GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 5, 1995, 547-558

FIXED POINTS OF SEMIGROUPS OF LIPSCHITZIAN MAPPINGS DEFINED ON NONCONVEX DOMAINS

KOK-KEONG TAN AND HONG-KUN XU

Abstract. Certain fixed point theorems are established for nonlinear semigroups of Lipschitzian mappings defined on nonconvex domains in Hilbert and Banach spaces. Some known results are thus generalized.

1. Introduction

Let X be a Banach space and C be a nonempty subset of X. A mapping T : C → C is said to be a Lipschitzian mapping if, for each integer n ≥ 1, there exists a constant k_{n} ≥ 0 such that ∥T^{n}x−T^{n}y∥ ≤ k_{n}∥x−y∥ ∀x, y ∈ C. A Lipschitzian mapping T is said to be uniformly Lipschitzian if k_{n} = k for all n ≥ 1, nonexpansive if k_{n} = 1 for all n ≥ 1, and asymptotically nonexpansive if lim_{n→∞} k_{n} = 1, respectively. Goebel and Kirk [1] initiated in 1973 the study of the fixed point theory for Lipschitzian mappings. They showed that if X is uniformly convex and C is a bounded closed convex subset of X, then every uniformly k-Lipschitzian mapping T : C → C with k < γ has a fixed point, where γ > 1 is the unique solution of the equation

γ [1 − δ_{X} (\frac{1}{γ} )] = 1,

with δ_{X} the modulus of convexity of X. Since then, much effort has been devoted to the existence theory for fixed points of Lipschitzian mappings in both Hilbert and Banach spaces; see [2], [4], [5], [7], [10], [11], and refereces cited there. Usually the domain C on which T is defined is assumed to be convex. Recently, Ishihara [3] and Takahashi [9] studied in Hilbert spaces the existence theory for fixed points of Lipschitzian mappings which are defined on nonconvex domains. However, their methods do not work outside Hilbert spaces.

547

1072-947X/95/0900-0547$7.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

1991 Mathematics Subject Classification. 47H10, 47H09 .

Key words and phrases. Fixed point, semigroup, Lipschitzian mapping, nonconvex domain.

The purpose of the present paper is to investigate the existence theory for the fixed point theory of semigroups of Lipschitzian mappings defined on nonconvex domains in both Hilbert and Banach spaces.

2. Preliminaries

Let G be a semitopological semigroup, i.e., G is a semigroup with a Hausdorff topology such that for each a ∈ G, the mappings t → at and t → ta from G into itself are continuous. Let C be a nonempty subset of a Banach space X. Then a family F = {T_{t} : t ∈ G} of self-mappings of C is said to be a Lipschitzian semigroup on C if the following properties are satisfied:

(1) T_{ts}x = T_{t}T_{s}x for all t, s ∈ G and x ∈ C;

(2) for each x ∈ C, the mapping t → T_{t}x is continuous on G;

(3) for each t ∈ G, there is a constant k_{t} > 0 such that ∥T_{t}x − T_{t}y∥ ≤

k_{t}∥x − y∥ ∀x, y ∈ C. A Lipschitzian semigroup F is called uniformly k-Lipschitzian if k_{t} = k for all t ∈ G and in particular, nonexpansive if k_{t} = 1 for all t ∈ G. We shall denote by F (F) the set of common fixed points of F = {T_{t} : t ∈ G}.

Recall that a semitopological semigroup G is said to be left reversible if any two closed right ideals of G have nonvoid intersection. In this case, (G, ≤) is a directed system when the binary relation “≤” on G is defined by a ≤ b if and only if \{a\} ∪ \overline{aG} ⊇ \{b\} ∪ \overline{bG}. Let B(G) be the Banach space of all bounded real-valued functions on G with the supremum norm and let X be a subspace of B(G) containing constants. Then an element μ of X^{∗}, the dual space of X, is said to be a mean on X if ∥μ∥ = μ(1) = 1. It is known that μ ∈ X^{∗} is a mean on X if and only if the inequalty

inf{f(t) : t ∈ G} ≤ μ(f) ≤ sup{f(t) : t ∈ G}

holds for all f ∈ X. For a mean μ on X^{∗} and an element f ∈ X, we use either μ_{t}(f(t)) or μ(f) to denote the value of μ at f. For each s ∈ G, we define the left transformation ℓ_{s} from B(G) into itself by (ℓ_{s}f)(t) = f(st), t ∈ G, for all f ∈ B(G). The right transformation r_{s} is defined similarly. Let X be a subspace of B(G) containing constants which is ℓ_{G}-invariant (r_{G}- invariant), i.e., ℓ_{s}(X) ⊆ X (r_{s}(X) ⊆ X) for all s ∈ G. Then a mean μ on X is said to be left invariant (right invariant) if μ(f) = μ(ℓ_{s}f) (μ(f) = μ(r_{s}f)) for all f ∈ X and s ∈ G. An invariant mean is a mean that is both left and right invariant.

