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We consider weak convergence and weak compactness in the space L¹(m) of real valued integrable functions with respect to a Banach space valued measure m equipped with its natural norm. We give necessary and sufficient conditions for a sequence in L¹(m) to be weak Cauchy, and we give necessary and sufficient conditions for a subset of L¹(m) to be conditionally sequentially weakly compact. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the exponential-type inequalities for maximal partial sums of negatively superadditive dependent (NSD) random variables, which extends the corresponding results for independent and negatively associated (NA) random variables. Using these inequalities, we further investigate the weak convergence of the M-estimators in the generalized linear model with NSD errors, which generalize and improve the corresponding results of the independent random errors to that of NSD random errors. [ABSTRACT FROM AUTHOR]

In this paper, we study to approximate solutions of quasi-equilibrium problems in Banach spaces, which generalizes equilibrium problems and quasi-variational inequalities. In fact, we prove weak convergence of the sequence generated by the proximal point method to a solution of the quasi-equilibrium problem. Then we show that the generated sequence converges strongly to a solution of the quasi-equilibrium problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider a split equality fixed point problem for asymptotically quasi-pseudo contractive operators which includes split feasibility problem, split equality problem, split fixed point problem etc, as special cases. Furthermore we propose a new algorithm for solving the split equality fixed point problem, and prove a weak and strong convergence theorem. The results obtained in this paper generalize and improve the recent ones announced by many others. [ABSTRACT FROM AUTHOR]

In this paper, we first show that for a countable family of random elements taking values in a partially ordered Polish space with a closed partial order (POP space), association (both positive and negative) of all finite dimensional marginals implies that of the infinite sequence. Our proof proceeds via Strassen's theorem for stochastic domination and thus avoids the assumption of normally ordered on the product space as needed for positive association in Lindqvist (1988). We use these results to show on POP spaces that finite dimensional negative association implies negative association of the random measure and negative association is preserved under weak convergence of random measures. The former provides a simpler proof in the most general setting of Polish spaces complementing the recent proofs in Poinas et al. (2019) and Lyons (2014) which restrict to point processes in ℝd and locally compact Polish spaces respectively. We also provide some examples of associated random measures which shall illustrate our results as well. [ABSTRACT FROM AUTHOR]

Nourdin and Peccati (2009a) established a neat characterization of Gamma approximation on a fixed Wiener chaos in terms of convergence of only the third and fourth cumulants. In this paper, we provide an optimal rate of convergence in the d2-distance in terms of the maximum of the third and fourth cumulants analogous to the result for normal approximation in Nourdin and Peccati (2015). In order to achieve our goal, we introduce a novel operator theory approach to Stein's method. The recent development in Stein's method for the Gamma distribution of Döbler and Peccati (2018) plays a pivotal role in our analysis. Several examples in the context of quadratic forms are considered to illustrate our optimal bound. [ABSTRACT FROM AUTHOR]

It is well known that the Riemann zeta-function is universal in the Voronin sense, i.e., its shifts ζ(s + iт), т ∊ R, approximate a wide class of analytic functions. The universality of ζ(s) is called discrete if т take values from a certain discrete set. In the paper, we obtain a weighted discrete universality theorem for ζ(s) when т takes values from the arithmetic progression fkh : k ∊ Ng with arbitrary fixed h > 0. For this, two types of h are considered. [ABSTRACT FROM AUTHOR]

In medical and epidemiological studies, it is often interest to study time-to-event distributions under competing risks that involve two or more failure types. Nonparametric analysis of competing risks is typically focused on the cumulative incidence function or nonparametric quantile function. However, the existing estimators may be very unstable due to their unsmoothness. In this paper, we propose a kernel nonparametric quantile estimator for right-censored competing risks data, which is a smoothed version of Peng and Fine's nonparametric quantile estimator. We establish the Bahadur representation of the proposed estimator. The convergence rate of the remainder term for the proposed estimator is substantially faster than Peng and Fine's quantile estimator. The pointwise confidence intervals and simultaneous confidence bands of the quantile functions are also derived. Simulation studies illustrate the good performance of the proposed estimator. The methodology is demonstrated with two applications of the Supreme Court Judge data and AIDSSI data. [ABSTRACT FROM AUTHOR]

