Modified Inertial-Type Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Finite Bregman Relatively Nonexpansive and Demicontractive Mappings

Abstract

In this paper, we design two inertial-type subgradient extragradient algorithms with the linear-search process for resolving the two pseudomonotone variational inequality problems (VIPs) of and the common fixed point problem (CFPP) of finite Bregman relatively nonexpansive operators and Bregman relatively demicontractive operators in Banach spaces of both p-uniform convexity and uniform smoothness, which are more general than Hilbert ones. By the aid of suitable restrictions, it is shown that the sequences fabricated by the suggested schemes converge weakly and strongly to a solution of a pair of VIPs with a CFPP constraint, respectively. Additionally, the illustrative instance is furnished to back up the practicability and implementability of the suggested methods. This paper reveals the competitive advantage of the proposed algorithms over the existing algorithms; that is, the existing hybrid projection method for a single VIP with an FPP constraint is extended to develop the modified inertial-type subgradient extragradient method for a pair of VIPs with an CFPP constraint.

1. Introduction

Suppose that the real Hilbert space H has both inner product $\langle ·,·\rangle$ and induced norm $\parallel ·\parallel$, and let $P_C$ be the metric projection of H onto a nonempty, convex and closed $C\subset H$ given a nonlinear operator $S:C\to C$. We denote by $\mathrm{Fix}\left(S\right)$ the fixed-point set of S. Also, the $\mathbf{R},\rightharpoonup$ and → are used to represent the real-number set, the weak convergence and the strong convergence, respectively. A mapping $S:C\to C$ is said to be strictly pseudocontractive (see [1]) if $\exists \xi \in [0,1)$ s.t. ${\parallel Sx-Sy\parallel }^2\le {\parallel x-y\parallel }^2+\xi \parallel$${(I-S)x-(I-S)y\parallel }^2\forall x,y\in C$. In particular, in case $\xi =0$, S reduces to a nonexpansive mapping. Moreover, S is said to be demicontractive if $\mathrm{Fix}\left(S\right)\ne \varnothing$ and $\exists \xi \in [0,1)$ s.t. ${\parallel Sx-y\parallel }^2\le {\parallel x-y\parallel }^2+\xi {\parallel x-Sx\parallel }^2\forall x\in C,y\in \mathrm{Fix}\left(S\right)$. In particular, in case $\xi =0$, S reduces to a quasi-nonexpansive mapping. During the past few decades, the fixed point theory has played a vital part in solving many problems arising in nonlinear analysis and optimization theory, such as differential hemivariational inequalities (see [2]), monotone bilevel equilibrium problems (see [3]), fractional set-valued projected dynamical systems (see [4]) and so on.

Let $A:H\to H$ be a mapping. Consider the classical variational inequality problem (VIP) of finding $u\in C$ s.t. $\langle Au,v-u\rangle \ge 0\forall v\in C$. The solution set of the VIP is denoted by $\mathrm{VI}(C,A)$. In 1976, to seek a point in $\mathrm{VI}(C,A)$, via relative weak conditions, Korpelevich [5] put forward an extragradient approach below; i.e., for any initial $x_0\in C$, the sequence $\left\{x_n\right\}$ is generated by

$$\left\{\begin{array}{cc}& y_n=P_C(x_n-\tau A{x}_n),\hfill \\ & x_{n+1}=P_C(x_n-\tau A{y}_n)\forall n\ge 0,\hfill \end{array}\right.$$

with $\tau \in (0,\frac{1}{L})$. If $\mathrm{VI}(C,A)\ne \varnothing$, then the sequence $\left\{x_n\right\}$ converges weakly to an element in $\mathrm{VI}(C,A)$. To the best of our knowledge, the Korpelevich extragradient approach is one of the most effective methods for solving the VIP at present. The literature on the VIP is vast and the Korpelevich extragradient approach has attained wide attention paid by many scholars, who ameliorated it in various forms; see, e.g., [1,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26].

Furthermore, in 2018, Thong and Hieu [18] first put forward the inertial subgradient extragradient method; that is, for any initial $x_0,x_1\in H$, the sequence $\left\{x_n\right\}$ is generated by

$$\left\{\begin{array}{cc}& g_n=x_n+{\alpha }_n(x_n-x_{n-1}),\hfill \\ & y_n=P_C(g_n-\ell A{g}_n),\hfill \\ & C_n=\{v\in H:\langle g_n-\ell A{g}_n-y_n,v-y_n\rangle \le 0\},\hfill \\ & x_{n+1}=P_{C_n}(g_n-\ell A{y}_n)\forall n\ge 1,\hfill \end{array}\right.$$

with constant $\ell \in (0,\frac{1}{L})$. Under suitable conditions, they proved the weak convergence of $\left\{x_n\right\}$ to an element of $\mathrm{VI}(C,A)$. Subsequently, Ceng et al. [14] proposed the modification of an inertial subgradient extragradient approach for settling the VIP of pseudomonotonicity and common fixed-point problem (CFPP) of finite nonexpansive mappings. Let $S_i:H\to H$ be nonexpansive for $i=1,\dots ,N$, $A:H\to H$ be of both L-Lipschitz continuity and pseudomonotonicity on H, and of sequentially weak continuity on C, s.t. $\Omega ={\bigcap }_{j=1}^N\mathrm{Fix}\left(S_j\right)\cap \mathrm{VI}(C,A)\ne \varnothing$. Let f be $\delta$-contractive self-mapping on H for $\delta \in [0,1)$ and the self-mapping F on H be of both $\eta$-strong monotone and $\kappa$-Lipschitz continuity s.t. $\tau :=1-\sqrt{1-\rho (2\eta -\rho {\kappa }^2)}>\delta$ with $\rho \in (0,2\eta /{\kappa }^2)$. Presume that $\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\},\left\{{\tau }_n\right\}$ are positive sequences s.t. ${\beta }_n+{\gamma }_n<1,{\sum }_{n=1}^{\infty }{\beta }_n=\infty ,{lim}_{n\to \infty }{\beta }_n=0,{lim\; inf}_{n\to \infty }{\gamma }_n(1-{\gamma }_n)>0$ and ${\tau }_n=o\left({\beta }_n\right)$. Moreover, one writes $S_n:=S_{n\mathrm{mod}N}$ for integer $n\ge 1$ with the mod function taking values in the set $\{1,\dots ,N\}$; i.e., if $n=jN+m$ for some integers $j\ge 0$ and $0\le m<N$, then $S_n=S_N$ if $m=0$ and $S_n=S_m$ if $0<m<N$.

Under appropriate conditions, they proved the strong convergence of $\left\{x_n\right\}$ to an element of $\Omega ={\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\cap \mathrm{VI}(C,A)$. In addition, combining the subgradient extragradient method and Halpern’s iteration method, Kraikaew and Saejung [19] proposed the Halpern subgradient extragradient rule for solving the VIP in 2014. They proved the strong convergence of the proposed method to an element in $\mathrm{VI}(C,A)$. In 2021, Reich et al. [23] invented two gradient-projection schemes for settling the VIP for uniformly continuous pseudomonotone mapping. In particular, they used a novel Armijo-type line search to acquire a hyperplane that strictly separates the current iterate from the solutions of the VIP under consideration. They proved that the sequences generated by two schemes converge weakly and strongly to a point in $\mathrm{VI}(C,A)$ for uniformly continuous pseudomonotone mapping A, respectively.

On the other hand, let $1/p+1/q=1$ for $p,q>1$, and suppose that E is a Banach space of both p-uniform convexity and uniform smoothness and the nonempty $C\subset E$ is of both convexity and closedness. The dual space of E is denoted by $E^{*}$. The norm and the duality pairing between E and $E^{*}$ are denoted by $\parallel ·\parallel$ and $\langle ·,·\rangle$, respectively. Let $J_E^p$ and $J_{E^{*}}^q$ be the duality mappings of E and $E^{*}$, respectively. Let $f_p\left(u\right)={\parallel u\parallel }^p/p\forall u\in E$, $D_{f_p}$ be the Bregman distance with respect to (w.r.t) $f_p$ and the surjective ${\Pi }_C:E\to C$ be the Bregman’s projection w.r.t. $f_p$, and presume that $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ s.t. ${lim}_{n\to \infty }{\alpha }_n=0$, ${lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)>0$ and ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$. Assume that $A:E\to E^{*}$ is uniformly continuous and pseudo-monotone mapping and $S:C\to C$ is Bregman relatively nonexpansive mapping. Very recently, inspired by the research outcomes in [23], Eskandani et al. [25] invented the hybrid projection method with linear search term in order to seek a solution of a VIP with an FPP constraint of S.

By the aid of mild restrictions, it was proven in [23] that the sequence $\left\{u_n\right\}$ converges strongly to ${\Pi }_{\Omega }u$. Motivated by the existing outcomes as above, we design two inertial-type subgradient extragradient algorithms with a linear-search process for resolving the two pseudomonotone VIPs and the CFPP of finite Bregman’s relative nonexpansivity operators and a Bregman’s relative demicontractivity operator in Banach spaces of both p-uniform convexity and uniform smoothness. With the help of appropriate assumptions, it is proven that the sequences fabricated by the suggested algorithms converge weakly and strongly to a solution of a pair of VIPs with a CFPP constraint, respectively. Additionally, an illustrative instance is furnished to back up the practicability and implementability of the proposed approaches.

The structure for the paper is built as follows: Section 2 releases certain terminologies and preliminary results. In Section 3, we discuss the convergent behavior of the sequences generated by the proposed approaches. In Section 4, the major outcomes are employed to deal with a pair of VIPs with a CFPP constraint in an illustrative instance. Our algorithms are of both advantage and flexibility over Algorithms 1 and 2 as above due to their solving a pair of VIPs with a CFPP constraint. Our outcomes are the improvement and extension of the existing ones in the literature; see, e.g., [14,23,25].

Algorithm 1: ([14], Algorithm 3)

Inertial subgradient extragradient method. Initialization: Given ${\lambda }_1>0,\alpha >0,\mu \in (0,1)$. Let $x_0,x_1\in H$ be arbitrary. Iterative steps: Calculate $x_{n+1}$ as follows: Step 1. Given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${\alpha }_n$ s.t. $0\le {\alpha }_n\le \overline{{\alpha }_n}$, where $$\overline{{\alpha }_n}=\left\{\begin{array}{cc}& min\{\alpha ,\frac{{\tau }_n}{\parallel x_n-x_{n-1}\parallel }\}\mathrm{if}{x}_n\ne x_{n-1},\hfill \\ & \alpha \mathrm{otherwise}.\hfill \end{array}\right.$$ Step 2. Compute $w_n=S_n{x}_n+{\alpha }_n(S_n{x}_n-S_n{x}_{n-1})$ and $y_n=P_C(w_n-{\lambda }_{n}A{w}_n)$. Step 3. Construct the half-space $C_n:=\{z\in H:\langle w_n-{\lambda }_{n}A{w}_n-y_n,z-y_n\rangle \le 0\}$, and compute $z_n=P_{C_n}(w_n-{\lambda }_{n}A{y}_n)$. Step 4. Calculate $x_{n+1}={\beta }_{n}f\left(x_n\right)+{\gamma }_n{x}_n+((1-{\gamma }_n)I-{\beta }_n\rho F)z_n$, and update $${\lambda }_{n+1}=\left\{\begin{array}{cc}& min\{\mu \frac{\parallel w_n-y_n{\parallel }^2+{\parallel z_n-y_n\parallel }^2}{2\langle A{w}_n-A{y}_n,z_n-y_n\rangle },{\lambda }_n\}\langle A{w}_n-A{y}_n,z_n-y_n\rangle >0,\hfill \\ & {\lambda }_n\mathrm{otherwise}.\hfill \end{array}\right.$$ Set $n:=n+1$ and go to Step 1.

Algorithm 2: ([25])

Hybrid projection method. Initial step: Let positive $l<1,\nu >0,\lambda \in (0,\frac{1}{\nu })$, and put $u,u_1\in C$ arbitrarily. Iterations: Compute $u_{n+1}$ below: Step 1. Calculate $v_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{u}_n-\lambda A{u}_n)\right)$ and $r_{\lambda }\left(u_n\right):=u_n-v_n$. If $r_{\lambda }\left(u_n\right)=0$ and $S{u}_n=u_n$, then stop; $u_n\in \Omega =\mathrm{Fix}\left(S\right)\cap \mathrm{VI}(C,A)$. Otherwise, Step 2. Compute $t_n=u_n-{\tau }_n{r}_{\lambda }\left(u_n\right)$, where ${\tau }_n:=l^{k_n}$ and $k_n$ is the smallest nonnegative integer k satisfying $\langle A{u}_n-A(u_n-l^k{r}_{\lambda }\left(u_n\right)),r_{\lambda }\left(u_n\right)\rangle \le \frac{\nu }{2}{D}_{f_p}(u_n,v_n)$. Step 3. Compute $w_n=J_{E^{*}}^q({\beta }_n{J}_E^p{u}_n+(1-{\beta }_n)J_E^p\left(S{\Pi }_{C_n}{u}_n\right))$ and $u_{n+1}={\Pi }_C\left(J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{w}_n)\right)$, with $C_n:=\{v\in C:{\overline{h}}_n\left(v\right)\le 0\}$ and ${\overline{h}}_n\left(v\right)=\langle A{t}_n,v-u_n\rangle +\frac{{\tau }_n}{2\lambda }{D}_{f_p}(u_n,v_n)$. Again set $n:=n+1$ and go to Step 1.

In the end, it is worthy to mention that the existing method in [25] is most closely related to our proposed method; that is, the hybrid projection method for resolving a single VIP with an FPP constraint in [25] is extended to develop our modified inertial-type subgradient extragradient method for resolving a pair of VIPs with a CFPP constraint. Compared with the corresponding results in [25], our results improve, extend and develop them in the two aspects below: (i) the problem of finding a solution of a single VIP with an FPP constraint (involving a Bregman relatively nonexpansive mapping) in [25] is extended to develop our problem of finding a solution of a pair of VIPs with a CFPP constraint (involving finite Bregman relatively nonexpansive mappings and a Bregman relatively demicontractive mapping); (ii) the hybrid projection method with line-search process in [25] is extended to develop our modified inertial-type subgradient extragradient method with the line-search process.

Algorithm 3: The 1st modified inertial-type subgradient extragradient method

The 1st modified inertial-type subgradient extragradient method. Initialization: Given $x_0,x_1\in C$ arbitrarily and let $ϵ>0,{\mu }_i>0,{\lambda }_i\in (0,\frac{1}{{\mu }_i}),l_i\in (0,1)$ for $i=1,2$. Choose $\left\{{\ell }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ and $\left\{{\alpha }_n\right\}\subset (\xi ,1)$ s.t. ${\sum }_{n=1}^{\infty }{\ell }_n<\infty$, ${lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)>0$ and ${lim\; inf}_{n\to \infty }({\alpha }_n-\xi )(1-{\alpha }_n)>0$. Moreover, given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ s.t. $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where $$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{\ell }_n}{\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel }\}\mathrm{if}{S}_n{x}_n\ne S_n{x}_{n-1},\hfill \\ & ϵ\hspace{0.17em}\mathrm{otherwise}.\hfill \end{array}\right.$$ Iterative steps: Calculate $x_{n+1}$ as follows: Step 1. Calculate $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_n{x}_n+{ϵ}_n{J}_E^p(2S_n{x}_n-S_n{x}_{n-1}))$ and calculate $u_n=J_{E^{*}}^q({\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)$, $y_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{u}_n-{\lambda }_1{A}_1{u}_n)\right)$, $r_{{\lambda }_1}\left(u_n\right):=u_n-y_n$ and $s_n=u_n-{\tau }_n{r}_{{\lambda }_1}\left(u_n\right)$, with ${\tau }_n:=l_1^{k_n}$ and $k_n$ being the smallest $k\ge 0$ s.t. $$\langle A_1{u}_n-A_1(u_n-l_1^k{r}_{{\lambda }_1}\left(u_n\right)),u_n-y_n\rangle \le \frac{{\mu }_1}{2}{D}_{f_p}(u_n,y_n).$$ (1) Step 2. Calculate $w_n={\Pi }_{C_n}\left(u_n\right)$, with $C_n:=\{y\in C:h_n\left(y\right)\le 0\}$ and $$h_n\left(y\right)=\langle A_1{s}_n,y-u_n\rangle +\frac{{\tau }_n}{2{\lambda }_1}{D}_{f_p}(u_n,y_n).$$ (2) Step 3. Calculate ${\tilde{y}}_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{w}_n-{\lambda }_2{A}_2{w}_n)\right)$, $r_{{\lambda }_2}\left(w_n\right):=w_n-{\tilde{y}}_n$ and $t_n=w_n-{\tilde{\tau }}_n{r}_{{\lambda }_2}\left(w_n\right)$, with ${\tilde{\tau }}_n:=l_2^{j_n}$ and $j_n$ is the smallest $j\ge 0$ s.t. $$\langle A_2{w}_n-A_2(w_n-l_2^j{r}_{{\lambda }_2}\left(w_n\right)),w_n-{\tilde{y}}_n\rangle \le \frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\tilde{y}}_n).$$ (3) Step 4. Calculate $v_n=J_{E^{*}}^q({\alpha }_n{J}_E^p{w}_n+(1-{\alpha }_n)J_E^p\left(S_0{w}_n\right))$ and $x_{n+1}={\Pi }_{{\tilde{C}}_n\cap Q_n}\left(w_n\right)$, with $Q_n:=\{y\in C:D_{f_p}(y,v_n)\le D_{f_p}(y,w_n)\}$, ${\tilde{C}}_n:=\{y\in C:{\tilde{h}}_n\left(y\right)\le 0\}$ and $${\tilde{h}}_n\left(y\right)=\langle A_2{t}_n,y-w_n\rangle +\frac{{\tilde{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n).$$ (4) Again set $n:=n+1$ and go to Step 1.

