Open Access

Convergence rates in regularization for a system of nonlinear ill-posed equations with m-accretive operators

Journal of Inequalities and Applications20142014:440

https://doi.org/10.1186/1029-242X-2014-440

Received: 18 April 2014

Accepted: 21 October 2014

Published: 3 November 2014

Abstract

In this paper, we study an operator version of the modified Browder-Tikhonov regularization method for finding a common solution for a system of ill-posed operator equations involving m-accretive operators A i , i = 0 , , N , in a reflexive Banach space. The convergence rates of the regularized solutions are estimated not only in the infinite-dimensional space, but also in connection with its finite-dimensional approximations without the weakly sequential continuity of the dual mapping.

MSC:47H17, 47H20.

Keywords

accretive operatorsdemicontinuousconvex Banach spacemodified Browder-Tikhonov regularizationFréchet differentiability

1 Introduction

Let X be a real reflexive Banach space with the property of approximations and its dual space X be strictly convex. The norms of X and X are denoted by the symbol . We write x , x instead of x ( x ) for x X and x X .

Definition 1.1 A Banach space X is said to be strictly convex if for x , y S X with x y , then
( 1 λ ) x + λ y < 1 λ ( 0 , 1 ) ,

where S X is the unit sphere S X = { x X : x = 1 } .

Definition 1.2 A mapping j from X onto X is called the normalized dual mapping of X, if it satisfies the condition
x , j ( x ) = x 2 , j ( x ) = x x X .

It is well known that if X is strictly convex then j is single-valued.

Definition 1.3 An operator A from X to X is said to be accretive, if
A ( x ) A ( y ) , j ( x y ) 0 x , y D ( A ) ,

where D ( A ) denotes the domain of A. An accretive operator A is said to be an m-accretive, if R ( A + λ I ) = X for λ > 0 where R ( A ) and I denote the range of A and the identity mapping of X, respectively.

Definition 1.4 An operator A from X to X is said to be
  1. (i)

    demicontinuous if x n x in X implies A ( x n ) A ( x ) ,

     
  2. (ii)

    weakly continuous if x n x implies A ( x n ) A ( x ) .

     

It is well known that if A is accretive, and is continuous, demicontinuous, or weakly continuous, then it is m-accretive [13].

Definition 1.5 A mapping A from X to X is called Fréchet differentiable at a point x D ( A ) , if
A ( x + h ) A ( x ) = B ( x ) h + o ( h ) x + h D ( A ) ,

where B ( x ) is a bounded linear mapping from X to X. And the Fréchet derivative of A at x D ( A ) is denoted by A ( x ) .

Let { A i } i = 0 N be a family of N + 1 accretive operators in X and satisfy one of the above mentioned three continuities.

Our problem is to find a common solution of the following operator equations:
A i ( x ) = f i , f i R ( A i ) , i = 0 , , N .
(1.1)
Set
S = i = 0 N S i ,

where S i is the solution set of (1.1), that is, S i = { x : A i ( x ) = f i } .

Suppose that S .

For m-accretive operators, some results of the approximating solution for each equation in (1.1) under suitable different conditions are investigated in [410], and [11].

The system of equations (1.1) is ill-posed, because each one of the system is ill-posed. By ill-posedness, we mean that its solutions do not depend continuously on the data ( A i , f i ) . Therefore, we have to use the stable methods in order to solve the problem. Some stable methods of approximating solution for each equation in (1.1) with m-accretive operator are investigated in [12, 13], and [14] having the weakly sequentially continuous duality mapping j. In [1519], the authors considered the modified Browder-Tikhonov regularization method with the regularization parameter choice without the property for j, for the case of demicontinuous or weakly continuous accretive operators A i satisfying the condition
A i ( y ) A i ( x 0 i ) j A i ( x 0 i ) j ( y x 0 i ) τ ˜ y x 0 i A i ( x ) j ( y x 0 i )
(1.2)

for y in some neighborhood of S i , where A i ( x 0 i ) is the Fréchet derivative of A i at x 0 i S i , τ ˜  is some positive constant, and j is the normalized duality mapping of X .

