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This chapter concerns random sequences of functions on metric spaces. The main issue is the distinction between convergence at all points of the space (pointwise) and uniform convergence, where limit points are also taken into account. The role of the stochastic equicontinuity property is highlighted. Generic uniform convergence conditions are given and linked to the question of uniform laws of large numbers.

We study criteria for the uniform convergence of trigonometric series with general monotone coefficients. We also obtain necessary and sufficient conditions for a given rate of convergence of partial Fourier sums of such series.

We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron-Frobenius operator. Our main result states that strong convergence of the powers of the Perron-Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by uniform mixing-like properties of the system.

The sign dynamical system for unconstrained minimization of a continuously differentiable function f is examined in this paper. This dynamical system has a discontinuous right hand side and it is interpreted here as neural network. Asymptotic convergence is proven (by using Filippov's approach) finite-time convergence of its solutions is established and an improved upper bound for convergence time is given. A first contribution of this paper is a detailed calculation of Filippov set-valued map for the sign dynamical system, in the general case, i.e. without any restrictive assumptions on the function f to be minimized. Convergence of its solutions to stationary points of f follows by using standard results, i.e. a generalized version of LaSalle's invariance principle. Next, in order to prove finite-time convergence of solutions, the applicability of standard results is extended so that they can be applied to the sign dynamical system. Finally, while establishing finite-time convergence, a novel proving procedure is introduced which (i) allows for milder assumptions to be made on the function f, and (ii) results in an improved upper bound for the convergence time. Numerical experiments confirm both the effectiveness and finite-time convergence of the sign neural network.

The holomorphic embedding load flow method (HELM) is an application for solving the power-flow problem based on a novel method developed by Dr. Trias. The advantage of the method is that it comes with a theoretical guarantee of convergence to the high-voltage (operable) solution, if it exists, provided the equations are suitably framed. While theoretical convergence is guaranteed by Stahl’s theorem, numerical convergence is not; it depends on the analytic continuation algorithm chosen. Since the holomorphic embedding method (HEM) has begun to find a broader range of applications (it has been applied to nonlinear structure-preserving network reduction, weak node identification and saddle-node bifurcation point determination), examining which algorithms provide the best numerical convergence properties, which do not, why some work and not others, and what can be done to improve these methods, has become important. The numerical Achilles heel of HEM is the calculation of the Pade approximant, which is needed to provide both the theoretical convergence guarantee and accelerated numerical convergence. In the past, only two ways of obtaining Pade approximants applied to the power series resulting from power-system-type problems have been discussed in detail: the matrix method and the Viskovatov method. This paper explores several methods of accelerating the convergence of these power series and/or providing analytic continuation and distinguishes between those that are backed by the theoretical convergence guarantee of Stahl’s theorem (i.e., those computing Pade approximants), and those that are not. For methods that are consistent with Stahl’s theoretical convergence guarantee, we identify which methods are computationally less expensive, which have better numerical performance and what remedies exist when these methods fail to converge numerically.

We prove the chain rule in the more general framework of the Wiener-Poisson space, allowing us to obtain the so-called Nourdin-Peccati bound. From this bound we obtain a second-order Poincare-type inequality that is useful in terms of computations. For completeness we survey these results on the Wiener space, the Poisson space, and the Wiener-Poisson space. We also give several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated fields (where the subordinated field can be processes like fractional Brownian motion or the solution of the Ornstein-Uhlenbeck SDE driven by fractional Brownian motion), Poisson functionals in the first Poisson chaos restricted to infinitely many \small" jumps (particularly fractional Levy processes) and the product of two Ornstein-Uhlenbeck processes (one in the Wiener space and the other in the Poisson space). We also obtain bounds for their rate of convergence to normality., arXiv admin note: substantial text overlap with arXiv:1104.1837

This paper is devoted to the study of a multi-step method with divided differences for solving nonlinear equations in Banach spaces. In earlier studies, hypotheses on the Fréchet derivative up to the sixth order of the operator under consideration is used to prove the convergence of the method. That restricts the applicability of the method. In this paper we extended the applicability of the sixth-order multi-step method by using only hypotheses on the first derivative of the operator involved. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.

