A general hybrid relaxed CQ algorithm for solving the fixed-point problem and split-feasibility problem

1.   Introduction

The split feasibility problem (SFP) in Euclidean spaces was introduced by Censor and Elfving^[1] in 1994. It is depicted as finding a point $x$ such that

where $C\subset H_{1}$ and $Q\subset H_{2}$ are nonempty closed convex sets; $A$ is a bounded linear operator from a Hilbert space $H_{1}$ onto a Hilbert space $H_{2}$ with $A\neq0$. Denote the set of solutions for (1.1) by $\Delta$. The SFP plays an important role in the study of medical image reconstruction, signal processing, etc. Many scholars regard the SFP and its generalizations, such as the multiple-set SFP and the split common-fixed-point problem as their research direction (see ^[2,3,4,5,6]).

It is known that Byrne ^[7,8] proposed the famous $CQ$ algorithm (CQA) for solving the problem (1.1). The proposed method is given as follows:

where $\tau_{n}\in\left(0, \dfrac{2}{\|A\|^{2}}\right)$ is the step size, $A^{*}$ is the adjoint operator of $A$, $P_{C}$ is the projection onto $C$ and $P_{Q}$ is the projection onto $Q$. Regrettably, there are two drawbacks of applying (1.2). On one hand, estimating the operator norm can often be challenging. On the other hand, computing projections onto both $C$ and $Q$ can be very difficult.

To overcome the first drawback, López et al. ^[9] proposed a novel approach for selecting the step size, denoted by $\tau_{n}$, which is defined as follows:

where $f(x_{n}): = \dfrac{1}{2}\|(I-P_{Q})Ax_{n}\|^{2}$, $\nabla f(x_{n}) = A^{*}(I-P_{Q})Ax_{n}$, $\rho_{n}\in(0, 4)$ and $\inf_{n\in\mathbb{N}}\rho_{n}(4-\rho_{n}) > 0$.

To overcome the second drawback, Fukushima ^[10] introduced the level sets $C$ and $Q$:

where $\phi:H_{1}\rightarrow \mathbb{R}$ and $\varphi:H_{2}\rightarrow \mathbb{R}$ are convex, subdifferential and weakly lower semi-continuous functions. Motivated by Fukushima's method, Yang ^[11] introduced a new relaxed CQA by replacing $P_{C}$ with $P_{C_{n}}$ and $P_{Q}$ with $P_{Q_{n}}$. The algorithm has the form

where $\alpha_{n}\in (0, \frac{2}{\|AA^{*}\|})$, $A^{*}$ is the adjoint operator of $A$ and $C_{n}$ and $Q_{n}$ are defined as follows:

and

where $\partial \phi$ and $\partial \varphi$ are bounded operators.

Qin and Wang ^[12] introduced (1.4) to find a common solution of the SFP (1.1) and fixed-point problem in 2019. Their algorithm has the form

where $g$ is a $k$-contraction from $C$ onto $C$, S is nonexpansive from $C$ onto $C$ and $\rm Fix$$(S): = \{x\in H:Sx = x\}$. $\{\alpha_{n}\}$, $\{\beta_{n}\}$, $\{\gamma_{n}\}$, $\{\delta_{n}\}$ and $\{\tau_{n}\}$ are included in $(0, 1)$; they satisfy the following:

(i) $\sum_{n = 1}^{\infty}\alpha_{n} = \infty$, $\lim\limits_{n\rightarrow \infty}\alpha_{n} = 0$;

(ii) $0 < \liminf\limits_{n\rightarrow \infty} \beta_{n} \leq \limsup\limits_{n\rightarrow \infty} \beta_{n} < 1$;

(iii) $\alpha_{n}+\beta_{n}+\gamma_{n} = 1$;

(iv) $\lim\limits_{n \rightarrow \infty}|\tau_{n}-\tau_{n+1}| = 0$, $0 < \liminf\limits_{n\rightarrow \infty}\tau_{n} \leq \limsup\limits_{n\rightarrow \infty}\tau_{n} < \frac{2}{\|A\|^{2}}$;

(v) $\lim\limits_{n\rightarrow \infty}|\delta_{n}-\delta_{n+1}| = 0$, $0 < \liminf\limits_{n\rightarrow \infty}\delta_{n}\leq \limsup\limits_{n\rightarrow \infty}\delta_{n} < 1$.

