Suppose that:

k=0

λk(2−λk)Ui_kx^k−x^k2<∞. (9.74)

Note that the series above is absolutely convergent, thus, m

i=1

k∈Ki

λk(2−λk)Uix^k−x^k2= ∞ k=0

λk(2−λk)Ui_kx^k−x^k2<∞. (9.75)

Therefore,

k∈Ki

λk(2−λk)Uix^k−x^k2<∞for all i∈I (9.76) and

k→∞,k∈Klim i

λk(2−λk)Uix^k−x^k2=0 for all i∈I. (9.77) Since lim inf_k→∞λk(2−λk)>0,we have lim_{k→∞,k∈Ki}Uix^k−x^k=0 for all i∈I, consequently,

lim inf

k→∞ Uix^k−x^k=0 (9.78)

for all i∈I, and {ik}^∞_k=0 is approximately semi-regular. One can prove that the approximate semi-regularity also holds for sequences generated by (9.46), where J_k=I andV =U and the sequence of weight functions{w^k}^∞_k=0has the property

i∈Ikw^k_i =1 for I_k⊂I, k≥0 and w^k_i >δ>0 for i∈I_k, and i∈I_k for infinitely many k, i∈I. Note that a repetitive control is a special case of a sequence{w^k}^∞k=0having the above property.

Theorem 9.27. Suppose that:

• Ui:H →H, i∈I, are cutters having a common fixed point,

• Ui−Id are demiclosed at 0, i∈I,

• V^k={V_i^k}i∈Jk are families of cutters V_i^k:H →H, i∈Jk, with the property

i∈JkFixV_i^k⊃

i∈IFixUi, k≥0,

• {w^k}^∞_k=0:H →Δ_|Jk|is a sequence of appropriate weight functions,

• lim inf_k→∞λk(2−λk)>0,

• {x^k}^∞_k=0is generated by the recurrence (9.46).

If the sequence of weight functions {w^k}^∞_k=0 applied to the sequence of familiesV^k:

(i) Is approximately regular with respect to the familyU ={Ui}i∈I then{x^k}^∞_k=0 converges weakly to a common fixed point of Ui, i∈I;

(ii) Is approximately semi-regular with respect to the familyU ={Ui}i∈IandH is finite-dimensional, then{x^k}^∞_k=0converges to a common fixed point of Ui, i∈I.

Proof. Let V^k:H →H be defined by V^kx=

i∈Jk

w^k_i(x)Vi^kx (9.79)

and let T_kbe theλk-relaxation of the operator V^k, i.e., T_kx=V_λ^k

kx=x+λk(V^kx−x). (9.80) The operators V^kare cutters,

Fix Tk=FixV^k=

i∈Jk

FixV_i^k⊃

i∈I

FixUi

and Tk are strongly quasi-nonexpansive, k≥0, (see Lemma9.22), consequently, _∞

k=0Fix Tk⊃

i∈IFixUi. Letε>0 be such that lim inf

k→∞ λk≥ε and lim inf

k→∞ (2−λk)≥ε (9.81) and let z∈

i∈IFixUi. For sufficiently large k we have 2−λk≥ε/2 and 2−λk

λk ≥ ε/4. Now, it follows from Lemma9.22that, for sufficiently large k,

x^k+1−z²=Tkx^k−z²

≤ x^k−z²−λk(2−λk)

i∈Jk

w^k_i(x^k)V_i^kx^k−x^k2

≤ x^k−z²−λk(2−λk)V^kx^k−x^k2

=x^k−z²−2−λk

λk Tkx^k−x^k2

≤ x^k−z²−ε

4Tkx^k−x^k2. (9.82)

Therefore, {x^k−z}^∞_k=0 decreases and

i∈Jkw^k_i(x^k)V_i^kx^k−x^k2→0. Conse-quently,

w^k_i

k(x^k)V_i^k

kx^k−x^k2→0 (9.83)

for arbitrary ik∈Jk.

(i) Suppose that{w^k}^∞_k=0is approximately regular with respect to the familyU = {Ui}i∈I. Let ik∈Jk, k≥0, be such that the implication (9.64) holds. Then (9.83) yields lim_k→∞Uix^k−x^k=0 for all i∈I. Let x^∗be a weak cluster point of {x^k}^∞_k=0and{x^nk}^∞_k=0be a subsequence of{x^k}^∞_k=0such that x^nkx^∗as k→∞.

The demiclosedness of Ui−Id at 0, i∈I, yields that x^∗∈

i∈IFixUi. Since x^∗is an arbitrary weak cluster point of{x^k}^∞_k=0and{x^k}^∞_k=0is Fej´er-monotone with

respect to

i∈IFixUi, the weak convergence of the whole sequence{x^k}^∞_k=0to x^∗follows from [7, Lemma 6] (see also [4, Theorem 2.16 (ii)]).

