NONLINEAR ERGODIC THEOREMS FOR ASYMPTOTICALLY ALMOST NONEXPANSIVE CURVES IN A HILBERT SPACE

We introduce the notion of asymptotically almost nonexpansive curves which include almost-orbits of commutative semigroups of asymptotically nonexpansive type mappings and study the asymptotic behavior and prove nonlinear ergodic theorems for such curves. As applications of our main theorems, we obtain the results on the asymptotic behavior and ergodicity for a commutative semigroup of non-Lipschitzian mappings with nonconvex domains in a Hilbert space.


Introduction
Let H be a real Hilbert space with norm • and inner product (•, •).Let C be a nonempty subset of H and G be a commutative semitopological semigroup with identity.In this case, (G, ) is a directed system when the binary relation " " on G is defined by b a if and only if there is c ∈ G such that a + c = b.Let = {T (t) : t ∈ G} be a semigroup acting on C, that is, T (t + s)x = T (t)T (s)x for all t, s ∈ G and x ∈ C. Recall that a semigroup on C is said to be (a) nonexpansive if T (t)x − T (t)y ≤ x − y for x, y ∈ C and t ∈ G, (b) asymptotically nonexpansive, [9], if there exists a function k : G → [0, ∞) with lim sup t∈G k t ≤ 1 such that for x, y ∈ C and t ∈ G, (c) of asymptotically nonexpansive type, [9], if for each x ∈ C, there is a function r(•, x) : G → [0, ∞) with lim t∈G r(t, x) = 0 such that where lim t∈G α(t) denotes the limit of a net α(•) on the directed system (G, ).
In 1975, Baillon [1] proved the first nonlinear mean ergodic theorem for nonexpansive mappings in a Hilbert space: let C be a nonempty closed convex subset of a Hilbert space H and let T be a nonexpansive mapping of C into itself.If the set F (T ) of fixed points of T is nonempty, then the Cesáro means converge weakly as n → ∞ to a fixed point y of T for each x ∈ C. In this case, letting y = P x for each x ∈ C, P is a nonexpansive retraction of C onto the fixed point set F (T ) of T such that P T = T P = P and P x ∈ conv{T n x : n = 0, 1, 2, . . .} for each x ∈ C, where convA denotes the closure of the convex hull of A. The analogous results are given for nonexpansive semigroups by Baillon and Brézis [2] and Brézis and Browder [3].In [13], Mizoguchi and Takahashi proved a nonlinear ergodic retraction theorem for Lipschitzian semigroups by using the notion of submean.
In this paper, we introduce the notion of asymptotically almost nonexpansive curves which include almost-orbits of commutative semigroups of asymptotically nonexpansive type mappings, and we prove nonlinear ergodic theorems for such curves.As applications of our main theorems, we obtain the results on the asymptotic behavior and ergodicity for a commutative semigroup of non-Lipschitzian mappings with nonconvex domains in a Hilbert space.Our results generalize and improve the previously known results of Baillon [1], Baillon and Brézis [2], Hirano and Takahashi [6], Ishihara and Takahashi [7], Lau, Nishiura, and Takahashi [10], Li and Ma [11,12], Mizoguchi and Takahashi [13], Takahashi [14,15], Takahashi and Zhang [16], and Tan and Xu [17] in many directions.

Preliminaries and notations
Throughout this paper, let H be a real Hilbert space with norm • and inner product (•, •).Let G be a commutative semitopological semigroup with identity and let m(G) be the Banach space of all bounded real-valued functions on G with the supremum norm.For each s ∈ G and f ∈ m(G), we define r s f in m(G) given by Let X be a subspace of m(G) and µ be an element of X * (the dual space of X).Then, we denote by µ(f ) the value of µ at f ∈ X.To specify the variable t, we write the value µ(f ) by µ(t) f (t) or f (t)dµ(t).When X contains a constant 1, an element µ of X * is called a mean on X if µ = µ(1) = 1.Further, let X be invariant under r s for all s ∈ G.Then, a mean µ on X is said to be invariant if µ(r s f ) = µ(f ) for all s ∈ G and f ∈ X.For s ∈ G, we can define a point evaluation δ s by δ s (f ) = f (s) for every f ∈ m(G).A convex combination of point evaluations is called a finite mean on G.
Recently, the notion of the almost nonexpansive curve was introduced by Rouhani [5] and Kada and Takahashi [8].
In the case ε(s, t) = 0 for all s, t ∈ G, u is called a nonexpansive curve.Now, we define the concept of the asymptotically almost nonexpansive curve.(2) for an arbitrary ε > 0 there exists t 0 ∈ G, and for each t t 0 there exists where ε 1 (t, s, h) satisfies the same condition (2) as ε(t, s, h).We denote by L(u) the following subset (possibly empty) of H : Throughout the rest of this paper, u(•) is a bounded asymptotically almost nonexpansive curve and X is a subspace of m(G) containing constants invariant under r s for each s ∈ G. Furthermore, suppose that for each x ∈ H , the function t → u(t) − x 2 is in X.Then by Riesz theorem, there exists a unique element u µ in H such that We denote by ω w (u) the set of all weak limits of subnets of the net {u(t) : t ∈ G}.

