EQUATIONS WITH ACCRETIVE OPERATORS
YA. I. ALBER, C. E. CHIDUME, AND H. ZEGEYEReceived 11 October 2004
We study the regularization methods for solving equations with arbitrary accretive op-erators. We establish the strong convergence of these methods and their stability with respect to perturbations of operators and constraint sets in Banach spaces. Our research is motivated by the fact that the fixed point problems with nonexpansive mappings are namely reduced to such equations. Other important examples of applications are evolu-tion equaevolu-tions and co-variaevolu-tional inequalities in Banach spaces.
1. Introduction
LetEbe a real normed linear space with dualE∗. Thenormalized duality mapping j:E→
2E∗
is defined by
j(x) :=x∗∈E∗:x,x∗ = x2,x∗ ∗= x
, (1.1)
wherex,φ denotes the dual product (pairing) between vectorsx∈Eandφ∈E∗. It is well known that ifE∗is strictly convex, then jis single valued. We denote the single valued normalized duality mapping byJ.
A mapA:D(A)⊆E→2Eis calledaccretiveif for allx,y∈D(A) there existsJ(x−y)∈
j(x−y) such that
u−v,J(x−y)≥0, ∀u∈Ax,∀v∈Ay. (1.2)
IfAis single valued, then (1.2) is replaced by
Ax−Ay,J(x−y)≥0. (1.3)
Ais calleduniformly accretiveif for allx,y∈D(A) there existJ(x−y)∈j(x−y) and a strictly increasing functionψ:R+:=[0,∞)→R+,ψ(0)=0 such that
Ax−Ay,J(x−y)≥ψx−y. (1.4)
It is calledstrongly accretiveif there exists a constantk >0 such that in (1.4)ψ(t)=kt2. If Eis a Hilbert space, accretive operators are also calledmonotone. An accretive operatorA is said to behemicontinuousat a pointx0∈D(A) if the sequence{A(x0+tnx)}converges
weakly toAx0 for any elementx such thatx0+tnx∈D(A), 0≤tn≤t(x0) and tn→0,
n→ ∞. An accretive operatorAis said to bemaximal accretiveif it is accretive and the inclusionG(A)⊆G(B), withBaccretive, whereG(A) andG(B) denote graphs ofAand B, respectively, implies thatA=B. It is known (see, e.g., [14]) that an accretive hemicon-tinuous operatorA:E→Ewith a domainD(A)=Eis maximal accretive. In a smooth Banach space, a maximal accretive operator is strongly-weakly demiclosed onD(A). An accretive operatorAis said to bem-accretiveifR(A+αI)=Efor allα >0, whereIis the identity operator inE.
Interest in accretive maps stems mainly from their firm connection with fixed point problems, evolution equations and co-variational inequalites in a Banach space (see, e.g. [6,7,8,9,10,11,12,26]). Recall that each nonexpansive mapping is a continuous ac-cretive operator [7,19]. It is known that many physically significant problems can be modeled by initial-value problems of the form (see, e.g., [10,12,26])
x(t) +Ax(t)=0, x(0)=x0, (1.5)
whereAis an accretive operator in an appropriate Banach space. Typical examples where such evolution equations occur can be found in the heat, wave, or Schr¨odinger equations. One of the fundamental results in the theory of accretive operators, due to Browder [11], states that ifAis locally Lipschitzian and accretive, thenAism-accretive. This result was subsequently generalized by Martin [23] to the continuous accretive operators. Ifx(t) in (1.5) is independent oft, then (1.5) reduces to the equation
Au=0, (1.6)
whose solutions correspond to the equilibrium points of the system (1.5). Consequently, considerable research efforts have been devoted, especially within the past 20 years or so, to iterative methods for approximating these equilibrium points.
The two well-known iterative schemes for successive approximation of a solution of the equationAx= f, whereAis either uniformly accretive or strongly accretive, arethe Ishikawa iteration process(see, e.g., [20]) andthe Mann iteration process(see, e.g., [22]). These iteration processes have been studied extensively by various authors and have been successfully employed to approximate solutions of several nonlinear operator equations in Banach spaces (see, e.g., [13,15,17]). But all efforts to use the Mann and the Ishikawa schemes to approximate the solution of the equationAx=f, whereAis an accretive-type mapping (not necessarily uniformly or strongly accretive), have not provided satisfactory results. The major obstacle is that for this class of operators the solution is not, in general, unique.
(see, e.g., [4]). Furthermore,the stabilityof our methods with respect to perturbation of the operators and constraint sets is also studied.
2. Preliminaries
LetEbe a real normed linear space of dimension greater than or equal to 2, andx,y∈E. Themodulus of smoothnessofEis defined by
ρE(τ) :=sup
x+y+x−y
2 −1 :x =1, y =τ . (2.1)
A Banach spaceEis calleduniformly smoothif
lim
τ→0hE(τ) :=limτ→0 ρE(τ)
τ =0. (2.2)
Examples of uniformly smooth spaces are the LebesgueLp, the sequencep, and the
SobolevWm
p spaces for 1< p <∞andm≥1 (see, e.g., [2]).
IfEis a real uniformly smooth Banach space, then the inequality
x2≤ y2+ 2x−y,Jx
≤ y2+ 2x−y,J y+ 2x−y,Jx−J y (2.3)
holds for every x,y∈E. A further estimation ofx2 needs one of the following two lemmas.
Lemma2.1 [5]. LetEbe a uniformly smooth Banach space. Then forx,y∈E,
x−y,Jx−J y ≤8x−y2+Cx,yρ
E
x−y, (2.4)
where
Cx,y≤4 max2L,x+y (2.5)
andLis the Figiel constant,1< L <1.7 [18,24].