Let C(G) be the Banach space of all bounded continuous real-valued functions on G, let RUC(G) be the space of all bounded right uniformly continuous functions on G, i.e., all f ∈ C(G) for which the mapping s → r_{s}f is continuous, and let AP (G) be the space of all f ∈ C(G) for which {ℓ_{s}f : s ∈ G} is relatively norm compact. Then RUC(G) is a closed subalgebra of C(G) containing constants and both ℓ_{G}- and r_{G}-invariant; see [7] for details. If {x_{s} : s ∈ G} is a bounded family of elements of a Banach space E, then for all x ∈ E and p ≥ 1, the functions g(s) := ∥x_{s} − x∥^{p} and h(s) := ⟨x_{s}, x⟩ (if E is a Hilbert space) are in RUC(G).

3. The Hilbert Space Setting

In this section, we prove fixed point theorems for Lipschitzian semigroups defined on nonconvex domains in Hilbert spaces.

Theorem 3.1. Let C be a nonempty subset of a real Hilbert space H, let G be a semitopological semigroup such that RUC(G) has a left invariant mean μ, and let F = {T_{t} : t ∈ G} be a Lipschitzian semigroup on C such that μ_{t}(k_{t}^{2}) < 2. Suppose that \{T_{t}x : t ∈ G\} is bounded and ∩_{s∈G} \overline{co}\{T_{st}x : t ∈ G} is contained in C for all x ∈ C. Then there exists a point z ∈ C for which T_{t}z = z for all t ∈ G.

Proof. Let x_{0} ∈ C. It is easily seen that the functional μ_{t}⟨T_{t}x_{0}, x⟩, x ∈ H, is a continuous linear functional on H. By Riesz’s representation theorem, there is a unique element x_{1} ∈ H satisfying μ_{t}⟨T_{t}x_{0}, x⟩ = ⟨x_{1}, x⟩, ∀x ∈ H. By a routine argument (cf. [3] and [9]) via the separation theorem, we have x_{1} ∈ ∩_{s∈G}\overline{co}\{T_{st}x_{0} : t ∈ G\} and hence by assumption, x_{1} does remain in

C. Therefore, we can continue the above procedure to obtain a sequence {x_{n}}_{n=1}^{∞} in C satisfying the following property:

μ_{t}⟨T_{t}x_{n−1}, x⟩ = ⟨x_{n}, x⟩, ∀x ∈ H, ∀n ≥ 1.  (3.1)

Noting the fact that for all u, v ∈ H, the function h(t) := ∥T_{t}u − v∥^{2} is in RUC(G), it follows from (3.1) that

μ_{t}∥T_{t}x_{n−1} − x∥^{2} =μ_{t}∥(T_{t}x_{n−1} − x_{n}) + (x_{n} − x)∥^{2} =μ_{t}(∥T_{t}x_{n−1} − x_{n}∥^{2}+

+ ∥x_{n} − x∥^{2} + 2 ⟨T_{t}x_{n−1} − x_{n}, x_{n} − x⟩) = μ_{t}∥T_{t}x_{n−1} − x_{n}∥^{2} +

+ ∥x_{n} − x∥^{2} + 2 μ_{t}⟨T_{t}x_{n−1} − x_{n}, x_{n} − x⟩ = μ_{t}∥T_{t}x_{n−1} − x_{n}∥^{2} +

+ ∥x_{n} − x∥^{2} + 2 ⟨x_{n} − x_{n}, x_{n} − x⟩ = μ_{t}∥T_{t}x_{n−1} − x_{n}∥^{2} + ∥x_{n} − x∥^{2}.

This shows that x_{n} is the unique minimizer of the convex function μ_{t}∥T_{t}x_{n−1} − x∥^{2} over H and in particular, taking x = T_{s}x_{n} and noting that μ is left invariant, we get

μ_{t}∥T_{t}x_{n−1} − x_{n}∥^{2} + ∥x_{n} − T_{s}x_{n}∥^{2} = μ_{t}∥T_{t}x_{n−1} − T_{s}x_{n}∥^{2} =

= μ_{t}∥T_{st}x_{n−1} − T_{s}x_{n}∥^{2} = μ_{t}∥T_{s}T_{t}x_{n−1} − T_{s}x_{n}∥^{2} ≤ k_{s}^{2}μ_{t}∥T_{t}x_{n−1} − x_{n}∥^{2}.

