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Our study in this paper is focused on the split equality fixed-point problem with firmly quasi-non-expansive operators in infinite-dimensional Hilbert spaces. A self-adaptive simultaneous scheme is introduced, and its weak convergence is established under mild and standard assumptions. The new proposed scheme generalizes and extends some related works in the literature, and its simple structure makes it easy for implementation and numerical testing. Primary experiments presented in this paper, in finite- and infinite-dimensional spaces, emphasize their practical advantages over existing results. [ABSTRACT FROM AUTHOR]

In this work, we consider the stochastic Cahn-Hilliard/Allen-Cahn equation with fractional noise, which is fractional in time and white in space. We obtain the existence, uniqueness, and Hölder regularity of the solution. Also, using a weak convergence approach, we prove that the law of solution associated with the above equation with a small perturbation satisfies the large deviation principle in the Hölder norm. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the difference between the deterministic and stochastic models of nonlocal diffusion. We use a nonlocal reaction-diffusion equation and a multi-dimensional jump Markov process to analyze these mathematical models. First, we demonstrate that the difference converges to 0 in probability with a supremum norm for a sizeable network. Next, we consider the rescaled difference and show that it converges to a stochastic process in distribution on the Skorokhod space. [ABSTRACT FROM AUTHOR]

This paper derived the homogenization of the heat equation with random convolutional potential. By Tartar's method of oscillating test function, the solution of the heat equation with random convolutional potential was shown to converge in distribution to the solution of the effective equation with determined convolutional potential. [ABSTRACT FROM AUTHOR]

This paper is concerned with the large deviation principle of the stochastic reaction-diffusion lattice systems defined on the $ N $-dimensional integer set, where the nonlinear drift term is locally Lipschitz continuous with polynomial growth of any degree and the nonlinear diffusion term is locally Lipschitz continuous with linear growth. We first prove the convergence of the solutions of the controlled stochastic lattice systems, and then establish the large deviations by the weak convergence method based on the equivalence of the large deviation principle and the Laplace principle. [ABSTRACT FROM AUTHOR]

We study homogenization for a class of non-symmetric pure jump Feller processes. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different scales. It turns out that the Feller process converges in distribution, as the scaling parameter goes to zero, to a Lévy process. As special cases of our result, some homogenization problems studied in previous works can be recovered. We also generalize the approach to the homogenization of symmetric stable-like processes with variable order. Moreover, we present some numerical experiments to demonstrate the usage of our homogenization results in the numerical approximation of first exit times. [ABSTRACT FROM AUTHOR]

In this paper, we find the common solution of a split variational inclusion problem and a fixed point problem of a finite collection of demicontractive mappings in real Hilbert spaces. We introduce two modified inertial-type iterative methods based on Mann and viscosity schemes to solve the considered problem. We prove weak and strong convergence results for the proposed methods under reasonable conditions on the control parameters. With the help of some numerical examples, we show that our proposed methods give better convergence results than some existing methods in the literature. [ABSTRACT FROM AUTHOR]

The projected reflected gradient method has been shown to be a simple and elegant method for solving variational inequalities. The method involves one projection onto the feasible set and one evaluation of the cost operator per iteration and has been shown numerically to be more efficient than most available methods for solving variational inequalities. Convergence results for methods with similar elegant structures of projected reflected gradient method are still rare. In this paper, we present weak and linear convergence of a projected reflected gradient method with an inertial extrapolation step and give some applications arising from optimal control problems. We first obtain weak convergence result for the projected reflected gradient method with an inertial extrapolation step for solving variational inequalities under standard assumptions with self-adaptive step sizes. We further obtain a linear convergence rate when the cost operator is strongly monotone and Lipschitz continuous. Finally, we give some numerical applications arising from optimal control. Preliminary results show that our method is effective and efficient when compared to other related state-of-the-art methods in the literature and show the advantage gained by incorporating inertial terms into the projected reflected gradient methods. [ABSTRACT FROM AUTHOR]

