambda Jy+\left(1-\lambda \right)Jz\right)\le \lambda \varphi \left(x,y\right)+\left(1-\lambda \right)\varphi \left(x,z\right),\forall x,y\in E.$$

(2.3)

Following Alber [31], the generalized projection Π _C : E → C is defined by

$${\mathrm{\Pi }}_C\left(x\right)=arg\underset{y\in C}{\text{inf}}\varphi \left(y,x\right),\forall x\in E.$$

Lemma 2.1[31]. Let E be a smooth, strictly convex, and reflexive Banach space and C be a nonempty closed convex subset of E. Then the following conclusions hold:

• (a) ϕ (x,Π _C y) +ϕ (Π _C y, y) ≤ ϕ (x, y) for all x ∈ C and y ∈ E;

• (b) If x ∈ E and z ∈ C, then z = Π _C x ⇔ 〈z - y, Jx - Jz〉 ≥ 0, ∀y ∈ C;

• (c) For x, y ∈ E, ϕ (x, y) = 0 if and only if x = y;

Remark 2.2. If E is a real Hilbert space H, then ϕ (x, y) = ||x - y||² and Π _C = P _C (the metric projection of H onto C).

Definition 2.3. A mapping T : C → C is said to be closed if, for any sequence {x _n }⊂ C with x _n → x and Tx _n → y, then Tx = y.

Definition 2.4. (1) A mapping T : C → C is said to be quasi-ϕ-nonexpansive, if F(T) ≠ ∅ and

$$\varphi \left(p,Tx\right)\le \varphi \left(p,x\right),\forall x\in C,p\in F\left(T\right).$$

(2) A mapping T : C → C is said to be quasi-ϕ-asymptotically nonexpansive, if F(T) ≠ ∅ and there exists a real sequence {k _n } ⊂ [1, ∞), k _n → 1 such that

$$\varphi \left(p,T^{n}x\right)\le k_n\varphi \left(p,x\right),\forall n\ge 1,x\in C,p\in F\left(T\right).$$

Remark 2.5[23]. (1) From the definitions, it is obvious that a quasi-ϕ-nonexpansive mapping is a quasi-ϕ-asymptotically nonexpansive mapping. However, the converse is not true.

• (2) Especially, if E is a real Hilbert space, than a quasi-ϕ-nonexpansive mapping is a quasi-nonexpansive mapping and a quasi-ϕ-asymptotically nonexpansive mapping is a quasi-asymptotically nonexpansive mapping.

Example 2.6[24]. Let E be a uniformly smooth and strictly convex Banach space and A : E → E* be a maximal monotone mapping such that A^-10 ≠ ∅, then J _r = (J + rA)^-1J is closed and quasi-ϕ-nonexpansive from E onto D(A);

Example 2.7[23]. Let Π _C be the generalized projection from a smooth, reflexive, and strictly convex Banach space E onto a nonempty closed convex subset C of E, then Π _C is a closed and quasi-ϕ -nonexpansive from E onto C.

Lemma 2.8 Let E be a uniformly convex and smooth Banach space and let {x _n } and {y _n } be two sequences of E. If ϕ (x _n , y _n ) → 0 and either {x _n } or {y _n } is bounded, then ||x _n - y _n || → 0.

Lemma 2.9[23]. Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and C be a nonempty closed and convex subset of E. Let T : C → C be a closed and quasi-ϕ-asymptotically nonexpansive mapping, then F(T) is a closed convex subset of C.

Definition 2.10. (I) Let E be a real Banach space, C be a nonempty closed convex subset of E. $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$ be one parameter family of mappings from C into C. $\mathcal{J}$ is said to be

• (1) quasi-ϕ-nonexpansive semi-group, if $\mathcal{F}={\bigcap }_{t\ge 0}F\left(T\left(t\right)\right)\ne \mathrm{\varnothing }$ and the following conditions are satisfied

• (i) T(0)x = x for all x ∈ C;

• (ii) T(s + t) = T(s)T(t) ∀ s, t ≥ 0;

• (iii) for each x ∈ C, the mapping t ↦ T(t)x is continuous;

• (iv) ϕ(p, T(t)x) ≤ ϕ (p, x), ∀t ≥ 0,$p\in \mathcal{F},$ x ∈ C.

