Nonlinear Mean Ergodic Theorems for Semigroups in Hilbert Spaces

Volume 2007, Article ID 73246,9pages doi:10.1155/2007/73246

Research Article

Nonlinear Mean Ergodic Theorems for Semigroups in

Hilbert Spaces

Seyit Temir and Ozlem Gul

Received 26 December 2006; Accepted 4 April 2007

Recommended by Nan-Jing Huang

LetKbe a nonempty subset (not necessarily closed and convex) of a Hilbert space and letΓ= {T(t); t≥0}be a semigroup onK and letα(·) : [0,∞)→K be an almost orbit ofΓ. In this paper, we prove that every almost orbit ofΓis almost weakly and strongly convergent to its asymptotic center.

Copyright © 2007 S. Temir and O. Gul. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetK be a nonempty subset of a Hilbert spaceᏴ, whereK is not necessarily closed and convex. A familyΓ= {T(t); t≥0}of mappingsT(t) is called a semigroup onKif

(S1)T(t) is a mapping fromKinto itself fort≥0,

(S2)T(0)x=xandT(t+s)x=T(t)T(s)xforx∈Kandt,s≥0,

(S3) for eachx∈K,T(·)xis strongly measurable and bounded on every bounded subinterval of [0,∞).

LetΓbe a semigroup onK. ThenF= {x∈K:T(t)x=x,t≥0}is said to be fixed-points set ofΓ. We state a condition introduced by Miyadera [1]. If, for every bounded setB⊂K,v∈K, ands≥0, there exists aδs(B,v)≥0 with lims→∞δs(B,v)=0 such that

_{T(s)u−T(s)v≤ u−v+δs(B,v) (1.1)}

Definition 1.1. A functiona(·) : [0,∞)→Kis called almost-orbit ofΓifa(·) : [0,∞)→K is strongly measurable and bounded on every bounded subinterval of [0,∞) and if

lim

t→ ∞sups≥0

_a(s+t)−Tk_(s)a(t)p_{=0. (1.2)}

Using these conditions, we prove that every almost-orbit ofΓis weakly and strongly convergent to its asymptotic center (see [1]). Xu [2] studied strong asymptotic behavior of almost-orbits of both of nonexpansive and asymptotically nonexpansive semigroups. Takahashi [3] generalized the nonlinear ergodic theorems for general semigroups of non-expansive mappings. Kada and Takahashi [4] proved a strong ergodic theorem for general semigroups of nonexpansive mappings. Oka [5] proved nonlinear ergodic theorems for commutative semigroups of asymptotically nonexpansive mappings. All of the above-mentioned authors studied, except Miyadera’s works,Kas a closed and convex subset of a Hilbert space. Miyadera [1] studied almost convergence of almost-orbits of semigroup of non-Lipschitzian mappings in Hilbert spaces. Miyadera [1] proved the following the-orem. IfΓis asymptotically nonexpansive in the weak sense andFis nonempty set, then the following conditions holds:

(a1)a(·) is weakly almost convergent to its asymptotic centery,

(a2) ifyis an element ofKand ifT(t0) :K→Kis continuous for somet0>0, theny is a fixed point ofΓ, that is,ybelongs toF.

There are some conditions in the discrete case in [6–8]. Miyadera [7,8] showed that the condition in [6] could be replaced by a weaker condition introduced in [7, 8]. Miyadera [7,8] and Wittmann [6] proved nonlinear ergodic theorems where the closed-ness and convexity ofKand the asymptotically nonexpansivity ofTwere not assumed. In this paper, in the light of these papers we establish weak ergodic theorem for semigroups of mappings onKsatisfying condition (I) given in the statement ofTheorem 3.1. We also establish strong ergodic theorem for semigroups of mappings onK satisfying condition (II) given before statement ofTheorem 4.1. This paper is organized as follows.

InSection 2, we prove the covering lemmas we need for establishing weakly conver-gence result. InSection 3, we deal witha(·) almost-orbit weakly almost-convergent to its asymptotic center with respect to condition (I). In the last section, we investigate strong convergence using condition (II). We establish that every almost-orbit of Γis strongly almost-convergent to its asymptotic center.

2. Lemmas

Leta(·) : [0,∞)→Ᏼbe a function strongly measurable and bounded on every bounded subinterval of [0,∞) and leta(t)be convergent ast→ ∞.

