For this paper, we consider the almost sure exponential stability of uncertain stochastic Hopfield neural networks based on subadditive measures. Firstly, we deduce two corollaries, using the Itô–Liu formula. Then, we introduce the concept of almost sure exponential stability for uncertain stochastic Hopfield neural networks. Next, we investigate the almost sure exponential stability of uncertain stochastic Hopfield neural networks, using the Lyapunov method, Liu inequality, the Liu lemma, and exponential martingale inequality. In addition, we prove two sufficient conditions for almost sure exponential stability. Furthermore, we consider stabilization with linear uncertain stochastic perturbation and present some exceptional examples. Finally, our paper provides our conclusion.

No previous study has involved uncertain fractional differential equation (FDE, for short) with jump. In this paper, we propose the uncertain FDEs with jump, which is driven by both an uncertain V-jump process and an uncertain canonical process. First of all, for the one-dimensional case, we give two types of uncertain FDEs with jump that are symmetric in terms of form. The next, for the multidimensional case, when the coefficients of the equations satisfy Lipschitz condition and linear growth condition, we establish an existence and uniqueness theorems of uncertain FDEs with jump of Riemann-Liouville type by Banach fixed point theorem. A symmetric proof in terms of form is suitable to the Caputo type. When the coefficients do not satisfy the Lipschitz condition and linear growth condition, we just prove an existence theorem of the Caputo type equation by Schauder fixed point theorem. In the end, we present an application about uncertain interest rate model.

Green retrofitting is a potent strategy to improve energy efficiency and support the cultural conservation of historical buildings. China has many historical buildings with significant potential for green retrofitting. Nevertheless, the Chinese green retrofit industry remains in its early stages due to the lack of a comprehensive promotion mechanism. Drawing upon evolutionary game theory, an evolutionary game model was formulated to analyze the interplay among governmental regulators, Energy Service Companies (ESCOs), the owners of historical buildings, and tourists. At an extreme disturbance intensity of 100%, a striking decline is observed in the likelihood of ESCOs maintaining the “green retrofit” strategy, with the probability dropping to a minimum of 24.68%. Under specific conditions however this interplay attains an optimal equilibrium state, denoted as (1,1,1,1). It is concluded that governmental regulators exhibit a greater degree of stability compared to other stakeholders, as they maintain their strategies despite external disturbances. Therefore, it is recommended that effective incentives and regulatory frameworks should consider the sensitivities of non-regulatory stakeholders to enhance the success of green retrofitting projects for historical buildings.

The existing literature lacks a study on age-dependent population equations based on subadditive measures. In this paper, we propose a hybrid age-dependent population dynamic system (referred to as APDS) that incorporates uncertain random perturbations driven by both the well-known Wiener process and the Liu process associated with belief degree, which have similar symmetry in terms of form. Firstly, we redefine the Liu integral in a mean square sense and then extend Liu’s lemma and the Itô-Liu formula. We then utilize the extensions of the Itô-Liu formula, Barkholder-Davis-Gundy (BDG) inequality, the Liu’s lemma, the Gronwall’s lemma and the symmetric nature of calculus itself to establish the uniqueness of a strong solution for the hybrid APDS. Additionally, we prove the existence of the hybrid APDS by combining the proof of uniqueness with some important lemmas. Finally, under appropriate assumptions, we demonstrate the exponential stability of the hybrid system.

A convex combined expectations regularized gap function with uncertain variable is presented to deal with uncertain nonlinear variational inequality problems (UNVIP). The UNVIP is transformed into a minimization problem through an uncertain weighted expected residual function. Moreover, the convergence of the global optimal solutions of the uncertain weighted expected residual minimization model is given through the integration by parts method under the compact space of the uncertain event. The limiting behaviors of the transformed model are analyzed. Furthermore, a compact approximation method is proposed in the unbounded uncertain event space. Through analysis of the convergence of UWERM model and reasonable hypothesis, the compact approximation method is verified under the circumstance of Holder continuity.