Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications

In this paper, an exponential inequality for the maximal partial sums of negatively superadditive-dependent (NSD, in short) random variables is established. By using the exponential inequality, we present some general results on the complete convergence for arrays of rowwise NSD random variables, which improve or generalize the corresponding ones of Wang et al. [28] and Chen et al. [2]. In addition, some sufficient conditions to prove the complete convergence are provided. As an application of the complete convergence that we established, we further investigate the complete consistency and convergence rate of the estimator in a nonparametric regression model based on NSD errors.     

Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications

The purpose of this article is to prove strong convergence theorems for common fixed points of two countable families of weak relatively nonexpansive mappings in Banach spaces. In order to get the strong convergence theorems, the monotone hybrid algorithms are presented and are used to approximate the common fixed points. Using this result, we also discuss the problem of strong convergence concerning the maximal monotone operators in a Banach space. The results of this article modify and improve the results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257-266] and the results of Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space, J. Approx. Theory 149 (2007) 103-115] and the results of Su et al. [Y. Su, Z. Wang and H. Xu, Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings, Nonlinear Anal. 71 (2009) 5616-5628], and many others.     

Fractional variational integrators for fractional Euler–Lagrange equations with holonomic constraints

In this paper, the fractional variational integrators developed by Wang and Xiao (2012) [28] are extended to the fractional Euler–Lagrange (E–L) equations with holonomic constraints. The corresponding fractional discrete E–L equations are derived, and their local convergence is discussed. Some fractional variational integrators are presented. The suggested methods are shown to be efficient by some numerical examples.     

A modified halpern-type iteration algorithm for totally quasi-?-asymptotically nonexpansive mappings with applications

The purpose of this article is to modify the Halpern-type iteration algorithm for total quasi-?-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding results of [X.L. Qin, Y.J. Cho, S.M. Kang, H. Y. Zhou, Convergence of a modified Halpern-type iterative algorithm for quasi-?-nonexpansive mappings, Appl. Math. Lett. 22 (2009) 1051-1055], [Z.M. Wang, Y.F. Su, D.X. Wang, Y.C. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011) 2364-2371], [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3906], [C. Martinez-Yanes, H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400-2411] and others.     

Convergence rate of Cesàro means of Fourier-Laplace series

The convergence rate of Fourier-Laplace series in logarithmic subclasses of L²(Σd) defined in terms of moduli of continuity is of interest. Lin and Wang [C. Lin, K. Wang, Convergence rate of Fourier-Laplace series of L²-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 Cesàro means for Fourier-Laplace series of the logarithmic subclasses are obtained. The strong approximation order of the Cesàro means and the partial summation operators are also presented.     

Two new families of sixth-order methods for solving non-linear equations

In this paper, we developed two new families of sixth-order methods for solving simple roots of non-linear equations. Per iteration these methods require two evaluations of the function and two evaluations of the first-order derivatives, which implies that the efficiency indexes of our methods are 1.565. These methods have more advantages than Newton’s method and other methods with the same convergence order, as shown in the illustration examples. Finally, using the developing methodology described in this paper, two new families of improvements of Jarratt method with sixth-order convergence are derived in a straightforward manner. Notice that Kou’s method in [Jisheng Kou, Yitian Li, An improvement of the Jarratt method, Appl. Math. Comput. 189 (2007) 1816-1821] and Wang’s method in [Xiuhua Wang, Jisheng Kou, Yitian Li, A variant of Jarratt method with sixth-order convergence, Appl. Math. Comput. 204 (2008) 14-19] are the special cases of the new improvements.     

Discontinuous Galerkin Methods for Solving Two Membranes Problem

The two membranes problem is known as a free boundary problem, which arises from a variety of applications. In this article we extend the ideas in the article [Wang, Han, and Cheng, SIAM J. Numer. Anal. 48 (2010), 703–733] to use discontinuous Galerkin methods to solve the two membranes problem. A priori error estimates are established, which reach optimal convergence order for linear elements.     

