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Some results on zero points of m-accretive operators in reflexive Banach spaces
Fixed Point Theory and Applications volume 2014, Article number: 118 (2014)
Abstract
A modified proximal point algorithm is proposed for treating common zero points of a finite family of m-accretive operators. A strong convergence theorem is established in a reflexive, strictly convex Banach space with the uniformly Gâteaux differentiable norm.
1 Introduction and preliminaries
Let E be a Banach space and let be the dual of E. Let denote the pairing between E and . The normalized duality mapping is defined by
A Banach space E is said to strictly convex if and only if for and implies that . Let . The norm of E is said to be Gâteaux differentiable if the limit exists for each . In this case, E is said to be smooth. The norm of E is said to be uniformly Gâteaux differentiable if for each , the limit is attained uniformly for all . The norm of E is said to be Fréchet differentiable if for each , the limit is attained uniformly for all . The norm of E is said to be uniformly Fréchet differentiable if the limit is attained uniformly for all . It is well known that (uniform) Fréchet differentiability of the norm of E implies (uniform) Gâteaux differentiability of the norm of E.
Let be the modulus of smoothness of E by
A Banach space E is said to be uniformly smooth if as . It is well known that if the norm of E is uniformly Gâteaux differentiable, then the duality mapping J is single valued and uniformly norm to weak∗ continuous on each bounded subset of E.
Recall that a closed convex subset C of a Banach space E is said to have a normal structure if for each bounded closed convex subset K of C which contains at least two points, there exists an element x of K which is not a diametral point of K, i.e., , where is the diameter of K.
Let D be a nonempty subset of a set C. Let . Q is said to be
-
(1)
sunny if for each and , we have ;
-
(2)
a contraction if ;
-
(3)
a sunny nonexpansive retraction if is sunny, nonexpansive, and a contraction.
D is said to be a nonexpansive retract of C if there exists a nonexpansive retraction from C onto D. The following result, which was established in [1–3], describes a characterization of sunny nonexpansive retractions on a smooth Banach space.
Let E be a smooth Banach space and let C be a nonempty subset of E. Let be a retraction and be the duality mapping on E. Then the following are equivalent:
-
(1)
is sunny and nonexpansive;
-
(2)
, , ;
-
(3)
, .
It is well known that if E is a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from E onto C. Let C be a nonempty closed convex subset of a smooth Banach space E, let , and let . Then we have from the above that if and only if for all , where is a sunny nonexpansive retraction from E onto C. For more additional information on nonexpansive retracts, see [4] and the references therein.
Let C be a nonempty closed convex subset of E. Let be a mapping. In this paper, we use to denote the set of fixed points of T. Recall that T is said to be an α-contractive mapping iff there exists a constant such that , . The Picard iterative process is an efficient method to study fixed points of α-contractive mappings. It is well known that α-contractive mappings have a unique fixed point. T is said to be nonexpansive iff , . It is well known that nonexpansive mappings have fixed points if the set C is closed and convex, and the space E is uniformly convex. The Krasnoselski-Mann iterative process is an efficient method for studying fixed points of nonexpansive mappings. The Krasnoselski-Mann iterative process generates a sequence in the following manner:
It is well known that the Krasnoselski-Mann iterative process only has weak convergence for nonexpansive mappings in infinite-dimensional Hilbert spaces; see [5–7] for more details and the references therein. In many disciplines, including economics, image recovery, quantum physics, and control theory, problems arise in infinite-dimensional spaces. In such problems, strong convergence (norm convergence) is often much more desirable than weak convergence, for it translates the physically tangible property that the energy of the error between the iterate and the solution x eventually becomes arbitrarily small. To improve the weak convergence of a Krasnoselski-Mann iterative process, so-called hybrid projections have been considered; see [8–22] for more details and the references therein. The Halpern iterative process was initially introduced in [23]; see [23] for more details and the references therein. The Halpern iterative process generates a sequence in the following manner:
where is an initial and u is a fixed element in C. Strong convergence of Halpern iterative process does not depend on metric projections. The Halpern iterative process has recently been extensively studied for treating accretive operators; see [24–31] and the references therein.
Let I denote the identity operator on E. An operator with domain and range is said to be accretive if for each and , , there exists such that . An accretive operator A is said to be m-accretive if for all . In this paper, we use to denote the set of zero points of A. For an accretive operator A, we can define a nonexpansive single valued mapping by for each , which is called the resolvent of A.
Now, we are in a position to give the lemmas to prove main results.
Lemma 1.1 [32]
Let , , , and be four nonnegative real sequences satisfying , , where is some positive integer, is a number sequence in such that , is a number sequence such that , and is a positive number sequence such that . Then .
Lemma 1.2 [33]
Let C be a closed convex subset of a strictly convex Banach space E. Let be some positive integer and let be a nonexpansive mapping for each . Let be a real number sequence in with . Suppose that is nonempty. Then the mapping is defined to be nonexpansive with .
