This paper investigates a Bregman projection algorithm for solving split variational inequality problems with multiple output sets in real Hilbert spaces, where the underlying operators are assumed to be pseudomonotone and not necessarily Lipschitz continuous. This relaxation considerably broadens the applicability of the proposed method. The algorithm, inspired by the Halpern iteration, the CQ algorithm, and Tseng's extragradient technique, incorporates a two-step inertial strategy to accelerate convergence. A strong convergence theorem is established without requiring prior knowledge of the operator norms. Numerical experiments, together with graphical illustrations, are provided to demonstrate the efficiency of the proposed algorithm in comparison with existing methods. In particular, an application to a signal recovery problem is included to highlight the practical relevance of the approach.

This paper proves a convergence theorem for the push-forward Wiener measures on holonomy groups via stochastic parallel transports along convergent metric connections.

In this paper, we study iterative approximation of attractive points for finitely many families of generalized nonexpansive mappings in a uniformly convex Banach space. We introduce a new algorithm combining a viscosity step with an inertial extrapolation. Under suitable control of the inertial and viscosity parameters and standard conditions on the mappings, we prove that the generated sequence is bounded. We then show that every weak cluster point of the sequence is an attractive point common to all mapping families. The main result establishes that the sequence converges weakly to a unique such attractive point. This extends earlier results confined to two mappings by considering finite family. These findings confirm that the proposed viscosity-inertial iteration successfully approximates the common attractive point under the stated hypotheses. Overall, the work broadens convergence theory in Banach spaces by enabling new classes of algorithms for approximating solution points of generalized nonlinear problems.

Abstract We investigate a novel first-passage percolation model, referred to as the Brochette first-passage percolation model, where the passage times associated with edges lying on the same line are equal. First, we establish a point-to-point convergence theorem, identifying the time constant. In particular, we explore the case where the time constant vanishes and demonstrate the existence of a wide range of possible behaviours. Next, we prove a shape theorem, showing that the limiting shape is the $L^1$ diamond. Finally, we extend the analysis by proving a point-to-point convergence theorem in the setting where passage times are allowed to be infinite.

This paper develops an informational convergence theorem for large dynamic games with imperfect public monitoring. I show that equilibrium differences between finite-agent and continuum-agent models are governed not by cardinality but by informational resolution- the precision with which individual deviations are statistically detectable. In an n-agent economy where unilateral deviations shift aggregates by 1/n and public signals are observed with noise σ_n, equilibrium selection depends on the scaling of n σ_n.If n σ_n → 0, finite-agent extraction equilibria survive; if n σ_n → ∞, the economy converges to the continuum equilibrium. The result provides a general convergence principle linking population size and monitoring precision and yields institutional implications for takeover markets, redevelopment holdouts, and fiscal capacity.

The traditional Fault Tree Analysis (FTA) has limitations in its capacity to process imprecise, incomplete, and subjective failure data that is usually found in railway door safety systems. To overcome this weakness, this paper proposed a new Fuzzy Fault Tree Analysis (FFTA) model based on Interval Type-2 Fuzzy Sets (IT2FS). A two-parameter Weibull distribution is used to model failure probabilities of basic events in a realistic way for the representation of time-dependent failure behaviour based on expert elicitation and maintenance data. Fuzzy logic gates are used to construct the fault tree, and fuzzy probability propagation is carried out through α-cut decomposition, which is backed by a convexity theorem that guarantees valid interval behaviour in the decomposition. To prioritize critical events, an Adapted Wasserstein Distance (AWD) based Fuzzy Importance Index (FII) is proposed, along with a convergence theorem proving the stability of fuzzy failure estimates with the increase in expert information. The actionable numerical values are derived through the Weighted Divided Search Enhanced Karnik-Mendel (WDEKM) algorithm to transform system-level fuzzy risk outputs into actionable numerical values. The framework is implemented in Python 3.11, using standard scientific libraries. Findings indicate the proposed method’s high performance with a Fuzzy Priority Index of 0.0214, Fuzzy Criticality Index of 0.0186, and better sensitivity values (SI = 0.69, NSC = 0.63, RCR = 16.8) than the existing models. These results prove that the proposed framework provides more informative uncertainty-aware risk estimates and sensitivity indicators for safety and maintenance prioritization in railway door systems.

In this work we introduce a novel modified subgradient extragradient method for solving equilibrium problems involving pseudomonotone and Lipschitz-type bifunctions in Hilbert spaces. We replace the second minimization problem regarding a closed and convex subset in the extragradient method with a subgradient projection onto a half-space and propose a new method for obtaining strong convergence results. A strong convergence theorem for the generated sequence is analysed and applied to finding a minimum solution under appropriate conditions. In addition, we present several numerical experiments in order to highlight the superior performance of our proposed algorithm in practice.