In this paper, we study the limit distribution functions of the (lower-lower), (upper-upper) and (lower-upper) extreme and central-central m-generalised order statistics (m–GOS) of stationary Gaussian sequences under an equi-correlated set up, when the random sample size is assumed to converge weakly and independent of the basic variables. Moreover, sufficient conditions for a weak convergence of generalised quasi-range with random indices are obtained. [ABSTRACT FROM AUTHOR]

Let B(x) be a multiple power series with nonnegative coefficients which is convergent for all x ∈ (0, 1)n and diverges at the point 1 = (1, ..., 1). Random vectors (r.v.)ξx such that ξx has distribution of the power series B(x) type is studied. The integral limit theorem for r.v. ξx as x ↑ 1 is proved under the assumption that B(x) is regularly varying at this point. Also local version of this theorem is obtained under the condition that the coefficients of the series B(x) are one-sided weakly oscillating at infinity. [ABSTRACT FROM AUTHOR]

We show how to pass to the limit in some nonlinear elliptic problems, despite the weak convergence of the right hand sides (and the weak convergence of the solutions). [ABSTRACT FROM AUTHOR]

By the Voronin theorem, the set of shifts of the Riemann zeta-function ζ (s + i τ) , s = σ + i t , τ ∈ R , that approximate any given non-vanishing analytic function defined on { s ∈ C : 1 2 < σ < 1 } has a positive lower density. In the paper, it is proved that the same property of the above shifts remains valid in the intervals [ T , T + H ] with T 1 / 3 (log ⁡ T) 26 / 15 ⩽ H ⩽ T. [ABSTRACT FROM AUTHOR]

In this paper, we use the dual variable to propose a self-adaptive iterative algorithm for solving the split common fixed point problems of averaged mappings in real Hilbert spaces. Under suitable conditions, we get the weak convergence of the proposed algorithm and give applications in the split feasibility problem and the split equality problem. Some numerical experiments are given to illustrate the efficiency of the proposed iterative algorithm. Our results improve and extend the corresponding results announced by many others. [ABSTRACT FROM AUTHOR]

The mass transport processes always accompany the flow phenomena and have attracted many researchers. A lot of numerical methods have been developed to study them. These numerical methods can be classified into the Eulerian and the Lagrangian approaches. The Lagrangian approach has advantages in high stability and simplicity over the Eulerian approach, but suffers from heavy computational cost. In this paper, we are mainly concerned with the trade-offs between the accuracy and computational cost when applying the random walk method, which is a Lagrangian approach for examining the mass transport scenario. We introduce a linear model to assess the accuracy of the random walk method in several computational configurations. Studies on computational parameters, i.e. the size of time step and number of particles, are conducted with the focus on estimation of the longitudinal dispersion coefficient D L in steady flows. The results show that the proposed linear model can satisfactorily explain the computational accuracy, both in sample and out-of-sample. Furthermore, we find a constant dimensionless parameter, which quantifies a generic relationship between the accuracy and the number of particles regardless of the flow and diffusion conditions. This dimensionless parameter is of theoretic value and offers guidelines for choosing the correct computational parameters to achieve the required numerical accuracy. Image 1 • We propose a linear model for assessing the accuracy of the random walk method. • A dimensionless parameter is proposed to link computational accuracy and particle number. • The present computational results are found to be insensitive to the time step in Couette flow. [ABSTRACT FROM AUTHOR]

The conditional copula of a random pair (Y 1 , Y 2) given the value taken by some covariate X ∈ R is the function C x : [ 0 , 1 ] 2 → [ 0 , 1 ] such that P (Y 1 ≤ y 1 , Y 2 ≤ y 2 | X = x) = C x { P (Y 1 ≤ y 1 | X = x) , P (Y 2 ≤ y 2 | X = x) } . In this note, the weak convergence of the two estimators of C x proposed by Gijbels et al. (Comput Stat Data Anal 55(5):1919–1932, 2011) is established under α -mixing. It is shown that under appropriate conditions on the weight functions and on the mixing coefficients, the limiting processes are the same as those obtained by Veraverbeke et al. (Scand J Stat 38(4):766–780, 2011) under the i.i.d. setting. The performance of these estimators in small sample sizes is investigated with simulations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a CQ-type algorithm for solving the split feasibility problem (SFP) in real Hilbert spaces. The algorithm is designed such that the step-sizes are directly computed at each iteration. We will show that the sequence generated by the proposed algorithm converges in norm to the minimum-norm solution of the SFP under appropriate conditions. In addition, we give some numerical examples to verify the implementation of our method. Our result improves and complements many known related results in the literature. [ABSTRACT FROM AUTHOR]

