Modified subgradient extragradient algorithms for systems of generalized equilibria with constraints

1.   Introduction

Let $H$ be a real Hilbert space with its inner product $\langle\cdot, \cdot\rangle$ and induced norm $\|\cdot\|$. Let $\emptyset\neq C\subset H$ be a closed convex set, and $P_C$ be the metric projection (or nearest point) from $H$ onto $C$. Let $S:C\to H$ be a nonlinear mapping. Let the ${\rm Fix}(S)$ and ${\bf R}$ indicate the fixed-point set of $S$ and the real-number set, respectively. We denote by the $\to$ and $\rightharpoonup$ the strong convergence and weak convergence in $H$, respectively. A mapping $S:C\to C$ is referred to as being asymptotically nonexpansive if $\exists\{\theta_r\}^\infty_{r = 1}\subset[0, +\infty)$ s.t. $\lim_{r\to\infty}\theta_r = 0$ and $\theta_r\|u-v\|+\|u-v\|\geq\|S^r u-S^r v\|, \; \forall r\geq1, \; u, v\in C$. In particular, if $\theta_r = 0, \; \forall r\geq1$, then $S$ is known as being nonexpansive.

Let $\Theta:C\times C\to{\bf R}$ be a bifunction. The equilibrium problem (EP) for $\Theta$ is to determine its equilibrium points, that is, the set ${\bf EP}(\Theta) = \{u\in C:\Theta(u, v)\geq0, \; \forall v\in C\}$. Under the theory framework of equilibrium problems, there is a unified way to investigate a wide number of problems arising in the physics, optimization, structural analysis, transportation, finance and economics. In order to find an element in ${\bf EP}(\Theta)$, one assumes that the following hold:

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

(H2) $\Theta$ is monotone, that is, $\Theta(u, v)+\Theta(v, u)\leq0, \quad\forall u, v\in C$;

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

(H4) $v\mapsto\Theta(u, v)$ is convex and lower semicontinuous (l.s.c.) for each $u\in C$.

In 1994, Blum and Oettli ^[34] gave the following lemma, which plays a key role in solving the equilibrium problems.

Lemma 1.1 (^[34]). Let $\Theta:C\times C\to{\bf R}$ satisfy the hypotheses (H1)–(H4). For any $u\in C$ and $\ell > 0$, let $S_\ell:H\to C$ be the mapping formulated below:

Then $T^\Theta_\ell$ is well defined and the following hold: (i) $T^\Theta_\ell$ is single-valued, and firmly nonexpansive, that is, $\|T^\Theta_\ell u-T^\Theta_\ell v\|^2\leq\langle T^\Theta_\ell u-T^\Theta_\ell v, u-v\rangle, \; \forall u, v\in H$; and (ii) ${\rm Fix}(T^\Theta_\ell) = {\rm EP}(\Theta)$, and ${\rm EP}(\Theta)$ is convex and closed.

It is worth pointing out that the variational inequality problem (VIP) is a special case of the EP. In particular, if $\Theta(u, v) = \langle Au, v-u\rangle, \; \forall u, v\in C$, then the EP reduces to the classical VIP of finding $u\in C$ s.t. $\langle Au, v-u\rangle\geq0, \; \; \forall v\in C$, where $A$ is a self-mapping on $H$. The solution set of the VIP is denoted by VI($C, A$). It is well known that, one of the most popular techniques for solving the VIP is the extragradient one put forth by Korpelevich ^[26] in 1976, that is, for any starting point $u_0\in C$, let $\{u_p\}$ be the sequence constructed below

where $\mu\in(0, \frac{1}{L})$ and $L$ is Lipschitz constant of $A$. Whenever ${\rm VI}(C, A)\neq\emptyset$, the sequence $\{u_p\}$ converges weakly to a point in ${\rm VI}(C, A)$. Till now, the vast literature on the Korpelevich extragradient technique shows that many authors have paid great attention to it and enhanced it in different manners; see e.g., ^{[1,2,3,4,5,6,7,9,10,12,13,14,15,16,17,18,20,21,22,23,24,25,27,28,29,30,31,36,37,38,39,40,41]} and references therein.

Let $\Theta_1, \Theta_2:C\times C\to{\bf R}$ be two bifunctions and let $B_1, B_2:C\to H$ be two nonlinear mappings. In 2010, Ceng and Yao ^[35] considered the following problem of finding $(u^*, v^*)\in C\times C$ such that

with $\mu_1, \mu_2 > 0$, which is called a system of generalized equilibrium problems (SGEP). In particular, if $\Theta_1 = \Theta_2 = 0$, then the SGEP reduces to the following general system of variational inequalities (GSVI) considered in ^[6]: Find $(u^*, v^*)\in C\times C$ such that

Note that SGEP (1.1) can be transformed into the fixed-point problem.

Lemma 1.2 (^[35]). Let $\Theta_1, \Theta_2:C\times C\to{\bf R}$ be two bifunctions satisfying the hypotheses (H1)–(H4) and let the mappings $B_1, B_2:C\to H$ be $\alpha$-inverse-strongly monotone and $\beta$-inverse-strongly monotone, respectively. Let $\mu_1\in(0, 2\alpha)$ and $\mu_2\in(0, 2\beta)$, respectively. Then, for given $u^*, v^*\in C$, $(u^*, v^*)$ is a solution of SGEP (1.1) if and only if $x^*\in {\rm Fix}({\mathcal G})$, where ${\rm Fix}({\mathcal G})$ is the fixed point set of the mapping ${\mathcal G}: = T^{\Theta_1}_{\mu_1}(I-\mu_1B_1)T^{\Theta_2} _{\mu_2}(I-\mu_2B_2)$, and $v^* = T^{\Theta_2}_{\mu_2}(I-\mu_2B_2)u^*$.

On the other hand, suppose that the mappings $B_1, B_2:C\to H$ are $\alpha$-inverse-strongly monotone and $\beta$-inverse-strongly monotone, respectively. Let $f:C\to C$ be contractive with constant $\delta\in[0, 1)$ and $F:C\to H$ be $\kappa$-Lipschitzian and $\eta$-strongly monotone with constants $\kappa, \eta > 0$ such that $\delta < \zeta: = 1-\sqrt{1-\mu(2\eta-\mu\kappa^2)}\in(0, 1]$ for $\mu\in(0, \frac{2\eta}{\kappa^2})$. Let $S:C\to C$ be an asymptotically nonexpansive mapping with a sequence $\{\theta_r\}$ such that $\Omega: = {\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\neq\emptyset$, where ${\rm Fix}({\mathcal G})$ is the fixed-point set of the mapping ${\mathcal G}: = P_C(I-\mu_1B_1)P_C(I-\mu_2B_2)$ for $\mu_1\in(0, 2\alpha)$ and $\mu_2\in(0, 2\beta)$. Recently, Cai, Shehu and Iyiola ^[13] proposed the modified viscosity implicit rule for finding an element of $\Omega$, that is, for any starting point $x_1\in C$, let $\{x_p\}$ be the sequence constructed below

where $\{\alpha_p\}, \{\beta_p\}\subset(0, 1]$ s.t. (ⅰ) $\sum^\infty_{p = 1}|\alpha_{p+1}-\alpha_p| < \infty, \; \sum^\infty_{p = 1}\alpha_p = \infty$; (ⅱ) $\lim_{p\to\infty}\alpha_p = 0, \; \lim_{p\to\infty}\frac{\theta_p}{\alpha_p} = 0$; (ⅲ) $0 < \varepsilon\leq\beta_p\leq1, \; \sum^\infty_{p = 1}|\beta_{p+1}-\beta_p| < \infty$; and (ⅳ) $\sum^\infty_{p = 1}\|T^{p+1}y_p-T^p y_p\| < \infty$. It was proved in ^[13] that the sequence $\{x_p\}$ converges strongly to an element $u^*\in\Omega$, which is a unique solution of the hierarchical variational inequality (HVI): $\langle(\mu F-f)u^*, u-u^*\rangle\geq 0, \; \forall u\in\Omega$. Very recently, Reich et al. ^[29] suggested the modified projection-type method for solving the pseudomonotone VIP with uniform continuity mapping $A$. Let $\{\alpha_p\}\subset(0, 1)$ and $f:C\to C$ be contractive with constant $\delta\in[0, 1)$. Given any initial $x_1\in C$.

