In this paper, we introduce the iterative scheme for finding a common fixed point of a countable family of G-nonexpansive mappings by the shrinking projection method which generalizes Takahashi Takeuchi and Kubota’s theorem in a Hilbert space with a directed graph. Simultaneously, we give examples and numerical results for supporting our main theorems and compare the rate of convergence of some examples under the same conditions.

Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of bi-derivations and bihomomorphisms in Banach algebras, associated with the bi-additive functional inequality \[ \| f(x+y, z+w) + f(x+y, z-w) + f(x-y, z+w) + f(x-y, z-w) -4f(x,z)\| \\ \quad \le \left \|s \left(2f\left(x+y, z-w\right) + 2f\left(x-y, z+w\right) - 4f(x,z )+ 4 f(y, w)\right)\right\| , \] where \(s\) is a fixed nonzero complex number with \(|s |< 1\).

‎This paper investigates scaled consensus problems of hybrid multi-agent systems(HMASs)‎, ‎which consist of a group of continuous-time(CT) and discrete-time(DT) dynamics agents‎. ‎Three kinds of consensus protocols have been proposed to solve scaled consensus problems‎, ‎respectively‎. ‎The first two consensus protocols are designed for solving the scaled consensus problems where the CT-agents can only communicate with their neighbors in the sampling time \(t_k\)‎. ‎In addition‎, ‎the impulsive consensus protocols are designed for solving scaled consensus problems of HMASs‎. ‎Finally‎, ‎some numerical examples are provided to illustrate the effectiveness of the theoretical results‎.

In contrast to existing research on consensus problems that primarily addresses the node dynamics of multi-agent systems, this study focuses on achieving consensus among the states of edges within a network. Initially, the scaled consensus of edge dynamics in hybrid multi-agent systems is investigated for both undirected and directed topologies. Subsequently, necessary and sufficient conditions are derived for both directed and undirected topologies to facilitate the design of consensus protocols aimed at resolving edge consensus challenges. Finally, numerical examples are provided to substantiate the effectiveness of the theoretical results.

This study addresses the challenge of achieving scaled consensus in multi-agent systems with edge dynamics, a critical aspect for practical applications such as traffic networks and distributed energy systems. Utilizing graph theory, matrix analysis, and Lyapunov stability, we propose impulsive consensus protocols designed for both directed and undirected topologies. Theseprotocols enable agents to converge proportionally to desired ratios among states, generalizing classical consensus approaches while significantly reducing communication and control costs. We establish sufficient conditions for achieving scaled consensus, demonstrating that it requires the line graph of the communication network to be connected and contain a spanning tree. Numerical simulations validate the robustness and efficiency of the proposed protocols, highlighting their scalability and adaptability in dynamic and resource-constrained environments. The findings provide a solid foundation for real-world applications, including autonomous vehicles, distributed energy systems, and robotic coordination, where proportional state alignment is essential.