We consider the weak convergence of iterates of so-called centred quadratic stochastic operators. These iterations allow us to study the discrete time evolution of probability distributions of vector-valued traits in populations of inbreeding or hermaphroditic species, whenever the offspring's trait is equal to an additively perturbed arithmetic mean of the parents' traits. It is shown that for the existence of a weak limit, it is sufficient that the distributions of the trait and the perturbation have a finite variance or have tails controlled by a suitable power function. In particular, probability distributions from the domain of attraction of stable distributions have found an application, although in general the limit is not stable. [ABSTRACT FROM AUTHOR]

We develop and analyze a numerical method for stochastic time-fractional diffusion driven by additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative of order α ∈ (0,1), and fractionally integrated Gaussian noise (with a Riemann-Liouville fractional integral of order γ ∈ [0,1] in the front). The numerical scheme approximates the model in space by the standard Galerkin method with continuous piecewise linear finite elements and in time by the classical Grünwald-Letnikov method (for both Caputo fractional derivative and Riemann-Liouville fractional integral), and the noise by the L2-projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the discrete solution operators for the deterministic inhomogeneous problem. One- and two-dimensional numerical results are presented to support the theoretical findings. [ABSTRACT FROM AUTHOR]

We consider the time-fractional nonlinear stochastic fourth-order reaction diffusion equation driven by the fractional noise (FSRDE). The FSRDE is discretized by using the mixed finite element method in space and the FBT- θ method in time. Some good techniques are proposed to prove the important lemmas in the proof of the strong convergence and the weak convergence. By proving the error estimates on the strong and weak convergence, we find the strong convergence and the weak convergence have the same convergence rate, and the convergence order is related to the fractional noise derivative β but not to the time fractional derivative α and the discrete format coefficient θ. Finally, the theorems on the strong and weak convergence are verified by the numerical tests. [ABSTRACT FROM AUTHOR]

In this work, we investigate a single projection method with double inertial extrapolation steps and self-adaptive step sizes for solving pseudomonotone variational inequality problems in a real Hilbert space. Weak and linear convergence are presented under suitable conditions. Some simple numerical examples are given to illustrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this note we provide a systematic reasoning to arrive at the reflexivity of the underlying Banach space as a sufficient condition for guaranteeing that any compact operator transforms weak convergence in strong convergence. Our starting point is an adaptation of the proof for the analogue result holding in the case of the weak convergence. Then, along the way, and as a by-product of the analysis, we characterize the existence of what we call the inverse-adjoint operator. [ABSTRACT FROM AUTHOR]

The present paper represents a continuation of Migórski et al. (J Elast 127:151–178, 2017). There, the analysis of a new class of elliptic variational–hemivariational inequalities in reflexive Banach spaces, including existence and convergence results, was provided. An inequality in the class is governed by a nonlinear operator, a convex set of constraints and two nondifferentiable functionals, among which at least one is convex. In the current paper we complete this study with new results, including a convergence result with respect the set of constraints. Then we formulate two optimal control problems for which we prove the existence of optimal pairs, together with some convergence results. Finally, we exemplify our results in the study of a one-dimensional mathematical model which describes the equilibrium of an elastic rod in unilateral contact with a foundation, under the action of a body force. [ABSTRACT FROM AUTHOR]

In this work, we establish a common fixed point result for mappings satisfying a controllable punctual inequality and we study the convergence (resp. weak convergence) of the generalized Kirk's process associated with them. In addition, our results are applied to investigate the convergence (resp. weak convergence) of Kuhfittig's iterative process to the solution of a nonlinear system of functional equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the limit distributions of upper and lower record values of a stationary Gaussian sequence under an equi-correlated set up. Moreover, the class of limit distribution functions (df's) of the joint upper (and the lower) record values of a stationary Gaussian sequence is fully characterized. As an application of this result, the sufficient conditions for the weak convergence of the record quasi-range, record quasi-mid-range, record extremal quasi-quotient and record extremal quasi-product are obtained. Moreover, the classes of the non-degenerate limit df's of these statistics are derived. [ABSTRACT FROM AUTHOR]