• (2) $\mathcal{J}$ is said to be quasi-ϕ-asymptotically nonexpansive semi-group, if the set $\mathcal{F}={\bigcap }_{t\ge 0}F\left(T\left(t\right)\right)$ is nonempty, and there exists a sequence {k _n } ⊂ [1, ∞), with k _n → 1 such that the conditions (i)-(iii) and the following conditions (v) are satisfied:

• (v) ϕ (p, Tⁿ (t)x) ≤ k _n ϕ(p, x), ∀t ≥ 0, $p\in \mathcal{F},$ n ≥ 1, x ∈ C.

• (II) A quasi-ϕ-asymptotically nonexpansive semi-group $\mathcal{J}$ is said to be uniformly Lips-chitzian, if there exists a bounded measurable function L : [0, ∞) → (0, ∞) such that

  $$\left|\right|T^n\left(t\right)x-T^n\left(t\right)y\left|\right|\le L\left(t\right)\left|\right|x-y\left|\right|,\forall x,y\in C,\forall n\ge 1,t\ge 0.$$

3. Main results

Theorem 3.1. Let C be a nonempty closed convex subset of a real uniformly convex and uniformly smooth Banach space E. Let $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$ be a closed, uniformly L-Lipschitz and quasi-ϕ-asymptotically nonexpansive semi-group with sequence {k _n } ⊂ [1, ∞), k _n → 1. Let {↦ _n } be a sequence in [0,1] and {β _n } be a sequence in (0, 1) satisfying the following conditions:

• (i) lim_n→∞ α _n = 0;

• (ii) 0 < lim inf _n→∞ β _n ≤ lim sup _n→∞ β _n < 1.

Let {x _n } be a sequence generated by

$$\begin{array}{c}\left\{\begin{array}{c}\hfill x_1\in Echosenarbitrarily;C_1=C,\hfill \\ \hfill y_{n,t}=J^{-1}\left[{\alpha }_{n}J{x}_1+\left(1-{\alpha }_n\right)\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{T}^n\left(t\right)x_n\right)\right],t\ge 0,\hfill \\ \hfill C_{n+1}=\left\{z\in C_n:\underset{t\ge 0}{\text{sup}}\varphi \left(z,y_{n,t}\right)\le {\alpha }_n\varphi \left(z,x_1\right)+\left(1-{\alpha }_n\right)\varphi \left(z,x_n\right)+{\xi }_n\right\}\hfill \\ \hfill x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,\forall n\ge 1,\hfill \end{array}\right.\end{array}$$

(3.1)

where $\mathcal{F}:={\bigcap }_{t\ge 0}^{\infty }F\left(T\left(t\right)\right),{\xi }_n=\left(k_n-1\right){\mathsf{\text{sup}}}_{p\in \mathcal{F}}\varphi \left(p,x_n\right),{\mathrm{\Pi }}_{C_{n+1}}$ is the generalized projection of E onto C_n+1. If $\mathcal{F}$ is bounded in C, then {x _n } converges strongly to ${\mathrm{\Pi }}_{\mathcal{F}}{x}_1.$

Proof. (I) First we prove that$\mathcal{F}$and C _n , n ≥ 1 all are closed and convex subsets in C.

In fact, it follows from Lemma 2.9 that F(T(t)), t ≥ 0 is a closed and convex subset of C. Therefore, $\mathcal{F}$ is closed and convex in C.

