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

Abstract

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

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

“…It can be easily seen from the proof of Theorem 2 that if the assumption lim n→∞ n α n = 0 is used in place of the assumption lim n→∞ n = 0 and sup n≥1 n α n < ∞, 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.…”

“…(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.…”

“…With the help of suitable conditions, it was proven in [14] that {t n } converges strongly to Π Ω 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.…”

“…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.…”