FIXED POINT AND NON-LINEAR ERGODIC THEOREMS FOR SEMIGROUP OF NON-LINEAR MAPPINGS

ANTHONY TO-MING LAU^{†} AND WATARU TAKAHASHI

1. Introduction

Let S be a semigroup, ℓ^{∞}(S) be the Banach space of bounded real valued functions on S with the supremum norm. There is a strong relation between the existence of an invariant mean (or submean) on an invariant subspace of ℓ^{∞}(S) and fixed point or ergodic properties of S when S is represented as a semigroup of nonexpansive mappings on a closed convex subset of a Banach space. It is the purpose of this chapter of the Handbook to exhibit on some recent results on such relations. Since this handbook is intended for researchers and graduate students, detailed proofs for central results, historical remarks, open problems and many references will be included. It is our hope that our effort will generate further research in this direction of non-linear analysis which depends on the ideal theory of S, and existence of an invariant mean (or submean) on a subspace of ℓ^{∞}(S) of a semigroup S.

This chapter is organized as follows: In section 3, we study the notion of non-linear “mean” (called submean) on a subspace of ℓ^{∞}(S), and its relation with left reversibility of S (i.e. S has finite intersection property of S). In section 4, we prove among other things that (Theorem 4.4) if ℓ^{∞}(S) has a left invariant submean μ, and S = {T_{s} : s ∈ S} is a representation of S as nonexpansive maps on a weakly compact convex subset C of a Banach space E with more than one point, and C has normal structure, then the set {z ∈ C; μ_{t}∥T_{t}x − z∥ = inf{μ_{t}∥T_{t}x − y∥; y ∈ C}} is a proper subset of C. Theorem 4.4 is then used to obtain an improvement of a classical fixed point theorem of Lim [66] in section 5 (Theorem 5.1) for left reversible semigroup of nonexpansive maps. In section 6, we study the relation between left ideal orbit and the fixed point set F (S) when S is a representation of S as non-expansive mappings on a closed convex subset C of E into C. The most important result of this section is Theorem 6.7 which shows that if the norm of E is Fréchet differentiable, then for each x ∈ C, F (S) ∩ Q(x) consists of at most one point, where Q(x) is the intersection of all left ideal orbits of x. Theorem 6.7 is crucial for some of the ergodic theorems (Theorem 7.4 and Theorem 7.10) which we study in section 7. In section 8, we shall discuss some related results.

2. Some preliminaries

All topologies in this paper are assumed to be Hausdorff. If E is a Banach space and A ⊆ E, then \overline{A} and \overline{co}A will denote the closure of A and the closed convex hull of A in E, respectively.

Given a non-empty set S, we denote by ℓ^{∞}(S) the Banach space of bounded real-valued functions on S with the supremum norm. Let S be a semigroup. Then a subspace X of ℓ^{∞}(S) is left (resp. right) translation invariant if ℓ_{a}(X) ⊆ X(resp. r_{a}(X) ⊆ X) for all a ∈ S, where (ℓ_{a}f)(s) = f(as) and (r_{a}f)(s) = f(sa), s ∈ S.

A semitopological semigroup S is a semigroup with Hausdorff topology such that for each a ∈ S, the mappings s ↦ a · s and s ↦ s · a from S into S is continuous. Examples of semitopological semigroups include all topological groups, the set M(n, ℂ) of all n × n matrices with complex entries, matrix multiplication and the usual topology, the unit ball of ℓ^{∞} with weak^{∗}-topology and pointwise multiplication, or B(H) ( = the space of bounded linear operators on a Hilbert space H) with the weak^{∗}-topology and composition.

† This research is supported by NSERA7679.

If S is a semitopological semigroup, we denote by CB(S) the closed subalgebra of ℓ^{∞}(S) consisting of continuous functions. Let LUC(S) (resp. RUC(S)) be the space of left (resp. right) uniformly continuous functions on S, i.e. all f ∈ CB(S) such that the mapping from S into CB(S) defined by s → ℓ_{s}f (resp. s → r_{s}f) is continuous when CB(S) has the sup norm topology. Then as is known (see [12]), LUC(S) and RUC(S) are left and right translation invariant closed subalgebras of CB(S) containing constants. Note that when S is a topological group, then LUC(S) is precisely the space of right uniformly continuous functions on S defined in [36]. Also let AP (S) (resp. W AP (S)) denote the space of almost periodic (resp. weakly almost periodic) functions f in CB(S), i.e. all f ∈ CB(S) such that {ℓ_{a}f; a ∈ S} is relatively compact in the norm (resp. weak) topology of CB(S), or equivalently {r_{a}f; a ∈ S} is relatively compact in the norm (resp. weak) topology of CB(S). Then as is known [12, p.164], AP (S) ⊆ LUC(S) ∩ RUC(S), and AP (S) ⊆ W AP (S). When S is a group, then W AP (S) ⊆ LUC(S) ∩ RUC(S) (see [12, p.167]).

Let E be a real Banach space and let E^{∗} be its topological dual. Then, the value of x^{∗} ∈ E^{∗} at x ∈ E will be denoted by ⟨x, x^{∗}⟩ or x^{∗}(x). With each x ∈ E, we associate the set

J(x) = {x^{∗} ∈ E^{∗}; ⟨x, x^{∗}⟩ = ∥x∥^{2} = ∥x^{∗}∥^{2}}.

Using the Hahn-Banach theorem, it is immediately clear that J(x) ≠ ϕ for each x ∈ E. The multivalued operator J : E → E^{∗} is called the duality mapping of E. Let S(E) = {x ∈ E; ∥x∥ = 1} be the unit sphere of E. Then the norm of E is said to be Gâteaux differentiable (and E is said to be smooth) if for each x, y ∈ S(E),

lim_{λ→0} \frac{∥x + λy∥ − ∥x∥}{λ}

exists. It is said to be Fréchet differentiable if for each x ∈ S(E), this limit is attained uniformly for y ∈ S(E). It is said to be uniformly Gâteaux differentiable if for each y ∈ S(E), this limit is attained uniformly for x ∈ S(E). It is well known that if E has a Fréchet differentiable norm, then J is single value and norm to norm continuous; see [14] or [23] for more details.

3. Submean and reversibility

Let S be a non-empty set and X be a subspace of ℓ^{∞}(S) containing constants. Then μ ∈ X^{∗} is called a mean on X if ∥μ∥ = μ(1) = 1. As well known, μ is a mean on X if and only if

inf_{s∈S} f(s) ≤ μ(f) ≤ sup_{s∈S} f(s)

for each f ∈ X.

By a submean on X, we shall mean a real-valued function μ on X satisfying the following properties :

(1) μ(f + g) ≤ μ(f) + μ(g) for every f, g ∈ X;

(2) μ(αf) = αμ(f) for every f ∈ X and α ≥ 0;

(3) for f, g ∈ X, f ≤ q implies μ(f) ≤ μ(g);

(4) μ(c) = c for every constant function c.

If S is a semigroup, and X ⊆ ℓ^{∞}(S) is a left translation invariant subspace of ℓ^{∞}(S) containing contants, a submean μ on X is left [right] invariant if μ(ℓ_{a}f) = μ(f) [μ(r_{a}f) = μ(f)] for each a ∈ S, f ∈ X.

We abbreviate left invariant submean = LISM and left invariant mean = LIM.

Depending on time and circumstances, the value of a submean (or mean) μ at f ∈ X will also be denoted by μ(f), ⟨μ, f⟩ or μ_{t}f(t).

A semitopological semigroup S is left reversible if any two closed right ideals of S have non-void intersection.

Lemma 3.1. Let S be a semitopological semigroup and X be a left translation invariant subspace of CB(S) containing constants and which separates closed subsets of S. If X has a LISM, then S is left reversible.

Proof: Let μ be a LISM of X, I_{1} and I_{2} be disjoint nonempty closed right ideals of S. By assumption, there exists f ∈ X such that f ≡ 1 on I_{1} and f ≡ 0 on I_{2}. Now if a_{1} ∈ I_{1}, then ℓ_{a_{1}}f = 1. So,

μ(f) = μ(ℓ_{a_{1}}f) = 1.

But if a_{2} ∈ I_{2} , then ℓ_{a_{2}}f ≡ 0 . So

μ(f) = μ(ℓ_{a_{2}}f) = 0,

which is impossible. □

Corollary 3.2. If S is normal and CB(S) has a LISM, then S is left reversible.

Corollary 3.3. If S is normal and CB(S) has a LISM, then AP (S) has a LIM.

Proof: This follows from Corollary 3.2 and [47, Corollary 3.3]. □

If S is a left reversible semitopological semigroup, then (S, ⪯) is a directed system when the binary relation ” ⪯ ” on S is defined by a ⪯ b if and only if \{a\} ∪ \overline{aS} ⊇ \{b\} ∪ \overline{bS}, a, b ∈ S.

Lemma 3.4. Let S be a semitopological semigroup, J be a non-empty subset of S and f ∈ LUC(S). If sup{f(t); t ⪰ u} ≥ β for each u ∈ J, then sup{f(t); t ⪰ p} ≥ β for each p ∈ \overline{J}.

Proof: Let p ∈ \overline{J} and sup\{f(t); t ⪰ p\} ≤ β − δ, δ > 0. Then

f(ps) ≤ β − δ for each s ∈ S ∪ {e},

where xe = ex = e. Let u_{α} ∈ J be a net such that u_{α} → p. Hence ∥ℓ_{u_{α}}f − ℓ_{p}f∥ → 0. Consequently there exists α_{0} such that

f(u_{α}s) ≤ β − δ/2 for each s ∈ S ∪ {e}, α ≥ α_{0}.

Hence for α ≥ α_{0}, we have sup{f(t); t ⪰ u_{α}} ≤ β −δ/2, which contradicts the assumption. □

A semitopological semigroup S is left amenable [resp. subamenable] if LUC(S) has a LIM [resp. LISM].

Proposition 3.5. Let S be a semitopological semigroup. If S is left reversible, then S is left subamenable.

Proof: For each f ∈ CB(S), define

μ(f) = inf_{s} sup_{t⪰s} f(t).

Then μ is a submean on CB(S). Indeed, if f, g ∈ CB(S), and ϵ > 0, choose a, b ∈ S such that

sup_{t⪰a} f(t) ≤ μ(f) + ϵ and sup_{t⪰b} g(t) ≤ μ(g) + ϵ.

Let c ∈ \overline{aS} ∩ \overline{bS} (which is non-empty by left reversibility). Then c ⪰ a, and c ⪰ b. Hence

sup_{t⪰c} f(t) ≤ μ(f) + ϵ and sup_{t⪰c} g(t) ≤ μ(g) + ϵ.

So

sup_{t⪰c}(f(t) + g(t)) ≤ sup_{t⪰c} f(t) + sup_{t⪰c} g(t)
≤ μ(f) + μ(g) + 2ϵ.

Consequently

μ(f + g) ≤ μ(f) + μ(g) + 2ϵ.

Since ϵ > 0 is arbitrary, condition (1) for submean holds. Proofs of conditions (2), (3) and (4) are routine.

To see that μ is left invariant, let f ∈ LUC(S) and a ∈ S. Then

μ(ℓ_{a}f) = inf_{s} sup_{t⪰s} f(at)
= inf_{s}\{sup\{f(at); t ∈ \overline{sS} ∪ \{s\}\}
= inf_{s}{sup{f(at); t ∈ sS ∪ {s}}

(by continuity of f and multiplication in S)

= inf_{s}{sup{f(ast); t ∈ S ∪ {e}}

(where se = s)

= inf_{s}{sup{f(t); t ∈ asS ∪ {as}}}
= inf_{s} sup_{t⪰as} f(t)
≥ μ(f).

To prove the reverse inequality, let α = μ(f) and β = μ(ℓ_{a}f), f ∈ LUC(S). Then for each s ∈ S, sup_{t⪰as} f(t) ≥ β. Hence, by Lemma 3.4,

sup_{t⪰p} f(t) ≥ β for all p ∈ \overline{aS}.  (3.1)

If α < β, let ϵ = (β − α)/2. Choose s_{0} such that sup_{t⪰s_{0}} f(t) < α + ϵ. Then for each s ⪰ s_{0}, sup_{t⪰s} f(t) < α + ϵ. Let p ∈ \overline{s_{0}S} ∩ \overline{aS}. Then p ⪰ s_{0} ; so sup_{t⪰p} f(t) < α + ϵ, contradicting (3.1). □

Corollary 3.6. Let S be a discrete semigroup. Then S is left reversible if and only if S is left subamenable. In this case W AP (S) has a LIM.

Proof: The first statement follows from Corollary 3.2 and Proposition 3.5, and the last statement follows from [39] (see also [48]). □

Notes and remarks (i) When S is a discrete semigroup, the following implications hold:

S left subamenable  A(S) has LIM A(S) left reversible A(S) left amenable  W (S) has LIM

The implication : S is left reversible ⇒ A(S) has a LIM for any semitopological semigroup was established in [48]. During the 1984 Richmond, Virginia, conference on analysis on semigroup, T. Mitchell [72] gave two examples to show that for discrete semigroups A(S) has a LIM ⇒ S left reversible (see [50]). The implication : S is left reversible ⇒ W (S) has a LIM for discrete semigroups was proved by Hsu [39]. Results and their proofs presented in this section are essentially taken from our work in [61].

(ii) Clearly every mean is a submean. The notion of submean was first introduced by Mizoguchi and Takahashi in [74].

Let SM denote the set of submeans on X. For each ϕ ∈ SM, −∥f∥ ≤ ϕ(f) ≤ ∥f∥ by (3) and (4). Hence SM may be identified as a subset of the product space ∏_{f∈X}[−∥f∥, ∥f∥], which is compact by Tychonoff’s Theorem. Hence SM is a compact convex subset of the product topological vector space ∏_{f∈X} ℝ_{f}, where each ℝ_{f} = ℝ.

(iii) Corollary 3.2 is false without normality. Indeed, let S be the topological space which is regular and Hausdorff and CB(S) consists of constant functions only ([35]). Define on S the multiplication st = s for s, t ∈ S. Let a ∈ S be fixed. Define

μ(f) = f(a) for all f ∈ CB(S).

Then μ is a LISM on CB(S), but S is not left reversible.