U-statistics represent a fundamental class of statistics arising from modeling quantities of interest defined by multi-subject responses. U-statistics generalize the empirical mean of a random variable X to sums over every m-tuple of distinct observations of X. Stute [182] introduced a class of so-called conditional U-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to: r(m)(φ, t) ≔ 피[φ(F1,..., Ym)|(X1,..., Xm) = t], for t ∈ Xm. In this paper, we are mainly interested in the study of the kNN conditional U-processes in a functional mixing data framework. More precisely, we investigate the weak convergence of the conditional empirical process indexed by a suitable class of functions and of the kNN conditional U-processes when the explicative variable is functional. We treat the uniform central limit theorem in both cases when the class of functions is bounded or unbounded satisfying some moment conditions. The second main contribution of this study is the establishment of a sharp almost complete Uniform consistency in the Number of Neighbors of the constructed estimator. Such a result allows the number of neighbors to vary within a complete range for which the estimator is consistent. Consequently, it represents an interesting guideline in practice to select the optimal bandwidth in nonparametric functional data analysis. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. The theoretical results established in this paper are (or will be) key tools for further functional data analysis developments. Potential applications include the set indexed conditional U-statistics, Kendall rank correlation coefficient, the discrimination problems and the time series prediction from a continuous set of past values. [ABSTRACT FROM AUTHOR]

In 1643 P. de Fermat introduced the problem of finding a point in the plane such that the sum of its Euclidean distances to the three given points is minimal. Recently, Reich and Tuyen (J Optim Theory Appl 196(1): 78–97, 2023) extended this problem in Hilbert space setting and named it ‘generalized Fermat–Torricelli problem’. They introduced some iterative methods for approximating the solution of this problem. This paper aims to formulate some novel iterative methods with inertial effect for approximating the solution of this generalized Fermat–Torricelli problem in real Hilbert spaces. First, we prove the weak and strong convergence of the proposed methods with the mild conditions on the control parameters. Then, we establish the weak and strong convergence results for approximating the solution of a variant of the split feasibility problem with multiple output sets. Finally, we provide some numerical examples to justify the validity and efficiency of the proposed iterative methods. The results in this paper improve and generalize the work of Reich and Tuyen (J Optim Theory Appl 196(1): 78–97, 2023). [ABSTRACT FROM AUTHOR]

We consider the Ginzburg–Landau equation in the perforated domain, with rapidly oscillating coefficients. We derive the homogenized Ginzburg–Landau equation with a 'strange term' (potential) and prove that the trajectory attractors of the given equation tend in a weak sense to the trajectory attractors of the homogenized one. Assuming additional conditions to be satisfied for the coefficients, we provide also a convergence of the global attractor. [ABSTRACT FROM AUTHOR]

We establish a large deviation principle (LDP) for a class of stochastic porous media equations driven by Lévy-type noise on a σ-finite measure space (E,B(E),μ), with the Laplacian replaced by a negative definite self-adjoint operator. One of the main contributions of this paper is that we do not assume the compactness of embeddings in the corresponding Gelfand triple, and to compensate for this generalization, a new procedure is provided. This is the first paper to deal with LDPs for stochastic evolution equations with Lévy noise without compactness conditions. The coefficient Ψ is assumed to satisfy nondecreasing Lipschitz nonlinearity, so an important physical problem covered by this case is the Stefan problem. Numerous examples of negative definite self-adjoint operators are applicable to our results, for example, for open E⊂Rd, L= Laplacian or fractional Laplacians, i.e., L=-(-Δ)α, α∈(0,1], generalized Schrödinger operators, i.e., L=Δ+2∇ρ/ρ⋅∇, Laplacians on fractals is also included. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Krasnosel'skiĭ-Mann iteration for finding a fixed point of nonexpansive operators in real Hilbert spaces. By introducing new, concise, and self-contained techniques, we propose new inertial factors with desirable upper bounds better than or comparable to existing ones. [ABSTRACT FROM AUTHOR]

This paper extends the notion of weak convergence in metric spaces to the case of S-metric spaces. Moreover, some results on the weak convergence of fixed point iterations of Banach’s, Kannan’s, Chatterjea’s, Reich’s, Hardy and Roger’s types of contractions on S-metric spaces are obtained. In addition, an example is presented to demonstrate our primary result. [ABSTRACT FROM AUTHOR]

