Abstract
In this paper, we consider a generalized mixed variational-like inequality problem and prove a Minty-type lemma for its related auxiliary problems in a real Banach space. We prove the existence of a solution of these auxiliary problems and also prove some properties for the solution set of generalized mixed variational-like inequality problem. Furthermore, we introduce and study an inertial hybrid iterative method for solving the generalized mixed variational-like inequality problem involving Bregman relatively nonexpansive mapping in Banach space. We study the strong convergence for the proposed algorithm. Finally, we list some consequences and computational examples to emphasize the efficiency and relevancy of the main result.
Highlights
In this paper, we consider a generalized mixed variational-like inequality problem and prove a Minty-type lemma for its related auxiliary problems in a real Banach space
We prove the existence of a solution of these auxiliary problems and prove some properties for the solution set of generalized mixed variational-like inequality problem
We introduce and study an inertial hybrid iterative method for solving the generalized mixed variational-like inequality problem involving Bregman relatively nonexpansive mapping in Banach space
Summary
Assume g: X ⟶ (− ∞, +∞] is a proper, convex, and lower semicontinuous mapping and g∗: X∗ ⟶ (− ∞, +∞] is a Fenchel conjugate of g, defined as g∗ u0 sup〈u0, u〉 − g(u): u ∈ Y, u0 ∈ Y∗. (10). Let T: C ⟶ int(domg) be a mapping and F(T) {u ∈ C: Tu u}, where F(T) is the set of fixed points of T. en, we have the following:. Let X be a Banach space, r > 0 be a constant, and g: X ⟶ R be a convex function which is uniformly convex on bounded subsets. Let g: X ⟶ (− ∞, +∞] be a Gateaux differentiable and totally convex function on int(domg). Let b: C × C ⟶ R satisfy the following: (i) b is skew-symmetric, i.e., b(u, u) − b(u, v) − b(v, u) + b(v, v) ≥ 0, ∀u, v ∈ C (ii) b is convex in the second argument (iii) b is continuous
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