Multidirectional hybrid algorithm for the split common fixed point problem and application to the split common null point problem.

In this article, a new multidirectional monotone hybrid iteration algorithm for finding a solution to the split common fixed point problem is presented for two countable families of quasi-nonexpansive mappings in Banach spaces. Strong convergence theorems are proved. The application of the result is to consider the split common null point problem of maximal monotone operators in Banach spaces. Strong convergence theorems for finding a solution of the split common null point problem are derived. This iteration algorithm can accelerate the convergence speed of iterative sequence. The results of this paper improve and extend the recent results of Takahashi and Yao (Fixed Point Theory Appl 2015:87, 2015) and many others .

Introduction and preliminaries

Let and be two real Hilbert spaces, let and be nonempty closed, and convex subsets, let be a bounded linear operator. Then the split feasibility problem (Censor and Elfving 1994) is to find such that . Defining in the split feasibility problem, we see that is an inverse strongly monotone operator (Alsulami and Takahashi 2014), where is the adjoint operator of A and is the metric projection of onto Q. Furthermore, if is nonempty, thenwhere and is the metric projection of onto D. Using such results regarding nonlinear operators and fixed points, many authors have studied the split feasibility problem in Hilbert spaces; see, for instance, Alsulami and Takahashi (2014), Byrne et al. (2012), Censor and Segal (2009), Moudafi (2010), Takahashi et al. (2015). Recently, Takahashi (2014) and Takahashi (2015) extended an equivalent relation as in (1) in Hilbert spaces to Banach spaces and then obtained strong convergence theorems for finding a solution of the split feasibility problem in Banach spaces. Very recently, using the hybrid method by Nakajo and Takahashi (2003) in mathematical programming, Alsulami et al. (2015) proved strong convergence theorems for finding a solution of the split feasibility problem in Banach spaces; see also Ohsawa and Takahashi (2003), Solodov and Svaiter (2000). Takahashi (2015) also obtained a result for finding a solution of the split feasibility problem in Banach space from the idea of the shrinking projection method by Takahashi et al. (2008). Takahashi and Yao (2015) presented the following hybrid iteration algorithm in a Hilbert space H: for ,They proved the following strong convergence theorem:

Theorem TY

Let H be a Hilbert space and let F be a uniformly convex and smooth Banach space. Let be the duality mapping on F. Let A and B be maximal monotone operators of H into and F into . such that and , respectively. Let be the resolvent of A for and let be the metric resolvent of B for . Let be a bounded linear operator such that and let be the adjoint operator of T. Suppose that . Let and let be a sequence generated by (TY), where and satisfy the condition such that for some . Then converges strongly to a point .

In this article, a new multidirectional monotone hybrid iteration algorithm for finding a solution to the split common fixed point problem is presented for two countable families of quasi-nonexpansive mappings in Banach spaces. Strong convergence theorems are proved. The application of the result is to consider the split common null point problem of maximal monotone operators in Banach spaces. Strong convergence theorems for finding a solution of the split common null point problem are derived. This iteration algorithm can accelerate the convergence speed of iterative sequence.

Let E be a real Banach space with norm and let be the dual space of E. We denote the value of at by . A Banach space E is uniformly convex if for any two sequences and in E such that holds. A uniformly convex Banach space is reflexive.

The duality mapping J from E into is defined byfor every . Let . The norm of E is said to be Gateaux differentiable if for each , the limitexists. In the case, E is called smooth. We know that E is smooth if and only if J is a single- valued mapping of E into . We also know that E is reflexive if and only if J is surjective, and E is strictly convex if and only if J is one-to-one. Therefore, if E is a smooth, strictly convex and reflexive Banach space, then J is a single-valued bijection and in this case, the inverse mapping coincides with the duality mapping on . For more details, see Takahashi (2009) and Takahashi (2000).

Let C be a nonempty, closed and convex subset of a strictly convex and reflexive Banach space E. Then we know that for any , there exists a unique element such that for all . Putting , we call the metric projection of E onto C.

Definition 1

Let E be a metric space, let be a mapping with the domain D(T) and the range R(T). The mapping T is said to be quasi-nonexpansive ifwhere F(T) is the nonempty fixed point set of T.

Definition 2

Let E be a smooth Banach space, let be a mapping with the domain D(T) and the range R(T). The mapping S is said to be second-type quasi-nonexpansive, ifwhere F(S) is the nonempty fixed point set of T.