(iv) The theory of amenability began in 1904 when Lebesgue asked if the uniqueness of the Lebesgue integral is still preserved if the Monotone Convergence Theorem (which is equivalent to countable additivity) is replaced by finite additivity. Banach in 1923 [6] showed that there exists a mean on the bounded real-valued functions on the integers which is invariant under all translations, i.e. the group of integers under addition is amenable. This constrasted the result of Hausdorff [34] who showed that there does not exist any mean on the bounded real-valued functions on the sphere in three dimensions, which is invariant under all rotations. Von Neumann [106] introduced and studied the class of discrete amenable groups. The term “amenable” was due to Day [19]. Since then the theory has grown in many directions to include amenability of discrete and topological semigroups, locally compact groups and Banach algebras (in particular C^{∗}-algebras and von Neumann algebras). Interested reader should consult the books [32], [77] and [76], and the survey article [51].

(v) The class 𝕊 of all left reversible semitopological semigroups includes trivially all topological semigroups which are algebraically groups, and all commuting topological semigroups.

The class 𝕊 is closed under the following operations.

(a) If S ∈ 𝕊 and S′ is a continuous homomorphic image of S, then S′ ∈ 𝕊.

(b) Let S_{α} ∈ 𝕊, α ∈ I and S be the topological semigroup consisting of the set of all functions f on I such that f(α) ∈ S_{α}, α ∈ I, the binary operation defined by fg(α) = f(α)g(α) for all α ∈ I and f, g ∈ S, and the product topology. Then S ∈ 𝕊.

(c) Let S be a topological semigroup and S_{α}, α ∈ I, topological sub-semigroups of S with the property that S = ∪S_{α} and, if α_{1}, α_{2} ∈ I, then there exists α_{3} ∈ I such that S_{α_{3}} ⊇ S_{α_{1}} ∪ S_{α_{2}}. If S_{α} ∈ 𝕊 for each α ∈ I, S ∈ 𝕊.

4. Submean and normal structure

A convex subset C of a Banach space is said to have normal structure [13] if every bounded convex subset D of C with |D| > 1 contains a point x such that sup{∥x − y∥ : y ∈ D} < δ(D), where δ(D) = sup{∥x − y∥ : x, y ∈ D} (the diameter of D).

In their original paper [13], Brodskii and Milman characterized normal structure as follows: A convex subset of a Banach space has normal structure if and only if it contains no diametral sequences. (A diametral sequence is a nonconstant bounded sequence {x_{n}}_{n=1}^{∞} such that d(x_{n+1}, co(x_{1}, …, x_{n})) → δ({x_{n}}_{n=1}^{∞}).) The following lemma is a simple variation of the above characterization.

Lemma 4.1. A convex subset C of a Banach space has normal structure if and only if it does not contain a sequence \{x_{n}\} such that for some c > 0, ∥x_{n} − x_{m}∥ ≤ c, ∥x_{n+1} − \overline{x}_{n}∥ ≥ c − \frac{1}{n^{2}} for all n ≥ 1, m ≥ 1, where \overline{x}_{n} = \frac{1}{n} ∑_{i=1}^{n} x_{i}.

Proof. If C contains a bounded convex subset D such that |D| > 1 and sup{∥x − y∥ : y ∈ D} = δ(D) for every x ∈ D, then it is easy to choose, by induction, a nonconstant sequence {x_{n}} ⊆ D satisfying the condition in the Lemma with c = δ(D). On the other hand, assume that {x_{n}} ⊆ C is a sequence satisfying the condition in the Lemma. If x ∈ co(x_{1}, …, x_{n}), it is not difficult to show that x = λ_{1}\overline{x}_{n} + λ_{2}x_{i_{2}} + ⋯ + λ_{n}x_{i_{n}} for some i_{2}, …, i_{n} ∈ {1, …, n} and λ_{i}, 1 ≤ i ≤ n, with ∑_{i=1}^{n} λ_{i} = 1, 0 < λ_{1} ≤ n, and λ_{j} ≤ 0 for 2 ≤ j ≤ n. It follows that c ≥ ∥x_{n+1} − x∥ ≥ c − \frac{1}{n} for every n ≥ 1 and every x ∈ co(x_{1}, …, x_{n}). Hence d(x_{n+1}, co(x_{1}, …, x_{n})) → c and c is necessarily equal to δ({x_{n}}). □

Let S be a non-empty set and f be a function from S into C such that {f(s); s ∈ S} is bounded, where C is a closed convex subset of a Banach space E with more than one point. For each x ∈ C, define f_{x}(s) = ∥f(s) − x∥, s ∈ S. Let X be a closed subspace of ℓ^{∞}(S) containing constants such that f_{x} ∈ X for each x ∈ X and let μ be a submean on X. Define r : C → ℝ by r(x) = μ(f_{x}).

Lemma 4.2. The function r : C → ℝ is continuous, convex on C and if ∥x_{n}∥ → ∞, then r(x_{n}) → ∞.

Proof. Let x_{n} → x, and x_{n}, x be in C. Then since

∥f(t) − x_{n}∥ − ∥f(t) − x∥ ≤ ∥x_{n} − x∥

and

∥f(t) − x∥ − ∥f(t) − x_{n}∥ ≤ ∥x_{n} − x∥,

it follows that

μ(f_{x_{n}}) ≤ ∥x_{n} − x∥ + μ(f_{x})

and

μ(f_{x}) ≤ ∥x_{n} − x∥ + μ(f_{x_{n}}).

Hence

|μ(f_{x_{n}}) − μ(f_{x})| ≤ ∥x_{n} − x∥ → 0.

This implies that r is continuous.

If α, β ≥ 0, α + β = 1 and x, y ∈ C, then since

∥f(t) − (αx + βy)∥ ≤ α∥f(t) − x∥ + β∥f(t) − y∥,

we have

r(αx + βy) ≤ αr(x) + βr(y).

Finally, if ∥x_{n}∥ → ∞, then since ∥x_{n}∥ ≤ ∥x_{n} − f(t)∥ + ∥f(t)∥ for each t ∈ S, we have

∥x_{n}∥ ≤ r(x_{n}) + M → ∞, where M = sup_{t∈S}{∥f(t)∥}. □

Let ρ_{0} = inf{r(x); x ∈ C}. Then we have the following:

Lemma 4.3. If C is non-empty weakly compact and convex, then the set A = {x; r(x) = ρ_{0}} is non-empty, closed and convex. Furthermore, if C = A, then

ρ_{0} = inf_{y∈C} sup_{x∈C}{∥x − y∥}.

Proof. Since r : C → ℝ is continuous and convex (Lemma 4.2), r is weakly lower semicontinuous. So, A is non-empty, closed and convex. Suppose C = A, let ϵ > 0 and σ = {x_{1}, …, x_{n}} ⊆ C. Consider a system {h_{i}}_{i=1}^{n} of convex weakly lower semicontinuous functions on C defined by

h_{i}(x) = ∥x − x_{i}∥, for x ∈ C and i = 1, …, n.

We shall show that the set D_{σ} = {z ∈ C; h_{i}(z) ≤ ρ_{0} + ϵ, i = 1, …, n} is non-empty. In fact, from A = C, we have r(x_{i}) = ρ_{0} for i = 1, 2, …, n. So for any {α_{i}}_{i=1}^{n} ⊆ ℝ with α_{1}, α_{2}, …, α_{n} ≥ 0 and ∑_{i=1}^{n} α_{i} = 1, we have

μ_{t}(∑_{i=1}^{n} α_{i}∥f(t) − x_{i}∥) ≤ ∑_{i=1}^{n} α_{i}r(x_{i}) = ρ_{0} < ρ_{0} + ϵ.

Then, there exists t_{0} ∈ S such that

∑_{i=1}^{n} α_{i}∥f(t_{0}) − x_{i}∥ ≤ ρ_{0} + ϵ

and hence ∑_{i=1}^{n} α_{i}h_{i}(f(t_{0})) ≤ ρ_{0} + ϵ. Hence by Fan’s Theorem for convex inequalities on a topological vector space [26], there exists z ∈ C such that

h_{i}(z) = ∥z − x_{i}∥ ≤ ρ_{0} + ϵ for each i = 1, …, n.

This implies that the set D_{σ} is non-empty. Hence by weak compactness of C, the set D = {z ∈ C; ∥x − z∥ ≤ ρ_{0} + ϵ for all x ∈ C} is also non-empty. So we have

ρ_{0} = min{r(y); y ∈ C}
≤ min_{y∈C}{sup_{t∈S} ∥f(t) − y∥}
≤ min_{y∈C}{sup_{x∈C} ∥x − y∥} ≤ ρ_{0} + ϵ.

Since ϵ > 0 is arbitrary, we have

ρ_{0} = inf_{y∈C}{sup_{x∈C} ∥x − y∥}. □

If S is a semigroup and X is left translation invariant, a submean μ on X is left subinvariant if μ(ℓ_{a}f) ≥ μ(f) for each f ∈ X and a ∈ S. A representation S = {T_{s}; s ∈ S} as mappings from a subset C of a Banach space into C is called X-admissible if for each x, y ∈ C, the function t ↦ ∥T_{t}x − y∥ belongs to X.

Theorem 4.4. Let C be a non-empty weakly compact convex subset of a Banach space E. If C has more than one point and normal structure, then C satisfies: (P) Whenever S is a semigroup, Φ is a closed left translation invariant subspace of ℓ^{∞}(S) containing constants with a left subinvariant submean μ, S = {T_{s}; s ∈ S} is a Φ-admissible representation of S as non-expansive mappings from C into C, then the set A_{x} = {y ∈ C; μ_{t}∥T_{t}x − y∥ = ρ_{x}} is a proper subset of C for some x ∈ C, where ρ_{x} = inf{μ_{t}∥T_{t}x − y∥; y ∈ C}. Furthermore, for each x ∈ C the set A_{x} is non-empty, closed, convex and T_{s}-invariant.

Proof . If A_{x} = C for all x ∈ C, then by Lemma 4.3,

ρ_{x} = ρ_{0} = inf_{x∈C}{sup_{y∈C} ∥x − y∥} = μ_{t}∥T_{t}x − y∥  (4.1)

for all x, y ∈ C. So let

A_{0} = {z ∈ C; sup_{t∈S} ∥T_{t}x − z∥ ≤ ρ_{0}, ∀x ∈ C}.

By weak compactness of C and (4.1), there exists z_{0} ∈ C such that

sup_{y∈C} ∥z_{0} − y∥ = ρ_{0}.

Hence A_{0} is a non-empty set. Let z_{0} ∈ A_{0} and s ∈ S. Then for any x ∈ C,

ρ_{0} = μ_{t}∥T_{t}x − z_{0}∥ ≤ μ_{t}∥T_{st}x − z_{0}∥ ≤ sup_{t∈S} ∥T_{st}x − z_{0}∥
≤ sup_{t∈S} ∥T_{t}x − z_{0}∥ ≤ ρ_{0}

and

ρ_{0} = μ_{t}∥T_{t}x − T_{s}z_{0}∥ ≤ μ_{t}∥T_{st}x − T_{s}z_{0}∥ ≤ sup_{t∈S} ∥T_{st}x − T_{s}z_{0}∥
≤ sup_{t∈S} ∥T_{t}x − z_{0}∥ ≤ ρ_{0}

by left subinvariance of μ. For using Lemma 4.1, fix z_{0} ∈ A_{0}. Then since ρ_{0} = μ_{t}∥T_{t}z_{0}−z_{0}∥, there exists s_{1} ∈ S such that ∥T_{s_{1}}z_{0} − z_{0}∥ ≥ ρ_{0} − 1. Let x_{1} = z_{0}, x_{2} = T_{s_{1}}z_{0} and

\overline{x_{2}} = \frac{1}{2}z_{0} + \frac{1}{2}T_{s_{1}}z_{0}.

Since ρ_{0} = μ_{t}∥T_{t}z_{0} − \overline{x_{2}}∥ ≤ μ_{t}∥T_{s_{1}t}z_{0} − \overline{x_{2}}∥, there exists s_{2} ∈ S such that ∥T_{s_{1}s_{2}}z_{0} − \overline{x_{2}}∥ ≥

ρ_{0} − \frac{1}{2^{2}} . So, let x_{3} = T_{s_{1}s_{2}}z_{0}. Then, we have

∥x_{1} − x_{2}∥ = ∥z_{0} − T_{s_{1}}z_{0}∥ ≤ sup_{t∈S} ∥z_{0} − T_{t}z_{0}∥ = ρ_{0},

∥x_{2} − x_{3}∥ = ∥T_{s_{1}}z_{0} − T_{s_{1}s_{2}}z_{0}∥ ≤ ∥z_{0} − T_{s_{1}}z_{0}∥ ≤ ρ_{0},

and

∥x_{3} − x_{1}∥ = ∥T_{s_{1}s_{2}}z_{0} − z_{0}∥ ≤ sup_{t∈S} ∥T_{t}z_{0} − z_{0}∥ = ρ_{0}.

Similarly, let

\overline{x_{3}} = \frac{1}{3}x_{1} + \frac{1}{3}x_{2} + \frac{1}{3}x_{3}.

Then, ρ_{0} = μ_{t}∥T_{t}z_{0} − \overline{x_{3}}∥ ≤ μ_{t}∥T_{s_{1}s_{2}t}z_{0} − \overline{x_{3}}∥, there exists s_{3} ∈ S such that ∥T_{s_{1}s_{2}s_{3}}z_{0} −

\overline{x_{3}}∥ ≥ ρ_{0} − \frac{1}{3^{2}} . So, let x_{4} = T_{s_{1}s_{2}s_{3}}z_{0}. Then, we have

∥x_{4} − x_{1}∥ = ∥T_{s_{1}s_{2}s_{3}}z_{0} − z_{0}∥ ≤ sup_{t∈S} ∥T_{t}z_{0} − z_{0}∥ = ρ_{0},

∥x_{4} − x_{2}∥ = ∥T_{s_{1}s_{2}s_{3}}z_{0} − T_{s_{1}}z_{0}∥ ≤ ∥T_{s_{2}s_{3}}z_{0} − z_{0}∥ ≤ sup_{t∈S} ∥T_{t}z_{0} − z_{0}∥ = ρ_{0},

and

∥x_{4} − x_{3}∥ = ∥T_{s_{1}s_{2}s_{3}}z_{0} − T_{s_{1}s_{2}}z_{0}∥ ≤ ∥T_{s_{3}}z_{0} − z_{0}∥ ≤ sup_{t∈S} ∥T_{t}z_{0} − z_{0}∥ = ρ_{0}.

By mathematical induction, let x_{5} = T_{s_{1}s_{2}s_{3}s_{4}}z_{0}, x_{6} = T_{s_{1}s_{2}s_{3}s_{4}s_{5}}z_{0}, · · ·. Then we have

∥x_{n} − x_{m}∥ ≤ ρ_{0}, ∀n, m and ∥x_{n+1} − \overline{x_{n}}∥ ≥ ρ_{0} − \frac{1}{n^{2}} .

