Fixed point theorems and convergence theorems for generalized nonspreading mappings in Banach spaces

Fixed point theorems and convergence theorems for generalized nonspreading mappings in Banach spaces

Wataru Takahashi, Ngai-Ching Wong and Jen-Chih Yao

Dedicated to Professor Dick Palais

Abstract. In this paper, we first prove a general fixed point theorem for nonlinear mappings in a Banach space. Then we prove a nonlinear mean convergence theorem of Baillon’s type and a weak convergence theorem of Mann’s type for 2-generalized nonspreading mappings in a Banach space.

Mathematics Subject Classification (2010). Primary 47H10; Secondary 47H05.

Keywords. Banach space, nonexpansive mapping, nonspreading map- ping, hybrid mapping, fixed point.

1. Introduction

LetEbe a real smooth Banach space and letJ be the duality mapping ofE.

LetC be a nonempty closed convex subset of E. Let T be a mapping of C into itself. Then we denote byF(T) the set of fixed points of T. Recently, Kohsaka and Takahashi [14] introduced the following nonlinear mapping: A mappingT :C→C is said to be nonspreading if

φ(T x, T y) +φ(T y, T x)≤φ(x, T y) +φ(y, T x) for allx, y∈C, where

φ(x, y) =x²−2x, Jy+y²

forx, y ∈E; see also [13]. Such a mapping is deduced from the resolvent of a maximal monotone operator in a Banach space; see [14, 27, 23]. A non- spreading mapping defined by [14] is as follows in a Hilbert space: LetH be DOI 10.1007/s11784-012-0074-3

Published online April 3, 2012

© Springer Basel AG 2012

Journal of Fixed Point Theory and Applications

a real Hilbert space and letC be a nonempty closed convex subset of H. A mappingT :C→C is said to be nonspreading [14] if

2T x−T y²≤ T x−y²+T y−x² (1.1) for allx, y∈C. Takahashi [22] also defined another nonlinear mapping in a Hilbert space: A mappingT :C→C is said to behybrid [22] if

3T x−T y²≤ x−y²+T x−y²+T y−x² (1.2) for allx, y ∈C. Furthermore, we know that a mappingT :C→Cis said to benonexpansive if

T x−T y ≤ x−y

for allx, y∈C; see [4, 21]. The classes of nonexpansive mappings, nonspread- ing mappings and hybrid mappings in a Hilbert space are deduced from the class of firmly nonexpansive mappings; see [22]. A mapping F : C → C is said to befirmly nonexpansive if

F x−F y²≤ x−y, F x−F y

for allx, y∈C; see [3, 4]. Recently, Kocourek, Takahashi and Yao [10] intro- duced a class of nonlinear mappings called generalized hybrid containing the classes of nonexpansive mappings, nonspreading mappings and hybrid map- pings in a Hilbert space: A mappingT :C →C is calledgeneralized hybrid [10] if there existα, β∈Rsuch that

αT x−T y²+ (1−α)x−T y²≤βT x−y²+ (1−β)x−y² for allx, y∈C. Kocourek, Takahashi and Yao [11] also extended the class of generalized hybrid mappings in a Hilbert space to Banach spaces: LetE be a smooth Banach space and letCbe a nonempty closed convex subset ofE.

Then a mappingT :C→Cis calledgeneralized nonspreading if there exist α, β, γ, δ∈Rsuch that

αφ(T x, T y) + (1−α)φ(x, T y) +γ

φ(T y, T x)−φ(T y, x)

≤βφ(T x, y) + (1−β)φ(x, y) +δ

φ(y, T x)−φ(y, x) (1.3) for allx, y ∈C. Very recently, Maruyama, Takahashi and Yao [16] introduced a broad class of nonlinear mappings containing the class of generalized hybrid mappings defined by [10] in a Hilbert space: A mappingT :C→C is called 2-generalized hybrid if there exist α₁, α₂, β₁, β₂∈Rsuch that

α₁T²x−T y²+α₂T x−T y²+ (1−α₁−α₂)x−T y²

≤β₁T²x−y²+β₂T x−y²+ (1−β₁−β₂)x−y² for allx, y∈C. Motivated by [16, 11], we introduced three classes of nonlin- ear mappings in Banach spaces [26] which contain the class of 2-generalized hybrid mappings in a Hilbert space. Then we proved fixed point theorems for these classes of nonlinear mappings in Banach spaces.

In this paper, we deal with the class of 2-generalized nonspreading map- pings which is one of the three classes of nonlinear mappings defined in [26]

in Banach spaces. We first prove a general fixed point theorem of nonlinear

mappings in a Banach space. Using this result, we give another proof of our fixed point theorem [26] for 2-generalized nonspreading mappings in Banach spaces. Then we prove a nonlinear mean convergence theorem of Baillon’s type [2] and a weak convergence theorem of Mann’s type[15] for such nonlin- ear mappings in a Banach space.

