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In this paper, we study the full compressible magnetohydrodynamic (MHD) flows with the viscous stress tensor given by the Maxwell law in R 3 . First, local existence of solutions for general large initial data and global existence of solutions for small initial data are shown. Then we obtain the convergence of solutions for the system with the Maxwell law to the corresponding solution for the classical system with the Newtonian law when the relaxation time goes to 0.

A recent article of the first author presented a constructive proof for the Weierstrass approximation theorem for a wide class of weighted spaces of continuous functions defined on [ 0 , ∞ ) , using Chlodovsky's generalization of the Bernstein polynomial operators. Here, we investigate necessary and sufficient conditions related to the uniform boundedness of a sequence of such operators and to their convergence properties when the weight is a classical Freud weight of the form w ( x ) = e − x α , α ≥ 1 .

We study the following fourth order quasilinear Schrodinger equations { Δ 2 u − Δ u + V ( x ) u − 1 2 u Δ ( u 2 ) = f ( u ) , u ∈ H 2 ( R N ) , where Δ 2 u = Δ ( Δ u ) and 3 ≤ N ≤ 6 . For some suitable assumptions on V ∈ C 1 ( R N ) and f ∈ C ( R ) , by developing a new Brezis-Lieb type convergence result, we prove the existence of Nehari-Pohozaev type ground state solutions. Our main results generalize and improve the existing ones, such as Chen et al. (2014) [10] , Liu and Zhao (2019) [26] , and some other related literature.

Our goal is to study fractional powers of a cascade system of partial differential equations. We explicitly calculate the fractional powers of linear operators associated to this type of system and we discuss local solvability of the fractional equation with subcritical nonlinearity. As an example, a cascade system of Schrodinger equation is analyzed. A connection between the fractional system and the original system is established and we prove the convergence of the linear semigroups obtained by the fractional power operator to the original linear semigroup, as the power α approaches 1.

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect condition. The continuity may fail otherwise., Comment: 13 pages

Maximal estimates for Schrodinger means and convergence almost everywhere of sequences of Schrodinger means are studied.

In this paper, we establish approximate representations of solutions to stochastic Volterra integral equations (SVIEs, for short), by virtue of solutions to a family of stochastic differential equations. As an application, we present two algorithms for solving SVIEs: stochastic theta method and splitting method, and prove the global 1/2 order convergence rates in L 2 norm.

In the present paper we introduce a new method to compute approximate analytical solutions for nonlinear differential equations, called the Least Squares Homotopy Perturbation Method. The method is based on the well-known Homotopy Perturbation Method and its main feature is an accelerated convergence compared to the regular Homotopy Perturbation Method. The comparison with previous results emphasizes the high accuracy of the method.

In this article, we are concerned with averaging principle for stochastic hyperbolic–parabolic equations driven by Poisson random measures with slow and fast time-scales. We first establish the existence and uniqueness of weak solutions of the stochastic hyperbolic–parabolic equations. Then, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the stochastic wave equation is an average with respect to the stationary measure of the fast varying process. Finally, we derive the rate of strong convergence for the slow component towards the solution of the averaged equation.

The averaging principle is an important method to extract effective macroscopic dynamic from complex systems with slow component and fast component. This paper concerns the L p -strong convergence of the averaging principle for two-time-scales stochastic partial differential equations (SPDEs) driven by Wiener processes and Poisson jumps. To achieve this, a key step is to show the existence for an invariant measure with exponentially ergodic property for the fast equation, where the dissipative conditions are needed. Furthermore, it is shown that under suitable assumptions the slow component L p -strongly converges to the solution of the averaged equation. The rate of the convergence is also obtained as a byproduct.

A modified alternating direction implicit scheme for the time integration of linear isotropic Maxwell equations with strictly positive conductivity on cuboids is constructed. A key feature of the proposed scheme is its uniform exponential stability, being achieved by coupling the Maxwell system with an additional damped PDE and adding artificial damping to the scheme. The implicit steps in the resulting time integrator further decouple into essentially one-dimensional elliptic problems, requiring only linear complexity. The convergence of the scheme to the solution of the original Maxwell system is analyzed in the abstract time-discrete setting, providing an error bound in a space related to H − 1 .