This paper investigates the event-triggered secure consensus problem for stochastic multi-agent systems (MASs) subject to bilateral false data injection attacks (FDIAs). To achieve reliable secure consensus while reducing resource consumption, an event-triggered defense scheme incorporated with a configurable waiting period is proposed. By introducing an adjustable time interval between consecutive trigger events, the developed scheme not only rigorously eliminates Zeno behavior but also alleviates the computational and sensing burdens. Notably, the analysis of event-triggered secure consensus for stochastic MASs is more challenging compared to conventional deterministic scenarios, due to the coupling effects of stochastic disturbances, event-triggered mechanisms, and bilateral FDIAs. To address this critical challenge, a stochastic convergence theorem is adopted in this study. Distinct from the traditional Lyapunov theorem for stochastic stability analysis, this theorem exhibits inherent similarities to the deterministic Barbalat lemma, which offers a more flexible analytical framework. A key advantage of the proposed approach is that it relaxes the positive definiteness constraint on the candidate Lyapunov function, thereby significantly enhancing the flexibility in constructing Lyapunov functions for stochastic MASs under bilateral FDIAs. Finally, two numerical simulation examples are presented to verify the correctness and effectiveness of the proposed control protocol and key theoretical results.

Abstract This paper investigates the approximation of fixed points for quasi-nonexpansive mappings in Banach spaces using the general Picard-Mann (GPM) algorithm. Under mild conditions such as the demiclosedness of $$I - \digamma $$ I - ϝ at zero and the Opial property, we derive weak and strong convergence theorems for the iterative sequences generated by the GPM method. We establish several stability results for the GPM scheme, including summably almost stability property for quasi-contractive mappings. The theoretical findings are applied to the classical relaxation method for solving systems of linear inequalities, demonstrating the practical relevance of our approach. Numerical examples in $$\mathbb {R}^4$$ R 4 and $$\ell ^2$$ ℓ 2 are provided to illustrate the efficiency and convergence behavior of the proposed algorithm. The results presented herein extend and complement existing work in fixed point theory and iterative approximation methods.

We investigate the asymptotic behaviour of fast rotating incompressible fluids with vanishing viscosity, in a three dimensional domain with topography including the case of land area. Assuming the initial data is well-prepared, we prove a convergence theorem of the velocity fields to a two-dimensional vector field solving a linear, damped ordinary differential equation. The proof is based on a weak-strong uniqueness argument, combined with an abstract result implying that the weak convergence of a family of weak solutions to the Navier-Stokes-Coriolis system can be translated into a form of uniform-in-time convergence. This argument yields strong convergence of the velocity fields, without a precise rate though.

In this paper, we introduce a novel class of mappings, referred to as noncyclic generalized θ-contractions. By employing the geometric concept of $WUC$ property in metric spaces, we establish new existence and convergence theorems for the fixed points associated with these mappings. The results presented herein generalize and improve several existing fixed point theorems related to generalized φ-contractions. Furthermore, we address the issue of error estimation and derive both a priori and a posteriori error bounds for the fixed points obtained via the Picard iterative process applied to a noncyclic generalized θ-contraction mapping defined on a uniformly convex Banach space. A distinctive feature of our analysis lies in avoiding the use of geometric progression techniques. Consequently, the resulting error estimates hold unconditionally in uniformly convex Banach spaces, thereby removing the need for any restrictive power-type condition on the modulus of convexity. We then present a comprehensive example to illustrate and validate the applicability and robustness of the main theoretical results. Finally, we apply the existence and convergence results for optimal pairs of fixed points to a system of differential equations.

This is an attempt to construct the well-posed hyperbolic heat conduction model based on the Caputo fractional derivative and to study the corresponding coupled thermoelastic problem. The continuous dependence on initial data and energy supply, and the uniqueness of the solutions are mathematically proved. The general closed-form solution of the time fractional conduction model for the initial Dirichlet boundary value problem is obtained analytically by applying the Laplace transform and finite Fourier sine transform in one-dimensional case. The application of theoretical study for heat propagation in the wire is considered. As a special case, two different examples have been discussed to study the analysis of the temperature distributions in the spatial geometry. The influence of the fractional orders on the speed of heat conductivity in the model is discussed. The physical behavior of the temperature distribution has been graphically represented for different fractional orders. Furthermore, the thermal stress analysis is studied using the coupled thermoelasticity theory. In the Laplace domain, the analytical solutions have been obtained. The Gaver–Stehfest technique was employed to numerically perform time domain inversions of the Laplace transforms, which satisfied Kuznetsov’s convergence theorem.