2. Preliminaries

LetE be a real Banach space with norm · and letE^∗ be the topological dual space ofE. We denote the value ofy^∗∈E^∗at x∈E byx, y^∗. When {x_n}is a sequence in E, we denote the strong convergence of{x_n}to x∈E byx_n→xand the weak convergence byx_n x. The modulusδof convexity ofE is defined by

δ() = inf

1−x+y

2 :x ≤1,y ≤1,x−y ≥

for everywith 0≤≤2. A Banach spaceEis said to beuniformly convex if δ()>0 for every >0. A uniformly convex Banach space is strictly convex and reflexive. LetCbe a nonempty subset of a Banach spaceE. A mapping T :C→CisnonexpansiveifT x−T y ≤ x−yfor allx, y∈C. A mapping T :C →C isquasi-nonexpansive if F(T)=∅ and T x−y ≤ x−y for allx∈C andy ∈F(T), whereF(T) is the set of fixed points ofT. IfC is a nonempty closed convex subset of a strictly convex Banach space E and T :C →C is quasi-nonexpansive, thenF(T) is closed and convex; see Itoh and Takahashi [8].

LetE be a Banach space. The duality mapping J from E into 2^E∗ is defined by

Jx=

x^∗∈E^∗:x, x^∗=x²=x^∗2

for everyx∈ E. Let U ={x∈E : x = 1}. The norm ofE is said to be Gˆateaux differentiable if for eachx, y∈U, the limit

t→0lim

x+ty − x

t (2.1)

exists. In this case, E is called smooth. We know that E is smooth if and only if J is a single-valued mapping of E into E^∗. We also know that E is reflexive if and only if J is surjective, and E is strictly convex if and only ifJ is one-to-one. Therefore, if E is a smooth, strictly convex and reflexive Banach space, thenJis a single-valued bijection. The norm ofEis said to be uniformly Gˆateaux differentiable if for eachy∈U, the limit (2.1) is attained uniformly for x ∈ U. It is also said to be Fr´echet differentiable if for each x∈U, the limit (2.1) is attained uniformly fory∈U. A Banach spaceE is calleduniformly smooth if the limit (2.1) is attained uniformly forx, y ∈U. It is known that if the norm ofE is uniformly Gˆateaux differentiable, thenJ is uniformly norm-to-weak^∗ continuous on each bounded subset ofE, and if the norm ofE is Fr´echet differentiable, thenJ is norm-to-norm continuous.

IfE is uniformly smooth, J is uniformly norm-to-norm continuous on each

bounded subset ofE. For more details, see [19, 20]. The following result is also in [19, 20].

Lemma 2.1. LetEbe a smooth Banach space and letJ be the duality mapping on E. Then x−y, Jx−Jy ≥0 for allx, y ∈E. Further, if E is strictly convex andx−y, Jx−Jy= 0, then x=y.

LetE be a smooth Banach space. The functionφ:E×E→(−∞,∞) is defined by

φ(x, y) =x²−2x, Jy+y² (2.2) forx, y∈E, where J is the duality mapping ofE; see [1, 9]. We have from the definition ofφthat

φ(x, y) =φ(x, z) +φ(z, y) + 2x−z, Jz−Jy (2.3) for all x, y, z∈E. From (x − y)² ≤φ(x, y) for allx, y∈E, we can see thatφ(x, y)≥0. Further, we can obtain the following equality:

2x−y, Jz−Jw=φ(x, w) +φ(y, z)−φ(x, z)−φ(y, w) (2.4) for x, y, z, w ∈ E. If E is additionally assumed to be strictly convex, then from Lemma 2.1 we have

φ(x, y) = 0⇐⇒x=y. (2.5)

The following lemmas are in Xu [29] and Kamimura and Takahashi [9], respectively.

Lemma 2.2. Let E be a uniformly convex Banach space and let r >0. Then there exists a strictly increasing, continuous and convex functiong: [0,∞)→ [0,∞)such that g(0) = 0 and

λx+ (1−λ)y²≤λx²+ (1−λ)y²−λ(1−λ)g(x−y) for allx, y∈B_r andλwith0≤λ≤1, whereB_r={z∈E:z ≤r}. Lemma 2.3. Let E be a smooth and uniformly convex Banach space and let r >0. Then there exists a strictly increasing, continuous and convex function g: [0,2r]→Rsuch that g(0) = 0and

g(x−y)≤φ(x, y) for allx, y∈B_r, whereB_r={z∈E:z ≤r}.

LetEbe a smooth Banach space and letCbe a nonempty subset ofE.

Then a mappingT :C→Cis calledgeneralized nonexpansive [5] ifF(T)=∅ and

φ(T x, y)≤φ(x, y)

for allx∈Candy∈F(T). LetDbe a nonempty subset of a Banach spaceE.

A mappingR:E→D is said to be sunny if R(Rx+t(x−Rx)) =Rx

for allx ∈E and t ≥ 0. A mapping R : E → D is said to be aretraction or a projection ifRx=xfor all x∈D. A nonempty subset D of a smooth Banach spaceEis said to be ageneralized nonexpansive retract(resp.,sunny

generalized nonexpansive retract) of E if there exists a generalized nonex- pansive retraction (resp., sunny generalized nonexpansive retraction)Rfrom E onto D; see [5] for more details. The following results are in Ibaraki and Takahashi [5].

