On Mann-type accelerated projection methods for pseudomonotone variational inequalities and common fixed points in Banach spaces

1.   Introduction

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot\rangle$ and induced norm $\|\cdot\|$. Let $\emptyset\neq C\subset H$ be a convex and closed set. Let ${\rm Fix}(S)$ be the set of fixed points of a mapping $S:C\to C$, i.e., ${\rm Fix}(S): = \{x\in C:x = Sx\}$. $S$ is said to be asymptotically nonexpansive if $\exists\{\theta_n\}\subset[0, +\infty)$ s.t. $\lim_{n\to\infty}\theta_n = 0$ and for all $n\ge 1$,

$S$ is nonexpansive when $\theta_n\equiv0, \; \forall n\geq1$.

Recall that the variational inequality (VIP) pursues to search $z\in C$ such that

where $F:H\to H$ is an operator. Use Ⅵ($C, F$) to denote the solution set of VIP.

Korpelevich ^[11] invented an extragradient method for solving VIP: The sequence $\{w_n\}$ is derived from an initial point $w_0\in C$ and

where $\ell\in(0, \frac{1}{L})$ with $L$ being the Lipschitz constant of $F$. If ${\rm VI}(C, F)\neq\emptyset$, then $\{w_n\}$ is convergent weakly to $w^*\in {\rm VI}(C, F)$. For solving VIP, many algorithms were introduced and adapted, see ^{[1,2,3,5,7,8,10,12,13,14,15,16,17,18,20,23,28,29,31,32]}. Within the extragradient method, one needs to compute two projections onto $C$ per iteration. If $C$ is a general convex and closed set, this might result in a prohibitive amount of computation time. To overcome this drawback, Censor et al. ^[2] presented a subgradient extragradient algorithm in which a half-space is constructed. Reich et al. ^[13] suggested an iterate for solving the pseudomonotone variational inequality by constructing a hyperplane.

Let $C$ be a nonempty, closed and convex subset of a $p$-uniformly convex and uniformly smooth Banach space $E$ with $p, q\in(1, \infty)$ and $\frac{1}{p}+\frac{1}{q} = 1$. Let $E^*$ be the dual space of $E$. Let $J^p_E$ and $J^q_{E^*}$ be the duality mappings of $E$ and $E^*$, respectively. Set $f_p(x) = \|x\|^p/p, \; \forall x\in E$. Use $D_{f_p}$ and $\Pi_C$ to denote the Bregman distance and the Bregman projection from $E$ onto $C$ with respect to (w.r.t) $f_p$, respectively. Eskandani et al. ^[18] introduced the hybrid projection method for finding a common solution of the VIP for uniformly continuous pseudomonotone mapping $F:E\to E^*$ and the FPP of Bregman relatively nonexpansive mapping $T$. Their algorithm is formulated as follows.

Algorithm 1.1 (^[18]). Let $\mu > 0, \; l\in(0, 1), \; \lambda\in(0, \frac{1}{\mu})$ be three constants. Let $x_1\in C$ be an initial point.

Step 1. Calculate $y_n = \Pi_C(J^q_{E^*}(J^p_Ex_n-\lambda Fx_n))$ and $r_\lambda(x_n): = x_n-y_n$. If $Tx_n = x_n$ and $r_\lambda(x_n) = 0$, then stop (in this case $x_n\in{\Omega} = {\rm Fix}(T)\cap{\rm VI}(C, F)$); otherwise, continue to the next step.

Step 2. Calculate $t_n = x_n-\tau_nr_\lambda(x_n)$ in which $\tau_n: = l^{j_n}$ with $j_n$ being the smallest nonnegative integer such that

Step 3. Calculate $v_n = J^q_{E^*}(\beta_nJ^p_Ex_n+(1-\beta_n)J^p_E(T\Pi_{C_n}x_n))$ and $x_{n+1} = \Pi_C(J^q_{E^*}(\alpha_nJ^p_Eu+(1-\alpha_n)J^p_Ev_n))$, where $C_n: = \{x\in C:h_n(x_n)\leq0\}$ and $h_n(x) = \langle Ft_n, x-x_n\rangle+\frac{\tau_n}{2\lambda}D_{f_p}(x_n, y_n)$.

Let $n: = n+1$ and return to Step 1.

Under suitable conditions, they proved the strong convergence of Algorithm 1.1 to $\Pi_{\Omega}u$. Inspired by the above research works, the main purpose of this paper is to introduce two Mann-type accelerated projection methods for solving the VIP for a uniformly continuous pseudomonotone operator and the CFPP of finitely many Bregman relatively nonexpansive mappings and a Bregman relatively asymptotically nonexpansive mapping in $p$-uniformly convex and uniformly smooth Banach spaces. Under mild conditions, we prove weak and strong convergence of the proposed algorithms to a common solution of the VIP and CFPP, respectively. An illustrated example is provided to demonstrate the applicability and implementability of our suggested method. Our algorithms are more advantageous and more flexible than the above Algorithm 1.1 because they involve solving the VIP for uniformly continuous pseudomonotone operator and the CFPP of finitely many Bregman relatively nonexpansive mappings and a Bregman relatively asymptotically nonexpansive mapping. The main theorems presented in this paper are the improvement and extension of the corresponding theorems obtained in ^[13,17,18].

2.   Preliminaries

Let $\{x_n\}$ be a sequence of a real Banach space $E$. Let $\omega_w(x_n)$ be the set of all weak cluster points of $\{x_n\}$, i.e., $\omega_w(x_n) = \{x^\dagger\in E:x_{n_k}\rightharpoonup x^\dagger \; {\rm for\; some}\; \{x_{n_k}\}\subset\{x_n\}\}$.

Let $E$ be a Banach space and $U: = \{u\in E:\|u\| = 1\}$. (ⅰ) $E$ is strictly convex if $\|u+v\|/2 < 1, \forall u, v\in U$ and $u\neq v$. (ⅱ) $E$ is uniformly convex if $\forall\varepsilon\in(0, 2]$, $\exists\delta > 0$ such that $\|u+v\|/2\leq1-\delta, \forall u, v\in U$ when $\|u-v\|\geq \varepsilon$.