Using Lemma 4.1, C does not have normal structure. This is a contradiction. To see that A_{x} is T_{s}-invariant, x ∈ C, s ∈ S, let y ∈ A_{x}. Then

ρ_{x} ≤ μ_{t}(∥T_{t}x − T_{s}y∥) ≤ μ_{t}(∥T_{st}x − T_{s}y∥) ≤ μ_{t}(∥T_{t}x − y∥) = ρ_{x}.

Hence

μ_{t}(∥T_{t}x − T_{s}y∥) = ρ_{x}, i.e., T_{s}y ∈ A_{x}. □

Proposition 4.5. Let K be a closed convex subset of a Banach space E. If K satisfies the following property (Q), then K has normal structure: (Q) Whenever S is a non-empty set and f is a function from S into C, where C is a bounded closed convex subset of K with more than one point, and for each x ∈ C, the function f_{x}(s) = ∥f(s) − x∥ belongs to a closed subspace X of ℓ^{∞}(S) containing constants, and μ be a submean on X, r(x) = μ(f_{x}), ρ_{0} = inf{r(x); x ∈ C}, then the set A = {x ∈ C; r(x) = ρ_{0}} is a proper subset of C.

Proof: Suppose K does not have normal structure. Then there exists a bounded closed convex subset C of K such that δ(C) > 0 and sup_{y∈C} ∥y − x∥ = δ(C) for every x ∈ C. Let

S = C, X = ℓ^{∞}(S), f : C → C, f(s) = s. Then for each x ∈ C, f_{x}(s) = ∥s − x∥, s ∈ C is in ℓ^{∞}(C) (since C is bounded).

Let μ(h) = sup h, h ∈ ℓ^{∞}(C), and r(x) = μ(f_{x}), x ∈ C, r = inf{r(x); x ∈ C}. Then since r(x) = δ(C) for every x ∈ C, we have r = δ(C), i.e., A = {x ∈ C : r(x) = r} = C. This completes the proof. □

Notes and remarks

It is an immediate consequence of Brodskii and Milman [13]’s characterization of normal structure that all compact convex set have normal structure. Using a straight forward transfinite induction argument, they showed that a weakly compact convex set K which has normal structure always contains a point z which is fixed under every linear isometry. In [28] (see also [GK]), Kirk proved the following fundamental existence theorem for nonexpansive mappings.

Theorem 4.6. Let K be a nonempty, weakly compact, convex subset of a Banach space, and suppose K has normal structure. Then every nonexpansive mapping T : K → K has a fixed point.

In [1] Alspach gave the first example of a weakly compact convex set (in L^{1}[0, 1]) without normal structure.

Lemma 4.1 is due to Lim [67]. The other results presented in this section are taken from [62].

5. Fixed point theorem

We are now ready to apply Theorem 4.4 to prove a fixed point theorem for a semigroup of nonexpansive mappings on weakly compact convex subset of a Banach space with normal structure. This is the first known application of Fan’s Theorem for convex inequalities on a topological vector space (see proof of Theorem 4.4) in fixed point theory for semigroup of nonexpansive mappings.

Theorem 5.1. Let S be a semitopological semigroup, let D be a nonempty weakly compact convex subset of a Banach space E which has normal structure and let S = {T_{s}; s ∈ S} be a continuous representation of S as nonexpansive self mappings on D. Suppose RUC(S) has a left subinvariant submean. Then S has a common fixed point in D.

Proof. We first prove that for any x ∈ D and y ∈ E, a function h defined by h(t) = ∥T_{t}x − y∥ for all t ∈ S is in RUC(S). In fact, we have, for s, u ∈ S,

∥r_{s}h − r_{u}h∥ = sup_{t∈S} |(r_{s}h)(t) − (r_{u}h)(t)| = sup_{t∈S} |h(ts) − h(tu)|
= sup_{t∈S} |∥T_{ts}x − y∥ − ∥T_{tu}x − y∥| ≤ sup_{t∈S} ∥T_{ts}x − T_{tu}x∥
≤ ∥T_{s}x − T_{u}x∥.

Let

U = {K ⊂ D : K is nonempty, closed, convex, and T_{s}-invariant }.

Then by Zorn’s Lemma, there exists a minimal element C of U. Let δ(C) > 0 and let μ be a left subinvariant submean. Then, for any x ∈ C,

A_{x} = {z ∈ C : μ_{t}∥T_{t}x − z∥ = min_{y∈C} μ_{t}∥T_{t}x − y∥}

is nonempty, closed, convex, and T_{s}-invariant (see Theorem 4.4). So, we have A_{x} = C by minimality of C. Hence by Theorem 4.4, C cannot have normal structure which is a contradiction. □

Corollary 5.2. Let S be a left reversible semitopological semigroup. Let D be a nonempty weakly compact convex subset of a Banach space E which has normal structure and let S = {T_{s}; s ∈ S} be a continuous representation of S as nonexpansive self mappings on D. Then S has a fixed point in D.

Proof. If S is left reversible, define μ(f) = inf_{s} sup_{t∈sS} f(t). Then the proof of Lemma 3.6

in [12] shows that μ is a submean on CB(S) such that μ(ℓ_{a}f) ≥ μ(f) for all f ∈ CB(S) and a ∈ S, i.e., μ is left subinvariant. □

Notes and remarks

(i) Belluce and Kirk [11] first proved that if K is a nonempty weakly compact convex subset of a Banach space and if K has complete normal structure, then every family of commuting nonexpansive self-maps on K has a common fixed point. Later Lim [66, Theorem 3] extended this theorem to a continuous representation of a left reversible semitopological semigroup S as nonexpansive mappings on a weakly compact convex set K with normal structure (Corollary 5.2).

Theorem 5.1 was proved by us in [62]. It answers affirmatively a problem posed during the Conference on Fixed Point Theory and Applications held at CIRM, Marseille-Luminy, 1989 (see [51, p.307, Problem 5].)

(ii) It has been proved by Hsu [39] (see also [9]) that if S is discrete and left reversible, and S = {T_{s}; s ∈ S} is a representation of S as weakly continuous nonexpansive mappings on a weakly compact convex subset C of a Banach space, then C has a common fixed point for S. Note that it follows from Alspach’s example [1] that there exists a commutative semigroup of nonexpansive mappings on a weakly compact convex subset of L^{1}[0, 1] with no common fixed point. But as is well known, ℓ^{∞}(S) always has an invariant mean when S is commutative. Also Schechtman [80] has shown that there exists a weakly compact convex subset W of L^{1}[0, 1] and a sequence T_{1}, T_{2}, … of commuting nonexpansive operators of W into itself such that any finite number of them have a common fixed point but there is no common fixed point for the entire sequence.

(iii) (see [51]). Consider on a pair (S, E), where S is a semitopological semigroup and E is a Banach space, the following fixed point properties:

(P) Whenever S = {T_{s}; s ∈ S} is a representation of S as nonexpansive mappings from a non-empty closed convex subset C of E and for each x ∈ C, the orbit O(x) = {T_{s}x; s ∈ S} is relatively compact, then closed convex hull of O(x) contains a common fixed point of S.

(Q) Whenever S = {T_{s}; s ∈ S} is a continuous representation of S as nonexpansive mappings on a non-empty closed convex subset C of E into C and C contains an element with bounded orbit, then C contains a common fixed point for S.

Of course any pair (S, E) satisfying (Q) must satisfy (P). However as shown by R. Sine [86], (P) is not true for arbitrary finite dimensional space E and commutative S (see [49, Problem 2]).

Theorem 5.3 ([9]). If S is left reversible (or more generally A(S) has a LIM) and E is strictly convex, then (P) holds.

Theorem 5.4 ([59] [60]). If RUC(S) has a LIM or S is left reversible and E is uniformly convex and uniformly smooth, then (Q) holds.

Problem 1. Does (Q) hold when W (S) has a LIM and E is a Hilbert space (see [49], Problem 1)?

A representation S = {T_{s}; s ∈ S} of S as self-maps on a subset C of a Banach space is called uniformly k-Lipschitzian, k > 0, if

∥T_{s}x − T_{s}y∥ ≤ k∥x − y∥

for all s ∈ S, x, y ∈ C. Uniformly k−Lipschitzian semigroups were introduced by Goebel, Kirk and Thele in [28] who proved the following (in slightly more general form) basic result:

Theorem 5.5. If E us a uniformly convex Banach space, then there is a k_{0} > 1 such that whenever K ⊆ E is a closed, bounded, convex set and S = {T_{s}; s ∈ S} is a representation from K into K with k < k_{0}, then K contains a common fixed point for S.

Precisely how large k_{0} may be taken remains, even in Hilbert space, an open question. The estimate provided for the Hilbert space case in [28] is \frac{\sqrt{5}}{2} with an upper bound of 2. Downing and Ray [24], by refining the technique in [28] shows that k_{0} may be taken to be \sqrt{2} in this case (see [41, Theorem 2] for a differennt proof). Ishihara and Takahashi [41, Theorem 1] shows that Downing and Ray’s result [24] remains valid when S is a semitopological semigroup such that RUC(S) has a LIM and the action of S on C is jointly continuous (see also [40]).

(iv) Locally convex spaces (see [48] and [51])

Let S be a semitopological semigroup and (E, Q), or simply E, be a separated locally convex space. Let S = {T_{s}; x ∈ S} be a representation of S as mapping on a subset X into X. We assume (throughout this discussion) that the map ψ : S × X → X, (s, x) ↦ T_{s}x, s ∈ S and x ∈ X is separately continuous, i.e. ψ is continuous in each of the two variables when the other is kept fixed. Then S is jointly continuous if the map ψ is continuous when S × X has the product topology; S is Q − nonexpansive if p(T_{s}x − T_{s}y) ≤ p(x − y) for all s ∈ S, p ∈ Q and x, y ∈ X.

Consider on S the following fixed point properties:

(G_{1}) Whenever S is a jointly continuous representation of S on a non-empty weak^{∗}- compact convex subset X of dual Banach space with the weak^{∗}-topology and S is norm nonexpansive, then X has a common fixed point in S.

(G_{2}) Whenever S is a representation of S as weakly continuous, Q-nonexpansive mappings on a non-empty weakly compact convex set X of a separated locally convex space (E, Q), then X contains a common fixed point for S.

(G_{3}) Whenever S is a representation of S as nonexpansive mappings on a non-empty compact convex subset X of a separated locally convex space (E, Q), then X contains a common fixed point for S.

DeMarr [22] proved that any commuting semigroup S has fixed point property (G_{3}). Later, W. Takahashi [87] proved that any discrete left amenable semigroup S has property (G_{3}). T. Mitchell [70] generalized both DeMarr and Takahashi’s result by showing that any discrete left reversible semigroup S has property (G_{3}). Finally, Lau proved in [47] the following:

Theorem 5.6. A(S) has LIM iff S has fixed point property (G_{3}).

Since the action of S on the weak^{∗}-compact convex set X in LUC(S)^{∗}:

X = {m ∈ LUC(S)^{∗}; m ≥ 0 and ∥m∥ = 1}

defined by ψ(a, m) = ℓ_{a}^{∗}m, a ∈ S, m ∈ X, is jointly continuous (see [71]) and norm non-expansive, it follows readily that

(G_{1}) ⇒ S is left amenable.

However the converse is unknown even for S commutative.

Problem 2. Does S left amenable ⇒ (G_{1})?

A similar argument as in the proof of the theorem in [48, p.123] shows that

(G_{2}) ⇒ W (S) has a LIM

However, the following problems are still open (see [48]):

Problem 3. Does S left reversible ⇒ (G_{2}) for semitopological semigroups.

Notice that weak^{∗}-continuity of the representation in (G_{1}) is important; (G_{1}) is not true even for commutative semigroup when the mappings {T_{s}; s ∈ S} are not weak^{∗}- weak^{∗} continuous. In fact, Alspach [1] has shown that there is a weakly compact convex subset K of L^{1}[0, 1] and a non-expansive mapping T : K → K without a fixed point. Since L^{1}[0, 1] may be embedded isometrically into (ℓ^{∞}(ℕ), ∥ · ∥_{∞}), K regarded as a subset of ℓ^{∞}(ℕ) is still weakly compact and convex. Hence K is weak^{∗}-compact. Let S = ℕ, the positive integers with addition. Then S = {T^{n}; n ∈ ℕ} is a representative of S as non-expansive mappings on K without fixed point.

(v) Fixed point property and Radon Nikodym property (see [52]).

A Banach space E is said to have fixed point property (f.p.p.) if for each non-empty weakly compact convex subset X ⊆ E and a nonexpansive mapping T : X → X, X contains a fixed point for T. As well known [27], a weakly compact convex subset of a Banach space E with normal structure has the f.p.p. On the other hand, L^{1}[0, 1] does not have the f.p.p.

Bruck [15] shows that if a Banach space E has the f.p.p., then E has the f.p.p. for commutative semigroup, i.e. whenever C is a non-empty weakly compact convex subset of E, and S is a commutative semigroup of non-expansive mappings from C into C, then C contains a common fixed point for S.

Let G be a locally compact group, and let A(G) denote the Fourier algebra of G, i.e. A(G) is the subalgebra of C_{0}(G) (complex-valued continuous functions on G vanishing at infinity) consisting of all funcitons ϕ of the form

ϕ(x) = ⟨ρ(s)h, k⟩, h, k ∈ L^{2}(G),

ρ(x)h(t) = ρ(x^{−1}t), x, t ∈ G. Then A(G) is a semi-simple commutative Banach algebra with norm

∥ϕ∥ = sup {|∑_{i=1}^{n} λ_{i}ϕ(x_{i})|; ∥∑_{i=1}^{n} λ_{i}ρ(x_{i})∥ ≤ 1} ,

and A(G)^{∗} = V N(G), the von Neumann algebra in B(L^{2}(G)) generated by {ρ(x); x ∈ G} (see [25] for details). It follows from Alspach [1] that if G = (Z, +), then A(Z) does not have the f.p.p.

A Banach space E is said to have the Radon-Nikodym property (RNP) if each closed bounded convex subset D of E is dentable, i.e. for any ε > 0, there exists x ∈ D such that x ∉ \overline{co}\{D\B_{ε}(x)\} where B_{ε}(x) = \{y ∈ E; ∥y − x∥ < ε\}.

Lemma 5.7 ([56]). Let G be a locally compact group. If A(G) has the RNP, then A(G) has the f.p.p.

The following problem is still open:

Problem 4. Does fixed point property for A(G) imply RNP?

