E ec i e esul s on composi ions o nonexpansi e
mappings
Lau en ¸iu Leu¸s ean1, Ad iana Nicolae2,3
1Simion S oilow Ins i u e o Ma hema ics o he Romanian Academy, Resea ch uni 5,
P. O. Box 1-764, RO-014700 Bucha es , Romania
2Depa men o Ma hema ics, Babe¸s-Bolyai Uni e si y,
Kog˘alniceanu 1, 400084 Cluj-Napoca, Romania
3Simion S oilow Ins i u e o Ma hema ics o he Romanian Academy,
Resea ch g oup o he p ojec PD-3-0152,
P. O. Box 1-764, RO-014700 Bucha es , Romania
E-mails: Lau en[email p o ec ed], [email p o ec ed]cluj. o
Sep embe 17, 2013
Abs ac
This pape p o ides uni o m bounds on he asymp o ic egula i y o
i e a ions associa ed o a ini e amily o nonexpansi e mappings. We ob-
ain ou quan i a i e esul s in he se ing o ( , δ)-con ex spaces, a class
o geodesic spaces which gene alizes me ic spaces wi h a con ex geodesic
bicombing.
MSC: 47J25; 47H09; 53C23; 03F10.
Keywo ds: E ec i e a es o asymp o ic egula i y; P oo mining; Families
o nonexpansi e mappings; Halpe n i e a ion; Con ex geodesic bicombing.
1 In oduc ion
Le Xbe a Hilbe space, C⊆Xa closed con ex subse , T1, . . . , TN:C→C
(whe e N∈Z+) a ini e amily o nonexpansi e mappings and (λn) a sequence
in [0,1]. Gi en u∈C, one can de ine an i e a ion s a ing om uby
x0=u, xn+1 =λn+1u+ (1 −λn+1)Tn+1xn,(1)
whe e Tn=Tnmod Nand he mod N unc ion akes alues in 1, . . . , N. Fo
he special case N= 1, his i e a ion coincides wi h he well-known Halpe n
i e a ion [8], whose s ong con e gence was p o ed by Wi mann [22] unde
1
a Xi :1306.5307 2 [ma h.FA] 14 Sep 2013
sui able condi ions on (λn), which a e sa is ied by he na u al candida e λn=
1
n+1 .
The gene al i e a ion de ined by (1) was i s s udied in Hilbe spaces
by Lions [18], who assumed di e en hypo heses on (λn), wi h he d awback
ha λn=1
n+1 does no sa is y hem. Bauschke [2] p o ed ha he i e a-
ion (xn) gi en by (1) con e ges s ongly o he common ixed poin o he
mappings T1, . . . , TNwhich is nea es o u, unde he assump ions ha F:=
N
i=1
Fix(Ti) is nonemp y, F= Fix(TNTN−1···T1) = . . . = Fix(T1TN···T2) =
Fix(TN−1···T1TN) and ha (λn) sa is ies
lim
n→∞ λn= 0,
∞
X
n=1 |λn+N−λn|<∞and
∞
X
n=1
λn=∞.(2)
Bauschke’s esul is a gene aliza ion o Wi manns’s heo em o a ini e
amily o mappings, since o N= 1 condi ions (2) coincide wi h he ones used
by Wi mann.
Ano he i e a ion ha will be conside ed in his pape is he ollowing one
associa ed o any nonexpansi e mapping T:C→C,
xn+1 =T(λn+1u+ (1 −λn+1)xn).(3)
This i e a ion, s udied by Xu [23], is a disc e e e sion o he app oxima ing
cu e z =T( u + (1 − )z ), ∈(0,1), analyzed by Combe es and Hi s oaga
[7]. S ong con e gence o he i e a ion (3) was es ablished by Xu in he se ing
o uni o mly smoo h Banach spaces unde app op ia e assump ions on (λn),
including hose gi en by (2). The s ong con e gence esul s o Xu and Com-
be es and Hi s oaga we e ex ended in [6] o mo e gene al app oxima ing cu es
and i e a ions. As abo e, one can de ine o he i e a ion (3) a cyclic algo i hm
associa ed o he ini e amily o nonexpansi e mappings T1, . . . , TN:C→C,
x0=u, xn+1 =Tn+1(λn+1u+ (1 −λn+1)xn).(4)
A e y impo an concep in he s udy o he asymp o ic beha io o nonlin-
ea i e a ions is he so-called asymp o ic egula i y, in oduced by B owde and
Pe yshyn [4] in hei s udy o solu ions o nonlinea unc ional equa ions using
Pica d i e a ions: T:C→Cis asymp o ically egula i lim
n→∞ kTnx−Tn+1xk=
0 o all x∈C. Mo e gene ally, an i e a ion (xn) associa ed o a mapping Tis
said o be asymp o ically egula i lim
n→∞ kxn−Txnk= 0 o all s a ing poin s
in C.
