Parallel Subgradient-like Extragradient Approaches for Variational Inequality and Fixed-Point Problems with Bregman Relatively Asymptotical Nonexpansivity

Abstract

In a uniformly smooth and p-uniformly convex Banach space, let the pair of variational inequality and fixed-point problems (VIFPPs) consist of two variational inequality problems (VIPs) involving two uniformly continuous and pseudomonotone mappings and two fixed-point problems implicating two uniformly continuous and Bregman relatively asymptotically nonexpansive mappings. This article designs two parallel subgradient-like extragradient algorithms with an inertial effect for solving this pair of VIFPPs, where each algorithm consists of two parts which are of a mutually symmetric structure. With the help of suitable registrations, it is proven that the sequences generated by the suggested algorithms converge weakly and strongly to a solution of this pair of VIFPPs, respectively. Lastly, an illustrative instance is presented to verify the implementability and applicability of the suggested approaches.

1. Introduction

Let $Ø\ne C\subset H$ and $P_C$ be the metric projection from H onto C with C being convex and closed in a real Hilbert space H. Suppose that the $⟨·,·⟩$, and $\parallel ·\parallel$ are the inner product and induced norm in H, respectively, and given a nonlinear operator $S:C\to C$. We denote by $\mathrm{Fix}(S)$ the fixed-point set of S. Furthermore, $\mathbf{R},\rightharpoonup$, and → are used to represent the real number set, weak convergence, and strong convergence, respectively. A self-mapping S on C is known to possess asymptotical nonexpansivity if ∃ (nonnegative real sequence) $\left\{{\theta }_n\right\}$ such that

$$({\theta }_n+1)\parallel v-u\parallel \ge \parallel S^{n}v-S^{n}u\parallel \forall v,u\in C,n\ge 1,$$

(1)

with ${\theta }_n\to 0$. In particular, in case ${\theta }_n=0\forall n\ge 1$, S reduces to a nonexpansive mapping. Let $A:H\to H$ be a mapping. Recall that the so-called variational inequality problem (VIP) is to find $v\in C$ such that

$$⟨Av,x-v⟩\ge 0\forall x\in C,$$

(2)

Here, $\mathrm{VI}(C,A)$ denotes the set of solutions of the VIP. In 1976, under weaker assumptions, Korpelevich [1] put forward the extragradient rule for approximating an element of $\mathrm{VI}(C,A)$, i.e., for any starting $t_0\in C$, $\left\{t_n\right\}$ is the sequence generated by

$$\left\{\begin{array}{cc}& s_n=P_C(t_n-ϵA{t}_n),\hfill \\ & t_{n+1}=P_C(t_n-ϵA{s}_n)\forall n\ge 0,\hfill \end{array}\right.$$

with $ϵ\in (0,\frac{1}{L})$. If $\mathrm{VI}(C,A)\ne Ø$, then $\left\{t_n\right\}$ converges weakly to an element in $\mathrm{VI}(C,A)$. To the best of our understanding, the Korpelevich extragradient rule has been one of the most effective approaches for solving the VIP until now. The literature on the VIP is vast and the Korpelevich extragradient rule has acquired the extensive attention of numerous scholars, who ameliorated it in various ways (see, e.g., [2,3,4,5] for more details).

Recently, Thong and Hieu [6] put forth the inertial subgradient extragradient rule, i.e., for any starting $t_0,t_1\in H$, $\left\{t_n\right\}$ is the sequence generated by

$$\left\{\begin{array}{cc}& y_n=t_n+{\alpha }_n(t_n-t_{n-1}),\hfill \\ & s_n=P_C(y_n-𝓁A{y}_n),\hfill \\ & C_n=\{v\in H:⟨y_n-𝓁A{y}_n-s_n,v-s_n⟩\le 0\},\hfill \\ & t_{n+1}=P_{C_n}(y_n-𝓁A{s}_n)\forall n\ge 1,\hfill \end{array}\right.$$

with constant $𝓁\in (0,\frac{1}{L})$ and $\left\{{\alpha }_n\right\}\subset (0,1)$ (see, e.g., [7,8,9,10] for more details). Under suitable assumptions, they proved the weak convergence of $\left\{t_n\right\}$ to an element of $\mathrm{VI}(C,A)$. Subsequently, Thong and Hieu [11] introduced two inertial subgradient extragradient algorithms with a line-search process for solving the VIP with Lipschitz-continuous monotone mapping A and the fixed-point problem (FPP) of a quasi-nonexpansive mapping S with a demiclosedness property in H, that is, the following Algorithms 1 and 2, which are specified concretely.

Algorithm 1: (The 1st inertial subgradient extragradient algorithm. See Algorithm 1 [11]).

Initialization: Given $\gamma >0,l\in (0,1),\mu \in (0,1)$ and $\left\{{\alpha }_n\right\}\subset (0,1)$. Choose any initial $t_0,t_1\in H$. Iterations: Compute $t_{n+1}$ below: Step 1. Put $s_n=t_n+{\alpha }_n(t_n-t_{n-1})$ and calculate $y_n=P_C(s_n-{𝓁}_{n}A{s}_n)$, wherein ${𝓁}_n$ is picked as the largest $𝓁\in \{\gamma ,\gamma l,\gamma l^2,\dots \}$ such that $𝓁\parallel A{s}_n-A{y}_n\parallel \le \mu \parallel s_n-y_n\parallel$. Step 2. Calculate $u_n=P_{C_n}(s_n-{𝓁}_{n}A{y}_n)$, where $C_n:=\{u\in H:⟨s_n-{𝓁}_{n}A{s}_n-y_n,u-y_n⟩\le 0\}$. Step 3. Calculate $t_{n+1}=(1-{\beta }_n)s_n+{\beta }_{n}S{u}_n$. When $s_n=u_n=t_{n+1}$, we have $s_n\in \mathrm{Fix}(T)\cap \mathrm{VI}(C,A)$. Put $n:=n+1$ and return to Step 1.

Algorithm 2: (The 2nd inertial subgradient extragradient algorithm. See Algorithm 2 [11]).

Initialization: Given $\gamma >0,l\in (0,1),\mu \in (0,1)$ and $\left\{{\alpha }_n\right\}\subset (0,1)$. Choose any initial $t_0,t_1\in H$. Iterative steps: Compute $t_{n+1}$ below: Step 1. Putt $s_n=t_n+{\alpha }_n(t_n-t_{n-1})$ and calculate $y_n=P_C(s_n-{𝓁}_{n}A{s}_n)$, wherein ${𝓁}_n$ is picked as the largest $𝓁\in \{\gamma ,\gamma l,\gamma l^2,\dots \}$ such that $𝓁\parallel A{s}_n-A{y}_n\parallel \le \mu \parallel s_n-y_n\parallel$. Step 2. Calculate $u_n=P_{C_n}(s_n-{𝓁}_{n}A{y}_n)$, where $C_n:=\{u\in H:⟨s_n-{𝓁}_{n}A{s}_n-y_n,u-y_n⟩\le 0\}$. Step 3. Calculate $t_{n+1}=(1-{\beta }_n)t_n+{\beta }_{n}S{u}_n$. When $s_n=u_n=t_n=t_{n+1}$, we have $t_n\in \mathrm{Fix}(T)\cap \mathrm{VI}(C,A)$. Put $n:=n+1$ and return to Step 1.

With the help of suitable assumptions, it was proven in [11] that the sequences generated by the suggested algorithms converge weakly to a point in $\mathrm{VI}(C,A)\cap \mathrm{Fix}(S)$. Besides, exploiting the subgradient extragradient approach along with Halpern’s iterative technique, Kraikaew and Saejung [12] designed Halpern’s subgradient extragradient approach for settling the VIP, and showed that the sequence generated by the proposed rule converges strongly to a point in $\mathrm{VI}(C,A)$. Recently, Reich et al. [13] put forward both gradient-projection schemes for handling the VIP for a mapping of both uniform continuity and pseudomonotonicity. Particularly, they employed a new Armijo-like linesearch term to achieve a hyper-plane strictly separating the present iterate from the solution set of the VIP considered. It was proven in [13] that the sequences constructed by both schemes are convergent weakly and strongly to an element of $\mathrm{VI}(C,A)$, respectively.

On the other hand, let $Ø\ne C\subset E$ where C is convex and closed in a uniformly smooth and p-uniformly convex Banach space E for $p,q>1$ satisfying $\frac{1}{p}+\frac{1}{q}=1$. Let $J_E^p$ be the duality mapping of E, and let $E^{\ast }$ be the dual of E with the duality $J_{E^{\ast }}^q$. Suppose that the norm and the duality pairing between E and $E^{\ast }$ are denoted by $\parallel ·\parallel$ and $⟨·,·⟩$, respectively. Let $f_p(u)={\parallel u\parallel }^p/p\forall u\in E$, $D_{f_p}$ be the Bregman distance with respect to $f_p$ and ${\Pi }_C$ be Bregman’s projection with respect to $f_p$ from E onto C, and presume that $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ such that ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$, ${lim}_{n\to \infty }{\alpha }_n=0$ and $0<{lim\; inf}_{n\to \infty }{\beta }_n\le {lim\; sup}_{n\to \infty }{\beta }_n<1$. Assume that $A:E\to E^{\ast }$ is uniformly continuous and pseudomonotone operator and S is Bregman relatively nonexpansive self-mapping on C. Very recently, inspired by the research works in [13], Eskandani et al. [14] proposed the hybrid projection approach with a line-search process for approximating a point in $\mathrm{VI}(C,A)\cap \mathrm{Fix}(S)$ (see also [15,16,17,18]), that is, the following Algorithm 3, which is specified concretely.

Algorithm 3: (Hybrid projection approach. See [14]).

Initialization: Given $l\in (0,1),\nu >0,\lambda \in (0,\frac{1}{\nu })$ and choose $u,t_1\in C$ randomly. Iterations: Compute $t_{n+1}(n\ge 1)$ below: Step 1. Calculate $y_n={\Pi }_C(J_{E^{\ast }}^q(J_E^p{t}_n-\lambda A{t}_n))$ and $r_{\lambda }(t_n):=t_n-y_n$. If $r_{\lambda }(t_n)=0$ and $S{t}_n=t_n$, then stop; $t_n\in \Omega =\mathrm{VI}(C,A)\cap \mathrm{Fix}(S)$. If this case does not occur, then, Step 2. Calculate $s_n=t_n-{ϵ}_n{r}_{\lambda }(t_n)$, with both ${ϵ}_n:=l^{k_n}$ and $k_n$ being the smallest $k\ge 0$ such that $⟨A{t}_n-A(t_n-l^k{r}_{\lambda }(t_n)),r_{\lambda }(t_n)⟩\le \frac{\nu }{2}{D}_{f_p}(t_n,y_n)$. Step 3. Calculate $u_n=J_{E^{\ast }}^q({\beta }_n{J}_E^p{t}_n+(1-{\beta }_n)J_E^p(S{\Pi }_{C_n}{t}_n))$ and $t_{n+1}={\Pi }_C(J_{E^{\ast }}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{u}_n))$, with $C_n:=\{y\in C:0\ge h_n(y)\}$ and $h_n(y)=⟨A{s}_n,y-t_n⟩+\frac{{ϵ}_n}{2\lambda }{D}_{f_p}(t_n,y_n)$. Again put $n:=n+1$ and return to Step 1.

With the help of suitable conditions, it was proven in [14] that $\left\{t_n\right\}$ converges strongly to ${\Pi }_{\Omega }u$. Meanwhile, Wang et al. [19] put forward modified inertial-type subgradient extragradient method with linear-search process for handling the two pseudomonotone variational inequality problems (VIPs) and the common fixed point problem (CFPP) of finite Bregman relatively nonexpansive operators and Bregman relatively demicontractive operator in E. By the virtue of suitable restrictions, it was shown in [19] that the sequences constructed by both suggested schemes converge weakly and strongly to a solution of a pair of VIPs with CFPP constraint, respectively. In this article, two parallel subgradient-like extragradient algorithms (with an inertial effect) are designed for resolving a pair of variational inequality and fixed-point problems (VIFPPs) in E. Here, two variational inequality problems (VIPs) involve two uniformly continuous pseudomonotone operators and two fixed point problems implicate two uniformly continuous Bregman relatively asymptotically nonexpansive mappings. Furthermore, each algorithm consists of two parts which are of symmetric structure mutually. It is worth mentioning that the hybrid projection method with linear-search process for resolving a single VIFPP in [14] is extended to develop our parallel subgradient-like extragradient method with an inertial effect for resolving a pair of VIFPPs. Moreover, the modified inertial-type subgradient extragradient method with linear-search process for resolving a pair of VIPs with CFPP constraint (involving finite Bregman relatively nonexpansive operators and Bregman relatively demicontractive operator) in [19] is extended to develop our parallel subgradient-like extragradient method with an inertial effect for resolving a pair of VIFPPs (involving two Bregman relatively asymptotically nonexpansive operators). Additionally, with the help of appropriate registrations, it is proven that the sequences generated by the suggested algorithms converge weakly and strongly to a solution of this pair of VIFPPs, respectively. Lastly, an illustrative instance is presented to verify the implementability and applicability of the proposed approaches.

The structure of the article is as follows: Section 2 presents certain terminologies and preliminary results for later use. Section 3 is focused on discussing the convergence of the suggested algorithms. In Section 4, the major outcomes are utilized to deal with the CFPP and VIPs in an illustrative instance. Our results improve and develop the relevant results obtained previously in [11,13,14].

2. Preliminaries

Let ($E,\parallel ·\parallel$) be a real Banach space, whose dual is denoted by $E^{\ast }$. We use the $y_n\to y$ and $y_n\rightharpoonup y$ to represent the strong and weak convergence of $\left\{y_n\right\}$ to $y\in E$, respectively. Moreover, the set of weak cluster points of $\left\{y_n\right\}$ is denoted by ${\omega }_w(y_n)$, i.e., ${\omega }_w(y_n)=\{y^{†}\in E:y_{n_k}\rightharpoonup y^{†}\mathrm{for}\mathrm{some}\left\{y_{n_k}\right\}\subset \left\{y_n\right\}\}$. Let $U=\{y\in E:\parallel y\parallel =1\}$ 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 $y,x\in U$ with $y\ne x$, we have $\parallel y+x\parallel /2<1$. E is referred to as being uniformly convex if $\forall \varsigma \in (0,2]$, $\exists \delta >0$ such that $\forall y,x\in U$ with $\parallel y-x\parallel \ge \varsigma$ we have $\parallel y+x\parallel /2\le 1-\delta$. It is clear that the uniform convexity of a Banach space implies its reflexivity and strict convexity. The modulus of convexity of E is the function $\delta :[0,2]\to [0,1]$ defined by $\delta (\varsigma )=inf\{1-\parallel y+x\parallel /2:y,x\in U\mathrm{with}\parallel y-x\parallel \ge \varsigma \}$. It is also known that E is uniformly convex if and only if $\delta (\varsigma )>0\forall \varsigma \in (0,2]$. Moreover, E is referred to as being p-uniformly convex if $\exists c>0$ such that $\delta (\varsigma )\ge c{\varsigma }^p$ for $0\le \varsigma \le 2$.