A nonmonotone trust-region method of conic model for unconstrained optimization

In this paper, we present a nonmonotone trust-region method of conic model for unconstrained optimization. The new method combines a new trust-region subproblem of conic model proposed in [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231] with a nonmonotone technique for solving unconstrained optimization. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments are conducted to compare this method with the method of [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231].     

Approximation theorems for total quasi-?-asymptotically nonexpansive mappings with applications

The purpose of this article is to prove some approximation theorems of common fixed points for countable families of total quasi-?-asymptotically nonexpansive mappings which contain several kinds of mappings as its special cases in Banach spaces. In order to get the approximation theorems, the hybrid algorithms are presented and are used to approximate the common fixed points. Using this result, we also discuss the problem of strong convergence concerning the maximal monotone operators in a Banach space. The results of this article extend and improve the results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 134 (2005) 257-266], Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 149 (2007) 103-115], Li, Su [H. Y. Li, Y. F. Su, Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems, Nonlinear Anal. 72(2) (2010) 847-855], Su, Xu and Zhang [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3960], Wang et al. [Z.M. Wang, Y.F. Su, D.X. Wang, Y.C. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011) 2364-2371], Chang et al. [S.S. Chang, H.W. Joseph Lee, Chi Kin Chan, A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications, Nonlinear Anal. 73 (2010) 2260-2270], Chang et al. [S.S. Chang, C.K. Chan, H.W. Joseph Lee, Modified block iterative algorithm for quasi-?-asymptotically nonexpansive mappings and equilibrium problem in Banach spaces, Appl. Math. Comput. 217 (2011) 7520-7530], Ofoedu and Malonza [E.U. Ofoedu, D.M. Malonza, Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type, Appl. Math. Comput. 217 (2011) 6019-6030] and Yao et al. [Y.H. Yao, Y.C. Liou, S.M. Kang, Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings, Appl. Math. Comput. 208 (2009) 211-218].     

Discontinuous Galerkin methods for solving double obstacle problem

In this article the ideas in Wang et al. [SIAM J Numec Anal 48 (2010), 708–73] are extended to solve the double obstacle problem using discontinuous Galerkin methods. A priori error estimates are established for these methods, which reach optimal order for linear elements. We present a test example, and the numerical results on the convergence order match the theoretical prediction. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Optimal control of the primitive equations of the ocean with state constraints

We investigate in this article Pontryagin’s maximum principle for a class of control problems associated with the primitive equations (PEs) of the ocean. These optimal problems involve a state constraint similar to that considered in Wang (2002) [7] for the three-dimensional Navier-Stokes (NS) equations. The main difference between this work and Wang (2002) [7] is that the nonlinearity in the PEs is stronger than in the three-dimensional NS systems.     

Numerical integration over polygons using an eight-node quadrilateral spline finite element

In this paper, a cubature formula over polygons is proposed and analysed. It is based on an eight-node quadrilateral spline finite element [C.-J. Li, R.-H. Wang, A new 8-node quadrilateral spline finite element, J. Comp. Appl. Math. 195 (2006) 54–65] and is exact for quadratic polynomials on arbitrary convex quadrangulations and for cubic polynomials on rectangular partitions. The convergence of sequences of the above cubatures is proved for continuous integrand functions and error bounds are derived. Some numerical examples are given, by comparisons with other known cubatures.     

An iterative approximation method for solving a general system of variational inequality problems and mixed equilibrium problems

In this paper, we introduce an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a general system of variational inequalities for a cocoercive mapping in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets. Our results extend and improve the corresponding results of Ceng, Wang, and Yao [L.C. Ceng, C.Y. Wang, J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res. 67 (2008) 375–390], Ceng and Yao [L.C. Ceng, J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. doi:10.1016/j.cam.2007.02.022], Takahashi and Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515] and many others.     

