IN BANACH SPACES

ITERATIVE METHODS FOR SOLVING FIXED-POINT PROBLEMS WITH NONSELF-MAPPINGS

IN BANACH SPACES

YAKOV ALBER, SIMEON REICH, AND JEN-CHIH YAO Received 21 January 2002

We study descent-like approximation methods and proximal methods of the re- traction type for solving fixed-point problems with nonself-mappings in Hilbert and Banach spaces. We prove strong and weak convergences for weakly contrac- tive and nonexpansive maps, respectively. We also establish the stability of these methods with respect to perturbations of the operators and the constraint sets.

1. Preliminaries

LetBbe a real uniformly convex and uniformly smooth Banach space [12] with norm · , letB^∗ be its dual space with the dual norm · ∗and, as usual, denote the duality pairing ofBandB^∗byx, ϕ, wherex∈Bandϕ∈B^∗(in other words,x, ϕis the value ofϕatx).

We recall that the uniform convexity of the spaceBmeans that, for any given >0, there exists δ >0 such that for all x, y∈B withx ≤1,y ≤1, and x−y =, the inequality

x+y ≤2(1−δ) (1.1)

holds. The function

δ_B()=inf1−2⁻¹x+y:x =1,y =1,x−y =

(1.2) is called the modulus of convexity of the spaceB.

The uniform smoothness of the spaceBmeans that, for any given>0, there existsδ >0 such that the inequality

2⁻¹x+y+x−y

−1≤y (1.3)

Copyright©2003 Hindawi Publishing Corporation Abstract and Applied Analysis 2003:4 (2003) 193–216

2000 Mathematics Subject Classification: 47H06, 47H09, 47H10, 47H14, 49M25 URL:http://dx.doi.org/10.1155/S1085337503203018

holds for allx, y∈Bwithx =1 andy ≤δ. The function ρB(τ)=sup2⁻¹x+y+x−y

−1 :x =1,y =τ (1.4)

is called the modulus of smoothness of the spaceB.

We observe that the spaceBis uniformly convex if and only if

δB()>0 ∀>0, (1.5)

and that it is uniformly smooth if and only if

limτ→0hB(τ)=lim

τ→0

ρB(τ)

τ ⁼0. (1.6)

Recall that the nonlinear, in general, operatorJ:B→B^∗defined by

Jx∗= x, x, Jx = x² (1.7)

is called the normalized duality mapping.

The following estimates, established in [2,3], will be used in the proofs of the convergence and stability theorems below (e.g., Theorems3.2and4.9). Define C(s, t) :R⁺×R⁺→R⁺by

C(s, t)=

s²+t²

2 . (1.8)

IfBis a uniformly smooth Banach space, then for allx, y∈B,

x−y, Jx−J y ≤2C²x,y ρB

4y−x Cx,y

, (1.9)

and ifBis a uniformly convex Banach space, then for allx, y∈B,

x−y, Jx−J y ≥8C²x,y δ_B

y−x Cx,y

. (1.10)

Ifx ≤Randy ≤R, then, respectively, x−y, Jx−J y ≤2LR²ρ_B

4y−x R

, (1.11)

x−y, Jx−J y ≥(2L)⁻¹R²δB

y−x 2R

, (1.12)

where 1< L <1.7 is the Figiel constant [10,17]. Furthermore, in a uniformly smooth Banach space, we also have for allx, y∈B,

Jx−J y∗≤8Rh_B16LR⁻¹x−y

. (1.13)

Now we recall the definitions of nonexpansive and weakly contractive map- pings (see, e.g., [4,6,11]).

Definition 1.1. A mapping A:G→Bis said to be nonexpansive on the closed convex subsetGof a Banach spaceBif for allx, y∈G,

Ax−Ay ≤ x−y. (1.14)

Definition 1.2. A mappingAis said to be weakly contractive of classC_ψ(t) on a closed convex subsetG of a Banach spaceBif there exists a continuous and increasing function ψ(t) defined onR⁺ such that ψ is positive on R⁺\ {0}, ψ(0)=0, limt→+∞ψ(t)=+∞, and for allx, y∈G,

Ax−Ay ≤ x−y −ψx−y

. (1.15)

We also use the concept of a sunny nonexpansive retraction [9,11,13].