In Section 4, we have provided a numerical example to show the competitive advantage of our proposed algorithms over the existing algorithms, e.g., the ones in [25]. In fact, we have provided the illustrative example of a pair of VIPs with a CFPP constraint in Section 4. Note that the existing method in [25] is only utilized for solving a single VIP with an FPP constraint. So, there is no way for this method to handle the numerical example in Section 4; that is, it is invalid for a pair of VIPs with a CFPP constraint. However, our suggested method can settle the illustrative example in Section 4. This ensures the competitive advantage of our proposed algorithms over the existing algorithms.

2. Preliminaries

Let the real Banach space ($E,\parallel ·\parallel$) possess the dual space $E^{*}$. The $g_n\to g$ (resp., $g_n\rightharpoonup g$) is used to stand for the strong (resp., weak) convergence of $\left\{g_n\right\}$ to $g\in E$. Moreover, the set of weak cluster points of $\left\{g_n\right\}$ is denoted by ${\omega }_w\left(g_n\right)$, i.e., ${\omega }_w\left(g_n\right)=\{g^{†}\in E:g_{n_k}\rightharpoonup g^{†}\mathrm{for}\mathrm{some}\left\{g_{n_k}\right\}\subset \left\{g_n\right\}\}$. Let $\left\{\mathcal{U}\right\}=\{g\in E:\parallel g\parallel =1\}$ and $1<q\le 2\le p$ with $\frac{1}{p}+\frac{1}{q}=1$. A Banach space E is referred to as being strictly convex if, for each $y,g\in U$ with $y\ne g$, one has $\parallel y+g\parallel /2<1$. E is referred to as being of uniform convexity if $\forall ϵ\in (0,2]$, $\exists \delta >0$ s.t. $\forall y,g\in \left\{\mathcal{U}\right\}$ with $\parallel y-g\parallel \ge ϵ$, one has $\parallel y+g\parallel /2\le 1-\delta$. It is known that a uniformly convex Banach space is reflexive and strictly convex. The modulus of convexity of E is the function $\delta :[0,2]\to [0,1]$ defined by $\delta \left(ϵ\right)=inf\{1-\parallel y+g\parallel /2:y,g\in \{\mathcal{U}\}\mathrm{with}\parallel y-g\parallel \ge ϵ\}$. It is also known that E is uniformly convex if and only if $\delta \left(ϵ\right)>0\forall ϵ\in (0,2]$. Moreover, E is referred to as being p-uniformly convex if $\exists c>0$ s.t. $\delta \left(ϵ\right)\ge c{ϵ}^p\forall ϵ\in [0,2]$.

A function ${\rho }_E:[0,\infty )\to [0,\infty )$ is the modulus of smoothness iff it is written as ${\rho }_E\left(\tau \right)=sup\left\{\right(\parallel y+\tau g\parallel +\parallel y-\tau g\parallel )/2-1:y,g\in \left\{\mathcal{U}\right\}\}$. E is said to be uniformly smooth if ${lim}_{\tau \to 0}{\rho }_E\left(\tau \right)/\tau =0$, and q-uniformly smooth if $\exists C_q>0$ s.t. ${\rho }_E\left(\tau \right)\le C_q{\tau }^q\forall \tau >0$. It is known that E is p-uniformly convex if and only if $E^{*}$ is q-uniformly smooth. For example, see [27] for more details. Putting $B(0,r)=\{g\in E:\parallel g\parallel \le r\}$ for each $r>0$, we say that $f:E\to \mathbf{R}$ is uniformly convex on bounded sets (see [25]) if ${\rho }_r\left(t\right)>0\forall r,t>0$, where ${\rho }_r\left(t\right):[0,\infty )\to [0,\infty ]$ is specified below

$$\begin{array}{cc}{\rho }_r\left(t\right)\hfill & =inf\left\{\right[\alpha f\left(y\right)+(1-\alpha )f\left(g\right)-f(\alpha y+(1-\alpha \left)g\right)]/\alpha (1-\alpha ):\hfill \\ & \alpha \in (0,1)\mathrm{and}y,g\in B(0,r)\mathrm{with}\parallel y-g\parallel =t\},\hfill \end{array}$$

for all $t\ge 0$. ${\rho }_r$ is called the gauge function of f with uniform convexity. It is obvious that the function ${\rho }_r$ is nondecreasing.

Let $f:E\to \mathbf{R}$ be a convex function. If the limit ${lim}_{t\to 0^{+}}\frac{f(y+tg)-f\left(y\right)}{t}$ exists for each $g\in E$, then f is referred to as being of Gâteaux differentiability at y. In this case, the gradient of f at y is the linear function $\nabla f\left(y\right)$, which is defined by $\langle \nabla f\left(y\right),g\rangle :={lim}_{t\to 0^{+}}\frac{f(y+tg)-f\left(y\right)}{t}$ for each $g\in E$. The function f is referred to as being of Gâteaux’s differentiability if it is of Gâteaux’s differentiability at each $y\in E$. Whenever ${lim}_{t\to 0^{+}}\frac{f(y+tg)-f\left(y\right)}{t}$ is achieved uniformly for any $g\in \{\mathcal{U}$, one says that f is of Fréchet’s differentiability at y. Furthermore, f is termed as being of uniform Fréchet differentiability on $K\subset E$ if ${lim}_{t\to 0^{+}}\frac{f(y+tg)-f\left(y\right)}{t}$ is achieved uniformly for $(y,g)\in K\times \mathcal{U}$. A Banach space E is called smooth if its norm is Gâteaux differentiable.

Let $1/p+1/q=1$ for $p,q>1$. The duality mapping $J_E^p:E\to E^{*}$ is specified as follows

$$J_E^p\left(y\right)=\{\varphi \in E^{*}:\langle \varphi ,y\rangle ={\parallel y\parallel }^p\mathrm{and}\parallel \varphi \parallel ={\parallel y\parallel }^{p-1}\}\forall y\in E.$$

Recall that E is of smoothness iff the duality mapping $J_E^p$ is single-valued. Also, E is of reflexivity iff $J_E^p$ is of surjectivity, and E is of strict convexity iff $J_E^p$ is an injection. So, it follows that, if E is smooth, strictly convex and reflexive Banach space, then $J_E^p$ is a single-valued bijection, and, in this case, $J_E^p={\left(J_{E^{*}}^q\right)}^{-1}$. Furthermore, E is of uniform smoothness if and only if function $f_p\left(y\right)={\parallel y\parallel }^p/p$ is of uniform Fréchet differentiability on any bounded set if and only if the single-valued $J_E^p$ is of uniform continuity on any bounded set. It is easy to see that E is of uniform convexity if and only if function $f_p$ is of uniform convexity (see [27]).

Let the convex function $f:E\to \mathbf{R}$ be of Gâteaux’s differentiability. Bregman’s distance w.r.t. f is specified below

$$D_f(y,g):=f\left(y\right)-f\left(g\right)-\langle \nabla f\left(g\right),y-g\rangle \forall y,g\in E.$$

It is worth mentioning that Bregman’s distance does not become a metric in the common sense of the terminology. Evidently, $D_f(y,y)=0$, but $D_f(y,g)=0$ cannot yield $y=g$. Generally, $D_f$ is of no symmetry and does not fulfill the triangle inequality. But, $D_f$ fulfills the three point identity

$$D_f(y,g)+D_f(g,w)=D_f(y,w)-\langle \nabla f\left(g\right)-\nabla f\left(w\right),y-g\rangle .$$

See [28] for more details on Bregman functions and distances.

It is noteworthy that, if E is a smooth Banach space, then $J_E^p$ is Gâteaux’s derivative of $f_p$. Then, Bregman’s distance w.r.t. $f_p$ is specified below

$$\begin{array}{cc}{D}_{f_p}(y,g)\hfill & ={\parallel y\parallel }^p/p-{\parallel g\parallel }^p/p-\langle J_E^p\left(g\right),y-g\rangle \hfill \\ & ={\parallel y\parallel }^p/p+{\parallel g\parallel }^p/q-\langle J_E^p\left(g\right),y\rangle \hfill \\ & ={(\parallel g\parallel }^p-{\parallel y\parallel }^p)/q-\langle J_E^p\left(g\right)-J_E^p\left(y\right),y\rangle .\hfill \end{array}$$

In the Banach space E of both smoothness and p-uniformly convexity for $p\ge 2$, there holds the following relationship between the metric and Bregman distance:

$${\tau \parallel y-g\parallel }^p\le D_{f_p}(y,g)\le \langle J_E^p\left(y\right)-J_E^p\left(g\right),y-g\rangle ,$$

(5)

where $\tau >0$ is some fixed number (see [29]). Using (5), one knows that, for any bounded $\left\{g_n\right\}\subset E$,

$$g_n\to g\iff D_{f_p}(g,g_n)\to 0(n\to \infty ).$$

Let the Banach space E be of reflexivity, smoothness and strict convexity. Let $\varnothing \ne C\subset E$ with C be of convexity and closedness. Bregman’s projections are formulated as minimizers of Bregman’s distances. The Bregman’s projection of $y\in E$ onto C w.r.t. $f_p$ is only a point ${\Pi }_{C}y\in C$ s.t. $D_{f_p}({\Pi }_{C}y,y)={min}_{g\in C}{D}_{f_p}(g,y)$. In Hilbert spaces, the Bregman projection w.r.t. $f_2$ reduces to the metric projection.

Using Corollary 4.4 in [30] and Theorem 2.1 [31] in Banach spaces of uniform convexity, Bregman projections can be featured as the relation below:

$$\langle J_E^p\left(y\right)-J_E^p\left({\Pi }_{C}y\right),g-{\Pi }_{C}y\rangle \le 0\forall g\in C.$$

(6)

Moreover, this inequality is equivalent to the descent property

$$D_{f_p}(g,{\Pi }_{C}y)+D_{f_p}({\Pi }_{C}y,y)\le D_{f_p}(g,y)\forall g\in C.$$

(7)

In case $p=2$, the duality mapping $J_E^p$ reduces to the normalized duality mapping and is denoted by J. The function $\varphi :E^2\to \mathbf{R}$ is formulated below

$$\varphi (y,g)={\parallel y\parallel }^2-2\langle Jg,y\rangle +{\parallel g\parallel }^2\forall y,g\in E,$$

and ${\Pi }_C\left(y\right)={\mathrm{argmin}}_{g\in C}\varphi (g,y)\forall y\in E$.

In terms of [25], the $V_{f_p}:E\times E^{*}\to [0,\infty )$ w.r.t. $f_p$ is specified below

$$V_{f_p}(y,y^{*})={\parallel y\parallel }^p/p-\langle y^{*},y\rangle +{\parallel y^{*}\parallel }^q/q\forall (y,y^{*})\in E\times E^{*}.$$

(8)

So, $V_{f_p}(y,y^{*})=D_{f_p}(y,J_{E^{*}}^q\left(y^{*}\right))\forall (y,y^{*})\in E\times E^{*}$. Moreover, by the subdifferential inequality, we obtain

$$V_{f_p}(y,y^{*})+\langle g^{*},J_{E^{*}}^q\left(y^{*}\right)-y\rangle \le V_{f_p}(y,y^{*}+g^{*})\forall y\in E,y^{*},g^{*}\in E^{*}.$$

(9)

In addition, $V_{f_p}$ is convex in the second variable. Thus, one has

$$D_{f_p}(y,J_{E^{*}}^q(\sum _{j=1}^n{t}_j{J}_E^p\left(g_j\right)))\le \sum _{j=1}^n{t}_j{D}_{f_p}(y,g_j)\forall y\in E,{\left\{g_j\right\}}_{j=1}^n\subset E,\sum _{j=1}^n{t}_j=1\mathrm{for}{\left\{t_j\right\}}_{j=1}^n\subset [0,1].$$

(10)

Lemma 1

([31]). Suppose that the Banach space E is of uniform convexity and $\left\{y_n\right\},\left\{g_n\right\}$ are two sequences in E such that the first one is bounded. If ${lim}_{n\to \infty }{D}_{f_p}(g_n,y_n)=0$, then ${lim}_{n\to \infty }\parallel g_n-y_n\parallel =0$.

Let $S:C\to C$ be a mapping. We denote by $\mathrm{Fix}\left(S\right)$ the set of fixed points of S; that is, $\mathrm{Fix}\left(S\right)=\{y\in C:y=Sy\}$. A point $g\in C$ is referred to as an asymptotic fixed point of S if $\exists \left\{g_n\right\}\subset C$ s.t. $g_n\rightharpoonup g$ and $g_n-S{g}_n\to 0$. We denote by $\widehat{\mathrm{Fix}}\left(S\right)$ the set of asymptotic fixed points of S. The notion of asymptotic fixed points was invented in Reich [32]. A self-mapping S on C is termed as Bregman’s relatively $\xi$-demicontractive operator w.r.t. $f_p$ iff $\mathrm{Fix}\left(S\right)=\widehat{\mathrm{Fix}}\left(S\right)\ne \varnothing$, and $\exists \xi \in [0,1)$ s.t. for each bounded $\left\{g_n\right\}\subset C$ satisfying ${sup}_{n\ge 1}\parallel S{g}_n\parallel <\infty$; the following holds:

$$D_{f_p}(y,S{g}_n)\le D_{f_p}(y,g_n)+\xi {\rho }_b^{*}\parallel J_E^p{g}_n-J_E^{p}S{g}_n\parallel \forall y\in \mathrm{Fix}\left(S\right),$$

with $b={sup}_{n\ge 1}\{\parallel g_n{\parallel }^{p-1},\parallel S{g}_n{\parallel }^{p-1}\}<\infty$. In particular, putting $b_x={max\{\parallel x\parallel }^{p-1}{,\parallel Sx\parallel }^{p-1}\}$ for each $x\in C$, one has

$$D_{f_p}(y,Sx)\le D_{f_p}(y,x)+\xi {\rho }_{b_x}^{*}\parallel J_E^{p}x-J_E^{p}Sx\parallel \forall y\in \mathrm{Fix}\left(S\right).$$

In addition, if $\xi =0$, then S reduces to a mapping of Bregman’s relative nonexpansivity w.r.t. $f_p$; that is, S is named as a mapping of Bregman’s relative nonexpansivity w.r.t. $f_p$ if $\mathrm{Fix}\left(S\right)=\widehat{\mathrm{Fix}}\left(S\right)\ne \varnothing$ and $D_{f_p}(y,Sg)\le D_{f_p}(y,g)\forall g\in C,y\in \mathrm{Fix}\left(S\right)$.

Definition 1.

Let C be a nonempty closed convex subset of E. A mapping $A:C\to E^{*}$ is known as being

(i) 

of monotonicity iff $\langle Ay-Ag,y-g\rangle \ge 0\forall y,g\in C$;

(ii) 

of pseudo-monotonicity iff $\langle Ay,g-y\rangle \ge 0\Rightarrow \langle Ag,g-y\rangle \ge 0\forall y,g\in C$;

(iii) 

ℓ-Lipschitz continuous or ℓ-Lipschitzian iff $\exists \ell >0$ s.t. $\parallel Ay-Ag\parallel \le L\parallel y-g\parallel \forall y,g\in C$;

(iv) 

of weakly sequential continuity iff $\forall \left\{g_n\right\}\subset C$; the relation holds: $g_n\rightharpoonup g\Rightarrow A{g}_n\rightharpoonup Ag$.

