A strong convergence theorem for solving pseudo-monotone variational inequalities and fixed point problems using subgradient extragradient method in Banach spaces

1.   Introduction

In 1959, A. Signorini ^[1] proposed an interesting contact problem which was well known as Signorini Problem. Since then, many researchers have carried on the research to this problem and reformulated as the variational inequality problem (the VI in short) ^[2]. A key step for the solution of the VI was introduced by Hartman and Stampacchia ^[3] in 1966, which produce the VI as an important tool in studying optimization theory, engineering mechanics, economics and applied sciences in a unified and general framework (see ^[4,5]).

Under appropriate conditions, there are two general methods for solving the VI problem: The projection method and the regularized method. Many projection-type algorithms for solving the VI problem can be found in ^{[6,7,8,9,10,11]}. The gradient method is the simplest algorithm in which only one projection on feasible set is performed, but a strongly monotonicity is required to obtain the convergence of the method. To avoid the hypothesis of the strongly monotonicity, Korpelevich ^[6] proposed a decisive algorithm for solving the variational inequalities in Euclidean space, which was called the extragradient-type method. In 2011, the subgradient extragradient-type method was introduced by Censor et al. ^[7], which for solving variational inequalities in real Hilbert space. Very recently, Liu ^[11] proposed an inertial Tseng's extragradient algorithm for solving pseudomonotone variational inequalities.

It is natural to consider the algorithm for solving the variational inequalities in the setting of Banach spaces or Hilbert spaces. Several results were obtained in the case of various iterative algorithms for finding a common element of the fixed points set and the set of solutions of the variational inequality problem in Hilbert spaces or Banach spaces (see ^{[12,13,14,15,16,17,18,19,20,21,22,23,24]}). Especially, the fixed point technique was introduced by Browder ^[12] in 1967. Then Liu and Kong ^[19] provided a algorithm for finding a common element of fixed points set and variational inequality in Banach space. Recently, Ceng ^[24] introduced two subgradient extragradient methods for solving pseudomonotone variational inequalities and fixed point problems.

Motivated by the works mentioned, in the present paper, we extend subgradient extragradient algorithm proposed by ^[22] for solving a common solution of variational inequalities and fixed point problems in Banach spaces. It is worth stressing that our algorithm has a simple structure and the convergence of algorithms is not required to know the Lipschitz constant of the mapping.

The paper is organized as follows. In Section 2, we present some preliminaries that will be needed in the sequel. In Section 3, we propose an algorithm and analyze its convergence. Finally, in Section 4 we present a numerical example and comparison.

2.   Mathematical preliminaries

Assume that $X$ is a real Banach space with its dual $X^{\ast}$, $\parallel\cdot\parallel$ and $\parallel\cdot\parallel_{\ast}$ denote the norms of $X$ and $X^{\ast}$, respectively, $\langle x^{\ast}, x \rangle$ is the duality coupling in $X^{\ast}\times X$, and $x_{n}\longrightarrow x \ (x_n\rightharpoonup x)$ is called a sequence $\{ x_{n}\}$ convergence to $x$ strongly (weakly). Let $C$ be a nonempty closed convex subset of $X$, and $F: C\longrightarrow X^{\ast}$ be a continuous mapping. Consider with the following variational inequality (for short, $VI(F, C)$) which consists in finding a point $x \in C$ such that

Let $S$ be the solution set of (2.1).

Definition 2.1. A mapping $F: C\longrightarrow X^{\ast}$ is said as follows:

$(A1)$ Monotone, if $\langle F(x)-F(y), x-y\rangle\geq0, \ \ \forall x, y\in C;$

$(A2)$ Pseudomonotone, if $\langle F(y), x-y\rangle\geq0 \Rightarrow \langle F(x), x-y\rangle\geq0, \ \ \forall x, y\in C;$

$(A3)$ Lipschitz-continuous with constant $L > 0$, if there exists $L > 0$ such that $\parallel F(x)-F(y) \parallel\leq L\parallel x-y \parallel, \ \ \forall x, y\in C.$

Recall that a point $x\in C$ is called fixed point of an operator $T : C \rightarrow C,$ if $Tx = x$. We shall denote the set of fixed points of $T$ by $F(T)$. It is well known that in a real Hilbert space, $x$ is the solution of the $VI(F, C)$ if and only if $x$ is the solution of the fixed point equation $x = P_C(x-\lambda F(x)),$ where $\lambda$ is an arbitrary positive constant. Therefore, fixed point algorithms can be used to solve $VI(F, C)$. The mapping $T : C \rightarrow C$ is called nonexpansive, if,

The normalized duality mapping $J_X$ (usually write by $J$) of $X$ into $2^{X^*}$ is defined by

for all $x\in X$. Let $q \in (0, 2]$. The generalized duality mapping $J_q:X \rightarrow 2^{X^*}$ is defined (the definitions and properties, see ^[15]) by

for all $x\in X$. More details, can be found in ^[25].

