Abstract
In this paper, we study a hybrid split problem (HSP for short) for equilibrium problems and fixed point problems of nonlinear operators. Some strong and weak convergence theorems are established.
MSC:47J25, 47H09, 65K10.
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1 Introduction
Throughout this paper, we assume that H is a real Hilbert space with zero vector θ, whose inner product and norm are denoted by and , respectively. Let C be a nonempty subset of H and be a mapping. Denote by the set of fixed points of T. The symbols ℕ and ℝ are used to denote the sets of positive integers and real numbers, respectively.
Let H be a Hilbert space and C be a closed convex subset of H. Let be a bi-function. The classical equilibrium problem (EP for short) is defined as follows.
The symbol is used to denote the set of all solutions of the problem (EP), that is,
It is known that the problem (EP) contains optimization problems, complementary problems, variational inequalities problems, saddle point problems, fixed point problems, bilevel problems, semi-infinite problems and others as special cases and have many applications in physics and economics problems; for detail, one can refer to [1–3] and references therein.
In last ten years or so, the problem (EP) has been generalized and improved to find a common element of the set of fixed points of a nonlinear operator and the set of solutions of the problem (EP). More precisely, many authors have studied the following problem (FTEP) (see, for instance, [4–9]):
where C is a closed convex subset of a Hilbert space H, is a bi-function and is a nonlinear operator.
In this paper, motivated by the problems (EP) and (FTEP), we study the following mathematical model about a hybrid split problem for equilibrium problems and fixed point problems of nonlinear operators (HSP for short).
Let and be two real Banach spaces. Let C be a closed convex subset of and K be a closed convex subset of . Let and be two bi-functions, be a bounded linear operator. Let and be two nonlinear operators with and . The mathematical model about a hybrid split problem for equilibrium problems and fixed point problems of nonlinear operators (HSP for short) is defined as follows:
In fact, (HSP) contains several important problems as special cases. We give some examples to explain about it.
Example A If T is an identity operator on C, then (HSP) will reduce to the following problem (P1):
(P1) Find such that , , and satisfying , , .
Example B If S is an identity operator on K, then (HSP) will reduce to the following problem (P2):
(P2) Find such that , , , and satisfying , .
Example C If T, S are all identity operators, then (HSP) will reduce to the following split equilibrium problem (P3) which has been considered in [10]:
(P3) Find such that , , and satisfying , .
Example D If S is an identity operator and for all , then (HSP) will reduce to the following problem (P4) which has been studied in [11]:
(P4) Find such that and satisfying , .
In this paper, we introduce some new iterative algorithms for (HSP) and some strong and weak convergence theorems for (HSP) will be established.
2 Preliminaries
In what follows, the symbols ⇀ and → will symbolize weak convergence and strong convergence as usual, respectively. A Banach space is said to satisfy Opial’s condition if for each sequence in X which converges weakly to a point , we have
It is well known that any Hilbert space satisfies Opial’s condition. Let K be a nonempty subset of real Hilbert spaces H. Recall that a mapping is said to be nonexpansive if for all .
Let and be two Hilbert spaces. Let and be two bounded linear operators. B is called the adjoint operator (or adjoint) of A if for all , , B satisfies . It is known that the adjoint operator of a bounded linear operator on a Hilbert space always exists and is bounded linear and unique. Moreover, it is not hard to show that if B is an adjoint operator of A, then .
Example 2.1 ([10])
Let with the standard norm and with the norm for some . denotes the inner product of for some and denotes the inner product of for some . Let for and for , then B is an adjoint operator of A.
Example 2.2 ([10])
Let with the norm for some and with the norm for some . Let and denote the inner product of and , respectively, where , . Let for and for . Obviously, B is an adjoint operator of A.
Let K be a closed convex subset of a real Hilbert space H. For each point , there exists a unique nearest point in K, denoted by , such that . The mapping is called the metric projection from H onto K. It is well known that has the following characteristics:
-
(i)
for every ;
-
(ii)
for and , , ;
-
(iii)
for and ,
(2.1)
Lemma 2.1 (see [1])
Let K be a nonempty closed convex subset of H and F be a bi-function of into ℝ satisfying the following conditions:
-
(A1)
for all ;
-
(A2)
F is monotone, that is, for all ;
-
(A3)
for each , ;
-
(A4)
for each , is convex and lower semi-continuous.
Let and . Then there exists such that for all .
Lemma 2.2 (see [12])
Let K be a nonempty closed convex subset of H and let F be a bi-function of into ℝ satisfying (A1)-(A4). For , define a mapping as follows:
for all . Then the following hold:
-
(i)
is single-valued and for and is closed and convex;
-
(ii)
is firmly non-expansive, that is, for any , .
Lemma 2.3 (see, e.g., [6])
Let H be a real Hilbert space. Then the following hold:
-
(a)
;
-
(b)
for all ;
-
(c)
for all and .
Lemma 2.4 Let be the same as in Lemma 2.2. If , then for any and , .