Again by the assumption that C₁ = C is closed and convex. Suppose that C _n is closed and convex for some n ≥ 2. In view of the definition of ϕ we have that

$$\begin{array}{c}{C}_{n+1}=\{z\in C_n:\underset{t\ge 0}{\sup }\varphi (z,y_{n,t})\le {\alpha }_n\varphi (z,x_1)+(1-{\alpha }_n)\varphi (z,x_n)+{\xi }_n\}\\ ={\displaystyle \underset{t\ge 0}{\cap }\{}z\in C:\varphi (z,y_{n,t})\le {\alpha }_n\varphi (z,x_1)+(1-{\alpha }_n)\varphi (z,x_n)+{\xi }_n\}{\displaystyle \cap C_n}\\ ={\displaystyle \underset{t\ge 0}{\cap }\{}z\in C:2{\alpha }_n〈z,J{x}_1〉+2(1-{\alpha }_n)〈z,J{x}_n〉-2〈z,J{y}_{n,t}〉\\ \le {\alpha }_n\left|\right|x_1|{|}^2+(1-{\alpha }_n)|\left|x_n\right|{|}^2-\left|\right|y_{n,t}\left|{|}^2\right\}{\displaystyle \cap C_n}.\end{array}$$

This shows that C_n+1is closed and convex. The conclusion is proved.

(II) Now we prove that$\mathcal{F}\subset C_{n,}$, ∀n ≥ 1.

In fact, it is obvious that $\mathcal{F}\subset C_1=C.$ Suppose that $\mathcal{F}\subset C_{n,}$ for some n ≥ 2. Letting

$$w_{n,t}=J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{T}^n\left(t\right)x_n\right),t\ge 0,$$

it follows from (2.3) that for any u ∈ $\mathcal{F}\subset C_{n,}$, we have

$$\begin{array}{ll}\hfill \varphi \left(u,y_{n,t}\right)& =\varphi \left(u,J^{-1}\left({\alpha }_{n}J{x}_1+\left(1-{\alpha }_n\right)J{w}_{n,t}\right)\right)\\ \le {\alpha }_n\varphi \left(u,x_1\right)+\left(1-{\alpha }_n\right)\varphi \left(u,w_{n,t}\right),\end{array}$$

(3.2)

and

$$\begin{array}{ll}\hfill \varphi \left(u,w_{n,t}\right)& =\varphi (u,J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{T}^n\left(t\right)x_n\right)\\ \le {\beta }_n\varphi \left(u,x_n\right)+\left(1-{\beta }_n\right)\varphi \left(u,T^n\left(t\right)x_n\right)\\ \le {\beta }_n\varphi \left(u,x_n\right)+\left(1-{\beta }_n\right)k_n\varphi \left(u,x_n\right)\\ =\varphi \left(u,x_n\right)+\left(1-{\beta }_n\right)\left(k_n-1\right)\varphi \left(u,x_n\right).\end{array}$$

(3.3)

Therefore, we have

$$\begin{array}{ll}\hfill \underset{t\ge 0}{\text{sup}}\varphi \left(u,y_{n,t}\right)& \le {\alpha }_n\varphi \left(u,x_1\right)+\left(1-{\alpha }_n\right)\left\{\varphi \left(u,x_n\right)+\left(1-{\beta }_n\right)\left(k_n-1\right)\varphi \left(u,x_n\right)\right\}\\ \le {\alpha }_n\varphi \left(u,x_1\right)+\left(1-{\alpha }_n\right)\varphi \left(u,x_n\right)+\left(k_n-1\right)\underset{p\in \mathcal{F}}{\text{sup}}\varphi \left(p,x_n\right)\\ ={\alpha }_n\varphi \left(u,x_1\right)+\left(1-{\alpha }_n\right)\varphi \left(u,x_n\right)+{\xi }_n,\end{array}$$

where ${\xi }_n=\left(k_n-1\right)\underset{p\in \mathcal{F}}{\text{sup}}\varphi \left(p,x_n\right).$ This shows that u ∈ C_n+1, and so $\mathcal{F}\subset C_{n+1}.$ The conclusion is proved.

(III) Next we prove that {x _n } is a Cauchy sequence in C.

