Modified inertial subgradient extragradient algorithms for generalized equilibria systems with constraints of variational inequalities and fixed points

1.   Introduction

Throughout, assume $H$ is a real Hilbert space with the inner product $\langle\cdot, \cdot\rangle$ and the norm $\|\cdot\|$. Assume $\emptyset\neq C\subset H$ is a closed and convex set. Let $S:C\to H$ be an operator and $\Theta:C\times C\to{\bf R}$ be a bifunction. Use ${\rm Fix}(S)$ to mean the fixed-point set of $S$.

Recall that the equilibrium problem (EP) is to search an equilibrium point in ${\rm EP}(\Theta)$, where

Under the theory framework of equilibrium problems, there exists a unified way for exploring a broad number of problems originating in structural analysis, transportation, physics, optimization, finance, and economics ^{[1,8,10,17,20,24,25,26,35]}. In order to find an element in ${\rm EP}(\Theta)$, one needs to make the hypotheses below:

(H1) $\Theta(v, v) = 0, \; \; \forall v\in C$;

(H2) $\Theta(w, v)+\Theta(v, w)\leq0, \; \; \forall v, w\in C$;

(H3) $\lim_{\lambda\to0^+}\Theta((1-\lambda)v+\lambda u, w)\leq\Theta(v, w), \; \; \forall u, v, w\in C$;

(H4) For every $v\in C$, $\Theta(v, \cdot)$ is convex and lower semicontinuous (l.s.c.).

In order to solve the equilibrium problems, in 1994, Blum and Oettli ^[1] obtained the following valuable lemma:

Lemma 1.1. ^[1] Assume that $\Theta:C\times C\to{\bf R}$ fulfills the hypotheses (H1)–(H4). If $\forall x\in H$ and $\ell > 0$, let $T^\Theta_\ell:H\to C$ be an operator formulated below:

Then, (i) $T^\Theta_\ell$ is single-valued and satisfies $\|T^\Theta_\ell v-T^\Theta_\ell w\|^2\leq\langle T^\Theta_\ell v-T^\Theta_\ell w, v-w\rangle, \; \; \forall v, w\in H$; and (ii) ${\rm Fix}(T^\Theta_\ell) = {\rm EP}(\Theta)$, and ${\rm EP}(\Theta)$ is convex and closed.

In particular, in case of $\Theta(x, y) = \langle Ax, y-x\rangle, \; \forall x, y\in C$, and the EP reduces to the classical variational ineqiality problem (VIP) of seeking $x\in C$ such that

The solution set of the VIP is denoted by VI($C, A$).

An effective approach for settling EP and VIP is the Korpelevich's algorithm ^[15]. The Korpelevich extragradient technique has been adapted and applied extensively; see e.g., the modified extragradient method ^[11,29,34], subgradient extragradient method ^{[3,13,16,28,31,32]}, relaxed extragradient method ^[7], Tseng-type method ^[22,23,33], inertial extragradient method ^[14,27], and so on.

In 2010, Ceng and Yao ^[6] investigated the system generalized equilibrium problems (SGEP) of finding $(x, y)\in C\times C$ satisfying

where $B_1, B_2:H\to H$ are two nonlinear operators, $\Theta_1, \Theta_2:C\times C\to{\bf R}$ are two bifunctions, and $\alpha_1, \alpha_2 > 0$ are two constants.

If $\Theta_1 = \Theta_2 = 0$, then the SGEP comes down to the generallized variational inequalities considered in ^[5]: Find $(x, y)\in C\times C$ satisfying

with constants $\alpha_1, \alpha_2 > 0$.

To solve problem (1.1), the authors in ^[6] used a fixed point technique. In fact, the SGEP (1.1) can be transformed into the fixed-point problem.

Lemma 1.2. ^[6] Suppose that the bifunctions $\Theta_1, \Theta_2:C\times C\to{\bf R}$ satisfy the hypotheses (H1)–(H4) and $B_1, B_2:H\to H$ are $\rho$-ism and $\sigma$-ism, respectively. Then, $(u^*, v^*)\in C\times C$ is a solution of SGEP (1.1) if and only if $u^{\dagger}\in {\rm Fix}(G)$, where $G: = T^{\Theta_1}_{\alpha_1}(I-\alpha_1B_1)T^{\Theta_2}_{\alpha_2}(I-\alpha_2B_2)$ and $v^* = T^{\Theta_2}_{\alpha_2}(I-\alpha_2B_2)u^*$ in which $\alpha_1\in(0, 2\rho)$ and $\alpha_2\in(0, 2\sigma)$.

On the other hand, in 2018, Cai, Shehu, and Iyiola ^[2] proposed the modified viscosity implicit rule for solving EP and a fixed-point problem: for $x_1\in C$, let $\{x_k\}$ be the sequence constructed below:

Under suitable conditions, Cai, Shehu, and Iyiola ^[2] proved $x_k\to u^*\in{\rm Fix}(S)\cap{\rm Fix}(G)$, which solves the hierarchical variational inequality (HVI):

Moreover, Ceng and Shang ^[4] suggested an algorithm for solving the common fixed-point problem (CFPP) of finite nonexpansive mappings $\{S_r\}^N_{r = 1}$, an asymptotically nonexpansive mapping $S$ and VIP.