The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. The notion of noncentral moderate deviations is used when the weak convergence is towards a non-Gaussian distribution. In this paper, noncentral moderate deviation results are presented for two fractional Skellam processes known in the literature (see [20]). It is established that, for the fractional Skellam process of type 2 (for which one can refer to the recent results for compound fractional Poisson processes in [3]), the convergences to zero are usually faster because one can prove suitable inequalities between rate functions. [ABSTRACT FROM AUTHOR]

This paper discusses the large deviations for stochastic predator–prey model driven by multiplicative Lévy noise. Using Galerkin approximation, we initially prove the existence and uniqueness of solution. Due to the equivalence between Laplace principle and large deviation principle under a Polish space, the method of weak convergence has been followed in order to establish our results for this coupled system of equations.

The Ginzburg-Landau equation with rapidly oscillating terms in the equation and boundary conditions in a perforated domain was considered. Proof was given that the trajectory attractors of this equation converge weakly to the trajectory attractors of the homogenized Ginzburg-Landau equation. To do this, we use the approach from the articles and monographs of V.V. Chepyzhov and M.I. Vishik about trajectory attractors of evolutionary equations, and we also use homogenization methods that appeared at the end of the 20th century. First, we use asymptotic methods to construct asymptotics formally, and then we justify the form of the main terms of the asymptotic series using functional analysis and integral estimates. By defining the corresponding auxiliary function spaces with weak topology, we derive a limit (homogenized) equation and prove the existence of a trajectory attractor for this equation. Then, we formulate the main theorems and prove them by using auxiliary lemmas. We prove that the trajectory attractors of this equation tend in a weak sense to the trajectory attractors of the homogenized Ginzburg-Landau equation in the subcritical case, and they disappear in the supercritical case.

The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. The notion of noncentral moderate deviations is used when the weak convergence is towards a non-Gaussian distribution. In this paper, noncentral moderate deviation results are presented for two fractional Skellam processes known in the literature (see [20]). It is established that, for the fractional Skellam process of type 2 (for which one can refer to the recent results for compound fractional Poisson processes in [3]), the convergences to zero are usually faster because one can prove suitable inequalities between rate functions.

This paper extends the notion of weak convergence in metric spaces to the case of S-metric spaces. Moreover, some results on the weak convergence of fixed point iterations of Banach's, Kannan's, Chatterjee's, Reich's, Hardy and Roger's types of contractions on S-metric spaces are obtained. In addition, an example is presented to demonstrate our primary result.

In this paper, we introduce a new fixed point iterative scheme called the AG iterative scheme that is used to approximate the fixed point of a contraction mapping in a uniformly convex Banach space. The iterative scheme is used to prove some convergence result. The stability of the new scheme is shown. Furthermore, weak convergence of Suzuki's generalized non-expansive mapping satisfying condition (C) is shown. The rate of convergence result is proved and it is demonstrated via an illustrative example which shows that our iterative scheme converges faster than the Picard, Mann, Noor, Picard-Mann, M and Thakur iterative schemes. Data dependence results for the iterative scheme are shown. Finally, our result is used to approximate the solution of a nonlinear fractional differential equation of Caputo type.

Abstract In this paper, we propose two inertial algorithms with a new self-adaptive step size for approximating a solution of the split common null-point problem in the framework of Banach spaces. The step sizes are adaptively updated over each iteration by a simple process without the prior knowledge of the operator norm of the bounded linear operator. Under suitable conditions, we prove the weak-convergence results for the proposed algorithms in p-uniformly convex and uniformly smooth Banach spaces. Finally, we give several numerical results in both finite- and infinite-dimensional spaces to illustrate the efficiency and advantage of the proposed methods over some existing methods. Also, data classifications of heart diseases and diabetes mellitus are presented as the applications of our methods.

By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function ζ(s) by shifts ζ(s+iτ). In this paper, it is determined that the Riemann hypothesis is equivalent to the positivity of density of the set of the above shifts approximating ζ(s) with all but at most countably many accuracies ε>0. Also, the analogue of an equivalent in terms of positive density in short intervals is discussed.

