Search

The aim of this paper is to study the difference gap (in short, D-gap) function and error bounds for a class of the random mixed equilibrium problems in real Hilbert spaces. Firstly, we consider regularized gap functions of the Fukushima type and Moreau-Yosida type. Then difference gap functions are established by using these terms of regularized gap functions. Finally, the global error bounds for random mixed equilibrium problems are also developed. The results obtained in this paper are new and extend some corresponding known results in literatures. Some examples are given for the illustration of our results.

In this paper, we study the bifurcation of a class of two-dimensional stochastic delay differential equations. Firstly, we translate the original system into an It? limiting diffusion system by applying stochastic Taylor expansion, small time delay expansion, polar coordinate transformation, and stochastic averaging procedure. Then we discuss the dynamical bifurcation by analyzing the qualitative changes of invariant measures, and investigate the phenomenological bifurcation by utilizing Fokker-Planck equation. The obtained conclusions are completely new, which generalize and improve some existing results.

In this paper, we obtain Hartman-Wintner and Lyapunov-type inequalities for the three-point fractional boundary value problem of the fractional Liouville-Caputo differential equation of order ? 2 (2; 3]. The results presented in this work are sharper than the existing results in the literature. As an application of the results, the fractional Sturm-Liouville eigenvalue problems have also been presented. Moreover, we examine the nonexistence of the nontrivial solution of the fractional boundary value problem.

It is important to be able to estimate the quadrature error in Gauss rules. Several approaches have been developed, including the evaluation of associated Gauss-Kronrod rules (if they exist), or the associated averaged Gauss and generalized averaged Gauss rules. Integrals with certain integrands can be approximated more accurately by rational Gauss rules than by Gauss rules. This paper introduces associated rational averaged Gauss rules and rational generalized averaged Gauss rules, which can be used to estimate the error in rational Gauss rules. Also rational Gauss-Kronrod rules are discussed. Computed examples illustrate the accuracy of the error estimates determined by these quadrature rules.

Integral equations under uncertainty are utilized to describe different formulations of physical phenomena in nature. This paper aims to obtain analytical and approximate solutions for a class of integral equations under uncertainty. The scheme presented here is based upon the reproducing kernel theory and the fuzzy real-valued mappings. The solution methodology transforms the linear fuzzy integral equation to crisp linear system of integral equations. Several reproducing kernel spaces are defined to investigate the approximate solutions, convergence and the error estimate in terms of uniform continuity. An iterative procedure has been given based on generating the orthonormal bases that rely on Gram-Schmidt process. Effectiveness of the proposed method is demonstrated using numerical experiments. The gained results reveal that the reproducing kernel is a systematic technique in obtaining a feasible solution for many fuzzy problems.

In this paper, we investigate the approximate controllability of retarded impulsive stochastic integro-differential equations driven by fractional Brownian motion(fBm) in Hilbert space. With the help of the resolvent operators, the fixed-point theorem, stochastic analysis and methods of controllability theory, a new set of sufficient conditions for approximate controllability of the integro-differential equations are formulated and proven. An example is provided to illustrate the obtained theory.

In the present paper we use the generalized Euler-Maclaurin summation formula to study the convergence and to solve weakly singular Fredholm and Volterra integral equations. Since these equations have different nature, the proposed convergence analysis for each equation has a different structure. Moreover, as an application of this summation formula, we consider the numerical solution of the fractional ordinary differential equations (FODEs) by transforming FODEs into the associated weakly singular Volterra integral equations of the first kind. Some numerical illustrations are designed to depict the accuracy and versatility of the idea.

The paper studies the problem of construction of optimal interpolation formulas with derivative in the Sobolev space L(m)2 (0,1). Here the interpolation formula consists of the linear combination of values of the function at nodes and values of the first derivative of that function at the end points of the interval [0,1]. For any function of the space L(m)2 (0, 1) the error of the interpolation formulas is estimated by the norm of the error functional in the conjugate space L(m)* 2 (0,1). For this, the norm of the error functional is calculated. Further, in order to find the minimum of the norm of the error functional, the Lagrange method is applied and the system of linear equations for coefficients of optimal interpolation formulas is obtained. It is shown that the order of convergence of the obtained optimal interpolation formulas in the space L(m)2 (0,1) is O(hm). In order to solve the obtained system it is suggested to use the Sobolev method which is based on the discrete analog of the differential operator d2m= dx2m. Using this method in the cases m = 2 and m = 3 the optimal interpolation formulas are constructed. It is proved that the order of convergence of the optimal interpolation formula in the case m = 2 for functions of the space C4(0,1) is O(h4) while for functions of the space L(2)2 (0,1) is O(h2). Finally, some numerical results are presented.