Theorem 2.4. Let C be a nonempty closed sunny generalized nonexpansive retract of a smooth and strictly convex Banach space E. Then the sunny generalized nonexpansive retraction fromE ontoC is uniquely determined.

Theorem 2.5. Let C be a nonempty closed subset of a smooth and strictly convex Banach spaceE such that there exists a sunny generalized nonexpan- sive retractionR from E ontoC and let (x, z)∈E×C. Then the following hold:

(i) z=Rx if and only ifx−z, Jy−Jz ≤0 for ally∈C;

(ii) φ(Rx, z) +φ(x, Rx)≤φ(x, z).

In 2007, Kohsaka and Takahashi [12] proved the following results.

Theorem 2.6. Let E be a smooth, strictly convex and reflexive Banach space and letCbe a nonempty closed subset ofE. Then the following are equivalent:

(a) C is a sunny generalized nonexpansive retract of E;

(b) C is a generalized nonexpansive retract ofE;

(c) JC is closed and convex.

Theorem 2.7. Let E be a smooth, strictly convex and reflexive Banach space and letC be a nonempty closed sunny generalized nonexpansive retract ofE.

Let R be the sunny generalized nonexpansive retraction from E ontoC and let(x, z)∈E×C. Then the following are equivalent:

(i) z=Rx;

(ii) φ(x, z) = min_y∈Cφ(x, y).

Very recently, Ibaraki and Takahashi [7] also obtained the following re- sult concerning the set of fixed points of a generalized nonexpansive mapping.

Theorem 2.8. Let E be a reflexive, strictly convex and smooth Banach space and let T be a generalized nonexpansive mapping from E into itself. Then F(T)is closed and JF(T)is closed and convex.

The following theorem is proved using Theorems 2.6 and 2.8.

Theorem 2.9 (see[7]). LetEbe a reflexive, strictly convex and smooth Banach space and let T be a generalized nonexpansive mapping from E into itself.

ThenF(T)is a sunny generalized nonexpansive retract of E.

Letl^∞be the Banach space of bounded sequences with supremum norm.

Let μ be an element of (l^∞)^∗ (the dual space of l^∞). Then, we denote by μ(f) the value of μ at f = (x₁, x₂, x₃, . . .) ∈ l^∞. Sometimes, we denote by μ_n(x_n) the value μ(f). A linear functional μ on l^∞ is called a mean if μ(e) =μ= 1, wheree= (1,1,1, . . .). A meanμis called aBanach limit on

l^∞ ifμ_n(x_n+1) =μ_n(x_n). We know that there exists a Banach limit on l^∞. Ifμis a Banach limit onl^∞, then forf = (x₁, x₂, x₃, . . .)∈l^∞,

lim inf

n→∞ x_n≤μ_n(x_n)≤lim sup

n→∞ x_n.

In particular, if f = (x₁, x₂, x₃, . . .) ∈ l^∞ and x_n → a ∈ R, then we have μ(f) =μ_n(x_n) =a. For the proof of existence of a Banach limit and its other elementary properties, see [19].

3. Fixed point theorems

Let E be a smooth Banach space and let C be a nonempty closed convex subset of E. Let n∈N. Then a mapping T :C →C is calledn-generalized nonspreading [26] if there exist α₁, α₂, . . . , α_n, β₁, β₂, . . . , β_n, γ₁, γ₂, . . . , γ_n, δ₁, δ₂, . . . , δ_n∈Rsuch that for allx, y∈C,

n k=1

α_kφ(T^n+1−kx, T y) +

1−ⁿ

k=1

α_k

φ(x, T y) +

n k=1

γ_k

φ(T y, T^n+1−kx)−φ(T y, x)

≤ⁿ

k=1

β_kφ(T^n+1−kx, y) +

1−ⁿ

k=1

β_k

φ(x, y) +

n k=1

δ_k

φ(y, T^n+1−kx)−φ(y, x) .

(3.1)

Such a mapping is called (α₁, α₂, . . . , α_n, β₁, β₂, . . . , β_n, γ₁, γ₂, . . . , γ_n, δ₁, δ₂, . . . , δ_n)-generalized nonspreading. For example, an (α₁, α₂, β₁, β₂, γ₁, γ₂, δ₁, δ₂)-generalized nonspreading mapping is as follows:

α₁φ(T²x, T y) +α₂φ(T x, T y) + (1−α₁−α₂)φ(x, T y) +γ₁

φ(T y, T²x)−φ(T y, x) +γ₂

φ(T y, T x)−φ(T y, x)

≤β₁φ(T²x, y) +β₂φ(T x, y) + (1−β₁−β₂)φ(x, y) +δ₁

φ(y, T²x)−φ(y, x) +δ₂

φ(y, T x)−φ(y, x)

(3.2)

for all x, y ∈ C. This is also called a 2-generalized nonspreading mapping;

see [26]. We know that an (α₁, α₂, β₁, β₂, γ₁, γ₂,δ₁, δ₂)-generalized nonspread- ing mapping is nonspreading in the sense of [14] forα₁=β₁ =γ₁ =δ₁ = 0, α₂=β₂=γ₂= 1 andδ₂= 0 in (3.2).

Now we state and prove the main result in this section.