The nonnegative function ${\rho }_E(·)$ on $[0,\infty )$ is called the modulus of smoothness of E if ${\rho }_E(\varsigma ):=sup\{(\parallel y+\varsigma x\parallel +\parallel y-\varsigma x\parallel )/2-1:y,x\in U\}\forall \varsigma \in [0,\infty )$. E is said to be uniformly smooth if ${lim}_{\varsigma \to 0}{\rho }_E(\varsigma )/\varsigma =0$, and q-uniformly smooth if $\exists C_q>0$ such that ${\rho }_E(\varsigma )\le C_q{\varsigma }^q\forall \varsigma >0$. Recall that E has p-uniform convexity if and only if $E^{\ast }$ has q-uniform smoothness (see, e.g., [20] for more details). Substituting $B(0,r)=\{y\in E:\parallel y\parallel \le r\}$ for each $r>0$, we say that $f:E\to \mathbf{R}$ is uniformly convex on bounded sets (see [14]) if ${\rho }_r(\varsigma )>0\forall r,\varsigma >0$, where ${\rho }_r(\varsigma ):[0,\infty )\to [0,\infty ]$ is specified below

$$\begin{array}{cc}{\rho }_r(\varsigma )\hfill & =inf\left\{\right[ϵf(y)+(1-ϵ)f(x)-f(ϵy+(1-ϵ)x)]/ϵ(1-ϵ):\hfill \\ & ϵ\in (0,1)\mathrm{and}y,x\in B(0,r)\mathrm{with}\parallel y-x\parallel =\varsigma \}\forall \varsigma \ge 0.\hfill \end{array}$$

The ${\rho }_r$ is known as the gauge function of f with uniform convexity. It is clear that the gauge ${\rho }_r$ is nondecreasing.

Let $f:E\to \mathbf{R}$ be a convex function. If the limit ${lim}_{\varsigma \to 0^{+}}\frac{f(y+\varsigma x)-f(y)}{\varsigma }$ exists for each $x\in E$, then f is referred to as being Gâteaux differentiable at y. In this case, the gradient $\nabla f(y)$ of f at y has linearity and is formulated as $⟨\nabla f(y),x⟩:={lim}_{\varsigma \to 0^{+}}\frac{f(y+\varsigma x)-f(y)}{\varsigma }\forall x\in E$. f is referred to as having Gâteaux differentiability if it has Gâteaux differentiability at any $y\in E$. In case ${lim}_{\varsigma \to 0^{+}}\frac{f(y+\varsigma x)-f(y)}{\varsigma }$ is achieved uniformly for any $x\in U$, we say that f has Fréchet differentiability at y. Additionally, f is referred to as having uniform Fréchet differentiability on a subset $K\subset E$ if ${lim}_{\varsigma \to 0^{+}}\frac{f(y+\varsigma x)-f(y)}{\varsigma }$ is achieved uniformly for $(y,x)\in K\times U$. When the norm of E has Gâteaux differentiability, E is said to possess smoothness.

Let $\frac{1}{p}+\frac{1}{q}=1$ for $p,q>1$. The $J_E^p:E\to E^{\ast }$ is specified below

$$J_E^p(y)=\{\phi \in E^{\ast }:⟨\phi ,y⟩={\parallel y\parallel }^p\mathrm{and}\parallel \phi \parallel ={\parallel y\parallel }^{p-1}\}\forall y\in E.$$

It is known that E has smoothness if and only if $J_E^p$ has a single value from E into $E^{\ast }$. Furthermore, E has reflexivity if and only if $J_E^p$ has surjectivity, and E is strictly convex if and only if $J_E^p$ has a one-to-one property. So, it follows that when the smooth Banach space E has both strict convexity and reflexivity, $J_E^p$ is the bijection and in this case, $J_{E^{\ast }}^q={(J_E^p)}^{-1}$. Furthermore, recall that E has uniform smoothness if and only if the function $f_p(y)={\parallel y\parallel }^p/p$ has uniform Fréchet differentiability on bounded sets if and only if $J_E^p$ has uniform continuity on bounded sets. Moreover, E has uniform convexity if and only if the function $f_p$ has uniform convexity (see [20]).

Let the function $f:E\to \mathbf{R}$ possess both Gâteaux differentiability and convexity. Bregman’s distance with respect to f is specified below

$$D_f(t,s):=f(t)-f(s)-⟨\nabla f(s),t-s⟩\forall t,s\in E.$$

It is worth mentioning that $D_f(·,·)$ is not a metric in the common usage of the terminology. Evidently, $D_f(t,t)=0$, but $D_f(t,s)=0$ cannot lead to $t=s$. Generally, $D_f$ does not possess symmetry and possess fulfill any triangle inequality. However, $D_f$ fulfills the three-point equality

$$D_f(t,s)+D_f(s,u)=D_f(t,u)-⟨\nabla f(s)-\nabla f(u),t-s⟩.$$

See [21] for more details.

It is remarkable that the $J_E^p$ on the smooth E is Gâteaux’s derivative of $f_p$. Thus, Bregman’s distance with respect to $f_p$ is specified below

$$\begin{array}{cc}{D}_{f_p}(y,x)\hfill & ={\parallel y\parallel }^p/p-{\parallel x\parallel }^p/p-⟨J_E^p(x),y-x⟩\hfill \\ & ={\parallel y\parallel }^p/p+{\parallel x\parallel }^p/q-⟨J_E^p(x),y⟩\hfill \\ & ={(\parallel x\parallel }^p-{\parallel y\parallel }^p)/q-⟨J_E^p(x)-J_E^p(y),y⟩.\hfill \end{array}$$

In the p-uniformly convex and smooth Banach space E for $2\le p<\infty$, there holds the following relationship between the metric and Bregman distance:

$${\tau \parallel y-x\parallel }^p\le D_{f_p}(y,x)\le ⟨J_E^p(y)-J_E^p(x),y-x⟩,$$

(3)

where $\tau >0$ is some fixed number (see [22]). Via (3), it can be easily seen that for each $\left\{y_n\right\}\subset E$ of boundedness, the relation is valid:

$$y_n\to y\iff D_{f_p}(y_n,y)\mathrm{converges}\mathrm{to}0\mathrm{as}n\to \infty .$$

Let $Ø\ne C\subset E$, with C being convex and closed in a strictly convex, smooth, and reflexive Banach space E. Bregman’s projection is formulated as minimizers of Bregman’s distance. Bregman’s projection of $y\in E$ onto C with respect to $f_p$ indicates a unique point ${\Pi }_{C}y\in C$ such that $D_{f_p}({\Pi }_{C}y,y)={min}_{x\in C}{D}_{f_p}(x,y)$. In the case of Hilbert space, Bregman’s projection with respect to $f_2$ reduces to the metric projection. Using Theorem 2.1 [23] and Corollary 4.4 [24] in a uniformly convex Banach space, the characterization of Bregman’s projection is formulated by:

$$⟨J_E^p(y)-J_E^p({\Pi }_{C}y),x-{\Pi }_{C}y⟩\le 0\forall x\in C.$$

(4)

Additionally, (4) is equivalent to the descent property

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

(5)

When $p=2$, $J_E^p$ reduces to the normalized duality mapping and is written as J. The $\varphi :E^2\to \mathbf{R}$ is formulated below

$$\varphi (t,s)={\parallel t\parallel }^2-2⟨Js,t⟩+{\parallel s\parallel }^2\forall t,s\in E,$$

and ${\Pi }_C(t)={\mathrm{argmin}}_{s\in C}\varphi (s,t)\forall t\in E$.

In terms of [14], the function $V_{f_p}:E\times E^{\ast }\to [0,\infty )$ associated with $f_p$ is specified below

$$V_{f_p}(y,y^{\ast })={\parallel y\parallel }^p/p-⟨y^{\ast },y⟩+{\parallel y^{\ast }\parallel }^q/q\forall (y,y^{\ast })\in E\times E^{\ast }.$$

(6)

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

$$V_{f_p}(y,y^{\ast })+⟨x^{\ast },J_{E^{\ast }}^q(y^{\ast })-y⟩\le V_{f_p}(y,y^{\ast }+x^{\ast })\forall y\in E,y^{\ast },x^{\ast }\in E^{\ast }.$$

(7)

In addition, $V_{f_p}$ is convex in the second variable. Hence, we have

$$D_{f_p}(z,J_{E^{\ast }}^q(\sum _{j=1}^n{\varsigma }_j{J}_E^p(y_j))\le \sum _{j=1}^n{\varsigma }_j{D}_{f_p}(z,y_j)\forall z\in E,{\left\{y_j\right\}}_{j=1}^n\subset E,{\left\{{\varsigma }_j\right\}}_{j=1}^n\subset [0,1]\mathrm{with}\sum _{j=1}^n{\varsigma }_j=1.$$

(8)

In what follows, the hybrid projection method with linear-search process for resolving a single VIFPP in [14] is extended to develop our parallel subgradient-like extragradient method with inertial effect for resolving a pair of VIFPPs. So, we will conduct convergence analysis of our proposed algorithms in p-uniformly convex and uniformly smooth Banach spaces, which are more general than Hilbert spaces. To successfully perform convergence analysis, we need to make use of Lemmas 1–7 below. In addition, it was shown in [14] that Lemma 5 holds. However, for convenience, we still give its proof.

Lemma 1

([23]). Let E be a uniformly convex Banach space and $\left\{s_n\right\},\left\{t_n\right\}$ be two sequences in E such that the first is bounded. If ${lim}_{n\to \infty }{D}_{f_p}(t_n,s_n)=0$, then ${lim}_{n\to \infty }\parallel t_n-s_n\parallel =0$.

Assume that S is a self-mapping on C. Let $\mathrm{Fix}(S)$ indicate the set of fixed points of S, that is, $\mathrm{Fix}(S)=\{y\in C:y=Sy\}$. A point $y^{†}\in C$ is referred to as an asymptotic fixed point of S if $\exists \left\{y_n\right\}\subset C$ such that $y_n\rightharpoonup y^{†}$ and $(I-S)y_n\to 0$. Let $\widehat{\mathrm{Fix}}(S)$ denote the asymptotic fixed-point set of S. The terminology of asymptotic fixed points was invented in [25]. A self-mapping S on C is known to have Bregman’s relatively asymptotical nonexpansivity with respect to $f_p$ if $\mathrm{Fix}(S)=\widehat{\mathrm{Fix}}(S)\ne Ø$, and $\exists \left\{{\theta }_n\right\}\subset [0,\infty )$ with both ${\theta }_n\to 0(n\to \infty )$ and

$$({\theta }_n+1)D_{f_p}(y,x)\ge D_{f_p}(y,S^{n}x)\forall y\in \mathrm{Fix}(S),x\in C,n\ge 1.$$

In particular, if ${\theta }_n=0\forall n\ge 1$, then S reduces to having Bregman’s relatively nonexpansivity with respect to $f_p$, that is, $\mathrm{Fix}(S)=\widehat{\mathrm{Fix}}(S)\ne Ø$ and $D_{f_p}(y,Sx)\le D_{f_p}(y,x)\forall y\in \mathrm{Fix}(S),x\in C$. In addition, a mapping $A:C\to E^{\ast }$ is known as being

(i)

monotone on C if $⟨Av-Ay,v-y⟩\ge 0\forall v,y\in C$;

(ii)

pseudo-monotone if $⟨Ay,v-y⟩\ge 0\Rightarrow ⟨Av,v-y⟩\ge 0\forall v,y\in C$;

(iii)

ℓ-Lipschitz continuous or ℓ-Lipschitzian if $\exists 𝓁>0$ such that $\parallel At-Ay\parallel \le 𝓁\parallel t-y\parallel \forall t,y\in C$;

(iv)

weakly sequentially continuous if $\forall \left\{t_n\right\}\subset C$, the relation holds: $t_n\rightharpoonup t\Rightarrow A{t}_n\rightharpoonup At$.

Lemma 2

([14]). Let $r>0$ be a constant and suppose that $f:E\to \mathbf{R}$ is a uniformly convex function on any bounded subset of a Banach space E. Then,

$$f(\sum _{k=1}^n{\alpha }_k{t}_k)\le \sum _{k=1}^n{\alpha }_{k}f(t_k)-{\alpha }_i{\alpha }_j{\rho }_r(\parallel t_i-t_j\parallel ),$$

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

Proof. 

It is easy to show the conclusion.    □

Lemma 3

([5]). Let $E_i$ be a Banach space for $i=1,2$ and suppose that $A:E_1\to E_2$ has uniform continuity on any bounded subset of $E_1$ and $D\subset E_1$ has boundedness. Then, $A(D)\subset E_2$ has boundedness.

Lemma 4

([26]). Assume $Ø\ne C\subset E$ with C being convex and closed, and let $A:C\to E^{\ast }$ be of both pseudomonotonicity and continuity. Given $y^{†}\in C$. Then $⟨A{y}^{†},y-y^{†}⟩\ge 0\forall y\in C\iff ⟨Ay,y-y^{†}⟩\ge 0\forall y\in C$.

Lemma 5.

Suppose that E is a smooth and p-uniformly convex Banach space for $p\ge 2$, where $J_E^p$ has weakly sequential continuity. Assume $\left\{s_n\right\}\subset E$ and $Ø\ne \Omega \subset E$. If ${\omega }_w(s_n)\subset \Omega$, then $\left\{D_{f_p}(z,s_n)\right\}$ converges for each $z\in \Omega$. Then we have the weak convergence of $\left\{s_n\right\}$ to an element of Ω.

Proof. 

First, we have $\tau \parallel z-s_n{\parallel }^p\le D_{f_p}(z,s_n)\forall z\in \Omega$ by (3). Thus, we obtain that $\left\{s_n\right\}$ possesses boundedness. Because E is reflexive, we obtain ${\omega }_w(s_n)\ne Ø$. Furthermore, we claim that $\left\{s_n\right\}$ converges weakly to an element of $\Omega$. Indeed, let $\overline{s},\widehat{s}\in {\omega }_w(s_n)$ with $\overline{s}\ne \widehat{s}$. Then, $\exists \left\{s_{n_k}\right\}\subset \left\{s_n\right\}$ and $\exists \left\{s_{m_k}\right\}\subset \left\{s_n\right\}$ such that $s_{n_k}\rightharpoonup \overline{s}$ and $s_{m_k}\rightharpoonup \widehat{s}$. Because $J_E^p$ is weakly sequentially continuous, we obtain both $J_E^p(s_{n_k})\rightharpoonup J_E^p\overline{s}$ and $J_E^p(s_{m_k})\rightharpoonup J_E^p\widehat{s}$. Note that $D_{f_p}(\overline{s},\widehat{s})+D_{f_p}(\widehat{s},s_n)=D_{f_p}(\overline{s},s_n)-⟨J_E^p\widehat{s}-J_E^p{s}_n,\overline{s}-\hat{s}⟩$. So, utilizing the convergence of the sequences $\left\{D_{f_p}(\overline{s},s_n)\right\}$ and $\left\{D_{f_p}(\widehat{s},s_n)\right\}$, we conclude that

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

which hence yields $⟨J_E^p\overline{s}-J_E^p\widehat{s},\overline{s}-\widehat{s}⟩=0$. From (3) we obtain $0<\tau \parallel \overline{s}-\widehat{s}{\parallel }^p\le D_{f_p}(\overline{s},\widehat{s})\le ⟨J_E^p\overline{s}-J_E^p\widehat{s},\overline{s}-\widehat{s}⟩=0$. This arrives at a contradiction. Therefore, this means that $\left\{s_n\right\}$ converges weakly to an element of $\Omega$.    □

It was proven in [27] that the following lemma holds in ${\mathbf{R}}^m$. It is not difficult to check that it still holds in a Banach space.