Lemma 1.3 [34]
Let and be bounded sequences in a Banach space E and let be a sequence in with . Suppose that for all and
Then .
Lemma 1.4 [35]
Let E be a real reflexive Banach space with the uniformly Gâteaux differentiable norm and the normal structure, and let C be a nonempty closed convex subset of E. Let be α-contractive mapping and let be a nonexpansive mapping with a fixed point. Let be a sequence generated by the following: , where . Then converges strongly as to a fixed point of T, which is the unique solution in to the following variational inequality: , .
2 Main results
Theorem 2.1 Let E be a real reflexive, strictly convex Banach space with the uniformly Gâteaux differentiable norm. Let be some positive integer. Let be an m-accretive operator in E for each . Assume that is convex and has the normal structure. Let be an α-contractive mapping. Let , , and be real number sequences in with the restriction . Let be a real number sequence in with the restriction . Let be a positive real numbers sequence and a sequence in E for each . Assume that is not empty. Let be a sequence generated in the following manner:
where . Assume that the control sequences , , , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
;
-
(c)
;
-
(d)
.
Then the sequence converges strongly to , which is the unique solution to the following variational inequality: , .
Proof Put . Fixing , we have
Hence, we have
This proves that the sequence is bounded, and so is . Since
we have
where is an appropriate constant such that
Define a sequence by , that is, . It follows that
where is an appropriate constant such that
This implies that
From the restrictions (a), (b), (c), and (d), we find that
Using Lemma 1.4, we find that . This further shows that . Put . It follows from Lemma 1.3 that T is nonexpansive with . Note that
This implies that
It follows from the restrictions (a), (b), and (d) that
Now, we are in a position to prove that , where , and solves the fixed point equation
It follows that
This implies that
Since , we find that . Since J is strong to weak∗ uniformly continuous on bounded subsets of E, we find that
Since , as , we have
For , there exists such that , we have
This implies that .
Finally, we show that as . Since is convex, we see that
It follows that
Hence, we have
Using Lemma 1.1, we find as . This completes the proof. □
Remark 2.2 There are many spaces satisfying the restriction in Theorem 2.1, for example , where .
Corollary 2.3 Let E be a Hilbert space and let be some positive integer. Let be a maximal monotone operator in E for each . Assume that is convex and has the normal structure. Let be an α-contractive mapping. Let , , and be real number sequences in with the restriction . Let be a real number sequence in with the restriction . Let be a positive real numbers sequence and a sequence in E for each . Assume that is not empty. Let be a sequence generated in the following manner:
where . Assume that the control sequences , , , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
;
-
(c)
;
-
(d)
.
Then the sequence converges strongly to , which is the unique solution to the following variational inequality: , .
3 Applications
In this section, we consider a variational inequality problem. Let be a single valued monotone operator which is hemicontinuous; that is, continuous along each line segment in C with respect to the weak∗ topology of . Consider the following variational inequality:
The solution set of the variational inequality is denoted by . Recall that the normal cone for C at a point is defined by
Now, we are in a position to give the convergence theorem.
Theorem 3.1 Let E be a real reflexive, strictly convex Banach space with the uniformly Gâteaux differentiable norm. Let be some positive integer and let C be nonempty closed and convex subset of E. Let a single valued, monotone and hemicontinuous operator. Assume that is not empty and C has the normal structure. Let be an α-contractive mapping. Let , , and be real number sequences in with the restriction . Let be a real number sequence in with the restriction . Let be a positive real numbers sequence and a sequence in E for each . Let be a sequence generated in the following manner:
Assume that the control sequences , , , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
;
-
(c)
;
-
(d)
.
Then the sequence converges strongly to , which is the unique solution to the following variational inequality: , .
Proof Define a mapping by
From Rockafellar [36], we find that is maximal monotone with . For each , and , we see that there exists a unique such that , where . Notice that
which is equivalent to
that is, . This implies that . Using Theorem 2.1, we find the desired conclusion immediately. □
From Theorem 3.1, the following result is not hard to derive.
Corollary 3.2 Let E be a real reflexive, strictly convex Banach space with the uniformly Gâteaux differentiable norm. Let C be nonempty closed and convex subset of E. Let a single valued, monotone and hemicontinuous operator with . Assume that C has the normal structure. Let be an α-contractive mapping. Let , , and be real number sequences in with the restriction . Let be a sequence generated in the following manner:
Assume that the control sequences , , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
.
Then the sequence converges strongly to , which is the unique solution to the following variational inequality: , .
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Wu, C.Q., Lv, S. & Zhang, Y. Some results on zero points of m-accretive operators in reflexive Banach spaces. Fixed Point Theory Appl 2014, 118 (2014). https://doi.org/10.1186/1687-1812-2014-118
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DOI: https://doi.org/10.1186/1687-1812-2014-118