In many papers, for each i, the regularized solution of (1.1) is constructed by the following operator equation:
A i h ( x ) + α x = f i δ ,
where ( A i h , f i δ ) is the approximation of ( A i , f i ) satisfying the conditions:
A i h ( x ) A i ( x ) h g ( x ) , f i f i δ δ , h , δ 0 ,
(1.3)

g ( t ) is a nonnegative bounded (image of bounded set is bounded) real function, and A i h is also accretive and the same continuity as A i .

The system of equations (1.1) can be written in the form
A ( x ) = f ,
(1.4)

where A : X X : = X × × X is defined by A ( x ) : = ( A 0 ( x ) , , A N ( x ) ) , and f : = ( f 0 , , f N ) .

Note that (1.4) can be seen as a special case of (1.1) with N = 0 . However, one potential advantage of (1.1) over (1.4) can be that it might better reflect the structure of the underlying information ( f 0 , , f N ) leading to the couplet system, than a plain concatenation into one single data element f could. In particular, the second advantage is that in estimating convergence rates of regularization solution, which is showed later, we need only the smooth property for one among A i , while for (1.4) we need the property for A i , i = 0 , , N .

When for each i, A i is the nonlinear Fréchet differentiable operator from the Hilbert space X to the Hilbert space Y i with derivative being uniformly bounded in a neighborhood of a common solution, a stable method for problem (1.1) is considered in [20].

In this paper, we show that a common solution of (1.1) involving m-accretive operators  A i , without the weakly sequentially continuous property of j, can be approximated by the modified Browder-Tikhonov regularization method which is described by the following operator equation:
A 0 h ( x ) + α 1 + μ 0 i = 1 N ( A i h ( x ) f i δ ) + α x = f 0 δ , μ 0 0 ,
(1.5)
where α > 0 is a small regularization parameter. Since the operator
A 0 h + α 1 + μ 0 i = 1 N ( A i h f i δ )

has the same properties as each A i h , it is also m-accretive. Therefore, (1.5) has a unique solution denoted by x α τ , τ = ( δ , h ) , for every value α > 0 .

In the following section, the convergence rates of the regularized solution x α τ and its finite-dimensional approximations x α , n τ are established under an assumption similar to (1.2).

The symbols ‘→’ and ‘’ denote strong and weak convergence, respectively, and the notation a b means that a = o ( b ) and b = o ( a ) .

2 Main results

Assumption A There exists a constant τ 0 > 0 such that
A 0 ( x ) A 0 ( x 0 ) j A 0 ( x 0 ) j ( x x 0 ) τ 0 A 0 ( x ) A 0 ( x 0 ) , x X .

Now, we are in a position to introduce the main theorem.

Theorem 2.1 Let X be a real reflexive Banach space with the property of approximations and its dual space X be strictly convex. Let { A i } i = 0 N be a family of N + 1 accretive operators in X and satisfy one of the above mentioned continuities. Assume that the following conditions hold:
  1. (i)

    A 0 is Fréchet differentiable at x 0 with Assumption A.

     
  2. (ii)
    There exists an element z X such that
    A 0 ( x 0 ) z = x 0 .
     
  3. (iii)

    The parameter α is chosen such that α ( δ + h ) μ , 0 < μ < 1 .

     
Then, for 0 < δ + h < 1 , we have
x α τ x 0 = o ( ( δ + h ) θ ) , θ = min { 1 μ , μ / 2 } .
Proof From the property of j, A i h , (1.1), (1.3), (1.5), and condition (ii), it follows that
x α τ x 0 2 = x α τ x 0 , j ( x α τ x 0 ) = 1 α f 0 δ A 0 h ( x α τ ) + α 1 + μ 0 i = 1 N ( f i δ A i h ( x α τ ) ) α x 0 , j ( x α τ x 0 ) 1 + N α 1 + μ 0 α [ δ + h g ( x 0 ) ] x α τ x 0 + z , A 0 ( x 0 ) j ( x α τ x 0 ) .
(2.1)
Therefore, { x α τ } is a bounded set. Since
z , A 0 ( x 0 ) j ( x α τ x 0 ) z A 0 ( x 0 ) j ( x α τ x 0 ) ,
by virtue of Assumption A, we have
A 0 ( x 0 ) j ( x α τ x 0 ) = j A 0 ( x 0 ) j ( x α τ x 0 ) ( τ 0 + 1 ) A 0 ( x α τ ) f 0 ( τ 0 + 1 ) [ A 0 h ( x α τ ) f 0 δ + δ + h g ( x α δ ) ] ( τ 0 + 1 ) [ α 1 + μ 0 i = 1 N A i h ( x α τ ) f i δ + α x α δ + δ + h g ( x α δ ) ] .
Since α ( δ + h ) μ , 0 < μ < 1 , and g ( t ) is a bounded function, from (2.1) and the last inequality, we obtain
x α τ x 0 2 C 1 ( δ + h ) 1 μ x α τ x 0 + C 2 ( δ + h ) μ , 0 < δ + h < 1 ,
where C 1 and C 2 are positive constants. Now, by using the implication
a , b , c 0 , p > q , a p b a q + c a p = o ( b p / ( p q ) + c ) ,
we obtain
x α τ x 0 = o ( ( δ + h ) θ ) , θ = min { 1 μ , μ / 2 } .