A locally compact group G is called an [IN]-group if there is a compact neighbourhood U of the identity e such that x^{−1}Ux = U for all x ∈ G. Examples of [IN]-groups include all compact groups, discrete groups and abelian groups.

Theorem 5.8 ([56]). If G is either an abelian group or a connected IN-group, then the following are equivalent:

(a) G compact.

(b) A(G) has the f.p.p.

(c) A(G) has RNP.

Theorem 5.9 ([56]). If G is discrete and A(G) has the f.p.p., then G cannot contain an infinite abelian subgroup. In particular, elements of G must have finite order.

Note that there are noncompact groups G (e.g. the “ax + b” group) such that A(G) has the RNP and hence the f.p.p. (see [105] or [65] for details). Also in the case G is compact, then A(G) = B(G) (the Fourier Stieltjes algebra of G) is the dual Banach space of C^{∗}(G), the group C^{∗}-algebra of G. In this case, A(G) even has the weak^{∗}-f.p.p., i.e. for every non-empty weak^{∗}-compact convex subset X of A(G) and every non-expansive mapping T : X → X, X contains a fixed point for T (see [13]).

When G is a locally compact abelian group, then the following are equivalent:

(i) G is comapct; (ii) B(G) has the weak^{∗}-f.p.p.; (iii) The weak^{∗} and norm topology agree on {ϕ ∈ B(G); ∥ϕ∥ = ϕ(e) = 1}.

When G is not abelian, we still have (i) ⟹ (ii) ⟺ (iii). However (ii) does not imply (i) in general (see [55], [56] and [10] for details).

6. Fixed point sets and left ideal orbits

Let S be a semitopological semigroup, E be a uniformly convex Banach space and S = {T_{s}; s ∈ S} be a continuous representation of S as nonexpansive mappings on a closed convex subset C of E into C. In this section, we shall discuss the relation between the left ideal orbits and fixed the point set F (S). We begin with a simple lemma:

Lemma 6.1. Let E be a strictly convex Banach space and let C be a convex subset of E. Let T be a nonexpansive mapping of C into itself. Then the set F (T) of fixed points of T is convex.

Proof. Let x, y ∈ F (T) and 0 ≤ α ≤ 1. Then putting z = αx + (1 − α)y, we have

∥x − T z∥ = ∥T x − T z∥ ≤ ∥x − z∥

and

∥y − T z∥ = ∥T y − T z∥ ≤ ∥y − z∥.

Hence,

∥x − y∥ ≤ ∥x − T z∥ + ∥T z − y∥ ≤ ∥x − z∥ + ∥z − y∥ = ∥x − y∥.

This implies that ∥x − z∥ = ∥x − T z∥ and ∥y − z∥ = ∥y − T z∥. Since E is strictly convex, we have z = T z. Therefore F (T) is convex. □

Let L(S) denote the collection of closed left ideals in S. Assume that F (S) ≠ ∅. For each x ∈ C and L ∈ L(S), define the real-valued function q_{x,L} on F (S) by

q_{x,L}(f) = inf{∥T_{t}x − f∥^{2} : t ∈ L}

and let

q_{x}(f) = sup{q_{x,L} : L ∈ L(S)}.

Then

q_{x}(f) = sup_{s} inf_{t} ∥T_{ts}x − f∥^{2}

as readily checked.

Lemma 6.2. Let C be a nonempty closed convex subset of a Banach space E. If F (S) ≠ ∅, then for each x ∈ C, q_{x} is a continuous real-valued function on F (S) such that 0 ≤ q_{x}(f) ≤ ∥x − f∥^{2} for each f ∈ F (S) and q_{x}(f_{n}) → ∞ if ∥f_{n}∥ → ∞. Further, if F (S) is convex, then q_{x} is a convex function on F (S).

Proof. Since 0 ≤ ∥T_{t}x − f∥^{2} = ∥T_{t}x − T_{t}f∥^{2} ≤ ∥x − f∥^{2} for every f ∈ F (S) and t ∈ S, it follows readily that 0 ≤ q_{x}(f) ≤ ∥x − f∥^{2}. Also if f ∈ F (S) and t ∈ S, then ∥T_{t}x − f∥ ≤ ∥x − f∥. Hence ∥T_{t}x∥ ≤ ∥T_{t}x − f∥ + ∥f∥ ≤ ∥x − f∥ + ∥f∥, i.e., M = sup{∥T_{t}x∥ : t ∈ S} < ∞. Let {f_{n}} be a sequence in F (S) such that ∥f_{n}∥ → ∞. Then we have for each t ∈ S,

∥T_{t}x − f_{n}∥^{2} ≥ (∥T_{t}x∥ − ∥f_{n}∥)^{2}
= ∥f_{n}∥^{2} − 2∥T_{t}x∥∥f_{n}∥ + ∥T_{t}x∥^{2}
≥ ∥f_{n}∥^{2} − 2M∥f_{n}∥
= ∥f_{n}∥^{2} (1 − \frac{2M}{∥f_{n}∥})

and hence for each L ∈ L(S),

q_{x,L}(f_{n}) ≥ ∥f_{n}∥^{2} (1 − \frac{2M}{∥f_{n}∥}) → ∞.

So we have q_{x}(f_{n}) → ∞. To see that q_{x} is continuous, let {f_{n}} be a sequence in F (S) converging to some f ∈ F (S) and

M′ = sup{∥T_{t}x − f_{n}∥ + ∥T_{t}x − f∥ : n = 1, 2, ⋯ and t ∈ S}.

Then since

∥T_{t}x − f_{n}∥^{2} − ∥T_{t}x − f∥^{2} ≤ (∥T_{t}x − f_{n}∥ + ∥T_{t}x − f∥) | ∥T_{t}x − f_{n}∥ − ∥T_{t}x − f∥ |
≤ M′∥f_{n} − f∥,

we have for each L ∈ L(S),

q_{x,L}(f_{n}) ≤ q_{x,L}(f) + M′∥f_{n} − f∥.

Similarly, we have

q_{x,L}(f) ≤ q_{x,L}(f_{n}) + M′∥f_{n} − f∥.

So we obtain

|q_{x}(f_{n}) − q_{x}(f)| ≤ M′∥f_{n} − f∥.

This implies that q_{x} is continuous on F (S). If F (S) is convex, for each f, g ∈ F (S) and α, β ≥ 0 with α + β = 1, αf + βg ∈ F (S). Let ε > 0. Then there exists L_{0} ∈ L(S) such that

sup_{L∈L(S)} inf_{t∈L}(α∥T_{t}x − f∥^{2} + β∥T_{t}x − g∥^{2}) < inf_{t∈L_{0}}(α∥T_{t}x − f∥^{2} + β∥T_{t}x − g∥^{2}) + \frac{ε}{2}.

Let u ∈ L_{0}. Then Su ⊆ L_{0} and hence

sup_{L∈L(S)} inf_{t∈L}(α∥T_{t}x − f∥^{2} + β∥T_{t}x − g∥^{2}) < inf_{t∈S}(α∥T_{tu}x − f∥^{2} + β∥T_{tu}x − g∥^{2}) + \frac{ε}{2}.

Moreover, there exist v, w ∈ S such that

∥T_{vu}x − f∥^{2} < inf_{t∈S} ∥T_{tu}x − f∥^{2} + \frac{ε}{2}

and

∥T_{wvu}x − f∥^{2} < inf_{t∈S} ∥T_{tvu}x − f∥^{2} + \frac{ε}{2}.

Therefore we obtain

q_{x}(αf + βg) = sup_{L∈L(S)} inf_{t∈L} ∥T_{t}x − (αf + βg)∥^{2}
≤ sup_{L∈L(S)} inf_{t∈L}(α∥T_{t}x − f∥^{2} + β∥T_{t}x − g∥^{2})
< inf_{t∈S}(α∥T_{tu}x − f∥^{2} + β∥T_{tu}x − g∥^{2}) + \frac{ε}{2}
≤ α∥T_{wvu}x − f∥^{2} + β∥T_{wvu}x − g∥^{2} + \frac{ε}{2}
≤ α∥T_{vu}x − f∥^{2} + β∥T_{wvu}x − g∥^{2} + \frac{ε}{2}
< α inf_{t∈S} ∥T_{tu}x − f∥^{2} + β inf_{t∈S} ∥T_{tvu}x − g∥^{2} + \frac{αε}{2} + \frac{βε}{2} + \frac{ε}{2}
= α inf_{t∈L_{1}} ∥T_{t}x − f∥^{2} + β inf_{t∈L_{2}} ∥T_{t}x − g∥^{2} + ε
( where L_{1} = \overline{Su} and L_{2} = \overline{Svu})
≤ αq_{x}(f) + βq_{x}(g) + ε.

Since ε > 0 is arbitrary, we have

q_{x}(αf + βg) ≤ αq_{x}(f) + βq_{x}(g). □

Theorem 6.3. Let C be a nonempty closed convex subset of a uniformly convex Banach space E. Assume that F (S) ≠ ∅. Then for any x ∈ C, there exists a unique element h ∈ F (S) such that

q_{x}(h) = inf{q_{x}(f) : f ∈ F (S)}.

Proof. Since E is uniformly convex, the fixed point set F (S) in C is closed and convex (see Lemma 6.1). Hence it follows from Lemma 6.2 and [8] that there exists h ∈ F (S) such that

q_{x}(h) = inf{q_{x}(f) : f ∈ F (S)}.

To see that h is unique, let k ∈ F (S). Then by [107], there exists a strictly increasing and convex function (depending on h and k) g : [0, ∞) → [0, ∞) such that g(0) = 0 and

∥T_{t}x − (λh + (1 − λ)k)∥^{2} = ∥λ(T_{t}x − h) + (1 − λ)(T_{t}x − k)∥^{2}
≤ λ∥T_{t}x − h∥^{2} + (1 − λ)∥T_{t}x − k∥^{2} − λ(1 − λ)g(∥h − k∥)

for each t ∈ S and λ with 0 ≤ λ ≤ 1. So we have for each λ with 0 ≤ λ ≤ 1,

q_{x}(h) ≤ q_{x}(λh + (1 − λ)k)
≤ λq_{x}(h) + (1 − λ)q_{x}(k) − λ(1 − λ)g(∥h − k∥)

and hence

q_{x}(h) ≤ q_{x}(k) − λg(∥h − k∥).

It follows that

q_{x}(h) ≤ q_{x}(k) − g(∥h − k∥) as λ → 1.

Since g is strictly increasing, it follows that if q_{x}(h) = q_{x}(k), then h = k. □

We call the unique element h ∈ F (S) in Theorem 6.3 the minimizer of q_{x} in F (S).

For each x ∈ C, let

Q(x) = ⋂_{L∈L(S)} \overline{co}\{T_{t}x : t ∈ L\}(= ⋂_{s∈S} \overline{co}\{T_{ts}x : t ∈ S\}).

Theorem 6.4. Let C be a nonempty closed convex subset of a Hilbert space H. Let S = {T_{s} : s ∈ S} be a continuous representation of S as nonexpansive mappings from C into C. Then for any x ∈ C, any element in Q(x) ∩ F (S) is the unique minimizer of q_{x} in F (S). In particular, Q(x) ∩ F (S) contains at most one point.

Proof. Let z ∈ F (S) be the minimizer of q_{x} in F (S) and y ∈ Q(x) ∩ F (S). Then for some ε > 0, there exists u ∈ S such that

sup_{s} inf_{t}(∥T_{ts}x − z∥^{2} + 2⟨T_{ts}x − z, z − y⟩ + ∥z − y∥^{2})

< inf_{t}(∥T_{tu}x − z∥^{2} + 2⟨T_{tu}x − z, z − y⟩ + ∥z − y∥^{2}) + \frac{ε}{4}.

Moreover there exist v, w ∈ S such that

∥T_{vu}x − z∥^{2} < inf_{t} ∥T_{tu}x − z∥^{2} + \frac{ε}{4}

and

⟨T_{wvu}x − z, z − y⟩ < inf_{t}⟨T_{tvu}x − z, z − y⟩ + \frac{ε}{4}.

Therefore we obtain

q_{x}(y) = sup_{s} inf_{t} ∥T_{ts}x − y∥^{2}
= sup_{s} inf_{t}(∥T_{ts}x − z∥^{2} + 2⟨T_{ts}x − z, z − y⟩ + ∥z − y∥^{2})
< inf_{t}(∥T_{tu}x − z∥^{2} + 2⟨T_{tu}x − z, z − y⟩ + ∥z − y∥^{2}) + \frac{ε}{4}
≤ ∥T_{wvu}x − z∥^{2} + 2⟨T_{wvu}x − z, z − y⟩ + ∥z − y∥^{2} + \frac{ε}{4}
≤ ∥T_{vu}x − z∥^{2} + 2⟨T_{wvu}x − z, z − y⟩ + ∥z − y∥^{2} + \frac{ε}{4}
< inf_{t} ∥T_{tu}x − z∥^{2} + 2 inf_{t}⟨T_{tvu}x − z, z − y⟩

+ ∥z − y∥^{2} + \frac{ε}{4} + \frac{ε}{2} + \frac{ε}{4}

≤ sup_{s} inf_{t} ∥T_{ts}x − z∥^{2} + 2 sup_{s} inf_{t}⟨T_{ts}x − z, z − y⟩

+ ∥z − y∥^{2} + ε

= q_{x}(z) + 2 sup_{s} inf_{t}⟨T_{ts}x − z, z − y⟩ + ∥z − y∥^{2} + ε.

This implies

2 sup_{s} inf_{t}⟨T_{ts}x − z, z − y⟩ > q_{x}(y) − q_{x}(z) − ∥z − y∥^{2} − ε
≥ −∥z − y∥^{2} − ε.

So, there exists a ∈ S such that

2⟨T_{ta}x − z, z − y⟩ > −∥z − y∥^{2} − ε

for every t ∈ S. From y ∈ \overline{co}\{T_{ta}x : t ∈ S\}, we have

2⟨y − z, z − y⟩ ≥ −∥z − y∥^{2} − ε.

This inequality implies ∥z − y∥^{2} ≤ ε. Since ε > 0 is arbitrary, we have z = y. □

Lemma 6.5. For each x ∈ C, y, z ∈ F (S) and λ with 0 ≤ λ ≤ 1,

inf_{s∈S} sup_{t∈S} ∥λT_{ts}x + (1 − λ)y − z∥ ≤ sup_{s∈S} inf_{t∈S} ∥λT_{ts}x + (1 − λ)y − z∥.

Further, if S is semitopological and right reversible and t ↦ T_{t}x is continuous for each x ∈ C, then equality holds.

