Applications to the Common Fixed Point Problem

LetU ={Ui}i∈I, where I :={1,2,...,m}, be a finite family of cutters Ui:H →H, having a common fixed point. The common fixed point problem is to find x^∗∈

i∈IFixUi. In this section, we study the convergence properties of sequences gen-erated by the recurrence

x^k+1=x^k+λkσk(x^k)

⎛

⎝

i∈Jk

w^k_i(x^k)Vi^kx^k−x^k

⎞

⎠, (9.44)

whereλk∈[0,2],σk:H →(0,+∞)are step-size functions,V^k={V_i^k}i∈J_kis a fam-ily of cutters V_i^k:H →H, i∈J_k={1,2,...,m_k}with the property

i∈J_kFixV_i^k⊃

i∈IFixUiand w^k:H →Δm_kare weight functions

w^k(x) = (w^k1(x),w^k2(x),...,w^k_mk(x)) (9.45)

(the subsetΔm denotes here the standard simplex, i.e.,Δm={u∈R^m: ui≥0, i= 1,2,...,m, andm

i=1ui=1}). Ifσk(x) =1 for all x∈H and for all k≥0, then the method defined by the recurrence (9.44) takes the form

x^k+1=x^k+λk

⎛

⎝

i∈J_k

w^k_i(x^k)Vi^kx^k−x^k

⎞

⎠, (9.46)

and is called the simultaneous cutter method. Ifσk(x)≥1 for all x∈H and for all k≥0, then method (9.44) is called the extrapolated simultaneous cutter method.

The recurrence (9.44) can be written in the form x^k+1=x^k+λkσk(x^k)

V^kx^k−x^k

, (9.47)

where V^k=

i∈Jkw^k_iV_i^k,or in the form x^k+1=V_σ^k

k,λkx^k. (9.48)

Remark 9.20. The sequence of weight functions {w^k}^∞_k=0 induces a control se-quence. This notion is usually applied in the literature if the values of w^k are extremal points of a standard simplex (see, e.g., [13, Definition 3.2] or [17, Definition 5.1.1]). One can recognize special cases of a sequence of weight func-tions w^k as known control sequences. In particular, the weight functions{w^k}^∞_k=0 can be constant, i.e., w^k(x) = (w^k₁,w^k₂,...,w^k_mk)∈Δm_kfor all x∈H, k≥0. A simple example of such a control sequence is the cyclic control (see [27, Equality (2)], [13, (3.3)] or [17, Definition 5.1.1]) The sequence{w^k}^∞_k=0can also be a constant sequence, i.e., Jk=J and w^k=w :H →Δmfor all k≥0. A simple example of such a control is the remotest set control (see [27, Equality (3’)] or [13, (3.5)] or [17, Definition 5.1.1]). Sequences of weights depending on x∈H enable, however, a more general model and demonstrate the importance of assumptions on the weight functions control.

Definition 9.21. Let Vi:H →H, i∈J={1,2,...,l}. We say that a weight func-tion w :H →Δl is appropriate with respect to the familyV ={Vi}i∈Jor, shortly, appropriate if for any x∈/

i∈JFixVithere exists a j∈J such that

wj(x)Vjx−x =0. (9.49)

Lemma 9.22. Let Vi:H →H, i∈J={1,2,...,l}, be cutters having a common fixed point and let V=

i∈JwiVi, where w :H →Δlis appropriate with respect to the familyV ={Vi}i∈J. Then

(i) FixV=

i∈JFixVi,

(ii) V is a cutter, consequently, for allλ∈(0,2), the operator V_λ is 2−λ

λ ^-strongly quasi-nonexpansive,

(iii) The following inequalities hold

i∈JFixVi=H then the inclusion is clear. Otherwise, sup-pose that x∈FixV , x∈/

i∈JFixVi and that z∈

i∈JFixVi. Since a cutter is strongly quasi-nonexpansive (see Lemma9.4(ii)) we haveVix−z<x−z for any i∈J such that x∈/FixVi. The convexity of the norm, the strict quasi-nonexpansivity of Viand the fact that the weight function w is appropriate yield

V x−z=

We get a contradiction, which shows that FixV ⊂

i∈JFixVi.

Applying again Lemma9.4(i) we deduce that V is a cutter. By Lemma9.4(ii) the operator V_λ is 2−λ

λ -strongly quasi-nonexpansive for anyλ∈(0,2).

(iii) Letλ∈[0,2], x∈H and z∈FixV . The convexity of · ²and Lemma9.4(i)

≤ x−z²+λ²

i∈J

wi(x)Vix−x²−2λ

i∈J

wi(x)Vix−x²

=x−z²−λ(2−λ)

i∈J

wi(x)Vix−x², (9.54) i.e., the inequality (9.50) holds. Inequality (9.51) follows from the convexity of

the function · ².