This completes the proof. □

Now, we consider the problem of approximating (1.5) by the sequence of finite-dimensional problems
A 0 , n h ( x ) + α 1 + μ 0 i = 1 N ( A i , n h ( x ) f i , n δ ) + α x = f 0 , n δ , x X n ,
(2.2)
where f i , n δ = P n f i δ , A i , n h = P n A i h P n , P n is the linear projection from X onto X n , P n x x for all x X , P n C 0 , C 0 is some positive constant, and { X n } is the sequence of finite-dimensional subspaces of X such that
X 1 X 2 X .

It is easy to see that A i , n h are also m-accretive. The aspects of existence and convergence of the solution x α , n τ of problem (2.2), as n , to the solution x α τ of the operator equation (1.5) for each α > 0 has been studied in [21]. The question under which conditions the sequence { x α , n τ } converges to a solution x 0 , as α , δ , h 0 and n , and the convergence rates of { x α , n τ } are subject of our further investigations.

In addition, suppose that j satisfies the following inequality:
j ( x ) j ( y ) C ( R ) x y ν , 0 < ν < 1 ,
(2.3)

where C ( R ) , R > 0 is positive increasing function on R = max { x , y } (see [11]).

Set
γ n = ( I P n ) x 0 .
Theorem 2.2 Let X be a real reflexive Banach space with the property of approximations and its dual space X be strictly convex. Let { A i } i = 0 N be a family of N + 1 accretive operators in X and satisfy one of the above mentioned continuities. Suppose that the following conditions hold:
  1. (i)

    A 0 is Fréchet differentiable with Assumption A and the derivative A 0 being uniformly bounded at x 0 .

     
  2. (ii)
    There exists an element z X such that
    A 0 ( x 0 ) z = x 0 .
     
  3. (iii)

    The parameter α is chosen such that α ( δ + h + γ n ) μ , 0 < μ < 1 .

