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Synchronal and cyclic algorithms for fixed point problems and variational inequality problems in Banach spaces
Fixed Point Theory and Applications volume 2013, Article number: 202 (2013)
Abstract
In this paper, we study synchronal and cyclic algorithms for finding a common fixed point of a finite family of strictly pseudocontractive mappings, which solve the variational inequality
where f is a contraction mapping, G is an η-strongly accretive and L-Lipschitzian operator, is a positive integer, are arbitrary fixed constants, and are N-strict pseudocontractions. Furthermore, we prove strong convergence theorems of such iterative algorithms in a real q-uniformly smooth Banach space. The results presented extend, generalize and improve the corresponding results recently announced by many authors.
MSC:47H06, 47H09, 47H10, 47J05, 47J20, 47J25.
1 Introduction
Let E be a real Banach space, and let be the dual of E. For some real number q (), the generalized duality mapping is defined by
where denotes the duality pairing between elements of E and those of . In particular, is called the normalized duality mapping and for . If E is a real Hilbert space, then , where I is the identity mapping. It is well known that if E is smooth, then is single-valued, which is denoted by .
Let C be a nonempty closed convex subset of E, and let be a nonlinear map. Then the variational inequality problem with respect to C and G is to find a point such that
We denote by the set of solutions of this variational inequality problem.
If , a real Hilbert space, the variational inequality problem reduces to the following: Find a point such that
A mapping is said to be a contraction if, for some ,
The map T is said to be nonexpansive if
The map T is said to be L-Lipschitzian if there exists such that
A point is called a fixed point of the map T if . We denote by the set of all fixed points of the mapping T, that is,
We assume that in the sequel. It is well known that above is closed and convex (see, e.g., Goebel and Kirk [1]).
An operator is said to be accretive if , there exists such that
For some positive real numbers η, λ, the mapping F is said to be η-strongly accretive if for any , there exists such that
and it is called λ-strictly pseudocontractive if
It is clear that (1.9) is equivalent to the following:
where I denotes the identity operator.
In Hilbert spaces, accretive operators are called monotone where inequality (1.7) holds with replaced by the identity map of H.
A bounded linear operator A on H is called strongly positive with coefficient γ if there is a constant with the property
Let K be a nonempty closed convex and bounded subset of a Banach space E, and let the diameter of K be defined by . For each , let , and let denote the Chebyshev radius of K relative to itself. The normal structure coefficient of E (see, e.g., [2]) is defined by . A space E, such that , is said to have a uniform normal structure.
It is known that all uniformly convex and uniformly smooth Banach spaces have a uniform normal structure (see, e.g., [3, 4]).
Let μ be a continuous linear functional on and . We write instead of . We call μ a Banach limit if μ satisfies and for all . If μ is a Banach limit, then
Let denote the unit sphere of a real Banach space E.
The space E is said to have a Gâteaux differentiable norm if the limit
exists for each . In this case, E is called smooth. E is said to be uniformly smooth if the limit (1.11) exists and is attained uniformly in . E is said to have a uniformly Gâteaux differentiable norm if, for any , the limit (1.11) exists uniformly for all .
The modulus of smoothness of E, with , is a function defined by
A Banach space E is said to be uniformly smooth if , and for , E is said to be q-uniformly smooth if there exists a fixed constant such that , .
It is well known (see, e.g., [6]) that Hilbert spaces, (or ) spaces () and Sobolev spaces, () are all uniformly smooth. More precisely, Hilbert spaces are 2-uniformly smooth, while
Also, it is well known (see, e.g., [7]) that q-uniformly smooth Banach spaces have a uniformly Gâteaux differentiable norm.
The variational inequality problem was initially introduced and studied by Stampacchia [8] in 1964. In the recent years, variational inequality problems have been extended to study a large variety of problems arising in structural analysis, economics and optimization. Thus, the problem of solving a variational inequality of the form (1.2) has been intensively studied by numerous authors. Iterative methods for approximating fixed points of nonexpansive mappings and their generalizations, which solve some variational inequality problems, have been studied by a number of authors (see, for example, [9–13] and the references therein).