Algorithm 1.1. ^[4] Let $x_1, x_0\in H$ be arbitrary. Let $\gamma > 0, \; \ell\in(0, 1), \; \nu\in(0, 1)$, and $x_k$ be known. Calculate $x_{k+1}$ via the following iterative steps:

Step 1. Set $p_k = S_kx_k+\varepsilon_k(S_kx_k-S_kx_{k-1})$ and calculate $y_k = P_C(p_k-\zeta_kAp_k)$, where $\zeta_k$ is the largest $\zeta\in\{\gamma, \gamma\ell, \gamma\ell^2, ...\}$ fulfilling $\zeta\|Ap_k-Ay_k\|\leq\nu\|p_k-y_k\|$.

Step 2. Compute $z_k = P_{C_k}(p_k-\zeta_kAy_k)$ where $C_k: = \{y\in H:\langle p_k-\zeta_kAp_k-y_k, y_k-y\rangle\geq0\}$.

Step 3. Compute $x_{k+1} = \rho_kf(x_k)+\sigma_kx_k+((1-\sigma_k)I-\rho_k\alpha F)S^kz_k$.

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

Motivated and inspired by the work in the literature, the main purpose of this article was to design two modified inertial composite subgradient extragradient implicit rules for solving the SGEP with the VIP and CFPP constraints. The suggested algorithms consisted of the subgradient extragradient rule, inertial iteration approach, and hybrid deepest-descent technique. We proved that the proposed algorithms converge to a solution of the SGEP with the VIP and CFPP constraints, which also solved some HVI.

2.   Preliminaries

Let $C$ be a nonempty, convex, and closed subset of a real Hilbert space $H$. For all $v, w\in C$, an operator $T:C\to H$ is called

● asymptotically nonexpansive if $\exists\{\varpi_m\}^\infty_{m = 1}\subset [0, + \infty)$ satisfying $\varpi_m\to0\; (m\to\infty)$ and

In particular, in the case of $\varpi_m = 0, \; \forall m\geq1$, and $T$ is known as being nonexpansive.

● $\alpha$-Lipschitzian if $\exists \alpha > 0$ such that $\|Tv-Tw\|\leq \alpha\|v-w\|$;

● monotone if $\langle Tv-Tw, v-w\rangle\geq0$;

● strongly monotone if there is $\rho > 0$ such that $\langle Tv-Tw, v-w\rangle\geq\rho\|v-w\|^2$;

● pseudomonotone if $\langle Tv, w-v\rangle\geq0\Rightarrow \langle Tw, w-v\rangle\geq0$;

● $\sigma$ inverse strongly monotone ($\sigma$-ism) if there is $\sigma > 0$ such that $\langle Tv-Tw, v-w\rangle\geq\sigma\|Tv-Tw\|^2$;

● sequentially weakly continuous if $\forall\{v_l\}\subset C$, $v_l\rightharpoonup v\Rightarrow Tv_l\rightharpoonup Tv$.

Recall that the metric (or nearest point) projection from $H$ onto $C$ is the mapping $P_C: H\to C$ which assigns to each point $x\in C$ the unique point $P_C(x)\in C$ satisfying the property

The following results are well-known (^[12]):

(a) $\|P_C(y)-P_C(z)\|^2\le \langle y-z, P_C(y)-P_C(z)\rangle, \; \; \forall y, z\in H$;

(b) $z = P_C(y)\Leftrightarrow\langle x-z, y-z\rangle\leq0, \; \; \forall y\in H, x\in C$;

(c) $\|y-z\|^2\geq\|z-P_C(y)\|^2+\|y-P_C(y)\|^2, \; \; \forall y\in H, z\in C$;

(d) $\|y-z\|^2 = \|y\|^2-2\langle y-z, z\rangle-\|z\|^2, \; \; \forall y, z\in H$;

(e) $\|ty+(1-t)x\|^2 = t\|y\|^2+(1-t)\|x\|^2-t(1-t)\|y-x\|^2,$ $\forall x, y\in H, t\in[0, 1]$.

Lemma 2.1. ^[6] Suppose $B:H\to H$ is an $\eta$-ism. Then,

When $0\leq\alpha\leq2\eta$, we have that $I-\alpha B$ is nonexpansive.

Lemma 2.2. ^[6] Let $B_1, B_2:H\to H$ be $\rho$-ism and $\sigma$-ism, respectively. Suppose that the bifunctions $\Theta_1, \Theta_2:C\times C\to{\bf R}$ satisfy the hypotheses (H1)–(H4). Then, $G: = T^{\Theta_1}_{\alpha_1}(I- \alpha_1B_1)T^{\Theta_2}_{\alpha_2}(I-\alpha_2B_2)$ is nonexpansive when $0 < \alpha_1\leq2\rho$ and $0 < \alpha_2\leq2\sigma$.

In particular, if $\Theta_1 = \Theta_2 = 0$, using Lemma 1.1, we deduce that $T^{\Theta_1}_{\alpha_1} = T^{\Theta _2}_{\alpha_2} = P_C$.

Corollary 2.1. ^[5] Let $B_1:H\to H$ be $\rho$-ism and $B_2:H\to H$ $\sigma$-ism. Define an operator $G:H\to C$ by $G: = P_C(I-\alpha_1B_1) P_C(I-\alpha_2B_2)$. Then $G$ is nonexpansive when $0 < \alpha_1\leq2\rho$ and $0 < \alpha_2\leq2\sigma$.

Lemma 2.3. ^[9] If the operator $A:C\to H$ is continuous pseudomonotone, then $v\in VI(C, A)$ if and only if $\langle Aw, w-v\rangle\geq0, \; \forall w\in C$.