Lemma2.2 [2]. In a uniformly smooth Banach spaceE, forx,y∈E,
x−y,Jx−J y ≤R2x,yρ
E 4
x−y
Rx,y
, (2.6)
where
Rx,y=2−1x2+y2. (2.7)
Ifx ≤Randy ≤R, then
x−y,Jx−J y ≤2LR2ρ
E
4x−y
R
, (2.8)
We will need the following lemma on the recursive numerical inequalities.
Lemma2.3 [1]. Let{λk}and{γk}be sequences of nonnegative numbers and let{αk}be a
sequence of positive numbers satisfying the conditions
∞
1
αn= ∞, γαn
n−→0 asn−→ ∞. (2.9)
Let the recursive inequality
λn+1≤λn−αnφ
λn
+γn, n=1, 2,..., (2.10)
be given whereφ(λ)is a continuous and nondecreasing function fromR+toR+such that it is positive onR+\ {0},φ(0)=0,lim
t→∞φ(t)≥c >0. Thenλn→0asn→ ∞.
We will also use the concept of a sunny nonexpansive retraction [19].
Definition 2.4. LetGbe a nonempty closed convex subset ofE. A mappingQG:E→Gis
said to be
(i) a retraction ontoGifQ2
G=QG;
(ii) a nonexpansive retraction if it also satisfies the inequality
QGx−QGy≤ x−y, ∀x,y∈E; (2.11)
(iii) a sunny retraction if for allx∈Eand for all 0≤t <∞,
QGQGx+tx−QGx=QGx. (2.12)
Definition 2.5. IfQGsatisfies (i)–(iii) ofDefinition 2.4, then the elementx=QGxis said
to be a sunny nonexpansive retractor ofx∈EontoG.
Proposition2.6. LetEbe a uniformly smooth Banach space, and letG be a nonempty
closed convex subset ofE. A mappingQG:E→Gis a sunny nonexpansive retraction if and
only if for allx∈Eand for allξ∈G,
x−QGx,J
QGx−ξ
≥0. (2.13)
Denote byᏴE(G1,G2) the Hausdorffdistance between setsG1andG2in the spaceE, that is,
ᏴE
G1,G2
=max
sup
z1∈G1 inf
z2∈G2
z1−z2, sup z1∈G2
inf
z2∈G1
z1−z2 . (2.14)
Lemma2.7 [7]. LetEbe a uniformly smooth Banach space, and letΩ1 andΩ2be closed
convex subsets ofEsuch that the HausdorffdistanceᏴE(Ω1,Ω2)≤σ. IfQΩ1 andQΩ2 are the sunny nonexpansive retractions onto the subsetsΩ1andΩ2, respectively, then
QΩ
1x−QΩ2x 2
≤16R(2r+q)hE16LR−1σ, (2.15)
3. Operator regularization method
We will deal with accretive operatorsA:E→Eand operator equation
Ax=f (3.1)
given on a closed convex subsetG⊂D(A)⊆E, whereD(A) is a domain ofA.
In the sequel, we understand a solution of (3.1) in the sense of a solution of the co-variational inequality (see, e.g., [9])
Ax−f,J(y−x)≥0, ∀y∈G,x∈G. (3.2)
The following statement is a motivation of this approach [25].
Theorem3.1. Suppose thatEis a reflexive Banach space with strictly convex dual spaceE∗. LetA:E→Ebe a hemicontinuous operator. If for fixedx∗∈Eand f ∈Ethe co-variational inequality
Ax−f,Jx−x∗≥0, ∀x∈E, (3.3)
holds, thenAx∗= f.
In fact, the following more general theorem was proved in [8].
Theorem3.2. LetEbe a smooth Banach space and letA:E→2Ebe an accretive operator.
Then the following statements are equivalent: (i)x∗satisfies the covariational inequality
z−f,Jx−x∗≥0, ∀z∈Ax,∀x∈E; (3.4)
(ii) 0∈R(Ax∗−f).
We present the following two definitions of a solution of the operator equation (3.1) onG.
Definition 3.3. An elementx∗∈G is said to be a generalized solution of the operator equation (3.1) onGif there existsz∈Ax∗such that
z−f,Jy−x∗≥0, ∀y∈G. (3.5)
Definition 3.4. An elementx∗∈Gis said to be a total solution of the operator equation (3.1) onGif
z−f,Jy−x∗≥0, ∀y∈G,∀z∈Ay. (3.6)
Lemma3.6 [6]. Suppose thatEis a reflexive Banach space with strictly convex dual space E∗. Let an operatorAbe either hemicontinuous or maximal accretive. IfG⊂intD(A), then Definitions3.3and3.4are equivalent.
Lemma3.7. Under the conditions ofLemma 3.6, the set of solutions of the operator equation (3.1) onGis closed.
The proof follows from the fact thatJis continuous in smooth reflexive Banach spaces and any hemicontinuous or maximal accretive operator is demiclosed in such spaces.
For finding a solutionx∗of (3.1), we consider the regularized equation
Azα+αzα=f, (3.7)
whereαis a positive parameter. Letz0
αbe a generalized solution of (3.7) onG, that is, there existsζα0∈Az0αsuch that
ζα0+αz0α−f,J
x−zα0
≥0, ∀x∈G. (3.8)
Theorem3.8. Assume thatEis a reflexive Banach space with strictly convex dual space
E∗and with originθ,Ais a hemicontinuous or maximal accretive operator with domain D(A)⊆E,G⊂intD(A)is convex and closed, (3.1) has a nonempty generalized solution setN⊂G. Thenz0
α ≤2x¯∗, wherex¯∗is an element ofN with minimal norm. If the
normalized duality mappingJis sequentially weakly continuous onE, thenz0
α→x∗asα→0,
wherex∗∈Nis a sunny nonexpansive retractor ofθontoN, that is, a (necessarily unique) solution of the inequality
x∗,Jx∗−x∗≥0, ∀x∗∈N. (3.9)
Proof. First, we show thatz0
αis theuniquesolution of (3.7). Suppose thatu0αis another
solution of this equation. Then along with (3.8), we have for someξ0
α∈Au0αthat
ξα0+αu0α−f,J
x−u0α
≥0, ∀x∈G. (3.10)
Sincez0
α∈Gandu0α∈G, we have the inequalities
ζ0
α+αz0α−f,J
u0
α−z0α
≥0,
ξ0
α+αu0α−f,J
z0
α−u0α
≥0. (3.11)
Summing these inequalities, we obtain
0≥ξα0−ζα0,Jzα0−u0α≥αzα0−u0α,Jzα0−u0α=αzα0−u0α2. (3.12)
From this the claim follows.