Let H be a real Hilbert space. In 2001, Yamada [13] proposed a hybrid steepest descent method for solving variational inequality as follows: Let be chosen arbitrarily and define a sequence by
where T is a nonexpansive mapping on H, F is L-Lipschitzian and η-strongly monotone with , , . If is a sequence in satisfying the following conditions:
-
(C1)
,
-
(C2)
,
-
(C3)
either or ,
then he proved that the sequence converges strongly to the unique solution of the variational inequality
Besides, he also proposed the cyclic algorithm
where ; he also proved strong convergence theorems for the cyclic algorithm.
In 2006, Marino and Xu [10] considered the following general iterative method: Starting with an arbitrary initial point , define a sequence by
where T is a nonexpansive mapping of H, f is a contraction, A is a linear bounded strongly positive operator, and is a sequence in satisfying the following conditions:
-
(M1)
;
-
(M2)
;
-
(M3)
or .
They proved that the sequence converges strongly to a fixed point of T, which solves the variational inequality
In 2010, Tian [11] combined the iterative method (1.13) with Yamada’s iterative method (1.12) and considered the following general iterative method:
where T is a nonexpansive mapping on H, f is a contraction, F is k-Lipschitzian and η-strongly monotone with , , . He proved that if the sequence of parameters satisfies conditions (M1)-(M3), then the sequence generated by (1.14) converges strongly to a fixed point of T, which solves the variational inequality
Very recently, in 2011, Tian and Di [12] studied two algorithms, based on Tian’s [11] general iterative algorithm, and proved the following theorems.
Theorem 1.1 (Synchronal algorithm)
Let H be a real Hilbert space, and let be a -strictly pseudocontraction for some () such that , and f be a contraction with coefficient and be a positive constant such that . Let be an η-strongly monotone and L-Lipschitzian operator with , . Assume that , . Let be chosen arbitrarily, and let and be sequences in satisfying the following conditions:
-
(N1)
, ;
-
(N2)
, ;
-
(N3)
, .
Let be a sequence defined by the composite process
Then converges strongly to a common fixed point of , which solves the variational inequality
Theorem 1.2 (Cyclic algorithm)
Let H be a real Hilbert space, and let be a -strictly pseudocontraction for some () such that , and f be a contraction with coefficient . Let be an η-strongly monotone and L-Lipschitzian operator with , . Assume that , . Let be chosen arbitrarily, and let and be sequences in satisfying the following conditions:
-
(N1′) , ;
-
(N2′) or ;
-
(N3′) , , where .
Let be a sequence defined by the composite process
where , with , , namely is one of cyclically. Then converges strongly to a common fixed point of , which solves the variational inequality (1.15).
In this paper, we study the synchronal and cyclic algorithms for finding a common fixed point of finite strictly pseudocontractive mappings, which solves the variational inequality
where f is a contraction mapping, G is an η-strongly accretive and L-Lipschitzian operator, is a positive integer, are arbitrary fixed constants, and are N-strict pseudocontractions defined on a closed convex subset C of a real q-uniformly smooth Banach space E whose norm is uniformly Gâteaux differentiable.
Let T be defined by
where such that . We will show that a sequence generated by the following synchronal algorithm:
converges strongly to a solution of problem (1.16).
Another approach to problem (1.16) is the cyclic algorithm. For each , let , where the constant satisfies . Beginning with , define a sequence cyclically by
Indeed, the algorithm can be written in a compact form as follows:
where , with , , namely is one of cyclically. We will show that (1.18) is also strongly convergent to a solution of problem (1.16) if the sequences and of parameters are appropriately chosen.
Motivated by the results of Tian and Di [12], in this paper we aim to continue the study of fixed point problems and prove new theorems for the solution of variational inequality problems in the framework of a real Banach space, which is much more general than that of Hilbert.
Throughout this research work, we will use the following notations:
-
1.
⇀ for weak convergence and → for strong convergence.
-
2.
denotes the weak ω-limit set of .
2 Preliminaries
In the sequel we shall make use of the following lemmas.
Lemma 2.1 (Zhang and Guo [14])
Let C be a nonempty closed convex subset of a real Banach space E. Given an integer , for each , is a -strict pseudocontraction for some such that . Assume that is a sequence of positive numbers such that . Then is a λ-strict pseudocontraction with , and
Lemma 2.2 (Zhou [15])
Let E be a uniformly smooth real Banach space, and let C be a nonempty closed convex subset of E. Let be a k-strict pseudocontraction. Then is demiclosed at zero. That is, if satisfies and , as , then .