Lemma 2.4. ^[30] Suppose $\{a_l\}\subset [0, \infty)$ s.t. $a_{l+1}\leq(1-\omega_l)a_l+ \omega_l\nu_l, \; \forall l\geq1$, where $\{\omega_l\}$ and $\{\nu_l\}$ satisfy: (i) $\{\omega_l\}\subset[0, 1]$; (ii) $\sum^\infty_{l = 1}\omega_l = \infty$, and (iii) $\limsup_{l\to\infty}\nu_l\leq0$ or $\sum^\infty_{l = 1}|\omega_l\nu_l| < \infty$. Then, we have $\lim_{l\to\infty}a_l = 0$.

Lemma 2.5. ^[18] Let $X$ be a Banach space with a weakly continuous duality mapping. Let $C$ be a nonempty closed convex subset of $X$ and $T:C\to C$ an asymptotically nonexpansive mapping such that ${\rm Fix}(T)\neq\emptyset$. Then $I-T$ is demiclosed at zero.

Lemma 2.6. ^[19] Suppose that the real number sequence $\{\Gamma_m\}$ is not decreasing at infinity: $\exists\{\Gamma_{m_k}\}\subset\{\Gamma_m\}$ s.t. $\Gamma_{m_k} < \Gamma_{m_k+1}, \; \forall k\geq1$. Let $\{\phi(m)\}_{m\geq m_0}$ be an integer sequence defined by

Then,

(i) $\phi(m_0)\leq\phi(m_0+1)\leq\cdots$ and $\phi(m)\to\infty$ as $m\to \infty$;

(ii) For all $m\geq m_0$, $\Gamma_{\phi(m)}\leq\Gamma_{\phi(m)+1}$ and $\Gamma_m\leq\Gamma_{\phi(m)+1}$.

Lemma 2.7. ^[30] Let $\lambda\in(0, 1]$, $S:C\to C$ be a nonexpansive operator, and $F:C\to H$ be a $\kappa$-Lipschitzian and $\eta$-strongly monotone operator. Set $S^\lambda v: = (I-\lambda\alpha F)Sv, \; \forall v\in C$. If $0 < \alpha < \frac{2\eta} {\kappa^2}$, then $\|S^\lambda v-S^\lambda w\|\leq(1-\lambda\tau)\|v-w\|, \; \forall v, w\in C$, where $\tau = 1-\sqrt{1-\alpha(2\eta-\alpha\kappa^2)}\in(0, 1]$.

3.   Algorithms and convergence theorems

Let the operator $S_r$ be nonexpansive on $H$ for all $r = 1, ..., N$ and $S:H\to H$ be a $\varpi_n$-asymptotically nonexpansive operator. Let $A:H\to H$ be an $L$-Lipschitz pseudomonotone operator satisfying $\|Ax\|\leq\liminf_{n\to\infty}\|Ax_n\|$ when $x_n\rightharpoonup x$. Let $\Theta_1, \Theta_2:C\times C\to{\bf R}$ be two bifunctions fulfilling the hypotheses (H1)–(H4). Let $B_1:H\to H$ be $\rho$-ism and $B_2:H\to H$ be $\sigma$-ism. Let $f:H\to H$ be $\delta$-contractive and $F:H\to H$ be $\kappa$-Lipschitz $\eta$-strongly monotone with $\delta < \tau: = 1-\sqrt{1-\alpha(2\eta-\alpha\kappa^2)}$ for $\alpha\in(0, \frac{2\eta}{\kappa^2})$. Suppose that the sequences $\{\varepsilon_n\}\subset[0, 1], \; \{\xi_n\}\subset(0, 1]$, and $\{\rho_n\}, \{\sigma_n\}\subset(0, 1)$ satisfy

(ⅰ) $\lim_{n\to\infty}\rho_n = 0$ and $\sum^\infty_{n = 1}\rho_n = \infty$;

(ⅱ) $\lim_{n\to\infty}\frac{\varpi_n}{\rho_n} = 0$ and $\sup_{n\geq1}\frac{\varepsilon_n}{\rho_n} < \infty$;

(ⅲ) $0 < \liminf_{n\to\infty}\sigma_n\leq\limsup_{n\to\infty}\sigma_n < 1$;

(ⅳ) $\limsup_{n\to\infty}\xi_n < 1$.

Let $\gamma > 0, \; \nu\in(0, 1), \; \ell\in(0, 1)$, $\alpha_1\in(0, 2\rho)$, and $\alpha_2\in (0, 2\sigma)$ be five constants. Set $S_0: = S$ and $G: = T^{\Theta_1}_{\alpha_1}(I-\alpha_1B_1)T^{\Theta_2}_{\alpha_2}(I-\alpha_2B_2)$. Suppose that $\Delta: = \bigcap^N_{r = 0}{\rm Fix}(S_r)\cap{\rm Fix}(G)\cap{\rm VI}(C, A)\neq\emptyset$.

Algorithm 3.1. Let $x_1, x_0\in H$ be arbitrary. Let $x_n$ be known and compute $x_{n+1}$ below:

Step 1. Set $q_n = S^nx_n+\varepsilon_n(S^nx_n-S^nx_{n-1})$ and calculate

Step 2. Compute $y_n = P_C(p_n-\zeta_nAp_n)$, where $\zeta_n$ is the largest $\zeta\in\{\gamma, \gamma \ell, \gamma\ell^2, ...\}$ s.t.

Step 3. Compute $t_n = \sigma_nx_n+(1-\sigma_n)z_n$ with $z_n = P_{C_n}(p_n-\zeta_nAy_n)$ and

Step 4. Compute

where $S_n$ is constructed as in Algorithm 1.1. Let $n: = n+1$ and return to Step 1.

