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An almost Pontryagin space A is an inner product space which admits a direct and orthogonal decomposition of the form A = A > [ + ˙ ] A ≤ with a Hilbert space A > and a finite-dimensional negative semidefinite space A ≤ . A reproducing kernel almost Pontryagin space is an almost Pontryagin space of functions (defined on some nonempty set and taking values in some Krein space), with the property that all point evaluation functionals are continuous. We address two problems. 1° In the presence of degeneracy, it is not possible to reproduce function values as inner products with a kernel function in the usual way. We obtain a natural substitute for a kernel function, and study the relation between spaces and kernel functions in detail. 2° Given an inner product space L of functions, does there exist a reproducing kernel almost Pontryagin space A which contains L isometrically? We characterise those spaces for which the answer is “yes”. We show that, in case of existence, there is a unique such space A which contains L isometrically and densely. Its geometry, in particular its degree of degeneracy, is an important invariant of L . It plays a role in connection with Krein's formula describing generalised resolvents and, thus, in several concrete problems related with the extension theory of symmetric operators. [ABSTRACT FROM AUTHOR]

The reproducing kernel element method is a numerical technique that combines finite element and meshless methods to construct shape functions of arbitrary order and continuity, yet retains the Kronecker- $$\delta $$ property. Central to constructing these shape functions is the construction of global partition polynomials on an element. This paper shows that asymmetric interpolations may arise due to such things as changes in the local to global node numbering and that may adversely affect the interpolation capability of the method. This issue arises due to the use of incomplete polynomials that are subsequently non-affine invariant. This paper lays out the new framework for generating general, symmetric, truly minimal and complete affine invariant global partition polynomials for triangular and tetrahedral elements. Optimal convergence rates were observed in the solution of Kirchhoff plate problems with rectangular domains. [ABSTRACT FROM AUTHOR]

We introduce a reproducing kernel structure for Hilbert spaces of functions where differences of point evaluations are bounded. The associated reproducing kernels are characterized in terms of conditionally negative functions. [ABSTRACT FROM AUTHOR]

This paper presents vibration and buckling analysis of functionally gradedbeams with different boundary conditions, using reproducing kernel particle method(RKPM). Vibration of simple Euler-Bernoullibeam using RKPM is already developed and reported in the literature. Modeling of FGM beams using theoretical method or finite element technique is not evolved with accurate results for power law form of FGM with large power of "n" value so far. Accuracy of the RKPM results is very good and is not sensitive to n value. System of equations of motion is derived using Lagrange's method under the assumption of Euler-Bernoulli beam theory. Boundary conditions of the beam are taken into account using Lagrange multipliers. It is assumed that material properties of the beam vary continuously in the thickness direction according to the power-law form. RKPM is implemented to obtain the equation of motion and consequently natural frequencies and buckling loads of the FGM beam are evaluated. Results are verified for special cases reported in the literature. Considering the displacement of the neutral axis, buckling loads with respect to length and material distribution are evaluated. For the special case of homogenous beam, RKPM matches theoretical evaluation with less than one percent error. [ABSTRACT FROM AUTHOR]

A point-wise evolutionarily coupled reproducing kernel (RK)/finite element (FE) formulation was developed to model fragmentation processes induced by blast and impact loadings. In this coupled approach, each integration point defines a coupling zone so that very localized deformations and damage can be treated. A triggering criterion, such as a damage index for concrete or a critical effective plastic strain for steel, is employed to initiate the fragmentation process, which is realized by changing the morphology of the zone where fragmentation is indicated from a FE model to a meshfree model. The evolutionary coupling, whereby the FE zones can morph into meshfree zones, provides a convenient and more accurate domain from which to track the debris mass and velocity distributions, which is of great interest in many applications. To facilitate the fragmentation modeling, a flexible-to-flexible meshfree contact algorithm was implemented so that high velocity (up to 3000 m/s) impact penetration problems can be effectively analyzed by the coupled Lagrangian RK/FE formulation. Numerical results are shown for structures subjected to quasi-static, blast, and high velocity impact loadings using this coupled formulation and employing widely accepted constitutive models for concrete and steel. The examples show that the coupled formulation can be used to model material fragmentation and the evolution of debris using damage indicators provided by any validated material constitutive model. [ABSTRACT FROM PUBLISHER]

