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In this paper we prove a generalized version of the Ekeland variational principle, which is a common generalization of Zhong variational principle and Borwein Preiss Variational principle. Therefore in a particular case, from this variational principle we get a Zhong type variational principle, and a Borwein-Preiss variational principle. As a consequence, we obtain a Caristi type fixed point theorem.

In this paper, we consider stochastic optimal control of systems driven by stochastic differential equations with irregular drift coefficient. We establish a necessary and sufficient stochastic maximum principle. To achieve this, we first derive an explicit representation of the first variation process (in the Sobolev sense) of the controlled diffusion. Since the drift coefficient is not smooth, the representation is given in terms of the local time of the state process. Then we construct a sequence of optimal control problems with smooth coefficients by an approximation argument. Finally, we use Ekeland’s variational principle to obtain an approximating adjoint process from which we derive the maximum principle by passing to the limit. The work is notably motivated by the optimal consumption problem of investors paying wealth tax.

Chikungunya is a mosquito-borne viral infection caused by the chikungunya virus (CHIKV). It is characterized by acute onset of high fever, severe polyarthralgia, myalgia, headache, and maculopapular rash. The virus is rapidly spreading and may establish in new regions where competent mosquito vectors are present. This research analyzes the regulatory dynamics of a stochastic differential equation (SDE) model describing the transmission of the CHIKV, incorporating seasonal variations, immunization efforts, and environmental fluctuations modeled through Poisson random measure noise under demographic heterogeneity. The model guarantees the existence of a global positive solution and demonstrates periodic dynamics driven by environmental factors. A key contribution of this study is the formulation of a stochastic threshold parameter, , which characterizes the conditions for disease persistence or extinction under random environmental influences. Although our analysis highlights age-specific heterogeneities to illustrate differential transmission risks, the framework is general and can incorporate other vulnerable demographic groups, ensuring broader applicability of the results. Using the Monte Carlo Markov Chain (MCMC) method, we estimate = 1.4978 (95% CI: 1.4968–1.5823) based on CHIKV data from Florida, USA, spanning 2005 to 2017, suggesting that the outbreak remains active and requires targeted control strategies. The effectiveness of immunization, screening, and treatment strategies varies depending on the prioritized demographic groups, due to substantial differences in CHIKV incidence across age categories in the USA. Numerical simulations were conducted using the truncated Euler–Maruyama method to robustly capture the stochastic dynamics of CHIKV transmission with Poisson-driven jumps. Employing an iterative approach and assuming mild convexity conditions, we formulated and solved a parameterized near-optimality problem using the Ekeland variational principle. Our findings indicate that vaccination campaigns are significantly more effective when focused on vulnerable adults over the age of 66, as well as individuals aged 21 to 25. Furthermore, enhancements in vaccine efficacy, diagnostic screening, and treatment protocols all contribute substantially to minimizing infection rates compared to current standard approaches. These insights support the development of targeted, age-specific public health interventions that can significantly improve the management and control of future CHIKV outbreaks.

In this paper, we introduce the generalized Ekeland’s variational principle in several forms. The general setting of our results includes a graphical metric structure and also employs a generalized w-distance. We then applied the proposed variational principles to obtain existence theorems for a class of quasi-equilibrium problems whose constraint maps are induced from the graphical structure. The conditions used in our existence results are based on a very general concept called a convergence class. Finally, we deduce the existence of a generalized Nash equilibrium via its quasi-equilibrium reformulation. A validating example is also presented.