Numerical Analysis of the Stochastic Moving Boundary Problem

We extend and generalize some recent results on complete convergence (cf. Hu, Moricz, and Taylor [14], Gut [11], Wang, Bhaskara Rao, and Yang [26], Kuczmaszewska and Szynal [17], and Sung [23]) for arrays of rowwise independent Banach space valued random elements. In the main result, no assumptions are made concerning the existence of expected values or absolute moments of the random elements and no assumptions are made concerning the geometry of the underlying Banach space. Some well-known results from the literature are obtained easily as corollaries. The corresponding convergence rates are also established     

Existence of partially regular weak solutions to Landau-Lifshitz-Maxwell equations

In this paper, the authors establish the existence of partially regular weak solutions to the Landau-Lifshitz equations coupling with static Maxwell systems in 3 dimensions by Ginzburg-Landau approximation. It is proved that the Hausdorff measure of the singular set is locally finite. This extends the similar results of Ding and Guo [S. Ding, B. Guo, Hausdorff measure of the singular set of Landau-Lifshitz equations with a nonlocal term, Comm. Math. Phys. 250 (1) (2004) 95-117] from the stationary solutions to weak solutions and the results of Wang [C. Wang, On Landau-Lifshitz equations in dimensions at most four, Indiana Univ. Math. J. 55 (5) (2006) 1615-1644] from Landau-Lifshitz equations to Landau-Lifshitz-Maxwell equations.     

Two valued measure and summability of double sequences in asymmetric context

We extend the ideas of convergence and Cauchy condition of double sequences extended by a two valued measure (called ??-statistical convergence/Cauchy condition and convergence/Cauchy condition in ??-density, studied for real numbers in our recent paper [7]) to a very general structure like an asymmetric (quasi) metric space. In this context it should be noted that the above convergence ideas naturally extend the idea of statistical convergence of double sequences studied by Móricz [15] and Mursaleen and Edely [17]. We also apply the same methods to introduce, for the first time, certain ideas of divergence of double sequences in these abstract spaces. The asymmetry (or rather, absence of symmetry) of asymmetric metric spaces not only makes the whole treatment different from the real case [7] but at the same time, like [3], shows that symmetry is not essential for any result of [7] and in certain cases to get the results, we can replace symmetry by a genuinely asymmetric condition called (AMA).     

离散参数集值（上下）鞅的停时定理

本文建立了更广泛的各种集值（上下）鞅的停时定理；推广并改进了N．S．Papageoriou[10]和张，汪，高[13]中的结果。     

A matrix inequality on Schur complements

We investigate a matrix inequality on Schur complements defined by {1}-generalized inverses, and obtain simultaneously a necessary and sufficient condition under which the inequality turns into an equality. This extends two existing matrix inequalities on Schur complements defined respectively by inverses and Moore-Penrose generalized inverses (see Wang et. al. [Lin. Alg. Appl., 302–303 (1999) 163–172] and Liu and Wang [Lin. Alg. Appl., 293(1999) 233–241]). Moreover, the non-uniqueness of {1}-generalized inverses yields the complicatedness of the extension.     

Existence and continuation of periodic solutions of autonomous Newtonian systems

In this article, we study the existence and the continuation of periodic solutions of autonomous Newtonian systems. To prove the results we apply the infinite-dimensional version of the degree for SO(2)-equivariant gradient operators defined by the third author in Nonlinear Anal. Theory Methods Appl. 23(1) (1994) 83-102 and developed in Topol. Meth. Nonlinear Anal. 9(2) (1997) 383-417. Using the results due to Rabier [Symmetries, Topological degree and a Theorem of Z.Q. Wang, J. Math. 24(3) (1994) 1087-1115] and Wang [Symmetries and calculation of the degree, Chinese Ann. Math. 10 (1989) 520-536] we show that the Leray-Schauder degree is not applicable in the proofs of our theorems, because it vanishes.