$E$ is uniformly convex $\Leftrightarrow$ for all $\varepsilon\in(0, 2]$, $\delta(\varepsilon) > 0$ where $\delta(\varepsilon) = \inf\{1-\|u+v\|/2:u, v\in U\; {\rm with}\; \|u-v\|\geq\varepsilon\}$ is the modulus of convexity of $E$. Moreover, $E$ is $p$-uniformly convex if $\exists c > 0$ s.t. $\delta(\varepsilon)\geq c\varepsilon^p$ for all $\varepsilon\in[0, 2]$. $E$ is uniformly smooth if $\lim_ {\tau\to0}\rho_E(\tau)/\tau = 0$ where $\rho_E(\tau) = \sup\{(\|u+\tau v\|+\|u-\tau v\|)/2-1:u, v\in U\}$ is the modulus of smoothness of $E$. $E$ is $q$-uniformly smooth if $\exists C_q > 0$ s.t. $\rho_E(\tau)\leq C_q\tau^q, \; \forall\tau > 0$. $E$ is $p$-uniformly convex $\Leftrightarrow$ $E^*$ is $q$-uniformly smooth. For more details, please refer to ^[19].

Let $r > 0$ and set $B(0, r) = \{x\in E:\|x\|\le r\}$. Let $f:E\to{\bf R}$ be a function. For $\alpha\in(0, 1)$ and $u, v\in B(0, r)$ with $\|u-v\| = t$. Define

$E$ is uniformly convex on bounded set $B(0, r)$ if $\rho_r(t) > 0$ for all $r, t > 0$ (see ^[9,18]).

Set $\frac{1}{p}+\frac{1}{q} = 1$ where $p, q\in(1, \infty)$. The duality mapping $J^p_E:E\to E^*$ is formulated below

(ⅰ) $E$ is smooth $\Leftrightarrow$ $J^p_E$ is single-valued. (ⅱ) $E$ is reflexive $\Leftrightarrow$ $J^p_E$ is surjective. (ⅱ) $E$ is strictly convex $\Leftrightarrow$ $J^p_E$ is one-to-one.

Let $f:E\to{\bf R}$ be a convex function. $f$ is said to be Gâteaux differentiable at $x$ if for each $y\in E$, $\lim_{t\to0^+}\frac{f(u+tv)-f(u)}{t}$ exists. In this case, define $\langle\nabla f(u), v\rangle = \lim_{t\to0^+}\frac{f(u+tv)-f(u)}{t}$ for each $v\in E$. Suppose $f:E\to {\bf R}$ is Gâteaux differentiable. The Bregman distance (^[21]) w.r.t. $f$ is formulated as

The Bregman distance ensures existence and uniqueness of the Bregman projection and it has also been used to generate generalized proximal point methods for convex optimization and variational inequalities, see ^[30]. It is easy to check that

Note that the Bregman distance w.r.t. $f_p$ is formulated by $\forall u, v\in E$,

If $E$ is $p$-uniformly convex and smooth Banach space $E$, then (see ^[26])

From (2.1) it is readily known that for any bounded sequence $\{x_n\}\subset E$, the following holds:

Let $E$ be a reflexive, smooth and strictly convex Banach space and $C\subset E$ a nonempty closed convex set. For every $u\in E$, there exists the unique element denoted by $\Pi_Cu\in C$ such that $D_{f_p}(\Pi_Cu, u) = \min_{v\in C}D_{f_p}(v, u)$. $\Pi_C$ is called the Bregman projection w.r.t. $f_p$. Furthermore, if $E$ is uniformly convex, then (^[22,24])

which equivalent to

Let $V_{f_p}:E\times E^*\to[0, \infty)$ (^[18]) be a function defined by

For all $u\in E$, $u^*\in E^*$ and $v^*\in E^*$, we have $V_{f_p}(u, u^*) = D_{f_p}(u, J^q_{E^*}(u^*))$ and

In addition, $V_{f_p}(x, \cdot)$ is convex. Then, for all $w\in E, \{u_i\}^n_{i = 1}\subset E$, $\{t_i\}^n_{i = 1}\subset[0, 1]$ and $\sum^n_{i = 1}t_i = 1$, we have

Lemma 2.1 (^[24]). Let $E$ be a uniformly convex Banach space. Let $\{u_n\}\subset E, \{v_n\}\subset E$ be two sequences and $\{u_n\}$ is bounded. Then, $\lim_{n\to\infty}D_{f_p}(v_n, u_n) = 0\Rightarrow \lim_{n\to\infty}\|v_n-u_n\| = 0$.

Let $T:C\to C$ be an operator. A point $x\in C$ is an asymptotic fixed point of $T$ (^[25]) if $\exists\{x_n\}\subset C$ s.t. $x_n\rightharpoonup x$ and $x_n-Tx_n\to0$. Let ${\rm Fix}(T)$ and $\widehat{\rm Fix}(T)$ be the set of fixed points of $T$ and the set of asymptotic fixed points of $T$, respectively. $T$ is said to be Bregman relatively asymptotically nonexpansive w.r.t. $f_p$ if ${\rm Fix}(T) = \widehat{\rm Fix}(T)\neq\emptyset$, and $\exists\{\theta_n\}\subset[0, \infty)$ s.t.

for all $v\in C$ and $u\in{\rm Fix}(T)$.

Recall that an operator $F:C\to E^*$ is said to be

(ⅰ) monotone on $C$ if $\langle Fu-Fv, u-v\rangle\geq0, \; \forall u, v\in C$;

(ⅱ) pseudomonotone if $\langle Fu, v-u\rangle\geq0\Rightarrow \langle Fv, v-u\rangle\geq0, \; \forall u, v\in C$;

(ⅲ) $L$-Lipschitz continuous if $\exists L > 0$ s.t. $\|Fu-Fv\|\leq L\|u-v\|, \forall u, v\in C$;

(ⅳ) weakly sequentially continuous if for any $\{x_n\}\subset C$, $x_n\rightharpoonup x\Rightarrow Fx_n\rightharpoonup Fx$.

Lemma 2.2 (^[18]). Let $E$ be a Banach space and $f:E\to{\bf R}$ be a uniformly convex function on $B(0, r)$. Let $\{x_k\}^n_{k = 1}$ be a sequence in $B(0, r)$ and $\{\alpha_k\}^n_{k = 1}$ be a real number sequence in $(0, 1)$ such that $\sum^n_{k = 1}\alpha_k = 1$. Then,

Lemma 2.3 (^[15]). Let $E_1$ and $E_2$ be two Banach spaces. Let $D\subset E_1$ be a bounded set. If $F:E_1\to E_2$ is uniformly continuous on $D$, then $F(D)$ is bounded.