Definition 1.3. LetGbe a nonempty closed convex subset ofB. A mappingQG: B→Gis said to be

(i) a retraction ontoGifQ_G²=Q_G;

(ii) a nonexpansive retraction if it also satisfies the inequality

Q_Gx−Q_Gy≤ x−y ∀x, y∈B; (1.16) (iii) a sunny retraction if for allx∈Band for all 0≤t <∞,

QG

QGx+tx−QGx=QGx. (1.17) Proposition 1.4. Let G be a nonempty closed convex subset of B. A mapping QG:B→Gis a sunny nonexpansive retraction if and only if for allx∈Band for allξ∈G,

x−Q_Gx, JQ_Gx−ξ≥0. (1.18)

More information regarding sunny nonexpansive retractions can be found in [11,16].

2. Recursive inequalities

We will often use the following facts concerning numerical recursive inequalities (see [5,7,8]).

Lemma2.1. Let{λk}be a sequence of nonnegative numbers and{αk}a sequence of positive numbers such that

∞ 0

αn= ∞. (2.1)

Let the recursive inequality

λn+1≤λn−αnψλn

, n=0,1,2, . . . , (2.2)

hold, whereψ(λ)is a continuous strictly increasing function for all λ≥0 with ψ(0)=0. Then

(1)λn→0asn→ ∞;

(2)the estimate of convergence rate

λ_n≤Φ⁻¹

Φλ0

−

n−1 0

α_j

(2.3)

is satisfied, whereΦis defined by the formulaΦ(t)=

dt/ψ(t)andΦ⁻¹is its inverse function.

Lemma2.2. Let{λk}and{γk}be sequences of nonnegative numbers and{αk}a sequence of positive numbers satisfying conditions (2.1) and

γn

αn −→0 asn−→ ∞. (2.4)

Let the recursive inequality

λn+1≤λn−αnψλn

+γn, n=0,1,2, . . . , λ0=λ,¯ (2.5) be given, whereψ(λ)is a continuous strictly increasing function for allλ≥0with ψ(0)=0. Then

(1)λ_n→0asn→ ∞;

(2)there exists a subsequence{λnl} ⊂ {λn},l=1,2, . . ., such that λnl≤ψ⁻¹

1 _nl

0 αm +γnl

αnl

,

λ_nl+1≤ψ⁻¹ 1

nl

0 α_m +γ_nl α_nl

+γ_nl, λ_n≤λ_nl+1−

n−1 nl+1

αm

Ꮽm

, n_l+ 1< n < n_l+1,Ꮽm= m

0

α_i,

λn+1≤λ¯− n

0

α_m

Ꮽm ≤λ,¯ 1≤n≤n1−1, 1≤n1≤smax=max

s:

s 0

αm

Ꮽm ≤λ¯

.

(2.6)

Lemma2.3. Let{µk},{αk},{βk}, and{γk}be sequences of nonnegative real num- bers satisfying the recurrence inequality

µ_k+1≤µ_k−α_kβ_k+γ_k. (2.7) Assume that

∞ k=0

αk= ∞, ∞ k=1

γk<∞. (2.8)

Then

(i)there exists an infinite subsequence{βk} ⊂ {βk}such that βk≤ 1

_k

j=1αj

, (2.9)

and, consequently,limk→∞βk=0;

(ii)if limk→∞α_k=0and there existsκ >0such that

βk+1−βk≤καk (2.10)

for allk≥0, thenlimk→∞βk=0.

Lemma2.4. Let{µk},{αk},{βk}, and{γk}be sequences of nonnegative real num- bers satisfying the recurrence inequality (2.7). Assume that

∞ k=0

αk= ∞, lim

k→∞

γk

αk =0. (2.11)

Then there exists an infinite subsequence{β_k} ⊂ {β_k}such thatlim_k→∞β_k=0.

3. Retraction methods for weakly contractive mappings

First of all, we consider the convergence of the retraction descent-like approxi- mation method

x_n+1=Q_Gx_n−ω_nx_n−Ax_n, n=0,1,2, . . . , (3.1) whereQ_Gis a nonexpansive retraction ofBonto the setG.

Theorem3.1. Let{ω_n}be a sequence of positive numbers such that^∞₀ ω_n= ∞. LetGbe a closed convex subset of a Banach spaceB, and letAbe a weakly con- tractive mapping fromGintoBof the classCψ(t)with a strictly increasing function ψ(t). Suppose that the mappingAhas a (unique) fixed pointx^∗∈G. Then

(i)the iterative sequence generated by (3.1), starting atx0∈G, converges in norm tox^∗asn→ ∞;

(ii)xn−Axn →0asn→ ∞;

(iii)the following estimate of the convergence rate holds xn−x^∗≤Φ⁻¹

Φx0−x^∗−

n−1 0

ωn

, (3.2)

whereΦ(t)is defined by the formulaΦ(t)=

dt/ψ(t)andΦ⁻¹is its inverse function.

