Strong convergence for total quasi- ϕ -asymptotically nonexpansive semigroups in Banach spaces

Jing Quan; Shih-sen Chang; Xiongrui Wang

doi:10.1186/1687-1812-2012-142

Research
Open Access

Strong convergence for total quasi-ϕ-asymptotically nonexpansive semigroups in Banach spaces

Jing Quan¹Email author,
Shih-sen Chang² and
Xiongrui Wang¹

Fixed Point Theory and Applications20122012:142

https://doi.org/10.1186/1687-1812-2012-142

Received: 8 May 2012

Accepted: 22 August 2012

Published: 5 September 2012

Abstract

The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-ϕ-asymptotically nonexpansive semigroups to prove strong convergence in Banach spaces. The main results presented in this paper extend and improve the corresponding results of many authors.

MSC:47H05, 47H09, 49M05.

Keywords

strong convergence
total quasi-ϕ-asymptotically nonexpansive semigroups
generalized projection

1 Introduction

Throughout this article, we assume that E is a real Banach space with norm

∥ \cdot ∥

,

E^{*}

is the dual space of E;

〈 \cdot, \cdot 〉

is the duality pairing between E and

E^{*}

; C is a nonempty closed convex subset of E;

N

and

R

denote the natural number set and the set of nonnegative real numbers respectively. The mapping

J : E \to 2^{E^{*}}

defined by

J (x) = {f^{*} \in E^{*} : 〈 x, f^{*} 〉 = {∥ x ∥}^{2}; ∥ f^{*} ∥ = ∥ x ∥, x \in E}

is called the normalized duality mapping. Let $T : C \to C$ be a nonlinear mapping; $F (T)$ denotes the set of fixed points of mapping T.

Alber et al.[1] introduced a more general class of asymptotically nonexpansive mappings called total asymptotically nonexpansive mappings and studied the methods of approximation of fixed points. They are defined as follows.

Definition 1.1 Let $T : C \to C$ be a mapping. T is said to be total asymptotically nonexpansive if there exist sequences ${μ_{n}}$ , ${ν_{n}}$ with $μ_{n}, ν_{n} \to 0$ as $n \to \infty$ and a strictly increasing continuous function $ψ : R \to R$ with $ψ (0) = 0$ such that $∥ T^{n} x - T^{n} y ∥ \leq ∥ x - y ∥ + μ_{n} ψ (∥ x - y ∥) + ν_{n}$ holds for all $x, y \in C$ and all $n \in N$ .

T is said to be total asymptotically quasi-nonexpansive if $F (T) \neq \emptyset$ , there exist sequences ${μ_{n}}$ , ${ν_{n}}$ with $μ_{n}, ν_{n} \to 0$ as $n \to \infty$ and a strictly increasing continuous function $ψ : R \to R$ with $ψ (0) = 0$ such that $∥ T^{n} x - p ∥ \leq ∥ x - p ∥ + μ_{n} ψ (∥ x - p ∥) + ν_{n}$ holds for all $x \in C$ , $p \in F (T)$ and all $n \in N$ .

Chidume and Ofoedu [2] introduced an iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings and total asymptotically quasi-nonexpansive mappings in Banach spaces. Chidume et al.[3] gave a new iterative sequence and necessary and sufficient conditions for this sequence to converge to a common fixed point of finite total asymptotically nonexpansive mappings. Chang [4] established some new approximation theorems of common fixed points for a countable family of total asymptotically nonexpansive mappings in Banach spaces.

Recently, many researchers have focused on studying the convergence of iterative algorithms for quasi-ϕ-asymptotically nonexpansive (see [5–9]) and total quasi-ϕ-asymptotically nonexpansive (see [10–12]) mappings. Ye et al.[13] used a new hybrid projection algorithm to obtain strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. Kim [14] used hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-ϕ-nonexpansive mappings to prove the strong convergence theorems. Saewan [15] used the shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-ϕ-nonexpansive mappings.