Let $U = \{x\in X: \|x\| = 1 \}$, and the norm of $X$ is called G $\hat{\hbox{a}}$ teaux differentiable if for each $x, y\in U,$ the limit

exists. In this case, the space $X$ is also called smooth. It is well known that if $X$ is a smooth, strictly convex and reflexive Banach space, then $J$ is a single-valued bijection, and furthermore, there exists inverse mapping $J^{-1}$ which coincides with the duality mapping $J^*$ on $X^*$. $X$ is said to be uniformly smooth if (2.2) converges uniformly for $x, y\in U$. It is strictly convex if $\parallel \frac{x+ y}{2}\parallel < 1$ for all $x, y\in U$ and $x\neq y$. The modulus $\delta_X$ of convexity is defined by

for all $\varepsilon \in [0, 2],$ where $B_X$ is the closed unit ball of $X$. A Banach space $X$ is called uniformly convex if $\delta_X(\varepsilon) > 0.$ A Banach space $X$ is uniformly convex iff for any two sequences $\{x_{n}\}, \ \{y_{n}\}\in X,$

hold. Moreover, $X$ is called 2-uniformly convex if there exists $c > 0$ such that for all $\varepsilon \in [0, 2],$ $\delta_X(\varepsilon) > c\varepsilon^2$. Obviously, every 2-uniformly convex Banach space is uniformly convex.

Alber ^[25] introduces a functional $V(x^*, y): X^* \times X \longrightarrow R$ by

The operator $P_C: X^*\longrightarrow C\subseteq X$ is called the generalized projection operator if it associates to an arbitrary fixed point $x^*\in X^*,$ where $x^*$ is the solution to the minimization problem

and $\tilde{x^*} = P_{C}x^*\in C \subset X$ is called a generalized projection of the point $x^*.$ For more results about $P_{C}$ refer to ^[25]. The next lemma can describe the properties of $P_{C}$.

Lemma 2.1. ^[25] Let $C$ be a nonempty closed convex set in $X$ and $x^*, y^* \in X^*$, $\tilde{x^*} = P_{C}x^*$. Then

(1) $\langle J\tilde{x^*}-x^*, y-\tilde{x^*}\rangle\geq0, \ \ \forall y\in C;$

(2) $V(J\tilde{x^*}, y)\leq V(x^*, y)-V(x^*, \tilde{x^*}), \ \ \forall y\in C;$

(3) $V(x^*, z)+2\langle y^*, J^{-1}x^*-z\rangle\leq V(x^*+ y^*, z), \forall z\in X.$

By the definition of $V$, it is easy to check the following lemma.

Lemma 2.2. For any $x, y, z \in X$ and $\alpha \in (0, 1),$

(1) $(\parallel x \parallel-\parallel y \parallel)^{2}\leq V(Jx, y) \leq (\parallel x \parallel+\parallel y \parallel)^{2};$

(2) $V(\alpha Jx+(1-\alpha)Jy, z)\leq \alpha V(Jx, z)+(1-\alpha)V(Jy, z);$

(3) $V(Jx, z) = V(Jx, y)+V(Jy, z)+2\langle Jz-Jy, y-x\rangle;$

(4) $V(Jx, y)\leq \parallel x \parallel \parallel Jx-Jy\parallel+\parallel y \parallel \parallel x-y\parallel.$

In ^[26], they prove following lemma.

Lemma 2.3. ^[26] Let $X$ be a real 2-uniformly convex Banach space. Then, there exists $\mu \geq 1$ such that for all $x, y\in X,$

The minimum value of the set of all $\tau$ is denoted by $\tau_{X}$ (also write by $\tau$) and is called the 2-uniform convexity constant of $X$.