Next, we prove that the sequence{z0
α}is bounded. Observe that the covariational
in-equality (3.8) implies that
becausex∗∈G. At the same time, sincex∗is a generalized solution of (3.1), there exists ξ∗∈Ax∗such that
ξ∗−f,Jz0
α−x∗
≥0. (3.14)
Then (3.13) and (3.14) together give
ζ0
α−ξ∗+αz0α,J
x∗−z0
α
=ζ0
α−ξ∗,J
x∗−z0
α
+αz0
α,J
x∗−z0
α
≥0. (3.15)
By accretiveness ofA, one gets
zα0,J
x∗−z0α
≥0. (3.16)
The obtained inequality yields the estimates
x∗−z0
α
2
≤x∗,Jx∗−z0α
≤x∗x∗−z0α. (3.17)
Hence,z0
α ≤2x∗for allx∗∈N, that is,zα0 ≤2x¯∗. Note that ¯x∗exists becauseN
is closed andEis reflexive. Show now thatz0
α−x∗ →0 asα→0. Since{zα0}is bounded, there exist a
subse-quencez0β⊂z0
αand an elementx∈Esuch thatzβ0xasβ→0. Sincezβ0∈GandG is
weakly closed (since it is closed and convex), we conclude thatx∈G. Due toLemma 3.6, the inequality (3.8) is equivalent to the following one:
w+αx−f,Jx−z0
α
≥0, ∀x∈G,∀w∈Ax. (3.18)
Therefore
w+βx−f,Jx−z0β≥0, ∀x∈G,∀w∈Ax. (3.19)
Passing to the limit in (3.19) asβ→0 and using the weak continuity ofJ, one gets
w−f,J(x−x)≥0, ∀x∈G,∀w∈Ax. (3.20)
ByLemma 3.6again, it follows thatxis a total (consequently, generalized) solution of (3.1) onG.
We now show thatx=x∗=QNθandx∗is unique. This will mean thatz0αx∗as we
presumed above. Consider (3.17) on{z0
β}withx∗=x. It is clear thatx−z0β →0. Then
we deduce from (3.16) that
x,Jx∗−x≥0, ∀x∗∈N. (3.21)
This means thatx=QNθ.
We prove thatxis a unique solution of the last inequality. Suppose thatx1∈Nis its another solution. Then
x1,Jx∗−x1
We have
x,Jx1−x≥0,
x1,Jx−x1≥0.
(3.23)
Their combination gives
x−x1,Jx1−x≥0, (3.24)
which contradicts the fact thatx−x1 ≥0. Thus, the claim is true.
Finally, the first inequality in (3.17) implies the strong convergence of{z0
α}to ¯x∗. The
proof is accomplished. In particular, the theorem is valid ifNis a singleton. Next we will study an operator regularization method for (3.1) with a perturbed right-hand side, perturbed constraint set, and perturbed operator. Assume that, instead of f, G, andA, we have the sequences{fδ} ∈E,{G
σ} ∈E, and{Aω},Aω:Gσ→E, such that fδ−f≤δ,
ᏴE
Gσ,G
≤σ, (3.25)
whereᏴE(G1,G2) is the Hausdorffdistance (2.14), and
Aωx−Ax≤ωζx, ∀x∈D, (3.26)
whereζ(t) is a positive and bounded function defined onR+andD=D(A)=D(Aω).
Thus, in reality, the equations
Aωy=fδ (3.27)
are given onGσ,σ≥0. Consider the following regularized equation onGσ:
Aωz+αz=fδ. (3.28)
Letzαγwithγ=(δ,σ,ω) be its (unique) generalized solution. This means that there exists
yαγ∈Aωzγαsuch that
yγα+αzγα−fδ,Jx−zγα≥0, ∀x∈Gσ. (3.29)
Theorem3.9. Assume that
(i)in real uniformly smooth Banach spaceEwith the modulus of smoothnessρE(τ), all
the conditions ofTheorem 3.8are fulfilled;
(ii)(3.28) has bounded generalized solutionszγαfor allδ≥0,σ≥0,ω≥0, andα >0;
(iii)the operatorsAω are accretive and bounded (i.e., they carry bounded sets of E to
bounded sets ofE);
(iv)G⊂DandGσ⊂Dare convex and closed sets;
If
δ+ω+hE(σ)
α −→0 asα−→0, (3.30)
thenzαγ→x¯∗, wherex¯∗is a sunny nonexpansive retractor ofθontoN.
Proof. Write the obvious inequality
zγ
α−x¯∗≤zα0−x¯∗+zγα−zα0, (3.31)
wherez0
αis a generalized solution of (3.7). The limit relationz0α−x¯∗ →0 has been
al-ready established inTheorem 3.8. At the same time, the resultzαγ−z0α →0 immediately
follows fromLemma 4.1proved in the next section. The condition (3.30) is sufficient for
this conclusion.
Remark 3.10. We do not suppose that in the operator equation (3.28) every operator Aωhas been defined on every setG
σ. Only possibility for the parametersωandσ to be
simultaneously rushed to zero is required.
