The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generali...The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .展开更多
In this paper,we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit.As an application,we obtain the convergence of random attractors ...In this paper,we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit.As an application,we obtain the convergence of random attractors for non-autonomous stochastic reactiondiffusion equations on unbounded domains,when the density of stochastic noises approaches zero.The weak convergence of solutions is proved by means of Alaoglu weak compactness theorem.A differentiability condition on nonlinearity is omitted,which implies that the existence conditions for random attractors are sufficient to ensure their upper semi-continuity.These results greatly strengthen the upper semi-continuity notion that has been developed in the literature.展开更多
L_r convergence and convergence in probability for weighted sums of L_q-mixingale arrays have been discussed and the Marcinkiewicz type weak law of large numbers for L_q-mixingale arrays has been obtained.
The paper develops the local convergence of Inexact Newton-Like Method(INLM)for approximating solutions of nonlinear equations in Banach space setting.We employ weak Lipschitz and center-weak Lipschitz conditions to p...The paper develops the local convergence of Inexact Newton-Like Method(INLM)for approximating solutions of nonlinear equations in Banach space setting.We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis.The obtained results compare favorably with earlier ones such as[7,13,14,18,19].A numerical example is also provided.展开更多
In this paper, the complete convergence and weak law of large numbers are established for ρ-mixing sequences of random variables. Our results extend and improve the Baum and Katz complete convergence theorem and the ...In this paper, the complete convergence and weak law of large numbers are established for ρ-mixing sequences of random variables. Our results extend and improve the Baum and Katz complete convergence theorem and the classical weak law of large numbers, etc. from independent sequences of random variables to ρ-mixing sequences of random variables without necessarily adding any extra conditions.展开更多
In this article, a class of weak Orlicz function spaces is defined and their basic properties are discused. In particular, for the sequences in weak Orlicz space, we establish several basic convergence theorems includ...In this article, a class of weak Orlicz function spaces is defined and their basic properties are discused. In particular, for the sequences in weak Orlicz space, we establish several basic convergence theorems including bounded convergence theorem, control convergence theorem and Vitali-type convergence theorem and so on. Moreover, the conditional compactness of its subsets is also discussed.展开更多
In this paper, we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-stro...In this paper, we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-strongly monotone operator and the set of common fixed points of two infinite families of relatively nonexpansive mappings or the set of common fixed points of an infinite family of relatively quasi-nonexpansive mappings in Banach spaces. Then we study the weak convergence of the two iterative sequences. Our results improve and extend the results announced by many others.展开更多
Let E be a real uniformly convex and smooth Banach space, and K be a nonempty closed convex subset of E with P as a sunny nonexpansive retraction. Let T1, T2 ： K → E be two weakly inward nonself asymptotically nonex...Let E be a real uniformly convex and smooth Banach space, and K be a nonempty closed convex subset of E with P as a sunny nonexpansive retraction. Let T1, T2 ： K → E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with a sequence {kn^（i）} [1, ∞） （i = 1, 2）, and F ：= F（T1）∩ F（T2） ≠ 0. An iterative sequence for approximation common fixed points of the two nonself asymptotically nonexpansive mappings is discussed. If E has also a Frechet differentiable norm or its dual E^＊ has Kadec-Klee property, then weak convergence theorems are obtained.展开更多
The purpose of this paper is to prove a new weak convergence theorem for a finite family of asymptotically nonexpansive mappings in uniformly convex Banach space.
A special approximation to Rosenblatt process with the finite-time interval representation was obtained. The construction of approximation family was based on the Poisson process. The proof to the approximation was di...A special approximation to Rosenblatt process with the finite-time interval representation was obtained. The construction of approximation family was based on the Poisson process. The proof to the approximation was divided into two aspects. Firstly, the approximation family was tight using the methods given by Billingsley; secondly, the finite-dimension distributions of approximation family converged weakly to the Rosenblatt process by proving the convergence of the corresponding characteristic functions.展开更多
We consider a generalization of Baum-Katz theorem for random variables satisfying some cover conditions.Consequently,we get the results for many dependent structures,such as END,ϱ^(*)mixing,ϱ^(-)mixing andφ-mixing,etc.