Algorithm 1.3 (^[29]). Initial step: Let $\nu > 0, \; \ell\in(0, 1), \; \lambda\in(0, \frac{1}{\nu})$.

Iterations: Given the current iterate $x_p$, calculate $x_{p+1}$ as follows:

Step 1. Compute $y_p = P_C(x_p-\lambda Ax_p)$ and $R_\lambda(x_p): = x_p-y_p$. If $R_\lambda(x_p) = 0$, then stop; $x_p$ is a solution of ${\rm VI}(C, A)$. Otherwise,

Step 2. Compute $w_p = x_p-\tau_p R_\lambda(x_p)$, where $\tau_p: = \ell^{j_p}$ and $j_p$ is the smallest nonnegative integer $j$ s.t. $\frac{\nu}{2}\|R_\lambda(x_p)\|^2\geq\langle Ax_p-A(x_p-\ell^jR_\lambda(x_p)), R_\lambda(x_p) \rangle$.

Step 3. Compute $x_{p+1} = \alpha_p f(x_p)+(1-\alpha_p)P_{C_p}(x_p)$, where $C_p: = \{x\in C:\hbar_p(x_p)\leq0\}$ and $\hbar_p(x) = \langle Aw_p, x-x_p\rangle+\frac{\tau_p}{2\lambda}\|R_\lambda(x_p)\|^2$. Again set $p: = p+1$ and go to Step 1.

It was proven in ^[29] that under mild conditions, $\{x_p\}$ converges strongly to an element of ${\rm VI}(C, A)$. In a real Hilbert space $H$, we always assume that the SGEP, VIP, HVI and FPP represent a system of generalized equilibrium problems, a pseudomonotone variational inequality problem, a hierarchical variational inequality and a fixed-point problem of an asymptotically nonexpansive mapping, respectively. We introduce the modified Mann-like subgradient-like extragradient implicit rules with linear-search process for finding a common solution of the SGEP, VIP and FPP. The proposed algorithms are based on the subgradient extragradient rule with linear-search process, Mann implicit iteration approach, and hybrid deepest-descent technique. Under mild restrictions, we demonstrate the strong convergence of the proposed algorithms to a common solution of the SGEP, VIP and FPP, which is a unique solution of a certain HVI defined on their common solution set. In addition, an illustrated example is provided to illustrate the feasibility and implementability of our suggested rules.

The architecture of this article is constituted below: In Section 2, we present some concepts and basic tools for further use. Section 3 treats the convergence analysis of the suggested algorithms. Last, Section 4 applies our main results to solve the SGEP, VIP and FPP in an illustrated example. Our results improve and extend the ones associated with very recent literature, e.g., ^[13,17,29].

2.   Preliminaries

Let $H$ be a real Hilbert space and $\emptyset\neq C\subset H$ be a convex and closed set. Given a sequence $\{u_k\}\subset H$. We denote by the $u_k\to u^*$ (resp., $u_k\rightharpoonup u^*$) the strong (resp., weak) convergence of $\{u_k\}$ to $u^*$. For all $u, v\in C$, an operator $\Psi:C\to H$ is referred to as being

(a) $L$-Lipschitzian (or $L$-Lipschitz continuous) if $\exists L > 0$ s.t. $\|\Psi u- \Psi v\|\leq L\|u-v\|$;

(b) pseudomonotone if $\langle \Psi u, v-u\rangle\geq0\Rightarrow \langle\Psi v, v-u\rangle\geq0$;

(c) monotone if $\langle\Psi u-\Psi v, u-v\rangle\geq0$;

(d) $\alpha$-strongly monotone if $\exists\alpha > 0$ s.t. $\langle\Psi u-\Psi v, u-v\rangle\geq\alpha\|u-v\|^2$;

(e) $\beta$-inverse-strongly monotone if $\exists\beta > 0$ s.t. $\langle\Psi u-\Psi v, u-v\rangle\geq\beta\|\Psi u-\Psi v\|^2$;

(f) sequentially weakly continuous if $\forall\{v_k\}\subset C$, the relation holds: $v_k\rightharpoonup v\Rightarrow \Psi v_k\rightharpoonup\Psi v$. It is clear that each monotone mapping is pseudomonotone but the converse is not true. Also, $\forall v\in H$, $\exists|$ (nearest point) $P_C(v)\in C$ s.t. $\|v-P_C(v)\|\leq\|v-w\|\; \forall w\in C$. $P_C$ is called a nearest point (or metric) projection of $H$ onto $C$. The following conclusions hold (see ^[19]):

(a) $\langle v-w, P_C(v)-P_C(w)\rangle\geq\|P_C(v)-P_C(w)\|^2, \; \; \forall v, w\in H$;

(b) $w = P_C(v)\Leftrightarrow\langle v-w, u-w\rangle\leq0, \; \; \forall v\in H, u\in C$;

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

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

(e) $\|sv+(1-s)w\|^2 = s\|v\|^2+(1-s)\|w\|^2-s(1-s)\|v-w\|^2, \; \; \forall v, w\in H, s\in[0, 1]$.

The following inequality is an immediate consequence of the subdifferential inequality of the function $\frac{1}{2}\|\cdot\|^2$:

The following lemmas will be used for demonstrating our main results in the sequel.

Lemma 2.1. Let the mapping $B:C\to H$ be $\gamma$-inverse-strongly monotone. Then, for a given $\lambda\geq0$,

In particular, if $0\leq\lambda\leq2\gamma$, then $I-\lambda B$ is nonexpansive.

Using Lemma 1.1 and Lemma 2.1, we immediately derive the following lemma.

Lemma 2.2 (^[35]). Let $\Theta_1, \Theta_2:C\times C\to{\bf R}$ be two bifunctions satisfying the hypotheses (H1)–(H4), and the mappings $B_1, B_2:C\to H$ be $\alpha$-inverse-strongly monotone and $\beta$-inverse-strongly monotone, respectively. Let the mapping ${\mathcal G}:C\to C$ be defined as ${\mathcal G}: = T^{\Theta_1}_{\mu_1}(I- \mu_1B_1)T^{\Theta_2}_{\mu_2}(I-\mu_2B_2)$. Then ${\mathcal G}:C\to C$ is a nonexpansive mapping provided $0 < \mu_1 \leq2\alpha$ and $0 < \mu_2\leq2 \beta$.

In particular, if $\Theta_1 = \Theta_2 = 0$, using Lemma 1.1 we deduce that $T^{\Theta_1}_{\mu_1} = T^{\Theta _2}_{\mu_2} = P_C$. Thus, from Lemma 2.2 we obtain the corollary below.

Corollary 2.3 (^[6]). Let the mappings $B_1, B_2:C\to H$ be $\alpha$-inverse-strongly monotone and $\beta$-inverse-strongly monotone, respectively. Let the mapping ${\mathcal G}:C\to C$ be defined as ${\mathcal G}: = P_C(I-\mu_1B_1)P_C(I-\mu_2 B_2)$. If $0 < \mu_1\leq2\alpha$ and $0 < \mu_2\leq2\beta$, then ${\mathcal G}:C\to C$ is nonexpansive.