Asymptotic behavior of curves
We begin with the following lemmas and proposition which play an important role in the proof of our main theorems.
Lemma 3.1.Let u(•) be a bounded asymptotically almost nonexpansive curve.Then the set L(u) (possibly empty) is closed and convex.
Proof.We can show the closedness from this inequality, And also, the convexity follows from the equality Proposition 3.2.The set s∈G conv{u(t) : t s}∩L(u) consists of at most one point.
Proof.Suppose that L(u) = ∅.Let p be the unique asymptotic center of {u(t) : t ∈ G} in L(u) and x ∈ s∈G conv{u(t) : t s} ∩ L(u).We conclude the proof by showing that x = p.Since we have For any ε > 0, there exists that is, p −x 2 ≤ ε.Since ε > 0 is arbitrary, we have x = p.This completes the proof.
Since G is commutative, there exists a net {λ α : α ∈ I } of finite means on G such that where I is a directed set and r * s is the conjugate of r s (see [4]).Then for all t t and h h t , where M = sup t∈G u(t) .Note that λ α β (t) u(t + t α β + t ε ) converges weakly to z.For fixed t t ε and h h t , taking the limit for β ∈ J , we have and hence Suppose that Since {µ α : α ∈ A} is strongly regular, there exists where M = sup{ u(t) : t ∈ G}.Since for all α α 1 , h ∈ G, for fixed t t ε and h h t .Taking the limit for α ∈ J , we have This implies z ∈ L(u) in the same way as in Lemma 3.3.

Asymptotic behavior of almost-orbits
In this section, using the main results in Section 3, we prove the ergodic theorems and weak convergence theorems for almost-orbits of commutative semigroups of asymptotically nonexpansive type mappings with nonconvex domains.
Let C be a nonempty subset of a Hilbert space H and = {T (t) : t ∈ G} be a family of mappings from C into itself.Recall that is said to be a commutative semigroup of asymptotically nonexpansive type mappings on C if the following conditions are satisfied: (a) T (t + s)x = T (t)T (s)x for all t, s ∈ G and x ∈ C; (b) for each x ∈ C and t ∈ G, there exists α(t, x) ≥ 0 such that Throughout the rest of this section, = {T (t) : t ∈ G} is a commutative semigroup of asymptotically nonexpansive type mappings on C, u(•) : G → C is a bounded almost-orbit of = {T (t) : t ∈ G}, and X is a subspace of m(G) containing constants invariant under r s for each s ∈ G. Furthermore, suppose that for each x ∈ H , the function t → u(t) − x 2 is in X. Denote by F ( ) the set of common fixed points of = {T (t) : t ∈ G}.We begin with the following lemmas.Proof.Put ϕ(t) = sup h∈G u(h + t) − T (h)u(t) .Then lim t∈G ϕ(t) = 0. Since for every h, t, s ∈ G.It is easily seen that u(•) is an asymptotically almost nonexpansive curve.
Lemma 4.2.If u(•) and v(•) are almost-orbits of , then lim t∈G u(t) − v(t) exists.Furthermore, we have F ( ) ⊆ L(u). Proof.Set G. Li and J. K. Kim 155 Then, lim t∈G ϕ(t) = lim t∈G ψ(t) = 0. Since for each t, s ∈ G, which complete the proof of the first part.The second part is obvious.
We can prove the following proposition from Lemma 4.2 and Proposition 3.2.
Proposition 4.3.The set s∈G conv{u(t) : t s}∩F ( ) consists of at most one point.
From Theorem 3.4, we can prove the following theorem which is an extension of the result of Tan and Xu [17]  Let AO( ) be the set of all almost-orbits of .Then for each h ∈ G and u ∈ AO( ), the function v : G → C, defined by v(t) = T (h)u(t), is also an almost-orbit of .In fact, as before, we set ϕ Using Theorem 4.4, we have the following ergodic retraction theorem.
Theorem 4.5.Let C be a nonempty bounded subset of a Hilbert space H and let be a commutative semigroup of asymptotically nonexpansive type mappings on C such that each T (t) is continuous.Then for an invariant mean µ, the mapping P : u → u µ is a unique retraction from the set AO( ) onto F ( ) such that (1) P is nonexpansive in the sense that (2) P T (h)u = T (h)P u = P u for u ∈ AO( ) and h ∈ G; (3) P u ∈ s∈G conv{u(t) : t s} for u ∈ AO( ).
As a direct consequence of Theorem 3.5, we can prove the following theorem which is an extension of the Takahashi and Zhang [16].Note that we do not assume F ( ) to be nonempty.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Lemma 4 . 1 .
Let u(•) be a bounded almost-orbit of the commutative semigroup = {T (t) : t ∈ G} of asymptotically nonexpansive type mappings on C. Then it is an asymptotically almost nonexpansive curve.