In Hashim and Harfash(Appl.Math.Comput.2021),using a finite element method,the attraction-repulsion chemotaxis model(P)in space is discretised;finite differences were used to do the same in time.Furthermore,the existe...In Hashim and Harfash(Appl.Math.Comput.2021),using a finite element method,the attraction-repulsion chemotaxis model(P)in space is discretised;finite differences were used to do the same in time.Furthermore,the existence of a global weak solution to the system(PΔt M)was demonstrated by means of analysis of the convergence of the fully discrete approximate problem(P h,Δt M, ).Moreover,the functions{U,Z,V}were proved to represent a global weak solution to the system(PΔt M)by means of a passage to the limit ,h→0 of the approximate system.This paper’s purpose is to demonstrate that the solutions can be bounded,independent of M.The analysis contained in this paper illustrates the idea of the existence of weak solutions to the model(P),that requires passing to the limits,Δt→0+and M→∞.The time stepΔt is subsequently linked to the cutoff parameter M>1 by positing a demand thatΔt=o(M−1),as M→∞,with the result that the cutoff parameter becomes the only parameter in the problem(PΔt M).The solutions can be bounded,independ-ent of M,with the use of special energy estimates,as demonstrated herein.Then,these M-independent bounds on the relative entropy are employed with the purpose of deriving M-independent bounds on the time-derivatives.Additionally,compactness arguments were utilised to explore the convergence of the finite element approximate problem.The conclu-sion was that a weak solution for(P)existed.Finally,we introduced the error estimate and the implicit scheme was used to perform simulations in one and two space dimensions.展开更多
In this paper,a finite element scheme for the attraction-repulsion chemotaxis model is analyzed.We introduce a regularized problem of the truncated system.Then we obtain some a priori estimates of the regularized func...In this paper,a finite element scheme for the attraction-repulsion chemotaxis model is analyzed.We introduce a regularized problem of the truncated system.Then we obtain some a priori estimates of the regularized functions,independent of the regularization parameter,via deriving a well-defined entropy inequality of the regularized problem.Also,we propose a practical fully discrete finite element approximation of the regularized problem.Next,we use a fixed point theorem to show the existence of the approximate solutions.Moreover,a discrete entropy inequality and some stability bounds on the solutions of regularized problem are derived.In addition,the uniqueness of the fully discrete approximations is preformed.Finally,we discuss the convergence to the fully discrete problem.展开更多
The aim of this paper is to study the weak integral convergence of Kergin interpolation. The results of the weighted integral convergence and the weighted (partial) derivatives integral convergence of Kergin interpola...The aim of this paper is to study the weak integral convergence of Kergin interpolation. The results of the weighted integral convergence and the weighted (partial) derivatives integral convergence of Kergin interpolation polynomial for the smooth functions on the unit disk were obtained in the paper. Those generalized Liang's main results were acquired in 1998 to the more extensive situation. At the same time, the estimation of convergence rate of Kergin interpolation polynomial is given by means of introducing a new kind of smooth norm.展开更多
The main aim of this article is to prove that the maximal operator σ^k＊ of the Marcinkiewicz-Fejer means of the two-dimensional Fourier series with respect to Walsh- Kaczmarz system is bounded from the Hardy space H...The main aim of this article is to prove that the maximal operator σ^k＊ of the Marcinkiewicz-Fejer means of the two-dimensional Fourier series with respect to Walsh- Kaczmarz system is bounded from the Hardy space H2/3 to the space weak-L2/3.展开更多
The almost convergent function which was introduced by Raimi [6] and discussed by Ho [4], Das and Nanda [2, 3], is the continuous analogue of almost convergent sequences （see [5]）. In this paper, we establish the Ta...The almost convergent function which was introduced by Raimi [6] and discussed by Ho [4], Das and Nanda [2, 3], is the continuous analogue of almost convergent sequences （see [5]）. In this paper, we establish the Tauberian conditions and the Cauchy criteria for weak almost convergent functions on R2＋ .展开更多
The Rayleigh–Taylor instability（RTI） in cylindrical geometry is investigated analytically through a second-order weakly nonlinear（WN） theory considering the Bell–Plesset（BP） effect. The governing equations for...The Rayleigh–Taylor instability（RTI） in cylindrical geometry is investigated analytically through a second-order weakly nonlinear（WN） theory considering the Bell–Plesset（BP） effect. The governing equations for the combined perturbation growth are derived. The WN solutions for an exponentially convergent cylinder are obtained. It is found that the BP and RTI growths are strongly coupled, which results in the bubble-spike asymmetric structure in the WN stage. The large Atwood number leads to the large deformation of the convergent interface. The amplitude of the spike grows faster than that of the bubble especially for large mode number m and large Atwood number A. The averaged interface radius is small for large mode number perturbation due to the mode-coupling effect.展开更多
In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transforma...In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1,1],so that the Jacobi orthogonal polynomial theory can be applied conveniently.In order to obtain high order accuracy for the approximation,the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules.In the end,we provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both the L^(∞)-norm and the weighted L^(2)-norm.展开更多
文摘The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .
文摘In this paper,we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit.As an application,we obtain the convergence of random attractors for non-autonomous stochastic reactiondiffusion equations on unbounded domains,when the density of stochastic noises approaches zero.The weak convergence of solutions is proved by means of Alaoglu weak compactness theorem.A differentiability condition on nonlinearity is omitted,which implies that the existence conditions for random attractors are sufficient to ensure their upper semi-continuity.These results greatly strengthen the upper semi-continuity notion that has been developed in the literature.
文摘L_r convergence and convergence in probability for weighted sums of L_q-mixingale arrays have been discussed and the Marcinkiewicz type weak law of large numbers for L_q-mixingale arrays has been obtained.
文摘The paper develops the local convergence of Inexact Newton-Like Method(INLM)for approximating solutions of nonlinear equations in Banach space setting.We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis.The obtained results compare favorably with earlier ones such as[7,13,14,18,19].A numerical example is also provided.
文摘In this paper, the complete convergence and weak law of large numbers are established for ρ-mixing sequences of random variables. Our results extend and improve the Baum and Katz complete convergence theorem and the classical weak law of large numbers, etc. from independent sequences of random variables to ρ-mixing sequences of random variables without necessarily adding any extra conditions.
基金Supported by Hubei Research Center for Financial Development and Financial Security(2008D029)
文摘In this article, a class of weak Orlicz function spaces is defined and their basic properties are discused. In particular, for the sequences in weak Orlicz space, we establish several basic convergence theorems including bounded convergence theorem, control convergence theorem and Vitali-type convergence theorem and so on. Moreover, the conditional compactness of its subsets is also discussed.
文摘In this paper, we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-strongly monotone operator and the set of common fixed points of two infinite families of relatively nonexpansive mappings or the set of common fixed points of an infinite family of relatively quasi-nonexpansive mappings in Banach spaces. Then we study the weak convergence of the two iterative sequences. Our results improve and extend the results announced by many others.
文摘Let E be a real uniformly convex and smooth Banach space, and K be a nonempty closed convex subset of E with P as a sunny nonexpansive retraction. Let T1, T2 ： K → E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with a sequence {kn^（i）} [1, ∞） （i = 1, 2）, and F ：= F（T1）∩ F（T2） ≠ 0. An iterative sequence for approximation common fixed points of the two nonself asymptotically nonexpansive mappings is discussed. If E has also a Frechet differentiable norm or its dual E^＊ has Kadec-Klee property, then weak convergence theorems are obtained.
文摘The purpose of this paper is to prove a new weak convergence theorem for a finite family of asymptotically nonexpansive mappings in uniformly convex Banach space.
基金National Natural Science Foundation of China(No. 11171062)Innovation Program of Shanghai Municipal Education Commission,China(No. 12ZZ063)Natural Science Foundation of Bengbu College,China(No. 2010ZR10)
文摘A special approximation to Rosenblatt process with the finite-time interval representation was obtained. The construction of approximation family was based on the Poisson process. The proof to the approximation was divided into two aspects. Firstly, the approximation family was tight using the methods given by Billingsley; secondly, the finite-dimension distributions of approximation family converged weakly to the Rosenblatt process by proving the convergence of the corresponding characteristic functions.
基金Supported by the National Natural Science Foundation of China(11701403).
文摘We consider a generalization of Baum-Katz theorem for random variables satisfying some cover conditions.Consequently,we get the results for many dependent structures,such as END,ϱ^(*)mixing,ϱ^(-)mixing andφ-mixing,etc.
