Convergence Theorem Research Articles

A New Extragradient-Viscosity Method for a Variational Inequality, an Equilibrium Problem, and a Fixed Point Problem

In this research article, we introduce a novel iterative approach that builds upon a two-step extragradient-viscosity method. This method aims to find a common element among the solution set of a variational inequality, an equilibrium problem, and the set of common fixed points from a countable family of demicontractive mappings in a Hilbert space. We offer a robust convergence theorem for the proposed iterative scheme, considering certain well-conditioned parameters. Our findings represent an improvement over similar results already available in the existing literature. Furthermore, we demonstrate the applicability of our main result to W-mappings. Lastly, we present two numerical examples to exhibit the consistency and accuracy of our devised scheme.

The sequential Henstock-Kurzweil delta integral on time scales

Abstract In this study, the basic theory of the sequential Henstock-Kurzweil delta integral on time scales will be discussed. First, we give the notion and the elementary properties of this integral; then we show the equivalence of the Henstock-Kurzweil delta integral and the sequential Henstock-Kurzweil delta integral on time scales. In addition, we consider the Cauchy criterion and the Fundamental Theorems of Calculus. Finally, we prove Henstock’s lemma and give some convergence theorems. As an application, we consider the existence theorem of a kind of functional dynamic equations.

COMONOTONICITY AND PARETO OPTIMALITY, WITH APPLICATION TO COLLABORATIVE INSURANCE

Two by-now folkloric results in the theory of risk sharing are that (i) any feasible allocation is convex-order-dominated by a comonotonic allocation; and (ii) an allocation is Pareto optimal for the convex order if and only if it is comonotonic. Here, comonotonicity corresponds to the so-called no-sabotage condition, which aligns the interests of all parties involved. Several proofs of these two results have been provided in the literature, all based on a version of the comonotonic improvement algorithm of Landsberger and Meilijson (1994) and a limit argument based on the Martingale Convergence Theorem. However, no proof of (i) is explicit enough to allow for an easy algorithmic implementation in practice; and no proof of (ii) provides a closed-form characterization of Pareto optima. In addition, while all of the existing proofs of (i) are provided only for the case of a two-agent economy with the observation that they can be easily extended beyond two agents, such an extension is far from being trivial in the context of the algorithm of Landsberger and Meilijson (1994) and it has never been explicitly implemented. In this paper, we provide novel proofs of these foundational results. Our proof of (i) is based on the theory of majorization and an extension of a result of Lorentz and Shimogaki (1968), which allows us to provide an explicit algorithmic construction that can be easily implemented beyond the case of two agents. In addition, our proof of (ii) leads to a crisp closed-form characterization of Pareto-optimal allocations in terms of α-quantiles (mixed quantiles). An application to peer-to-peer insurance, or collaborative insurance, illustrates the relevance of these results.Acknowledgements: The authors thank the Editor and two anonymous Referees for numerous suggestions which helped to improve the text compared to its previous versions. We are grateful to K.C. Cheung for comments on an earlier version of this paper, and to Christopher Blier-Wong for his excellent research assistance in the numerical example. We thank the Associate Editor and two anonymous reviewers for comments and suggestions. Michel Denuit and Jan Dhaene gratefully acknowledge funding from the FWO and F.R.S.-FNRS under the Excellence of Science (EOS) programme, project ASTeRISK (40007517). Jan Dhaene acknowledges the financial support of BOF KU Leuven (project C14/21/089). Mario Ghossoub acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada NSERC Grant No. 2018-03961 and 2024-03744).

Refined Iterative Method for a Common Variational Inclusion and Common Fixed-Point Problem with Practical Applications

This paper introduces a novel parallel method for solving common variational inclusion and common fixed-point (CVI-CFP) problems. The proposed algorithm provides a strong convergence theorem established under specific conditions associated with the CVI-CFP problem. Numerical simulations demonstrate the algorithm’s efficacy in the context of signal recovery problems involving various types of blurred filters. The results highlight the algorithm’s potential for practical applications in image processing and other fields.