Lemma2.1 [1]. Forr,s,t≥0, the following statements are mutually equivalent: (i) lims→∞limt→∞limr→∞[(a(t+r),a(t))−(a(s+r),a(s))]≤0;

(ii) lims→∞limt→∞limr→∞[a(t+r) +a(t)2− a(s+r) +a(s)2]≤0; (iii) lims→∞limt→∞limr→∞[a(s+r)−a(s)2− a(t+r)−a(t)2]≤0.

Lemma2.2 [1]. Leta(·) : [0,∞)→Ᏼbe a function strongly measurable and bounded on every bounded subinterval of[0,∞)and leta(t)be convergent ast→ ∞. Then, one has that the following statements are mutually equivalent:

(i) lims→∞limt→∞supr≥0[(a(t+r),a(t))−(a(s+r),a(s))]≤0; (ii) lims→∞limt→∞supr≥0[a(t+r) +a(t)2− a(s+r) +a(s)2]≤0; (iii) lims→∞limt→∞supr≥0[a(s+r)−a(s)2− a(t+r)−a(t)2]≤0.

a(t)is convergent ast→ ∞. Moreover, ifa(·)satisfies the equivalent conditions (i), (ii), and (iii), thena(·)is strongly almost-convergent to its asymptotic centery.

Remark 2.3. We can take the following conditions instead of (ii) and (iii) inLemma 2.2, forA,C >0,

(ii) lims→∞limt→∞supr≥0[a(t+r) +a(t)2−Aa(s+r) +a(s)2]≤0; (iii) lims→∞limt→∞supr≥0[a(s+r)−a(s)2−Aa(t+r)−a(t)2]≤0. We can obtain

lim

s→ ∞tlim→ ∞supr≥0

_{a(t+r) +a(t)}2

−Aa(s+r) +a(s)2

≤ lim

s→ ∞tlim→ ∞supr≥0

_{a(t+r) +a(t)}2

−a(s+r) +a(s)2≤0, (2.1)

and

lim

s→ ∞tlim→ ∞supr≥0

_{a(s+r)−a(s)}2

−Aa(t+r)−a(t)2

≤lim

s→ ∞tlim→ ∞supr≥0

_{a(s+r)−a(s)}2

−a(t+r)−a(t)2≤0. (2.2)

Moreover, we can write

lim

s→ ∞tlim→ ∞supr≥0

_{a(s+r)−a(s)}2

−Aa(t+r)−a(t)2

−C≤0. (2.3)

Note thatLemma 2.2holds for this condition.

3. Weak ergodic theorems

LetᏴbe a Hilbert space with inner product (·,·) and · norm, and letKbe a nonempty subset ofᏴ, whereK is not necessarily closed and convex. LetΓ= {T(t); t≥0} be a semigroup acting onK.

Theorem3.1. Suppose that for every bounded setB⊂K,v∈K,u∈Bandr≥0, there existsδr(B,v)≥0withlimr→∞δr(B,v)=0such that

_Tk_(r)u−Tk_(r)vp

≤λru−vp+cλrup−Tk(r)up+λrvp−Tk(r)vp+δr(B,v),

whereλr,care nonnegative constants such thatlimr→∞λr=1, andp≥1. IfF= ∅orc >0,

thena(·)is almost weakly convergent to its asymptotic center, which isy.

Proof. SupposeF= ∅andc=0 for the semigroupΓ= {T(t); t∈R+_{}. Then foru=x} and f ∈F, we can takeB= {x}. If we writeu=xandv=f in (I), then we have

_Tk_(r)x−Tk_(r)fp_=Tk_(r)x−fp_≤λ

rx−fp+δr(B,f). (3.1)

Thus, for everyx∈K, the sequence{Tk_{(r)x−f + f} = {T}k_{(r)x}is bounded. Let}

a(·) : [0,∞)→Kbe almost-orbit ofΓ.

FromDefinition 1.1, we have limt→∞sups≥0[a(t+s)−Tk(s)a(t)p]=0. There ist0= t0(ε)>0 forε >0,t≥t0, ands≥0 such thata(t+s)−Tk(s)a(t)p< ε.