Lemma 2.4 (^[6]). Let $C$ be a nonempty closed convex subset of a real Banach space $E$. Let $F:C\to E^*$ be a continuous pseudomonotone operator. Then $u\in \mathit{\text{VI}}(C, F)\Leftrightarrow\langle Fv, v-u\rangle\geq0, \; \forall v\in C$.

Lemma 2.5. Let $2\leq p < \infty$ and let $E$ be a smooth and $p$-uniformly convex Banach space with weakly sequentially continuous duality mapping $J^p_E$. Let $\{x_n\}$ be a sequence in $E$ and $C$ be a nonempty subset of $E$. Suppose that $\{D_{f_p}(x, x_n)\}$ converges for every $x\in C$, and $\omega_w(x_n)\subset C$. Then $\{x_n\}$ converges weakly to a point in $C$.

Proof. Since the inequality (2.1) leads to $\tau\|x-x_n\|^p\leq D_{f_p}(x, x_n), \; \forall x\in C$, we know that $\{x_n\}$ is bounded. Hence from the reflexivity of $E$ it follows that $\omega_w(x_n)\neq\emptyset$. In what follows, we claim that $\omega_w(x_n)$ is a single-point set. Indeed, let $x, y\in\omega_w(x_n)\subset C$ with $x\neq y$. Then, $\exists\{x_{n_k}\}\subset \{x_n\}$ and $\exists\{x_{m_k}\}\subset\{x_n\}$ s.t. $x_{n_k}\rightharpoonup x$ and $x_{m_k}\rightharpoonup y$. By the weakly sequential continuity of $J^p_E$ one has that $J^p_E(x_{n_k}) \rightharpoonup x$ and $J^p_E(x_{m_k})\rightharpoonup y$. Note that $D_{f_p}(x, y)+D_{f_p}(y, x_n) = D_{f_p}(x, x_n)-\langle J^p_Ey-J^p_Ex_n, x-y\rangle$. Since $\{D_{f_p}(x, x_n)\}$ and $\{D_{f_p}(y, x_n)\}$ are convergent, we obtain

which immediately yields $\langle J^p_Ex-J^p_Ey, x-y\rangle = 0$. Again from (2.1) we have $0 < \tau\|x-y\|^p\leq D_{f_p}(x, y)\leq\langle J^p_Ex-J^p_Ey, x-y\rangle = 0$. It is impossible. So, $\omega_w(x_n)$ is a single-point set. □

Lemma 2.6 (^[16]). Let $C$ be a nonempty closed convex subset of a Banach space $E$. Define $D: = \{x\in C:g(x)\leq0\}$ where $g$ is a real-valued function on $E$. If $D\ne \emptyset$ and $g$ is Lipschitz continuous on $C$ with modulus $\theta > 0$, then ${\rm dist}(x, D)\geq\theta^{-1}\max\{g(x), 0\}, \; \forall x\in C$.

Lemma 2.7 (^[4]). Let $\{a_n\}$ be a sequence of nonnegative numbers such that $a_{n+1}\leq(1-\lambda_n)a_n+\lambda_n\mu_n+\nu_n\; \forall n\geq1$, where the following hold for sequences $\{\lambda_n\}, \{\mu_n\}, \{\nu_n\}\subset{\bf R}$:

(i) $\{\lambda_n\}\subset[0, 1]$ and $\sum^\infty_{n = 1}\lambda_n = \infty$;

(ii) $\limsup_{n\to \infty}\mu_n\leq0$ and $\sum^\infty_{n = 1}|\nu_n| < \infty$.

Then $\lim_{n\to\infty}a_n = 0$.

Lemma 2.8 (^[27]). Let $\{\Phi_n\}$ be a sequence of real numbers that does not decrease at infinity in the sense that, $\exists\{\Phi_{n_k}\}\subset\{\Phi_n\}$ s.t. $\Phi_{n_k} < \Phi_{n_k+1}, \; \forall k\geq1$. Let $n_0\geq1$ and $\{\psi(n)\}_{n\geq n_0}$ be integers sequence defined by $\psi(n) = \max\{k\leq n:\Phi_k < \Phi_{k+1}\}$ satisfying $\{k\leq n_0:\Phi_k < \Phi_{k+1}\}\neq\emptyset$. Then,

(i) $\psi(n_0)\leq\psi(n_0+1)\leq\cdots$ and $\psi(n)\to\infty$;

(ii) for all $n\geq n_0$, $\Phi_{\psi(n)}\leq\Phi_{\psi(n)+1}$ and $\Phi_n\leq\Phi_{\psi(n)+1}$.

3.   Main results

Let $C$ be a nonempty closed convex subset of of a $p$-uniformly convex and uniformly smooth Banach space $E$. Suppose that

(C1) the mapping $T:C\to C$ is Bregman relatively asymptotically nonexpansive with $\{\theta_n\}$ and uniformly continuous.

(C2) the mapping $T_i:C\to C$($i = 1, ..., N$) is Bregman relatively nonexpansive and uniformly continuous and $T_n: = T_{n{\rm mod}N}$ for integer $n\geq 1$ with the mod function taking values in the set $\{1, 2, ..., N\}$.

(C3) the mapping $F:E\to E^*$ is uniformly continuous and pseudomonotone such that $\|Fz\|\leq\liminf_{n\to\infty}$ $\|Fx_n\|$ for any $\{x_n\}\subset C$ with $x_n\rightharpoonup z$.

(C4) ${\Omega} = \bigcap^N_{i = 0}{\rm Fix}(T_i)\cap{\rm VI}(C, F)\neq\emptyset$ where $T_0: = T$.

Let $\mu > 0$, $\lambda\in(0, \frac{1}{\mu})$ and $l\in(0, 1)$ be three constants. Let $\{\sigma_n\}, \{\alpha_n\}$ be two sequences in $(0, 1)$ s.t. $\liminf_{n\to \infty}\sigma_n(1-\sigma_n) > 0$ and $\liminf_{n\to\infty}\alpha_n(1-\alpha_n) > 0$.

Algorithm 3.1. Let $x_1\in C$ be an initial point.

Step 1. Calculate $w_n = J^q_{E^*}(\sigma_nJ^p_Ex_n+(1-\sigma_n)J^p_E(T_nx_n))$, $y_n = \Pi_C(J^q_{E^*}(J^p_Ew_n-\lambda Fw_n))$ and $r_\lambda(w_n): = w_n-y_n$.

