On split common solution problems: new nonlinear feasible algorithms, strong convergence results and their applications
© He and Du; licensee Springer. 2014
Received: 12 July 2014
Accepted: 26 September 2014
Published: 22 October 2014
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.
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 every .
For and , for all .
- (iii)For and ,(1.1)
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.
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
- (i)pseudocontractive ifor, equivalently,
- (ii)demicontractive if, for all and ,
- (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
nonexpansive if it is Lipschitzian with ;
contractive if it is Lipschitzian with .
It is well known that any Hilbert space satisfies Opial’s condition.
Definition 2.1 (see )
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 .
The following concept of zero-demiclosedness was introduced as follows.
Definition 2.2 (see [, 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 , but we give the proof for the sake of completeness and the reader’s convenience.
Theorem 2.1 (see [, Proposition 2.4])
Let T be a mapping from K into K. Then T is zero-demiclosed if and only if is demiclosed at θ.
Let T be a nonexpansive mapping from H into itself. If there is a bounded sequence such that as , then T is zero-demiclosed.
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.
T is not quasi-nonexpansive.
T is a -demicontractive mapping. Moreover, T is also a pseudocontractive mapping.
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 )
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 .
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.
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 .
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.
T is discontinuous -demicontractive.
T is demiclosed at θ.
T is not pseudocontractive.
T is not quasi-nonexpansive.
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.
Proof We will show the conclusion by proceeding with the following steps.
and our desired result is proved.
Step 3. We show that is a nonempty closed convex set for any .
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 .
By the completeness of C, there exists such that as .
Hence we confirm . The proof is completed. □
Strong convergence algorithms for the split common solution problem for Lipschitzian pseudocontractive mappings and nonexpansive mappings (see Corollary 3.1 below).
Strong convergence algorithms for the split common solution problem for Lipschitzian pseudocontractive mappings and quasi-nonexpansive mappings (see Corollary 3.2 below).
Proof Since the mapping S is nonexpansive, it is 0-demicontractive. Hence the desired conclusion follows from Theorem 3.1 immediately by taking . □
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
where θ is the zero vector of H.
Strong convergence algorithms for the split common solution problem for Lipschitzian accretive mappings and demicontractive nonexpansive mappings (see Theorem 4.1 below).
Strong convergence algorithms for the split common solution problem for Lipschitzian accretive mappings and nonexpansive mappings (see Corollary 4.1 below).
Strong convergence algorithms for the split common solution problem for Lipschitzian accretive mappings and quasi-nonexpansive mappings (see Corollary 4.2 below).
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.
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.
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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