Lemma 6.

Assume $Ø\ne C\subset E$ with C is convex and closed. Suppose that $K:=\{y\in C:h(y)\le 0\}$ where $h:E\to \mathbf{R}$ is defined on E. If $K\ne Ø$ and h are Lipschitz continuous on C with modulus $\theta >0$, then $\theta \mathrm{dist}(x,K)\ge max\{h(x),0\}\forall x\in C$, where $\mathrm{dist}(x,K)$ represents the distance of x to K.

Lemma 7

([28]). Let $\left\{{\mathbf{\Gamma }}_n\right\}$ be a sequence of real numbers that does not decrease at infinity in the sense that, $\exists \left\{{\mathbf{\Gamma }}_{n_k}\right\}\subset \left\{{\mathbf{\Gamma }}_n\right\}$ such that ${\mathbf{\Gamma }}_{n_k}<{\mathbf{\Gamma }}_{n_k+1}$ for all k. Assume that ${\left\{\phi (n)\right\}}_{n\ge n_0}$ is defined as $\phi (n)=max\{k\le n:{\mathbf{\Gamma }}_k<{\mathbf{\Gamma }}_{k+1}\}$, with integer $n_0\ge 1$ satisfying $\{k\le n_0:{\mathbf{\Gamma }}_k<{\mathbf{\Gamma }}_{k+1}\}\ne Ø$. Then, the following hold:

(i) 

$\phi (n_0)\le \phi (n_0+1)\le \cdots$ and $\phi (n)\to \infty$;

(ii) 

${\mathbf{\Gamma }}_{\phi (n)}\le {\mathbf{\Gamma }}_{\phi (n)+1}$ and ${\mathbf{\Gamma }}_n\le {\mathbf{\Gamma }}_{\phi (n)+1}\forall n\ge n_0$.

Lemma 8

([29]). Let $\left\{{\sigma }_n\right\}$ be a sequence in $[0,\infty )$ satisfying ${\sigma }_{n+1}\le (1-{\mu }_n){\sigma }_n+{\mu }_n{c}_n\forall n\ge 1$, with $\left\{{\mu }_n\right\}$ and $\left\{c_n\right\}$ being real sequences satisfying the conditions: (i) $\left\{{\mu }_n\right\}\subset [0,1]$ and ${\sum }_{n=1}^{\infty }{\mu }_n=\infty$; and (ii) ${lim\; sup}_{n\to \infty }{c}_n\le 0$ or ${\sum }_{n=1}^{\infty }\left|{\mu }_n{c}_n\right|<\infty$. Then, ${\sigma }_n\to 0$ as $n\to \infty$.

Lemma 9

([30]). 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 ${\sum }_{n=1}^{\infty }{b}_n<\infty$, then ${lim}_{n\to \infty }{a}_n$ exists.

3. Main Results

In this section, let $Ø\ne C\subset E$ with C being convex and closed in uniformly smooth and p-uniformly convex Banach space E for $p\ge 2$. We are now in a position to present and analyze our iterative algorithms for approximating a common solution of a pair of VIFPPs, where each algorithm consists of two parts of a mutually symmetric structure. Assume always that the following conditions hold:

(C1)

$S_1,S_2:C\to C$ are the mappings of both uniform continuity and Bregman’s relatively asymptotical nonexpansivity with sequences ${\left\{{\theta }_n\right\}}_{n=1}^{\infty }$ and ${\left\{{\overline{\theta }}_n\right\}}_{n=1}^{\infty }$, respectively.

(C2)

For $i=1,2$, $A_i:E\to E^{\ast }$ has both uniform continuity and pseudomonotonicity on C such that $\parallel A_i{y}^{†}\parallel \le {lim\; inf}_{n\to \infty }\parallel A_i{y}_n\parallel \forall \left\{y_n\right\}\subset C$ with $y_n\rightharpoonup y^{†}$.

(C3)

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

The following lemmas are used in the proofs of our main results in the following.

Lemma 10.

Suppose that $\left\{x_n\right\}$ is the sequence constructed in Algorithm 4. Then, the following hold:  $\frac{1}{{\lambda }_1}{D}_{f_p}(s_n,y_n)\le ⟨A_1{s}_n,e_{{\lambda }_1}(s_n)⟩$ and $\frac{1}{{\lambda }_2}{D}_{f_p}(w_n,{\overline{y}}_n)\le ⟨A_2{w}_n,e_{{\lambda }_2}(w_n)⟩$.

Proof. 

Note that the former inequality is analogous to the latter. So, it suffices to show that the latter holds. Indeed, using the definition of ${\overline{y}}_n$ and properties of ${\Pi }_C$, we have

$$0\ge ⟨J_E^p{w}_n-{\lambda }_2{A}_2{w}_n-J_E^p{\overline{y}}_n,y-{\overline{y}}_n⟩\forall y\in C.$$

Setting $y=w_n$ in the last inequality, from (3) we obtain

$${\lambda }_2⟨A_2{w}_n,w_n-{\overline{y}}_n⟩\ge ⟨J_E^p{w}_n-J_E^p{\overline{y}}_n,w_n-{\overline{y}}_n⟩\ge D_{f_p}(w_n,{\overline{y}}_n),$$

which completes the proof.    □

Algorithm 4: (The 1st parallel subgradient-like extragradient approach.)

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 $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ and $\left\{{𝓁}_n\right\}\subset (0,\infty )$ such that $0<{lim\; inf}_{n\to \infty }{\alpha }_n(1-{\alpha }_n)$, $0<{lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)$ and ${\sum }_{n=1}^{\infty }{𝓁}_n<\infty$. Moreover, assume ${\sum }_{n=1}^{\infty }{\theta }_n<\infty$, and given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ such that $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where                   $\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{𝓁}_n}{\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel }\}\mathrm{if}{x}_n\ne x_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$ Iterations: Compute $x_{n+1}$ below: Step 1. Put $g_n=J_{E^{\ast }}^q((1-{ϵ}_n)J_E^p{S}_1^n{x}_n+{ϵ}_n{J}_E^p(S_1^n{x}_n+x_n-x_{n-1}))$ and calculate $s_n=J_{E^{\ast }}^q({\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)$, $y_n={\Pi }_C(J_{E^{\ast }}^q(J_E^p{s}_n-{\lambda }_1{A}_1{s}_n))$, $e_{{\lambda }_1}(s_n):=s_n-y_n$ and $t_n=s_n-{\tau }_n{e}_{{\lambda }_1}(s_n)$, with ${\tau }_n:=l_1^{k_n}$ and $k_n$ being the smallest $k\ge 0$ such that $$\frac{{\mu }_1}{2}{D}_{f_p}(s_n,y_n)\ge ⟨A_1{s}_n-A_1(s_n-l_1^k{e}_{{\lambda }_1}(s_n)),s_n-y_n⟩.$$ (9) Step 2. Calculate $w_n={\Pi }_{C_n}(s_n)$, with $C_n:=\{y\in C:h_n(y)\le 0\}$ and $$h_n(y)=⟨A_1{t}_n,y-s_n⟩+\frac{{\tau }_n}{2{\lambda }_1}{D}_{f_p}(s_n,y_n).$$ (10) Step 3. Calculate ${\overline{y}}_n={\Pi }_C(J_{E^{\ast }}^q(J_E^p{w}_n-{\lambda }_2{A}_2{w}_n))$, $e_{{\lambda }_2}(w_n):=w_n-{\overline{y}}_n$ and ${\overline{t}}_n=w_n-{\overline{\tau }}_n{e}_{{\lambda }_2}(w_n)$, with ${\overline{\tau }}_n:=l_2^{j_n}$ and $j_n$ being the smallest $j\ge 0$ such that $$\frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\overline{y}}_n)\ge ⟨A_2{w}_n-A_2(w_n-l_2^j{e}_{{\lambda }_2}(w_n)),w_n-{\overline{y}}_n⟩.$$ (11)

Step 4. Calculate $v_n=J_{E^{\ast }}^q({\alpha }_n{J}_E^p{w}_n+(1-{\alpha }_n)J_E^p(S_2^n{w}_n))$ and $x_{n+1}={\Pi }_{{\overline{C}}_n\cap Q_n}(w_n)$, with $Q_n:=\{y\in C:(1+{\overline{\theta }}_n)D_{f_p}(y,w_n)\ge D_{f_p}(y,v_n)\}$, ${\overline{C}}_n:=\{y\in C:{\overline{h}}_n(y)\le 0\}$ and $${\overline{h}}_n(y)=⟨A_2{\overline{t}}_n,y-w_n⟩+\frac{{\overline{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\overline{y}}_n).$$ (12) Again set $n:=n+1$ and go to Step 1.

Lemma 11.

Linesearch rules (9) and (11) of Armijo-type and sequence $\left\{x_n\right\}$ constructed in Algorithm 4 are well defined.

Proof. 

Observe that rule (9) is analogous to the (11). So, it suffices to show that the latter is valid. Using the uniform continuity of $A_2$ on C, from $l_2\in (0,1)$ one obtains ${lim}_{j\to \infty }⟨A_2{w}_n-A_2(w_n-l_2^j{e}_{{\lambda }_2}(w_n)),e_{{\lambda }_2}(w_n)⟩=0$. In the case of $e_{{\lambda }_2}(w_n)=0$, it is explicit that $j_n=0$. In the case of $e_{{\lambda }_2}(w_n)\ne 0$, we obtain that $\exists j_n\ge 0$ such that (11) holds.

It is not hard to verify that $Q_n$ and ${\overline{C}}_n$ are convex and closed for all n. Let us show that $Q_n\cap {\overline{C}}_n\supset \Omega$. Choose a $z\in \Omega$ arbitrarily. Because $S_2$ is a Bregman’s relatively asymptotically nonexpansive mapping, by Lemma 2 we obtain

$$\begin{array}{cc}& D_{f_p}(z,v_n)\le {\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)D_{f_p}(z,S_2^n{w}_n)-{\alpha }_n(1-{\alpha }_n){\rho }_{b_{w_n}}^{\ast }\parallel J_E^P{w}_n-J_E^P{S}_2^n{w}_n\parallel \hfill \\ & \le {\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)(1+{\overline{\theta }}_n)D_{f_p}(z,w_n)-{\alpha }_n(1-{\alpha }_n){\rho }_{b_{w_n}}^{\ast }\parallel J_E^P{w}_n-J_E^P{S}_2^n{w}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)D_{f_p}(z,w_n)-{\alpha }_n(1-{\alpha }_n){\rho }_{b_{w_n}}^{\ast }\parallel J_E^P{w}_n-J_E^P{S}_2^n{w}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)D_{f_p}(z,w_n),\hfill \end{array}$$

which hence leads to $z\in Q_n$. Additionally, from Lemma 4, we obtain $⟨A_2{\overline{t}}_n,{\overline{t}}_n-z⟩\ge 0$. Thus,

$$\begin{array}{cc}{\overline{h}}_n(z)\hfill & =⟨A_2{\overline{t}}_n,z-w_n⟩+\frac{{\overline{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\overline{y}}_n)\hfill \\ \hfill & =-⟨A_2{\overline{t}}_n,w_n-{\overline{t}}_n⟩-⟨A_2{\overline{t}}_n,{\overline{t}}_n-z⟩+\frac{{\overline{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\overline{y}}_n)\hfill \\ \hfill & \le -{\overline{\tau }}_n⟨A_2{\overline{t}}_n,e_{{\lambda }_2}(w_n)⟩+\frac{{\overline{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\overline{y}}_n).\hfill \end{array}$$

(13)

So, it follows from (11) that

$$\frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\overline{y}}_n)\ge ⟨A_2{w}_n-A_2{\overline{t}}_n,e_{{\lambda }_2}(w_n)⟩.$$

By Lemma 10 we have

$$(\frac{1}{{\lambda }_2}-\frac{{\mu }_2}{2})D_{f_p}(w_n,{\overline{y}}_n)\le ⟨A_2{w}_n,e_{{\lambda }_2}(w_n)⟩-\frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\overline{y}}_n)\le ⟨A_2{\overline{t}}_n,e_{{\lambda }_2}(w_n)⟩,$$

which together with (13), attains

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

Therefore, $\Omega \subset {\overline{C}}_n\cap Q_n$. As a result, the sequence $\left\{x_n\right\}$ is well defined.    □

Lemma 12.

Suppose that $\left\{y_n\right\}$ and $\left\{{\overline{y}}_n\right\}$ are the sequences generated by Algorithm 4. If ${lim}_{n\to \infty }\parallel s_n-y_n\parallel =0$ and ${lim}_{n\to \infty }\parallel w_n-{\overline{y}}_n\parallel =0$, then ${\omega }_w(s_n)\subset \mathrm{VI}(C,A_1)$ and ${\omega }_w(w_n)\subset \mathrm{VI}(C,A_2)$.

Proof. 