Step 2. Calculate $t_n = w_n-\tau_nr_\lambda(w_n)$, where $\tau_n: = l^{j_n}$ with $j_n$ being the smallest nonnegative integer $j$ such that

Step 3. Calculate $v_n = J^q_{E^*}(\alpha_nJ^p_Ew_n+(1-\alpha_n)J^p_E(T^nw_n))$ and $x_{n+1} = \Pi_{C_n\cap Q_n}(w_n)$, where $Q_n: = \{x\in C:D_{f_p}(x, v_n)\leq(1+\theta_n)D_{f_p}(x, w_n)\}$, $C_n: = \{x\in C:h_n(x)\leq0\}$ and

Set $n: = n+1$ and go to Step 1.

Lemma 3.2. Suppose that the sequence $\{x_n\}$ is constructed in Algorithm 3.1. Then the inequality holds: $\langle Fw_n, r_\lambda(w_n)\rangle\geq\frac{1}{\lambda}D_{f_p}(w_n, y_n)$.

Proof. Using the property of $\Pi_C$, we obtain

It follows from (2.1) that

□

Lemma 3.3. The rule (3.1) and $\{x_n\}$ generated by Algorithm 3.1 are well defined.

Proof. Note that $\lim_{j\to\infty}\langle Fw_n- F(w_n-l^jr_\lambda(w_n)), r_\lambda(w_n)\rangle = 0$. If $r_\lambda(w_n) = 0$, then it is obvious that $j_n = 0$. If $r_\lambda(w_n)\neq0$, then there is $j_n\geq0$ fulfilling (3.1).

It is easy to see that for every $n\geq1, \; C_n$ and $Q_n$ are closed and convex. Next, we show ${\Omega}\subset C_n\cap Q_n$. Take any $z\in{\Omega}$. By (2.6) and the Bregman relatively asymptotical nonexpansivity of $T$, we have

which immediately yields $z\in Q_n$. Moreover, using Lemma 2.4, we have $\langle Ft_n, t_n-z\rangle\geq0$. Hence

Thanks to (3.1), we have

Using this and Lemma 3.2, we have

Combining this and (3.3) to deduce

Consequently, ${\Omega}\subset C_n\cap Q_n$. Therefore, $\{x_n\}$ is well defined. □

Lemma 3.4. Let the sequence $\{w_n\}$ be defined by Algorithm 3.1. Then $\lim_{n\to\infty}\|w_n-y_n\| = 0$ implies that $\omega_w(w_n)\subset{\rm VI}(C, F)$.

Proof. Let $z\in\omega_w(w_n)$. Then, $\exists\{w_{n_k}\}\subset\{w_n\}$, s.t. $w_{n_k}\rightharpoonup z$ and $\lim_{n\to\infty}\|w_{n_k}-y_{n_k}\| = 0$. Hence, it is known that $y_{n_k} \rightharpoonup z$. Since $C$ is convex and closed and $\{y_n\}\subset C$, $z\in C$. Next, we consider two cases. If $Fz = 0$, then $z\in{\rm VI}(C, F)$. If $Fz\neq0$, using the assumption on $F$, instead of the weakly sequential continuity of $F$, we get $0 < \|Fz\|\leq\liminf_{k\to\infty} \|Fw_{n_k}\|$. So, we might assume that $\|Fw_{n_k}\|\neq0, \forall k\geq1$. Using (2.2), we obtain

and hence

By Lemma 2.3, $\{Fw_{n_k}\}$ is bounded. Note that $\{y_{n_k}\}$ is also bounded as well. From (3.4) we get

Let $\{\epsilon_k\}$ be a sequence in $(0, 1)$ fulfilling $\epsilon_k\downarrow0$ as $k\to\infty$. Let $l_k$ be the smallest positive integer satisfying

Since $\{\epsilon_k\}$ is decreasing, $\{l_k\}$ is increasing. For convenience, we denote $\{Fw_{n_{l_k}}\}$ by $\{Fw_{l_k}\}$. Note that $Fw_{l_k}\neq0, \; \forall k \geq1$. Put $\upsilon_{l_k} = \frac{Fw_{l_k}}{\|Fw_{l_k}\|^{\frac{q}{q-1}}}$. We have $\langle Fw_{l_k}, J^q_{E^*}\upsilon_{l_k}\rangle = 1, \; \forall k\geq1$. Indeed, it is clear that $\langle Fw_{l_k}, J^q_{E^*}\upsilon_{l_k}\rangle = \langle Fw_{l_k}, (\frac{1}{{\|Fw_{l_k}\|^{\frac{q}{q-1}}}})^{q-1}J^q_{E^*}Fw_{l_k}\rangle = (\frac{1} {{\|Fw_{l_k}\|^{\frac{q}{q-1}}}})^{q-1}\|Fw_{l_k}\|^q = 1, \; \forall k\geq1$. So, using (3.6) one has $\langle Fw_{l_k}, y+\epsilon_kJ^q_{E^*}\upsilon_{l_k}-w_{l_k}\rangle\geq0, \; \forall k\geq1$. Since $F$ is pseudomonotone, we have

We claim that $\lim_{k\to\infty}\epsilon_kJ^q_{E^*}\upsilon_{l_k} = 0$. In fact, since $\{w_{l_k}\}\subset\{w_{n_k}\}$ and $\epsilon_k\downarrow0$, we have

Hence one gets $\epsilon_kJ^q_{E^*}\upsilon_{l_k}\to0$ as $k\to\infty$. Thus, letting $k\to\infty$ in (3.7) and from (C3), we have $\langle Fy, y-z\rangle\geq0, \; \forall y \in C$. According to Lemma 2.4 one has $z\in{\rm VI}(C, F)$. □

Lemma 3.5. Let the sequence $\{w_n\}$ be generated by Algorithm 3.1. Then,

Proof. Suppose that $\liminf_{n\to\infty}\tau_n > 0$. In this case, assume that $\exists\tau > 0$ s.t. $\tau_n\geq\tau > 0, \; \forall n\geq1$. Then,

This together with $\lim_{n\to\infty}\tau_nD_{f_p}(w_n, y_n) = 0$, leads to $\lim_{n\to\infty}D_{f_p}(w_n, y_n) = 0$.