Lemma 3.1. ^[21] $\min\{\gamma, \nu\ell /L\}\leq\zeta_n\leq\gamma$.

Lemma 3.2. Let $p\in\Delta$ and $q = T^{\Theta_2}_{\alpha_2}(p-\alpha_2B_2p)$. Then,

where $v_n = T^{\Theta_2}_{\alpha_2}(p_n-\alpha_2B_2p_n)$.

Proof. According to Lemma 2.2, there exists the unique point $p_n\in H$ satisfying $p_n = \xi_nq_n+(1-\xi_n)Gp_n.$ Since $p\in C_n$, we have

which implies that

Noting that $z_n = P_{C_n}(p_n-\zeta_nAy_n)$, we have $\langle p_n-\zeta_nAp_n-y_n, z_n-y_n\rangle\leq0$. Owing to the pseudomonotonicity of $A$, by (3.1), we get

Observe that $u_n = T^{\Theta_1}_ {\alpha_1}(v_n-\alpha_1B_1v_n)$, $v_n = T^{\Theta_2}_{\alpha_2}(p_n-\alpha_2B_2p_n)$, and $q = T^{\Theta_2}_{\alpha_2}(p-\alpha_2B_2p)$. Then $u_n = Gp_n$. Applying Lemma 2.1 to get

and

Then,

Besides, thanks to $p_n = \xi_nq_n+(1-\xi_n)u_n$, we get $\|p_n-p\|^2\leq\xi_n\langle q_n-p, p_n-p\rangle +(1-\xi_n)\|p_n-p\|^2$, which results in $\|p_n-p\|^2\leq\langle q_n-p, p_n-p\rangle = \frac{1}{2}[\|q_n-p\|^2 +\|p_n-p\|^2-\|q_n-p_n\|^2]$. So,

Then,

which, together with (3.3), yields

This ensures that the conclusion holds. □

Lemma 3.3. Assume that

$(i)$ the sequences $\{p_n\}, \{q_n\}, \{y_n\}$, and $\{z_n\}$ are bounded;

$(ii)$ $\lim_{n\to \infty}(x_{n+1}-x_n) = \lim_{n\to \infty}(q_n-z_n) = \lim_{n\to \infty}(x_n-y_n)$ $= \lim_{n\to \infty}(S^{n+1}x_n-S^nx_n) = 0$.

Then $\omega_w(x_n)\subset \Delta$, where $\omega_w(x_n) = \{z\in H, {\rm and\; there\; is\; some}\; \{x_{n_i}\}\subset\{x_n\}\; {\rm such \; that}\; x_{n_i}\rightharpoonup z\}$.

Proof. Take an arbitrary fixed $z\in\omega_w(\{x_n\})$. Then, there is some $\{n_i\}\subset\{n\}$ such that $x_{n_i} \rightharpoonup z$ and $y_{n_i} \rightharpoonup z\in H$. Next, we show $z\in\Delta$. Using Lemma 3.2, we deduce

Because $q_n-z_n\to0, \; \nu\in(0, 1), \; \alpha_1\in(0, 2\rho), \; \alpha_2\in(0, 2\sigma)$, and $0 < \liminf_{n\to\infty}(1-\xi_n)$, we deduce that

and

Hence,

and

Note that

Therefore,

Note that

Observe that

which arrives at

Similarly, we get

Combining the last two inequalities, we deduce that

Hence,

This immediately implies that

Since $q_n-z_n\to0,$ and $0 < \liminf_{n\to\infty}(1-\xi_n)$, from (3.5) and the boundedness of $\{u_n\}, \{v_n\}$, $\{q_n\}$, and $\{z_n\}$ we get that

which hence yields

This immediately implies that

Next, we prove $\lim_{n\to\infty}\|x_n-S_nx_n\| = 0$. In fact, we have

and

Hence,

Now, we show $x_n-S_rx_n\to0, \forall r\in \{1, ..., N\}$. For $1\le l\le N$, it holds that

This, together with assumptions, implies that $x_n-S_{n+l}x_n\to0, 1\le l\le N$. So,

Now, we show $z\in{\rm VI}(C, A)$. If $Az = 0$, then $z\in{\rm VI}(C, A)$. Next, we suppose that $Az\neq0$. By the condition, we conclude that $0 < \|Az\|\leq\liminf_{i\to\infty}\|Ay_{n_i}\|$ because $y_{n_i}\rightharpoonup z$. Observe that $y_n = P_C(p_n-\zeta_nAp_n)$. It follows that $\langle p_n-\zeta_n Ap_n-y_n, y-y_n\rangle\leq0, \; \forall y\in C$. Therefore,

Since $\{Ap_n\}$ and $\{y_n\}$ are all bounded, from (3.9) and Lemma 3.1, we obtain $\liminf_{i\to\infty}\langle Ap_{n_i},$ $y-p_{n_i}\rangle\geq0, \; \forall y\in C$. Meanwhile, $\langle Ay_n, y-y_n\rangle = \langle Ay_n-Ap_n, y-p_n\rangle+\langle Ap_n, y-p_n\rangle+\langle Ay_n, p_n-y_n\rangle$. Using $p_n-y_n\to0$ and the uniform continuity of $A$, we get $Ap_n-Ay_n\to0$, which hence attains $\liminf_{i\to\infty}\langle Ay_{n_i}, y-y_{n_i}\rangle\geq0, \; \forall y\in C$.

