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On split common solution problems: new nonlinear feasible algorithms, strong convergence results and their applications
Fixed Point Theory and Applications volume 2014, Article number: 219 (2014)
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.
1 Introduction and preliminaries
Let K be a closed convex subset of a real Hilbert space H with the inner product \u3008\cdot ,\cdot \u3009 and the norm \parallel \cdot \parallel. The following inequalities are known and useful.

{\parallel x+y\parallel}^{2}\le {\parallel y\parallel}^{2}+2\u3008x,x+y\u3009;

{\parallel xy\parallel}^{2}={\parallel x\parallel}^{2}+{\parallel y\parallel}^{2}2\u3008x,y\u3009 for all x,y\in H;

{\parallel \alpha x+(1\alpha )y\parallel}^{2}=\alpha {\parallel x\parallel}^{2}+(1\alpha ){\parallel y\parallel}^{2}\alpha (1\alpha ){\parallel xy\parallel}^{2} for all x,y\in H and \alpha \in [0,1].
For each point x\in H, there exists a unique nearest point in K, denoted by {P}_{K}x, such that
The mapping {P}_{K} is called the metric projection from H onto K. It is well known that {P}_{K} has the following properties:

(i)
\u3008xy,{P}_{K}x{P}_{K}y\u3009\ge {\parallel {P}_{K}x{P}_{K}y\parallel}^{2} for every x,y\in H.

(ii)
For x\in H and z\in K, z={P}_{K}x\iff \u3008xz,zy\u3009\ge 0 for all y\in K.

(iii)
For x\in H and y\in K,
{\parallel y{P}_{K}x\parallel}^{2}+{\parallel x{P}_{K}x\parallel}^{2}\le {\parallel xy\parallel}^{2}.(1.1)
Let {H}_{1} and {H}_{2} be two Hilbert spaces. Let A:{H}_{1}\to {H}_{2} and {A}^{\ast}:{H}_{2}\to {H}_{1} be two bounded linear operators. {A}^{\ast} 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 {A}^{\ast} is an adjoint operator of A, then \parallel A\parallel =\parallel {A}^{\ast}\parallel. The symbols ℕ and ℝ are used to denote the sets of positive integers and real numbers, respectively.
Let {H}_{1} and {H}_{2} be two real Hilbert spaces. Let C be a closed convex subset of {H}_{1} and K be a closed convex subset of {H}_{2} . Let T:C\to C with \mathcal{F}(T)\ne \mathrm{\varnothing} and S:K\to K with \mathcal{F}(S)\ne \mathrm{\varnothing} be two mappings. Let A:{H}_{1}\to {H}_{2} 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 kdemicontractive 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 \mathcal{D}(T) and range \mathcal{R}(T) in a Hilbert space H. Recall that T is said to be

(i)
pseudocontractive if
\u3008TxTy,xy\u3009\le {\parallel xy\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in \mathcal{D}(T),or, equivalently,
{\parallel TxTy\parallel}^{2}\le {\parallel xy\parallel}^{2}+{\parallel (IT)x(IT)y\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in \mathcal{D}(T); 
(ii)
demicontractive if, for all x\in \mathcal{D}(T) and p\in \mathcal{F}(T),
\u3008Txp,xp\u3009\le {\parallel xp\parallel}^{2}
or, equivalently,

(iii)
kdemicontractive if there exists a constant k\in [0,1) such that
{\parallel Txp\parallel}^{2}\le {\parallel xp\parallel}^{2}+k{\parallel (IT)x\parallel}^{2}\phantom{\rule{1em}{0ex}}\text{for all}x\in \mathcal{D}(T)\text{and}p\in \mathcal{F}(T); 
(iv)
quasinonexpansive if it is 0demicontractive, that is,
\parallel Txp\parallel \le \parallel xp\parallel \phantom{\rule{1em}{0ex}}\text{for all}x\in \mathcal{D}(T)\text{and}p\in \mathcal{F}(T); 
(v)
Lipschitzian if there exists L>0 such that
\parallel TxTy\parallel \le L\parallel xy\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in \mathcal{D}(T); 
(vi)
nonexpansive if it is Lipschitzian with L=1;