Lemma 2

([25]). Given a constant $r>0$. If the function $f:E\to \mathbf{R}$ is of uniform convexity on any bounded subset of a Banach space E, then

$$f(\sum _{k=1}^n{\alpha }_k{g}_k)\le \sum _{k=1}^n{\alpha }_{k}f\left(g_k\right)-{\alpha }_i{\alpha }_j{\rho }_r(\parallel g_i-g_j\parallel ),$$

$\forall i,j\in \{1,2,\dots ,n\},{\left\{g_k\right\}}_{k=1}^n\subset B(0,r)$ and ${\left\{{\alpha }_k\right\}}_{k=1}^n\subset (0,1)$ with ${\sum }_{k=1}^n{\alpha }_k=1$, where ${\rho }_r$ is the gauge of uniform convexity of f.

Proof. 

It is easy to show the conclusion.    □

Lemma 3

([24]). Let $E_1$ and $E_2$ be two Banach spaces. Suppose that $A:E_1\to E_2$ is uniformly continuous on bounded subsets of $E_1$ and D is a bounded subset of $E_1$. Then, $A\left(D\right)$ is bounded.

Lemma 4

([33]). Let $\varnothing \ne C\subset E$ with C being closed and convex in a Banach space E and let $A:C\to E^{*}$ be of both pseudo-monotonicity and continuity. Given $x^{†}\in C$. Then, $\langle A{x}^{†},y-x^{†}\rangle \ge 0\forall y\in C\iff \langle Ay,y-x^{†}\rangle \ge 0\forall y\in C$.

Lemma 5.

Given $p\ge 2$. Suppose that the Banach space E is of both smoothness and p-uniform convexity s.t. the duality mapping $J_E^p$ is of sequentially weak continuity. Let $\left\{q_n\right\}\subset E$ and $\varnothing \ne \Omega \subset E$. If ${lim}_{n\to \infty }\{D_{f_p}(g,q_n)\}$ exists for each $g\in \Omega$, and ${\omega }_w\left(q_n\right)\subset \Omega$. Then, $\left\{q_n\right\}$ is weakly convergent to an element of Ω.

Proof. 

Using (5), we obtain $\tau \parallel y-q_n{\parallel }^p\le D_{f_p}(y,q_n)\forall y\in \Omega$. This ensures that $\left\{q_n\right\}$ is of boundedness. Hence, from the reflexivity of E, we have ${\omega }_w\left(q_n\right)\ne \varnothing$. Also, let us show the weak convergence of $\left\{q_n\right\}$ to a point in $\Omega$. Indeed, let $\overline{q},\widehat{q}\in {\omega }_w\left(q_n\right)$ with $\overline{q}\ne \widehat{q}$. Then, $\exists \left\{q_{n_k}\right\}\subset \left\{q_n\right\}$ and $\exists \left\{q_{m_k}\right\}\subset \left\{q_n\right\}$ s.t. $q_{n_k}\rightharpoonup \overline{q}$ and $q_{m_k}\rightharpoonup \widehat{q}$. From the sequentially weak continuity of $J_E^p$, we obtain $J_E^p\left(q_{n_k}\right)\rightharpoonup J_E^p\overline{q}$ and $J_E^p\left(q_{m_k}\right)\rightharpoonup J_E^p\widehat{q}$. Note that $D_{f_p}(\overline{q},\widehat{q})+D_{f_p}(\widehat{q},q_n)=D_{f_p}(\overline{q},q_n)-\langle J_E^p\widehat{q}-J_E^p{q}_n,\overline{q}-\widehat{q}\rangle$. So, exploiting the convergence of the sequences $\{D_{f_p}(\overline{q},q_n)\}$ and $\{D_{f_p}(\widehat{q},q_n)\}$, we deduce that

$$\begin{array}{cc}& -\langle J_E^p\widehat{q}-J_E^p\overline{q},\overline{q}-\widehat{q}\rangle ={\displaystyle \underset{k\to \infty }{lim}}[-\langle J_E^p\widehat{q}-J_E^p{q}_{n_k},\overline{q}-\widehat{q}\rangle ]\hfill \\ & ={\displaystyle \underset{n\to \infty }{lim}}[D_{f_p}(\overline{q},\widehat{q})+D_{f_p}(\widehat{q},q_n)-D_{f_p}(\overline{q},q_n)]\hfill \\ & ={\displaystyle \underset{k\to \infty }{lim}}[-\langle J_E^p\widehat{q}-J_E^p{q}_{m_k},\overline{q}-\widehat{q}\rangle ]=-\langle J_E^p\widehat{q}-J_E^p\widehat{q},\overline{q}-\widehat{q}\rangle =0,\hfill \end{array}$$

which, hence, yields $\langle J_E^p\overline{q}-J_E^p\widehat{q},\overline{q}-\widehat{q}\rangle =0$. From (5), we obtain $0<\tau \parallel \overline{q}-\widehat{q}{\parallel }^p\le D_{f_p}(\overline{q},\widehat{q})\le \langle J_E^p\overline{q}-J_E^p\widehat{q},\overline{q}-\widehat{q}\rangle =0$. This arrives at a contradiction. Consequently, the sequence $\left\{q_n\right\}$ converges weakly to a point in $\Omega$.    □

The lemma below was put forth in ${\mathbf{R}}^m$ by [34]. It is easy to verify that the proof of the lemma in a Banach space E is actually the same as in ${\mathbf{R}}^m$. Here, we present the lemma but drop its demonstration.

Lemma 6.

Assume the nonempty $C\subset E$ with C being convex and closed in E. Suppose that $K:=\{y\in C:h(y)\le 0\}$, where $h:E\to \mathbf{R}$ is real-valued. If $K\ne \varnothing$ and h is Lipschitz continuous on C with modulus $\theta >0$, then $\theta \mathrm{dist}(y,K)\ge max\left\{h\right(y),0\}\forall y\in C$, where $\mathrm{dist}(y,K)$ stands for the distance of y to K.

Lemma 7

([35]). Let $\left\{{\mathbf{\Gamma }}_m\right\}$ be a sequence of real numbers that does not decrease at infinity in the sense that $\exists \left\{{\mathbf{\Gamma }}_{m_i}\right\}\subset \left\{{\mathbf{\Gamma }}_m\right\}$ s.t. ${\mathbf{\Gamma }}_{m_i}<{\mathbf{\Gamma }}_{m_i+1}\forall i\ge 1$. Let the sequence ${\left\{\psi \left(m\right)\right\}}_{m\ge m_0}$ of integers be defined as $\psi \left(m\right)=max\{i\le m:{\mathbf{\Gamma }}_i<{\mathbf{\Gamma }}_{i+1}\}$, with integer $m_0\ge 1$ satisfying $\{i\le m_0:{\mathbf{\Gamma }}_i<{\mathbf{\Gamma }}_{i+1}\}\ne \varnothing$. Then, the following holds:

(i) 

$\psi \left(m_0\right)\le \psi (m_0+1)\le \cdots$ and $\psi \left(m\right)\to \infty$;

(ii) 

${\mathbf{\Gamma }}_{\psi \left(m\right)}\le {\mathbf{\Gamma }}_{\psi \left(m\right)+1}$ and ${\mathbf{\Gamma }}_m\le {\mathbf{\Gamma }}_{\psi \left(m\right)+1}\forall m\ge m_0$.

Lemma 8

([36]). Let $\left\{a_n\right\}$ be a sequence in $[0,\infty )$ satisfying $a_{n+1}\le (1-{\mu }_n)a_n+{\mu }_n{\nu }_n\forall n\ge 1$, where $\left\{{\mu }_n\right\}$ and $\left\{{\nu }_n\right\}$ both are real sequences such that (i) $\left\{{\mu }_n\right\}\subset [0,1]$ and ${\sum }_{n=1}^{\infty }{\mu }_n=\infty$, and (ii) ${lim\; sup}_{n\to \infty }{\nu }_n\le 0$ or ${\sum }_{n=1}^{\infty }\left|{\mu }_n{\nu }_n\right|<\infty$. Then, ${lim}_{n\to \infty }{a}_n=0$.

Lemma 9

([37]). Let $\left\{a_n\right\},\left\{b_n\right\}$ and $\left\{{\delta }_n\right\}$ be sequences of nonnegative real numbers satisfying the inequality $a_{n+1}\le (1+{\delta }_n)a_n+b_n\forall n\ge 1$. If ${\sum }_{n=1}^{\infty }{\delta }_n<\infty$ and ${\sum }_{n=1}^{\infty }{b}_n<\infty$, and then ${lim}_{n\to \infty }{a}_n$ exists.

3. Main Results

In this section, let E be a p-uniformly convex and uniformly smooth Banach space with $2\le p<\infty$. Let $\varnothing \ne C\subset E$ with C be closed and convex in E. We are now in a position to state and analyze our iterative algorithms for settling a pair of VIPs with CFPP constraint, where the pair of VIPs implicates two mappings of both uniform continuity and pseudomonotonicity and the CFPP involves finite mappings of Bregman’s relative nonexpansivity and a mapping of Bregman’s relative demicontractivity in E. Assume always that the conditions hold below:

(C1)

For $i=1,\dots ,N$, the self-mapping $S_i$ on C is of both uniform continuity and Bregman’s relative nonexpansivity and self-mapping $S_0$ on C is of both uniform continuity and Bregman’s relative $\xi$-demicontractivity.

(C2)

$\left\{S_n\right\}$ is defined as $S_n:=S_{n\mathrm{mod}N}$ for integer $n\ge 1$ with the mod function taking values in the set $\{1,\dots ,N\}$; i.e., if $n=jN+m$ for some integers $j\ge 0$ and $0\le m<N$, then $S_n=S_N$ if $m=0$ and $S_n=S_m$ if $0<m<N$.

(C3)

For $i=1,2$, $A_i:E\to E^{*}$ is pseudomonotone and uniformly continuous on C, s.t. $\parallel A_i{x}^{†}\parallel \le {lim\; inf}_{n\to \infty }\parallel A_i{x}_n\parallel \forall \left\{x_n\right\}\subset C$ with $x_n\rightharpoonup x^{†}$.

(C4)

$\Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)\right)\ne \varnothing$.

We will make use of Lemmas 10–13 below to derive our major outcomes in this paper.

Lemma 10.

Let $\left\{x_n\right\}$ be the constructed sequence in Algorithm 3. Then, the relations hold: $\frac{1}{{\lambda }_1}{D}_{f_p}(u_n,y_n)\le \langle A_1{u}_n,r_{{\lambda }_1}\left(u_n\right)\rangle$ and $\frac{1}{{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\le \langle A_2{w}_n,r_{{\lambda }_2}\left(w_n\right)\rangle$.

Proof. 

Observe that the last two relations are similar. Then, it suffices to show that the latter relation holds. In fact, using the definition of ${\tilde{y}}_n$ and properties of ${\Pi }_C$, one has

$$\langle J_E^p{w}_n-{\lambda }_2{A}_2{w}_n-J_E^p{\tilde{y}}_n,y-{\tilde{y}}_n\rangle \le 0\forall y\in C.$$

Setting $y=w_n$ in the last inequality, from (5), we obtain

$$D_{f_p}(w_n,{\tilde{y}}_n)\le \langle J_E^p{w}_n-J_E^p{\tilde{y}}_n,w_n-{\tilde{y}}_n\rangle \le {\lambda }_2\langle A_2{w}_n,w_n-{\tilde{y}}_n\rangle .$$

This completes the proof.    □

Lemma 11.

The linear-search rules (1), (3) and the sequence $\left\{x_n\right\}$ constructed in Algorithm 3 are well defined.

Proof. 

Observe that the rules (1) and (3) are similar. Then, it suffices to show that the latter rule (3) is valid. Using the uniform continuity of $A_2$ on C, from $l_2\in (0,1)$, one obtains ${lim}_{j\to \infty }\langle A_2{w}_n-A_2(w_n-l_2^j{r}_{{\lambda }_2}\left(w_n\right)),r_{{\lambda }_2}\left(w_n\right)\rangle =0$. In case $r_{{\lambda }_2}\left(w_n\right)=0$, it is evident that $j_n=0$. In case $r_{{\lambda }_2}\left(w_n\right)\ne 0$, we know that $\exists j_n\ge 0$ s.t. (3) holds.

It is evident to see that, for each $n,{\tilde{Q}}_n$ and $C_n$ are convex and closed. In what follows, we assert that $\Omega$ lies in $Q_n\cap {\tilde{C}}_n$. Let $z\in \Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)\right)$. Using Lemma 2 and the Bregman relative $\xi$-demicontractivity of $S_0$, from $\left\{{\alpha }_n\right\}\subset (\xi ,1)$, we obtain

$$\begin{array}{cc}& D_{f_p}(z,v_n)\le {\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)D_{f_p}(z,S_0{w}_n)\hfill \\ & -{\alpha }_n(1-{\alpha }_n){\rho }_{b_{w_n}}^{*}\parallel J_E^P{w}_n-J_E^P{S}_0{w}_n\parallel \hfill \\ & \le {\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)[D_{f_p}(z,w_n)+\xi {\rho }_{b_{w_n}}^{*}\parallel J_E^P{w}_n-J_E^P{S}_0{w}_n\parallel ]\hfill \\ & -{\alpha }_n(1-{\alpha }_n){\rho }_{b_{w_n}}^{*}\parallel J_E^P{w}_n-J_E^P{S}_0{w}_n\parallel \hfill \\ & =D_{f_p}(z,w_n)-({\alpha }_n-\xi )(1-{\alpha }_n){\rho }_{b_{w_n}}^{*}\parallel J_E^P{w}_n-J_E^P{S}_0{w}_n\parallel \hfill \\ & \le D_{f_p}(z,w_n),\hfill \end{array}$$

which, hence, leads to $z\in Q_n$. Meanwhile, by Lemma 4, one obtains $\langle A_2{t}_n,z-t_n\rangle \le 0$. Therefore,

$$\begin{array}{cc}{\tilde{h}}_n\left(z\right)\hfill & =\langle A_2{t}_n,z-w_n\rangle +\frac{{\tilde{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\hfill \\ \hfill & =-\langle A_2{t}_n,w_n-t_n\rangle -\langle A_2{t}_n,t_n-z\rangle +\frac{{\tilde{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\hfill \\ \hfill & \le -{\tilde{\tau }}_n\langle A_2{t}_n,r_{{\lambda }_2}\left(w_n\right)\rangle +\frac{{\tilde{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n).\hfill \end{array}$$

(11)

So, it follows from (3) that

$$\langle A_2{w}_n-A_2{t}_n,r_{{\lambda }_2}\left(w_n\right)\rangle \le \frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\tilde{y}}_n).$$

Using Lemma 10, we have

$$\begin{array}{cc}\langle A_2{t}_n,r_{{\lambda }_2}\left(w_n\right)\rangle \hfill & \ge \langle A_2{w}_n,r_{{\lambda }_2}\left(w_n\right)\rangle -\frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\tilde{y}}_n)\hfill \\ & \ge (\frac{1}{{\lambda }_2}-\frac{{\mu }_2}{2})D_{f_p}(w_n,{\tilde{y}}_n).\hfill \end{array}$$

This together with (11) arrives at

$${\tilde{h}}_n\left(z\right)\le -\frac{{\tilde{\tau }}_n}{2}(\frac{1}{{\lambda }_2}-{\mu }_2)D_{f_p}(w_n,{\tilde{y}}_n)\le 0.$$

Consequently, $\Omega$ lies in $Q_n\cap {\tilde{C}}_n$. So, $\left\{x_n\right\}$ is well defined.    □

Lemma 12.

Suppose that $\left\{y_n\right\}$ and $\left\{{\tilde{y}}_n\right\}$ are the sequences generated by Algorithm 3. If ${lim}_{n\to \infty }\parallel u_n-y_n\parallel =0$ and ${lim}_{n\to \infty }\parallel w_n-{\tilde{y}}_n\parallel =0$, then ${\omega }_w\left(u_n\right)\subset \mathrm{VI}(C,A_1)$ and ${\omega }_w\left(w_n\right)\subset \mathrm{VI}(C,A_2)$.

Proof. 