Note that the former inclusion is analogous to the latter. So it suffices to show that the latter is valid. Indeed, taking a $z\in {\omega }_w(w_n)$ arbitrarily, we know that $\exists \left\{w_{n_k}\right\}\subset \left\{w_n\right\}$, such that $w_{n_k}-{\overline{y}}_{n_k}\to 0$ and $w_{n_k}\rightharpoonup z$. So, we have ${\overline{y}}_{n_k}\rightharpoonup z$. Noticing the convexity and closedness of C, according to ${\overline{y}}_{n_k}\rightharpoonup z$ and $\left\{{\overline{y}}_n\right\}\subset C$, we obtain $z\in C$. In what follows, ones consider two aspects. If $A_{2}z=0$, then $z\in \mathrm{VI}(C,A_2)$ (due to $⟨A_{2}z,x-z⟩\ge 0$ for all $x\in C$). If $A_{2}z\ne 0$, by the condition on $A_2$, we obtain $0<\parallel A_{2}z\parallel \le {lim\; inf}_{k\to \infty }\parallel A_2{w}_{n_k}\parallel$. So, we might assume that $\parallel A_2{w}_{n_k}\parallel \ne 0\forall k\ge 1$. From (4), we obtain

$$⟨J_E^p{w}_{n_k}-{\lambda }_2{A}_2{w}_{n_k}-J_E^p{\overline{y}}_{n_k},y-{\overline{y}}_{n_k}⟩\le 0\forall y\in C,$$

and hence

$$\frac{1}{{\lambda }_2}⟨J_E^p{w}_{n_k}-J_E^p{\overline{y}}_{n_k},y-{\overline{y}}_{n_k}⟩+⟨A_2{w}_{n_k},{\overline{y}}_{n_k}-w_{n_k}⟩\le ⟨A_2{w}_{n_k},y-w_{n_k}⟩\forall y\in C.$$

(14)

Because $A_2$ is uniformly continuous, using Lemma 3 we deduce that $\left\{A_2{w}_{n_k}\right\}$ has boundedness. Observe that $\left\{{\overline{y}}_{n_k}\right\}$ also has boundedness. So, using the uniform continuity of $J_E^p$ on any bounded subset of E, from (14) we have

$$\underset{k\to \infty }{lim\; inf}⟨A_2{w}_{n_k},y-w_{n_k}⟩\ge 0\forall y\in C.$$

(15)

To prove that z lies in $\mathrm{VI}(C,A_2)$, we select $\left\{{\kappa }_k\right\}$ in $(0,1)$ such that ${\kappa }_k\downarrow 0$. For any k, we choose the smallest $m_k>0$ such that for all $j\ge m_k$,

$$⟨A_2{w}_{n_j},y-w_{n_j}⟩+{\kappa }_k\ge 0.$$

(16)

Because $\left\{{\kappa }_k\right\}$ is decreasing, we obtain the increasing property of $\left\{m_k\right\}$. For the sake of simplicity, $\left\{A_2{w}_{n_{m_k}}\right\}$ is still written as $\left\{A_2{w}_{m_k}\right\}$. It is known that $A_2{w}_{m_k}\ne 0$ for all k (due to $\left\{A_2{w}_{m_k}\right\}\subset \left\{A_2{w}_{n_k}\right\}$). Then, substituting ${\overline{g}}_{m_k}=\frac{A_2{w}_{m_k}}{\parallel A_2{w}_{m_k}{\parallel }^{\frac{q}{q-1}}}$, we obtain $⟨A_2{w}_{m_k},J_{E^{\ast }}^q{\overline{g}}_{m_k}⟩=1\forall k\ge 1$. Indeed, it is evident that $⟨A_2{w}_{m_k},J_{E^{\ast }}^q{\overline{g}}_{m_k}⟩=$$⟨A_2{w}_{m_k},{(\frac{1}{\parallel A_2{w}_{m_k}{\parallel }^{\frac{q}{q-1}}})}^{q-1}{J}_{E^{\ast }}^q{A}_2{w}_{m_k}⟩={(\frac{1}{\parallel A_2{w}_{m_k}{\parallel }^{\frac{q}{q-1}}})}^{q-1}{\parallel A_2{w}_{m_k}\parallel }^q=1\forall k\ge 1$. So, by (16) we have $⟨A_2{w}_{m_k},y+{\kappa }_k{J}_{E^{\ast }}^q{\overline{g}}_{m_k}-w_{m_k}⟩\ge 0\forall k\ge 1$. Again, from the pseudomonotonicity of $A_2$ we have

$$⟨A_2(y+{\kappa }_k{J}_{E^{\ast }}^q{\overline{g}}_{m_k}),y+{\kappa }_k{J}_{E^{\ast }}^q{\overline{g}}_{m_k}-w_{m_k}⟩\ge 0\forall y\in C.$$

(17)

Let us show that ${lim}_{k\to \infty }{\kappa }_k{J}_{E^{\ast }}^q{\overline{g}}_{m_k}=0$. Indeed, noticing ${\kappa }_k\downarrow 0$ and $\left\{w_{m_k}\right\}\subset \left\{w_{n_k}\right\}$, we obtain that

$$0\le \underset{k\to \infty }{lim\; sup}\parallel {\kappa }_k{J}_{E^{\ast }}^q{\overline{g}}_{m_k}\parallel =\underset{k\to \infty }{lim\; sup}\frac{{\kappa }_k}{\parallel A_2{w}_{m_k}\parallel }\le \frac{{lim\; sup}_{k\to \infty }{\kappa }_k}{{lim\; inf}_{k\to \infty }\parallel A_2{w}_{n_k}\parallel }=0.$$

Hence, we obtain ${\kappa }_k{J}_{E^{\ast }}^q{\overline{g}}_{m_k}\to 0$ as $k\to \infty$. Thus, taking the limit as $k\to \infty$ in (17), from (C3) we have $⟨A_{2}y,y-z⟩\ge 0$ for all $y\in C$. In terms of Lemma 2.4 we conclude that z lies in $\mathrm{VI}(C,A_2)$.    □

Lemma 13.

Suppose that $\left\{y_n\right\}$ and $\left\{{\overline{y}}_n\right\}$ are the sequences generated by Algorithm 4. Then, the following hold:

(i) 

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

(ii) 

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

Proof. 

Note that claim (i) is analogous to claim (ii). So, it suffices to show that the second is valid. To verify the second claim, we discuss two cases. In case ${lim\; inf}_{n\to \infty }{\overline{\tau }}_n>0$, we may presume that $\exists \overline{\tau }>0$ satisfying ${\overline{\tau }}_n\ge \overline{\tau }>0$ for all n, which immediately leads to

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

(18)

This, together with ${lim}_{n\to \infty }{\overline{\tau }}_n{D}_{f_p}(w_n,{\overline{y}}_n)=0$, arrives at ${lim}_{n\to \infty }{D}_{f_p}(w_n,{\overline{y}}_n)=0$.

In case ${lim\; inf}_{n\to \infty }{\overline{\tau }}_n=0$, we assume that ${lim\; sup}_{n\to \infty }{D}_{f_p}(w_n,{\overline{y}}_n)=\widehat{a}>0$. This ensures that $\exists \left\{m_j\right\}\subset \left\{n\right\}$ satisfying

$$\underset{j\to \infty }{lim}{\overline{\tau }}_{m_j}=0\mathrm{and}\underset{j\to \infty }{lim}{D}_{f_p}(w_{m_j},{\overline{y}}_{m_j})=\widehat{a}>0.$$

We define $\widehat{t_{m_j}}=\frac{1}{l_2}{\overline{\tau }}_{m_j}{\overline{y}}_{m_j}+(1-\frac{1}{l_2}{\overline{\tau }}_{m_j})w_{m_j}\forall j\ge 1$. Noticing ${lim}_{j\to \infty }{\overline{\tau }}_{m_j}{D}_{f_p}(w_{m_j},{\overline{y}}_{m_j})=0$, from (3) we obtain ${lim}_{j\to \infty }{\overline{\tau }}_{m_j}{\parallel w_{m_j}-{\overline{y}}_{m_j}\parallel }^p=0$ and hence

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

(19)

Because $A_2$ is uniformly continuous on bounded subsets of C, we obtain

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

(20)

From the step size rule (11) and the definition of $\widehat{t_{m_j}}$, it follows that

$$⟨A_2{w}_{m_j}-A_2\widehat{t_{m_j}},w_{m_j}-{\overline{y}}_{m_j}⟩>\frac{{\mu }_2}{2}{D}_{f_p}(w_{m_j},{\overline{y}}_{m_j}).$$

(21)

Now, taking the limit as $j\to \infty$, from (20) we have ${lim}_{j\to \infty }{D}_{f_p}(w_{m_j},{\overline{y}}_{m_j})=0$. This, however, yields a contradiction. As a result, $D_{f_p}(w_n,{\overline{y}}_n)\to 0$ as $n\to \infty$.    □

In what follows, we show the first main result.

Theorem 1.

Suppose that E is uniformly smooth and p-uniformly convex, where $J_E^p$ has weakly sequential continuity. If under Algorithm 4, $S_1^{n+1}{x}_n-S_1^n{x}_n\to 0$ and $S_2^{n+1}{w}_n-S_2^n{w}_n\to 0$, then $x_n\rightharpoonup z\in \Omega \iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

Proof. 

Note that necessity is valid, so we need to only show sufficiency. Presume ${sup}_{n\ge 0}\parallel x_n\parallel <\infty$. Choose a $z\in \Omega$ arbitrarily. Clearly, $x_{n-1}\ne x_n\iff J_E^p{S}_1^n{x}_n\ne J_E^p(S_1^n{x}_n-x_{n-1}+x_n)$. Using the definition of ${ϵ}_n$, we obtain ${ϵ}_n\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \le {𝓁}_n\forall n\ge 1$. From (3) and (8) and the three-point identity of $D_{f_p}$, we obtain

$$\begin{array}{cc}& D_{f_p}(z,g_n)\le (1-{ϵ}_n)D_{f_p}(z,S_1^n{x}_n)+{ϵ}_n{D}_{f_p}(z,S_1^n{x}_n+x_n-x_{n-1})\hfill \\ & =D_{f_p}(z,S_1^n{x}_n)+{ϵ}_n[D_{f_p}(z,S_1^n{x}_n+x_n-x_{n-1})-D_{f_p}(z,S_1^n{x}_n)]\hfill \\ & =D_{f_p}(z,S_1^n{x}_n)+{ϵ}_n[D_{f_p}(S_1^n{x}_n,S_1^n{x}_n+x_n-x_{n-1})\hfill \\ & +⟨J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1}),z-S_1^n{x}_n⟩]\hfill \\ & \le (1+{\theta }_n)D_{f_p}(z,x_n)+{ϵ}_n⟨J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1}),z+x_{n-1}-x_n-S_1^n{x}_n⟩\hfill \\ & \le (1+{\theta }_n)D_{f_p}(z,x_n)+{ϵ}_n\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \parallel z+x_{n-1}-x_n-S_1^n{x}_n\parallel \hfill \\ & \le (1+{\theta }_n)D_{f_p}(z,x_n)+{𝓁}_{n}M,\hfill \end{array}$$

where ${sup}_{n\ge 1}\parallel z+x_{n-1}-x_n-S_1^n{x}_n\parallel \le M$ for some $M>0$. By Lemma 2 we obtain

$$\begin{array}{cc}& D_{f_p}(z,s_n)=V_{f_p}(z,{\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)\hfill \\ & \le \frac{1}{p}{\parallel z\parallel }^p-{\beta }_n⟨J_E^p{x}_n,z⟩-(1-{\beta }_n)⟨J_E^p{g}_n,z⟩+\frac{{\beta }_n}{q}{\parallel J_E^p{x}_n\parallel }^q\hfill \\ & +\frac{(1-{\beta }_n)}{q}\parallel J_E^p{g}_n{\parallel }^q-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & =\frac{1}{p}{\parallel z\parallel }^p-{\beta }_n⟨J_E^p{x}_n,z⟩-(1-{\beta }_n)⟨J_E^p{g}_n,z⟩+\frac{{\beta }_n}{q}{\parallel x_n\parallel }^p\hfill \\ & +\frac{(1-{\beta }_n)}{q}\parallel g_n{\parallel }^p-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & ={\beta }_n{D}_{f_p}(z,x_n)+(1-{\beta }_n)D_{f_p}(z,g_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & \le {\beta }_n{D}_{f_p}(z,x_n)+(1-{\beta }_n)[(1+{\theta }_n)D_{f_p}(z,x_n)+{𝓁}_{n}M]-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ & \le (1+{\theta }_n)D_{f_p}(z,x_n)+{𝓁}_{n}M-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel .\hfill \end{array}$$

Noticing $w_n={\Pi }_{C_n}{s}_n$, by (3) and (5) we obtain

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

Because $x_{n+1}={\Pi }_{{\overline{C}}_n\cap Q_n}{w}_n$, by (3) and (5) we have

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

Combining these inequalities and (21) leads to

$$\begin{array}{cc}& D_{f_p}(z,x_{n+1})\le D_{f_p}(z,w_n)-D_{f_p}({\Pi }_{{\overline{C}}_n\cap Q_n}{w}_n,w_n)\hfill \\ \hfill & \le D_{f_p}(z,s_n)-D_{f_p}(w_n,s_n)-D_{f_p}(x_{n+1},w_n)\hfill \\ \hfill & \le D_{f_p}(z,s_n)-\tau {\left[\mathrm{dist}(C_n,s_n)\right]}^p-\tau {\left[\mathrm{dist}({\overline{C}}_n,w_n)\right]}^p\hfill \\ \hfill & \le (1+{\theta }_n)D_{f_p}(z,x_n)+{𝓁}_{n}M-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ \hfill & -\tau {\left[\mathrm{dist}(C_n,s_n)\right]}^p-\tau {\left[\mathrm{dist}({\overline{C}}_n,w_n)\right]}^p,\hfill \end{array}$$

(22)

which hence leads to

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

Because ${\sum }_{n=1}^{\infty }{𝓁}_n<\infty$ and ${\sum }_{n=1}^{\infty }{\theta }_n<\infty$, 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 $\left\{g_n\right\},\left\{s_n\right\},\left\{v_n\right\},\left\{w_n\right\},\left\{y_n\right\},\left\{{\overline{y}}_n\right\}$, $\left\{t_n\right\},\left\{{\overline{t}}_n\right\},\left\{S_1^n{x}_n\right\}$ and $\left\{S_2^n{w}_n\right\}$ are also bounded. From (22) we obtain

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

which immediately yields

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

Because ${lim}_{n\to \infty }{𝓁}_n=0$, ${lim}_{n\to \infty }{\theta }_n=0$, ${lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)>0$ and ${lim}_{n\to \infty }{D}_{f_p}(z,x_n)$ exists, it follows that ${lim}_{n\to \infty }{D}_{f_p}(w_n,s_n)=0$, ${lim}_{n\to \infty }{D}_{f_p}(x_{n+1},w_n)=0$, and ${lim}_{n\to \infty }{\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel =0$, which hence yields ${lim}_{n\to \infty }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel =0$. From $s_n=J_{E^{\ast }}^q({\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)$, it is readily known that ${lim}_{n\to \infty }\parallel J_E^p{s}_n-J_E^p{x}_n\parallel =0$. Noticing $g_n=J_{E^{\ast }}^q((1-{ϵ}_n)J_E^p{S}_1^n{x}_n+{ϵ}_n{J}_E^p(S_1^n{x}_n+x_n-x_{n-1}))$, we obtain from ${lim}_{n\to \infty }{𝓁}_n=0$ and the definition of ${ϵ}_n$ that

$$\parallel J_E^p{g}_n-J_E^p{S}_1^n{x}_n\parallel ={ϵ}_n\parallel J_E^p(S_1^n{x}_n+x_n-x_{n-1})-J_E^p{S}_1^n{x}_n\parallel \le {𝓁}_n\to 0(n\to \infty ).$$

Hence, using (3) and uniform continuity of $J_{E^{\ast }}^q$ on bounded subsets of $E^{\ast }$, we conclude that ${lim}_{n\to \infty }\parallel g_n-S_1^n{x}_n\parallel =0$ and

$$\underset{n\to \infty }{lim}\parallel w_n-s_n\parallel =\underset{n\to \infty }{lim}\parallel x_{n+1}-w_n\parallel =\underset{n\to \infty }{lim}\parallel x_n-S_1^n{x}_n\parallel =\underset{n\to \infty }{lim}\parallel s_n-x_n\parallel =0.$$

