Convergence Theorem Research Articles

On the iterative scheme generating methods using mean-valued sequences

Using the Mann method and the shrinking projection method, we present generalized forms of iterative scheme generating methods and compared them with prior frameworks. To this end, the properties of mean-valued sequences are leveraged. Subsequently, we establish a convergence theorem similar to that developed by Martinez-Yanes and Xu. This approach highlights the difference between the conventional shrinking projection method and the Martinez-Yanes and Xu variant. The proposed frameworks yield various types of iterative schemes for finding common fixed points, including a three-step iterative scheme. The class of mappings considered incorporate general types, including nonexpansive mappings.

Asymptotically ɑ hemicontractive mappings in Hilbert spaces and a new algorithm for solving associated split common fixed point problem

We introduce a novel class of asymptotically $\alpha-$hemicontractive mappings and demonstrate its relationship with the existing related families of mappings. We establish certain interesting properties of the fixed point set of the new class of mappings. Furthermore, we propose and investigate a new iterative algorithm for solving split common fixed point problem for the new class of mappings. In particular, weak and strong convergence theorems for solving split common fixed point problem for our new class of mappings in Hilbert spaces are proved. Moreover, using our method, we require no prior knowledge of norm of the transfer operator. The results presented in the paper extend and improve the results of Censor and Segal [Censor, Y.; Segal, A. The split common fixed point problem for directed operators. {\it J. Convex Anal.} {\bf 16 } (2009), no. 2, 587–600.], Moudafi [Moudafi, A. The split common fixed-point problem for demicontractive mappings. {\it Inverse Problems} {\bf 26} (2010), no. 5:055007.; Moudafi, A. A note on the split common fixed-point problem for quasi-nonexpansive operators. {\it Nonlinear Anal.} {\bf 74} (2011), no. 12, 4083–4087.], Chima and Osilike [Chima, E. E.; Osilike, M. O. Split common fixed point problem for class of asymptotically hemicontractive mappings. {\it J. Nigerian Math. Soc.} {\bf 38} (2019), no. 3, 363--390.], Fan \textit{et al} [Fan, Q.; Peng, J.; He, H. Weak and strong convergence theorems for the split common fixed point problem with demicontractive operators. {\it Optimization} {\bf 70} (2021), no. 5-6, 1409--1423.] and host of other related results in literature.

Fixed point results involving a finite family of enriched strictly pseudocontractive and pseudononspreading mappings

In this study, we introduce a method for finding common fixed points of a finite family of (ηi,ki)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\eta _{i}, k_{i})$\\end{document}-enriched strictly pseudocontractive (ESPC) maps and (ηi,βi)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\eta _{i}, \\beta _{i})$\\end{document}-enriched strictly pseudononspreading (ESPN) maps in the setting of real Hilbert spaces. Further, we prove the strong convergence theorem of the proposed method under mild conditions on the control parameters. Our main results are also applied in proving strong convergence theorems for ηi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\eta _{i}$\\end{document}-enriched nonexpansive, strongly inverse monotone, and strictly pseudononspreading maps. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature.

Strong convergence theorem for split variational inequality problems involving non-lipschitizian pseudomonotone and finite family of $$\eta $$-demimetric operators

Strong convergence theorem for split variational inequality problems involving non-lipschitizian pseudomonotone and finite family of $$\eta $$-demimetric operators

Correction: Convergence theorem for fixed point and split generalized variational inclusion problems with multiple output sets in Banach spaces

Correction: Convergence theorem for fixed point and split generalized variational inclusion problems with multiple output sets in Banach spaces

A Jump-Gain Integral Recurrent Neural Network for Solving Noise-Disturbed Time-Variant Nonlinear Inequality Problems.

Nonlinear inequalities are widely used in science and engineering areas, attracting the attention of many researchers. In this article, a novel jump-gain integral recurrent (JGIR) neural network is proposed to solve noise-disturbed time-variant nonlinear inequality problems. To do so, an integral error function is first designed. Then, a neural dynamic method is adopted and the corresponding dynamic differential equation is obtained. Third, a jump gain is exploited and applied to the dynamic differential equation. Fourth, the derivatives of errors are substituted into the jump-gain dynamic differential equation, and the corresponding JGIR neural network is set up. Global convergence and robustness theorems are proposed and proved theoretically. Computer simulations verify that the proposed JGIR neural network can solve noise-disturbed time-variant nonlinear inequality problems effectively. Compared with some advanced methods, such as modified zeroing neural network (ZNN), noise-tolerant ZNN, and varying-parameter convergent-differential neural network, the proposed JGIR method has smaller computational errors, faster convergence speed, and no overshoot when disturbance exists. In addition, physical experiments on manipulator control have verified the effectiveness and superiority of the proposed JGIR neural network.

