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Neural networks, mimicking the structural and functional aspects of the human brain, have found widespread applications in diverse fields such as pattern recognition, control systems, and information processing. A critical phenomenon in these systems is synchronization, where multiple neurons or neural networks harmonize their dynamic behaviors to a common rhythm, contributing significantly to their efficient operation. However, the inherent complexity and nonlinearity of neural networks pose significant challenges in understanding and controlling this synchronization process. In this paper, we focus on the synchronization of a class of fractional-order, delayed, and non-autonomous neural networks. Fractional-order dynamics, characterized by their ability to capture memory effects and non-local interactions, introduce additional layers of complexity to the synchronization problem. Time delays, which are ubiquitous in real-world systems, further complicate the analysis by introducing temporal asynchrony among the neurons. To address these challenges, we propose a straightforward yet powerful global synchronization framework. Our approach leverages novel state feedback control to derive an analytical formula for the synchronization controller. This controller is designed to adjust the states of the neural networks in such a way that they converge to a common trajectory, achieving synchronization. To establish the asymptotic stability of the error system, which measures the deviation between the states of the neural networks, we construct a Lyapunov function. This function provides a scalar measure of the system’s energy, and by showing that this measure decreases over time, we demonstrate the stability of the synchronized state. Our analysis yields sufficient conditions that guarantee global synchronization in fractional-order neural networks with time delays and Caputo derivatives. These conditions provide a clear roadmap for designing neural networks that exhibit robust and stable synchronization properties. To validate our theoretical findings, we present numerical simulations that demonstrate the effectiveness of our proposed approach. The simulations show that, under the derived conditions, the neural networks successfully synchronize, confirming the practical applicability of our framework.

Charles proved the convergence of Picard-type iteration for generalized Φ−accretive nonself-mappings in a real uniformly smooth Banach space. Based on the theorems of the zeros of strongly Φ−quasi-accretive mappings and fixed points of strongly Φ−hemi-contractions, we extend the results to Noor iterative process and SP iterative process for generalized Φ−hemi-contractive mappings. Finally, we analyze the rate of convergence of four iterative schemes, namely, Noor iteration, iteration of Corollary 2, SP iteration, and iteration of Corollary 4.

In this paper, the convergence to minimizers of a convex function of a modified proximal point algorithm involving a single-valued nonexpansive mapping and a multivalued nonexpansive mapping in CAT(0) spaces is studied and a numerical example is given to support our main results.

In this manuscript, based on extended b-metric spaces, we propose a new structure of cyclic mapping, that is, BSS-type cyclic mapping. The fixed-point theorem for BSS-type cyclic mapping is established. A concrete example of BSS-type cyclic mapping is given to show the reasonability and correctness of the obtained results. The result proposed by this manuscript extends known conclusions in some references.

In this manuscript, based on extended b-metric spaces, we propose a new structure of cyclic mapping, that is ϕ δ -type cyclic mapping. The fixed-point theorem for ϕ δ -type cyclic mapping is established. A concrete example of ϕ δ -type cyclic mapping is given to show the reasonability and correctness of the obtained results. The result proposed by this manuscript extends known conclusions in some references.

In this manuscript, based on extended b-metric spaces, we propose a new structure of cyclic mapping, that is ϕδ-type cyclic mapping. The fixed-point theorem for ϕδ-type cyclic mapping is established. A concrete example of ϕδ-type cyclic mapping is given to show the reasonability and correctness of the obtained results. The result proposed by this manuscript extends known conclusions in some references.

The purpose of this paper is to study the split common fixed point problems (SCFP) involved in nonexpansive mappings in real Hilbert space. We introduce two iterative algorithms for finding a solution of the SCFP involved in nonexpansive mappings, where one is a Mann-type iterative algorithm and another is a Halpern-type iterative algorithm.

We introduce the modified iterations of Mann's type for nonexpansive mappings and asymptotically nonexpansive mappings to have the strong convergence in a uniformly convex Banach space. We study approximation of common fixed point of asymptotically nonexpansive mappings in Banach space by using a new iterative scheme. Applications to the accretive operators are also included.

In irrigation and water diversion projects, the sediment wear in the pipeline after the pump seriously restricts the safety and stability of the project, leading to. Pipeline leakage requires frequent maintenance and replacement, which increases the maintenance cost of operation. Therefore, this study combines numerical simulation and orthogonal experimental methods to comprehensively consider the effects of multiple factors, such as flow rate, rotation speed, and bend angle behind the pump, on the sediment wear intensity and wear area of the pipeline. Research indicates that flow rate plays a dominant role in determining wear intensity, while the bend angle of the pipeline downstream of the pump primarily influences the extent of wear. The most severe erosion is concentrated at the bend, closely associated with the high-pressure region acting on this section. These findings offer useful insights for forecasting sediment-induced wear and potential leakage in water supply networks.

In this paper, by the classic Mann-type and Halpern-type algorithms, on the basis of monotone operators with firmly nonexpansive property, we build Mann-Halpern type and Halpern-Mann type proximal point algorithms about a zero of monotone operators in Hadamard space, and prove strong convergence and Δ-convergence to a zero of monotone operators, respectively.