Lemma 2.4 (^[6]). Let $A:C\to H$ be pseudomonotone and continuous. Then $u\in C$ is a solution to the VIP $\langle Au, \upsilon-u\rangle\geq0, \; \forall\upsilon\in C$, if and only if $\langle A\upsilon, \upsilon-u\rangle \geq0, \; \forall\upsilon\in C$.

Lemma 2.5 (^[8]). Let $\{a_l\}$ be a sequence of nonnegative numbers satisfying the conditions: $a_{l+1} \leq(1-\lambda_l)a_l+\lambda_l\gamma_l\; \forall l\geq1$, where $\{\lambda_l\}$ and $\{\gamma_l\}$ are sequences of real numbers such that (i) $\{\lambda_l\}\subset[0, 1]$ and $\sum^\infty_{l = 1}\lambda_l = \infty$, and (ii) $\limsup_{l\to \infty}\gamma_l\leq0$ or $\sum^\infty_{l = 1}|\lambda_l\gamma_l| < \infty$. Then $\lim_{l\to\infty}a_l = 0$.

Later on, we will make use of the following lemmas to demonstrate our main results.

Lemma 2.6 (^[32]). Let $H_1$ and $H_2$ be two real Hilbert spaces. Suppose that $A:H_1\to H_2$ is uniformly continuous on bounded subsets of $H_1$ and $M$ is a bounded subset of $H_1$. Then, $A(M)$ is bounded.

Lemma 2.7 (^[33]). Let $h$ be a real-valued function on $H$ and define $K: = \{x\in C:h(x)\leq0\}$. If $K$ is nonempty and $h$ is Lipschitz continuous on $C$ with modulus $\theta > 0$, then ${\rm dist}(x, K)\geq\theta^{-1}\max\{ h(x), 0\}\; \forall x\in C$, where ${\rm dist}(x, K)$ denotes the distance of $x$ to $K$.

Lemma 2.8 (^[11]). Let $X$ be a Banach space which admits a weakly continuous duality mapping, $C$ be a nonempty closed convex subset of $X$, and $T:C\to C$ be an asymptotically nonexpansive mapping with ${\rm Fix} (T)\neq\emptyset$. Then $I-T$ is demiclosed at zero, i.e., if $\{u_k\}$ is a sequence in $C$ such that $u_k \rightharpoonup u\in C$ and $(I-T)u_k\to0$, then $(I-T)u = 0$, where $I$ is the identity mapping of $X$.

The following lemmas are very crucial to the convergence analysis of the proposed algorithms.

Lemma 2.9 (^[30]). Let $\{\Lambda_m\}$ be a sequence of real numbers that does not decrease at infinity in the sense that, $\exists\{\Lambda_{m_k}\}\subset\{\Lambda_m\}$ s.t. $\Lambda_{m_k} < \Lambda_{m_k+1}\; \forall k\geq1$. Let the sequence $\{\phi(m)\}_{m\geq m_0}$ of integers be formulated below:

with integer $m_0\geq1$ satisfying $\{k\leq m_0:\Lambda_k < \Lambda_{k+1}\}\neq\emptyset$. Then there hold the statements below:

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

(ii) $\Lambda_{\phi(m)}\leq\Lambda_{\phi(m)+1}$ and $\Lambda_m\leq\Lambda_{\phi(m)+1}, \; \forall m\geq m_0$.

Lemma 2.10 (^[8]). Let $\lambda\in(0, 1]$ and Let $S:C\to C$ be a nonexpansive mapping. Let $S^\lambda:C\to H$ be the mapping formulated by $S^\lambda u: = (I-\lambda\mu F)Su\; \forall u\in C$ with $F:C\to H$ being $\kappa$-Lipschitzian and $\eta$-strongly monotone. Then $S^\lambda$ is a contraction provided $0 < \mu < \frac{2\eta} {\kappa^2}$, i.e., $\|S^\lambda u-S^\lambda v\|\leq(1-\lambda\tau)\|u-v\|, \; \forall u, v\in C$, where $\tau = 1-\sqrt{1-\mu(2\eta-\mu\kappa^2)}\in(0, 1]$.

3.   Main results

In this section, let the feasible set $C$ be a nonempty closed convex subset of a real Hilbert space $H$, and assume always that the following conditions hold:

(1) $S:C\to C$ is an asymptotically nonexpansive mapping with a sequence $\{\theta_n\}$, and $A:H\to H$ is pseudomonotone and uniformly continuous on $C$, s.t. $\|Az\|\leq\liminf_{n\to\infty}$ $\|Au_n\|$ for each $\{u_n\}\subset C$ with $u_n\rightharpoonup z$.

(2) $\Theta_1, \Theta_2:C\times C\to{\bf R}$ are two bifunctions satisfying the hypotheses (H1)–(H4), and $B_1, B_2:C\to H$ are $\alpha$-inverse-strongly monotone and $\beta$-inverse-strongly monotone, respectively.

(3) $\Omega = {\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A)\neq\emptyset$ where ${\mathcal G}: = T^{\Theta_1}_{\mu_1}(I-\mu_1B_1) T^{\Theta_2}_{\mu_2}(I-\mu_2B_2)$ for $\mu_1\in(0, 2\alpha)$ and $\mu_2\in(0, 2\beta)$.

(4) $f:C\to H$ is a contraction with constant $\delta\in[0, 1)$, and $F:C\to H$ is $\eta$-strongly monotone and $\kappa$-Lipschitzian such that $\delta < \tau: = 1-\sqrt{1-\mu(2\eta-\mu\kappa^2)}$ for $\mu\in(0, \frac{2\eta}{\kappa^2})$.

(5) $\{\sigma_n\}, \{\alpha_n\}\subset(0, 1]$ and $\{\beta_n\}\subset[0, 1]$ are three real number sequences satisfying

(ⅰ) $\sum^\infty_{n = 1}\alpha_n = \infty, \; \lim_{n\to\infty}\alpha_n = 0$ and $\lim_{n\to\infty}\frac{\theta_n}{\alpha_n} = 0$;

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

(ⅲ) $\limsup_{n\to\infty}\sigma_n < 1$.

Algorithm 3.1. Initial step: Given $\nu > 0, \; \ell\in(0, 1), \; \lambda\in(0, \frac{1}{\nu})$. Let $x_1\in C$ be arbitrary.

Iterations: Given the current iterate $x_n$, calculate $x_{n+1}$ below:

Step 1. Calculate $w_n = \sigma_nx_n+(1-\sigma_n)u_n$ with

Step 2. Calculate $y_n = P_C(w_n-\lambda Aw_n)$ and $R_\lambda(w_n): = w_n-y_n$.

Step 3. Calculate $t_n = w_n-\tau_nR_\lambda(w_n)$, where $\tau_n: = \ell^{j_n}$ and $j_n$ is the smallest nonnegative integer $j$ satisfying

Step 4. Compute $z_n = P_{C_n}(w_n)$ and $x_{n+1} = \beta_nx_n+(1-\beta_n)P_C[\alpha_nf(x_n)+(I-\alpha_n\mu F)S^nz_n]$, where $C_n: = \{u\in C:\hbar_n(u)\leq0\}$ and

Again put $n: = n+1$ and return to Step 1.

Lemma 3.2. The Armijo-type search approach (3.1) is well formulated, and the relation holds: $\lambda^{-1}\|R_\lambda(w_n)\|^2\leq\langle R_\lambda(w_n), Aw_n\rangle$.