In this paper, we study the three‐dimensional‐simplified Ericksen‐Leslie system for the steady compressible flow of nematic liquid crystals in a bounded domain. It is proved that the existence of a weak solution for the adiabatic exponent γ > 1 provided the initial direction field in the upper hemisphere. [ABSTRACT FROM AUTHOR]

Our principal aim in this article is to show the equivalence of two concepts used recently in the theory of Banach algebras. The result we present here solves an open problem raised by Jeribi and Krichen in their 2015 book. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with modeling the dependence structure of two (or more) time series in the presence of a (possibly multivariate) covariate which may include past values of the time series. We assume that the covariate influences only the conditional mean and the conditional variance of each of the time series but the distribution of the standardized innovations is not influenced by the covariate and is stable in time. The joint distribution of the time series is then determined by the conditional means, the conditional variances and the marginal distributions of the innovations, which we estimate nonparametrically, and the copula of the innovations, which represents the dependency structure. We consider a nonparametric and a semiparametric estimator based on the estimated residuals. We show that under suitable assumptions, these copula estimators are asymptotically equivalent to estimators that would be based on the unobserved innovations. The theoretical results are illustrated by simulations and a real data example. [ABSTRACT FROM AUTHOR]

It is shown that in general the popular coefficient alpha estimator for reliability of multi-component measuring instruments converges almost surely to a quantity that is not equal to the population reliability coefficient. This convergence with probability 1 is a stronger statement than convergence in probability (consistency) and convergence in distribution for the alpha estimator, which have been studied in the past. In the special case of congeneric measures with uncorrelated errors and equal loadings on the common true score, the alpha estimator converges almost surely to the population reliability coefficient that equals population alpha, which implies also its consistency as a reliability estimator. When the loadings are unequal but sufficiently high and similar, the alpha estimator converges almost surely to population alpha that is essentially indistinguishable from the population reliability coefficient, which implies alpha's approximate consistency then. For the general case, the results entail that the alpha estimator is not a consistent estimator of reliability. The findings add to the critical literature on coefficient alpha in the general case, as well as to the justification of its use as a dependable measuring instrument reliability estimator in special cases and settings resulting under appropriate restrictive conditions, and are illustrated using a numerical example. [ABSTRACT FROM AUTHOR]

Abstract A fundamental algorithm for selecting ranks from a finite subset of an ordered set is Radix Selection. This algorithm requires the data to be given as strings of symbols over an ordered alphabet, e.g., binary expansions of real numbers. Its complexity is measured by the number of symbols that have to be read. In this paper the model of independent data identically generated from a Markov chain is considered. The complexity is studied as a stochastic process indexed by the set of infinite strings over the given alphabet. The orders of mean and variance of the complexity and, after normalization, a limit theorem with a centered Gaussian process as limit are derived. This implies an analysis for two standard models for the ranks: uniformly chosen ranks, also called grand averages, and the worst case rank complexities which are of interest in computer science. For uniform data and the asymmetric Bernoulli model (i.e. memoryless sources), we also find weak convergence for the normalized process of complexities when indexed by the ranks while for more general Markov sources these processes are not tight under the standard normalizations. [ABSTRACT FROM AUTHOR]

Douglas-Rachford method is a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Each of its iterations requires the sequential solution of two proximal subproblems. The aim of this work is to present a fully inexact version of Douglas-Rachford method wherein both proximal subproblems are solved approximately within a relative error tolerance. We also present a semi-inexact variant in which the first subproblem is solved exactly and the second one inexactly. We prove that both methods generate sequences weakly convergent to the solution of the underlying inclusion problem, if any. [ABSTRACT FROM AUTHOR]

The Sierpinski gasket is known to support an exotic stochastic process called the asymptotically one-dimensional diffusion. This process displays local anisotropy, as there is a preferred direction of motion which dominates at the microscale, but on the macroscale we see global isotropy in that the process will behave like the canonical Brownian motion on the fractal. In this paper we analyse the microscale behaviour of such processes, which we call non-fixed point diffusions, for a class of fractals and show that there is a natural limit diffusion associated with the small scale asymptotics. This limit process no longer lives on the original fractal but is supported by another fractal, which is the Gromov-Hausdorff limit of the original set after a shorting operation is performed on the dominant microscale direction of motion. We establish the weak convergence of the rescaled diffusions in a general setting and use this to answer a question raised in [14] about the ultraviolet limit of the asymptotically one-dimensional diffusion process on the Sierpinski gasket. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to investigate a split feasibility problem based on a fixed point algorithm. A weak convergence theorem of solutions is established in the framework of infinite dimensional Hilbert spaces. As an application, a split equality problem is also investigated. [ABSTRACT FROM AUTHOR]