The following Lemma which will be useful to our subsequent convergence analysis.

Lemma 2.4. ^[27] Let $\{a_n\}$ be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence $\{a_{n_j}\}$ of $\{a_n\}$ which satisfies $a_{n_j} < a_{{n_j}+1}$ for all $j\in\mathbb{N}$. Define the sequence $\{\tau(n)\}_{n\geq n_0}$ of integers as follows:

where $n_0 \in \mathbb{N}$ such that $\{k\leq n_0: a_k < a_{k+1}\}$ is nonempty. Then the following hold:

(1) $\tau(n)\leq\tau(n+1) < \cdots,$ and $\tau(n)\longrightarrow \infty;$

(2) $a_{\tau(n)}\leq a_{{\tau(n)}+1} \ \hbox{and} \ a_n\leq a_{{\tau(n)}+1}.$

3.   Main results

In this section, we introduce a new subgradient extragradient algorithm for solving pseudomonotone variational inequality and fixed point problems in Banach spaces. At first, let's make the following assumptions.

Assumption 3.1:

(a) $X$ is a real 2-uniformly convex Banach space and $C$ is its nonempty closed convex subset.

(b) $F:X\rightarrow X^{\ast}$ is pseudomonotone on $C$, $L$ -Lipschitz continuous on $X$, and $T$ is a nonexpansive mapping of $C$ into itself such that $S\bigcap F(T)\neq \emptyset$.

(c) The mapping $F$ is sequentially weakly continuous, i.e., for each sequence $\{x_n\} \in C$ : if $x_n \rightharpoonup x$, then $F(x_n)\rightharpoonup F(x)$.

Our algorithm has the following forms:

Algorithm 3.1:

(Step 0)    Take $\lambda_0 > 0$, $x_0\in X$, $\mu\in (0, 1)$. Choose a nonnegative real sequence $\{\theta_{n}\}$ such that $\sum_{n = 0}^{\infty}\theta_{n} < \infty.$

(Step 1)    Given the current iterate $x_n$, compute

If $x_n = y_n$, and $Tx_n = x_n,$ then stop: $x_n$ is a solution. Otherwise,

(Step 2)    Construct $T_{n} = \{x\in X|\langle Jx_n-\lambda_n F(x_n)-Jy_{n}, x-y_{n} \rangle\leq0\}$ and compute

(Step 3)    Compute

Set $n : = n + 1$ and return to step 1.

We prove the strong convergence theorem for Algorithm 3.1. Firstly, we give the following lemma, which plays a crucial role in the proof of the main theorem.

Lemma 3.1. Assume that $x_n, \ y_n, \ \lambda_n$ are the sequences generated by Algorithm 3.1 and Assumption 3.1 holds, then

(1) If $x_n = y_n$ and $Tx_n = x_n,$ for some $n\in N$, then $x_n \in S\bigcap F(T);$

(2) $\lim\limits_{n\rightarrow \infty}\lambda_n = \lambda \in [min\{\frac{\mu}{L }, \lambda_0 \}, \lambda_0+\theta],$ where $\theta = \sum_{n = 0}^{\infty}\theta_{n}.$

Proof. (1) If $x_n = y_n$, by Algorithm 3.1, we have $x_{n} = P_C(Jx_n-\lambda_n F(x_n))$, and thus $x_{n} \in C.$ By the definition of $P_C$, we have

Therefore,

Since $\lambda_n > 0,$ we have $x_n \in S.$ Combining $Tx_n = x_n,$ we obtain $x_n \in S\bigcap F(T).$

(2) Since $F$ is a Lipschitz-continuous mapping with positive constant $L$, in the case of $\langle F(x_n)-F(y_n), z_{n}-y_n \rangle > 0,$ we get

Thus, $\{\lambda_n\}$ has the upper bound $\lambda_0+\theta,$ and lower bound $min\{{\frac{\mu}{L }, \lambda_0 }\}$. Similar to the proof of Lemma 3.1 in ^[21], we have that

The proof is complete.

Theorem 3.1. Assume that Assumption 3.1 holds, the sequence $\{\alpha_n\}$ satisfies $\{\alpha_n\}\subset(0, 1)$, $\sum_{n = 0}^\infty\alpha_n = \infty$, $\lim\limits_{n\rightarrow \infty}\alpha_n = 0$ and $\beta_n \in (0, 1)$. Let $\{x_n\}$ be a sequence generated by Algorithm 3.1. Then $\{x_n\}$ strongly converges to a solution $x^* = P_{S\bigcap F(T)}Jx_0$.