A na u al ques ion is o compu e a es o asymp o ic egula i y o he i e -
a ion (xn), i.e. a es o con e gence o (kxn−Txnk) owa ds 0. Fo he Halpe n
i e a ion his was done in a se ies o pape s [15,16,13,14,17], co esponding
o di e en classes o spaces. Fo he i e a ion gi en by (3), such a es we e
ob ained in [6].
2
The no ion o asymp o ic egula i y can be ex ended o sequences (xn) as-
socia ed o a amily o mappings T1, . . . , TN:C→C, as i is he case in
his pape . Thus, we say ha (xn) is asymp o ically egula i lim
n→∞ kxn−
Tn+N···Tn+1xnk= 0 o all s a ing poin s in C. The ollowing asymp o ic
egula i y esul is con ained in Bauschke’s s ong con e gence p oo o he
i e a ion (1).
Theo em 1.1. Le Xbe a Hilbe space, C⊆Xcon ex, T1, . . . , TN:C→C
nonexpansi e mappings and (λn)a sequence in [0,1] sa is ying (2). Le (xn)be
gi en by (1)and assume ha (xn)is bounded. Then,
lim
n→∞ kxn−Tn+N···Tn+1xnk= 0.
The main esul o his pape is a quan i a i e e sion o Theo em 1.1 o
bo h i e a ions (1) and (4). In o de o ge his esul we apply me hods o
p oo mining de eloped by Kohlenbach [12] wi h he aim o ob aining e ec i e
and uni o m bounds om p oo s whe e such in o ma ion is no eadily a ailable.
As a consequence, we p o ide o he i s ime e ec i e and uni o m a es o
asymp o ic egula i y o he i e a ions (1) and (4).
Ac ually, we ob ain ou quan i a i e esul s in a se ing mo e gene al han
he one o no med space. Mo e p ecisely, we in oduce ( , δ)-con ex spaces, a
class o me ic spaces which also includes Busemann spaces (and, hence, CAT(0)
spaces), hype con ex spaces, CAT(κ) spaces wi h κ > 0, as well as he so-called
W-hype bolic spaces (see [11]). Consequen ly, e en when N= 1 and so (1)
educes in ac o he Halpe n i e a ion, ou esul s gene alize ones ob ained
p e iously by he au ho s o CAT(κ) spaces wi h κ > 0 [17] and by he i s
au ho o no med [15] o W-hype bolic spaces [16].
2( , δ)-con ex spaces
Le (X, d) be a me ic space. We ecall i s basic ac s in geodesic geome y.
Gi en x, y ∈X, a cons an speed geodesic om x o yis a mapping γ: [0,1] →X
such ha γ(0) = x,γ(1) = yand d(γ(s), γ( )) = |s− |d(x, y) o all s, ∈[0,1].
The image γ([0,1]) o γis a geodesic segmen which joins xand y. No e ha
a geodesic segmen om x o yis no necessa ily unique. Gi en ∈(0,∞], we
say ha (X, d) is a (uniquely) -geodesic space i e e y wo poin s x, y ∈Xwi h
d(x, y)≤ can be joined by a (unique) geodesic segmen . Fo =∞, we say
simply ha Xis a (uniquely) geodesic space.