4. Proximity lemma
We further presentthe proximity lemmabetween solutions of two regularized equations
T1z1+α1z1=f1, α1>0, (4.1) T2z2+α2z2=f2, α2>0, (4.2)
onG1andG2, respectively, provided their intersectionG1
G2is not empty.
Lemma4.1 (cf. [3]). Suppose that
(i)Eis a real uniformly smooth Banach space with the modulus of smoothnessρE(τ);
(ii)the solution sequences{z1}and{z2}of (4.1) and (4.2), respectively, are bounded, that is, there exists a constantM1>0such thatz1 ≤M1andz2 ≤M1;
(iii)the operatorsT1andT2are accretive and bounded on the sequences{z1}and{z2}, that is, there exist constantsM2>0andM3>0such thatT1z1 ≤M2andT2z2 ≤ M3;
(iv)G1⊂DandG2⊂Dare convex and closed subsets ofEandD=D(T1)=D(T2); (v)the estimatesf1−f2 ≤δ,ᏴE(G1,G2)≤σ, andT1z−T2z ≤ωζ(z),∀z∈D,
are fulfilled. Then
z1−z2≤ζ
M1
ω α1+
δ α1+M1
α1−α2
α1 +
c1hE
c2σ
α1 , (4.3)
where
c1=8R
Proof. Solutionsz1∈G1andz2∈G2of the operator equations (4.1) and (4.2) are defined by the following co-variational inequalities, respectively:
T1z1+α1z1−f1,J
x−z1
≥0, ∀x∈G1,α1>0, (4.5)
T2z2+α2z2−f2,J
x−z2
≥0, ∀x∈G2,α2>0. (4.6)
Estimate a dual product
B=T1z1+α1z1−f1−T2z2−α2z2+ f2,J
z1−z2
. (4.7)
Obviously,
B=T1z1−T1z2+α1z1−z2+T1z2−T2z2+α1−α2z2+f2−f1,Jz1−z2. (4.8)
The operatorT1is accretive, therefore,
T1z1−T1z2,J
z1−z2
≥0. (4.9)
Then
B≥α1z1−z2 2
−T1z2−T2z2+α1−α2z2+f1−f2z1−z2. (4.10)
Sincez2 ≤M1, we conclude in conformity with (v) that
B≥ −cz1−z2+α1z1−z2 2
, (4.11)
where
c=ωζM1
+M1α1−α2+δ. (4.12)
Next, ifᏴE(G1,G2)≤σ, then for everyz2∈G2there existsz∈G1such thatz2−z ≤σ and
T1z1+α1z1−f1,J
z1−z2
=T1z1+α1z1−f1,J
z1−z2
+Jz1−z
−Jz1−z
=T1z1+α1z1−f1,J
z1−z
+T1z1+α1z1−f1,Jz1−z2−Jz1−z.
(4.13)
By (4.5),
T1z1+α1z1−f1,Jz1−z≤0. (4.14)
Estimate the last term in (4.13). For this recall that ifx ≤Randy ≤R, then (see [2, page 38])
J(x)−J(y)
∗≤8RhE
Forz1−z2 ≤2M1andz1−z ≤2M1+σ=R, this implies that
T1z1+α1z1−f1,J
z1−z2
−Jz1−z
≤T1z1+α1z1−f1J
z1−z2
−Jz1−z∗
≤8Rα1z1+T1z1+f1hE16LR−1z2−z
≤8Rα1M1+M2+f1hE
16LR−1σ.
(4.16)
Analogously to the previous chain of inequalities,
T2z2+α2z2−f2,Jz2−z1≤8Rα1M1+M3+f2hE16LR−1σ. (4.17)
Therefore,
B≤c1hEc2σ. (4.18)
Finally, combining (4.11) with (4.18), one gets
c1hE
c2σ
+cz1−z2≥α1z1−z2 2
. (4.19)
This quadratic inequality gives
z1−z2≤c+c2+ 4α 1c1hE
c2σ
2α1 ≤
c α1+
c1hE
c2σ
α1 , (4.20)
because√a+b≤√a+√bfor alla,b≥0. Thus, (4.3) holds. From Theorem (3.10) andLemma 4.1we obtain the following corollary.
Corollary4.2. If, in the conditions ofLemma 4.1,ω=δ=σ=0, that is,T1=T2,f1=f2, andG1=G2, then
z1−z2≤2x∗α1−α2 α1 .
(4.21)
5. Iterative regularization methods
5.1. We begin by considering iterative regularization with exact given data.
Theorem5.1. LetEbe a real uniformly smooth Banach space with the modulus of
smooth-ness ρE(τ), let A:E→E be a bounded accretive operator withD(A)⊆E, and let G⊂
intD(A)be a closed convex set. Suppose that (3.1) has a generalized solutionx∗onG. Let
{n}and{αn}be real sequences such thatn≤1,αn≤1. Starting from arbitraryx0∈G define the sequence{xn}as follows:
xn+1:=QG
xn−n
Axn+αnxn−f
whereQG is a nonexpansive retraction ofEonto G. Then there exists1> d >0 such that
whenever
n≤d, ρE
n
nαn ≤d
2 (5.2)
for alln≥0, the sequence{xn}is bounded.
Proof. Denote byBr(x∗) the closed ball of radiusrwith the center inx∗. Chooser >0
sufficiently large such thatr≥2x∗andx0∈Br(x∗). Construct the setS=Br(x∗)∩G
and let
M:=3
2r+f+ sup
Ax:x∈S. (5.3)
We claim that{xn}is bounded in our circumstances. Show by induction thatxn∈Sfor
all positive integers. Actually,x0∈Sby the assumption. Hence, for givenn >0, we may presume the inclusionxn∈Sand prove thatxn+1∈S. Suppose thatxn+1does not belong toS. Sincexn+1∈G, this means thatxn+1−x∗> r. By (5.1) and due to the nonexpan-siveness ofQG, we have
xn+1−x∗=Q
G
xn−n
Axn+αnxn−f
−QGx∗ ≤xn−x∗−n
Axn+αnxn−f ≤xn−x∗+nAxn+αnxn−f
≤xn−x∗+Axn+xn−x∗+x∗+f ≤r+ sup
x∈SAx
+r+1
2r+f =r+M=M.
