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Convergence rates in regularization for a system of nonlinear illposed equations with maccretive operators
Journal of Inequalities and Applications volume 2014, Article number: 440 (2014)
Abstract
In this paper, we study an operator version of the modified BrowderTikhonov regularization method for finding a common solution for a system of illposed operator equations involving maccretive 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 infinitedimensional space, but also in connection with its finitedimensional approximations without the weakly sequential continuity of the dual mapping.
MSC:47H17, 47H20.
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 singlevalued.
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 maccretive, 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 maccretive [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.
Our problem is to find a common solution of the following operator equations:
Set
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 maccretive operators, some results of the approximating solution for each equation in (1.1) under suitable different conditions are investigated in [4–10], and [11].
The system of equations (1.1) is illposed, because each one of the system is illposed. By illposedness, 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 maccretive operator are investigated in [12, 13], and [14] having the weakly sequentially continuous duality mapping j. In [15–19], the authors considered the modified BrowderTikhonov 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}.
In many papers, for each i, the regularized solution of (1.1) is constructed by the following operator equation:
where ({A}_{i}^{h},{f}_{i}^{\delta}) is the approximation of ({A}_{i},{f}_{i}) satisfying the conditions:
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
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].
In this paper, we show that a common solution of (1.1) involving maccretive operators {A}_{i}, without the weakly sequentially continuous property of j, can be approximated by the modified BrowderTikhonov regularization method which is described by the following operator equation:
where \alpha >0 is a small regularization parameter. Since the operator
has the same properties as each {A}_{i}^{h}, it is also maccretive. 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 finitedimensional 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
Therefore, \{{x}_{\alpha}^{\tau}\} is a bounded set. Since
by virtue of Assumption A, we have
Since \alpha \sim {(\delta +h)}^{\mu}, 0<\mu <1, and g(t) is a bounded function, from (2.1) and the last inequality, we obtain
where {C}_{1} and {C}_{2} are positive constants. Now, by using the implication
we obtain
This completes the proof. □
Now, we consider the problem of approximating (1.5) by the sequence of finitedimensional problems
where {f}_{i,n}^{\delta}={P}_{n}{f}_{i}^{\delta}, {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\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 finitedimensional subspaces of X such that
It is easy to see that {A}_{i,n}^{h} are also maccretive. 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.
In addition, suppose that 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]).
Set
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
Clearly,
Due to condition (i) and {x}_{0}^{n}\to {x}_{0} as n\to \mathrm{\infty}, we have
where {C}_{0}^{\prime} is a positive constant such that
for x in a neighborhood of {x}_{0}. Thus, we have
Each term of the sum in (2.4) is estimated as follows:
By virtue of the continuity of {A}_{i}, there exists a positive constant {C}^{\prime} such that
From (2.4)(2.6), we see that
Consequently, \{{x}_{\alpha ,n}^{\tau}\} is bounded as \delta ,h,\alpha \to 0 and n\to \mathrm{\infty}. Obviously, from (2.3), Assumption A, and condition (ii), it follows that
where {R}_{1} is a positive constant with {R}_{1}\ge max\{\parallel {x}_{0}\parallel ,\parallel {x}_{\alpha ,n}^{\tau}\parallel \}.