Theorem 3.1. Let E be a smooth, strictly convex and reflexive Banach space with the duality mappingJ and letC be a nonempty closed convex subset of E. LetT be a mapping ofCinto itself. Let{x_n} be a bounded sequence ofC

and letμbe a mean on l^∞. Suppose that

μ_nφ(x_n, T y)≤μ_nφ(x_n, y) for ally∈C. ThenT has a fixed point inC.

Proof. Using a meanμ and a bounded sequence{x_n}, we define a function g:E^∗→Ras follows:

g(x^∗) =μ_nx_n, x^∗

for allx^∗∈E^∗. Sinceμis linear,gis also linear. Furthermore, we have

|g(x^∗)|=|μ_nx_n, x^∗|

≤ μsup

n∈N|x_n, x^∗|

≤ μsup

n∈Nx_nx^∗

= sup

n∈Nx_nx^∗

for allx^∗∈E^∗. Thengis a linear and continuous real-valued function onE^∗. SinceE is reflexive, there exists a unique elementz ofE such that

g(x^∗) =μ_nx_n, x^∗=z, x^∗

for allx^∗∈E^∗. Such an elementz is inC. In fact, ifz /∈C, then there exists y^∗∈E^∗ by the separation theorem [19] such that

z, y^∗< inf

y∈Cy, y^∗. So, from{x_n} ⊂Cwe have

z, y^∗< inf

y∈Cy, y^∗ ≤ inf

n∈Nx_n, y^∗ ≤μ_nx_n, y^∗=z, y^∗.

This is a contradiction. Then we have z ∈C. From (2.3) we have that for y∈C andn∈N,

φ(x_n, y) =φ(x_n, T y) +φ(T y, y) + 2x_n−T y, JT y−Jy. So, we have that fory∈C,

μ_nφ(x_n, y) =μ_nφ(x_n, T y) +μ_nφ(T y, y) + 2μ_nx_n−T y, JT y−Jy

=μ_nφ(x_n, T y) +φ(T y, y) + 2z−T y, JT y−Jy. Since, by assumption,μ_nφ(x_n, T y)≤μ_nφ(x_n, y) for ally∈C, we have

μ_nφ(x_n, y)≤μ_nφ(x_n, y) +φ(T y, y) + 2z−T y, JT y−Jy. This implies that

0≤φ(T y, y) + 2z−T y, JT y−Jy.

We know thatz is an element ofC. Puttingy=z, we have that 0≤φ(T z, z) + 2z−T z, JT z−Jz.

Thus we have from (2.4) that

0≤φ(T z, z) +φ(z, z) +φ(T z, T z)−φ(z, T z)−φ(T z, z).

So, we have 0≤ −φ(z, T z) and hence 0 =φ(z, T z). SinceEis strictly convex,

we haveT z=z. This completes the proof.

Using Theorem 3.1, we prove a fixed point theorem for n-generalized nonspreading mappings in a Banach space; see [26, Remark 2].

Theorem 3.2. Let Cbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE and let T :C→C be ann-generalized nonspreading mapping. ThenT has a fixed point in C if and only if {T^mz} is bounded for somez∈C.

Proof. IfF(T)=∅, then{T^mz}={z}forz∈F(T). So,{T^mz}is bounded.

Conversely, letT :C→C ben-generalized nonspreading. Then there exist α₁, α₂, . . . , α_n, β₁, β₂, . . . , β_n, γ₁, γ₂, . . . , γ_n, δ₁, δ₂, . . . , δ_n∈R such that for allx, y∈C,

n k=1

α_kφ(T^n+1−kx, T y) +

1−ⁿ

k=1

α_k

φ(x, T y) +

n k=1

γ_k

φ(T y, T^n+1−kx)−φ(T y, x)

≤ⁿ

k=1

β_kφ(T^n+1−kx, y) +

1−ⁿ

k=1

β_k

φ(x, y) +

n k=1

δ_k

φ(y, T^n+1−kx)−φ(y, x) .

(3.3)

By assumption, we can takez ∈C such that{T^mz} is bounded. Replacing xbyT^mz in (3.3), we have that for anyy∈C andm∈N∪ {0},

n k=1

α_kφ(T^n+1−kT^mz, T y) +

1−ⁿ

k=1

α_k

φ(T^mz, T y) +

n k=1

γ_k

φ(T y, T^n+1−kT^mz)−φ(T y, T^mz)

≤ n k=1

β_kφ(T^n+1−kT^mz, y) +

1− n k=1

β_k

φ(T^mz, y) +

n k=1

δ_k

φ(y, T^n+1−kT^mz)−φ(y, T^mz) .

Since{T^mz}is bounded, we can apply a Banach limitμto both sides of the

above inequality. Then we have μ_m

_n

k=1

α_kφ(T^m+n+1−kz, T y) +

1−ⁿ

k=1

α_k

φ(T^mz, T y) +

n k=1

γ_k

φ(T y, T^m+n+1−kz)−φ(T y, T^mz)

≤μ_{m n}

k=1

β_kφ(T^m+n+1−kz, y) +

1− n k=1

β_k

φ(T^mz, y) +

n k=1

δ_k

φ(y, T^m+n+1−kz)−φ(y, T^mz) .

