In this work, we introduced new notions of a new contraction named $S$ -weakly contraction; after that, we obtained the p-common best proximity point results for different types of contractions in the setting of complete metric spaces by using weak $P_p$ -property and proved the uniqueness of these points. Also, we presented some examples to prove the validity of our results.

The basic purpose of this article is to define new so-called \(\mathcal{S}\)-contractions and discuss the presence of common best proximity point theorems for such contractions in the setting of Cauchy metric spaces. We also calculate some common optimal approximate solutions of some fixed point equations when there does not exist any common fixed point. We also introduce the notions of \(\mathcal{G_{P}}\)-functions and \(\mathcal {G_{P}}\)-contractions with the help of \(\mathcal{P}\)-functions and prove the existence of a unique best proximity point in partially order metric spaces. We give some examples that justify the validity of our results. These results extend and unify many existing results in the literature.

The recently introduced triple exponentially weighted moving average (TEWMA) chart is the extended form of the classical EWMA and double EWMA (DEWMA) charts. On the other hand, the auxiliary information-based (AIB) homogeneously weighted moving average () chart is used for the monitoring of process location shifts efficiently as compared to the HWMA chart. Combining TEWMA with HWMA features is a new idea and is not seen in the literature until now. So, the objective of this study is to combine the idea of and TEWMA charts and propose an AIB triple HWMA, symbolized as chart to further improve the process location shift monitoring. The proposed chart is developed by combining the AIB double HWMA plotting statistic into the other HWMA chart. The numerical results are computed using the Monte Carlo simulation method. Average run length (), relative , performance comparison index, and extra quadratic loss are used as comparison tools for the proposed chart with the classical EWMA, TEWMA, mixed HWMA-CUSUM (MHC), AIB EWMA (), HWMA, double HWMA (DHWMA), , AIB mixed EWMA-CUSUM (), and AIB mixed CUSUM-EWMA () charts. Finally, a practical application is also provided for users to demonstrate the proposed study’s vitality.

This work introduces a novel modified Halpern-type proximal point algorithm, designed for a finite family of k-demimetric mappings, resolvents of monotone

This paper addresses the challenges of achieving robust coordination in discrete-time multi-robot systems subject to uncertainties and Byzantine attacks affecting both actuator and sensor channels. Such adversarial disruptions degrade system performance by corrupting control inputs and state measurements, ultimately threatening stability and consensus in networked robotic systems. To overcome these limitations, a novel discrete-time adaptive control framework is proposed that ensures reliable tracking and stability under both uncoupled and coupled robot dynamics. The approach integrates a modified graph-theoretic structure with node-dependent weighting to capture heterogeneous robot interactions, while explicitly modeling attack effects within the system dynamics. An adaptive control law is developed using a nonlinear basis function approximation to handle unknown system uncertainties, along with a dynamic weight update mechanism that compensates for adversarial disturbances in real time. For the uncoupled case, stability is established through a composite Lyapunov function incorporating logarithmic and quadratic terms, guaranteeing boundedness of all closed-loop signals and asymptotic convergence of the tracking error. This framework is further extended to systems with coupled dynamics by introducing an auxiliary estimation mechanism to reconstruct unmeasurable interactions, leading to a unified adaptive controller capable of mitigating both internal uncertainties and external attacks. Rigorous Lyapunov-based analysis demonstrates that the proposed method ensures asymptotic tracking performance despite the presence of Byzantine disturbances. Numerical simulations validate the theoretical results, showing improved resilience, accurate trajectory tracking, and enhanced robustness compared to existing approaches.