Proof. We divide the proof into two steps.

Step 1. The sequences $\{x_n\}$, $\{y_n\}$, $\{z_n\}$, and $\{t_n\}$ generated by Algorithm 3.1 are bounded.

To observe this, take $u\in S\bigcap F(T).$ Noting that $y_n\in C,$ we have $\langle F(u), y_n-u\rangle\geq0$, for all $n\in \mathbb{N}.$ Since $F$ is pseudomonotone, we have $\langle F(y_n), y_n-u\rangle\geq 0, \ \ \ \forall n\in \mathbb{N}.$ Then,

This implies that

By the definition of $T_n$, we know $\langle Jx_n-\lambda_n F(x_n)-Jy_{n}, z_n-y_{n} \rangle\leq0.$ Then

By Lemma 2.1(2), the definition of $\lambda_{n+1}$ and combining (3.1), (3.2), we obtain

From Lemma 3.1(2), we obtain $\lim\limits_{n\rightarrow \infty}\lambda_n\frac{\mu}{\lambda_{n+1}} = \mu(0 < \mu < 1)$. It means that there exists a positive integer number $N_0$, such that for all $n > N_0$, $0 < \lambda_n\frac{\mu}{\lambda_{n+1}} < 1$. Combining Lemma 2.3 and (3.3), we know that there exits a 2-uniformly convex constant $\tau$, such that when $n > N_0$,

Then, by Lemma 2.2(2) and the definition of $x_{n+1}$, we obtain for every $n > N_0$,

Thus, $\{V(Jx_n, u) \}$ is bounded. Combining $V(Jx_n, u)\geq \frac{1}{\tau}\|x_n-u \|^2,$ we get $\{x_n\}$ is bounded. Furthermore, from (3.3), we have the fact that $\{y_n\}$, $\{z_n\}$, and $\{t_n\}$ are bounded.

Step 2.$\{x_n\}$ strongly converges to a point $x^* = P_{S\bigcap F(T)}Jx_0$.

Let $x^* = P_{S\bigcap F(T)}Jx_0$. From Lemma 2.1(1), we can obtain

From Step 1, we know that there exists $\exists N_0\geq0,$ such that $\forall n\geq N_0,$ $V(Jz_n, x^*)\leq V(Jx_n, x^*),$ and the sequences $\{x_n\}$, $\{y_n\}$ $\{z_n\}$ and $\{t_n\}$ are bounded. Moreover, by Lemma 2.1(3) and Lemma 2.2, exists $N_0\geq0,$ such that for every $n\geq N_0,$

By (3.3), (3.4) and Lemma 2.3, Lemma 3.1(2), we can obtain that for every $n\geq N_0,$

Two cases arise:

Case 1. From the result of Lemma 2.5 in ^[28], set $a_n = \varphi(x_n, x^*) = V(Jx_{n}, x^*)$. By the proof of Step 1, there exists $N_1\in\mathbb{N}$ ($N_1\geq N_0$), such that $\{\varphi(x_n, x^*)\}^{\infty}_{n = N_1}$ is nonincreasing sequence. Then $\{a_n\}^{\infty}_{n = 1}$ converges. By using this in (3.5), when $n > N_1\geq N_0$, we have

By $V(Jx_0-Jx_n, x^*)$ is bounded and $\{a_n\}^{\infty}_{n = 1}$ converges, we have that when $n\longrightarrow \infty,$

Combining $\varphi(x_n, y_n)\geq 0$ and $0 < \mu, \alpha_n < 1,$ we have that when $n\longrightarrow \infty,$

Thus, when $n\longrightarrow \infty,$

for some $M_1 > 0.$ By (3.7), we also can see that $\|J(Tt_n)-Jz_{n}\|\longrightarrow 0, \|Jt_n-Jz_{n}\|\longrightarrow 0$. From $\|J(Tt_n)-Jt_{n}\|\leq \|J(Tt_n)-Jz_{n}\|+\|J(z_n)-Jt_{n}\|,$ we have $\|J(Tt_n)-Jt_{n}\|\longrightarrow 0.$

