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Split feasibility problems for total quasiasymptotically nonexpansive mappings
Fixed Point Theory and Applications volume 2012, Article number: 151 (2012)
Abstract
The purpose of this paper is to propose an algorithm for solving the split feasibility problems for total quasiasymptotically nonexpansive mappings in infinitedimensional Hilbert spaces. The results presented in the paper not only improve and extend some recent results of Moudafi [Nonlinear Anal. 74:40834087, 2011; Inverse Problem 26:055007, 2010], but also improve and extend some recent results of Xu [Inverse Problems 26:105018, 2010; 22:20212034, 2006], Censor and Segal [J. Convex Anal. 16:587600, 2009], Censor et al. [Inverse Problems 21:20712084, 2005], Masad and Reich [J. Nonlinear Convex Anal. 8:367371, 2007], Censor et al. [J. Math. Anal. Appl. 327:12441256, 2007], Yang [Inverse Problem 20:12611266, 2004] and others.
MSC:47J05, 47H09, 49J25.
1 Introduction
Throughout this paper, we always assume that {H}_{1}, {H}_{2} are real Hilbert spaces, ‘→’, ‘⇀’ denote strong and weak convergence, respectively, and F(T) is a fixed point set of a mapping T.
The split feasibility problem (SFP) in finitedimensional spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph and radiation therapy treatment planning [3–5]. The split feasibility problem in an infinitedimensional real Hilbert space can be found in [2, 4, 6–10].
The purpose of this paper is to introduce and study the following split feasibility problem for total quasiasymptotically nonexpansive mappings in the framework of infinitedimensional real Hilbert spaces:
where A:{H}_{1}\to {H}_{2} is a bounded linear operator, S:{H}_{1}\to {H}_{1} and T:{H}_{2}\to {H}_{2} are mappings; C:=F(S) and Q:=F(T). In the sequel, we use Γ to denote the set of solutions of (SFP)(1.1), i.e.,
2 Preliminaries
We first recall some definitions, notations and conclusions which will be needed in proving our main results.
Let E be a Banach space. A mapping T:E\to E is said to be demiclosed at origin if for any sequence \{{x}_{n}\}\subset E with {x}_{n}\rightharpoonup {x}^{\ast} and \parallel (IT){x}_{n}\parallel \to 0, {x}^{\ast}=T{x}^{\ast}.
A Banach space E is said to have the Opial property, if for any sequence \{{x}_{n}\} with {x}_{n}\rightharpoonup {x}^{\ast},
Remark 2.1 It is well known that each Hilbert space possesses the Opial property.
Definition 2.2 Let H be a real Hilbert space.

(1)
A mapping G:H\to H is said to be a (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total quasiasymptotically nonexpansive mapping if F(G)\ne \mathrm{\varnothing}; and there exist nonnegative real sequences \{{\nu}_{n}\}, \{{\mu}_{n}\} with {\nu}_{n}\to 0 and {\mu}_{n}\to 0 and a strictly increasing continuous function \zeta :{\mathcal{R}}^{+}\to {\mathcal{R}}^{+} with \zeta (0)=0 such that for each n\ge 1,
{\parallel p{G}^{n}x\parallel}^{2}\le {\parallel px\parallel}^{2}+{\nu}_{n}\zeta (\parallel px\parallel )+{\mu}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall}p\in F(G),x\in H.(2.1)
Now, we give an example of total quasiasymptotically nonexpansive mapping.
Let C be a unit ball in a real Hilbert space {l}^{2}, and let T:C\to C be a mapping defined by
where \{{a}_{i}\} is a sequence in (0, 1) such that {\prod}_{i=2}^{\mathrm{\infty}}{a}_{i}=\frac{1}{2}.
It is proved in Goebal and Kirk [17] that

(i)
\parallel TxTy\parallel \le 2\parallel xy\parallel, \mathrm{\forall}x,y\in C;

(ii)
\parallel {T}^{n}x{T}^{n}y\parallel \le 2{\prod}_{j=2}^{n}{a}_{j}\parallel xy\parallel, \mathrm{\forall}x,y\in C, \mathrm{\forall}n\ge 2.
Denote by {k}_{1}^{\frac{1}{2}}=2, {k}_{n}^{\frac{1}{2}}=2{\prod}_{j=2}^{n}{a}_{j}, n\ge 2, then
Letting {\nu}_{n}=({k}_{n}1), \mathrm{\forall}n\ge 1, \zeta (t)=t, \mathrm{\forall}t\ge 0 and \{{\mu}_{n}\} be a nonnegative real sequence with {\mu}_{n}\to 0, from (i) and (ii), \mathrm{\forall}x,y\in C, n\ge 1, we have
Again, since 0\in C and 0\in F(T), this implies that F(T)\ne \mathrm{\varnothing}. From (2.2), we have
This shows that the mapping T defined as above is a total quasiasymptotically nonexpansive mapping.