We have developed the generalized quasilinearization method (QM) for an initial value problem (IVP) of dynamic equations on time scales by using comparison theorems with a coupled lower solution (LS) and upper solution (US) of the natural type. Under some conditions, we observed that the solutions converged to the unique solution of the problem uniformly and monotonically, and the rate of convergence was investigated.

The convective Brinkman-Forchheimer (CBF) equations are employed to characterize the motion of incompressible fluid in a saturated porous medium. This work investigates the small noise asymptotic of two-time-scale stochastic CBF equations in two and three dimensional bounded domains. More precisely, we establish a Wentzell-Freidlin type large deviation principle for stochastic partial differential equations that have slow and fast time-scales. The slow component is the stochastic CBF equations in two or three dimensions perturbed by a small multiplicative Gaussian noise, while the fast component is a stochastic reaction-diffusion equation with damping. The results are obtained by using a variational method (based on weak convergence approach) developed by Budhiraja and Dupuis, Khasminkii's time discretization approach and stopping time arguments. In particular, the findings from this study are also applicable to two-dimensional stochastic Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

A modular string averaging procedure (MSA, for short) for a finite number of operators was first introduced by Reich and Zalas in 2016. The MSA concept provides a flexible algorithmic framework for solving various feasibility problems such as common fixed point and convex feasibility problems. In 2001 Bauschke and Combettes introduced the notion of coherence and applied it to proving weak and strong convergence of many iterative methods. In 2019 Barshad, Reich and Zalas proposed a stronger variant of coherence which provides a more convenient sufficient convergence condition for such methods. In this paper we combine the ideas of both modular string averaging and coherence. Focusing on extending the above MSA procedure to an infinite sequence of operators with admissible controls, we establish strong coherence of its output operators. Various applications of these concepts are presented with respect to weak and strong convergence. They also provide important generalizations of known results, where the weak convergence of sequences of operators generated by the MSA procedure with intermittent controls was considered. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an inertial accelerated algorithm for solving the split equality common fixed-point problem of directed operators in real Hilbert space. Our algorithm includes the simultaneous iterative algorithm as special case which has been proposed by Moudafi and Al-Shemas for solving the split equality common fixed-point problem. We establish a weak convergence theorem for the proposed iterative algorithm, which combines the primal-dual method and the inertial technique. In our algorithm, the step sizes are chosen self-adaptively so that the implementation of the algorithm does not need any prior information about bounded linear operator norms. The efficiency of the proposed algorithm is illustrated by some numerical experiments. [ABSTRACT FROM AUTHOR]

A new numerical approximation method for a class of Gaussian random fields on compact connected oriented Riemannian manifolds is introduced. This class of random fields is characterized by the Laplace–Beltrami operator on the manifold. A Galerkin approximation is combined with a polynomial approximation using Chebyshev series. This so-called Galerkin–Chebyshev approximation scheme yields efficient and generic sampling algorithms for Gaussian random fields on manifolds. Strong and weak orders of convergence for the Galerkin approximation and strong convergence orders for the Galerkin–Chebyshev approximation are shown and confirmed through numerical experiments. [ABSTRACT FROM AUTHOR]

This paper presents a fully adaptive iterative method without linesearch procedure to solve variational inequalities in the setting of quasi-monotone which is locally Lipschitz. Our method involves one functional evaluation alongside one projection per iteration and our stepsizes are not restricted to be nonincreasing as opposed to relevant methods without linesearch procedure but with nonincreasing stepsizes. We give weak and strong convergence analysis of the sequences generated by our proposed method under some mild conditions. We also give R -linear rate convergence under an error bound condition and give some numerical implementations of our method. Numerical results show that our proposed method is competitive with other related methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new fixed point iterative scheme called the AG iterative scheme that is used to approximate the fixed point of a contraction mapping in a uniformly convex Banach space. The iterative scheme is used to prove some convergence result. The stability of the new scheme is shown. Furthermore, weak convergence of Suzuki's generalized non-expansive mapping satisfying condition (C) is shown. The rate of convergence result is proved and it is demonstrated via an illustrative example which shows that our iterative scheme converges faster than the Picard, Mann, Noor, Picard-Mann, M and Thakur iterative schemes. Data dependence results for the iterative scheme are shown. Finally, our result is used to approximate the solution of a nonlinear fractional differential equation of Caputo type. [ABSTRACT FROM AUTHOR]