Proof. Let x ∈ C, let y, z ∈ F (S) and let 0 ≤ λ ≤ 1. Let D = {u ∈ C : ∥u − y∥ ≤ ∥x−y∥}. Then by Lemma 1.1 in [16], there exists a strictly increasing, continuous, convex function γ from [0, ∞) to [0, ∞) such that γ(0) = 0 and

γ(∥ηT_{t}u + (1 − η)T_{t}v − T_{t}(ηu + (1 − η)v)∥) ≤ ∥u − v∥ − ∥T_{t}u − T_{t}v∥  (6.1)

for each t ∈ S, u, v ∈ D and η with 0 ≤ η ≤ 1. Let ε > 0. Since each T_{t} is nonexpansive, inf_{s} sup_{t} ∥T_{ts}x − y∥ ≤ inf_{s} ∥T_{s}x − y∥. So there exists s_{0} ∈ S such that ∥T_{ts_{0}}x − y∥ ≤ ∥T_{s}x − y∥ + ε for each t, s ∈ S. Putting η = λ, u = T_{ws_{0}}x and v = y in (6.1), we get

∥λT_{tws_{0}}x + (1 − λ)y − z∥

≤ ∥λT_{tws_{0}}x + (1 − λ)y − T_{t}(λT_{ws_{0}}x + (1 − λ)y)∥ + ∥T_{t}(λT_{ws_{0}}x + (1 − λ)y) − z∥

≤ γ^{−1}(∥T_{ws_{0}}x − y∥ − ∥T_{tws_{0}}x − y∥) + ∥λT_{ws_{0}}x + (1 − λ)y − z∥
≤ γ^{−1}(ε) + ∥λT_{ws_{0}}x + (1 − λ)y − z∥

for each t, w ∈ S. Hence we have

inf_{s∈S} sup_{t∈S} ∥λT_{ts}x + (1 − λ)y − z∥ ≤ γ^{−1}(ε) + inf_{w∈S} ∥λT_{ws_{0}}x + (1 − λ)y − z∥

≤ γ^{−1}(ε) + sup_{s∈S} inf_{w∈S} ∥λT_{ws}x + (1 − λ)y − z∥.

Since ε > 0 is arbitrary, we obtain the desired inequality.

Suppose now S is semitopological and right reversible. Set

a = inf_{s∈S} sup_{t∈S} ∥λT_{ts}x + (1 − λ)y − z∥ and b = sup_{s∈S} inf_{t∈S} ∥λT_{ts}x + (1 − λ)y − z∥.

Let ε > 0. Then there exist s_{0}, s_{1} ∈ S such that

sup_{t∈S} ∥λT_{ts_{0}}x + (1 − λ)y − z∥ ≤ a + ε and inf_{t∈S} ∥λT_{ts_{1}}x + (1 − λ)y − z∥ ≥ b − ε.

Since \overline{Ss_{0}} ∩ \overline{Ss_{1}} is nonempty, there exists s_{2} ∈ \overline{Ss_{0}} ∩ \overline{Ss_{1}}. Let t ∈ S. ¿¿¿¿¿From s_{2} ∈ \overline{Ss_{0}}, there is a net {t_{α}} of S such that t_{α}s_{0} → s_{2}. Since ∥λT_{tt_{α}s_{0}}x + (1 − λ)y − z∥ ≤ a + ε, we have ∥λT_{ts_{2}}x + (1 − λ)y − z∥ ≤ a + ε. Similarly, we have ∥λT_{ts_{2}}x + (1 − λ)y − z∥ ≥ b − ε. Hence

a + ε ≥ sup_{t∈S} ∥λT_{ts_{2}}x + (1 − λ)y − z∥ ≥ inf_{t∈S} ∥λT_{ts_{2}}x + (1 − λ)y − z∥ ≥ b − ε.

¿¿¿¿¿From the arbitrariness of ε > 0, we have a ≥ b and so a = b by above. □

The following inequality will be useful for us:

Lemma 6.6. Assume that the norm of E is Fréchet differentiable. Then for each x ∈ C, y ∈ Q(x) ∩ F (S) and z ∈ F (S),

sup_{s∈S} inf_{t∈S}⟨T_{ts}x − y, J(y − z)⟩ ≥ 0.

Proof. Let x ∈ C, let y ∈ Q(x)∩F (S) and let z ∈ F (S). Put g(λ) = inf_{s∈S} sup_{t∈S} ∥λT_{ts}x+ (1 − λ)y − z∥ for 0 ≤ λ ≤ 1. Let 0 ≤ λ ≤ 1 and let ε > 0. Then there exists s_{0} ∈ S such that

∥λT_{ts_{0}}x + (1 − λ)y − z∥ ≤ g(λ) + ε for each t ∈ S.

¿¿¿¿¿From y ∈ \overline{co}\{T_{ts_{0}}x : t ∈ S\}, we have ∥y − z∥ ≤ g(λ) + ε. Hence we get ∥y − z∥ ≤ g(λ) for each λ with 0 ≤ λ ≤ 1. Set

δ(λ) = sup_{∥k∥=1}(\frac{1/2∥y − z + λk∥^{2} − 1/2∥y − z∥^{2}}{λ} − ⟨k, J(y − z)⟩)

for 0 < λ ≤ 1. Then δ is an increasing function and δ(λ) → 0 as λ → 0 from (6.1). Let M = sup_{t} ∥T_{t}x − y∥. By ∥y − z∥ ≤ g(λ) for 0 < λ ≤ 1 and Lemma 6.5, we have

0 ≤ \frac{1}{2} inf_{s∈S} sup_{t∈S} ∥λT_{ts}x + (1 − λ)y − z∥^{2} − \frac{1}{2}∥y − z∥^{2}

≤ \frac{1}{2} sup_{s∈S} inf_{t∈S} ∥λT_{ts}x + (1 − λ)y − z∥^{2} − \frac{1}{2}∥y − z∥^{2}

≤ sup_{s∈S} inf_{t∈S}(λ⟨T_{ts}x − y, J(y − z)⟩ + λ∥T_{ts}x − y∥δ(λ∥T_{ts}x − y∥))

≤ λ sup_{s∈S} inf_{t∈S}⟨T_{ts}x − y, J(y − z)⟩ + λMδ(λM)

for 0 < λ ≤ 1, and hence the assertion follows. □

Theorem 6.7. Assume that the norm of E is Fréchet differentiable. Then for each x ∈ C, F (S) ∩ Q(x) consists of at most one point.

Proof. Let y, z ∈ Q(x)∩F (S) and let ε > 0. ¿¿¿¿¿From Lemma 6.6, there exists s_{0} ∈ S such that ⟨T_{ts_{0}}x − y, J(y − z)⟩ > −ε for each t ∈ S. Since z ∈ \overline{co}\{T_{ts_{0}}x : t ∈ S\}, we have ⟨z − y, J(y − z)⟩ ≥ −ε, i.e., ∥y − z∥^{2} ≤ ε. ¿¿¿¿¿From the arbitrariness of ε > 0, we obtain y = z. This completes the proof. □

Notes and remarks

(i) Theorem 6.7 was originally proved by us in [59, Theorem 1] when S is right reversible (i.e. S has finite intersection property for closed left ideals). This was improved to arbitrary semitopological semigroup by Lau, Nishiura and Takahashi in [57, Theorem 3.11]. The proof given here is taken from [58]. A study of the relation between left ideal orbits and fixed point set was first done [57] where Theorem 6.3 and Theorem 6.4 were proved.

(ii) In view of theorem 6.3, it is natural to ask the following questions.

Problem 5. If E is a uniformly convex Banach space, x ∈ C and y ∈ Q(x) ∩ F (S), is y always the minimizer of q_{x} in F (S)?

Problem 6. If E is a uniformly convex Banach space, does Q(x) ∩ F (S) contain at most one point for each x ∈ C?

Clearly, by Theorem 6.3, an affirmative answer for problem 5 gives an affirmative answer to problem 6; Theorem 6.7 gives an affirmative answer for problem 6 when E has a Fréchet differentiable norm.

(iii) It should be noted that in general, ∩_{s∈S}\overline{co}\{T_{ts}x; t ∈ S\} cannot be replaced by ∩_{s∈S}\overline{co}\{T_{st}x; t ∈ S\} in Theorem 6.7. Indeed, let n > 1 and let S = \{a_{1}, …, a_{n}\} with multiplication defined by s · t = t for each t, s ∈ S. Let E = ℓ^{2}(S), i.e., the n-dimensional Hilbert space. For t ∈ S, define a mapping T_{t} from E into itself by (T_{t}f)(s) = f(st) for each f ∈ E and s ∈ S. Then S = {T_{t}; t ∈ S} is a nonexpansive semigroup on E such that each T_{t} is linear. Let f ∈ ℓ^{2}(S) such that f(a_{i}) = i for i = 1, …, n. In this case, ∩_{s∈S}\overline{co}\{T_{st}f; t ∈ S\} = \overline{co}\{T_{t}f; t ∈ S\} and \{T_{t}f; t ∈ S\} has n common fixed points.

7. Ergodic theorems

Let S be a semitopological semigroup, μ be an element in CB(S)^{∗}. Let E be a reflexive Banach space. Let f be a bounded and continuous function on S into E. Then the function ϕ on E^{∗} into the set ℝ of real number given by

ϕ(x^{∗}) = μ_{s}⟨f(s), x^{∗}⟩ for every x^{∗} ∈ E^{∗}

is linear and continuous. Hence there exists an element f(μ) in E such that

⟨f(μ), x^{∗}⟩ = μ_{s}⟨f(s), x^{∗}⟩ for every x^{∗} ∈ E^{∗}.

If S = {T_{s}; s ∈ S} be a continuous representation of S as nonexpansive mappings on a closed convex subset C of E, {T_{s}x; s ∈ S} is bounded for some x ∈ C and f(x) = T_{s}x for s ∈ S, we shall denote f(μ) by T_{μ}x.

Lemma 7.1. Let S be a semitopological semigroup having a right invatiant mean μ on CB(S), let E be a reflexive Banach space and let f be a bounded and continuous function from S to E. Then f(μ) ∈ ∩_{s∈S}\overline{co}\{f(ts); t ∈ S\}.

Proof. Let μ be a right invariant mean on CB(S). Then there exists a net {ϕ_{α}} of finite means such that

⟨h, ϕ_{α} − μ⟩ → 0 for every h ∈ CB(S);

see [19]. So putting ϕ_{α} = ∑_{i=1}^{n_{α}} λ_{i}^{α}δ(t_{i}^{α}) with ∑_{i=1}^{n_{α}} λ_{i}^{α} = 1 and λ_{i}^{α} ≥ 0 (i = 1, 2, …, n_{α}), we have, for each s ∈ S and y^{∗} ∈ E^{∗}.

⟨∑_{i=1}^{n_{α}} λ_{i}^{α}f(t_{i}^{α}s), y^{∗}⟩ = ∑_{i=1}^{n_{α}} λ_{i}^{α}⟨f(t_{i}^{α}s), y^{∗}⟩
= (ϕ_{α})_{t}⟨f(ts), y^{∗}⟩
→ μ_{t}⟨f(ts), y^{∗}⟩ = μ_{t}⟨f(t), y^{∗}⟩ = ⟨f(μ), y^{∗}⟩.

This implies that for each s ∈ S, ∑_{i=1}^{n_{α}} λ_{i}^{α}f(t_{i}^{α}s) converge weakly to f(μ) and hence f(μ) ∈ \overline{co}\{f(ts); t ∈ S\}. Therefore we have

f(μ) ∈ ∩_{s∈S}\overline{co}\{f(ts); t ∈ S\}. □

Lemma 7.2. Let S be a semitopological semigroup, let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E, and let S = {T_{s}; s ∈ S} be a nonexpansive semigroup on C. Let μ be a mean on CB(S). Then:

(a) T_{μ} is nonexpansive on C.

(b) If T_{μ}x ∈ F (S) for each x ∈ C, then T_{μ}^{2} = T_{μ}, i.e. T_{μ} is a retraction on C.

(c) If S is right reversible and μ is a right invariant mean, then

T_{μ}T_{s} = T_{μ} for each s ∈ S.

Proof. (a) Let J : E → E^{∗} be the duality map, and let x^{∗} ∈ J(T_{μ}x − T_{μ}y). Then

∥T_{μ}x − T_{μ}y∥^{2} = ⟨T_{μ}x − T_{μ}y, x^{∗}⟩ = μ_{t}⟨T_{t}x − T_{t}y, x^{∗}⟩
≤ sup_{t} ∥T_{t}x − T_{t}y∥∥T_{μ}x − T_{μ}y∥
≤ ∥x − y∥∥T_{μ}x − T_{μ}y∥,

so that T_{μ} is nonexpansive.

(b) From

⟨T_{μ}^{2}x, x^{∗}⟩ = μ_{t}⟨T_{t}T_{μ}x, x^{∗}⟩ = μ_{t}⟨T_{μ}x, x^{∗}⟩ = ⟨T_{μ}x, x^{∗}⟩,

we obtain T_{μ}^{2} = T_{μ}.

(c) Since μ is right invariant, we have, for any x^{∗} ∈ E^{∗} and s ∈ S,

⟨T_{μ}T_{s}x, x^{∗}⟩ = μ_{t}⟨T_{t}T_{s}x, x^{∗}⟩ = μ_{t}⟨T_{ts}x, x^{∗}⟩
= μ_{t}⟨T_{t}x, x^{∗}⟩ = ⟨T_{μ}x, x^{∗}⟩,

and hence T_{μ}T_{s} = T_{μ} for every s ∈ S. Also, if x ∈ C, then T_{μ}x ∈ ∩_{s∈S}\overline{co}\{T_{t}x; t ≥ s\} by Lemma 7.1. □

Theorem 7.3. Let C be a closed, convex subset of a uniformly convex Banach space E, let S be a semigroup, let S = {T_{t} : t ∈ S} be a nonexpansive semigroup on C with F (S) ≠ ∅, and let X be a subspace of ℓ^{∞}(S) such that 1 ∈ X, the mapping t ↦ ⟨T_{t}x, x^{∗}⟩ is an element of X for each x ∈ C and x^{∗} ∈ E^{∗} and X is l_{s}-and r_{s}-invariant for each s ∈ S. If X is amenable, then there exists a nonexpansive retraction P from C onto F (S) such that P T_{t} = T_{t}P = P for each t ∈ S and P x ∈ \overline{co}\{T_{t}x : t ∈ S\} for each x ∈ C.