Le ∈(0,∞] and Xbe an -geodesic space. We conside an -geodesic
bicombing Γ on X, ha is, a choice o a cons an speed geodesic γx,y joining x
and y o each pai o poin s x, y ∈Xwi h d(x, y)≤ . When =∞, Γ is called
a geodesic bicombing on X. I Xis uniquely -geodesic, hen clea ly one can
de ine an -geodesic bicombing in a unique way. Fo γx,y ∈Γ, we deno e by [x, y]
he geodesic segmen γx,y([0,1]). A subse Co Xis -con ex i [x, y]∈C o all
x, y ∈Cwi h d(x, y)≤ . Gi en ∈[0,1], we use he no a ion (1 − )x+ y o
3
γx,y( ). Then, d(x, (1− )x+ y) = d(x, y) and d(y, (1− )x+ y) = (1− )d(x, y).
The -geodesic bicombing is con ex i i sa is ies
d((1 − )x+ y, (1 − )x+ z)≤ d(y, z) (5)
o all x, y, z ∈Xwi h d(x, y), d(x, z), d(z, y)≤ and all ∈[0,1].
No med spaces a e ob iously geodesic spaces wi h a con ex geodesic bi-
combing. Ano he na u al example a e Busemann spaces, which we e used o
he i s ime by Busemann [5] o gi e a de ini ion o nonposi i e cu a u e in
geodesic spaces. Thus, geodesic spaces wi h he p ope y ha each poin has
a con ex neighbo hood which is a Busemann space a e ‘nonposi i ely cu ed’
spaces in he sense o Busemann, who called hem G-spaces. We e e o [20]
o a nice exposi ion o his e y impo an class o geodesic spaces. I u ns
ou ha Busemann spaces a e uniquely geodesic spaces wi h a (unique) con ex
geodesic bicombing.
A ela ed example o me ic spaces wi h a con ex geodesic bicombing a e he
so-called W-hype bolic spaces, de ined in [11] as me ic spaces oge he wi h a
con exi y mapping W:X×X×[0,1] →Xsa is ying sui able p ope ies.
As i was ema ked in [1], Busemann spaces a e exac ly he uniquely geodesic
W-hype bolic spaces.
I is well-known ha a hype con ex space Xalso admi s a con ex geodesic
bicombing ob ained by embedding Xisome ically in o `∞(X) and using he
exis ence o a nonexpansi e e ac ion om `∞(X) in o X o de ine he geodesic
bicombing h ough he con ex linea geodesic bicombing on `∞(X) ( o mo e
de ails see, o ins ance, Chap e 13 in [10]).
In he ollowing we de ine a na u al gene aliza ion o me ic spaces wi h a
con ex -geodesic bicombing.
De ini ion 2.1. Le ∈(0,∞]and δ∈[0,1]. A me ic space (X, d)wi h
an -geodesic bicombing is said o be ( , δ)-con ex i o all x, y, z ∈Xwi h
d(x, y), d(x, z), d(z, y)≤ and all ∈[0,1],
d((1 − )x+ y, (1 − )x+ z)≤( +δ(1 − ))d(y, z).
I =∞, we say ha Xis δ-con ex.
An example o such spaces a e CAT(κ) spaces wi h κ > 0. CAT(κ) spaces
a e de ined in e ms o compa isons wi h he model spaces M2
κ(see [3] o mo e
de ails). Deno e Dκ=π/√κ.
P oposi ion 2.2. A CAT(κ)space Xis µDκ
2,1−cos µπ
2-con ex o any
µ∈(0,1].
P oo . Le x, y, z ∈Xwi h d(x, y), d(x, z), d(z, y)≤µDκ
2and ∈[0,1]. By [21,
Lemma 3.3] (see also [17, Lemma 4.1]) we ha e ha d((1− )x+ y, (1− )x+ z)≤
4
sin µπ
2
sin µπ
2
d(y, z).No e ha
1−sin µπ
2
sin µπ
2
=2 cos (1+ )µπ
4sin (1− )µπ
4
sin µπ
2≥2 cos µπ
2sin (1− )µπ
4
sin µπ
2
≥(1 − ) cos µπ
2,since sin (1 − )µπ
4≥1−
2sin µπ
2.
Hence,
sin µπ
2
sin µπ
2≤1−(1 − ) cos µπ
2= +1−cos µπ
2(1 − ).