To attain $z\in{\rm VI}(C, A)$, let $\{\lambda_i\}\subset(0, 1)$ be a sequence such that $\lambda_i\downarrow0(i\to\infty)$. For every $i\geq1$, let $k_i$ be the smallest positive integer satisfying

Put $\upsilon_{k_i} = \frac{Ay_{k_i}} {\|Ay_{k_i}\|^2}$, and hence get $\langle Ay_{k_i}, \upsilon_{k_i}\rangle = 1, \; \forall i\geq1$. So, from (3.10), one gets $\langle Ay_{k_i}, y+\lambda_i\upsilon_{k_i}-y_{k_i}\rangle\geq0$, $\forall i\geq1$. Since $A$ is pseudomonotone, we have $\langle A(y+\lambda_i\upsilon_{k_i}), y+\lambda_i\upsilon_{k_i}-y_{k_i}\rangle\geq0, \; \forall i\geq1$. So,

Let us show that $\lim_{i\to\infty}\lambda_i\upsilon_{k_i} = 0$. In fact, note that $\{y_{k_i}\}\subset\{y_{n_i}\}$ and $\lambda_i\downarrow0$ as $i\to\infty$. So it follows that $0\leq\limsup_{i\to\infty}\|\lambda_i \upsilon_{k_i}\| = \limsup_{i\to\infty}\frac{\lambda_i}{\|Ay_{k_i}\|} \leq\frac{\limsup_{i\to\infty}\lambda_i}{\liminf_{i\to\infty}\|Ay_{n_i}\|} = 0$. Therefore, one gets $\lambda_i \upsilon_{k_i}\to0$ as $i\to\infty$. Thus, letting $i\to\infty$ in (3.11), we deduce that $\langle Ay, y-z\rangle = \liminf_{i\to\infty}\langle Ay, y-y_{k_i}\rangle\geq0, \; \forall y\in C$. We apply Lemma 2.3 to conclude $z\in{\rm VI}(C, A)$.

Finally, we prove $z\in\Delta$. Note that $x_{n_i}\rightharpoonup z$ and $x_{n_i}-S_rx_{n_i}\to0, \; \forall r\in\{1, ..., N\}$ (due to (3.8)). Since $I-S_r(1\le r\le N)$ is demiclosed by Lemma 2.5, we attain $z\in\bigcap^N_{r = 1}{\rm Fix}(S_r)$. By (3.6) and (3.7), we have $x_{n_i}-Sx_{n_i} \to0$ and $x_{n_i}-Gx_{n_i}\to0$, respectively. Similarly, $I-S$ and $I-G$ are all demiclosed at zero, and we have $z\in{\rm Fix}(S)\cap{\rm Fix}(G)$. Therefore, $z\in\bigcap ^N_{r = 0}{\rm Fix}(S_r)\cap{\rm Fix}(G)\cap{\rm VI}(C, A) = \Delta$. □

Theorem 3.1. We have the following equivalent relation:

where $u^{\dagger}\in\Delta$ solves the HVI: $\langle(\alpha F-f)u^{\dagger}, p-u^{\dagger}\rangle\geq0, \; \; \forall p\in\Delta$.

Proof. According to the condition, we assume that $\varpi_n\leq\frac{\rho_n(\tau-\delta)}{2}$ and $\{\sigma_n\}\subset[a, b]\subset (0, 1)$ for all $n\geq1$. $\forall x, y\in H$, by Lemma 2.7, we obtain

which implies that $P_\Delta(I-\alpha F+f)$ is contractive. Set $u^{\dagger} = P_\Delta(I-\alpha F+f)(u^{\dagger})$. Therefore, there is the unique solution $u^{\dagger}\in\Delta = \bigcap^N_{r = 0}{\rm Fix}(S_r)\cap{\rm Fix}(G)\cap{\rm VI}(C, A)$ of the HVI:

If $x_n\to u^{\dagger}\in\Delta$, then we know that $u^{\dagger} = Su^{\dagger}$ and

Note that

Now, we prove the sufficiency.

Step 1. Let $p\in\Delta$. Then $Gp = p$, $p = P_C(p-\zeta_nAp)$, $S_np = p, \; \forall n\geq0$, and the inequalities (3.3) and (3.4) hold, i.e.,

and

Combining (3.13) and (3.14) guarantees that

Observe that

Since $\sup_{n\geq1}\frac{\varepsilon_n}{\rho_n}\|x_n-x_{n-1}\| < \infty$, there is a constant $M_1 > 0$ satisfying

Combining (3.15)–(3.17), we get

Also, it is readily known that

Thus, using (3.19) and $\varpi_n\leq\frac{\rho_n(\tau-\delta)}{2}\; \forall n\geq1$, from Lemma 2.7, we receive

Hence,

We deduce that the sequences $\{x_n\}$, $\{p_n\}, \{q_n\}, \{y_n\}, \{z_n\}, \{t_n\}, \{f(x_n)\}$, $\{S_nt_n\}$, and $\{S^nx_n\}$ are bounded.

Step 2. Observe that $t_n-p = \sigma_n(x_n-p)+(1-\sigma_n)(z_n-p)$ and

Utilizing Lemma 2.7, we attain

where $M_2 > 0$ is a constant such that $\sup_{n\geq1}2\|(f-\alpha F)p\|\|x_n-p\|\leq M_2$. By (3.20) and Lemma 3.2, we have

Taking (3.18) into account, we obtain

where $M_3 > 0$ is a constant such that $\sup_{n\geq1}\{(2\|x_n-p\|+\rho_nM_1)M_1+\frac{\varpi_n(2+\varpi_n)}{\rho_n}(\|x_n-p\|+\rho_nM_1)^2\}\leq M_3$. Based on (3.21) and (3.22), we get

where $M_4: = M_3+M_2$. This immediately implies that

Step 3. Note that

Combining (3.18), (3.20), and (3.24), we receive

where $M > 0$ is a constant such that $\sup_{n\geq1}\{\|x_n-p\|, \varepsilon_n\|x_n-x_{n-1}\|, \|x_n-p\|^2\}\leq M$.