Proof
By (ii) of Lemma 2.2 and (b) of Lemma 2.3,
which shows that . □
Let the mapping be defined as in Lemma 2.2. Then, for and ,
In particular, for any and , that is, is nonexpansive for any .
Remark 2.1 In fact, Lemma 2.5 is motivated by a proof of [[5], Theorem 3.2]. In order to the sake of convenience for proving, we restated the fact and gave its proof in Lemma 2.5 [10, 11].
Lemma 2.6 ([13])
Let be a nonnegative real sequence satisfying the following condition:
where is some nonnegative integer, is a sequence in and is a sequence in R such that
-
(i)
;
-
(ii)
or is convergent. Then .
Lemma 2.7 ([14])
Let and be bounded sequences in a Banach space E and let be a sequence in with . Suppose for all integers and , then .
3 Weak convergence iterative algorithms for (HSP)
In this section, we will introduce some weak convergence iterative algorithms for the hybrid split problem.
Theorem 3.1 Let and be two real Hilbert spaces. Let and be two nonempty closed convex sets. Let and be non-expansive mappings and and be bi-functions satisfying the conditions (A1)-(A4). Let be a bounded linear operator with its adjoint B. Let , and be sequences generated by
where , and with , is a projection operator from into C. Suppose that , then and .
Proof Let , the following several inequalities can be proved easily:
By Lemma 2.4, , hence
By (b) of Lemma 2.3 and (3.3), for each , we have
On the other hand, , so from (3.1)-(3.4), we have
Since , , by (3.2) and (3.5), we have
and
The inequality (3.6) implies exists. Further, from (3.6) and (3.7), we get
The inequality (3.8) also implies that
Using Lemma 2.4 and (3.8), we have
Notice that
hence,
From (3.10) and (3.11), we also have
The existence of implies that is bounded, hence has a weak convergence subsequence . Assume that for some , then , and by (3.12) and (3.8).
We say , in other words, and . By (3.10), we also obtain . If , then, by Opial’s condition and (3.11), we get
which is a contradiction. Hence or . On the other hand, from Lemma 2.2, we know for any . Hence, if for , then by Opial’s condition and (3.10) and Lemma 2.5, we have
which is also a contradiction. So, for each , , namely . Thus, we have proved . Similarly, we can also prove . Hence, .
Finally, we prove converges weakly to . Otherwise, if there exists another subsequence of , which is denoted by , such that with , then by Opial’s condition,
This is a contradiction. Hence converges weakly to an element . Together with (see (3.10)), we also get .
Finally, we prove converges weakly to . From (3.12), we have , so . Thus, from (3.8) we have . The proof is completed. □
If or , where I denotes an identity operator, then the following corollaries follow from Theorem 3.1.
Corollary 3.1 Let and be two real Hilbert spaces. Let and be two nonempty closed convex sets. Let be a non-expansive mapping and and be bi-functions satisfying the conditions (A1)-(A4). Let be a bounded linear operator with its adjoint B. Let , and be sequences generated by
where and with , is a projection operator from into C. Suppose that , then and .
Corollary 3.2 Let and be two real Hilbert spaces. Let and be two nonempty closed convex sets. Let be a non-expansive mapping and and be bi-functions satisfying the conditions (A1)-(A4). Let be a bounded linear operator with its adjoint B. Let , and be sequences generated by
where , and with , is a projection operator from into C. Suppose that , then and .
Corollary 3.3 Let and be two nonempty closed convex sets. Let and be bi-functions satisfying the conditions (A1)-(A4). Let be a bounded linear operator with its adjoint B. Let , and be sequences generated by
where and with , is a projection operator from into C. Suppose that , then and .
4 Strong convergence iterative algorithms for (HSP)
In this section, we introduce two strong convergence algorithms for (HSP); see Theorem 4.1 and Theorem 4.2.
Theorem 4.1 Let and be two real Hilbert spaces. Let and be two nonempty closed convex sets. Let and be non-expansive mappings and and be bi-functions satisfying the conditions (A1)-(A4). Let be a bounded linear operator with its adjoint B. Let , and be sequences generated by
where , and with , is a projection operator from into C. Suppose that , then and .
Proof We claim that is a nonempty closed convex set for . In fact, let , it follows from (3.4) that
By (3.2), (4.1) and (4.2), we obtain
Notice , . It follows from (4.3) that
hence , which yields that and for .
It is not hard to verify that is closed for , so it suffices to verify is convex for . Indeed, let and , we have
namely . Similarly, , which implies and is a convex set, .
Notice that and , then for . It follows that exists. Hence is bounded, which yields that and are bounded. For some with , from and (2.1), we have
By exists and (4.4), we have , so is a Cauchy sequence.
Let , then . Firstly, by , from (4.1) we have
Setting , by (4.3) again, we have
So,
Notice that and , so
Further, from (4.5) and (4.8),
Since , we have by (4.9). Thus
namely and . On the other hand, for , by Lemma 2.5, we have
which yields . We have verified .