In one-dimensional regression models, we establish a rate for the rth moment convergence (r⩾1) of the ordinary least-squares estimator involving explicitly the regressors, answering to an open question raised lately by Afendras and Markatou (Test 25:775–784, 2016). An extension of the classic Theorem 2.6.1 of Anderson (The statistical analysis of time series, Wiley, New York, 1971) is also presented.

Statistical convergence has recently attracted the wide-spread attention of researchers due mainly to the fact that it is more general than the classical convergence. Furthermore, the notion of equi-statistical convergence is stronger than that of the statistical uniform convergence. Such concepts were introduced and studied by Balcerzak et al. (J Math Anal Appl 328:715–729, 2007). In this paper, we have used the notion of equi-statistical convergence, statistical point-wise convergence and statistical uniform convergence in conjunction with the deferred Norlund statistical convergence in order to establish several inclusion relations between them. We have also applied our presumably new concept of the deferred Norlund equi-statistical convergence to prove a Korovkin type approximation theorem and demonstrated that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems which were proven by earlier authors. Finally, we consider a number of interesting cases in support of our definitions and results presented in this paper.

We establish sufficient conditions that should be satisfied by functions of a general orthonormal system (ONS) {φ n (x)} in order that the Fourier series in this system be convergent almost everywhere on [0, 1] for any function from the class Lip 1. It is shown that the obtained conditions are best possible in a certain sense.

In this work, the complete moment convergence and Lp convergence for asymptotically almost negatively associated (AANA, in short) random variables are investigated. As an application, the complete convergence theorem for weighted sums of AANA random variables is obtained. These theorems obtained extend and improve some earlier results.

Korovkin type approximation theory is concerned with the convergence of the sequences of positive linear operators to the identity operator. In this paper, we deal with the Korovkin type approximation properties of the Cheney–Sharma operators by using A-statistical convergence and Abel convergence that are some well known methods of summability theory. We also study the rate of convergence. Finally, we show that the results obtained in this paper are stronger than previous ones and we support our results with particular examples and graphs.

Using our new idea of restricted convergence domains, a robust convergence theorem for inexact Newton’s method is presented to find a solution of nonlinear inclusion problems in Banach space. Using this technique, we obtain tighter majorizing functions. Consequently, we get a larger convergence domain and tighter error bounds on the distances involved. Moreover, we obtain an at least as precise information on the location of the solution than in earlier studies. Furthermore, a numerical example is presented to show that our results apply to solve problems in cases earlier studies cannot.

There are generalizations and complementations of results from our joint papers with Filipow ( [15] and [16] ). We study relationships between ideal equal convergence and various kinds of ideal convergences of sequences of real functions. Furthermore we consider ideal version of the bounding number on sets from coideals.

By means of a nice idea, a Newton–Kurchatov type iterative process is constructed for solving nonlinear equations in Banach spaces. We analyze the local convergence of this iterative process. This study have an important and novel feature, since it is applicable to non-differentiable operators. So far, most of the local convergence results considered by other authors may apply only to differentiable operators due to the conditions that are required on the solution of the nonlinear equation.

In the present work, we define a new type of statistical convergence by using the notion of the relatively uniform convergence. We prove a Korovkin-type approximation theorem with the help of this new definition. Then, we construct a strong example that satisfies our theory. Finally, we compute the rate of statistical relatively equal convergence.

The problems of absolute convergence of multiple Fourier series with respect to both general orthonormal systems and multiple Haar series are studied.

In this paper, we analyze some properties of quasi-polynomial dynamical systems. This general class can be related to the class of quadratic Lotka-Volterra systems through a suitable mapping and change of variables. We analyze a criterion for convergence of solutions and the structure of ω-limit sets.

