We study the proximal point method for solving quasi-equilibrium problems in Banach spaces, which generalizes the proximal point method for equilibrium problems and quasi-variational inequalities. We propose a regularization procedure which ensures strong convergence of the generated sequence to a solution of the quasi-equilibrium problem, under standard assumptions on the problem without assuming neither pseudo-monotonicity nor any weak continuity assumption of the bifunction in its arguments that in many well-known methods have been used. Also, we show that the boundedness of the generated sequences implies that the solution set of the quasi-equilibrium problem is nonempty and prove the strong convergence of the generated sequence to a solution of the quasi-equilibrium problem when these conditions are satisfied.

We study the extragradient method for solving quasi-equilibrium problems in Banach spaces, which generalizes the extragradient method for equilibrium problems and quasi-variational inequalities. We propose a regularization procedure which ensures strong convergence of the generated sequence to a solution of the quasi-equilibrium problem, under standard assumptions on the problem assuming neither any monotonicity assumption on the bifunction nor any weak continuity assumption of f in its arguments that in the many well-known methods have been used. Also, we give a necessary and sufficient condition for the solution set of the quasi-equilibrium problem to be nonempty and we show that, in this case, this iterative sequence converges strongly to a solution of the quasi-equilibrium problem. In other words, we prove strong convergence of the generated sequence to a solution of the quasi-equilibrium problem without assuming existence of a solution of the problem. Finally, we give an application of our main result to a generalized Nash equilibrium problem.

Influenza, often referred to as the flu, is an extremely contagious respiratory illness caused by influenza viruses, impacting populations globally with significant health consequences annually. A hallmark of influenza is its seasonal patterns, influenced by a mix of geographic, evolutionary, immunological, and environmental factors. Understanding these seasonal trends is crucial for informing public health decisions, including the planning of vaccination campaigns and their formulation. In our study, we introduce a genotype-structured infectious disease model for influenza transmission, immunity, and evolution. In this model, the population of infected individuals is structured according to the virus they harbor. It considers a symmetrical fitness landscape where the influenza A and B variants are considered. The model incorporates the effects of population immunity, climate, and epidemic heterogeneity, which makes it suitable for investigating influenza seasonal dynamics. We parameterize the model to the genomic surveillance data of flu in the US and use numerical simulations to elucidate the scenarios that result in the alternating or consecutive prevalence of flu variants. We show that the speed of virus evolution determines the alternation and co-circulation patterns of seasonal influenza. Our simulations indicate that slow immune waning reduces how often variants change, while cross-immunity regulates the co-circulation of variants. The framework can be used to predict the composition of future influenza outbreaks and guide the development of cocktail vaccines and antivirals that mitigate influenza in both the short and long term.

The aim of this paper is to propose an inertial-type AA-viscosity algorithm for approximating the common solutions of the split variational inclusion problem, the generalized equilibrium problem and the common fixed-point problem of nonexpansive mappings. The strong convergence of an iterative sequence obtained through the proposed method is proved under some mild assumptions. Consequently, approximations of the solution of the split feasibility problem, the relaxed split feasibility problem, the split common null point problem and the split minimization problem are given. The applicability of our proposed algorithm has been illustrated with the help of a numerical example. Our iterative method was then compared graphically with different comparable methods in the existing literature.