Proof. Assume that X is amenable. Then there is an invariant mean μ on X. Let t ∈ S and let x ∈ C. Let z be an arbitrary element of F (S). Set D = {y ∈ C : ∥y−z∥ ≤ ∥x−z∥}. We remark that x ∈ D and T_{t}(D) ⊂ D. Let ε > 0. By [17, Theorem 1.2], there exists δ > 0 such that \overline{co}F_{δ}(T_{t}; D) ⊂ F_{ε}(T_{t}; D). By [17, Corollary 1.1], there also exists a natural number N such that ∥\frac{1}{N+1} ∑_{i=0}^{N}(T_{t})^{i}y − T_{t} (\frac{1}{N+1} ∑_{i=0}^{N}(T_{t})^{i}y)∥ ≤ δ for all y ∈ D. So we have

∥\frac{1}{N + 1} ∑_{i=0}^{N}(T_{t})^{i} (T_{s}x) − T_{t}(\frac{1}{N + 1} ∑_{i=0}^{N}(T_{t})^{i} (T_{s}x))∥ ≤ δ

for all s ∈ S. Since μ is left invariant, we get

T_{μ}x = ∫ T_{s}x dμ(s) = \frac{1}{N + 1} ∑_{i=0}^{N} ∫ T_{t^{i}s}x dμ(s) = ∫ \frac{1}{N + 1} ∑_{i=0}^{N}(T_{t})^{i} (T_{s}x) dμ(s)

∈ \overline{co}\{\frac{1}{N + 1} ∑_{i=0}^{N}(T_{t})^{i} (T_{s}x) : s ∈ S\} ⊂ \overline{co}F_{δ}(T_{t}; D) ⊂ F_{ε}(T_{t}; D),

where t^{0}s represents s. ¿¿¿¿¿From the arbitrariness of ε > 0, we have T_{t}T_{μ}x = T_{μ}x. Hence we obtain T_{t}T_{μ} = T_{μ} for each t ∈ S. The other assertions now follow from Lemma 7.2 with P = T_{μ}. □

Theorem 7.4. Let C, E, S, S and X be as in Theorem 1. Assume that the norm of E is Fréchet differentiable and X is amenable. Then there exists a unique nonexpansive retraction P from C onto F (S) such that P T_{t} = T_{t}P = P for each t ∈ S and P x ∈ \overline{co}\{T_{t}x : t ∈ S\} for each x ∈ C. Further, if \{μ_{α}\} is an asymptotically invariant net of means on X, then for each x ∈ C, {T_{μ_{α}}x} converges weakly to P x.

Proof. From Theorem 7.3, there exists a nonexpansive retraction P from C onto F (S) such that P T_{t} = T_{t}P = P for each t ∈ S and P x ∈ \overline{co}\{T_{t}x : t ∈ S\} for each x ∈ C. For each x ∈ C and s ∈ S, we have P x = P T_{s}x ∈ \overline{co}\{T_{ts}x : t ∈ S\}. So by Theorem 6.7, we get {P x} = F (S) ∩ Q(x) for each x ∈ C. Hence such a retraction P is unique. Let {μ_{α}} be an asymptotically invariant net of means on X. We shall show {T_{μ_{α}}x} converges weakly to P x for each x ∈ C. Let x ∈ C and let {μ_{α_{β}}} be a subnet of {μ_{α}} such that {μ_{α_{β}}} converges to an invariant mean μ on X with respect to the weak star topology on X^{∗}. Then for each x^{∗} ∈ E^{∗},

lim_{β}⟨T_{μ_{α_{β}}} x, x^{∗}⟩ = lim_{β}(μ_{α_{β}})_{t}⟨T_{t}x, x^{∗}⟩ = μ_{t}⟨T_{t}x, x^{∗}⟩
= ⟨T_{μ}x, x^{∗}⟩ = ⟨P x, x^{∗}⟩.

Hence {T_{μ_{α}}x} converges weakly to P x. □

In the case where S is commutative, we have the following:

Corollary 7.5. Let E, C, S, S = {T_{t} : t ∈ S} and X be as in Theorem 7.3. Assume S is commutative and the norm of E is Fréchet differentiable. Then the same conclusion in Theorem 7.4 holds. Moreover, if {μ_{α}} is a strongly regular net of means on X, then for each x ∈ C, {T_{μ_{α}}T_{t}x} converges weakly to P x uniformly in t ∈ S.

Proof. We remark that l_{s} = r_{s} for each s ∈ S. Let {μ_{α}} be a net of means on X such that lim_{α} ∥r_{s}^{∗}μ_{α} − μ_{α}∥ = 0 for each s ∈ S. We shall show that for each x ∈ C, {T_{μ_{α}}T_{t}x} converges weakly to P x uniformly in t ∈ S. Suppose not. Then there exist x ∈ C, x^{∗} ∈ E^{∗} and ε > 0 such that for each α, there exist β_{α} ≥ α and t_{α} ∈ S satisfying |⟨T_{μ_{βα}} T_{t_{α}}x − P x, x^{∗}⟩| ≥ ε. Since S is commutative, we have

lim_{α}|(r_{s}^{∗}r_{t_{α}}^{∗}μ_{β_{α}} − r_{t_{α}}^{∗}μ_{β_{α}})(f)| = lim_{α}|(r_{t_{α}}^{∗}r_{s}^{∗}μ_{β_{α}} − r_{t_{α}}^{∗}μ_{β_{α}})(f)| ≤ lim_{α} ∥r_{s}^{∗}μ_{β_{α}} − μ_{β_{α}}∥∥f∥ = 0

for each s ∈ S and f ∈ X. So {r_{t_{α}}^{∗}μ_{β_{α}}} is an asymptotically invariant net of means on X. By Theorem 7.4, {T_{μ_{βα}} T_{t_{α}}x} converges weakly to P x, which contradicts that |⟨T_{μ_{βα}} T_{t_{α}}x − P x, x^{∗}⟩| ≥ ε for all α. This completes the proof. □

The following three results are analogues of Day’s ergodic theorem [18] and [21] for an amenable semigroup of nonexpansive mappings on a uniformly convex Banach space.

Theorem 7.6. Let C, E, S, S and X be as in Theorem 7.3. Let {μ_{α}} be a strongly regular net of means on X. Then for each t ∈ S and for each bounded subset B of C, lim_{α} ∥T_{μ_{α}}T_{t}x − T_{μ_{α}}x∥ = 0 and lim_{α} ∥T_{t}T_{μ_{α}}x − T_{μ_{α}}x∥ = 0 uniformly for x ∈ B.

Proof Let t ∈ S and let B be a bounded subset of C. Let ε > 0 and set M = sup{∥T_{s}x∥; s ∈ S, x ∈ B}. Since {μ_{α}} is strongly regular, there exists α_{0} such that ∥r_{t}^{∗}μ_{α} − μ_{α}∥ ≤ ε/M for α ≥ α_{0}. Then for α ≥ α_{0}, we have

sup_{x∈B} ∥T_{μ_{α}}T_{t}x − T_{μ_{α}}x∥ = sup_{x∈B} sup_{∥x^{∗}∥=1}|(μ_{α})_{s}⟨T_{s}T_{t}x, x^{∗}⟩ − (μ_{α})_{s}⟨T_{s}x, x^{∗}⟩|
≤ ∥r_{t}^{∗}μ_{α} − μ_{α}∥M ≤ ε.

Hence we obtain lim_{α} ∥T_{μ_{α}}T_{t}x − T_{μ_{α}}x∥ = 0 uniformly for x ∈ B. It has been established in [84, Lemma 1] that lim_{α} ∥T_{t}T_{μ_{α}}x − T_{μ_{α}}x∥ = 0 uniformly for x ∈ B. For the sake of completeness, we give here a sketch of its proof. Let ε > 0 and let M be as above. ¿¿¿¿¿From F (S) ≠ ∅, there exists a bounded, convex subset D of C satisfying {T_{s}x : s ∈ S} ⊂ D. We can choose δ > 0 and a natural number N such that

(\overline{co}F_{δ}(T_{t}; D) + \{y ∈ E : ∥y∥ ≤ δ\}) ∩ C ⊂ F_{ε}(T_{t}; C)  (7.1)

and

∥\frac{1}{N + 1} ∑_{i=0}^{N} T_{t^{i}s}x − T_{t}(\frac{1}{N + 1} ∑_{i=0}^{N} T_{t^{i}s}x)∥ ≤ δ for all x ∈ B and s ∈ S.  (7.2)

Since {μ_{α}} is strongly regular, there exists α_{1} such that ∥μ_{α} − l_{t^{i}}^{∗}μ_{α}∥ ≤ δ/M for α ≥ α_{1} and i = 1, …, N. Then we have

sup_{x∈B}∥T_{μ_{α}} − ∫ \frac{1}{N + 1} ∑_{i=0}^{N} T_{t^{i}s}x dμ_{α}(s)∥

= sup_{x∈B} sup_{∥x^{∗}∥=1}|(μ_{α})_{s}⟨T_{s}x, x^{∗}⟩ − (μ_{α})_{s}⟨\frac{1}{N + 1} ∑_{i=0}^{N} T_{t^{i}s}x, x^{∗}⟩|

≤ \frac{1}{N + 1} ∑_{i=0}^{N} sup_{x∈B} sup_{∥x^{∗}∥=1}|(μ_{α})_{s}⟨T_{s}x, x^{∗}⟩ − (l_{t^{i}}^{∗}μ_{α})_{s}⟨T_{s}x, x^{∗}⟩|

≤ max_{i=1,... ,N} ∥μ_{α} − l_{t^{i}}^{∗}μ_{α}∥ · M ≤ δ

for α ≥ α_{1}. ¿¿¿¿¿From (7.1), (7.2) and the inequality above, we have sup_{x∈B} ∥T_{t}T_{μ_{α}}x − T_{μ_{α}}x∥ ≤ ε for α ≥ α_{1}. Hence we obtain lim_{α} ∥T_{t}T_{μ_{α}}x − T_{μ_{α}}x∥ = 0 uniformly for x ∈ B, and we complete the proof. □

As a direct consequence of Theorem 7.6, we have the following by using [19]:

Theorem 7.7. Let C, E, S and S be as in Theorem 1. Assume that ℓ^{∞}(S) is amenable. Then there exists a net {A_{α}} of finite averages of S such that for each t ∈ S and for each bounded subset B of C, lim_{α} ∥A_{α}T_{t}x − A_{α}x∥ = 0 and lim_{α} ∥T_{t}A_{α}x − A_{α}x∥ = 0 uniformly for x ∈ B.

Let ⟨F, G⟩ be a dual pair of vector spaces, let σ(F, G) be the weak topology on F generated by G.

Lemma 7.8. Let X be a translation invariant closed subspace of ℓ^{∞}(S) containing constants, and assume that X has an invariant mean. Then there exists a net {ϕ_{α}} of finite means such that (ℓ_{s}^{∗}ϕ_{α}−ϕ_{α})(f) → 0 for each s ∈ S uniformly on σ(X, ℓ^{1}(S))-compact absolutely convex subsets of X.

Proof. We follow ideas of Day [19] and Namioka [75]. Consider the dual pair ⟨ℓ^{1}(S), X⟩ and the Mackey topology τ on ℓ^{1}(S) with respect to this dual pair. By Mackey-Arens theorem, the continuous dual of (ℓ^{1}(S), τ) is also X. Let E = (ℓ^{1}(S), τ)^{S}. Define a map T : Φ → E given by

T (ϕ)(S) = r_{s}^{∗}ϕ − ϕ for every s ∈ S,

where Φ is the set of finite means. Then T (Φ) is a convex subset of E. Now since X has a left invariant mean, there exists a net {λ_{α}} ⊂ Φ such that (r_{s}^{∗}λ_{α} − λ_{α})(f) → 0 for each f ∈ X. Hence T (λ_{α}) → 0 in the weak topology of E, i.e., 0 ∈ the weak closure of T (Φ). Hence by Mazur theorem, 0 ∈ the strong closure of T (Φ), that is, there exists a net {ϕ_{α}} of finite means such that (r_{s}^{∗}ϕ_{α} − ϕ_{α})(f) → 0 uniformly on σ(X, ℓ^{1}(S))-compact absolutely convex subsets of X. □

Theorem 7.9. Let C and E be as in Theorem 7.3. Let S be a semitopological semigroup such that CB(S) is amenable and CB(S) separates each distinct two points of S, and let S = {T_{s}; s ∈ S} be a nonexpansive semigroup on C such that F (S) ≠ ∅ and t ↦ T_{t}x is continuous for each x ∈ C. Then there exists a net {A_{α}} of finite averages of S such that for each t ∈ S and x ∈ C, lim_{α} ∥A_{α}T_{t}x − A_{α}x∥ = 0 and lim_{α} ∥T_{t}A_{α}x − A_{α}x∥ = 0.

Proof. By Lemma 7.8, there exists a net {μ_{α}} of finite means on CB(S) such that for each s ∈ S, both ℓ_{s}^{∗}μ_{α} − μ_{α} and r_{s}^{∗}μ_{α} − μ_{α} converge to 0 uniformly on absolutely convex, σ(CB(S), ℓ^{1}(S))-compact subsets of CB(S). We know from [42, Lemma 3.2] that for each x ∈ C, {t ↦ ⟨T_{t}x, x^{∗}⟩; ∥x^{∗}∥ = 1} is contained in an absolutely convex, σ(CB(S), ℓ^{1}(S))- compact subset of CB(S). Hence by the similar argument in the proof of Theorem 7.6, we obtain the conclusion. □

Theorem 7.10. Let C, E, S and S be as in Theorem 7.3. Assume that the norm of E is Fréchet differentiable and for each x ∈ C, Q(x) ∩ F (S) is nonempty. Then there exists a nonexpansive retraction P from C onto F (S) such that P T_{t} = T_{t}P = P for each t ∈ S and P x ∈ \overline{co}\{T_{t}x; t ∈ S\} for each x ∈ C.

Proof. From Theorem 6.7 and the assumption, Q(x) ∩ F (S) consists of exactly one point for each x ∈ C. So we can define a retraction P from C onto F (S) such that P T_{t} = T_{t}P = P for each t ∈ S and P x ∈ \overline{co}\{T_{t}x : t ∈ S\} for each x ∈ C. We show that P is nonexpansive. Let ε > 0 and let x, y ∈ C. By Lemma 6.6, there exists s ∈ S such that

⟨T_{ts}x − P x, J(P x − P y)⟩ > −ε

for each t ∈ S. For such an element s, we have

sup_{u∈S} inf_{t∈S}⟨T_{tu}T_{s}y − P T_{s}y, J(P T_{s}y − P x)⟩ ≥ 0

from Lemma 6.6. So there also exists u ∈ S such that

⟨T_{tus}y − P T_{s}y, J(P T_{s}y − P x)⟩ > −ε

for each t ∈ S. Then from P T_{s}y = P y, we have

−2ε < ⟨T_{sus}x − P x, J(P x − P y)⟩ + ⟨T_{sus}y − P y, J(P y − P x)⟩
= ⟨T_{sus}x − T_{sus}y, J(P x − P y)⟩ − ∥P x − P y∥^{2}
≤ ∥x − y∥∥P x − P y∥ − ∥P x − P y∥^{2}.