(vii)
contractive if it is Lipschitzian with L<1.
A Banach space (X,\parallel \cdot \parallel ) is said to satisfy Opial’s condition if, for each sequence \{{x}_{n}\} in X which converges weakly to a point x\in X, 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 \{{x}_{n}\} which weakly converges to y, and if the sequence \{T{x}_{n}\} strongly converges to z, then Ty=z.
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 z=\theta, the zero vector of H; for more details, one can refer to [2].
The following concept of zerodemiclosedness 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 zerodemiclosed if \{{x}_{n}\} in K satisfying \parallel {x}_{n}T{x}_{n}\parallel \to 0 and {x}_{n}\rightharpoonup z\in K implies Tz=z.
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 zerodemiclosed if and only if IT is demiclosed at θ.

(b)
Let T be a nonexpansive mapping from H into itself. If there is a bounded sequence \{{x}_{n}\}\subset H such that \parallel {x}_{n}T{x}_{n}\parallel \to 0 as n\to 0, then T is zerodemiclosed.
Proof Obviously, the conclusion (a) holds. To see (b), since \{{x}_{n}\} is bounded, there is a subsequence \{{x}_{{n}_{k}}\}\subset \{{x}_{n}\} and z\in H such that {x}_{{n}_{k}}\rightharpoonup z. One can claim Tz=z. Indeed, if Tz\ne z, it follows from Opial’s condition that
which is a contradiction. So Tz=z and hence T is zerodemiclosed. □
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 H=\mathbb{R} with the absolutevalue norm \cdot  and C=[2,0]. Let T:C\to C be defined by
Then \mathcal{F}(T)=\{1\}. Since
we know that T is a \frac{1}{2}demicontractive mapping. However, due to
T is not quasinonexpansive.
Example B Let H=\mathbb{R} with the absolutevalue norm \cdot  and C=[\frac{1}{2},2]. Let T:C\to C be defined by
Then \mathcal{F}(T)=\{1\}. Since
T is a \frac{3}{4}demicontractive mapping. Moreover, T is also a pseudocontractive mapping.
Example C Let H=\mathbb{R} with the absolutevalue norm \cdot . Let T:H\to H be defined by
It is easy to see that
So T is continuous nonexpansive with \mathcal{F}(T)=\mathrm{\varnothing}.
The following example shows that there exists a continuous quasinonexpansive mapping which is not nonexpansive.
Example D (see [8])
Let H=\mathbb{R} with the absolutevalue norm \cdot  and C=[0,+\mathrm{\infty}). Define T:C\to C by
Obviously, \mathcal{F}(T)=\{2\}. It is easy to see that
and
Hence T is a continuous quasinonexpansive mapping but not nonexpansive.
The following example shows that there exists a demicontractive mapping which is neither pseudocontractive nor kdemicontractive for all k\in [0,1).
Example E Let H=\mathbb{R} with the absolutevalue norm \cdot . Let T:H\to H be defined by
Then \mathcal{F}(T)=\{1\}. Since
T is a demicontractive mapping. However, T is not a pseudocontractive mapping due to the fact that when x=3 and y=2.5, we have
It is easy to see that T is not a kdemicontractive mapping for all k\in [0,1).
The following example shows that there exists a discontinuous pseudocontractive mapping which is not a demicontractive mapping.
Example F Let H=\mathbb{R} with the absolutevalue norm \cdot . Let T:H\to H be defined by
Then \mathcal{F}(T)=\mathrm{\varnothing}. 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 kdemicontractive for all k\in [0,1).
Example G Let H=\mathbb{R} with the absolutevalue norm \cdot . Let T:H\to H be defined by
Then \mathcal{F}(T)=\{1\}. Since
T is a pseudocontractive mapping. It is easy to see that T is not a kdemicontractive mapping for all k\in [0,1).
The following example shows that there exists a discontinuous kdemicontractive mapping for some k\in [0,1) as well as being demiclosed at θ which is neither pseudocontractive nor quasinonexpansive.
Example H Let H=\mathbb{R} with the absolutevalue norm \cdot  and C=[2,0]. Let T:C\to C be defined by
Then the following statements hold.