     
Then, for 0 < δ + h < 1 , we have
x α , n τ x 0 = o ( ( δ + h + γ n ) θ + γ n ν / 2 ) , θ = min { 1 μ , μ / 2 } .
Proof Set x 0 n = P n x 0 . From (2.2) and the property j n ( x ) = j ( x ) for all x X n , where j n = P n j P n is the dual mapping of X n (see [13]), it follows that
x α , n τ x 0 n 2 = x α , n τ x 0 n , j n ( x α , n τ x 0 n ) = 1 α f 0 , n δ A 0 , n h ( x α , n τ ) , j n ( x α , n τ x 0 n ) + x 0 n , j n ( x α , n τ x 0 n ) + α μ 0 i = 1 N f i , n δ A i , n h ( x α , n τ ) , j n ( x α , n τ x 0 n ) .
(2.4)
Clearly,
f 0 , n δ A 0 , n h ( x α , n τ ) , j n ( x α , n τ x 0 n ) = f 0 δ f 0 + A 0 ( x 0 ) A 0 ( x 0 n ) + A 0 ( x 0 n ) A 0 h ( x 0 n ) , j n ( x α , n τ x 0 n ) + A 0 h ( x 0 n ) A 0 h ( x α , n τ ) , j n ( x α , n τ x 0 n ) [ δ + A 0 ( x 0 ) A 0 ( x 0 n ) + h g ( x 0 n ) ] x α , n τ x 0 n .
Due to condition (i) and x 0 n x 0 as n , we have
A 0 ( x 0 n ) A 0 ( x 0 ) C 0 γ n ,
where C 0 is a positive constant such that
A 0 ( x 0 ) C 0
for x in a neighborhood of x 0 . Thus, we have
f 0 , n δ A 0 , n h ( x α , n τ ) , j n ( x α , n τ x 0 n ) [ δ + h g ( x 0 n ) + C 0 γ n ] x α , n τ x 0 n .
(2.5)
Each term of the sum in (2.4) is estimated as follows:
f i , n δ A i , n h ( x α , n τ ) , j n ( x α , n τ x 0 n ) = f i δ A i h ( x α , n τ ) , j n ( x α , n τ x 0 n ) = f i δ A i h ( x 0 n ) + A i h ( x 0 n ) A i h ( x α , n τ ) , j n ( x α , n τ x 0 n ) f i δ A i h ( x 0 n ) , j n ( x α , n τ x 0 n ) [ δ + h g ( x 0 n ) + A i ( x 0 ) A i ( x 0 n ) ] x α , n τ x 0 n .
(2.6)
By virtue of the continuity of A i , there exists a positive constant C such that
A i ( x 0 ) A i ( x 0 n ) C , i = 1 , , N .
From (2.4)-(2.6), we see that
x α , n τ x 0 n 2 [ 1 α ( δ + h + C 0 γ n ) + α μ 0 ( δ + h g ( x 0 n ) + C ) ] x α , n τ x 0 n + x 0 , j ( x α , n τ x 0 n ) .
(2.7)
Consequently, { x α , n τ } is bounded as δ , h , α 0 and n . Obviously, from (2.3), Assumption A, and condition (ii), it follows that
x 0 , j ( x α , n τ x 0 n ) = z , A 0 ( x 0 ) [ j ( x α , n τ x 0 n ) j ( x α , n τ x 0 ) ] + z , A 0 ( x 0 ) j ( x α , n τ x 0 ) C ( R 1 ) A 0 ( x 0 ) z γ n ν + z A 0 ( x 0 ) j ( x α , n τ x 0 ) ,

where R 1 is a positive constant with R 1 max { x 0 , x α , n τ } .

On the other hand,
A 0 ( x 0 ) j ( x α , n τ x 0 ) ( τ 0 + 1 ) A 0 ( x α , n τ ) f 0 ( τ 0 + 1 ) [ A 0 h ( x α , n τ ) f 0 h + δ + h g ( x α , n τ ) ] .
By virtue of the Hahn-Banach theorem, there exists an element y X with y = 1 such that
A 0 h ( x α , n τ ) f 0 h = A 0 h ( x α , n τ ) f 0 h , y .
Since
A 0 h ( x α , n τ ) f 0 h , y = A 0 h ( x α , n τ ) f 0 h , ( I P n ) y + A 0 h ( x α , n τ ) f 0 h , P n y
and
( I P n ) y 1 / 2 ,
for sufficiently large n, where I is the identity operator in X , we have
A 0 h ( x α , n τ ) f 0 h 2 A 0 , n h ( x α , n τ ) f 0 , n h .
Therefore,
A 0 ( x 0 ) j ( x α , n τ x 0 ) ( τ 0 + 1 ) C 0 [ α 1 + μ 0 i = 1 N A i h ( x α , n τ ) f i δ + α x α , n τ + δ + h g ( x α , n τ ) ] .
Thus, (2.7) has the form
x α , n τ x 0 n 2 C ˜ 1 ( δ + h + γ n ) 1 μ x α , n τ x 0 n + C ˜ 2 [ ( δ + h + γ n ) μ + γ n ν ] ,
where C ˜ i > 0 ( i = 1 , 2 ). Consequently, we have
x α , n τ x 0 = O ( ( δ + h + γ n ) θ + γ n ν / 2 ) , θ = min { 1 μ , μ / 2 } .

This completes the proof. □

Declarations

Acknowledgements

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042138).

Authors’ Affiliations

(1)
Department of Mathematics Education, Kyungnam University, Changwon, Gyeongnam
(2)
Institute of Information Technology, Vietnamese Academy of Science and Technology, Ha Noi, Vietnam

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© Kim and Buong; licensee Springer. 2014

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