A new fixed-time stability of neural network to solve split convex feasibility problems

Abstract

In this paper, we propose a novel neural network that achieves stability within the fixed time (NFxNN) based on projection to solve the split convex feasibility problems. Under the bounded linear regularity assumption, the NFxNN admits a solution of the split convex feasibility problem. We introduce the relationships between NFxNN and the corresponding neural networks. Additionally, we also prove the fixed-time stability of the NFxNN. The convergence time of the NFxNN is independent of the initial states. The effectiveness and superiority of the NFxNN are also demonstrated by numerical experiments compared with the other methods.

1 Introduction

Since the split convex feasibility problem (SCFP) was introduced in [1] in 1994, the SCFP was widely used in retrieval problems [1], image restoration [2], signal processing [3], IMRT [4], systems biology [5]. Many numerical algorithms have been proposed to solve the split convex feasibility problem (see, e.g., [5–11]). The SCFP is as follows:

$$\begin{aligned} \text{find } x\in C, y\in Q \quad \text{such that}\quad Ax=y, \end{aligned}$$

(1)

where $C \subset \mathbb{R}^{n}$ and $Q \subset \mathbb{R}^{m}$ are two non-empty, closed and convex sets, and $A:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ is a linear mapping. Among them, some classical and semi-alternating CQ algorithms were proposed to solve the split convex feasible problems in [3, 7]. In fact, the CQ algorithm is a projected gradient method, and its constrained minimization problem is as follows:

$$\begin{aligned} \min_{x\in C}\frac{1}{2} \Vert Ax-P_{Q}Ax \Vert ^{2}. \end{aligned}$$

(2)

As we know, projection method only is related to the orthogonal projection that is easy to compute, and it does not require matrix inversion. The CQ algorithm is an efficient and popular method to solve SCFP. The split convex feasibility problem can be converted into the CQ algorithm with the following iterative format:

$$\begin{aligned} x_{k+1}=P_{C}\bigl[x_{k}- \lambda A^{T}(I-P_{Q}Ax_{k}\bigr],\quad k=0,1,2,\ldots, \end{aligned}$$

(3)

where $P_{C}$, $P_{Q}$ are the orthogonal projections onto C, Q, respectively, λ is suitable positive constant, $A^{T}$ is the transpose of A, and I is the identical operator,

In addition, a wide variety of continuous-time algorithms attracted increasing attention in areas such as optimization, artificial intelligence, image restoration, and optimal control (see, e.g., [12–15]). The dynamic system method was first proposed in [12] to solve the linear programming problem of the electronic analog computer. Subsequently, a recurrent neural network was presented to deal with the nonlinear projection formulation in [13]. To further solve the optimization problem, a new continuous dynamical system of a gradient-projection type was proposed and studied in [14]. Some neural network models were introduced to solve projection formulation in [15]. The continuation of the split convex feasibility problem has recently been studied. A projection dynamical system was proposed to solve the split equality problem in [16]. Continuous CQ algorithms were proposed, and an exponential convergence was obtained under the bounded linear regularity property in [17]. Although neural networks and their variants have been widely adopted, discrete algorithms generally focus on the analysis of convergence and complexity, and continuous algorithms mainly focus on the convergence rate, and theoretical research mainly deals with the asymptotic or exponential stability (see, e.g., [17–21]). Among the existing works on the CQ algorithm and variations for the SCFP, the convergence and convergence rate of algorithms have received great attention (see, e.g., [5, 22]), but a few papers deal with convergence time.

In practical production, more attention is paid to the convergence time of the method. As discussed in [23], the design of finite-time convergence based on asymptotic or exponential stability was of more practical significance than the classical asymptotic and exponential stability. Nevertheless, the finite-time terminating neural network method is highly dependent on the initial conditions. To overcome this drawback, the notion of convergence within a fixed time was introduced in [24]. Recently, some neural networks, which achieve convergence within a finite time or fixed time, were proposed to solve some mathematical models, such as $\ell _{1}$-minization problem, absolute value equation, and mixed variational inequalities (see, e.g., [25–31]). First, a novel proximal dynamical system, which achieves the fixed-time stability, was presented to deal with mixed variational inequality problems in [26]. For different problems, a new proximal neurodynamic network that is fixed-time converging (FXPNN) was proposed to solve equilibrium problems (EPs) in [28]. Recently, a dynamical system was released to deal with the split feasibility problem within the fixed time in [30]. To further decrease the convergence time of the dynamical system, a novel dynamical system that can update automatically, dynamical stepsize was established to deal with pseudo-monotone mixed variational inequalities in [31]. As far as we know, few papers exist for solving split convex feasible problems in fixed-time stability. We want to combine continuous algorithms with convergence time to make the algorithm more intuitive and efficient.

Motivated by the above works, we investigate a novel fixed-time stability neural network (NFxNN) to deal with the SCFP. Further, the fixed-time stability of the NFxNN is obtained under the bounded linear regularity assumption. Under mild conditions, we prove that the equilibrium point of the NFxNN globally converges to a solution of the SCFP. Besides, the convergence time of the NFxNN is uniformly bounded and is independent of any initial condition. Numerical examples present the efficiency and competitiveness of the NFxNN compared with the neural network systems.

Compared with the existing related work, the contributions of this paper can be summarized as follows:

• (i) To solve the SCFP, we propose a new fixed-time neural network model. The stability of the neural network is proved. It is proved that the NFxNN converges to the solution within a fixed time. The convergence time is uniformly bounded and is independent of the initial condition.

• (ii) The convergence results are compared with the relative methods (such as [17, 30]). We provide an upper bound formula for the convergence time independent of the initial condition. Compared with other methods, the NFxNN is faster to solve the SCFP. Examples in the paper show that the convergence time of the NFxNN is slightly faster than the fixed-time system in [30] and is faster than the neural network in [17].

• (iii) The relations between the parameters of the NFxNN and the fixed-time stability are discussed through the combination of theory and simulation. In this paper, the influence of parameters γ, $\alpha _{1}$, $\alpha _{2}$ on the NFxNN is presented. This provides a reference for us to choose the appropriate parameters when solving the SCFP with this model.

The rest of this article is structured as follows: Sect. 2 reviews useful notions and definitions. A new fixed-time stability neural network model and the consistent discretization of the NFxNN are presented, and its global convergence is proved in Sect. 3. Numerical examples also confirm the effectiveness of the method in Sect. 4. Section 5 gives a conclusion.

2 Preliminaries

In this paper, let $\mathbb{R}^{n}$ be the n-dimensional Euclidean space with the inner products $\langle \cdot ,\cdot \rangle$ and the Euclidean norm $\Vert \cdot \Vert$. Θ denotes the solution set of the split convex feasibility problem. For the sets $U\subseteq \mathbb{R}^{n}$ and $V \subseteq \mathbb{R}^{m}$, the family of k-times continuously differentiable functions from U to V is denoted by $C^{k}(U,V)$, i.e., $C^{k}(U,V):=\{f : f: U\to V \text{ is }k\text{-times continuously differentiable mapping}\}$. The supremum of a set S, denoted as $\sup (S)$, is the smallest real number M such that M is greater than or equal to every element in S.

2.1 Definitions and notions

We first review some common definitions and facts.

Definition 1

[32] Let $F:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ be a mapping. F is called

• (i) μ-Lipschitz continuous on $\mathbb{R}^{n}$ if there is a constant $\mu >0$ that satisfies

  $$\begin{aligned} \mu \Vert x-y \Vert \geq \bigl\Vert F(x)-F(y) \bigr\Vert ,\quad \forall x,y\in \mathbb{R}^{n}, \end{aligned}$$

  and F is said to be nonexpansive if it is 1-Lipschitz continuous.