This paper studies the law of large numbers (LLN) and central limit theorem (TCL) for a class of scaled pure jump Markov processes where the processes state variables are in R d and where jump amplitudes depend on the state variables. Non-explosion property and semi-martingale decomposition are studied first for a class of stochastic processes, allowing to study in a second step these same properties for the scaled pure jump Markov processes we consider characterized by infinitely small jumps and rapid jumps rates. Then, asymptotic behavior is derived and convergence results are obtained.

Using the convex process theory and the majorizing technique, we study the convergence issues of the iterative sequences generated by the extended Newton method for solving the conic inequality system F ≥ C 0 defined by a cone C and a Frechet differentiable function F satisfying the (extended) weak γ-condition. Convergence criterion of the extended Newton method is presented. As an application, we establish, when F is analytic, an extended α-theory similar to Smale's α-theory for nonlinear analytic equations.

Considered in this report is the existence of multi-speed solitary waves of a Klein–Gordon–Schrodinger system with cubic interaction. These solutions behave at large time as a couple of single solitary waves at different speeds. This type of solutions has been investigated for nonlinear Schrodinger system in [15] , [27] . It is obtained by solving the system backward in time around a sequence of approximate multi-speed solitary waves and showing convergence to a solution with the desired property. The new ingredients of the proof are coercivity of the Hessian of the action around each component of the multi-speed solitary waves, energy estimate and finite speed of propagation.

We study convergence almost everywhere of sequences of Schrodinger means. We also replace sequences by uncountable sets.

In the paper we develop some two-level finite element iterative methods and use these methods to solve the stationary incompressible magnetohydrodynamics (MHD) with different Reynolds numbers under some different uniqueness conditions. Firstly, we use the Stokes type iterative method, Newton type iterative method and Oseen type iterative method of m times on a coarse mesh with mesh size H and then solve a linear problem with the Stokes type, Newton type and Oseen type correction of one time on a fine grid with mesh sizes h ≪ H . Furthermore, we analyze the uniform stability and convergence of these methods with respect to Reynolds numbers, mesh sizes h and H and iterative times m. Finally, the stationary incompressible MHD equations with large Reynolds number are researched by the one-level finite element method based on the Oseen type iteration on a fine mesh under a weak uniqueness condition.

This paper studies the singular limits of the non-isentropic compressible magnetohydrodynamic equations for viscous and heat-conductive ideal polytropic flows with magnetic diffusions in a three-dimensional bounded domain as the Mach number goes to zero. Provided that the initial data are well-prepared, we establish the uniform estimates with respect to the Mach number, which gives the convergence from the full compressible magnetohydrodynamic equations to isentropic incompressible magnetohydrodynamic equations.

The Cauchy problem for the one-dimensional generalized Ginzburg–Landau (GGL) equation is considered and the inviscid limit behavior of its solution is proved. That is, the solution of Cauchy problem for the GGL equation converges to the solution of Cauchy problem for the derivative nonlinear Schrodinger equation in the natural space C ( [ 0 , T ] ; H s ) with s > 1 2 , for some T > 0 , if some coefficients tend to zero. Moreover, the convergence holds in C ( [ 0 , T ] ; H 1 ) for any T > 0 if initial data belong to H 2 . Furthermore, the convergence also holds in C ( [ 0 , T ] ; H s ) for any T > 0 if initial data belong to H s ( s > 4 7 ) with some coefficients conditions.

The combined non-relativistic and quasi-neutral limit of two-fluid Euler–Maxwell equations for plasmas is rigorously justified in this paper. For well-prepared initial data, the convergence of the two-fluid Euler–Maxwell system to the compressible Euler equations is proved in the time interval where a smooth solution of the limit problem exists.

The main objective of the paper is to describe asymptotic behaviour of Fourier–Haar coefficients of functions from Marcinkiewicz spaces. We also discuss the Cesaro summability and the almost convergence of sequences related to Fourier–Haar coefficients and generalise a result from [17] . Some analogues of Mercer's theorem for Fourier coefficients are proved.