In this paper, an online learning algorithm is proposed as sequential stochastic approximation of a regularization path converging to the regression function in reproducing kernel Hilbert spaces (RKHSs). We show that it is possible to produce the best known strong (RKHS norm) convergence rate of batch learning, through a careful choice of the gain or step size sequences, depending on regularity assumptions on the regression function. The corresponding weak (mean square distance) convergence rate is optimal in the sense that it reaches the minimax and individual lower rates in this paper. In both cases, we deduce almost sure convergence, using Bernstein-type inequalities for martingales in Hilbert spaces. To achieve this, we develop a bias-variance decomposition similar to the batch learning setting; the bias consists in the approximation and drift errors along the regularization path, which display the same rates of convergence, and the variance arises from the sample error analyzed as a (reverse) martingale difference sequence. The rates above are obtained by an optimal tradeoff between the bias and the variance. [ABSTRACT FROM AUTHOR]

Causal discovery via the asymmetry between the cause and the effect has proved to be a promising way to infer the causal direction from observations. The basic idea is to assume that the mechanism generating the cause distribution p(x) and that generating the conditional distribution p(y|x) correspond to two independent natural processes and thus p(x) and p(y|x) fulfill some sort of independence condition. However, in many situations, the independence condition does not hold for the anticausal direction; if we consider p(x, y) as generated via p(y)p(x|y), then there are usually some contrived mutual adjustments between p(y) and p(x|y). This kind of asymmetry can be exploited to identify the causal direction. Based on this postulate, in this letter, we define an uncorrelatedness criterion between p(x) and p(y|x) and, based on this uncorrelatedness, show asymmetry between the cause and the effect in terms that a certain complexity metric on p(x) and p(y|x) is less than the complexity metric on p(y) and p(x|y). We propose a Hilbert space embedding-based method EMD (an abbreviation for EMbeDding) to calculate the complexity metric and show that this method preserves the relativemagnitude of the complexity metric. Based on the complexity metric, we propose an efficient kernelbased algorithm for causal discovery. The contribution of this letter is threefold. It allows a general transformation from the cause to the effect involving the noise effect and is applicable to both one-dimensional and high-dimensional data. Furthermore it can be used to infer the causal ordering for multiple variables. Extensive experiments on simulated and real-world data are conducted to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The mathematical theory of reproducing kernel Hilbert spaces (RKHSs) provides powerful tools for minimum variance estimation (MVE) problems. Here, we extend the classical RKHS-based analysis of MVE in several directions. We develop a geometric formulation of five known lower bounds on the estimator variance (Barankin bound, Cramér-Rao bound, constrained Cramér-Rao bound, Bhattacharyya bound, and Hammersley-Chapman–Robbins bound) in terms of orthogonal projections onto a subspace of the RKHS associated with a given MVE problem. We show that, under mild conditions, the Barankin bound (the tightest possible lower bound on the estimator variance) is a lower semicontinuous function of the parameter vector. We also show that the RKHS associated with an MVE problem remains unchanged if the observation is replaced by a sufficient statistic. Finally, for MVE problems conforming to an exponential family of distributions, we derive novel closed-form lower bounds on the estimator variance and show that a reduction of the parameter set leaves the minimum achievable variance unchanged. [ABSTRACT FROM AUTHOR]

Abstract: Using reproducing kernel particle method (RKPM), concentrically and eccentrically functionally graded stiffened plates (FGSPs) are analyzed based on first order shear deformation theory (FSDT). The plates are subjected to uniformly distributed loads with simply supported and clamped boundary conditions. The interactions between the plate and stiffeners are imposed by compatibility equations. Metal-ceramic composition is assumed as the functionally graded material (FGM). Material properties vary through the thickness direction according to the power law of volume fraction. Mori–Tanaka scheme is used to obtain effective material properties. Poisson’s ratios of plates and stiffeners are taken to be constant. Full transformation approach is used to enforce essential boundary conditions. Effects of eccentricity of the stiffeners, dimensionless support domain parameter, dimensionless thickness, boundary conditions and the volume fractions of the constituents on the behavior of the stiffened plates are investigated. [Copyright &y& Elsevier]