• (ii) β strongly monotone on $\mathbb{R}^{n}$ if there is a constant $\beta >0$ to satisfy

  $$\begin{aligned} \bigl\langle F(x)-F(y), x-y\bigr\rangle \geq \beta \bigl\Vert F(x)-F(y) \bigr\Vert ^{2},\quad \forall x,y\in \mathbb{R}^{n} . \end{aligned}$$

  F is said to be firmly nonexpansive if it is 1-inverse strongly monotone.

• (iii) ϱ-averaged mapping if it is reformulated into

  $$\begin{aligned} F(x)=(1-\varrho )I+\varrho T, \end{aligned}$$

  where $\varrho \in (0,1)$, I is the identity mapping, and $T:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ denotes a nonexpansive mapping.

• (iv) Firmly nonexpansive iff

  $$\begin{aligned} \Vert x-y \Vert ^{2}- \bigl\Vert (I-F)x-(I-F)y \bigr\Vert ^{2}\geq \bigl\Vert F(x)-F(y) \bigr\Vert ^{2},\quad \forall x,y\in \mathbb{R}^{n} . \end{aligned}$$

Property 1

We give the nonexpansive projection mapping:

• (i) Both $P_{C}$ and $I-P_{C}$ are firmly nonexpansive, iff

  $$\begin{aligned} \bigl\langle x-P_{C}(x),y-P_{C}(x)\bigr\rangle \leq 0, \quad \forall x\in \mathbb{R}^{n}, y\in C, \end{aligned}$$

  and

  $$\begin{aligned} \bigl\langle P_{C}(x)-P_{C}(y),x-y\bigr\rangle \geq \bigl\Vert P_{C}(x)-P_{C}(y) \bigr\Vert ^{2}, \quad \forall x,y\in \mathbb{R}^{n} . \end{aligned}$$

• (ii) A mapping is said to be nonexpansive if it is averaged or firmly nonexpansive.

Definition 2

[17] Suppose that the solution set Θ of the SCFP is non-empty, the SCFP is called to satisfy the bounded linear regularity property if, for any $r > 0$, such that $\overline{\mathbb{B}(0,r)}\cap \Theta \neq \varnothing$, there is a $\varpi _{r}>0$ to satisfy

$$\begin{aligned} \operatorname{dis}(x,\Theta )\leq \varpi _{r} \operatorname{dis}(Ax,Q), \quad \forall x \in \overline{\mathbb{B}(0,r)}\cap C, \end{aligned}$$

(4)

where Θ is the solution set of the SCFP, and $\operatorname{dis}(\cdot ,\cdot )$ is the distance function, $\mathbb{B}(x,r)$ and $\overline{\mathbb{B}(x,r)}$ denote the open and closed ball with center at x and radius r, respectively.

We consider the following differential equation:

$$\begin{aligned} \dot{y}=f(y), \end{aligned}$$

(5)

where $f:\mathbb{R}^{n}\to \mathbb{R}^{n}$ is a vector valued function, and $f(\mathbf{0})=\mathbf{0}$. Suppose that the solution of differential equation (5) exists. Without loss of generality, we suppose that the origin is the unique equilibrium point of equation (5).

Definition 3

[33] The origin is called to be a finite-time stability equilibrium of equation (5) if equation (5) is Lyapunov stable and finite-time convergent, i.e., for any $y(0)\in \mathcal{O}\setminus \{\mathbf{0}\}$, $\lim_{t \rightarrow T} y(t)=\mathbf{0}$, where $\mathcal{O}$ is some open neighborhood of the origin and $T=T(y(0)) < \infty$. The origin is called to be a globally finite-time stability equilibrium of equation (5) whenever $\mathcal{O}=\mathbb{R}^{n}$.

Definition 4

[24] The origin is called to be a fixed-time stability equilibrium of equation (5) if it is globally finite-time stability, and the setting time $T(y(0))$ is uniformly bounded, i.e., there exists $\bar{T}<\infty$ such that $\sup_{x(0)\in \mathbb{R}^{n}}T(y(0))\leqslant \bar{T}$.

Lemma 1

[33] Suppose that there is a positive definite function $V \in C^{1}(D,\mathbb{R})$ to satisfy

$$\begin{aligned} \dot{V}(y)+cV(y)^{\alpha}\leqslant 0,\quad \forall y\in \mathcal{V} \setminus \{\mathbf{0}\}, \end{aligned}$$

(6)

where $D\subseteq \mathbb{R} ^{n}$ is a neighborhood of the origin, an open neighborhood $\mathcal{V}\subseteq D$ of the origin, $c>0$ and $\alpha \in (0,1)$ are constants. Then, the origin is a finite-time stability equilibrium point of equation (5) with the setting time $T \leqslant \frac{V(y(0))^{1-\alpha}}{c (1-\alpha )}$.

Lemma 2

[24] Suppose that there exists a positive definite function $V\in C^{1}(\mathbb{R}^{n},\mathbb{R})$ to satisfy

$$\begin{aligned} \dot{V}(y)\leqslant -mV(y)^{a}-nV(y)^{b},\quad \forall y\in V\backslash \{0\}, \end{aligned}$$

(7)

where $m,n>0$, $0< a<1$ and $b >1$. Then, equation (5) is a fixed-time stability with settling time $T\leqslant \frac{1}{m(1-a)}+\frac{1}{n(b-1)}$.

2.2 Neural network approach

In this subsection, we present a projection-based neural network approach to solve the SCFP (1). We give some decision methods for solving the SCFP (1).

It follows that $x\in C$ is a solution of the SCFP (1) iff it is a solution of the following:

$$\begin{aligned} \text{Find } x\in C\quad \text{such that} \quad y=Ax\in Q. \end{aligned}$$

Based on CQ algorithm (3), the paper [17] proposed a continuous neural network (NN) to deal with the SCFP (1). For simplicity, let us use the following auxiliary functions:

$$\begin{aligned} U=I-\gamma A^{T}(I-P_{Q})A, \end{aligned}$$

(8)

where $x\in \mathbb{R}^{n}$ and $\gamma \in (0,\frac{2}{\Vert A\Vert ^{2}})$. The neural network (NN) method is as follows:

$$\begin{aligned} \dot{x}= -\lambda \{x-P_{C}\circ Ux\}, \end{aligned}$$

(9)

where the function U is defined by (8) for $\gamma \in (0,\frac{2}{\Vert A\Vert ^{2}})$ and $\lambda >0$. As a convenience, we let $\Psi (x)=x-P_{C}\circ Ux$.

Lemma 3

[34] Assume that the solution set Θ of the SCFP is non-empty. The following statements are the same:

• (i) $x^{*}$ solves the SCFP.

• (ii) $P_{C}\circ Ux^{*}=x^{*}$, i.e., $\Theta =\{x\vert \Psi (x)=0\}$.

• (iii) $x^{*}\in C$ and $Ux^{*}=x^{*}$.

• (iv) $x^{*}\in C$ and $Ax^{*}=P_{Q}Ax^{*}$.

2.3 Blanket assumptions

Assumption 1

The SCFP is satisfied with bound linear condition with a constant ϖ.

Assumption 2

The parameter $\gamma \in (0,\frac{2}{L})$ with $L=\Vert A\Vert ^{2}$.