We investigate the mixed joint discrete value distribution and the mixed joint discrete universality for the pair consisting of a rather general form of zeta-function with an Euler product and a periodic Hurwitz zeta-function with transcendental parameter. The common differences of relevant arithmetic progressions are not necessarily the same. Also some generalizations are given. For this purpose, certain arithmetic conditions on the common differences are used. [ABSTRACT FROM AUTHOR]

We prove that every collection of analytic functions (f1(s),... , fr(s)) defined on the right-hand side of the critical strip can be simultaneously approximated by shifts of Hurwitz zeta-functions (ζ(s + iγκh, α1), ... , ζ(s + iγκh, αr)), h > 0, where 0 < γ1 ≤ γ2 ≤ ... are the imaginary parts of nontrivial zeros of the Riemann zeta-function ζ(s). We use the weak form of the Montgomery pair correlation conjecture and the linear independence over ℚ of the set {log(m + αj) : m ∈ ℕ0, j = 1, ... , r}. [ABSTRACT FROM AUTHOR]

We consider reaction-diffusion systems with random rapidly oscillating coefficient. We do not assume any Lipschitz condition for the nonlinear function in the system, so, the uniqueness theorem for the corresponding initial-value problem may not hold for the considered reaction-diffusion system. Under the assumption that the random function is ergodic and statistically homogeneous in space variables we prove that the trajectory attractors of these systems tend in a weak sense to the trajectory attractors of the homogenized reaction-diffusion systems whose coefficient is the average of the corresponding term of the original systems. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some results for the weak convergence of semi-implicit split-step (SISS)methods which are recently developed to solve a class of nonlinear stochastic differential equation with non-Lipschitz drift term. First, we present some moment estimates based on the actual and numerical solutions of stochastic Ginzburg-Landau equations by SISS methods. Then, we show that our theoretical results are consistent with the numerical results that are obtained by performing simulations. Finally, we present the weak convergence rate of SISS methods is approximately 1 with respect to the numerical results. [ABSTRACT FROM AUTHOR]

We provide a process on the space of collections of coalescing cadlag stable paths and show convergence in an appropriate topology for coalescing stable random walks on the integer lattice. [ABSTRACT FROM AUTHOR]

We consider sequences of random variables living in a finite sum of Wiener chaoses. We find necessary and sufficient conditions for convergence in law to a target variable living in the sum of the first two Wiener chaoses. Our conditions hold notably for sequences of multiple Wiener integrals. Malliavin calculus and in particular the Γ-operators are used. Our results extend previous findings by Azmoodeh et al. (2015) and are applied to central and non-central convergence situations. Our methods are applied to investigate stable convergence and we exclude certain classes of random variables as target variables for sequences living in a fixed Wiener chaos. [ABSTRACT FROM AUTHOR]

The estimation of a survival function and its asymptotic behaviour under the random left truncation and right censor model have been extensively studied over the past few decades. In application, the assumption of independence between cases is often not true. To the best of our knowledge, there is no study in the current literature addressing weak convergence of the estimator of the survival function under this model. The current article studies the weak convergence of the Tsai-Jewell-Wang estimator under the positive or negative association, the main results of which are used to analyse a real data set. [ABSTRACT FROM AUTHOR]

The Lerch zeta-function L(λ, α, s) with a transcendental parameter α, or with rational parameters α and λ is universal, i.e., a wide class of analytic functions is approximated by shifts L(λ, α, s + iτ), τ ∈ R. The case of an algebraic irrational α is an open problem. In the paper, it is proved that for all parameters α, 0 < α < 1, and λ, 0 < λ 6 1, including an algebraic irrational α, there exists a closed non-empty set of analytic functions Fα,λ such that every function f ∈ Fα,λ can be approximated by shifts L(λ, α, s + iτ). [ABSTRACT FROM AUTHOR]

Sufficient conditions for the weak convergence of the distributions of the random variables (1 - x)ξx as x → 1- to the limiting gamma-distribution are put forward. The random variable ξx has power-series distribution with radius of convergence 1 and parameter x ∈ (0, 1). Limit theorems for the probabilities P{ξx = k} are proposed. Asymptotic expansions of local probabilities are derived for sums of independent identically distributed variables with the same distribution as ξx in a triangular array with x → 1-. For the corresponding general allocation scheme, local limit theorems for the joint distributions of the occupancies of the cells are obtained. [ABSTRACT FROM AUTHOR]

