Iterative Algorithms for Common Solutions of Nonlinear Problems in Banach Spaces

Abstract

The purpose of this manuscript is to construct an iterative algorithm for approximating a common solution of variational inequality problem and g-fixed point problem of pseudomonotone and Bregman relatively g-nonexpansive mappings, respectively, and prove strong convergence of a sequence generated by the proposed method to a common solution of the problems in real reflexive Banach spaces. The assumption that the mapping is Lipschitz monotone mapping is dispensed with. In addition, we give an application of our main result to find a minimum point of a convex function in real reflexive Banach spaces. Finally, we provide a numerical example to validate our result. Our results extend and generalize many results in the literature.

1. Introduction

Let $E$ be a real Banach space with its dual space $E^{\ast }$. Let $C$ be a nonempty, closed, and convex subset of $E$. A mapping $G:C\to E^{\ast }$ is said to be monotone provided that for all points $p$ and $q$ in $C$,

$$⟨\mathit{Gp}-\mathit{Gq},p-q⟩\ge 0.$$E1

It is called $\alpha$-strongly monotone if there exists a positive real number $\alpha$ such that for all points $p$ and $q$ in $C$,

$$⟨\mathit{Gp}-\mathit{Gq},p-z⟩\ge {‖\mathit{Gp}-\mathit{Gq}‖}^2.$$E2

We remark that $\alpha$-strongly monotone is ${\alpha }^{-1}-$Lipschitz monotone mapping. A mapping $G:C\to E^{\ast }$ is called pseudomonotone mapping provided that for all points $p$ and $q$ in $C$,

$$⟨\mathit{Gp},p-q⟩\ge 0\text{implies}⟨\mathit{Gq},p-q⟩\ge 0.$$E3

From inequalities (1)-(3) above, we can observe that the class of pseudomonotone mappings contains the classes of monotone and $\alpha$-strongly monotone mappings. Let $G:C\to E^{\ast }$ be a mapping. The variational inequality problem (VIP) introduced by Hartman and Stampacchia [1] in 1966 is mathematically formulated as the problem of finding a point $z$ in $C$ such that for all points $p$ in $C$,

$$⟨\mathit{Gz},p-z⟩\ge 0.$$E4

We denote the solution set of problem (4) by $\mathit{VIP}\left(C,V\right)$. This problem contains, as special cases, many problems in the fields of applied mathematics, such as mechanics, physics, engineering, the theory of convex programming, and the theory of control. Consequently, considerable research efforts have been devoted to methods of finding approximate solutions of variational inequality problems in several directions for different classes of mappings (see, e.g., [2, 3, 4, 5, 6, 7, 8, 9, 10]).

Several authors have also studied, different iterative algorithms for approximating a common solution of VIP and fixed point problem of Lipschitz monotone and nonexpansive mappings, respectively (see, e.g., [4, 7, 11, 12, 13, 14, 15]).

In 2003, Takahashi and Tododa [13] introduced an iterative algorithm for finding a common solution for VIP and fixed point problem of $\alpha$-strongly monotone and nonexpansive mappings, respectively, in Hilbert spaces setting. Under certain conditions, they proved that the sequence generated by their proposed method converges weakly to a common solution.

In 2005, Iiduka and Takashi [3] studied an iterative scheme for finding a common solution of VIP and fixed point problem of $\alpha$-strongly monotone and nonexpansive mappings, respectively, in Hilbert spaces setting. They proved that the sequence generated by their proposed scheme converges strongly to a common solution provided that the control sequences satisfy appropriate conditions.

In 2016, Zhang and Yuan [16] established an algorithm for approximating a common solution of VIP and fixed point problem for a finite family of $\alpha$-inverse strongly monotone and nonexpansive mappings, respectively, in the Hilbert spaces setting. They proved strong convergence of the sequence proposed by their method.

In space, more general than Hilbert spaces, Tufa and Zegeye [17] introduced an iterative algorithm for approximating a common solution of VIP and fixed point problem of Lipschitz monotone and relatively nonexpansive mappings, respectively in real $2$-uniformly convex and uniformly smooth Banach spaces. They proved that the sequence generated by their algorithm converges strongly to a common solution of the problems. A mapping $T:C\to E^{\ast }$ is said to be relatively nonexpansive if $F\left(T\right)\ne \varnothing$, $\varphi \left(z,\mathit{Tu}\right)\le \varphi \left(z,u\right)\forall u\in C,z\in F\left(T\right)$ and $\widehat{F}\left(T\right)=F\left(T\right)$, where $F\left(T\right)$, is the set of fixed points of $T$ and $\widehat{F}\left(T\right)$ is the set of asymptotical fixed point of $T$.

Recently, Wega and Zegeye [18] introduced an iterative scheme for approximating a common solution of VIP and $g$-fixed point problem (GFP) of Lipschitz monotone and Bregman relatively $g$-nonexpansive mappings, respectively in real reflexive Banach spaces and obtained strong convergence results. A mapping $T:C\to E^{\ast }$ is said to be Bregman relatively $g$-nonexpansive (BRGN) if $F\left(T\right)\ne \varnothing$, $D_g\left(z,\mathit{Tu}\right)\le D_g\left(z,u\right)\forall u\in C,z\in F\left(T\right)$ and ${\widehat{F}}_g\left(T\right)=F_g\left(T\right)$, where $F_g\left(T\right)$, is the set of $g$-fixed points of $T$ and ${\widehat{F}}_g\left(T\right)$ is the set of asymptotical $g$-fixed point of $T$, where $g$ is a convex function of $E$ satisfies certain conditions. A point $z$ in $C$ is said to be $g$-fixed point of $T$ provided that $\mathit{Tz}=\nabla \mathit{gz}$.

Motivated and inspired by the above results, it is our purpose in this book chapter to construct an iterative algorithm, which converge strongly to a common element of the set of VIP solutions of continuous pseudomonotone and the set GFPP of BRGN mappings in real reflexive Banach spaces. In addition, we give an application of our main result to find a minimum point of a convex function and provide a numerical example to validate our main result. Our results extend and generalize many results in the literature.

Now, we recall some definitions that we will need in the sequel.

Hereafter in this paper let $E$ be a real reflexive Banach space with its dual space $E^{\ast }$, $C$ be a nonempty, convex and closed subset of $E$ and let $\mathcal{G}$ be a family of proper, lower semi-continuous and convex functions on $E$.

Let $g$ be an element of $\mathcal{G}$. The domain of $g$, $\mathit{dom}g$, is given by $\mathit{dom}g=\left\{p\in E:g\left(p\right)<\infty \right\}$, the Fenchel conjugate of $g$ at $p^{\ast }$, $g^{\ast }\left(p^{\ast }\right)$, is given by $g^{\ast }\left(p^{\ast }\right)=\mathit{sup}\left\{⟨p^{\ast },p⟩-g\left(p\right):p\in E\mathit{\text{and}}{p}^{\ast }\in E^{\ast }\right\}$, the subdifferential of $g$ at $p$, $\partial g\left(p\right)$, is given by $\partial g\left(p\right)=\left\{p^{\ast }\in E^{\ast }:g\left(q\right)\ge g\left(p\right)+⟨p^{\ast },q-p⟩,\forall p\in E\right\}$, the right-hand derivative of $g$ at $u$ in the direction of $q$, $g^{\prime }\left(p,q\right)$, is given by:

$$g^{\prime }\left(p,q\right)=\underset{s\to 0_{+}}{lim}\frac{g\left(p+\mathit{sq}\right)-g\left(p\right)}{s},$$E5

and the gradient of $g$, at $p$ is a linear function, $\nabla g$, is given by $⟨\nabla g\left(p\right),q⟩=g^{\prime }\left(p,q\right)$.

