Abstract
In this paper, we introduce a generalized system (for short, GS) in real Banach spaces. Using Brouwer’s fixed point theorem, we establish some existence theorems for the generalized system without monotonicity. Further, we extend the concept of C-strong pseudomonotonicity and extend Minty’s lemma for the generalized system. And using the Minty lemma and KKM-Fan lemma, we establish an existence theorem for the generalized system with monotonicity in real reflexive Banach spaces. As the continuation of existing studies, our paper present a series of extended results based on existing corresponding results.
Highlights
Variational inequality theory has played a fundamental and important role in the study of a wide range of problems arising in physics, mechanics, differential equations, contact problems in elasticity, optimization, economics and engineering sciences, etc
We introduce a generalized system in real Banach spaces
We extend the concept of C-strong pseudomonotonicity and extend Minty’s lemma for the generalized system
Summary
Variational inequality theory has played a fundamental and important role in the study of a wide range of problems arising in physics, mechanics, differential equations, contact problems in elasticity, optimization, economics and engineering sciences, etc. Using the Minty lemma and KKM-Fan lemma, we establish an existence theorem for the generalized system with monotonicity in real reflexive Banach spaces.
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