Proof. Consider the sequence{xn}generated by (3.1). We have x_n+1−x^∗=Q_Gx_n−ω_nx_n−Ax_n−Q_Gx^∗

≤xn−ωn

xn−Axn

−x^∗

=1−ωn

xn+ωnAxn− 1−ωn

x^∗−ωnx^∗

≤

1−ωnxn−x^∗+ωnAxn−Ax^∗

≤

1−ωnxn−x^∗+ωnxn−x^∗−ωnψxn−x^∗. (3.3)

Thus, for alln≥0,

xn+1−x^∗≤xn−x^∗−ωnψxn−x^∗. (3.4)

Now the claims (i), (ii), and (iii) follow from (3.4) and Lemma 2.1 because x_n−Ax_n ≤2x_n−x^∗.

The following theorem provides other estimates of the convergence rate.

Theorem3.2. Let{ω_n}be a sequence of positive numbers such that^∞0 ω_n= ∞, ω_n≤ω, and lim_n→∞ω_n=0. LetGbe a closed convex subset of Banach spaceB,

and letAbe a weakly contractive mapping fromGintoBof the classCψ(t)with a strictly increasing functionψ(t). Suppose that the mappingAhas a (unique) fixed pointx^∗∈G. Then the iterative sequence generated by (3.1), starting atx0∈G, converges in norm tox^∗. Moreover, there exist a subsequence {x_nl} ⊂ {x_n},l= 1,2, . . . ,and constantsK >0andR >0such that

x_nl−x^∗2≤ψ₁⁻¹ 1

nl

0ωj + γnl

ωnl

, (3.5)

where

ψ1(t)=√

tψ^√t, γ_n=2LR²ρ_B8KR⁻¹ω_n, (3.6) x_nl+1−x^∗2≤ψ₁⁻¹

1

nl

0 ωj+ γnl

ωnl

+γ_nl, (3.7) xn−x^∗2≤xnl+1−x^∗2−

n−1 nl+1

ω_m

Ꮽm, nl+ 1< n < nl+1,Ꮽm= m

0

ωi, (3.8) xn+1−x^∗2≤x0−x^∗2−

n 0

ωm

Ꮽm, 0≤n≤n1−1, (3.9) 0≤n1≤smax=max

s:

s 0

ωm

Ꮽm ≤x0−x^∗2

. (3.10)

Proof. By (3.4),

x_n+1−x^∗≤x_n−x^∗≤x0−x^∗=K (3.11)

for alln≥0. Thusxn−Axn ≤2Kand

xn≤xn−x^∗+x^∗≤K+x^∗=C. (3.12)

Setφ_n=x_n−ω_n(x_n−Ax_n). SinceQ_Gis a nonexpansive retraction andx²is a convex functional, we have

x_n+1−x^∗2≤x_n−ω_nx_n−Ax_n−x^∗2=φ_n−x^∗2

≤xn−x^∗2−2ωn xn−Axn, Jφn−x^∗

=xn−x^∗2−2ωn xn−Axn, Jxn−x^∗ + 2 φn−xn, Jφn−x^∗−Jxn−x^∗.

(3.13)

By (1.11), ifx ≤Randy ≤R, then x−y, Jx−J y ≤2LR²ρB

4R⁻¹x−y

. (3.14)

Therefore,

xn+1−x^∗2≤xn−x^∗2−2ωnψxn−x^∗xn−x^∗+ 2γn, (3.15)

where

γ_n= φ_n−x_n, Jφ_n−x^∗−Jx_n−x^∗≤2LR²ρ_B8KR⁻¹ω_n. (3.16) Here we used the estimatesφn ≤C+ 2ωK=Randxn ≤C≤R. It is obvious that

γ_n

ω_n ^−→0 asn−→ ∞, (3.17)

becauseBis a uniformly smooth space. Thus, we get forλn= xn−x^∗2 the following recursive inequality:

λn+1≤λn−2ωnψ1

λn

+ 2γn. (3.18)

The strong convergence of{xn}tox^∗and the estimates (3.5), (3.6), (3.7), (3.8),

(3.9), and (3.10) now follow fromLemma 2.2.