In particular, fors≥0, we havea(s+t0)−Tt0(s)a(t0)p< ε. If we consider both this inequality and boundness of sequence{Tk_{(s)x}, we have}

_a

s+t0

−Tt0(s)at

0

+Tt0(s)at

0p <2p−1_as+t

0

−Tt0₍s₎at

0p+Tt0(s)a

t0p

<2p−1ε+ 2p−1Tt0(s)pat

0p,

(3.2)

then{a(s);s∈R+_{}is bounded.}

If we take in (I),B= {a(t); t∈R+_{}, andv= f, then we obtain}

_Tk_(r)a(t)−Tk_(r)fp_≤λ

ra(t)−fp. (3.3)

Thus

_a(r+t)−fp_≤a(r

+t)−Tk(r)a(t) +Tk(r)a(t)−fp

≤2p−1_a(r+t)−Tk_(r)a(t)p_+Tk_(r)a(t)−fp <2p−1ε+λra(t)−fp

(3.4)

(sinceTk_(r)a(t)−Tk_(r)fp_{≤λra(t)−f}p_).

Taking limit asr→ ∞, because of limr→∞λr=1, for arbitraryε,

lim

r→∞a(r+t)−f

p_<

2p−1_rlim_→∞a(t)−fp. (3.5)

Therefore{a(t)−f}is convergent.

forh≥0, we have

_Tk_(t−s)Tk_(h)a(s)−Tk_(t−s)a(s)p_≤λ

t−sTk(h)a(s)−a(s)p+δt−sB,a(s).

(3.6)

Consequently,

_{a(t+h)−a(t)}p

≤a(t+h)−Tk_{(t+h−s)a(s) +T}k_{(t+h−s)a(s)−a(t)}p

≤2p−1a(t+h)−Tk_{(t+h−s)a(s)}p_+Tk_{(t+h−s)a(s)−a(t)}p

=2p−1_a(t+h)−Tk_{(t+h−s)a(s)}p

+Tk(t+h−s)a(s) +Tk_{(t−s)a(s)−T}k_{(t−s)a(s)−a(t)}p

≤2p−1_{a(t+h−s) +s−T}k_{(t+h−s)a(s)}p + 22(p−1)Tk(t−s)Tk_(h)a(s)−Tk_(t−s)a(s)p

+ 22(p−1)_Tk_{(t−s)a(s)−a(t)}p <2p−1ε+ 22(p−1)λt−sTk(h)a(s)−a(s)p

+ 22(p−1)_Tk_{(t−s)a(s)−a(t−s+s)}p_+δ

t−sB,a(s)

<2p−1_{ε1 + 2}p−1_{+ 2}2(p−1)_λ

t−sTk(h)a(s)−a(s+h)

+a(s+h)−a(s)p+δt−sB,a(s)

<2p−1ε1 + 2p−1+ 22(p−1)ελt−s

+ 22(p−1)_λ

t−sa(s+h)−a(s)p+δt−sB,a(s).

(3.7)

Then

_{a(t+h)−a(t)}p₋₂2(p−1)_λ

t−sa(s+h)−a(s)p<2p−1ε1+2p−1+2p−1λt−s+δt−sB,a(s).

(3.8)

Taking limit ast,s→ ∞, forh≥0 and arbitraryε, from the last inequality, we obtain

lim

t→ ∞slim→ ∞suph≥0

_{a(t+h)−a(t)}p

−22(p−1)_{a(s+h)−a(s)}p_{≤0. (3.9)}

FromRemark 2.3,a(·) is weakly almost convergent to its asymptotic center. Now, we investigate the caseF= ∅andc >0.

Forx∈K, if we writeB= {x}andv=xin (I), then we obtain

and from this we can write

_Tk_(r)xp_≤λ

rxp+δr(B,_{2c .}x) (3.11)

Then for everyx∈K,{Tk(t)x;t∈R+_{}is bounded. ByDefinition 1.1, forε >0 taking} t≥t0, s≥0, there exists t0=t0(ε) such that a(t0+s)−Tk(s)a(t0)p < ε. Since

{Tk_(t)x;t∈R+_{}is bounded,}

_a

t0+s−Tk(s)at0+Tk(s)at0p

≤2(p−1)at0+s

−Tk_(s)at

0p+Tk(s)pa

t0p

(3.12)

{a(t); t∈R+_{}is bounded. We can takeB= {T}k_{(h)a(s) :h≥0}, if we writev=a(s) and}

r=t−sin (I), then we obtain

_Tk_(t−s)Tk_(h)a(s)−Tk_(t−s)a(s)p

≤λt−sTk(h)a(s)−a(s)p+cλt−sTk(h)a(s)p

−Tk_(t−s)Tk_(h)a(s)p_+λ

t−sa(s)p

−Tk_(t−s)a(s)p_+δ

t−sB,a(s).