(5.4)
In the next calculations, we applyLemma 2.2withx=xn+1−x∗and y=xn−x∗. It is
easy to see that
x =xn+1−x∗≤M, y =xn−x∗≤r, x−y =xn+1−xn≤εnAxn+αnxn−f≤εnM.
(5.5)
Thus, max{x,y} ≤M, and we have
xn+1−xn,J
xn+1−x∗
−Jxn−x∗
≤2LM2ρE
4MM−1n
, (5.6)
because the functionρE(τ) is nondecreasing [18,21]. Besides, the functionρE(τ) is
con-vex, therefore,ρE(cτ)≤cρE(τ), for allc≤1. SinceMM−
1
≤1, (5.6) yields
xn+1−xn,Jxn+1−x∗
−Jxn−x∗≤2LMMρE4εn. (5.7)
Then using the facts thatρE(τ) is continuous, 0≤εn≤1, and by [16],
2≤lim
τ→0 ρE(4τ)
ρE(2τ)≤
we conclude that there is a constantC >1 such that
xn+1−xn,Jxn+1−x∗−Jxn−x∗≤8LCMMρEεn. (5.9)
Moreover, by (2.3), (5.1), (5.6) and by the inclusionxn∈S, one gets xn+1−x∗2
≤xn−x∗−n
Axn+αnxn−f2 ≤xn−x∗
2
−2n
Axn−f,J
xn−x∗
−2nαnxn,Jxn−x∗+ 16LCMMρEεn.
(5.10)
Sincex∗is a generalized solution of (3.1) onGandxn∈Gfor alln≥0, we can write
Axn−f,J
xn−x∗
≥0. (5.11)
Then (5.10) implies the inequality
xn+1−x∗2
≤xn−x∗
2
−2nαn
xn,J
xn−x∗
+ 16LCMρE
εn
. (5.12)
ChooseK >0 such that
K≤ r2
4(√D+M)2, (5.13)
whereD=8LCMM. Setd:=√K. It is not difficult to verify that 1> d >0. By virtue of our assumption,xn+1−x∗>xn−x∗. This allows us to deduce from (5.12) the
following estimate:
nαn
xn,J
xn−x∗
≤8LCMMρE
n
. (5.14)
It gives the inequality
xn,J
xn−x∗
≤DK (5.15)
because of the assumption that
ρEn
αnn ≤K=d
2, ∀n≥0. (5.16)
Now adding−x∗,J(xn−x∗)to both sides of (5.15), we get xn−x∗2
≤KD+−x∗,Jxn−x∗
≤KD+x∗xn−x∗≤KD+r2xn−x∗.
(5.17)
Solving this quadratic inequality forxn−x∗and using the estimate
r2
16+KD≤ r 4+
√
we derive that
x
n−x∗≤r2+ √
KD. (5.19)
In any case,
xn+1−x∗≤x
n−x∗+nAxn+αnxn−f, (5.20)
so that
xn+1−x∗≤r 2+
√
KD+nM < r, (5.21)
by the original choices ofKandn, and this contradicts the assumption thatxn+1is not inS. Thereforexn∈Sfor any integersn≥0. Thus{xn}is bounded, say,xn ≤C.
In what follows, we suppose that the normalized duality mappingJis continuous and sequentially weakly continuous in the ballBr(θ) withr=C. We show thatxn→x¯∗, where
¯
x∗is a unique solution of (3.9).
Theorem5.2. Assume that all the conditions of Theorems3.8and5.1are fulfilled. In addi-tion, letαn→0asn→ ∞,
n
αn −→0,
αn−αn+1
nα2n
−→0, ρE
n
nαn −→0. (5.22)
Then the sequence{xn}generated by (5.1) converges strongly tox¯∗asn→ ∞.
Proof. So, byTheorem 5.1,{xn}is bounded by a constantC. Letznandzn+1be general-ized solutions of the equation
Az+αkz=f (5.23)
onGfork=nandk=n+ 1, respectively. It follows from (4.3) and (5.22) that there exists a constantd >0 such thatzn−zn+1 ≤d. Put
pn=xn−n
Axn+αnxn−f
. (5.24)
Then by (5.1) and by convexity of the functionalx2, we have that
xn+1−zn+12
=QGpn−QGzn+1 2
≤pn−zn+1 2
≤pn−zn2+ 2zn+1−zn,Jzn+1−pn ≤pn−zn2+ 2zn+1−zn,Jzn−pn
+ 2zn+1−zn,J
zn+1−pn
−Jzn−pn
.
We continue the estimation of (5.25) using Lemma 2.1. It is easy to see that if H is a Hilbert space andτ≤τ¯, then [18,21]
ρE(τ)≥ρH(τ)=
1 +τ21/2−1≥cτ¯ 2, (5.26)
where
¯
c=1 + ¯τ2+ 1−1. (5.27)
One gets
xn+1−zn+12
≤pn−zn2+ 2zn−pn·zn+1−zn
+ 16zn+1−zn2+C1(n)ρEzn+1−zn,
(5.28)
where
C1(n)=8 max2L,zn−pn+zn+1−pn
≤8 max2L,C+M+ 2x∗=C1, (5.29)
whereMis defined by (5.3). Therefore, due toCorollary 4.2,
xn+1−zn+12
≤pn−zn2+ 4
αn−αn+1
αn ·
x∗·p
n−zn
+16 ¯c−1+C1
ρE 2
αn−αn+1
αn
x∗,
(5.30)
and in (5.27) ¯τ=d≥ zn+1−zn =τ.
