一类具变时滞细胞神经网络的全局渐近稳定性

讨论一类具变时滞细胞神经网络平衡点的唯一性和全局渐近稳定性,利用矩阵不等式,通过构造合适的Lyapunov泛函,得到了具变时滞细胞神经网络全局渐近稳定性的新的充分条件．     

ON JENSEN’S INEQUALITY FOR g-EXPECTATION

Briand et al. gave a counterexample showing that given g, Jensen's inequalityfor g-expectation usually does not hold in general. This paper proves that Jensen'sinequality for g-expectation holds in general if and only if the generator g(t,z) issuper-homogeneous in z. In particular, g is not necessarily convex in z.     

Generalized statistics variational perturbation approximation using q-deformed calculus

A principled framework to generalize variational perturbation approximations (VPAs) formulated within the ambit of the nonadditive statistics of Tsallis statistics, is introduced. This is accomplished by operating on the terms constituting the perturbation expansion of the generalized free energy (GFE) with a variational procedure formulated using q-deformed calculus. A candidate q-deformed generalized VPA (GVPA) is derived with the aid of the Hellmann-Feynman theorem. The generalized Bogoliubov inequality for the approximate GFE are derived for the case of canonical probability densities that maximize the Tsallis entropy. Numerical examples demonstrating the application of the q-deformed GVPA are presented. The qualitative distinctions between the q-deformed GVPA model vis-á-vis prior GVPA models are highlighted.     

Converses of the Jensen inequality derived from the Green functions with applications in information theory

Jensen integral inequality has got much importance regarding their applications in different fields of mathematics. In this paper, two converses of Jensen integral inequality for convex function are obtained. The results are applied to establish converses of Hölder and Hermite-Hadamard inequalities as well. At the end, some useful applications in information theory of the obtained results are given. The idea used in this paper may inculcate further research.     

Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space

In a previous work (Adimurthi and Yang, 2010 [2]), Adimurthi–Yang proved a singular Trudinger–Moser inequality in the entire Euclidean space ${\mathbb{R}}^N$$(N\ge 2)$. Precisely, if $0\le \beta <1$ and $0<\gamma \le 1?\beta$, then there holds for any $\tau >0$,

$\underset{u\in W^{1,N}({\mathbb{R}}^N),{\int }_{{\mathbb{R}}^N}(|\mathrm{?}u{|}^N+\tau |u{|}^N)dx\le 1}{\sup }?\underset{{\mathbb{R}}^N}{\int }\frac{1}{|x{|}^{N\beta }}(e^{{\alpha }_N\gamma |u{|}^{\frac{N}{N?1}}}?\sum _{k=0}^{N?2}\frac{{{\alpha }_N}^k{\gamma }^k|u{|}^{\frac{kN}{N?1}}}{k!})dx<\infty ,$

where ${\alpha }_N=N{\omega }_{N?1}^{1/(N?1)}$ and ${\omega }_{N?1}$ is the area of the unit sphere in ${\mathbb{R}}^N$. The above inequality is sharp in the sense that if $\gamma >1?\beta$, all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case $\beta =0$ has been considered by Li and Ruf (2008) [12] and Ishiwata (2011) [11]. We shall investigate the singular case $0<\beta <1$ and prove that for all $\tau >0$, $0<\beta <1$ and $0<\gamma \le 1?\beta$, extremal functions for the above inequalities exist. The proof is based on blow-up analysis.     

A generalized rotationally symmetric case of the centroaffine Minkowski problem

In this paper the centroaffine Minkowski problem, a critical case of the $L_p$-Minkowski problem in the $n+1$ dimensional Euclidean space, is studied. By its variational structure and the method of blow-up analyses, we obtain two sufficient conditions for the existence of solutions, for a generalized rotationally symmetric case of the problem.     

Several logarithmic Caffarelli–Kohn–Nirenberg inequalities and applications

In this paper, based on the Caffarelli–Kohn–Nirenberg inequalities on the Euclidean space and the weighted Hölder inequality, we establish the logarithmic Caffarelli–Kohn–Nirenberg inequalities and parameter type logarithmic Caffarelli–Kohn–Nirenberg inequalities, and give applications for the weighted ultracontractivity of positive strong solutions to a kind of evolution equations. We also prove corresponding logarithmic Caffarelli–Kohn–Nirenberg inequalities and parameter type logarithmic Caffarelli–Kohn–Nirenberg inequalities on the Heisenberg group and related to generalized Baouendi–Grushin vector fields. Some applications are provided.     

Gap Functions and Error Bounds for Weak Generalized Ky Fan Inequalities

In this paper, we study a weak generalized Ky Fan inequality with cone constraints through image space analysis. First, we characterize the separation for the weak generalized Ky Fan inequality with cone constraints using the saddle points of generalized Lagrangian function. Then, we use regular weak separation functions to construct gap functions and regularized gap functions for the weak generalized Ky Fan inequality with cone constraints in a general way, and establish its error bounds in terms of these gap functions.     

Convergence Theorem for Variational Inequality in Hilbert Spaces with Applications

In this paper, we study variational inequality over the set of the common fixed points of a countable family of quasi-nonexpansive mappings. To tackle this problem, we propose an algorithm and use it to prove a strong convergence theorem under suitable conditions. As applications, we study variational inequality over the solution set of different nonlinear or linear problems, like minimization problems, split feasibility problems, convexly pseudoinverse problems, convex linear inverse problems, etc.     

A note on quasi-equilibrium problems

The purpose of this paper is to prove the existence of solutions of quasi-equilibrium problems without any generalized monotonicity assumption. Additionally, we give some applications.