Step 4. Taking $p = u^{\dagger}$, by (3.25), we have

Set $\Gamma_n = \|x_n-u^{\dagger}\|^2$.

Case 1. There is an integer $n_0\geq1$ such that $\{\Gamma_n\}$ is nonincreasing. In this case, $\lim_{n\to\infty}\Gamma_n = \hbar < +\infty$. From (3.23), we have

Noticing $0 < \liminf_{n\to\infty}(1-\xi_n)$, $\rho_n\to0$ and $\Gamma_n-\Gamma_{n+1}\to0$, for $\nu\in(0, 1)$ one has $\lim_{n\to\infty}\|q_n-p_n\| = \lim_{n\to\infty}\|y_n-z_n\| = 0$, and $\lim_{n\to\infty}\|y_n-p_n\| = \lim_{n\to\infty}\|x_n-z_n\| = 0$. Thus, we get

and

Since $\{x_n\}$ is bounded, there is a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ satisfying $x_{n_k}\rightharpoonup\widetilde x$ and

In the light of (3.29), one gets

Since $x_{n+1}-x_n\to0, \; y_n-x_n\to0, \; z_n-q_n\to0$ and $S^{n+1}x_n-S^nx_n\to0$, applying Lemma 3.3, we conclude that $\widetilde x\in\omega_w(\{x_n\})\subset\Delta$. Combining (3.12) and (3.30), we get

Since $x_n-x_{n+1}\to0$, we have

Note that

According to (3.26) and Lemma 2.4, we deduce that $\lim_{n\to\infty}\|x_n-u^{\dagger}\|^2 = 0$.

Case 2. Suppose that $\exists\{\Gamma_{n_k}\}\subset\{\Gamma_n\}$ s.t. $\Gamma_{n_k} < \Gamma_{n_k+1}, \; \forall k\in{\bf N}$. Let $\phi:{\bf N}\to{\bf N}$ be a mapping defined by

Based on Lemma 2.6, we have

Putting $p = u^{\dagger}$, from (3.23), we have

which immediately yields $\lim_{n\to\infty}\|q_{\phi(n)}-p_{\phi(n)}\| = \lim_{n\to\infty}\|y_{\phi(n)}-z_{\phi(n)}\| = 0$ and $\lim_{n\to\infty}\|y_{\phi(n)}$ $-p_{\phi(n)}\| = \lim_{n\to\infty}\|x_{\phi(n)}-z_{\phi(n)}\| = 0$. Therefore,

and

At the same time, by (3.26), we known that

which hence arrives at

Thus, $\lim_{n\to\infty}\|x_{\phi(n)}-u^{\dagger}\|^2 = 0$. Also, note that

Since $\Gamma_n\leq\Gamma_{\phi(n)+1}$, we have

So, $x_n\to u^{\dagger}$. □

According to Theorem 3.1, we have the following corollary.

Corollary 3.1. Suppose that $S:C\to C$ is a nonexpansive mapping. For two fixed points $x_1, x_0\in H$, let the sequence $\{x_n\}$ be defined by

where $C_n$ and $\zeta_n$ have the same form as in Algorithm 3.1. Then, $x_n\to u^{\dagger}\in\Delta\; \Leftrightarrow\; x_{n+1}-x_n\to0,$ where $u^{\dagger}\in\Delta$ is the unique solution of the HVI: $\langle(\alpha F-f)u^{\dagger}, p-u^{\dagger}\rangle\geq0, \; \; \forall p\in\Delta$.

Next, we put forth another modification of the inertial composite subgradient extragradient implicit rule with line-search process.

Algorithm 3.2. Let $x_1, x_0\in H$ be two fixed points. Let $x_n$ be given. Compute $x_{n+1}$ via the following iterative steps:

Step 1. Set $q_n = S^nx_n+\varepsilon_n(S^nx_n-S^nx_{n-1})$ and calculate

Step 2. Compute $y_n = P_C(p_n-\zeta_nAp_n)$, with $\zeta_n$ being chosen to be the largest $\zeta\in\{\gamma, \gamma \ell, \gamma\ell^2, ...\}$ s.t.

Step 3. Compute $t_n = \sigma_nz_n+(1-\sigma_n)S_nt_n$ with $z_n = P_{C_n}(p_n-\zeta_nAy_n)$ and

Step 4. Compute

where $S_n$ is constructed as in Algorithm 1.1. Set $n: = n+1$ and go to Step 1

Theorem 3.2. Let the sequence $\{x_n\}$ be generated by Algorithm 3.2. Then

where $u^{\dagger}\in\Delta$ is the unique solution of the HVI: $\langle(\alpha F-f)u^{\dagger}, p-u^{\dagger}\rangle\geq0, \; \; \forall p\in\Delta$.

Proof. The necessity is obvious. Next, we prove the sufficiency.

Note that

This, together with (3.18), ensures that

By (3.37) and Lemma 2.7, we have

It follows that $\|x_n-p\|\leq\max\{\|x_1-p\|, \frac{2(2M_1+\|(f-\alpha F)p\|)}{\tau-\delta}\}\; \forall n\geq1$. Therefore, $\{x_n\}$, $\{p_n\}$, $\{q_n\}$, $\{y_n\}$, $\{z_n\}$, $\{t_n\}$, $\{f(x_n)\}$, $\{S_nt_n\}$, and $\{S^nx_n\}$ are bounded.