(ii) Suppose that H is finite-dimensional and {w^k}^∞_k=0 is approximately semi-regular with respect to the familyU ={Ui}i∈I. Let ik∈Jk, k≥0, be such that Remark 9.28. Bauschke and Borwein [4, Sect. 3, page 378] consider algorithms which are similar to (9.46), where J_k=I, λk=1, V_i^k is replaced by a firmly non-expansive operator U_i^kwith FixU_i^k⊃FixUi, i∈I, k≥0, and

i∈IFixUi=/0. They assumed that these algorithms are focusing, strongly focusing or linearly focusing (see [4, Definitions 3.7 and 4.8]). These assumptions differ from the assumptions on the regularity, approximate regularity or approximate semi-regularity, but they play a similar role in the proof of convergence of sequences generated by the considered algorithms. The recurrence considered by Bauschke and Borwein has the form

x^k+1= Δm is a sequence of weight vectors (see [4, page 378]). Note that (9.84) can be written in the form sat-isfies this assumption then lim inf_k→∞λk(2−λk)>0. Furthermore, if the sequence of weight vectors{v^k}^∞_k=0applied to the recurrence (9.84) is regular (approximately regular, approximately semi-regular) and lim inf_k→∞μ_i^k(2−μ_i^k)>0, i∈I, then the sequence of weight vectors{w^k}^∞_k=0 applied to the recurrence (9.85) is reg-ular (approximately regreg-ular, approximately semi-regreg-ular). Bauschke and Borwein proved the weak convergence of sequences{x^k}^∞_k=0generated by (9.84) to a point x∈

i∈IFixUi under the assumptions that (i) the algorithm is focusing and inter-mittent and (ii) that lim inf_k→∞,vk

i>0v^k_i >0 for all i∈I (see [4, Theorem 3.20]).

Assumption (ii) applied to sequences generated by (9.84) is equivalent to the following assumption (ii)’ lim inf_k→∞,wk

i>0w^k_i >0 applied to sequences generated by (9.85). Note, however, that assumptions (i) as well as (ii)’ do not appear in Theorem9.27. Assumptions similar to those in [4, Theorem 3.20] can be also found in [19, equalities (15)–(17)].

In the following examples we suppose that Ci⊂H, i∈I, are closed and convex (see Example9.24(b)), it is approximately regular and it follows from Theorem 9.27(i) that x^kx^∗∈C. This convergence was proved by Iusem and De Pierro [28, Corollary 4] forH =Rⁿ. Note, however, that in finite-dimensional case the convergence holds for any sequence{w^k}^∞_k=0containing a subsequence ofβ-regular weight functions, whereβ >0, e.g., if w^k=w for infinitely many k≥0.

Example 9.30. Aharoni and Censor [2, Theorem 1] consider the recurrence (9.46), whereH =Rⁿ, J_k=I for all k≥0, V_i^k=PC_i, i∈I,λk∈[ε,2−ε], whereε∈(0,1), w^k is approximately semi-regular. Theorem9.27(ii) yields now the convergence x^k→x^∗∈C.

Example 9.31. Butnariu and Censor [8, Theorem 4.4] consider the recurrence

Therefore, {w^k}^∞_k=0 is approximately semi-regular. Theorem9.27(ii) yields now lim_k→∞x^k=x^∗∈C. If we suppose that all cluster points of{w^k}^∞_k=0belong to riΔm

then{w^k}^∞_k=0is approximately regular, consequently the weak convergence x^kx^∗ holds in general Hilbert spaces.

Example 9.32. Consider the recurrence (9.46), where J_k=I for all k≥0, lim inf that the sequence of weight vectors w^ksatisfies the following conditions:

(i) lim sup_k→∞w^k_i >0, i∈I,

(ii) w^k_iPC_ix^k−x^k =0 implies w^k_i >δ >0.

Inequality (9.92) and (i) and (ii) guarantee that the sequence of weights{w^k}^∞_k=0is regular and thus all assumptions of Theorem9.27(i) are satisfied. Therefore, x^k x^∗∈C. This convergence was proved by Fl˚am and Zowe [25, Theorem 1] in case H =Rⁿ. Actually, they have considered a recurrence which can be reduced to (9.46). We omit the details.

Results similar to Theorem9.27also hold for sequences generated by extrap-olated simultaneous cutters. Before formulating our next theorem, we prove some auxiliary results. The following lemma is an extension of Lemma9.22. A part of this lemma can be found in [21, Proposition 2.4], where w is a constant weight function with positive coordinates.

