This paper is concerned with exponential synchronization for a class of coupled neural networks with hybrid delays and stochastic distributed delayed impulses. First of all, based on the average impulsive interval method, total probability formula and ergodic theory, several novel impulsive Halanay differential inequalities are established. Two types of stochastic impulses, i.e., stochastic distributed delayed impulses with dependent property and Markov property have been taken into account, respectively. Secondly, some criteria on exponential synchronization in the mean square of a class of coupled neural networks with stochastic distributed delayed impulses are acquired by combining the proposed lemmas and graph theory. The validity of the theoretical results is demonstrated by several numerical simulation examples.

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.

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.

In this article, we delve into the exponential stability of uncertainty systems characterized by stochastic differential equations driven by G-Brownian motion, where coefficient uncertainty exists. To stabilize the system when it is unstable, we consider incorporating a delayed stochastic term. By employing linear matrix inequalities (LMI) and Lyapunov–Krasovskii functions, we derive a sufficient condition for stabilization. Our findings demonstrate that an unstable system can be stabilized with a control interval within $({\theta }^{*},1)$. Some numerical examples are provided at the end to validate the correctness of our theoretical results.