It follows that

∥x_{n} − T_{s}x_{n}∥^{2} ≤ (k_{s}^{2} − 1)μ_{t}∥T_{t}x_{n−1} − x_{n}∥^{2}.  (3.2)

Set A = μ_{s}(k_{s}^{2} −1), r_{n} = μ_{t}∥T_{t}x_{n} −x_{n+1}∥^{2} and R_{n} = μ_{t}∥T_{t}x_{n} −x_{n}∥^{2}. Then from (3.2) we have

R_{n} ≤ Ar_{n−1} ≤ AR_{n−1} ≤ ⋯ ≤ A^{n}R_{0}.  (3.3)

It then follows from (3.3) that

∥x_{n+1} − x_{n}∥^{2} = μ_{t}∥(x_{n+1} − T_{t}x_{n}) + (T_{t}x_{n} − x_{n})∥^{2} ≤
≤ 2 μ_{t}(∥T_{t}x_{n} − x_{n+1}∥^{2} + ∥T_{t}x_{n} − x_{n}∥^{2}) =
= 2(r_{n} + R_{n}) ≤ 4 R_{n} ≤ 4 A^{n}R_{0}.

Since A < 1, we see that {x_{n}} is Cauchy and hence convergent in norm. Let z be the limit of {x_{n}}. We claim that z is a common fixed point of F. In fact, for any s ∈ G, since μ is left invariant, we have

∥T_{s}z − z∥^{2} = μ_{t}(∥(T_{s}z − T_{st}x_{n}) + (T_{st}x_{n} − z)∥^{2} ≤
≤ 2 μ_{t}(∥T_{st}x_{n} − T_{s}z∥^{2} + ∥T_{st}x_{n} − z∥^{2}) ≤
≤ 2(k_{s}^{2} μ_{t}∥T_{t}x_{n} − z∥^{2} + μ_{t}∥T_{st}x_{n} − z∥^{2}) =
= 2(1 + k_{s}^{2}) μ_{t}∥T_{t}x_{n} − z∥^{2} =
= 2(1 + k_{s}^{2}) μ_{t}∥(T_{t}x_{n} − x_{n}) + (x_{n} − z)∥^{2} ≤
≤ 4(1 + k_{s}^{2})(μ_{t}∥T_{t}x_{n} − x_{n}∥^{2} + ∥x_{n} − z∥^{2}) =
= 4(1 + k_{s})(R_{n} + ∥x_{n} − z∥^{2}) → 0 as n → ∞.

Therefore, T_{s}z = z.

Corollary 3.1 (Theorem 3.5 [4]). If F = {T_{t} : t ∈ G} is a nonexpansive semigroup on a closed convex subset C of a Hilbert space H, RUC(G) has a left invariant mean, and there exists an x ∈ C such that {T_{s}(x) : s ∈ G} is bounded, then F has a common fixed point in C.

Let X be a subspace of B(G) containing constants. Following Mizoguchi and Takahashi [7T], we say that a real valued function on X is a submean on X if the following conditions are fulfilled:

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

(2) μ(αf) ≤ α μ(f), ∀f ∈ X, ∀α ≥ 0;

(3) ∀ f, g ∈ X, f ≤ g ⟹ μ(f) ≤ μ(g);

(4) μ(c) = c for all constants c.

Theorem 3.2. Let H be a real Hilbert space, C a nonempty subset of H, X an ℓ_{G}-invariant subspace of B(G) containing constants that has a left invariant submean μ on X, and F = {T_{t} : t ∈ G} a Lipschitzian semigroup on C. Suppose that {T_{t}x : t ∈ G} is bounded for some x ∈ C and ∩_{s∈G} \overline{co}\{T_{st}x : t ∈ G\} ⊂ C for all x ∈ C. Suppose also that for all u, v ∈ C, the function f on G defined by f(t) = ∥T_{t}u − v∥^{2} and the function h on G defined by h(t) = k_{t}^{2} belong to X and μ_{t}(k_{t}^{2}) < 2. Then there is a point z ∈ C such that T_{t}z = z for all t ∈ G.

Proof. Let x_{0} ∈ C and define the function r_{0} on H by r_{0}(x) = μ_{t}∥T_{t}x_{0}−x∥^{2}, x ∈ H. Note that r_{0} is well defined since by assumption, the function t ↦ ∥T_{t}x_{0} − x∥^{2} is in X for every x ∈ H. As r_{0} is strictly convex and continuous and r_{0}(x) → ∞ as ∥x∥ → ∞, there is a unique element x_{1} ∈ H such that r_{0} := r_{0}(x_{1}) = inf{r_{0}(x) : x ∈ H}. We claim that this x_{1} belongs to ∩_{s∈G} \overline{co}\{T_{st}x_{0} : t ∈ G\} and thus to C by our hypothesis. Indeed, if we denote by P_{s} the nearest point projection of H onto the set \overline{co}\{T_{st}x_{0}; t ∈ G\}, then, as P_{s} is nonexpansive and μ is left invariant, we get