Convergence Theorems for Derivatives of a New Baskakov -Type Operators

The motive of this paper is to introduce and investigate the derivatives of some Baskakov operators. Firstly, we treated with the convergence theorems on some certain type of Baskakov operators and then we found the derivative of these operators. We found the convergence of the derivative of Hl Finally we established and proved a Voronovskaya type theorem for the derivative of Hl

Using the Ellipsoid Method for Sylvester's Problem and its Generalization

Sylvester's problem or the problem of the smallest bounding circle is the problem of constructing a circle of the smallest radius that contains a finite set of points on the plane. In n-dimensional space, it corresponds to the problem of the smallest bounding hypersphere, which can be formulated as the problem of minimizing a convex piecewise quadratic function. The article is dedicated to study of the ellipsoid method application for solving this problem and the minimax convex programming problem, which is equivalent to the generalized problem of the smallest bounding hypersphere. The generalized problem consists in finding the center of a sphere in an n-dimensional space that has a minimal radius and contains a finite set of n-dimensional spheres given by their centers and radii. The article consists of 3 sections. Section 1 describes the emshor algorithm – the algorithm of the ellipsoid method for problem of minimization of an arbitrary convex function, proves its convergence theorems, gives a geometric interpretation of the algorithm, which is based on the use of a minimum volume ellipsoid. Octave implementation of the emshor algorithm is presented here, which can be successfully applied for non-smooth convex functions minimization if the number of variables is n = 2 - 30. In Section 2, the sylvester1 algorithm is built, which is an application of the emshor algorithm for solving the problem of minimizing a convex piecewise quadratic function, which is equivalent to the problem of finding a sphere of minimum radius for a finite set of points. In Section 3, the sylvester2 algorithm is built, which is an application of the emshor algorithm for solving the problem of minimization of a convex function, which is equivalent to the generalized problem of finding a sphere of minimum radius for a finite set of spheres with given centers and radii. The results of testing the sylvester1 and sylvester2 algorithms demonstrate high working speed for modern computers and high accuracy in terms of optimal value of the objective function when solving problems in n-dimensional spaces for small values n = 2 - 30. Keywords: ellipsoid method, convex function, Sylvester's problem, piecewise smooth function, minimax problem.

Convergence Theorems for Common Solutions of Nonlinear Problems and Applications

Convergence Theorems for Common Solutions of Nonlinear Problems and Applications

On the intermixed method for mixed variational inequality problems: another look and some corrections

We explore the intermixed method for finding a common element of the intersection of the solution set of a mixed variational inequality and the fixed point set of a nonexpansive mapping. We point out that Khuangsatung and Kangtunyakarn’s statement [J. Inequal. Appl. 2023:1, 2023] regarding the resolvent utilized in their paper is not correct. To resolve this gap, we employ the epigraphical projection and the product space approach. In particular, we obtain a strong convergence theorem with a weaker assumption.

Strong convergence of split equality variational inequality, variational inclusion, and multiple sets fixed point problems in Hilbert spaces with application

This paper introduces an innovative inertial simultaneous cyclic iterative algorithm designed to address a range of mathematical problems within the realm of split equality variational inequalities. Specifically, the algorithm accommodates finite families of split equality variational inequality problems, infinite families of split equality variational inclusion problems, and multiple-sets split equality fixed point problems involving demicontractive operators in infinite-dimensional Hilbert spaces. The algorithm integrates well-established methods, including the cyclic method, the inertial method, the viscosity approximation method, and the projection method. We establish the strong convergence of this proposed algorithm, demonstrating its applicability in various scenarios and unifying disparate findings from existing literature. Additionally, a numerical example is presented to validate the primary convergence theorem.

On fixed point convergence results for class of nonexpansivemappings in hyperbolic spaces via PJ iteration process

In this paper, we provide certain fixed point results for a mean nonexpansive mapping, as well as a new iterative algorithm called PJ-iterationfor approximating the fixed point of this class of mappings in the setting of hyperbolic spaces. Furthermore, we establish strong and∆-convergence theorem for mean nonexpansive mapping in hyperbolic space. Finally, we present a numerical example to illustrate ourmain result and then display the efficiency of the proposed algorithm compared to different iterative algorithms in the literature. Our resultsobtained in this paper improve, extend and unify some related results in the literature.