This paper focuses on sparse and irregular longitudinal data with a scalar response. The predictor consists of sparse and irregular observations on predictor trajectories, potentially contaminated with measurement errors. For this type of data, Yao, Müller, & Wang (2005a) proposed a principal components analysis through conditional expectation (PACE) approach, which is capable of predicting each predictor trajectory based on sparse and irregular observations. Nonparametric functional data analysis provides an attractive alternative due to its high flexibility. Early work includes functional additive models as in Müller & Yao (2008) and Ferraty & Vieu (2006), which are mainly based on kernel smoothing methods. In this work, we propose a new functional nonparametric regression framework based on reproducing kernel Hilbert spaces (RKHS). The proposed method involves two steps. The first step is to estimate each predictor trajectory based on sparse and irregular observations using PACE. The second step is to conduct a RKHS-based nonparametric regression using the estimated predictor trajectories. Our approach shows improvement over existing methods in simulation studies as well as in a real data example. The Canadian Journal of Statistics 42: 204-216; 2014 © 2014 Statistical Society of Canada [ABSTRACT FROM AUTHOR]

Functional additive models provide a flexible yet simple framework for regressions involving functional predictors. The utilization of a data-driven basis in an additive rather than linear structure naturally extends the classical functional linear model. However, the critical issue of selecting non-linear additive components has been less studied. In this work, we propose a new regularization framework for structure estimation in the context of reproducing kernel Hilbert spaces. The approach proposed takes advantage of functional principal components which greatly facilitates implementation and theoretical analysis. The selection and estimation are achieved by penalized least squares using a penalty which encourages the sparse structure of the additive components. Theoretical properties such as the rate of convergence are investigated. The empirical performance is demonstrated through simulation studies and a real data application. [ABSTRACT FROM AUTHOR]

A new method of the reproducing kernel Hilbert space is applied to a two-dimensional parabolic inverse source problem with the final overdetermination. The exact and approximate solutions are both obtained in a reproducing kernel space. The approximate solution and its partial derivatives are proved to converge to the exact solution and its partial derivatives, respectively. A technique is proposed to improve some existing methods. Numerical results show that the method is of high precision, and confirm the robustness of our method for reconstructing source parameter. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce the notion of reproducing kernel Hilbert spaces for graphs and the Gram matrices associated with them. Our aim is to investigate the Gram matrices of reproducing kernel Hilbert spaces. We provide several bounds on the entries of the Gram matrices of reproducing kernel Hilbert spaces and characterize the graphs which attain our bounds. [Copyright &y& Elsevier]

This paper is concerned with a technique for solving a class of nonlinear systems of partial differential equations (PDEs) in the reproducing kernel Hilbert space. The analytical solution is represented in the form of series. An iterative method is given to obtain the approximate solution. The convergence analysis is established theoretically. The proposed method is successfully used for solving a coupled system of viscous Burgers’ equations and a nonlinear hyperbolic system. Performance of the method is tested in terms of various error norms. In the case of non-availability of exact solution, it is compared with the existing methods. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, a support vector regression algorithm in the sum of reproducing kernel Hilbert spaces (SVRSS) is proposed for multivariate calibration. In SVRSS, the target regression function is represented as the sum of several single kernel decision functions, where each single kernel function with specific scale can approximate certain component of the target function. For sum spaces with two Gaussian kernels, the proposed method is compared, in terms of RMSEP, to traditional chemometric PLS calibration methods and recent promising SVR, GPR and ELM methods on a simulated data set and four real spectroscopic data sets. Experimental results demonstrate that SVR methods outperform PLS methods for spectroscopy regression problems. Moreover, SVRSS method with multi-scale kernels improves the single kernel SVR method and shows superiority over GPR and ELM methods. [Copyright &y& Elsevier]

This paper is devoted to a new numerical method for fractional Riccati differential equations. The method combines the reproducing kernel method and the quasilinearization technique. Its main advantage is that it can produce good approximations in a larger interval, rather than a local vicinity of the initial position. Numerical results are compared with some existing methods to show the accuracy and effectiveness of the present method. [ABSTRACT FROM AUTHOR]

A numerical method is presented for solving the singularly perturbed multipantograph delay equations with a boundary layer at one end point. The original problem is reduced to boundary layer and regular domain problems. The regular domain problem is solved by combining the asymptotic expansion and the reproducing kernel method (RKM). The boundary layer problem is treated by the method of scaling and the RKM. Two numerical examples are provided to illustrate the effectiveness of the present method. The results from the numerical example show that the present method can provide very accurate analytical approximate solutions. [ABSTRACT FROM AUTHOR]

In this article, we proposed a collocation method based on reproducing kernels to solve a modified anomalous subdiffusion equation problem. We give constructively the [ABSTRACT FROM AUTHOR]