Various iterative stochastic optimization schemes can be represented as discrete-time Markov processes defined by the nonautonomous equation X t + 1 = T t ( X t , Y t ) , where Y t is an independent sequence and T t is a sequence of mappings. This paper presents a general framework for the study of the stability and convergence of such optimization processes. Some applications are given: the mathematical convergence analysis of two optimization methods, the elitist evolution strategy ( μ + λ ) and the grenade explosion method, is presented.

This paper is concerned with the vanishing capillarity–viscosity limit for the one-dimensional compressible Navier–Stokes–Korteweg system to the Riemann solution of the Euler system that consists of the supposition of two rarefaction waves and a contact discontinuity. It is shown that there exists a family of smooth solutions to the compressible Navier–Stokes–Korteweg system which converge to the Riemann solution away from the initial time t = 0 and the contact discontinuity located at x = 0 , as the coefficients of capillarity, viscosity and heat conductivity tend to zero. Moreover, a uniform convergence rate in terms of the above physical parameters is also obtained. Here, the strengths of both the rarefaction waves and the contact discontinuity are not required to be small.

In this paper, we introduce the notion of generalized hybrid sequences, extending the notion of nonexpansive sequences introduced and studied in our previous work Djafari Rouhani (1981, 1990, 1990, 1997, 2002, 2004, 2002) [2] , [3] , [4] , [5] , [6] , [7] , [8] , and prove ergodic and convergence theorems for such sequences in a Hilbert space H . Subsequently, we apply our results to prove new fixed point theorems for generalized hybrid mappings, first introduced in Kocourek et al. (2010) [14] , Takahashi and Takeuchi (2011) [20] , defined on arbitrary nonempty subsets of H .

In a Hilbert space, we study the convergence of the subgradient method to a common solution of a finite family of variational inequalities and of a finite family of fixed point problems under the presence of computational errors. Most results known in the literature establish the convergence of algorithms, when computational errors are summable. In the present paper, the convergence of the subgradient method is established for nonsummable computational errors. We show that the subgradient method generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant.

General conditions for normalized, time-scaled stochastic integrals of independently scattered Levy random measures to converge to a limit are described. The idea is to provide general conditions to bypass the use of characteristic functions, which can sometimes have tedious calculations, to simplify and shorten the proofs of convergence of infinitely divisible processes, which have previously been done on a case by case basis. It is of particular interest to study both small and large scale asymptotics, since there are many applications, such as modeling internet traffic. The purpose of this paper is to generalize previous results to the greater class of all stochastically continuous (or, more generally, separable in probability) infinitely divisible processes with no Gaussian part and to expand the results to the multi-dimensional case.

This paper studies the average control problem of discrete-time Markov Decision Processes (MDPs for short) with general state space, Feller transition probabilities, and possibly non-compact control constraint sets A ( x ) . Two hypotheses are considered: either the cost function c is strictly unbounded or the multifunctions A r ( x ) = { a ∈ A ( x ) : c ( x , a ) ≤ r } are upper-semicontinuous and compact-valued for each real r . For these two cases we provide new results for the existence of a solution to the average-cost optimality equality and inequality using the vanishing discount approach. We also study the convergence of the policy iteration approach under these conditions. It should be pointed out that we do not make any assumptions regarding the convergence and the continuity of the limit function generated by the sequence of relative difference of the α -discounted value functions and the Poisson equations as often encountered in the literature.

We consider sequential iterative processes for the common fixed point problem of families of cutter operators on a Hilbert space. These are operators that have the property that, for any point x\inH, the hyperplane through Tx whose normal is x-Tx always "cuts" the space into two half-spaces one of which contains the point x while the other contains the (assumed nonempty) fixed point set of T. We define and study generalized relaxations and extrapolation of cutter operators and construct extrapolated cyclic cutter operators. In this framework we investigate the Dos Santos local acceleration method in a unified manner and adopt it to a composition of cutters. For these we conduct convergence analysis of successive iteration algorithms., Comment: Journal of Mathematical Analysis and Applications, accepted for publication

Ostrowski showed that there are intimate connections between the gap structure of a Taylor series and the behaviour of its partial sums outside the disk of convergence. This paper investigates the corresponding problem for the homogeneous polynomial expansion of a harmonic function. The results for harmonic functions display new features in the case of higher dimensions. Science Foundation Ireland This research was supported by Science Foundation Ireland under Grant 09/RFP/MTH2149 DG 23/11/12

Using a geometric approach (in particular, angles between subspaces), we establish useful estimates regarding convergence of infinite products involving orthogonal projections and other (possibly nonlinear) nonexpansive operators in Hilbert space.

