Newton-Kantorovich Iterative Regularization for Nonlinear Ill-Posed Equations Involving Accretive Operators
The Newton-Kantorovich iterative regularization for nonlinear ill-posed equations involving monotone operators in Hilbert spaces is developed for the case of accretive operators in Banach spaces. An estimate for the convergence rates of the method is established.
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2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509712290676736 |
|---|---|
| author | Vu, Quang Hung Nguen, Byong Ву, Куанг Хунг Нгуєн, Бионг |
| author_facet | Vu, Quang Hung Nguen, Byong Ву, Куанг Хунг Нгуєн, Бионг |
| author_sort | Vu, Quang Hung |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:59:22Z |
| description | The Newton-Kantorovich iterative regularization for nonlinear ill-posed equations involving monotone operators in Hilbert spaces is developed for the case of accretive operators in Banach spaces. An estimate for the convergence rates of the method is established. |
| first_indexed | 2026-03-24T02:45:27Z |
| format | Article |
| fulltext |
UDC 517.9
Nguyen Buong (Nat. Centre Nat. Sci. and Technol. Vietnam, Inst. Inform. Technol.),
Vu Quang Hung (Milit. Techn. Univ., Vietnam)
NEWTON – KANTOROVICH ITERATIVE REGULARIZATION
FOR NONLINEAR ILL-POSED EQUATIONS
INVOLVING ACCRETIVE OPERATORS
ITERACIJNA REHULQRYZACIQ
N|GTONA – KANTOROVYÇA
DLQ NELINIJNYX NEKOREKTNYX RIVNQN|
Z AKRETYVNYMY OPERATORAMY
The Newton – Kantorovich iterative regularization for nonlinear ill-posed equation involving monotone
operator in Hilbert spaces is developed for the case of accretive operator in Banach spaces. Estimate for
convergence rates of the method is established.
Rozrobleno iteracijnu rehulqryzacig N\gtona – Kantorovyça dlq nelinijnyx nekorektnyx
rivnqn\ z monotonnym operatorom u hil\bertovyx prostorax dlq vypadku akretyvnoho operatora
v banaxovyx prostorax. Vstanovleno ocinky ßvydkostej zbiΩnosti metodu.
1. Introduction. Consider the operator equation of the first kind
A ( x ) = f, f ∈ R( A ) ⊂ X, (1)
where the operator A : D ( A ) = X → X is a nonlinear and m-accretive operator, X
together with X* and the adjoint space of X are uniformly convex Banach spaces, X
possesses the approximations (see [1]), and D ( A ) , R( A ) denote the domain and the
range of A, respectively. For the sake of simplicity, norms of X and X* will be
denoted by the symbol || ⋅ ||, and we write 〈x, x*〉 instead of x* ( x ) for x* ∈ X* and
x ∈ X. If the normalized dual mapping J of X is continuous and sequential weak
continuous, then the condition
〈A ( x ) – f, J ( x )〉 > 0, x r> ˜ , (2)
with some positive constant r̃ is sufficient for the existence of a solution of (1) (see
[1]). It is well known that problem (1) is ill-posed (see [2]). For the case where A is a
monotone operator in the Hilbert space X , the operator version of Tikhonov
regularization method was considered in [3]. Later, the results were generalized for the
equations involving monotone operator in Banach spaces (see [4]).
Meantimes, the Newton – Kantorovich iterative regularization
x0 ∈ X, A ( xn ) + αn
xn + (A ′( xn ) + αn I)( x n + 1 – x n ) = f (3)
was investigated in [5] for the monotone operator A in the Hilbert space X, where I
denotes the identity operator in X and {αn} is the sequence of positive numbers.
Then, algorithm (3) was developed for the case of the monotone operator A from
Banach space X into X* with some modification in the form
x0 ∈ X, A ( xn ) + A ′( xn )( x n + 1 – x n ) + αn
J
s( x n + 1 ) = f, (4)
where J
s
: X → X* satisfies the condition (see [6])
〈x, J
s( x )〉 = || x ||s, || J
s( x ) || = || x ||s
–
1
, s ≥ 2.