Abstract: In the previous works, the authors presented the reproducing kernel method (RKM) for solving various differential equations. However, to the best of our knowledge, there exist no results for functional differential equations. The aim of this paper is to extend the application of reproducing kernel theory to nonlocal functional differential equations with delayed or advanced arguments, and give the error estimation for the present method. Some numerical examples are provided to show the validity of the present method. [Copyright &y& Elsevier]

Abstract: The objective of this paper is to present a numerical method for solving singularly perturbed turning point problems exhibiting an interior layer. The method is based on the asymptotic expansion technique and the reproducing kernel method (RKM). The original problem is reduced to interior layer and regular domain problems. The regular domain problems are solved by using the asymptotic expansion method. The interior layer problem is treated by the method of stretching variable and the RKM. Four numerical examples are provided to illustrate the effectiveness of the present method. The results of numerical examples show that the present method can provide very accurate approximate solutions. [Copyright &y& Elsevier]

Abstract: Singular integral equations (SIEs) are often encountered in certain contact and fracture problems in solid mechanics. Numerical methods for solving SIEs have been the focus of much research, including reproducing kernel methods. However, there are no reports on reproducing kernel methods for solving differential–integral equations with weakly singular kernels. We developed a reproducing kernel method for solving Fredholm integro-differential equations with weakly singular kernels in reproducing kernel Hilbert space. This involves changing a weakly singular kernel to a logarithm kernel to a Kalman kernel. Weak singularity is removed by applying a smooth transformation to the Kalman kernel. Solution representations are obtained in reproducing kernel Hilbert space. Numerical experiments show that our reproducing kernel method is efficient. [Copyright &y& Elsevier]

We first introduce some related definitions of the bounded linear operator L in the reproducing kernel space Wm2 (D). Then we showspectral analysis of L and derive several property theorems. [ABSTRACT FROM AUTHOR]

The complex variable reproducing kernel particle method (CVRKPM) of solving two-dimensional variable coefficient advection—diffusion problems is presented in this paper. The advantage of the CVRKPM is that the shape function of a two-dimensional problem is formed with a one-dimensional basis function. The Galerkin weak form is employed to obtain the discretized system equation, and the penalty method is used to apply the essential boundary conditions. Then the corresponding formulae of the CVRKPM for two-dimensional variable coefficient advection—diffusion problems are obtained. Two numerical examples are given to show that the method in this paper has greater accuracy and computational efficiency than the conventional meshless method such as reproducing the kernel particle method (RKPM) and the elementfree Galerkin (EFG) method. [ABSTRACT FROM AUTHOR]

Various statements on the distribution of eigenvalues of random matrices are obtained by considering the limiting behavior of the reproducing kernels of a certain naturally associated sequence of orthogonal polynomials. We establish a universal limiting behavior of this type in the case when the underlying measure does not have finite moments. In this case the orthogonal polynomials are replaced by a nested family of de Branges spaces of entire functions. [ABSTRACT FROM AUTHOR]

In this work, we investigate a sequence of approximations converging to the existing unique solution of a multi-point boundary value problem(BVP) given by a linear fourth-order ordinary differential equation with variable coeffcients involving nonlocal integral conditions by using reproducing kernel method(RKM). Obtaining the reproducing kernel of the reproducing kernel space by using the original conditions given directly by RKM may be troublesome and may introduce computational costs. Therefore, in these cases, initially considering more admissible conditions which will allow the reproducing kernel to be computed more easily than the original ones and then taking into account the original conditions lead us to satisfactory results. This analysis is illustrated by a numerical example. The results demonstrate that the method is still quite accurate and effective for the cases with both derivative and integral conditions even if the accuracy is less compared to the cases with just derivative conditions. [ABSTRACT FROM PUBLISHER]

A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions includes the Szegö and the Bergman kernel on the symmetrized polydisc. [ABSTRACT FROM AUTHOR]

We propose a framework for adapting text mining models that discovers low-rank shared concept space. Our major characteristic of this concept space is that it explicitly minimizes the distribution gap between the source domain with sufficient labeled data and the target domain with only unlabeled data, while at the same time it minimizes the empirical loss on the labeled data in the source domain. Our method is capable of conducting the domain adaptation task both in the original feature space as well as in the transformed Reproducing Kernel Hilbert Space (RKHS) using kernel tricks. Theoretical analysis guarantees that the error of our adaptation model can be bounded with respect to the embedded distribution gap and the empirical loss in the source domain. We have conducted extensive experiments on two common text mining problems, namely, document classification and information extraction, to demonstrate the efficacy of our proposed framework. [ABSTRACT FROM AUTHOR]