Since $J^{-1}$ is also uniformly norm-to-norm continuous on bounded subset of $X^*,$ we have $\|x_{n+1}-z_{n}\|\longrightarrow 0.$ Therefore, we get that when $n\longrightarrow \infty,$

Thus, when $n\longrightarrow \infty,$

and

Since $\{x_n\}$ is bounded, then there exists a subsequence $\{x_{n_k}\}$ that converges weakly to some $z_0\in X$, such that $x_{n_k}\rightharpoonup z_0$. By (3.9), we also have $\{t_{n_k}\}$ converges weakly to $z_0$. It follows from (3.8) and the definition of the nonexpansive mapping $T$ that $z_0\in F(T).$

Now, we show that $z_0\in S.$

Since $\{x_{n_k}\}$ converges weakly to $z_0,$ we have

Since $\|x_n-y_n\|^2\longrightarrow0$, we know that $y_{n_k}\rightharpoonup z_0$ and $z_0\in C.$ Since $y_{n_k} = P_C(x_{n_k}-\lambda_{n_k} F(x_{n_k}))$, by Lemma 2.1(1), we have that for all $z\in C,$ $\langle Jx_{n_k}-\lambda_{n_k}F(x_{n_k})-Jy_{n_k}, z-y_{n_k}\rangle\leq 0.$ This implies that

Therefore, we have that for all $z \in C,$

Fixing $z \in C,$ according to (3.6), and considering that $\{x_{n_k}\}$ is bounded, we can obtain

Choose a decreasing nonnegative sequence $\{\varepsilon_{k}\}$, such that $\lim_{k\rightarrow \infty}\varepsilon_{k} = 0.$ By definition of the lower limit, for each $\varepsilon_{k},$ there exists a smallest positive integer $M_k$ such that for all $k\geq M_k,$

Clearly, as $\{\varepsilon_{k}\}$ is decreasing, $\{M_{k}\}$ is increasing.

If there exists a subsequence $\{x_{n_{k_i}}\}$ of $\{x_{n_k}\}$, such that for every $i,$ $F(x_{n_{k_i}}) = 0,$ then

It means $z_0 \in S.$

If there exists a positive integer $N_2\in \mathbb{N}$ such that for all positive integer $n_{k_i} \geq N_2,$ $F(x_{n_{k_i}})\neq0.$ Let $u_{n_{k_i}} = F(x_{n_{k_i}})/ \|F(x_{n_{k_i}})\|^2.$ Then for each positive integer $n_{k_i} \geq N_2,$ $\langle F(x_{n_{k_i}}), u_{n_{k_i}}\rangle = 1.$ Thus, from (3.11), we have that for all positive integer $n_{k_i} \geq N_2,$

Since $F$ is pseudomonotone, then we have from (3.12) that $\langle F(z +\varepsilon_{k}u_{n_{k_i}}), z+\varepsilon_{k}u_{n_{k_i}}-x_{n_{k_i}}\rangle\geq 0.$ This implies that

Since $\{x_{n_k}\}$ converges weakly to $z_0\in C,$ and $F$ is sequentially weakly continuous on $C$, we get $F(x_{n_k})$ converges weakly to $F(z_0).$ If $F(z_0) = 0$, then $z_0 \in S.$ Now, assume that $F(z_0)\neq 0$. Combining $\|F(z_0)\|\leq\liminf\limits_{k\rightarrow \infty}\| F(x_{n_k})\|$ and $\lim_{k\rightarrow \infty}\varepsilon_{k} = 0,$ we get that the right-hand of (3.13) tends to zero. Thus, we obtain that for all $z\in C,$

By the result of Lemma 3.1 in ^[29], we also have $z_0\in S$. combine Lemma 2.1(1) and (3.10), we can obtain,

By Lemma 2.1(3) and (3.6), we have that for all positive integer $n > max\{N_1, N_2\},$

It follows from the result of [14, Lemma 3.3] and [28, Lemma 2.5], we obtain $\lim\limits_{n\rightarrow \infty}\varphi(x_n, x^*) = 0$, that means

Case 2. Assume that there exists a subsequence $\{x_{m_j}\}$ of $\{x_n\}$ such that for all $j\in\mathbb{N},$ $\varphi(x_{m_j}, x^*) < \varphi(x_{m_{j}+1}, x^*)$. Then, from Lemma 2.4, we know that there exists a nondecreasing sequence $m_k \in \mathbb{N}$ such that $\lim\limits_{k\rightarrow \infty}m_k = \infty$ and for every $k\in\mathbb{N}$ :