So, we obtain n k=1

α_kμ_mφ(T^m+n+1−kz, T y) +

1− n k=1

α_k

μ_mφ(T^mz, T y) +

n k=1

γ_k

μ_mφ(T y, T^m+n+1−kz)−μ_mφ(T y, T^mz)

≤ n k=1

β_kμ_mφ(T^m+n+1−kz, y) +

1− n k=1

β_k

μ_mφ(T^mz, y) +

n k=1

δ_k

μ_mφ(y, T^m+n+1−kz)−μ_mφ(y, T^mz)

and hence n k=1

α_kμ_mφ(T^mz, T y) +

1−ⁿ

k=1

α_k

μ_mφ(T^mz, T y) +

n k=1

γ_k

μ_mφ(T y, T^mz)−μ_mφ(T y, T^mz)

≤ⁿ

k=1

β_kμ_mφ(T^mz, y) +

1−ⁿ

k=1

β_k

μ_mφ(T^mz, y) +

n k=1

δ_k

μ_mφ(y, T^mz)−μ_mφ(y, T^mz) .

This implies

μ_mφ(T^mz, T y)≤μ_mφ(T^mz, y)

for ally∈C. By Theorem 3.1,T has a fixed point inC.

4. Some properties of generalized nonspreading mappings

In this section, we obtain fundamental properties for 2-generalized nonspread- ing mappings in a Banach space.

Proposition 4.1. Let E be a smooth Banach space and let C be a nonempty closed convex subset of E. Let α₁, α₂, β₁, β₂, γ₁, γ₂, δ₁, δ₂ ∈R. Then a map- ping T : C → C is (α₁, α₂, β₁, β₂, γ₁, γ₂, δ₁, δ₂)-generalized nonspreading if and only ifT satisfies that for anyx, y∈C,

0≤(β₁−α₁)

φ(T²x, T y)−φ(x, T y) + (β₂−α₂)

φ(T x, T y)−φ(x, T y)

+φ(T y, y) + 2x−T y+β₁(T²x−x) +β₂(T x−x), JT y−Jy

−γ₁

φ(T y, T²x)−φ(T y, x)

−γ₂

φ(T y, T x)−φ(T y, x) +δ₁

φ(y, T²x)−φ(y, x) +δ₂

φ(y, T x)−φ(y, x) .

Proof. Since a mappingT :C→Cis (α₁, α₂, β₁, β₂, γ₁, γ₂, δ₁, δ₂)-generalized nonspreading, we have that for anyx, y ∈C,

α₁φ(T²x, T y) +α₂φ(T x, T y) + (1−α₁−α₂)φ(x, T y) +γ₁

φ(T y, T²x)−φ(T y, x) +γ₂

φ(T y, T x)−φ(T y, x)

≤β₁φ(T²x, y) +β₂φ(T x, y) + (1−β₁−β₂)φ(x, y) +δ₁

φ(y, T²x)−φ(y, x) +δ₂

φ(y, T x)−φ(y, x) . Then we have from (2.3) that for anyx, y∈C,

0≤β₁φ(T²x, y) +β₂φ(T x, y) + (1−β₁−β₂)φ(x, y) +δ₁

φ(y, T²x)−φ(y, x) +δ₂

φ(y, T x)−φ(y, x)

−α₁φ(T²x, T y)−α₂φ(T x, T y)−(1−α₁−α₂)φ(x, T y)

−γ₁

φ(T y, T²x)−φ(T y, x)

−γ₂

φ(T y, T x)−φ(T y, x)

=β₁

φ(T²x, T y) +φ(T y, y) + 2T²x−T y, JT y−Jy +β₂

φ(T x, T y) +φ(T y, y) + 2T x−T y, JT y−Jy + (1−β₁−β₂)

φ(x, T y) +φ(T y, y) + 2x−T y, JT y−Jy +δ₁

φ(y, T²x)−φ(y, x) +δ₂

φ(y, T x)−φ(y, x)

−α₁φ(T²x, T y)−α₂φ(T x, T y)−(1−α₁−α₂)φ(x, T y)

−γ₁

φ(T y, T²x)−φ(T y, x)

−γ₂

φ(T y, T x)−φ(T y, x)

= (β₁−α₁)

φ(T²x, T y)−φ(T x, T y) + (β₂−α₂)

φ(T x, T y)−φ(x, T y)

+φ(T y, y) + 2β₁T²x+β₂T x+ (1−β₁−β₂)x−T y, JT y−Jy

−γ₁

φ(T y, T²x)−φ(T y, x)

−γ₂

φ(T y, T x)−φ(T y, x) +δ₁

φ(y, T²x)−φ(y, x) +δ₂

φ(y, T x)−φ(y, x) .

Hence we have that for anyx, y∈C, 0≤(β₁−α₁)

φ(T²x, T y)−φ(x, T y) + (β₂−α₂)

φ(T x, T y)−φ(x, T y)

+φ(T y, y) + 2x−T y+β₁(T²x−x) +β₂(T x−x), JT y−Jy

−γ₁

φ(T y, T²x)−φ(T y, x)

−γ₂

φ(T y, T x)−φ(T y, x) +δ₁

φ(y, T²x)−φ(y, x) +δ₂

φ(y, T x)−φ(y, x) .

