Extended new generalized mixed vector variational-like inequalities in reflexive Banach spaces

In this paper, we introduce and study an extended new generalized mixed vector variational-like inequality (ENGMVVLI) problem in real reflexive Banach spaces. This problem is more general than many existing equilibrium problems and variational inequalities available in the literature. We prove the existence results of (ENGMVVLI) by utilizing the KKM lemma and Nadlar’s results with relaxed semi- α-monotone mappings. Also, by using the Brouwer’s fixed point theorem, we prove the existence of solutions for (ENGMVVLI) without using the monotonicity concept. Finally, we give an example to justify the validity of our results. © 2016, Operational Research Society of India.