r_{0}(P_{s}x_{1}) = μ_{t}∥T_{t}x_{0} − P_{s}x_{1}∥^{2} = μ_{t}∥T_{st}x_{0} − P_{s}x_{1}∥^{2} =
= μ_{t}∥P_{s}T_{st}x_{0} − P_{s}x_{1}∥^{2} ≤ μ_{t}∥T_{st}x_{0} − x_{1}∥^{2} =
= μ_{t}∥T_{t}x_{0} − x_{1}∥^{2} = r_{0},

which shows that P_{s}x_{1} is also a minimizer of r_{0} and hence by uniqueness, P_{s}x_{1} = x_{1}, i.e., x_{1} ∈ ∩_{s∈G} \overline{co}\{T_{st}x_{0} : t ∈ G\}. This proves the claim. Repeating the above process, we obtain a sequence {x_{n}} in C with the following property:

x_{n} ∈ ⋂_{s∈G} \overline{co}\{T_{st}x_{n−1} : t ∈ G\} ∀n ≥ 1

and x_{n} is the unique minimizer over H of the functional r_{n}(·) defined by r_{n}(x) = μ_{t}∥T_{t}x_{n−1}−x∥^{2}, x ∈ H. Now by the same argument as in the proof of Theorem 3.1, we conclude that {x_{n}} converges strongly to a common fixed point z ∈ C.

Corollary 3.2 (Theorem 1 [8]). Let C be a closed convex subset of a Hilbert space H and X be an ℓ_{G}-invariant subspace of B(G) containing constants which has a left invariant submean μ. Let F = {T_{t} : t ∈ G} be a Lipschitzian semigroup on C such that {T_{s}x : s ∈ G} is bounded for some x ∈ C. If for each u, v ∈ C, the function f(t) := ∥T_{t}u−v∥^{2} and the function g(t) := k_{t}^{2} (t ∈ G) belong to X and μ_{s}(k_{s}^{2}) < 2, then there is z ∈ C such that T_{s}z = z for all s ∈ G.

We now extend Theorem 3.3 of Lau [4] to a wider class of Lipschitzian semigroups which are defined on nonconvex domains.

Theorem 3.3. Suppose H is a real Hilbert space, C is a nonempty subset of H, and F = {T_{t} : t ∈ G} is a Lipschitzian semigroup on C. Suppose also AP (G) has a left invariant mean μ. If μ_{t}(k_{t}^{2}) ≤ 1 and if there exists an x ∈ C such that \{T_{t}x : t ∈ G\} is relatively compact in norm and ∩_{s∈G} \overline{co}\{T_{st}x : t ∈ G} is contained in C, then F has a common fixed point.

Proof. Since {T_{t}x : t ∈ G} is relatively compact, by Lemma 3.1 of Lau [4], for all y ∈ H, the functions h and g defined on G by h(t) = ⟨y, T_{t}x⟩ and g(t) = ∥y − T_{t}x∥^{2} are both in AP (G). So we have a unique z ∈ H such that μ_{t}⟨T_{t}x, y⟩ = ⟨z, y⟩, ∀y ∈ H. As seen before, we have (i) z ∈ ∩_{s∈G} \overline{co}\{T_{st}x : t ∈ G} and hence z ∈ C and (ii) μ_{t}∥T_{t}x − y∥^{2} = μ_{t}∥T_{t}x − z∥^{2} + ∥y − z∥^{2} ∀y ∈ H. In particular, we have for all s ∈ G,

μ_{t}∥T_{t}x − T_{s}z∥^{2} = μ_{t}∥T_{t}x − z∥^{2} + ∥T_{s}z − z∥^{2}.

Noting that

μ_{t}∥T_{t}x − T_{s}z∥^{2} = μ_{t}∥T_{st}x − T_{s}z∥^{2} = μ_{t}∥T_{s}T_{t}x − T_{s}z∥^{2} ≤ k_{s}^{2}μ_{t}∥T_{t}x − z∥^{2},

we get for all s ∈ G, ∥T_{s}z − z∥^{2} ≤ (k_{s}^{2} − 1)μ_{t}∥T_{t}x − z∥^{2}. Hence

μ_{s}∥T_{s}z − z∥^{2} ≤ (μ_{s}(k_{s}^{2} − 1))μ_{t}∥T_{t}x − z∥^{2} ≤ 0

for μ_{s}(k_{s}^{2}) ≤ 1; namely, μ_{s}∥T_{s}z − z∥^{2} = 0. Now for a ∈ G, we have

∥T_{a}z − z∥^{2} = μ_{s}∥(T_{a}z − T_{s}z) + (T_{s}z − z)∥^{2} ≤ 2 μ_{s}(∥T_{a}z − T_{s}z∥^{2} +
+ ∥T_{s}z − z∥^{2}) = 2(μ_{s}∥T_{s}z − T_{a}z∥^{2} + μ_{s}∥T_{s}z − z∥^{2}) =
= 2 μ_{s}∥T_{as}z − T_{a}z∥^{2} = 2 μ∥T_{a}T_{s}z − T_{a}z∥^{2} ≤
≤ 2 k_{a}^{2}μ_{s}∥T_{s}z − z∥^{2} = 0.