Since ε > 0 is arbitrary, this implies ∥P x − P y∥ ≤ ∥x − y∥. □

Notes and remarks

(i)In 1975, Baillon [5] originally studied the nonlinear ergodic theorem in the framework of Hilbert spaces: If C is a closed, convex subset of a Hilbert space and T is a nonexpansive mapping from C into itself such that the set F (T) of fixed points of T is nonempty, then for each x ∈ C, the Cesàro mean

S_{n}(x) = \frac{1}{n} ∑_{k=1}^{n} T^{k}x

converges weakly to some y ∈ F (T). In this case, putting y = P x for each x ∈ C, P is a nonexpansive retraction from C onto F (T) such that P T = T P = P and P x ∈ \overline{co}\{T^{n}x : n = 1, 2, … \} for each x ∈ C. In [88], Takahashi proved the existence of such an retraction for an amenable semigroup of nonexpansive mappings on a Hilbert space: If S is an amenable semigroup, C is a closed, convex subset of a Hilbert space H and S = {T_{t} : t ∈ S} is a nonexpansive semigroup on C such that the set F (S) of common fixed points of S is nonempty, then there exists a nonexpansive retraction P from C onto F (S) such that P T_{t} = T_{t}P = P for each t ∈ S and P x ∈ \overline{co}\{T_{t}x : t ∈ S\} for each x ∈ C. Rodé [79] also found a sequence of means on a semigroup, generalizing the Cesàro means, and extended Baillon’s theorem as follows: If S, C, H and S are as above and {μ_{α}} is an asymptotically invariant net of means, then for each x ∈ C, {T_{μ_{α}}x} converges weakly to an element of F (S). Further, for each x ∈ C, the limit point of {T_{μ_{α}}x} is the same for all asymptotically invariant nets {μ_{α}} of means. ¿¿¿¿¿From their results, we know that for each x ∈ C, {T_{μ_{α}}x} converges weakly to P x for all asymptotically invariant nets {μ_{α}} of means; see [93]. These results were extended to a uniformly convex Banach space whose norm is Fréchet differentiable in the case when S is commutative by Hirano, Kido and Takahashi [37]. However, it has been an open problem whether Takahashi’s result and Rodé’s result can be fully extended to such a Banach space for an amenable semigroup; see [94]. On the other hand, Day [18] proved the following ergodic theorem for an amenable semigroup of bounded linear operators on a Banach space: If S is an amenable semigroup and S = {T_{t} : t ∈ S} is a bounded representation of S as bounded linear operators on a Banach space E, then there exists a net {A_{α}} of finite averages of S such that lim_{α} ∥A_{α}(T_{t} − I)∥ = 0 and lim_{α} ∥(T_{t} − I)A_{α}∥ = 0 for each t ∈ S. In this case, there is also a projection P from E onto F (S) such that P T_{t} = T_{t}P = P for each t ∈ S and P x ∈ \overline{co}\{T_{t}x : t ∈ S\} for each x ∈ E; see also [21] and [45].

Results in this section are essentially taken from [58]. Theorems 7.3 and 7.4 gave an answer to the open problem mentioned above by extending Takahashi’s result and Rhodé’s result. The proofs of our ergodic theorems depend on a deep convex approximation property for nonexpansive mappings due to Bruck [17]. Theorem 7.7 was proved in [57, Theorem 4.2] when E is a Hilbert space, and [65, Theorem 2] where S is assumed to be right reversible.

(ii) Ergodic Sequences (see [52])

Let S be a semigroup and ℓ^{1}(S) denote the Banach space of all f : S → ℝ such that ∥f∥_{1} = ∑ |f(x)| < ∞. Let (ℓ^{1}(S))_{1}^{+} = all θ ∈ ℓ^{1}(S) such that θ ≥ 0 and ∥θ∥_{1} = 1 (countable means).

Let H be a Hilbert space, C be a closed convex subset of H, and S = {T_{s}; s ∈ S} be a representation of S as nonexpansive mappings from C into C such F (S) ≠ ϕ.

Let x ∈ C. For each y ∈ H, consider the bounded real-valued function on S : s ↦ ⟨T_{s}x, y⟩. Let θ be a mean on ℓ^{∞}(S), define

⟨T_{θ}(x), y⟩ = θ_{s}(⟨T_{s}x, y⟩)
= ∑{⟨T_{s}x, y⟩θ(s); s ∈ S} if θ ∈ ℓ^{1}(S)_{1}^{+}.

Then T_{θ} is a nonexpansive mapping of C into C (see [79]).

Call a sequence (net) {θ_{n}} of means on S and ergodic sequence (net) for nonexpansive mappings if for any representation S = {T_{s}; s ∈ S} of S as nonexpansive mappings on a closed convex subset C of a Hilbert space into C such that F (S) ≠ ϕ, then for each x ∈ C, the sequence (net) T_{θ_{n}}(x) converges weakly to a fixed point of S.

A net of means {μ_{α}} on ℓ^{∞}(S) is called “asymptotically invariant” if

lim_{α}(μ_{α}(ℓ_{s}f) − μ_{α}(f)) = 0 and
lim_{α}(μ_{α}(r_{s}f) − μ_{α}(f)) = 0 for all s ∈ S.

Theorem 7.11 ([79]). Let S be an amenable semigroup. Then any “asymptotically invariant net” of means is an ergodic net for nonexpansive mappings.

Notes:

1. Every invariant mean on ℓ^{∞}(S) is asymptotically invariant.

2. If m is an invariant mean on ℓ^{∞}(S), then there is a net θ_{α} ∈ (ℓ^{1}(S))_{1}^{+} such that θ_{α} has finite support, i.e., θ_{α} = ∑_{i=1}^{n} λ_{i}δ_{s_{i}} (convex combination) such that θ_{α} ⟶^{w^{∗}} m. In particular the net {θ_{α}} is asymptotically invariant. Hence {θ_{α}} is an ergodic net of finite means on S for nonexpansive mappings.

Example([79]): If S = ({0, 1, 2, … }, +) and

θ_{n} = \frac{1}{n} ∑_{k=0}^{n−1} δ_{k},

then {θ_{n}} is an asymptotically invariant sequence of finite means on S. Consequently, {θ_{n}} is an ergodic sequence of finite means on S for nonexpansive mappings.

Problem 7. Given an amenable semigroup S, when does there exist a ergodic sequence of countable (or finite) means on S for nonexpansive mappings?

Problem 8. When can the net {A_{α}} of finite average of S in Theorem 7.9 be chosen to be a sequence dependent on the semigroup S?

8. Related results

Let C be a nonempty closed convex subset of a Banach space E. Let S = {S(t) : t ≥ 0} be a family of nonexpansive mappings of C into itself such that S(0) = I, S(t + s) = S(t)S(s) for all t, s ∈ [0, ∞) and S(t)x is continuous in t ∈ [0, ∞) for each x ∈ C. Then S is said to be a one-parameter nonexpansive semigroup on C. Miyadera and Kobayasi[73] introduced the notion of an almost-orbit of a one-parameter nonexpansive semigroup on C. A continuous function u : [0, ∞) → C is called an almost-orbit of S = {S(t) : t ≥ 0} if

lim_{s→∞}(sup_{t≥0} ∥u(t + s) − S(t)u(s)∥) = 0.  (8.1)

For example, consider the initial value problem:

\frac{du(t)}{dt} + Au(t) ∋ f(t), t > 0,  (8.2)

u(0) = x,

where A is an m-accretive operator in E, f ∈ L^{1}(0, ∞; E) and x ∈ \overline{D(A)}. Then it is well known that (8.2) has a unique integral solution [7] and the integral solution u(t) of (8.2) is an almost-orbit a one-parameter nonexpansive semigroup on \overline{D(A)} generated by −A. In fact, let A ⊂ E × E be an m-accretive operator and let u(t) be an integral solution of (8.2). Then we know that

∥u(t) − u∥^{2} − ∥u(s) − u∥^{2} ≤ 2 ∫_{s}^{t}⟨f(τ) − v, u(τ) − u⟩_{s}dτ

holds for each (u, v) ∈ A and 0 ≤ s ≤ t < +∞, where for x and y in E,

⟨y, x⟩_{s} = max{⟨y, j⟩ : j ∈ J(x)}.

We also know that if v(t) is another integral solution of (8.2) corresponding to g ∈ L^{1}(0, ∞; E) and y ∈ \overline{D(A)}, then

∥u(t) − v(t)∥ − ∥u(s) − v(s)∥ ≤ ∫_{s}^{t} ∥f(τ) − g(τ)∥dτ,  (8.3)

whenever 0 ≤ s ≤ t < +∞. If A^{−1}0 = {x : x ∈ D(A), Ax ∋ 0} is nonempty, we can take z ∈ A^{−1}0 and g(t) = 0 in (8.3) to obtain

∥u(t) − z∥ − ∥u(s) − z∥ ≤ ∫_{s}^{t} |f(τ)|dτ.  (8.4)

Consequently, u(t) is bounded on [0, ∞) and the function t ↦ ∥u(t) − z∥ − ∫_{0}^{t} ∥f(τ)∥dτ is nonincreasing on [0, ∞). Then, since f ∈ L^{1}(0, ∞; E), we deduce that

lim_{t→∞} ∥u(t) − z∥ = ρ(z)

exists for each z ∈ A^{−1}0. Finally, let {S(t) : t ≥ 0} be a one-parameter nonexpansive semigroup generated by −A and let u(t) be the integral solution of (8.2). Then we obtain that

∥S(t)u(s) − u(t + s)∥ ≤ ∫_{s}^{t+s} ∥f(τ)∥dτ,  (8.5)

whenever 0 ≤ s, t < +∞. In fact, the function t ↦ v(t) = u(t + s) is an integral solution of (8.2) corresponding to g(t) = f(t + s) and u(s) ∈ \overline{D(A)}. ¿¿¿¿¿From (8.5), we obtain that

lim_{s→∞} sup_{t≥0} ∥S(t)u(s) − u(t + s)∥ = 0.

This implies that the integral solution of (8.2) is an almost-orbit of the one-parameter nonexpansive semigroup generated by −A. Using a fixed point theorem for a commutative family of nonexpansive mappings, we can prove the following lemma.

Lemma 8.1. Let C be a closed convex subset of a real reflexive Banach space which has normal structure and let {u(t) : t ≥ 0} be an almost-orbit of S = {S(t) : t ≥ 0}. Then F (S) ≠ ϕ if and only if {u(t) : t ≥ 0} is bounded.

Proof. Suppose that {u(t) : t ≥ 0} is bounded. Then, since {u(t) : t ≥ 0} is an almost-orbit of S = {S(t) : t ≥ 0}, there exists t_{1} such that {S(t)u(t_{1}) : t ≥ 0} is bounded. Therefore by [90], F (S) is nonempty. Conversely, if F (S) ≠ ϕ, then since the limit of ∥u(t) − z∥ exists for each z ∈ F (S), it follows that {u(t) : t ≥ 0} is bounded. □

As in the proof of Theorem 6.7, we can prove the following theorem. See [73, 100] for details.

Theorem 8.2. Let E be a uniformly convex Banach space E with a Fréchet differentiable norm and let {u(t) : t ≥ 0} be an almost-orbit of S = {S(t) : t ≥ 0}. Suppose F (S) ≠ ϕ. Then the set

⋂_{s≥0} \overline{co}\{u(t + s) : t ≥ 0\} ⋂ F (S)

consists of at most one point.

Using Theorems 7.4 and 8.2, we also have the following theorems.

Theorem 8.3. Let E be a uniformly convex Banach space E with a Fréchet differentiable norm and let {u(t) : t ≥ 0} be an almost-orbit of S = {S(t) : t ≥ 0}. Suppose F (S) ≠ ϕ. Then, there exists an element y ∈ F (S) such that

w − lim_{s→∞} \frac{1}{s} ∫_{0}^{s} u(h + t)dt = y uniformly in h ≥ 0.

Proof. Let S = [0, ∞), S = {S(t) : t ≥ 0} and let X be the Banach space CB(S) of all bounded continuous functions on S with the supremum norm. Define λ_{s}(f) = \frac{1}{s} ∫_{0}^{s} f(t)dt for every s > 0 and f ∈ X. Then, we obtain

∥λ_{s} − r_{k}^{∗}λ_{s}∥ = sup_{∥f∥≤1} ∥\frac{1}{s} ∫_{0}^{s} f(t)dt − \frac{1}{s} ∫_{0}^{s} f(t + k)dt∥

= \frac{1}{s} sup_{∥f∥≤1} ∥∫_{0}^{s} f(t)dt − ∫_{k}^{s+k} f(t)dt∥ = \frac{1}{s} sup_{∥f∥≤1} ∥∫_{0}^{k} f(t)dt − ∫_{s}^{s+k} f(t)dt∥

≤ \frac{1}{s} sup_{∥f∥≤1} (∫_{0}^{k} ∥f(t)∥dt + ∫_{s}^{s+k} ∥f(t)∥dt) = \frac{2k}{s} → 0,

as s → ∞. Therefore, using Theorem 7.4, we obtain Theorem 8.3. □

Theorem 8.4. Let E, C, S and u be as in Theorem 8.3. Then,

r ∫_{0}^{∞} e^{−rt}u(t + k)dt converges weakly to some y ∈ F (S), as r → 0,

uniformly in k ≥ 0.

Proof. Let S = [0, ∞), S = {S(t) : t ≥ 0} and X = CB(S). Define λ_{r}(f) = r ∫_{0}^{∞} e^{−rt}f(t)dt for each r > 0 and f ∈ X. Then, for each s ∈ [0, ∞), we have

∥λ_{r} − r_{s}^{∗}λ_{r}∥ = sup_{∥f∥≤1} |r ∫_{0}^{∞} e^{−rt}f(t)dt − r ∫_{0}^{∞} e^{−rt}f(s + t)dt|

= sup_{∥f∥≤1} |r ∫_{0}^{s} e^{−rt}f(t)dt + r(1 − e^{rs}) ∫_{s}^{∞} e^{−rt}f(t)dt|
≤ rs + |1 − e^{rs}| → 0,

as r → 0. Therefore, using Theorem 7.4, we obtain Theorem 8.4. □

Using some ideas in the nonlinear ergodic theory, we can also study the iteration procedure of Mann’s type [69] for a family of nonexpansive mappings in a Banach space. The following proposition is crucial to prove our result; see [78, 99].