Then, there is a point $q$ of $\Delta \cap {\rm Fix}(S)$ with $x_{n}\rightarrow q$. Furthermore, it satisfies (1.5):

Motivated by López et al. ^[9], Wang et al. ^[13] improved the selection of step size in (1.4). In order to accelerate the convergence rate, they introduced (1.6) to solve the SFP (1.1) and fixed-point problem. Given that $x_{0}, x_{1}\in H$, their algorithm has the form

where $\mu\geq0$, $\tau_{n} = \dfrac{\rho_{n}f(w_{n})}{\|\nabla f(w_{n})\|^{2}}$, $0 < \rho_{n} < 4$, $f(w_{n}): = \dfrac{1}{2}\|(I-P_{Q})Aw_{n}\|^{2}$ and $\nabla f(w_{n}) = A^{*}(I-P_{Q})Aw_{n}$. Assume that $S:H_{1}\rightarrow H_{1}$ is quasi-nonexpansive, $I-S$ is demiclosed at zero and $g:H_{1}\rightarrow H_{1}$ is a $k$-contraction. $\{\alpha_{n}\}$, $\{\beta_{n}\}$, $\{\gamma_{n}\}$ and $\{\delta_{n}\}$ are included in $[0, 1]$, which satisfy the following conditions:

(i) $\sum_{n = 1}^{\infty}\alpha_{n} = \infty$, $\lim\limits_{n\rightarrow \infty}\alpha_{n} = 0$;

(ii) $\limsup\limits_{n\rightarrow \infty} \beta_{n} < 1$;

(iii) $\alpha_{n}+\beta_{n}+\gamma_{n} = 1$;

(iv) $0 < \liminf\limits_{n\rightarrow \infty}\delta_{n}\leq \limsup\limits_{n\rightarrow \infty}\delta_{n} < 1$;

(v) $\varepsilon_{n} > 0$, $\lim\limits_{n \rightarrow \infty} \frac{\varepsilon_{n}}{\alpha_{n}} = 0$.

Then, there is a point $q$ of ${\rm Fix}(S)\cap \Delta$ with $x_{n}\rightarrow q$. Furthermore, it satisfies (1.7):

In addition, Tian ^[14] introduced an iterative form:

Suppose that $\rm Fix$$(S)\neq\emptyset$, where $\rm Fix$$(S): = \{x\in H:Sx = x\}$. Then, there is a point $q$ in $\rm Fix$$(S)$ with $x_{n}\rightarrow q$. Furthermore, it satisfies the following:

where $B$ is $\overline{\lambda}$-strongly monotone and $M$-Lipschitz continuous from $H$ onto $H$ with $\overline{\lambda}, M > 0$, $S$ is nonexpansive from $H$ onto $H$ and $g$ is a $k$-contraction from $H$ onto $H$ with $0 < k < 1$. Let $\iota, \lambda\in\mathbb{R}$ and $\{\alpha_{n}\}\subset(0, 1)$ satisfy the following:

(i) $0 < \iota < \dfrac{2\overline{\lambda}}{M^{2}}$;

(ii) $0 < \lambda < \iota\dfrac{\overline{\lambda}-\frac{M^{2}\iota}{2}}{k}$;

(iii) $\sum_{n = 1}^{\infty}|\alpha_{n+1}-\alpha_{n}| < \infty$, $\sum_{n = 1}^{\infty}\alpha_{n} = \infty$, $\lim\limits_{n\rightarrow \infty}\alpha_{n} = 0$.