Definition 9.23. Let V ={Vi}i∈J be a finite family of operators Vi:H →H, i∈J, and letβ∈(0,1]be a constant. We say that a weight function w :H →Δ|J|

isβ-regular with respect to the family of cuttersU ={Ui}i∈I, or, shortly, regular if for any x∈H there exists a j∈J such that

wj(x)Vjx−x²≥βmax

Uix−x²|i∈I

. (9.55)

If

i∈JFixVi⊃

i∈IFixUithen a weight function which is regular with respect to the familyU ={Ui}i∈Iis appropriate with respect to the familyV ={Vi}i∈J. Example 9.24. LetV =U and let I(x) ={i∈I|x∈/FixUi}and let m(x) =|I(x)|

be the cardinality of I(x), for x∈H. The following weight functions w :H →Δm, where w(x) = (w1(x),...,wm(x)), are regular:

(a) Positive constant weights, i.e.,

w(x) =w∈riΔm (9.56)

for all x∈H, where riΔm={w∈R^m|w>0 ande,w=1}is the relative interior ofΔm. A specific example is furnished by equal weights, i.e., wi(x) = 1/m, i∈I. To verify that w is regular set j∈Argmax_i∈IUix−x andβ = min_i∈Iwiin Definition9.23.

(b) Constant weights for violated constraints, i.e.,

wi(x):=

⎧⎪

⎪⎨

⎪⎪

⎩ wi

j∈I(x)

wj

, for i∈I(x),

0, fori∈/I(x),

(9.57)

where w= (w1,w2,...,wm)∈riΔm. A specific example is wi(x):=

1/m(x), for i∈I(x),

0, fori∈/I(x). (9.58)

To verify that w is regular set j∈Argmax_i∈IUix−x andβ =min_i∈Iwi in Definition9.23.

(c) Weights proportional toUix−x, i.e.,

wi(x) =

⎧⎪

⎪⎨

⎪⎪

⎩

Uix−x

j∈I

Ujx−x, for x∈/

i∈IFixUi,

0, for x∈

i∈IFixUi.

(9.59)

To verify, set j∈Argmax_i∈IUix−xandβ=1/m in Definition9.23.

(d) Weight functions w :H →Δmsatisfying the condition

wi(x)≥δ for i∈I(x) (9.60)

for some constantδ>0. To verify, choose j(x)∈Argmax_i∈IUix−xand set β=δin Definition9.23. These weight functions were applied by Combettes in [19, Sect. III] and in [20, Sect. 1]. Observe that the weight functions defined by (9.56) and by (9.57) satisfy (9.60).

(e) Weight functions w :H →Δm for which wi(x) =0 for all x∈H and for all i∈/J_γ(x), where

J_γ(x) ={j∈I| Ujx−x ≥γmax

i∈I Uix−x}, (9.61) for someγ ∈(0,1]. To verify, set j=j(x)∈J_γ(x)withωj(x)≥1/m andβ = γ²/m in Definition 9.23. The existence of such j follows from the fact that wi(x)≥0 for all i∈J_γ(x)and

i∈Jγ(x)wi(x) =1. Specific examples are obtained as follows:

(i) When Ui=PCi for a closed convex subset Ci⊂H, i∈I, and wi(x) =

1, if i=argmax_j∈IUjx−x

0, otherwise. (9.62)

In this case, w defines a remotest set control (for the definition, see [27, (3’)]

or [17, Sect. 5.1]).

(ii) When Ui=PCi for a closed convex subset Ci⊂H, i∈I, and wi(x) =

1, if i=j(x)

0, otherwise, (9.63)

where j(x)∈J_γ(x)for someγ∈(0,1]. In this case w is an approximately remotest set control (for the definition, see [27, (3)] or [17, Sect. 5.1]).

Definition 9.25. LetV^k={V_i^k}i∈Jk be a sequence of cutters V_i^k:H →H, i∈ J_k={1,2,...,m_k}, k≥0, and let the sequence{x^k}^∞_k=0be generated by the recur-rence (9.44). We say that a sequence of appropriate weight functions w^k:H →Δm_k

(applied to the sequence of familiesV^k) is

• Regular (with respect to the familyU ={Ui}i∈I) if there is a constantβ∈(0,1]

such that w^kareβ-regular for all k≥0,

• Approximately regular (with respect to the familyU ={Ui}i∈I) if there exists a sequence ik∈Jksuch that the following implication holds

k→∞limw^k_ik(x^k)V_i^k_kx^k−x^k2=0=⇒ lim

k→∞Uix^k−x^k=0 for all i∈I, (9.64)

• Approximately semi-regular (with respect to the familyU ={Ui}i∈I) if there exists a sequence ik∈Jksuch that the following implication holds

k→∞limw^k_i

k(x^k)V_i^k

kx^k−x^k2=0=⇒lim inf

k→∞ Uix^k−x^k=0 for all i∈I. (9.65) Example 9.26. Here are examples of weight functions which are approximately reg-ular or approximately semi-regreg-ular with respect to the familyU ={Ui}i∈I. (a) A regular sequence of weight functions is approximately regular.