Observe that the last two inclusions are similar. Then, it suffices to show that the latter inclusion is valid. In fact, let $z\in {\omega }_w\left(w_n\right)$. Whereby, we know that there exists a subsequence $\left\{w_{n_k}\right\}$ of $\left\{w_n\right\}$, satisfying both $w_{n_k}\rightharpoonup z$ and $w_{n_k}-{\tilde{y}}_{n_k}\to 0$. Thus, one obtains ${\tilde{y}}_{n_k}\rightharpoonup z$. Noticing the convexity and closedness of C, from $\left\{{\tilde{y}}_n\right\}\subset C$ and ${\tilde{y}}_{n_k}\rightharpoonup z$, we obtain $z\in C$. In what follows, one discusses two aspects. In case $A_{2}z=0$, one has $z\in \mathrm{VI}(C,A_2)$ because $\langle A_{2}z,y-z\rangle \ge 0\forall y\in C$. In case $A_{2}z\ne 0$, by the condition on $A_2$, one has ${lim\; inf}_{k\to \infty }\parallel A_2{w}_{n_k}\parallel \ge \parallel A_{2}z\parallel >0$. So, we could assume that $\parallel A_2{w}_{n_k}\parallel \ne 0\forall k\ge 1$. From (6), we obtain

$$\langle J_E^p{w}_{n_k}-{\lambda }_2{A}_2{w}_{n_k}-J_E^p{\tilde{y}}_{n_k},x-{\tilde{y}}_{n_k}\rangle \le 0\forall x\in C,$$

and hence

$$\frac{1}{{\lambda }_2}\langle J_E^p{w}_{n_k}-J_E^p{\tilde{y}}_{n_k},x-{\tilde{y}}_{n_k}\rangle +\langle A_2{w}_{n_k},{\tilde{y}}_{n_k}-w_{n_k}\rangle \le \langle A_2{w}_{n_k},x-w_{n_k}\rangle \forall x\in C.$$

(12)

Note that $A_2$ is uniformly continuous. Then, we know that $\left\{A_2{w}_{n_k}\right\}$ is of boundedness by Lemma 3. Since $\left\{{\tilde{y}}_{n_k}\right\}$ is of boundedness as well, using the uniform continuity of $J_E^p$ on any bounded subset of E, from (12), we deduce that, for all x in C,

$$\underset{k\to \infty }{lim\; inf}\langle A_2{w}_{n_k},x-w_{n_k}\rangle \ge 0.$$

(13)

To show $z\in \mathrm{VI}(C,A_2)$, one picks a positive $\left\{{\tilde{\epsilon }}_k\right\}\subset (0,1)$ s.t. ${\tilde{\epsilon }}_k\downarrow 0$. For every k, $l_k$ is denoted as the smallest $j\ge 1$ satisfying

$$\langle A_2{w}_{n_j},y-w_{n_j}\rangle +{\tilde{\epsilon }}_k\ge 0\forall j\ge l_k.$$

(14)

Since $\left\{{\tilde{\epsilon }}_k\right\}$ is decreasing, it is explicit that $\left\{l_k\right\}$ is increasing. For simplicity, $\{A_2{w}_{n_{l_k}}\}$ is still written as $\left\{A_2{w}_{l_k}\right\}$. Noticing $A_2{w}_{l_k}\ne 0\forall k\ge 1$ (due to $\left\{A_2{w}_{l_k}\right\}\subset \left\{A_2{w}_{n_k}\right\}$), we put ${\tilde{g}}_{l_k}=\frac{A_2{w}_{l_k}}{\parallel A_2{w}_{l_k}{\parallel }^{\frac{q}{q-1}}}$ and hence obtain $\langle A_2{w}_{l_k},J_{E^{*}}^q{\tilde{g}}_{l_k}\rangle =1\forall k\ge 1$. Indeed, it is evident that $\langle A_2{w}_{l_k},J_{E^{*}}^q{\tilde{g}}_{l_k}\rangle =\langle A_2{w}_{l_k},{(\frac{1}{\parallel A_2{w}_{l_k}{\parallel }^{\frac{q}{q-1}}})}^{q-1}{J}_{E^{*}}^q{A}_2{w}_{l_k}\rangle ={(\frac{1}{\parallel A_2{w}_{l_k}{\parallel }^{\frac{q}{q-1}}})}^{q-1}{\parallel A_2{w}_{l_k}\parallel }^q=1\forall k\ge 1$. So, by (14), one has $\langle A_2{w}_{l_k},y+{\tilde{\epsilon }}_k{J}_{E^{*}}^q{\tilde{g}}_{l_k}-w_{l_k}\rangle \ge 0\forall k\ge 1$. Again, from the pseudomonotonicity of $A_2$, one has that, for all $y\in C$,

$$\langle A_2(y+{\tilde{\epsilon }}_k{J}_{E^{*}}^q{\tilde{g}}_{l_k}),y+{\tilde{\epsilon }}_k{J}_{E^{*}}^q{\tilde{g}}_{l_k}-w_{l_k}\rangle \ge 0.$$

(15)

Let us show ${lim}_{k\to \infty }{\tilde{\epsilon }}_k{J}_{E^{*}}^q{\tilde{g}}_{l_k}=0$. Indeed, since $\left\{w_{l_k}\right\}\subset \left\{w_{n_k}\right\}$ and ${\tilde{\epsilon }}_k\downarrow 0$, we obtain that

$$0\le \underset{k\to \infty }{lim\; sup}\parallel {\tilde{\epsilon }}_k{J}_{E^{*}}^q{\tilde{g}}_{l_k}\parallel =\underset{k\to \infty }{lim\; sup}\frac{{\tilde{\epsilon }}_k}{\parallel A_2{w}_{l_k}\parallel }\le \frac{{lim\; sup}_{k\to \infty }{\tilde{\epsilon }}_k}{{lim\; inf}_{k\to \infty }\parallel A_2{w}_{n_k}\parallel }=0.$$

Hence, one obtains ${\tilde{\epsilon }}_k{J}_{E^{*}}^q{\tilde{g}}_{l_k}\to 0$ as $k\to \infty$. Thus, taking the limit as $k\to \infty$ in (15), by condition (C2), one has $\langle A_{2}y,y-z\rangle \ge 0\forall y\in C$. This implies that $z\in \mathrm{VI}(C,A_2)$.    □

Lemma 13.

Suppose that $\left\{y_n\right\}$ and $\left\{{\tilde{y}}_n\right\}$ are the sequences generated by Algorithm 3. Then, the following hold:

(i) 

${lim}_{n\to \infty }{\tau }_n{D}_{f_p}(u_n,y_n)=0\Rightarrow {lim}_{n\to \infty }{D}_{f_p}(u_n,y_n)=0$;

(ii) 

${lim}_{n\to \infty }{\tilde{\tau }}_n{D}_{f_p}(w_n,{\tilde{y}}_n)=0\Rightarrow {lim}_{n\to \infty }{D}_{f_p}(w_n,{\tilde{y}}_n)=0$.

Proof. 

Observe that the assertions (i) and (ii) are similar. Then, it suffices to show that assertion (ii) is valid. To verify assertion (ii), we consider two cases. In case ${lim\; inf}_{n\to \infty }{\tilde{\tau }}_n>0$, we might presume that $\exists \tilde{\tau }>0$ s.t. ${\tilde{\tau }}_n\ge \tilde{\tau }>0\forall n\ge 1$, which, hence, arrives at

$$D_{f_p}(w_n,{\tilde{y}}_n)=\frac{1}{{\tilde{\tau }}_n}{\tilde{\tau }}_n{D}_{f_p}(w_n,{\tilde{y}}_n)\le \frac{1}{\tilde{\tau }}·{\tilde{\tau }}_n{D}_{f_p}(w_n,{\tilde{y}}_n).$$

(16)

Combining (16) and ${lim}_{n\to \infty }{\tilde{\tau }}_n{D}_{f_p}(w_n,{\tilde{y}}_n)=0$ attains ${lim}_{n\to \infty }{D}_{f_p}(w_n,{\tilde{y}}_n)=0$.

In case ${lim\; inf}_{n\to \infty }{\tilde{\tau }}_n=0$, we presume ${lim\; sup}_{n\to \infty }{D}_{f_p}(w_n,{\tilde{y}}_n)=a_2>0$. Whereby, one knows that $\exists \left\{n_k\right\}\subset \left\{n\right\}$ s.t.

$$\underset{k\to \infty }{lim}{\tilde{\tau }}_{n_k}=0\mathrm{and}\underset{k\to \infty }{lim}{D}_{f_p}(w_{n_k},{\tilde{y}}_{n_k})=a_2>0.$$

One puts $\widehat{t_{n_k}}=\frac{1}{l_2}{\tilde{\tau }}_{n_k}{\tilde{y}}_{n_k}+(1-\frac{1}{l_2}{\tilde{\tau }}_{n_k})w_{n_k}\forall k$. From (5) and ${lim}_{k\to \infty }{\tilde{\tau }}_{n_k}{D}_{f_p}(w_{n_k},{\tilde{y}}_{n_k})=0$, we have ${lim}_{k\to \infty }{\tilde{\tau }}_{n_k}{\parallel w_{n_k}-{\tilde{y}}_{n_k}\parallel }^p=0$ and hence

$$\underset{k\to \infty }{lim}\parallel \widehat{t_{n_k}}-w_{n_k}{\parallel }^p=\underset{k\to \infty }{lim}\frac{{\tilde{\tau }}_{n_k}^{p-1}}{l_2^p}·{\tilde{\tau }}_{n_k}{\parallel w_{n_k}-{\tilde{y}}_{n_k}\parallel }^p=0.$$

(17)

Because $A_2$ is of uniform continuity on any bounded subset of C, one obtains

$$\underset{k\to \infty }{lim}\parallel A_2{w}_{n_k}-A_2\widehat{t_{n_k}}\parallel =0.$$

(18)

From the step size rule (3) and the definition of $\widehat{t_{n_k}}$, it follows that

$$\langle A_2{w}_{n_k}-A_2\widehat{t_{n_k}},w_{n_k}-{\tilde{y}}_{n_k}\rangle >\frac{{\mu }_2}{2}{D}_{f_p}(w_{n_k},{\tilde{y}}_{n_k}).$$

(19)

Now, taking the limit as $k\to \infty$, from (18), we have ${lim}_{k\to \infty }{D}_{f_p}(w_{n_k},{\tilde{y}}_{n_k})=0$. This, however, reaches a contradiction. So, it follows that ${lim}_{n\to \infty }{D}_{f_p}(w_n,{\tilde{y}}_n)$ is zero.    □

In what follows, we intend to demonstrate the first convergence result in this paper.

Theorem 1.

Suppose that the Banach space E is of both p-uniform convexity and uniform smoothness s.t. $J_E^p$ is of sequentially weak continuity. If $\left\{x_n\right\}$ is the constructed sequence in Algorithm 3, then $x_n\rightharpoonup z\in \Omega \iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

Proof. 

It is clear that the necessity of Theorem 1 is valid. Next, it suffices to show that the sufficiency is valid. Assume that ${sup}_{n\ge 0}\parallel x_n\parallel <\infty$. Let $z\in \Omega$. It is clear that $S_n{x}_n\ne S_n{x}_{n-1}\iff J_E^p{S}_n{x}_n\ne J_E^p(2S_n{x}_n-S_n{x}_{n-1})$. Using the definition of ${ϵ}_n$, we obtain ${ϵ}_n\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \le {\ell }_n\forall n\ge 1$. From (5) and (10) and the three point identity of $D_{f_p}$, we obtain

$$\begin{array}{cc}& D_{f_p}(z,g_n)\le (1-{ϵ}_n)D_{f_p}(z,S_n{x}_n)+{ϵ}_n{D}_{f_p}(z,2S_n{x}_n-S_n{x}_{n-1})\hfill \\ & =D_{f_p}(z,S_n{x}_n)+{ϵ}_n[D_{f_p}(z,2S_n{x}_n-S_n{x}_{n-1})-D_{f_p}(z,S_n{x}_n)]\hfill \\ & =D_{f_p}(z,S_n{x}_n)+{ϵ}_n[D_{f_p}(S_n{x}_n,2S_n{x}_n-S_n{x}_{n-1})\hfill \\ & +\langle J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1}),z-S_n{x}_n\rangle ]\hfill \\ & \le D_{f_p}(z,x_n)+{ϵ}_n\langle J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1}),z+S_n{x}_{n-1}-2S_n{x}_n\rangle \hfill \\ & \le D_{f_p}(z,x_n)+{ϵ}_n\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \parallel z+S_n{x}_{n-1}-2S_n{x}_n\parallel \hfill \\ & \le D_{f_p}(z,x_n)+{\ell }_{n}M,\hfill \end{array}$$

where ${sup}_{n\ge 1}\parallel z+S_n{x}_{n-1}-2S_n{x}_n\parallel \le M$ for some $M>0$. Using Lemma 2, we obtain

$$\begin{array}{cc}& D_{f_p}(z,u_n)=V_{f_p}(z,{\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)\hfill \\ & \le \frac{1}{p}{\parallel z\parallel }^p-{\beta }_n\langle J_E^p{x}_n,z\rangle -(1-{\beta }_n)\langle J_E^p{g}_n,z\rangle +\frac{{\beta }_n}{q}{\parallel J_E^p{x}_n\parallel }^q\hfill \\ & +\frac{(1-{\beta }_n)}{q}\parallel J_E^p{g}_n{\parallel }^q-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & =\frac{1}{p}{\parallel z\parallel }^p-{\beta }_n\langle J_E^p{x}_n,z\rangle -(1-{\beta }_n)\langle J_E^p{g}_n,z\rangle +\frac{{\beta }_n}{q}{\parallel x_n\parallel }^p\hfill \\ & +\frac{(1-{\beta }_n)}{q}\parallel g_n{\parallel }^p-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & ={\beta }_n{D}_{f_p}(z,x_n)+(1-{\beta }_n)D_{f_p}(z,g_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & \le D_{f_p}(z,x_n)+{\ell }_{n}M-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel .\hfill \end{array}$$

Since $w_n={\Pi }_{C_n}{u}_n$, by (5) and (7), we obtain

$$\begin{array}{cc}{D}_{f_p}(z,w_n)\hfill & \le D_{f_p}(z,u_n)-D_{f_p}(w_n,u_n)\hfill \\ & =D_{f_p}(z,u_n)-D_{f_p}({\Pi }_{C_n}{u}_n,u_n)\hfill \\ & \le D_{f_p}(z,u_n)-\tau {\parallel {\Pi }_{C_n}{u}_n-u_n\parallel }^p\hfill \\ & \le D_{f_p}(z,u_n)-\tau {\parallel P_{C_n}{u}_n-u_n\parallel }^p\hfill \\ & =D_{f_p}(z,u_n)-\tau {\left[\mathrm{dist}(C_n,u_n)\right]}^p.\hfill \end{array}$$

Because $x_{n+1}={\Pi }_{{\tilde{C}}_n\cap Q_n}{w}_n$, from (5) and (7), we obtain

$$\begin{array}{cc}{D}_{f_p}(z,x_{n+1})\hfill & \le D_{f_p}(z,w_n)-D_{f_p}(x_{n+1},w_n)\hfill \\ & \le D_{f_p}(z,w_n)-D_{f_p}({\Pi }_{{\tilde{C}}_n}{w}_n,w_n)\hfill \\ & \le D_{f_p}(z,w_n)-\tau {\parallel P_{{\tilde{C}}_n}{w}_n-w_n\parallel }^p\hfill \\ & =D_{f_p}(z,w_n)-\tau {\left[\mathrm{dist}({\tilde{C}}_n,w_n)\right]}^p.\hfill \end{array}$$

This along with (19) arrives at

$$\begin{array}{cc}& D_{f_p}(z,x_{n+1})\le D_{f_p}(z,w_n)-D_{f_p}(x_{n+1},w_n)\hfill \\ \hfill & \le D_{f_p}(z,u_n)-D_{f_p}(w_n,u_n)-D_{f_p}(x_{n+1},w_n)\hfill \\ \hfill & \le D_{f_p}(z,u_n)-\tau {\left[\mathrm{dist}(C_n,u_n)\right]}^p-\tau {\left[\mathrm{dist}({\tilde{C}}_n,w_n)\right]}^p\hfill \\ \hfill & \le D_{f_p}(z,x_n)+{\ell }_{n}M-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ \hfill & -\tau {\left[\mathrm{dist}(C_n,u_n)\right]}^p-\tau {\left[\mathrm{dist}({\tilde{C}}_n,w_n)\right]}^p,\hfill \end{array}$$