Motivated by the algorithms developed by Qin and Wang ^[12], Wang et al. ^[13], Tian ^[14] and Kwari et al. ^[15], we propose a new half-space relaxation projection method for solving the SFP and fixed-point problem of demicontractive mappings by expanding the scope of operator $S$, modifying the iteration format and optimizing the selection of step sizes. Our result in this article extends and improves many recent correlative results of other authors. Particularly, our method extends and improves the methods in some papers of other authors in the following ways: (1) We have considered and studied a new modified half-space hybrid method to solve the fixed-point problem and SFP of nonlinear operators at the same time. And, our method can be used more widely; (2) We extend the nonexpansive mapping and the quasi-nonexpansive mapping to the demicontractive mapping, which expands the scope of the study; (3) The inertia is added to accelerate the convergence rate further; (4) The selection of the step size is a multistep and self-adaptive process, so it no longer depends on the operator norm; (5) The projection onto a half-space greatly facilitates the calculation, and under some weaker conditional assumptions, the iterative sequence generated by our new algorithm converges strongly to the common solution of the two problems. At last, we present some numerical experiments to show the effectiveness and feasibility of our new iterative method.

2.   Preliminaries

In this paper, let $H$ be a real Hilbert space; the inner product be denoted by $\langle\cdot, \cdot\rangle$ and the norm be denoted by $\|\cdot\|$. We use the symbol $\rightharpoonup$ to indicate weak convergence and $\rightarrow$ to indicate strong convergence.

Definition 2.1. A self-mapping $P$ is said to be

${\rm (i)}$ nonexpansive if

${\rm (ii)}$ quasi-nonexpansive if ${\rm Fix}(P)\neq\emptyset$ such that

${\rm (iii)}$ $\alpha$ -demicontractive ($0\leq \alpha < 1$) if ${\rm Fix}(P)\neq\emptyset$ such that

${\rm (iv)}$ firmly nonexpansive if

${\rm (v)}$ $t$-contractive ($0\leq t < 1$) if

${\rm (vi)}$ $M$-Lipschitz continuous ($M > 0$) if

${\rm (vii)}$ $\beta$-strongly monotone ($\beta\geq0$) if

When $\rm Fix(P)\neq\emptyset$, we obtain (i)$\Rightarrow$(ii)$\Rightarrow$(iii).

Definition 2.2. When the function $f:H\rightarrow \mathbb{R}$ is convex and subdifferentiable, an element $d\in H$ is called a subgradient of $f(x_{0})$ if

The set of subgradients of $f$ at $x_{0}$ is denoted by $\partial f(x_{0})$.

Definition 2.3. The function $f:H\rightarrow \mathbb{R}$ is called weakly lower semi-continuous if $x_{n}\rightharpoonup x_{0}$ implies that

Lemma 2.1 (^[16,17,18]). Let $D\subset H$ be a nonempty, convex and closed set. For any $x\in H$, we have

${\rm (i)}$ $\|P_{D}x-y\|^{2}\leq\|x-y\|^{2}-\|x-P_{D}x\|^{2}$, $\quad \forall y\in D$;

${\rm (ii)}$ $\langle x-P_{D}x, y-P_{D}x\rangle\leq0$, $\quad \forall y\in D$;

${\rm (iii)}$ $\|P_{D}x-P_{D}y\|^{2}\leq\langle P_{D}x-P_{D}y, x-y\rangle$, $\quad \forall y\in H$;

${\rm (iv)}$ $\|(I-P_{D})x-(I-P_{D})y\|^{2}\leq\langle(I-P_{D})x-(I-P_{D})y, x-y\rangle$, $\quad \forall y\in H$.

Lemma 2.2 (^[19,20]). For any $x, y\in H$, the following results hold

${\rm (i)}$ $\|x+y\|^{2}\leq 2\langle y, x+y\rangle+\|x\|^{2}$;

${\rm (ii)}$ $\|(1-m)x+my\|^{2} = m\|y\|^{2}+(1-m)\|x\|^{2}-(1-m)m\|x-y\|^{2}$, $\quad$ $m\in \mathbb{R}$.