A Banach space E is said to be strictly convex if $\frac{∥ x + y ∥}{2} < 1$ for $∥ x ∥ = ∥ y ∥ = 1$ and $x \neq y$ ; it is also said to be uniformly convex if ${lim}_{n \to \infty} ∥ x_{n} - y_{n} ∥ = 0$ for any two sequences ${x_{n}}$ , ${y_{n}}$ in E such that $∥ x_{n} ∥ = ∥ y_{n} ∥ = 1$ and ${lim}_{n \to \infty} \frac{∥ x_{n} + y_{n} ∥}{2} = 1$ . Let $U = {x \in E : ∥ x ∥ = 1}$ be the unit sphere of E, then the Banach space E is said to be smooth provided ${lim}_{t \to 0} \frac{∥ x + t y ∥ - ∥ y ∥}{t}$ exists for each $x, y \in U$ . It is also said to be uniformly smooth if the limit is attained uniformly for each $x, y \in U$ . It is well known that if E is reflexive and smooth, then the duality mapping J is single valued. A Banach space E is said to have the Kadec-Klee property if a sequence ${x_{n}}$ of E satisfies that $x_{n} ⇀ x \in E$ and $∥ x_{n} ∥ \to x$ , then $x_{n} \to x$ . It is known that if E is uniformly convex, then E has the Kadec-Klee property.

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. We use

ϕ : E \times E \to R^{+}

to denote the Lyapunov functional defined by

ϕ (x, y) = {∥ x ∥}^{2} - 2 〈 x, J y 〉 + {∥ y ∥}^{2}, \forall x, y \in E .

It is obvious that

{(∥ x ∥ - ∥ y ∥)}^{2} \leq ϕ (x, y) \leq {(∥ x ∥ + ∥ y ∥)}^{2}, \forall x, y \in E,

(1)

and

ϕ (x, J^{- 1} (λ J y + (1 - λ) J z)) \leq λ ϕ (x, y) + (1 - λ) ϕ (x, z) .

(2)

Following Alber [16], the generalized projection

Π_{C} x : E \to C

is defined by

Π_{C} x = arg inf_{y \in C} ϕ (y, x), \forall x \in E .

The quasi-ϕ-asymptotically nonexpansive and total quasi-ϕ-asymptotically nonexpansive mappings are defined as follows.

Definition 1.2 A mapping

T : C \to C

is said to be quasi-ϕ-asymptotically nonexpansive, if

F (T) \neq \emptyset

, there exist sequences

{k_{n}} \subset [1, + \infty)

with

k_{n} \to 1

as

n \to \infty

such that

ϕ (p, T^{n} x) \leq k_{n} ϕ (p, x)

holds for all $x \in C$ , $p \in F (T)$ and all $n \in N$ .

A mapping

T : C \to C

is said to be total quasi-ϕ-asymptotically nonexpansive, if

F (T) \neq \emptyset

, there exist sequences

{μ_{n}}

,

{ν_{n}}

with

μ_{n}, ν_{n} \to 0

as

n \to \infty

and a strictly increasing continuous function

ψ : R \to R

with

ψ (0) = 0

such that

ϕ (p, T^{n} x) \leq ϕ (p, x) + μ_{n} ψ (ϕ (p, x)) + ν_{n}

holds for all $x \in C$ , $p \in F (T)$ and all $n \in N$ .

In recent years, many researchers have considered the convergence of asymptotically nonexpansive semigroups [17, 18]. The asymptotically nonexpansive semigroups are defined as follows.

Definition 1.3[17]

One-parameter family

T : = {T (t) : t \geq 0}

of mappings from C into itself is said to be an asymptotically nonexpansive semigroup on C, if the following conditions are satisfied:

(a)

$T (0) x = x$ for each $x \in C$ ;
(b)

$T (t + s) x = T (s) T (t)$ for any $t, s \in R^{+}$ and $x \in C$ ;
(c)

For any $x \in C$ , the mapping $t \to T (t) x$ is continuous;
(d)

There exist sequences ${k_{n}} \subset [1, + \infty)$ with $k_{n} \to 1$ as $n \to \infty$ such that
$∥ T^{n} (t) x - T^{n} (t) y ∥ \leq k_{n} ∥ x - y ∥$

holds for all $x, y \in C$ , $n \in N$ .

We use $F (T)$ to denote the common fixed point set of the semigroup T, i.e., $F (T) = ⋂_{t \geq 0} F (T (t))$ .

Chang [19] used the modified Halpern-Mann type iteration algorithm for quasi-ϕ-asymptotically nonexpansive semigroups to prove the strong convergence in the Banach space. The quasi-ϕ-asymptotically nonexpansive semigroups are defined as follows.

