Abstract
In this paper, we study and give examples for classes of generalized contractive mappings. We establish some new strong convergence theorems of feasible iterative algorithms for the split common solution problem (SCSP) and give some applications of these new results.
MSC:47H06, 47J25, 47H09, 65K10.
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1 Introduction and preliminaries
Let K be a closed convex subset of a real Hilbert space H with the inner product and the norm . The following inequalities are known and useful.
-
;
-
for all ;
-
for all and .
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 properties:
-
(i)
for every .
-
(ii)
For and , for all .
-
(iii)
For and ,
(1.1)
Let and be two Hilbert spaces. Let and be two bounded linear operators. is called the adjoint operator (or adjoint) of A if
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 is an adjoint operator of A, then . The symbols ℕ and ℝ are used to denote the sets of positive integers and real numbers, respectively.
Let and be two real Hilbert spaces. Let C be a closed convex subset of and K be a closed convex subset of . Let with and with be two mappings. Let be a bounded linear operator. The mathematical model of the split common solution problem (SCSP in short) is defined as follows:
In fact, SCSP contains several important problems as special cases and many authors have studied and introduced some new iterative algorithms for SCSP and presented some strong and weak convergence theorems for SCSP; see, for instance, [1–24] and the references therein. Motivated and inspired by their works, in this paper, we study and establish new strong convergence results by using new iterative algorithms of SCSP for pseudocontractive mappings and k-demicontractive mappings in Hilbert spaces.
The paper is divided into four sections. In Section 2, we study and give examples for classes of generalized contractive mappings. Some new strong convergence theorems of feasible iterative algorithms for SCSP are established in Section 3. Finally, some applications and further remarks for our new results are given in Section 4. Consequently, in this paper, some of our results are original and completely different from these known related results in the literature.
2 Classes of generalized contractive mappings and their examples
Let T be a mapping with domain and range in a Hilbert space H. Recall that T is said to be
-
(i)
pseudocontractive if
or, equivalently,
-
(ii)
demicontractive if, for all and ,
or, equivalently,
-
(iii)
k-demicontractive if there exists a constant such that
-
(iv)
quasi-nonexpansive if it is 0-demicontractive, that is,
-
(v)
Lipschitzian if there exists such that
-
(vi)
nonexpansive if it is Lipschitzian with ;
-
(vii)
contractive if it is Lipschitzian with .
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.
Definition 2.1 (see [2])
Let K be a nonempty closed convex subset of a real Hilbert space H and T be a mapping from K into K. The mapping T is said to be demiclosed if, for any sequence which weakly converges to y, and if the sequence strongly converges to z, then .
Remark 2.1 In Definition 2.1, the particular case of demiclosedness at zero is frequently used in some iterative convergence algorithms, which is the particular case when , the zero vector of H; for more details, one can refer to [2].
The following concept of zero-demiclosedness was introduced as follows.
Definition 2.2 (see [[25], Definition 2.3])
Let K be a nonempty closed convex subset of a real Hilbert space and T be a mapping from K into K. The mapping T is called zero-demiclosed if in K satisfying and implies .
The following result was essentially proved in [25], but we give the proof for the sake of completeness and the reader’s convenience.
Theorem 2.1 (see [[25], Proposition 2.4])
Let K be a nonempty closed convex subset of a real Hilbert space with zero vector θ. Then the following statements hold.
-
(a)
Let T be a mapping from K into K. Then T is zero-demiclosed if and only if is demiclosed at θ.
-
(b)
Let T be a nonexpansive mapping from H into itself. If there is a bounded sequence such that as , then T is zero-demiclosed.
Proof Obviously, the conclusion (a) holds. To see (b), since is bounded, there is a subsequence and such that . One can claim . Indeed, if , it follows from Opial’s condition that
which is a contradiction. So and hence T is zero-demiclosed. □
Now, we give some examples to show the existence of these generalized contractive mappings (i)-(vi) which also expound the relation between them.
Example A Let with the absolute-value norm and . Let be defined by
Then . Since
we know that T is a -demicontractive mapping. However, due to
T is not quasi-nonexpansive.
Example B Let with the absolute-value norm and . Let be defined by
Then . Since
T is a -demicontractive mapping. Moreover, T is also a pseudocontractive mapping.
Example C Let with the absolute-value norm . Let be defined by
It is easy to see that
So T is continuous nonexpansive with .
The following example shows that there exists a continuous quasi-nonexpansive mapping which is not nonexpansive.