Proof. Since $\ell\in(0, 1)$ and $A$ is of uniform continuity on $C$, it is clear that $\lim_{j\to\infty}\langle Aw_n-A(w_n-\ell^jR_\lambda(w_n)), R_\lambda(w_n)\rangle = 0$. If $R_\lambda(w_n) = 0$, one gets $j_n = 0$. Otherwise, from $R_\lambda(w_n)\neq0$, it follows that $\exists$ (integer) $j_n\geq0$ fulfilling (3.1). It is readily known that the firm nonexpansivity of $P_C$ implies $\langle u-P_Cv, u-v\rangle\geq\|u-P_Cv\|^2, \; \forall u\in C, v\in H$. Setting $v = w_n-\lambda Aw_n$ and $u = w_n$, one has $\lambda\langle w_n-P_C(w_n-\lambda Aw_n), Aw_n\rangle\geq\|w_n-P_C (w_n-\lambda Aw_n)\|^2$. Hence the relation holds.

Lemma 3.3. Suppose that $\hbar_n$ is the function formulated in (3.2). Then, $\hbar_n(\upsilon)\leq0\; \forall \upsilon\in\Omega$. In addition, when $R_\lambda(w_n)\neq0$, one has $\hbar_n(w_n) > 0$.

Proof. It suffices to show the former claim of Lemma 3.3 because the latter claim is clear. In fact, pick an arbitrary $\upsilon\in\Omega$. By Lemma 2.4 one gets $\langle At_n, t_n-\upsilon\rangle\geq0$. Thus, one has

Meanwhile, from (3.1) it follows that $\frac{\nu}{2}\|R_\lambda(w_n)\|^2\geq\langle Aw_n-At_n, R_\lambda(w_n)\rangle$. So, from Lemma 3.2 one gets

This along with (3.3), arrives at

Therefore, we derive the desired result.

Lemma 3.4. Let $\{w_n\}, \{x_n\}, \{y_n\}, \{z_n\}$ be the bounded sequences constructed in Algorithm 3.1. Assume that $x_n-x_{n+1}\to0, \; x_n-{\mathcal G}w_n\to0, \; w_n-y_n\to0$ and $x_n-z_n\to0$. If $S^nx_n-S^{n+1}x_n\to0$ and $\exists\{x_{n_k}\}\subset\{x_n\}$ such that $x_{n_k}\rightharpoonup z\in C$, then $z\in\Omega$.

Proof. From Algorithm 3.1, we get $w_n-x_n = (1-\sigma_n)(u_n-x_n)\; \forall n\geq1$, and hence $\|w_n-x_n\| = (1-\sigma_n)\|u_n-x_n\|\leq\|u_n-x_n\|$. Utilizing the assumption $u_n-x_n\to0$, we have

Putting $q_n: = \alpha_nf(x_n)+(I-\alpha_n\mu F)S^nz_n$, by Algorithm 3.1 we know that $x_{n+1} = \beta_nx_n+(1- \beta_n)P_C(q_n)$ and $q_n-S^nz_n = \alpha_nf(x_n)-\alpha_n\mu FS^nz_n$. Hence one gets

This immediately ensures that

Since $x_n-x_{n+1}\to0, \; \alpha_n\to0$ and $\liminf_{n\to\infty}(1-\beta_n) > 0$, by the boundedness of $\{x_n\}, \{z_n\}$ we obtain

We claim that $\lim_{n\to\infty}\|x_n-Sx_n\| = 0$. In fact, using the asymptotical nonexpansivity of $S$, one deduces that

Since $x_n-z_n\to0$, $x_n-S^nz_n\to0$ and $S^nx_n-S^{n+1}x_n\to0$, we obtain

Also, let us show that $\lim_{n\to\infty}\|x_n-{\mathcal G}x_n\| = 0$. In fact, by Lemma 2.2 we know that ${\mathcal G}:C\to C$ is nonexpansive for $\mu_1\in(0, 2\alpha)$ and $\mu_2\in(0, 2\beta)$. Again from Algorithm 3.1, we have $u_n = {\mathcal G}w_n$. Since

Noticing $u_n-x_n\to0$ and $x_n-w_n\to0$ (due to (3.6)), we obtain

Next, let us show $z\in{\rm VI}(C, A)$. Indeed, noticing $x_n-w_n\to0$ and $x_{n_k}\rightharpoonup z$, we know that $w_{n_k}\rightharpoonup z$. Since $C$ is convex and closed, from $\{w_n\}\subset C$ and $w_{n_k}\rightharpoonup z$ we get $z\in C$. In what follows, we consider two cases. In the case of $Az = 0$, it is clear that $z\in{\rm VI}(C, A)$ because $\langle Az, y-z\rangle\geq0, \; \forall y\in C$. In the case of $Az\neq0$, it follows from $w_n-x_n\to0$ and $x_{n_k}\rightharpoonup z$ that $w_{n_k}\rightharpoonup z$ as $k\to\infty$. Utilizing the assumption on $A$, instead of the sequentially weak continuity of $A$, we get $0 < \|Az\|\leq\liminf_{k\to\infty}\|Aw_{n_k}\|$. So, we might assume that $\|Aw_{n_k}\|\neq0\; \forall k\geq1$. On the other hand, from $y_n = P_C(w_n-\lambda Aw_n)$, one has $\langle w_n-\lambda Aw_n-y_n, x-y_n\rangle\leq0, \; \forall x\in C$, and hence

In the light of the uniform continuity of $A$ on $C$, one knows that $\{Aw_n\}$ is bounded (due to Lemma 2.6). Note that $\{y_n\}$ is bounded as well. Thus, from (3.9) we get $\liminf_{k\to\infty}\langle Aw_{n_k}, x-w_{n_k}\rangle \geq0\; \forall x\in C$.

To show that $z\in{\rm VI}(C, A)$, we now choose a sequence $\{\gamma_k\}\subset(0, 1)$ satisfying $\gamma_k\downarrow0$ as $k\to\infty$. For each $k\geq1$, we denote by $l_k$ the smallest positive integer such that

Because $\{\gamma_k\}$ is decreasing, it is readily known that $\{l_k\}$ is increasing. Note that $Aw_{l_k} \neq0\; \forall k\geq1$ (due to $\{Aw_{l_k}\}\subset\{Aw_{n_k}\}$). Then one puts $\upsilon_{l_k} = \frac{Aw_{l_k}}{\|Aw_{l_k}\|^2}$, one gets $\langle Aw_{l_k}, \upsilon_{l_k}\rangle = 1, \; \forall k\geq1$. So, using (3.10) one has $\langle Aw_{l_k}, x+\gamma_k\upsilon_{l_k}-w_{l_k}\rangle\geq0, \; \forall k\geq1$. Again from the pseudo-monotonicity of $A$ one has $\langle A(x+\gamma_k\upsilon_{l_k}), x+\gamma_k\upsilon_{l_k}-w_{l_k}\rangle\geq0, \; \forall k\geq1$. This immediately arrives at

We claim that $\lim_{k\to\infty}\gamma_k\upsilon_{l_k} = 0$. In fact, from $x_{n_k}\rightharpoonup z\in C$ and $w_n-x_n\to0$, we obtain $w_{n_k}\rightharpoonup z$. Note that $\{w_{l_k}\}\subset\{w_{n_k}\}$ and $\gamma_k\downarrow0$ as $k\to\infty$. So it follows that

Hence one gets $\gamma_k\upsilon_{l_k}\to0$ as $k\to\infty$. Thus, letting $k\to\infty$, we deduce that the right-hand side of (3.11) tends to zero by the uniform continuity of $A$, the boundedness of $\{w_{l_k}\}, \{\upsilon_{l_k}\}$ and the limit $\lim_{k\to\infty}\gamma_k\upsilon_{l_k} = 0$. Therefore, $\langle Ax, x-z\rangle = \liminf_{k\to\infty}\langle Ax, x-w_{l_k}\rangle\geq0\; \forall x\in C$. Using Lemma 2.4 one has $z\in{\rm VI}(C, A)$.