Definition 1.4[19]

One-parameter family

T : = {T (t) : t \geq 0}

of mappings from C into itself is said to be a quasi-ϕ-asymptotically nonexpansive semigroup on C if the conditions (a), (b), (c) in Definition 1.3 and following condition (e) are satisfied:

(e)

For all $x, y \in C$ , $p \in F ((T))$ , $t \geq 0$ , there exist sequences ${k_{n}} \subset [1, + \infty)$ with $k_{n} \to 1$ as $n \to \infty$ , such that
$ϕ (p, T^{n} (t) x) \leq k_{n} ϕ (p, x)$

holds for all $n \in N$ .

2 Preliminaries

This section contains some definitions and lemmas which will be used in the proofs of our main results in the following section.

Definition 2.1 One-parameter family

T : = {T (t) : t \geq 0}

of mappings from C into itself is said to be a total quasi-ϕ-asymptotically nonexpansive semigroup on C if conditions (a), (b), (c) in Definition 1.3 and following condition (f) are satisfied:

(f)

If $F (T) \neq \emptyset$ , there exist sequences ${μ_{n}}$ , ${ν_{n}}$ with $μ_{n}, ν_{n} \to 0$ as $n \to \infty$ and a strictly increasing continuous function $ψ : R \to R$ with $ψ (0) = 0$ such that
$ϕ (p, T^{n} (t) x) \leq ϕ (p, x) + μ_{n} ψ (ϕ (p, x)) + ν_{n}$

holds for all $x \in C$ , $p \in F (T)$ and all $n \in N$ .

A total quasi-ϕ-asymptotically nonexpansive semigroup T is said to be uniformly Lipschitzian if there exists a bounded measurable function

L : [0, \infty) \to (0, + \infty)

such that

∥ T^{(n)} (t) x - T^{(n)} (t) y ∥ \leq L (t) ∥ x - y ∥, \forall x, y \in C, t \geq 0, n \in N .

The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-ϕ-nonexpansive asymptotically semigroups to prove the strong convergence in Banach spaces. The results presented in the article improve and extend the corresponding results of [5, 6, 9–12, 14, 15, 19] and many others.

In order to prove the results of this paper, we shall need the following lemmas:

Lemma 2.1 (See [16])

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:

(i)

$ϕ (x, Π_{C} y) + ϕ (Π_{C} y, y) \leq ϕ (x, y)$ for all $x \in C$ , $y \in E$ ;
(ii)

If $x \in E$ and $z \in C$ , then $z = Π_{C} x \Leftrightarrow 〈 z - y, J x - J z 〉 \geq 0$ , $\forall y \in C$ ;
(iii)

For $x, y \in E$ , $ϕ (x, y) = 0$ if and only if $x = y$ .

Lemma 2.2[19]

Let E be a uniformly convex and smooth Banach space and let ${x_{n}}$ and ${y_{n}}$ be two sequences of E. If $ϕ (x_{n}, y_{n}) \to 0$ and either ${x_{n}}$ or ${y_{n}}$ is bounded, then $∥ x_{n} - y_{n} ∥ \to 0$ .

Lemma 2.3[10]

Let E be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, and C be a nonempty closed convex subset of E. Let $T : C \to C$ be a closed and total quasi-ϕ-asymptotically nonexpansive mapping defined by Definition 1.2. If $ν_{1} = 0$ , then the fixed point set $F (T)$ of T is a closed and convex subset of C.

3 Main results

Theorem 3.1 Let E be a real uniformly convex and uniformly smooth Banach space and C be a nonempty closed convex subset of E. Let

T : = {T (t) : t \geq 0}

be a total quasi-ϕ-asymptotically nonexpansive semigroup from C into itself defined by Definition 2.1. Suppose

T : = {T (t) : t \geq 0}

is closed, uniformly L-Lipschitz and

F (T) : = ⋂_{t \geq 0} F (T (t)) \neq \emptyset

. Suppose there exists

M^{*} > 0

such that

ψ (η_{n}) \leq M^{*} η_{n}

. Let

α_{n}

be a sequence in

[0, 1]

and

β_{n}

be a sequence in

(0, 1)

satisfying the following conditions:

{lim}_{n \to \infty} α_{n} = 0

,

0 < {lim inf}_{n \to \infty} β_{n} < {lim sup}_{n \to \infty} β_{n} < 1

. Let

x_{n}

be a sequence generated by

{\begin{matrix} x_{1} \in E, chosen arbitrarily; C_{1} = C, \\ l_{n, t} = β_{n} J x_{n} + (1 - β_{n}) J T^{n} (t) x_{n}, \\ y_{n, t} = J^{- 1} [α_{n} J x_{1} + (1 - α_{n}) l_{n, t}], t \geq 0, \\ C_{n + 1} = {z \in C_{n} : {sup}_{t \geq 0} ϕ (z, y_{n, t}) \leq α_{n} ϕ (z, x_{1}) + (1 - α_{n}) ϕ (z, x_{n}) + ξ_{n}}, \\ x_{n + 1} = Π_{C_{n + 1}} x_{1}, \forall n \geq 1, \end{matrix}

(3)

where $ξ_{n} = μ_{n} M^{*} {sup}_{p \in F (T)} ϕ (p, x_{n})$ . If $ν_{1} = 0$ and $F (T)$ is bounded in C, then the iterative sequence ${x_{n}}$ converges strongly to a common fixed point $x^{*} \in F (T)$ in C.

Proof (I) We prove $F (T)$ and $C_{n} (n \in N)$ all are closed and convex subsets in C.

It follows from Lemma 2.3 that

F (T (t))

,

t \geq 0

is a closed and convex subset of C. So

F (T)

is closed and convex in C. By the assumption we know that

C_{1} = C

is closed and convex. We suppose that

C_{n}

is closed and convex for some

n \geq 2

. By the definition of ϕ, we have that

\begin{array}{rcl} C_{n + 1} & = & {z \in C_{n} : sup_{t \geq 0} ϕ (z, y_{n, t}) \leq α_{n} ϕ (z, x_{1}) + (1 - α_{n}) ϕ (z, x_{n}) + ξ_{n}} \\ = & ⋂_{t \geq 0} {z \in C : ϕ (z, y_{n, t}) \leq α_{n} ϕ (z, x_{1}) + (1 - α_{n}) ϕ (z, x_{n}) + ξ_{n}} \cap C_{n} \\ = & ⋂_{t \geq 0} {z \in C : 2 α_{n} 〈 z, J x_{1} 〉 + 2 (1 - α_{n}) 〈 z, J x_{n} 〉 - 2 〈 z, J y_{n, t} 〉 \\ \leq α_{n} {∥ x_{1} ∥}^{2} + (1 - α_{n}) {∥ x_{n} ∥}^{2} - {∥ y_{n, t} ∥}^{2}} \cap C_{n} . \end{array}

This shows that

C_{n + 1}

is closed and convex.

(II)

We prove that $F (T) \subset C_{n}$ .

In fact

F (T) \subset C_{1} = C

. Suppose that

F (T) \subset C_{n}

,

n \geq 2

. Let

ω_{n, t} = J^{- 1} (β_{n} J x_{n} + (1 - β_{n}) J T^{n} (t) x_{n}), t \geq 0 .

It follows from (2) that for any

u \in F (T) \subset C_{n}

, we have

\begin{array}{rcl} ϕ (u, y_{n, t}) & = & ϕ (u, J^{- 1} (α_{n} J x_{1} + (1 - α_{n}) J ω_{n, t})) \\ \leq & α_{n} ϕ (u, x_{1}) + (1 - α_{n}) ϕ (u, ω_{n, t}), \end{array}

and

\begin{array}{rcl} ϕ (u, ω_{n, t}) & = & ϕ (u, J^{- 1} (β_{n} J x_{n} + (1 - β_{n}) J T^{n} (t) x_{n})) \\ \leq & β_{n} ϕ (u, x_{n}) + (1 - β_{n}) ϕ (u, T^{n} (t) x_{n}) \\ \leq & β_{n} ϕ (u, x_{n}) + (1 - β_{n}) [ϕ (u, x_{n}) + μ_{n} ψ (ϕ (u, x_{n})) + ν_{n}] \\ \leq & ϕ (u, x_{n}) + (1 - β_{n}) (μ_{n} M^{*} ϕ (u, x_{n}) + ν_{n}) . \end{array}

Therefore, we have

\begin{array}{rcl} sup_{t \geq 0} ϕ (u, y_{n, t}) & \leq & α_{n} ϕ (u, x_{1}) + (1 - α_{n}) [ϕ (u, x_{n}) + (1 - β_{n}) (μ_{n} M^{*} ϕ (u, x_{n}) + ν_{n})] \\ \leq & α_{n} ϕ (u, x_{1}) + (1 - α_{n}) ϕ (u, x_{n}) + μ_{n} M^{*} sup_{p \in F (T)} ϕ (p, x_{n}) + ν_{n} \\ = & α_{n} ϕ (u, x_{1}) + (1 - α_{n}) ϕ (u, x_{n}) + ξ_{n} . \end{array}

Where

ξ_{n} = μ_{n} M^{*} {sup}_{p \in F (T)} ϕ (p, x_{n}) + ν_{n}

. This shows that

u \in C_{n + 1}

, so

F (T) \subset C_{n + 1}

.

