Abstract: Abstract Using blow-up analysis, the author proves the existence of extremal functions for Trudinger–Moser inequalities with nonnegative weights on bounded Euclidean domains or compact Riemannian surfaces. This extends recent results of Yang (J. Differ. Equ. 258:3161–3193, 2015) and Yang–Zhu (Proc. Am. Math. Soc. 145:3953–3959, 2017). PubDate: 2018-05-25

Abstract: Abstract The purpose of this paper is to propose a modified proximal point algorithm for solving minimization problems in Hadamard spaces. We then prove that the sequence generated by the algorithm converges strongly (convergence in metric) to a minimizer of convex objective functions. The results extend several results in Hilbert spaces, Hadamard manifolds and non-positive curvature metric spaces. PubDate: 2018-05-24

Abstract: Abstract In this paper, we introduce a new problem, the modified split generalized equilibrium problem, which extends the generalized equilibrium problem, the split equilibrium problem and the split variational inequality problem. We introduce a new method of an iterative scheme \(\{x_{n}\}\) for finding a common element of the set of solutions of variational inequality problems and the set of common fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of the modified split generalized equilibrium problem without assuming a demicloseness condition and \(T_{\omega }:= (1-\omega)I+ \omega T\) , where T is a quasi-nonexpansive mapping and \(\omega \in ( 0,\frac{1}{2} ) \) ; a difficult proof in the framework of Hilbert space. In addition, we give a numerical example to support our main result. PubDate: 2018-05-22

Abstract: Abstract Consider the linear regression model $$y_{i}=x_{i}^{T}\beta+e_{i},\quad i=1,2, \ldots,n, $$ where \(e_{i}=g(\ldots,\varepsilon_{i-1},\varepsilon_{i})\) are general dependence errors. The Bahadur representations of M-estimators of the parameter β are given, by which asymptotically the theory of M-estimation in linear regression models is unified. As applications, the normal distributions and the rates of strong convergence are investigated, while \(\{\varepsilon_{i},i\in{Z}\}\) are m-dependent, and the martingale difference and \((\varepsilon,\psi)\) -weakly dependent. PubDate: 2018-05-22

Abstract: Abstract We provide several sharp upper and lower bounds for the generalized Euler–Mascheroni constant. As consequences, some previous bounds for the Euler–Mascheroni constant are improved. PubDate: 2018-05-18

Abstract: Abstract In this paper we prove the Hadamard and the Fejér–Hadamard inequalities for the extended generalized fractional integral operator involving the extended generalized Mittag-Leffler function. The extended generalized Mittag-Leffler function includes many known special functions. We have several such inequalities corresponding to special cases of the extended generalized Mittag-Leffler function. Also there we note the known results that can be obtained. PubDate: 2018-05-18

Abstract: Abstract In this paper, we introduce an iterative scheme using the gradient projection method with a new step size, which is not depend on the related matrix inverses and the largest eigenvalue (or the spectral radius of the self-adjoint operator) of the related matrix, based on Moudafi’s viscosity approximation method for solving the split feasibility problem (SFP), which is to find a point in a given closed convex subset of a real Hilbert space such that its image under a bounded linear operator belongs to a given closed convex subset of another real Hilbert space. We suggest and analyze this iterative scheme under some appropriate conditions imposed on the parameters such that another strong convergence theorems for the SFP are obtained. The results presented in this paper improve and extend the main results of Tian and Zhang (J. Inequal. Appl. 2017:Article ID 13, 2017), and Tang et al. (Acta Math. Sci. 36B(2):602–613, 2016) (in a single-step regularized method) with a new step size, and many others. The examples of the proposed SFP are also shown through numerical results. PubDate: 2018-05-18

Abstract: Abstract This paper studies the admissibility of simultaneous prediction of actual and average values of the regressand in the generalized linear regression model under the quadratic loss function. Necessary and sufficient conditions are derived for the simultaneous prediction to be admissible in classes of homogeneous and nonhomogeneous linear predictors, respectively. PubDate: 2018-05-16

Abstract: Abstract We present a variant of the classical integration by parts to introduce a new type of Taylor series expansion and to present some closed forms for integrals involving Jacobi and Laguerre polynomials, which cannot be directly obtained by usual symbolic computation programs, i.e., only some very specific values can be computed by the mentioned programs. An error analysis is given in the sequel for the introduced expansion. PubDate: 2018-05-15

Abstract: Abstract For large-scale unconstrained optimization problems and nonlinear equations, we propose a new three-term conjugate gradient algorithm under the Yuan–Wei–Lu line search technique. It combines the steepest descent method with the famous conjugate gradient algorithm, which utilizes both the relevant function trait and the current point feature. It possesses the following properties: (i) the search direction has a sufficient descent feature and a trust region trait, and (ii) the proposed algorithm globally converges. Numerical results prove that the proposed algorithm is perfect compared with other similar optimization algorithms. PubDate: 2018-05-11

Abstract: Abstract Using weight coefficients, a complex integral formula, and Hermite–Hadamard’s inequality, we give an extended reverse Hardy–Hilbert’s inequality in the whole plane with multiparameters and a best possible constant factor. Equivalent forms and a few particular cases are considered. PubDate: 2018-05-11