Assumption 3

If $\varpi ^{2}L<1$, we take $\gamma \in (0,\frac{1-\sqrt{1-\varpi ^{2}L}}{L})$ or $\gamma \in (\frac{1+\sqrt{1-\varpi ^{2}L}}{L},\frac{2}{L})$.

Assumption 4

There exists $x\in \mathbb{R}^{n}$, such that $\Psi (x)=0$.

Remark 1

Assumption 2 guarantees that the NFxNN (10) has a unique solution. Assumptions 1 and 3 guarantee that the NFxNN (10) achieves stability within the fixed time. Assumption 4 guarantees a consistent discretization of the NFxNN (10).

Lemma 4

[3] If Assumption 2holds, then $U:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is an averaged mapping.

Lemma 5

Assumption 2holds. Then,

$$\begin{aligned} \bigl\Vert \Psi (x)- \Psi (y) \bigr\Vert \leqslant 2 \Vert x-y \Vert , \quad \forall x,y\in \mathbb{R}^{n}. \end{aligned}$$

Proof

It is proved in [17, Theorem 1, p. 2993]. □

3 Fixed-time stability of the NFxNN and its convergence time estimation

A novel neural network is introduced to deal with the split convex feasibility problem. The fixed-time stability of the proposed neural network is obtained, and the upper bound forum of its convergence time is uniformly bounded. Moreover, it is irrelevant to the initial states.

3.1 A fixed-time converging neural network to deal with the SCFP

Inspired by [17, 31], we now present a new fixed-time converging neural network (NFxNN) to solve the SCFP (1):

$$\begin{aligned} \dot{x}=-\rho (x) (x-P_{C}\circ Ux ), \end{aligned}$$

(10)

where $\lambda >0$, $\gamma \in (0,\frac{2}{\Vert A\Vert ^{2}})$, U is defined by (8), and

$$ \rho (x):= \textstyle\begin{cases} \rho _{1}\frac{1}{ \Vert \Psi (x) \Vert ^{1-\alpha _{1}}}+\rho _{2} \frac{1}{ \Vert \Psi (x) \Vert ^{1-\alpha _{2}}}+\rho _{3} \frac{1}{ \Vert \Psi (x) \Vert },& \text{if } x\in \mathbb{R}^{n}\setminus \text{E}( \Psi ), \\ 0,& \text{else}. \end{cases} $$

Where $\rho _{1},\rho _{2},\rho _{3}>0$, $\alpha _{1}\in (0,1)$, $\alpha _{2}>1$ are tunable parameters, $\Psi (x)=x-P_{C}\circ Ux$, and $\text{E}(\Psi )=\{x \in \mathbb{R}^{n} \vert \Psi (x)=\mathbf{0}\}$ is the equilibrium points set of Ψ.

Carefully observe NFxNN (10), which can also be expressed as:

$$\begin{aligned} \dot{x}=-\omega (x)\rho _{1} \biggl( \bigl\Vert \Psi (x) \bigr\Vert ^{\alpha _{1}}\biggl(1+ \frac{\rho _{2}}{\rho _{1}} \bigl\Vert \Psi (x) \bigr\Vert ^{\alpha _{2}-\alpha _{1}}\biggr)+ \frac{\rho _{3}}{\rho _{1}} \biggr), \end{aligned}$$

(11)

where $-\omega (x)=\frac{\Psi (x)}{\Vert \Psi (x)\Vert }= \frac{x-P_{C}\circ Ux}{\Vert \Psi (x)\Vert }$. From the perspective of algorithm design, $\omega (x)$ can be deemed as a search direction, and $\alpha _{t}:=\rho _{1} (\Vert \Psi (x)\Vert ^{\alpha _{1}}(1+ \frac{\rho _{2}}{\rho _{1}}\Vert \Psi (x)\Vert ^{\alpha _{2}-\alpha _{1}})+ \frac{\rho _{3}}{\rho _{1}} )$ can be deemed as stepsize at the current iterate $x = x(t)$ in NFxNN (10). The stepsize $\alpha _{t}$ can be self-update with respect to t whenever $\rho _{1}$, $\rho _{2}$, $\rho _{3}$ and $\alpha _{1}$, $\alpha _{2}$ are given. If $\rho _{3}=1$, the NFxNN (10) is reduced to

$$ \dot{x}=-\omega (x)\rho _{1} \biggl( \bigl\Vert \Psi (x) \bigr\Vert ^{\alpha _{1}}\biggl(1+ \frac{\rho _{2}}{\rho _{1}} \bigl\Vert \Psi (x) \bigr\Vert ^{\alpha _{2}-\alpha _{1}}\biggr)+ \frac{1}{\rho _{1}} \biggr). $$

If $\rho _{1}=\rho _{2}=0$ in the NFxNN (10), the NFxNN (10) is reduced to neural network (NN) in [17]

$$ \dot{x}=-\rho _{3}\omega (x). $$

(12)

If $\rho _{2}=\rho _{3}=0$, the NFxNN (10) is reduced to the following finite-time converging neural network (FiNN):

$$ \dot{x}=-\rho _{1}\omega (x) (x-P_{C} \circ Ux)^{\alpha _{1}}. $$

(13)

If $\rho _{3}=0$ in, specially the tunable parameter $\alpha =1$ in [30], the NFxNN (10) degenerates into the fixed-time dynamical system (FxDS) in [30].

$$ \dot{x}=-\omega (x) \bigl(\rho _{1}(x-P_{C} \circ Ux)^{\alpha _{1}}+ \rho _{2}(x-P_{C}\circ Ux)^{\alpha _{2}} \bigr). $$

(14)

Besides, the stepsize $\alpha _{t}$ in NFxNN (10) when $\rho _{3}>0$ is larger than that when $\rho _{3}=0$. Generally, the larger stepsize can speed up the convergence rate of the numerical algorithms. It will be showed in Sect. 5. So, the NFxNN (10) is distinct of the fixed-time dynamical system in [30].

3.2 Convergence analysis of the NFxNN with the fixed time

We establish the globally convergence of the NFxNN. Futhermore, its convergence time estimation is obtained under the bounded linear regularity assumption and the Lipschitz continuity assumption. For proving the convergence of the NFxNN (10), we give the following auxiliary lemmas.

Lemma 6

$x^{*}\in \Theta$ is a solution of the NN (9) iff it is a solution of the NFxNN (10).

Proof

If $x^{*}\in \Theta$ is a solution of the NFxNN (10), then

$$\begin{aligned}& \rho \bigl( x^{*}\bigr) \bigl(x^{*}-P_{C} \circ Ux^{*}\bigr)=0 \\& \quad \Rightarrow \quad \rho _{1} \frac{\Psi ( x^{*})}{ \Vert \Psi ( x^{*}) \Vert ^{1-\alpha _{1}}}+\rho _{2} \frac{\Psi ( x^{*})}{ \Vert \Psi ( x^{*}) \Vert ^{1-\alpha _{2}}}+\rho _{3} \frac{\Psi ( x^{*})}{ \Vert \Psi ( x^{*}) \Vert }=0\quad \text{or}\quad \rho \bigl( x^{*}\bigr)=0 \\& \quad \Rightarrow \quad \Psi \bigl( x^{*}\bigr) \biggl(\rho _{1} \frac{1}{ \Vert \Psi ( x^{*}) \Vert ^{1-\alpha _{1}}}+\rho _{2} \frac{1}{ \Vert \Psi ( x^{*}) \Vert ^{1-\alpha _{2}}}+\rho _{3} \frac{1}{ \Vert \Psi ( x^{*}) \Vert } \biggr)=0\quad \text{or}\quad x^{*}\in E(\Psi ) \\& \quad \Rightarrow\quad \Psi \bigl( x^{*}\bigr)=0 \quad \text{or} \quad x^{*}\in E(\Psi ), \end{aligned}$$

it means that $x^{*}$ is a solution of the equation (9).

