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Variable KMlike algorithms for fixed point problems and split feasibility problems
Fixed Point Theory and Applications volume 2014, Article number: 211 (2014)
Abstract
The convergence analysis of a variable KMlike method for approximating common fixed points of a possibly infinitely countable family of nonexpansive mappings in a Hilbert space is proposed and proved to be strongly convergent to a common fixed point of a family of nonexpansive mappings. Our variable KMlike technique is applied to solve the split feasibility problem and the multiplesets split feasibility problem. Especially, the minimum norm solutions of the split feasibility problem and the multiplesets split feasibility problem are derived. Our results can be viewed as an improvement and refinement of the previously known results.
MSC:47H10, 65J20, 65J22, 65J25.
1 Introduction
Problems of image reconstruction from projections can be represented by a system of linear equations
In practice, the system (1.1) is often inconsistent, and one usually seeks a point which minimizes x\in {\mathbb{R}}^{n} by some predetermined optimization criterion. The problem is frequently illposed and there may be more than one optimal solution. The standard approach to dealing with that problem is via regularization. The wellknown convex feasibility problem is to find a point {x}^{\ast} satisfying the following:
where m\ge 1 is an integer, and each {C}_{i} is a nonempty closed convex subset of a Hilbert space H. A special case of the convex feasibility problem is the split feasibility problem given by:
Let C, Q be nonempty closed convex subsets of Hilbert spaces {H}_{1} and {H}_{2}, respectively, and let A:{H}_{1}\to {H}_{2} be a bounded linear operator. The split feasibility problem (SFP) is
The SFP is said to be consistent if (1.2) has a solution. It is easy to see that SFP (1.2) is consistent if and only if the following fixed point problem has a solution:
where {P}_{C} and {P}_{Q} are the projections onto C and Q, respectively, and {A}^{\ast} is the adjoint of A. Let L denote the spectral radius of {A}^{\ast}A. It is well known that if \gamma \in (0,2/L), the operator T={P}_{C}(I\gamma {A}^{\ast}(I{P}_{Q})A) in the operator equation (1.3) is nonexpansive [1].
It has been extensively studied during the last decade because of its applications in modeling inverse problems which arise in phase retrievals and in medical image reconstruction. It has also been applied to modeling intensitymodulated radiation therapy; see, for example [2–7] and the references therein.
Several iterative methods have been proposed and analyzed to solve the SFP (1.2); see, for example [1, 3, 6, 8–14] and the references therein. Byrne [3] introduced the CQ algorithm
and proved that the sequence \{{x}_{n}\} generated by the CQ algorithm (1.4) converges weakly to a solution of SFP (1.2), where T={P}_{C}(I\gamma {A}^{\ast}(I{P}_{Q})A) and 0<\gamma <2/L.
In view of the fixed point formulation (1.3) of the SFP (1.2), Xu [1] and Yang [14] applied the following perturbed Krasnosel’skiĭMann CQ algorithm to solve the SFP (1.2):
Here \{{T}_{n}\} is a sequence of operators defined by
where \{{C}_{n}\} and \{{Q}_{n}\} are sequences of nonempty closed convex subsets in {H}_{1} and {H}_{2}, respectively, which obey the following assumption:
(C0) {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}{d}_{\rho}({C}_{n},C)<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}{d}_{\rho}({Q}_{n},Q)<\mathrm{\infty} for each \rho >0, where {d}_{\rho} is the ρdistance between {Q}_{n} and Q (see Section 3.2).
It is not very easy to verify condition (C0) for each \rho >0. Thus, the condition (C0) is quite restrictive even for weak convergence of the sequence \{{x}_{n}\} defined by (1.5). One of our objectives is to relax the condition (C0).
Many practical problems can be formulated as a fixed point problem (FPP): finding an element x such that
where T is a nonexpansive selfmapping defined on a closed convex subset C of a Hilbert space H. The solution set of FPP (1.6) is denoted by F(T). It is well known that if F(T)\ne \mathrm{\varnothing}, then F(T) is closed and convex. The fixed point problem (1.6) is illposed (it may fail to have a solution, nor uniqueness of solution) in general. Regularization by contractions can removed such illness. We replace the nonexpansive mapping T by a family of contractions {T}_{t}^{f}:=tf+(1t)T, with t\in (0,1) and f:C\to C a fixed contraction. We call f an anchoring function. The regularized problem of fixed point for T is the fixed point problem for {T}_{t}^{f}. The mapping {T}_{t}^{f} has a unique fixed point, namely, {x}_{t}\in C. Therefore, {x}_{t} is the solution of the fixed point problem
We now discretize the regularization (1.7) to define an explicit iterative algorithm:
The iterative algorithm (1.8) is due to Moudafi [15], by generalizing Browder’s and Halpern’s methods, who introduced viscosity approximation methods. Suzuki [16] established a strong convergence theorem by using Halpern’s method to averaged mapping {T}_{\lambda}=\lambda I+(1\lambda )T, \lambda \in (0,1) for nonexpansive mappings T in certain Banach spaces. Takahashi [17] proved a strong convergence theorem of the following iterative algorithm for countable families of nonexpansive mappings in certain Banach spaces:
Recently, Yao and Xu [18] introduced and studied strong convergence of the following modified methods:
where f:C\to H is a fixed nonself contraction and \{{t}_{n}\} is a sequence in (0,1) satisfying the conditions:
(S1) {lim}_{n\to \mathrm{\infty}}{t}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{t}_{n}=\mathrm{\infty},
(S2) either {\sum}_{n=1}^{\mathrm{\infty}}{t}_{n+1}{t}_{n}<\mathrm{\infty} or {t}_{n+1}/{t}_{n}\to 0 as n\to \mathrm{\infty}.
One can easily see that (1.10) is a regularized iterative algorithm.
Motivated by [1, 11, 14], we study the following more general nonregularized algorithm, called variable KMlike algorithm which generates a sequence \{{x}_{n}\} according to the recursive formula:
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are sequences in (0,1), \{{T}_{n}\} is a sequence of nonexpansive selfmappings of C and \{{f}_{n}\} is a sequence of (not necessarily contraction) mappings from C into H.
In the present paper, we will study the strong convergence of the proposed variable KMlike algorithm in the framework of Hilbert spaces. The paper is organized as follows. The next section contains preliminaries. In Section 3, we will study the convergence analysis of our variable KMlike algorithm for fixed point problem (1.6) without the assumption (S2). This result will be applied to prove convergence of some perturbed algorithms for the SFP (1.2) and the multiplesets split feasibility problem under some weaker assumptions. As special cases, we obtain algorithms which converge strongly to the minimum norm solutions of the split feasibility problem and the multiplesets split feasibility problem. Our results are new and interesting in the following contexts:

(i)
Our algorithm (3.1) is not regularized by contractions.

(ii)
{f}_{n} is not necessarily contraction. In the existing literature, anchoring function f is a fixed contraction mapping [15, 17–19] or strongly pseudocontraction mapping [20].

(iii)
In the convergence analysis of (3.1) for fixed point problem (1.6), the assumption (S2) is not required.

(iv)
A fixed \rho >0 for a (C0)like condition is adopted.
2 Preliminaries
Let C be a nonempty subset of a Hilbert space H. Throughout the paper, we denote by {B}_{r}[x] the closed ball defined by {B}_{r}[x]=\{y\in C:\parallel yx\parallel \le r\}. Let {T}_{1},{T}_{2}:C\to H be two mappings. We denote by \mathcal{B}(C) the collection of all bounded subsets of C. The deviation between {T}_{1} and {T}_{2} on B\in \mathcal{B}(C) [21], denoted by {\mathcal{D}}_{B}({T}_{1},{T}_{2}), is defined by
Let T:C\to H be a mapping. Then T is said to be a κcontraction if there exists \kappa \in [0,1) such that \parallel TxTy\parallel \le \kappa \parallel xy\parallel for all x,y\in C. Furthermore, it is called nonexpansive if \parallel TxTy\parallel \le \parallel xy\parallel for all x,y\in C.
Let \{{f}_{n}\} be a sequence of mappings from C into H. Following [20–22], we say \{{f}_{n}\} is a sequence of nearly contraction mappings with sequence \{({k}_{n},{a}_{n})\} if there exist a sequence \{{k}_{n}\} in [0,1) and a sequence \{{a}_{n}\} in [0,\mathrm{\infty}) with {a}_{n}\to 0 such that
One can observe that a sequence of contraction mappings is essentially a sequence of nearly contraction mappings.
We now construct a sequence of nearly contractions.
Example 2.1 Let H=\mathbb{R} and C=[0,1]. Let \{{f}_{n}\} be a sequence of mappings {f}_{n}:C\to H defined by
Set {k}_{n}:=\frac{1}{n+1}. We consider the following cases:
Case 1: If x,y\in [0,\frac{1}{2}], then
Case 2: If x,y\in (\frac{1}{2},1], then
Case 3: If x\in [0,\frac{1}{2}] and y\in (\frac{1}{2},1], then
Therefore, for all x,y\in [0,1], we have
where {a}_{n}:=\frac{5}{2(n+1)}. Therefore, \{{f}_{n}\} is a sequence of nearly contraction mappings with sequence \{({k}_{n},{a}_{n})\}.
Let C be a nonempty closed convex subset of a Hilbert space H. We use {P}_{C} to denote the (metric) projection from H onto C; namely, for x\in H, {P}_{C}(x) is the unique point in C with the property
The following is a useful characterization of projections.
Lemma 2.2 Let C be a nonempty closed convex subset of a real Hilbert space H and let {P}_{C} be the metric projection from H onto C. Given x\in H and z\in C. Then z={P}_{C}(x) if and only if
Lemma 2.3 [[23], Corollary 5.6.4]
Let C be a nonempty closed convex subset of H and T:C\to C a nonexpansive mapping. Then IT is demiclosed at zero, that is, if \{{x}_{n}\} is a sequence in C weakly converging to x and if \{(IT){x}_{n}\} converges strongly to zero, then (IT)x=0.
Lemma 2.4 [24]
Let \{{a}_{n}\} and \{{c}_{n}\} be two sequences of nonnegative real numbers and let \{{b}_{n}\} be a sequence in ℝ satisfying the following condition:
where \{{\alpha}_{n}\} is a sequence in (0,1]. Assume that {\sum}_{n=1}^{\mathrm{\infty}}{c}_{n}<\mathrm{\infty}. Then the following statements hold:

(a)
If {b}_{n}\le K{\alpha}_{n} for all n\in \mathbb{N} and for some K\ge 0, then
{a}_{n+1}\le {\delta}_{n}{a}_{1}+(1{\delta}_{n})K+\sum _{j=1}^{n}{c}_{j}\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}n\in \mathbb{N},where {\delta}_{n}={\mathrm{\Pi}}_{j=1}^{n}(1{\alpha}_{j}) and hence \{{a}_{n}\} is bounded.

(b)
If {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty} and {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}({b}_{n}/{\alpha}_{n})\le 0, then {\{{a}_{n}\}}_{n=1}^{\mathrm{\infty}} converges to zero.
3 Convergence analysis of a variable KMlike algorithm
First, we prove the following.
Proposition 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H, T:C\to C a nonexpansive mapping with F(T)\ne \mathrm{\varnothing} and f:C\to H a κcontraction. Then there exists a unique point {x}^{\ast}\in C such that {x}^{\ast}={P}_{F(T)}f({x}^{\ast}).
Proof Since f:C\to H is a κcontraction, it follows that {P}_{F(T)}f is a κcontraction from C onto itself. Then there exists a unique point {x}^{\ast}\in C such that {x}^{\ast}={P}_{F(T)}f({x}^{\ast}). □
3.1 A variable KMlike algorithm
Let C be a nonempty closed convex subset of a real Hilbert space H. Let \{{f}_{n}\} be a sequence of nearly contractions from C into H such that \{{f}_{n}\} converges pointwise to f and let \{{T}_{n}\} be a sequence of nonexpansive selfmappings of C which are viewed as perturbations. For computing a common fixed point of the sequence \{{T}_{n}\} of nonexpansive mappings, we propose the following variable KMlike algorithm:
where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are sequences in [0,1].
We investigate the asymptotic behavior of the sequence \{{x}_{n}\} generated, from an arbitrary {x}_{1}\in C, by the algorithm (3.1) to a common fixed point of the sequence \{{T}_{n}\}.
Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H, T:C\to C be a nonexpansive mapping such that F(T)\ne \mathrm{\varnothing}, and let f:C\to H be a κcontraction with \kappa \in [0,1) such that {P}_{F(T)}f({x}^{\ast})={x}^{\ast}\in F(T). Let \{{f}_{n}\} be a sequence of nearly contraction mappings from C into H with the sequence \{({k}_{n},{a}_{n})\} in [0,1)\times [0,\mathrm{\infty}) such that {k}_{n}\to \kappa, and let \{{T}_{n}\} be a sequence of nonexpansive mappings from C into itself. For given {x}_{1}\in C, let \{{x}_{n}\} be a sequence in C generated by (3.1), where \{{\alpha}_{n}\} is a sequence in (0,1] and \{{\beta}_{n}\} is a sequence in (0,1). Assume that the following conditions are satisfied:
(C1) {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty},
(C2) 0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\beta}_{n}\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\beta}_{n}<1,
(C3) {lim}_{n\to \mathrm{\infty}}{f}_{n}{x}^{\ast}=f{x}^{\ast},
(C4) {\sum}_{n=1}^{\mathrm{\infty}}(1{\alpha}_{n})\parallel {T}_{n}{x}^{\ast}{x}^{\ast}\parallel <\mathrm{\infty}.
Define
Then the following statements hold:

(a)
The sequence \{{x}_{n}\} generated by (3.1) remains in the closed ball {B}_{R}[{x}^{\ast}].