This completes the proof.

LetE be a Banach space and letC be a nonempty subset ofE. LetT : C→Cbe a mapping. Thenp∈Cis an asymptotic fixed point ofT (see [17]) if there exists{x_n} ⊂C such thatx_n pand lim_n→∞x_n−T x_n= 0. We denote by ˆF(T) the set of asymptotic fixed points of T. Motivated by the concept of asymptotic fixed points, we have the following result. This result is used in Section 6.

Proposition 4.2. Let E be a strictly convex Banach space with a uniformly Gˆateaux differentiable norm, let C be a nonempty closed convex subset ofE and letT :C→C be a2-generalized nonspreading mapping with F(T)=∅. Suppose that{x_n}is a sequence inCsuch thatx_n p,lim_n→∞x_n−T x_n= 0 andlim_n→∞x_n−T²x_n= 0. Thenp∈F(T).

Proof. SinceT :C→Cis a 2-generalized nonspreading mapping, there exist α₁, α₂, β₁, β₂, γ₁, γ₂, δ₁, δ₂∈Rsuch that for anyx, y∈C,

α₁φ(T²x, T y) +α₂φ(T x, T y) + (1−α₁−α₂)φ(x, T y) +γ₁

φ(T y, T²x)−φ(T y, x) +γ₂

φ(T y, T x)−φ(T y, x)

≤β₁φ(T²x, y) +β₂φ(T x, y) + (1−β₁−β₂)φ(x, y) +δ₁

φ(y, T²x)−φ(y, x) +δ₂

φ(y, T x)−φ(y, x) .

(4.1)

Let {x_n} be a sequence in C such that x_n p, lim_n→∞x_n−T x_n = 0 and lim_n→∞x_n −T²x_n = 0. Since the norm of E is uniformly Gˆateaux differentiable, the duality mappingJ onE is uniformly norm-to-weak^∗ con- tinuous on each bounded subset ofE; see [20]. Using lim_n→∞x_n−T x_n= 0 and lim_n→∞x_n−T²x_n = 0, we have lim_n→∞w, JT x_n−Jx_n = 0 and lim_n→∞w, JT²x_n−Jx_n= 0 for allw∈E. On the other hand, replacingx byx_n andy bypin (4.1), we obtain that

α₁φ(T²x_n, T p) +α₂φ(T x_n, T p) + (1−α₁−α₂)φ(x_n, T p) +γ₁

φ(T p, T²x_n)−φ(T p, x_n) +γ₂

φ(T p, T x_n)−φ(T p, x_n)

≤β₁φ(T²x_n, p) +β₂φ(T x_n, p) + (1−β₁−β₂)φ(x_n, p) +δ₁

φ(p, T²x_n)−φ(p, x_n) +δ₂

φ(p, T x_n)−φ(p, x_n) .

(4.2)

We have from Proposition 4.1 and (4.2) that 0≤(β₁−α₁)

φ(T²x_n, T p)−φ(x_n, T p) + (β₂−α₂)

φ(T x_n, T p)−φ(x_n, T p)

+φ(T p, p)

+ 2x_n−T p+β₁(T²x_n−x_n) +β₂(T x_n−x_n), JT p−Jp

−γ₁

φ(T p, T²x_n)−φ(T p, x_n)

−γ₂

φ(T p, T x_n)−φ(T p, x_n) +δ₁

φ(p, T²x_n)−φ(p, x_n) +δ₂

φ(p, T x_n)−φ(p, x_n)

= (β₁−α₁)

T²x_n²− x_n²−2T²x_n−x_n, JT p + (β₂−α₂)

T x_n²− x_n²−2T x_n−x_n, JT p

+φ(T p, p) + 2x_n−T p+β₁(T²x_n−x_n) +β₂(T x_n−x_n), JT p−Jp

−γ₁

T²x_n²− x_n²−2T p, JT²x_n−Jx_n

−γ₂

T x_n²− x_n²−2T p, JT x_n−Jx_n +δ₁

T²x_n²− x_n²−2p, JT²x_n−Jx_n +δ₂

T x_n²− x_n²−2p, JT x_n−Jx_n .

(4.3)

From

|T²x_n²− x_n²|= (T²x_n+x_n)|T²x_n − x_n|

≤(T²x_n+x_n)T²x_n−x_n and

|T x_n²− x_n²|= (T x_n+x_n)|T x_n − x_n|

≤(T x_n+x_n)T x_n−x_n,

we have T²x_n²− x_n² → 0 and T x_n²− x_n² → 0 as n → ∞. So, lettingn→ ∞in (4.3), we have that

0≤φ(T p, p) + 2p−T p, JT p−Jp

=φ(T p, p) +φ(p, p) +φ(T p, T p)−φ(p, T p)−φ(T p, p)

=−φ(p, T p).

Thus φ(p, T p) ≤ 0 and then φ(p, T p) = 0. Since E is strictly convex, we

obtainp=T p. This completes the proof.