On the other hand,
By virtue of the HahnBanach theorem, there exists an element {y}^{\ast}\in {X}^{\ast} with \parallel {y}^{\ast}\parallel =1 such that
Since
and
for sufficiently large n, where {I}^{\ast} is the identity operator in {X}^{\ast}, we have
Therefore,
Thus, (2.7) has the form
where {\tilde{C}}_{i}>0 (i=1,2). Consequently, we have
This completes the proof. □
References
Browder FE: Nonlinear mapping of nonexpansive and accretive type in Banach spaces. Bull. Am. Math. Soc. 1967, 73: 875–882. 10.1090/S000299041967118238
Fitzgibbon WE: Weak continuous accretive operators. Bull. Am. Math. Soc. 1973, 79: 473–474. 10.1090/S000299041973132240
Martin RH Jr.: A global existence theorem for autonomous differential equations in Banach spaces. Proc. Am. Math. Soc. 1970, 26: 307–314. 10.1090/S00029939197002641956
Aoyama K, Iiduka H, Takahashi W: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. 2006., 2006: Article ID 35390
Chen R, Zhu Z: Viscosity approximation fixed points for nonexpansive and m accretive operators. Fixed Point Theory Appl. 2006., 2006: Article ID 81325
Chidume CE, Zegeye H: Iterative solution of 0\in Ax for an m accretive operator A in certain Banach spaces. J. Math. Anal. Appl. 2002, 269: 421–430. 10.1016/S0022247X(02)00015X
Kim JK: Convergence of Ishikawa iterative sequence for accretive Lipschitzian mappings in Banach spaces. Taiwan. J. Math. 2006, 10: 553–561.
Kim JK, Tuyen TM: Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2011. 10.1186/16871812201152
Kim JK, Buong N: Regularization inertial proximal point algorithm for monotone hemicontinuous mapping and inverse strongly monotone mappings in Hilbert spaces. J. Inequal. Appl. 2010., 2010: Article ID 451916
Miyake H, Takahashi W: Approximating zero points of accretive operators with compact domains in general Banach spaces. Fixed Point Theory Appl. 2005,2005(1):93–102.
Moudafi A: A remark on recent results for finding zeros of accretive operators. J. Appl. Math. Stoch. Anal. 2006., 2006: Article ID 56704
Al’ber YI: On solution by the method of regularization for operator equation of the first kind involving accretive mappings in Banach spaces. Differ. Equ. 1975, 11: 2242–2248. (in Russian)
Ryazanseva IP: On nonlinear operator equations involving accretive mappings. Izv. Vysš. Učebn. Zaved., Mat. 1985, 1: 42–46. (in Russian)
Ryazantseva IP: Regularization proximal algorithm for nonlinear equations of monotone type in Banach space. Zh. Vychisl. Mat. Mat. Fiz. 2002, 42: 1295–1303. (in Russian)
Buong N: Convergence rates in regularization for nonlinear illposed equations under accretive perturbations. Zh. Vychisl. Mat. Mat. Fiz. 2004, 44: 397–402.
Buong N: On nonlinear illposed accretive equations. Southeast Asian Bull. Math. 2004, 28: 595–600.
Buong N: Generalized discrepancy principle and illposed equations involving accretive operators. Nonlinear Funct. Anal. Appl. 2004, 9: 73–78.
Buong N, Hung VQ: NewtonKantorovich iterative regularization for nonlinear illposed equations involving accretive operators. Ukr. Mat. Zh. 2005, 57: 271–276.
Buong N, Phuong NTH: Regularization methods for nonlinear illposed equations involving m accretive mappings in Banach spaces. Russ. Math. (Izv. VUZ) 2013,57(2):58–64.
Burger M, Katenbacher B: Regularizing NewtonKaczmarz methods for nonlinear illposed problems. SIAM J. Numer. Anal. 2006, 44: 153–182. 10.1137/040613779
Vainberg MM: Variational Method and Method of Monotone Operators. Nauka, Moscow; 1972. (in Russian); Wiley (1974) Translated from Russian by A. Libin, Translation edited by D. Louvish
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).
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The main idea of this paper was proposed by JKK. JKK and NB prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the final manuscript.
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Kim, J.K., Buong, N. Convergence rates in regularization for a system of nonlinear illposed equations with maccretive operators. J Inequal Appl 2014, 440 (2014). https://doi.org/10.1186/1029242X2014440
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DOI: https://doi.org/10.1186/1029242X2014440
Keywords
 accretive operators
 demicontinuous
 convex Banach space
 modified BrowderTikhonov regularization
 Fréchet differentiability