(b)
If the following assumption holds:
(C5) {lim}_{n\to \mathrm{\infty}}\parallel {T}_{n}{v}_{n}T{v}_{n}\parallel =0 for all \{{v}_{n}\} in {B}_{R}[{x}^{\ast}],
then \{{x}_{n}\} converges strongly to {x}^{\ast}.
Proof (a) Set {z}_{n}:={P}_{C}[{y}_{n}]. Observe that
From (3.1), we have
Since {\sum}_{n=1}^{\mathrm{\infty}}(1{\alpha}_{n})\parallel {T}_{n}{x}^{\ast}{x}^{\ast}\parallel <\mathrm{\infty}, by Lemma 2.4, we find that \{\parallel {x}_{n}{x}^{\ast}\parallel \} is bounded. Moreover,
Therefore, \{{x}_{n}\} is well defined in the ball {B}_{R}[{x}^{\ast}].
(b) Assume that {lim}_{n\to \mathrm{\infty}}\parallel {T}_{n}{v}_{n}T{v}_{n}\parallel =0 for all \{{v}_{n}\} in {B}_{R}[{x}^{\ast}]. Set {\gamma}_{n}:=\u3008{x}^{\ast}f{x}^{\ast},{x}^{\ast}{z}_{n}\u3009. We now proceed with the following steps:
Step 1: \{{f}_{n}{x}_{n}\} and \{{T}_{n}{x}_{n}\} are bounded.
Without loss of generality, we may assume that \beta \le {\beta}_{n} for all n\in \mathbb{N} for some \beta >0. From (C3), we have
which implies that {lim}_{n\to \mathrm{\infty}}(1{\alpha}_{n})\parallel {T}_{n}{x}^{\ast}{x}^{\ast}\parallel =0. Since {\alpha}_{n}\to 0, it follows that
Since
and \{\parallel {T}_{n}{x}^{\ast}{x}^{\ast}\parallel \} converges to 0, we conclude that \{{T}_{n}{x}_{n}\} is bounded. Moreover,
it follows that \{{f}_{n}{x}_{n}\} is bounded.
Step 2: {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{x}_{n+1}\parallel =0.
Set {u}_{n}:={f}_{n}{x}_{n}. We write
Observe that
which gives
As we have shown in Step 1, \{{T}_{n}{x}_{n}\} and \{{u}_{n}\} are bounded. Observe that
and
Thus, \{{f}_{n+1}{x}_{n}\} and \{{T}_{n+1}{x}_{n}\} are bounded. Hence,
By [[25], Lemma 2.2], we obtain
which implies that
Step 3: {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0.
Note
and hence
which implies that
Note {\alpha}_{n}\to 0 and \parallel {T}_{n}{x}_{n}T{x}_{n}\parallel \to 0, we conclude that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0.
Step 4: {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\gamma}_{n}\le 0.
Note that
We take a subsequence \{{z}_{{n}_{i}}\} of \{{z}_{n}\} such that
Since \{{z}_{n}\} is in C and \parallel {z}_{n}T{z}_{n}\parallel \to 0, we conclude, from Lemma 2.3 that z\in F(T). Since {x}^{\ast}={P}_{F(T)}f({x}^{\ast}), we obtain from Lemma 2.2 that
Step 5: {x}_{n}\to {x}^{\ast}.
Since \{\parallel {z}_{n}{x}^{\ast}\parallel \} is bounded, there exists {R}_{1}>0 such that \parallel {z}_{n}{x}^{\ast}\parallel \le {R}_{1} for all n\in \mathbb{N}. Noting that {z}_{n}={P}_{C}[{y}_{n}]. Hence, from (3.1), we have
which implies that
From (3.1), we have
Since {lim}_{n\to \mathrm{\infty}}\frac{\parallel {f}_{n}{x}^{\ast}f{x}^{\ast}\parallel}{1{k}_{n}}=0 and {\sum}_{n=1}^{\mathrm{\infty}}(1{\alpha}_{n})\parallel {T}_{n}{x}^{\ast}T{x}^{\ast}\parallel <\mathrm{\infty}, we conclude from Lemma 2.4(b) that {x}_{n}\to {x}^{\ast}. □
Remark 3.3 Theorem 3.2 has the following characterization for convergence analysis of (3.1):

(a)
Iterates of (3.1) remains in the closed ball {B}_{R}[{x}^{\ast}].

(b)
The assumption (S2) is not required.

(c)
(C4) is adopted for only for {x}^{\ast}\in F(T). In particular, the condition ‘{\sum}_{n=1}^{\mathrm{\infty}}\parallel {T}_{n}zTz\parallel <\mathrm{\infty} for all z\in F(T)’ is adopted in [[26], Theorem 3.1].
Thus, Theorem 3.2 is more general by nature. Therefore, Theorem 3.2 significantly extends and improves [[26], Theorem 3.1] and [[18], Theorem 3.2].
Theorem 3.2 remains true if condition (C4) is replaced with the condition that the mappings \{{T}_{n}\} and T have common fixed points. In fact, we have
Theorem 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H, T:C\to C a nonexpansive mapping such that F(T)\ne \mathrm{\varnothing}, and f:C\to H be a κcontraction with \kappa \in [0,1) such that {P}_{F(T)}f({x}^{\ast})={x}^{\ast}\in F(T). Let \{{f}_{n}\} be a sequence of nearly contraction mappings from C into H with sequence \{({k}_{n},{a}_{n})\} in [0,1)\times [0,\mathrm{\infty}) such that {k}_{n}\to \kappa. Let \{{T}_{n}\} be a sequence of nonexpansive mappings from C into itself such that F(T)={\bigcap}_{n\in \mathbb{N}}F({T}_{n}). For given {x}_{1}\in C, let \{{x}_{n}\} be a sequence in C generated by (3.1), where \{{\alpha}_{n}\} is a sequence in (0,1] and \{{\beta}_{n}\} is a sequence in (0,1) satisfying (C1), (C2), and (C3). Then the following statements hold:

(a)
The sequence \{{x}_{n}\} generated by (3.1) remains in the closed ball {B}_{R}[{x}^{\ast}], where R=max\{\parallel {x}_{1}{x}^{\ast}\parallel ,{K}^{\ast}\} and {K}^{\ast} is given in (3.2).