This paper discusses a class of impulsive nonlocal stochastic functional integrodifferential inclusions in a real separable Hilbert space. The existence of mild solutions of these inclusions is determined under the mixed continuous and Carathéodory conditions by using Bohnenblust-Karlin's fixed point theorem and fractional operators combined with approximation techniques. An example is provided to illustrate the theory. [ABSTRACT FROM AUTHOR]

We investigate the Pick problem for the polydisk and unit ball using dual algebra techniques. Some factorization results for Bergman spaces are used to describe a Pick theorem for any bounded region in Cd. [ABSTRACT FROM AUTHOR]

An evolutionarily coupled finite-element (FE) and reproducing kernel (RK) formulation is implemented in the Karagozian and Case-finite element/mesh-free (KC-FEMFRE) code for modeling concrete behaviors under blast and impact loadings. Mesh-free methods, such as the RK particle method, have the capacity to overcome regularization requirements and numerical instabilities that encumber finite-element methods in large deformation problems, and they are also more naturally suited for problems involving material perforation and fragmentation. To enhance efficiency, a novel approach is developed by coupling the FE approximation with the RK approximation in a controllable and evolutionary fashion. A unique domain integration, stabilized conforming nodal integration, is applied to both FE and RK domains, and therefore the state variables are stored at nodal points directly, and thus no state variable transition is required when mesh conversion is performed. This has a wide range of utility, not only related to efficiency of RK simulations, but by providing a consistent numerical approach for element-to-particle conversion. Moreover, the Karagozian and Case concrete (KCC) model is implemented into the KC-FEMFRE framework, and the internal damage variable of the KCC model is used to evolve mesh-free particles. Quasi-static, blast, and high-speed impact problems are simulated by the coupled formulation, and realistic responses are obtained. [ABSTRACT FROM AUTHOR]

In this paper, a dynamical memory control strategy based on projection technique is proposed for kernel-based online regression. Namely, when an instance is removed from the memory, its contribution will be kept by projecting the regression function onto the subspace expanded instead of throwing it away cheaply. This strategy is composed of incremental and decremental controls. To the former, a new example will be added to the memory if it brings a significant change to the regression function, otherwise discarded by the projection technique. The latter is applied when a new instance is added to the memory, or the memory size has reached a predefined budget. The proposed method is analyzed theoretically and its performance is tested on four benchmark data sets. [ABSTRACT FROM AUTHOR]

In this paper, a new technique is presented to measure dissimilarity in kernel space providing scaling and translation invariance. The motivation comes from signal/image processing, where classifiers are often required to ensure invariance against linear transforms, since in many cases linear transforms do not affect the content of a signal/image for a human observer. We examine the theoretical background of linear invariance in the polynomial kernel space, introduce the centered correlation and centered Euclidean dissimilarity in kernel space, deduce formulas to compute it efficiently and test the proposed dissimilarity measures with the kNN classifier. The experimental results show that the presented techniques are highly competitive in similarity or dissimilarity based classification methods. [ABSTRACT FROM AUTHOR]

Abstract: The theory of symmetric operators has several deep applications to the function theory of certain reproducing kernel Hilbert spaces of analytic functions, as well as to the study of ordinary differential operators in mathematical physics. Examples of simple symmetric operators include multiplication operators on various spaces of analytic functions (model subspaces of Hardy spaces, de Branges–Rovnyak spaces, Herglotz spaces), Sturm–Liouville and Schrodinger differential operators, Toeplitz operators, and infinite Jacobi matrices. In this paper we develop a general representation theory of simple symmetric operators with equal deficiency indices, and obtain a collection of results which refine and extend classical works of Krein and Livsic. In particular, we provide an alternative proof of a theorem of Livsic which characterizes when two simple symmetric operators with equal deficiency indices are unitarily equivalent. Moreover, we provide a new, more easily computable formula for the Livsic characteristic function of a simple symmetric operator. [Copyright &y& Elsevier]

We investigate the effectiveness of reproducing kernel method (RKM) in solving partial differential equations. We propose a reproducing kernel method for solving the telegraph equation with initial and boundary conditions based on reproducing kernel theory. Its exact solution is represented in the form of a series in reproducing kernel Hilbert space. Some numerical examples are given in order to demonstrate the accuracy of this method. The results obtained from this method are compared with the exact solutions and other methods. Results of numerical examples show that this method is simple, effective, and easy to use. [ABSTRACT FROM AUTHOR]