Thurston conjectured that hexagonal circle packings can be used to approximate the Riemann mapping. The corresponding convergence was proven by Rodin and Sullivan. He and Schramm showed that for hexagonal circle packings the convergence is C ∞ . Here the C ∞ -convergence is generalized to the case of non-hexagonal circle packings with bounded degree. Furthermore, the estimation of the convergence rate is obtained for arbitrary order derivatives.

In this paper, we establish the strong convergence of possible solutions to the following nonhomogeneous second order evolution system { u ″ ( t ) + c u ′ ( t ) ∈ A u ( t ) + f ( t ) a.e. t ∈ ( 0 , + ∞ ) , u ( 0 ) = u 0 , sup t ⩾ 0 | u ( t ) | + ∞ to an element of A − 1 ( 0 ) , with an exponential rate of convergence when f ≡ 0 , where A is a general maximal monotone operator in a real Hilbert space H , c > 0 is a real constant and f : R + → H is a given function. We show also that the curve u is almost nonexpansive, and present some applications of our result.

International audience; Two splitting procedures for solving equilibrium problems involving the sum of two bifunctions are proposed and their convergence is established under mild assumptions.

The present paper is concerned with the convergence problem of Newton's method to solve singular systems of equations with constant rank derivatives. Under the hypothesis that the derivatives satisfy a type of weak Lipschitz condition, a convergence criterion based on the information around the initial point is established for Newton's method for singular systems of equations with constant rank derivatives. Applications to two special and important cases: the classical Lipschitz condition and the Smale's assumption, are provided; the latter, in particular, extends and improves the corresponding result due to Dedieu and Kim in [J.P. Dedieu, M. Kim, Newton's method for analytic systems of equations with constant rank derivatives, J. Complexity 18 (2002) 187–209].

In this paper, we investigate a multidimensional nonisentropic hydrodynamic (Euler–Poisson) model for semiconductors. We study the convergence of the nonisentropic Euler–Poisson equation to the incompressible nonisentropic Euler type equation via the quasi-neutral limit. The local existence of smooth solutions to the limit equations is proved by an iterative scheme. The method of asymptotic expansion and energy methods are used to rigorously justify the convergence of the limit.

Versions of the Banach Principle for different types of convergence ‘with respect to an ideal’ are established both in the commutative and noncommutative (von Neumann algebraic) context.

In this paper, we prove a strong convergence theorem by the hybrid method for a family of nonexpansive mappings which generalizes Nakajo and Takahashi's theorems [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379], simultaneously. Furthermore, we obtain another strong convergence theorem for the family of nonexpansive mappings by a hybrid method which is different from Nakajo and Takahashi. Using this theorem, we get some new results for a single nonexpansive mapping or a family of nonexpansive mappings in a Hilbert space.

We discuss here the convergence of quantum systems on grids embedded in R d and generalize the earlier results found for scalar-valued potentials to the case of matrix-valued potentials. We also discuss the essential self-adjointness of Schrodinger operators for a large class of matrix potentials and give a Feynman–Kac formula for their associated imaginary time Schrodinger semigroups when the matrix potential is positive and continuous. Furthermore, we establish an operator kernel estimate for the semigroups.

Some pathological properties of the first-return integrals are explored. In particular it is proved that there exist Riemann improper integrable functions which are first-return recoverable almost everywhere, but not first-return integrable, with respect to each trajectory. It is also proved that the usual convergence theorems fail to be true for the first-return integrals.

Mann's algorithm is proved robust in the sense that appropriately small perturbations do not alter the convergence of the algorithm. We prove this for nonexpansive mappings in a Banach space setting, which extends the initial work of Combettes [P.L. Combettes, On the numerical robustness of the parallel projection method in signal synthesis, IEEE Signal Process. Lett. 8 (2001) 45–47] where projections are considered in the Hilbert space framework.

