A. Ali

In this paper, we extends the classical theory of variational inequalities to the hyperbolic scalar setting using the structure of $\mathbb{D}$-Hilbert spaces. We introduce and analyze a new class of variational inequalities, termed general mildly $\mathbb{D}$-nonlinear variational inequalities, which generalize classical formulations by incorporating $\mathbb{D}$-nonlinear and product-type mappings. We characterize these problems in terms of their idempotent components and demonstrate that several known variational inequality problems, including Stampacchia-type and complementarity problems, emerge as special cases.

In this paper we prove the bicomplex version of the Alaoglu theorem for a topological $\mathbb{BC}$-module $X$. The concept of a $\mathbb{BC}$-dual pair and a product-type open cover in bicomplex is also introduced.

Доведено бікомплексну версію теореми Alaoglu у топологічному $\mathbb{BC}$-модулі $X.$ Також наведено поняття про $\mathbb{BC}$-дуальну пару та відкрите покриття типу добутку в бікомплексі.