Abstract: This paper proposes a new method for online identification of a nonlinear system modelled on Reproducing Kernel Hilbert Space (RKHS). The proposed SVD–KPCA method uses the Singular Value Decomposition (SVD) technique to update the principal components. Then we use the Reduced Kernel Principal Component Analysis (RKPCA) to approach the principal components which represent the observations selected by the KPCA method. [Copyright &y& Elsevier]

In this paper, we study the Dunkl–Gabor transform on ℝ d . We also give the practical real inversion formulas for this transform using the theory of reproducing kernels. [ABSTRACT FROM PUBLISHER]

In this paper, we obtain a common random fixed point theorem for six weakly compatible random operators defined on a nonempty closed subset of a separable Hilbert space under some conditions. [ABSTRACT FROM AUTHOR]

In this paper, we analyse the equal width (EW) wave equation by using the mesh-free reproducing kernel particle Ritz (kp-Ritz) method. The mesh-free kernel particle estimate is employed to approximate the displacement field. A system of discrete equations is obtained through the application of the Ritz minimization procedure to the energy expressions. The effectiveness of the kp-Ritz method for the EW wave equation is investigated by numerical examples in this paper. [ABSTRACT FROM AUTHOR]

Let K be a smooth Calderón–Zygmund convolution kernel on R2 and suppose that we are given a function . The two-dimensional commutator was shown to be bounded on Lp(R2), p>1 by Christ and Journé [2]. In this article, we show that this operator is also of weak type (1,1). [ABSTRACT FROM PUBLISHER]

We give in terms of Berezin symbols some characterizations for the operators belonging to the Schatten-von Neumann classes , , which were motivated by the questions posed by Nordgren and Rosenthal [7]. Some other problems related with the multiplicative equality for Berezin symbols are also discussed. We also prove in terms of Berezin symbols a Gohberg-Krein type theorem on the weak limit of compact operators. [ABSTRACT FROM AUTHOR]

We show that a modulation space of type Mm²(R) is a reproducing kernel Hilbert space (RKHS). In particular, we explore the special cases of variable bandwidth spaces Aceska and Feichtinger (2011) with a suitably chosen weight to provide strong enough decay in the frequency direction. The reproducing kernel property is valid even if Mm²(R) does not coincide with any of the classical Sobolev spaces because unbounded bandwidth (globally) is allowed. The reproducing kernel will be described explicitly. [ABSTRACT FROM AUTHOR]

For solving electroencephalographic forward problem, coupled method of finite element method (FEM) and fast moving least square reproducing kernel method (FMLSRKM) which is a kind of meshfree method is proposed. Current source modeling for FEM is complicated, so source region is analyzed usingmeshfreemethod. First order of shape function is used for FEM and second order for FMLSRKM because FMLSRKM adopts point collocation scheme. Suggested method is tested using simple equation using 1-, 2-, and 3-dimensional models, and error tendency according to node distance is studied. In addition, electroencephalographic forward problem is solved using spherical head model. Proposed hybrid method can produce well-approximated solution. [ABSTRACT FROM AUTHOR]

We provide a new definition for reproducing kernel space with weighted integral and present a method to construct and calculate the reproducing kernel for the space. The new reproducing kernel space is an enlarged reproducing kernel space, which contains the traditional reproducing kernel space. The proposed method of this paper is a universal method and is suitable for the case of that the weight is variable. Obviously, this new method will generalize a number of applications of reproducing kernel theory to many areas. [ABSTRACT FROM AUTHOR]

In (Wang et al., 2011), we give an iterative reproducing kernel method (IRKM). The main contribution of this paper is to use an IRKM (Wang et al., 2011), in singular perturbation problems with boundary layers. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate that the method is simple and effective. [ABSTRACT FROM AUTHOR]

We present a numerical method to solve the linear Fredholm integro-differential equation in reproducing kernel space. A simple algorithm is given to obtain the approximate solutions of the equation. Through the comparison of approximate and true solution, we can find that the method can effectively solve the linear Fredholm integro-differential equation. At the same time the numerical solution of the equation is stable. [ABSTRACT FROM AUTHOR]

The object of this paper is to obtain a common fixed point theorem for continuous random operators defined on a non-empty closed subset of a separable Hilbert space. [ABSTRACT FROM AUTHOR]