Split Common Fixed Point Problem for a Class of Total Asymptotic Pseudocontractions

Abstract

We study the split common fixed point problem (SCFP) for a class of total asymptotically pseudocontractive mappings. We obtain some important properties of our class of mappings including the demiclosedness property and the closedness and convexity of the fixed point set. We then propose an algorithm and prove weak and strong convergence theorems for the approximation of solutions of the SCFP for certain class of these mappings.

1. Introduction

Let and be two real Hilbert spaces, and nonempty closed convex subsets of and , respectively, and a bounded linear operator. The split feasibility problem (SFP) (see, e.g., [1–9]) is If , a singleton, we have the convexly constrained linear inverse problem (CCLIP): The split feasibility problem (SFP) has various important applications in several disciplines (see, e.g., [2–9]).

Let and be mappings such that and . The split common fixed point problem (SCFP) for and is to find a point such that and . In sequel we use to denote the set of solutions of (SCFP); that is,

Definition 1. Let be a real Hilbert space, and let be a nonempty closed convex subset of .
A mapping is said to be ()-total asymptotically -strictly pseudocontractive (see, e.g., [3]) if there exist a constant , a continuous and strictly increasing function with , and sequences and with and such that, for all , is said to be ()-total asymptotically pseudocontractive if in (4).
Observe that if and in (4), we obtain where and Mappings satisfying (5) are the well-known class of -strictly asymptotically pseudocontractive mappings, while mappings satisfying (5) for are called asymptotically pseudocontractive mappings. These classes of mappings are generalizations of the well-known important class of asymptotically nonexpansive mappings introduced by Goebel and Kirk [10] (i.e., mappings which satisfy , and for some sequence with ).

We consider the following examples.

Example 2 (see [10]). In the real Hilbert space , let denote the closed unit ball, and define by , where is a real sequence in such that Then is asymptotically nonexpansive and hence asymptotically strictly pseudocontractive. It follows that is total asymptotically strictly pseudocontractive and hence total asymptotically pseudocontractive.

Example 3 (see [6, 7]). Let be an orthogonal subspace of the Euclidean space , and for each , define by Then is asymptotically nonexpansive (see, e.g., [7]) and hence total asymptotically strictly pseudocontractive (see, e.g., [6]) and hence it is total asymptotically pseudocontractive.

Example 4 (see [11]). Let denote the reals with the usual norm, , and define by It is shown in [11] that and , . It is easy to observe that, for all integers , we have Thus we easily obtain Hence is asymptotically pseudocontractive and hence total asymptotically pseudocontractive. Furthermore, , so that is uniformly -Lipschitzian.

The following is an example of a total asymptotically pseudocontractive map which is not total asymptotically strictly pseudocontractive.

Example 5. Let denote the reals with the usual norm, , and define by Then for all integers and for all we have Thus and hence is total asymptotically pseudocontractive. is not total asymptotically strictly pseudocontractive since, in every real Hilbert space , every total asymptotically strictly pseudocontractive mapping satisfies In [3] the authors studied the split common fixed point problem (SCFP) for a class of total asymptotically strictly pseudocontractive mappings in real Hilbert spaces. They proposed an algorithm and proved weak and strong convergence theorems for finding solutions of SCFP for the class of mappings studied.

It is our purpose in this work to study the split common fixed point problem (SCFP) for a class of total asymptotically pseudocontractive mappings which is much more general than the class of mappings studied in [3]. We obtain some important properties of our class of mappings including the demiclosedness property and then propose an algorithm and prove weak and strong convergence theorems for the approximation of solutions of the SCFP.

2. Preliminaries

In what follows, we will need the following.

Let be a real Banach space and a nonempty closed convex subset of . A mapping is said to be if, for any bounded sequence with , there exists a subsequence such that converges strongly to some point .

is said to be uniformly -Lipschitzian if there exists a constant , such that, for all , is said to have the Opial property if, for any sequence with , we have It is well known that every Hilbert space satisfies the Opial condition.

Lemma 6 (see [10]). Let be a real Hilbert space. If is a sequence in weakly convergent to , then

Lemma 7 (see [11]). Let , , and be sequences of nonnegative real numbers satisfying the inequality If and , then exists. If in addition has a subsequence which converges to zero, then

3. Main Results

We start with the following important properties of ()-total asymptotically pseudocontractive mappings.

Proposition 8. Let be a real Hilbert space, a nonempty closed convex subset of , and a uniformly -Lipschitzian -total asymptotically pseudocontractive mapping with . Let ; and set . Then for each and each , the following equivalent inequalities hold:

Proof. For arbitrary and we have It follows from (20) that

Proposition 9. Let be a real Hilbert space, and let be a nonempty closed convex subset of . Let be a uniformly -Lipschitzian -total asymptotically pseudocontractive mapping with , . Then (i) is demiclosed at 0;(ii) is closed and convex.