On the other hand, it is well known that the theory of accretive operators is a
direction of developing the theory of monotone operator in Hilbert space. Therefore,
© NGUYEN BUONG, VU QUANG HUNG, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 271
272 NGUYEN BUONG, VU QUANG HUNG
the consideration of Newton – Kantorovich iterative regularization for equations
involving accretive operator is a necessary problem. The purpose of the paper is to
fulfil the task, and to give an estimate for convergence rates of the method.
Below, the notation a ∼ b means that a = O( b ) and b = O( a ).
In the following section, we suppose that all above conditions are satisfied.
2. Main results. First, consider the operator equation
A x f
nα ( ) = , A A I
n nα α= + . (5)
Equation (5) for every fixed αn possesses a unique solution denoted by x
nα , because
the operator A is m-accretive and αn > 0.
Theorem 1. Assume that the following conditions hold:
i) A is Frechet differentiable with
A x A x J A x J x x A x A x( ) ( ˜) ( ˜) ( ˜) ( ) ( ˜)* *− − ′ − ≤ −τ ∀ x ∈ X, (6)
where τ > 0 is some constant, x̃ S∈ 0 is the set of solutions of (1), and J*
denotes the normalized dual mapping of X*;
ii) there exists an element z ∈ X such that ′ = −A x z x( ˜) ˜ .
Then
x x O
n nα α− = ( )˜ .
Proof. Since
x x x x J x x
n n nα α α− = − −˜ ˜, ( ˜)
2
=
=
1
α α α α
n
f A x J x x x J x x
n n n
− − − −( ), ( ˜) ˜, ( ˜) ≤
≤ z A x J x x
n
, ( ˜) ( ˜)*′ −α ≤ z A x J x x
n
′ −( ˜) ( ˜)*
α =
= z J A x J x x
n
* *( ˜) ( ˜)′ −α ≤ ( )τ α α+1 n z x
n
,
we have the boundedness of x
nα{ }. Therefore, we obtain from the last inequality that
x x O
n nα α− = ( )˜ . The theorem is proved.
The conclusion that the sequence x
nα{ } converges to x̃ as n → ∞ if (1) has
only unique solution and J is sequential weak continuous and continuous, was
considered in [7]. Unfornately, the class of proper Banach spaces having the dual
mapping J with sequential weak continuity is small (only the finite-dimensional
Banach spaces and lp). Moreover, the problem of convergence rates for x
nα{ } is still
opened upto now. Hence, the results obtained in the present paper are essential has
because they permit us to use the method in the spaces of type Lp , W p , and, in
particular, even if (1) possesses nonunique solution.
Note that the requirement of uniqueness for solution of (1) will be redundant, when
we have (6) for some x̃ S∈ 0. This condition can be replaced by
A x A x J A x J x x x x( ) ( ˜) ( ˜) ( ˜) ˜* *− − ′ − ≤ −τ ′ −A x J x x( ˜) ( ˜) ∀ x ∈ X,
which is proposed in [8] for the case where X is a Hilbert space. Then it is applied for
estimating the convergence rates of the regularized solutions for nonlinear ill-posed
problems involving monotone operators in Banach spaces (see [9]).
We now return to the problem of convergence of the iteration sequence {xn}
obtained by (3) if X is a Banach space. As in [5], assume that A is twice
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
NEWTON – KANTOROVICH ITERATIVE REGULARIZATION… 273
differentiable in the sense of Gateau, ′′ +A x th h J h( ) , ( )2 is summable for each fixed
x and h, and ′′ ≤A x N( ) ∀ x ∈ X.
Theorem 2. Assume that the conditions of Theorem 1 are satisfied, the sequence
{αn} , αn > 0 and x0 satisfy the following conditions:
lim
n
n
→∞
=α 0, 1 1≤ ≤−α
α
n
n
r , n = 1, 2, … ,
N x x
q
r
0 0
0
2
1−
≤ <α
α
,
α α
α
n n
n
r q q
Nd
−
−
− ≤ −1
1
2
1( / )
,
d x≥ ˜ , x̃ S∈ 0.
Then we have
lim ˜
n
nx x
→∞
− = 0.