We poin ou ha CAT(κ) spaces wi h κ > 0 do no ha e in gene al a con ex
-geodesic bicombing o < Dκ( o see his i su ices o conside he sphe ical
space S2).
Le us ecall ano he no ion o con exi y o me ic spaces, in oduced by
Oh a [19]. Gi en L1, L2∈[0,∞), a geodesic space Xis said o be L-con ex o
(L1, L2) i o any x, y, z ∈X, any cons an speed geodesics γ, ξ : [0,1] →X
wi h γ(0) = ξ(0) = x,γ(1) = y,ξ(1) = zand o e e y ∈[0,1],
d(γ( ), ξ( )) ≤1 + L1
min{d(x, y) + d(x, z),2L2}
2 d(y, z).
An addi ional ela ed no ion says ha an -geodesic bicombing on a me ic
space Xis weakly con ex i he e exis s a cons an C≥1 such ha
d((1 − )x+ y, (1 − )x+ z)≤C d(y, z),
o all ∈[0,1] and all x, y, z ∈Xas in De ini ion 2.1. One can easily see ha
an L-con ex space o (L1, L2) has a weakly con ex -geodesic bicombing, whe e
∈(0,∞] and C:= 1 + L1min{ , L2}.
Rema k 2.3. We ema k ha we could ha e de ined an e en mo e gene al
no ion: gi en > 0and η: [0,1] →[0,∞), a me ic space wi h an -geodesic
bicombing is ( , η)-con ex i
d((1 − )x+ y, (1 − )x+ z)≤( +η( ))d(y, z),
o all ∈[0,1] and all x, y, z ∈Xas in De ini ion 2.1.
This e y gene al de ini ion has he ad an age ha i co e s he case o
me ic spaces wi h a weakly con ex -geodesic bicombing and, hus, o L-con ex
spaces.
Howe e , we use in his pape De ini ion 2.1, as his is he no ion which
allows us o ge he e ec i e esul s om he nex sec ion.
5

3 E ec i e a es o asymp o ic egula i y
Le ∈(0,∞], δ ∈[0,1), Xbe an ( , δ)-con ex space, C⊆Xa con ex subse
and T1, . . . , TN:C→Cbe nonexpansi e mappings, whe e N∈Z+. I (λn) is
a sequence in [0,1] and u∈C, one can, ob iously, de ine he i e a ions (1) and
(4) s a ing wi h uin his se ing, oo:
x0=u, xn+1 =λn+1u+ (1 −λn+1)Tn+1xn,(6)
x0=u, xn+1 =Tn+1(λn+1u+ (1 −λn+1)xn),(7)
whe e Tn=Tnmod Nand he mod N unc ion akes alues in 1, . . . , N. We use
he ollowing no a ion Tn,N := Tn+N···Tn+1.
The main heo em o he pape is a quan i a i e esul on he asymp o ic
egula i y o he abo e i e a ions.
Theo em 3.1. Le ε > 0,M > 0be such ha M≤
2,α, γ : (0,∞)→Z+and
θ:Z+→Z+. Suppose ha
(i)
∞
X
n=1
λn=∞wi h a e o di e gence θ;
(ii)
∞
X
n=1 |λn+N−λn|con e ges wi h Cauchy modulus γ.
Le
˜
Φ(ε, M, γ, θ, δ) = θ 1
1−δγε
4M+ max ln 4M
ε,1,
Φ(ε, M, γ, θ, δ, N, α) = max n˜
Φε
2, M, γ, θ, δ, α ε
4MN o.
Assume ei he
(i) (xn)is gi en by (6)wi h d(xn, u)≤M o all n≥1and d(u, Tiu)≤M
o each i= 1, . . . , N, o
(ii) (xn)is gi en by (7)wi h d(xn, u)≤M o all n≥1.
Then lim
n→∞ d(xn, xn+N)=0wi h a e o con e gence ˜
Φ. Fu he mo e, i
lim
n→∞ λn+1 = 0 wi h a e o con e gence α, hen lim
n→∞ d(xn, Tn,N (xn)) = 0 wi h
a e o con e gence Φ.
