In this article, we consider the following new Kirchhoff-type problem: $\begin{array}{rl}-\left(a-b{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)\mathrm{\Delta }u& =|u{|}^{p-2},\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u& =0,\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}$ where a and b are positive constants, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with $C^1$ boundary $\mathrm{\partial }\mathrm{\Omega }$, $p\in [2,2^{\ast })$ with $2^{\ast }=2N/\left(N-2\right)$ if $N\ge 3$, and $2^{\ast }=+\mathrm{\infty }$ if N = 1, 2. We show that the problem possesses infinitely many sign-changing solutions by using combination of invariant sets of descent flow and the Ljusternik–Schnirelman type minimax method. And an example for p = 2 is illustrated our results.

The distributed training of federated machine learning, referred to as federated learning (FL), is discussed in models by multiple participants using local data without compromising data privacy and violating laws. In this paper, we consider the training of federated machine models with uncertain participation attitudes and uncertain benefits of each federated participant, and to encourage all participants to train the desired FL models, we design a fuzzy Shapley value incentive mechanism with supervision. In this incentive mechanism, if the supervision of the supervised mechanism detects that the payoffs of a federated participant reach a value that satisfies the Pareto optimality condition, the federated participant receives a distribution of federated payoffs. The results of numerical experiments demonstrate that the mechanism successfully achieves a fair and Pareto optimal distribution of payoffs. The contradiction between fairness and Pareto-efficient optimization is solved by introducing a supervised mechanism.

This paper establishes the existence of unique and multiple solutions to two nonlocal equations with fractional operators. The main results are obtained using the variational method and algebraic analysis. The conclusion is that there exists a constant ${\lambda }^{*}>0$ such that the equations have only three, two, and one solution, respectively, for $\lambda \in (0,{\lambda }^{*})$, $\lambda ={\lambda }^{*}$, and $\lambda >{\lambda }^{*}$. The main conclusions fill the gap in the knowledge of this kind of fractional-order problem.

In this article, we consider a nonlocal problem with a strong singular term and a general weight function. By using Ekeland’s variational principle, we prove a necessary and sufficient condition for the existence of a positive solution. Moreover, a method of algebraic analysis is used to deal with the multiplicity of solutions. Compared with the existing literature, our problems and results are novel.