(20)

which, hence, arrives at

$$D_{f_p}(z,x_{n+1})\le D_{f_p}(z,x_n)+{\ell }_{n}M.$$

Since ${\sum }_{n=1}^{\infty }{\ell }_n<\infty$, by Lemma 9, we deduce that ${lim}_{n\to \infty }{D}_{f_p}(z,x_n)$ exists. In addition, by the boundedness of $\left\{x_n\right\}$, we conclude that $\left\{g_n\right\},\left\{u_n\right\},\left\{v_n\right\},\left\{w_n\right\},\left\{y_n\right\},\left\{{\tilde{y}}_n\right\}$, $\left\{s_n\right\},\left\{t_n\right\},\left\{S_n{x}_n\right\}$ and $\left\{S_0{w}_n\right\}$ are also bounded. Using (20), we obtain

$$\begin{array}{cc}& D_{f_p}(w_n,u_n)+D_{f_p}(x_{n+1},w_n)\le D_{f_p}(z,u_n)-D_{f_p}(z,x_{n+1})\hfill \\ & \le D_{f_p}(z,x_n)+{\ell }_{n}M-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel -D_{f_p}(z,x_{n+1}),\hfill \end{array}$$

which immediately yields

$$\begin{array}{cc}& D_{f_p}(w_n,u_n)+D_{f_p}(x_{n+1},w_n)+{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & \le D_{f_p}(z,x_n)-D_{f_p}(z,x_{n+1})+{\ell }_{n}M.\hfill \end{array}$$

Since ${lim}_{n\to \infty }{\ell }_n=0$, ${lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)>0$ and ${lim}_{n\to \infty }{D}_{f_p}(z,x_n)$ exist, it follows that ${lim}_{n\to \infty }{D}_{f_p}(w_n,u_n)=0$, ${lim}_{n\to \infty }{D}_{f_p}(x_{n+1},w_n)=0$ and ${lim}_{n\to \infty }{\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel =0$, which, hence, yields ${lim}_{n\to \infty }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel =0$. From $u_n=J_{E^{*}}^q({\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)$, it can be readily seen that ${lim}_{n\to \infty }\parallel J_E^p{u}_n-J_E^p{x}_n\parallel =0$. Noticing $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_n{x}_n+{ϵ}_n{J}_E^p(2S_n{x}_n-S_n{x}_{n-1}))$, we obtain from ${lim}_{n\to \infty }{\ell }_n=0$ and the definition of ${ϵ}_n$ that

$$\parallel J_E^p{g}_n-J_E^p{S}_n{x}_n\parallel ={ϵ}_n\parallel J_E^p(2S_n{x}_n-S_n{x}_{n-1})-J_E^p{S}_n{x}_n\parallel \le {\ell }_n\to 0(n\to \infty ).$$

Hence, using (5) and uniform continuity of $J_E^p$ on bounded subsets of E, we conclude that ${lim}_{n\to \infty }\parallel g_n-S_n{x}_n\parallel =0$ and

$$\underset{n\to \infty }{lim}\parallel w_n-u_n\parallel =\underset{n\to \infty }{lim}\parallel x_{n+1}-w_n\parallel =\underset{n\to \infty }{lim}\parallel x_n-S_n{x}_n\parallel =\underset{n\to \infty }{lim}\parallel u_n-x_n\parallel =0.$$

(21)

Since $\left\{x_n\right\}$ is bounded and E is reflexive, then we know that ${\omega }_w\left(x_n\right)\ne \varnothing$. In what follows, we claim that ${\omega }_w\left(x_n\right)$ lies in $\Omega$. Let $z\in {\omega }_w\left(x_n\right)$. Whereby, one knows that there exists a subsequence $\left\{x_{n_k}\right\}$ of $\left\{x_n\right\}$ such that $x_{n_k}$ converges weakly to z. From (21), one obtains $w_{n_k}\rightharpoonup z$. Since $\left\{A_1{s}_n\right\}$ is of boundedness, one infers that $\exists L_1>0$ such that $\parallel A_1{s}_n\parallel \le L_1$. So, it follows that, for each $y,g\in C_n$,

$$|h_n\left(y\right)-h_n\left(g\right)|=|\langle A_1{s}_n,y-g\rangle |\le \parallel A_1{s}_n\parallel \parallel y-g\parallel \le L_1\parallel y-g\parallel ,$$

which implies that $h_n\left(y\right)$ is $L_1$-Lipschitzian. Using Lemma 6, one obtains

$$\mathrm{dist}(C_n,u_n)\ge \frac{1}{L_1}{h}_n\left(u_n\right)=\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n).$$

(22)

Noticing $x_{n+1}\in Q_n$, from the definition of $Q_n$ and (20), we have

$$\begin{array}{cc}{D}_{f_p}(x_{n+1},v_n)\hfill & \le D_{f_p}(x_{n+1},w_n)\hfill \\ & \le D_{f_p}(z,w_n)-D_{f_p}(z,x_{n+1})\hfill \\ & \le D_{f_p}(z,u_n)-D_{f_p}(z,x_{n+1})\hfill \\ & \le D_{f_p}(z,x_n)-D_{f_p}(z,x_{n+1})+{\ell }_{n}M.\hfill \end{array}$$

Since ${lim}_{n\to \infty }{\ell }_n=0$ and ${lim}_{n\to \infty }{D}_{f_p}(z,x_n)$ exists, we have ${lim}_{n\to \infty }{D}_{f_p}(x_{n+1},v_n)$$=0$, which immediately yields $x_{n+1}-v_n\to 0$. Hence, from (21), we have

$$\underset{n\to \infty }{lim}\parallel w_n-v_n\parallel =0.$$

(23)

Furthermore, by Lemma 2, we obtain that

$$\begin{array}{cc}& D_{f_p}(z,v_n)=V_{f_p}(z,{\alpha }_n{J}_E^p{w}_n+(1-{\alpha }_n)J_E^p{S}_0{w}_n)\hfill \\ & \le \frac{1}{p}{\parallel z\parallel }^p-{\alpha }_n\langle J_E^p{w}_n,z\rangle -(1-{\alpha }_n)\langle J_E^p{S}_0{w}_n,z\rangle +\frac{{\alpha }_n}{q}{\parallel J_E^p{w}_n\parallel }^q\hfill \\ & +\frac{(1-{\alpha }_n)}{q}\parallel J_E^p{S}_0{w}_n{\parallel }^q-{\alpha }_n(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel \hfill \\ & =\frac{1}{p}{\parallel z\parallel }^p-{\alpha }_n\langle J_E^p{w}_n,z\rangle -(1-{\alpha }_n)\langle J_E^p{S}_0{w}_n,z\rangle +\frac{{\alpha }_n}{q}{\parallel w_n\parallel }^p\hfill \\ & +\frac{(1-{\alpha }_n)}{q}\parallel S_0{w}_n{\parallel }^p-{\alpha }_n(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel \hfill \\ & ={\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)D_{f_p}(z,S_0{w}_n)-{\alpha }_n(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel \hfill \\ & \le {\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)[D_{f_p}(z,w_n)+\xi {\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel ]\hfill \\ & -{\alpha }_n(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel \hfill \\ & =D_{f_p}(z,w_n)-({\alpha }_n-\xi )(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel .\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}({\alpha }_n-\xi )(1-{\alpha }_n){\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel \hfill & \le D_{f_p}(z,w_n)-D_{f_p}(z,v_n)\hfill \\ & \le D_{f_p}(z,w_n)-D_{f_p}(z,v_n)+D_{f_p}(w_n,v_n)\hfill \\ & =\langle J_E^p{v}_n-J_E^p{w}_n,z-w_n\rangle .\hfill \end{array}$$

Taking the limit in the last inequality as $n\to \infty$, and using uniform continuity of $J_E^p$ on bounded subsets of E, (23) and ${lim\; inf}_{n\to \infty }({\alpha }_n-\xi )(1-{\alpha }_n)>0$, we obtain ${lim}_{n\to \infty }{\rho }_b^{*}\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel =0$ and hence ${lim}_{n\to \infty }\parallel J_E^p{w}_n-J_E^p{S}_0{w}_n\parallel =0$. Since $J_{E^{*}}^q$ is uniformly continuous on any bounded subset of $E^{*}$, we have

$$\underset{n\to \infty }{lim}\parallel w_n-S_0{w}_n\parallel =0.$$

(24)

Now, let us show $z\in {\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)$. By the boundedness of $\left\{A_2{t}_n\right\}$, one knows that there exists $L_2>0$, satisfying $\parallel A_2{t}_n\parallel \le L_2$. So, it follows that, for each $y,g\in {\tilde{C}}_n$,

$$|{\tilde{h}}_n\left(y\right)-{\tilde{h}}_n\left(g\right)|=|\langle A_2{t}_n,y-g\rangle |\le \parallel A_2{t}_n\parallel \parallel y-g\parallel \le L_2\parallel y-g\parallel ,$$

which guarantees that ${\tilde{h}}_n\left(y\right)$ is $L_2$-Lipschitzian. By Lemma 6, one obtains

$$\mathrm{dist}({\tilde{C}}_n,w_n)\ge \frac{1}{L_2}{\tilde{h}}_n\left(w_n\right)=\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n).$$

(25)

Combining (20), (22) and (25), we obtain

$$\begin{array}{cc}& D_{f_p}(z,x_n)-D_{f_p}(z,x_{n+1})+{\ell }_{n}M\hfill \\ \hfill & \ge D_{f_p}(z,u_n)-D_{f_p}(z,x_{n+1})\hfill \\ \hfill & \ge \tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p+\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p.\hfill \end{array}$$

(26)

Thus,

$$\underset{n\to \infty }{lim}{\tilde{\tau }}_n{D}_{f_p}(w_n,{\tilde{y}}_n)=\underset{n\to \infty }{lim}{\tau }_n{D}_{f_p}(u_n,y_n)=0.$$

By Lemma 13, we obtain

$$\underset{n\to \infty }{lim}\parallel w_n-{\tilde{y}}_n\parallel =\underset{n\to \infty }{lim}\parallel u_n-y_n\parallel =0.$$

Moreover, noticing $x_{n_k}\rightharpoonup z$ and (21), we obtain that $u_{n_k}\rightharpoonup z$ and $w_{n_k}\rightharpoonup z$. By Lemma 12, we deduce that $z\in {\omega }_w\left(u_n\right)\subset \mathrm{VI}(C,A_1)$ and $z\in {\omega }_w\left(w_n\right)\subset \mathrm{VI}(C,A_2)$. Consequently,

$$z\in \bigcap _{i=1}^2\mathrm{VI}(C,A_i).$$

Next, we claim that $z\in {\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)$. Indeed, by (21), we immediately obtain

$$\parallel x_{n+1}-x_n\parallel \le \parallel x_{n+1}-w_n\parallel +\parallel w_n-u_n\parallel +\parallel u_n-x_n\parallel \to 0(n\to \infty ).$$

(27)

We first claim that ${lim}_{n\to \infty }\parallel x_n-S_r{x}_n\parallel =0\forall r\in \{1,\dots ,N\}$. Indeed, by the formulation of $S_n$, one obtains $S_n\in \{S_1,\dots ,S_N\}$, which, hence, leads to $S_{n+k}\in \{S_1,\dots ,S_N\}\forall n\ge 1,k\in \{1,\dots ,N\}$. Note that

$$\begin{array}{cc}\parallel x_n-S_{n+k}{x}_n\parallel \hfill & \le \parallel x_n-x_{n+k}\parallel +\parallel x_{n+k}-S_{n+k}{x}_{n+k}\parallel +\parallel S_{n+k}{x}_{n+k}-S_{n+k}{x}_n\parallel \hfill \\ & \le \parallel x_n-x_{n+k}\parallel +\parallel x_{n+k}-S_{n+k}{x}_{n+k}\parallel +{\displaystyle \sum _{m=1}^N}\parallel S_m{x}_{n+k}-S_m{x}_n\parallel .\hfill \end{array}$$

Utilizing the uniform continuity of each $S_m$, one deduces from (21) and (27) that $\forall k,m\in \{1,\dots ,N\}$, $x_{n+k}-S_{n+k}{x}_{n+k}\to 0$ and $S_m{x}_{n+k}-S_m{x}_n\to 0$. Hence, one obtains ${lim}_{n\to \infty }\parallel x_n-S_{n+k}{x}_n\parallel =0$. So, it follows that

$$\underset{n\to \infty }{lim}\parallel S_r{x}_n-x_n\parallel =0\forall r\in \{1,\dots ,N\}.$$

This along with $x_{n_k}\rightharpoonup z$, leads to $z\in \widehat{\mathrm{Fix}}\left(S_r\right)=\mathrm{Fix}\left(S_r\right)\forall r\in \{1,\dots ,N\}$. Accordingly, $z\in {\bigcap }_{k=1}^N\mathrm{Fix}\left(S_k\right)$. Also, using $x_{n_k}\rightharpoonup z$ and (21), one has $w_{n_k}\rightharpoonup z$. As a result, from (24), we obtain $z\in \widehat{\mathrm{Fix}}\left(S_0\right)=\mathrm{Fix}\left(S_0\right)$. Therefore, $z\in {\bigcap }_{k=0}^N\mathrm{Fix}\left(S_k\right)$, and thus $z\in \Omega$. Consequently, ${\omega }_w\left(x_n\right)\subset \Omega$. Hence, by Lemma 5, one concludes that $\left\{x_n\right\}$ converges weakly to z.    □

On the other hand, let us show the strongly convergent result for a pair of VIPs with CFPP constraint, where the two VIPs implicate two mappings of both uniform continuity and pseudomonotonicity and the CFPP involves finite mappings of Bregman’s relative nonexpansivity and a mapping of Bregman’s relative demicontractivity.

Theorem 2.

Suppose that the conditions (C1)–(C3) hold. If $\left\{x_n\right\}$ is the constructed sequence in Algorithm 4, then $x_n\to {\Pi }_{\Omega }u\iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

Proof. 

It is clear that the necessity of the theorem is true. Next, it suffices to show that the sufficiency is valid. Assume that ${sup}_{n\ge 0}\parallel x_n\parallel <\infty$. In what follows, we divide our proof into four claims.

Claim 1. We show that

$$\begin{array}{cc}& (1-{\alpha }_n){\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(z,x_n)-D_{f_p}(\widehat{u},x_{n+1})+{\ell }_{n}M,\hfill \end{array}$$

for some $M>0$. Indeed, put $\widehat{u}={\Pi }_{\Omega }u$. Noticing $w_n={\Pi }_{C_n}{u}_n$ and $z_n={\Pi }_{{\tilde{C}}_n}{w}_n$, we deduce from (5) and (7) that

$$\begin{array}{cc}{D}_{f_p}(\widehat{u},w_n)\hfill & \le D_{f_p}(\widehat{u},u_n)-D_{f_p}(w_n,u_n)\hfill \\ & \le D_{f_p}(\widehat{u},u_n)-\tau {\left[\mathrm{dist}(C_n,u_n)\right]}^p,\hfill \end{array}$$

and

$$\begin{array}{cc}{D}_{f_p}(\widehat{u},z_n)\hfill & \le D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n)\hfill \\ & \le D_{f_p}(\widehat{u},w_n)-\tau {\left[\mathrm{dist}({\tilde{C}}_n,w_n)\right]}^p.\hfill \end{array}$$

Using the same inferences as in the proof of Theorem 1, we know that

$$\begin{array}{cc}{D}_{f_p}(\widehat{u},g_n)\hfill & \le D_{f_p}(\widehat{u},x_n)+{ϵ}_n\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel \le D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M,\hfill \end{array}$$

where ${sup}_{n\ge 1}\parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel \le M$ for some $M>0$. This ensures that $\left\{g_n\right\}$ is bounded.

Algorithm 4: The 2nd modified inertial-type subgradient extragradient method

The 2nd modified inertial-type subgradient extragradient method. Initialization: Given $x_0,x_1\in C$ arbitrarily and let $ϵ>0,l_k\in (0,1),{\mu }_k>0$ and ${\lambda }_k\in (0,\frac{1}{{\mu }_k})$ for $k=1,2$. Choose $\left\{{\ell }_n\right\},\left\{{\gamma }_n\right\},\left\{{\alpha }_n\right\}\subset (0,1)$ and $\left\{{\beta }_n\right\}\subset (\xi ,1)$ s.t.