(2)
A mapping G:H\to H is said to be (\{{k}_{n}\})quasiasymptotically nonexpansive if F(G)\ne \mathrm{\varnothing}; and there exists a sequence \{{k}_{n}\}\subset [1,\mathrm{\infty}) with {k}_{n}\to 1 such that for all n\ge 1,
{\parallel p{G}^{n}x\parallel}^{2}\le {k}_{n}{\parallel px\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}p\in F(G),x\in H.(2.4) 
(3)
A mapping G:H\to H is said to be quasinonexpansive if F(G)\ne \mathrm{\varnothing} such that
\parallel pGx\parallel \le \parallel px\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}p\in F(G),x\in H.(2.5)
Remark 2.3 It is easy to see that every quasinonexpansive mapping is a (\{1\})quasiasymptotically nonexpansive mapping and every \{{k}_{n}\}quasiasymptotically nonexpansive mapping is a (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total quasiasymptotically nonexpansive mapping with \{{\nu}_{n}={k}_{n}1\}, \{{\mu}_{n}=0\} and \zeta (t)={t}^{2}, t\ge 0.
Definition 2.4 (1) A mapping G:H\to H is said to be uniformly LLipschitzian if there exists a constant L>0 such that

(2)
A mapping G:H\to H is said to be semicompact if for any bounded sequence \{{x}_{n}\}\subset H with {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}G{x}_{n}\parallel =0, there exists a subsequence \{{x}_{{n}_{i}}\}\subset \{{x}_{n}\} such that {x}_{{n}_{i}} converges strongly to some point {x}^{\ast}\in H.
Proposition 2.5 Let G:H\to H be a (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta )total quasiasymptotically nonexpansive mapping. Then for each q\in F(G) and for each x\in H, the following inequalities are equivalent: for each n\ge 1
Proof (I) (2.1) ⇔ (2.6) In fact, since
from (2.1) we have that
Simplifying it, inequality (2.6) is obtained.
Conversely, from (2.6) the inequality (2.1) can be obtained immediately.

(II)
(2.6) ⇔ (2.7) In fact, since
\begin{array}{rcl}\u3008x{G}^{n}x,xq\u3009& =& \u3008x{G}^{n}x,x{G}^{n}x+{G}^{n}xq\u3009\\ =& {\parallel x{G}^{n}x\parallel}^{2}+\u3008x{G}^{n}x,{G}^{n}xq\u3009\end{array}
it follows from (2.6) that
Simplifying it, the inequality (2.7) is obtained.
Conversely, from (2.7) the inequality (2.6) can be obtained immediately.
This completes the proof of Proposition 2.5. □
Lemma 2.6 [11]
Let \{{a}_{n}\}, \{{b}_{n}\} and \{{\delta}_{n}\} be sequences of nonnegative real numbers satisfying
If {\sum}_{i=1}^{\mathrm{\infty}}{\delta}_{n}<\mathrm{\infty} and {\sum}_{i=1}^{\mathrm{\infty}}{b}_{n}<\mathrm{\infty}, then the limit {lim}_{n\to \mathrm{\infty}}{a}_{n} exists.
3 Split feasibility problem
For solving the split feasibility problem (1.1), let us assume that the following conditions are satisfied:

1.
{H}_{1} and {H}_{2} are two real Hilbert spaces, A:{H}_{1}\to {H}_{2} is a bounded linear operator;

2.
S:{H}_{1}\to {H}_{1} and T:{H}_{2}\to {H}_{2} are two uniformly LLipschitzian and (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta)total quasiasymptotically nonexpansive mappings satisfying the following conditions:

(i)
T and S both are demiclosed at origin;

(ii)
{\sum}_{n=1}^{\mathrm{\infty}}({\mu}_{n}+{\nu}_{n})<\mathrm{\infty};

(iii)
there exist positive constants M and {M}^{\ast} such that \zeta (t)\le \zeta (M)+{M}^{\ast}{t}^{2}, \mathrm{\forall}t\ge 0.
We are now in a position to give the following result.
Theorem 3.1 Let {H}_{1}, {H}_{2}, A, S, T, L, \{{\mu}_{n}\}, \{{\nu}_{n}\}, ζ be the same as above. Let \{{x}_{n}\} be the sequence generated by:
where \{{\alpha}_{n}\} is a sequence in [0,1], and \gamma >0 is a constant satisfying the following conditions:

(iv)
0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1; and \gamma \in (0,\frac{1}{{\parallel A\parallel}^{2}}),

(I)
If \mathrm{\Gamma}\ne \mathrm{\varnothing} (where Γ is the set of solutions to ((SFP)(1.1)), then \{{x}_{n}\} converges weakly to a point {x}^{\ast}\in \mathrm{\Gamma}.