(a)
T is discontinuous \frac{3}{4}demicontractive.

(b)
T is demiclosed at θ.

(c)
T is not pseudocontractive.

(d)
T is not quasinonexpansive.
Proof Clearly, \mathcal{F}(T)=\{1\}. Since
T is a discontinuous \frac{3}{4}demicontractive mapping and (a) is proved. Now, we verify (b). In fact, let \{{x}_{n}\}\subset [2,0] with {x}_{n}\to z and {x}_{n}T{x}_{n}\to 0 as n\to \mathrm{\infty}. If all {x}_{n}\in [1,0], we can prove Tz=z and z=1\in F(T) easily. If there exists a subsequence \{{x}_{{n}_{k}}\}\subset [2,1], then, from {x}_{n}T{x}_{n}\to 0 as n\to \mathrm{\infty}, we can find a subsequence \{{x}_{{n}_{{k}_{i}}}\} of \{{x}_{{n}_{k}}\} such that {x}_{{n}_{{k}_{i}}}\ne \frac{3}{2} for all i. Hence we have
which implies z=1\in \mathcal{F}(T). To see (c) and (d), note that
and
so T is neither pseudocontractive nor quasinonexpansive. 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 {H}_{1} and {H}_{2} be two real Hilbert spaces and {\theta}_{i} be the zero vector of {H}_{i} for i=1,2. Let C be a nonempty closed convex subset of {H}_{1} and A:{H}_{1}\to {H}_{2} be a bounded linear operator with its adjoint B. Let T:C\to C be a Lipschitzian pseudocontractive mapping with Lipschitz constant L>0 and \mathcal{F}(T)\ne \mathrm{\varnothing}, and let S:{H}_{2}\to {H}_{2} be a kdemicontractive mapping with \mathcal{F}(S)\ne \mathrm{\varnothing} which is demiclosed at {\theta}_{2}. Let {C}_{1}=C and \{{x}_{n}\} be a sequence generated by the following algorithm:
where 0<1\beta <\alpha <\frac{1}{2\sqrt{1+{L}^{2}}}, \xi \in (0,\frac{1k}{{\parallel B\parallel}^{2}}) and {P}_{{C}_{n}} is the projection operator from {H}_{1} into {C}_{n} for n\in \mathbb{N}. Suppose that
Then there exists q\in \mathrm{\Omega} such that

(a)
{x}_{n}\to q as n\to \mathrm{\infty},

(b)
A{x}_{n}\to Aq as n\to \mathrm{\infty}.
Proof We will show the conclusion by proceeding with the following steps.
Step 1. For any p\in \mathrm{\Omega}, we prove
Indeed, since
and
we get
and our desired result is proved.
Step 2. We prove
For any n\in \mathbb{N}, by (3.1), we have
Since \alpha +\beta >1 and \alpha <\frac{1}{2\sqrt{1+{L}^{2}}}, from (3.4), we have {\parallel {z}_{n}p\parallel}^{2}\le {\parallel {x}_{n}p\parallel}^{2}, or, equivalently,
Step 3. We show that {C}_{n} is a nonempty closed convex set for any n\in \mathbb{N}.
For any p\in \mathrm{\Omega}, by taking into account (3.2) and (3.5), we obtain
So we know \mathrm{\Omega}\subset {C}_{n} and hence {C}_{n}\ne \mathrm{\varnothing} for all n\in \mathbb{N}. It is easy to verify that {C}_{n} is closed and convex for all n\in \mathbb{N}.
Step 4. We prove that \{{x}_{n}\} is a Cauchy sequence in C and {x}_{n}\to q as n\to \mathrm{\infty} for some q\in C.
Since \mathrm{\Omega}\subset {C}_{n+1}\subset {C}_{n} and {x}_{n+1}={P}_{{C}_{n+1}}({x}_{1})\subset {C}_{n}, we get
and
which show that \{{x}_{n}\} is bounded and \{\parallel {x}_{n}{x}_{1}\parallel \} is nondecreasing in [0,\mathrm{\infty}). So
exists. For any m,n\in \mathbb{N} with m>n, from {x}_{m}={P}_{{C}_{m}}({x}_{1})\subset {C}_{n} and (1.1), we have
Inequality (3.6) implies
So \{{x}_{n}\} is a Cauchy sequence. Clearly,
By the completeness of C, there exists q\in C such that {x}_{n}\to q as n\to \mathrm{\infty}.
Step 5. Finally, we show that the following hold:

(i)
q\in \mathrm{\Omega},

(ii)
A{x}_{n}\to Aq as n\to \mathrm{\infty}.
For any n\in \mathbb{N}, since {x}_{n+1}={P}_{{C}_{n+1}}({x}_{1})\in {C}_{n+1}\subset {C}_{n}, 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 {x}_{n}\to q as n\to \mathrm{\infty}, from (3.12) and the continuity of the norm \parallel \cdot \parallel and the Lipschitzian pseudocontractive mapping T, we can deduce that Tq=q, namely q\in \mathcal{F}(T). On the other hand, from (3.2) and (3.11), we have
which yields that
Since the kdemicontractive mapping S is demiclosed at {\theta}_{2}, taking into account {x}_{n}\to q, A{x}_{n}\to Aq, \parallel {z}_{n}{x}_{n}\parallel \to 0 and (3.13), we have
and
Hence we confirm q\in \mathrm{\Omega}. 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 quasinonexpansive mappings (see Corollary 3.2 below).
Corollary 3.1 Let {H}_{1} and {H}_{2} be two real Hilbert spaces and {\theta}_{i} be the zero vector of {H}_{i} for i=1,2. Let C be a nonempty closed convex subset of {H}_{1} and A:{H}_{1}\to {H}_{2} be a bounded linear operator with its adjoint B. Let T:C\to C be a Lipschitzian pseudocontractive mapping with Lipschitz constant L>0 and \mathcal{F}(T)\ne \mathrm{\varnothing}, and let S:{H}_{2}\to {H}_{2} be a nonexpansive mapping with \mathcal{F}(S)\ne \mathrm{\varnothing}. Let {C}_{1}=C and \{{x}_{n}\} be a sequence generated by the following algorithm:
where 0<1\beta <\alpha <\frac{1}{2\sqrt{1+{L}^{2}}}, \xi \in (0,\frac{1}{{\parallel B\parallel}^{2}}) and {P}_{{C}_{n}} is the projection operator from {H}_{1} into {C}_{n} for n\in \mathbb{N}. Suppose that
Then there exists q\in \mathrm{\Omega} such that

(a)
{x}_{n}\to q as n\to \mathrm{\infty},

(b)
A{x}_{n}\to Aq as n\to \mathrm{\infty}.
Proof Since the mapping S is nonexpansive, it is 0demicontractive. Hence the desired conclusion follows from Theorem 3.1 immediately by taking k=0. □
Corollary 3.2 Let {H}_{1} and {H}_{2} be two real Hilbert spaces and {\theta}_{i} be the zero vector of {H}_{i} for i=1,2. Let C be a nonempty closed convex subset of {H}_{1} and A:{H}_{1}\to {H}_{2} be a bounded linear operator with its adjoint B. Let T:C\to C be a Lipschitzian pseudocontractive mapping with Lipschitz constant L>0 and \mathcal{F}(T)\ne \mathrm{\varnothing}, and let S:{H}_{2}\to {H}_{2} be a quasinonexpansive mapping with \mathcal{F}(S)\ne \mathrm{\varnothing} which is demiclosed at {\theta}_{2}. Let {C}_{1}=C and \{{x}_{n}\} be a sequence generated by the following algorithm:
where 0<1\beta <\alpha <\frac{1}{2\sqrt{1+{L}^{2}}}, \xi \in (0,\frac{1}{{\parallel B\parallel}^{2}}) and {P}_{{C}_{n}} is the projection operator from {H}_{1} into {C}_{n} for n\in \mathbb{N}. Suppose that
Then there exists q\in \mathrm{\Omega} such that