Approximating Fixed Point of (α, β, γ)-Nonexpansive Mapping Using Picard-Thakur Iterative Scheme

In this paper, we explore the fixed point approximation of (α, β, γ)–nonexpansive mappings using the Picard-Thakur iterative scheme. We identify the efficiency of Picard-Thakur iterative scheme through a numerical example by comparing it with other iterative schemes. We also proved strong and weak fixed point convergence theorems for (α, β, γ)–nonexpansive mappings by using Picard-Thakur iterative scheme.

Approximation by Schurer Type λ-Bernstein–Bézier Basis Function Enhanced by Shifted Knots Properties

In this article, a novel Schurer form of λ-Bernstein operators augmented by Bézier basis functions is presented by utilizing the features of shifted knots. The shifted knots form of Bernstein operators and the Schurer form of the Bézier basis function are used in this article, then, new operators, the Schurer type λ-Bernstein shifted knots operators are constructed in terms of the Bézier basis function. First, the test functions are calculated and the central moments for these operators are obtained. Then, Korovkin’s type approximation properties are studied by the use of a modulus of continuity of orders one and two. Finally, the convergence theorems for these new operators are obtained by using Peetre’s K-functional and Lipschitz continuous functions. In the end, some direct approximation theorems are also obtained.

Exponential Sampling Type Kantorovich Max-Product Neural Network Operators

This article deals with the study of approximation behavior of the exponential sampling type Kantorovich max-product neural network operators based on sigmoidal activation function. We study point-wise and uniform convergence theorems in the space of continuous functions for these operators and determine the rate of convergence by means of logarithmic modulus of continuity. We also investigate the approximation of p -th ( 1 ≤ p < ∞ ) Lebesgue integrable functions by the aforementioned operators and study the rate of convergence exploiting Peetre’s K − functional.

Strong Convergence of an Iterative Method for Solving Generalized Mixed Equilibrium Problems and Split Feasibility Problems

In this paper, we investigate iterative methods for solving generalized mixed equilibrium problems, split feasibility problems, and fixed point problems in Banach spaces. We introduce a new extragradient algorithm using the generalized metric projection and prove a strong convergence theorem for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions to the split feasibility problem, the set of fixed points of a resolvent operator, and the set of solutions of the generalized mixed equilibrium problem. The algorithm is analyzed in a real 2-uniformly convex and uniformly smooth Banach space, taking into account computational errors. A numerical example is provided to illustrate the applicability and performance of the proposed method.

Chebyshev’s Method for Multiple Zeros of Analytic Functions: Convergence, Dynamics and Real-World Applications

This paper deals with the convergence and dynamics of Chebyshev’s method for simple and multiple zeros of analytic functions. We establish a local convergence theorem that provides error estimates and exact domains of initial approximations to guarantee the Q-cubic convergence of the method right from the first iteration. Applications to some real-world problems as well as theoretical and numerical comparison with the famous Halley’s method are also provided.

Machine Learning in Small and Medium-Sized Enterprises, Methodology for the Estimation of the Production Time

Data mining (DM) and machine learning (ML) are widely used in production planning and scheduling. Their application to production time estimation leads to improved planning and scheduling accuracy, resulting in increased overall efficiency. Small and medium-sized enterprises (SMEs) often have a small amount of data, which results in the limited adoption of DM and ML. Instead, production time estimation is still performed using rough approximations, which are inaccurate and non-reproducible. Therefore, this article proposes an ML methodology for production time estimation. It is adapted to the needs of SMEs and is applied with limited data. The methodology is based on the categorization of four job types (from A to D), the partitioning of data according to the limit theorem of data convergence, and the definition of risk based on metrics of probability and statistics. ML was applied by WEKA Workbench (Waikato Environment for Knowledge Analysis). It is also integrated into the Cross Industry Standard Process for DM. The methodology was implemented on data from a medium-sized company, Schoepstal Maschinenbau GmbH, for job types A and B to estimate machine/job cycle time, manufacturing cycle time, and lead time. Different accuracies were obtained for individual estimation models, confirming the strong dependence of the models on data quality. Suitable models were found for the implementation of the estimation of the manufacturing cycle time and the machine/job cycle time. The modeling of lead time estimation was unsuccessful. This was due to the weak dependence between the learning values and the values of the selected model attributes. The implementation of the methodology for job types C and D is the subject of further research.