(23)

Because $\left\{x_n\right\}$ has boundedness and E has reflexivity, we obtain that ${\omega }_w(x_n)$ is nonempty. Next, let us show that $\Omega \supset {\omega }_w(x_n)$. Choose a z in ${\omega }_w(x_n)$ arbitrarily. It is known that $\exists \left\{x_{n_k}\right\}\subset \left\{x_n\right\}$ satisfying $x_{n_k}\rightharpoonup z$. By (23) we obtain $w_{n_k}\rightharpoonup z$. Because $\left\{A_1{t}_n\right\}$ has boundedness, we know that $\exists L_1>0$ satisfying $\parallel A_1{t}_n\parallel \le L_1$. So, it follows that for all $u,v\in C_n$,

$$|h_n(u)-h_n(v)|=|⟨A_1{t}_n,u-v⟩|\le \parallel A_1{t}_n\parallel \parallel u-v\parallel \le L_1\parallel u-v\parallel ,$$

which implies that $h_n(y)$ is $L_1$-Lipschitz continuous on $C_n$. Using Lemma 6, we obtain

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

(24)

Because $x_{n+1}$ lies in $Q_n$, by (22) we have

$$\begin{array}{cc}{D}_{f_p}(x_{n+1},v_n)\hfill & \le (1+{\overline{\theta }}_n)[D_{f_p}(z,w_n)-D_{f_p}(z,x_{n+1})]\hfill \\ \hfill & \le (1+{\overline{\theta }}_n)[D_{f_p}(z,s_n)-D_{f_p}(z,x_{n+1})]\hfill \\ \hfill & \le (1+{\overline{\theta }}_n)[(1+{\theta }_n)D_{f_p}(z,x_n)-D_{f_p}(z,x_{n+1})+{𝓁}_{n}M].\hfill \end{array}$$

(25)

Because ${lim}_{n\to \infty }{\theta }_n=0$, ${lim}_{n\to \infty }{\overline{\theta }}_n=0$, ${lim}_{n\to \infty }{𝓁}_n=0$ and ${lim}_{n\to \infty }{D}_{f_p}(z,x_n)$ exists, we have $D_{f_p}(x_{n+1},v_n)\to 0$ and thus $x_{n+1}-v_n\to 0$. By (23) we obtain

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

Furthermore, by Lemma 2, we have

$$\begin{array}{cc}& D_{f_p}(z,v_n)=V_{f_p}(z,{\alpha }_n{J}_E^p{w}_n+(1-{\alpha }_n)J_E^p{S}_2^n{w}_n)\hfill \\ & \le \frac{1}{p}{\parallel z\parallel }^p-{\alpha }_n⟨J_E^p{w}_n,z⟩-(1-{\alpha }_n)⟨J_E^p{S}_2^n{w}_n,z⟩+\frac{{\alpha }_n}{q}{\parallel J_E^p{w}_n\parallel }^q\hfill \\ & +\frac{(1-{\alpha }_n)}{q}\parallel J_E^p{S}_2^n{w}_n{\parallel }^q-{\alpha }_n(1-{\alpha }_n){\rho }_b^{\ast }\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel \hfill \\ & =\frac{1}{p}{\parallel z\parallel }^p-{\alpha }_n⟨J_E^p{w}_n,z⟩-(1-{\alpha }_n)⟨J_E^p{S}_2^n{w}_n,z⟩+\frac{{\alpha }_n}{q}{\parallel w_n\parallel }^p\hfill \\ & +\frac{(1-{\alpha }_n)}{q}\parallel S_2^n{w}_n{\parallel }^p-{\alpha }_n(1-{\alpha }_n){\rho }_b^{\ast }\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel \hfill \\ & ={\alpha }_n{D}_{f_p}(z,w_n)+(1-{\alpha }_n)D_{f_p}(z,S_2^n{w}_n)-{\alpha }_n(1-{\alpha }_n){\rho }_b^{\ast }\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)D_{f_p}(z,w_n)-{\alpha }_n(1-{\alpha }_n){\rho }_b^{\ast }\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel .\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}& {\alpha }_n(1-{\alpha }_n){\rho }_b^{\ast }\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel \le (1+{\overline{\theta }}_n)D_{f_p}(z,w_n)-D_{f_p}(z,v_n)\hfill \\ & \le D_{f_p}(z,w_n)-D_{f_p}(z,v_n)+D_{f_p}(w_n,v_n)+{\overline{\theta }}_n{D}_{f_p}(z,w_n)\hfill \\ & =⟨J_E^p{v}_n-J_E^p{w}_n,z-w_n⟩+{\overline{\theta }}_n{D}_{f_p}(z,w_n).\hfill \end{array}$$

Taking the limit in the last inequality as $n\to \infty$ and using the uniform continuity of $J_E^p$ on bounded subsets of E, (25) and ${lim\; inf}_{n\to \infty }{\alpha }_n(1-{\alpha }_n)>0$, we obtain ${lim}_{n\to \infty }{\rho }_b^{\ast }\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel =0$ and hence ${lim}_{n\to \infty }\parallel J_E^p{w}_n-J_E^p{S}_2^n{w}_n\parallel =0$. Because $J_{E^{\ast }}^q$ is uniformly continuous on any bounded subset of $E^{\ast }$, we deduce that

$$\underset{n\to \infty }{lim}\parallel w_n-S_2^n{w}_n\parallel =0.$$

(26)

Now, let us show $z\in {\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)$. Because $\left\{A_2{\overline{t}}_n\right\}$ has boundedness, it follows that $\exists L_2>0$ satisfying $\parallel A_2{\overline{t}}_n\parallel \le L_2$. Thus, we deduce that for all $u,v\in {\overline{C}}_n$,

$$|{\overline{h}}_n(u)-{\overline{h}}_n(v)|=|⟨A_2{\overline{t}}_n,u-v⟩|\le \parallel A_2{\overline{t}}_n\parallel \parallel u-v\parallel \le L_2\parallel u-v\parallel ,$$

which guarantees that ${\overline{h}}_n(y)$ is $L_2$-Lipschitz continuous on ${\overline{C}}_n$. By Lemma 6, we obtain

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

(27)

Combining (22), (24) and (27), we have

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

(28)

Thus,

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

According to Lemma 13, we have

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

In addition, from (23) and $x_{n_k}\rightharpoonup z$, we infer that $s_{n_k}\rightharpoonup z$ and $w_{n_k}\rightharpoonup z$. By Lemma 12 we obtain that $z\in {\omega }_w(s_n)\subset \mathrm{VI}(C,A_1)$ and $z\in {\omega }_w(w_n)\subset \mathrm{VI}(C,A_2)$. Consequently,

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

Next, we claim that $z\in {\bigcap }_{i=1}^2\mathrm{Fix}(S_i)$. Indeed, by (23) we immediately obtain

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

(29)

We first claim that ${lim}_{n\to \infty }\parallel x_n-S_1{x}_n\parallel =0$ and ${lim}_{n\to \infty }\parallel w_n-S_2{w}_n\parallel =0$. In fact, using (23), (26) and uniform continuity of $S_i$ on C for $i=1,2$, we obtain that $S_1{x}_n-S_1^{n+1}{x}_n\to 0$ and $S_2{w}_n-S_2^{n+1}{w}_n\to 0$. Thus, from $S_1^{n+1}{x}_n-S_1^n{x}_n\to 0$ and $S_2^{n+1}{w}_n-S_2^n{w}_n\to 0$ (due to the assumptions) we deduce that

$$\parallel x_n-S_1{x}_n\parallel \le \parallel x_n-S_1^n{x}_n\parallel +\parallel S_1^n{x}_n-S_1^{n+1}{x}_n\parallel +\parallel S_1^{n+1}{x}_n-S_1{x}_n\parallel \to 0(n\to \infty )$$

and

$$\parallel w_n-S_2{w}_n\parallel \le \parallel w_n-S_2^n{w}_n\parallel +\parallel S_2^n{w}_n-S_2^{n+1}{w}_n\parallel +\parallel S_2^{n+1}{w}_n-S_2{w}_n\parallel \to 0(n\to \infty ).$$

These, together with $x_{n_k}\rightharpoonup z$ and $w_{n_k}\rightharpoonup z$, ensure that $z\in {\bigcap }_{i=1}^2\widehat{\mathrm{Fix}}(S_i)={\bigcap }_{i=1}^2\mathrm{Fix}(S_i)$; therefore, $z\in \Omega$. This means that ${\omega }_w(x_n)\subset \Omega$. As a result, by Lemma 5 we obtain the desired conclusion.    □

In what follows, we prove the second main outcome for finding a solution of a pair of VIFPPs for two operators of both uniform continuity and pseudomonotonicity and two mappings of both uniform continuity and Bregman’s relatively asymptotical nonexpansivity in E.

Theorem 2.

Suppose that the conditions (C1)–(C3) hold. If under Algorithm 5, $S_1^{n+1}{x}_n-S_1^n{x}_n\to 0$ and $S_2^{n+1}{z}_n-S_2^n{z}_n\to 0$, then $x_n\to {\Pi }_{\Omega }u\iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

Proof. 

It is explicit that the necessity of Theorem 2 holds. Hence, we need to only prove sufficiency. 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

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

for some $M>0$. In fact, substitute $\widehat{u}={\Pi }_{\Omega }u$. Noticing $w_n={\Pi }_{C_n}{s}_n$ and $z_n={\Pi }_{{\overline{C}}_n}{w}_n$, we obtain from (3) and (5) that

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

and

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

From similar reasonings to those in the proof of the above theorem, we obtain

$$\begin{array}{cc}{D}_{f_p}(\widehat{u},g_n)\hfill & \le D_{f_p}(\widehat{u},S_1^n{x}_n)+{ϵ}_n\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel \le (1+{\theta }_n)D_{f_p}(\widehat{u},x_n)+{𝓁}_{n}M,\hfill \end{array}$$

where ${sup}_{n\ge 1}\parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel \le M$ for some $M>0$. This ensures that $\left\{g_n\right\}$ is bounded.

Using (8) and the last two inequalities, from $\left\{{\gamma }_n\right\}\subset (0,1)$ and $\left\{{\beta }_n\right\}\subset (0,1)$ we obtain

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)(1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},z_n)\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)(1+{\overline{\theta }}_n)[{\gamma }_n{D}_{f_p}(\widehat{u},x_n)+(1-{\gamma }_n)D_{f_p}(\widehat{u},g_n)\hfill \\ & -{\gamma }_n(1-{\gamma }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel ]\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1+{\overline{\theta }}_n)[(1+{\theta }_n)D_{f_p}(\widehat{u},x_n)+{𝓁}_{n}M]\hfill \\ & -(1-{\alpha }_n)(1+{\overline{\theta }}_n){\gamma }_n(1-{\gamma }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel ,\hfill \end{array}$$

which immediately arrives at the desired claim. In addition, it is easily known that $\left\{s_n\right\},\left\{v_n\right\}$, $\left\{w_n\right\},\left\{y_n\right\},\left\{{\overline{y}}_n\right\},\left\{z_n\right\},\left\{t_n\right\},\left\{{\overline{t}}_n\right\}$ and $\left\{S_2^n{z}_n\right\}$ are of boundedness.

Claim 2. We show that

$$\begin{array}{cc}& ({\overline{\theta }}_n+1)[D_{f_p}(w_n,s_n)+D_{f_p}(z_n,w_n)]\hfill \\ & \le {\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩-D_{f_p}(\widehat{u},x_{n+1})+({\overline{\theta }}_n+1)D_{f_p}(\widehat{u},s_n).\hfill \end{array}$$

Indeed, set $b={sup}_{n\ge 1}\{\parallel x_n{\parallel }^{p-1},\parallel g_n{\parallel }^{p-1},\parallel z_n{\parallel }^{p-1},\parallel S_2^n{z}_n{\parallel }^{p-1}\}$. By Lemma 2 we obtain

$$\begin{array}{cc}& D_{f_p}(\widehat{u},s_n)=V_{f_p}(\widehat{u},{\gamma }_n{J}_E^p{x}_n+(1-{\gamma }_n)J_E^p{g}_n)\hfill \\ \hfill & \le \frac{1}{p}\parallel \widehat{u}{\parallel }^p-{\gamma }_n⟨J_E^p{x}_n,\widehat{u}⟩-(1-{\gamma }_n)⟨J_E^p{g}_n,\widehat{u}⟩+\frac{{\gamma }_n}{q}{\parallel J_E^p{x}_n\parallel }^q\hfill \\ \hfill & +\frac{(1-{\gamma }_n)}{q}\parallel J_E^p{g}_n{\parallel }^q-{\gamma }_n(1-{\gamma }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ \hfill & =\frac{1}{p}\parallel \widehat{u}{\parallel }^p-{\gamma }_n⟨J_E^p{x}_n,\widehat{u}⟩-(1-{\gamma }_n)⟨J_E^p{g}_n,\widehat{u}⟩+\frac{{\gamma }_n}{q}{\parallel x_n\parallel }^p\hfill \\ \hfill & +\frac{(1-{\gamma }_n)}{q}\parallel g_n{\parallel }^p-{\gamma }_n(1-{\gamma }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ \hfill & ={\gamma }_n{D}_{f_p}(\widehat{u},x_n)+(1-{\gamma }_n)D_{f_p}(\widehat{u},g_n)-{\gamma }_n(1-{\gamma }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ \hfill & \le {\gamma }_n{D}_{f_p}(\widehat{u},x_n)+(1-{\gamma }_n)[(1+{\theta }_n)D_{f_p}(z,x_n)+{𝓁}_{n}M]\hfill \\ \hfill & -{\gamma }_n(1-{\gamma }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel \hfill \\ \hfill & \le (1+{\theta }_n)D_{f_p}(\widehat{u},x_n)+{𝓁}_{n}M-{\gamma }_n(1-{\gamma }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel ,\hfill \end{array}$$

(30)

and

$$\begin{array}{cc}& D_{f_p}(\widehat{u},v_n)=V_{f_p}(\widehat{u},{\beta }_n{J}_E^p{z}_n+(1-{\beta }_n)J_E^p{S}_2^n{z}_n)\hfill \\ \hfill & \le {\beta }_n{D}_{f_p}(\widehat{u},z_n)+(1-{\beta }_n)D_{f_p}(\widehat{u},S_2^n{z}_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel \hfill \\ \hfill & \le {\beta }_n{D}_{f_p}(\widehat{u},z_n)+(1-{\beta }_n)(1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},z_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel \hfill \\ \hfill & \le (1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},z_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel \hfill \\ \hfill & \le (1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},w_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel .\hfill \end{array}$$

(31)

Set ${\zeta }_n=J_{E^{\ast }}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)$. From (7), we have