${lim}_{n\to \infty }{\ell }_n=0,{\sum }_{n=1}^{\infty }{\alpha }_n=\infty$, ${lim}_{n\to \infty }{\alpha }_n=0$, ${lim\; inf}_{n\to \infty }({\beta }_n-\xi )(1-{\beta }_n)>0$ and ${lim\; inf}_{n\to \infty }{\gamma }_n(1-{\gamma }_n)>0$. Moreover, given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ s.t. $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where ${sup}_{n\ge 1}\frac{{ϵ}_n}{{\alpha }_n}<\infty$ and $$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{\ell }_n}{\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel }\}\mathrm{if}{S}_n{x}_n\ne S_n{x}_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$$

Iterations: Compute $x_{n+1}$ below:

Step 1. Put $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_n{x}_n+{ϵ}_n{J}_E^p(2S_n{x}_n-S_n{x}_{n-1}))$, and calculate $u_n=J_{E^{*}}^q({\gamma }_n{J}_E^p{x}_n+(1-{\gamma }_n)J_E^p{g}_n)$, $y_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{u}_n-{\lambda }_1{A}_1{u}_n)\right)$, $r_{{\lambda }_1}\left(u_n\right):=u_n-y_n$ and $s_n=u_n-{\tau }_n{r}_{{\lambda }_1}\left(u_n\right)$, with ${\tau }_n:=l_1^{k_n}$ and $k_n$ being the smallest $k\ge 0$ s.t. $$\langle A_1{u}_n-A_1(u_n-l_1^k{r}_{{\lambda }_1}\left(u_n\right)),u_n-y_n\rangle \le \frac{{\mu }_1}{2}{D}_{f_p}(u_n,y_n).$$

Step 2. Calculate $w_n={\Pi }_{C_n}\left(u_n\right)$, with $C_n:=\{y\in C:h_n\left(y\right)\le 0\}$ and $$h_n\left(y\right)=\langle A_1{s}_n,y-u_n\rangle +\frac{{\tau }_n}{2{\lambda }_1}{D}_{f_p}(u_n,y_n).$$

Step 3. Calculate ${\tilde{y}}_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{w}_n-{\lambda }_2{A}_2{w}_n)\right)$, $r_{{\lambda }_2}\left(w_n\right):=w_n-{\tilde{y}}_n$ and $t_n=w_n-{\tilde{\tau }}_n{r}_{{\lambda }_2}\left(w_n\right)$, with ${\tilde{\tau }}_n:=l_2^{j_n}$ and $j_n$ being the smallest $j\ge 0$ s.t. $$\langle A_2{w}_n-A_2(w_n-l_2^j{r}_{{\lambda }_2}\left(w_n\right)),w_n-{\tilde{y}}_n\rangle \le \frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\tilde{y}}_n).$$

Step 4. Set $z_n={\Pi }_{{\tilde{C}}_n}\left(w_n\right)$, and calculate $v_n=J_{E^{*}}^q({\beta }_n{J}_E^p{z}_n+(1-{\beta }_n)J_E^p\left(S_0{z}_n\right))$ and $x_{n+1}={\Pi }_C(J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)$, with ${\tilde{C}}_n:=\{y\in C:{\tilde{h}}_n\left(y\right)\le 0\}$ and $${\tilde{h}}_n\left(y\right)=\langle A_2{t}_n,y-w_n\rangle +\frac{{\tilde{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n).$$

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

Using (10) and the last two inequalities, from $\left\{{\gamma }_n\right\}\subset (0,1)$ and $\left\{{\beta }_n\right\}\subset (\xi ,1)$, we obtain

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)[{\beta }_n{D}_{f_p}(\widehat{u},z_n)+(1-{\beta }_n)D_{f_p}(\widehat{u},S_0{z}_n)\hfill \\ & -{\beta }_n(1-{\beta }_n){\rho }_{b_{z_n}}^{*}\parallel z_n-S_0{z}_n\parallel ]\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)[D_{f_p}(\widehat{u},z_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_{b_{z_n}}^{*}\parallel z_n-S_0{z}_n\parallel ]\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)D_{f_p}(\widehat{u},u_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)[{\gamma }_n{D}_{f_p}(\widehat{u},x_n)+(1-{\gamma }_n)D_{f_p}(\widehat{u},g_n)\hfill \\ & -{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel ]\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(z,x_n)+{\ell }_{n}M-(1-{\alpha }_n){\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel ,\hfill \end{array}$$

which immediately arrives at the desired claim. In addition, it is easily known that $\left\{u_n\right\},\left\{v_n\right\}$, $\left\{w_n\right\},\left\{y_n\right\},\left\{{\tilde{y}}_n\right\},\left\{z_n\right\},\left\{s_n\right\},\left\{t_n\right\}$ and $\left\{S_0{z}_n\right\}$ are also bounded.

   Claim 2. We show that

$$\begin{array}{cc}& D_{f_p}(w_n,u_n)+D_{f_p}(z_n,w_n)\hfill \\ & \le D_{f_p}(\widehat{u},u_n)-D_{f_p}(\widehat{u},x_{n+1})+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle .\hfill \end{array}$$

Indeed, take $b={sup}_{n\ge 1}\{\parallel x_n{\parallel }^{p-1},\parallel g_n{\parallel }^{p-1},\parallel z_n{\parallel }^{p-1},\parallel S_0{z}_n{\parallel }^{p-1}\}$. Using Lemma 2, one obtains

$$\begin{array}{cc}& D_{f_p}(\widehat{u},u_n)=V_{f_p}(\widehat{u},{\gamma }_n{J}_E^p{x}_n+(1-{\gamma }_n)J_E^p{g}_n)\hfill \\ \hfill & \le \frac{1}{p}\parallel \widehat{u}{\parallel }^p-{\gamma }_n\langle J_E^p{x}_n,\widehat{u}\rangle -(1-{\gamma }_n)\langle J_E^p{g}_n,\widehat{u}\rangle +\frac{{\gamma }_n}{q}{\parallel J_E^p{x}_n\parallel }^q\hfill \\ \hfill & +\frac{(1-{\gamma }_n)}{q}\parallel J_E^p{g}_n{\parallel }^q-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ \hfill & =\frac{1}{p}\parallel \widehat{u}{\parallel }^p-{\gamma }_n\langle J_E^p{x}_n,\widehat{u}\rangle -(1-{\gamma }_n)\langle J_E^p{g}_n,\widehat{u}\rangle +\frac{{\gamma }_n}{q}{\parallel x_n\parallel }^p\hfill \\ \hfill & +\frac{(1-{\gamma }_n)}{q}\parallel g_n{\parallel }^p-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ \hfill & ={\gamma }_n{D}_{f_p}(\widehat{u},x_n)+(1-{\gamma }_n)D_{f_p}(\widehat{u},g_n)-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ \hfill & \le {\gamma }_n{D}_{f_p}(\widehat{u},x_n)+(1-{\gamma }_n)[D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M]-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ \hfill & \le D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel ,\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}& D_{f_p}(\widehat{u},v_n)=V_{f_p}(\widehat{u},{\beta }_n{J}_E^p{z}_n+(1-{\beta }_n)J_E^p{S}_0{z}_n)\hfill \\ \hfill & \le {\beta }_n{D}_{f_p}(\widehat{u},z_n)+(1-{\beta }_n)D_{f_p}(\widehat{u},S_0{z}_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ \hfill & \le {\beta }_n{D}_{f_p}(\widehat{u},z_n)+(1-{\beta }_n)[D_{f_p}(\widehat{u},z_n)+\xi {\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel ]\hfill \\ \hfill & -{\beta }_n(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ \hfill & =D_{f_p}(\widehat{u},z_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ \hfill & \le D_{f_p}(\widehat{u},w_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel .\hfill \end{array}$$

(29)

Set ${\zeta }_n=J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)$. From (9), we have

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le V_{f_p}(\widehat{u},{\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)\hfill \\ \hfill & \le V_{f_p}(\widehat{u},{\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n-{\alpha }_n(J_E^{p}u-J_E^p\widehat{u}))+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ \hfill & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ \hfill & \le (1-{\alpha }_n)[D_{f_p}(\widehat{u},w_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel ]+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ \hfill & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},w_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle .\hfill \end{array}$$

(30)

Furthermore, from (29), one has

$$\begin{array}{cc}{D}_{f_p}(\widehat{u},v_n)\hfill & \le D_{f_p}(\widehat{u},z_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ & \le D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ & \le D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n).\hfill \end{array}$$

This, along with (30), leads to

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le (1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le D_{f_p}(\widehat{u},u_n)-D_{f_p}(w_n,u_n)-D_{f_p}(z_n,w_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle .\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}& D_{f_p}(w_n,u_n)+D_{f_p}(z_n,w_n)\hfill \\ \hfill & \le D_{f_p}(\widehat{u},u_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle -D_{f_p}(\widehat{u},x_{n+1}).\hfill \end{array}$$

(31)

Claim 3. Let us show

$$\begin{array}{cc}& {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},x_n)-D_{f_p}(\widehat{u},x_{n+1})+{\ell }_{n}M\hfill \\ & \ge (1-{\alpha }_n)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p+\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\}.\hfill \end{array}$$

Indeed, by the analogous reasonings to these of (26), one obtains

$$\begin{array}{cc}{D}_{f_p}(\widehat{u},z_n)\hfill & \le D_{f_p}(\widehat{u},w_n)-\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\hfill \\ \hfill & \le D_{f_p}(\widehat{u},u_n)-\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p-\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p.\hfill \end{array}$$

(32)

Applying (28), (29) and (32), we have

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)\hfill \\ \hfill & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)[D_{f_p}(\widehat{u},z_n)-({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel z_n-S_0{z}_n\parallel ]\hfill \\ \hfill & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)\{D_{f_p}(\widehat{u},u_n)-\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p-\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\}\hfill \\ \hfill & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},u_n)-(1-{\alpha }_n)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p+\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\}\hfill \\ \hfill & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ \hfill & -(1-{\alpha }_n)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p+\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\}\hfill \\ \hfill & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M\hfill \\ \hfill & -(1-{\alpha }_n)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p+\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\}.\hfill \end{array}$$

(33)

Claim 4. We show that $x_n\to \widehat{u}$ as $n\to \infty$. Indeed, since E is reflexive and $\left\{x_n\right\}$ is bounded, we know that ${\omega }_w\left(x_n\right)$ is nonempty. Let $z\in {\omega }_w\left(x_n\right)$. Whereby, there exists a subsequence $\left\{x_{n_k}\right\}$ of $\left\{x_n\right\}$ such that $\left\{x_{n_k}\right\}$ converges weakly to z. One defines ${\mathbf{\Gamma }}_n:=D_{f_p}(\widehat{u},x_n)\forall n$. In what follows, let us demonstrate ${\mathbf{\Gamma }}_n\to 0(n\to \infty )$ in both possible aspects.

    Aspect 1. Presume that there exists $n_0\ge 1$ s.t. $\left\{{\mathbf{\Gamma }}_n\right\}$ is non-increasing. Whereby, ${\mathbf{\Gamma }}_n\to d<+\infty$ and, hence, ${\mathbf{\Gamma }}_n-{\mathbf{\Gamma }}_{n+1}\to 0$. From (28) and (31), we obtain

$$\begin{array}{cc}& D_{f_p}(w_n,u_n)+D_{f_p}(z_n,w_n)\hfill \\ & \le D_{f_p}(\widehat{u},u_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle -D_{f_p}(\widehat{u},x_{n+1})\hfill \\ & \le D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M-{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel -D_{f_p}(\widehat{u},x_{n+1})\hfill \\ & +{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle ,\hfill \end{array}$$

which, hence, yields

$$\begin{array}{cc}& D_{f_p}(w_n,u_n)+D_{f_p}(z_n,w_n)+{\gamma }_n(1-{\gamma }_n){\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & \le {\mathbf{\Gamma }}_n-{\mathbf{\Gamma }}_{n+1}+{\ell }_{n}M+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle .\hfill \end{array}$$

Because ${\ell }_n\to 0,{\alpha }_n\to 0,0<{lim\; inf}_{n\to \infty }{\gamma }_n(1-{\gamma }_n),{\mathbf{\Gamma }}_n-{\mathbf{\Gamma }}_{n+1}\to 0$ and the sequence $\left\{{\zeta }_n\right\}$ is of boundedness, one deduces that $D_{f_p}(w_n,u_n)\to 0$, $D_{f_p}(z_n,w_n)\to 0$ and ${\rho }_b^{*}\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \to 0$, which, hence, yields $\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \to 0$. From $u_n=J_{E^{*}}^q({\gamma }_n{J}_E^p{x}_n+(1-{\gamma }_n)J_E^p{g}_n)$, it is easily known that ${lim}_{n\to \infty }\parallel J_E^p{u}_n$ $-J_E^p{x}_n\parallel =0$. Noticing $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_n{x}_n+{ϵ}_n{J}_E^p(2S_n{x}_n-S_n{x}_{n-1}))$, we deduce from ${lim}_{n\to \infty }{\ell }_n=0$ and the definition of ${ϵ}_n$ that

$$\parallel J_E^p{g}_n-J_E^p{S}_n{x}_n\parallel ={ϵ}_n\parallel J_E^p(2S_n{x}_n-S_n{x}_{n-1})-J_E^p{S}_n{x}_n\parallel \le {\ell }_n\to 0(n\to \infty ).$$

Hence, using (5) and uniform continuity of $J_E^p$ on bounded subsets of E, we conclude that ${lim}_{n\to \infty }\parallel g_n-x_n\parallel =0$ and

$$\underset{n\to \infty }{lim}\parallel w_n-u_n\parallel =\underset{n\to \infty }{lim}\parallel z_n-w_n\parallel =\underset{n\to \infty }{lim}\parallel x_n-S_n{x}_n\parallel =\underset{n\to \infty }{lim}\parallel u_n-x_n\parallel =0.$$

(34)

Furthermore, from (28) and (30), we have

$$\begin{array}{cc}& (1-{\alpha }_n)({\beta }_n-\xi )(1-{\beta }_n){\rho }_b^{*}\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel \hfill \\ & \le D_{f_p}(\widehat{u},u_n)-D_{f_p}(\widehat{u},x_{n+1})+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ & \le D_{f_p}(\widehat{u},x_n)-D_{f_p}(\widehat{u},x_{n+1})+{\ell }_{n}M+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle .\hfill \end{array}$$

According to the analogous reasonings, one obtains ${lim}_{n\to \infty }\parallel J_E^p{z}_n-J_E^p{S}_0{z}_n\parallel =0$, which, hence, leads to ${lim}_{n\to \infty }\parallel J_E^p{v}_n-J_E^p{z}_n\parallel =0$ (due to $v_n=J_{E^{*}}^q({\beta }_n{J}_E^p{z}_n+(1-{\beta }_n)J_E^p{S}_0{z}_n)$). Using uniform continuity of $J_{E^{*}}^q$ on any bounded subset of $E^{*}$, we obtain

$$\underset{n\to \infty }{lim}\parallel z_n-S_0{z}_n\parallel =\underset{n\to \infty }{lim}\parallel v_n-z_n\parallel =0.$$

(35)

This together with (34) implies that

$$\parallel v_n-x_n\parallel \le \parallel v_n-z_n\parallel +\parallel z_n-w_n\parallel +\parallel w_n-u_n\parallel +\parallel u_n-x_n\parallel \to 0(n\to \infty ).$$

It is clear that

$$\underset{n\to \infty }{lim}\parallel z_n-x_n\parallel =0.$$

(36)

Let us show that $z\in {\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)$. Indeed, since ${\zeta }_n=J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)$, it can be readily seen that

$$\underset{n\to \infty }{lim}\parallel {\zeta }_n-x_n\parallel =0.$$

(37)

In addition, using (7), (28) and (29), we have

$$\begin{array}{cc}{D}_{f_p}(\widehat{u},x_{n+1})\hfill & \le D_{f_p}(\widehat{u},J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)-D_{f_p}(x_{n+1},{\zeta }_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},w_n)-D_{f_p}(x_{n+1},{\zeta }_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},u_n)-D_{f_p}(x_{n+1},{\zeta }_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+D_{f_p}(\widehat{u},x_n)+{\ell }_{n}M-D_{f_p}(x_{n+1},{\zeta }_n),\hfill \end{array}$$

which, hence, arrives at

$$D_{f_p}(x_{n+1},{\zeta }_n)\le {\alpha }_n{D}_{f_p}(\widehat{u},u)+{\mathbf{\Gamma }}_n-{\mathbf{\Gamma }}_{n+1}+{\ell }_{n}M.$$

So, it follows that ${lim}_{n\to \infty }{D}_{f_p}(x_{n+1},{\zeta }_n)=0$ and thus $x_{n+1}-{\zeta }_n\to 0$. This along with (37) arrives at

$$\parallel x_{n+1}-x_n\parallel \le \parallel x_{n+1}-{\zeta }_n\parallel +\parallel {\zeta }_n-x_n\parallel \to 0(n\to \infty ).$$

(38)

Observe that $\forall k\in \{1,...,N\}$,

$$\begin{array}{cc}\parallel x_n-S_{n+k}{x}_n\parallel \hfill & \le \parallel x_n-x_{n+k}\parallel +\parallel x_{n+k}-S_{n+k}{x}_{n+k}\parallel +\parallel S_{n+k}{x}_{n+k}-S_{n+k}{x}_n\parallel \hfill \\ & \le \parallel x_n-x_{n+k}\parallel +\parallel x_{n+k}-S_{n+k}{x}_{n+k}\parallel +{\displaystyle \sum _{m=1}^N}\parallel S_m{x}_{n+k}-S_m{x}_n\parallel .\hfill \end{array}$$