Suppose that $\liminf_{n\to\infty}\tau_n = 0$. In this case, assume that $\limsup_{n\to\infty}D_{f_p}(w_n, y_n) = a > 0$. Then we know that $\exists\{n_k\}\subset\{n\}$ such that

We define $\overline{t_{n_k}} = \frac{1}{l}\tau_{n_k}y_{n_k}+(1-\frac{1}{l}\tau_{n_k})w_{n_k}$ for each $k\geq1$. Applying (2.1) and noticing that $\lim_{k\to\infty}\tau_{n_k}D_{f_p}(w_{n_k}, y_{n_k}) = 0$, we have $\lim_{k\to\infty}\tau_{n_k}\|w_{n_k}-y_{n_k}\|^p = 0$ and hence

It follows that

So,

Now, letting $k\to\infty$ and from (3.10) we have $\lim_{k\to\infty}D_{f_p}(w_{n_k}, y_{n_k}) = 0$. It is a contradiction. Therefore, $\lim_{n\to\infty}D_{f_p}(w_n, y_n) = 0$. □

Theorem 3.6. Suppose that $E$ is a $p$-uniformly convex and uniformly smooth Banach space with weakly sequentially continuous duality mapping $J^p_{E}$. Let the sequence $\{x_n\}$ be defined by Algorithm 3.1. Then $\{x_n\}$ is convergent weakly to a point in ${\Omega}$ provided $T^nw_n-T^{n+1}w_n\to0$.

Proof. Take any $z\in{\Omega}$. Using Lemma 2.2, we get

From (2.1) and (2.3), we obtain

Combining the last two inequalities, we obtain

This indicates that $\lim_{n\to\infty}D_{f_p}(z, x_n)$ exists and the sequence $\{x_n\}$ is bounded. It is easy to check that $\{Fw_n\}, \{y_n\}, \{t_n\}, \{v_n\}, \{T_nx_n\}$ and $\{T^n w_n\}$ are also bounded. Note that $\omega_w(x_n)\neq\emptyset$. Next, we show $\omega_w(x_n)\subset{\Omega}$. Let $z^*\in\omega_w(x_n)$. Then, $\exists\{x_{n_k}\}\subset\{x_n\}$ s.t. $x_{n_k}\rightharpoonup z^*$. From (3.12), we obtain

This implies that $\lim_{n\to\infty}D_{f_p}(x_{n+1}, w_n) = \lim_{n\to\infty}D_{f_p}(x_{n+1}, v_n) = 0$ and hence

Hence,

Using Lemma 2.2, we get

Therefore

By (3.13), we get $\lim_{n\to\infty}\rho^*_b\|J^p_Ew_n-J^p_ET^nw_n\| = 0$ and hence $\lim_{n\to\infty}\|J^p_Ew_n-J^p_ET^nw_n\| = 0$. So,

In addition, from (3.12) we get $\sigma_n(1-\sigma_n)\rho^*_b\|J^p_Ex_n-J^p_ET_nx_n\|\leq D_{f_p} (z, x_n)-D_{f_p}(z, x_{n+1})$. Noticing the existence of $\lim_{n\to\infty}D_{f_p}(z, x_n)$ and $\liminf_{n\to\infty}\sigma_n(1-\sigma_n) > 0$, we have $\lim_{n\to\infty}\rho^*_b\|J^p_Ex_n -J^p_ET_nx_n\| = 0$ and hence $\lim_{n\to\infty}\|J^p_Ex_n-J^p_ET_nx_n\| = 0$. Thus,

Since $w_n = J^q_{E^*}(\sigma_nJ^p_Ex_n+(1-\sigma_n)J^p_ET_nx_n)$, we deduce that

It follows that

Now, we prove $z^*\in{\rm VI}(C, F)$. Since $\{Ft_n\}$ is bounded, we know that $\exists L > 0$ s.t. $\|Ft_n\|\leq L$. This ensures that for any $x, y\in C_n$,

This indicates that $h_n(x)$ is $L$-Lipschitz in $C_n$. Applying Lemma 2.6, we have

Using (3.12) and (3.17), we have

Hence $\lim_{n\to\infty}\tau_nD_{f_p}(w_n, y_n) = 0$. By Lemma 3.5, we get $\lim_{n\to\infty}\|w_n-y_n\| = 0$. Besides, combining (3.16) and $x_{n_k}\rightharpoonup z^*$ leads to $w_{n_k} \rightharpoonup z^*$. According to Lemma 3.4, we conclude that $z^*\in\omega_w(w_n)\subset{\rm VI}(C, F)$.

Next, we show $z^*\in\bigcap^N_{i = 0}{\rm Fix}(T_i)$ with $T_0: = T$. Indeed, we first show that $\lim_{n\to\infty}\|x_n-T_rx_n\| = 0$ for $r = 1, ..., N$. Actually, according to the definition of $T_n$, we obtain that $T_n\in\{T_1, ..., T_N\}\; \forall n\geq1$, which hence leads to $T_{n+i}\in\{T_1, ..., T_N\}\; \forall n\geq1, i = 1, ..., N$. Observe that

Thanks to (3.15) and (3.16), we have $x_{n+i}-T_{n+i}x_{n+i}\to0$ and $T_jx_{n+i}-T_jx_n\to0$ for $i, j = 1, ..., N$. Thus, we get $\lim_{n\to\infty}\|x_n-T_{n+i}x_n\| = 0$ for $i = 1, ..., N$. This immediately implies that

So it follows from (3.19) and $x_{n_k}\rightharpoonup z^*$ that $z\in\widehat{{\rm Fix}}(T_r) = {\rm Fix}(T_r)$ for $r = 1, ..., N$. Therefore, $z\in\bigcap^N_{i = 1}{\rm Fix}(T_i)$. In addition, observe also that

Noticing the uniform continuity of $T$ on $C$, we conclude from (3.14) that $Tw_n-T^{n+1}w_n\to0$. Thus, using the assumption $T^nw_n-T^{n+1}w_n\to0$, from (3.20) we get $\lim_{n\to\infty} \|w_n-Tw_n\| = 0$. Again from (3.16) and $x_{n_k}\rightharpoonup z^*$, one has that $w_{n_k}\rightharpoonup z^*$. Hence, we obtain $z^*\in\widehat{{\rm Fix}}(T) = {\rm Fix}(T)$. Consequently, $z^*\in\bigcap^N_{i = 0}{\rm Fix}(T_i)$, and hence $z^*\in{\Omega} = \bigcap^N_{i = 0}{\rm Fix}(T_i)\cap{\rm VI}(C, F)$. This means that $\omega_w(x_n)\subset{\Omega}$. Accordingly, applying Lemma 2.5 we conclude that $x_n\rightharpoonup z^*$. □

Next, we show a strong convergence result.