A qualitative investigation of a system of third-order difference equations with multiplicative reciprocal terms

In this paper, we study the system of third-order difference equations \begin{equation*} x_{n+1}=a+\frac{a_{1}}{y_{n}}+\frac{a_{2}}{y_{n-1}}+\frac{a_{3}}{y_{n-2}}% ,\quad y_{n+1}=b+\frac{b_{1}}{x_{n}}+\frac{b_{2}}{x_{n-1}}+\frac{b_{3}}{% x_{n-2}},\quad n\in \mathbb{N}_{0}, \end{equation*}% where the parameters $a$, $a_{i}$, $b$, $b_{i}$, $i=1,2,3$, and the initial values $x_{-j}$, $y_{-j}$, $j=0,1,2$, are positive real numbers. We first prove a general convergence theorem. By applying this convergence theorem to the system, we show that positive equilibrium is a global attractor. We also study the local asymptotic stability of the equilibrium and show that it is globally asymptotically stable. Finally, we study the invariant set of solutions.

Weak and strong convergence theorems for a new class of enriched strictly pseudononspreading mappings in Hilbert spaces

Let Ω be a nonempty closed convex subset of a real Hilbert space H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{H}$\\end{document}. Let ℑ be a nonspreading mapping from Ω into itself. Define two sequences {ψn}n=1∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\{\\psi _{{n}}\\}_{n=1}^{\\infty}$\\end{document} and {ϕn}n=1∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\{\\phi _{{n}}\\}_{n=1}^{\\infty}$\\end{document} as follows: {ψn+1=πnψn+(1−πn)ℑψn,ϕn=1n∑nt=1ψt,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} \ extstyle\\begin{cases} \\psi _{n+1}=\\pi _{n}\\psi _{{n}}+(1-\\pi _{n})\\Im \\psi _{{n}}, \\\\ \\phi _{{n}}=\\dfrac{1}{n}\\underset{t=1}{\\overset{n}{\\sum}}\\psi _{t}, \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} for n∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n\\in \\mathit{N}$\\end{document}, where 0≤πn≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0\\leq \\pi _{n}\\leq 1$\\end{document}, and πn→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\pi _{n} \\rightarrow 0$\\end{document}. In 2010, Kurokawa and Takahashi established weak and strong convergence theorems of the sequences developed from the above Baillion-type iteration method (Nonlinear Anal. 73:1562–1568, 2010). In this paper, we prove weak and strong convergence theorems for a new class of (η,β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\eta ,\\beta )$\\end{document}-enriched strictly pseudononspreading ((η,β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\eta ,\\beta )$\\end{document}-ESPN) maps, more general than that studied by Kurokawa and W. Takahashi in the setup of real Hilbert spaces. Further, by means of a robust auxiliary map incorporated in our theorems, the strong convergence of the sequence generated by Halpern-type iterative algorithm is proved thereby resolving in the affirmative the open problem raised by Kurokawa and Takahashi in their concluding remark for the case in which the map ℑ is averaged. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature.

Approximation by Subsequences of Matrix Transform Means of Some Two-Dimensional Rectangle Walsh–Fourier Series

In the present paper we discuss the rate of the approximation by the matrix transform of special partial sums of some two-dimensional rectangle (decreasing diagonal) Walsh-Fourier series in Lp(G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p(G^{2})$$\\end{document} space (1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p <\\infty $$\\end{document}) and in C(G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C(G^{2})$$\\end{document}. It implies in some special case norm convergence. We also show an application of our results for Lipschitz functions. At the end of the paper we show the most important result, the almost everywhere convergence theorem. We note that T summation is a common generalization of the following known summation methods Cesàro, Weierstrass, Riesz and Picar and Bessel methods.