(3.13)

Consequently,

_{a(t+h)−a(t)}p

=a(t+h)−Tk_{(t+h−s)a(s) +T}k_{(t+h−s)a(s)−a(t)}p

≤2p−1a(t+h)−Tk(t+h−s)a(s)p+Tk(t+h−s)a(s)−a(t)p

≤2p−1_a(t+h)−Tk_{(t+h−s)a(s)}p + 2p−1Tk(t+h−s)a(s)−Tk(t−s)a(s)p

+Tk(t−s)a(s)−a(t)p≤2p−1_a(t+h)−Tk_{(t+h−s)a(s)}p

+ 22(p−1)Tk(t−s)Tk_(h)a(s)−Tk_(t−s)a(s)p

+Tk(t−s)a(s)−a(t−s+s)p <2p−1ε+ 22(p−1)λt−sTk(h)a(s)−a(s)p

+cλt−sTk(h)a(s)p−Tk(t−s)Tk(h)a(s)p

+λt−sa(s)p−Tk(t−s)a(s)p

+δt−sB,a(s)+ε22(p−1).

Takinga(s) ≤MandTk_{(h) ≤N,}

<2p−1_{ε1 + 2}p−1_{+ 2}2(p−1)_λ

t−sTk(h)a(s)−a(s)p

+cλt−sMpNp−M2Np+λt−sMp−MpNp+δt−sB,a(s)

<2p−1ε1 + 2p−1+cλt−sMpNp+ 1−cMpNpMp−1

+ 22(p−1)λt−sTk(h)a(s)−a(s+h) +a(s+h)−a(s)p+δt−sB,a(s)

<2p−1ε1 + 2p−1+cλt−sMpNp+ 1−cMpNpMp−1 + 22(p−1)_λ

t−s2p−1Tk(h)a(s)−a(s+h)p

+ 22(p−1)λt−s2p−1a(s+h)−a(s)p+δt−sB,a(s)

<2p−1ε1 + 2p−1+cλt−sMpNp+ 1−cMpNpMp−1+ 23(p−1)λt−sε

+ 23(p−1)_λ

t−sa(s+h)−a(s)p+δt−sB,a(s).

(3.15)

Taking limit ast,s→ ∞, forh≥0,

lim

t→ ∞ lim

s→ ∞ sup

h≥0

_{a(t+h)−a(t)}p

−23(p−1)_{a(s+h)−a(s)}p_{≤A.´ (3.16)}

That is,

lim

t→ ∞ lim

s→ ∞ sup

h≥0

_{a(t+h)−a(t)}p

−23(p−1)a(s+h)−a(s)p−_A´_{≤0. (3.17)}

Then fromRemark 2.3,a(·) is weakly almost convergent to its asymptotic center. Thus,

the proof is completed.

4. Strong ergodic theorems

LetΓ= {T(t);t≥0}be semigroup onK. Suppose that for every bounded setB⊂Kand integerk≥0, there exists aδr(B,v)≥0 with limr→∞δr(B,v)=0 such that

_Tk_(r)u+Tk_(r)vp

≤λru+vp+cλrup−Tk(r)up+λrvp−Tk(r)vp+δr(B) (II)

foru,v∈B, whereλr,c, and p are nonnegative constants such that limr→∞λr=1 and

p≥1.

Theorem4.1. IfΓ= {T(t);t≥0}is a semigroup onKsatisfying condition (II), then every almost-orbit ofΓis strongly almost convergent to its asymptotic center.

Proof. Leta(·) : [0,∞)→Kbe almost-orbit ofΓ. Fort≥0, we set

ϕ(s)=sup

t≥0

_a(t+s)−Tk_{(t)a(s). (4.1)}

Whens→ ∞,ϕ(s)→0 and from condition (II) by takingB= {x}andv=x, we have

_Tk_(r)xp_≤λ

rxp+δr

{x}

Thus forx∈K,{Tk_{(r)x:r≥0}is bounded. ByDefinition 1.1, since{T}k_{(r)x:r≥0}is}

bounded, we have

_a(s+t)−Tk_(s)a(t)p_{≤ε. (4.3)}

Therefore{a(s) :s≥0}is bounded. Letr > h≥0. Since{Tk_{(h)x:h≥0}and{a(s) :s≥0}}

are bounded then{Tk_{(h)a(s) :h≥0}is bounded, by using (II) withB= {T}k_{(h)a(s) :h≥}

0},v=a(s) andr=t−swe have

_Tk_(t−s)Tk_{(h)a(s) +T}k_(t−s)a(s)p_≤λ

t−sTk(h)a(s) +a(s)p

+cλt−sTk(h)a(s)p−Tk(t−s)Tk(h)a(s)p

+λt−sa(s)p−Tk(t−s)a(s)p+δt−s(B).