Last, we claim that $z\in\Omega$. In fact, because (3.7) yields $x_{n_k}-Sx_{n_k}\to0$. By Lemma 2.8 one knows that $I-S$ is demiclosed at zero. So, from $x_{n_k}\rightharpoonup z$ it follows that $(I-S)z = 0$, i.e., $z\in{\rm Fix}(S)$. Besides, let us claim that $z\in{\rm Fix}({\mathcal G})$. Actually, by Lemma 2.8 we deduce that $I-{\mathcal G}$ is demiclosed at zero. Thus, from (3.8) and $x_{n_k}\rightharpoonup z$ one has $(I-{\mathcal G})z = 0$, i.e., $z\in{\rm Fix}({\mathcal G})$. Accordingly, $z\in{\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A) = \Omega$. This completes the proof.

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

Proof. We claim that $\limsup_{n\to\infty}\|R_\lambda(w_n)\| = 0$. Conversely, suppose that $\limsup_{n\to\infty}$ $\|R_\lambda(w_n)\| = d > 0$. Then, $\exists\{n_p\}\subset\{n\}$ s.t. $\lim_{p\to\infty}\|R_\lambda(w_{n_p})\| = d > 0$. Note that $\lim_{p\to\infty}\tau_{n_p}\|R_\lambda(w_{n_p})\|^2 = 0$. First, if $\liminf_{p\to\infty}\tau_{n_p} > 0$, we might assume that $\exists\xi > 0$ s.t. $\tau_{n_p}\geq\xi > 0, \; \forall p\geq1$. So it follows that

which immediately leads to

So, this reaches at a contradiction.

If $\liminf_{p\to\infty}\tau_{n_p} = 0$, there exists a subsequence of $\{\tau_{n_p}\}$, still denoted by $\{\tau_{n_p}\}$, s.t. $\lim_{p\to\infty}\tau_{n_p} = 0$. We now set

Then, from $\lim_{p\to\infty}\tau_{n_p}\|R_\lambda(w_{n_p})\|^2 = 0$ we infer that

Using the stepsize rule (3.1), one gets $\langle Aw_{n_p}-A\upsilon_{n_p}, R_\lambda(w_{n_p})\rangle > \frac{\nu}{2}\|R_\lambda(w_{n_p})\|^2$. Since $A$ is uniformly continuous on bounded subsets of $C$, (3.14) guarantees that

which hence attains $\lim_{p\to\infty}\|R_\lambda(w_{n_p})\| = 0$. So, this reaches a contradiction. Therefore, $R_\lambda(w_n)\to0$ as $n\to\infty$. This completes the proof.

Theorem 3.6. Suppose that $\{x_n\}$ is the sequence constructed in Algorithm 3.1. Then $x_n\to x^*\in \Omega$ provided $S^nx_n-S^{n+1}x_n\to0$, with $x^*\in\Omega$ being only a solution to the HVI

Proof. First of all, noticing $0 < \liminf_{n\to\infty}\beta_n\leq\limsup_{n\to\infty}\beta_n < 1$ and $\lim_{n\to \infty}\frac{\theta_n}{\alpha_n} = 0$, we may assume, without loss of generality, that $\{\sigma_n\}\subset[a, b]\subset(0, 1)$ and $\theta_n\leq\frac{\alpha_n(\tau-\delta)}{2}, \; \forall n\geq1$. We claim that $P_\Omega(I-\mu F+f):C\to C$ is contractive map. In fact, using Lemma 2.10, one has

This ensures that $P_\Omega(I-\mu F+f)$ is contractive. Banach's Contraction Mapping Principle guarantees that there exists a unique fixed point of $P_\Omega(I-\mu F+f)$ in $C$. Say $x^*\in C$ s.t. $x^* = P_\Omega(I-\mu F+f)x^*$. That is, $\exists\mid$ (solution) $x^*\in\Omega = {\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A)$ of the HVI

Next we demonstrate the conclusion of the theorem. To the goal, we divide the remainder of the proof into several aspects.

Aspect 1. We assert that $\{x_n\}$ is of boundedness. Indeed, for $x^*\in\Omega = {\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A)$ we have $Sx^* = x^*, \; {\mathcal G}x^* = x^*$ and $P_C(x^*-\lambda Ax^*) = x^*$. We observe that

which hence leads to

Using Lemma 2.2, one knows that ${\mathcal G} = T^{\theta_1}_{\mu_1}(I-\mu_1B_1)T^{\theta_2}_{\mu_2}(I-\mu_2B_2)$ is nonexpansive for $\mu_1\in(0, 2\alpha)$ and $\mu_2\in(0, 2\beta)$. Thus, by the definition of $w_n$, one gets

which immediately yields

This together with (3.18), yields

Thus, using (3.19), from Lemma 2.10 we obtain

By induction, we get

Thus, $\{x_n\}$ is bounded, and so are the sequences $\{w_n\}, \{y_n\}, \{z_n\}, \{f(x_n)\}, \{At_n\}, \{{\mathcal G}w_n\}, \{S^nz_n\}$.

Aspect 2. We assert that

for some $M_1 > 0$. In fact, noticing $z_n = P_{C_n}(w_n)$ and $w_n = \sigma_nx_n+(1-\sigma_n)u_n$, we obtain

Since $x_{n+1} = \beta_nx_n+(1-\beta_n)P_C(q_n)$ where $q_n = \alpha_nf(x_n)+(I-\alpha_n\mu F)S^nz_n$, by Lemma 2.10 and the convexity of the function $h(s) = s^2, \; \forall s\in{\bf R}$ we deduce that

(due to $\alpha_n\delta+(1-\alpha_n\tau)+\theta_n\leq1-\alpha_n(\tau-\delta)+\frac{\alpha_n(\tau-\delta)}{2} = 1-\frac{\alpha_n(\tau-\delta)}{2}\leq1$), which together with $u_n = {\mathcal G}w_n$ and (3.19), ensures that

where $\sup_{n\geq1}2\|(f-\mu F)x^*\|\|q_n-x^*\|\leq M_1$ for some $M_1 > 0$. This attains the desired assertion.

Aspect 3. We assert that

In fact, we claim that for some $\bar L > 0$,

Noticing the boundedness of $\{At_n\}$, one knows that $\exists\bar L > 0$ s.t. $\|At_n\|\leq\bar L, \; \forall n\geq1$. This implies that

which hence guarantees that $\hbar_n(\cdot)$ is $\bar L$-Lipschitz continuous on $C_n$. By Lemmas 2.7 and 3.3, we have

Combining (3.17) and (3.23) yields

From (3.20), (3.19) and (3.22) it follows that

This hence leads to

Aspect 4. We assert that

for some $M > 0$. In fact, from Lemma 2.10 and (3.19), one obtains

where $\sup_{n\geq1}\|x_n-x^*\|^2\leq M$ for some $M > 0$.

Aspect 5. We assert that $x_n\to x^*\in\Omega$, which is only a solution of the HVI (3.16).

In fact, from (3.24), we have

Setting $\Lambda_n = \|x_n-x^*\|^2$, we demonstrate the convergence of $\{\Lambda_n\}$ to zero by the following two situations.