Now we evaluatepn−zn2. The convexity inequality (2.3) yields pn−zn2
=xn−n
Axn+αnxn−f
−zn
2
≤xn−zn
2
−2n
Axn+αnxn−f,J
pn−zn
=xn−zn2−2nAxn+αnxn−f,Jxn−zn
+ 2pn−xn,Jpn−zn−Jxn−zn.
(5.31)
Since
Azn+αnzn−f,J
xn−zn
≥0, (5.32)
and by the accretiveness property ofA,
Axn−Azn,J
xn−zn
we deduce
p n−zn
2
≤xn−zn
2
−2nαnxn−zn
2
+ 2pn−xn,Jpn−zn−Jxn−zn
≤xn−zn2−2nαnxn−zn2+ 162nAxn+αnxn−f2
+C2(n)ρE
nAxn+αnxn−f,
(5.34)
where
C2(n)=4 max2L,zn−pn+xn−zn ≤4 max2L, 2C+M+ 2x∗=C2.
(5.35)
Substituting (5.34) for (5.30) and using the fact thatρE(τ)≤τ, we obtain xn+1−zn+12
≤xn−zn
2
−2nαnxn−zn
2
+ 162nM2
+C2ρE
nM
+C3
αn−αn+1
αn ,C1
ρE
αn−αn+1
αn M
, (5.36)
where
C3=MC1+ 4x∗ C+M+ 2x∗+ 32Mc¯−1. (5.37)
Therefore, byLemma 2.3and by hypothesis (5.22), we conclude thatxn−zn →0. In
addition, byTheorem 3.8,
xn−x¯∗≤x
n−zn+zn−x¯∗−→0 asn−→ ∞, (5.38)
which implies that{xn}converges strongly to ¯x∗.
5.2. In this subsection, we study an iterative regularization method for (3.1) with a per-turbed operator and perper-turbed right-hand side. Assume that, instead of f andA, we have the sequences{fn},fn∈E, and{An},An:D(An)⊆E→E, such that
f
n−f≤δn, A
nx−Ax≤ωnζ
x+µn, ∀x∈G,
(5.39)
whereζ(t) is a positive and bounded function defined onR+,G⊂D(A
n) andG⊂D(A).
Thus, in reality, the following equations are given:
Any=fn, (5.40)
which may not have a solution. Consider the regularizing iterative algorithm
yn+1=QG
yn−n
Anyn+αnyn−fn
, n=0, 1, 2,..., (5.41)
Theorem5.3. Assume that all the conditions of Theorems3.8and5.1are fulfilled. Suppose that there exist sequences of positive numbersωn,µn, andδnconverging to zero asn→ ∞,
such that (5.39) holds. Suppose that
αn−→0, ωn
+µn+δn
αn −→
0, ρE
n
nαn −→
0, αn−αn+1 nα2n
−→0 (5.42)
asn→ ∞. Starting from arbitrary y0∈Gdefine the sequence{yn}by (5.41). Then there exists1> d >0such that whenever
n≤d, ρE
n
nαn ≤d
2 (5.43)
for alln≥0, the sequence{yn}is bounded and converges in norm to the solutionx¯∗, where
¯
x∗is the unique solution of inequality (3.9).
Proof. First of all, as in the proof ofTheorem 5.1, we aim at showing that{yn}is bounded.
To this end, introduce again a closed ballBr(x∗) with sufficiently large radiusr >0 such
thatr≥2x∗and y0∈Br(x∗). And construct again the setS=Br(x∗)∩G. Without
loss of generality, according to (5.42), putωn≤ω¯,µn≤µ¯,δn≤δ¯, and
M:=3
2r+f+ sup
Ay:y∈S+ ¯ωMζ+ ¯µ+ ¯δ, (5.44)
where
Mζ=supζy:y∈S. (5.45)
We claim that{yn}is bounded.
Assume thatyn∈Sand show thatyn+1∈S. We denote
pn=yn−n
Anyn+αnyn−fn
. (5.46)
The operatorQGis nonexpansive, therefore, by (5.41) we have yn+1−x∗≤Q
Gpn−QGx∗
≤yn−x∗−nAnyn+αnyn−fn ≤yn−x∗+nAnyn+αnyn−fn ≤yn−x∗+Ayn+Anyn−Ayn
+yn−x∗+x∗+f+fn−f.
(5.47)
Then
yn+1−x∗≤r+ sup
y∈SAy
+ωnsup y∈Sζ
y+µn+r+12r+f+δn
≤r+M=M,
(5.48)
Moreover, by (2.3), (2.6), and (5.41) one gets similarly to (5.10) that
y
n+1−x∗ 2
≤yn−x∗−n
Anyn+αnyn−fn
2
≤yn−x∗
2
−2n
Anyn−fn,J
yn−x∗
−2nαn
yn,J
yn−x∗
+ 16LCMMρE
εn
≤yn−x∗
2
−2n
Ayn−f,J
yn−x∗
−2nAnyn−Ayn,J
yn−x∗) −2n
fn−f,J
yn−x∗
−2nαn
yn,J
yn−x∗
+ 16LCMMρE
εn
.
(5.50)
Sincex∗is a solution of the equationAx=f onGandyn∈Gfor alln≥0, we can write
Ayn−f,J
yn−x∗
≥0. (5.51)
Then (5.50) implies the inequality
yn+1−x∗2
≤yn−x∗2−2nαn
yn,J
yn−x∗
+ 2nωnMζ+µn+δnyn−x∗+ 16LMCMρEεn.
(5.52)
DenotingD=8LCMM+rM+
ζ, whereMζ+=max{1,Mζ}, we obtain
yn,J
yn−x∗
≤8LCMMρE
εn
nαn +
rωnMζ+µn+δn
αn ≤DK, (5.53)
because of the given inequalities
ρE
n
αnn ≤K=d
2, ωn+µn+δn
αn ≤K=d
2, ∀n≥0. (5.54)
The rest of the boundedness proof of{yn}follows as in the proof ofTheorem 5.1. Thus, there existsCsuch thatyn ≤C.