By (3.5) and Lemma 3.1(2), we have

Since $\{x_{n}\}$ is bounded, then exists a subsequence $\{x_{m_k}\}$ of $\{x_{n}\}$ which converges weakly to $z_0 \in X.$ Use the same argument as shown in Case 1, and Combining (3.15) we obtain

Similarly we can obtain

It follows from (3.15) and the proof of case 1, for all $m_k\geq N_0,$ we have

Since $0 < \alpha_n, \beta_n < 1,$ this implies that $\forall \ m_k\geq N_1,$ we have

And then

we obtain $\limsup\limits_{k\rightarrow \infty}\varphi(x_{m_{k}}, x^*) = 0,$ that means $\lim\limits_{k\rightarrow \infty}\|x_{m_k}-x^*\|^2 = 0$. Since $\|x_k-x^*\|\leq\|x_{{m_k}+1}-x^*\|$, we have $\lim\limits_{k\rightarrow \infty}\|x_k-x^*\| = 0$. Therefore $x_k\rightarrow x^*.$ The proof is complete.

4.   Numerical experiments

In this section, we give a numerical experiment to demonstrate the convergence and efficiency of the proposed algorithm. We will compare Algorithm 3.1 with a strongly convergent algorithms as HSEGM proposed in [30, Theorem 4.2].

Example 4.1 Let $H = L^{2}([0, 1]).$ We apply our problem in $H$ with norm $\|x\| = (\int_{0}^{1}|x(t)|^{2}dt)^{\frac{1}{2}}$ and inner product $\langle x, y\rangle = \int_{0}^{1}x(t)y(t)dt,$ $x, y\in H.$ The operator $F: H\rightarrow H$ is of form

for all $x\in H.$ Clearly, $F$ is Lipschitz-continuous and monotone (so is also pseudomonotone). The feasible set is $C = \{x\in H: \|x\|\leq 1 \}.$ Observe that $0\in S$ and so $S\neq \emptyset.$ Let $T: H\rightarrow H$ is defined by

Clearly, $0 \in F(T)$ and so $F(T)\neq \emptyset.$ Since

This means that $T$ is nonexpansive and therefore

Hence, the solution of the problem is $x^* = 0.$ To terminate the algorithms, we use the condition $\|x_n-x^*\|\leq \varepsilon$ and $\varepsilon = 10^{-3}$ for all the algorithms. We take $\alpha_n = \frac{1}{100n}$, $\beta_n = \frac{1}{2n+1}$, $\theta_n = 0$, $\lambda_0 = 0.7$ and $\mu = 0.9$ for Algorithm 3.1. For Theorem 4.2 in ^[30], we take $\alpha_n = \frac{1}{100n}$, $\beta_n = \frac{1}{2n+1}$ and $\tau = 0.7$. The numerical results are showed in Table 1 and Figure 1.

Remark 4.1. Observing from the numerical results of the example presented above, the conclusion that our algorithm is consistent, stable, effective and easy to implement is obtained. This example shows that Algorithm 3.1 converges slightly slower than that of Theorem 4.2 in ^[30], but it has some advantages. First, it is done without any information of the Lipschitz constant of the cost operator $F$. Second, the step size is variable in Algorithm 3.1. Third, it can be applied to solve fixed point problem for non-expansive mappings in a real 2-uniformly convex Banach space, and furthermore, it is more useful in the infinite-dimensional spaces.

5.   Conclusions

In this paper, we consider a strong convergence result for solving a common solution of pseudomonotone Variational Inequalities and fixed point problems in convex Banach spaces. Our algorithm is based on the subgradient extragradient methods with a new step size, the convergence of algorithm is established without the knowledge of the Lipschitz constant of the mapping. Finally, some numerical experiments are given to illustrate the convergence of our algorithms and compared with other known methods.

Acknowledgments

The Project Supported by "Qinglan talents" Program of Xianyang Normal University of China (No. XSYQL201801), The Educational Science Foundation of Shaanxi of China (No. 18JK0830), Scientific research plan projects of Xianyang Normal University of China (No. 14XSYK003).

Conflict of interest

The authors declare no conflicts of interest.