Lemma 2.3 (^[14]). Assume that $B$ is an $M$-Lipschitz continuous and $\overline{\lambda}$-strongly monotone operator with $\overline{\lambda}, M > 0$. For $\mu, \nu > 0$, they satisfy the following conditions:

Let $\{\gamma_{n}\}\subset(0, 1)$ and $\lim\limits_{n\rightarrow \infty}\gamma_{n} = 0$. If $n$ is a sufficiently large number, we have

Proof.

Combining $\lim\limits_{n\rightarrow \infty}\gamma_{n} = 0$ and $\mu-\gamma_{n} > 0$ ($n$ sufficiently large), we have

Since $1-\gamma_{n}\nu > 0$ ($n$ sufficiently large), we have

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Lemma 2.4 (^[21]). Let the self-mapping $S$ be $\alpha$-demicontractive in $H$ and ${\rm Fix}(S)\neq\emptyset$. Define $S_{\sigma} = (1-\sigma)I+\sigma$S, $\sigma\in(0, 1-\alpha)$; then, we have the following:

${\rm (i)} {\rm Fix}(S) = {\rm Fix}(S_{\sigma})\subset H$ is convex and closed;

${\rm (ii)}$ $\|S_{\sigma}x-q\|^{2}\leq\|x-q\|^{2}-\dfrac{1}{\sigma}(1-\alpha-\sigma)\|(I-S_{\sigma})x\|^{2} \quad \forall x\in H$, $q\in{\rm Fix}(S)$.

Lemma 2.5 (^[8]). Let $f(z) = \frac{1}{2}\|(I-P_{Q})Az\|^{2}$. We can know that $\nabla f$ is $M$-$Lipschitz$ continuous with $M = \|A\|^{2}$.

Lemma 2.6 (^[22]). Let $\{a_{n}\}$ be a sequence of nonnegative numbers such that

where $n_{0}$ is a sufficiently large integer, the sequence $\{\Theta_{n}\}$ is contained in $(0, 1)$, the real sequence $\{\Pi_{n}\}$ is nonnegative $\{\Upsilon_{n}\}$ and $\{\Gamma_{n}\}$ are two sequences which are included in $\mathbb{R}$. If the following conditions hold

$(\rm i) \sum_{n = 0}^{\infty} \Theta_{n} = \infty$; $(\rm ii) \lim\limits_{n\rightarrow \infty} \Gamma_{n} = 0$; $(\rm iii) \lim\limits_{i\rightarrow \infty}\Pi_{n_{i}} = 0$ $\Rightarrow$ $\limsup\limits_{i \rightarrow \infty}\Upsilon_{n_{i}} \leq 0$, $\forall \{n_{i}\}\subseteq \{n\}$, then $\lim\limits_{n\rightarrow \infty} a_{n} = 0$.

3.   Results

In this section, $C$ and $Q$ are defined by (1.3). We give some conditions for the convergence analysis of our Algorithm 1.

$({\rm C}_{1})$ $\sum_{n = 1}^{\infty}\alpha_{n} = \infty$, $\lim\limits_{n\rightarrow \infty}\alpha_{n} = 0$, $0 < \alpha_{n} < 1$;

$({\rm C}_{2})$ $\beta_{n}\in [0, 1]$, $\limsup\limits_{n\rightarrow \infty} \beta_{n} < 1$;

$({\rm C}_{3})$ $\varepsilon_{n} > 0$, $\lim\limits_{n \rightarrow \infty} \frac{\varepsilon_{n}}{\alpha_{n}} = 0$;

$({\rm C}_{4})$ $0 < \liminf\limits_{n\rightarrow \infty}\delta_{n}\leq\limsup\limits_{n\rightarrow \infty}\delta_{n} < 1$;

$({\rm C}_{5})$ $\sigma_{n}\in(0, 1-\alpha]$, $\liminf\limits_{n\rightarrow \infty}\sigma_{n} > 0$;

$({\rm C}_{6})$ $\inf_{n\in\mathbb{N}}\rho_{n}(4-\rho_{n}) > 0$, $\{\rho_{n}\}\subset(0, 4)$.