We present next the convergence analysis of (5.41). By convexity ofx2, we obtain as in (5.25) the following:
y
n+1−zn+1 2
≤yn+1−zn
2
+ 2zn+1−zn,J
zn+1−yn+1
≤yn+1−zn2+ 2zn+1−zn,Jzn−yn+1
+ 2zn+1−zn,Jzn+1−yn+1−Jzn−yn+1
≤yn+1−zn2+ 2zn−yn+1·zn+1−zn
+16 ¯c−1+C1
ρEzn+1−zn,
whereC1is defined by (5.29). Moreover,
yn+1−zn2
=QGpn−QGzn2≤pn−zn2 ≤yn−zn2+ 2pn−yn,Jpn−zn =yn−zn2+ 2
pn−yn,J
yn−zn
+ 2pn−yn,J
pn−zn
−Jyn−zn
=yn−zn
2
−2n
Anyn+αnyn−fn,J
yn−zn
+ 2pn−yn,J
pn−zn
−Jyn−zn
≤yn−zn2−2nAzn+αnzn−f,Jyn−zn
+ 2nAnyn−Ayn+fn−fyn−zn−2nαnyn−zn2
+ 2pn−yn,Jpn−zn−Jyn−zn.
(5.56)
Now (5.39), (5.46), and (5.56) yield
yn+1−zn2
≤yn−zn2−2nαnyn−zn2+ 2nyn−znωnζyn+µn+δn
+ 2pn−yn,J
pn−zn
−Jyn−zn
≤yn−zn2−2nαnyn−zn2+ 2nyn−znωnζyn+µn+δn
+ 16pn−yn
2
+C4(n)ρEpn−yn,
(5.57)
where
C4(n)=8 max2L,zn−yn+zn−pn. (5.58)
Since{yn}and{zn}are bounded andAis a bounded operator, there exists a constant d1>0 such that
pn−yn=nAnyn+αnyn−fn≤nd1,
C4(n)≤8 max2L,C+M+ 2x∗=C4.
(5.59)
Then
162
nd12+C4(n)ρE
nd1
≤C5ρE
nd1
, (5.60)
where
C5=
16 ¯d−1+C 4
, d¯=1 +d2 1+ 1
−1
Substituting (5.57) for (5.55), we deduce that
yn+1−zn+12
≤1−2nαnyn−zn2+ 2nyn−znωnζyn+δn+µn
+C5ρE
nd1
+C6ρEzn+1−zn
+ 2zn−yn+1·zn+1−zn,
(5.62)
whereC6=16 ¯c−1+C1. Finally, there existsC >0 such thatyn−zn ≤Cand then yn+1−zn+12
≤yn−zn2−2nαnyn−zn2+ 2nCωnζ(C) +δn+µn
+C5ρEnd1
+ 2x∗C6+C
αn−αn+1
αn ,
(5.63)
because ρE(τ)≤τ. Now, the conclusion yn−zn →0 follows from Lemma 2.3. By
Theorem 3.8,
yn−x∗≤y
n−zn+zn−x∗−→0 asn−→ ∞. (5.64)
Thus,{yn}converges strongly tox∗. The proof is accomplished.
5.3. Next we study the iterative regularization method for (3.1) defined by wn+1:=QGn+1
wn−nAwn+αnwn−f, n=0, 1, 2,..., (5.65)
on approximately given setsGn, where, for eachn,QGnis a sunny nonexpansive retraction ofEontoGn.
Theorem5.4. Assume that all the conditions of Theorems3.8and5.1are fulfilled andGn⊆
intD(A),n=1, 2,..., are closed convex sets such that the HausdorffdistanceᏴE(Gn,G)≤
σn≤σ¯,σn+1≤σn. DenoteG=G G, whereG=
Gn. Assume that an operatorA:G→E
is accretive and bounded. Assume that conditions (5.22) hold and that
hE
σn
nαn −→0 asn−→ ∞. (5.66)
If the iterative sequence{wn}generated by (5.65) is bounded, then it converges strongly to ¯
x∗, wherex¯∗is the unique solution of inequality (3.9). Proof. Denote
Zx=x−nAx+αnx−f. (5.67)
Since{wn}is bounded and, hence,{Awn}is also bounded, then there exists a constant
d >0 such that
Zw
n=wn−n
Awn+αnwn−f≤d. (5.68)
By analogy,
Zxn=xn−n
where{xn}is the bounded sequence generated by (5.1) (seeTheorem 5.2). From (5.65)
and (5.1), we have
wn+1−xn+1=QG
n+1Zwn−QGZxn
≤QGZwn−QGZxn+QGn+1Zwn−QGZwn.
(5.70)
Estimate the first term of the right-hand side of the previous inequality:
QGZwn−QGZxn2
≤Zwn−Zxn2
≤wn−xn2−2nAwn−Axn+αnwn−xn,JZwn−Zxn ≤wn−xn2−2nαnwn−xn2−2n
Awn−Axn,J
wn−xn
−2n
Awn−Axn+αn
wn−xn
,JZwn−Zxn
−Jwn−xn
≤wn−xn
2
−2nαnwn−xn
2
+ 162nAwn−Axn+αn
wn−xn
2
+C7(n)ρE
nAwn−Axn+αn
wn−xn,
(5.71)
where
C7(n)=8 max2L,Zwn−Zxn+wn−xn. (5.72)
Thus, there exists a constantC7>0 such thatC7(n)≤C7. UsingLemma 2.7, we come to the following inequality:
Q
Gn+1Zwn−QGZwn≤16(R+ ¯σ)(2d+q)hB
8q−1σ
n+1
, (5.73)
whereR=2(2d+q),q=max{q1,q2},q1=dist(θ,G1), q2=max{dist(θ,Gn)},n=0, 1,
2,..., andθis the origin of the Banach spaceE. Hence, from (5.70), (5.71), and (5.73) we get
w
n+1−xn+1 2
≤wn−xn
2
−2nαnwn−xn
2 + 162
nC+C7ρE
nC
+Ch¯ B
8q−1σ
n+1
1/2
, (5.74)
for ¯C=16(R+ ¯σ)(2d+q) and someC >0.