It is well known that the law of a one-dimensional diffusion on natural scale is fully characterized by its speed measure. Stone proved a continuous dependence of such diffusions on their speed measures. In this paper we establish the converse direction, i.e., we prove a continuous dependence of the speed measures on their diffusions. Furthermore, we take a topological point of view on the relation. More precisely, for suitable topologies, we establish a homeomorphic relation between the set of regular diffusions on natural scale without absorbing boundaries and the set of locally finite speed measures. [ABSTRACT FROM AUTHOR]

This paper studies transition probabilities from a Borel subset of a Polish space to a product of two Borel subsets of Polish spaces. For such transition probabilities it introduces and studies the property of semi-uniform Feller continuity. This paper provides several equivalent definitions of semi-uniform Feller continuity and establishes its preservation under integration. The motivation for this study came from the theory of Markov decision processes with incomplete information, and this paper provides the fundamental results useful for this theory. [ABSTRACT FROM AUTHOR]

We consider infinitely wide multi-layer perceptrons (MLPs) which are limits of standard deep feed-forward neural networks. We assume that, for each layer, the weights of an MLP are initialized with independent and identically distributed (i.i.d.) samples from either a light-tailed (finite-variance) or a heavy-tailed distribution in the domain of attraction of a symmetric $\alpha$ -stable distribution, where $\alpha\in(0,2]$ may depend on the layer. For the bias terms of the layer, we assume i.i.d. initializations with a symmetric $\alpha$ -stable distribution having the same $\alpha$ parameter as that layer. Non-stable heavy-tailed weight distributions are important since they have been empirically seen to emerge in trained deep neural nets such as the ResNet and VGG series, and proven to naturally arise via stochastic gradient descent. The introduction of heavy-tailed weights broadens the class of priors in Bayesian neural networks. In this work we extend a recent result of Favaro, Fortini, and Peluchetti (2020) to show that the vector of pre-activation values at all nodes of a given hidden layer converges in the limit, under a suitable scaling, to a vector of i.i.d. random variables with symmetric $\alpha$ -stable distributions, $\alpha\in(0,2]$. [ABSTRACT FROM AUTHOR]

In this paper, a general splitting method with alternated inertia and its relaxed version are proposed for solving the split equality problem in Hilbert spaces. The proposed methods combine both the relaxation and alternated inertial techniques to speed up the rate of convergence. Furthermore, the methods employ a simple self-adaptive stepsize, which does not require any prior information about the operator norm. Four options of inertial parameters and relaxation parameters are discussed. The weak convergence of the proposed algorithms is analyzed under mild conditions. Finally, two numerical experiments and an application in signal recovery problem are provided to demonstrate the advantages of the proposed algorithms compared to a recent related one. [ABSTRACT FROM AUTHOR]

We first establish weak convergence results regarding an inertial Krasnosel'skiĭ-Mann iterative method for approximating common fixed points of countable families of nonexpansive mappings in real Hilbert spaces with no extra assumptions on the considered countable families of nonexpansive mappings. The method of proof and the imposed conditions on the iterative parameters are different from those already available in the literature. We then present some applications to the Douglas–Rachford splitting method and image restoration problems, and compare the performance of our method with that of other popular inertial Krasnosel'skiĭ-Mann methods which can be found in the literature. [ABSTRACT FROM AUTHOR]

Let a = { a m : m ∈ N } be a periodic multiplicative sequence of complex numbers and L (s ; a) , s = σ + i t a Dirichlet series with coefficients a m . In the paper, we obtain a theorem on the approximation of non-vanishing analytic functions defined in the strip 1 / 2 < σ < 1 via discrete shifts L (s + i h t k ; a) , h > 0 , k ∈ N , where { t k : k ∈ N } is the sequence of Gram points. We prove that the set of such shifts approximating a given analytic function is infinite. This result extends and covers that of [Korolev, M.; Laurinčikas, A. A new application of the Gram points. Aequat. Math. 2019, 93, 859–873]. For the proof, a limit theorem on weakly convergent probability measures in the space of analytic functions is applied. [ABSTRACT FROM AUTHOR]