We present a local convergence analysis of a two-step and derivative-free Kung–Traub’s method, which is based on a parameter and has fourth order of convergence. Using basins of attraction of the method, dynamical behavior of the scheme is studied and the best choice of the parameter is found in the sense of reliability and stability. Some illustrative examples show that as the parameter gets close to zero, radius of convergence of the method becomes larger.

A family of Jarratt-type methods has been proposed for solving nonlinear equations of almost sixth convergence order. Moreover, the method has been extended to the multidimensional case by preserving the order of convergence. Theoretical and computational properties have also been investigated along with the order of convergence. In this study, using our idea of restricted convergence domain, we extend the applicability of these methods.

We consider the notion of generalized density, namely, the natural density of weight g recently introduced in [M. Balcerzak, P. Das, M. Filipczak, and J. Swaczyna, Acta Math. Hung., 147, No. 1, 97–115 (2015)] and primarily study some sufficient and almost converse necessary conditions for the generalized statistically convergent sequence under which the subsequence is also generalized statistically convergent. Some results are also obtained in a more general form by using the notion of ideals. The entire investigation is performed in the setting of general metric spaces extending the recent results of M. Kücükaslan, U. Deger, and O. Dovgoshey, [Ukr. Math. J., 66, No. 5, 712–720 (2014)]. © 2017, Springer Science+Business Media, LLC.

Super Halley’s method is one of the most important iterative methods for solving nonlinear equations in Banach spaces. It’s local and semilocal convergence analysis is established using either majorizing sequences or recurrence relations under various continuity conditions such as Lipschitz or Holder using first/second order Frechet derivatives. In this paper, an attempt is made to establish it’s local convergence analysis under weaker continuity conditions on first order Frechet derivative. This work generalizes the earlier work in this direction and it is observed that it is applicable to cases whether they either fail to converge or give smaller balls of convergence.

The purpose of this paper is to establish the rate of convergence in terms of the weighted modulus of continuity and Lipschitz type maximal function for the q-Szász-beta operators. We also study the rate of A-statistical convergence. Lastly, we modify these operators using King type approach to obtain better approximation.

In this paper, we study the convergence and the convergence rates of an inexact Newton–Landweber iteration method for solving nonlinear inverse problems in Banach spaces. Opposed to the traditional methods, we analyze an inexact Newton–Landweber iteration depending on the Holder continuity of the inverse mapping when the data are not contaminated by noise. With the namely Holder-type stability and the Lipschitz continuity of DF, we prove convergence and monotonicity of the residuals defined by the sequence induced by the iteration. Finally, we discuss the convergence rates.

The local convergence of generalized Mann iteration is investigated in the setting of a real Hilbert space. As application, we obtain an algorithm for estimating the local radius of convergence for some known iterative methods. Numerical experiments are presented showing the performances of the proposed algorithm. For a particular case of the Ezquerro-Hernandez method (Ezquerro and Hernandez, J. Complex., 25:343-361: 2009), the proposed procedure gives radii which are very close to or even identical with the best possible ones.

The proximal point algorithm (PPA) is a classical method for finding zeros of maximal monotone operators. It is known that the algorithm only has weak convergence in a general Hilbert space. Recently, Wang, Wang and Xu proposed two modifications of the PPA and established strong convergence theorems on these two algorithms. However, these two convergence theorems exclude an important case, namely, the over-relaxed case. In this paper, we extend the above convergence theorems from under-relaxed case to the over-relaxed case, which in turn improve the performance of these two algorithms. Preliminary numerical experiments show that the algorithm with over-relaxed parameter performs better than that with under-relaxed parameter.

We present a local convergence analysis of fourth-order methods in order to approximate a locally unique solution of a nonlinear equation in Banach space setting. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the fifth derivative although only the first derivative appears in these methods. We only show convergence using hypotheses on the first derivative. We also provide computable: error bounds, radii of convergence as well as uniqueness of the solution with results based on Lipschitz constants not given in earlier studies. The computational order of convergence is also used to determine the order of convergence. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.