Abstract: Abstract In this paper, we concern stability of numerical methods applied to stochastic delay integro-differential equations. For linear stochastic delay integro-differential equations, it is shown that the mean-square stability is derived by the split-step backward Euler method without any restriction on step-size, while the Euler–Maruyama method could reproduce the mean-square stability under a step-size constraint. We also confirm the mean-square stability of the split-step backward Euler method for nonlinear stochastic delay integro-differential equations. The numerical experiments further verify the theoretical results. PubDate: 2018-05-11

Abstract: Abstract We consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a smooth metric measure space \((M, g,e^{-f}\,dv)\) : $$\Delta_{f} u+au\log u+bu=0, $$ where a, b are two real constants. When the ∞-Bakry–Émery Ricci curvature is bounded from below, we obtain a global gradient estimate which is not dependent on \( \nabla f \) . In particular, we find that any bounded positive solution of the above equation must be constant under some suitable assumptions. PubDate: 2018-05-10

Abstract: Abstract In this paper, we first introduce the definition of triple Diamond-Alpha integral for functions of three variables. Therefore, we present the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral on time scales, and then we obtain some new generalizations of the Hölder and reverse Hölder inequalities for the triple Diamond-Alpha integral. Moreover, using the obtained results, we give a new generalization of the Minkowski inequality for the triple Diamond-Alpha integral on time scales. PubDate: 2018-05-10

Abstract: Abstract The present study is concerned with the following fractional p-Laplacian equation involving a critical Sobolev exponent of Kirchhoff type: $$\biggl[a+b \biggl( \int_{\mathbb {R}^{2N}}\frac{ u(x)-u(y) ^{p}}{ x-y ^{N+ps}}\,dx\,dy \biggr)^{\theta-1} \biggr](-\Delta)_{p}^{s}u = u ^{p_{s}^{*}-2}u+\lambda f(x) u ^{q-2}u \quad\text{in } \mathbb {R}^{N}, $$ where \(a,b>0\) , \(\theta=(N-ps/2)/(N-ps)\) and \(q\in(1,p)\) are constants, and \((-\Delta)_{p}^{s}\) is the fractional p-Laplacian operator with \(0< s<1<p<\infty\) and \(ps< N\) . For suitable \(f(x)\) , the above equation possesses at least two nontrivial solutions by variational method for any \(a,b>0\) . Moreover, we regard \(a>0\) and \(b>0\) as parameters to obtain convergent properties of solutions for the given problem as \(a\searrow0^{+}\) and \(b\searrow0^{+}\) , respectively. PubDate: 2018-05-10

Abstract: Abstract Based on the extreme value conditions of a multiple variables function, a new class of Wirtinger-type double integral inequality is established in this paper. The proposed inequality generalizes and refines the classical Wirtinger-based integral inequality and has less conservatism in comparison with Jensen’s double integral inequality and other double integral inequalities in the literature. Thus, the stability criteria for delayed control systems derived by the proposed refined Wirtinger-type integral inequality are less conservative than existing results in the literature. PubDate: 2018-05-09

Abstract: Abstract This paper focuses on a class of nonlinear optimization subject to linear inequality constraints with unavailable-derivative objective functions. We propose a derivative-free trust-region methods with interior backtracking technique for this optimization. The proposed algorithm has four properties. Firstly, the derivative-free strategy is applied to reduce the algorithm’s requirement for first- or second-order derivatives information. Secondly, an interior backtracking technique ensures not only to reduce the number of iterations for solving trust-region subproblem but also the global convergence to standard stationary points. Thirdly, the local convergence rate is analyzed under some reasonable assumptions. Finally, numerical experiments demonstrate that the new algorithm is effective. PubDate: 2018-05-09

Abstract: Abstract In this paper, based on the Rosenthal-type inequality for asymptotically negatively associated random vectors with values in \(\mathbb{R}^{d}\) , we establish results on \(L_{p}\) -convergence and complete convergence of the maximums of partial sums are established. We also obtain weak laws of large numbers for coordinatewise asymptotically negatively associated random vectors with values in \(\mathbb{R}^{d}\) . PubDate: 2018-05-08

Abstract: Abstract In this paper, new bounds for the exponential function with cotangent are found by using the recurrence relation between coefficients in the expansion of power series of the function \(\ln (1-2x^{2}/15-px^{6})\) and a new criterion for the monotonicity of the quotient of two power series. PubDate: 2018-05-08

Abstract: Abstract We consider a kind of nonsmooth optimization problems with \(l_{1}\) -norm minimization, which has many applications in compressed sensing, signal reconstruction, and the related engineering problems. Using smoothing approximate techniques, this kind of nonsmooth optimization problem can be transformed into a general unconstrained optimization problem, which can be solved by the proposed smoothing modified three-term conjugate gradient method. The smoothing modified three-term conjugate gradient method is based on Polak–Ribière–Polyak conjugate gradient method. For the Polak–Ribière–Polyak conjugate gradient method has good numerical properties, the proposed method possesses the sufficient descent property without any line searches, and it is also proved to be globally convergent. Finally, the numerical experiments show the efficiency of the proposed method. PubDate: 2018-05-03