The function $g$ is called:

1. G$\widehat{a}$teaux differentiable at $p$ element of $E$ if the limit in (5) exists for any $q$ in $E$ as $s\to 0$.

2. G$\widehat{a}$teaux differentiable if it is G$\widehat{a}$teaux differentiable at every element $u$ in $\mathit{int}\mathit{\text{domg}}$_.

3. Uniformity Fréchet differentiable on $C$ if the limit as $s\to 0$ in (5) attained uniformly for $p\in C$ and $\Vert q\Vert =1$.

4. Strongly coercive if ${lim}_{\Vert p\Vert \to \infty }\frac{g\left(p\right)}{\Vert p\Vert }=\infty$.

G$\widehat{a}$teaux differentiable function $g$ is called Legendre if $g^{\ast }$ is G$\widehat{a}$teaux differentiable, both $\mathit{int}\mathit{dom}g$ and $\mathit{int}\mathit{dom}{g}^{\ast }$ are nonempty, $\mathit{dom}\nabla g=\mathit{int}\mathit{dom}g$ and $\mathit{dom}\nabla g^{\ast }=\mathit{int}\mathit{dom}{g}^{\ast }$.

Remark 1.1 $\nabla g^{\ast }={\left(\nabla g\right)}^{-1}$ (see, [19]) provided that $g$ is Legendre function and the gradient of Legendre function $g$ defined by $g\left(u\right)=\frac{{‖u‖}^p}{p}$ is coinciding with the generalized duality map, that is, $\nabla g=J_p$, where $\left(1<p,q<\infty \right)$ and $q$ is a conjugate of $p$ (see, e.g., [20]).

The Bregman distance with respect to $g$ (see, e.g., [21]) is a function $D_g:\mathit{dom}g\times \mathit{int}\mathit{dom}g\to \left[0,\infty \right)$ defined by:

$$D_g\left(q,p\right)=g\left(q\right)-g\left(p\right)-⟨\nabla g\left(p\right),q-p⟩,$$E6

where $g$ is G$\widehat{a}$teaux differentiable. The Bregman projection with respect to $g$ at $p$ in $\mathit{int}\mathit{dom}g$ onto $C$ is denoted by $P_C^{g}p$ defined by $D_g\left(P_C^{g}p,p\right)=inf\left\{D_g\left(q,p\right):\forall q\in C\right\}$.

Remark 1.2 We note that the Bregiman distance is not distance in the usual sense. However, it has the following properties (see, e.g., [22, 23, 24]):

1. The three point identity:

   $$D_g\left(p,q\right)+D_g\left(q,w\right)-D_g\left(p,q\right)=⟨\nabla g\left(w\right)-\nabla g\left(q\right),p-q⟩$$E7

   for all $q\in \mathit{dom}g$ and $p,w\in \mathit{int}\mathit{dom}g$.

2. The four point identity:

$$D_g\left(q,p\right)+D_g\left(q,z\right)-D_g\left(w,p\right)+D_g\left(w,z\right)-⟨\nabla g\left(z\right)-\nabla g\left(p\right),q-w⟩,$$E8

for all $q,w\in \mathit{dom}g$ and $p,z\in \mathit{int}\mathit{dom}g$.

Lemma 1.3 Let $g$ be a totally convex and Gáteaux differentiable on $\mathit{int}\mathit{\text{domg}}$. Let $p\in \mathit{int}\mathit{\text{domg}}$. Then, the $P_c^g$ from $E$ onto $C$ is a unique point with the following properties [25]:

1. $⟨\nabla g\left(p\right)-\nabla g\left(z\right),q-z\le 0$ if and only if $z=P_C^{g}p$, $\forall q\in C$.

2. $D_g\left(p,q\right)\ge D_g\left(q,P_C^{g}p\right)+D_g\left(P_C^{g}p,p\right)$, $\forall q\in C$.

Let $g$ be a Legendre and $V_g:E\times E^{\ast }\to \left[0,\infty \right)$ be a function defined by:

$$V_g\left(p,p^{\ast }\right)=g\left(p\right)-⟨p^{\ast },p⟩+\nabla g^{\ast }\left(q^{\ast }\right),\forall p\in E,p^{\ast }\in E^{\ast }.$$E9

Then, $V_g$ is nonnegative which satisfies (see, e.g., [26])

$$V_g\left(p,p^{\ast }\right)=D_g\left(p,\nabla g^{\ast }\left(p^{\ast }\right)\right)$$E10

and

$$V_g\left(p,p^{\ast }\right)\le V_g\left(p,p^{\ast }+q^{\ast }\right)-⟨q^{\ast },\nabla g^{\ast }\left(p^{\ast }\right)-p⟩,$$E11

for all $p\in E$ and $p^{\ast }\in E^{\ast }$.

Lemma 1.4 If $g$ is lower, convex, semi-convex proper function, then $g^{\ast }$ is a weak${}^{\ast }$ lower semi-convex and proper function and hence, we have

$$D_g\left(w,\nabla g^{\ast }\left(\sum _{i=1}^N{s}_i\nabla g\left(p_i\right)\right)\right)\le \sum _{i=1}^N{s}_i{D}_g\left(w,p_i\right),$$E12

for all $w$ in $E$, where $\left\{p_i\right\}\subseteq E$ and $\left\{s_i\right\}\subseteq \left(0,1\right)$ with ${\sum }_{i=1}^N{s}_i=1$ [27].

A G$\widehat{a}$teaux differentiable function $g$ is called.

1. Uniformly convex function (see, [28]), provided that for all $p$ and $q$ $\mathit{dom}g$ $s\in \left[0,1\right]$, we have

$$g\left(\mathit{sp}+\left(1-s\right)q\right)\le \mathit{sg}\left(p\right)+\left(1-s\right)g\left(p\right)-\left(1-s\right)\mathit{s\varphi }\left(\Vert p-q\Vert \right),$$E13

where $\varphi$ is a function that is increasing and vanishes only at zero.

1. Strongly convex with constant $\alpha >0$ for all $u$ and $q$ elements of $\mathit{\text{domg}}$ (see, [29])

   $$⟨\nabla g\left(p\right)-\nabla g\left(p\right),p-q⟩\ge \alpha {‖p-q‖}^2.$$E14

2. Totaly convex if ${\nu }_g\left(p,s\right)={inf}_{\left\{p\in E:\Vert p-q\Vert =s\right\}}{D}_g\left(q,p\right)>0$, for all $p\in E$ and $s>0$.

We note that $g$ is uniformly convex if and only if $g$ is totally convex on bounded subsets of $E$ (see, [25], Theorem 2.10 p. 9). Moreover, the class of uniformly convex function functions contains the class of strongly convex functions.

Lemma 1.5 Let $E$ be a Banach space and $r>0$ be a constant. Let $g:E\to \mathbb{R}$ be a continuous convex function that is uniformly convex on bounded subsets of $E$. Then,

$$g\left(\sum _{k=0}^n{\beta }_k{u}_k\right)\le \sum _{k=0}^n{\beta }_{k}g\left(u_k\right)-{\beta }_i{\beta }_j{\rho }_r\left(\Vert u_i-u_j\Vert \right),$$E15

$\forall 0\le i,j\le n$, $u_k\in B_r$, ${\beta }_k\in \left(0,1\right)$ with ${\sum }_{k=0}^n{\beta }_k=1$, where ${\rho }_r$ is the gauge of uniform convexity of $g$ [30].