Exploiting the uniform continuity of each $S_m$, one deduces from (34) and (38) that $\forall k,m\in \{1,\dots ,N\}$, $x_{n+k}-S_{n+k}{x}_{n+k}\to 0$ and $S_m{x}_{n+k}-S_m{x}_n\to 0$. Hence, one has that $x_n-S_{n+k}{x}_n\to 0\forall k\in \{1,\dots ,N\}$. So, it follows that $x_n-S_r{x}_n\to 0\forall r\in \{1,\dots ,N\}$. This along with $x_{n_k}\rightharpoonup z$, attains $z\in \widehat{\mathrm{Fix}}\left(S_r\right)=\mathrm{Fix}\left(S_r\right)\forall r\in \{1,...,N\}$. Consequently, z lies in ${\bigcap }_{k=1}^N\mathrm{Fix}\left(S_k\right)$. Additionally, from (36) and $x_{n_k}\rightharpoonup z$, one has that $z_{n_k}\rightharpoonup z$. Thus, using (35), we obtain $z\in \widehat{\mathrm{Fix}}\left(S_0\right)=\mathrm{Fix}\left(S_0\right)$. Consequently, $z\in {\bigcap }_{k=0}^N\mathrm{Fix}\left(S_k\right)$,

In what follows, we show that $z\in {\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)$. From (33), we have

$$(1-{\alpha }_n)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)\right]}^p+\tau {\left[\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)\right]}^p\}\le {\alpha }_n{D}_{f_p}(\widehat{u},u)+{\mathbf{\Gamma }}_n-{\mathbf{\Gamma }}_{n+1}+{\ell }_{n}M.$$

So, it follows that ${lim}_{n\to \infty }\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(u_n,y_n)={lim}_{n\to \infty }\frac{{\tilde{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\tilde{y}}_n)=0$, and, hence,

$$\underset{n\to \infty }{lim}{\tau }_n{D}_{f_p}(u_n,y_n)=\underset{n\to \infty }{lim}{\tilde{\tau }}_n{D}_{f_p}(w_n,{\tilde{y}}_n)=0.$$

(39)

By Lemma 13, one has that

$$\underset{n\to \infty }{lim}\parallel u_n-y_n\parallel =\underset{n\to \infty }{lim}\parallel w_n-{\tilde{y}}_n\parallel =0.$$

(40)

Applying (40) and Lemma 12, one obtains that z lies in ${\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)$. Thus, we obtain ${\omega }_w\left(x_n\right)\subset {\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)$. Consequently, ${\omega }_w\left(x_n\right)\subset \Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)\right)$. Lastly, we show that ${lim\; sup}_{n\to \infty }\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \le 0$. We can pick a subsequence $\left\{x_{n_j}\right\}$ of $\left\{x_n\right\}$ such that

$$\underset{n\to \infty }{lim\; sup}\langle J_E^{p}u-J_E^p\widehat{u},x_n-\widehat{u}\rangle =\underset{j\to \infty }{lim}\langle J_E^{p}u-J_E^p\widehat{u},x_{n_j}-\widehat{u}\rangle .$$

Because E is reflexive and $\left\{x_n\right\}$ is bounded, we might assume that $\left\{x_{n_j}\right\}$ converges weakly to $\tilde{z}$. This, along with $\tilde{z}\in \Omega$ and (6), arrives at

$$\underset{n\to \infty }{lim\; sup}\langle J_E^{p}u-J_E^p\widehat{u},x_n-\widehat{u}\rangle =\underset{j\to \infty }{lim}\langle J_E^{p}u-J_E^p\widehat{u},x_{n_j}-\widehat{u}\rangle =\langle J_E^{p}u-J_E^p\widehat{u},\tilde{z}-\widehat{u}\rangle \le 0.$$

(41)

In terms of (37), we have

$$\underset{n\to \infty }{lim\; sup}\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \le 0.$$

From (28) and (30), we obtain

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le (1-{\alpha }_n)D_{f_p}(\widehat{u},w_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ \hfill & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},u_n)+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ \hfill & \le (1-{\alpha }_n)[D_{f_p}(\widehat{u},x_n)+{ϵ}_n\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \hfill \\ \hfill & \times \parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel ]+{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ \hfill & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},x_n)+{ϵ}_n\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \hfill \\ \hfill & \times \parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel +{\alpha }_n\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \hfill \\ \hfill & =(1-{\alpha }_n)D_{f_p}(\widehat{u},x_n)+{\alpha }_n\{\frac{{ϵ}_n}{{\alpha }_n}\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \hfill \\ \hfill & \times \parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel +\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \}.\hfill \end{array}$$

(42)

Using uniform continuity of each $S_i(1\le i\le N)$ on C and uniform continuity of $J_E^p$ on bounded subsets of E, from (38) and the boundedness of $\left\{x_n\right\}$, we obtain

$$\underset{n\to \infty }{lim}\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel =0.$$

Noticing ${sup}_{n\ge 1}\frac{{ϵ}_n}{{\alpha }_n}<\infty$ and ${lim\; sup}_{n\to \infty }\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \le 0$, we infer that

$$\begin{array}{cc}& {\displaystyle \underset{n\to \infty }{lim\; sup}}\{\frac{{ϵ}_n}{{\alpha }_n}\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+S_n{x}_{n-1}-2S_n{x}_n\parallel +\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}\rangle \}\le 0.\hfill \end{array}$$

Because $\left\{{\alpha }_n\right\}$ lies in $(0,1)$ and ${\sum }_{n=1}^{\infty }{\alpha }_n$ diverges, by the application of Lemma 8 to (42), one obtains $D_{f_p}(\widehat{u},x_n)\to 0$, and, hence, $x_n\to \widehat{u}$.

    Aspect 2. Presume that $\exists \left\{{\mathbf{\Gamma }}_{n_k}\right\}\subset \left\{{\mathbf{\Gamma }}_n\right\}$ s.t. ${\mathbf{\Gamma }}_{n_k}<{\mathbf{\Gamma }}_{n_k+1}$ for all $k\ge 1$. Let the $\mathcal{N}$ indicate the natural-number set and $\psi :\mathcal{N}\to \mathcal{N}$ be formulated below

$$\psi \left(n\right):=max\{k\le n:{\mathbf{\Gamma }}_k<{\mathbf{\Gamma }}_{k+1}\}.$$

In terms of Lemma 7, one obtains

$$max\{{\mathbf{\Gamma }}_{\psi \left(n\right)},{\mathbf{\Gamma }}_n\}\le {\mathbf{\Gamma }}_{\psi \left(n\right)+1}.$$

(43)

From (28) and (31), it follows that

$$\begin{array}{cc}& D_{f_p}(w_{\psi \left(n\right)},u_{\psi \left(n\right)})+D_{f_p}(z_{\psi \left(n\right)},w_{\psi \left(n\right)})+{\gamma }_{\psi \left(n\right)}(1-{\gamma }_{\psi \left(n\right)}){\rho }_b^{*}\parallel J_E^p{x}_{\psi \left(n\right)}-J_E^p{g}_{\psi \left(n\right)}\parallel \hfill \\ & \le {\mathbf{\Gamma }}_{\psi \left(n\right)}-{\mathbf{\Gamma }}_{\psi \left(n\right)+1}+{\ell }_{\psi \left(n\right)}M+{\alpha }_{\psi \left(n\right)}\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\psi \left(n\right)}-\widehat{u}\rangle .\hfill \end{array}$$

Noticing $g_{\psi \left(n\right)}=J_{E^{*}}^q((1-{ϵ}_{\psi \left(n\right)})J_E^p{S}_{\psi \left(n\right)}{x}_{\psi \left(n\right)}+{ϵ}_{\psi \left(n\right)}{J}_E^p(2S_{\psi \left(n\right)}{x}_{\psi \left(n\right)}-S_{\psi \left(n\right)}{x}_{\psi \left(n\right)-1}))$ and $u_{\psi \left(n\right)}=J_{E^{*}}^q({\gamma }_{\psi \left(n\right)}{J}_E^p{x}_{\psi \left(n\right)}+(1-{\gamma }_{\psi \left(n\right)})J_E^p{g}_{\psi \left(n\right)}))$, we obtain that ${lim}_{n\to \infty }\parallel g_{\psi \left(n\right)}-x_{\psi \left(n\right)}\parallel =0$ and

$$\underset{n\to \infty }{lim}\parallel w_{\psi \left(n\right)}-u_{\psi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel z_{\psi \left(n\right)}-w_{\psi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel x_{\psi \left(n\right)}-S_{\psi \left(n\right)}{x}_{\psi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel u_{\psi \left(n\right)}-x_{\psi \left(n\right)}\parallel =0.$$

(44)

Also, from (28) and (30), one has

$$\begin{array}{cc}& (1-{\alpha }_{\psi \left(n\right)})({\beta }_{\psi \left(n\right)}-\xi )(1-{\beta }_{\psi \left(n\right)}){\rho }_b^{*}\parallel J_E^p{z}_{\psi \left(n\right)}-J_E^p{S}_0{z}_{\psi \left(n\right)}\parallel \hfill \\ & \le {\mathbf{\Gamma }}_{\psi \left(n\right)}-{\mathbf{\Gamma }}_{\psi \left(n\right)+1}+{\ell }_{\psi \left(n\right)}M+{\alpha }_{\psi \left(n\right)}\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\psi \left(n\right)}-\widehat{u}\rangle .\hfill \end{array}$$

Noticing $v_{\psi \left(n\right)}=J_{E^{*}}^q({\beta }_{\psi \left(n\right)}{J}_E^p{z}_{\psi \left(n\right)}+(1-{\beta }_{\psi \left(n\right)})J_E^p{S}_0{z}_{\psi \left(n\right)})$ and using the analogous reasonings to those in Aspect 1, one obtains

$$\underset{n\to \infty }{lim}\parallel z_{\psi \left(n\right)}-S_0{z}_{\psi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel v_{\psi \left(n\right)}-z_{\psi \left(n\right)}\parallel =0.$$

This together with (44) implies that

$$\underset{n\to \infty }{lim}\parallel v_{\psi \left(n\right)}-x_{\psi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel z_{\psi \left(n\right)}-x_{\psi \left(n\right)}\parallel =0.$$

(45)

Noticing ${\zeta }_{\psi \left(n\right)}=J_{E^{*}}^q({\alpha }_{\psi \left(n\right)}{J}_E^{p}u+(1-{\alpha }_{\psi \left(n\right)})J_E^p{v}_{\psi \left(n\right)})$, from (45), we obtain

$$\underset{n\to \infty }{lim}\parallel {\zeta }_{\psi \left(n\right)}-x_{\psi \left(n\right)}\parallel =0.$$

(46)

Using the analogous reasonings to those in Aspect 1, one obtains $x_{\psi \left(n\right)+1}-x_{\psi \left(n\right)}\to 0$,

$$\underset{n\to \infty }{lim}\parallel u_{\psi \left(n\right)}-y_{\psi \left(n\right)}\parallel =\underset{n\to \infty }{lim}\parallel w_{\psi \left(n\right)}-{\tilde{y}}_{\psi \left(n\right)}\parallel =0,$$

(47)

and

$$\underset{n\to \infty }{lim\; sup}\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\psi \left(n\right)}-\widehat{u}\rangle \le 0.$$

(48)

Using (42), we obtain

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{\psi \left(n\right)+1})\le (1-{\alpha }_{\psi \left(n\right)})D_{f_p}(\widehat{u},x_{\psi \left(n\right)})\hfill \\ \hfill & +{\alpha }_{\psi \left(n\right)}\{\frac{{ϵ}_{\psi \left(n\right)}}{{\alpha }_{\psi \left(n\right)}}\parallel J_E^p{S}_{\psi \left(n\right)}{x}_{\psi \left(n\right)}-J_E^p(2S_{\psi \left(n\right)}{x}_{\psi \left(n\right)}-S_{\psi \left(n\right)}{x}_{\psi \left(n\right)-1})\parallel \hfill \\ \hfill & \times \parallel \widehat{u}+S_{\psi \left(n\right)}{x}_{\psi \left(n\right)-1}-2S_{\psi \left(n\right)}{x}_{\psi \left(n\right)}\parallel +\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\psi \left(n\right)}-\widehat{u}\rangle \},\hfill \end{array}$$

(49)

which, together with (43), hence yields

$$\begin{array}{cc}& {\mathbf{\Gamma }}_{\psi \left(n\right)}\le \frac{{ϵ}_{\psi \left(n\right)}}{{\alpha }_{\psi \left(n\right)}}\parallel J_E^p{S}_{\psi \left(n\right)}{x}_{\psi \left(n\right)}-J_E^p(2S_{\psi \left(n\right)}{x}_{\psi \left(n\right)}-S_{\psi \left(n\right)}{x}_{\psi \left(n\right)-1})\parallel \hfill \\ & \times \parallel \widehat{u}+S_{\psi \left(n\right)}{x}_{\psi \left(n\right)-1}-2S_{\psi \left(n\right)}{x}_{\psi \left(n\right)}\parallel +\langle J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\psi \left(n\right)}-\widehat{u}\rangle .\hfill \end{array}$$

As a result, from (48), we deduce that

$$\underset{n\to \infty }{lim}{\mathbf{\Gamma }}_{\psi \left(n\right)}=0.$$

(50)

From (48)–(50), one has that

$$\underset{n\to \infty }{lim}{\mathbf{\Gamma }}_{\psi \left(n\right)+1}=0.$$

(51)

Again, using (43), one obtains ${\mathbf{\Gamma }}_n\to 0$. Thus, $x_n\to \widehat{u}$. This completes the proof.    □

Remark 1.

It can be easily seen from the proof of Theorem 2 that, if the assumption that ${lim}_{n\to \infty }\frac{{\ell }_n}{{\alpha }_n}=0$ is used in place of the one that ${lim}_{n\to \infty }{\ell }_n=0$ and ${sup}_{n\ge 1}\frac{{ϵ}_n}{{\alpha }_n}<\infty$, then Theorem 2 is still valid. It can be easily seen that the existing method in [25] is most closely related to our proposed method; that is, the hybrid projection method for resolving a single VIP with FPP constraint in [25] is extended to develop our modified inertial-type subgradient extragradient method for resolving a pair of VIPs with CFPP constraint. Compared with the corresponding results in [25], our results exhibit the novelty below:

First, the problem of finding a solution of a single VIP with FPP constraint (involving a Bregman relatively nonexpansive mapping) in [25] is extended to develop our problem of finding a solution of a pair of VIPs with CFPP constraint (involving finite Bregman relatively nonexpansive mappings and a Bregman relatively demicontractive mapping).

Second, the hybrid projection method with line-search process in [25] is extended to develop our modified inertial-type subgradient extragradient method with line-search process.

If we set $A_2=0$, then Algorithm 3 reduces to the iterative algorithm for finding an element of $\Omega =\mathrm{VI}(C,A_1)\cap \left({\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)\right)$.

Corollary 1.