Example D (see [8])
Let with the absolute-value norm and . Define by
Obviously, . It is easy to see that
and
Hence T is a continuous quasi-nonexpansive mapping but not nonexpansive.
The following example shows that there exists a demicontractive mapping which is neither pseudocontractive nor k-demicontractive for all .
Example E Let with the absolute-value norm . Let be defined by
Then . Since
T is a demicontractive mapping. However, T is not a pseudocontractive mapping due to the fact that when and , we have
It is easy to see that T is not a k-demicontractive mapping for all .
The following example shows that there exists a discontinuous pseudocontractive mapping which is not a demicontractive mapping.
Example F Let with the absolute-value norm . Let be defined by
Then . Due to
we know that T is a discontinuous pseudocontractive mapping but not a demicontractive mapping.
The following example shows that there exists a pseudocontractive mapping which is not k-demicontractive for all .
Example G Let with the absolute-value norm . Let be defined by
Then . Since
T is a pseudocontractive mapping. It is easy to see that T is not a k-demicontractive mapping for all .
The following example shows that there exists a discontinuous k-demicontractive mapping for some as well as being demiclosed at θ which is neither pseudocontractive nor quasi-nonexpansive.
Example H Let with the absolute-value norm and . Let be defined by
Then the following statements hold.
-
(a)
T is discontinuous -demicontractive.
-
(b)
T is demiclosed at θ.
-
(c)
T is not pseudocontractive.
-
(d)
T is not quasi-nonexpansive.
Proof Clearly, . Since
T is a discontinuous -demicontractive mapping and (a) is proved. Now, we verify (b). In fact, let with and as . If all , we can prove and easily. If there exists a subsequence , then, from as , we can find a subsequence of such that for all i. Hence we have
which implies . To see (c) and (d), note that
and
so T is neither pseudocontractive nor quasi-nonexpansive. The proof is completed. □
3 New feasible iterative algorithms for SCSP and strong convergence theorems
In this section, we establish some new strong convergence theorems by using feasible iterative algorithms for SCSP.
Theorem 3.1 Let and be two real Hilbert spaces and be the zero vector of for . Let C be a nonempty closed convex subset of and be a bounded linear operator with its adjoint B. Let be a Lipschitzian pseudocontractive mapping with Lipschitz constant and , and let be a k-demicontractive mapping with which is demiclosed at . Let and be a sequence generated by the following algorithm:
where , and is the projection operator from into for . Suppose that
Then there exists such that
-
(a)
as ,
-
(b)
as .
Proof We will show the conclusion by proceeding with the following steps.
Step 1. For any , we prove
Indeed, since
and
we get
and our desired result is proved.
Step 2. We prove
For any , by (3.1), we have
Since and , from (3.4), we have , or, equivalently,
Step 3. We show that is a nonempty closed convex set for any .
For any , by taking into account (3.2) and (3.5), we obtain
So we know and hence for all . It is easy to verify that is closed and convex for all .
Step 4. We prove that is a Cauchy sequence in C and as for some .
Since and , we get
and
which show that is bounded and is nondecreasing in . So
exists. For any with , from and (1.1), we have
Inequality (3.6) implies
So is a Cauchy sequence. Clearly,
By the completeness of C, there exists such that as .
Step 5. Finally, we show that the following hold:
-
(i)
,
-
(ii)
as .
For any , since , from (3.1), we have
and
From inequalities (3.7), (3.8) and (3.9), we deduce
and hence
By taking into account (3.4) and (3.10), we get
So, we obtain
Since as , from (3.12) and the continuity of the norm and the Lipschitzian pseudocontractive mapping T, we can deduce that , namely . On the other hand, from (3.2) and (3.11), we have
which yields that
Since the k-demicontractive mapping S is demiclosed at , taking into account , , and (3.13), we have
and
Hence we confirm . The proof is completed. □
By virtue of Theorem 3.1, we can establish the following:
-
(i)
Strong convergence algorithms for the split common solution problem for Lipschitzian pseudocontractive mappings and nonexpansive mappings (see Corollary 3.1 below).
-
(ii)
Strong convergence algorithms for the split common solution problem for Lipschitzian pseudocontractive mappings and quasi-nonexpansive mappings (see Corollary 3.2 below).
Corollary 3.1 Let and be two real Hilbert spaces and be the zero vector of for . Let C be a nonempty closed convex subset of and be a bounded linear operator with its adjoint B. Let be a Lipschitzian pseudocontractive mapping with Lipschitz constant and , and let be a nonexpansive mapping with . Let and be a sequence generated by the following algorithm:
where , and is the projection operator from into for . Suppose that
Then there exists such that
-
(a)
as ,
-
(b)
as .
