# Variable KM-like algorithms for fixed point problems and split feasibility problems

- Abdul Latif
^{1}Email author, - Daya R Sahu
^{2}and - Qamrul H Ansari
^{3, 4}

**2014**:211

https://doi.org/10.1186/1687-1812-2014-211

© Latif et al.; licensee Springer. 2014

**Received: **9 May 2014

**Accepted: **16 September 2014

**Published: **14 October 2014

## Abstract

The convergence analysis of a variable KM-like 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 KM-like technique is applied to solve the split feasibility problem and the multiple-sets split feasibility problem. Especially, the minimum norm solutions of the split feasibility problem and the multiple-sets 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.

## Keywords

## 1 Introduction

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:

*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

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 intensity-modulated radiation therapy; see, for example [2–7] and the references therein.

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 $0<\gamma <2/L$.

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 *ρ*-distance 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).

*fixed point problem*(FPP): finding an element

*x*such that

*T*is a nonexpansive self-mapping 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 ill-posed (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 ${T}_{t}^{f}:=tf+(1-t)T$, 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

*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:

where $f:C\to H$ is a fixed non-self 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.

*variable KM-like 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 self-mappings of *C* and $\{{f}_{n}\}$ is a sequence of (not necessarily contraction) mappings from *C* into *H*.

- (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 pseudo-contraction 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

*C*be a nonempty subset of a Hilbert space

*H*. Throughout the paper, we denote by ${B}_{r}[x]$ the closed ball defined by ${B}_{r}[x]=\{y\in C:\parallel y-x\parallel \le r\}$. 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 *κ*-*contraction* if there exists $\kappa \in [0,1)$ such that $\parallel Tx-Ty\parallel \le \kappa \parallel x-y\parallel $ for all $x,y\in C$. Furthermore, it is called *nonexpansive* if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel $ for all $x,y\in C$.

*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:

where ${a}_{n}:=\frac{5}{2(n+1)}$. Therefore, $\{{f}_{n}\}$ is a sequence of nearly contraction mappings with sequence $\{({k}_{n},{a}_{n})\}$.

*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* $I-T$ *is demiclosed at zero*, *that is*, *if* $\{{x}_{n}\}$ *is a sequence in* *C* *weakly converging to* *x* *and if* $\{(I-T){x}_{n}\}$ *converges strongly to zero*, *then* $(I-T)x=0$.

**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 KM-like 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* *κ*-*contraction*. *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 *κ*-contraction, it follows that ${P}_{F(T)}f$ is a *κ*-contraction 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 KM-like algorithm

*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 self-mappings 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 KM-like 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* *κ*-*contraction 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

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*.

it follows that $\{{f}_{n}{x}_{n}\}$ is bounded.

Step 2: ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-{x}_{n+1}\parallel =0$.

Step 3: ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-T{x}_{n}\parallel =0$.

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: ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\gamma}_{n}\le 0$.

*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 ${lim}_{n\to \mathrm{\infty}}\frac{\parallel {f}_{n}{x}^{\ast}-f{x}^{\ast}\parallel}{1-{k}_{n}}=0$ 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 ‘${\sum}_{n=1}^{\mathrm{\infty}}\parallel {T}_{n}z-Tz\parallel <\mathrm{\infty}$ 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*

*κ*-

*contraction 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*

*κ*-

*contraction 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. □

*H*with ${u}_{n}\to u\in H$, define ${f}_{n}:C\to H$ by

**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 *ρ*-distance:

**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

*ρ-distance*between

*C*and

*Q*is defined by

By employing Theorem 3.5, we present a variable KM-like 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*

*κ*-

*contraction 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 KM*-

*like 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).

Hence condition (C6) in Theorem 3.5 holds. Therefore, Theorem 3.9(b) follows from Theorem 3.5(b). □

*variable KM-like CQ algorithm*:

We now present strong convergence of the variable KM-like 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 KM*-

*like 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 multiple-sets split feasibility problem

In this section, we consider the following *multiple-sets split feasibility problem* which models the intensity-modulated radiation therapy [6] and has recently been investigated by many researchers, see, for example, [1, 3, 6, 8–14] and the references therein.

*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) multiple-sets 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 multiples-set split feasibility problem (MSSFP) model image retrieval [28] and intensity-modulated radiation therapy [6], and they have recently been investigated by many researchers.

*ψ*of a convex and continuously differentiable function $\psi :{H}_{1}\to \mathbb{R}$ defined by

*B*is a Lipschitzian mapping with Lipschitz constant

Thus, variable KM-like 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*

*κ*-

*contraction 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

*ψ*and ${\psi}_{n}$ are given by

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).

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 KM-like 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 KM*-

*like 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 self-mapping on

*C*defined by $Tx=1-x$ 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:

*C*can be expressed as

*T*. The convergence of each iteration is also shown in Figure 1 for comparison.

**The numerical results for initial guess**
${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{0.2}\mathbf{,}\mathbf{0.8}\mathbf{,}\mathbf{1}$

n | ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0}$ | ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0.2}$ | ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0.8}$ | ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{1}$ |
---|---|---|---|---|

5 | 0.452291666666667 | 0.452604166666667 | 0.514843750000000 | 0.514947916666667 |

10 | 0.476041066961784 | 0.476041067553836 | 0.476047944746260 | 0.476047944894273 |

15 | 0.483829591828978 | 0.483829591828978 | 0.483829591829845 | 0.483829591829845 |

20 | 0.487787940564059 | 0.487787940564059 | 0.487787940564059 | 0.487787940564059 |

25 | 0.490187554131032 | 0.490187554131032 | 0.490187554131032 | 0.490187554131032 |

30 | 0.491798400218960 | 0.491798400218960 | 0.491798400218960 | 0.491798400218960 |

35 | 0.492954698188619 | 0.492954698188619 | 0.492954698188619 | 0.492954698188619 |

40 | 0.493825130132048 | 0.493825130132048 | 0.493825130132048 | 0.493825130132048 |

45 | 0.494504074844059 | 0.494504074844059 | 0.494504074844059 | 0.494504074844059 |

50 | 0.495048473664881 | 0.495048473664881 | 0.495048473664881 | 0.495048473664881 |

## Declarations

### Acknowledgements

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.

## Authors’ Affiliations

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