The multistage stochastic variational inequality is reformulated into a variational inequality with separable structure through introducing a new variable. The prediction–correction ADMM which was originally proposed in He et al. (J Comput Math 24:693–710, 2006) for solving deterministic variational inequalities in finite-dimensional spaces is adapted to solve the multistage stochastic variational inequality. Weak convergence of the sequence generated by that algorithm is proved under the conditions of monotonicity and Lipschitz continuity. When the sample space is a finite set, the corresponding multistage stochastic variational inequality is actually defined on a finite-dimensional Hilbert space and the strong convergence of the algorithm naturally holds true. Some numerical examples are given to show the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

Fixed point theory is a branch of mathematics that studies solutions that remain unchanged under a given transformation or operator, and it has numerous applications in fields such as mathematics, economics, computer science, engineering, and physics. In the present article, we offer a quicker iteration technique, the D * * iteration technique, for approximating fixed points in generalized α -nonexpansive mappings and nearly contracted mappings. In uniformly convex Banach spaces, we develop weak and strong convergence results for the D * * iteration approach to the fixed points of generalized α -nonexpansive mappings. In order to demonstrate the effectiveness of our recommended iteration strategy, we provide comprehensive analytical, numerical, and graphical explanations. Here, we also demonstrate the stability consequences of the new iteration technique. We approximately solve a fractional Volterra–Fredholm integro-differential problem as an application of our major findings. Our findings amend and expand upon some previously published results. [ABSTRACT FROM AUTHOR]

This paper derives a free analogue of the Euler-Maruyama method (fEMM) to numerically approximate solutions of free stochastic differential equations (fSDEs). Simply speaking, fSDEs are SDEs in the context of noncommutative random variables (e.g., large random matrices). By applying the theory of multiple operator integrals, we derive a free Itô formula from Taylor expansion of operator-valued functions. Iterating the free Itô formula allows us to motivate and define fEMM. Then we consider weak and strong convergence in the fSDE setting and prove strong convergence order of 1/2 and weak convergence order of 1. Numerical examples support the theoretical results and show solutions for equations where no analytical solution is known. [ABSTRACT FROM AUTHOR]

Bregman distance iterative methods for solving optimization problems are important and interesting because of the numerous applications of Bregman distance techniques. In this paper, for solving a split feasibility problem, we introduce a new Bregman projection algorithm and construct two selection strategies of stepsizes. Moreover, a relaxed Bregman projection algorithm is proposed with two selection strategies of stepsizes, where the two closed and convex sets are both level sets of convex functions. Weak convergence results of the proposed algorithms are obtained under suitable assumptions. In addition, using the proposed algorithms with different Bregman distances, a numerical experiment solving signal processing problem is also given to demonstrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we propose two inertial algorithms with a new self-adaptive step size for approximating a solution of the split common null-point problem in the framework of Banach spaces. The step sizes are adaptively updated over each iteration by a simple process without the prior knowledge of the operator norm of the bounded linear operator. Under suitable conditions, we prove the weak-convergence results for the proposed algorithms in p-uniformly convex and uniformly smooth Banach spaces. Finally, we give several numerical results in both finite- and infinite-dimensional spaces to illustrate the efficiency and advantage of the proposed methods over some existing methods. Also, data classifications of heart diseases and diabetes mellitus are presented as the applications of our methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two modified Tseng's extragradient algorithms with a new generalized adaptive stepsize for solving monotone variational inequalities (VI) in reflexive Banach spaces. The advantage of our methods is that stepsizes do not require prior knowledge of the Lipschitz constant of the cost mapping. Based on Bregman projection-type methods, we prove weak and strong convergence of the proposed algorithms to a solution of VI. Some numerical experiments to show the efficiency of our methods including a comparison with related methods are provided. [ABSTRACT FROM AUTHOR]