We study the Pringsheim pointwise convergence of multiple trigonometric series. We obtain a condition on the coefficients of the series that guarantees its Pringsheim convergence and prove the unimprovability of this condition.

We consider sequences fn of continuous functions on [0,∞) that converge to a continuous monotone limit function. We prove a convergence theorem for the iterative map xn+1=fn(xn). This result is used to solve an open problem in the field concerning the convergence properties of first-order rational difference equations that are linear in numerator and denominator with variable coefficients. We also establish convergence properties for a certain class of systems of first-order difference equations.

We study the local convergence of a sixth-order Newton–Traub method to approximate a locally-unique solution of a system of nonlinear equations. Earlier studies show convergence under hypotheses on the sixth derivative or even higher, although only the first derivatives are used in the method. The convergence in this study is shown under hypotheses only on the first derivative. Our analysis avoids the usual Taylor expansions requiring higher order derivatives but uses generalized Lipschitz–Holder-type conditions only on the first derivative. Moreover, our new approach provides computable radius of convergence as well as error bounds on the distances involved and estimates on the uniqueness of the solution based on some functions appearing in these generalized conditions. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives which may not exist or may be very expensive or impossible to compute. Therefore, we do not know how close to the solution the initial point should be for convergence of the method. That is the initial point is a shot in the dark with the approaches using Taylor expansions. The convergence order is computed using computational order of convergence or approximate computational order of convergence which do not require usage of higher derivatives. This technique can be applied to any iterative method using Taylor expansions involving high order derivatives. In this sense the applicability of the method is expanded. Finally, numerical examples are provided to show that the present results apply to solve equations in cases where earlier results cannot apply.

In this paper we find the order of convergence and semilocal convergence of three step Kurchatov-type method. We also analyze the efficiency index and computational efficiency of this method. The semilocal convergence analysis of method has been established by using recurrence relations under the assumption of first order divided difference operators satisfy Lipschitz condition. The convergence theorem and domain of parameters of the method has also been included. The applicability of the proposed convergence analysis is illustrated by solving some numerical examples.

In this paper, we propose piecewise homotopy analysis method (PHAM) to extend the convergence region of the homotopy analysis solution for well-posed initial value problems. A convergence theorem of the homotopy analysis solution in the expression of polynomials in the form of Cauchy-Kowalevskaya theorem is given for the nonlinear equation which is analytical near the initial point. We also discuss the influence of the convergence-control parameter h. Our convergence result supports Liao’s conjecture (1): the convergence region increases with the increasing of h(

We present a new semilocal convergence analysis for Newton-like methods using restricted convergence domains in a Banach space setting. The main goal of this study is to expand the applicability of these methods in cases not covered in earlier studies. The advantages of our approach include, under the same computational cost as previous studies, a more precise convergence analysis under the same computational cost on the Lipschitz constants involved. Numerical studies including a chemical application are also provided in this study.

We present new sufficient convergence conditions for the semilocal convergence of Newton’s method to a locally unique solution of a nonlinear equation in a Banach space. We use Holder and center Holder conditions, instead of just Holder conditions, for the first derivative of the operator involved in combination with our new idea of restricted convergence domains. This way, we find a more precise location where the iterates lie, leading to at least as small Holder constants as in earlier studies. The new convergence conditions are weaker, the error bounds are tighter and the information on the solution at least as precise as before. These advantages are obtained under the same computational cost. Numerical examples show that our results can be used to solve equations where older results cannot.

In this paper, the convergence of iterative learning control with initial state error for some fractional equation is studied. According to the Laplace transform and the M-L function, the concept of mild solutions is showed. The sufficient conditions of convergence for the open and closed P-type iterative learning control are obtained. Some examples are given to illustrate our main results.

We study an ordinary differential operator of third order and absolute and uniform convergence of spectral expansion of the function from the class W1p(G), G = (0,1), p > 1, in eigenfunctions of the operator. Uniform convergence rate of this expansion is estimated.