Theorem 3.1. Let $S: H_{1}\rightarrow H_{1}$ be $\alpha$-demicontractive, $I-S$ be demiclosed at zero and $g:H_{1}\rightarrow H_{1}$ be $k$-contractive. Moreover, let operator $B$ be $L$-Lipschitz continuous and $\overline{\lambda}$-strongly monotone. Assume that $\{\alpha_{n}\}$, $\{\beta_{n}\}$, $\{\delta_{n}\}$, $\{\sigma_{n}\}$, $\{\rho_{n}\}$ are sequences satisfying $(C_{1})-(C_{6})$ and ${\rm Fix}(S)\cap \Delta \neq\emptyset$. Then, the $\{x_{n}\}$ generated by Algorithm 1 converges strongly to $x^{*}\in{\rm Fix}(S)\cap \Delta$. Furthermore, it is the unique solution of (3.1):

Proof. Let $r \in\Delta\cap{\rm Fix}(S)$. Since $C\subset C_{n}$ and $Q\subset Q_{n}$, we have that $r\in C_{n}$, $Ar\in Q_{n}$ and $r\in\rm {Fix}(S)$. By combining Lemma 2.1 with Lemma 2.2, we get

Set

according to Lemma 2.4, we have

From the definition of $\nabla f_{n}(d_{n})$, we have

which means that when $\|\nabla f_{n}(d_{n})\| = 0$, we get that $f_{n}(d_{n}) = 0$. Substituting (3.3) into (3.2), we have

Noting that $\{\delta_{n}\}$ is a sequence in $(0, 1)$ and $\rho_{n} \in (0, 4)$, we derive that

From the definition of $d_{n}$, we have

In the rest of the proof, $n_{0}$ is assumed to be a sufficiently large positive integer and $n\geq n_{0}$.

Now, let $\gamma_{n}: = \frac{\alpha_{n}}{1-\beta_{n}}$. By Lemma 2.3, (3.5) and (3.6), we estimate $\|x_{n+1}-r\|$:

We can assume that $M$ is a suitable positive number such that $\frac{\varepsilon_{n}}{\alpha_{n}}\leq M$ since $\lim\limits_{n\rightarrow \infty}\frac{\varepsilon_{n}}{\alpha_{n}} = 0$. Next, we have

Therefore, the sequence $\{x_{n}\}$ is bounded.

From the definition of $x_{n+1}$,

set

then, we obtain

In view of the arbitrariness of $r$, Lemma 2.2, Lemma 2.3, (3.5) and (3.7), we have

According to the definition of $\{d_{n}\}$ and Lemma 2.2, we have

Next, we can estimate $\|x_{n+1}-x^{*} \|^{2}$. On the one hand, in view of (3.6), (3.8), (3.9), (3.10) and Lemma 2.2, we obtain

On the other hand, using the definition of $\{z_{n}\}$, Lemma 2.2, (3.4) and (3.10), we get

Apply the following:

Thus, we can rewrite (3.11) and (3.12):

It is easy for us to know that $\lim\limits_{n\rightarrow \infty}\Theta_{n} = 0$, $\sum_{n = 1}^{\infty}\Theta_{n} = \infty$ and$\lim\limits_ {n\rightarrow \infty}\Gamma_{n} = 0$. If we prove that $\limsup\limits_{i\rightarrow \infty}\Upsilon_{n_{i}} \leq 0$ when $\lim\limits_{i\rightarrow \infty}\Pi_{n_{i}} = 0$ for any subsequence $\{n_{i}\} \subset \{n\}$, we can get that $\lim\limits_{n\rightarrow \infty}\|x_{n}-x^{*}\| = 0$ by Lemma 2.6.

