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

- Jong Kyu Kim
^{1}Email author and - Nguyen Buong
^{2}

**2014**:440

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

© Kim and Buong; licensee Springer. 2014

**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 ${A}_{i}$, $i=0,\dots ,N$, 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 operators
- demicontinuous
- convex Banach space
- modified Browder-Tikhonov regularization
- Fréchet differentiability

## 1 Introduction

Let *X* be a real reflexive Banach space with the property of approximations and its dual space ${X}^{\ast}$ be strictly convex. The norms of *X* and ${X}^{\ast}$ are denoted by the symbol $\parallel \cdot \parallel $. We write $\u3008x,{x}^{\ast}\u3009$ instead of ${x}^{\ast}(x)$ for ${x}^{\ast}\in {X}^{\ast}$ and $x\in X$.

**Definition 1.1**A Banach space

*X*is said to be strictly convex if for $x,y\in {S}_{X}$ with $x\ne y$, then

where ${S}_{X}$ is the unit sphere ${S}_{X}=\{x\in X:\parallel x\parallel =1\}$.

**Definition 1.2**A mapping

*j*from

*X*onto ${X}^{\ast}$ is called the normalized dual mapping of

*X*, if it satisfies the condition

It is well known that if ${X}^{\ast}$ is strictly convex then *j* is single-valued.

**Definition 1.3**An operator

*A*from

*X*to

*X*is said to be accretive, if

where $D(A)$ denotes the domain of *A*. An accretive operator *A* is said to be an *m*-accretive, if $\mathcal{R}(A+\lambda I)=X$ for $\lambda >0$ where $\mathcal{R}(A)$ 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

- (i)
demicontinuous if ${x}_{n}\to x$ in

*X*implies $A({x}_{n})\rightharpoonup A(x)$, - (ii)
weakly continuous if ${x}_{n}\rightharpoonup x$ implies $A({x}_{n})\rightharpoonup A(x)$.

It is well known that if *A* is accretive, and is continuous, demicontinuous, or weakly continuous, then it is *m*-accretive [1–3].

**Definition 1.5**A mapping

*A*from

*X*to

*X*is called Fréchet differentiable at a point $x\in D(A)$, if

where $B(x)$ is a bounded linear mapping from *X* to *X*. And the Fréchet derivative of *A* at $x\in D(A)$ is denoted by ${A}^{\prime}(x)$.

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.

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

Suppose that $S\ne \mathrm{\varnothing}$.

For *m*-accretive operators, some results of the approximating solution for each equation in (1.1) under suitable different conditions are investigated in [4–10], and [11].

*m*-accretive operator are investigated in [12, 13], and [14] having the weakly sequentially continuous duality mapping

*j*. In [15–19], 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

for *y* in some neighborhood of ${S}_{i}$, where ${A}_{i}^{\prime}({x}_{0}^{i})$ is the Fréchet derivative of ${A}_{i}$ at ${x}_{0}^{i}\in {S}_{i}$, $\tilde{\tau}$ is some positive constant, and ${j}^{\ast}$ is the normalized duality mapping of ${X}^{\ast}$.

*i*, the regularized solution of (1.1) is constructed by the following operator equation:

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

where $\mathcal{A}:\mathcal{X}\to \mathcal{X}:=X\times \cdots \times X$ is defined by $\mathcal{A}(x):=({A}_{0}(x),\dots ,{A}_{N}(x))$, and $f:=({f}_{0},\dots ,{f}_{N})$.

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 $({f}_{0},\dots ,{f}_{N})$ 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 ${A}_{i}$, $i=0,\dots ,N$.

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].

*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:

has the same properties as each ${A}_{i}^{h}$, it is also *m*-accretive. Therefore, (1.5) has a unique solution denoted by ${x}_{\alpha}^{\tau}$, $\tau =(\delta ,h)$, for every value $\alpha >0$.

In the following section, the convergence rates of the regularized solution ${x}_{\alpha}^{\tau}$ and its finite-dimensional approximations ${x}_{\alpha ,n}^{\tau}$ are established under an assumption similar to (1.2).

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

## 2 Main results

**Assumption A**There exists a constant ${\tau}_{0}>0$ such that

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*${X}^{\ast}$

*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*:

- (i)
${A}_{0}$

*is Fréchet differentiable at*${x}_{0}$*with Assumption*A. - (ii)
*There exists an element*$z\in X$*such that*${A}_{0}^{\prime}({x}_{0})z=-{x}_{0}.$ - (iii)
*The parameter**α**is chosen such that*$\alpha \sim {(\delta +h)}^{\mu}$, $0<\mu <1$.

*Then*,

*for*$0<\delta +h<1$,

*we have*

*Proof*From the property of

*j*, ${A}_{i}^{h}$, (1.1), (1.3), (1.5), and condition (ii), it follows that

This completes the proof. □

*X*onto ${X}_{n}$, ${P}_{n}x\to x$ for all $x\in X$, $\parallel {P}_{n}\parallel \le {C}_{0}$, ${C}_{0}$ is some positive constant, and $\{{X}_{n}\}$ is the sequence of finite-dimensional subspaces of

*X*such that

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

*j*satisfies the following inequality:

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

**Theorem 2.2**

*Let*

*X*

*be a real reflexive Banach space with the property of approximations and its dual space*${X}^{\ast}$

*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*:

- (i)
${A}_{0}$

*is Fréchet differentiable with Assumption*A*and the derivative*${A}_{0}^{\prime}$*being uniformly bounded at*${x}_{0}$. - (ii)
*There exists an element*$z\in X$*such that*${A}_{0}^{\prime}({x}_{0})z=-{x}_{0}.$ - (iii)
*The parameter**α**is chosen such that*$\alpha \sim {(\delta +h+{\gamma}_{n})}^{\mu}$, $0<\mu <1$.

*Then*,

*for*$0<\delta +h<1$,

*we have*

*Proof*Set ${x}_{0}^{n}={P}_{n}{x}_{0}$. From (2.2) and the property ${j}^{n}(x)=j(x)$ for all $x\in {X}_{n}$, where ${j}^{n}={P}_{n}^{\ast}j{P}_{n}$ is the dual mapping of ${X}_{n}$ (see [13]), it follows that

*x*in a neighborhood of ${x}_{0}$. Thus, we have

where ${R}_{1}$ is a positive constant with ${R}_{1}\ge max\{\parallel {x}_{0}\parallel ,\parallel {x}_{\alpha ,n}^{\tau}\parallel \}$.

*n*, where ${I}^{\ast}$ is the identity operator in ${X}^{\ast}$, we have

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

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