Proposition 8.5. Let C be a nonempty, closed, convex subset of a uniformly convex Banach space E whose norm is Fréchet differentiable and let {T_{i}; i ∈ ℕ} be a sequence of nonexpansive mappings from C into itself such that ⋂_{n=0}^{∞} F (T_{n}) is not empty. Let S_{n}y = T_{n}T_{n−1} ⋯ T_{0}y for each y ∈ C and n ∈ ℕ. Then for each x ∈ C, the set ⋂_{n=0}^{∞} \overline{co}\{S_{m}x; m ≥ n} ∩ ⋂_{n=0}^{∞} F (T_{n}) consists of at most one point.

Let C be a nonempty closed, convex subset of a Banach space E. A family S = {T_{t}; t ∈ S} is said to be a nonexpansive semigroup on C if

(i) for each t ∈ S, T_{t} is a nonexpansive mapping from C into itself; (ii) T_{ts} = T_{t}T_{s} for each t, s ∈ S.

Theorem 8.6. Let C be a nonempty, closed, convex subset of a Banach space E. Let S be a semigroup and let S = {T_{t}; t ∈ S} be a nonexpansive semigroup on C such that F (S) is not empty. Let X be a subspace of ℓ^{∞}(S) such that 1 ∈ X, it is l_{s}-invariant for each s ∈ S, and the function t ↦ ⟨T_{t}x, x^{∗}⟩ is an element of X for each x ∈ C and x^{∗} ∈ E^{∗}. Let {μ_{n}} be a sequence of means on X such that ∥μ_{n} − ℓ_{s}^{∗}μ_{n}∥ → 0 for every s ∈ S and let {α_{n}} be a real sequence such that 0 ≤ α_{n} ≤ 1 for each n ∈ ℕ and ∑_{n=0}^{∞}(1 − α_{n}) = ∞. Let x ∈ C and let {x_{n}} be the sequence defined by

x_{0} = x, x_{n+1} = α_{n}x_{n} + (1 − α_{n})T_{μ_{n}}x_{n} for each n ∈ ℕ.  (8.6)

If E is a uniformly convex Banach space whose norm is Fréchet differentiable, then {x_{n}} converges weakly to an element of F (S).

For proving Theorem 8.6, we need the following lemmas.

Lemma 8.7. Let D be a bounded closed convex subset of C such that D is T_{t}-invariant for each t ∈ S. Then

lim_{n→∞} sup_{y∈D} ∥T_{μ_{n}}y − T_{t}T_{μ_{n}}y∥ = 0 for each t ∈ S.

Proof. Let t ∈ S and ε > 0. Set M = sup{∥T_{s}y∥; s ∈ S, y ∈ D}. As in the proof of Theorem 7.6, we can choose δ > 0 and N ∈ ℕ such that

(\overline{co}(F_{δ}(T_{t}) ∩ D) + \{y ∈ E; ∥y∥ ≤ δ\}) ∩ C ⊂ F_{ε}(T_{t}) for all t ∈ S  (8.7)

and

∥\frac{1}{N + 1} ∑_{i=0}^{N} T_{t^{i}s}y − −T_{t}(\frac{1}{N + 1} ∑_{i=0}^{N} T_{t^{i}s}y)∥ ≤ δ for all y ∈ D and s ∈ S.  (8.8)

Since {μ_{n}} is strongly left regular, there exists n_{0} ∈ ℕ such that ∥μ_{n} − l_{t^{i}}^{∗}μ_{n}∥ ≤ δ/M for n ≥ n_{0} and i = 1, …, N. Then we have

sup_{y∈D}∥T_{μ_{n}}y − ∫ \frac{1}{N + 1} ∑_{i=0}^{N} T_{t^{i}s}y dμ_{n}(s)∥

= sup_{y∈D} sup_{∥y^{∗}∥=1}|(μ_{n})_{s}⟨T_{s}y, y^{∗}⟩ − −(μ_{n})_{s}⟨\frac{1}{N + 1} ∑_{i=0}^{N} T_{t^{i}s}y, y^{∗}⟩|

≤ \frac{1}{N + 1} ∑_{i=0}^{N} sup_{y∈D} sup_{∥y^{∗}∥=1}|(μ_{n})_{s}⟨T_{s}y, y^{∗}⟩ − −(l_{t^{i}}^{∗}μ_{n})_{s}⟨T_{s}y, y^{∗}⟩|

≤ max_{i=1,... ,N} ∥μ_{n} − l_{t^{i}}^{∗}μ_{n}∥ · M ≤ δ

for n ≥ n_{0}. ¿¿¿¿¿From (8.7), (8.8) and the inequality above, we have \overline{lim}_{n} sup_{y∈D} ∥T_{t}T_{μ_{n}}y− T_{μ_{n}}y∥ ≤ ε. Since ε > 0 is arbitrary, we obtain the conclusion. □

Lemma 8.8. Let {α_{n}} be a real sequence such that 0 ≤ α_{n} ≤ 1 for each n ∈ ℕ and ∑_{n=0}^{∞}(1 − α_{n}) = ∞. Let x be an element of C and let {x_{n}} be the sequence defined by (8.6). Then

lim_{n→∞} ∥x_{n} − T_{t}x_{n}∥ = 0 for each t ∈ S.

Proof. Fix ε > 0 and t ∈ S. Set r = max{sup_{n} ∥x_{n}∥, sup_{n} sup_{t∈S} ∥T_{t}x_{n}∥}. Then there exists δ > 0 which satisfies (8.7). ¿¿¿¿¿From Lemma 8.7, there exists n ∈ ℕ such that ∥T_{μ_{n+k}}y − T_{t}T_{μ_{n+k}}y∥ ≤ δ for each y ∈ D and k ∈ ℕ. ¿¿¿¿¿From ∑_{k=0}^{∞}(1 − α_{k}) = ∞, there exists m ∈ ℕ such that α_{n}α_{n+1} ⋯ α_{n+k−1} < δ/(2r) for all k > m. ¿¿¿¿¿From (3.1), it follows that for each k ≥ 1, there exists y_{k} ∈ co{T_{μ_{n}}x_{n}, ⋯, T_{μ_{n+k−1}}x_{n+k−1}} such that

x_{n+k} = α_{n+k−1} ⋯ α_{n}x_{n} + (1 − α_{n+k−1} ⋯ α_{n})y_{k}.

So for each k > m, T_{μ_{i}}x_{i} ∈ F_{δ}(T_{t}) for i = n, …, n + k − 1, and

∥x_{n+k} − y_{k}∥ ≤ α_{n+k−1} ⋯ α_{n} · ∥x_{n} − y_{k}∥ < \frac{δ}{2r} · 2r ≤ δ.

Since y_{k} ∈ co(F_{δ}(T_{t}) ∩ D)g and x_{n+k} − y_{k} ∈ {y ∈ E; ∥y∥ ≤ δ}, (8.7) yields ∥x_{n+k} − T_{t}x_{n+k}∥ ≤ ε for each k ≥ m. Hence we get \overline{lim}_{n} ∥x_{n} − T_{t}x_{n}∥ ≤ ε. Since ε > 0 is arbitrary, we obtain the conclusion. □

Proof of Theorem 8.6 Let w ∈ F (S). Since

∥x_{n+1} − w∥ ≤ α_{n}∥x_{n} − w∥ + (1 − α_{n})∥T_{μ_{n}}x_{n} − w∥ ≤ ∥x_{n} − w∥,

lim_{n} ∥x_{n} − w∥ exists. Assume that the norm of E is Fréchet differentiable. Set

T_{n}y = α_{n}y + (1 − α_{n})T_{μ_{n}}y and S_{n}y = T_{n}T_{n−1}T_{n−2} ⋯ T_{0}y

for each y ∈ D and n ∈ ℕ. ¿¿¿¿¿From Lemma 8.8 and Browder [14] , the set W of weak limit points of \{x_{n}\} is contained in ⋂_{n=0}^{∞} \overline{co}\{x_{m}; m ≥ n\} ∩ F (S). Since x_{n+1} = S_{n}x and F (S) ⊂ ⋂_{n=0}^{∞} F (T_{n}), the set W is contained in ⋂_{n=0}^{∞} \overline{co}\{S_{m}x; m ≥ n\} ∩ ⋂_{n=0}^{∞} F (T_{n}). Hence, from Proposition 8.5, {x_{n}} converges weakly to an element of F (S). □

Using Theorem 8.6, we prove some results.

Theorem 8.9. Let T be a nonexpansive mapping from C into itself such that F (T) is not empty. Let {α_{n}} be a real sequence such that 0 ≤ α_{n} ≤ 1 for each n ∈ ℕ and ∑_{n=0}^{∞}(1 − α_{n}) = ∞. Let x ∈ C and let {x_{n}} be the sequence defined by

x_{0} = x, x_{n+1} = α_{n}x_{n} + (1 − α_{n})\frac{1}{n + 1} ∑_{i=0}^{n} T^{i}x_{n} for each n ∈ ℕ.

If E is a uniformly convex Banach space whose norm is Fréchet differentiable, then {x_{n}} converges weakly to a fixed point of T.

Theorem 8.10. Let S = {T (t); t ∈ [0, ∞)} be a one-parameter nonexpansive semigroup on C such that F (S) is not empty and the mapping t ↦ ⟨T (t)u, u^{∗}⟩ is measurable for each u ∈ C and u^{∗} ∈ E^{∗}. Let {α_{n}} be a real sequence such that 0 ≤ α_{n} ≤ 1 for each n ∈ ℕ and ∑_{n=0}^{∞}(1 − α_{n}) = ∞ and let {s_{n}} be a positive real sequence with s_{n} → ∞. Let x be an element of C and let {x_{n}} be the sequence defined by

x_{0} = x, x_{n+1} = α_{n}x_{n} + (1 − α_{n}) \frac{1}{s_{n}} ∫_{0}^{s_{n}} T (t)x_{n} dt for each n ∈ ℕ.

If E is a uniformly convex Banach space whose norm is Fréchet differentiable, then {x_{n}} converges weakly to a common fixed point of S.

Proof Let X be the space of all bounded measurable functions from [0, ∞) into the set of real numbers. We remark that an element f in X is not an equivalence class with the usual equivalence relation, where the usual equivalence relation g ∼ h means the Lebesgue measure of the set {t ∈ [0, ∞); g(t) ≠ h(t)} is zero. The reason is that we consider that X is a subspace of B([0, ∞)) with the supremum norm. For each n ∈ ℕ, define a mean μ_{n} on X by μ_{n}(f) = \frac{1}{s_{n}} ∫_{0}^{s_{n}} f(t) dt for f ∈ X. Wee see that \{μ_{n}\} is strongly regular and T_{μ_{n}}y = \frac{1}{s_{n}} ∫_{0}^{s_{n}} T (t)y dt for y ∈ C and n ∈ ℕ; see [37]. Hence by Theorem 8.6, we obtain the conclusion. □

Notes and Remark

(i) The notion of almost-orbit was introduced by Miyadera and Kobayasi [73] and Theorem 8.2 was proved by them. A similar result for commutative semigroups of nonexpansive mappings was also proved by Takahashi and Park [100].

(ii) Mann [69] introduced an iteration procedure for approximating fixed points of a mapping T as follows:

x_{0} = x ∈ C, x_{n+1} = α_{n}x_{n} + (1 − α_{n})T x_{n} for each n ∈ ℕ,

where {α_{n}} is a sequence in [0, 1]. Later, Shimizu and Takahashi [82] considered iterative method for approximation of common fixed points; see also [81]. Motivated by Shimizu and Takahashi [82], Atsushiba and Takahashi [3] studied the iteration procedures of Mann’s type for a family of nonexpansive mappings in Hilbert space. Theorem 8.6 was proved by Atsushiba, Shioji Takahashi [2]. (iii) Shioji and Takahashi [85] studied the iterative procedure of Halpern’s type [33] and proved the following theorem.

Theorem 8.11. Let E be a uniformly convex Banach space E with a uniformly Gâteaux differentiable norm. Let C be a nonempty closed convex subset of E and let S = {T_{t}; t ∈ S} be a nonexpansive semigroup on C such that F (S) ≠ ϕ. Let X be a subspace of ℓ^{∞}(S) such that 1 ∈ X, X is ℓ_{s}-invariant for each s ∈ S and t ↦ ⟨T_{t}x, x^{∗}⟩ is an element of X for each x ∈ C and x^{∗} ∈ E^{∗}. Let {μ_{n}} be a sequence of means on X such that ∥μ_{n} −ℓ_{s}^{∗}μ_{n}∥ = 0 for every s ∈ S. Suppose x, y_{1} ∈ C and {y_{n}} is given by

y_{n+1} = β_{n}x + (1 − β_{n})T_{μ_{n}}y_{n} for every n ≥ 1,

where {β_{n}} is a sequence in [0, 1]. If {β_{n}} is chosen so that lim_{n→∞} β_{n} = 0 and ∑_{n=1}^{∞} β_{n} = ∞, then {y_{n}} converges strongly to the element of F (S) which is nearest to x in F (S).

Using Theorem 8.11, we can prove the following theorem.

Theorem 8.12. Let E be a uniformly convex Banach space E with a uniformly Gâteaux differentiable norm. Let C be a nonempty closed convex subset of E and let S = {S(t); t ≥ 0} be a one-parameter nonexpansive semigroup on C such that F (S) ≠ ϕ. Suppose x, y_{1} ∈ C and {y_{n}} is given by

y_{n+1} = β_{n}x + (1 − β_{n})\frac{1}{λ_{n}} ∫_{0}^{λ_{n}} S(t)y_{n}dt for every n ≥ 1,

where {β_{n}} is a sequence in [0, 1]. If {β_{n}} and {λ_{n}} are chosen so that lim_{n→∞} β_{n} = 0, ∑_{n=1}^{∞} β_{n} = ∞ and λ_{n} → ∞, then {y_{n}} converges strongly to the element of F (S) which is nearest to x in F (S).

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(Anthony To-Ming Lau) Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G-2G1

E-mail address, Anthony To-Ming Lau: tlau@math.ualberta.ca

(Wataru Takahashi) Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ohokayama, Meguro-ku, Tokyo 152-8552, Japan

E-mail address, Wataru Takahashi: wataru@is.titech.ac.jp