In fact, since $x_n={\prod }_{C_n}{x}_1$, from Lemma 2.1(b) we have

$$⟨x_n-y,J{x}_1-J{x}_n⟩\ge 0,\forall y\in C_n.$$

Again since $\mathcal{F}\subset C_n\forall n\ge 1,$ we have

$$⟨x_n-u,J{x}_1-J{x}_n⟩\ge 0,\forall u\in \mathcal{F}.$$

It follows from Lemma 2.1(a) that for each $u\in \mathcal{F}$ and for each n ≥ 1

$$\varphi \left(x_n,x_1\right)=\varphi \left({\mathrm{\Pi }}_{C_n}{x}_1,x_1\right)\le \varphi \left(u,x_1\right)-\varphi \left(u,x_n\right)\le \varphi \left(u,x_1\right).$$

(3.4)

Therefore {ϕ(x _n , x₁)} is bounded. By virtue of (2.2), {x _n } is also bounded. Since $x_n={\prod }_{C_n}{x}_1$ and $x_{n+1}={\prod }_{C_n}{x}_1\in C_{n+1}\subset C_n$, we have ϕ(x _n , x₁) ≤ϕ (x_n+1, x₁), ∀n ≥ 1. This implies that {ϕ(x _n , x₁)} is nondecreasing. Hence, the limit lim_n→∞ϕ (x _n , x₁) exists. By the construction of {C _n }, for any positive integer m ≥ n, we have C _m ⊂ C _n and x _m = Π _Cm x₁ ∈ C _n . This show that

$$\varphi \left(x_m,x_n\right)=\varphi \left(x_m,{\mathrm{\Pi }}_{C_n}{x}_1\right),\le \varphi \left(x_m,x_1\right)-\varphi \left(x_n,x_1\right)\to 0asn,m\to \mathrm{\infty }.$$

It follows from Lemma 2.8 that lim_n,m→∞||x _m - x _n || = 0. Hence {x _n } is a Cauchy sequence in C. Since C is complete, without loss of generality, we can assume that x _n → p* (some point in C).

By the assumption, it is easy to see that

$$\underset{n\to \mathrm{\infty }}{\text{lim}}{\xi }_n=\underset{n\to \mathrm{\infty }}{\text{lim}}\left(k_n-1\right)\underset{p\in \mathcal{F}}{\text{sup}}\varphi \left(p,x_n\right)=0.$$

(3.5)

(IV) Now we prove that$p^{*}\in \mathcal{F}$.

In fact, since x_n+1∈ C_n+1and α _n → 0, follows from (3.1) and (3.5) that

$$\underset{t\ge 0}{\text{sup}}\varphi \left(x_{n+1},y_{n\mathsf{\text{,}}t}\right)\le {\alpha }_n\varphi \left(x_{n+1},x_1\right)+\left(1-{\alpha }_n\right)\varphi \left(x_{n+1},x_n\right)+{\xi }_n\to 0\left(asn\to \mathrm{\infty }\right).$$

Since x _n → p*, by virtue of Lemma 2.8 for each t ≥ 0

$$\underset{n\to \mathrm{\infty }}{\text{lim}}{y}_{n,t}=p^{*}.$$

(3.6)

Since {x _n } is bounded, and $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$ is a quasi-ϕ-asymptotically nonexpansive semi-group with sequence {k _n }⊂ [1,∞), k _n →1, for any given $p\in \mathcal{F},$, we have

$$\varphi \left(p,T^n\left(t\right)x_n\right)\le k_n\varphi \left(p,x_n\right),\forall t\ge 0.$$

This implies that {Tⁿ(t)x _n }_t≥0is uniformly bounded. Since for each t ≥ 0,

$$\begin{array}{cc}\hfill \left|\right|w_{n,t}\left|\right|\hfill & \hfill =\left|\right|J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{T}^n\left(t\right)x_n\right)\left|\right|\hfill \\ \hfill \le {\beta }_n\left|\right|x_n\left|\right|+\left(1-{\beta }_n\right)\left|\right|T^n\left(t\right)x_n\left|\right|\hfill \\ \hfill \le \text{max}\left\{\left|\right|x_n\left|\right|,\left|\right|T^n\left(t\right)x_n\left|\right|\right\}.\hfill \end{array}$$

This implies that {w_n,t}_t≥0is also uniformly bounded.