Definition 3

Let E, F be two normed spaces and T be a linear operator from E into F. The adjoint operator of T is defined bywhere and are the adjoint spaces of E and F, respectively.

The adjoint spaces and adjoint operators are very important in the theory of functional analysis and applications. Not only is it an important theoretical subject but it is also a very useful tool in the functional analysis and topological theory.

Definition 4

Let E be a Banach space, let C be a nonempty, closed, and convex subset of E. Let be sequence of mappings from C into itself with nonempty common fixed point set . The is said to be uniformly closed if for any convergent sequence such that as , the limit of belong to F.

Main results

Lemma 5

Let H be a Hilbert space, let C be a closed convex subset of H, and let be a uniformly closed family of countable quasi-nonexpansive mappings from C into itself. Then the common fixed point set F is closed and convex.

Proof

Let and as , we haveas . Since is uniformly closed, we know that , therefore F is closed. Next we show that F is also convex. For any , let for any , we havefor all n. This implies , therefore F is convex. This completes the proof.

Lemma 6

Let E be a smooth Banach space, let C be a closed convex subset of E, and let be a uniformly closed family of countable second-type quasi-nonexpansive mappings from C into itself. Then the common fixed point set F is closed and convex.

Let and as , we haveas . Since is uniformly closed, we know that , therefore F is closed. Next we show that F is also convex. For any , let for any , we haveFrom the two inequalities given above, we have thatwhich impliesTherefore , that is , so that Therefore F is convex. This completes the proof.

Lemma 7

(Alber 1996) Let H be a Hilbert space, let C be a nonempty closed convex subset of H and let Then

Next, we present a new hybrid algorithm so-called the multidirectional hybrid algorithm for finding the common fixed point of a uniformly closed family of countable quasi-nonexpansive mappings and a uniformly closed family of countable second-type quasi-nonexpansive mappings.

Theorem 8

Let H be a Hilbert space and let E be a uniformly convex and smooth Banach space. Let J be the duality mapping on E. Let be a uniformly closed family of countable quasi-nonexpansive mappings with the nonempty common fixed point set and be a uniformly closed family of countable second-type quasi-nonexpansive mappings with the nonempty common fixed point sets . Suppose that . Let be a bounded linear operator such that and let be the adjoint operator of T. Let and let and be two sequences generated by where satisfy the condition such that for some constants a, b and is a constant. Then the following conclusions hold:

(1) and converge strongly to a point ;

(2) the limits ;

(3) .

It is not hard to see that, is closed and convex for all . Let us show that, for all . For any , we haveSo, , which implies that for all .

Let for all . Since F is nonempty, closed, and convex, there exist such thatThis means that is bounded for all .

From and , we have thatfor all . This implies that is bounded and nondecreasing for all . Then there exist the limits of . PutOn the other hand, , by using Lemma 7, we have, for any positive integer m, thatSo that is Cauchy sequences in C for all , therefore there exit two points such thatThat isThereforeSince , we havewhich impliesFrom (2), we have, for any , thatas . This impliesSinceand the sequence is uniformly closed, so thatThat isOn the other hand, fromwe haveThis together with (3) implies thatSinceand the sequence is uniformly closed, so thatFrom above two hands, we have .

Finally, we prove . From Lemma 7, we haveOn the other hand, since and , for all n. Also from Lemma 7, we haveSinceCombining (4), (5) and (6), we know that . Therefore, it follows from the uniqueness of that . This completes the proof.

By using Theorem  8 and setting , we can get the following result.

Theorem 9

Let H be a Hilbert space and let E be a uniformly convex and smooth Banach space. Let J be the duality mapping on E. Let be a uniformly closed family of countable quasi-nonexpansive mappings with the nonempty common fixed point set and be a uniformly closed family of countable second-type quasi-nonexpansive mappings with the nonempty common fixed point sets . Suppose that . Let be a bounded linear operator such that and let be the adjoint operator of T. Let and let be a sequence generated by where satisfy the condition such that for some constants a, b. Then converges strongly to a point .

Application for common null point problem

Let E be a Banach space, let A be a multi-valued operator from E to with domain and range . An operator A is said to be monotone iffor each and . A monotone operator A is said to be maximal if it’s graph is not properly contained in the graph of any other monotone operator. We know that if A is a maximal monotone operator, then is closed and convex. The following result is also well-known.

Theorem 10

(Rockafellar 1970). Let E be a reflexive, strictly convex and smooth Banach space and let A be a monotone operator from E to . Then A is maximal if and only if . for all .