Therefore T_{a}z = z.

Corollary 3.3 (Theorem 3.2 [4]). If C is a closed convex subset of a Hilbert space H, F = {T_{t} : t ∈ G} is a nonexpansive semigroup on C, AP (G) has a left invariant mean, and x ∈ C such that {T_{s}x : s ∈ G} is relatively compact, then C contains a common fixed point for F.

By the same proof as in Theorem 3.1, we get immediately the following result.

Theorem 3.4. Suppose H and C are as in Theorem 3.3, suppose AP (G) has a left invariant mean μ, and suppose F = {T_{t} : t ∈ G} is a Lipschitzian semigroup on C such that μ_{t}(k_{t}^{2}) < 2. If, for every x ∈ C, {T_{t}x : t ∈ G} is relatively compact in norm and ∩_{s∈G} \overline{co}\{T_{st}x : t ∈ G\} is contained in C, then F has a common fixed point.

4. The Banach Space Setting

In this section we study the existence of fixed points for Lipschitzian mappings defined on nonconvex domains in Banach spaces. So suppose C is a nonempty subset of a Banach space X and F = {T_{t} : t ∈ G} is a Lipschitzian semigroup defined on C. (Here G is as in Section 3 a semitopological semigroup.) We will employ the following condition introduced by Goebel, Kirk, and Thele in [2]: A nonempty subset E of C is said to satisfy the property

(P): For every x ∈ E and ε > 0, there exists s ∈ G such that dist(T_{t}x, E) < ε for all t ≥ s.

Recall that the modulus of convexity of a Banach space X is defined as the function

δ_{X}(ε) = inf \{1 − \frac{1}{2}∥x + y∥ : x, y ∈ B_{X} with ∥x − y∥ ≥ ε\} , 0 ≤ ε ≤ 2,

where B_{X} is the closed unit ball of X. A Banach space X is said to be uniformly convex if δ_{X}(ε) > 0 for all 0 < ε ≤ 2. It is said to be p-uniformly convex for some p ≥ 2 if there exists a constant d > 0 such that δ_{X}(ε) ≥ d ε^{p} for 0 ≤ ε ≤ 2. It is known that a Hilbert space is 2-uniformly convex and an L^{p} (1 < p < ∞) space is max(2,p)-uniformly convex. We shall need the following characterization of a p-uniformly convex Banach space.

Proposition ([cf. [12]). Given a number p ≥ 2. A Banach space X is p-uniformly convex if and only if there exists a constant d = d_{p} > 0 such that

∥λx + (1 − λ)y∥^{p} ≤ λ∥x∥^{p} + (1 − λ)∥y∥^{p} − dW_{p}(λ)∥x − y∥^{p} (4.1)

for all x, y ∈ X and 0 ≤ λ ≤ 1, where W_{p}(λ) = λ^{p}(1 − λ) + λ(1 − λ)^{p}.

Theorem 4.1. Suppose X is a p-uniformly convex Banach space for some p ≥ 2, C is a nonempty subset of X, G is a semitopological semigroup that is left reversible, and F = {T_{t} : t ∈ G} is a uniformly k-Lipschitzian semigroup on C with k < (1 + d)^{\frac{1}{p}}, d being the constant appearing in (4.1). Suppose also there exist an x̄ ∈ C such that {T_{t}x̄ : t ∈ G} is bounded and a nonempty bounded closed convex subset E of C with Property (P). Then there exists a point z ∈ E such that T_{t}z = z for all t ∈ G.

Proof. Since {T_{t}x̄ : t ∈ G} is bounded, it is easily seen that for all x ∈ C, {T_{t}x : t ∈ G} is bounded. Now choose any x_{0} ∈ E and define a functional f on E by f(x) = inf_{s} sup_{t≥s} ∥T_{t}x_{0} − x∥^{p}, x ∈ E. By Lemma 3 of [11], we have a unique x_{1} ∈ E such that f(x_{1}) ≤ f(x) − d∥x − x_{1}∥^{p}, ∀x ∈ E.