We gi e he p oo o he heo em in he nex sec ion. Le us s a e now some
immedia e consequences.
Co olla y 3.2. Le ε, M, (λn), α, γ, θ, ˜
Φ,Φbe as abo e. Assume mo eo e ha
Cis bounded and Mis an uppe bound on i s diame e .
I (xn)is gi en by ei he (6)o (7), hen lim
n→∞ d(xn, xn+N)=0wi h a e o
con e gence ˜
Φand lim
n→∞ d(xn, Tn,N (xn)) = 0 wi h a e o con e gence Φ.
6
Thus, o Cbounded we ob ain a highly uni o m a e o asymp o ic egula -
i y Φ which does no depend a all on he s a ing poin uand he nonexpansi e
mappings T1, . . . , TN. Mo eo e , he dependence on he se Cand he space X
is e y weak: ia δand a bound M≤
2on he diame e o C.
Co olla y 3.3. Suppose ha λn=1
n+ 1, n ≥1. Then lim
n→∞ d(xn, xn+N) =
lim
n→∞ d(xn, Tn,N (xn)) = 0 wi h a common a e o con e gence
Ψ(ε, M, N, δ) = exp  1
1−δ8M(N+ 1)
ε+ 2ln 4.
P oo . We can ake θ(n) = exp(nln 4), γ(ε) = N
εand α(ε) = 1
ε.
As men ioned be o e, in he case N= 1, he i e a i e scheme de ined by
(6) yields he usual Halpe n i e a ion, o which a es o asymp o ic egula i y
ha e al eady been compu ed in he se ing o CAT(κ) spaces [17]. Ou main
heo em eco e s (wi h a sligh ly modi ied a e) [17, P oposi ion 3.2] since, o
M < Dκ
2, one akes µ=2M
Dk
in P oposi ion 2.2 o ge ha any CAT(κ) space
is (M, 1−cos(M√κ))-con ex and hen apply Co olla y 3.2. Fu he mo e, we
gene alize wi h basically he same bounds he esul s ob ained o he Halpe n
i e a ion in no med spaces [15] and, mo e gene al, W-hype bolic spaces [16].
As we ha e al eady poin ed ou , in his pape we ob ain o he i s ime,
e en o Banach spaces, e ec i e bounds on he asymp o ic egula i y o he
i e a ions (6) and (7). Recen ly, using p oo mining me hods as well, Khan and
Kohlenbach [9] ob ained in he se ing o uni o mly con ex Busemann spaces
e ec i e esul s on he asymp o ic beha io o a di e en i e a ion associa ed o
a ini e amily o nonexpansi e mappings which ex ends he K asnoselski-Mann
i e a ion o a single nonexpansi e mapping.
4 P oo o he main esul
Assume he hypo hesis o Theo em 3.1. As in he case o he Halpe n i e a ion
associa ed o a single mapping [15,16,14,17], we shall apply he ollowing
quan i a i e lemma.
Lemma 4.1. [17,14] Le (αn)n≥1be a sequence in [0,1] and (an)n≥1,(bn)n≥1
be sequences in R+such ha
an+1 ≤(1 −αn+1)an+bn o all n∈Z+.(8)
Assume ha
∞
X
n=1
bnis con e gen wi h Cauchy modulus γand
∞
X
n=1
αn+1 di e ges
wi h a e o di e gence θ.
7
Then, lim
n→∞ an= 0 wi h a e o con e gence Σgi en by
Σ(ε, P, γ, θ) = θγε
2+ max ln 2P
ε,1+ 1 (9)
whe e P > 0is an uppe bound on (an).
The nex lemma is he second main ool o he p oo o ou main esul .
Lemma 4.2. Le M > 0sa is y 2M≤ .
(i) Assume ha (xn)is gi en by (6),d(u, Tiu)≤M o each i= 1, . . . , N
and d(xn, u)≤M o all n∈N. Then, o all n≥1,
d(xn+1, Tn+1xn)≤2Mλn+1,
d(xn, xn+N)≤(1 −(1 −δ)λn)d(xn−1, xn+N−1)+2M|λn+N−λn|.
