A subgradient extragradient algorithm for solving monotone variational inequalities in Banach spaces

Abstract

In this paper, we introduce an algorithm for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in Banach space. We modify the subgradient extragradient methods with a new and simple iterative step size, the strong convergence of algorithm is established without the knowledge of the Lipschitz constant of the mapping. Finally, a numerical experiment is presented to show the efficiency and advantage of the proposed algorithm. Our results generalize some of the work in Hilbert spaces to Banach spaces.

Inertial Method for Solving Pseudomonotone Variational Inequality and Fixed Point Problems in Banach Spaces

“…(a) Theorem 1 improves, extends, and generalizes the corresponding results [12,13,[30][31][32][33] in the sense that either our method requires an inertial term to improve the convergence rate and/or the space considered is more general. (b) We observe that the result in Corollary 1 improves, and extends the results in [7,[34][35][36] from Hilbert space to a p-uniformly convex and uniformly smooth real Banach space as well as from solving the monotone variational inequality problem to the pseudomonotone variational inequality problem.…”

“…However, we understand from the definition that y n := Since {x n s } is bounded, it follows that there exists a subsequence {x n s k } of {x n s } that converges weakly to some point z in E. By using (33), we obtain v n s z; from (27) and Definition 2, we conclude that z ∈ F(T). Furthermore, from (32), we obtain that y n s z. This together with lim s→∞ ||y n s − w n s || = 0 in (28) and Lemma 4, we conclude that z ∈ V I(C, A), therefore z ∈ Γ.…”

A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings

“…In the early 1960's, Stampacchia [43] and Fichera [12] introduced the theory of variational inequality problem. The Problem (1) is a fundamental problem which has a wide range of applications in applied field of mathematics such as network equilibrium problems, complementary problems, optimization theory and systems of nonlinear equations (see [5,13,19,24,27,28,40,48]). Under suitable conditions, there are generally two main approaches to finding the solutions of VIP (1).…”