Lemma 2.3 (Petryshyn [16])
Let E be a real Banach space, and let be the generalized duality mapping. Then, for any and ,
Lemma 2.4 (Lim and Xu [4])
Suppose that E is a Banach space with a uniform normal structure, K is a nonempty bounded subset of E, and let be a uniformly k-Lipschitzian mapping with . Suppose also that there exists a nonempty bounded closed convex subset C of K with the following property (P):
where is the ω-limit set of T at x, i.e., the set
Then T has a fixed point in C.
Lemma 2.5 (Xu [17])
Let , and let E be a real q-uniformly smooth Banach space, then there exists a constant such that for all and ,
Lemma 2.6 Let E be a real q-uniformly smooth Banach space with constant , , and let C be a nonempty closed convex subset of E. Let be an η-strongly accretive and L-Lipschitzian operator with , . Assume that , and . Then, for any , the following inequality holds:
That is, is a contraction with coefficient .
Proof For any , we have, by Lemma 2.5, (1.6) and (1.8),
From , and , we have . It then follows that
□
Lemma 2.7 Let E be a real q-uniformly smooth Banach space with constant , , and let C be a nonempty closed convex subset of E. Suppose that is a λ-strict pseudocontraction such that . For any , we define by for each . Then, as , , is a nonexpansive mapping such that .
Proof For any , we have, by Lemma 2.5 and (1.10),
which shows that is a nonexpansive mapping.
It is clear that . This proves our assertions. □
Lemma 2.8 (Xu [18])
Let be a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence in ℝ such that
-
(i)
and ;
-
(ii)
or .
Then .
Lemma 2.9 Let E be a real q-uniformly smooth Banach space with constant , , and let C be a nonempty closed convex subset of E. Suppose that are -strict pseudocontractions for (). Let , (). If , then, as , , we have
Proof We prove it by induction. For , set , , , . Obviously,
Now we prove
For all , , if , then , the conclusion holds. In fact, we can claim that . From Lemma 2.7, we know that is nonexpansive and .
Take , then, by Lemma 2.5 and (1.10), we have
So, we get
Namely , that is,
Suppose that the conclusion holds for , we prove that
It suffices to verify
For all , . Using Lemma 2.5 and (1.10) again, take , then
So, we get
Thus, , that is, . Namely,
From (2.1) and inductive assumption, we get
that is,
Substituting it into (2.1), we obtain , , that is, , , and hence
□
Lemma 2.10 (Ali et al. [9])
Let E be a real q-uniformly smooth Banach space with constant , . Let be a contraction mapping with constant . Let be a nonexpansive mapping such that , and let be an η-strongly accretive mapping which is also k-Lipschitzian. Let and . For each and , the path defined by
converges strongly as to a fixed point of T, which solves the variational inequality
Lemma 2.11 (Chang et al. [19])
Let E be a real Banach space with a uniformly Gâteaux differentiable norm. Then the generalized duality mapping is single-valued and uniformly continuous on each bounded subset of E from the norm topology of E to the topology of .
Lemma 2.12 (Zhou et al. [20])
Let α be a real number, and let a sequence satisfy the condition for all Banach limit μ. If , then .
Lemma 2.13 (Mitrinović [21])
Suppose that . Then, for any arbitrary positive real numbers , the following inequality holds:
3 Synchronal algorithm
Theorem 3.1 Let E a real q-uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let be -strict pseudocontractions for () such that . Let f be a contraction with coefficient , and let be a sequence of positive numbers such that . Let be an η-strongly accretive and L-Lipschitzian operator with , . Assume that , . Let and be sequences in satisfying the following conditions:
-
(K1)
, ;
-
(K2)
, ;
-
(K3)
, where ;
-
(K4)
, where .
Let be a sequence defined by algorithm (1.17), then converges strongly to a common fixed point of , which solves the variational inequality (1.16).
Proof Let , then by Lemma 2.1 we conclude that T is a k-strict pseudocontraction and . We can rewrite algorithm (1.17) as follows:
Furthermore, by Lemma 2.7 we have that is a nonexpansive mapping and . From condition (K1) we may assume, without loss of generality, that . Let , then the sequence satisfies
We prove this by mathematical induction as follows.