An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces

Abstract In this article, we study the split variational inclusion and fixed point problems using Bregman weak relatively nonexpansive mappings in the p p -uniformly convex smooth Banach spaces. We introduce an inertial shrinking projection self-adaptive iterative scheme for the problem and prove a strong convergence theorem for the sequences generated by our iterative scheme under some mild conditions in real p p -uniformly convex smooth Banach spaces. The algorithm is designed to select its step size self-adaptively and does not require the prior estimate of the norm of the bounded linear operator. Finally, we provide some numerical examples to illustrate the performance of our proposed scheme and compare it with other methods in the literature.

On discrete models of Boltzmann-type kinetic equations

The known nonlinear kinetic equations, in particular, the wave kinetic equation and the quantum Nordheim–Uehling–Uhlenbeck equations are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables \(F(x,y;
 v,w)\). The function \(F\) is assumed to satisfy certain simple relations. The main properties of this kinetic equation are studied. It is shown that the above mentioned specific kinetic equations correspond to different polynomial forms of the function \(F\). Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas similar to those used for construction of discrete velocity models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similarly to the Boltzmann \(H\)-function. The theorem of existence, uniqueness and convergence to equilibrium of solutions to the Cauchy problem with any positive initial conditions is formulated and discussed. The differences in long time behaviour between solutions of the wave kinetic equation and solutions of its discrete models are also briefly discussed.

On a Faster Iterative Method for Solving Fractional Delay Differential Equations in Banach Spaces

In this paper, we consider a faster iterative method for approximating the fixed points of generalized α-nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our findings, we present some nontrivial examples of the considered mappings. Furthermore, we show that the class of mappings considered is more general than some nonexpansive-type mappings. Also, we show numerically that the method studied in our article is more efficient than several existing methods. Lastly, we use our main results to approximate the solution of a delay fractional differential equation in the Caputo sense. Our results generalize and improve many well-known existing results.

Strong convergence theorem for split null point, split variational inequality and fixed point problems for a non-expansive semigroup

The primary goal of this paper is to find a common solution of a split null point problem, split variational inequality problem and fixed point problem for a non-expansive semigroup with the help of new type of iterative algorithm. We construct that the sequences induced by the proposed iterative method to solve the above stated problems in the setting of real Hilbert spaces and obtain a strong convergence theorem. We also deduce certain consequences from the main convergence result. At the end, we provide a numerical experiment to demonstrate the convergence analysis of proposed iterative method. The methodology and conclusion described in this work extend and unify previously published findings in this field.

An improved asynchronous batch gradient method for ridge polynomial neural network

The ridge polynomial neural network composed of pi-sigma modules is a typical higher-order feedforward network, which has good non-linear mapping capabilities. Due to the strong coupling of the network structure, the synchronous gradient method can easily result in significant fluctuations in updated weights. This will reduce the generalization ability of the ridge polynomial neural network for solving classification problems. In this paper, the proposed method is a novel improved asynchronous batch gradient method, and combined with the adaptive parameters of the activation function are trained synchronously. We strictly prove the convergence theorem of the improved method. The numerical experimental results also indicate that our method for training ridge polynomial neural networks can help the weight changes smoothly and the network has good generalization ability. The feasibility and effectiveness of our method are verified from both theoretical and experimental perspectives.

Strong Convergence Theorems for Split Common Fixed Point Problem of Bregman Generalized Asymptotically Nonexpansive Mappings in Banach Spaces

In this paper, a new iterative scheme is introduced and also strong convergence theorems for solving split common fixed point problem for uniformly continuous Bregman generalized asymptotically nonexpansive mappings in uniformly convex and uniformly smooth Banach spaces are presented. The results are proved without the assumption of semicompactness property and or Opial condition

Numerical study of distributed-order Bessel fractional derivative with application to Euler–Poisson–Darboux equation

The present paper introduces the distributed-order (DO) Bessel fractional derivative for study of the Euler–Poisson–Darboux (EPD) equation including the spatial Riesz fractional derivative (RFD). For this purpose, we discretize the integral term of the DO fractional derivative and approximate the RFD derivative. We thereafter apply an implicit difference method (IDM) for numerical analysis and solvability of the reduced system from the IDM is discussed. The stability and convergent theorems are stated and the numerical tests are given to show efficiency of the proposed technique and to verify good agreement with the theoretical concepts.

Iterative Stability Analysis for Generalized α-Nonexpensive Mappings with Fixed Points

This article introduces a novel iterative process, denoted as F★, designed for the class of generalized α-Nonexpensive mappings. The study establishes strong and weak convergence theorems within the context of Banach spaces, supported by carefully chosen assumptions. The convergence results contribute to the theoretical foundation of iterative processes in functional analysis. The presented framework is applied to address nonlinear integral equations, showcasing the versatility and applicability of the proposed F* for the class of generalized iteration process. Additionally, the article includes numerical examples that not only validate the theoretical findings but also provide insights into the practical utility of the developed methodology.