(a)
{x}_{n}\to q as n\to \mathrm{\infty},

(b)
A{x}_{n}\to Aq as n\to \mathrm{\infty}.
Example 3.1 Let {H}_{1}=\mathbb{R} with the absolutevalue norm \cdot . Let {H}_{2}={[\frac{1}{\sqrt{2}},\sqrt{2}]}^{2} with the norm \parallel \alpha \parallel ={({a}_{1}^{2}+{a}_{2}^{2})}^{\frac{1}{2}} for \alpha =({a}_{1},{a}_{2})\in {H}_{2} and the inner product \u3008\alpha ,\beta \u3009={\sum}_{i=1}^{2}{a}_{i}{b}_{i} for \alpha =({a}_{1},{a}_{2}) and \beta =({b}_{1},{b}_{2})\in {H}_{2}. Let A:{H}_{1}\to {H}_{2} be defined by Ax=(x,x) for x\in \mathbb{R}. Then A is a bounded linear operator with its adjoint operator Bz={z}_{1}+{z}_{2} for z=({z}_{1},{z}_{2})\in {H}_{2}. Clearly, \parallel A\parallel =\parallel B\parallel =\sqrt{2}. Let C=[\frac{1}{\sqrt{2}},\sqrt{2}]. Let T:C\to C and S:{H}_{2}\to {H}_{2} be defined by
and
respectively. It is easy to see that

\mathcal{F}(T)=\{1\};

\mathcal{F}(S)=\{(1,1)\};

\mathrm{\Omega}=\{p\in \mathcal{F}(T):Ap\in \mathcal{F}(S)\}=\{1\}\ne \mathrm{\varnothing};

T is a Lipschitzian pseudocontractive mapping with Lipschitz constant L=\sqrt{2};

T and S both are \frac{3}{4}demicontractive mappings.
By using algorithm (3.1) with 0<1\beta <\alpha <\frac{1}{2\sqrt{3}} and \xi \in (0,\frac{1}{8}), we can verify {x}_{n}\to 1 and A{x}_{n}\to A(1)=(1,1)\in \mathcal{F}(S) as n\to \mathrm{\infty}.
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 U:C\to C is said to be accretive if
Obviously, U:C\to C is accretive if and only if IU:C\to C 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 quasinonexpansive mappings (see Corollary 4.2 below).
Theorem 4.1 Let {H}_{1} and {H}_{2} be two real Hilbert spaces and {\theta}_{i} be the zero vector of {H}_{i} for i=1,2. Let A:{H}_{1}\to {H}_{2} be a bounded linear operator with its adjoint B and U:{H}_{1}\to {H}_{1} be a Lipschitzian accretive mapping with Lipschitz constant L>0 and {U}^{1}({\theta}_{1})\ne \mathrm{\varnothing}. Let S:{H}_{2}\to {H}_{2} be a kdemicontractive mapping with \mathcal{F}(S)\ne \mathrm{\varnothing} which is demiclosed at {\theta}_{2}. Let \{{x}_{n}\} be a sequence generated by the following algorithm:
where 0<1\beta <\alpha <\frac{1}{2\sqrt{1+{L}^{2}}}, \xi \in (0,\frac{1k}{{\parallel B\parallel}^{2}}) and {P}_{{C}_{n}} is the projection operator from {H}_{1} into {C}_{n} for n\in \mathbb{N}. Suppose that
Then there exists q\in \mathrm{\Omega} such that