Proof. Set
˜ ( ) ( ) ( ) ( )A x A x A x I x x
n n n n n nα α α
− −
≡ + ′ +( ) −− − − −1 1 1 1 1 1 . (7)
On the basis of (3) and the accretive property of A′ ( x ) , x ∈ X (see [1]), we obtain
˜ ( ) ( ), ( )A x A x J x x
n n n n n nα α α α α− − − − −
− −
1 1 1 1 1
=
= A x A x J x x
n n n nn nα α α α− − − −− − −
1 1 1 11( ) ( ), ( ) +
+ ′ +( ) − −− − −− −
A x I x x J x xn n n nn n
( ) ( ), ( )1 1 11 1
α α α =
= A x A x I x x f J x x
n nn n n n n nα αα
− −− − − −+ ′ +( ) − − −
1 11 1 1 1( ) ( ) ( ) , ( ) +
+ ′ +( ) − −− − − −
A x I x x J x xn n n nn n
( ) ( ), ( )1 1 1 1
α α α ≥ α αn nx x
n− −
−1
2
1
.
On the other hand, (7) and equality
A x A x A x x x
n nn n n( ) ( ) ( )( )α α− −
= + ′ −− − −1 11 1 1 +
′′ − −
− −− −
A c
x x x x
n nn n
( )
!
( )( )
2 1 11 1α α ,
where c is some element in X, imply that
˜ ( ) ( ), ( )A x A x J x x
n n n n n nα α α α α− − − − −
− −
1 1 1 1 1
≤
N
x x x x
n nn n2 1 11
2
! α α− −
− −− .
Therefore,
∆n n nx x x x x x
n n n n
:= − ≤ − + −
− −α α α α1 1
≤
N
x x
n
n n n2 1
1
2
1α α α
−
− + −
−
∆ .
(8)
By virtue of (5), we have
A x A x J x x x x J x x
n n n n n n n nn( ) ( ), ( ) , ( )α α α α α α α αα− − + − −
− − − −−1 1 1 11 =
= ( ) , ( )α α α α αn n x J x x
n n n− − −
− −1 1 1
or
x x x x
n n n
n n
n
n n
n
α α α
α α
α
α α
α− −
− ≤ − ≤ −−
−
−
−
1 1
1
1
1
1
2 ˜ .
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
274 NGUYEN BUONG, VU QUANG HUNG
The last inequality follows from x x x
nα − ≤˜ ˜ ∀ n (see [7]). Now inequality (8)
has the form
∆ ∆n
n
n
n n
n
N
x≤ + −
−
−
−
−2
2
1
1
2 1
1α
α α
α
˜ .
As in [5], we have N qn n∆ / 2α ≤ ∀ n. Thus,
∆n
nO x x O q
n n
= −( ) +
−α α1
( ) = O x x O q O O q
n
n
n
n
α α
−
−( ) + = ( ) +−1 1˜ ( ) ( ) .
The theorem is proved.
For an accretive operator A, algorithms (3) and (4) are alike because, in this case,
we need to replace J in (4) by the identity operator I . Then (4) is completely
transformed into (3) without any change.
Suppose that instead of ( A, f ) we have the approximations ( , )A fn n such that
A x A x h g xn n( ) ( )− ≤ ( ) , f fn n− ≤ δ ,
hn n, δ → 0 as n → +∞ , where An is also m-accretive and continuous, and g( t ) is a
continuous, bounded, and nonnegative function. The Newton – Kantorovich iterative
regularization is defined as follows:
x X0 ∈ , A x x A x I x x fn n n n n n n n n n( ) ( ) ( )+ + ′ +( ) − =+α α 1 . (9)
In addition, suppose that An possesses the same properties as A has. For each fixed
n, an element xn+1 is well defined, because ′ +A x In n n( ) α is strongly accretive.
The convergence and convergence rates of the method defined by (9) will be
established on the basis of the following result:
Theorem 3. Assume that the conditions of Theorem 1 hold, and δ αn n/ → 0,
hn n/ α → 0, αn → 0 as n → 0. Then the solution x̃n of the equation
A x fn n( ) = , A A In n n= + α , (10)
converges to x̃ . Moreover, if αn is chosen such that α δn n n
ph∼ +( ) , 0 < p < 1,
then, for 0 < δn + hn < 1, we have
˜ ˜ ( )x x O hn n n− = +( )δ θ , θ = −{ }min , /1 2p p .