(III)

We prove that ${x_{n}}$ is a Cauchy sequence in C.

Since

x_{n} = Π_{C_{n}} x_{1}

, from Lemma 2.1(ii), we have

〈 x_{n} - y, J x_{1} - J x_{n} 〉 \geq 0, \forall y \in C_{n} .

Again since

F (T) \subset C_{n}

,

n \geq 1

, we have

〈 x_{n} - u, J x_{1} - J x_{n} 〉 \geq 0, \forall u \in F (T) .

It follows from Lemma 2.1(i) that for each

u \in F (T)

,

n \geq 1

,

ϕ (x_{n}, x_{1}) = ϕ (Π_{C_{n}} x_{1}, x_{1}) \leq ϕ (u, x_{1}) - ϕ (u, x_{n}) \leq ϕ (u, x_{1}) .

Therefore,

ϕ (x_{n}, x_{1})

is bounded. By virtue of (1),

x_{n}

is also bounded. Since

x_{n} = Π_{C_{n}} x_{1}

and

x_{n + 1} = Π_{C_{n + 1}} x_{1} \in C_{n + 1} \subset C_{n}

, we have

ϕ (x_{n}, x_{1}) \leq ϕ (x_{n + 1}, x_{1})

. This implies that

{ϕ (x_{n}, x_{1})}

is nondecreasing. Hence, the limit

{lim}_{n \to \infty} ϕ (x_{n}, x_{1})

exists. By the construction of

C_{n}

, for any positive integer

m \geq n

, we have

C_{m} \subset C_{n}

and

x_{m} = Π_{C_{1}} x_{1} \in C_{n}

. This shows that

\begin{array}{rcl} ϕ (x_{m}, x_{n}) & = & ϕ (x_{m}, Π_{C_{n}} x_{1}) \\ \leq & ϕ (x_{m}, x_{1}) - ϕ (x_{n}, x_{1}) \to 0, as m, n \to \infty . \end{array}

It follows from Lemma 2.2 that

{lim}_{n, m \to \infty} ∥ 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} \to p^{*}

(some point in C). By the assumption, we have that

lim_{n \to \infty} ξ_{n} = lim_{n \to \infty} [μ_{n} M^{*} sup_{p \in F (T)} ϕ (p, x_{n}) + ν_{n}] = 0 .

(4)

(IV)

Now we prove $p^{*} \in F (T)$ .

Since

x_{n + 1} \in C_{n + 1}

and

α_{n} \to 0

, it follows from (3) and (4) that

sup_{t \geq 0} ϕ (x_{n + 1}, y_{n, t}) \leq α_{n} ϕ (x_{n + 1}, x_{1}) + (1 - α_{n}) ϕ (x_{n + 1}, x_{n}) + ξ_{n} \to 0 as n \to \infty .

Since

x_{n} \to p^{*}

, by Lemma 2.2, for each

t \geq 0

, we have

lim_{n \to \infty} y_{n, t} = p^{*} .

(5)

Since

x_{n}

is bounded, and

T = {T (t), t \geq 0}

is a total quasi-ϕ-asymptotically nonexpansive semigroup with sequence

μ_{n}, ν_{n}, p \in F (T)

, we have

ϕ (p, T^{n} (t) x) \leq ϕ (p, x) + μ_{n} ψ (ϕ (p, x)) + ν_{n} \leq ϕ (p, x) + μ_{n} M^{*} ϕ (p, x) + ν_{n} .