According to Lemma 2.7, we get

where $M_2 > 0$ is a constant such that $\sup_{n\geq1}2\|(f-\alpha F)p\|\|x_n-p\|\leq M_2$. Using Lemma 3.2, from (3.37) and (3.38), we have

Also, using the same inferences as those of (3.22) of Theorem 3.1, we have

where $\sup_{n\geq1}\{M_1(2\|x_n-p\|+\rho_nM_1)+\frac{\varpi_n(2+\varpi_n)}{\rho_n}(\|x_n-p\| +\rho_nM_1)^2\}\leq M_3$ for some constant $M_3$. By (3.39) and (3.40), we attain

where $M_4: = M_3+M_2$. Hence, we attain the assertion.

By the same argument as those of (3.24), we have

By (3.37), (3.38), and (3.41), we obtain

where $\sup_{n\geq1}\{\|x_n-p\|, \varepsilon_n\|x_n-x_{n-1}\|, \|x_n-p\|^2\}\leq M$ for some constant $M$.

Setting $p = u^{\dagger}$, by (3.42), we have

Set $\Gamma_n = \|x_n-u^{\dagger}\|^2$.

Case 1. Assume $\{\Gamma_n\}$ is nonincreasing when $n\ge n_0$. Then, $\lim_{n\to\infty}\Gamma_n = \hbar < +\infty$. Choosing $p = u^{\dagger}$, from (3.38), we have

Since $\Gamma_n-\Gamma_{n+1}\to0$ for $\nu\in(0, 1)$, $\lim_{n\to\infty}\|q_n-p_n\| = \lim_{n\to\infty}\|y_n-z_n\| = 0$, and $\lim_{n\to\infty}\|y_n-p_n\| = \lim_{n\to\infty}\|z_n-S_nt_n\| = 0$. Observe that

By the similar arguments as those in Theorem 3.1, we deduce $\lim_{n\to\infty}\|x_n-u^{\dagger}\|^2 = 0$.

Case 2. Assume $\exists\{\Gamma_{n_k}\}\subset\{\Gamma_n\}$ s.t. $\Gamma_{n_k} < \Gamma _{n_k+1}, \; \forall k\in{\bf N}$. Let $\phi:{\bf N}\to{\bf N}$ be a mapping defined by

By Lemma 2.6, we have

Set $p = u^{\dagger}$. Then,

which immediately yields $\lim_{n\to\infty}\|q_{\phi(n)}-p_{\phi(n)}\| = \lim_{n\to\infty}\|y_{\phi(n)}-z_{\phi(n)}\| = 0$ and $\lim_{n\to\infty}\|y_{\phi(n)}$ $-p_{\phi(n)}\| = \lim_{n\to\infty}\|z_{\phi(n)}-S_{\phi(n)}t_{\phi(n)}\| = 0$. Therefore, $\lim_{n\to\infty}\|z_{\phi(n)}-x_{\phi(n)}\| = 0$. Finally, using the similar arguments to those in Theorem 3.1, we get the conclusion. □

Remark 3.1. Compared with the corresponding results in Cai, Shehu, and Iyiola ^[2], Ceng and Shang ^[4], and Thong and Hieu ^[28], our results improve and extend them in the following aspects:

(ⅰ) The problem of finding an element of ${\rm Fix}(S)\cap{\rm Fix}(G)$ (with $G = P_C(I-\mu_1B_1)P_C(I-\mu_2B_2)$) in ^[2] is extended to develop our problem of finding an element of $\bigcap^N_{r = 0}{\rm Fix}(S_r)\cap{\rm Fix}(G) \cap{\rm VI}(C, A)$ where $G = T^{\Theta_1}_{\mu_1}(I-\mu_1B_1)T^{\Theta_2}_{\mu_2}(I-\mu_2B_2)$ and $S$ is an asymptotically nonexpansive mapping. The modified viscosity implicit rule for finding an element of ${\rm Fix}(S)\cap{\rm Fix}(G)$ in ^[2] is extended to develop our modified inertial composite subgradient extragradient implicit rules with line-search process for finding an element of $\bigcap^N_{r = 0}{\rm Fix}(S_r)\cap{\rm Fix} (G)\cap{\rm VI}(C, A)$, which is on the basis of the subgradient extragradient rule with line-search process, inertial iteration approach, viscosity approximation method, and hybrid deepest-descent technique.

(ⅱ) The problem of finding an element of ${\rm Fix}(S)\cap{\rm VI}(C, A)$ with a quasi-nonexpansive mapping $S$ in ^[4] is extended to develop our problem of finding an element of $\bigcap^N_{r = 0}{\rm Fix}(S_r)\cap{\rm Fix}(G)\cap{\rm VI}(C, A)$ with an asymptotically nonexpansive mapping $S$. The inertial subgradient extragradient method with line-search process for finding an element of ${\rm Fix}(S)\cap{\rm VI}(C, A)$ in ^[28] is extended to develop our modified inertial composite subgradient extragradient implicit rules with line-search process for finding an element of $\bigcap^N_{r = 0}{\rm Fix}(S_r)\cap{\rm Fix}(G)\cap{\rm VI}(C, A)$, which is on the basis of the subgradient extragradient rule with line-search process, inertial iteration approach, viscosity approximation method, and hybrid deepest-descent technique.