Lemma 1.6 Let $g$ be a total convex G$\widehat{a}$ teaux differentiable such that $\mathit{dom}g=E$. Then, for each $x^{\ast }\in E^{\ast }\left\{0\right\},\tilde{y}\in E,x\in H^{+}$ and $\tilde{x}\in H^{-}$, it holds that

$$D_g\left(\tilde{x},x\right)\ge D_g\left(\tilde{x},z\right)+D_g\left(z,x\right),$$E16

where $z={\mathit{\text{argmin}}}_{y\in H}{D}_g\left(y,x\right)$ and $H^{-}=\left\{y\in E:⟨x^{\ast },y-\tilde{y}⟩\le 0\right\}$, $H=\left\{y\in E:⟨x^{\ast },y-\tilde{y}⟩=0\right\}$ and $H^{+}=\left\{y\in E:⟨x^{\ast },y-\tilde{y}⟩\ge 0\right\}$.

2. An iterative algorithm for a common solution of variational inequality and $g-$fixed problems

In this section, let $E$ be a real reflexive Banach space with its dual space $E^{\ast }$. Let $C$ be a nonempty, closed, and convex subset of $E$. Let $g:E\to \left(-\infty ,+\infty \right]\in \mathcal{G}$ be a uniformly Fr$\widehat{e}$chet differentiable Legendre which is bounded, uniformly convex, and strongly coercive on bounded subsets of $E$. We denote the family of such functions by $\mathcal{G}\left(E\right)$.

In the sequel, we shall make use of the following assumptions.

Assumption:

1. A1) Let $l\in \left(0,1\right)$, $\mu >0$ and $\left[\beta \in \underset{¯}{\beta },\overline{\beta }\right]\subset \left(0,\frac{1}{\mu }\right)$.

2. A2) Let $\left\{{\alpha }_n\right\}\subset \left(0,c\right)$ with the properties ${lim}_{n\to \infty }{\alpha }_n=0\mathit{\text{and}}{\sum }_{n=1}^{\infty }{\alpha }_n=\infty$, where $c>0$.

Algorithm 1: For any $x_0,v\in C$, define an algorithm by.

Step 1. Compute

$$y_n=\nabla g^{\ast }\left[\nabla {\mathit{gx}}_n-\beta {\mathit{Gx}}_n\right]\text{and}d\left(y_n\right)=x_n-P_C^g{y}_n.$$E17

If $d\left(y_n\right)=0$ and $\nabla {\mathit{gx}}_n-{\mathit{Tx}}_n=0$, then stop and $x_n\in \mathrm{\Omega }$. Otherwise,

Step 2. Compute $p_n=x_n-{\tau }_{n}d\left(y_n\right)$,

where ${\tau }_n=l^{j_n}$ and $j_n$ is the smallest nonnegative integer $j$ satisfying

$$⟨{\mathit{Gx}}_n-{\mathit{Gp}}_n,d\left(y_n\right)⟩\le \mu D_g\left(P_C^g{y}_n,x_n\right).$$E18

Step 3. Compute

$$\left(\begin{array}{l}{a}_n=P_{P_n}^f\nabla g^{\ast }\left(\nabla {\mathit{gx}}_n-\beta {\mathit{Gp}}_n\right),\\ r_n=\nabla g^{\ast }\left({\eta }_{n,1}\nabla {\mathit{gx}}_n+{\eta }_{n,2}{\mathit{Tx}}_n+{\eta }_{n,3}\nabla {\mathit{gu}}_n\right),\\ x_{n+1}=P_C^g\nabla g^{\ast }\left({\alpha }_n\nabla \mathit{gv}+\left(1-{\alpha }_n\right)\nabla {\mathit{gr}}_n\right),\end{array}\right.$$E19

where $g\in \mathcal{G}\left(E\right)$, $P_n=\left\{p\in C:⟨{\mathit{Gp}}_n,p-p_n⟩=0\right\}$, $u_n=P_C^g{a}_n$ and $\left\{{\eta }_{n,i}\right\}\subset \left[\epsilon ,1\right)\subset \left(0,1\right)$, for $i=\mathrm{1,2,3}$ such that ${\sum }_{i=1}^3{\eta }_{n,i}=1$, $\forall n\ge 0$.

Step 4. Set $n≔n+1$ and go to Step 1.

We shall need the following Lemmas in the sequel.

Lemma 1.7 Assume that $\left\{x_n\right\}$ and $\left\{y_n\right\}$ are sequences generated by Algorithm 1. Then, the search rule in Step 2 is well defined.

Proof: Since $l\in \left(0,1\right)$ and $G$ is continuous on $C$, we have

$$⟨{\mathit{Gx}}_n-{\mathit{Gp}}_n,d\left(y_n\right)⟩\to 0$$E20

as $j\to \infty$. On the other hand, the fact that $D_g\left(P_C^g{y}_n,x_n\right)>0$, there exists a nonnegative integer $j_n$ satisfying the inequality in Step 2, and the claim holds.

Lemma 1.8 Assume that $\left\{x_n\right\}$ and $\left\{y_n\right\}$ are sequences generated by Algorithm 1. Then, we have:

$$⟨{\mathit{Gx}}_n,d\left(y_n\right)⟩\ge \frac{1}{\beta }{D}_g\left(P_C^g{y}_n,x_n\right)$$E21

Proof: From (17), we have:

$$\nabla {\mathit{gy}}_n=\nabla {\mathit{gx}}_n-\beta {\mathit{Gx}}_n,$$E22

which implies:

$$\nabla {\mathit{gx}}_n-\nabla {\mathit{gy}}_n=\beta {\mathit{Gx}}_n.$$E23

Thus, from (23), (17), and (7), we get:

$$⟨{\mathit{Gx}}_n,d\left(y_n\right)⟩=\frac{1}{\beta }⟨\nabla {\mathit{gx}}_n-\nabla {\mathit{gy}}_n,x_n-P_C^g{y}_n⟩$$E24

$$=\frac{1}{\beta }\left[D_g\left(P_C^g{y}_n,x_n\right)+D_g\left(x_n,y_n\right)-D_g\left(P_C^g{y}_n,y_n\right)\right]$$E25

$$\ge \frac{1}{\beta }{D}_g\left(P_C^g{y}_n,x_n\right),$$E26

and hence the assertion hold.

Lemma 1.9 Suppose the assumption (A1) holds. Let $G:C\to E^{\ast }$ be a continuous pseudomonotone mapping. Then, $⟨{\mathit{Gp}}_n,x_n-p_n⟩\ge {\tau }_n\left(\frac{1}{\beta }-\mu \right)D_g\left(P_C^g{y}_n,x_n\right)$. In particular, if $d\left(y_n\right)\ne 0$, then $⟨{\mathit{Gp}}_n,x_n-p_n⟩>0$.

Proof: Using Step 2 of the algorithm we know that

$$⟨{\mathit{Gp}}_n,x_n-p_n⟩=⟨{\mathit{Gp}}_n,x_n-\left(x_n-{\tau }_{n}d\left(y_n\right)\right)⟩$$E27

$$={\tau }_n⟨{\mathit{Gp}}_n,d\left(y_n\right)⟩.$$E28

On the other hand, from (18), we have:

$$⟨{\mathit{Gx}}_n-{\mathit{Gp}}_n,d\left(y_n\right)⟩\le \mu D_g\left(P_C^g{y}_n,x_n\right)$$E29

which implies that

$$⟨{\mathit{Gp}}_n,d\left(y_n\right)⟩\ge ⟨{\mathit{Gx}}_n,d\left(y_n\right)⟩-\mu D_g\left(P_C^g{y}_n,x_n\right).$$E30

From (30) and Lemma 8, we get:

$$⟨{\mathit{Gp}}_n,d\left(y_n\right)⟩\ge \left(\frac{1}{\beta }-\mu \right)D_g\left(P_C^g{y}_n,x_n\right).$$E31

Combining (28) and (31), we obtain:

$$⟨{\mathit{Gp}}_n,x_n-p_n⟩\ge {\tau }_n\left(\frac{1}{\beta }-\mu \right)D_g\left(P_C^g{y}_n,x_n\right),$$E32

and the proof is complete.

Theorem 1.10 Suppose the Assumptions (A1) and (A2) hold. Let $G:C\to E^{\ast }$ and $T:C\to E^{\ast }$ be continuous pseudomonotone and BRGN mappings, respectively, with $\mathrm{\Omega }=\mathit{VI}\left(C,G\right)\cap F_g\left(T\right)\ne \varnothing$. Then, the sequens $\left\{x_n\right\}$ generated by Algorithm 1 is bounded.

Proof: Let $x^{\ast }=P_{\mathrm{\Omega }}^g\left(v\right)$ and $w_n=\nabla g^{\ast }\left({\alpha }_n\nabla \mathit{gv}+\left(1-{\alpha }_n\right)\nabla {\mathit{gr}}_n\right)$. We note that from Lemma 1.3 (i), we obtain

$$⟨u-x^{\ast },\nabla \mathit{gv}-\nabla {\mathit{gx}}^{\ast }⟩\le 0,\forall u\in \mathrm{\Omega }.$$E33

Now, for each $n\ge 0$, define the sets: $P_n^{-}=\left\{p\in C:⟨{\mathit{Gx}}_n,p-x_n⟩\le 0\right\}$, $P_n=\left\{p\in C:⟨{\mathit{Gx}}_n,p-x_n⟩=0\right\}$, and $P_n^{+}=\left\{p\in C:⟨{\mathit{Gx}}_n,p-x_n⟩\ge 0\right\}$. Let $x^{\ast }\in \mathrm{\Omega }$, from definition of $G$, we have $⟨{\mathit{Gx}}^{\ast },y-x^{\ast }⟩\ge 0$, which implies that $⟨\mathit{Gy},y-x^{\ast }⟩\ge 0$ for all $y\in C$, and hence, $x^{\ast }\in P_n^{-}$ for all $n\ge 0$. Moreover, from Lemma 9, we have $⟨G{p}_n,x_n-p_n⟩>0$, which implies that $x_n\in P_n^{+}$ and $x_n\notin P_n^{-}$ for all $n\ge 0$. Now, from Lemma 1.6, we get:

$$D_g\left(x^{\ast },a_n\right)+D_g\left(a_n,x_n\right)\le D_g\left(x^{\ast },x_n\right).$$E34

Since $u_n=P_C^g{a}_n$, from Lemma 1.3, we get:

$$D_g\left(x^{\ast },u_n\right)+D_g\left(u_n,a_n\right)\le D_g\left(x^{\ast },a_n\right).$$E35

Substituting (35) into (34), we obtain:

$$D_g\left(x^{\ast },u_n\right)+D_g\left(u_n,a_n\right)+D_g\left(a_n,x_n\right)\le D_g\left(x^{\ast },x_n\right),$$E36

which implies that

$$D_g\left(x^{\ast },u_n\right)\le D_g\left(x^{\ast },x_n\right)-D_g\left(u_n,a_n\right)-D_g\left(a_n,x_n\right).$$E37

Using the same techniques of proof of Theorem 3.2 pp. 64 of [18] from (19), (9), (10), and Lemma 1.5, we get:

$$D_g\left(x^{\ast },r_n\right)\le {\eta }_{n,1}{D}_g\left(x^{\ast },x_n\right)+{\eta }_{n,2}{D}_g\left(x^{\ast },\nabla g^{\ast }{\mathit{Tx}}_n\right)+{\eta }_{n,3}{D}_g\left(x^{\ast },u_n\right)$$E38

$$-{\eta }_{n,1}{\eta }_{n,2}{\rho }_r^{\ast }\left(\Vert \nabla {\mathit{gx}}_n-{\mathit{Tx}}_n\Vert \right).$$E39

From inequalities (37) and (39) above and assumption on $T$, we get:

$$D_g\left(x^{\ast },r_n\right)\le {\eta }_{n,1}{D}_g\left(x^{\ast },x_n\right)+{\eta }_{n,2}{D}_g\left(x^{\ast },x_n\right)+{\eta }_{n,3}{D}_g\left(x^{\ast },u_n\right)$$E40

$$\le D_g\left(x^{\ast },x_n\right)-{\eta }_{n,3}\left[D_g\left(u_n,a_n\right)+D_g\left(a_n,x_n\right)\right]$$E41

$$-{\eta }_{n,1}{\eta }_{n,2}{\rho }_r^{\ast }\left(\Vert \nabla {\mathit{gx}}_n-{\mathit{Tx}}_n\Vert \right)$$E42

$$\le D_g\left(x^{\ast },x_n\right).$$E43

Now, from (19), Lemma 1.3 (ii), Lemma 1.4, and (44), we obtain:

$$D_g\left(x^{\ast },x_{n+1}\right)\le D_g\left(x^{\ast },\nabla g^{\ast }\left({\alpha }_n\nabla \mathit{gv}+\left(1-{\alpha }_n\right)\nabla {\mathit{gr}}_n\right)\right)$$E44

$$\le {\alpha }_n{D}_g\left(x^{\ast },v\right)+\left(1-{\alpha }_n\right)D_g\left(x^{\ast },x_n\right)$$E45

$$\le max\left\{D_g\left(x^{\ast },v\right),D_g\left(x^{\ast },x_n\right)\right\},$$E46

and by induction, we get:

$$D_g\left(x^{\ast },x_n\right)\le max\left\{D_g\left(x^{\ast },v\right),D_g\left(x^{\ast },x_0\right)\right\}.$$E47

Hence, the sequence $\left\{D_g\left(x^{\ast },x_n\right)\right\}$ is bounded. Thus, by Lemma 7 in ref. [31], the sequence $\left\{x_n\right\}$ is bounded and so are $\left\{a_n\right\}$, $\left\{u_n\right\}$, $\left\{r_n\right\}$, $\left\{{\mathit{Gp}}_n\right\}$, and $\left\{{\mathit{Tx}}_n\right\}$.

Theorem 1.11 Suppose the Assumptions (A1) and (A2) hold. Let $G:C\to E^{\ast }$ and $T:C\to E^{\ast }$ be continuous pseudomonotone and BRGN mappings, respectively with $\mathrm{\Omega }=\mathit{VI}\left(C,G\right)\cap F_g\left(T\right)\ne \varnothing$. Then, the sequens $\left\{x_n\right\}$ generated by Algorithm 1 converge strongly to an element $x^{\ast }=P_{\mathrm{\Omega }}^g\left(v\right)$.

Proof: From Theorem 1.10 above, we know that the sequence $\left\{x_n\right\}$ is bounded. Let $x^{\ast }=P_{\mathrm{\Omega }}^g\left(v\right)$. Now, using the same techniques of proof of Theorem 2 of ref. [32], we get:

$$D_g\left(x^{\ast },x_{n+1}\right)\le \left(1-{\alpha }_n\right)D_g\left(x^{\ast },x_n\right)+{\alpha }_n\Vert \nabla \mathit{gv}-\nabla {\mathit{gx}}^{\ast }\Vert \Vert x_n-w_n\Vert$$E48

$$+{\alpha }_n⟨\nabla \mathit{gv}-\nabla {\mathit{gx}}^{\ast },x_n-x^{\ast }⟩.$$E49

Furthermore, from (19), Lemma 1.3 (ii), and Lemma 1.4, we have:

$$D_g\left(x^{\ast },x_{n+1}\right)\le D_g\left(x^{\ast },\nabla g^{\ast }\left({\alpha }_n\nabla \mathit{gv}+\left(1-{\alpha }_n\right)\nabla {\mathit{gr}}_n\right)\right)$$E50

$$\le {\alpha }_n{D}_g\left(x^{\ast },v\right)+\left(1-{\alpha }_n\right)D_g\left(x^{\ast },r_n\right).$$E51

Thus, from (51) and (43), we get:

$$D_g\left(x^{\ast },x_{n+1}\right)\le {\alpha }_n{D}_g\left(x^{\ast },v\right)+\left(1-{\alpha }_n\right)D_g\left(x^{\ast },x_n\right)$$E52

$$-\left(1-{\alpha }_n\right){\eta }_{n,3}\left[D_g\left(u_n,a_n\right)+D_g\left(a_n,x_n\right)\right]$$E53

$$-\left(1-{\alpha }_n\right){\eta }_{n,1}{\eta }_{n,2}{\rho }_r^{\ast }\left(\Vert \nabla {\mathit{gx}}_n-{\mathit{Tx}}_n\Vert \right),$$E54

Now, to complete the proof we use the following two cases:

Case 1. Assume that there exists $n_0\in \mathrm{\mathbb{N}}$ such that the sequence $D_g\left(x^{\ast },x_n\right)$ is decreasing for all $n\ge n_0$. It then follows that the sequence $D_g\left(x^{\ast },x_n\right)$ converges, and hence, $D_g\left(x^{\ast },x_n\right)-D_g\left(x^{\ast },x_{n+1}\right)\to 0$ as $n\to \infty$. Thus, from (53), we obtain:

$$\underset{n\to \infty }{lim}\left[D_g\left(u_n,a_n\right)+D_g\left(a_n,x_n\right)\right]=0,$$E55

and

$$\underset{n\to \infty }{lim}{\rho }_r^{\ast }(\Vert {\mathit{Tx}}_n-\nabla {\mathit{gx}}_n\Vert =0.$$E56

Hence, from (55) and Lemma 2.4 of [33] p. 15, we get:

$$\underset{n\to \infty }{lim}\Vert u_n-a_n\Vert =\underset{n\to \infty }{lim}\Vert x_n-a_n\Vert =0.$$E57

From (56) and property of ${\rho }_r^{\ast }$, we obtain:

$$\underset{n\to \infty }{lim}\Vert {\mathit{Tx}}_n-\nabla {\mathit{gx}}_n\Vert =0.$$E58

From (58) and the fact that $\nabla g^{\ast }$ is uniformly continuous on bounded subsets of $E^{\ast }$, we obtain:

$$\underset{n\to \infty }{lim}\Vert \nabla g^{\ast }{\mathit{Tx}}_n-x_n\Vert =0.$$E59

Moreover, from (19) and Lemma 1.4, we get:

$$D_g\left(x_n,w_n\right)=D_g\left(x_n,\nabla g^{\ast }\left[{\alpha }_n\nabla \mathit{gv}+\left(1-{\alpha }_n\right)\nabla {\mathit{gr}}_n\right]\right)$$E60

$$\le {\alpha }_n{D}_g\left(x_n,v\right)+\left(1-{\alpha }_n\right)D_g\left(x_n,r_n\right)$$E61

$$={\alpha }_n{D}_g\left(x_n,v\right)+\left(1-{\alpha }_n\right)\left[{\eta }_{n,1}{D}_g\left(x_n,x_n\right)+{\eta }_{n,2}{D}_g\left(x_n,\nabla g^{\ast }{\mathit{Tx}}_n\right)\right]$$E62

$$+\left(1-{\alpha }_n\right)\left[{\eta }_{n,3}{D}_g\left(x_n,u_n\right)+{\eta }_{n,4}{D}_g\left(x_n,v_n\right)\right]$$E63

Thus, from Lemma 2.4 of [33] p. 15, (57), (59), and (61), we get:

$$\underset{n\to \infty }{lim}{D}_G\left(x_n,w_n\right)=0,$$E64

which implies that

$$\underset{n\to \infty }{lim}\Vert x_n-w_n\Vert =0.$$E65

Now, since $\left\{x_n\right\}$ is bounded in $C$ there exists $u\in C$ and a subsequence $\left\{x_{n_k}\right\}$ such that $\left\{x_{n_k}\right\}$ converges weakly to $u$ and

$$\underset{n\to \infty }{limsup}⟨x_n-x^{\ast },\nabla \mathit{gv}-\nabla {\mathit{gx}}^{\ast }⟩=\underset{k\to \infty }{lim}⟨x_{n_k}-x^{\ast },\nabla \mathit{gv}-\nabla {\mathit{gx}}^{\ast }⟩.$$E66

From (58) and definition of $T$, we have $u\in F_g\left(T\right)$.

Next, we prove that $u\in \mathit{VI}\left(C,G\right)$. Since $a_n\in P_n$then we can get:

$$0=⟨{\mathit{Gp}}_{n_k},a_{n_k}-p_{n_k}⟩$$E67

$$=⟨{\mathit{Gp}}_{n_k},a_{n_k}-x_{n_k}⟩+⟨{\mathit{Gp}}_{n_k},x_{n_k}-p_{n_k}⟩$$E68

which implies that

$$⟨{\mathit{Gp}}_{n_k},x_{n_k}-p_{n_k}⟩=⟨{\mathit{Gp}}_{n_k},x_{n_k}-a_{n_k}⟩\le \Vert {\mathit{Gp}}_{n_k}\Vert \Vert x_{n_k}-a_{n_k}\Vert .$$E69

From (57), (69) and the fact that the sequence $\left\{{\mathit{Gp}}_n\right\}$ is bounded, we get:

$$\underset{k\to \infty }{lim}⟨{\mathit{Gp}}_{n_k},x_{n_k}-p_{n_k}⟩=0.$$E70

Now, we prove

$$\underset{k\to \infty }{lim}\Vert P_C^g{y}_{n_k}-x_{n_k}\Vert =0.$$E71

From (70), Lemma 1.9 and Lemma 2.4 of [33] p. 15, we get:

$$\underset{k\to \infty }{lim}{\tau }_{n_k}\Vert P_C^g{y}_{n_k}-x_{n_k}\Vert =0.$$E72

First, consider the case when ${liminf}_{k\to \infty }{\tau }_{n_k}>0$. In this case, there is a constant $\tau >0$ such that ${\tau }_{n_k}\ge \tau >0$ for all $k\in \mathrm{\mathbb{N}}$. Thus, we have:

$$\Vert P_C^g{y}_{n_k}-x_{n_k}\Vert =\frac{1}{{\tau }_{n_k}}{\tau }_{n_k}\Vert P_C^g{y}_{n_k}-x_{n_k}\Vert \le \frac{1}{\tau }{\tau }_{n_k}\Vert P_C^g{y}_{n_k}-x_{n_k}\Vert .$$E73

Thus, from (72) and (73), we obtain:

$$\underset{k\to \infty }{lim}\Vert P_C^g{y}_{n_k}-x_{n_k}\Vert =0.$$E74

Second, we consider the case when ${liminf}_{k\to \infty }{\tau }_{n_k}=0$. In this case, we take a subsequence $\left\{n_{k_j}\right\}$ of $\left\{n_k\right\}$, if necessary, we assume without loss of generality that

$$\underset{k\to \infty }{lim}{\tau }_{n_k}=0\mathit{\text{and}}\underset{k\to \infty }{lim}\Vert x_{n_k}-P_C^g{y}_{n_k}\Vert =a>0.$$E75

Consider $p_{n_k}^{\prime }=\frac{1}{l}{\tau }_{n_k}{P}_C^g{y}_{n_k}+\left(1-\frac{1}{l}{\tau }_{n_k}\right)x_{n_k}$. Then, from (75), we have:

$$\underset{k\to \infty }{lim}\Vert x_{n_k}-p_{n_k}^{\prime }\Vert =\underset{k\to \infty }{lim}\frac{1}{l}{\tau }_{n_k}\Vert x_{n_k}-P_C^g{y}_{n_k}\Vert =0.$$E76

From the search rule in Step 2 and the definition of $p_{n_k}^{\prime }$, we get:

$$⟨{\mathit{Gx}}_{n_k}-G{p}_{n_k}^{\prime },x_{n_k}-P_C^g{y}_{n_k}⟩>\mu D_g\left(P_C^g{y}_{n_k},x_{n_k}\right).$$E77

Using (76), (77), and Lemma 2.4 of [33] p. 15, and the fact that $G$ is uniformly continuous on bounded subsets of $C$, we obtain:

$$\underset{k\to \infty }{lim}\Vert P_C^g{y}_{n_k}-x_{n_k}\Vert =0,$$

which is a contradiction to (75). Therefore, the equality in (71) holds. Combining Lemma 1.3 and 17, we get:

$$⟨{\mathit{Gx}}_{n_k},z-P_C^g{z}_{n_k}⟩\ge ⟨\nabla {\mathit{gx}}_{n_k}-\nabla {\mathit{gP}}_C^g{y}_{n_k},z-P_C^g{y}_{n_k}⟩,\forall z\in C,$$E78

which implies that

$$⟨{\mathit{Gx}}_{n_k},y-x_{n_k}⟩\ge ⟨{\mathit{Gx}}_{n_k},P_C^g{y}_{n_k}-x_{n_k}⟩$$E79

$$+⟨\nabla {\mathit{gx}}_{n_k}-\nabla {\mathit{gP}}_C^g{y}_{n_k},z-P_C^g{y}_{n_k}⟩,\forall z\in C.$$E80

Thus, from (80), (78) and the fact that $\nabla g$ is uniformly continuous, we obtain:

$$\underset{k\to \infty }{liminf}⟨{\mathit{Gx}}_{n_k},z-x_{n_k}⟩\ge 0,\forall z\in C.$$E81

Moreover, let $\left\{{\xi }_k\right\}$ be a sequence of decreasing numbers such that $\left\{{\xi }_k\right\}\to 0$ as $k\to \infty$ and $w$ be an arbitrary element of $C$. Using inequality (81), we can find a large enough $N_k$ such that

$$⟨{\mathit{Gx}}_{n_k},w-x_{n_k}⟩+{\xi }_k\ge 0,\forall k\ge N_k.$$E82

From (82) and the fact that ${\mathit{Gx}}_{n_k}\ne 0$, we get:

$$⟨{\mathit{Gx}}_{n_k},{\xi }_k{d}_k+w-x_{n_k}⟩\ge 0,\forall k\ge N_k,$$E83

for some $d_k\in C$ satisfying $⟨{\mathit{Gx}}_{n_k},d_k⟩=1$. In addition, from the definition of $G$ and inequality (83), we have:

$$⟨G\left(w+{\xi }_n{d}_{k}w\right),w+{\xi }_k{d}_{k}w-x_{n_k}⟩\ge 0,\forall k\ge N_k,$$E84

which implies that

$$⟨\mathit{Gw},w-x_{n_k}⟩\ge ⟨\mathit{Gw}-G\left(w+{\xi }_k{d}_{k}w\right),w+{\xi }_k{d}_{k}w-x_{n_k}⟩$$E85

$$-{\xi }_k⟨\mathit{Gw},d_k⟩,\forall k\ge N_k,$$E86

Since ${\xi }_k\to 0$ as $k\to \infty$ and $G$ is continuous, then from inequality (86), we obtain:

$$⟨\mathit{Gw},w-u⟩=\underset{k\to \infty }{liminf}⟨\mathit{Gw},w-x_{n_k}⟩\ge 0,\forall w\in C.$$E87

Thus, $u\in \mathit{VI}\left(C,G\right)$, and hence, $u\in \mathrm{\Omega }$. It follows Lemma 1.3 (i), that

$$\underset{n\to \infty }{limsup}⟨x_n-x^{\ast },\nabla \mathit{gv}-\nabla {\mathit{gx}}^{\ast }⟩=\underset{k\to \infty }{lim}⟨x_{n_k}-x^{\ast },\nabla \mathit{gv}-\nabla {\mathit{gx}}^{\ast }⟩$$E88

$$=⟨u-x^{\ast },\nabla \mathit{gv}-\nabla {\mathit{gx}}^{\ast }⟩\le 0.$$E89

Therefore, from (49), (65), (89), and Lemma 2.5 of [34] p. 243, we conclude that $D_g\left(x^{\ast },x_n\right)\to 0$ as $n\to \infty$. Hence, by Lemma 2.4 of [33] p. 15, $x_n\to x^{\ast }$ as $n\to \infty$.

Case 2. Suppose that there exists a subsequence $\left\{n_i\right\}$ of $\left\{n\right\}$ such that

$$D_g\left(x^{\ast },x_{n_i}\right)<D_g\left(x^{\ast },x_{n_i+1}\right),\forall i\in \mathrm{\mathbb{N}}.$$E90

Then, by Lemma 3.1 of [35] p. 904, there exists a nondecreasing sequence $\left\{m_k\right\}$ in the set of natural numbers such that $m_k\to \infty$ as $k\to \infty$, $D_g\left(x^{\ast },x_{m_k}\right)\le D_g\left(x^{\ast },x_{m_k+1}\right)$ and $D_g\left(x^{\ast },x_k\right)\le D_g\left(x^{\ast },x_{m_k+1}\right)$ for all $k$ elements of the set of natural numbers. Thus, from (53), we obtain:

$$\underset{k\to \infty }{lim}\Vert u_{m_k}-a_{m_k}\Vert =\underset{k\to \infty }{lim}\Vert {\mathit{Tx}}_{m_k}-\nabla {\mathit{gx}}_{m_k}\Vert =0.$$E91

Moreover, following the methods in Case 1 above, we get:

$$\underset{k\to \infty }{lim}\Vert x_{m_k}-w_{m_k}\Vert ,$$E92

and

$$\underset{k\to \infty }{limsup}⟨⟨x_{m_k}-x^{\ast },\nabla \mathit{gv}-\nabla {\mathit{gx}}^{\ast }⟩\le 0.$$E93

In addition, from (49) and inequality (90) above, we obtain:

$$D_g\left(x^{\ast },x_{m_k}\right)\le \Vert x_{m_k}-r_{m_k}\Vert \nabla \mathit{gv}-\nabla {\mathit{gx}}^{\ast }\Vert$$E94

$$+⟨x_{m_k}-x^{\ast },\nabla \mathit{gv}-\nabla {\mathit{gx}}^{\ast }⟩.$$E95

Therefore, from (92), (93), and (95), we obtain ${lim}_{k\to \infty }{D}_g\left(x^{\ast },x_{m_k}\right)=0$. But from inequality (53), we obtain that ${lim}_{k\to \infty }{D}_g\left(x^{\ast },x_{m_k+1}\right)=0$, which implies that ${lim}_{k\to \infty }{D}_g\left(x^{\ast },x_k\right)=0$. Thus, by Lemma 2.4 of [33] p. 15 $x_k\to x^{\ast }$ as $k\to \infty$.

We remark that the proof of Theorem 11 provides the following result for a common point in the solution set of VIP and the set of $g-$fixed point of continuous monotone and BRGN, mappings, respectively.

Theorem 1.12 Suppose the Assumptions (A1) and (A2) hold. Let $G:C\to E^{\ast }$ and $T:C\to E^{\ast }$ be continuous monotone and BRGN mappings, respectively with $\mathrm{\Omega }=\mathit{VI}\left(C,G\right)\cap F_g\left(T\right)\ne \varnothing$. Then, the sequens $\left\{x_n\right\}$ generated by Algorithm 1 converge strongly to an element $x^{\ast }=P_{\mathrm{\Omega }}^g\left(v\right)$.

If in Algorithm 1, we put $C=E$, then $P_C^g$ is reduced to the identity mapping in $E$ and $\mathit{VI}\left(C,G\right)=G^{-1}\left(0\right)$. Thus, we get the following Algorithm 2 for a common point in the set of zeros and the set of $g-$fixed point of continuous pseudomonotone and BRGN mappings, respectively.

Algorithm 2: For any $x_0,v\in E$, define an algorithm by.

Step 1. Compute

$$y_n=\nabla g^{\ast }\left[\nabla {\mathit{gx}}_n-\beta {\mathit{Gx}}_n\right]\text{and}d\left(y_n\right)=x_n-y_n.$$E96

If $d\left(y_n\right)=0$ and $\nabla {\mathit{gx}}_n-{\mathit{Tx}}_n=0$, then stop and $x_n\in \mathrm{\Omega }$. Otherwise,

Step 2. Compute $p_n=x_n-{\tau }_{n}d\left(y_n\right)$,

where ${\tau }_n=l^{j_n}$ and $j_n$ is the smallest nonnegative integer $j$ satisfying

$$⟨{\mathit{Gx}}_n-{\mathit{Gp}}_n,d\left(y_n\right)⟩\le \mu D_g\left(y_n,x_n\right).$$E97

Step 3. Compute

$$\left(\begin{array}{l}{u}_n=P_{P_n}^f\nabla g^{\ast }\left(\nabla {\mathit{gx}}_n-\beta {\mathit{Gp}}_n\right),\\ r_n=\nabla g^{\ast }\left({\eta }_{n,1}\nabla {\mathit{gx}}_n+{\eta }_{n,2}{\mathit{Tx}}_n+{\eta }_{n,3}\nabla {\mathit{gu}}_n\right),\\ x_{n+1}=\nabla g^{\ast }\left({\alpha }_n\nabla \mathit{gv}+\left(1-{\alpha }_n\right)\nabla {\mathit{gr}}_n\right),\end{array}\right.$$E98

where $g\in \mathcal{G}\left(E\right)$, $P_n=\left\{p\in C:⟨{\mathit{Gp}}_n,p-p_n⟩=0\right\}$, and $\left\{{\eta }_{n,i}\right\}\subset \left[\mathrm{ϵ},1\right)\subset \left(0,1\right)$, for $i=\mathrm{1,2,3}$ such that ${\sum }_{i=1}^3{\eta }_{n,i}=1$, $\forall n\ge 0$.

Step 4. Set $n≔n+1$ and go to Step 1.

Corollary 1.13 Suppose the Assumptions (A1) and (A2) hold. Let $G:E\to E^{\ast }$ and $T:E\to E^{\ast }$ be continuous pseudomonotone and BRGN mappings, respectively with $\mathrm{\Omega }=G^{-1}\left(0\right)\cap F_g\left(T\right)\ne \varnothing$. Then, the sequens $\left\{x_n\right\}$ generated by Algorithm 2 converge strongly to an element $x^{\ast }=P_{\mathrm{\Omega }}^g\left(v\right)$.

If in Algorithm 2, we put $T=\nabla g$, the identity mapping in $E$, then we get the following corollary for zero point of continuous pseudomonotone.

Corollary 1.14 Suppose the Assumptions (A1) and (A2) hold. Let $G:E\to E^{\ast }$ be a continuous pseudomonotone mapping with $G^{-1}\left(0\right)\ne \varnothing$. Then, the sequens $\left\{x_n\right\}$ generated by Algorithm 2 converge strongly to an element $x^{\ast }=P_{G^{-1}\left(0\right)}^g\left(v\right)$.

2.1 Application to convex minimization problem

In this section, we apply Corollary 1.14 to find the minimum point of the convex function in Banach Spaces.

Let $f:E\to \mathbb{R}$ be a convex smooth function. We consider the problem of finding a point $z\in E$ such that

$$f\left(z\right)=\underset{x\in E}{min}\left\{f\left(x\right)\right\}.$$E99

According to Fermat’s rule, this problem is equivalent to the problem of finding $z\in E$ such that

$$\nabla \mathit{fz}=0,$$E100

where $\nabla f$ is a gradient of $f$. We note that $\nabla f$ is monotone mapping (see, e.g., [36, 37]) and hence pseudomonotone mapping.

Now, if in Algorithm 2, we assume $G=\nabla f$, then we obtain the following Algorithm 3 for the minimum point problem of convex functions in real reflexive Banach spaces.

Algorithm 3: For any $x_0,v\in E$, define an algorithm by.

Step 1. Compute

$$y_n=\nabla g^{\ast }\left[\nabla {\mathit{gx}}_n-\beta \nabla {\mathit{fx}}_n\right]\text{and}d\left(y_n\right)=x_n-y_n.$$E101

If $d\left(y_n\right)=0$, then stop and $x_n\in \mathrm{\Omega }$. Otherwise,

Step 2. Compute $p_n=x_n-{\tau }_{n}d\left(y_n\right)$,

where ${\tau }_n=l^{j_n}$ and $j_n$ is the smallest nonnegative integer $j$ satisfying

$$⟨\nabla {\mathit{fx}}_n-\nabla {\mathit{fp}}_n,d\left(y_n\right)⟩\le \mu D_g\left(y_n,x_n\right).$$E102

Step 3. Compute

$$\left(\begin{array}{l}{u}_n=P_{P_n}^f\nabla g^{\ast }\left(\nabla {\mathit{gx}}_n-\beta \nabla {\mathit{fp}}_n\right),\\ r_n=\nabla g^{\ast }\left({\eta }_n\nabla {\mathit{gx}}_n+\left(1-{\eta }_n\right)\nabla {\mathit{gu}}_n\right),\\ x_{n+1}=\nabla g^{\ast }\left({\alpha }_n\nabla \mathit{gv}+\left(1-{\alpha }_n\right)\nabla {\mathit{gr}}_n\right),\end{array}\right.$$E103

where $g\in \mathcal{G}\left(E\right)$, $P_n=\left\{p\in C:⟨\nabla {\mathit{fp}}_n,p-p_n⟩=0\right\}$, and $\{{\eta }_n\subset [\mathrm{ϵ},1)\subset \left(0,1\right),\forall n\ge 0$.

Step 4. Set $n≔n+1$ and go to Step 1.

The method of proof Theorem 1.11 provides the proof of the following theorem of finding the minimum point of a convex function in reflexive Banach spaces.

Theorem 1.15 Suppose the Assumptions (A1) and (A2) hold. Let $f:E\to \mathbb{R}$ be a convex smooth function with $\nabla f$ is continuous and $\mathrm{\Omega }=\left\{z:f\left(z\right)={min}_{x\in E}f\left(x\right)\right\}\ne \varnothing$. Then, the sequens $\left\{x_n\right\}$ generated by Algorithm 3 converge strongly to an element $x^{\ast }=P_{\mathrm{\Omega }}^g\left(v\right)$.

2.2 Numerical example

In this section, we provide a numerical example to explain the conclusion of our main result. The following example verifies the conclusion of Theorem 1.11.

Example 1.16. Let $E=\mathbb{R}$ be with the standard topology. Define $g:\mathbb{R}\to \mathbb{R}$, by $g\left(x\right)=\frac{x^2}{2}$, then $g^{\ast }\left(x^{\ast }\right)=\frac{x^{\ast 2}}{2}$ and $\nabla g\left(x\right)=x=\nabla g^{\ast }\left(x^{\ast }\right)=x^{\ast }$, where $x=\left(x_1,x_2,x_3\right)\in \mathbb{R}$. Let $C=\left\{x\in \mathbb{R}:\Vert x\Vert \le 1\right\}$. Let $G,T:C\to \mathbb{R}$ be defined by $G\left(x_1,x_2,x_3\right)=\left(x_1,x_2,x_3\right)\left(1.8-\sqrt{x_1^2+x_2^2+x_3^2}\right)$ and $T\left(x_1,x_2,x_3\right)=\frac{\left(x_1,x_2,x_3\right)}{5}$, then $G$ is continuous pseudomonotone mapping and $T$ is BRGN mapping with $\mathrm{\Omega }=\mathit{VI}\left(C,G\right)=\left\{0\right\}=F_g\left(T\right)\ne \varnothing$. Now, if we assume $v=\left(v_1,v_2,v_3\right)=\left(\mathrm{0,0.5,0.5}\right)$, ${\alpha }_n=\frac{1}{n+10}$, ${\eta }_{n,1}={\eta }_{n,2}=0.001+\frac{1}{n+1000}$ and ${\eta }_{n,3}=0.998-\frac{2}{n+1000}$, $l=0.8$, $\mu =0.9$ and $\lambda =1$ for all $n\ge 0$, and take different initial points $x_0=\left(0,1,-1\right)$, $x_0^{\text{'}}=\left(\mathrm{1.2233,2},-1.4532\right)$ and $x_0^{{\text{'}}^{\prime }}=\left(\mathrm{1,2,3}\right)$, then in all cases the numerical example result using MATLAB provides that the sequence $\left\{x_n\right\}$, generated by Algorithm 1 converges strongly to $x^{\ast }=\left(\mathrm{0,0,0}\right)$ (see, Table 1). In addition, we have sketched the error term $\Vert x_n-x^{\ast }\Vert$ for each initial point. From the sketch, we observe that $\Vert x_n-x^{\ast }\Vert \to 0$ as $n\to \infty$ (see, Figure 1).

$n$	$x_n$	$x_n$	$x_n$
$0$	$\left(\mathrm{0.0000,1.0000},-1.0000\right)$	$\left(\mathrm{1.2233,2.0000},-1.4532\right)$	$\left(\mathrm{1.0000,2.0000,3.0000}\right)$
$1$	$\left(\mathrm{0.0000,0.2379,0.1881}\right)$	$\left(-\mathrm{0.1729,0.1325,0.9238}\right)$	$\left(\mathrm{0.0195,0.0622},-0.8500\right)$
$10$	$\left(\mathrm{0.0000,0.0335,0.0228}\right)$	$\left(-\mathrm{0.0868,0.0280,0.0797}\right)$	$\left(\mathrm{0.0110,0.0269},-0.2428\right)$
$100$	$\left(\mathrm{0.0000,0.0058,0.0040}\right)$	$\left(-\mathrm{0.0134,0.0049,0.0026}\right)$	$\left(\mathrm{0.0019,0.0047,0.0024}\right)$
$200$	$\left(\mathrm{0.0000,0.0030,0.0021}\right)$	$\left(-\mathrm{0.0069,0.0025,0.0014}\right)$	$\left(9.8943e^{-04}\mathrm{,0.0024,0.0012}\right)$
$300$	$\left(\mathrm{0.0000,0.0020,0.0014}\right)$	$\left(-\mathrm{0.0046,0.0017,9.1348}{e}^{-04}\right)$	$\left(6.6775e^{-04}\mathrm{,0.0017,8.4377}{e}^{-04}\right)$
$400$	$\left(\mathrm{0.0000,0.0016,0.0011}\right)$	$\left(-\mathrm{0.0035,0.0013,6.9030}{e}^{-04}\right)$	$\left(5.0352e^{-04}\mathrm{,0.0013,6.3737}{e}^{-04}\right)$
$500$	$\left(\mathrm{0.0000,0.0012,8.5023}{e}^{-04}\right)$	$\left(-\mathrm{0.0028,0.0010,5.5401}{e}^{-04}\right)$	$\left(4.0389e^{-04}\mathrm{,0.0010,5.1207}{e}^{-04}\right)$
	$⋮$	$⋮$	$⋮$
	$\downarrow$	$\downarrow$	$\downarrow$
	$\left(\mathrm{0,0,0}\right)$	$\left(\mathrm{0,0,0}\right)$	$\left(\mathrm{0,0,0}\right)$

Table 1.

Convergence of the sequence $\left\{x_n\right\}$ generated by Algorithm 1 for different choices of $x_0$.

The sequence $\left\{x_n\right\}$, generated by Algorithm 1 converges strongly to $x^{\ast }=\left(0,0,0\right)$.

Figure 1.

The graph of $\Vert x_n-x^{\ast }\Vert$ versus number of iterations with different choices of $x_0$.

3. Conclusions

In this, manuscript, we introduced an iterative method for approximating a common solution of VIP of continuous pseudomonotone and GFPP of BRGN mappings and proved strong convergence of the sequence generated by the method to a common solution in real reflexive Banach spaces. In addition, we gave an application of our main result to find a minimum point of convex functions in real reflexive Banach spaces. Finally, a numerical example that supports our main result is presented. Our results extend and generalize many results in the literature. In particular, Theorem 1.11 extends the results in [3, 4, 7, 13, 16, 17, 38] from real Hilbert spaces to real reflexive Banach spaces. Moreover, Theorem 1.11 extends the classes of mappings in Theorem 3.1 of Tufa and Zegeye [17] and Theorem 3.2 of Wega and Zegeye [18] from Lipschitz monotone mapping to continuous pseudomonotone mappings in reflexive real Banach spaces.

Conflict of interest

The authors declare that they have no competing interests.