文摘In Hashim and Harfash(Appl.Math.Comput.2021),using a finite element method,the attraction-repulsion chemotaxis model(P)in space is discretised;finite differences were used to do the same in time.Furthermore,the existence of a global weak solution to the system(PΔt M)was demonstrated by means of analysis of the convergence of the fully discrete approximate problem(P h,Δt M, ).Moreover,the functions{U,Z,V}were proved to represent a global weak solution to the system(PΔt M)by means of a passage to the limit ,h→0 of the approximate system.This paper’s purpose is to demonstrate that the solutions can be bounded,independent of M.The analysis contained in this paper illustrates the idea of the existence of weak solutions to the model(P),that requires passing to the limits,Δt→0+and M→∞.The time stepΔt is subsequently linked to the cutoff parameter M>1 by positing a demand thatΔt=o(M−1),as M→∞,with the result that the cutoff parameter becomes the only parameter in the problem(PΔt M).The solutions can be bounded,independ-ent of M,with the use of special energy estimates,as demonstrated herein.Then,these M-independent bounds on the relative entropy are employed with the purpose of deriving M-independent bounds on the time-derivatives.Additionally,compactness arguments were utilised to explore the convergence of the finite element approximate problem.The conclu-sion was that a weak solution for(P)existed.Finally,we introduced the error estimate and the implicit scheme was used to perform simulations in one and two space dimensions.
文摘In this paper,a finite element scheme for the attraction-repulsion chemotaxis model is analyzed.We introduce a regularized problem of the truncated system.Then we obtain some a priori estimates of the regularized functions,independent of the regularization parameter,via deriving a well-defined entropy inequality of the regularized problem.Also,we propose a practical fully discrete finite element approximation of the regularized problem.Next,we use a fixed point theorem to show the existence of the approximate solutions.Moreover,a discrete entropy inequality and some stability bounds on the solutions of regularized problem are derived.In addition,the uniqueness of the fully discrete approximations is preformed.Finally,we discuss the convergence to the fully discrete problem.
文摘The aim of this paper is to study the weak integral convergence of Kergin interpolation. The results of the weighted integral convergence and the weighted (partial) derivatives integral convergence of Kergin interpolation polynomial for the smooth functions on the unit disk were obtained in the paper. Those generalized Liang's main results were acquired in 1998 to the more extensive situation. At the same time, the estimation of convergence rate of Kergin interpolation polynomial is given by means of introducing a new kind of smooth norm.
基金supported by project TMOP-4.2.2.A-11/1/KONV-2012-0051,Shota Rustaveli National Science Foundation grant no.13/06(Geometry of function spaces,interpolation and embedding theorems)
文摘The main aim of this article is to prove that the maximal operator σ^k＊ of the Marcinkiewicz-Fejer means of the two-dimensional Fourier series with respect to Walsh- Kaczmarz system is bounded from the Hardy space H2/3 to the space weak-L2/3.
文摘The almost convergent function which was introduced by Raimi [6] and discussed by Ho [4], Das and Nanda [2, 3], is the continuous analogue of almost convergent sequences （see [5]）. In this paper, we establish the Tauberian conditions and the Cauchy criteria for weak almost convergent functions on R2＋ .
基金Supported by the National Natural Science Foundation of China under Grant Nos 11275031,11475034,11575033 and 11274026the National Basic Research Program of China under Grant No 2013CB834100
文摘The Rayleigh–Taylor instability（RTI） in cylindrical geometry is investigated analytically through a second-order weakly nonlinear（WN） theory considering the Bell–Plesset（BP） effect. The governing equations for the combined perturbation growth are derived. The WN solutions for an exponentially convergent cylinder are obtained. It is found that the BP and RTI growths are strongly coupled, which results in the bubble-spike asymmetric structure in the WN stage. The large Atwood number leads to the large deformation of the convergent interface. The amplitude of the spike grows faster than that of the bubble especially for large mode number m and large Atwood number A. The averaged interface radius is small for large mode number perturbation due to the mode-coupling effect.
基金supported by the State Key Program of National Natural Science Foundation of China(11931003)the National Natural Science Foundation of China(41974133,11671157)。
文摘In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1,1],so that the Jacobi orthogonal polynomial theory can be applied conveniently.In order to obtain high order accuracy for the approximation,the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules.In the end,we provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both the L^(∞)-norm and the weighted L^(2)-norm.