GRP – A direct Godunov extension

The classical paper [40] by S. K. Godunov had a revolutionary effect on the field of numerical simulations of compressible fluid flows. Yet, for some twenty years after its publication, it was mostly used by the mathematical community in the Soviet Union. The seminal paper of van Leer [85] has inaugurated the period of universal interest in high-resolution extensions of Godunov's scheme. The first step to take was to modify the (locally) self-similar solution to the Riemann Problem (at discontinuities) by allowing piecewise-polynomial (rather than piecewise-constant) initial data. The GRP (Generalized Riemann Problem) analysis [6] provided analytical solutions (for piecewise-linear data) that could be readily implemented in a high-resolution robust code. This review paper focuses on the evolution of some mathematical aspects of this method. It addresses the three concepts of consistency, stability and convergence in the context of compact finite difference (or finite volume) schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of “balance laws”, a common viewpoint in relevant physical conservation laws. The first significant observation is that under very mild conditions a weak solution is indeed a solution to the balance law (obtained by a formal application of the Gauss-Green formula). Since high-resolution schemes require the computation of several quantities per mesh cell (e.g., slopes), the notion of “consistency” must be extended to this framework. A combination of consistency hypothesis with stability of the scheme leads to a suitable convergence theorem, generalizing the classical convergence theorem of Lax and Wendroff [52]. Finally, the limit functions are shown to be entropy solutions by using a notion of “Godunov compatibility”, which serves as a substitute to the entropy condition.

Some convergence theorems of fuzzy concave integral on fuzzy σ-algebra

Some convergence theorems of fuzzy concave integral on fuzzy σ-algebra

Singularities of the hyperbolic elastic flow: convergence, quantization and blow-ups

We study the elastic flow of closed curves and of open curves with clamped boundary conditions in the hyperbolic plane. While global existence and convergence toward critical points for initial data with sufficiently small energy is already known, this study pioneers an investigation into the flow’s singular behavior. We prove a convergence theorem without assuming smallness of the initial energy, coupled with a quantification of potential singularities: Each singularity carries an energy cost of at least 8. Moreover, the blow-ups of the singularities are explicitly classified. A further contribution is an explicit understanding of the singular limit of the elastic flow of λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-figure-eights, a class of curves that previously served in showing sharpness of the energy threshold 16 for the smooth convergence of the elastic flow of closed curves.

Sobolev‐type regularization method for the backward diffusion equation with fractional Laplacian and time‐dependent coefficient

This work is concerned with an ill‐posed problem of reconstructing the historical distribution of a backward diffusion equation with fractional Laplacian and time‐dependent coefficient in multidimensional space. The investigated problem is regularized by a Sobolev‐type equation method. Unlike previous works, to prove the convergence of the regularized solution to the exact one, we only require a very weak and natural a priori condition that the solution belongs to the standard Lebesgue space . This is done by suitably employing the Lebesgue‐dominated convergence theorem. If we go further to impose a stronger a priori condition, one may know how fast the convergence is. Finally, some MATLAB‐based numerical examples are provided to confirm the efficiency of the proposed method.

New algorithms for solving pseudo-monotone variational inequalities in Banach spaces

In this paper, we introduce new algorithms for finding a solution of a variational inequality problem involving pseudo-monotone operator which is also a fixed point of a Bregman relatively inexpensive mapping in $p$-uniformly convex and uniformly smooth Banach spaces that are more general than Hilbert spaces. We prove weak and strong convergence theorems for proposed algorithms. Finally, we give some numerical experiments for supporting our main results.

Convergence theorems of implicit type iterations in geodesic spaces with negative curvature

Convergence theorems of implicit type iterations in geodesic spaces with negative curvature

A Robust Hybrid Iterative Learning Formation Strategy for Multi-Unmanned Aerial Vehicle Systems with Multi-Operating Modes

This paper investigates the formation control problem of multi-unmanned aerial vehicle (UAV) systems with multi-operating modes. While mode switching enhances the flexibility of multi-UAV systems, it also introduces dynamic model switching behaviors in UAVs. Moreover, obtaining an accurate dynamic model for a multi-UAV system is challenging in practice. In addition, communication link failures and time-varying unknown disturbances are inevitable in multi-UAV systems. Hence, to overcome the adverse effects of the above challenges, a hybrid iterative learning formation control strategy is proposed in this paper. The proposed controller does not rely on precise modeling and exhibits its learning ability by utilizing historical input–output data to update the current control input. Furthermore, two convergence theorems are proven to guarantee the convergence of state, disturbance estimation, and formation tracking errors. Finally, three simulation examples are conducted for a multi-UAV system consisting of four quadrotor UAVs under multi-operating modes, switching topologies, and external disturbances. The results of the simulations show the strategy’s effectiveness and superiority in achieving the desired formation control objectives.