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le V_{f_p}(\widehat{u},{\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)\hfill \\ \hfill & \le V_{f_p}(\widehat{u},{\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n-{\alpha }_n(J_E^{p}u-J_E^p\widehat{u}))+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\hfill \\ \hfill & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\hfill \\ \hfill & \le (1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},w_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel +{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\hfill \\ \hfill & \le (1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},w_n)+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩.\hfill \end{array}$$

(32)

Furthermore, from (31) we have

$$\begin{array}{cc}{D}_{f_p}(\widehat{u},v_n)\hfill & \le (1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},z_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)[D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n)]-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)[D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n)].\hfill \end{array}$$

This, together with (32), arrives at

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le (1-{\alpha }_n)D_{f_p}(\widehat{u},v_n)+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\hfill \\ & \le (1+{\overline{\theta }}_n)[D_{f_p}(\widehat{u},w_n)-D_{f_p}(z_n,w_n)]+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\hfill \\ & \le (1+{\overline{\theta }}_n)[D_{f_p}(\widehat{u},s_n)-D_{f_p}(w_n,s_n)-D_{f_p}(z_n,w_n)]+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩,\hfill \end{array}$$

which immediately yields

$$\begin{array}{cc}& (1+{\overline{\theta }}_n)[D_{f_p}(w_n,s_n)+D_{f_p}(z_n,w_n)]\hfill \\ \hfill & \le {\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩-D_{f_p}(\widehat{u},x_{n+1})+(1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},s_n).\hfill \end{array}$$

(33)

Claim 3. We show that

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

Indeed, by analogous reasonings to those of (28), we obtain

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

(34)

Applying (30), (31), and (34), we have

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le D_{f_p}(\widehat{u},J_{E^{\ast }}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n))\hfill \\ \hfill & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)[({\overline{\theta }}_n+1)D_{f_p}(\widehat{u},z_n)-{\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel z_n-S_2^n{z}_n\parallel ]\hfill \\ \hfill & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1-{\alpha }_n)({\overline{\theta }}_n+1)\{D_{f_p}(\widehat{u},s_n)-\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(s_n,y_n)\right]}^p\hfill \\ \hfill & -\tau {\left[\frac{{\overline{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\overline{y}}_n)\right]}^p\}\hfill \\ \hfill & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+({\overline{\theta }}_n+1)D_{f_p}(\widehat{u},s_n)-(1-{\alpha }_n)({\overline{\theta }}_n+1)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(s_n,y_n)\right]}^p\hfill \\ \hfill & +\tau {\left[\frac{{\overline{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\overline{y}}_n)\right]}^p\}\hfill \\ \hfill & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+({\overline{\theta }}_n+1)[({\theta }_n+1)D_{f_p}(\widehat{u},x_n)+{𝓁}_{n}M-{\gamma }_n(1-{\gamma }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel ]\hfill \\ \hfill & -(1-{\alpha }_n)({\overline{\theta }}_n+1)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(s_n,y_n)\right]}^p+\tau {\left[\frac{{\overline{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\overline{y}}_n)\right]}^p\}\hfill \\ \hfill & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+({\overline{\theta }}_n+1)[({\theta }_n+1)D_{f_p}(\widehat{u},x_n)+{𝓁}_{n}M]\hfill \\ \hfill & -(1-{\alpha }_n)({\overline{\theta }}_n+1)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(s_n,y_n)\right]}^p+\tau {\left[\frac{{\overline{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\overline{y}}_n)\right]}^p\}.\hfill \end{array}$$

(35)

Claim 4. We show that ${lim}_{n\to \infty }\parallel x_n-\widehat{u}\parallel =0$. Indeed, because E is reflexive and $\left\{x_n\right\}$ is bounded, we have ${\omega }_w(x_n)\ne Ø$. Choose a z in ${\omega }_w(x_n)$ arbitrarily. It is known that $\exists \left\{x_{n_k}\right\}\subset \left\{x_n\right\}$ satisfying $x_{n_k}\rightharpoonup z$. We write ${\mathbf{\Gamma }}_n=D_{f_p}(\widehat{u},x_n)$ for all n. In what follows, let us prove $\left\{{\mathbf{\Gamma }}_n\right\}\to 0(n\to \infty )$ in the two possible aspects below.

Aspect 1. Assume that $\exists n_0\ge 1$ such that ${\left\{{\mathbf{\Gamma }}_n\right\}}_{n=n_0}^{\infty }$ is non-increasing. It is known that ${\mathbf{\Gamma }}_n\to d<+\infty$ and hence ${\mathbf{\Gamma }}_n-{\mathbf{\Gamma }}_{n+1}\to 0$. From (30) and (33) we obtain

$$\begin{array}{cc}& ({\overline{\theta }}_n+1)[D_{f_p}(w_n,s_n)+D_{f_p}(z_n,w_n)]\hfill \\ & \le {\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩-D_{f_p}(\widehat{u},x_{n+1})+({\overline{\theta }}_n+1)D_{f_p}(\widehat{u},s_n)\hfill \\ & \le ({\overline{\theta }}_n+1)[({\theta }_n+1)D_{f_p}(\widehat{u},x_n)+{𝓁}_{n}M-{\gamma }_n(1-{\gamma }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel ]\hfill \\ & -D_{f_p}(\widehat{u},x_{n+1})+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩,\hfill \end{array}$$

which hence yields

$$\begin{array}{cc}& ({\overline{\theta }}_n+1)[D_{f_p}(w_n,s_n)+D_{f_p}(z_n,w_n)+{\gamma }_n(1-{\gamma }_n){\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel ]\hfill \\ & \le ({\theta }_n+1)({\overline{\theta }}_n+1)D_{f_p}(\widehat{u},x_n)-D_{f_p}(\widehat{u},x_{n+1})+({\overline{\theta }}_n+1){𝓁}_{n}M+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\hfill \\ & =({\theta }_n+1)({\overline{\theta }}_n+1){\mathbf{\Gamma }}_n-{\mathbf{\Gamma }}_{n+1}+({\overline{\theta }}_n+1){𝓁}_{n}M+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩.\hfill \end{array}$$

Because ${𝓁}_n\to 0,{\overline{\theta }}_n\to 0,{\theta }_n\to 0,{\alpha }_n\to 0,{lim\; inf}_{n\to \infty }{\gamma }_n(1-{\gamma }_n)>0,{\mathbf{\Gamma }}_n\to d$ and $\left\{{\zeta }_n\right\}$ has boundedness, we obtain ${lim}_{n\to \infty }{D}_{f_p}(w_n,s_n)=0$, ${lim}_{n\to \infty }{D}_{f_p}(z_n,w_n)=0$, and ${lim}_{n\to \infty }{\rho }_b^{\ast }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel =0$, which hence yields ${lim}_{n\to \infty }\parallel J_E^p{x}_n-J_E^p{g}_n\parallel =0$. From $u_n=J_{E^{\ast }}^q({\gamma }_n{J}_E^p{x}_n+(1-{\gamma }_n)J_E^p{g}_n)$, it is easily known that ${lim}_{n\to \infty }\parallel J_E^p{s}_n-J_E^p{x}_n\parallel =0$. Noticing $g_n=J_{E^{\ast }}^q((1-{ϵ}_n)J_E^p{S}_1^n{x}_n+{ϵ}_n{J}_E^p(S_1^n{x}_n+x_n-x_{n-1}))$, we infer from ${lim}_{n\to \infty }{𝓁}_n=0$ and the definition of ${ϵ}_n$ that

$$\parallel J_E^p{g}_n-J_E^p{S}_1^n{x}_n\parallel ={ϵ}_n\parallel J_E^p(S_1^n{x}_n+x_n-x_{n-1})-J_E^p{S}_1^n{x}_n\parallel \le {𝓁}_n\to 0(n\to \infty ).$$

Hence, using (3) and the uniform continuity of $J_{E^{\ast }}^q$ on any bounded subset of $E^{\ast }$, we conclude that ${lim}_{n\to \infty }\parallel g_n-S_1^n{x}_n\parallel =0$ and

$$\underset{n\to \infty }{lim}\parallel w_n-s_n\parallel =\underset{n\to \infty }{lim}\parallel z_n-w_n\parallel =\underset{n\to \infty }{lim}\parallel x_n-S_1^n{x}_n\parallel =\underset{n\to \infty }{lim}\parallel s_n-x_n\parallel =0.$$

(36)

Furthermore, from (30) and (32) we have

$$\begin{array}{cc}& (1-{\alpha }_n){\beta }_n(1-{\beta }_n){\rho }_b^{\ast }\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel \hfill \\ & \le (1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},w_n)-D_{f_p}(\widehat{u},x_{n+1})+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\hfill \\ & \le (1+{\overline{\theta }}_n)(1+{\theta }_n)D_{f_p}(\widehat{u},x_n)-D_{f_p}(\widehat{u},x_{n+1})+(1+{\overline{\theta }}_n){𝓁}_{n}M+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩.\hfill \end{array}$$

By similar reasonings, we deduce that ${lim}_{n\to \infty }\parallel J_E^p{z}_n-J_E^p{S}_2^n{z}_n\parallel =0$, which hence leads to ${lim}_{n\to \infty }\parallel J_E^p{v}_n-J_E^p{z}_n\parallel =0$ (due to $v_n=J_{E^{\ast }}^q({\beta }_n{J}_E^p{z}_n+(1-{\beta }_n)J_E^p{S}_2^n{z}_n)$). Using the uniform continuity of $J_{E^{\ast }}^q$ on bounded subsets of $E^{\ast }$, we obtain

$$\underset{n\to \infty }{lim}\parallel z_n-S_2^n{z}_n\parallel =\underset{n\to \infty }{lim}\parallel v_n-z_n\parallel =0.$$

(37)

This, together with (36), implies that

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

It is clear that

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

(38)

Let us show that $z\in {\bigcap }_{i=1}^2\mathrm{Fix}(S_i)$. Indeed, because ${\zeta }_n=J_{E^{\ast }}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)$, it can be readily seen that

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

(39)

In addition, using (5), (30), and (31), we have

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

which hence yields

$$\begin{array}{cc}{D}_{f_p}(x_{n+1},{\zeta }_n)\hfill & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)+(1+{\overline{\theta }}_n)(1+{\theta }_n)D_{f_p}(\widehat{u},x_n)-D_{f_p}(\widehat{u},x_{n+1})+(1+{\overline{\theta }}_n){𝓁}_{n}M\hfill \\ & ={\alpha }_n{D}_{f_p}(\widehat{u},u)-{\mathbf{\Gamma }}_{n+1}+({\theta }_n+1)({\overline{\theta }}_n+1){\mathbf{\Gamma }}_n+({\overline{\theta }}_n+1){𝓁}_{n}M.\hfill \end{array}$$

So, it follows that $D_{f_p}(x_{n+1},{\zeta }_n)\to 0$ and hence $\parallel x_{n+1}-{\zeta }_n\parallel \to 0$. Thus, from (39) we obtain

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

(40)

We now claim that ${lim}_{n\to \infty }\parallel x_n-S_1{x}_n\parallel =0$ and ${lim}_{n\to \infty }\parallel w_n-S_2{w}_n\parallel =0$. Indeed, using (36), (37) and the uniform continuity of $S_i$ on C for $i=1,2$, we obtain that $S_1{x}_n-S_1^{n+1}{x}_n\to 0$ and $S_2{z}_n-S_2^{n+1}{z}_n\to 0$. Thus, from $S_1^{n+1}{x}_n-S_1^n{x}_n\to 0$ and $S_2^{n+1}{z}_n-S_2^n{z}_n\to 0$ (due to the assumptions) we deduce that

$$\parallel x_n-S_1{x}_n\parallel \le \parallel x_n-S_1^n{x}_n\parallel +\parallel S_1^n{x}_n-S_1^{n+1}{x}_n\parallel +\parallel S_1^{n+1}{x}_n-S_1{x}_n\parallel \to 0(n\to \infty )$$

and

$$\parallel z_n-S_2{z}_n\parallel \le \parallel z_n-S_2^n{z}_n\parallel +\parallel S_2^n{z}_n-S_2^{n+1}{z}_n\parallel +\parallel S_2^{n+1}{z}_n-S_2{z}_n\parallel \to 0(n\to \infty ).$$

These, together with $x_{n_k}\rightharpoonup z$ and $z_{n_k}\rightharpoonup z$ (due to (38)), ensure that $z\in {\bigcap }_{i=1}^2\widehat{\mathrm{Fix}}(S_i)={\bigcap }_{i=1}^2\mathrm{Fix}(S_i)$.

In what follows, we show that $z\in {\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)$. From (35), we have

$$\begin{array}{cc}& ({\overline{\theta }}_n+1)(1-{\alpha }_n)\{\tau {\left[\frac{{\tau }_n}{2{\lambda }_1{L}_1}{D}_{f_p}(s_n,y_n)\right]}^p+\tau {\left[\frac{{\overline{\tau }}_n}{2{\lambda }_2{L}_2}{D}_{f_p}(w_n,{\overline{y}}_n)\right]}^p\}\hfill \\ & \le {\alpha }_n{D}_{f_p}(\widehat{u},u)-D_{f_p}(\widehat{u},x_{n+1})+({\theta }_n+1)({\overline{\theta }}_n+1)D_{f_p}(\widehat{u},x_n)+({\overline{\theta }}_n+1){𝓁}_{n}M\hfill \\ & ={\alpha }_n{D}_{f_p}(\widehat{u},u)-{\mathbf{\Gamma }}_{n+1}+({\theta }_n+1)({\overline{\theta }}_n+1){\mathbf{\Gamma }}_n+({\overline{\theta }}_n+1){𝓁}_{n}M.\hfill \end{array}$$

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

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

(41)

By Lemma 13, we obtain

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

(42)

Applying (42) and Lemma 12, we obtain $z\in {\bigcap }_{j=1}^2\mathrm{VI}(C,A_j)$. Thus, we have ${\omega }_w(x_n)\subset {\bigcap }_{i=1}^2\mathrm{VI}(C,A_i)$. Consequently, $\Omega \supset {\omega }_w(x_n)$. Finally, let us prove ${lim\; sup}_{n\to \infty }⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\le 0$. We can select $\left\{x_{n_j}\right\}\subset \left\{x_n\right\}$ such that

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

Because E is reflexive and $\left\{x_n\right\}$ is bounded, we might assume $x_{n_j}\rightharpoonup \overline{z}$. Using (4) and $\overline{z}\in \Omega$ we infer that

$$\underset{n\to \infty }{lim\; sup}⟨J_E^{p}u-J_E^p\widehat{u},x_n-\widehat{u}⟩=\underset{j\to \infty }{lim}⟨J_E^{p}u-J_E^p\widehat{u},x_{n_j}-\widehat{u}⟩=⟨J_E^{p}u-J_E^p\widehat{u},\overline{z}-\widehat{u}⟩\le 0,$$