In this paper, we introduce and study a new system of nonlinear A-monotone multivalued variational inclusions in Hilbert spaces. By using the concept and properties of A-monotone mappings, and the resolvent operator technique associated with A-monotone mappings due to Verma, we construct a new iterative algorithm for solving this system of nonlinear multivalued variational inclusions associated with A-monotone mappings in Hilbert spaces. We also prove the existence of solutions for the nonlinear multivalued variational inclusions and the convergence of iterative sequences generated by the algorithm. Our results improve and generalize many known corresponding results.

In this paper we study the mixed summation-integral type operators having Szasz and Beta basis functions. We extend the study of Gupta and Noor [V. Gupta, M.A. Noor, Convergence of derivatives for certain mixed Szasz–Beta operators, J. Math. Anal. Appl. 321 (1) (2006) 1–9] and obtain some direct results in local approximation without and with iterative combinations. In the last section are established direct global approximation theorems.

In this paper the sufficient conditions of existence and uniqueness of the solutions for stochastic pantograph equation are given, i.e., the local Lipschitz condition and the linear growth condition. Under the Lipschitz condition and the linear growth condition it is proved that the semi-implicit Euler method is convergence with strong order 1 2 .

The convergence rate of Fourier–Laplace series in logarithmic subclasses of L 2 ( Σ d ) defined in terms of moduli of continuity is of interest. Lin and Wang [C. Lin, K. Wang, Convergence rate of Fourier–Laplace series of L 2 -functions, J. Approx. Theory 128 (2004) 103–114] recently presented a characterization of those subclasses and provided the almost everywhere convergence rates of Fourier–Laplace series in those subclasses. In this note, the almost everywhere convergence rates of the Cesaro means for Fourier–Laplace series of the logarithmic subclasses are obtained. The strong approximation order of the Cesaro means and the partial summation operators are also presented.

In this paper we study the mixed summation–integral type operators having Szasz and Beta basis functions in summation and integration, respectively, we obtain the rate of point-wise convergence, a Voronovskaja type asymptotic formula and an error estimate in simultaneous approximation.

This work is concerned with a class of neutral functional differential equations with nonlocal history conditions in a Hilbert space. The approximation of solution to a class of such problems is studied. Moreover, the convergence of Faedo–Galerkin approximation of solution is shown. For illustration, an example is worked out.

This paper is for mixed Fourier–Jacobi approximation and its applications to numerical solutions of semiperiodic singular problems, semi-periodic problems on unbounded domains and axisymmetric domains, and exterior problems. The stability and convergence of proposed spectral schemes are proved. Numerical results demonstrate the efficiency of this new approach.  2005 Elsevier Inc. All rights reserved.

Let (M, ρ) be a metric space and τ a Hausdorff topology on M such that {M,τ } is compact. Let S be a right reversible semitopological semigroup and �={ T( s): s ∈ S} a representation of S as ρ-asymptotically nonexpansive type self-mappings of M and u a ρ-bounded almost-orbit of � . We study the τ -convergence of the net {u(s): s ∈ S} in M when the triplet {M,ρ,τ } satisfies various types of τ -Opial conditions. Our results extend and unify many previously known results.  2002 Elsevier Science (USA). All rights reserved.

A Mysovskii-type theorem for Newton's method under (k,p)-Hölder continuous derivative is considered in this paper. For the application studied by others, the new condition is weaker than ones in the literature. Also we prove that the optimal convergent order is p+1 for 0

A generalized version of the Tjon–Wu equation is considered. It describes the evolution of the energy distribution in a model of gas in which simultaneous collisions of many particles are permitted. Using the technique of Zolotarev metrics we show that the stationary solution is exponentially stable.

In this paper, we consider and analyze a new class of projection methods for solving pseudomonotone general variational inequalities using the Wiener–Hopf equations technique. The modified methods converge for pseudomonotone operators. Our proof of convergence is very simple as compared with other methods. The proposed methods include several known methods as special cases.