Let E be a reflexive, strictly convex and smooth Banach space, and let A be a maximal monotone operator from E to . Using Theorem  10 and strict convexity of E, we obtain that for every and , there exists a unique such thatThen we can define a single valued mapping by and such a is called the resolvent of A. We know that is a nonexpansive mapping and for all , see Takahashi (2000, 2009), Alber (1996).

Lemma 11

(Aoyama et al. 2009) Let E be a reflexive, strictly convex and smooth Banach space, and let A be a maximal monotone operator from E to . Then where is the resolvent of A.

From Lemma 11, we know that, is a second-type quasi-nonexpansive mapping, where is the resolvent of A with

Definition 12

Let E be a Banach space, let C be a nonempty, closed, and convex subset of E. Let be sequence of mappings from C into itself with nonempty common fixed point set . The is said to be uniformly weak closed if for any weak convergent sequence such that as , the weak limit of belong to F.

A uniformly weak closed family of countable quasi-nonexpansive mappings must be a uniformly closed family of countable quasi-nonexpansive mappings.

Theorem 13

Let , for some constant c, then is a uniformly weak closed family of countable quasi-nonexpansive mappings with the nonempty common fixed point sets .

It is well-known that, and is a family of countable nonexpansive mappings. Let be a sequence such that and . Since J is uniformly norm-to-norm continuous on bounded sets, we obtainIt follows fromand the monotonicity of A thatfor all and . Letting , we have for all and . Therefore from the maximality of A, we obtain . That is . This completes the proof.

Theorem 14

Let H be a Hilbert space and let F be a uniformly convex and smooth Banach space. Let be the duality mapping on F. Let A and B be maximal monotone operators of H into and F into such that and , respectively. Let be the resolvent of A for and let be the metric resolvent of B for . Let be a bounded linear operator such that and let be the adjoint operator of T. Suppose that . Let and let be a sequence generated by where satisfy the condition such that for some constants a, b, c. Then converges strongly to a point .

Let for all , then satisfy the all conditions of Theorem  8, andBy using Theorem  9, we obtain the conclusion of Theorem  14. This completes the proof.

Theorem 15

Let H be a Hilbert space and let F be a uniformly convex and smooth Banach space. Let be the duality mapping on F. Let A and B be maximal monotone operators of H into and F into such that and , respectively. Let be the resolvent of A for and let be the metric resolvent of B for . Let be a bounded linear operator such that and let be the adjoint operator of T. Suppose that . Let and let and be two sequences generated by where satisfy the condition such that for some constants a, b and is a constant. Then the following conclusions hold:

(1) and converge strongly to a point ;

(2) the limits ;

(3) .

Let for all , then satisfy the all conditions of Theorem  9, andBy using Theorem  8, we obtain the conclusion of Theorem  15. This completes the proof.

Examples

It is easy to see that, a uniformly weak closed family of countable quasi-nonexpansive mappings must be a uniformly closed family of countable quasi-nonexpansive mappings. Next we will give an example which is a uniformly closed family of countable quasi-nonexpansive mappings, but not a uniformly weak closed family of countable quasi-nonexpansive mappings.

Conclusion 16

Let H be a Hilbert space, be a sequence such that it converges weakly to a non-zero element and for any . Define a sequence of mappings as follows where and . Then is a uniformly closed family of countable quasi-nonexpansive mappings with the common fixed point set , but not a uniformly weak closed family of countable quasi-nonexpansive mappings.

It is obvious that, has a unique common fixed point 0. Next, we prove that, is uniformly closed. In fact that, for any strong convergent sequence such that and as , there exists sufficiently large nature number N such that , for any . Then for , it follows from that and hence . From the definition of , we haveso that is a uniformly closed family of countable quasi-nonexpansive mappings. Next, we prove the is not weak closed. Since converges weakly to andas , but is not a fixed point.

Conclusion

In the multidirectional iteration algorithm, the is a closed convex set, and for any . If we use one initial , the projection point belongs to the boundary of the . If we use N initials , the element belongs to the interior of the . In general, the distance is less than the distance , so the multidirectional iteration algorithm can accelerate the convergence speed of iterative sequence . We give a simple experimental example in the following.

Example

Let . Case 1, take only one initial , , then . Case 2, take two initials ,then . From the inequality “”, we can see that, the multidirectional iteration algorithm can accelerate the convergence speed of iterative sequence .