Inertial extrapolation method for a class of generalized variational inequality problems in real Hilbert spaces

Abstract

The purpose of this paper is to propose an inertial extrapolation method for solving a certain class of variational inequality problems more general than the classical variational inequality problems in real Hilbert spaces. Our proposed method is of viscosity-type and converges strongly to a solution of the aforementioned problem when the underlying/cost operator is pseudo-monotone and uniformly continuous; this makes our method to be potentially more applicable than most existing methods in the literature. To support our results numerically, we considered some examples in both finite and infinite dimensional Hilbert spaces and compared our results with other existing results in the literature.

Appendix A: Algorithms

Algorithm A.1

(The Algorithm in [48]) Let $x_1 \in C$, define the sequences $\{x_n\}$, $\{y_n\}$ and $\{t_n\}$ by

$$\begin{aligned} {} \left\{ \begin{array}{ll} y_n=P_C(x_n-\tau _n T^{*}(I-S)T x_n),\\ t_n=P_C(y_n-\lambda _n A(y_n)), \\ x_{n+1}=P_C(y_n-\lambda _n A(t_n)),\quad n\ge 1, \end{array} \right. \end{aligned}$$

(A.1)

where $\{\tau _n\}\subset [a, b]$ for some $a, b \in \left( 0, \dfrac{1}{\Vert T\Vert ^2}\right)$ and $\{\lambda _n\} \subset [c, d]$ for some $c,d \in \left( 0, \dfrac{1}{L}\right) , ~S:H_2 \rightarrow H_2$ is a nonexpansive mapping, and $A:C\rightarrow H_1$ is a monotone and L-Lipschitz continuous operator.

Algorithm A.2

(The Algorithm in Tian and Jiang [47]) Let $x_1 \in C$, define the sequences $\{x_n\}$, $\{y_n\}$, $\{w_n\}$ and $\{t_n\}$ by

$$\begin{aligned} {\left\{ \begin{array}{ll} y_n= P_C(x_n-\tau _n T^*(I-S)Tx_n),\\ t_n = P_C(y_n - \lambda _n A(y_n)),\\ w_n= P_C(y_n-\lambda _n A(t_n)),\\ x_{n+1}= \alpha _{n}g(x_n) + (1-\alpha _{n})w_n,~n\ge 1, \end{array}\right. } \end{aligned}$$

(A.2)

where $\{\alpha _{n}\}\subset (0,1)$ with $\lim \limits _{n\rightarrow \infty } \alpha _{n}=0$ and $\sum _{n=1}^{\infty }\alpha _{n}=\infty$,   $\{\tau _n\}\subset [a, b]$ for some $a, b \in \left( 0, \dfrac{1}{\Vert T\Vert ^2}\right)$ and $\{\lambda _n\} \subset [c, d]$ for some $c,d \in \left( 0, \dfrac{1}{L}\right) ,~ S:H_2 \rightarrow H_2$ is a nonexpansive mapping, $A:C\rightarrow H_1$ is a monotone and L-Lipschitz continuous operator and g is a contraction on C.

Algorithm A.3

(The Algorithm in Chidume and Nnakwe [13]) For $x_1=x\in H_1, C_1=H_1$ and $W_1=H_1$, define the sequence $\{x_n\}$ by

$$\begin{aligned} {\left\{ \begin{array}{ll} y_n^i=P_{C_i}\Big (x_n-{\tau }T^*(I-S_i)Tx_n \Big ), i=1,\cdots ,N\\ u_n^i=P_{C_i}\Big (y_n^i-\lambda _n A_i(y_n^i)\Big ), i=1,\cdots ,N\\ t_n^i=P_{C_i}\Big (y_n^i-\lambda _n A_i(u_n^i)\Big ), i=1,\cdots , N\\ C_n^i=\Big \{z\in H:\Vert t_n^i-z\Vert \le \Vert x_n-z\Vert \Big \}\\ W_n=\Big \{z\in H:\langle z-x_n,x-x_n\rangle \le 0\Big \}\\ x_{n+1}=P_{C_{n}}\cap W_nx, \forall n\ge 1, \end{array}\right. } \end{aligned}$$

(A.3)

where $C_n=\cap ^N_{i=1}C_n^i, {\tau }\in \Big (0,\frac{1}{\Vert A\Vert ^2}\Big )$, $\lambda _n\in \Big (0,\frac{1}{L}\Big ), C_i,i=1,\cdots ,N$ are nonempty closed and convex subsets of $H_1$ such that $C=\cap ^N_{i=1}C_i\ne \emptyset$,$T:H_1\rightarrow H_2$ is a bounded linear operator such that $T\ne 0$ and $T^*$ is the adjoint of T. $A_i:C_i\rightarrow H_1, i=1,\cdots ,N$ is a finite family of monotone and L-Lipschitz mappings and $S_i:H_2\rightarrow H_2, ~i=1,\cdots ,N$ is a finite family of nonexpansive mappings.