Lemma 9.33. Let Vi:H →H be cutters having a common fixed point, i∈J= {1,2,...,l}, let w :H →Δl be an appropriate weight function and letσ:H → (0,+∞)be a step-size function defined by

σ(x) =

and let V_σ:=Id+σ(l

i=1wiVi−Id)be a generalized relaxation of the simultaneous cutter V=_l

Proof. Lemma9.22(i) and the positivity of the step-size functionσ ^{yield FixV}σ= FixV=

i∈JFixVi. Let x∈H and z∈FixV_σ. We prove that

z−x,Vσx−x ≥ Vσx−x², (9.95) which is equivalent to V_σ being a cutter; see Lemma9.4(i). The inequality is clear for x∈FixV_σ. For x∈/FixV_σ we have which is equivalent to (9.95). By the convexity of the function·²we haveσ(x)≥ 1, i.e., V_σ is an extrapolation of V . Lemma9.4(ii) and the fact V_σ,λ = (V_σ)_λ yield now the 2−λ

λ -strong quasi-nonexpansivity of V_σ,λ. Inequality (9.94) follows from

the equality V_σ,λx−x=λσ(x)(V x−x).

For a family of cuttersV ={Vi}i∈J and for an appropriate weight function w : H →Δ_|J|denote andσw(x)is well-defined.

Definition 9.34. Let Vi:H →H, i∈J, be cutters with a common fixed point and let w :H →Δ_|J|be a weight function which is appropriate with respect to the family

V ={Vi}i∈J. We say that the step-size functionσ:H →(0,+∞)isα-admissible with respect to the familyV, whereα∈(0,1], or, shortly, admissible, if

ασw(x)≤σ(x)≤σw(x) (9.99)

for all x∈/

i∈JFixVi. Theorem 9.35. Suppose that:

• Ui:H →H, i∈I, are cutters having a common fixed point,

• Ui−Id, i∈I, are demiclosed at 0,

• V^k={V_i^k}i∈Jk are families of cutters V_i^k:H →H, i∈Jk, with the properties

i∈J_kFixV_i^k ⊃

i∈IFixUi, and maxi∈J_kV_i^kx−x ≤γmaxi∈IUix−x for all x∈H, k≥0, and for some constantγ>0,

• {w^k}^∞_k=0:H →Δ|Jk|is a sequence of appropriate weight functions,

• The step-sizeσk:H →(0,+∞)isα-admissible with respect toV^k, k≥0, for someα∈(0,1],

• lim infk→∞λk(2−λk)>0,

• {x^k}^∞_k=0is generated by the recurrence (9.44).

If the sequence of weight functions {w^k}^∞_k=0 applied to the sequence of familiesV^k:

(i) Is regular with respect to the familyU ={Ui}i∈Ithen{x^k}^∞_k=0converges weakly to a common fixed point of Ui, i∈I;

(ii) Contains a subsequence which is regular with respect to the familyU ={Ui}i∈I

andH is finite-dimensional, then{x^k}^∞_k=0converges to a common fixed point of Ui, i∈I.

Proof. Let V^k:H →H be defined by V^kx=

i∈Jk

w^k_i(x)V_i^kx (9.100)

and let T_kbe a generalized relaxation of the operator V^k, i.e., T_kx=V_σ^k

k,λkx=x+λkσk(x)(V^kx−x). (9.101) The operators V^kare cutters and Fix Tk=FixV^k=

i∈JkFixV_i^k(see Lemma9.22).

Consequently,_∞

k=0Fix Tk⊃

i∈IFixUi. Letε>0 and k0∈Nbe such thatλk∈ [ε,2−ε]for k≥k0. By Lemma9.33the operator V_σ^k_wk is a cutter. Now, the second inequality in (9.99) and (9.5) which remains true also for λ :H →[0,1] yield that V_σ^k_k is a cutter, consequently Tkis aλk-relaxed cutter, k≥0. Lemma9.33also implies that

x^k+1−z²≤ x^k−z²−2−λk

λk Tkx^k−x^k2 (9.102) for all z∈_m

i=1FixUi. Therefore,{x^k}^∞_k=0is bounded,{x^k−z}^∞_k=0is monotone and lim_k→∞Tkx^k−x^k=0.

(i) Letβ∈(0,1], k1≥k0and jk∈Jkbe such that wj_k(x)Vj^k_kx−x²≥βmax

i∈I Uix−x² (9.103)

for any x∈H and for k≥k1. Sinceσkisα-admissible, the norm is a convex function andV_j^kx^k−x^k ≤γmaxiUix^k−x^kfor all j∈Jk, we have

Tkx^k−x^k=λkσk(x^k)V^kx^k−x^k

≥λkα

i∈Jkw^k_i(x^k)V_i^kx^k−x^k2

i∈Jkw^k_i(x^k)V_i^kx^k−x^k

≥λkα ^w

k

j_k(x^k)V^k_jkx^k−x^k2

i∈Jkw^k_i(x^k)V_i^kx^k−x^k

≥λkα

γ β^maxi∈IUix^k−x^k2 i∈Jkw^k_i(x^k)

max_i∈IUix^k−x^k

=εαβ

γ ^maxi∈I Uix^k−x^k, (9.104) and limk→∞Uix^k−x^k=0 for all i∈I. Therefore, condition (9.16) is satisfied for Uk=Tkand S=Ui, i∈I. We have proved that all assumptions of Theorem 9.9(i) are satisfied for S=Ui, i∈I. Therefore,{x^k}^∞_k=0converges weakly to a common fixed point of Ui, i∈I.