On the other hand, if $x^{*}$ is a solution of the equation (9), $\Psi ( x^{*})=0$, we get

$$\begin{aligned} \rho \bigl( x^{*}\bigr) \bigl(P_{C}\circ U x^{*}- x^{*}\bigr)=0, \end{aligned}$$

which gets that $x^{*}$ is a solution of the NFxNN (10). So far, we have completed the proof. □

Remark 2

The significant idea of the projection neural approach is to establish the equivalence between optimization problems and Brouwer’s fixed-point problems. The SCFP (1) is equivalent to the solution of equation (9). Due to Lemma 6, the NFxNN (10) can solve the SCFP (1).

Lemma 7

Let $x(t)$ be the trajectory of system (10). Where $x(t_{0})=x_{0}\in C$ and $x^{*}\in \Theta$, which satisfies $x(t)\rightarrow x^{*}$. Let $x^{*}\in \Theta$ be a solution of the SCFP. Let Assumptions 1, 2, and 3hold. Then, $\Vert P_{C}\circ Ux- P_{\Theta }x\Vert \leq \tau \Vert x-P_{\Theta }x\Vert$, where $0<\tau =\sqrt{1-\frac{\gamma (2-\gamma L)}{\varpi ^{2}}}<1$, and U is defined by (8).

Proof

Due to nonexpansive property of projection mapping, we get

$$\begin{aligned} \bigl\Vert P_{C}\circ Ux-P_{C}\circ U(P_{\Theta }x) \bigr\Vert ^{2}\leq \bigl\Vert Ux- U(P_{ \Theta }x) \bigr\Vert ^{2}. \end{aligned}$$

(15)

Recalling the definition of U in (8), we have

$$\begin{aligned}& \bigl\Vert Ux- U(P_{\Theta }x) \bigr\Vert ^{2} \\& \quad = \bigl\Vert x-P_{\Theta }x-\gamma A^{T}(I-P_{Q})Ax+ \gamma A^{T}(I-P_{Q})AP_{ \Theta }x \bigr\Vert ^{2} \\& \quad = \Vert x-P_{\Theta }x \Vert ^{2}+ \bigl\Vert \gamma A^{T}(I-P_{Q})Ax-\gamma A^{T}(I-P_{Q})A(P_{ \Theta }x) \bigr\Vert ^{2} \\& \qquad{} -2\bigl\langle x-P_{\Theta }x,\gamma A^{T}(I-P_{Q})Ax- \gamma A^{T}(I-P_{Q})A(P_{ \Theta }x) \bigr\rangle . \end{aligned}$$

We now estimate the last two terms of the above formula. Using (iii) of Lemma 3 and the L definition of Assumption 2, we have that

$$\begin{aligned} \bigl\Vert \gamma A^{T}(I-P_{Q})Ax-\gamma A^{T}(I-P_{Q})A(P_{\Theta }x) \bigr\Vert ^{2} \leq \gamma ^{2}L \bigl\Vert (I-P_{Q})Ax \bigr\Vert ^{2}. \end{aligned}$$

From $(i)$ of Property 1 and $(\mathit{iv})$ of Lemma 3, we get

$$\begin{aligned}& -2\bigl\langle x-P_{\Theta }x,\gamma A^{T}(I-P_{Q})Ax- \gamma A^{T}(I-P_{Q})A(P_{ \Theta }x) \bigr\rangle \\& \quad =-2\gamma \bigl\langle Ax-A(P_{\Theta }x),(I-P_{Q})Ax-(I-P_{Q})A(P_{ \Theta }x) \bigr\rangle \\& \quad \leq -2\gamma \bigl\Vert (I-P_{Q})Ax \bigr\Vert ^{2}. \end{aligned}$$

Substitute the above two inequalities into (15), we get

$$\begin{aligned} \bigl\Vert P_{C}\circ Ux-P_{C}\circ U(P_{\Theta }x) \bigr\Vert ^{2}\leq \Vert x-P_{\Theta }x \Vert ^{2}-\gamma (2-\gamma L) \bigl\Vert (I-P_{Q})Ax \bigr\Vert ^{2}. \end{aligned}$$

Sine $x(t)\in C$ for all $t\geq t_{0}$ was proved in [17, Lemma 3.4, p. 2998], and the bounded linear regularity property holds, there is a $\varpi >0$ and $\bar{t}\geq t_{0}$ such that

$$\begin{aligned} \operatorname{dist}(x,\Theta )= \Vert x-P_{\Theta }x \Vert \leq \varpi \bigl\Vert (I-P_{Q})Ax \bigr\Vert , \quad \forall t\geq \bar{t}. \end{aligned}$$

So,

$$\begin{aligned} -\gamma (2-\gamma L) \bigl\Vert (I-P_{Q})Ax \bigr\Vert ^{2}\leq - \frac{\gamma (2-\gamma L)}{\varpi ^{2}} \Vert x-P_{\Theta }x \Vert ^{2}. \end{aligned}$$

If $\varpi ^{2}L\geq 1$, then $0<\frac{\gamma (2-\gamma L)}{\varpi ^{2}}\leq 1$. If $\varpi ^{2}L<1$, we take $\gamma \in (0,\frac{1-\sqrt{1-\varpi ^{2}L}}{L})$ or $\gamma \in (\frac{1+\sqrt{1-\varpi ^{2}L}}{L},\frac{2}{L})$, then $0<\frac{\gamma (2-\gamma L)}{\varpi ^{2}}<1$. We can get that

$$\begin{aligned} \bigl\Vert Ux- U(P_{\Theta }x) \bigr\Vert ^{2}\leq \biggl(1- \frac{\gamma (2-\gamma L)}{\varpi ^{2}}\biggr) \Vert x-P_{\Theta }x \Vert ^{2}. \end{aligned}$$

(16)

By Lemma 3, we have $P_{C}\circ U(P_{\Theta }x)=P_{\Theta }x$. It follows from (15) and (16) that

$$\begin{aligned} \Vert P_{C}\circ Ux- P_{\Theta }x \Vert \leq \tau \Vert x-P_{\Theta }x \Vert , \end{aligned}$$

with $\tau =\sqrt{1-\frac{\gamma (2-\gamma L)}{\varpi ^{2}}}\in (0,1)$. It completes the proof. □

Theorem 1

Let $x(t)$ be the trajectory of system (10). Where $x(t_{0})=x_{0}\in C$ and $x^{*}\in \Theta$ satisfy $x(t)\rightarrow x^{*}$. Let $x^{*}\in \Theta$ be a solution of the SCFP. Assumptions 1, 2, and 3hold. Then the solution of the SCFP is a globally fixed-time equilibrium point of the NFxNN, and the convergence time is given as

$$\begin{aligned} T(x_{0})\leq \frac{1}{N_{1}(1-p_{1})}+\frac{1}{N_{2}(p_{2}-1)}, \end{aligned}$$

where $x_{0}$ is an initial point of the NFxNN (10), and $p_{1}=\frac{1+\alpha _{1}}{2}$, $p_{2}=\frac{1+\alpha _{2}}{2}$, $N_{1}=2^{\frac{1+\alpha _{1}}{2}}\rho _{1} \frac{1-\tau}{(1+\tau )^{1-\alpha _{1}}}$, $N_{2}=2^{\frac{1+\alpha _{2}}{2}}(1-\tau )^{\alpha _{2}}(\rho _{2}+ \rho _{3}\frac{1}{\Vert \Psi (x)\Vert ^{\alpha _{2}}})$ for $x\in \mathbb{R}^{n}$ and $\tau =\sqrt{1-\frac{\gamma (2-\gamma L)}{\varpi ^{2}}}\in (0,1)$.