(43)

which along with (39), arrives at

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

From (30) and (32), we obtain

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{n+1})\le (1-{\alpha }_n)(1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},w_n)+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\hfill \\ & \le (1-{\alpha }_n)(1+{\overline{\theta }}_n)D_{f_p}(\widehat{u},s_n)+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\hfill \\ & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},s_n)+{\overline{\theta }}_n{D}_{f_p}(\widehat{u},s_n)+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\hfill \\ & \le (1-{\alpha }_n)[(1+{\theta }_n)D_{f_p}(\widehat{u},x_n)+{ϵ}_n\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel ]+{\overline{\theta }}_n{D}_{f_p}(\widehat{u},s_n)+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\hfill \\ & \le (1-{\alpha }_n)D_{f_p}(\widehat{u},x_n)+{ϵ}_n\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel +{\theta }_n{D}_{f_p}(\widehat{u},x_n)+{\overline{\theta }}_n{D}_{f_p}(\widehat{u},s_n)+{\alpha }_n⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\hfill \\ & =(1-{\alpha }_n)D_{f_p}(\widehat{u},x_n)+{\alpha }_n\{\frac{{ϵ}_n}{{\alpha }_n}\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \hfill \\ & \times \parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel +\frac{{\theta }_n}{{\alpha }_n}{D}_{f_p}(\widehat{u},x_n)+\frac{{\overline{\theta }}_n}{{\alpha }_n}{D}_{f_p}(\widehat{u},s_n)+⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\}.\hfill \end{array}$$

(44)

Using the uniform continuity of $J_E^p$ on any bounded subset of E, from (40) and the boundedness of $\left\{x_n\right\}$ we obtain

$$\underset{n\to \infty }{lim}\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel =0.$$

Noticing ${sup}_{n\ge 1}\frac{{ϵ}_n}{{\alpha }_n}<\infty ,{lim}_{n\to \infty }\frac{{\theta }_n+{\overline{\theta }}_n}{{\alpha }_n}=0$ and ${lim\; sup}_{n\to \infty }⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\le 0$, we deduce that

$$\begin{array}{cc}& {\displaystyle \underset{n\to \infty }{lim\; sup}}\{\frac{{ϵ}_n}{{\alpha }_n}\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel \parallel \widehat{u}+x_{n-1}-x_n-S_1^n{x}_n\parallel \hfill \\ & +\frac{{\theta }_n}{{\alpha }_n}{D}_{f_p}(\widehat{u},x_n)+\frac{{\overline{\theta }}_n}{{\alpha }_n}{D}_{f_p}(\widehat{u},s_n)+⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_n-\widehat{u}⟩\}\le 0.\hfill \end{array}$$

Thanks to $\left\{{\alpha }_n\right\}\subset (0,1)$ with ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$, utilizing Lemma 8 to (44) we obtain $D_{f_p}(\widehat{u},x_n)\to 0$, and hence $\parallel x_n-\widehat{u}\parallel \to 0$.

Aspect 2. Assume that $\exists \left\{{\mathbf{\Gamma }}_{n_k}\right\}\subset \left\{{\mathbf{\Gamma }}_n\right\}$ satisfying ${\mathbf{\Gamma }}_{n_k}<{\mathbf{\Gamma }}_{n_k+1}$ for all k, with $\mathcal{N}$ being the natural number set. Let $\phi :\mathcal{N}\to \mathcal{N}$ be formulated below

$$\phi (n):=max\{j\le n:{\mathbf{\Gamma }}_j<{\mathbf{\Gamma }}_{j+1}\}.$$

Using Lemma 7, we have

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

(45)

From (30) and (33) it follows that

$$\begin{array}{cc}& (1+{\overline{\theta }}_{\phi (n)})[D_{f_p}(w_{\phi (n)},s_{\phi (n)})+D_{f_p}(z_{\phi (n)},w_{\phi (n)})\hfill \\ & +{\gamma }_{\phi (n)}(1-{\gamma }_{\phi (n)}){\rho }_b^{\ast }\parallel J_E^p{x}_{\phi (n)}-J_E^p{g}_{\phi (n)}\parallel ]\hfill \\ & \le (1+{\overline{\theta }}_{\phi (n)})(1+{\theta }_{\phi (n)}){\mathbf{\Gamma }}_{\phi (n)}-{\mathbf{\Gamma }}_{\phi (n)+1}+(1+{\overline{\theta }}_{\phi (n)}){𝓁}_{\phi (n)}M\hfill \\ & +{\alpha }_{\phi (n)}⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\phi (n)}-\widehat{u}⟩.\hfill \end{array}$$

Noticing $g_{\phi (n)}=J_{E^{\ast }}^q((1-{ϵ}_{\phi (n)})J_E^p{S}_1^{\phi (n)}{x}_{\phi (n)}+{ϵ}_{\phi (n)}{J}_E^p(S_1^{\phi (n)}{x}_{\phi (n)}+x_{\phi (n)}-x_{\phi (n)-1}))$ and $s_{\phi (n)}=J_{E^{\ast }}^q({\gamma }_{\phi (n)}{J}_E^p{x}_{\phi (n)}+(1-{\gamma }_{\phi (n)})J_E^p{g}_{\phi (n)}))$, we obtain that ${lim}_{n\to \infty }\parallel g_{\phi (n)}-S_1^{\phi (n)}{x}_{\phi (n)}\parallel =0$ and

$$\underset{n\to \infty }{lim}\parallel w_{\phi (n)}-s_{\phi (n)}\parallel =\underset{n\to \infty }{lim}\parallel z_{\phi (n)}-w_{\phi (n)}\parallel =\underset{n\to \infty }{lim}\parallel x_{\phi (n)}-S_1^{\phi (n)}{x}_{\phi (n)}\parallel =\underset{n\to \infty }{lim}\parallel s_{\phi (n)}-x_{\phi (n)}\parallel =0.$$

(46)

Furthermore, from (30) and (32) we have

$$\begin{array}{cc}& (1-{\alpha }_{\phi (n)}){\beta }_{\phi (n)}(1-{\beta }_{\phi (n)}){\rho }_b^{\ast }\parallel J_E^p{z}_{\phi (n)}-J_E^p{S}_2^{\phi (n)}{z}_{\phi (n)}\parallel \hfill \\ & \le (1+{\overline{\theta }}_{\phi (n)})(1+{\theta }_{\phi (n)}){\mathbf{\Gamma }}_{\phi (n)}-{\mathbf{\Gamma }}_{\phi (n)+1}+(1+{\overline{\theta }}_{\phi (n)}){𝓁}_{\phi (n)}M+{\alpha }_{\phi (n)}⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\phi (n)}-\widehat{u}⟩.\hfill \end{array}$$

Noticing $v_{\phi (n)}=J_{E^{\ast }}^q({\beta }_{\phi (n)}{J}_E^p{z}_{\phi (n)}+(1-{\beta }_{\phi (n)})J_E^p{S}^{\phi (n)}{z}_{\phi (n)})$ and using the similar reasonings to those in Case 1, we obtain

$$\underset{n\to \infty }{lim}\parallel z_{\phi (n)}-S_2^{\phi (n)}{z}_{\phi (n)}\parallel =\underset{n\to \infty }{lim}\parallel v_{\phi (n)}-z_{\phi (n)}\parallel =0.$$

This together with (46) implies that

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

(47)

Noticing ${\zeta }_{\phi (n)}=J_{E^{\ast }}^q({\alpha }_{\phi (n)}{J}_E^{p}u+(1-{\alpha }_{\phi (n)})J_E^p{v}_{\phi (n)})$, by (47) we obtain

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

(48)

Applying the same reasonings as in Case 1, we have that ${lim}_{n\to \infty }\parallel x_{\phi (n)}-x_{\phi (n)+1}\parallel =0$,

$$\underset{n\to \infty }{lim}\parallel s_{\phi (n)}-y_{\phi (n)}\parallel =\underset{n\to \infty }{lim}\parallel w_{\phi (n)}-{\overline{y}}_{\phi (n)}\parallel =0,$$

(49)

and

$$\underset{n\to \infty }{lim\; sup}⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\phi (n)}-\widehat{u}⟩\le 0.$$

(50)

Using (44), we obtain

$$\begin{array}{cc}& D_{f_p}(\widehat{u},x_{\phi (n)+1})\hfill \\ \hfill & \le (1-{\alpha }_{\phi (n)})D_{f_p}(\widehat{u},x_{\phi (n)})+{\alpha }_{\phi (n)}\{\frac{{ϵ}_{\phi (n)}}{{\alpha }_{\phi (n)}}\parallel J_E^p{S}_1^{\phi (n)}{x}_{\phi (n)}-J_E^p(S_1^{\phi (n)}{x}_{\phi (n)}+x_{\phi (n)}-x_{\phi (n)-1})\parallel \hfill \\ \hfill & \times \parallel \widehat{u}+x_{\phi (n)-1}-x_{\phi (n)}-S_1^{\phi (n)}{x}_{\phi (n)}\parallel +\frac{{\theta }_{\phi (n)}}{{\alpha }_{\phi (n)}}{D}_{f_p}(\widehat{u},x_{\phi (n)})+\frac{{\overline{\theta }}_{\phi (n)}}{{\alpha }_{\phi (n)}}{D}_{f_p}(\widehat{u},s_{\phi (n)})\hfill \\ \hfill & +⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\phi (n)}-\widehat{u}⟩\},\hfill \end{array}$$

(51)

which together with (45), yields

$$\begin{array}{cc}& {\mathbf{\Gamma }}_{\phi (n)}\hfill \\ & \le \frac{{ϵ}_{\phi (n)}}{{\alpha }_{\phi (n)}}\parallel J_E^p{S}_1^{\phi (n)}{x}_{\phi (n)}-J_E^p(S_1^{\phi (n)}{x}_{\phi (n)}+x_{\phi (n)}-x_{\phi (n)-1})\parallel \parallel \widehat{u}+x_{\phi (n)-1}-x_{\phi (n)}-S_1^{\phi (n)}{x}_{\phi (n)}\parallel \hfill \\ & +\frac{{\theta }_{\phi (n)}}{{\alpha }_{\phi (n)}}{D}_{f_p}(\widehat{u},x_{\phi (n)})+\frac{{\overline{\theta }}_{\phi (n)}}{{\alpha }_{\phi (n)}}{D}_{f_p}(\widehat{u},s_{\phi (n)})+⟨J_E^{p}u-J_E^p\widehat{u},{\zeta }_{\phi (n)}-\widehat{u}⟩.\hfill \end{array}$$

As a result, from (50) we deduce that

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

(52)

From (50)–(52), we conclude that

$${\mathbf{\Gamma }}_{\phi (n)+1}\to 0(n\to \infty ).$$

(53)

Again using (45), we obtain ${\mathbf{\Gamma }}_n\to 0$. Therefore, $x_n-\widehat{u}\to 0$. This completes the proof.   □

Algorithm 5: (The 2nd parallel subgradient-like extragradient approach.)

Initialization: Given $x_0,x_1\in C$ arbitrarily and let $ϵ>0,{\mu }_{\iota }>0,l_{\iota }\in (0,1)$ and ${\lambda }_{\iota }\in (0,\frac{1}{{\mu }_{\iota }})$ for $\iota =1,2$. Choose $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\}\subset (0,1)$ and $\left\{{𝓁}_n\right\}\subset (0,\infty )$ such that ${lim}_{n\to \infty }{𝓁}_n=0,{\sum }_{n=1}^{\infty }{\alpha }_n=\infty$, ${lim}_{n\to \infty }{\alpha }_n=0$, ${lim}_{n\to \infty }\frac{{\theta }_n+{\overline{\theta }}_n}{{\alpha }_n}=0$, $0<{lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)$ and $0<{lim\; inf}_{n\to \infty }{\gamma }_n(1-{\gamma }_n)$. Moreover, given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ such that $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where ${sup}_{n\ge 1}\frac{{ϵ}_n}{{\alpha }_n}<\infty$ and $$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{𝓁}_n}{\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n-x_{n-1}+x_n)\parallel }\}\mathrm{if}{x}_{n-1}\ne x_n,\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$$ Iterations: Compute $x_{n+1}$ below: Step 1. Put $g_n=J_{E^{\ast }}^q((1-{ϵ}_n)J_E^p{S}_1^n{x}_n+{ϵ}_n{J}_E^p(S_1^n{x}_n+x_n-x_{n-1}))$, and calculate $s_n=J_{E^{\ast }}^q({\gamma }_n{J}_E^p{x}_n+(1-{\gamma }_n)J_E^p{g}_n)$, $y_n={\Pi }_C(J_{E^{\ast }}^q(J_E^p{s}_n-{\lambda }_1{A}_1{s}_n))$, $e_{{\lambda }_1}(s_n):=s_n-y_n$ and $t_n=s_n-{\tau }_n{e}_{{\lambda }_1}(s_n)$, where ${\tau }_n:=l_1^{k_n}$ and $k_n$ is the smallest $k\ge 0$ such that $$\frac{{\mu }_1}{2}{D}_{f_p}(s_n,y_n)\ge ⟨A_1{s}_n-A_1(s_n-l_1^k{e}_{{\lambda }_1}(s_n)),s_n-y_n⟩.$$ Step 2. Calculate $w_n={\Pi }_{C_n}(s_n)$, with $C_n:=\{y\in C:h_n(y)\le 0\}$ and $$h_n(y)=⟨A_1{t}_n,y-s_n⟩+\frac{{\tau }_n}{2{\lambda }_1}{D}_{f_p}(s_n,y_n).$$ Step 3. Calculate ${\overline{y}}_n={\Pi }_C(J_{E^{\ast }}^q(J_E^p{w}_n-{\lambda }_2{A}_2{w}_n))$, $e_{{\lambda }_2}(w_n):=w_n-{\overline{y}}_n$ and ${\overline{t}}_n=w_n-{\overline{\tau }}_n{e}_{{\lambda }_2}(w_n)$, where ${\overline{\tau }}_n:=l_2^{j_n}$ and $j_n$ is the smallest $j\ge 0$ such that $$\frac{{\mu }_2}{2}{D}_{f_p}(w_n,{\overline{y}}_n)\ge ⟨A_2{w}_n-A_2(w_n-l_2^j{e}_{{\lambda }_2}(w_n)),w_n-{\overline{y}}_n⟩.$$ Step 4. Set $z_n={\Pi }_{{\overline{C}}_n}(w_n)$, and calculate $v_n=J_{E^{\ast }}^q({\beta }_n{J}_E^p{z}_n+(1-{\beta }_n)J_E^p(S_2^n{z}_n))$ and $x_{n+1}={\Pi }_C(J_{E^{\ast }}^q({\alpha }_n{J}_E^{p}u+(1-{\alpha }_n)J_E^p{v}_n)$, where ${\overline{C}}_n:=\{y\in C:{\overline{h}}_n(y)\le 0\}$ and $${\overline{h}}_n(y)=⟨A_2{\overline{t}}_n,y-w_n⟩+\frac{{\overline{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\overline{y}}_n).$$ Again, set $n:=n+1$ and return to Step 1.

Remark 1.

