Abstract
We introduce two new concepts of weakly relaxedη-αmonotone mappings and weakly relaxedη-αsemimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxedη-αmonotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxedη-αsemimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem.
Highlights
The variational inequality theory provides us with a simple, natural, unified, and elegant framework to study a wide class of linear and nonlinear problems arising in many fields, such as mechanics, engineering sciences, elasticity, optimization, control, programming, economics, transportation, oceanography, and regional
Using KKM technique, we study and prove the existence of solutions for variational-like inequalities problems with mapping in such class in Banach spaces
Let E be an arbitrary Banach space with its dual space E∗, let E∗∗ denote the dual space of E∗, and let K be a nonempty closed convex subset of E∗∗
Summary
The variational inequality theory provides us with a simple, natural, unified, and elegant framework to study a wide class of linear and nonlinear problems arising in many fields, such as mechanics, engineering sciences, elasticity, optimization, control, programming, economics, transportation, oceanography, and regional. Because of their wide applicability, various extensions and generalizations of the classical variational inequality problem have been proposed and studied in recent years. In [16], Fang and Huang introduced a new concept of relaxed η-α monotonicity and obtained the existence of solutions for variational-like inequalities with relaxed η-α monotone mappings in reflexive Banach spaces. Abstract and Applied Analysis extend and improve the results of Fang and Huang [16] and many results in the literature
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