Algorithm 3.7. Let $x_1\in C$, $\mu > 0, \; l\in(0, 1)$ and $\lambda\in(0, \frac{1}{\mu})$. Choose $\{\sigma_n\}, \{\alpha_n\}, \{\beta_n\}\subset(0, 1)$ s.t. (i) $\liminf_ {n\to\infty}\sigma_n(1-\sigma_n) > 0$ and $\liminf_{n\to\infty}\beta_n(1-\beta_n) > 0$, and (ii) $\sum^\infty_{n = 1}\alpha_n = \infty$, $\lim_{n\to\infty}\alpha_n = 0, \; \lim_{n\to\infty}\theta_n/\alpha _n = 0$ and $\sum^\infty_{n = 1}\theta_n < \infty$.

Step 1. Set $w_n = J^q_{E^*}(\sigma_nJ^p_Ex_n+(1-\sigma_n)J^p_E(T_nx_n))$, and calculate $y_n = \Pi_C(J^q_{E^*}(J^p_Ew_n-\lambda Fw_n))$ and $r_\lambda(w_n): = w_n-y_n$.

Step 2. Calculate $t_n = w_n-\tau_nr_\lambda(w_n)$ in which $\tau_n: = l^{j_n}$ with $j_n$ being the smallest nonnegative integer $j$ fulfilling

Step 3. Set $z_n = \Pi_{C_n}(w_n)$, and compute $v_n = J^q_{E^*}(\beta_nJ^p_Ew_n+(1-\beta_n)J^p_E(T^nz_n))$ and $x_{n+1} = \Pi_C(J^q_{E^*}(\alpha_nJ^p_Eu+(1-\alpha_n)J^p_Ev_n)$, where $C_n: = \{x\in C:h_n(x)\leq0\}$ and

Let $n: = n+1$ and go to Step 1.

Theorem 3.8. Suppose that the conditions (C1)–(C4) are satisfied. Then, the sequence $\{x_n\}$ constructed in Algorithm 3.7 converges strongly to $\Pi_{\Omega}u$ provided $T^nz_n-T^{n+1}z_n\to0$.

Proof. We divide our proof into four claims.

Claim 1. The sequence $\{x_n\}$ is bounded. Indeed, set $\hat u = \Pi_{\Omega}u$. According to Theorem 3.6 and Lemma 2.2, we have

Using (2.3), (2.6) and (3.21), we deduce

From (3.22) that

Noticing $\sum^\infty_{n = 1}\theta_n < \infty$, we obtain that $\{D_{f_p}(\hat u, x_n)\}$ is bounded. This together with (2.1), implies that $\{x_n\}$ is bounded. Hence, $\{T_nx_n\}, \{Fw_n\}, \{w_n\}, \{y_n\}, \{t_n\}$, $\{z_n\}, \{T^nz_n\}$ and $\{v_n\}$ are also bounded.

Claim 2. We show $(1-\beta_n)(1+\theta_n)D_{f_p}(z_n, w_n)\leq(1+\theta_n)D_{f_p}(\hat u, w_n)-D_{f_p}(\hat u, x_{n+1})+\alpha_n\langle J^p_Eu-J^p_E\hat u, s_n-\hat u\rangle$. Set $b = \sup_{n\geq1}\{\|w_n|^{p-1}, \|T^nz_n\|^{p-1}\}$. By Lemma 2.2, we obtain

Set $s_n = J^q_{E^*}(\alpha_nJ^p_Eu+(1-\alpha_n)J^p_Ev_n$. Using (2.5), we have

On the other hand, we have

This together with (3.24) implies that

This immediately arrives at

Claim 3. We show that

Using the similar inferences to these of (3.18) in Theorem 3.6, we get

Applying (3.26), we get

Claim 4. Finally, we prove $x_n\to\hat u$ as $n\to\infty$. Note that $\omega_w(x_n)\neq\emptyset$. Let $z^\dagger \in\omega_w(x_n)$. Then, $\exists\{x_{n_k}\}\subset\{x_n\}$ s.t. $x_{n_k}\rightharpoonup z^\dagger$. For each $n\geq1$, we write $\Phi_n = D_{f_p}(\hat u, x_n)$. Put $\Phi_n = \|x_n-z^*\|^2$.

Case 1. Suppose that $\{\Phi_n\}^\infty_{n = n_0}$ is nonincreasing for some $n_0\ge 1$. Then $\lim_{n\to\infty}\Phi_n = d < +\infty$ and $\lim_{n\to\infty} (\Phi_n-\Phi_{n+1}) = 0$. By (3.21) and (3.25) we get

which immediately yields

Since $\lim_{n\to\infty}\theta_n = 0, \; \lim_{n\to\infty}\alpha_n = 0, \; \liminf_{n\to\infty}\beta_n(1-\beta_n) > 0, \; \liminf_{n\to\infty}\sigma_n(1-\sigma_n) > 0, \; \lim_{n\to\infty}\Phi_n = d$ and the sequences $\{s_n\}$ is bounded, we obtain that $\lim_{n\to\infty}D_{f_p}(z_n, w_n) = 0$ and $\lim_{n\to\infty}\|J^p_Ex_n-J^p_ET_nx_n\| = 0$. Noticing $w_n = J^q_{E^*}(\sigma_nJ^p_Ex_n+(1-\sigma_n) J^p_ET_nx_n)$, we also have $\lim_{n\to\infty}\|J^p_Ew_n-J^p_Ex_n\| = 0$. So it follows from (2.1) that

Furthermore, from (3.24) we have

By the similar inferences, we infer that $\lim_{n\to\infty}\|J^p_Ew_n-J^p_ET^nz_n\| = 0$, which hence leads to $\lim_{n\to\infty}\|J^p_Ev_n-J^p_Ew_n\| = 0$. Then,

Combining with (3.28), we have

Since $s_n = J^q_{E^*}(\alpha_nJ^p_Eu+(1-\alpha_n)J^p_Ev_n)$, it can be readily seen from (3.30) that