Proof

It follows from the global version of the Picard-Lindelöf theorem in [35, Theorem 2.2, p. 38] that the NFxNN (10) can achieve a unique solution under the Assumptions 1, 2, and 3. Since $x^{*}$ is a solution of the NFxNN (10), we deduce by the global version of the Picard-Lindelöf theorem and Lemma 6 that $x^{*}\in \mathbb{R}^{n}$ is a unique solution of the NFxNN (10). To simplify the proof, we derive from Lemma 7 that

$$\begin{aligned} \bigl\langle x-P_{\Theta }x,\omega (x)\bigr\rangle =& \frac{1}{ \Vert \Psi (x) \Vert } \langle x-P_{\Theta }x,x-P_{\Theta }x+P_{\Theta }x-P_{C} \circ U x \rangle \\ =& \frac{1}{ \Vert \Psi (x) \Vert } \bigl( \Vert x-P_{\Theta }x \Vert ^{2}-\langle x-P_{ \Theta }x,P_{C}\circ U x-P_{\Theta }x\rangle \bigr) \\ \geq & (1-\tau )\frac{1}{ \Vert \Psi (x) \Vert } \Vert x-P_{\Theta }x \Vert ^{2}, \end{aligned}$$

(17)

where the inequality can be obtained by Lemma 7, i.e.,

$$ \bigl\langle x-P_{\Theta }x,\Psi (x)\bigr\rangle \geq (1-\tau ) \Vert x-P_{\Theta }x \Vert ^{2}. $$

The solution of the NFxNN (10) exists uniquely for any initial state according to [26, Proposition 2]. Now, consider a candidate Lyapunov function:

$$\begin{aligned} V\bigl(x(t)\bigr)=\frac{1}{2}\operatorname{dist}^{2} \bigl(x(t),\Theta \bigr)=\frac{1}{2} \bigl\Vert x(t)-P_{ \Theta }x(t) \bigr\Vert ^{2}. \end{aligned}$$

(18)

The time-derivation of the candidate Lyapunov function V along the solution of (10), we get

$$\begin{aligned} \dot{V}(x)={}&(x-P_{\Theta }x )^{\top}\dot{x} \\ ={}&{}-\bigl(\rho _{1} \bigl\Vert \Psi ( x) \bigr\Vert ^{\alpha _{1}}+\rho _{2} \bigl\Vert \Psi (x) \bigr\Vert ^{ \alpha _{2}}+\rho _{3}\bigr)\bigl\langle x-P_{\Theta }x, \omega (x) \bigr\rangle \\ \leq{}&{}-\rho _{1}(1-\tau ) \frac{ \Vert x-P_{\Theta }x \Vert ^{2}}{ \Vert \Psi ( x) \Vert ^{\alpha _{1}}}-\rho _{2}(1- \tau )\frac{ \Vert x-P_{\Theta }x \Vert ^{2}}{ \Vert \Psi ( x) \Vert ^{\alpha _{2}}}- \rho _{3}(1-\tau ) \frac{ \Vert x-P_{\Theta }x \Vert ^{2}}{ \Vert \Psi ( x) \Vert } \\ \leq{}&{}-\rho _{1}\frac{1-\tau}{(1+\tau )^{1-\alpha _{1}}} \Vert x-P_{ \Theta }x \Vert ^{1+\alpha _{1}} \\ &{} -(1-\tau )^{\alpha _{2}} \bigl(\rho _{2}+ \rho _{3} \bigl\Vert \Psi (x) \bigr\Vert ^{\alpha _{2}} \bigr) \Vert x-P_{\Theta }x \Vert ^{1+ \alpha _{2}}, \end{aligned}$$

(19)

where the first inequality is established by inequality (17), and the second by Lemma 7. Then,

$$ \dot{V}\leq -M_{1} \Vert x-P_{\Theta }x \Vert ^{1+\alpha _{1}} -M_{2} \Vert x-P_{ \Theta }x \Vert ^{1+\alpha _{2}}, $$

(20)

where $M_{1}=\rho _{1}\frac{1-\tau}{(1+\tau )^{1-\alpha _{1}}}$ and $M_{2}=(1-\tau )^{\alpha _{2}} (\rho _{2}+\rho _{3}\Vert \Psi (x)\Vert ^{ \alpha _{2}} )$.

It can be seen from (18) and (21) that

$$\begin{aligned} \dot{V}(x)&\leq -M_{1} \Vert x-P_{\Theta }x \Vert ^{1+\alpha _{1}} -M_{2} \Vert x-P_{ \Theta }x \Vert ^{1+\alpha _{2}} \\ &=-M_{1}\bigl(2V(x)\bigr)^{\frac{1+\alpha _{1}}{2}}-M_{2} \bigl(2V(x)\bigr)^{ \frac{1+\alpha _{2}}{2}} \\ &\leq -\bigl(N_{1}V(x)^{p_{1}}+N_{2}V(x)^{p_{2}} \bigr), \end{aligned}$$

(21)

where $p_{i}=\frac{1+\alpha _{i}}{2}$, $i=(1,2)$, $N_{i}=M_{i}2^{p_{i}}$, $i=(1,2)$. And $N_{1}=M_{1}2^{p_{1}}>0$, $p_{1}=\frac{1+\alpha _{1}}{2}\in (0,0.5)$ since $\alpha _{1}\in (0,1)$, $N_{2}=M_{2}^{p_{2}}>0$, $p_{2}=\frac{1+\alpha _{2}}{2}\in (1,\infty )$ since $\alpha _{2}\in (1,\infty )$. It then follows from Lemma 2, (ii) and (iv) of Lemma 3, and the intersection of two convex sets is still convex, that the equilibrium point $x^{*}$ of the NFxNN (10) is globally fixed-time stability, and the settling time satisfies

$$ T\bigl(x(0)\bigr)\leq \frac{1}{N_{1}(1-p_{1})}+\frac{1}{N_{2}(p_{2}-1)}, $$

(22)

where $x(0)$ is an initial condition of the NFxNN (10) and $N_{1}=2^{\frac{1+\alpha _{1}}{2}}\rho _{1} \frac{1-\tau}{(1+\tau )^{1-\alpha _{1}}}$ and $N_{2}=2^{\frac{1+\alpha _{2}}{2}}(1-\tau )^{\alpha _{2}}(\rho _{2}+ \rho _{3}\frac{1}{\Vert \Psi (x)\Vert ^{\alpha _{2}}})$. Here, $\Vert \Psi (x)\Vert ^{\alpha _{2}}>0$, for $x\in \mathbb{R}^{n}$, and since $\rho _{3}>0$, $\rho _{3}\frac{1}{\Vert \Psi (x)\Vert ^{\alpha _{2}}}>0$. We can find that the NFxNN (10) reduces the upper bound of the convergence time, and the method is also independent of the initial state. The proof is completed. □

Theorem 2

Let $x(t)$ be the trajectory of the FiNN (13). Where $x(t_{0})=x_{0}\in C$, and $x^{*}\in \Theta$ satisfies $x(t)\rightarrow x^{*}$. Assumptions 1, 2, and 3hold. For each $\alpha _{1}\in (0,1)$, $\rho _{1}>0$, the FiNN (13) is finite-time stability with the convergence time

$$\begin{aligned} T \leqslant \frac{V(x(0))^{1-\beta}}{c (1-\beta )}, \end{aligned}$$

where $c=2^{\frac{1+\alpha _{1}}{2}}\rho _{1} \frac{1-\tau}{(1+\tau )^{1-\alpha _{1}}}>0$, $\beta =\frac{1+\alpha _{1}}{2}\in (0.5,1)$, $0\leq \tau =\sqrt{1-\frac{\gamma (2-\gamma L)}{\varpi ^{2}}}<1$ and $V(x(t))=\frac{1}{2}\operatorname{dist}^{2}(x(t),\Theta )$.