(a)
{x}_{n}\to q as n\to \mathrm{\infty},

(b)
A{x}_{n}\to Aq as n\to \mathrm{\infty}.
Proof Let {C}_{1}={H}_{1}. Then the iterative process (4.1) can be rewritten as follows:
Set T:=IU, then \mathcal{F}(T)={U}^{1}({\theta}_{1}) and T is a Lipschitzian pseudocontractive mapping with Lipschitz constant 1+L. Therefore the desired conclusion follows from Theorem 3.1 immediately. □
The following interesting results are immediate from Theorem 4.1.
Corollary 4.1 Let {H}_{1} and {H}_{2} be two real Hilbert spaces and {\theta}_{i} be the zero vector of {H}_{i} for i=1,2. Let A:{H}_{1}\to {H}_{2} be a bounded linear operator with its adjoint B and U:{H}_{1}\to {H}_{1} be a Lipschitzian accretive mapping with Lipschitz constant L>0 and {U}^{1}({\theta}_{1})\ne \mathrm{\varnothing}. Let S:{H}_{2}\to {H}_{2} be a quasinonexpansive mapping with \mathcal{F}(S)\ne \mathrm{\varnothing} which is demiclosed at {\theta}_{2}. Let \{{x}_{n}\} be a sequence generated by the following algorithm:
where 0<1\beta <\alpha <\frac{1}{2\sqrt{1+{L}^{2}}}, \xi \in (0,\frac{1}{{\parallel B\parallel}^{2}}) and {P}_{{C}_{n}} is the projection operator from {H}_{1} into {C}_{n} for n\in \mathbb{N}. Suppose that
Then there exists q\in \mathrm{\Omega} such that

(a)
{x}_{n}\to q as n\to \mathrm{\infty},

(b)
A{x}_{n}\to Aq as n\to \mathrm{\infty}.
Corollary 4.2 Let {H}_{1} and {H}_{2} be two real Hilbert spaces and {\theta}_{i} be the zero vector of {H}_{i} for i=1,2. Let A:{H}_{1}\to {H}_{2} be a bounded linear operator with its adjoint B. Let U:{H}_{1}\to {H}_{1} be a Lipschitzian accretive mapping with Lipschitz constant L>0 and {U}^{1}({\theta}_{1})\ne \mathrm{\varnothing}. Let S:{H}_{2}\to {H}_{2} be a nonexpansive mapping with \mathcal{F}(S)\ne \mathrm{\varnothing}. Let \{{x}_{n}\} be a sequence generated by the following algorithm:
where 0<1\beta <\alpha <\frac{1}{2\sqrt{1+{L}^{2}}}, \xi \in (0,\frac{1}{{\parallel B\parallel}^{2}}) and {P}_{{C}_{n}} is the projection operator from {H}_{1} into {C}_{n} for n\in \mathbb{N}. Suppose that
Then there exists q\in \mathrm{\Omega} such that

(a)
{x}_{n}\to q as n\to \mathrm{\infty},

(b)
A{x}_{n}\to Aq as n\to \mathrm{\infty}.
Remark 4.1 In Theorems 3.1 and 4.1, the control coefficients α and β can be respectively replaced with the sequences \{{\alpha}_{n}\} and \{{\beta}_{n}\} satisfying 0<\epsilon <1{\beta}_{n}<{\alpha}_{n}<\frac{1}{2\sqrt{1+{L}^{2}}} for some positive real number ε.
Remark 4.2 Obviously, all results in this paper are true if {H}_{1}={H}_{2}. 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 1032115M017001 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/10.1186/168718122014219
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DOI: https://doi.org/10.1186/168718122014219
Keywords
 Lipschitzian
 demicontractive mapping
 pseudocontractive mapping
 quasinonexpansive mapping
 nonexpansive mapping
 split common solution problem (SCSP)
 iterative algorithm
 strong convergence theorem