(b) A sequence containing a regular subsequence of weight functions is approxi-mately semi-regular.

(c) Let{x^k}^∞_k=0be a sequence generated by the recurrence (9.46), whereV^k=U and w^k =δi_k. We call the sequence {ik}^∞_k=0 a control sequence (see [13, Definition 3.2]). Recurrence (9.46) can be written as follows

x^k+1=x^k+λk

Ui_kx^k−x^k

. (9.66)

Implication (9.64) takes the form

k→∞limUikx^k−x^k=0=⇒lim

k→∞Uix^k−x^k=0 for all i∈I. (9.67) If (9.67) is satisfied we say that the control sequence{ik}^∞_k=0is approximately regular. If we set Ui=PCi for a closed convex subset Ci⊂H, i∈I, then impli-cation (9.67) can be written in the form

klim→∞PC_ikx^k−x^k=0=⇒lim

k→∞max

i∈I PCix^k−x^k=0. (9.68) A sequence{ik}^∞_k=0satisfying (9.68) is called approximately remotest set con-trol (see [27, Sect. 1]).

(d) (Combettes [19, Sect. II D]). Let Ikbe a nonempty subset of I, k≥0. Suppose that there is a constant s≥1 such that

I=Ik∪I_k+1∪...∪I_k+s−1for all k≥0. (9.69) Let Ui=PC_i, where Ci⊂H is closed and convex. Let{x^k}^∞_k=0be a sequence generated by the recurrence (9.46), whereV^k=U ={Ui}i∈I,λk∈[ε,2−ε] for someε∈(0,1), and w^k∈Δmis a weight vector such that

i∈Ikw^k_i =1 and

w^k_i ≥δ >0 for all i∈Ik∩I(x^k), k≥0, and I(x) ={i∈I|x∈/Ci}. Bauschke and Borwein called a sequence of weights satisfying (9.69) with Ik={i∈I| w^k_i >0}an intermittent control (see [4, Definition 3.18]). The recurrence (9.46) can be written in the form x^k+1=Tkx^k, where Tk=Id+λk(Vk−Id)and Vk=

i∈I_kw^k_iPC_i, or, equivalently, in the form

x^k+1=x^k+λk

⎛

⎝

i∈Ik

w^k_iPC_ix^k−x^k

⎞

⎠. (9.70)

One can show that Vkis a cutter. We show that{w^k}^∞_k=0is approximately regular.

Let i∈I be arbitrary and let rk∈ {0,1,...,s−1}be such that i∈I_k+rk, k≥0.

By the triangle inequality, we have x^k+rk−x^k ≤

r_k−1 i=0

x^k+i+1−x^k+i=

r_k−1 i=0

Tk+ix^k+i−x^k+i

≤

s−1

i=0

T_k+ix^k+i−x^k+i, (9.71)

for k≥0. Since Tkareλk-relaxed cutters andλk∈[ε,2−ε], Lemma9.22(iii) yields lim_k→∞Tk+ix^k+i−x^k+i=0, i=1,2,...,s−1, consequently,x^k+rk− x^k →0. Further, by the definition of the metric projection and by the triangle inequality, we have

PCix^k−x^k ≤ PCix^k+rk−x^k ≤ PCix^k+rk−x^k+rk+x^k+rk−x^k. (9.72) Let jk ∈ Ik be such that PC_jkx^k−x^k =max_j∈IkPCjx^k−x^k, k ≥0. Let lim_k→∞w^k_j

kPC_jkx^k−x^k2=0. Since w^k_j

k≥δ for jk∈I(x^k)we have

klim→∞PC_jkx^k−x^k=0. (9.73) Since i∈I_k+rkwe have

PCix^k+rk−x^k+rk ≤ PC_jk+

rkx^k+rk−x^k+rk.

consequently, lim_k→∞PC_ix^k+rk−x^k+rk=0. The inequalities (9.71) and (9.72) yield now lim_k→∞PC_ix^k−x^k=0, i.e.,{w^k}^∞_k=0is approximately regular.

(e) LetH =Rⁿ, let Ui:H →H, i∈I, be cutters having a common fixed point and let lim inf_k→∞λk(2−λk)>0. Consider a sequence generated by the recurrence (9.66) with a repetitive control{ik}^∞_k=0⊂I, i.e., a control for which the subset Ki={k≥0|ik=i}is infinite for any i∈I (see., e.g., [1, Sect. 3]). It is clear that N0={0,1,2,3,...}=K1∪K2∪...∪Kmand that Ki∩Jj=/0 for all i,j∈I,i=j.

The control{ik}^∞_k=0is approximately semi-regular. This follows from inequality (9.26) which guarantees 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.

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