(II)
In addition, if S is also semicompact, then \{{x}_{n}\} and \{{u}_{n}\} both converge strongly to {x}^{\ast}\in \mathrm{\Gamma}.
The proof of conclusion (I) (1) First, we prove that for each p\in \mathrm{\Gamma}, the following limits exist:
In fact, since p\in \mathrm{\Gamma}, we have p\in C:=F(S) and Ap\in Q:=F(T). It follows from (3.1) and (2.4) that
On the other hand, by condition (iii), we have
Substituting (3.4) into (3.3) and simplifying, we have
On the other hand,
and
and
In (2.5), taking x=A{x}_{n}, {G}^{n}={T}^{n}, q=Ap, and noting Ap\in F(T), from (2.7) and condition (iii), we have
Substituting (3.9) into (3.8) and simplifying it, we have
Substituting (3.7) and (3.10) into (3.6) after simplifying, we have
Substituting (3.11) into (3.5) and simplifying it, we have
where
By condition (iii), we have
By condition (iv), (1\gamma {\parallel A\parallel}^{2})>0. Hence, from (3.12), we have
By Lemma 2.6, the following limit exists:
Now, we rewrite (3.12) as follows:
This together with the condition (iv) implies that
and
It follows from (3.6), (3.14) and (3.15) that the limit {lim}_{n\to \mathrm{\infty}}\parallel {u}_{n}p\parallel exists and
The conclusion (3.2) is proved.

(2)
Next, we prove that
\underset{n\to \mathrm{\infty}}{lim}\parallel {x}_{n+1}{x}_{n}\parallel =0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{n\to \mathrm{\infty}}{lim}\parallel {u}_{n+1}{u}_{n}\parallel =0.(3.16)
In fact, it follows from (3.1) that
In view of (3.14) and (3.15), we have that
Similarly, it follows from (3.1), (3.15) and (3.17) that
The conclusion (3.16) is proved.

(3)
Next, we prove that
\parallel {u}_{n}S{u}_{n}\parallel \to 0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\parallel A{x}_{n}TA{x}_{n}\parallel \to 0\phantom{\rule{1em}{0ex}}(\text{as}n\to \mathrm{\infty}).(3.19)
In fact, from (3.14), we have
Since S is uniformly LLipschitzian continuous, it follows from (3.16) and (3.20) that
Similarly, from (3.15), we have
Since T is uniformly LLipschitzian continuous, by the same way as above, from (3.16) and (3.21), we can also prove that

(4)
Finally, we prove that {x}_{n}\rightharpoonup {x}^{\ast} and {u}_{n}\rightharpoonup {x}^{\ast}, which is a solution of (SFP)(1.1).
Since \{{u}_{n}\} is bounded, there exists a subsequence \{{u}_{{n}_{i}}\}\subset \{{u}_{n}\} such that {u}_{{n}_{i}}\rightharpoonup {x}^{\ast} (some point in {H}_{1}). From (3.19), we have
By the assumption that S is demiclosed at zero, we get that {x}^{\ast}\in F(S).
Moreover, from (3.1) and (3.15), we have
Since A is a linear bounded operator, we get A{x}_{{n}_{i}}\rightharpoonup A{x}^{\ast}. In view of (3.19), we have
Since T is demiclosed at zero, we have A{x}^{\ast}\in F(T). Summing up the above argument, it is clear that {x}^{\ast}\in \mathrm{\Gamma}, i.e., {x}^{\ast} is a solution to the (SFP)(1.1).
Now, we prove that {x}_{n}\rightharpoonup {x}^{\ast} and {u}_{n}\rightharpoonup {x}^{\ast}.
Suppose, to the contrary, that if there exists another subsequence \{{u}_{{n}_{j}}\}\subset \{{u}_{n}\} such that {u}_{{n}_{j}}\rightharpoonup {y}^{\ast}\in \mathrm{\Gamma} with {y}^{\ast}\ne {x}^{\ast}, then by virtue of (3.2) and the Opial property of Hilbert space, we have
This is a contradiction. Therefore, {u}_{n}\rightharpoonup {x}^{\ast}. By using (3.1) and (3.15), we have
□
The proof of conclusion (II) By the assumption that S is semicompact, it follows from (3.23) that there exists a subsequence of \{{u}_{{n}_{i}}\} (without loss of generality, we still denote it by \{{u}_{{n}_{i}}\}) such that {u}_{{n}_{i}}\to {u}^{\ast}\in H (some point in H). Since {u}_{{n}_{i}}\rightharpoonup {x}^{\ast}. This implies that {x}^{\ast}={u}^{\ast}, and so {u}_{{n}_{i}}\to {x}^{\ast}\in \mathrm{\Gamma}. By virtue of (3.2), we know that {lim}_{n\to \mathrm{\infty}}\parallel {u}_{n}{x}^{\ast}\parallel =0 and {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{x}^{\ast}\parallel =0, i.e., \{{u}_{n}\} and \{{x}_{n}\} both converge strongly to {x}^{\ast}\in \mathrm{\Gamma}.
This completes the proof of Theorem 3.1. □
Theorem 3.2 Let {H}_{1}, {H}_{2} and A be the same as in Theorem 3.1. Let S:{H}_{1}\to {H}_{1} and T:{H}_{2}\to {H}_{2} be two (\{{k}_{n}\})quasiasymptotically nonexpansive mappings with \{{k}_{n}\}\subset [1,\mathrm{\infty}), {k}_{n}\to 1 satisfying the following conditions:

(i)
T and S both are demiclosed at origin;

(ii)
{\sum}_{n=1}^{\mathrm{\infty}}({k}_{n}1)<\mathrm{\infty}.
Let \{{x}_{n}\} be the sequence generated by
where \{{\alpha}_{n}\} is a sequence in [0,1] and \gamma >0 is a constant satisfying the condition (iv) in Theorem 3.1. Then the conclusions in Theorem 3.1 still hold.
Proof By assumptions, S:{H}_{1}\to {H}_{1} and T:{H}_{2}\to {H}_{2} both are (\{{k}_{n}\})quasiasymptotically nonexpansive mappings with \{{k}_{n}\}\subset [1,\mathrm{\infty}), {k}_{n}\to 1; by Remark 2.3, S and T both are uniformly LLipschitzian (where L={sup}_{n\ge 1}{k}_{n}) and (\{{\nu}_{n}\},\{{\mu}_{n}\},\zeta)total quasiasymptotically nonexpansive mapping with \{{\nu}_{n}={k}_{n}1\}, \{{\mu}_{n}=0\} and \zeta (t)={t}^{2}, t\ge 0. Therefore, all conditions in Theorem 3.1 are satisfied. The conclusions of Theorem 3.2 can be obtained from Theorem 3.1 immediately. □
Theorem 3.3 Let {H}_{1}, {H}_{2} and A be the same as in Theorem 3.1. Let S:{H}_{1}\to {H}_{1} and T:{H}_{2}\to {H}_{2} be two quasinonexpansive mappings and demiclosed at origin. Let \{{x}_{n}\} be the sequence generated by
where \{{\alpha}_{n}\} is a sequence in [0,1] and \gamma >0 is a constant satisfying the condition (iv) in Theorem 3.1. Then the conclusions in Theorem 3.1 still hold.
Proof By the assumptions, S:{H}_{1}\to {H}_{1} and T:{H}_{2}\to {H}_{2} are quasinonexpansive mappings. By Remark 2.3, S and T both are uniformly LLipschitzian (where L=1) and (\{1\}) quasiasymptotically nonexpansive mappings. Therefore, all conditions in Theorem 3.2 are satisfied. The conclusions of Theorem 3.3 can be obtained from Theorem 3.2 immediately. □
Remark 3.4 Theorems 3.1, 3.2 and 3.3 not only improve and extend the corresponding results of Moudafi [12, 13], but also improve and extend the corresponding results of Censor et al. [4, 5], Yang [7], Xu [14], Censor and Segal [15], Masad and Reich [16] and others.
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Acknowledgements
This work was supported by the Scientific Research Fund of Science Technology Department of Sichuan Province (2011JYZ010) and the Natural Science Foundation of Yunnan Province (Grant No.2011FB074).
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Wang, X.R., Chang, Ss., Wang, L. et al. Split feasibility problems for total quasiasymptotically nonexpansive mappings. Fixed Point Theory Appl 2012, 151 (2012). https://doi.org/10.1186/168718122012151
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DOI: https://doi.org/10.1186/168718122012151