First
Round of ReviewsMay 1, 2009 Let u(•) : G → H be a function, in what follows we refer to such u(•) as a curve in H .A bounded function u is called an almost nonexpansive curve if there exists a function To show this, let {t α β : β ∈ J } be a subnet of {t α : α ∈ B} such that λ α β (t) u(t + t α β ) converges weakly to some z in H , where J is a directed set.For any ε > 0, there exists t ε ∈ G such that for any t t , there exists h t ∈ G such that Lemma 3.3.λα (t) u(t +h) converges weakly to an element p in s∈G conv{u(t) : t s} ∩L(u) uniformly in h ∈ G.Proof.For any {t α } ∈ G, let W be the set of all weak limit points of λ α (t) u(t + t α ) .
.13) Since ε > 0 is arbitrary, we have z ∈ L(u).Now, we show that z ∈ s∈G conv{u(t) : t s}.For each s ∈ G, since λ α β (t) u(t + t α β + s) ∈ conv{u(t) : t s}, we get z ∈ s∈G conv{u(t) : t s}.This completes the proof.Now, we can prove the ergodic convergence theorem for asymptotically almost nonexpansive curves.A net {µ α : α ∈ A} of continuous linear functionals on X is called strongly regular if it satisfies the following conditions: (a) sup α∈A µ α < +∞; (b) lim α∈A µ α (1) = 1; (c) lim α∈A µ α − r * s µ α = 0 for every s ∈ G. Theorem 3.4.Let {µ α : α ∈ A} be a strongly regular net of continuous linear functional on X.Then there exists p ∈ s∈G conv{u(t) : t s} ∩ L(u) such that Moreover, u µ = p for each invariant mean µ.Proof.By Lemma 3.3, there exists p ∈ s∈G conv{u(t) : t s} L(u) and for any ε > 0 and y 0 ∈ H with y 0 = 1, there exists α 0 ∈ B such that in many directions.Theorem 4.4.Let C be a nonempty subset of H , = {T (t) : t ∈ G} a commutative semigroup of asymptotically nonexpansive type mappings on C, and u(•) be a bounded almost-orbit of .If {µ α : α ∈ A} is a strongly regular net of continuous linear functional on X, then Proof.By Lemma 4.1 and Theorem 3.4, we need only to prove that if each T (t) is continuous and s∈G conv{u(t) : t s} ⊂ C, then p ∈ F ( ).By assumption, we have p ∈ C. Let 0 < ε ≤ 1.Then there exists t 1 ∈ G such that It follows that T (t)p is convergent strongly to p, therefore, p ∈ F ( ) by the continuity of {T (t) : t ∈ G}.This completes the proof.
2 = u s + t + t 1 − T (s)u t + t 1 2 + T (s)u t + t 1 − T (s)p 2 + 2 u s + t + t 1 − T (s)u t + t 1 , T (s)u t + t 1 − T (s)p ≤ u t + t 1 − p Theorem 4.6.Let C be a nonempty subset of a Hilbert space H and let be a commutative semigroup of asymptotically nonexpansive type mappings on C, and let G. Li and J. K. Kim 157 u(•) be a bounded almost-orbit of .Then w − lim t∈G u(t) exists (in L(u)) if and only if w − lim t∈G (u(h + t) − u(t)) = 0 for all h ∈ G.