Situation 1. $\exists\; ({\rm integer})\; n_0\geq1$ s.t. $\{\Lambda_n\}$ is nonincreasing. It is clear that the limit $\lim_{n\to\infty}\Lambda_n = k < +\infty$ and $\lim_{n\to\infty}(\Lambda_n-\Lambda_{n+1}) = 0$. From Aspect 2 and $\{\beta_n\}\subset[a, b]\subset(0, 1)$ we obtain

Thanks to the facts that $\alpha_n\to0$ and $\Lambda_n-\Lambda_{n+1}\to0$, from $\frac{\tau+\delta}{2}\in(0, 1)$ one deduces that

Hence it is readily known that

and

Next, we show that $\|x_n-u_n\|\to0$ as $n\to\infty$. Indeed, note that $y^* = T^{\Theta_2}_{\mu_2}(x^*-\mu_2B_2x^*)$, $v_n = T^{\Theta_2}_{\mu_2}(w_n-\mu_2B_2w_n)$ and $u_n = T^{\Theta_1}_{\mu_1}(v_n-\mu_1B_1v_n)$. Then $u_n = {\mathcal G}w_n$. By Lemma 2.1 we have

and

Combining the last two inequalities, from (3.19) we obtain

This together with (3.20), implies that

which immediately arrives at

Since $\beta_n\leq b < 1, \; \mu_1\in(0, 2\alpha), \; \mu_2\in(0, 2\beta), \; \lim_{n\to\infty}\alpha_n = 0$ and $\limsup_{n\to\infty}\sigma_n < 1$, we obtain from $\Lambda_n-\Lambda_{n+1}\to0$ that

On the other hand, by Lemma 1.1 one has

which hence leads to

Similarly, one gets

Combining the last two inequalities, from (3.19) we get

This together with (3.20), ensures that

which immediately leads to

Since $\beta_n\leq b < 1, \; \lim_{n\to\infty}\alpha_n = 0$, $\limsup_{n\to\infty}\sigma_n < 1$ and $\limsup_{n\to\infty}(\Lambda_n-\Lambda_{n+1}) = 0$, we deduce from (3.27) and the boundedness of $\{u_n\}, \{v_n\}$ that

Therefore,

Noticing $w_n = \sigma_nx_n+(1-\sigma_n)u_n$, we get

which immediately yields

So it follows that

which together with (3.20), leads to

This hence arrives at

Since $\beta_n\leq b < 1, \; \lim_{n\to\infty}\alpha_n = 0$, $\limsup_{n\to\infty}\sigma_n < 1$ and $\limsup_{n\to\infty}(\Lambda_n-\Lambda_{n+1}) = 0$, we deduce from $\frac{\tau+\delta}{2}\in(0, 1)$ that

So it follows from (3.26) and (3.28) that

and

Meanwhile, from Aspect 3 we obtain

Noticing $\beta_n\leq b < 1, \; \alpha_n\to0$ and $\Lambda_n-\Lambda_{n+1}\to0$, one gets

which together with Lemma 3.5, yields

By the boundedness of $\{x_n\}$, we know that $\exists\; {\rm subsequence}\; \{x_{n_p}\}\subset\{x_n\}$ s.t.

Since $H$ is reflexive and $\{x_n\}$ is bounded, we might assume that $x_{n_p}\rightharpoonup\bar x$. Thus, from (3.32) one has

Since $x_n-x_{n+1}\to0, \; x_n-{\mathcal G}w_n\to0, \; w_n-y_n\to0, \; x_n-z_n\to0$ and $x_{n_p}\rightharpoonup\bar x$, by Lemma 3.4 we infer that $\bar x\in\Omega$. Thus, using (3.16) and (3.33) one has

which together with (3.26), arrives at

Note that $\{\alpha_n(1-\beta_n)(\tau-\delta)\}\subset[0, 1], \; \sum^\infty_{n = 1}\alpha_n(1-\beta_n)(\tau-\delta) = \infty$, and

Consequently, applying Lemma 2.5 to (3.25), one has $\lim_{n\to\infty}\|x_n-x^*\|^2 = 0$.

Situation 2. $\exists\{\Lambda_{n_p}\}\subset\{\Lambda_n\}$ s.t. $\Lambda_{n_p} < \Lambda_{n_p+1}\; \forall p\in{\mathcal N}$, with ${\mathcal N}$ being the set of all natural numbers. Let $\phi:{\mathcal N}\to{\mathcal N}$ be formulated as

Using Lemma 2.9, we have

By Aspect 2 one gets

which immediately ensures that

By Aspect 3 we have

which hence leads to

Using the similar arguments to those of Situation 1, we infer that $\lim_{n\to\infty}\|x_{\phi(n)+1}-x_{\phi(n)}\| = 0$,

and

On the other hand, from (3.25) we obtain

which immediately attains

Thus, $\lim_{n\to\infty}\|x_{\phi(n)}-x^*\|^2 = 0$. In addition, observe that

Thanks to $\Lambda_n\leq\Lambda_{\phi(n)+1}$, one gets

That is, $x_n\to x^*$ as $n\to\infty$. This completes the proof.

Theorem 3.7. If $S:C\to C$ is nonexpansive and $\{x_n\}$ is the sequence constructed in the modified version of Algorithm 3.1, that is, for any initial $x_1\in C$,

where for each $n\geq1$, $C_n$ and $\tau_n$ are picked as in Algorithm 3.1, then $x_n\to x^*\in\Omega$ with $x^*\in\Omega$ being only a solution to the HVI: $\langle(\mu F-f)x^*, y-x^*\rangle\geq0, \; \; \forall y\in\Omega$.

Proof. We divide the proof of the theorem into several aspects.

Aspect 1. We assert the boundedness of $\{x_n\}$. Indeed, using the same reasonings as in Aspect 1 of the proof of Theorem 3.6, one derives the desired assertion.

Aspect 2. We assert that

for some $M_1 > 0$. In fact, putting $\theta_n = 0$, from (3.20) we get

where $\sup_{n\geq1}2\|(f-\mu F)x^*\|\|q_n-x^*\|\leq M_1$ for some $M_1 > 0$. This attains the desired assertion.

Aspect 3. We assert that

Indeed, using the same reasonings as in Aspect 3 of the proof of Theorem 3.6, one deduces the desired assertion.

Aspect 4. We assert that

Indeed, using the same reasonings as in Aspect 4 of the proof of Theorem 3.6, one obtains the desired assertion.

Aspect 5. We assert that $\{x_n\}$ converges strongly to the unique solution $x^*\in\Omega$ of the HVI (3.16). Indeed, using the same reasonings as in Aspect 5 of the proof of Theorem 3.6, one gets the desired assertion. On the other hand, we put forth another modification of Mann-like subgradient-like extragradient implicit rule with linear-search process.

Algorithm 3.8. Initial step: Given $\nu > 0, \; \ell\in(0, 1), \; \lambda\in(0, \frac{1}{\nu})$. Let $x_1\in C$ be arbitrary.

Iterations: Given the current iterate $x_n$, calculate $x_{n+1}$ below:

Step 1. Calculate $w_n = \sigma_nx_n+(1-\sigma_n)u_n$ with

Step 2. Calculate $y_n = P_C(w_n-\lambda Aw_n)$ and $R_\lambda(w_n): = w_n-y_n$.