Continuing this procedure, we construct a sequence {x_{n}}_{n=1}^{∞} in E such that for every integer n ≥ 1,

inf_{s} sup_{t≥s} ∥T_{t}x_{n−1} − x_{n}∥^{p} ≤ inf_{s} sup_{t≥s} ∥T_{t}x_{n−1} − x∥^{p} −
− d∥x − x_{n}∥^{p}, ∀x ∈ E.  (4.2)

Now by Property (P), we can find for each n ≥ 1 an a_{n} ∈ G such that

dist(T_{a}x_{n}, E) < A^{n} for all a ≥ a_{n},  (4.3)

where A = \frac{k^{p}−1}{d} < 1. For each fixed a ≥ a_{n}, from (4.3) we can thus find a y_{n} ∈ E such that

∥T_{a}x_{n} − y_{n}∥ < A^{n}.  (4.4)

Using the mean value theorem, it is easy to see that for all x, y ∈ X,

|∥y∥^{p} − ∥x∥^{p}| ≤ p [max{∥x∥, ∥y∥}]^{p−1}|∥y∥ − ∥x∥| ≤ p(∥x∥ + ∥y∥)^{p−1}∥y − x∥.

Since all the involved sequences are bounded, we can find a constant M big enough so that for all t, a ∈ G and n ≥ 1,

p(∥T_{t}x_{n−1} − y_{n}∥ + ∥T_{t}x_{n−1} − T_{a}x_{n}∥)^{p−1} ≤ M/2

and

p(∥T_{a}x_{n} − x_{n}∥ + ∥y_{n} − x_{n}∥)^{p−1} ≤ M/2.

It thus follows from (4.2) and (4.4) that

inf_{s} sup_{t≥s} ∥T_{t}x_{n−1} − x_{n}∥^{p} ≤ inf_{s} sup_{t≥s} ∥T_{t}x_{n−1} − y_{n}∥^{p} − d∥y_{n} − x_{n}∥^{p} ≤

≤ inf_{s} sup_{t≥s} ∥T_{t}x_{n−1} − T_{a}x_{n}∥^{p} − d∥T_{a}x_{n} − x_{n}∥^{p} + sup_{t}(∥T_{t}x_{n−1} −

− y_{n}∥^{p} − ∥T_{t}x_{n−1} − T_{a}x_{n}∥^{p}) + d(∥T_{a}x_{n} − x_{n}∥^{p} − ∥y_{n} − x_{n}∥^{p}) ≤

≤ inf_{s} sup_{t≥s} ∥T_{t}x_{n−1} − T_{a}x_{n}∥^{p} − d∥T_{a}x_{n} − x_{n}∥^{p} + M∥T_{a}x_{n} − y_{n}∥ ≤

≤ inf_{s} sup_{t≥s} ∥T_{t}x_{n−1} − T_{a}x_{n}∥^{p} − d∥T_{a}x_{n} − x_{n}∥^{p} + MA^{n} ≤

≤ inf_{s} sup_{t≥s} ∥T_{at}x_{n−1} − T_{a}x_{n}∥^{p} − d∥T_{a}x_{n} − x_{n}∥^{p} + MA^{n} ≤

≤ k_{a}^{p} inf_{s} sup_{t≥s} ∥T_{t}x_{n−1} − x_{n}∥^{p} − d∥T_{a}x_{n} − x_{n}∥^{p} + MA^{n}.

Hence for all a ≥ a_{n},

d∥T_{a}x_{n} − x_{n}∥^{p} ≤ (k^{p} − 1) inf_{s} sup_{t≥s} ∥T_{t}x_{n−1} − x_{n}∥^{p} + MA^{n},

which results in the conclusion

inf_{s} sup_{t≥s} ∥T_{t}x_{n} − x_{n}∥^{p} ≤ A inf_{s} sup_{t≥s} ∥T_{t}x_{n−1} − x_{n}∥^{p} + M′A^{n},

where M′ = M/d. Write R_{n} = inf_{s} sup_{t≥s} ∥T_{t}x_{n} − x_{n}∥^{p} and r_{n} = inf_{s} sup_{t≥s} ∥T_{t}x_{n} −x_{n+1}∥^{p}. Then r_{n} ≤ R_{n} by (4.2) and

R_{n} ≤ Ar_{n−1} + M′A^{n} ≤ AR_{n−1} + M′A^{n} ≤ A(AR_{n−2} + M′A^{n−1}) +
+ M′A^{n} = A^{2}R_{n−2} + 2M′A^{n} ≤ ⋯ ≤ (R_{0} + nM′)A^{n}.

Therefore,

∥x_{n} − x_{n−1}∥^{p} = inf_{s} sup_{t≥s} ∥(x_{n} − T_{t}x_{n−1}) + (T_{t}x_{n−1} − x_{n−1})∥^{p} ≤

≤ 2^{p−1} inf_{s} sup_{t≥s}(∥x_{n} − T_{t}x_{n−1}∥^{p} + ∥T_{t}x_{n−1} − x_{n−1}∥^{p}) ≤

≤2^{p−1}(r_{n−1}+R_{n−1})≤2^{p}R_{n−1} ≤2^{p}(R_{0}+(n − 1)M′)A^{n−1},

which shows that {x_{n}} is Cauchy and hence convergent to some z strongly. We now show that this z is a common fixed point of F. In fact, noting that the inequality (cf. [11])

inf_{s} sup_{t≥s} ∥T_{t}x − y∥^{p} ≤ inf_{s} sup_{t≥s} ∥T_{at}x − y∥^{p}

is valid for all x, y ∈ C and a ∈ G, we have for all a ∈ G,

∥z − T_{a}z∥^{p} ≤ inf_{s} sup_{t≥s}(∥z − x_{n}∥ + ∥x_{n} − T_{t}x_{n}∥ +

+ ∥T_{t}x_{n} − T_{a}x_{n}∥ + ∥T_{a}x_{n} − T_{a}z∥)^{p} ≤ 4^{p} inf_{s} sup_{t≥s}((1 + k^{p})∥z − x_{n}∥^{p} +

+ ∥x_{n} − T_{t}x_{n}∥^{p} + ∥T_{t}x_{n} − T_{a}x_{n}∥^{p}) ≤ 4^{p}((1 + k^{p})∥z − x_{n}∥^{p} + R_{n} +

+ inf_{s} sup_{t≥s} ∥T_{at}x_{n} − T_{a}x_{n}∥^{p}) ≤ 4^{p}(1 + k^{p})(∥z − x_{n}∥^{p} + R_{n}) → 0 (n → ∞).

Hence T_{a}z = z.

Theorem 4.2. Let X be a p-uniformly convex Banach space for some p ≥ 2, C a nonempty subset of X, G a semitopological semigroup for which RUC(G) has a left invariant mean μ, and F = {T_{t} : t ∈ G} a uniformly k-Lipschitzian semigroup on C such that k < (1 + d)^{\frac{1}{p}} with d being the constant appearing in (4.1). Suppose there is a point x̄ ∈ C for which the orbit {T_{t}x̄ : t ∈ G} is bounded. Suppose also there exists a nonempty bounded closed convex subset E of C with the following property:

(P^{∗}) For every x ∈ E and ε > 0, there exists s ∈ G such that dist(T_{st}x, E)

< ε, ∀t ∈ G. Then there exists some z ∈ E such that T_{t}z = z for all t ∈ G.

Proof. Choose an arbitrary x_{0} ∈ E. As the function t ↦ ∥T_{t}u − v∥^{p} is in RUC(G) for any u ∈ C and v ∈ X, we can inductively construct a sequence {x_{n}}_{n=1}^{∞} in E in the following way: For each integer n ≥ 1, x_{n} ∈ E is the unique minimizer of the functional μ_{t}∥T_{t}x_{n−1} − x∥^{2} over E. Then by Lemma 2 of [11], we have

μ_{t}∥T_{t}x_{n−1} − x_{n}∥^{p} ≤ μ_{t}∥T_{t}x_{n−1} − x∥^{p} − d∥x − x_{n}∥^{p}, ∀x ∈ E. (4.5)

Write r_{n} = inf{μ_{t}∥T_{t}x_{n} − x∥^{p} : x ∈ E} = μ_{t}∥T_{t}x_{n} − x_{n+1}∥^{p} and R_{n} = μ_{t}∥T_{t}x_{n} − x_{n}∥^{p}. Then for a fixed integer n ≥ 1, condition (P^{∗}) yields an s_{n} ∈ G such that

dist(T_{s_{n}r}x_{n}, E) < A^{n}, ∀r ∈ G,  (4.6)

where A = (k^{p} − 1)/d < 1. From (4.6) we can find a y_{n} ∈ E (depending on r) such that ∥T_{s_{n}r}x_{n} − y_{n}∥ < A^{n}. It then follows from (4.5) that

r_{n−1} ≤ μ_{t}∥T_{t}x_{n−1} − y_{n}∥^{p} − d∥y_{n} − x_{n}∥^{p} = μ_{t}∥(T_{t}x_{n−1} − T_{s_{n}r}x_{n}) +

+ (T_{s_{n}r}x_{n} − y_{n})∥^{p} − d∥(y_{n} − T_{s_{n}r}x_{n}) + (T_{s_{n}r}x_{n} − x_{n})∥^{p} ≤ ≤ μ_{t}∥T_{t}x_{n−1} − T_{s_{n}r}x_{n}∥^{p} − d∥T_{s_{n}r}x_{n} − x_{n}∥ + MA^{n} = = μ_{t}∥T_{s_{n}rt}x_{n−1} − T_{s_{n}r}x_{n}∥^{p} − d∥T_{s_{n}r}x_{n} − x_{n}∥^{p} + MA^{n} ≤ ≤ k^{p} μ_{t}∥T_{t}x_{n−1} − x_{n}∥^{p} − d∥T_{s_{n}r}x_{n} − x_{n}∥^{p} + MA^{n},

where M > 0 is some appropriate constant independent of r ∈ G, which can be found similarly to the proof of Theorem 4.2. Hence

∥T_{s_{n}r}x_{n} − x_{n}∥^{p} ≤ \frac{k^{p} − 1}{d}r_{n−1} + \frac{M}{d} A^{n} = Ar_{n−1} + M′A^{n},

where M′ = M/d, and

R_{n} = μ_{t}∥T_{t}x_{n} − x_{n}∥^{p} = μ_{r}∥T_{s_{n}r}x_{n} − x_{n}∥^{p} ≤
≤ Ar_{n−1} + M′A^{n} ≤ ⋯ ≤ (R_{0} + nM′)A^{n}.

Therefore, we have

∥x_{n} − x_{n−1}∥^{p} ≤ μ_{t}∥(x_{n} − T_{t}x_{n−1}) + (T_{t}x_{n−1} − x_{n−1})∥^{p} ≤
≤ 2^{p} μ(∥x_{n} − T_{t}x_{n−1}∥^{p} + ∥T_{t}x_{n−1} − x_{n−1}∥^{p}) =
= 2^{p}(r_{n−1} + R_{n−1}) ≤ 2^{p+1}R_{n−1} ≤
≤ 2^{p+1}(R_{0} + (n − 1)M′)A^{n−1},

showing that {x_{n}} is Cauchy and hence strongly convergent. Let z be the limit. Now for all a ∈ G, we have

∥z − T_{a}z∥^{p} ≤ μ_{t}(∥z − x_{n}∥ + ∥x_{n} − T_{t}x_{n}∥ + ∥T_{t}x_{n} − T_{a}x_{n}∥ + + ∥T_{a}x_{n} − T_{a}z∥)^{p} ≤ 4^{p} μ_{t}(∥z − x_{n}∥^{p} + ∥x_{n} − T_{t}x_{n}∥^{p} + ∥T_{t}x_{n} − T_{a}x_{n}∥^{p} + + ∥T_{a}x_{n} − T_{a}z∥^{p}) ≤ 4^{p}((1 + k^{p})∥x_{n} − z∥^{p} + R_{n} + μ_{t}∥T_{at}x_{n} − T_{a}z_{n}∥^{p}) ≤ ≤ 4^{p}(1 + k^{p})(∥x_{n} − z∥^{p} + R_{n}) → 0 (n → ∞).

Hence T_{a}z = z.

In L^{p} (1 < p < ∞), we have the following inequalities (cf. [5], [6], [12]):

∥λx + (1 − λ)y∥^{q} ≤ λ∥x∥^{q} + (1 − λ)∥y∥^{q} − d_{p}W_{q}(λ)∥x − y∥^{q}

for all x, y ∈ L^{p} and λ ∈ [0, 1], where q = max{2, p}, W_{q}(λ) = λ^{q}(1 − λ) + λ(1 − λ)^{q}, and

d_{p} = \begin{cases}\frac{(1+t_{p}^{p−1})}{(1+t_{p})^{p−1}} , & if 2 < p < ∞; \\ p − 1, & if 1 < p ≤ 2. \end{cases}

t_{p} is the unique solution of the equation (p − 2)t^{p−1} + (p − 1)t^{p−2} − 1 = 0, t ∈ (0, 1). Thus we have the following consequence of Theorems 4.1 and 4.2.

Corollary 4.1. Let C be a nonempty subset of L^{p} (1 < p < ∞), G a semitopological semigroup which is left reversible or for which the space RUC(G) has an invariant mean, and F = {T_{t} : t ∈ G} a uniformly k-Lipschitzian semigroup on C with k < \sqrt{p} if 1 < p ≤ 2 or k <1 + (1 + t_{p}^{p−1}) (1+t_{p})^{1−p}]^{\frac{1}{p}} if 2 < p < ∞. Suppose there exists an x ∈ C such that the orbit {T_{t}x : t ∈ G} is bounded. Suppose also there exists a nonempty bounded closed convex subset of C which possesses Property (P) in the case where G is left reversible or Property (P^{∗}) in the case RUC(G) has an invariant mean. Then there exists a z ∈ E such that T_{s}z = z for all s ∈ G.

Acknowledgement

The authors would like to thank the anonymous referee for a careful reading of the manuscript and helpful suggestions.

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(Received 15.07.1994)

Authors’ addresses: Kok-Keong Tan Department of Mathematics, Statistics and Computing Science Dalhousie University Halifax, Nova Scotia Canada B3H 3J5 Hong-Kun Xu Institute of Applied Mathematics, East China University of Science and Technology, Shanghai 200237, China