Proof. (i) Let be a sequence in which converges weakly to and converges strongly to 0. We prove that . Since converges weakly, it is bounded. For each , define by Observe that, for arbitrary but fixed integer , we have Set , where Then we obtain HenceFurthermore, Since is bounded we also obtain for some .
From Lemma 6, we obtain , . Thus , , and hence Observe that It follows thatand thusIt follows thatand henceIt now follows that . Since is continuous, we have that , and hence .
(ii) Let be such that . We prove that . ConsiderHence , and is closed.
Let and let be arbitrary. Set . We prove that . Observe that and . Set where Then , and . Observe that Observe thatwhere .
Similarly, Thusand it follows that Hence . Observe that Thus and hence . Since is continuous we have Thus

We now introduce our algorithm and prove weak and strong convergence theorems for solving the split common fixed point problem for a class of total asymptotically pseudocontractive mappings in real Hilbert spaces.

Let and be two real Hilbert spaces, a bounded linear operator, a uniformly -Lipschitzian -total asymptotically pseudocontractive mapping, with , and a uniformly -Lipschitzian -total asymptotically pseudocontractive mapping, with . We now introduce the following iterative algorithm for approximating solutions of split common fixed point problem: such that and (i.e., and .

For arbitrary , the sequence is given bywhere and are suitable sequences in and is a suitable parameter in . We prove the following.

Theorem 10. Let and be two real Hilbert spaces, a bounded linear operator, a uniformly -Lipschitzian -total asymptotically pseudocontractive mapping, with , and a uniformly -Lipschitzian -total asymptotically pseudocontractive mapping, with ; let , , , and such that there exist two positive real constants and such that , . Let and ; let and be sequences of real numbers satisfying the condition:where and . Let . Then for arbitrary , the sequence generated from by (44) converges weakly to a point in .
If in addition is semicompact, then and converge strongly to a point in .

Proof. We will divide the proof into four steps.
Step 1. We prove that, for each , the following limits exist and Since is a continuous and an increasing function, it follows that , and by hypothesis . In either case, we can obtain that Let , Then, for arbitrary , we obtainwhere and Observe that Using (48) in (50) we obtain Observe also that if we let , then, But Furthermore, Since , we set and in (19) to obtain Substituting (55) into (54) yields and it follows from (52) that where and .
Substituting (57) in (51) we obtain where and .
It follows from (58) and condition (45) that The conditions and imply that and . It now follows from Lemma 7 that exists. It now follows from (58) that Consequently,Hence This together with (61) implies that Since exists, it follows from (52) and (64) that exists andStep 2. We prove thatFrom (44) we obtainObserve that Using (68) in (67) we obtainand it follows from (60) and (61) thatSimilarly, it follows from (44), (61), and (70) thatStep 3. We prove that In fact, from (60), we haveSince is uniformly -Lipschitzian, it follows from (71) and (73) that Similarly, from (64), we obtain Since is uniformly -Lipschitzian, it follows from (70) and (75) that This implies thatStep 4. We prove that the sequences and converge weakly to .
Observe that since is bounded, then there exists a subsequence of which converges weakly to a point . Since , we obtain It follows from Proposition 9 that .
Furthermore, from (44) and (61), we obtain Since is linear and bounded, we obtain , and it follows from (72) that The demiclosedness of at zero now yields that , and thus Since every Hilbert space is an Opial space and has a subsequence which converges weakly to a point , it follows from a standard argument that converges weakly to .
If is semicompact, then since is bounded and , we have that there exists a subsequence of which converges strongly to a point . Since converges weakly to , we have . Thus and it follows from Lemma 7 that (and hence ) converges strongly to .

Corollary 11. Let and be two real Hilbert spaces, a bounded linear operator, a uniformly -Lipschitzian asymptotically pseudocontractive mapping with sequence such that and , and a uniformly -Lipschitzian asymptotically pseudocontractive mapping with sequence such that and ; let and . Let and be sequences of real numbers satisfying the condition: where and . Let . Then for arbitrary the sequence generated from by (44) converges weakly to a point in .
If in addition is semicompact, then and converge strongly to a point in .

Example 12. Let and be as in Example 2, and let and be as in Example 3. Then , and . Furthermore, and are nonempty closed convex subsets of and , respectively. Define by for each . Then is a bounded linear operator with adjoint operator for . Furthermore, . Thus using algorithm (44) with and satisfying the condition and , it follows from Theorem 10 that and .

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final paper.