Proof

Follow the proof of Theorem 1. Then, the time-derivation of the candidate Lyapunov function V along the solution of (13), one has

$$\begin{aligned} \dot{V}(x) =&(x-\bar{x} )^{\top}\dot{x} \\ =&\biggl\langle x-P_{\Theta }x,\rho _{1} \frac{P_{C}\circ Ux-x}{ \Vert \Psi (x) \Vert ^{1-\alpha _{1}}}\biggr\rangle \\ =&\biggl\langle x-P_{\Theta }x,\rho _{1} \frac{P_{C}\circ Ux-P_{\Theta }x+P_{\Theta }x-x}{ \Vert \Psi (x) \Vert ^{1-\alpha _{1}}} \biggr\rangle \\ =&-\rho _{1} \frac{ \Vert x-P_{\Theta }x \Vert ^{2}}{ \Vert \Psi (x) \Vert ^{1-\alpha _{1}}}+\biggl\langle x-P_{ \Theta }x,\rho _{1} \frac{P_{C}\circ Ux-P_{\Theta }x }{ \Vert \Psi (x) \Vert ^{1-\alpha _{1}}} \biggr\rangle \\ \leq & -\rho _{1} \frac{ \Vert x-P_{\Theta }x \Vert ^{2}}{ \Vert \Psi (x) \Vert ^{1-\alpha _{1}}} + \rho _{1} \frac{ \Vert x-P_{\Theta }x \Vert \Vert P_{C}\circ Ux-P_{\Theta }x \Vert }{ \Vert \Psi (x) \Vert ^{1-\alpha _{1}}} \\ \leq & -\rho _{1}(1-\tau ) \frac{ \Vert x-P_{\Theta }x \Vert ^{2}}{ \Vert \Psi (x) \Vert ^{1-\alpha _{1}}}, \end{aligned}$$

(23)

where the first inequality is established by the Cauchy-Schwarz inequality, and the second holds by Lemma 7, $\Psi (x)=P_{C}\circ Ux-x$ and $\psi (x)=P_{C}\circ Ux$. Using Lemma 7, there exists $0\leq \tau =\sqrt{1-\frac{\gamma (2-\gamma L)}{\varpi ^{2}}}<1$ such that

$$\begin{aligned}& (1-\tau )\Vert x-P_{\Theta }x)\Vert \leq \Vert x-P_{\Theta }x \Vert - \bigl\Vert \psi (x)-P_{ \Theta }x \bigr\Vert \leq \bigl\Vert x-\psi (x) \bigr\Vert ,\quad \forall x\in \mathbb{R}^{n}, \\& \bigl\Vert x-\psi (x) \bigr\Vert \leqslant \Vert x-P_{\Theta }x \Vert + \bigl\Vert \psi (x)-P_{\Theta }x \bigr\Vert \leqslant (1+\tau )\Vert x-P_{\Theta }x)\Vert ,\quad \forall x\in \mathbb{R}^{n}. \end{aligned}$$

(24)

Combining (23) with (24), we have

$$\begin{aligned} \dot{V}(x) &\leq -\rho _{1}\frac{1-\tau}{(1+\tau )^{1-\alpha _{1}}} \Vert x-P_{ \Theta }x \Vert ^{1+\alpha _{1}} \\ &= -2^{\frac{1+\alpha _{1}}{2}}\rho _{1} \frac{1-\tau}{(1+\tau )^{1-\alpha _{1}}}V^{\frac{1+\alpha _{1}}{2}} \\ &=-cV^{\beta}, \end{aligned}$$

where $c=2^{\frac{1+\alpha _{1}}{2}}\rho _{1} \frac{1-\tau}{(1+\tau )^{1-\alpha _{1}}}>0$, $\beta =\frac{1+\alpha _{1}}{2}\in (0.5,1)$. Therefore, it follows from Lemma 1 that for each $\alpha _{1}\in (0,1)$, the FiNN (13) is globally finite-time stability with the convergence time

$$\begin{aligned} T=T\bigl( x(0)\bigr) \leqslant \frac{V(x(0))^{1-\beta}}{c (1-\beta )}, \end{aligned}$$

which means that $x^{*}$ is a finite-time stability solution of the FiNN (13). □

Remark 3

(i) Theorem 1 presents the upper bound of the convergence time of the NFxNN (10), which is less, and the accuracy is higher. Besides, the the upper bound of the convergence time of the NFxNN (10) is less than that in [30].

(ii) We can see from Fig. 1 that the convergence time of the NFxNN (10) is about 0.01 second, and the convergence time of the FiNN (13) and the FxDS (14) is about 0.025 second. The convergence time of the NN (12) is about 0.5 second. It can thus be seen that the convergence time of the NFxNN (10) is less than that of the NN (12), the FiNN model (13), and the FxDS model (14), respectively. The convergence responses of the errors are different in the NN (12), FiNN (13), FxDS (14), and NFxNN (10). The NFxNN (10) can reach it quickly under the equal error condition, and NN (12) is much slower than them.

Fig. 1

(a) The trajectory of the NN (12) in [17] with initial condition $x_{0}=[1,1,1]^{T}$; (b) The trajectory of the FiNN (13) with initial condition $x_{0}=[1,1,1]^{T}$; (c) The trajectory of the FxDS (14) with initial condition $x_{0}=[1,1,1]^{T}$; (d) The trajectory of the NFxNN (10) with initial condition $x_{0}=[1,1,1]^{T}$; (e) Convergence responses of the errors $\log _{10}\Vert x(t)-x^{\ast}\Vert _{2}^{2}$ for the neural network (12) (red), the FiNN (13) (green), the FxDS (14) (purple) and the NFxNN (10) (blue)

3.3 Consistent discretization of the NFxNN

Continuous-time neural networks provide an effective way to accelerate the design of schemes for solving the split convex feasibility problems. Typically, discrete-time implementations use iterative methods to solve problems. We note that the fixed-time convergence of continuous-time neural networks may not hold in discrete cases. Uniform discretization refers to the discretization scheme that still has the convergence result of the continuous-time neural network under discrete conditions. In this section, we give the uniform discretization of the NFxNN (10). We next give the result of the forward-Euler discretization of the NFxNN (10):

$$\begin{aligned} \frac{x_{n+1}-x_{n}}{\eta}=-\rho (x_{n}) (x_{n}-P_{C}\circ Ux_{n}) ), \end{aligned}$$

(25)

where $\eta >0$ is the time-step and $\rho (\cdot )$ is given by (10), with $\rho _{1},\rho _{2},\rho _{3}>0$, and $x_{k}:=x(t)$ with $t=\eta n$ and $\dot{x}=\frac{dx}{dt}$.