Next we will study the iterative method (3.1) with perturbed mappingsA_n: G→B:

yn+1=QG

yn−ωn

yn−Anyn

, n=0,1,2, . . . , (3.19)

provided that the sequence{A_n}satisfies the following condition:

Anv−Av≤hngv

+δn ∀v∈G. (3.20)

Theorem3.3. Let{ω_n}be a sequence of positive numbers such that^∞0 ω_n= ∞ andω≥ωn>0. LetGbe a closed convex subset of the Banach spaceB, letAbe a weakly contractive mapping fromGintoBof the classCψ(t)with a strictly increasing functionψ(t), and letx^∗∈G be its unique fixed point. Suppose that there exist sequences of positive numbers{δ_n} and{h_n}converging to 0 as n→ ∞, and a

positive functiong(t)defined onR⁺and bounded on bounded subsets ofR⁺such that (3.20) is satisfied for alln≥0. If the sequence generated by (3.19) and starting at an arbitraryy0∈Gis bounded, then it converges in norm to the pointx^∗. If, in addition,lim_n→∞ω_n=0, then there exist a subsequence{y_nl} ⊂ {y_n},l=1,2, . . . , converging tox^∗asl→ ∞and a constantK >0such that

y_nl−x^∗2≤ψ₁⁻¹ 1

nl

1 ω_j+h_nlg(K) +δ_nl

. (3.21)

Proof. Similarly to the proof ofTheorem 3.1, we get, for alln≥0, the inequality yn+1−x^∗=QG

yn−ωn

yn−Anyn

−QGx^∗

≤yn−ωn

yn−Anyn

−x^∗

=1−ωn

yn+ωnAnyn− 1−ωn

x^∗−ωnx^∗

≤

1−ωnyn−x^∗+ωnAnyn−Ax^∗

≤

1−ω_ny_n−x^∗+ω_nAy_n−Ax^∗+ω_nA_ny_n−Ay_n

≤y_n−x^∗−ω_nψy_n−x^∗+ω_nh_ngy_n+δ_n.

(3.22)

By our assumptions, there existsK >0 such thaty_n≤K. Hence, yn+1−x^∗≤yn−x^∗−ωnψyn−x^∗+ωn

hng(K) +δn

(3.23) for some constantC1>0.Since

ω_nh_ng(K) +δ_n

ω_n ^−→0 asn−→ ∞, (3.24)

we conclude, again, byLemma 2.2thatyn→x^∗. The estimate (3.21) is obtained

as in the proof ofTheorem 3.2.

LetG1andG2be closed convex subsets ofB. The Hausdorffdistance between G1andG2is defined by the following formula:

ᏴG1, G2

=max

sup

z1∈G1

z2inf∈G2

z1−z2,sup

z1∈G2

z2inf∈G1

z1−z2

. (3.25) In order to prove the next theorem, we need the following lemma.

Lemma 3.4. If B is a uniformly smooth Banach space, and if Ω1 and Ω2 are closed convex subsets ofBsuch that the HausdorffdistanceᏴ(Ω1,Ω2)≤σ,h_B(τ)= ρ_B(τ)/τ, andQ_Ω1andQ_Ω2are the sunny nonexpansive retractions onto the subsets

Ω1andΩ2, respectively, then

Q_Ω1x−Q_Ω2x²≤16R(2r+d)h_B16LR⁻¹σ, (3.26)

whereLis the Figiel constant,r= x,d=max{d1, d2}, andR=2(2r+d) +σ.

Heredi=dist(θ,Ωi),i=1,2, andθis the origin of the spaceB.

Proof. Denote ˘x1=Q_Ω1xand ˘x2=Q_Ω2x. SinceᏴ(Ω1,Ω2)≤σ, there existsξ1∈ Ω1such thatx˘2−ξ1 ≤σ. Now

x−x˘1, Jx˘2−x˘1

= x−x˘1, Jξ1−x˘1

+ x−x˘1, Jx˘2−x˘1

−Jξ1−x˘1

≤x−x˘1Jx˘2−x˘1

−Jξ1−x˘1,

(3.27) becausex−x˘1, J(ξ1−x˘1) ≤0 by (1.18). It is obvious that

x−x˘1≤x−Q_Ω1θ+Q_Ω1θ−Q_Ω1x≤2x+Q_Ω1θ≤2r+d, x−x˘2≤x−Q_Ω2θ+Q_Ω2θ−Q_Ω2x≤2x+Q_Ω2θ≤2r+d, x˘1−x˘2≤x˘1−x+x−x˘2≤2(2r+d),

x˘1−ξ1≤x˘1−x˘2+x˘2−ξ1≤2(2r+d) +σ.

