The goal of this paper is to study quasi-monotone variational inequality problems in infinite dimensional Hilbert spaces and to prove that the iterative sequence generated by an extragradient algorithm converges weakly to a solution.

Senthil Kumar Prakasam, Arul Joseph Gnanaprakasam, Nasreen Kausar Gunaseelan Mani, Mohammed Munir, and Salahuddin. International Journal of Fuzzy Logic and Intelligent Systems 2022;22:287-95. https://doi.org/10.5391/IJFIS.2022.22.3.287

In this work, we suggest a differential variational inequality in reflexive Banach spaces and construct a sequence with a set of constraints and a penalty parameter. We use the penalty method to prove a unique solution to the problem and make suitable assumptions to prove the convergence of the sequence. The proof is based on arguments for compactness, symmetry, pseudomonotonicity, Mosco convergence, inverse strong monotonicity and Lipschitz continuity. Finally, we discuss the boundary value problem for the differential variational inequality problem as an application.

In this article, we introduce two new rough set models based on the concept of double fuzzy relations. These models are called optimistic and pessimistic multi-granulation double fuzzy rough sets. We discuss their properties and explore the relationship between these new models and double fuzzy rough sets. Our study focuses on the lower and upper approximations of these models, which generalize the conventional rough set model. In addition, we suggest that the development of the multi-granulation double fuzzy rough set model is significant for the generalization of the rough set model.

In this paper, our goal was to establish the relationship between solutions of local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, also Minty local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, under generalized approximate $\eta$-convexity conditions for locally Lipschitzian functions.