Since (5.66) holds, we conclude thatwn−xn →0 asn→ ∞. Thus, for alln≥0, one
has
wn+1−x¯∗≤w
n+1−xn+1+xn+1−x¯∗, (5.75) and, therefore,
w
n−x¯∗−→0 asn−→ ∞. (5.76)
Remark 5.5. Obviously, ifGnare bounded, then allwnare bounded too.
Now we are able to combine Theorems5.2,5.3, and5.4in order to investigate the iterative regularization method for (3.1) with perturbed dataA, f, andGdefined by the following algorithm:
un+1:=QGn+1
un−n
Anun+αnun−fn
, n=0, 1, 2,.... (5.77)
Theorem 5.6. Suppose that the conditions of Theorems 5.2,5.3, and5.4are fulfilled. If the iterative sequence{wn}generated by (5.77) is bounded, then it converges strongly tox¯∗,
wherex¯∗is the unique solution of inequality (3.9).
Acknowledgments
The first author thanks The Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, where he was a Visiting Professor while this paper was written. This work was supported in part by the Kamea Program of the Ministry of Absorption.
References
[1] Ya. I. Alber,On the solution of equations and variational inequalities with maximal monotone operators, Soviet Math. Dokl.20(1979), no. 4, 871–876.
[2] ,Metric and generalized projection operators in Banach spaces: Properties and applica-tions, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (A. Kartsatos, ed.), Lecture Notes in Pure and Appl. Math., vol. 178, Dekker, New York, 1996, pp. 15–50.
[3] Ya. I. Alber, A. G. Kartsatos, and E Litsyn,Iterative solution of unstable variational inequalities on approximately given sets, Abstr. Appl. Anal.1(1996), no. 1, 45–64.
[4] Ya. I. Alber and A. I. Notik,Geometric properties of Banach spaces and approximate methods for solving nonlinear operator equations, Soviet Math. Dokl.29(1984), no. 3, 611–616. [5] ,Parallelogram inequalities in Banach spaces and some properties of a duality mapping,
Ukrainian Math. J.40(1988), no. 6, 650–652.
[6] Ya. I. Alber, S. Reich, and I. Ryazantseva,Nonlinear problems with accretive andd-accretive mappings, preprint, 2003.
[7] Ya. I. Alber, S. Reich, and J.-C. Yao,Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces, Abstr. Appl. Anal.2003(2003), no. 4, 193–216. [8] Ya. I. Alber and I. Ryazantseva,Nonlinear Ill-posed problems of monotone type, Tech. Rep.,
Tech-nion, Haifa, 2003.
[9] Ya. I. Alber and J.-C. Yao,Algorithm for generalized multi-valued co-variational inequalities in Banach spaces, Funct. Differ. Equ.7(2000), no. 1-2, 5–13 (2001).
[10] F. E. Browder,Nonlinear functional analysis and nonlinear partial differential equations, Diff er-ential Equations and Their Applications, Bratislava, 1967, pp. 89–113.
[11] ,Nonlinear mappings of nonexpansive and accretive type in Banach spaces., Bull. Amer. Math. Soc.73(1967), 875–882.
[12] ,Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), American Mathematical Society, Rhode Island, 1976, pp. 1–308.
[14] I. Cioranescu,Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathe-matics and Its Applications, vol. 62, Kluwer Academic Publishers, Dordrecht, 1990. [15] L. Deng,Iteration processes for nonlinear Lipschitzian strongly accretive mappings inLp-spaces, J.
Math. Anal. Appl.188(1994), no. 1, 128–140.
[16] J. Diestel,Geometry of Banach Spaces—Selected Topics, Lecture Notes in Mathematics, vol. 485, Springer-Verlag, Berlin, 1975.
[17] W. G. Dotson,On the Mann iterative process, Trans. Amer. Math. Soc.149(1970), 65–73. [18] T. Figiel,On the moduli of convexity and smoothness, Studia Math.56(1976), no. 2, 121–155. [19] K. Goebel and S. Reich,Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings,
Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, New York, 1984.
[20] S. Ishikawa,Fixed points by a new iteration method, Proc. Amer. Math. Soc.44(1974), no. 1, 147–150.
[21] J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces. II. Function Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 97, Springer, Berlin, 1979.
[22] W. R. Mann,Mean value methods in iteration, Proc. Amer. Math. Soc.4(1953), 506–510. [23] R. H. Martin Jr.,A global existence theorem for autonomous differential equations in a Banach
space, Proc. Amer. Math. Soc.26(1970), 307–314.
[24] I. S¸erb,Estimates for the modulus of smoothness and convexity of a Banach space, Mathematica (Cluj)34(57)(1992), no. 1, 61–70.
[25] M. M. Vainberg,Variational Method and Method of Monotone Operators in the Theory of Non-linear Equations, Halsted Press (A division of John Wiley & Sons), New York, 1973. [26] E. Zeidler, Nonlinear Functional Analysis and Its Applications.—II. Monotone Operators,
Springer-Verlag, Berlin, 1985.
Ya. I. Alber: Department of Mathematics, Technion – Israel Institute of Technology, 32000 Haifa, Israel
E-mail address:[email protected]
C. E. Chidume: Mathematics Section, The Abdus Salam International Centre for Theoretical Phys-ics, 340314 Trieste, Italy
E-mail address:[email protected]
H. Zegeye: Mathematics Section, The Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy
