A partially block randomized extended Kaczmarz method for solving large overdetermined inconsistent linear systems

1.   Introduction

The singularity phenomenon exists in a large number of physical models and biological processes, for instance, viscoelasticity, aerodynamics, hydrodynamics, rheology, and infectious diseases. In nonlinear elastic fracture mechanics, there is a singular relationship between the range ${q}$ and the stress near the crack tip; the stress shows that the power singularity is ${q}^{-\frac{1}{2}}$, where ${q}$ is the range determined from the crack tip ^[1]. As is well known, fractional differential equations have significant advantages in describing local limitations and long-term, large-scale physical phenomena. Westerland ^[2] depicts the transmission of electromagnetic waves in the following model:

here $\vartheta, \varsigma, \chi$ are constants, and $\mathcal D^{\nu }_{s} \mathcal B(u, s) = \frac{\partial^\nu \mathcal B(u, s)}{\partial t^\nu}$ is a fractional derivative. The authors in article ^[3] constructed a fractional Maxwell model

with fractional parameters $\rho$ and $\varrho$ satisfying $0 < \rho\leq\varrho < 1$.

This work treats the singular fractional order system:

where $\lambda_i > 0$ $(i = 1, 2)$ is a parameter, and $\mathcal D^{\alpha_i}_{0^+}$ is the Riemann-Liouville derivative. $n_i-1 < \alpha_i\leq n_i$, $n_i\geq 3$, $1\leq p_i\leq n_i-2$, $0\leq q_i\leq p_i$, $\mu_i > 0$, $0 < \eta_i\leq 1$, $h\in L^1[0, 1]$ is nonnegative, $\Lambda_i = \Gamma(\alpha_i)/\Gamma(\alpha_i-p_i)(1-\mu_i\int_0^{\eta_i}h_1(s)s^{a_i-q_i-1}ds) > 0$, $f_i: (0, 1)\times[0, +\infty)^2\setminus\{(0, 0)\} \rightarrow [0, +\infty)$ is continuous, $f_i(t, x, y)$ may have singularity at $t = 0, \ 1$ and $(x, y) = (0, 0)$.

Fractional calculus has attracted widespread attention from scholars in various fields. Additionally, a large number of theories and arguments have been concentrated on that can be utilized to characterize various differential equations, such as operator theory ^[4,5,6], space theory ^[7,8,9,10], variational methods ^[11,12,13], fixed point theorems ^{[14,15,16,17]}, etc. For instance, employing the fixed index theory, Xu et al. ^[18] talked over the fractional boundary value problem (BVP):

where $\mathcal D^{\alpha}_{0^+}$ is the Riemann-Liouville derivative. $f\in C([0, 1]\times[0, +\infty), (0, +\infty))$, $h\in C(0, 1)\cap L(0, 1)$, $h$ is nonnegative. By adopting the famous Krasnosel'skii fixed-point theorem, the author in ^[19] also researched BVP (1.2) with $f: [0, 1]\times[0, +\infty)\rightarrow [0, +\infty))$ being continuous and $h\equiv1$.

Henderson and Luca ^[20] focus on the following equation:

with multi-point boundary value conditions of fractional order

where $\overline{ \mu} > 0$ is a parameter and $\mathcal D^{\alpha}_{0^+}$ is the Riemann-Liouville derivative. $0 < \xi_1 < \cdots < \xi_m < 1$, $1\leq p \leq n -2$, $0\leq q \leq p$, the sign of $f$ can be changed, and it can be singular when $t = 0, \ 1$. Existence results for at least one positive solution are given in ^[20] according to Krasnosel'skii fixed-point theorem.

Wang and Jiang ^[21] also studied Eq (1.3) with the parameter $\overline{ \mu} = 1$, and the boundary condition is

They determined that there exist two positive solutions on the basis of the Leray-Schauder nonlinear alternative and the fixed-point theory of cone tension and compression.

Motivated by the aforementioned works, in this article, we consider the system (1.1), under the argument of fixed-point theory, whose existence will be gained. What is more, we obtain the exact interval in which the positive solution exists under the confinement of the singularity in nonlinear term. The remaining parts of this article are arranged as below: In Section 2, we put forward some preliminary and necessary lemmas. The proof of system (1.1) will be constructed under the fixed-point theory in Section 3. In the following section, an example for demonstrating the applicability of the primary conclusions will be given. In Section 5, we draw conclusions from this article.

2.   Preliminaries and lemmas

Please refer to ^[22,23] for the related definitions and lemmas of fractional derivative and integral. We put in several lemmas for the rest of the paper. The following Lemmas 2.1 and 2.2 have been proved in ^[20,24,25].

Lemma 2.1. Assume that $\Lambda_i\neq 0$, $\verb"y"_i\in C[0, 1]\ (i = 1, 2)$, the boundary value problems

has the representation

in which

□

Lemma 2.2. $\mathcal G_i(t, s)\ (i = 1, 2)$ labeled (2.1) has the following properties:

(1) $\mathcal G_i(t, s)\in C([0, 1]\times[0, 1], [0, +\infty))$.

(2) $\mathcal G_i(t, s)\leq \xi_i (s), \ 0\leq t, s \leq 1$,

(3) $\widehat{\omega}(t)\xi_i (s)\leq \mathcal G_i(t, s) \leq \sigma\bar{\omega}(t), \ 0\leq t, s \leq 1$,

□

Denote the Banach space $X = C[0, 1]\times C[0, 1]$ with norms:

Let

then $\mathcal K\subset X$ is a positive cone. For any real number $0 < \overline{r} < \overline{R}$, let

For $i = 1, 2$, the assumption through the work must hold.

$(\boldsymbol{H}_1)$ $f_i: (0, 1)\times[0, +\infty)^2\setminus\{(0, 0)\} \rightarrow [0, +\infty)$ is continuous and

$\verb"b"_i\in C(0, 1)$, $\verb"b"_i$ is singular at $t = 0$ and/or $t = 1$, $\verb"b"_i(t)\not\equiv 0$ on $[0, +\infty)$, $\digamma_i:[0, 1]\times[0, +\infty)^2\setminus\{(0, 0)\}\rightarrow [0, +\infty)$ is continuous,

where $H(\widetilde{m}) = \left[0, \frac{1}{\widetilde{m}}\right]\cup\left [\frac{\widetilde{m}-1}{\widetilde{m}}, 1\right]$.

$(\boldsymbol{H}_2)$ $\int_0^1\xi_i(s)\verb"b"_i(s)ds < +\infty$.

Remark 2.1. $(\textbf{H}_1)$ can be inferred from $(\textbf{H}_2)$. As a matter of fact, by $(\textbf{H}_1)$, for $0 < r_1 < r_2 < +\infty$, choose $(w(t), v(t))\equiv\left(\frac{r_1}{2}, \frac{r_1}{2}\right)\in \mathcal K_{[r_1, r_2]}$, for any fixed $\widetilde{m} > 0$, we have

Since $\digamma_i:[0, 1]\times[0, +\infty)^2\setminus\{(0, 0)\}\rightarrow [0, +\infty)$ is continuous, we have

As $\verb"b"_i, \ \xi_i:\left[\frac{1}{\widetilde{m}}, \frac{\widetilde{m}-1}{\widetilde{m}}\right]\rightarrow [0, +\infty)$ is continuous, we see that

and so

According to conditions $(\textbf{H}_1)$ and $(\textbf{H}_2)$, for any $(w, v)\in \mathcal K\setminus\{(0, 0)\}$, define $\mathcal T: \mathcal K\setminus\{(0, 0)\}\to X$,

It can be declared that $\mathcal T(w, v)$ is well defined. Actually, for any fixed $(w_0, v_0)\in \mathcal K\setminus\{(0, 0)\}$, there exists $r > 0$, such that $\|(w_0, v_0)\| = r$. In $(\textbf{H}_1)$ and $(\textbf{H}_2)$, we will show that

By $(\textbf{H}_1)$, there exists $\widetilde{l}\in \text{N}$ (N represents the set of natural numbers) such that

For each $(w, v)\in \partial \mathcal K_{r}$, $t\in\left[\frac{1}{\widetilde{l}}, \frac{\widetilde{l}-1}{\widetilde{l}}\right]$,

Let

So, by $(\textbf{H}_1),$ $(\textbf{H}_2)$ and Lemma 2.2, for $0\leq t\leq 1, \ i = 1, 2$, we have

Then, we know (2.3) holds. In combination with the continuity of $\mathcal G_i(t, s)$, $\mathcal T_i(w_0, v_0)\in C[0, 1]$, $\mathcal T_i: \mathcal K\setminus\{(0, 0)\}\to C[0, 1]$ is well defined. Therefore, $\mathcal T = (\mathcal T_1, \mathcal T_2): \mathcal K\setminus\{(0, 0)\}\to X$ is well defined. The fixed point of operator $\mathcal T$ in $\mathcal K\setminus\{(0, 0)\}$ is the solution of the system (1.1).

Lemma 2.3. Assume that $(\boldsymbol{H}_1)$ $(\boldsymbol{H}_2)$ hold. Then $\mathcal T: \mathcal K_{[r_1, r_2]}\to \mathcal K$ is completely continuous.

Proof. Firstly, we illustrate $\mathcal T\left(\mathcal K_{[r_1, r_2]}\right)\subseteq \mathcal K$. For $i = 1, 2$, $(w, v)\in \mathcal K_{[r_1, r_2]},$ $0\leq t\leq1$, as for the proof of (2.3), we know

So, by $(\textbf{H}_2)$ and Lemma 2.2,

By $(\textbf{H}_2)$ and Lemma 2.2, for any $(w, v)\in \mathcal K_{[r_1, r_2]},$ $0\leq t\leq1$,

Then $\mathcal T_i(w, v)(t)\geq \widehat{\omega}(t) \| \mathcal T_i(w, v)\|$, $T\left(\mathcal K_{[r_1, r_2]}\right)\subseteq \mathcal K.$

Next, we explain $\mathcal T: \mathcal K_{[r_1, r_2]}\to \mathcal K$ is a continuous operator. Let $(w_n, v_n), (w, v)\in \mathcal K_{[r_1, r_2]}$, such that $\|(w_n, v_n)-(w, v)\|\rightarrow 0\ (n\rightarrow +\infty)$, we will prove that $\| \mathcal T(w_n, v_n)- \mathcal T(w, v)\|\rightarrow 0 \ (n\rightarrow +\infty)$. By $(\textbf{H}_1)$, for any $\varepsilon > 0$, there exists $\ell\in \text{N}$ satisfying

For each $(w, v)\in \mathcal K_{[r_1, r_2]}$, $t\in\left[\frac{1}{\ell}, \frac{\ell-1}{\ell}\right]$, we have

For $t\in\left[\frac{1}{\ell}, \frac{\ell-1}{\ell}\right]$, $\omega' r_1\leq w(t)+v(t)\leq r_2$, $w(t)$, $v(t)$ meet one of the following three cases:

(1) $\frac{\omega' r_1}{2}\leq w(t)\leq r_2$, $\frac{\omega' r_1}{2}\leq v(t)\leq r_2$.

(2) $\frac{\omega' r_1}{2}\leq w(t)\leq r_2$, $0\leq v(t)\leq\frac{\omega' r_1}{2}$.

(3) $0\leq w(t)\leq\frac{\omega' r_1}{2}$, $\frac{\omega' r_1}{2}\leq v(t)\leq r_2$.

Since for $(t, x, y)\in\left[\frac{1}{\ell}, \frac{\ell-1}{\ell}\right]\times\left[\frac{\omega' r_1}{2}, r_2\right]\times\left[\frac{\omega' r_1}{2}, r_2\right]$ or $(t, x, y)\in\left[\frac{1}{\ell}, \frac{\ell-1}{\ell}\right]\times\left[\frac{\omega' r_1}{2}, r_2\right]\times\left[0, \frac{\omega' r_1}{2}\right]$ or $(t, x, y)\in\left[\frac{1}{\ell}, \frac{\ell-1}{\ell}\right]\times\left[0, \frac{\omega' r_1}{2}\right]\times\left[\frac{\omega' r_1}{2}, r_2\right]$, $f_i(t, x, y)$ is uniformly continuous, we have

holds uniformly on $s\in\left[\frac{1}{\ell}, \frac{\ell-1}{\ell}\right]$. Then the Lebesgue-dominated convergence theorem yields that

So, for the above $\varepsilon > 0$, there exists a sufficiently large $N_0 (N_0\in \text{N})$, when $n > N_0$,

It follows from (2.4) and (2.5) that, when $n > N_0$,

That is, $\mathcal T: \mathcal K_{[r_1, r_2]}\rightarrow \mathcal K$ is a continuous operator.

Then, we show $\mathcal T: \mathcal K_{[r_1, r_2]}\rightarrow \mathcal K$ is compact. Let $\mathcal A\subset \mathcal K_{[r_1, r_2]}$ be any bounded set. For $(w, v)\in \mathcal A$, we have $r_1\leq \|(w, v)\|\leq r_2$. By $(\textbf{H}_1)$, there exists $\jmath\in \text{N}$ satisfying

For each $(w, v)\in \mathcal K_{[r_1, r_2]}$, $t\in\left[\frac{1}{\jmath}, \frac{\jmath-1}{\jmath}\right]$, we can see

Let

So, by $(\textbf{H}_1)$ and $(\textbf{H}_2)$, we have

Therefore, for any $(w, v)\in \mathcal A$, $0\leq t\leq 1$, we have

So, $\mathcal T(\mathcal A)$ is bounded in $X$.

Finally, we explain that $\mathcal T_i(\mathcal A)$ is equicontinuous. By $(\textbf{H}_1)$, for any $\varepsilon > 0$, there exists $\hbar\in \text{N}$, we have

Let

$\omega_0 = \min\left\{\widehat{\omega}(t): t\in\left[\frac{1}{\hbar}, \frac{\hbar-1}{\hbar}\right]\right\}.$ The uniform continuity of $\mathcal G_i(t, s)$ on $[0, 1]\times [0, 1]$ means, for the above $\varepsilon > 0$, there exists $\delta_0 > 0$ such that for any $t_1, t_2 \in [0, 1]$, $|t_1-t_2| < \delta_0$, $s\in \left[\frac{1}{\hbar}, \frac{\hbar-1}{\hbar}\right]$, we have

Thus, when $t_1, t_2 \in [0, 1]$, $|t_1-t_2| < \delta_0$, for any $(w, v)\in \mathcal A$,

This means that $T_i(\mathcal A)$ is equicontinuous. By the Arzela-Ascoli theorem, $T_i(\mathcal A)$ is a relatively compact set. So $\mathcal T = (\mathcal T_1, \mathcal T_2): \mathcal K_{[r_1, r_2]}\rightarrow \mathcal K$ is compact. Combining with the continuity of $T: \mathcal K_{[r_1, r_2]}\rightarrow \mathcal K$, $\mathcal T: \mathcal K_{[r_1, r_2]}\rightarrow \mathcal K$ is a completely continuous operator.□

The following Lemmas 2.4 and 2.5 can be used to explain the existence of the fixed point, that is, the existence of positive solutions to the system (1.1).

Lemma 2.4. ^[26] Let $\mathcal P$ be a positive cone in a Banach space $E$, $\Omega_1$ and $\Omega_2$ are bounded open sets in $E$, $\theta\in\Omega_1$, $\overline {\Omega}_1\subset\Omega_2$, $\verb"A" : \mathcal P\cap\overline {\Omega}_{2}\backslash\Omega_{1} \rightarrow \mathcal P$ is a completely continuous operator. If the following conditions are satisfied:

$\| \verb"A"x \| \leq \| x \|$, $\forall{x}\in{ \mathcal P\cap\partial\Omega_1}$, $\| \verb"A"x \| \geq \| x \|$, $\forall{x}\in{ \mathcal P\cap\partial\Omega_2}$,

or

$\| \verb"A"x \| \geq \| x \|$, $\forall{x}\in{ \mathcal P\cap\partial\Omega_1}$, $\| \verb"A"x \| \leq \| x \|$, $\forall{x}\in{ \mathcal P\cap\partial\Omega_2}$,

then $\verb"A"$ has at least one fixed point in $\mathcal P\cap(\overline{\Omega}_{2}\backslash\Omega_{1})$.

Lemma 2.5. ^[27] Let $\mathcal P$ be a positive cone in a real Banach space $E$. Denote $\mathcal P_r = \{x\in \mathcal P:\|x\| < r\}$, $\overline{ \mathcal P}_{r, R} = \{x\in \mathcal P: r\leq\|x\|\leq R\}$, $0 < r < R < +\infty$. Let $\verb"A":\overline{ \mathcal P}_{r, R}\rightarrow \mathcal P$ be a completely continuous operator. If the following conditions are satisfied:

(1) $\|\verb"A"x\|\leq \|x\|$, $\forall x \in \partial \mathcal P_{R}$,

(2) There exists a $x_0\in\partial \mathcal P_{1}$, such that $x\neq \verb"A"x+mx_0, \ \forall x\in \partial \mathcal P_{r}$, $m > 0$,

then $\verb"A"$ has fixed points in $\overline{ \mathcal P}_{r, R}$.