We propose an extension of the Cox-Ross-Rubinstein (CRR) model based on q-binomial (or Kemp) random walks, with application to default with logistic failure rates. This model allows us to consider time-dependent switching probabilities varying according to a trend parameter on a non-self-similar binomial tree. In particular, it includes tilt and stretch parameters that control increment sizes. Option pricing formulas are written using q-binomial coefficients, and we study the convergence of this model to a Black-Scholes type formula in continuous time. A convergence rate of order O(N-1/2) is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we obtain weak convergence results for a family of Gibbs measures depending on the parameter θ > 0 in the following form d P θ (x) = Z θ exp − H θ (x) / θ d Q (x) , where we show that the limit distribution is concentrated in the set of the global minima of the limit Gibbs potential. We also give an explicit calculus for the limit distribution. Here, we use the above as an alternative to Lyapunov's function or to direct methods for stationary probability convergence and apply it to the repairman problem. Finally, we illustrate this method with a numerical example. [ABSTRACT FROM AUTHOR]

In the paper we consider a problem for complex Ginzburg–Landau equations in a medium with locally periodic small obstacles. It is assumed that the obstacle surface can have different conductivity coefficients. We prove that the trajectory attractors of this system converge in a certain weak topology to the trajectory attractors of the homogenized Ginzburg–Landau equations with an additional potential (in the critical case), without an additional potential (in the subcritical case) in the medium without obstacles, or disappear (in the supercritical case). [ABSTRACT FROM AUTHOR]

In the paper, we prove that the set of discrete shifts of the Riemann zeta-function (ζ(s + 2πia1k), . . ., ζ(s + 2πiark)), k ∈ N, approximating analytic non-vanishing functions f1(s), . . ., fr(s) defined on {s ∈ C : 1/2 < Res < 1} has a positive density in the interval [N,N + M] with M = o(N), N → ∞, with real algebraic numbers a1, . . ., ar linearly independent over Q. A similar result is obtained for shifts of certain absolutely convergent Dirichlet series. [ABSTRACT FROM AUTHOR]

In real Hilbert spaces, for the purpose of trying to deal with the pseudo-monotone variational inequalities problem, we present a new extrapolation projection contraction algorithm based on the golden ratio in this study. Unlike ordinary inertial extrapolation, the algorithms are constructed based on a convex combined structure about the entire iterative trajectory. Extrapolation parameter ϕ is selected in a more relaxed range instead of only taking the golden ratio ϕ = √5+1/2 as the upper bound. Second, we propose an alternating extrapolation projection contraction algorithm to better increase the convergence effects of the extrapolation projection contraction algorithm based on the golden ratio. All our algorithms employ non-constantly decreasing adaptive step-sizes. The weak convergence results of the two algorithms are established for the pseudo-monotone variational inequalities. Additionally, the R-linear convergence results are investigated for strongly pseudo-monotone variational inequalities. Finally, we show the validity and superiority of the suggested methods with several numerical experiments. The numerical results show that alternating extrapolation does have obvious acceleration effect in practical application compared with no alternating extrapolation. Thus, the obvious effect of relaxing the selection range of parameter on our two algorithms is clearly demonstrated. [ABSTRACT FROM AUTHOR]

We establish the existence and uniqueness of fixed points of generalized contractions in the setting of Banach spaces and prove the convergence of Mann iteration for this general class of mappings. Also, we show the existence of fixed points and the convergence of Mann iteration as well for generalized nonexpansive mappings. Last but not least, we provide two applications, one from the field of numerical analysis of linear systems and another one dealing with functional equations. This new approach significantly extends the classes of enriched contractions and enriched nonexpansive mappings, and allows the use of Mann iteration as opposed to all papers on the subject, which necessarily have to rely on Krasnoselskij iteration.

This article analyzes the JK iteration process with the class of mappings that are essentially endowed with a condition called condition (E). The convergence of the iteration toward a fixed point of a specific mapping satisfying the condition (E) is obtained under some possible mild assumptions. It is worth mentioning that the iteration process JK converges better toward a fixed point compared to some prominent iteration processes in the literature. This fact is confirmed by a numerical example. Furthermore, it has been shown that the iterative scheme JK is stable in the setting of generalized contraction. The data dependence result is also established. Our results are new in the iteration theory and extend some recently announced results of the literature.