Since α _n → 0, from (3.1) we have

$$\underset{n\to \mathrm{\infty }}{\text{lim}}\left|\right|J{y}_{n,t}-J{w}_{n\mathsf{\text{,}}t}\left|\right|=\underset{n\to \mathrm{\infty }}{\text{lim}}{\alpha }_n\left|\right|J{x}_1-J{w}_{n\mathsf{\text{,}}t}\left|\right|=0,fort\ge 0.$$

(3.7)

Since E* is uniformly smooth, J^-1 is uniformly continuous on each bounded subset of E*, it follows from (3.6) and (3.7) that

$$\underset{n\to \mathrm{\infty }}{\text{lim}}{w}_{n\mathsf{\text{,}}t}=p^{*}foreacht\ge 0.$$

(3.8)

Since x _n → p* and J is uniformly continuous on each bounded subset of E, we have J x _n → Jp*, and so for each t ≥ 0

$$\begin{array}{ll}\hfill 0& =\underset{n\to \mathrm{\infty }}{\text{lim}}\left|\right|J{w}_{n,t}-J{p}^{*}\left|\right|=\underset{n\to \mathrm{\infty }}{\text{lim}}\left|\right|{\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{T}^n\left(t\right)x_n-J{p}^{*}\left|\right|\\ =\underset{n\to \mathrm{\infty }}{\text{lim}}\left|\right|{\beta }_n\left(J{x}_n-J{p}^{*}\right)+\left(1-{\beta }_n\right)\left(J{T}^n\left(t\right)x_n-J{p}^{*}\right)\left|\right|\\ =\underset{n\to \mathrm{\infty }}{\text{lim}}\left(1-{\beta }_n\right)\left|\right|J{T}^n\left(t\right)x_n-J{p}^{*}\left|\right|.\end{array}$$

By condition (ii), we have that

$$\underset{n\to \mathrm{\infty }}{\text{lim}}\left|\right|\left(J{T}^n\left(t\right)x_n-J{p}^{*}\right)\left|\right|=0\mathsf{\text{uniformly in}}t\ge 0.$$

Since J is uniformly continuous, this shows that lim_n→∞Tⁿ(t)x _n = p* uniformly in t ≥ 0.

Again by the assumptions that the semi-group $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$ is closed and uniformly L-Lipschitzian, thus we have

$$\begin{array}{ll}\hfill \left|\right|T^{n+1}\left(t\right)x_n-T^{n+1}\left(t\right)x_n& \le \left|\right|T^{n+1}\left(t\right)x_n-T^{n+1}\left(t\right)x_{n+1}\left|\right|+\left|\right|T^{n+1}\left(t\right)x_{n+1}-x_{n+1}\left|\right|\\ +\left|\right|x_{n+1}-x_n\left|\right|+\left|\right|x_n-T^n\left(t\right)x_n\left|\right|\\ \le \left(L\left(t\right)+1\right)\left|\right|x_{n+1}-x_n\left|\right|+\left|\right|T^{n+1}\left(t\right)x_{n+1}-x_{n+1}\left|\right|+\left|\right|x_n-T^n\left(t\right)x_n\left|\right|.\end{array}$$

(3.9)

Since lim_n→∞Tⁿ(t)x _n = p* uniformly in t ≥ 0, x _n → p* and L(t): [0, ∞) → [0, ∞) is a bounded and measurable function, these together with (3.9) imply that

$$\underset{n\to \mathrm{\infty }}{\text{lim}}\left|\right|T^{n+1}\left(t\right)x_n-T^n\left(t\right)x_n\left|\right|=0,\mathsf{\text{uniformly}}\mathsf{\text{in }}t\ge 0,$$

and so

$$\underset{n\to \mathrm{\infty }}{\text{lim}}{T}^{n+1}\left(t\right)x_n=p^{*},\mathsf{\text{uniformly in }}t\ge 0,$$

i.e.,

$$\underset{n\to \mathrm{\infty }}{\text{lim}}T\left(t\right)T^n\left(t\right)x_n=p^{*}\mathsf{\text{, uniformly in }}t\ge 0.$$