Remark 2.2. If (1) and (2) are satisfied for $x\in\partial \mathcal P_r$ and $x\in \partial \mathcal P_R$, respectively, then Lemma 2.6 still holds.

3.   Main results

Theorem 3.1. Assume that $(\boldsymbol{H}_1)$ $(\boldsymbol{H}_2)$ $(\boldsymbol{H}_3)$ hold.

$(\boldsymbol{H}_3)$

where $L_i = \left(4\int_{0}^{1}\xi_i(s)\verb"b"_i(s)ds\right)^{-1}$, $l_i = \left(2\overline{\omega} ^2\int_{a}^{b} \mathcal \xi_i(s)ds\right)^{-1}.$ $\overline{\omega} = \min_{t\in[a, b]}\widehat{\omega}(t)$. For any

system (1.1) has at least one positive solution.

Proof. For $i = 1, 2$, let $\lambda_i$ satisfy (3.1) and let $\varepsilon_0 > 0$ be chosen such that $L_i-\varepsilon_0 > 0$ and $\lambda_ig_i^\infty\leq L_i-\varepsilon_0.$ From the first inequality in $(\textbf{H}_3)$, it can be inferred that there exists $\overline{r} > 0$, making

Let

As for the proof of (2.3), we obtain $\overline{M} < +\infty$. Take

For any $(w, v)\in \partial \mathcal K_{R_1}$, let

So, for any $t\in D(w, v)$, we receive

Thus, by (3.2),

For any $(w, v)\in \partial \mathcal K_{R_1}$, let

then we have

So for any $(w, v)\in \partial \mathcal K_{R_1}$, by (3.3),

Thus

For $i = 1, 2$, let $\lambda_i$ satisfy (3.1) and let $\overline{\varepsilon} > 0$ be chosen so that $l_i+\overline{\varepsilon}\leq\lambda_if_{i0}.$ By the second inequality of $(\textbf{H}_3)$, there exists $R_2 < R_1$ meeting

Choose $\mathcal K_{R_2} = \left\{(w, v)\in \mathcal K: \|(w, v)\| < R_2\right\}$. For any $(w, v)\in \partial \mathcal K_{R_2}$,

Combining with (3.5), we gain

Therefore, for any $(w, v)\in \partial \mathcal K_{R_2}$, by (3.6),

Thus

As can be seen by (3.4), (3.7), and Lemmas 2.3 and 2.4, $\mathcal T$ has a fixed point $(w, v)$ with $0 < R_2\leq \|(w, v)\|\leq R_1$. Then system (1.1) has a positive solution $(w, v)$. □

Remark 3.1. From Theorem 3.1, the superlinear and sublinear conditions of $f_i(t, x, y)$ and $\digamma_i(t, x, y)$ are not necessary. In reality, Theorem 3.1 still applies to the following situations:

(1) $\digamma_i^\infty < L_i$, $f_{i0} = +\infty$, $\lambda_i\in\left(0, \frac{L_i}{\digamma_i^\infty}\right)$.

(2) $\digamma_i^\infty = 0$, $f_{i0} = +\infty$, $\lambda_i\in(0, +\infty)$.

(3) $\digamma_i^\infty = 0$, $f_{i0} > l_i > 0$, $\lambda_i\in\left(\frac{l_i}{f_{i0}}, +\infty\right)$.

Remark 3.2. For $i = 1, 2$, since $0 < \frac{l_i}{f_{i0}} < 1$, $\frac{L_i}{\digamma_i^{\infty}} > 1$, we have $1\in\left(\frac{l_i}{f_{i0}}, \frac{L_i}{\digamma_i^{\infty}}\right)$, so when $\lambda_1 = \lambda_2 = 1$, Theorem 3.1 is true.

Theorem 3.2. Assume that $(\boldsymbol{H}_1)$ $(\boldsymbol{H}_2)$ $(\boldsymbol{H}_4)$ hold.

$(\boldsymbol{H}_4)$

the definitions of $L_i$ is the same in Theorem 3.1: $l'_i = \left(2\min_{t\in[a, b]}\int_{a}^{b} \mathcal G_i(t, s)ds\right)^{-1}$. For any

system (1.1) has at least one positive solution.