This paper is concerned with a numerical procedure for fractional Volterra integro-differential equations with weakly singular kernels. The fractional derivative is in the Caputo sense. In this study, Bernoulli polynomial of first kind is used and its matrix form is given. Then, the matrix form based on the collocation points is constructed for each term of the problem. Hence, the proposed scheme simplifies the problem to a system of algebraic equations. Error analysis is also investigated. Numerical examples are announced to demonstrate the validity of the method.

This paper studies the existence, uniqueness and the exponential stability in p-th moment of the mild solution of neutral second order stochastic partial differential equations with infinite delay and Poisson jumps. The existence and uniqueness of the mild solution of neutral second order stochastic differential equation is first established by means of Banach fixed point principle and stochastic analysis. The exponential stability in the p-th moment for the mild solution to impulsive neutral stochastic integrodifferential equations with Poisson jump is obtained by establishing an integral inequality.

In this paper, by using a new comparison inequality for order statistics of Gaussian variables, we proved an almost sure central limit theorem for extreme order statistics of stationary Gaussian sequences with covariance rn under the condition rn log n(log log n)1+? = O(1) for some ? > 0. A similar result on intermediate order statistics is also proved for stationary Gaussian sequences. The obtained results improve some of the existing results.

In this paper, a new linear programming formulation of a 1-norm twin support vector regression is proposed whose solution is obtained by solving a pair of dual exterior penalty problems as unconstrained minimization problems using Newton method. The idea of our formulation is to reformulate TSVR as a strongly convex problem by incorporated regularization technique and then derive a new 1-norm linear programming formulation for TSVR to improve robustness and sparsity. Our approach has the advantage that a pair of matrix equation of order equals to the number of input examples is solved at each iteration of the algorithm. The algorithm converges from any starting point and can be easily implemented in MATLAB without using any optimization packages. The efficiency of the proposed method is demonstrated by experimental results on a number of interesting synthetic and real-world datasets.

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

In this paper we study the global dynamics of stochastic predator-prey models with non constant mortality rate and Holling type II response. Concretely, we establish sufficient conditions for the extinction and persistence in the mean of autonomous stochastic model and obtain a critical value between them. Then by constructing appropriate Lyapunov functions, we prove that there is a nontrivial positive periodic solution to the non-autonomous stochastic model. Finally, numerical examples are introduced to illustrate the results developed.

In this paper we first introduce the concept of the *w-product. We then proceed to show that the transform with respect to the Gaussian process for the functional F can be expressed in terms of the *w-product without the concept of the transform. We then proceed to obtain several relationships involving the *w-product.

In this paper we introduce a new algorithm based on viscosity approximation method for solving the generalized multiple-set split feasibility problem (GMSSFP)in an infinite dimensional Hilbert spaces . We establish the strong convergence for the algorithm to find a unique solution of the variational inequality which is the optimality condition for the minimization problem.

In this paper, the fractional optimal control problem for differential system is considered. The fractional time derivative is considered in Riemann-Liouville sense. Constraints on controls are imposed. Necessary and sufficient optimality conditions for the fractional Dirichlet and Neumann problems with the quadratic performance functional are derived. Some examples are analyzed in details.

The aim of this paper is to establish a new version of Ostrowski's type integral inequality. The results are obtained by using a new type of kernel with five sections which improves the approximation errors. Applications to a composite quadrature rule and to Cumulative Distributive Functions are considered

In this paper we provide existence and uniqueness results for linear operator equations of the form (I+Am) x = f , where A is a self-adjoint operator in Hilbert space. Some applications to the study of invertible matrices are also presented.

In this paper, fixed point and variational inclusion problems are investigated based on a proximal-type iterative algorithm. Strong convergence theorems are established in the framework of Hilbert spaces.

The purpose of this paper is to investigate variational inequalities, fixed point problems and generalized mixed equilibrium problems. Anextragradient iterative algorithm is investigated in the framework of Hilbert spaces. Weak convergence theorems for common solutions are established.

This paper is concerned with the relative controllability for a class of dynamical control systems described by semilinear fractional stochastic dierential equations with nonlocal conditions in Hilbert space. Sucient conditions for relative controllability results are obtained using Schaefer's xed point theorem. An example is provided to show the application of our result

The study of ill-posed abstract Volterra equations is a recent subject. In this paper, we investigate equations on the line, continue the research of (a, k)-regularized C-resolvent families, subordination principles, abstract semilinear Volterra integrodifferential equations, and provide several illustrative examples.

In this paper we get estimation of the absolute condition number a Hilbert space operator, which is related with the outer generalized inverse of a given operator.