Proof. Since An is m -accretive, then there exists a unique solution x̃n of (10).
From (5), (10) and the property of J, we have
˜ ˜ ˜ ˜, ( ˜ ˜)x x x x J x xn n n− = − −2 =
=
1
αn
n n n n nf A x J x x x J x x− − + − −( ˜ ), ( ˜ ˜) ˜, ( ˜ ˜) ≤
≤
1
α
δ
n
n n n nh g x x x z A x J x x+ ( )( ) − + ′ −˜ ˜ ˜ , ( ˜) ( ˜ ˜)* . (11)
Hence, x̃n{ } is bounded. On the other hand,
z A x J x x z J A x J x xn n, ( ˜) ( ˜ ˜) ( ˜) ( ˜ ˜)* * *′ − ≤ ′ − ≤ z A x fn( ) ( ˜ )τ + −1 ≤
≤ z A x f h g xn n n n n n( ) ( ˜ ) ˜τ δ+ − + + ( )( )1 ≤
≤ z x h g xn n n n n( ) ˜ ˜τ α δ+ + + ( )( )1 .
Therefore, (11) implies that
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
NEWTON – KANTOROVICH ITERATIVE REGULARIZATION… 275
˜ ˜ ˜ ˜ ( )x x C
h
x x C hn
n n
n
n n n
p− ≤ + − + +2
1 2
δ
α
δ ≤
≤ C h x x C hn n
p
n n n
p
1
1
2( ) ˜ ˜ ( )δ δ+ − + +− , 0 < δn + hn < 1,
where Ci are the positive constants. Hence, the conclusions of the theorem follow
from the last two inequalities (see [9]).
The theorem is proved.
Theorem 4. Assume that the following conditions hold:
i) conditions of Theorem 2;
ii) the sequence αn{ } , αn > 0, is chosen such that
lim
n
n
→∞
=α 0, 1 1≤ ≤−α
α
n
n
r , n = 1, 2, … ,
N x x
q
r
0
0
0
2
1−
≤ <α
α
, a
r q q
Nn ≤ −
2
1( / )
,
where
a
h h
x bn
n n
n
n n
n
n
n n
n
= − + +
+( ) + +−
−
−
−
−
−
α α
α α
δ δ
α
1
1
2
1
1
2
1
1
22 ˜ ,
b
h g x h g x
n
n n
n
n n
n
= + ( ) + ( )
− −
−
max
˜
,
˜δ
α
δ
α
1 1
1
.
Then, we have
lim ˜
n
nx x
→+∞
− = 0,
where xn is defined by (9).
Proof. Setting
˜ ( ) ( ) ( ) ( )A x A x A x I x xn n n n n n n− − − − − − −≡ + ′ +( ) −1 1 1 1 1 1 1α ,
we have
˜ ( ˜ ) ( ˜ ), ˜ ˜A x A x J x x x xn n n n n n n n n− − − − − − −− −( ) ≥ −1 1 1 1 1 1 1
2α .
Using the differential property of An−1, we can write
A x A x A x x xn n n n n n n n− − − − − − −= + ′ −( )1 1 1 1 1 1 1( ˜ ) ( ) ( ) ˜ +
+
′′ −( ) −( )− − − −
A c
x x x xn
n n n n
( ˜)
!
˜ ˜
2 1 1 1 1 ,
where c̃ is some element in X. Therefore,
˜ ( ˜ ) ( ˜ ), ˜A x A x J x xn n n n n n− − − − −− −( )1 1 1 1 1 ≤
N
x x x xn n n n2 1 1
2
1˜ ˜− − −− − .
Hence,
˜ : ˜ ˜ ˜ ˜∆ ∆n n n
n
n n nx x
N
x x= − ≤ + −
−
− −2 1
1
2
1α
.
Relation (10) implies that
A x A x J x x x x J x xn n n n n n n n n n n( ˜ ) ( ˜ ), ˜ ˜ ˜ ˜ , ˜ ˜− −( ) + − −( )− − − − − −1 1 1 1 1 1α =
= f f J x x x J x xn n n n n n n n n− −( ) + −( ) −( )− − − −1 1 1 1, ˜ ˜ ˜ , ˜ ˜α α
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
276 NGUYEN BUONG, VU QUANG HUNG
and
˜ ˜ ˜x x
h
g xn
n
n
n
n
≤ + + ( )2
δ
α α
∀ n.
