Nonlinear variational inequality in Reflexive Banach space with application in epidemiology

Abstract

This paper studies some nonlinear variational inequalities with operators defined on the reflexive and separable Banach spaces and verifies the pseudomonotone properties. This study come up with some new results, such as the existence and the strong convergence of solutions, all these results are based on the Galerkin method. As an application, the framework's conclusion has been used for epidemiology problems by taking the spatio-temporal SIR model, which is defined as a reaction-diffusion system with Signorini boundary condition to generate a variational inequality. In such application, the Signorini boundary condition is considered for infected individuals while the Neumann boundary condition is considered for susceptible and recovered individuals. The paper ended with a numerical analysis of the problem by applying a splitting method and the Uzawa algorithm based on the augmented Lagrangian method. Some tests are included to show the solutions.