In this paper, we design two inertial-type subgradient extragradient algorithms with line-search process for solving the pseudomonotone variational inequality problems (VIPs) and common fixed-point problem (CFPP) of finite Bregman relatively nonexpansive mapping and a Bregman relatively demicontractive mapping in p-uniformly convex and uniformly smooth Banach spaces, which are more general than Hilbert spaces. Under mild conditions, we derive weak and strong convergence of the suggested algorithms to a common solution of the VIPs and CFPP, respectively. Additionally, an illustrated example is furnished to back up the feasibility and implementability of our proposed methods.
In a p-uniformly convex and uniformly smooth 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 inertial effect for solving this pair of VIFPPs, where each algorithm consists of two parts which are of symmetric structure mutually. Under mild registrations, we prove weak and strong convergence of the suggested algorithms to a common solution of this pair of VIFPPs, respectively. Lastly, an illustrative example is furnished to verify the applicability and implementability of our proposed approaches.
In real Hilbert spaces, let the CFPP indicate a common fixed-point problem of asymptotically nonexpansive operator and countably many nonexpansive operators, and suppose that the HVI and VIP represent a hierarchical variational inequality and a variational inequality problem, respectively. We put forward Mann hybrid deepest-descent extragradient approach for solving the HVI with the CFPP and VIP constraints. The proposed algorithms are on the basis of Mann’s iterative technique, viscosity approximation method, subgradient extragradient rule with linear-search process, and hybrid deepest-descent rule. Under suitable restrictions, it is shown that the sequences constructed by the algorithms converge strongly to a solution of the HVI with the CFPP and VIP constraints.
In a real Hilbert space, let the CFPP, VIP, and HFPP denote the common fixed-point problem of countable nonexpansive operators and asymptotically nonexpansive operator, variational inequality problem, and hierarchical fixed point problem, respectively. With the help of the Mann iteration method, a subgradient extragradient approach with a linear-search process, and a hybrid deepest-descent technique, we construct two modified Mann-type subgradient extragradient rules with a linear-search process for finding a common solution of the CFPP and VIP. Under suitable assumptions, we demonstrate the strong convergence of the suggested rules to a common solution of the CFPP and VIP, which is only a solution of a certain HFPP.
In real Banach spaces, the concept of α-well-posedness is extended to the class of generalized hemivariational inequalities systems consisting of two parts which are of symmetric structure mutually. First, certain concepts of α-well-posedness for generalized hemivariational inequalities systems are put forward. Second, certain metric characterizations of α-well-posedness for generalized hemivariational inequalities systems are presented. Lastly, certain equivalence results between strong α-well-posedness of both the system of generalized hemivariational inequalities and its system of derived inclusion problems are established.
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