Proof. For $i = 1, 2$, for any $\lambda_i\in\left(\frac{l'_i}{f_{i\infty}}, \frac{L_i}{\digamma_i^0}\right)$, there exists $\varepsilon' > 0$ so that

According to the first inequality of $(\textbf{H}_4)$, there exists $r > 0$,

Set

From $(\textbf{H}_1)$, there exists $m'$ $(m'\in \text{N})$,

For any $(w, v)\in \partial \mathcal K_{r_1}$, we can acquire

where $\omega' = \min\left\{\widehat{\omega}(t): t\in\left[\frac{1}{m'}, \frac{m'-1}{m'}\right]\right\}$. From (3.8), we know

Equations (3.9) and (3.10) can be introduced so that, for any $(u, v)\in \partial \mathcal K_{r_1}$,

Thus

For $i = 1, 2$, for the above $\varepsilon' > 0$, on the basis of the second inequality of $(\textbf{H}_5)$,

satisfying

Let

Then, we demonstrate

Otherwise, there exists $(w_2, v_2)\in\partial \mathcal K_{r_2}$ and $\mu_2 > 0$ such that

Owing to the fact that

by (3.12), we can find that

Suppose

Then $w_2(t)+v_2(t)\geq \xi > 0$, $a\leq t\leq b$. Therefore, for any $a\leq t\leq b$, by (3.14),

Thus

Obviously, (3.16) and (3.15) yield contradiction, thus (3.13) holds. It follows from (3.11), (3.13), and Lemmas 2.3 and 2.5 that $\mathcal T$ has a fixed point $(w, v)$ with $0 < r_1\leq \|(w, v)\|\leq r_2$. Then system (1.1) has a positive solution $(w, v)$. □

Remark 3.3. Similar to Remark 3.1, for $i = 1, 2$, Theorem 3.2 holds to the situations:

(1) $f_{i\infty} = +\infty$, $\digamma_i^0 < L_i$, $\lambda_i\in\left(0, \frac{L_i}{\digamma_i^0}\right)$.

(2) $f_{i\infty} = +\infty$, $\digamma_i^0 = 0$, $\lambda_i\in(0, +\infty)$.

(3) $f_{i\infty} > l'_i$, $\digamma_i^0 = 0$, $\lambda_i\in\left(\frac{l'_i}{f_{i\infty}}, +\infty\right)$.

Remark 3.4. Since $0 < \frac{l'_i}{f_{i\infty}} < 1$, $\frac{L_i}{\digamma_i^{0}} > 1$, we have $1\in\left(\frac{l'_i}{f_{i\infty}}, \frac{L_i}{\digamma_i^{0}}\right)$, so when $\lambda_1 = \lambda_2 = 1$, Theorem 3.2 also holds.

4.   An example

Example 4.1. Take account of the fractional system

where

Obviously,

By computation

For $\widehat{\omega}(t) = t^ \frac{7}{2}$, define a cone

For any $0 < r^*_1 < r^*_2 < +\infty$ and $(w, v)\in \mathcal K_{[{r^*_1}, r^*_2]}$, $\widehat{\omega}(t)r^*_1\leq w(t)+v(t)\leq r^*_2$. So

The absolute continuity of the integral gives

By (4.2) (4.3), we obtain

By

we know

so, we have

Similar to (4.4), we also have

These imply that condition $(\textbf{H}_1)$ holds.

By calculation, $\begin{aligned} \digamma_1^\infty = \digamma_2^\infty = 0, \end{aligned}$ and $f_{10} = f_{20} = +\infty.$ Therefore, by Theorem 3.1, for each $\lambda_i\in (0, +\infty)$ $(i = 1, 2)$, we get that system (4.1) has at least one positive solution.

5.   Conclusions

In this article, we investigate a singular fractional differential system involving integral boundary value conditions. The existence and explicit interval can be acquired under the argument of fixed-point theory. Since $f_i \ (i = 1, 2)$ is the abstract function, in the actual world, there are quantities of functions that satisfy the requirements of this article, which proves the effectiveness and feasibility of these theorems.

Author contributions

Ying Wang: Writing original draft, methodology, proof of conclusions; Linmin Guo: Methodology, proof of conclusions; Yumei Zi: Validation, writing review, editing; Jing Li: Validation, writing review, editing. All the authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

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

Acknowledgments

We would like to thank you for following the instructions above very closely in advance. It will definitely save us lot of time and expedite the process of your paper's publication. This work is supported by NSFC (12271232, 12101086), the Science Research Foundation for Doctoral Authorities of Linyi University (LYDX2016BS080).

Conflict of interest

The authors declare no conflicts of interest.