(b)
If the assumption (C5) holds, then \{{x}_{n}\} converges strongly to {x}^{\ast}.
We now prove strong convergence of the sequence \{{x}_{n}\} generated by (3.1) under condition (C6).
Theorem 3.5 Let C be a nonempty closed convex subset of a real Hilbert space H, T:C\to C be a nonexpansive mapping such that F(T)\ne \mathrm{\varnothing}, and \{{T}_{n}\} be a sequence of nonexpansive mappings from C into itself. Let f:C\to H be a κcontraction with \kappa \in [0,1) such that {P}_{F(T)}f({x}^{\ast})={x}^{\ast}\in F(T) and \{{f}_{n}\} be a sequence of {k}_{n}contraction mappings from C into H such that {k}_{n}\to \kappa. For given {x}_{1}\in C, let \{{x}_{n}\} be a sequence in C generated by (3.1), where \{{\alpha}_{n}\} is a sequence in (0,1] and \{{\beta}_{n}\} is a sequence in (0,1) satisfying (C1), (C2), (C3), and (C4). Then the following statements hold:

(a)
The sequence \{{x}_{n}\} generated by (3.1) remains in the closed ball {B}_{R}[{x}^{\ast}], where
R=max\{\parallel {x}_{1}{x}^{\ast}\parallel ,\underset{n\in \mathbb{N}}{sup}\frac{\parallel {f}_{n}{x}^{\ast}{x}^{\ast}\parallel}{1{k}_{n}}\}+\sum _{n=1}^{\mathrm{\infty}}(1{\alpha}_{n})\parallel {T}_{n}{x}^{\ast}{x}^{\ast}\parallel . 
(b)
If the following assumption holds:
(C6) {\sum}_{n=1}^{\mathrm{\infty}}{\mathcal{D}}_{{B}_{R}[{x}^{\ast}]}({T}_{n},T)<\mathrm{\infty},
then \{{x}_{n}\} converges strongly to {x}^{\ast}.
Proof We show that {\sum}_{n=1}^{\mathrm{\infty}}{\mathcal{D}}_{{B}_{R}[{x}^{\ast}]}({T}_{n},T)<\mathrm{\infty} implies that {lim}_{n\to \mathrm{\infty}}\parallel {T}_{n}{v}_{n}T{v}_{n}\parallel =0 for all \{{v}_{n}\} in {B}_{R}[{x}^{\ast}]. Let \{{w}_{n}\} be a sequence in {B}_{R}[{x}^{\ast}]. Then
It follows that {lim}_{n\to \mathrm{\infty}}\parallel {T}_{n}{w}_{n}T{w}_{n}\parallel =0. Thus, the condition (C5) in Theorem 3.2 holds. Therefore, Theorem 3.5 follows from Theorem 3.2. □
For a sequence \{{u}_{n}\} in H with {u}_{n}\to u\in H, define {f}_{n}:C\to H by
Then each {f}_{n} is 0contraction with {f}_{n}x\to fx=u. In this case algorithm (3.1) with {T}_{n}=T reduces to
Corollary 3.6 Let C be a nonempty closed convex subset of a real Hilbert space H and T:C\to C be a nonexpansive mapping such that F(T)\ne \mathrm{\varnothing}. Let \{{u}_{n}\} be a sequence in H such that {u}_{n}\to u\in H and {P}_{F(T)}(u)={x}^{\ast}\in F(T). For given {x}_{1}\in C, let \{{x}_{n}\} be a sequence in C generated by (3.3), where \{{\alpha}_{n}\} is a sequence in (0,1] and \{{\beta}_{n}\} is a sequence in (0,1) satisfying (C1) and (C2). Then the following statements hold:

(a)
The sequence \{{x}_{n}\} generated by (3.3) remains in the closed ball {B}_{R}[{x}^{\ast}], where R=max\{\parallel {x}_{1}{x}^{\ast}\parallel ,{sup}_{n\in \mathbb{N}}\parallel {u}_{n}{x}^{\ast}\parallel \}.