(ii) Assume ha (xn)is gi en by (7)and d(xn, u)≤M o all n∈N. Then,
o all n≥1,
d(xn+1, Tn+1xn)≤Mλn+1,
d(xn, xn+N)≤(1 −(1 −δ)λn)d(xn−1, xn+N−1) + M|λn+N−λn|.
P oo . (i) Fi s , le us no e ha d(u, Tnxm)≤2M o all m, n ≥1. I ollows
ha o all n≥1,
d(xn+1, Tn+1xn) = λn+1d(u, Tn+1xn)≤2Mλn+1
and
d(xn, xn+N) = d(λnu+ (1 −λn)Tnxn−1, λn+Nu+ (1 −λn+N)Tnxn+N−1)
≤d(λnu+ (1 −λn)Tnxn−1, λnu+ (1 −λn)Tnxn+N−1)
+d(λnu+ (1 −λn)Tnxn+N−1, λn+Nu+ (1 −λn+N)Tnxn+N−1)
≤(1 −(1 −δ)λn)d(xn−1, xn+N−1)+2M|λn+N−λn|.
(ii) The p oo is simila , using ha
d(xn+1, Tn+1xn)≤d(λn+1u+ (1 −λn+1)xn, xn)≤Mλn+1
and
d(xn, xn+N) = d(Tn(λnu+ (1 −λn)xn−1), Tn(λn+Nu+ (1 −λn+N)xn+N−1))
≤d(λnu+ (1 −λn)xn−1, λn+Nu+ (1 −λn+N)xn+N−1).
o all n≥1.
8
4.1 P oo o Theo em 3.1
Le (xn) be gi en by ei he (6) o (7). Deno e n+1 = (1 −δ)λn∈[0,1]. As an
immedia e consequence o Lemma 4.2, we ge ha
d(xn, xn+N)≤(1 − n+1)d(xn−1, xn+N−1)+2M|λn+N−λn|.
No e ha
∞
X
n=1
2M|λn+N−λn|con e ges wi h Cauchy modulus ˜γ(ε) = γε
2M
and
∞
X
n=1
n+1 =∞wi h a e o di e gence ˜
θ(n) = θ 1
1−δn.
We apply Lemma 4.1 wi h αn:= n,P:= 2M,an:= d(xn−1, xn+N−1) and
bn:= 2M|λn+N−λn| o ob ain ha lim
n→∞ d(xn−1, xn+N−1) = 0 wi h a e o
con e gence ˜
Φ + 1. Hence, lim
n→∞ d(xn, xn+N) = 0 wi h a e o con e gence ˜
Φ.
Assume now ha lim
n→∞ λn+1 = 0 wi h a e o con e gence α. By Lemma 4.2,
i ollows ha d(xn+1, Tn+1(xn)) ≤2Mλn+1, hence
d(xn+1, Tn+1(xn)) ≤ε
2N o all n≥αε
4MN .
One can easily see ha
d(xn, Tn,N (xn)) ≤d(xn, xn+N) +
N
X
i=1
d(xn+i, Tn+i(xn+i−1)).
The e o e, d(xn, Tn,N (xn)) ≤ε o all n≥Φ. 
Acknowledgemen s:
Lau en ¸iu Leu¸s ean was suppo ed by a g an o he Romanian Na ional Au-
ho i y o Scien i ic Resea ch, CNCS - UEFISCDI, p ojec numbe PN-II-ID-
PCE-2011-3-0383.
Ad iana Nicolae was suppo ed by a g an o he Romanian Minis y o Educa-
ion, CNCS - UEFISCDI, p ojec numbe PN-II-RU-PD-2012-3-0152.
Re e ences
[1] D. A iza-Ruiz, L. Leu¸s ean, G. L´opez, Fi mly nonexpansi e mappings in
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lica ion in T ans. Ame . Ma h. Soc., 2012.
[2] H.H. Bauschke, The app oxima ion o ixed poin s o composi ions o non-
expansi e mappings in Hilbe spaces, J. Ma h. Anal. Appl. 202 (1996),
150-159.
[3] M. B idson, A. Hae lige , Me ic spaces o non-posi i e cu a u e, Sp inge -
Ve lag, Be lin, 1999.
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