Assume that

Using the conditions of $\{\beta_{n}\}$, $\{\delta_{n}\}$ and $\{\sigma_{n}\}$, we have

From (3.13) and the conditions of $\{\rho_{n}\}$, we deduce that

If $\|\nabla f_{n_{i}}(d_{n_{i}})\|\neq0$, we have that $f_{n_{i}}(d_{n_{i}})\neq0$ and

which means that $\inf_{i\in\mathbb{N}}\tau_{n_{i}} > 0$. Hence, we obtain $f_{n_{i}}(d_{n_{i}})\rightarrow0$ as $i\rightarrow \infty$. This implies that

Using (3.16) and the conditions of $\{\rho_{n}\}$, we have

Moreover, from (3.14) and the condition of $\{\sigma_{n}\}$, we get

Using (3.15) and the definition of $y_{n}$, we have

again using (3.15), we obtain

i.e.,

Due to (3.17) and (3.19), we get

i.e.,

Then, we have

combining (3.18) with (3.20), we obtain

Moreover, using the definition of $\{\mu_{n}\}$ and the boundedness of $\{x_{n}\}$, we can have

Next, combining (3.21) and (3.22), we have

It is easy for us to know that $\omega_{w}(d_{n_{i}}) \subset{\rm Fix}(S)$ by combining (3.18) and the case that $I-S$ is demiclosed at zero. Now, we choose a subsequence $\{d_{n_{i_{j}}}\}$ of $\{d_{n_{i}}\}$ to satisfy

Without loss of generality, we suppose that $d_{n_{i_{j}}}\rightharpoonup z'$. It is easy to get that $y_{n_{i_{j}}}\rightharpoonup z'$ and $x_{n_{i_{j}}}\rightharpoonup z'$ by using (3.21) and (3.23). Now, we show that $z'\in C$. We know that $y_{n_{i_{j}}}\in C_{n_{i_{j}}}$by the definition of $y_{n_{i_{j}}}$; this implies that

where $\vartheta_{n_{i_{j}}}\in \partial \phi(x_{n_{i_{j}}})$. From (3.23) and the boundedness of $\partial \phi$, we have

as $n\rightarrow \infty$. Because of the weakly lower semi-continuous nature of $\phi$, $x_{n_{i_{j}}}\rightharpoonup z'$ and (3.25), the following holds:

Hence $z'\in C$.

Next, we show that $Az'\in Q$. Since $P_{Q_{n_{i_{j}}}}(Ax_{n_{i_{j}}})\in Q_{n_{i_{j}}}$, we have

where $\chi_{n_{i_{j}}}\in\partial \varphi(Ax_{n_{i_{j}}})$. Given that $\lim\limits_{i\rightarrow \infty}\|(I-P_{Q_{n_{i}}})Ad_{n_{i}}\| = 0$ and (3.22), we get that $\lim\limits_{i\rightarrow \infty}\|(I-P_{Q_{n_{i}}})Ax_{n_{i}}\| = 0$. By (3.26) and the boundedness of $\partial \varphi$, we have

as $n\rightarrow \infty$. In view of $x_{n_{i_{j}}}\rightharpoonup z'$, $Ax_{n_{i_{j}}}\rightharpoonup Az'$, the weakly lower semi-continuous nature of $\varphi$ and (3.27), we can get

Hence $Az'\in Q$.

From the above proof process, we can derive that $z'\in{\rm Fix}(S)\cap\Delta$. Now, using (3.21), we get

It means that

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4.   Numerical experiments

Now, we present a comparison of Algorithm 1 with other algorithms by describing two numerical experiments. All of the programs were written in Matlab 9.5 and performed on a desktop PC with AMD Ryzen 7 4800U with Radeon Graphics 1.80 GHz, RAM 16.0GB.

Example 4.1. We assume that $H_{1} = H_{2} = C = \mathbb{R}^{4}$ and $Q = \{b\}$, where $b\in\mathbb{R}^{4}$; then, we have that $\Delta = \{x\in \mathbb{R}^{4}: Ax = b\}$. Considering the system of linear equations $Ax = b$, let

By calculation, we can find that $Ax = b$ has a unique solution $x^{*} = (-1, -2, 1, 2)^{\mathrm{T}}$. Let

By calculation, we can find that the mapping $S$ is nonexpansive and $x^{*} = (-1, -2, 1, 2)^{\mathrm{T}}\in{\rm Fix}(S)$.