(ii) Suppose thatH is finite-dimensional and{w^k}^∞_k=0contains an approximately β-regular subsequence{w^nk}^∞_k=0. Letβ∈(0,1], k1≥k0and jnk∈I be such that

vj_nk(x)V_jⁿ_nk^kx−x²≥βmax

i∈I Uix−x². (9.105) Similarly to (i), one can prove that

Tn_kx^nk−x^nk ≥ εαβ

γ ^maxi∈I Uix^nk−x^nk. (9.106) Therefore, lim inf_k→∞Uix^k−x^k=0 for all i∈I. If we set Uk=Tkand S=Ui, i∈I, in Theorem9.9(ii), we obtain the convergence of{x^k}^∞_k=0to a fixed point

of Uifor all i∈I.

Remark 9.36. Combettes considers an algorithm which is similar to (9.44) with Jk= I, w^k=w∈riΔm, V_i^k=P_Ck

i, where C_i^k⊃Ciare closed and convex, i∈I, k≥0, and with a constant sequence of step-size functionsσk=σwgiven by

σw(x) =

i∈IwiPC_ix−x²

i∈Iwi(PC_ix−x)² (9.107)

for x∈/C=

i∈ICi (see [19, (33)–(36)]). He proves there weak convergence of sequences generated by this algorithm to a point x∈C under the assumption that the algorithm is focusing (see [19, Theorem 2]). However, the assumption w∈riΔm

is a special case of a regular sequence of weight functions and the step-size function σw,given by (9.107) is a special case of a sequence ofα-admissible step-sizes which are considered in Theorem9.35.

Remark 9.37. Results closely related to Theorems9.27(ii) and9.35(ii) appear in Kiwiel [29, Theorem 5.1], for the case H =Rⁿ. Kiwiel applies some assump-tions on weights and on the operators [29, Assumption 3.10] which differ from the assumptions in Theorems9.27(ii) and9.35on the approximate semi-regularity.

Our Theorems9.27 and9.35 show the importance of the regularity, approximate regularity and the approximate semi-regularity in both the finite- and the infinite-dimensional cases.

Example 9.38. Dos Santos’ [24, Sect. 5] work is related to ours as follows. Let ci: H →R be continuous and convex, let Ci={x∈H |ci(x)≤0}, i∈I and let sub-gradient of the function ci at the point x, i∈I. This operator Ui is called the subgradient projection onto Ci, i∈I. It follows from the definition of the subgra-dient that Ui is a cutter. Note that FixUi=Ci, and thus_m

i=1FixUi=/0. Suppose that the subgradients giare bounded on bounded subsets, i∈I (this holds if, e.g., H =Rⁿ). Then the operator Ui−Id is demiclosed at 0, i∈I. Indeed, let x^kx^∗ The sequence{x^k}^∞_k=0is bounded due to its weak convergence. Condition (9.109) and the boundedness of gi(x^k)imply the convergence lim_k→∞ci(x^k)+=0. Since ci

is weakly lower semi-continuous, we have ci(x^∗) =0, i.e., Ui−Id is demiclosed at 0. Consider an extrapolated simultaneous subgradient projection method, i.e., a method which generates sequences{x^k}^∞_k=0defined by the recurrence (9.44) where V_i^k=Ui, w^kis a sequence of appropriate weight functions, lim inf_k→∞λk(2−λk)>0

Note that

Uix−x=−(ci(x))+

gi(x)²gi(x), (9.111)

and so,

σk(x) =σw(x) =

i∈Jw^k_i(x)Uix−x²

i∈Jw^k_i(x)Uix−x², (9.112) and σk are 1-admissible. If we suppose that the sequence of weight functions {w^k}^∞_k=0is regular then, by Theorem9.35(i) the sequence{x^k}^∞_k=0converges weakly to a point x^∗∈C. Dos Santos [24] considers positive constant weights w∈riΔmand proves the convergence in the finite-dimensional case.

Acknowledgements We thank two anonymous referees for their constructive comments. This work was partially supported by Award Number R01HL070472 from the National Heart, Lung and Blood Institute and by United States-Israel Binational Science Foundation (BSF) grant No. 2009012.

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