(b)
\{{x}_{n}\} converges strongly to {x}^{\ast}.
Remark 3.7 If u=0 in Corollary 3.6, then \{{x}_{n}\} generated by Algorithm 3.3 converges strongly to the minimum norm solution of the FPP (1.6). Corollary 3.6 also provides a closed ball in which \{{x}_{n}\} lies. Therefore, Corollary 3.6 significantly extends and improves [[27], Theorem 3.1].
3.2 The split feasibility problem
In this section we apply Theorem 3.5 to solve the SFP (1.2). We begin with the ρdistance:
Definition 3.8 Let C and Q be two closed convex subsets of a Hilbert space H and let ρ be a positive constant. The ρdistance between C and Q is defined by
By employing Theorem 3.5, we present a variable KMlike CQ algorithm (3.6) for finding solutions of the SFP (1.2) and prove its strong convergence.
Theorem 3.9 Let C and Q be two nonempty closed convex subsets of real Hilbert spaces {H}_{1} and {H}_{2}, respectively, and let \{{C}_{n}\} and \{{Q}_{n}\} be sequences of closed convex subsets of {H}_{1} and {H}_{2}, respectively. Let f:C\to {H}_{1} be a κcontraction and \{{f}_{n}\} be a sequence of {k}_{n}contraction mappings from C into {H}_{1} such that {k}_{n}\to \kappa. Let A:{H}_{1}\to {H}_{2} be a bounded linear operator with the adjoint {A}^{\ast}. For \gamma \in (0,2/L), define
and
Assume that SFP (1.2) is consistent with {P}_{F(T)}f{x}^{\ast}={x}^{\ast}\in F(T). For given {x}_{1}\in C, let \{{x}_{n}\} be a sequence in C generated by the following variable KMlike CQ algorithm:
where \{{\alpha}_{n}\} is a sequence in (0,1] and \{{\beta}_{n}\} is a sequence in (0,1) satisfying (C1), (C2), (C3), and (C4). Then the following statements hold:

(a)
The sequence \{{x}_{n}\} generated by (3.6) remains in the closed ball {B}_{R}[{x}^{\ast}], where
R=max\{\parallel {x}_{1}{x}^{\ast}\parallel ,\underset{n\in \mathbb{N}}{sup}(\parallel {f}_{n}{x}^{\ast}{x}^{\ast}\parallel /(1{k}_{n}))\}+\sum _{n=1}^{\mathrm{\infty}}(1{\alpha}_{n})\parallel {T}_{n}{x}^{\ast}{x}^{\ast}\parallel . 
(b)
If \overline{\rho}=max\{\parallel Ax\parallel ,\parallel (I\gamma {A}^{\ast}(I{P}_{Q})A)x\parallel :x\in {B}_{R}[{x}^{\ast}]\} and the following assumption holds:
(C7) {\sum}_{n=1}^{\mathrm{\infty}}{d}_{\overline{\rho}}({C}_{n},C)<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{d}_{\overline{\rho}}({Q}_{n},Q)<\mathrm{\infty},
then \{{x}_{n}\} converges strongly to {x}^{\ast}.
Proof (a) Since \gamma \in (0,2/L), T and {T}_{n} for all n\in \mathbb{N} are nonexpansive mappings and F(T)\ne \mathrm{\varnothing} because SFP (1.2) is consistent. Hence this part follows from Theorem 3.5(a).
(b) Assume that
Now, let x\in {H}_{1} be such that x\in {B}_{R}[{x}^{\ast}]. Since each {P}_{{C}_{n}} is the nonexpansive, we have
Thus,
Hence condition (C6) in Theorem 3.5 holds. Therefore, Theorem 3.9(b) follows from Theorem 3.5(b). □
For a sequence \{{u}_{n}\} in {H}_{1} with {u}_{n}\to 0\in {H}_{1}, define {f}_{n}:C\to {H}_{1} by
Then each {f}_{n} is 0contraction with {f}_{n}x\to fx=0. In this case variable KMlike CQ algorithm (3.6) reduces to the following variable KMlike CQ algorithm:
We now present strong convergence of the variable KMlike CQ algorithm (3.7) to the minimum norm solution of the SFP (1.2).
Corollary 3.10 Let C and Q be two nonempty closed convex subsets of real Hilbert spaces {H}_{1} and {H}_{2}, respectively, and let \{{C}_{n}\} and \{{Q}_{n}\} be sequences of closed convex subsets of {H}_{1} and {H}_{2}, respectively. Let A:{H}_{1}\to {H}_{2} be a bounded linear operator with the adjoint {A}^{\ast}. For \gamma \in (0,2/L), define T and {T}_{n} by (3.4) and (3.5), respectively. Assume that the SFP (1.2) is consistent with {P}_{F(T)}(0)={x}^{\ast}\in F(T). For given {x}_{1}\in C and a sequence \{{u}_{n}\} in {H}_{1} with {u}_{n}\to 0\in {H}_{1}, let \{{x}_{n}\} be a sequence in C generated by a variable KMlike CQ algorithm (3.7), \{{\alpha}_{n}\} is a sequence in (0,1] and \{{\beta}_{n}\} is a sequence in (0,1) satisfying (C1), (C2), and (C4). Then the following statements hold:

(a)
The sequence \{{x}_{n}\} generated by (3.7) remains in the closed ball {B}_{R}[{x}^{\ast}], where
R=max\{\parallel {x}_{1}{x}^{\ast}\parallel ,\parallel {x}^{\ast}\parallel \}+\sum _{n=1}^{\mathrm{\infty}}(1{\alpha}_{n})\parallel {T}_{n}{x}^{\ast}{x}^{\ast}\parallel . 
(b)
If \overline{\rho}=max\{\parallel Ax\parallel ,\parallel (I\gamma {A}^{\ast}(I{P}_{Q})A)x\parallel :x\in {B}_{R}[{x}^{\ast}]\} and the assumption (C7) holds, then \{{x}_{n}\} converges strongly to {x}^{\ast}.
Corollary 3.10 significantly extends and improves [[11], Theorem 3.1].
3.3 The constrained multiplesets split feasibility problem
In this section, we consider the following multiplesets split feasibility problem which models the intensitymodulated radiation therapy [6] and has recently been investigated by many researchers, see, for example, [1, 3, 6, 8–14] and the references therein.
Let {H}_{1} and {H}_{2} be two Hilbert spaces and let r and p be two natural numbers. For each i\in \{1,2,\dots ,p\}, let {C}_{i} be a nonempty closed convex subset of {H}_{1} and for each j\in \{1,2,\dots ,r\}, let {Q}_{j} be a nonempty closed convex subset of {H}_{2}. Further, for each j\in \{1,2,\dots ,r\}, let {A}_{j}:{H}_{1}\to {H}_{2} be a bounded linear operator and Ω be a closed convex subset of {H}_{1}. The (constrained) multiplesets split feasibility problem (MSSFP) is to find a point {x}^{\ast}\in \mathrm{\Omega} such that
When p=r=1, then the MSSFP (3.8) reduces to the SFP (1.2).
The split feasibility problem (SFP) and multiplesset split feasibility problem (MSSFP) model image retrieval [28] and intensitymodulated radiation therapy [6], and they have recently been investigated by many researchers.
For each i\in \{1,2,\dots ,p\} and j\in \{1,2,\dots ,r\}, let {\overline{\alpha}}_{i} and {\overline{\beta}}_{j} be two positive numbers. Let B:{H}_{1}\to {H}_{1} be the gradient ∇ψ of a convex and continuously differentiable function \psi :{H}_{1}\to \mathbb{R} defined by
Following [28], we see that
where {A}_{j}^{\ast} is the adjoint of {A}_{j}, j\in \{1,2,\dots ,r\}. The nonexpansivity of I{P}_{C} implies that B is a Lipschitzian mapping with Lipschitz constant
Thus, variable KMlike CQ algorithm can be developed to solve the MSSFP (3.8). Let \{{\mathrm{\Omega}}_{n}\}, \{{C}_{n,i}\} and \{{Q}_{n,j}\} be the sequences of closed convex sets, which are viewed as perturbations for the closed convex sets Ω, \{{C}_{i}\} and \{{Q}_{j}\}, respectively.
We now present an iterative algorithm for solving the MSSFP (3.8).
Theorem 3.11 Let f:\mathrm{\Omega}\to {H}_{1} be a κcontraction and let \{{f}_{n}\} be a sequence of {k}_{n}contraction mappings from Ω into {H}_{1} such that {k}_{n}\to \kappa. For \gamma \in (0,2/{L}^{\ast}), define
and
Assume that the MSSFP (3.8) is consistent with {P}_{F(T)}f{x}^{\ast}={x}^{\ast}\in F(T). For given {x}_{1}\in \mathrm{\Omega}, let \{{x}_{n}\} be a sequence in Ω generated by
where \{{\alpha}_{n}\} is a sequence in (0,1] and \{{\beta}_{n}\} is a sequence in (0,1) satisfying (C1), (C2), (C3), and (C4). Then the following statements hold:

(a)
The sequence \{{x}_{n}\} generated by (3.12) remains in the closed ball {B}_{R}[{x}^{\ast}], where
R=max\{\parallel {x}_{1}{x}^{\ast}\parallel ,\underset{n\in \mathbb{N}}{sup}(\parallel {f}_{n}{x}^{\ast}{x}^{\ast}\parallel /(1{k}_{n}))\}+\sum _{n=1}^{\mathrm{\infty}}(1{\alpha}_{n})\parallel {T}_{n}{x}^{\ast}{x}^{\ast}\parallel . 
(b)
If \overline{\rho}=max\{{max}_{1\le j\le p}\parallel {A}_{j}x\parallel ,\parallel (I\gamma \mathrm{\nabla}\psi )x\parallel :x\in {B}_{R}[{x}^{\ast}]\} and for each i\in \{1,2,\dots ,p\} and j\in \{1,2,\dots ,r\}, the following assumption holds:
(C8) {\sum}_{n=1}^{\mathrm{\infty}}{d}_{\overline{\rho}}({\mathrm{\Omega}}_{n},\mathrm{\Omega})<\mathrm{\infty}, {\sum}_{n=1}^{\mathrm{\infty}}{D}_{{B}_{R}[{x}^{\ast}]}({P}_{{C}_{i,n}},{P}_{{C}_{i}})<\mathrm{\infty} and {\sum}_{n=1}^{\mathrm{\infty}}{d}_{\overline{\rho}}({Q}_{n,i},Q)<\mathrm{\infty},
then \{{x}_{n}\} converges strongly to {x}^{\ast}.
Proof (a) Define
The gradients of ψ and {\psi}_{n} are given by
and
Hence, from (3.12) and (3.13), we have
and
Since \gamma \in (0,2/{\mathcal{L}}^{\ast}), T and {T}_{n}, for all n\in \mathbb{N}, are nonexpansive mappings, and F(T)\ne \mathrm{\varnothing} because the MSSFP (3.8) is consistent. Hence, this part follows from Theorem 3.5(a).
(b) Assume that
Let x\in {H}_{1} be such that x\in {B}_{R}[{x}^{\ast}]. Since each {P}_{{C}_{n}} is the nonexpansive, we have
By the assumptions, we have
Hence condition (C6) in Theorem 3.5 holds. Therefore, Theorem 3.9(b) follows from Theorem 3.5(b). □
Theorem 3.11 significantly extends and improves [[12], Theorem 1].
Finally, we present strong convergence of variable KMlike CQ algorithm (3.7) to the minimum norm solution of the MSSFP (3.8).
Corollary 3.12 Define T and {T}_{n} by (3.12) and (3.13), respectively. Assume that the MSSFP (3.8) is consistent with {P}_{F(T)}(0)={x}^{\ast}\in F(T). For given {x}_{1}\in C and a sequence \{{u}_{n}\} in {H}_{1} with {u}_{n}\to 0\in {H}_{1}, let \{{x}_{n}\} be a sequence in C generated by the following variable KMlike CQ algorithm:
where 0<\gamma <2/{\mathcal{L}}^{\ast}, \{{\alpha}_{n}\} is a sequence in (0,1] and \{{\beta}_{n}\} is a sequence in (0,1) satisfying (C1), (C2), and (C4). Then the following statements hold:

(a)
The sequence \{{x}_{n}\} generated by (3.12) remains in the closed ball {B}_{R}[{x}^{\ast}], where
R=max\{\parallel {x}_{1}{x}^{\ast}\parallel ,\parallel {x}^{\ast}\parallel \}+\sum _{n=1}^{\mathrm{\infty}}(1{\alpha}_{n})\parallel {T}_{n}{x}^{\ast}{x}^{\ast}\parallel . 
(b)
If \overline{\rho}=max\{{max}_{1\le j\le p}\parallel {A}_{j}x\parallel ,\parallel (I\gamma \mathrm{\nabla}\psi )x\parallel :x\in {B}_{R}[{x}^{\ast}]\} and for each i\in \{1,2,\dots ,p\} and j\in \{1,2,\dots ,r\}, the assumption (C8) holds, then \{{x}_{n}\} converges strongly to {x}^{\ast}.
4 Numerical examples
In order to demonstrate the effectiveness, realization, and convergence of algorithm of Theorem 3.2, we consider the following example.
Example 4.1 Let H=\mathbb{R} and C=[0,1]. Let T be a selfmapping on C defined by Tx=1x for all x\in C. Define \{{\alpha}_{n}\} in (0,1) by {\alpha}_{n}=\frac{1}{n+1} and \{{\beta}_{n}\} by {\beta}_{n}=\frac{1}{2} for all n\in \mathbb{N}. For each n\in \mathbb{N}, define {f}_{n}:C\to H by (2.1). It is shown in Example 2.1 that \{{f}_{n}\} is a sequence of nearly contraction mappings from C into H with sequence \{({k}_{n},{a}_{n})\}, where {k}_{n}=\frac{1}{n+1} and {a}_{n}=\frac{5}{2(n+1)}. It is easy to see that \{{f}_{n}\} converges pointwise to f, where f(x)=0 for all x\in C. Note {k}_{n}\to \kappa =0, F(T)=\{{x}^{\ast}\}=\{1/2\}, and {lim}_{n\to \mathrm{\infty}}{f}_{n}{x}^{\ast}=f{x}^{\ast}. It can be observed that all the assumptions of Theorem 3.2 are satisfied and the sequence \{{x}_{n}\} generated by (3.1) with {T}_{n}=T converges to \frac{1}{2}. In fact, under the above assumptions, the algorithm (3.1) can be simplified as follows:
The projection point of {y}_{n} onto C can be expressed as
The iterates of algorithm (4.1) for initial guess {x}_{1}=0,0.2,0.8,1 are shown in Table 1. From Table 1, we see that the iterations converge to 1/2 which is the unique fixed point of T. The convergence of each iteration is also shown in Figure 1 for comparison.
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This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the authors acknowledge with thanks DSR, for technical and financial support.
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Latif, A., Sahu, D.R. & Ansari, Q.H. Variable KMlike algorithms for fixed point problems and split feasibility problems. Fixed Point Theory Appl 2014, 211 (2014). https://doi.org/10.1186/168718122014211
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DOI: https://doi.org/10.1186/168718122014211
Keywords
 split feasibility problems
 fixed point problems
 regularized algorithms
 convergence analysis of algorithms