(4.4)

Forc=0, we have

_Tk_(t−s)Tk_{(h)a(s) +T}k_(t−s)a(s)p_≤λ

t−sTk(h)a(s) +a(s)p+δt−sB,a(s). (4.5)

Consequently,

_{a(t+h) +a(t)}p_≤2p−1_{a(t+r) +s−s−T}k_{(t+h−s)a(s)}p

+ 22(p−1)Tk(t+h−s)a(s) +Tk_(t−s)a(s)p

+a(t−s+s)−Tk_(t−s)a(s)p

≤2p−1ϕp(s) + 22(p−1)Tk(t−s)Tk_{(h)a(s) +T}k_(t−s)a(s)p_+ϕp_(s)

≤2p−1ϕp(s) + 22(p−1)λt−sTk(h)a(s) +a(s)p+ϕp(s)+δt−sB,a(s)

≤2p−1_ϕp_{(s) + 2}2(p−1)_λ

t−sϕp(s)

+ 22(p−1)λt−sTk(h)a(s) +a(s) +a(h+s)−a(h+s)p+δt−sB,a(s)

≤2p−1_ϕp_{(s)1 + 2}p−1_λ

t−s

+ 22(p−1)2p−1λt−sa(h+s) +a(s)p+Tk(h)a(s)−a(h+s)p +δt−sB,a(s).

(4.6)

Taking limit ass,t→ ∞, forh≥0,

lim

t→ ∞ sup

h≥0

_{a(t+h) +a(t)}p

−23(p−1)_λ

t−sa(h+s) +a(s)p

≤2p−1ϕp(s)1 + 2p−1λt−s+λt−s2p−1.

(4.7)

Sinceϕ(s)→0, we have

lim

s→ ∞ lim

t→ ∞ sup

h≥0

_{a(t+h) +a(t)}p

that is, condition (ii) inLemma 2.2is satisfied. Thus, every almost-orbit ofΓis strongly almost convergent to its asymptotic center.

Remark 4.2. Our results presented in this paper generalize the results of Miyadera [7,8] to the case ofF= ∅andc >0 for semigroups of asymptotically nonexpansive mappings in Hilbert spaces.

References

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[2] H.-K. Xu, “Strong asymptotic behavior of almost-orbits of nonlinear semigroups,”Nonlinear Analysis, vol. 46, no. 1, pp. 135–151, 2001.

[3] W. Takahashi, “A nonlinear ergodic theorem for an amenable semigroup of nonexpansive map-pings in a Hilbert space,”Proceedings of the American Mathematical Society, vol. 81, no. 2, pp. 253–256, 1981.

[4] O. Kada and W. Takahashi, “Strong convergence and nonlinear ergodic theorems for commu-tative semigroups of nonexpansive mappings,”Nonlinear Analysis, vol. 28, no. 3, pp. 495–511, 1997.

[5] H. Oka, “Nonlinear ergodic theorems for commutative semigroups of asymptotically nonex-pansive mappings,”Nonlinear Analysis, vol. 18, no. 7, pp. 619–635, 1992.

[6] R. Wittmann, “Mean ergodic theorems for nonlinear operators,”Proceedings of the American Mathematical Society, vol. 108, no. 3, pp. 781–788, 1990.

[7] I. Miyadera, “Nonlinear mean ergodic theorems,”Taiwanese Journal of Mathematics, vol. 1, no. 4, pp. 433–449, 1997.

[8] I. Miyadera, “Nonlinear mean ergodic theorems—II,”Taiwanese Journal of Mathematics, vol. 3, no. 1, pp. 107–114, 1999.

Seyit Temir: Department of Mathematics, Arts and Science Faculty, Harran University, 63200 Sanliurfa, Turkey

Email address:[email protected]

Ozlem Gul: Department of Mathematics, Arts and Science Faculty, Harran University, 63200 Sanliurfa, Turkey