Thus,
˜ ˜ ˜x x xn n
n n
n
n−
−
−
− ≤ −
1
1
1
α α
α
+
h h
x an n
n
n
n n
n
n n
+ + + ≤−
−
−
−
−
−
1
1
1
1
1
1α
δ δ
α
α˜ .
Consequently,
˜ ˜∆ ∆n
n
n n n
N
a≤ +
−
− −2 1
1
2
1α
α .
As in [5], we have N qn n
˜ /∆ 2α ≤ ∀n. Thus,
˜ ˜ ˜ ( ) ( )∆n n
n
n
nO x x O q O O q= −( ) + = ( ) +− −1 1α .
The theorem is proved.
Remark. For the given δn and hn
, we can chose α α δn n nh= ( , ) such that
δ
α
n
n
,
hn
nα
,
δ δ
α
n n
n
+ −
−
1
1
2 ,
h hn n
n
+ →−
−
1
1
2 0
α
as n → ∞.
Indeed, for example, if δn = hn = qn, then we can take αn
nq= /3 . Consequently,
for sufficiently large n = N0, we have αN r q q N
0
2 1≤ −( )/ / . Then we can chose
α α0 0
= N such that
N x x
q
r
N
N
0
0
2
1
0
−
≤ <
α
α
and replace n in (9) by n + N0
.
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Received 21.10.2003
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
|
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | Ukrainian English |
| last_indexed | 2026-03-24T02:45:27Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ea/2c60607845c65611e176b6d35b99e8ea.pdf |
| spelling | umjimathkievua-article-35952020-03-18T19:59:22Z Newton-Kantorovich Iterative Regularization for Nonlinear Ill-Posed Equations Involving Accretive Operators Ітераційна регуляризація Ньютона - Канторовича для нелінійних некоректних рівнянь з акретивними операторами Vu, Quang Hung Nguen, Byong Ву, Куанг Хунг Нгуєн, Бионг The Newton-Kantorovich iterative regularization for nonlinear ill-posed equations involving monotone operators in Hilbert spaces is developed for the case of accretive operators in Banach spaces. An estimate for the convergence rates of the method is established. Розроблено ітераційну регулярнзацію Ньютона - Канторовича для нелінійних некоректних рівнянь з монотонним оператором у гільбертових просторах для випадку акретивного оператора в банахових просторах. Встановлено оцінки швидкостей збіжності методу. Institute of Mathematics, NAS of Ukraine 2005-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3595 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 2 (2005); 271–276 Український математичний журнал; Том 57 № 2 (2005); 271–276 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3595/3921 https://umj.imath.kiev.ua/index.php/umj/article/view/3595/3922 Copyright (c) 2005 Vu Quang Hung; Nguen Byong |
| spellingShingle | Vu, Quang Hung Nguen, Byong Ву, Куанг Хунг Нгуєн, Бионг Newton-Kantorovich Iterative Regularization for Nonlinear Ill-Posed Equations Involving Accretive Operators |
| title | Newton-Kantorovich Iterative Regularization for Nonlinear Ill-Posed Equations Involving Accretive Operators |
| title_alt | Ітераційна регуляризація Ньютона - Канторовича для нелінійних некоректних рівнянь з акретивними операторами |
| title_full | Newton-Kantorovich Iterative Regularization for Nonlinear Ill-Posed Equations Involving Accretive Operators |
| title_fullStr | Newton-Kantorovich Iterative Regularization for Nonlinear Ill-Posed Equations Involving Accretive Operators |
| title_full_unstemmed | Newton-Kantorovich Iterative Regularization for Nonlinear Ill-Posed Equations Involving Accretive Operators |
| title_short | Newton-Kantorovich Iterative Regularization for Nonlinear Ill-Posed Equations Involving Accretive Operators |
| title_sort | newton-kantorovich iterative regularization for nonlinear ill-posed equations involving accretive operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3595 |
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