5. Nonlinear ergodic theorem

LetE be a smooth Banach space, letCbe a nonempty closed convex subset of E and let J be the duality mapping from E into E^∗. We know that a mappingT :C→C is called n-generalized nonspreading ifT satisfies (3.1).

Observe that if T : C → C is ann-generalized nonspreading mapping and F(T)=∅, then

φ(u, T y)≤φ(u, y)

for allu∈F(T) andy∈C. Indeed, puttingx=u∈F(T) in (3.1), we obtain n

k=1

α_kφ(u, T y) +

1− n k=1

α_k

φ(u, T y) +

n k=1

γ_k

φ(T y, u)−φ(T y, u)

≤ⁿ

k=1

β_kφ(u, y) +

1−ⁿ

k=1

β_k

φ(u, y) +

n k=1

δ_k

φ(y, u)−φ(y, u) . So, we have that

φ(u, T y)≤φ(u, y) (5.1)

for allu ∈ F(T) and y ∈ C. Similarly, putting y =u ∈ F(T) in (3.1), we obtain that forx∈C,

n k=1

α_kφ(T^n+1−kx, u) +

1−ⁿ

k=1

α_k

φ(x, u) +

n k=1

γ_k

φ(u, T^n+1−kx)−φ(u, x)

≤ n k=1

β_kφ(T^n+1−kx, u) +

1− n k=1

β_k

φ(x, u) +

n k=1

δ_k

φ(u, T^n+1−kx)−φ(u, x) and hence

n k=1

(α_k−β_k)

φ(T^n+1−kx, u)−φ(x, u)

+ n k=1

(γ_k−δ_k)

φ(u, T^n+1−kx)−φ(u, x)

≤0.

Ifα_k−β_k = 0 for all k= 1,2, . . . , n−1,γ_k ≤δ_k for all k= 1,2, . . . , nand α_n > β_n, then we have from (5.1) that

(α_n−β_n)

φ(T x, u)−φ(x, u)

≤ n k=1

(δ_k−γ_k)

φ(u, T^n+1−kx)−φ(u, x)

≤0.

So, we have that

φ(T x, u)≤φ(x, u) (5.2)

for all x ∈ C and u ∈ F(T). For example, let us consider a 2-generalized

nonspreading mappingT :C→C which satisfies (3.2); i.e., α₁φ(T²x, T y) +α₂φ(T x, T y) + (1−α₁−α₂)φ(x, T y)

+γ₁

φ(T y, T²x)−φ(T y, x) +γ₂

φ(T y, T x)−φ(T y, x)

≤β₁φ(T²x, y) +β₂φ(T x, y) + (1−β₁−β₂)φ(x, y) +δ₁

φ(y, T²x)−φ(y, x) +δ₂

φ(y, T x)−φ(y, x)

(5.3)

for allx, y ∈C. Then we have thatα₁ =β₁, α₂> β₂, γ₁ ≤δ₁ and γ₂≤δ₂ imply that

φ(T x, u)≤φ(x, u)

for all x ∈ C and u ∈ F(T). Now using the technique developed by [18, 25], we can prove the following nonlinear ergodic theorem for 2-generalized nonspreading mappings in a Banach space. For proving this result, we need the following lemma.

Lemma 5.1. Let E be a uniformly convex Banach space with a Fr´echet dif- ferentiable norm and let C be a nonempty closed convex sunny generalized nonexpansive retract ofE. LetT :C→Cbe a generalized nonexpansive map- ping, that is,F(T)=∅ andφ(T x, u)≤φ(x, u)for all x∈C andu∈F(T).

LetR be the sunny generalized nonexpansive retraction ofE ontoF(T). De- fine, for anyx∈C,

S_nx= 1 n

n−1

k=0

T^kx.

If each weak cluster point of {S_nx} belongs to F(T), then{S_nx} converges weakly to the strong limit of{RTⁿx}.

Proof. We know that since C is a sunny generalized nonexpansive retract of E, there exists the sunny generalized nonexpansive retraction P of E onto C. On the other hand, since a mapping T : C → C satisfies that F(T)=∅and

φ(T x, u)≤φ(x, u)

for all x ∈ C and u ∈ F(T), T is generalized nonexpansive. So, putting S =T P, we have that S is a generalized nonexpansive mapping of E into itself such thatF(S) =F(T). Indeed, we have

z∈F(T)⇐⇒z=T z⇐⇒z=P z=T P z⇐⇒z=Sz.

So, it follows that F(S) = F(T). We also have that for any x ∈ E and u∈F(S) =F(T),

φ(Sx, u) =φ(T P x, u)≤φ(P x, u)≤φ(x, u).

So,Sis a generalized nonexpansive mapping ofEinto itself such thatF(S) =

F(T). From Theorems 2.9 and 2.4, there exists the sunny generalized nonex- pansive retractionR of E ontoF(T). From Theorem 2.7, this retraction R is characterized by

Rx= argmin_u∈F(T)φ(x, u).

We also know from Theorem 2.5 that

0≤ v−Rv, JRv−Ju ∀u∈F(T), v∈C.

