The spread of rumors in online social networks (OSNs) has caused a serious threat to the normal social order. In order to describe the rumor-spreading dynamics in OSNs during emergencies, a novel model with consideration of fans versus ordinary people is proposed in this paper. In contrast to previous studies, we consider the case that two rumors exist simultaneously. It is assumed that one is an entertainment rumor that fans care about, and the other is a common rumor. First, we derive the mean-field equations that describe the dynamics of this dual rumor propagation model and obtain the threshold parameter. Secondly, after finding the necessary and sufficient conditions for the existence of equilibriums, we examine the equilibrium’s local and global stability. Finally, simulations are used to explain how various parameters affect the process of spreading rumors.

In this paper, we focus on a fractional differential equation ${\hspace{0.17em}}_0^C{D}^{\alpha }u\left(t\right)+q\left(t\right)u\left(t\right)=0$ with boundary value conditions $u\left(0\right)=\delta u\left(1\right),u^{\prime }\left(0\right)=\gamma u^{\prime }\left(1\right)$. The paper begins by pointing out the inadequacies of the study conducted by Ma and Yangin establishing Lyapunov-type inequalities. It then discusses the properties of its Green’s function and investigates extremum problems related to several linear functions. Finally, thorough classification and analysis of various cases for parameters $\delta$ and $\gamma$ are conducted. As a result, a comprehensive solution corresponding to the Lyapunov-type inequality is obtained.

The trust region method is an effective method for solving unconstrained optimization problems. Incorrectly updating the rules of the trust region radius will increase the number of iterations and affect the calculation efficiency. In order to obtain an effective radius for the trust region, an adaptive radius updating criterion is proposed based on the gradient of the current iteration point and the eigenvalue of the Hessian matrix which avoids calculating the inverse of the Hessian matrix during radius updating. This approach reduces the computation time and enhances the algorithm’s performance. On this basis, we apply adaptive radius and non-monotonic techniques to the trust region algorithm and propose an improved non-monotonic adaptive trust region algorithm. Under proper assumptions, the convergence of the algorithm is analyzed. Numerical experiments confirm that the suggested algorithm is effective.

Suppose each of $A_1,\dots ,A_n$ is a maximal monotone, and $\beta B$ is firmly nonexpansive with $\beta >0$. In this paper, we have two purposes: the first is finding the zeros of ${\sum }_{j=1}^n{A}_j+B$, and the second is finding the zeros of ${\sum }_{j=1}^n{A}_j$. To address the first problem, we produce fixed-point equations on the original Hilbert space as well as on the product space and find that these equations associate with crucial operators which are called generalized forward–backward splitting operators. To tackle the second problem, we point out that it can be reduced to a special instance of $n=2$ by defining new operators on the product space. Iterative schemes are given, which produce convergent sequences and these sequences ultimately lead to solutions for the last two problems.

This paper addresses the stability of stochastic delayed recurrent neural networks (SDRNNs), identifying challenges in existing scalar methods, which suffer from strong assumptions and limited applicability. Three key innovations are introduced: (1) weakening noise perturbation conditions by extending diagonal matrix assumptions to non-negative definite matrices; (2) establishing criteria for both mean-square exponential stability and almost sure exponential stability in the absence of input; (3) directly handling complex structures like time-varying delays through matrix analysis. Compared with prior studies, this approach yields broader stability conclusions under weaker conditions, with numerical simulations validating the theoretical effectiveness.