Convergence Results for Fixed Point Problems of Accretive Operators in Banach Spaces

Corresponding Author: Dennis Ferdinand Agbebaku Department of Mathematics, University of Nigeria, Nsukka, Nigeria Email: dennis.agbebaku@unn.edu.ng Abstract: This paper deals with the approximate solutions of accretive maps in a uniformly convex Banach space. A weak convergence of a three step iterative scheme involving the resolvents of accretive operators is proved. The main result is applied to a convex minimization problem in Hilbert spaces. In particular, the minimizer of a convex and proper lower semi-continuous function defined in a Hilbert space was obtained. Numerical illustration with graphical display of the convergence of the sequence obtained from the iterative scheme is also presented.


Introduction
Let  be a real Banach space. The norm on  is represented by |||| and  * represents the dual space of . Let      be an accretive operator with whose resolvent is ,0 r Jr   . Denote by   r FJ  , the set of fixed points of r J  . This paper deals with the iterative scheme for approximating the fixed point of the resolvent of an accretive operator. That is, it is concerned with a sequence n generated by a given iterative scheme which converges (weakly or strongly) to   . (1) Rockafellar (1976) proved that the sequence obtained from the iterative scheme (1) converges weakly. The convergence of the iterative scheme (1) in Hilbert spaces, have has been studied by many authors, see for instance (Brézis and Lions, 1978) and the references in the paper. Güler (1991) showed, through a counter example, that the iteration (1) does not converge strongly. This lead to the modification of the proximal point algorithm in various forms so that strong convergence can be archived, see (Takahashi, 2007;Kamimura and Takahashi, 2000a).
The following result for the Mann's type iterative scheme was proved.
x  (5) The solutions of Equation (5) are the equilibrium points of differential Equation (4). Many research works have been dedicated to constructing iterative scheme for approximating these equilibrium problems, see for example (Chen and Zhu, 2008) and (Zeidler, 1985) Recently, many researchers have inroduced interesting and important iterative schemes for obtaining the zeros of accretive maps in Banach spaces, see for instance (Chang et al., 2018;2019). Moreover, iterative scheme for finding approximate solution to differential equations have also been introduced, (Arqub, 2017;Arqub et al., 2017;2016). This paper considers the following three-step algorithm defined with respct to the resolvent map of an accretive operator . Given that D is a nonempty, closed and convex subset of a Banach space . Let 0  D: where the real sequences n, n, n  (0,1).

Preliminaries
This section is concern with some preliminary definitions and lemmas that are to be used in the proof of the main result of his paper.

Definition 1
'Let D be a closed convex and nonempty subset of . We say a mapping T: D  D is nonexpansive, if for all ,   D': Denote by F(T) the fixed point set of T. A point   , is a fixed point if T = . The set F(T) is closed and convex, see (Goebel and Kirk, 1990 A firmly nonexpansive map is also nonexpansive, (Ariza-Ruiz et al., 2014a;2014b;Bauschke and Combettes, 2011). However, the converse fails. Indeed, the mapping T = -I, where I is the identity mapping, is nonexpansive but it is not firmly nonexpansive. A characterization of firmly nonexpansive mapping is given by the following result, (Bauschke and Combettes, 2011;Goebel and Kirk, 1990).

Theorem 2
'Let  be a Banach space and D   a subset of .
Let T: D  be a mapping with J the duality mapping. For all ,   X the following are equivalent': i. T is firmly nonexpansive ii. For each ,   D and for all r > 0: iii. For each ,   D and for all 0  s  1: If  is a Hilbert space H with the inner product , then The following characterization of firmly nonexpansive maps hold in Hilbert space:

Theorem 3
'Let H be a Hilbert space and D   a subset of H. For the mapping T: H  H the following are equivalent for all ,   D, (Bauschke and Combettes, 2011;Goebel and Kirk, 1990) where, PD is the projection map from  onto D.
Definition 3 (Opial, 1967) 'A Banach space  is said satisfies the Opial's condition if for any sequence n: for all z   with' z  .
The mapping A is accretive if and only if for each , for all r > 0. A mapping is m-accretive if it is accretive and R(I + r) =  for all r > 0.

Definition 5
Let :   2  be a multi-valued map with domain D() and range R() in . The resolvent of , of order r > 0, is the multi-valued mapping If the operator  is accretive then its resolvent is firmly nonexpansive and single-valued. The proof is given in the following Theorem which characterizes, in Banach spaces, the resolvent mapping of an accretive operator (Bruck and Reich, 1977;Goebel and Reich, 1984

Definition 6
Let  be an accretive operator. The Yosida approximation r of  is defines as: The following Proposition gives the characterization of the Yosida approximation.

Remark
In any Hilbert space, the identity operator is the normalized duality mapping. The notions of accretive and monotone coincide in a Hilbert space. Similarly, the notions of m-accretive and maximal monotone also coincide; see (Barbu, 2010).
The following Theorems give characterizations of monotone operators in Hilbert spaces, (Minty, 1962;1964

T T I
  from the set of firmly nonexpansive mapping to the set of maximal monotone mapping is a bijection whose inverse is given as the mapping:

Main Result
The proof of the main result in this study, which is the weak convergence of the sequence obtained from (6), is presented in this section. Before stating and proving the Theorem some useful Lemmas are presented. Browder (1968) proved the following Lemma 1 (Browder, 1968) Let D be a closed, bounded and convex subset of a uniformly convex space E. Let T: D  D be a nonexpansive self-mapping. Suppose the sequence {n} converges weakly to zD and the sequence {n-Tn} converges strongly to 0, then Tz = z.
The main result of this paper is the following:

Theorem 7
Let  be a uniformly convex Banach space whose norm satisfies the Opial's condition. Let :   2  be an m-accretive operator and r J  :  D()   the resolvent of  of order r > 0. Assume that sequences n, n, n  (0,1) satisfy: That is: Also: So that by (17): Hence from the inequalities (17) and (18) the following inequality follows: Now set sn = ||n- * ||. Then sn+1 = ||n+1- * || so that inequality (19) leads to sn+1  sn which implies that sn is decreasing. Moreover, 0  sn+1  sn  s1 which implies sn is bounded. There exist a real number M > 0 and |sn|  M, hence: Thus: Therefore {n} is bounded. From (17) and (18), it is easy to see that the sequences {zn} and {yn} are also bounded.
The inequality: This leads to: n r n nn n n n r n n n n r n n n n n r n n n n n r n

Uniqueness of the Weak Limit
What remains is to prove that the weak limit is  Table 1 and the convergence of {n} for initial values 0 = 50 and 0 = -50 is shown in Fig. 1.

Conclusion
A three-step iterative scheme for finding approximate solution to equations involving an accretive operator is introduced. Weak convergence of the sequence obtained from the iterative scheme was proved. A numerical example to illustrate the validity of the main result is given. From the numerical example, it is clear that the convergence of the sequence obtained from the iterative scheme is independent of the initial condition. The results obtained in this study complements some of the existing results in the literature and can be applied to some real world problem involving accretive operators.