Proof Since the mapping S is nonexpansive, it is 0-demicontractive. Hence the desired conclusion follows from Theorem 3.1 immediately by taking . □
Corollary 3.2 Let and be two real Hilbert spaces and be the zero vector of for . Let C be a nonempty closed convex subset of and be a bounded linear operator with its adjoint B. Let be a Lipschitzian pseudocontractive mapping with Lipschitz constant and , and let be a quasi-nonexpansive mapping with which is demiclosed at . Let and be a sequence generated by the following algorithm:
where , and is the projection operator from into for . Suppose that
Then there exists such that
-
(a)
as ,
-
(b)
as .
Example 3.1 Let with the absolute-value norm . Let with the norm for and the inner product for and . Let be defined by for . Then A is a bounded linear operator with its adjoint operator for . Clearly, . Let . Let and be defined by
and
respectively. It is easy to see that
-
;
-
;
-
;
-
T is a Lipschitzian pseudocontractive mapping with Lipschitz constant ;
-
T and S both are -demicontractive mappings.
By using algorithm (3.1) with and , we can verify and as .
4 Some applications and further remarks for Theorem 3.1
Let C be a nonempty subset of a Hilbert space H. Recall that a mapping is said to be accretive if
Obviously, is accretive if and only if is pseudocontractive. Moreover,
where θ is the zero vector of H.
At the end of this paper, by applying Theorem 3.1, we obtain the following:
-
(i)
Strong convergence algorithms for the split common solution problem for Lipschitzian accretive mappings and demicontractive nonexpansive mappings (see Theorem 4.1 below).
-
(ii)
Strong convergence algorithms for the split common solution problem for Lipschitzian accretive mappings and nonexpansive mappings (see Corollary 4.1 below).
-
(iii)
Strong convergence algorithms for the split common solution problem for Lipschitzian accretive mappings and quasi-nonexpansive mappings (see Corollary 4.2 below).
Theorem 4.1 Let and be two real Hilbert spaces and be the zero vector of for . Let be a bounded linear operator with its adjoint B and be a Lipschitzian accretive mapping with Lipschitz constant and . Let be a k-demicontractive mapping with which is demiclosed at . Let be a sequence generated by the following algorithm:
where , and is the projection operator from into for . Suppose that
Then there exists such that
-
(a)
as ,
-
(b)
as .
Proof Let . Then the iterative process (4.1) can be rewritten as follows:
Set , then and T is a Lipschitzian pseudocontractive mapping with Lipschitz constant . Therefore the desired conclusion follows from Theorem 3.1 immediately. □
The following interesting results are immediate from Theorem 4.1.
Corollary 4.1 Let and be two real Hilbert spaces and be the zero vector of for . Let be a bounded linear operator with its adjoint B and be a Lipschitzian accretive mapping with Lipschitz constant and . Let be a quasi-nonexpansive mapping with which is demiclosed at . Let be a sequence generated by the following algorithm:
where , and is the projection operator from into for . Suppose that
Then there exists such that
-
(a)
as ,
-
(b)
as .
Corollary 4.2 Let and be two real Hilbert spaces and be the zero vector of for . Let be a bounded linear operator with its adjoint B. Let be a Lipschitzian accretive mapping with Lipschitz constant and . Let be a nonexpansive mapping with . Let be a sequence generated by the following algorithm:
where , and is the projection operator from into for . Suppose that
Then there exists such that
-
(a)
as ,
-
(b)
as .
Remark 4.1 In Theorems 3.1 and 4.1, the control coefficients α and β can be respectively replaced with the sequences and satisfying for some positive real number ε.
Remark 4.2 Obviously, all results in this paper are true if . They generalize and improve many results in the literature; see, for instance, [23, 24, 26–29].
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Acknowledgments
The first author was supported by the Candidate Foundation of Youth Academic Experts at Honghe University (2014HB0206); the second author was supported by Grant No. MOST 103-2115-M-017-001 of the Ministry of Science and Technology of the Republic of China.
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He, Z., Du, WS. On split common solution problems: new nonlinear feasible algorithms, strong convergence results and their applications. Fixed Point Theory Appl 2014, 219 (2014). https://doi.org/10.1186/1687-1812-2014-219
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DOI: https://doi.org/10.1186/1687-1812-2014-219