Step 3. Calculate $t_n = w_n-\tau_nR_\lambda(w_n)$, where $\tau_n: = \ell^{j_n}$ and $j_n$ is the smallest nonnegative integer $j$ satisfying

Step 4. Compute $z_n = P_{C_n}(w_n)$ and $x_{n+1} = \beta_nw_n+(1-\beta_n)P_C[\alpha_nf(z_n)+(I-\alpha_n\mu F)S^nz_n]$, where $C_n: = \{u\in C:\hbar_n(u)\leq0\}$ and

Again put $n: = n+1$ and return to Step 1.

It is worth mentioning that (3.16)–(3.19) and Lemmas 3.2–3.5 remain true for Algorithm 3.8.

Theorem 3.9. Suppose that $\{x_n\}$ is the sequence constructed in Algorithm 3.8. Then $x_n\to x^*\in\Omega$ provided $S^nx_n-S^{n+1}x_n\to0$, with $x^*\in\Omega$ being only a solution to the HVI: $\langle (\mu F-f)x^*, y-x^*\rangle\geq0, \; \; \forall y\in\Omega$.

Proof. In what follows, under the assumption $S^nx_n-S^{n+1}x_n\to0$, one divides the proof into several aspects.

Aspect 1. We assert that $\{x_n\}$ is of boundedness. Indeed, for $x^*\in\Omega = {\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A)$ we have $Sx^* = x^*, \; {\mathcal G}x^* = x^*$ and $P_C(x^*-\lambda Ax^*) = x^*$. Using (3.19), from Lemma 2.10 we obtain

By induction, we get $\|x_n-x^*\|\leq\max\{\|x_1-x^*\|, \frac{2\|(f-\mu F)x^*\|}{\tau-\delta}\}\; \forall n\geq1$. Thus, $\{x_n\}$ is bounded, and so are the sequences $\{w_n\}, \{y_n\}, \{z_n\}, \{f(z_n)\}, \{At_n\}, \{{\mathcal G}w_n\}, \{S^nz_n\}$.

Aspect 2. We assert that

for some $M_1 > 0$. In fact, it is clear that

Since $x_{n+1} = \beta_nw_n+(1-\beta_n)P_C(\bar q_n)$ where $\bar q_n = \alpha_nf(z_n)+(I-\alpha_n\mu F)S^nz_n$, Using Lemma 2.10 and the convexity of the function $h(s) = s^2\; \forall s\in{\bf R}$, from (3.19) we obtain that

(due to $\alpha_n\delta+(1-\alpha_n\tau)+\theta_n\leq1-\alpha_n(\tau-\delta)+\frac{\alpha_n(\tau-\delta)}{2} = 1-\frac{\alpha_n(\tau-\delta)}{2}\leq1$), which together with $u_n = {\mathcal G}w_n$, guarantees that

where $\sup_{n\geq1}2\|(f-\mu F)x^*\|\|\bar q_n-x^*\|\leq M_1$ for some $M_1 > 0$. This attains the desired assertion.

Aspect 3. We assert that

In fact, using the similar arguments to those of (3.22) in the proof of Theorem 3.56, we can deduce that for some $\bar L > 0$,

From (3.41), (3.19) and (3.43) it follows that

which hence yields the desired assertion.

Aspect 4. We assert that

for some $M > 0$. In fact, from Lemma 2.10 and (3.19), one obtains

where $\sup_{n\geq1}\|x_n-x^*\|^2\leq M$ for some $M > 0$.

Aspect 5. We assert that $x_n\to x^*\in\Omega$, which is only a solution of the HVI (3.16).

In fact, from (3.44), we have

Setting $\Lambda_n = \|x_n-x^*\|^2$, we demonstrate the convergence of $\{\Lambda_n\}$ to zero by the following two situations.

Situation 1. $\exists\; ({\rm integer})\; n_0\geq1$ s.t. $\{\Lambda_n\}$ is nonincreasing. It is clear that the limit $\lim_{n\to\infty}\Lambda_n = k < +\infty$ and $\lim_{n\to\infty}(\Lambda_n-\Lambda_{n+1}) = 0$. From Aspect 2 and $\{\beta_n\}\subset[a, b]\subset(0, 1)$ we obtain

Owing to the facts that $\alpha_n\to0$ and $\Lambda_n-\Lambda_{n+1}\to0$, from $\frac{\tau+\delta}{2}\in(0, 1)$ one deduces that

Hence it is readily known that

and

Next, we show that $\|x_n-u_n\|\to0$ and $\|x_n-x_{n+1}\|\to0$ as $n\to\infty$. Indeed, note that $y^* = T^{\Theta_2}_{\mu_2}(x^*-\mu_2B_2x^*)$, $v_n = T^{\Theta_2}_{\mu_2}(w_n-\mu_2B_2w_n)$ and $u_n = T^{\Theta_1}_{\mu_1}(v_n-\mu_1B_1v_n)$. Then $u_n = {\mathcal G}w_n$. Using Lemma 2.1, from (3.19) we have

This together with (3.41), implies that

which immediately arrives at

This hence ensures that

On the other hand, using Lemma 1.1, from (3.19) we get

This together with (3.41), implies that

which immediately leads to

This hence ensures that

Therefore,

Noticing $w_n = \sigma_nx_n+(1-\sigma_n)u_n$, we get

which immediately yields

This together with (3.41), arrives at

So it follows that

which immediately yields

So it follows from (3.46)–(3.48) that

and

Also, using the similar arguments to those of (3.31) in the proof of Theorem 3.6, we can obtain that

By the boundedness of $\{x_n\}$, we know that $\exists\; {\rm subsequence}\; \{x_{n_p}\}\subset\{x_n\}$ s.t.

Since $H$ is reflexive and $\{x_n\}$ is bounded, we might assume that $x_{n_p}\rightharpoonup\bar x$. Thus, from (3.53) one has

Since $x_n-x_{n+1}\to0, \; x_n-{\mathcal G}w_n\to0, \; w_n-y_n\to0, \; x_n-z_n\to0$ and $x_{n_p}\rightharpoonup\bar x$, by Lemma 3.4 we infer that $\bar x\in\Omega$. Thus, using (3.16) and (3.54) one has

which together with (3.46), arrives at

Note that $\{\alpha_n(1-\beta_n)(\tau-\delta)\}\subset[0, 1], \; \sum^\infty_{n = 1}\alpha_n(1-\beta_n)(\tau-\delta) = \infty$, and

Consequently, applying Lemma 2.5 to (3.45), one has $\lim_{n\to\infty}\|x_n-x^*\|^2 = 0$.

Situation 2. $\exists\{\Lambda_{n_p}\}\subset\{\Lambda_n\}$ s.t. $\Lambda_{n_p} < \Lambda_{n_p+1}\; \forall p\in{\mathcal N}$, with ${\mathcal N}$ being the set of all natural numbers. Let $\phi:{\mathcal N}\to{\mathcal N}$ be formulated as

Using Lemma 2.9, we get

In the remainder of the proof, using the same reasonings as in Situation 2 of Aspect 5 in the proof of Theorem 3.6, we obtain the desired assertion.

Theorem 3.10. If $S:C\to C$ is nonexpansive and $\{x_n\}$ is the sequence constructed in the modified version of Algorithm 3.8, that is, for any initial $x_1\in C$,

where for each $n\geq1$, $C_n$ and $\tau_n$ are picked as in Algorithm 3.8, then $x_n\to x^*\in\Omega$ with $x^*\in\Omega$ being only a solution to the HVI: $\langle(\mu F-f)x^*, y-x^*\rangle\geq0, \; \; \forall y\in\Omega$.

Proof. We divide the proof of the theorem into several aspects.

Aspect 1. We assert the boundedness of $\{x_n\}$. Indeed, using the same reasonings as in Aspect 1 of the proof of Theorem 3.9, one derives the desired assertion.