(3.28)

IfR=2(2r+d) +σ, then by (1.13) x−x˘1, Jx˘2−x˘1

≤8Rx−x˘1h_B16LR⁻¹x˘2−ξ1

≤8R(2r+d)hB

16LR⁻¹σ. (3.29)

In the same way, we see that there existsξ2∈Ω2such thatx˘1−ξ2 ≤σand x−x˘2, Jx˘1−x˘2

= x−x˘2, Jξ2−x˘2

+ x−x˘2, Jx˘1−x˘2

−Jξ2−x˘2

≤x−x˘2Jx˘1−x˘2

−Jξ2−x˘2,

(3.30) becausex−x˘2, J(ξ2−x˘2) ≤0. As above,

x˘2−ξ2≤x˘1−x˘2+x˘1−ξ2≤2(2r+d) +σ, (3.31)

and if, again,R=2(2r+d) +σ, we have x−x˘2, Jx˘1−x˘2

≤8Rx−x˘2hB

16LR⁻¹x˘1−ξ2

≤8R(2r+d)hB

16LR⁻¹σ. (3.32) Therefore, the estimate (3.26) holds by (3.29) and (3.32).Lemma 3.4is proved.

Next we will study the iterative method (3.1) with perturbed setsG_n:

z_n+1=Q_Gn+1z_n−ω_nz_n−Az_n, n=0,1,2, . . . . (3.33) Theorem3.5. LetG⊂D(A)andGn⊂D(A),n=0,1,2, . . . ,be closed convex sub- sets ofBsuch that the HausdorffdistanceᏴ(Gn, G)≤σn≤σ, and letAbe a weakly contractive mapping fromD(A)intoBof the classC_ψ(t)with a strictly increasing functionψ(t). Suppose that the mappingAhas a (unique) fixed pointx^∗∈G. As- sume that^∞₀ ωn= ∞,ωn≤ω, and that

hB

σn

ω_n ^−→0 asn−→ ∞. (3.34)

If the iterative sequence (3.33), starting at an arbitrary pointz0∈G0, is bounded, then it converges in norm tox^∗.

Proof. For alln_≥0, we have

z_n+1−x^∗≤z_n+1−x_n+1+x_n+1−x^∗, (3.35)

where the sequence{xn}is generated by (3.1), and therefore,

xn−x^∗−→0 asn−→ ∞. (3.36)

We will show that

z_n−x_n−→0 asn−→ ∞. (3.37)

We have

z_n+1−x_n+1=Q_Gn+1z_n−ω_nz_n−Az_n−Q_Gx_n−ω_nx_n−Ax_n

≤Q_Gz_n−ω_nz_n−Az_n−Q_Gx_n−ω_nx_n−Ax_n +QGn+1

zn−ωn

zn−Azn

−QG

zn−ωn

zn−Azn. (3.38)

It is easy to check that QG

zn−ωn

zn−Azn

−QG

xn−ωn

xn−Axn

≤

1−ω_nz_n−x_n+ω_nAz_n−Ax_n

≤z_n−x_n−ω_nψz_n−x_n.

(3.39)

Ifzn ≤K, then zn−Azn ≤2zn−x^∗ ≤2(K+x^∗) and there exists a constantr >0 such thatzn−ωn(zn−Azn) ≤r. ByLemma 3.4,

Q_Gn+1z_n−ω_nz_n−Az_n−Q_Gz_n−ω_nz_n−Az_n²

≤16R(2r+d)h_B16LR⁻¹σ_n+1, (3.40) whereR=2(2r+d) +σ,d=max{d1, d2},d1=dist(θ, G),d2=sup_n{dist(θ, Gn)}, andθis the origin of the spaceB. Hence,

z_n+1−x_n+1

≤z_n−x_n−ω_nψz_n−x_n+16R(2r+d)h_B16LR⁻¹σ_n+1^1/2. (3.41) Sinceω⁻_n¹hB(σn)→0, (3.37) is, indeed, true.Theorem 3.5is proved.

Next we study the method of successive approximations

xn+1=QGAxn, n=0,1,2, . . . , (3.42) whereQ_Gis the sunny nonexpansive retraction ofBonto its subsetG.

Theorem3.6. Suppose thatBis a uniformly smooth Banach space, andA:G→B is a weakly contractive mapping from a closed convex subsetGintoBof the class C_ψ(t) with a strictly increasing functionψ(t). Suppose that the mappingAhas a (unique) fixed pointx^∗∈G. Consider the sequence{x_n}generated by (3.42). Then

(i)the sequence{xn}is bounded;

(ii)the sequence{x_n}strongly converges tox^∗;

(iii)the following estimate of the convergence rate holds:

x_n−x^∗2≤Φ⁻¹Φx0−x^∗2−(n−1), (3.43)

where Φ(t) is defined by the formula Φ(t)=

dt/ψ1(t) with ψ1(t)=

√tψ(^√t)andΦ⁻¹is its inverse function.