(ⅲ) The problem of finding an element of ${\it\Omega} = \bigcap^N_{r = 0}{\rm Fix}(S_r)\cap{\rm VI}(C, A)$ is extended to develop our problem of finding an element of ${\it\Omega} = \bigcap^N_{r = 0}{\rm Fix}(S_r)\cap{\rm Fix}(G)\cap{\rm VI} (C, A)$ with $G = T^{\Theta_1} _{\mu_1}(I-\mu_1B_1)T^{\Theta_2}_{\mu_2}(I-\mu_2B_2)$. The hybrid inertial subgradient extragradient method with line-search process in ^[4] is extended to develop our modified inertial composite subgradient extragradient implicit rules with line-search process.

4.   Examples

In this section, we give an example to show the feasibility of our algorithms. Put $\Theta_1 = \Theta_2 = 0$, $\alpha = 2$, $\alpha_1 = \alpha_2 = \frac{1}{3}, \; \gamma = 1, \; \nu = \ell = \frac{1}{2}, \; \sigma_n = \xi_n = \frac{2}{3}$, and $\varepsilon_n = \rho_n = \frac{1}{3(n+1)}$, for all $n\ge 0$. Now, we construct an example of $\Delta = \bigcap^N _{r = 0}{\rm Fix}(S_r)\cap{\rm Fix}(G)\cap{\rm VI}(C, A)\neq\emptyset$ with $S_0: = S$ and $G = T^{\Theta_1}_{\alpha_1} (I-\alpha_1B_1)T^{\Theta_2}_{\alpha_2}(I-\alpha_2B_2) = P_C(I-\alpha_1B_1)P_C(I-\alpha_2B_2)$, where $A:H\to H$ is pseudomonotone and a Lipschitz continuous mapping, $B_1, B_2:H\to H$ are two inverse-strongly monotone mappings, $S:H\to H$ is asymptotical nonexpansive, and each $S_r:H\to H$ is nonexpansive for $r = 1, ..., N$.

Let $H = {\bf R}$ and use $\langle a, b\rangle = ab$ and $\|\cdot\| = |\cdot|$ to denote its inner product and induced norm, respectively. Set $C = [-2, 4]$ and the starting point $x_1$ is arbitrarily chosen in $C$. Let $f(x) = F(x) = \frac{1}{2}x$, $\forall x\in H$ with

Let $B_1x = B_2x: = Bx = x-\frac{1}{2}\sin x, \; \forall x\in C$. Let the operators $A, S, S_r:H\to H$ be defined by

We have the following assertions:

(ⅰ) $A$ is $2$-Lipschitz continuous, in fact, for each $x, y\in H$, we have

(ⅱ) $A$ is pseudomonotone, in fact, for each $x, y\in H$, if

then

(ⅲ) $B$ is $\frac{2}{9}$-inverse-strongly monotone. In fact, since $B$ is $\frac{1}{2}$-strongly monotone and $\frac{3}{2}$-Lipschitz continuous, we know that $B$ is $\frac{2}{9}$-inverse-strongly monotone with $\rho = \sigma = \frac{2}{9}$.

Moreover, it is easy to check that $S$ is asymptotically nonexpansive with $\varpi_n = (\frac{3}{4})^n, \; \forall n\geq1$, such that $\|S^{n+1}x_n-S^nx_n\|\to0$ as $n\to\infty$. In fact, note that

and

It is obvious that ${\rm Fix}(S) = \{0\}$ and

Accordingly, $\Delta = {\rm Fix}(S)\cap{\rm Fix}(S_1)\cap{\rm Fix}(G)\cap{\rm VI}(C, A) = \{0\}\neq\emptyset$. In this case, noticing $G = P_C(I-\alpha_1B_1)P_C(I-\alpha_2B_2) = [P_C(I-\frac{1}{3}B)]^2$, we rewrite Algorithm 3.1 as follows:

where $C_n$ and $\zeta_n$ are chosen as in Algorithm 3.1. Then, $x_n\to0\in \Delta$.

In particular, since $Sx: = \frac{3}{4}\sin x$ is also nonexpansive, we consider the modified version of Algorithm 3.1, that is,

where $C_n$ and $\zeta_n$ are chosen as above. Then, $x_n\to0\in \Delta$.

5.   Conclusions

In a real Hilbert space, we have put forward two modified inertial composite subgradient extragradient implicit rules with line-search process for settling a generalized equilibrium problems system with constraints of a pseudomonotone variational inequality problem and a common fixed-point problem of finite nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. Under the lack of the sequential weak continuity and Lipschitz constant of the cost operator $A$, we have demonstrated the strong convergence of the proposed algorithms to an element of the studied problem. In addition, an illustrated example was provided to demonstrate the feasibility of our proposed algorithms.

In the end, it is worthy to mention that part of our future research is aimed at acquiring the strong convergence results for the modifications of our proposed rules with a Nesterov inertial extrapolation step and adaptive stepsizes.

Use of AI tools declaration

The authors declare that they have not used artificial intelligence (AI) tools in the creation of this article.

Acknowledgement

Yeong-Cheng Liou is partially supported by a grant from the Kaohsiung Medical University Research Foundation (KMU-M113001). Lu-Chuan Ceng is partially supported by the 2020 Shanghai Leading Talents Program of the Shanghai Municipal Human Resources and Social Security Bureau (20LJ2006100), the Innovation Program of Shanghai Municipal Education Commission (15ZZ068) and the Program for Outstanding Academic Leaders in Shanghai City (15XD1503100).

Conflict of interest

The authors declare that there are no conflicts of interest.