Adding upφ(Rv, u) to both sides of this inequality, we have φ(Rv, u)≤φ(Rv, u) + 2v−Rv, JRv−Ju

=φ(Rv, u) +φ(v, u) +φ(Rv, Rv)−φ(v, Rv)−φ(Rv, u)

=φ(v, u)−φ(v, Rv).

(5.4)

Sinceφ(T z, u)≤φ(z, u) for anyu∈F(T) andz∈C, it follows that φ(Tⁿx, RTⁿx)≤φ(Tⁿx, RTⁿ⁻¹x)

≤φ(Tⁿ⁻¹x, RTⁿ⁻¹x).

Hence the sequenceφ(Tⁿx, RTⁿx) is nonincreasing. Puttingu=RTⁿxand v=T^mxwithn≤min (5.4), we have from Theorem 2.3 that

g(RT^mx−RTⁿx)≤φ(RT^mx, RTⁿx)

≤φ(T^mx, RTⁿx)−φ(T^mx, RT^mx)

≤φ(Tⁿx, RTⁿx)−φ(T^mx, RT^mx),

where g is a strictly increasing, continuous and convex real-valued function with g(0) = 0. From the properties of g, {RTⁿx} is a Cauchy sequence.

Therefore, {RTⁿx} converges strongly to a point q ∈ F(T) since F(T) is closed from Theorem 2.8. Next consider a fixed x ∈ C and an arbitrary subsequence{S_nix} of{S_nx} such thatS_nix v. By assumption, we know that v ∈F(T). Rewriting the characterization of the retraction R, we have that

0≤ T^kx−RT^kx, JRT^kx−Ju and hence

T^kx−RT^kx, Ju−Jq ≤ T^kx−RT^kx, JRT^kx−Jq

≤ T^kx−RT^kx · JRT^kx−Jq

≤KJRT^kx−Jq,

whereKis an upper bound forT^kx−RT^kx. Summing up these inequalities fork= 0,1, . . . , n−1, we get

S_nx− 1 n

n−1

k=0

RT^kx, Ju−Jq

≤K1 n

n−1

k=0

JRT^kx−Jq,

where S_nx = _n¹_n−1

k=0T^kx. Letting n_i → ∞ and remembering that J is

continuous, we get

v−q, Ju−Jq ≤0.

This holds for anyu∈F(T). Therefore Rv=q. But becausev ∈F(T), we havev=q. Thus the sequence{S_nx}converges weakly to the pointq, where

q= lim_n→∞RTⁿx.

Using Lemma 5.1, we obtain the following nonlinear ergodic theorems for 2-generalized nonspreading mappings in a Banach space.

Theorem 5.2. Let E be a uniformly convex Banach space with a Fr´echet differentiable norm and letC be a nonempty closed convex sunny generalized nonexpansive retract of E. Let T : C → C be a 2-generalized nonspreading mapping with F(T) = ∅ such that φ(T x, u) ≤ φ(x, u) for all x ∈ C and u∈F(T). LetR be the sunny generalized nonexpansive retraction ofE onto F(T). Then, for anyx∈C,

S_nx= 1 n

n−1

k=0

T^kx

converges weakly to an elementq ofF(T), whereq= lim_n→∞RTⁿx.

Proof. Since a 2-generalized nonspreading mappingT :C→C withF(T)=

∅ satisfies that φ(T x, u) ≤ φ(x, u) for all x ∈ C and u ∈ F(T), T is gen- eralized nonexpansive. Fix x ∈ C. To show the theorem, it is sufficient to show from Lemma 5.1 that each weak cluster point of {S_nx} belongs to F(T). SinceT :C→Cis a 2-generalized nonspreading mapping, there exist α₁, α₂, β₁, β₂, γ₁, γ₂, δ₁, δ₂∈Rsatisfying (5.3). SinceF(T)=∅, we have from (5.1) that φ(u, T y) ≤ φ(u, y) for all u ∈ F(T) and y ∈ C. Taking a fixed pointuofT, we have that forx∈C,φ(u, Tⁿx)≤φ(u, x) for alln∈N. Then {Tⁿx}is bounded. ReplacingxbyT^kxin (5.3), we have that for anyy∈C andk∈N∪ {0},

α₁φ(T^k+2x, T y) +α₂φ(T^k+1x, T y) + (1−α₁−α₂)φ(T^kx, T y) +γ₁

φ(T y, T^k+2x)−φ(T y, T^kx) +γ₂

φ(T y, T^k+1x)−φ(T y, T^kx)

≤β₁φ(T^k+2x, y) +β₂φ(T^k+1x, y) + (1−β₁−β₂)φ(T^kx, y) +δ₁

φ(y, T^k+2x)−φ(y, T^kx) +δ₂

φ(y, T^k+1x)−φ(y, T^kx)

=β₁

φ(T^k+2x, T y) +φ(T y, y) + 2T^k+2x−T y, JT y−Jy +β₂

φ(T^k+1x, T y) +φ(T y, y) + 2T^k+1x−T y, JT y−Jy + (1−β₁−β₂)

φ(T^kx, T y) +φ(T y, y) + 2T^kx−T y, JT y−Jy +δ₁

φ(y, T^k+2x)−φ(y, T^kx) +δ₂

φ(y, T^k+1x)−φ(y, T^kx) .

(5.5)