ABSTRACT In this paper, we study a splitting proximal method for minimizing the sum of convex functions defined on metric spaces with negative curvature. Our approach utilizes the resolvent operator and is tailored to the geometry of such spaces. We establish convergence rate theorems for the proposed splitting method by imposing additional conditions on the objective function. Finally, we apply our results to convex optimization problems arising in convex feasibility problems, the centroid problem, and, in particular, the computation of Karcher means.

This paper presents two inertial viscosity Mann-type extrapolated algorithms for finding a common solution to the variational inequality problem involving a monotone and Lipschitz continuous operator and the fixed-point problem for a demicontractive mapping in real Hilbert spaces. The proposed algorithms feature an adaptive step size strategy, computed iteratively, which circumvents the need for prior knowledge of the operator’s Lipschitz constant. Under appropriate assumptions, we establish two strong convergence theorems guaranteeing the robustness of the methods. Furthermore, we provide a comparative performance analysis of the proposed algorithms against some existing strongly convergent schemes, supported by numerical experiments with MATLAB-based graphical illustrations.

In this paper, we propose two new inexact projection algorithms, which can be easily implemented, for solving pseudomonotone variational inequality problems based on self-adaptive step sizes, viscosity technique, and inexact projections.We obtain two strongly convergent theorems of solutions in a real Hilbert space.Numerical experiments illustrate and compare the performances of the proposed algorithms with three other known results.

The behavior of a new modification of operators of the Baskakov–Schurer–Stancu variant is discussed in this study. First, we establish certain necessary moment and central moment estimates. We then demonstrate the weighted approximation result of the suggested operators using a Korovkin-type theorem in weighted spaces. We also give the rate at which these operators converge. Next, we establish theorems of pointwise convergence. Finally, we show several graphical representations to illustrate the accuracy and functionality of the operators.

Abstract In this paper, we utilize the AK-iteration procedure for hyperbolic metric spaces, ensuring the symmetry condition is met and establish the weak $w^{2}$ w 2 -stability and data dependence results for contraction mappings. Additionally, we establish several Δ-convergence and strong convergence theorems for the generalized $(\alpha ,\beta )$ ( α , β ) -nonexpansive (GABNE) mappings. To demonstrate it, we provide a numerical example of the GABNE-mappings, emphasizing the enhanced efficiency of the AK-iteration method relative to other iterative approaches. Additionally, the applications of the main results are demonstrated by solving 2 D and 3 D Volterra integral equations. These findings extend and improve numerous related results found in the current literature.

This paper introduces a new hybrid operator based on combining the Phillips concept with a sequence of lambda-Bernstein operators. This operator represents a qualitative improvement over classical Bernstein-Durrmeyer operators, which faced significant limitations in controlling the behavior of functions at critical points such as the zero point and suffered from a significantly slow rate of convergence. The developed operator overcomes these challenges, achieving a substantial improvement in the quality and accuracy of convergence. To demonstrate the effectiveness of this operator, the study proves a set of basic theoretical results. First, the paper proves the regular convergence theorem for the operator. This is followed by establishing the error estimation theorem using a continuum measure, which in turn confirms the achievement of first-order convergence. Finally, the study presents a precise Voronovskaya-type asymptotic formula that reveals the detailed behavior of the operator's approximation rate when studying functions regularly.

The practically fixed-time leaderless cluster synchronization is addressed for uncertain spatiotemporal networks (USTNs) with coopetition interactions, actuator faults and external disturbances. Firstly, by introducing sign-based coupling, a class of USTN is formulated to capture the dynamics of coopetition interactions among different clusters, which provides a more accurate representation compared to dynamical networks with unsigned coupling. Secondly, a practical fixed-time (PFT) convergence theorem is developed for a general partial differential system, which relaxes the constraints on the derivative of the Lyapunov function and provides a less conservative method for estimating the settling time. Subsequently, a distributed fault-tolerant control algorithm is designed to drive the cluster synchronization error to an adjustable attraction region in a fixed time. By exploring specific properties of the intra-cluster Laplacian matrix and proposing a new inter-degree balanced condition, several flexible synchronization criteria are derived and a quantitative relationship among control parameters, the settling time and the size of the attraction region is presented. Finally, the effectiveness of the developed controllers and criteria is validated through a coupled reaction-diffusion neural network.

In this paper, we study the forward and backward trajectories associated with multivalued mappings in a complete metric space. We analyze the convergence of sequences generated by multivalued function systems (SMFSs) toward an attractor under the Hausdorff metric. A generalized class of multivalued contractions defined via a comparison function is introduced, extending several existing contraction principles. Under appropriate assumptions, we establish sufficient conditions ensuring the convergence of trajectories to an attractor. Our results unify and generalize various known fixed point and convergence theorems in the setting of multivalued dynamics. MSC2020 Classification: 28A80, 47H10, 54E50 54H25