It can be easily seen from the proof of Theorem 2 that if the assumption ${lim}_{n\to \infty }\frac{{𝓁}_n}{{\alpha }_n}=0$ is used in place of the assumption ${lim}_{n\to \infty }{𝓁}_n=0$ and ${sup}_{n\ge 1}\frac{{ϵ}_n}{{\alpha }_n}<\infty$, then Theorem 2 is still valid. Moreover, in comparison with the associated outcomes with [14,19], our outcomes are the improvement, extension and development of them in the aspects below.

(i) The issue of seeking a solution of a single VIFPP (implicating a mapping of Bregman’s relative nonexpansivity) in [14] is developed into our issue of seeking a solution of a pair of VIFPPs (implicating both mappings of Bregman’s relatively asymptotical nonexpansivity); the hybrid-projection approach with linesearch term in [14] is developed into our parallel subgradient-like extragradient approach with inertial effect.

(ii) The issue of seeking a solution of a pair of VIPs with CFPP constraint (involving finite Bregman relatively nonexpansive mappings and Bregman relatively demicontractive mapping) in [19] is developed into our issue of seeking a solution of a pair of VIFPPs (involving both mappings of Bregman’s relatively asymptotical nonexpansivity); the modified inertial-type subgradient extragradient method with linear-search process in [19] is developed into our parallel subgradient-like extragradient approach with inertial effect.

Under Algorithm 4, setting $A_2=0$ we obtain the algorithm below for approximating a point in $\Omega =\mathrm{VI}(C,A_1)\cap ({\bigcap }_{i=1}^2\mathrm{Fix}(S_i))$.

Corollary 1.

Let the terms (C1) and (C2) with $A_2=0$ be valid, and assume $\Omega =\mathrm{VI}(C,A_1)\cap ({\bigcap }_{i=1}^2\mathrm{Fix}(S_i))\ne Ø$. If under Algorithm 6, $S_1^{n+1}{x}_n-S_1^n{x}_n\to 0$ and $S_2^{n+1}{w}_n-S_2^n{w}_n\to 0$, then $x_n\rightharpoonup z\in \Omega \iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$.

Algorithm 6: (The 3rd parallel subgradient-like extragradient approach.)

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 $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}\subset (0,1)$ and $\left\{{𝓁}_n\right\}\subset (0,\infty )$ such that ${lim\; inf}_{n\to \infty }{\alpha }_n(1-{\alpha }_n)>0$, ${lim\; inf}_{n\to \infty }{\beta }_n(1-{\beta }_n)>0$ and ${\sum }_{n=1}^{\infty }{𝓁}_n<\infty$. Moreover, assume ${\sum }_{n=1}^{\infty }{\theta }_n<\infty$, and given the iterates $x_{n-1}$ and $x_n(n\ge 1)$, choose ${ϵ}_n$ such that $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where $$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{𝓁}_n}{\parallel J_E^p{S}_1^n{x}_n-J_E^p(S_1^n{x}_n+x_n-x_{n-1})\parallel }\}\mathrm{if}{x}_n\ne x_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$$ Iterations: Compute $x_{n+1}$ below: Step 1. Put $g_n=J_{E^{\ast }}^q((1-{ϵ}_n)J_E^p{S}_1^n{x}_n+{ϵ}_n{J}_E^p(S_1^n{x}_n+x_n-x_{n-1}))$, and calculate $s_n=J_{E^{\ast }}^q({\beta }_n{J}_E^p{x}_n+(1-{\beta }_n)J_E^p{g}_n)$, $y_n={\Pi }_C(J_{E^{\ast }}^q(J_E^p{s}_n-{\lambda }_1{A}_1{s}_n))$, $e_{{\lambda }_1}(s_n):=s_n-y_n$ and $t_n=s_n-{\tau }_n{e}_{{\lambda }_1}(s_n)$, with ${\tau }_n:=l_1^{k_n}$ and $k_n$ being the smallest $k\ge 0$ such that $$\frac{{\mu }_1}{2}{D}_{f_p}(s_n,y_n)\ge ⟨A_1{s}_n-A_1(s_n-l_1^k{e}_{{\lambda }_1}(s_n)),s_n-y_n⟩.$$ Step 2. Calculate $w_n={\Pi }_{C_n}(s_n)$, with $C_n:=\{y\in C:h_n(y)\le 0\}$ and $$h_n(y)=⟨A_1{t}_n,y-s_n⟩+\frac{{\tau }_n}{2{\lambda }_1}{D}_{f_p}(s_n,y_n).$$ Step 3. Calculate $v_n=J_{E^{\ast }}^q({\alpha }_n{J}_E^p{w}_n+(1-{\alpha }_n)J_E^p(S_2^n{w}_n))$ and $x_{n+1}={\Pi }_{Q_n}(w_n)$, with $Q_n:=\{y\in C:D_{f_p}(y,v_n)\le ({\overline{\theta }}_n+1)D_{f_p}(y,w_n)\}$. Again, set $n:=n+1$ and return to Step 1.

Next, set $S_2=I$ as the identity mapping of E. Then, we obtain $\Omega =\mathrm{Fix}(S_1)\cap ({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i))$. In this case, Algorithm 5 can be rewritten as the iterative scheme below for settling a pair of VIPs and the FPP of $S_1$. By Theorem 2 we derive the strong convergence outcome below.

Corollary 2.

Suppose that the condition (C2) holds, and let $\Omega =({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i))\cap \mathrm{Fix}(S_1)\ne Ø$. For initial $x_0,x_1\in C$, choose ${ϵ}_n$ such that $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where

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

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

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

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

$$⟨A_1{s}_n-A_1(s_n-l_1^k(s_n-y_n)),s_n-y_n⟩\le \frac{{\mu }_1}{2}{D}_{f_p}(s_n,y_n),$$

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

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

(i) 

$C_n:=\{y\in C:h_n(y)\le 0\}$ and $h_n(y)=⟨A_1{t}_n,y-s_n⟩+\frac{{\tau }_n}{2{\lambda }_1}{D}_{f_p}(s_n,y_n)$;

(ii) 

${\overline{C}}_n:=\{y\in C:{\overline{h}}_n(y)\le 0\}$ and ${\overline{h}}_n(y)=⟨A_2{\overline{t}}_n,y-w_n⟩+\frac{{\overline{\tau }}_n}{2{\lambda }_2}{D}_{f_p}(w_n,{\overline{y}}_n)$.

Then, $x_n\to {\Pi }_{\Omega }u\iff {sup}_{n\ge 0}\parallel x_n\parallel <\infty$, provided $S_1^{n+1}{x}_n-S_1^n{x}_n\to 0$.

4. Implementability and Applicability

In this section, we provide an illustrative example to demonstrate the applicability and implementability of our suggested approaches. For $i=1,2$, we take $ϵ=\frac{1}{3}$, ${\mu }_i=1$ and $l_i={\lambda }_i=\frac{1}{3}$. First, we present an instance involving the mappings $A_1,A_2:E\to E^{\ast }$ of both uniform continuity and pseudomonotonicity and the mappings $S_1,S_2:C\to C$ of both uniform continuity and Bregman’s relatively asymptotical nonexpansivity satisfying $\Omega \ne Ø$. Set $C=[-2,2]$ and $E=H=\mathbf{R}$ with the inner product and induced norm being written as $⟨a,b⟩=ab$ and $\parallel ·\parallel =|·|$, respectively. The starting $x_0,x_1$ values are randomly chosen in C. For $i=1,2$, let $A_i:H\to H$ be defined as $A_{1}y:=\frac{1}{1+|siny|}-\frac{1}{1+\left|y\right|}$ and $A_{2}y:=y+siny$ for all $y\in H$. Next, let us prove that $A_1$ is the mapping of both Lipschitz continuity and pseudomonotonicity. In fact, for each $v,w\in H$ we have

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

Thus, $A_1$ has Lipschitz continuity. Furthermore, we show that $A_1$ has pseudomonotonicity. For any $v,w\in H$, it is easily known that

$$\begin{array}{cc}& ⟨A_{1}v,w-v⟩=(\frac{1}{1+|sinv|}-\frac{1}{1+\left|v\right|})(w-v)\ge 0\hfill \\ & \Rightarrow ⟨A_{1}w,w-v⟩=(\frac{1}{1+|sinw|}-\frac{1}{1+\left|w\right|})(w-v)\ge 0.\hfill \end{array}$$

It is easy to see that $A_2$ has both Lipschitz continuity and monotonicity. Indeed, we deduce that $\parallel A_{2}v-A_{2}y\parallel \le \parallel v-y\parallel +\parallel sinv-siny\parallel \le 2\parallel v-y\parallel$ and

$$⟨A_{2}v-A_{2}y,v-y⟩={\parallel v-y\parallel }^2+⟨sinv-siny,v-y⟩\ge {\parallel v-y\parallel }^2-{\parallel v-y\parallel }^2=0.$$

Now, let $S_1:C\to C$ and $S_2:C\to C$ be defined as $S_{1}y=S_{2}y:=Sy=\frac{4}{5}siny$. It is clear that $\mathrm{Fix}(S_i)=\mathrm{Fix}(S)=\left\{0\right\}$ for $i=1,2$.

Furthermore, $S:C\to C$ is the mapping of Bregman’s relatively asymptotical nonexpansivity with ${\theta }_n={(\frac{4}{5})}^n$, and $\forall \left\{{\varrho }_n\right\}\subset C$ we obtain $\parallel S^{n+1}{\varrho }_n-S^n{\varrho }_n\parallel \to 0$. In fact, note that

$$\parallel S^{n}v-S^n{w\parallel }^2\le {(\frac{4}{5})}^2\parallel S^{n-1}v-S^{n-1}{w\parallel }^2\le \cdots \le {(\frac{4}{5})}^{2n}{\parallel v-w\parallel }^2\le (1+{\theta }_n){\parallel v-w\parallel }^2,$$

$$\parallel S^{n+1}{\varrho }_n-S^n{\varrho }_n\parallel \le {(\frac{4}{5})}^{n-1}\parallel S^2{\varrho }_n-S{\varrho }_n\parallel ={(\frac{4}{5})}^{n-1}\parallel \frac{4}{5}sin(S{\varrho }_n)-\frac{4}{5}sin{\varrho }_n\parallel \le 2{(\frac{4}{5})}^n\to 0(n\to \infty ),$$

and

$$\underset{n\to \infty }{lim}\frac{{\theta }_n}{1/2(n+1)}=\underset{n\to \infty }{lim}\frac{{(4/5)}^n}{1/2(n+1)}=0.$$

Consequently,

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

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

$$\underset{n\to \infty }{lim}{\beta }_n(1-{\beta }_n)=\underset{n\to \infty }{lim}\frac{n+2}{2(n+1)}(1-\frac{n+2}{2(n+1)})=\frac{1}{2}(1-\frac{1}{2})=\frac{1}{4}>0$$

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

Example 1.

Let ${𝓁}_n=\frac{1}{2{(n+1)}^2}$ 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 ${ϵ}_n$ such that $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where

$$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{𝓁}_n}{\parallel x_n-x_{n-1}\parallel }\}\mathrm{if}{x}_n\ne x_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$$

Algorithm 4 is rewritten as follows:

$$\left\{\begin{array}{cc}& g_n=S^n{x}_n+{ϵ}_n(x_n-x_{n-1}),\hfill \\ \hfill & s_n=\frac{n+2}{2(n+1)}{x}_n+\frac{n}{2(n+1)}{g}_n,\hfill \\ \hfill & y_n=P_C(s_n-\frac{1}{3}{A}_1{s}_n),\hfill \\ \hfill & t_n=(1-{\tau }_n)s_n+{\tau }_n{y}_n,\hfill \\ \hfill & w_n=P_{C_n}{s}_n,\hfill \\ \hfill & {\overline{y}}_n=P_C(w_n-\frac{1}{3}{A}_2{w}_n),\hfill \\ \hfill & {\overline{t}}_n=(1-{\overline{\tau }}_n)w_n+{\overline{\tau }}_n{\overline{y}}_n,\hfill \\ \hfill & v_n=\frac{n}{2(n+1)}{S}^n{w}_n+\frac{n+2}{2(n+1)}{w}_n,\hfill \\ \hfill & Q_n=\{y\in C:\parallel v_n{-y\parallel }^2\le (1+{(\frac{4}{5})}^n)\parallel w_n-y{\parallel }^2\},\hfill \\ \hfill & x_{n+1}=P_{{\overline{C}}_n\cap Q_n}{w}_n,\hfill \end{array}\right.$$

(54)

with the sets $C_n,{\overline{C}}_n$ and the step sizes ${\tau }_n,{\overline{\tau }}_n$ being selected as in Algorithm 4. By Theorem 1, we obtain $x_n\to 0\in \Omega =({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i))\cap \mathrm{Fix}(S))$.

Example 2.

Let ${𝓁}_n=\frac{1}{2{(n+1)}^2},{\alpha }_n=\frac{1}{2(n+1)}$ 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 ${ϵ}_n$ such that $0\le {ϵ}_n\le \overline{{ϵ}_n}$, where

$$\overline{{ϵ}_n}=\left\{\begin{array}{cc}& min\{ϵ,\frac{{𝓁}_n}{\parallel x_n-x_{n-1}\parallel }\}\mathrm{if}{x}_n\ne x_{n-1},\hfill \\ & ϵ\mathrm{otherwise}.\hfill \end{array}\right.$$

Algorithm 5 is rewritten as follows:

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

(55)

with the sets $C_n,{\overline{C}}_n$ and the step sizes ${\tau }_n,{\overline{\tau }}_n$ being selected as in Algorithm 5. By Theorem 2, we deduce that $x_n\to 0\in \Omega =({\bigcap }_{i=1}^2\mathrm{VI}(C,A_i))\cap \mathrm{Fix}(S)$.

It is worth pointing out that the illustrating instance as above bears up the competitive strength of our suggested schemes over the existing schemes, e.g., the ones in [14]. Indeed, we have constructed the illustrating instance of a pair of VIFPPs as above. It is clear that the existing method in [14] is only utilized to solving a single VIFPP. So it follows that, there is no way for this approach to treat the above illustrating instance, that is, it is invalid for a pair of VIFPPs. But, our suggested approach can handle the above illustrating instance. This ensures the competitive strength of our suggested schemes over the existing schemes in the literature.

5. Conclusions

This article designs iterative algorithms for resolving a pair of VIFPPs in uniformly smooth and p-uniformly convex Banach spaces. With the help of parallel subgradient-like extragradient methods with both inertial effect and linesearch process, we fabricate two algorithms for approximating a common solution of the two pseudomonotone VIPs and the CFPP of two mappings of Bregman’s relatively asymptotical nonexpansivity. We are focused on discussing the weak and strong convergence of the proposed algorithms by using standard terms and novel maneuvers. Additionally, an illustrative example is presented to validate the implementability and applicability of our proposed approaches. Finally, it is worth mentioning that part of our future research is aimed at achieving the weak and strong convergence results for the modifications of our proposed approaches with Nesterov double inertial extrapolation steps (see [31]) and adaptive step sizes.