Assume the conditions (C1)–(C3) with $A_2=0$, hold, and $\Omega =\mathrm{VI}(C,A_1)\cap \left({\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)\right)\ne \varnothing$. If $\left\{x_n\right\}$ is the fabricated sequence in Algorithm 5, and then $x_n\rightharpoonup z\in \Omega \iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

Algorithm 5: The 3rd modified inertial-type subgradient extragradient method

The 3rd modified inertial-type subgradient extragradient method. Initialization: Given $x_0,x_1\in C$ arbitrarily and let $ϵ>0,{\mu }_1>0,{\lambda }_1\in (0,\frac{1}{{\mu }_1}),l_1\in (0,1)$. Choose $\left\{{\ell }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ and $\left\{{\alpha }_n\right\}\subset (\xi ,1)$ s.t. ${\sum }_{n=1}^{\infty }{\ell }_n<\infty$, ${lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)>0$ and ${lim\; inf}_{n\to \infty }({\alpha }_n-\xi )(1-{\alpha }_n)>0$. Moreover, given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ s.t. $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where $$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{\ell }_n}{\parallel J_E^p{S}_n{x}_n-J_E^p(2S_n{x}_n-S_n{x}_{n-1})\parallel }\}\mathrm{if}{S}_n{x}_n\ne S_n{x}_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$$ Iterative steps: Calculate $x_{n+1}$ as follows: Step 1. Set $g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_n{x}_n+{ϵ}_n{J}_E^p(2S_n{x}_n-S_n{x}_{n-1}))$, and calculate $u_n=J_{E^{*}}^q({\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)$, $y_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{u}_n-{\lambda }_1{A}_1{u}_n)\right)$, $r_{{\lambda }_1}\left(u_n\right):=u_n-y_n$ and $s_n=u_n-{\tau }_n{r}_{{\lambda }_1}\left(u_n\right)$, with ${\tau }_n:=l_1^{k_n}$ and $k_n$ being the smallest $k\ge 0$ s.t. $$\langle A_1{u}_n-A_1(u_n-l_1^k{r}_{{\lambda }_1}\left(u_n\right)),u_n-y_n\rangle \le \frac{{\mu }_1}{2}{D}_{f_p}(u_n,y_n).$$ Step 2. Calculate $w_n={\Pi }_{C_n}\left(u_n\right)$, with $C_n:=\{y\in C:h_n\left(y\right)\le 0\}$ and $$h_n\left(y\right)=\langle A_1{s}_n,y-u_n\rangle +\frac{{\tau }_n}{2{\lambda }_1}{D}_{f_p}(u_n,y_n).$$ Step 3. Calculate $v_n=J_{E^{*}}^q({\alpha }_n{J}_E^p{w}_n+(1-{\alpha }_n)J_E^p\left(S_0{w}_n\right))$ and $x_{n+1}={\Pi }_{Q_n}\left(w_n\right)$, with $Q_n:=\{y\in C:D_{f_p}(y,v_n)\le D_{f_p}(y,w_n)\}$. Again put $n:=n+1$ and return to Step 1.

Next, let $S_1:E\to C$ be a Bregman relatively nonexpansive mapping and $S_i=S=I$ the identity mapping of E for $i=2,\dots ,N$. Then, we obtain $\Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^N\mathrm{Fix}\left(S_i\right)\right)=\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \mathrm{Fix}\left(S_1\right)$. In this case, Algorithm 4 reduces to the following iterative scheme for solving a pair of VIPs and the FPP of $S_1$. By Theorem 2, we obtain the following strong convergence result.

Corollary 2.

Suppose that the condition (C3) holds, and let $\Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \mathrm{Fix}\left(S_1\right)\ne \varnothing$. For initial $x_0,x_1\in C$, choose ${ϵ}_n$ s.t. $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where

$$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{\ell }_n}{\parallel J_E^p{S}_1{x}_n-J_E^p(2S_1{x}_n-S_1{x}_{n-1})\parallel }\}\mathrm{if}{S}_1{x}_n\ne S_1{x}_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$$

Suppose that $\left\{x_n\right\}$ is the sequence constructed by

$$\left\{\begin{array}{cc}& g_n=J_{E^{*}}^q((1-{ϵ}_n)J_E^p{S}_1{x}_n+{ϵ}_n{J}_E^p(2S_1{x}_n-S_1{x}_{n-1})),\hfill \\ & u_n=J_{E^{*}}^q({\gamma }_n{J}_E^p{x}_n+(1-{\gamma }_n)J_E^p{g}_n),\hfill \\ & y_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{u}_n-{\lambda }_1{A}_1{u}_n)\right),\hfill \\ & s_n=(1-{\tau }_n)u_n+{\tau }_n{y}_n,\hfill \\ & w_n={\Pi }_{C_n}{u}_n,\hfill \\ & {\tilde{y}}_n={\Pi }_C\left(J_{E^{*}}^q(J_E^p{w}_n-{\lambda }_2{A}_2{w}_n)\right),\hfill \\ & t_n=(1-{\tilde{\tau }}_n)w_n+{\tilde{\tau }}_n{\tilde{y}}_n,\hfill \\ & z_n={\Pi }_{{\tilde{C}}_n}{w}_n,\hfill \\ & x_{n+1}={\Pi }_C(J_{E^{*}}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{z}_n)\forall n\ge 1,\hfill \end{array}\right.$$

where ${\tau }_n:=l_1^{k_n},{\tilde{\tau }}_n:=l_2^{j_n}$ and $k_n,j_n$ are the smallest nonnegative integers k and j satisfying, respectively,

$$\langle A_1{u}_n-A_1(u_n-l_1^k(u_n-y_n)),u_n-y_n\rangle \le \frac{{\mu }_1}{2}{D}_{f_p}(u_n,y_n),$$

$$\langle A_2{w}_n-A_2(w_n-l_2^j(w_n-{\tilde{y}}_n)),w_n-{\tilde{y}}_n\rangle \le \frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\tilde{y}}_n),$$

and the sets $C_n,{\tilde{C}}_n$ are constructed below:

(i) 

$C_n:=\{y\in C:h_n\left(y\right)\le 0\}$ and $h_n\left(y\right)=\langle A_1{s}_n,y-u_n\rangle +\frac{{\tau }_n}{2{\lambda }_1}{D}_{f_p}(u_n,y_n)$;

(ii) 

${\tilde{C}}_n:=\{y\in C:{\tilde{h}}_n\left(y\right)\le 0\}$ and ${\tilde{h}}_n\left(y\right)=\langle A_2{t}_n,y-w_n\rangle +\frac{{\tilde{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\tilde{y}}_n)$.

Then, $x_n\to {\Pi }_{\Omega }u\iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

4. Examples

In what follows, we furnish an illustrative instance to back up the practicability and implementability of the suggested approaches. Put $ϵ=\frac{1}{3}$, ${\mu }_i=1$ and $l_i={\lambda }_i=\frac{1}{3}$ for $i=1,2$. We first provide an example of uniformly continuous and pseudomonotone mappings $A_i:E\to E^{*},i=1,2$, Bregman relatively nonexpansive mapping $S_1:C\to C$ and Bregman relatively demicontractive mapping $S_0:C\to C$ with $\Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^1\mathrm{Fix}\left(S_i\right)\right)\ne \varnothing$. Let $C=[-2,2]$ and $E=H=\mathbf{R}$ with the inner product $\langle a,b\rangle =ab$ and induced norm $\parallel ·\parallel =|·|$. The initial points $x_0,x_1$ are randomly chosen in C. For $i=1,2$, let $A_i:H\to H$ be defined as $A_{1}x:=\frac{1}{1+|sinx|}-\frac{1}{1+\left|x\right|}$ and $A_{2}x:=x+sinx$ for all $x\in H$. Now, we first show that $A_1$ is Lipschitz continuous and pseudomonotone. In fact, for each $y,g\in H$, one has

$$\begin{array}{cc}\parallel A_{1}y-A_{1}g\parallel \hfill & =|\frac{1}{1+\parallel siny\parallel }-\frac{1}{1+\parallel y\parallel }-\frac{1}{1+\parallel sing\parallel }+\frac{1}{1+\parallel g\parallel }|\hfill \\ & \le |\frac{\parallel g\parallel -\parallel y\parallel }{(1+\parallel y\parallel )(1+\parallel g\parallel )}|+|\frac{\parallel sing\parallel -\parallel siny\parallel }{(1+\parallel siny\parallel )(1+\parallel sing\parallel )}|\hfill \\ & \le \parallel y-g\parallel +\parallel siny-sing\parallel \le 2\parallel y-g\parallel .\hfill \end{array}$$

This ensures that $A_1$ is of Lipschitz continuity. Also, let us show that $A_1$ is of pseudomonotonicity. For each $y,g\in H$, it can be easily seen that

$$\begin{array}{cc}& \langle A_{1}y,g-y\rangle =(\frac{1}{1+|siny|}-\frac{1}{1+\left|y\right|})(g-y)\ge 0\hfill \\ & \Rightarrow \langle A_{1}g,g-y\rangle =(\frac{1}{1+|sing|}-\frac{1}{1+\left|g\right|})(g-y)\ge 0.\hfill \end{array}$$

It is readily known that $A_2$ is Lipschitz continuous and monotone. Indeed, we deduce that $\parallel A_{2}x-A_{2}y\parallel \le \parallel x-y\parallel +\parallel sinx-siny\parallel \le 2\parallel x-y\parallel$ and

$$\langle A_{2}x-A_{2}y,x-y\rangle ={\parallel x-y\parallel }^2+\langle sinx-siny,x-y\rangle \ge {\parallel x-y\parallel }^2-{\parallel x-y\parallel }^2=0.$$

Now, let $S_1:C\to C$ and $S_0:C\to C$ be defined as $S_{1}x=sinx$ and $S_{0}x=\frac{1}{5}x+\frac{3}{5}sinx$. It is easy to verify that $\mathrm{Fix}\left(S_1\right)=\mathrm{Fix}\left(S_0\right)=\left\{0\right\}$ and $S_1:C\to C$ is Bregman relatively nonexpansive. Also, $S_0:C\to C$ is Bregman relatively $\xi$-demicontractive with $\xi =\frac{1}{5}$. Indeed, note that

$$\parallel S_{0}x-S_0{y\parallel }^2=\parallel \frac{1}{5}(x-y)+\frac{3}{5}{(sinx-siny)\parallel }^2\le {\parallel x-y\parallel }^2+\frac{2}{5}{\parallel (I-S_0)x-(I-S_0)y\parallel }^2.$$

Consequently,

$$\Omega =(\bigcap _{i=1}^2\mathrm{VI}(C,A_i))\cap (\bigcap _{i=0}^1\mathrm{Fix}\left(S_i\right))=\left\{0\right\}\ne \varnothing .$$

In addition, putting ${\beta }_n=\frac{n+2}{2(n+1)}\forall n\ge 1$, we obtain

$$\underset{n\to \infty }{lim}({\beta }_n-\xi )(1-{\beta }_n)=\underset{n\to \infty }{lim}(\frac{n+2}{2(n+1)}-\frac{1}{5})(1-\frac{n+2}{2(n+1)})=(\frac{1}{2}-\frac{1}{5})(1-\frac{1}{2})=\frac{3}{20}>0$$

In this case, the conditions (C1)–(C3) are satisfied.

Example 1.

Let ${\ell }_n=\frac{1}{2{(n+1)}^2}$ and ${\alpha }_n={\beta }_n=\frac{n+2}{2(n+1)}\forall n\ge 1$. Given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ s.t. $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where

$$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{\ell }_n}{\parallel S_1{x}_n-S_1{x}_{n-1}\parallel }\}\mathrm{if}{S}_1{x}_n\ne S_1{x}_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$$

Algorithm 3 is rewritten as follows:

$$\left\{\begin{array}{cc}& g_n=S_1{x}_n+{ϵ}_n(S_1{x}_n-S_1{x}_{n-1}),\hfill \\ \hfill & u_n=\frac{n+2}{2(n+1)}{x}_n+\frac{n}{2(n+1)}{g}_n,\hfill \\ \hfill & y_n=P_C(u_n-\frac{1}{3}{A}_1{u}_n),\hfill \\ \hfill & s_n=(1-{\tau }_n)u_n+{\tau }_n{y}_n,\hfill \\ \hfill & w_n=P_{C_n}{u}_n,\hfill \\ \hfill & {\tilde{y}}_n=P_C(w_n-\frac{1}{3}{A}_2{w}_n),\hfill \\ \hfill & t_n={\tilde{\tau }}_n{\tilde{y}}_n+(1-{\tilde{\tau }}_n)w_n,\hfill \\ \hfill & v_n=\frac{n}{2(n+1)}{S}_0{w}_n+\frac{n+2}{2(n+1)}{w}_n,\hfill \\ \hfill & Q_n=\{y\in C:\parallel y-v_n\parallel \le \parallel y-w_n\parallel \},\hfill \\ \hfill & x_{n+1}=P_{{\tilde{C}}_n\cap Q_n}{w}_n\forall n\ge 1,\hfill \end{array}\right.$$

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with the sets $C_n,{\tilde{C}}_n$ and the step-sizes ${\tau }_n,{\tilde{\tau }}_n$ being picked as in Algorithm 3 for each n. Then, by Theorem 1, we deduce that $\left\{x_n\right\}$ converges to $0\in \Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^1\mathrm{Fix}\left(S_i\right)\right)$.

Example 2.

Let ${\ell }_n=\frac{1}{2{(n+1)}^2},{\alpha }_n=\frac{1}{2(n+1)}$ and ${\beta }_n={\gamma }_n=\frac{n+2}{2(n+1)}\forall n\ge 1$. Given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ s.t. $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where

$$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{\ell }_n}{\parallel S_1{x}_n-S_1{x}_{n-1}\parallel }\}\mathrm{if}{S}_1{x}_n\ne S_1{x}_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$$

Algorithm 4 is rewritten as follows:

$$\left\{\begin{array}{cc}& g_n=S_1{x}_n+{ϵ}_n(S_1{x}_n-S_1{x}_{n-1}),\hfill \\ \hfill & u_n=\frac{n+2}{2(n+1)}{x}_n+\frac{n}{2(n+1)}{g}_n,\hfill \\ \hfill & y_n=P_C(u_n-\frac{1}{3}{A}_1{u}_n),\hfill \\ \hfill & s_n=(1-{\tau }_n)u_n+{\tau }_n{y}_n,\hfill \\ \hfill & w_n=P_{C_n}{u}_n,\hfill \\ \hfill & {\tilde{y}}_n=P_C(w_n-\frac{1}{3}{A}_2{w}_n),\hfill \\ \hfill & t_n=(1-{\tilde{\tau }}_n)w_n+{\tilde{\tau }}_n{\tilde{y}}_n,\hfill \\ \hfill & z_n=P_{{\tilde{C}}_n}{w}_n,\hfill \\ \hfill & v_n=\frac{n+2}{2(n+1)}{z}_n+\frac{n}{2(n+1)}{S}_0{z}_n,\hfill \\ \hfill & x_{n+1}=P_C(\frac{1}{2(n+1)}u+\frac{2n+1}{2(n+1)}{v}_n),\hfill \end{array}\right.$$

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with the sets $C_n,{\tilde{C}}_n$ and the step-sizes ${\tau }_n,{\tilde{\tau }}_n$ are picked as in Algorithm 4 for each n. Then, by Theorem 2, we deduce that $\left\{x_n\right\}$ converges to $0\in \Omega =\left({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)\right)\cap \left({\bigcap }_{i=0}^1\mathrm{Fix}\left(S_i\right)\right)$.

It is worthy to emphasize that the above numerical example shows the competitive advantage of our suggested algorithms over the existing algorithms, e.g., the ones in [25]. In fact, we have provided the illustrative example of a pair of VIPs with CFPP constraint as above. Note that the existing method in [25] is only utilized for solving a single VIP with an FPP constraint. Hence, there is no way for this method to handle the above numerical example; that is, it is invalid for a pair of VIPs with a CFPP constraint. However, our suggested method can settle the above illustrative example. This reveals the competitive advantage of our proposed algorithms over the existing algorithms in the literature.

5. Conclusions

Let $1<q\le 2\le p<\infty$ with $\frac{1}{p}+\frac{1}{q}=1$ and let E be a p-uniformly convex and uniformly smooth Banach space. Then, its dual space $E^{*}$ is q-uniformly smooth Banach space with $1<q\le 2$. Utilizing the geometric properties of E and $E^{*}$, we design two inertial-type subgradient extragradient algorithms with the line-search process for solving the pseudomonotone variational inequality problems (VIPs) and common fixed-point problem (CFPP), where the geometric properties involve the properties of the generalized duality mappings $J_E^p,J_{E^{*}}^q$ and Bregman projection operator ${\Pi }_C$. Here, the CFPP indicates the common fixed-point problem of finite Bregman’s relative nonexpansivity mappings and a Bregman’s relative demicontractivity mapping in E. Under the properties of the generalized duality mappings $J_E^p,J_{E^{*}}^q$ and Bregman projection operator ${\Pi }_C$, we have proved that the sequences generated by the suggested algorithms converge weakly and strongly to a solution of a pair of VIPs with a CFPP constraint, respectively. Additionally, an illustrated example is furnished to demonstrate the feasibility and implementability of our proposed approaches. Compared with the corresponding results in [25], our results reveal the novelty as follows: (a) the problem of finding a solution of a single VIP with an FPP constraint (involving Bregman relatively nonexpansive mapping) in [25] is extended to develop our problem of finding a solution of a pair of VIPs with a CFPP constraint (involving finite Bregman relatively nonexpansive mappings and a Bregman relatively demicontractive mapping); (b) the hybrid projection method with a line-search process in [25] is extended to develop our modified inertial-type subgradient extragradient method with the line-search process. In the end, it is worthy to point out that part of our future research is aimed at attaining the weak and strong convergence results for the modifications of our proposed approaches with Nesterov double inertial-type extrapolation steps (see [26]) and adaptive stepsizes.