In addition, using (2.3) and (3.23), we achieve

which hence arrives at

Then, $\lim_{n\to\infty}D_{f_p}(x_{n+1}, s_n) = 0$ and hence $\lim_{n\to\infty}\|x_{n+1}-s_n\| = 0$. This together with (3.31), leads to

Note that

By (3.30), we have $Tz_n-T^{n+1}z_n\to0$. This together with (3.33) implies that $\lim_{n\to\infty} \|z_n-Tz_n\| = 0$. Again from (3.28) and $x_{n_k}\rightharpoonup z^\dagger$, one has that $z_{n_k}\rightharpoonup z^\dagger$. Hence, we obtain $z^\dagger \in\widehat{{\rm Fix}}(T) = {\rm Fix}(T)$. In the meantime, let us show that $z\in\bigcap^N_{i = 1}{\rm Fix}(T_i)$. Indeed, by the definition of $T_n$, we know that $T_n\in\{T_1, ..., T_N\}, \; \forall n\geq1$, which hence leads to $T_{n+i} \in\{T_1, ..., T_N\}, \; \forall n\geq1, i = 1, ..., N$. Observe that

Since each $T_j$ is uniformly continuous on $C$, we deduce from (3.28) and (3.32) that $x_{n+i}-T_{n+i}x_{n+i}\to0$ and $T_jx_{n+i}-T_jx_n\to0$ for $i, j = 1, ..., N$. Thus, we get $\lim_{n\to \infty}\|x_n-T_{n+i}x_n\| = 0$ for $i = 1, ..., N$. Hence, $\lim_{n\to\infty}\|x_n-T_ix_n\| = 0(i = 1, ..., N)$ and so $z^\dagger \in\widehat{{\rm Fix}}(T_i) = {\rm Fix}(T_i)$ for $i = 1, ..., N$. Consequently, $z^\dagger \in\bigcap^N_{i = 0}{\rm Fix}(T_i)$ with $T_0: = T$.

Next, we prove $z^\dagger \in{\rm VI}(C, F)$. Using (3.21) and (3.27), we have

Then, $\lim_{n\to\infty}\frac{\tau_n}{2\lambda L}D_{f_p}(w_n, y_n) = 0$ and hence $\lim_{n\to\infty}\tau_nD_{f_p}(w_n, y_n) = 0$. Using Lemma 3.5, we infer that

By Lemma (3.34) and 3.3, we obtain that $z^\dagger \in{\rm VI}(C, F)$, and hence $z^\dagger \in{\Omega} = \bigcap^N_{i = 0}{\rm Fix}(T_i)\cap{\rm VI}(C, F)$ with $T_0: = T$. This means that $\omega _w(x_n)\subset{\Omega}$. Lastly, we show that $\limsup_{n\to\infty}\langle J^p_Eu-J^p_E\hat u, s_n-\hat u\rangle\leq0$. We can choose a subsequence $\{x_{n_j}\}$ of $\{x_n\}$ such that

Without loss of generality, assume that $x_{n_j}\rightharpoonup\tilde z$. So it follows from (2.2) and $\tilde z\in{\Omega}$ that

This together with (3.31) ensures that

Using (3.21) and (3.24), we get

Since $\{D_{f_p}(\hat u, x_n)\}$ is bounded and $\sum^\infty_{n = 1}\theta_n < \infty$, one has that $\sum^\infty_{n = 1}\theta_nD_{f_p}(\hat u, x_n) < \infty$. Noticing $\{\alpha_n\}\subset(0, 1)$ and $\sum^\infty_{n = 1}\alpha_n = \infty$, by Lemma 2.7 and (3.36), we conclude that $\lim_{n\to\infty}D_{f_p}(\hat u, x_n) = 0$ and $\lim_{n\to\infty}\|\hat u-x_n\| = 0$.

Case 2. Suppose that $\exists\{\Phi_{n_k}\}\subset\{\Phi_n\}$ s.t. $\Phi_{n_k} < \Phi_{n_k+1}, \; \forall k\in{\mathbb N}$. Define an operator $\psi:{\mathbb N}\to{\mathbb N}$ by

Based on Lemma 2.8, we have

From (3.21) and (3.25), we have

Noticing $w_{\psi(n)} = J^q_{E^*}(\sigma_{\psi(n)}J^p_Ex_{\psi(n)}+(1-\sigma_{\psi(n)})J^p_ET_{\psi(n)}x_{\psi(n)})$, we deduce that

Furthermore, by (3.21) and (3.24), we obtain

Since $v_{\psi(n)} = J^q_{E^*}(\beta_{\psi(n)}J^p_Ew_{\psi(n)}+(1-\beta_{\psi(n)})J^p_ET^{\psi(n)}z_{\psi(n)})$, by (3.38) we get

Noticing $s_{\psi(n)} = J^q_{E^*}(\alpha_{\psi(n)}J^p_Eu+(1-\alpha_{\psi(n)})J^p_Ev_{\psi(n)})$, from (3.39) we get

Using (3.37) and exploiting the similar inferences to those in the proof of Case 1, we conclude that $\lim_{n\to\infty}\|x_{\psi(n)+1}-x_{\psi(n)}\| = 0$, $\lim_{n\to\infty}\|x_{\psi(n)}-T_rx_ {\psi(n)}\| = 0$ for $r = 1, ..., N$,

and

Using (3.21) and (3.24) we have

Combining with (3.37), we have

Since $\frac{\theta_{\psi(n)}}{\alpha_{\psi(n)}}\to0$, from (3.42) we deduce that

From (3.42), (3.43) and (3.44), we get that

Now from (3.37), we deduce $\lim_{n\to\infty}D_{f_p}(\hat u, x_n) = \lim_{n\to\infty}\Phi_n = 0$. Hence $\lim_{n\to\infty}\|x_n-\hat u\| = 0$. □

Putting $F = 0$ in Algorithm 3.1, we have the following corollary.

Corollary 3.9. Let $E$ be a $p$-uniformly convex and uniformly smooth Banach space with weakly sequentially continuous duality mapping $J^p_E$. Let $T:C\to C$ be a uniformly continuous and Bregman relatively asymptotically nonexpansive mapping with and $T_i:C\to C$($i = 1, ..., N$) be a uniformly continuous and Bregman relatively nonexpansive mapping. Assume that ${\Omega}: = \bigcap^N_{i = 0}{\rm Fix}(T_i)\neq\emptyset$ with $T_0: = T$. For an initial $x_1\in C$, let $\{x_n\}$ be the sequence constructed by

where $\{\sigma_n\}, \{\alpha_n\}\subset(0, 1)$ s.t. $\liminf_{n\to\infty}\sigma_n(1-\sigma_n) > 0$ and $\liminf_{n\to\infty}\alpha_n(1-\alpha_n) > 0$. Then, $\{x_n\}$ converges weakly to a point in ${\Omega}$ provided $T^nw_n-T^{n+1}w_n\to0$.

Setting $T = I$ in Algorithm 3.7, we obtain the following algorithm and corollary.

Algorithm 3.10. Let $x_1\in C$, $\mu > 0, \; l\in(0, 1)$ and $\lambda\in(0, \frac{1}{\mu})$. Choose $\{\sigma_n\}, \{\alpha_n\}, \{\beta_n\}\subset(0, 1)$ s.t. (i) $\liminf_ {n\to\infty}\sigma_n(1-\sigma_n) > 0$ and $\liminf_{n\to\infty}\beta_n(1-\beta_n) > 0$, and (ii) $\sum^\infty_{n = 1}\alpha_n = \infty$ and $\lim_{n\to\infty}\alpha_n = 0$.

Step 1. Set $w_n = J^q_{E^*}(\sigma_nJ^p_Ex_n+(1-\sigma_n)J^p_E(T_nx_n))$, and compute $y_n = \Pi_C(J^q_{E^*}(J^p_Ew_n-\lambda Fw_n))$ and $r_\lambda(w_n): = w_n-y_n$.

Step 2. Calculate $t_n = w_n-\tau_nr_\lambda(w_n)$ in which $\tau_n: = l^{j_n}$ with $j_n$ being the smallest nonnegative integer $j$ fulfilling

Step 3. Compute $v_n = J^q_{E^*}(\beta_nJ^p_Ew_n+(1-\beta_n)J^p_E(\Pi_{C_n}w_n))$ and $x_{n+1} = \Pi_C(J^q_{E^*}(\alpha_nJ^p_Eu+(1-\alpha_n)J^p_Ev_n)$, where $C_n: = \{x\in C:h_n(x)\leq0\}$ and

Set $n: = n+1$ and go to Step 1.

Corollary 3.11. Suppose that (C1)–(C4) are satisfied. Then, $\{x_n\}$ constructed in Algorithm 3.7 converges strongly to $\Pi_{\Omega}u$.

4.   Examples

In this section, we provide an illustrated example to demonstrate the applicability and implementability of our proposed method. We first provide an example of Lipschitz continuous and pseudomonotone monotone mapping $F$, Bregman relatively asymptotically nonexpansive mapping $T$ and Bregman relatively nonexpansive mapping $T_1$ with $\Omega = {\rm Fix}(T_1)\cap{\rm Fix}(T)\cap{\rm VI}(C, F)\neq\emptyset$.

Let $C = [-3, 3]$ and $H = {\bf R}$ with the inner product $\langle a, b\rangle = ab$ and induced norm $\|\cdot\| = |\cdot|$. The initial point $x_1$ is randomly chosen in $C$. Put $\mu = 1, \; l = \lambda = \frac{1}{3}$ and $\sigma_n = \frac{1}{2}$.

Let $F:H\to H$ and $T, T_1:C\to C$ be defined as $Fx: = \frac{1}{1+|\sin x|}-\frac{1}{1+|x|}$, $Tx: = \frac{3}{4}\sin x$ and $T_1x: = \sin x$ for all $x\in C$. Now, we first show that $F$ is pseudomonotone and Lipschitz continuous. Indeed, for all $x, y\in H$ we have

This implies that $F$ is Lipschitz continuous. Next, we show that $F$ is pseudomonotone. For each $x, y\in H$, it is easy to see that

which implies that

Furthermore, it is easy to check that $T$ is asymptotically nonexpansive with $\theta_n = (\frac{3}{4})^n\; \forall n\geq1$, and for each $\{p_n\}\subset C$ one has $\|T^{n+1}p_n-T^np_n\|\to0$ as $n\to\infty$. Indeed, we observe that

and

It is clear that ${\rm Fix}(T) = \{0\}$ and $T$ is also Bregman relatively asymptotically nonexpansive with $\theta_n = (\frac{3}{4})^n\; \forall n\geq1$. In addition, it is clear that ${\rm Fix}(T_1) = \{0\}$ and $T_1$ is Bregman relatively nonexpansive. Therefore, $\Omega = {\rm Fix}(T_1)\cap{\rm Fix}(T)\cap{\rm VI}(C, A) = \{0\}\neq\emptyset$.

Example 4.1. Putting $\alpha_n = \frac{n}{2(n+1)}, \forall n\geq1$, we can rewrite Algorithm 3.1 as follows:

where for each $n\geq1$, $C_n$ and $\tau_n$ are chosen as in Algorithm 3.1. Then, by Theorem 3.6, we deduce that $\{x_n\}$ converges to $0\in \Omega = {\rm Fix}(T_1)\cap{\rm Fix}(T)\cap {\rm VI}(C, A)$.

Example 4.2. Putting $\alpha_n = \frac{1}{2(n+1)}$ and $\beta_n = \frac{n}{2(n+1)}, \forall n\geq1$, we can rewrite Algorithm 3.7 as follows:

where for each $n\geq1$, $C_n$ and $\tau_n$ are chosen as in Algorithm 3.7. Then, by Theorem 3.8, we obtain that $\{x_n\}$ converges to $0\in \Omega = {\rm Fix}(T_1)\cap{\rm Fix}(T)\cap {\rm VI}(C, A)$.

5.   Conclusions

In this paper, we investigate iterative algorithms for solving the variational inequality and the common fixed-point problem in $p$-uniformly convex and uniformly smooth Banach spaces. With the help of accelerated projection methods and line search technique, we construct two algorithms for finding a common solution of the pseudomonotone variational inequality and the common fixed-point problem of finitely many Bregman relatively nonexpansive mappings and a Bregman relatively asymptotically nonexpansive mapping. We provide convergence analysis of the proposed algorithms by using standard conditions and new techniques.

Use of AI tools declaration

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

Acknowledgments

Yeong-Cheng Liou was supported in part by the MOST Project in Taiwan (110-2410-H-037-001) and the grant from NKUST and KMU joint R & D Project (110KK002).

Conflict of interest

The authors declare no conflict of interest.