This implies that

{T^{n} (t) x_{n}}_{t \geq 0}

is uniformly bounded. Since for each

t \geq 0

,

\begin{array}{rcl} ∥ ω_{n, t} ∥ & = & ∥ J^{- 1} (β_{n} J x_{n} + (1 - β_{n}) J T^{n} (t) x_{n}) ∥ \\ \leq & β_{n} ∥ x_{n} ∥ + (1 - β_{n}) ∥ T^{n} (t) x_{n} ∥ \\ \leq & max {∥ x_{n} ∥, ∥ T^{n} (t) x_{n} ∥} . \end{array}

This implies that

{ω_{n, t}}

,

t \geq 0

is also uniformly bounded. Since

α_{n} \to 0

, from (3) we have

lim_{n \to \infty} ∥ J y_{n, t} - J ω_{n, t} ∥ = lim_{n \to \infty} α_{n} ∥ J x_{1} - J ω_{n, t} ∥ = 0, t \geq 0 .

(6)

Since E is uniformly smooth,

J^{- 1}

is uniformly continuous on each bounded subset of

E^{*}

, it follows from (5) and (6) that

lim_{n \to \infty} ω_{n, t} = p^{*}, \forall t \geq 0 .

Since

x_{n} \to p^{*}

and J is uniformly continuous on each bounded subset of E, we have

J x_{n} \to J p^{*}

, and for each

t \geq 0

,

\begin{array}{rcl} 0 & = & lim_{n \to \infty} ∥ J ω_{n, t} - J p^{*} ∥ = lim_{n \to \infty} ∥ β_{n} J x_{n} + (1 - β_{n}) J T^{n} (t) x_{n} - J p^{*} ∥ \\ = & lim_{n \to \infty} ∥ β_{n} (J x_{n} - J p^{*}) + (1 - β_{n}) (J T^{n} (t) x_{n} - J p^{*}) ∥ \\ = & lim_{n \to \infty} (1 - β_{n}) ∥ (J T^{n} (t) x_{n} - J p^{*}) ∥ . \end{array}

By condition

0 < {lim inf}_{n \to \infty} β_{n} < {lim sup}_{n \to \infty} β_{n} < 1

, we have that

lim_{n \to \infty} ∥ J T^{n} (t) x_{n} - J p^{*} ∥ = 0, uniformly for t \geq 0 .

Since J is uniformly continuous, this shows that

{lim}_{n \to \infty} T^{n} (t) x_{n} = p^{*} = 0

uniformly for

t \geq 0

. Again by the assumptions that the semigroup

T : = {T (t) : t \geq 0}

is closed and uniformly L-Lipschitzian, we have

(7)

By

{lim}_{n \to \infty} T^{n} (t) x_{n} = p^{*}

uniformly for

t \geq 0

,

x_{n} \to p^{*}

and

L (t)

is a bounded and measurable function, and from (7) we have that

lim_{n \to \infty} ∥ T^{n + 1} (t) x_{n} - T^{n} (t) x_{n} ∥ = 0 uniformly for t \geq 0,

and

lim_{n \to \infty} T^{n + 1} (t) x_{n} = p^{*} uniformly for t \geq 0,

so we get

lim_{n \to \infty} T (t) T^{n} (t) x_{n} = p^{*} uniformly for t \geq 0 .

By virtue of the closeness of semigroup T, we have that

T (t) p^{*} = p^{*}

, i.e.,

p^{*} \in F (T (t))

. By the arbitrariness of

t \geq 0

, we have

p^{*} \in F (T) = ⋂_{t \geq 0} F (T (t))

.

(V)

Finally, we prove $x_{n} \to p^{*} = Π_{F (T)} x_{1}$ .

Let

ω = Π_{F (T)} x_{1}

. Since

ω \in F (T) \subset C_{n}

and

x_{n} = Π_{C_{n}} x_{1}

, we get

ϕ (x_{n}, x_{1}) \leq ϕ (ω, x_{1})

,

n \geq 1

. This implies that

ϕ (p^{*}, x_{1}) = lim_{n \to \infty} ϕ (x_{n}, x_{1}) \leq ϕ (ω, x_{1}) .

(8)

In view of the definition of $Π_{F (T)} x_{1}$ , from (8), we have $p^{*} = ω$ . Therefore, $x_{n} \to p^{*} = Π_{F (T)} x_{1}$ . This completes the proof of Theorem 3.1. □

Declarations

Acknowledgements

This work was supported by National Research Foundation of Yibin University (No. 2011B07) and by Scientific Research Fund Project of Sichuan Provincial Education Department (No. 12ZB345 and No. 11ZA172).

Authors’ Affiliations

(1)

Department of Mathematics, Yibin University, Yibin, China

(2)

College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, China

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