The numerical approximation of the solution to a stochastic partial differential equation with additive spatial white noise on a bounded domain is considered. The differential operator is assumed to be a fractional power of an integer order elliptic differential operator. The solution is approximated by means of a finite element discretization in space and a quadrature approximation of an integral representation of the fractional inverse from the Dunford-Taylor calculus. For the resulting approximation, a concise analysis of the weak error is performed. Specifically, for the class of twice continuously Fréchet differentiable functionals with second derivatives of polynomial growth, an explicit rate of weak convergence is derived, and it is shown that the component of the convergence rate stemming from the stochasticity is doubled compared to the corresponding strong rate. Numerical experiments for different functionals validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we discuss weak convergence of continuous-time Markov chains to a non-symmetric pure jump process. We approach this problem using Dirichlet forms as well as semimartingales. As an application, we discuss how to approximate a given Markov process by Markov chains. [ABSTRACT FROM AUTHOR]

In this article, we first propose an extended split equality problem which is an extension of the convex feasibility problem, and then introduce a parameter w to establish the fixed point equation system. We show the equivalence of the extended split equality problem and the fixed point equation system. Based on the fixed point equation system, we present a simultaneous iterative algorithm and obtain the weak convergence of the proposed algorithm. Further, by introducing the concept of a G-mapping of a finite family of strictly pseudononspreading mappings {Ti}i=1N, we consider an extended split equality fixed point problem for G-mappings and give a simultaneous iterative algorithm with a way of selecting the stepsizes which do not need any prior information about the operator norms, and the weak convergence of the proposed algorithm is obtained. We apply our iterative algorithms to some convex and nonlinear problems. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we consider the averaging principle for one dimensional stochastic Burgers equation with slow and fast time-scales. Under some suitable conditions, we show that the slow component strongly converges to the solution of the corresponding averaged equation. Meanwhile, when there is no noise in the slow component equation, we also prove that the slow component weakly converges to the solution of the corresponding averaged equation with the order of convergence 1 − r , for any 0 < r < 1 . [ABSTRACT FROM AUTHOR]

Abstract In applications the properties of a stochastic feature often change gradually rather than abruptly, that is: after a constant phase for some time they slowly start to vary. In this paper we discuss statistical inference for the detection and the localization of gradual changes in the jump characteristic of a discretely observed Ito semimartingale. We propose a new measure of time variation for the jump behaviour of the process. The statistical uncertainty of a corresponding estimate is analysed by deriving new results on the weak convergence of a sequential empirical tail integral process and a corresponding multiplier bootstrap procedure. [ABSTRACT FROM AUTHOR]

In this paper, we present two penalty-splitting inspired iteration schemes (PFFSA) and (PFFSA) for hierarchical equilibrium problems in Hilbert space. Based on the Opial-Passty lemma, we propose weak ergodic convergence and weak convergence of the iterative sequences generated by the Forward-Forward algorithm (PFFSA) and the Forward-Backward algorithm (PFFSA), which are proved under quite mild conditions: the bifunction of the two level equilibrium problems are supposed pseudomonotone. For strong convergence, we first add a strong monotonicity condition on the objective bifunction. We present after, a strong convergence result of algorithm (PFFSA) by adding a topological assumption, i.e. the objective bifunction is of class . Some examples are given to illustrate our results. The first example deals with pseudomonotone variational inequalities and convex minimization problem in the upper level problem. In the second one, we propose a convex minimization in the lower-level problem, where strong convergence of (PFFSA) to a minimum point is insured under infcompactness condition for objective function. These convergence results are new and generalize some recent results in this field. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid proximal algorithm with inertial effect is introduced to solve a split variational inclusion problem in real Hilbert spaces. Under mild conditions on the parameters, we establish weak convergence results for the proposed algorithm. Unlike the earlier iterative methods, we do not impose any conditions on the sequence generated by the proposed algorithm. Also, we extend our results to find a common solution of a split variational inclusion problem and a fixed-point problem. Finally, some numerical examples are given to discuss the convergence and superiority of the proposed iterative methods. [ABSTRACT FROM AUTHOR]