Next, we prove . Since and by (4.8) and (4.9) and by (4.7), we have and and . So,
namely and . On the other hand, for , by Lemma 2.5 again, we have
which implies that . We have verified .
So, we have obtained and and , the proof is completed. □
If or , where I denotes an identity operator, then the following corollaries follow from Theorem 4.1.
Corollary 4.1 Let and be two real Hilbert spaces. Let and be two nonempty closed convex sets. Let and be bi-functions satisfying the conditions (A1)-(A4) and be a non-expansive mapping. Let be a bounded linear operator with its adjoint B. Let , and be sequences generated by
where and with , is a projection operator from into C. Suppose that , then and .
Corollary 4.2 Let and be two real Hilbert spaces. Let and be two nonempty closed convex sets. Let be a non-expansive mapping and and be bi-functions satisfying the conditions (A1)-(A4). Let be a bounded linear operator with its adjoint B. Let , and be sequences generated by
where, , and with , is a projection operator from into C. Suppose that , then and .
Corollary 4.3 Let and be two real Hilbert spaces. Let and be two nonempty closed convex sets. Let and be bi-functions satisfying the conditions (A1)-(A4). Let be a bounded linear operator with its adjoint B. Let , and be sequences generated by
where and with , is a projection operator from into C. Suppose that , then and .
It is well known that the viscosity iterative method is always applied to study the iterative solution for the fixed point problem of nonlinear operators, for example, [5, 6, 8, 15, 16]. Similarly, the viscosity iterative method can also be used to study the hybrid split problem (HSP). So, at the end of this paper, we introduce a viscosity iterative algorithm which can converge strongly to a solution of (HSP).
Theorem 4.2 Let and be two real Hilbert spaces. Let and be two nonempty closed convex sets. Let be a α-contraction mapping, and be non-expansive mappings and and be bi-functions satisfying the conditions (A1)-(A4). Let be a bounded linear operator with its adjoint B. Let , and be sequences generated by
where , and , is a projection operator from into C, and the coefficients and satisfy the following conditions:
-
(1)
, , ;
-
(2)
, .
Suppose that , then and , where .
Proof Let . The following inequalities are easily verified:
By Lemma 2.4,
From (4.10) and (4.12), we have
and
So, from (4.10)-(4.11) and (4.14), we have
We say is bounded. In fact, from (4.10) and (4.15), we have
which implies that
so is bounded. Further, , and are also bounded by (4.11).
By Lemma 2.5, from (4.10) we have
and
where is a constant satisfying
Proving as . Setting and , namely . Let be a constant satisfying for all . Then
From (4.18) and (4.19), we have
By the conditions (1) and (2) and (4.20), we obtain
Notice , hence from (4.21) we have
By Lemma 2.7 and (4.22), we have , which implies that
by the definition of . Since , together with (4.23), we have
Using (4.10), (4.12) and (4.15),
which yields
From (4.26) we have
Again, applying (4.25), (4.15) and (4.14), we have
which implies that
From (4.29) we have
and
Notice and for all , so
so
Further, from (4.27), (4.32) and (4.24), we have
and
Let . Choose a subsequence such that
Since is bounded, is bounded. Hence is a constant, namely exists, which implies (4.35) is well defined. Because is bounded, has a weak convergence subsequence which is still denoted by . Suppose , we say . When , from (4.30), (4.32) and (4.33), we have
If , then by (4.34) and (4.36) and Opial’s condition, we have
which is a contradiction, so and . Since for each , by Lemma 2.2, we have . Otherwise, if there exists such that , then by (4.27) and Lemma 2.5 and Opial’s condition, we have
which is also a contradiction, so and . Up to now, we have proved . Similarly, we can also prove . Hence , because of this, we can also obtain
Finally, we prove the conclusion of this theorem is right. For , from (4.10) we have
From (4.40) we have
by (4.41) and Lemma 2.6, we have . Again, from (4.27) and (4.30), we have and , respectively. The proof is completed. □
Remark
-
(1)
In this paper, the iterative coefficient α or r can be replaced with the sequence if satisfies , where ;
-
(2)
Obviously, if in this paper, these weak and strong convergence theorems are also true;
-
(3)
In this paper, if T is a nonexpansive mapping from into and is a bi-function from into ℝ with the conditions (A1)-(A4), S is a nonexpansive mapping from into and is a bi-function from into ℝ with the conditions (A1)-(A4), then we may obtain a series of similar algorithms.
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Acknowledgements
The first author was supported by the Natural Science Foundation of Yunnan Province (2010ZC152). The second author was supported partially by grant No. NSC 101-2115-M-017-001 of the National Science Council of the Republic of China.
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He, Z., Du, WS. On hybrid split problem and its nonlinear algorithms. Fixed Point Theory Appl 2013, 47 (2013). https://doi.org/10.1186/1687-1812-2013-47
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DOI: https://doi.org/10.1186/1687-1812-2013-47