Theorem 3

Let $x^{*}$ be a trajectory of the SCFP. All assumptions of Theorem 1hold. Then, for every $x_{0}\in \mathbb{R}^{n}$, $\varepsilon >0$ and $\gamma \in (0,\frac{2}{L})$, satisfying Assumptions 2, 3, and 4, there exist $\delta \in (0.5,1)$, $N_{1}$, $N_{2}$ and $\eta ^{*}>0$ satisfied for any $\eta \in (0,\eta ^{*}]$, the sequence $x_{n}$ generated by (25) holds:

$$ \bigl\Vert x_{n}-x^{*} \bigr\Vert < \textstyle\begin{cases} \sqrt{2} (\sqrt{\frac{N_{1}}{N_{2}}}\tan (\frac{\pi}{2}- \sqrt{N_{1}N_{2}} \delta \eta k) )^{\frac{1}{2-2\delta}}+\varepsilon , & n\leq n^{*}; \\ \varepsilon ,& \textit{otherwise}, \end{cases} $$

where $n^{*}=\lceil \frac{\pi}{2\sqrt{N_{1}N_{2}}\eta \delta}\rceil$, $\lceil x\rceil =[x]+1$. $[x]$ denotes the largest integer not exceeding x, and $x_{n}$ is a solution of (25) staring from the point $x_{0}$, and $x^{*}\in \mathbb{R}^{n}$ is the solution of (10). $\alpha _{1}$, $\alpha _{2}$, $N_{1}$, $N_{2}$ is given by Theorem 1.

Proof

Since $P_{C}\circ Ux$ is Lipschitz continuous, $\Psi (x)$ is continuous. The remainder of proof is like Theorem 4 in [26]. So, it is omitted. □

4 Numerical experiments

In this section, we aim to illustrate the analytical results with some simulations. We demonstrate the validity of the proposed approaches to deal with the split convex feasibility problem by several numerical examples. All programs are edited by Matlab 2018. All programs are executed on a personal computer. Unless otherwise stated, let $\alpha =0.2$, $\alpha _{1} =0.2$, $\alpha _{2} =1.2$, $\rho _{1} = \rho _{2} =100$ and $\gamma = \frac{1}{L}$.

Example 1

[6, 17] Consider the SCFP (1), where

$$\begin{aligned}& A= \begin{bmatrix} 2& -1 & 3 \\ 4& 2 & 5 \\ 2& 0 & 2 \end{bmatrix} ,\qquad B=I ,\qquad C =\bigl\{ x\in \mathbb{R}^{3}\vert x_{1}+x_{2}^{2}+2x_{3} \leq 0\bigr\} \quad \text{and} \\& Q= \bigl\{ y\in \mathbb{R}^{3}\vert y_{1}^{2}+y_{2}-y_{3}\leq 0\bigr\} . \end{aligned}$$

All requirements in Theorem 1 can be met. Numerical simulations of the NN system (12) in [17] achieve the equilibrium point $x^{\ast}=[ 0.2488, 0.0831, -0.2969]^{T}\in \Theta$. Numerical simulations of the FiNN system (13) obtain the solution $x^{\ast}=[ 0.2437, 0.0810, -0.2954]^{T}\in \Theta$. Numerical simulations of the FxDS system (14) converge to the point $x^{\ast}=[ 0.2129, 0.0703, -0.2914]^{T}\in \Theta$. Numerical simulations of the NFxNN system (10) achieve the solution $x^{\ast}=[ 0.2237, 0.0758, -0.2905]^{T}\in \Theta$. Figure 1 presents the convergence trajectories of the NN (12) in [17] , the FiNN model (13), the NFxNN (10) for initial conditions $x_{0}=[1,1,1]^{T}$. We can see from Fig. 1 that the convergence time of the NFxNN (10) is about 0.01 second, and the convergence time of the FiNN (13) and the FxDS (14) is about 0.025 second. The convergence time of the NN (12) is about 0.5 second. It can thus be seen that the convergence time of the NFxNN (10) is less than that of the NN (12), the FiNN model (13), and the FxDS model (14), respectively. The convergence responses of the errors are different in the NN (12), FiNN (13), FxDS (14), and NFxNN (10). The NFxNN (10) can reach it quickly under the equal error condition, and NN (12) is much slower than them.

Example 2

[17] We consider SCFP (1): the matrix $A=(a_{ij})_{2\times 2}$ and $a_{ij}\in (0,1)$ are generated randomly, and $C=\{u\in \mathbb{R}^{2}\vert c(u)\leq 0\}$ and $Q=\{v\in \mathbb{R}^{2}\vert q(v)\leq 0\}$ are the non-empty closed convex set with

$$\begin{aligned}& c(u)=-u_{1}+u_{2}^{2}, \\& q(v)=v_{1}+v_{2}^{2}; \end{aligned}$$

for all $u=(u_{1},u_{2})^{T}\in \mathbb{R}^{2}$ and $v=(v_{1},v_{2})^{T}\in \mathbb{R}^{2}$. We take

$$ A= \begin{bmatrix} 0.9572& 0.8013 \\ 0.4854& 0.1419 \end{bmatrix} \quad \text{and}\quad x_{0}= \begin{bmatrix} -10 \\ 4 \end{bmatrix} , $$

generated randomly. All conditions of Theorem 1 are satisfied. The trajectory $x(t)$ of the NN in [17] achieves the point $x^{\ast}=[ -0.0003956 , 0.0005248]^{T}\in \Theta$. Numerical simulations of the FiNN system (13) converge to the equilibrium point $x^{\ast}=[ 0.00001986 , -0.00003241]^{T}\in \Theta$. Numerical simulations of the FxDS (14) obtain the solution $x^{\ast}=[ 0.00001994 , -0.00002881]^{T}\in \Theta$. Numerical simulations of the NFxNN (10) converge to the point $x^{\ast}=[ 0.00001994 , -0.00002881]^{T}\in \Theta$.

Figure 2 demonstrates the convergence trajectories of the NN (12) in [17], the FiNN (13), the FxDS (14), and the NFxNN (10) for initial conditions $x_{0}$. We can observe from Fig. 2 that the convergence time of the NFxNN (10) is about 0.05 second. The NFxNN (10) takes about 0.06 second, and the convergence time of the NN (12) is much more than 0.5 second. The FiNN (13) takes about 0.09 second. We can observe the difference from the convergence responses of the errors in Fig. 2 (d). It also presents that the convergence rate of the NFxNN (10) is faster than that of the NN (12) and slightly faster than that of the FiNN (13) and FxDS (14). So, the convergence time of the NFxNN (10) is less than that of the NN (12), FiNN (13), and FxDS (14). From the error image, we can also see that the NN (12) achieves less accuracy than the FiNN (13), FxDS (14), and NFxNN (10).