Obviously, it is true for . Assume that it is true for for some .
From (1.17) and Lemma 2.6, we have
Hence the proof. Thus, the sequence is bounded and so are , and .
Observe that
so that
where is an appropriate constant such that .
On the other hand, we note that
where is an appropriate constant such that .
Now, substituting (3.2) into (3.1) yields
where is an appropriate constant such that .
By Lemma 2.8 and conditions (K1), (K2), we have
From (1.17) and condition (K1), we have
On the other hand,
which implies, by condition (K3), that
Hence, from (3.3) and (3.4), we have
From the boundedness of , without loss of generality, we may assume that . Hence, by Lemma 2.2 and (3.5), we obtain . So, we have
We now prove that , where is obtained in Lemma 2.10. Put , .
Define a map by
Then ϕ is continuous, convex, and as . Since E is reflexive, there exists such that . Hence the set
Therefore, applying Lemma 2.4, we have . Without loss of generality, assume . Let . Then it follows that , and using Lemma 2.3, we obtain that
This implies that
By Lemma 2.11, is norm-to-weak∗ uniformly continuous on a bounded subset of E, so we obtain, as , that
Hence, for all , there exists such that and for all ,
Consequently,
Since ϵ is arbitrary, we have
Thus, for any Banach limit μ.
Furthermore, by (3.3), as . We therefore conclude that
Hence, by Lemma 2.12 we obtain , that is,
From (1.17), Lemmas 2.3, 2.6 and 2.13, we have
This implies that
where and . From (K1), , . . So, . Hence, by Lemma 2.8, we have that as . This completes the proof. □
4 Cyclic algorithm
Theorem 4.1 Let E be a real q-uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let be -strict pseudocontractions for () such that , let f be a contraction with coefficient . Let be an η-strongly accretive and L-Lipschitzian operator with , . Assume that , . Let and be sequences in satisfying the following conditions:
-
(K1′) , ;
-
(K2′) or ;
-
(K3′) , , where ;
-
(K4′) , where .
Let be a sequence defined by algorithm (1.18), then converges strongly to a common fixed point of , which solves the variational inequality (1.16).
Proof From condition (K1′) we may assume, without loss of generality, that . Let , then the sequence satisfies
We prove this by mathematical induction as follows.
Obviously, it is true for . Assume it is true for for some .
From (1.18) and Lemma 2.6, we have
Hence the proof. Thus, the sequence is bounded and so are , , , and .
From (1.18) and Lemma 2.6, we have
where is an appropriate constant such that . By conditions (K1′), (K2′) and Lemma 2.8, we have
From (1.18) and condition (K1′), we have
Recursively,
By condition (K3′) and Lemma 2.7, we know that is nonexpansive, so we get
Proceeding accordingly, we have
Note that
From the above inequality, we obtain
Since
we conclude that
Take a subsequence , by (4.3) we get
Notice that for each , is some permutation of the mappings . Since are finite, all the finite permutations are N!, there must be some permutation appearing infinitely many times. Without loss of generality, suppose this permutation is , we can take a subsequence such that () and
By Lemma 2.7, we conclude that are all nonexpansive. It is clear that is nonexpansive, so is . By Lemma 2.2, we have . From Lemmas 2.7 and 2.9, we obtain
that is,
We now prove that , where is obtained in Lemma 2.10. Put , .
Define a map by
Then ϕ is continuous, convex, and as . Since E is reflexive, there exists such that . Hence the set
Therefore, applying Lemma 2.4, we have . Without loss of generality, assume . Let . Then it follows that , and using Lemma 2.3, we obtain that
This implies that
By Lemma 2.11, is norm-to-weak∗ uniformly continuous on a bounded subset of E, so we obtain, as , that
Hence, for all , there exists such that , and for all ,
Consequently,
Since ϵ is arbitrary, we have
Thus, for any Banach limit μ.
Furthermore, by (4.1) as , we therefore conclude that
Hence, by Lemma 2.12 we obtain , that is,
From (1.18), Lemmas 2.3, 2.6 and 2.13, we have
This implies that
where and