In view of the closeness of the semi-group $\mathcal{J}$, it yields that T(t)p* = p*, i.e., p* ∈ F(T(t)). By the arbitrariness of t ≥ 0, we have $p^{*}\in \mathcal{F}:={\bigcap }_{t\ge 0}F\left(T\left(t\right)\right).$

(V) Finally, we prove that $x_n\to p^{*}={\mathrm{\Pi }}_{\mathcal{F}}{x}_1.$

SHIH-SEN CHANG*

Let $w={\mathrm{\Pi }}_{\mathcal{F}}{x}_1.$ Since $w\in \mathcal{F}\subset C_n$ and $x_n={\prod }_{C_n}{x}_1$, we have ϕ(x _n , x₁) ≤ ϕ (w, x₁), ∀n ≥

• 1. This implies that

  $$\varphi \left(p^{*},x_1\right)=\underset{n\to \mathrm{\infty }}{\text{lim}}\varphi \left(x_n,x_1\right)\le \varphi \left(w,x_1\right).$$

  (3.10)

In view of the definition of ${\mathrm{\Pi }}_{\mathcal{F}}{x}_1,$ from (3.10) we have p* = w. Therefore, $x_n\to p^{*}={\mathrm{\Pi }}_{\mathcal{F}}{x}_1.$ This completes the proof of Theorem 3.1.

Theorem 3.2. Let E, C, {α _n }, {β _n } be the same as in Theorem 3.1. Let $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$ be a closed, quasi-ϕ- nonexpansive semi-group such that the set $\mathcal{F}:={\bigcap }_{t\ge 0}F\left(T\left(t\right)\right)$ is nonempty. Let {x _n } be the sequence generated by

$$\begin{array}{c}\left\{\begin{array}{c}{x}_1\in E\mathsf{\text{chosen}}\mathsf{\text{arbitrarily}};C_1=C,\\ y_{n,t}=J^{-1}\left[{\alpha }_{n}J{x}_1+\left(1-{\alpha }_n\right)\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)JT\left(t\right)x_n\right)\right],t\ge 0,\\ C_{n+1}=\left\{z\in C_n:\underset{t\ge 0}{\text{sup}}\varphi \left(z,y_{n,t}\right)\le {\alpha }_n\varphi \left(z,x_1\right)+\left(1-{\alpha }_n\right)\varphi \left(z,x_n\right)\right\}\\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,\forall n\ge 1.\end{array}\right.\end{array}$$

(3.11)

Then the sequence {x _n }converges strongly to ${\mathrm{\Pi }}_{\mathcal{F}}{x}_1,$

Proof. Since $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$ is a closed, quasi-ϕ-nonexpansive semi-groups, by Remark 2.5, it is a closed, uniformly Lipschitzian and quasi-ϕ- asymptotically nonexpansive semi-group with sequence{k _n = 1}.^.Hence ${\xi }_n=\left(k_n-1\right){\text{sup}}_{u\in \mathcal{F}}\varphi \left(u,x_n\right)=0.$ Therefore the conditions appearing in Theorem 3.1: "$\mathcal{F}$ is a bounded subset in C" and "$\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}$ is uniformly Lipschitzian" are no use here. Therefore all conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.2 can be obtained from Theorem 3.1 immediately.

Remark 3.3. Theorems 3.1 and 3.2 improve and extend the corresponding results of Suzuki [5], Xu [6], Chang et al. Chang et al. [7], Zhang [8], Chang et al. [9], Cho et al. [11], Thong [12], Buong [13], Mann [14], Halpern [15], Qin et al. [16], Nakajo and Takahashi [19], Kang et al. [23], Chang et al. [24], and others.