Fig. 2

(a) The trajectory of the NN (12) in [17] with initial condition $x_{0}=[-10,4]^{T}$; (b) The trajectory of the FiNN (13) with initial condition $x_{0}=[-10,4]^{T}$; (c) The trajectory of the the FxDS (14) with initial condition $x_{0}=[-10,4]^{T}$; (d) The trajectory of the NFxNN (10) with initial condition $x_{0}=[-10,4]^{T}$; (e) Convergence responses of the errors $\log _{10}\Vert x(t)-x^{\ast}\Vert _{2}^{2}$ for the neural network (12) (red), the FiNN (13) (green), the FxDS (14) (purple) and the NFxNN (10) (blue)

Example 3

[18] Consider linear equation problem (LEP), find $x\in \mathbb{R}^{n}$ to satisfy

$$\begin{aligned} Bx=b. \end{aligned}$$

When $C=\mathbb{R}^{n}$ and $Q=\{b\}$, the LEP can transform into a split feasible problem. In this case, the algorithm degrades into the following form:

$$\begin{aligned} x_{n+1}=x_{n}+\gamma B^{T}(b-Bx_{n}), \quad n=0,1,2,\ldots, \end{aligned}$$

for $x_{0}$ arbitrary. So, system (9) degrades into the following dynamical system:

$$\begin{aligned} \dot{x}=\gamma B^{T}(b-Bx),\quad n=0,1,2,\ldots. \end{aligned}$$

We consider LEP with

$$\begin{aligned} B= \begin{bmatrix} 2& -1 & 3 \\ 4& 2 & 5 \\ 2& 1 & 2 \end{bmatrix} ,\qquad b=[1,4,7]^{T}. \end{aligned}$$

By means of algebraic calculation, it is easy to verify that $x^{*}=[9,-1,-6]^{T}$ is a unique solution. All numerical simulations achieve the unique solution $x^{\ast}=[9,-1,-6]^{T}$. Figure 3 displays that the trajectories of dynamical system (12), (13), (14), and (10) with initial point $x_{0}=[-3,5,2]^{T}$. And all globally converge to $x^{*}$.

Fig. 3

(a) The trajectory of the NN (12) in [17] with initial condition $x_{0}=[-3,5,2]^{T}$; (b) The trajectory of the FiNN (13) with initial condition $x_{0}=[-3,5,2]^{T}$; (c) The trajectory of the FxDS (14) with initial condition $x_{0}=[-3,5,2]^{T}$; (d) The trajectory of the NFxNN (10) with initial condition $x_{0}=[-3,5,2]^{T}$; (e) Convergence errors $\log _{10}\Vert x(t)-x^{\ast}\Vert _{2}^{2}$ for the neural network (12) (red), the FiNN (13) (green), the FxDS (14) (purple) and the NFxNN (10) (blue)

Figure 3 presents the convergence trajectories of the NFxNN (10), FxDS (14), FiNN (13), and NN (12) in [17] for solving the split convex feasibility problem.

We can observe from Fig. 3 (a), (b), (c) the convergence times of the NFxNN (10), FxDS (14), FiNN (13), and NN (12). The NFxNN (10) takes about 0.1 second. The FxDS (14) and FiNN (13) are about 0.2 second, and the NN (12) is much more than 2 second. So, the convergence speed of the NFxNN (10) is faster than that of the FxDS (14), FiNN (13), and NN (12). The difference on the convergence error of the NFxNN (10), FxDS (14), FiNN (13), and NN (12) is also shown from the convergence errors in Fig. 3 (d).

Example 4

We use the continuous-time method to solve the problem of recovering noisy sparse signals from a limited number of samples. The problem was considered in [8]. Let $x\in \mathbb{R}^{n}$ be a $K-\mathit{sparse}$ signal and $K\ll n$. The sampling matrix $B\in \mathbb{R}^{m\times n}$ is stimulated by the standard Gaussian distribution and vector $b=Bx+e$ with e being an additional noisy. The main work is to recover the signal x from the data b. The recovering noisy sparse signals can be converted into the LASSO problem:

$$\begin{aligned}& \min_{x\in \mathbb{R}^{n}}\frac{1}{2} \Vert Bx-b \Vert _{2}^{2} \\& \text{s.t. }\Vert x \Vert _{1}\leq t, \end{aligned}$$

where $t>0$ is a given constant. We take $C=\{x:\Vert x\Vert _{1}\leq t\}$ and $Q=\{b\}$, the above LASSO problem is converted into SCFP (1). So, we can deal with the problem by dynamical system (12), (13), (14), and (10). In the process of using the system to solve (such as (12), (13), (14) and (10)), we take the following set of parameters: the initial point $x_{0}\in \mathbb{R}^{n}$ of system (12), (13), (14), (10) and the matrix $B\in \mathbb{R}^{m\times n}$ are generated randomly. The true signal $x\in \mathbb{R}^{n}$ with K non-zero elements is generated from uniform distribution in the interval [−2,2] and vector $b=Ax$. Let $m=n=100$, $t=K=10$, $\gamma =\frac{1}{\Vert A\Vert ^{2}}$ and $\lambda =5$.

We compare the NFxNN (10) with NN (12) in [17], the FiNN (13), and FxDS (14). The convergence trajectories of the NFxNN (10), the NN (12) in [17], the FiNN (13), the FxDS (14) are shown in Fig. 4. We can observe from Fig. 4 (a), (b), (c), (d) the convergence time of the NFxNN (10), FxDS (14), FiNN (12), and NN (9). The NFxNN (10) is about 0.02 second. The FxDS (14) takes about 0.025 second. The FiNN (12) is about 0.09 second. The NN (9) is much more than 0.45 second. So, the convergence rate of the NFxNN (10) is quicker than that of the FxDS (14), FiNN (13), and NN (12). The difference in the convergence rates of the NFxNN (10), FxDS (14), FiNN (13), and NN (12) is also shown from the convergence responses of the errors in Fig. 3 (d). Figure 4 means that the convergence rate of the NFxNN (10) is faster than that of NN (12) in [17] and the FxDS (14).

Fig. 4

(a) The trajectory of the NN (12) in [17]; (b) The trajectory of the FiNN (13); (c) The trajectory of the FxDS (14); (d) The trajectory of the NFxNN (10); (e) Convergence responses of the errors $\log _{10}\Vert x(t)-x^{\ast}\Vert _{2}^{2}$ for the NN (12) (red), the FiNN (13) (green), the FxDS (14) (purple) and the NFxNN (10) (blue)

Taking Example 4 as an example, we explore the influence of parameters on the NFxNN method. Figure 5 illustrates the influence of parameters γ, $\alpha _{1}$, and $\alpha _{2}$ on the NFxNN (10), i.e., it means that the convergence time of the NFxNN (10) is obtained much less as γ increase. When $\alpha _{1}$ tends to 0, the NFxNN (10) can achieve the solution faster. The NFxNN (10) can converge to the solution faster while $\alpha _{2}$ tends to 1.

Fig. 5

(a) The NFxNN (10) method plot of $\log _{10}\Vert x(t)-x^{\ast}\Vert _{2}^{2}$ vs. t (time) for various values of γ; ; (b) The NFxNN (10) method plot of $\log _{10}\Vert x(t)-x^{\ast}\Vert _{2}^{2}$ vs. t (time) for various values of $\alpha _{1}$; (c) The NFxNN (10) method plot of $\log _{10}\Vert x(t)-x^{\ast}\Vert _{2}^{2}$ vs. t (time) for various values of $\alpha _{2}$

5 Conclusions

The NFxNN for solving the split convex feasibility problem is proposed by using the projection method. Under the bound linear regularity assumption, the NFxNN obtains a solution of the SCFP. At the same time, we derive the relationship between the method and the corresponding neural network. We obtain the results of the global fixed-time stability of the NFxNN, i.e., the convergence time of the NFxNN is independent of the initial conditions, and the convergence time is uniformly bounded. The results obtained are different from the relevant neural network methods (such as [17, 30]). Numerical examples also present the superiority and availability of the method.