Aspect 2. We assert that

for some $M_1 > 0$. In fact, putting $\theta_n = 0$, from (3.41) we get

where $\sup_{n\geq1}2\|(f-\mu F)x^*\|\|\bar q_n-x^*\|\leq M_1$ for some $M_1 > 0$. This attains the desired assertion.

Aspect 3. We assert that

Indeed, using the same reasonings as in Aspect 3 of the proof of Theorem 3.9, one deduces the desired assertion.

Aspect 4. We assert that

Indeed, using the same reasonings as in Aspect 4 of the proof of Theorem 3.9, one obtains the desired assertion.

Aspect 5. We assert that $\{x_n\}$ converges strongly to the unique solution $x^*\in\Omega$ of the HVI (3.16). Indeed, using the same reasonings as in Aspect 5 of the proof of Theorem 3.9, one gets the desired assertion.

Remark 3.11. Compared with the corresponding results in Cai, Shehu and Iyiola ^[13], Thong and Hieu ^[17] and Reich et al. ^[29], our results improve and extend them in the following aspects.

(ⅰ) The problem of finding an element of ${\rm Fix}(S)\cap{\rm Fix}({\mathcal G})$ (with ${\mathcal G} = P_C(I-\mu_1B_1) P_C(I-\mu_2B_2)$) in ^[13] is extended to develop our problem of finding an element of ${\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A)$ where ${\mathcal G} = T^{\Theta_1}_{\mu_1}(I-\mu_1B_1)T^{\Theta_2}_{\mu_2} (I-\mu_2B_2)$ and $S$ is asymptotically nonexpansive mapping. The modified viscosity implicit rule for finding an element of ${\rm Fix}(S)\cap{\rm Fix}({\mathcal G})$ in ^[13] is extended to develop our modified Mann-like subgradient-like extragradient implicit rules with linear-search process for finding an element of ${\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A)$, which is on the basis of the subgradient extragradient rule with linear-search process, Mann implicit iteration approach, and hybrid deepest-descent technique.

(ⅱ) The problem of finding an element of ${\rm Fix}(S)\cap{\rm VI}(C, A)$ with quasi-nonexpansive mapping $S$ in ^[17] is extended to develop our problem of finding an element of ${\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A)$ with asymptotically nonexpansive mapping $S$. The inertial subgradient extragradient method with linear-search process for finding an element of ${\rm Fix}(S)\cap{\rm VI}(C, A)$ in ^[17] is extended to develop our modified Mann-like subgradient-like extragradient implicit rules with linear-search process for finding an element of ${\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A)$, which is on the basis of the subgradient extragradient rule with linear-search process, Mann implicit iteration approach, and hybrid deepest-descent technique.

(ⅲ) The problem of finding an element of ${\rm VI}(C, A)$ with pseudomonotone uniform continuity mapping $A$ is extended to develop our problem of finding an element of ${\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A)$ with both asymptotically nonexpansive mapping $S$ and nonexpansive mapping ${\mathcal G}$. The modified projection-type method with linear-search process in ^[29] is extended to develop our modified Mann-like subgradient-like extragradient implicit rule with linear-search process, e.g., the original projection step $y_n = P_C(x_n-\lambda Ax_n)$ in ^[29] is developed into the modified Mann-like implicit projection step $w_n = (1-\sigma_n)x_n+\sigma_n{\mathcal G}w_n$ and $y_n = P_C(w_n-\lambda Aw_n)$; meantime, the original viscosity step $x_{n+1} = \alpha_nf(x_n)+(1-\alpha_n)P_{C_n}(x_n)$ is developed into the composite viscosity iterative step $x_{n+1} = P_C[\alpha_nf(x_n)+(I-\alpha_n\mu F)S^nP_{C_n}(w_n)]$.

4.   Examples

In what follows, we provide an illustrated instance to show the feasibility and implementability of suggested rules. Put $\Theta_1 = \Theta_2 = 0$, $\mu = 2$, $\mu_1 = \mu_2 = \frac{1}{3}, \; \nu = 1, \; \lambda = \ell = \frac{1}{2}, \; \sigma_n = \beta_n = \frac{2}{3}$ and $\alpha_n = \frac{1}{3(n+1)}$. We first provide an example of two inverse-strongly monotone mappings $B_1, B_2:C\to H$, Lipschitz continuous and pseudomonotone mapping $A$ and asymptotically nonexpansive mapping $S$ with $\Omega = {\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A)\neq\emptyset$, where ${\mathcal G}: = T^{\Theta_1}_{\mu_1}(I-\mu_1B_1) T^{\Theta_2}_{\mu_2}(I-\mu_2B_2) = P_C(I-\mu_1B_1)P_C(I-\mu_2B_2)$. We set $H = {\bf R}$ and use the $\langle a, b\rangle = ab$ and $\|\cdot\| = |\cdot|$ to denote its inner product and induced norm, respectively. Moreover, we put $C = [-2, 3]$. The starting point $x_1$ is arbitrarily chosen in $C$. Let $f(x) = F(x) = \frac{1}{2}x\; \forall x\in C$ with

We let $B_1x = B_2x: = Bx = x-\frac{1}{2}\sin x, \; \forall x\in C$. Let $A:H\to H$ and $S:C\to C$ be formulated as $Ax: = \frac{1}{1+|\sin x|}-\frac{1}{1+|x|}$ and $Sx: = \frac{5}{6}\sin x$. We now assert that $B$ 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 $\alpha = \beta = \frac{2}{9}$. Let us assert that $A$ is pseudomonotone and Lipschitz continuous. In fact, for each $a, b\in H$ one has

This means that $A$ is Lipschitz continuous with $L = 2$. Next, we assert that $A$ is pseudomonotone. For each $a, b\in H$, it is readily known that

Moreover, it is easy to check that $S$ is asymptotically nonexpansive with $\theta_n = (\frac{5}{6})^n, \; \forall n\geq1$, 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, $\Omega = {\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A) = \{0\}\neq\emptyset$. In this case, noticing

we 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 know that $\{x_n\}$ converges to $0\in\Omega = {\rm Fix}(S)\cap\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A)$.

In particular, since $Sx: = \frac{5}{6}\sin x$ is also nonexpansive, we consider the modified version of Algorithm 3.1, that is,

where for each $n\geq1$, $C_n$ and $\tau_n$ are chosen as above. Then, by Theorem 3.7, we know that $\{x_n\}$ converges to $0\in\Omega = {\rm Fix}(S)\cap{\rm Fix}({\mathcal G})\cap{\rm VI}(C, A)$.

5.   Conclusions

In this paper, we introduce the modified Mann-like subgradient-like extragradient implicit rules with linear-search process for finding a common solution of the SGEP, VIP and FPP. The proposed algorithms are on the basis of the subgradient extragradient rule with linear-search process, Mann implicit iteration approach, and hybrid deepest-descent technique. Under mild restrictions, we demonstrate the strong convergence of the suggested algorithms to a common solution of the SGEP, VIP and FPP, which is a unique solution of a certain HVI defined on their common solution set. In addition, an illustrated example is provided to show the feasibility and implementability of our proposed rule.

Acknowledgments

This research was 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). Li-Jun Zhu was supported by the National Natural Science Foundation of China [grant number 11861003], the Natural Science Foundation of Ningxia province [grant number NZ17015], the Major Research Projects of NingXia [grant numbers 2021BEG03049]and Major Scientific and Technological Innovation Projects of YinChuan [grant numbers 2022RKX03 and NXYLXK2017B09].

Conflict of interest

The authors declare no conflicts of interest.