In Algorithm 1 and scheme (1.4), we let $\alpha_{n} = \frac{1}{10n}$, $\beta_{n} = 0.2$, $\delta_{n} = 0.5$, $x_{1} = (1, 2, 3, 4)^{\mathrm{T}}$ and $g = 0.2I$; in scheme (1.4), we let $\tau_{n} = \frac{7}{4\|A\|^{2}}$; in Algorithm 1, we let $\phi(x) = 0$, $\forall x\in \mathbb{R}^4$, $\varphi(y) = \|y-b\|$, $\forall y\in\mathbb{R}^4$, $B = I$, $\lambda = 1$, $\varepsilon_{n} = \frac{1}{n^{2}}$, $\mu = 1.5$, $\sigma_{n} = 1$, $\rho_{n} = 3.5$, $\tau = 1$ and $x_{0} = (1, 2, 3, 4)^{\mathrm{T}}$. The results of numercial experiments are revealed in Table 1 and Figure 1.

From Table 1, we can find that $x_{n}$ is closer to the exact solution $x^{*}$ with an increase of the number of iterations. We can also see that errors are closer to 0. Therefore, it can be concluded that Algorithm 1 is feasible. From Figure 1, we can find that the number of iterations for Algorithm 1 is less than that for scheme (1.4).

Example 4.2. Assume that $H_{1} = \mathbb{R}$, $H_{2} = \mathbb{R}^{3}$, $C = [-2, 6]$, $Q = \{(y_{a}, y_{b}, y_{c})^{\mathrm{T}}:|y_{a}|+|y_{b}|+|y_{c}|\leq3 \}$,

We can see that $S$ is not nonexpansive but quasi-nonexpansive ^[23]. By calculation, we can find that ${\rm Fix}(S) = \{0\}$. It is easy to obtain that $0\in\Delta$. We denote $x^{*} = 0$.

In Algorithm 1 and scheme (1.6), we let $x_{0} = x_{1} = 1$, $\varepsilon_{n} = \frac{1}{n^{2}}$, $\mu = 0.9$, $\beta_{n} = 0.5$, $\rho_{n} = 3.5$, $\delta_{n} = 0.4$, $g(x) = \frac{1}{3}\sin x$ and $\alpha_{n} = \frac{1}{8n^{p}}$, where $0 < p\leq1$; in scheme (1.6), we used the function quadprog to compute the projection over $Q$ by using Matlab 9.5 Optimization Toolbox; In Algorithm 1, we let $\phi(x) = (x-6)(x+2)$, $\forall x\in\mathbb{R}$, $\varphi(y) = |y_{a}|+|y_{b}|+|y_{c}|-3$, $\forall y = (y_{a}, y_{b}, y_{c})^\mathrm{T}\in \mathbb{R}^3$, $B = I$, $\lambda = 1$, $\sigma_{n} = 1$ and $\tau = 1$. We used $\|x_n-x^*\|\leq 10^{-8}$ as the stopping criterion. The results of numerical experiments for different values of $p$ are shown in Table 2. The convergence behavior for $p = 0.5$ is shown in Figures 2 and 3.

5.   Conclusions

In this work, we developed a method for solving the SFP and the fixed-point problem involving demicontractive mapping. In Algorithm 1, we have expanded the nonexpansive mapping to the demicontractive mapping, selected a new step size and added the inertial method based on scheme (1.4). Our Algorithm 1 exhibits better convergence behavior than scheme (1.4). Additionally, we have extended the quasi-nonexpansive mapping to the demicontractive mapping, applied projection onto a half-space and selected a new step size in our Algorithm 1 based on scheme (1.6). Our Algorithm 1 also demonstrated superior convergence behavior compared to scheme (1.6). From the above, our algorithm exhibits a superior convergence rate and a wider range for the operator, thus making it applicable in a broader range of scenarios. Furthermore, we have presented two numerical results that demonstrate the feasibility and effectiveness of our method. In the following work, we intend to extend the demicontractive mapping to the quasi-pseudocontractive mapping and extend the SFP to the multiple-set split feasibility problem.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 12171435).

Conflict of interest

The authors declare that they have no competing interests.