一类二阶差分方程边值问题解的存在性

本文研究了一个二阶差分方程边值问题解的存在性问题.利用临界点理论和变分方法,获得了几个解的存在性结果,推广了一些现有的结果.     

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.     

Vector solitary waves for a class of nonlocal nonlinear Schrödinger system

In this paper, we study a class of one-dimensional nonlocal nonlinear Schrödinger system, which describes two-color optical beams propagating through a cell with nematic liquid crystals. The existence of local and global solutions is derived first upon applying the Strichartz's estimates, conservation laws, and fixed points theorem. Then, we prove the existence of positive normalized vector solitary wave solutions by using variational approach and the concentration-compactness technique.     

New exact solutions of a nonlinear integrable equation

Analytic solutions of the partial differential equations are needed to explain many phenomena seen in thermodynamics, aerodynamics, plasma physics, and other fields. In this paper, variational principle is analyzed of the integrable nonlinear Korteweg–de Vries (KdV) typed equation. In addition, exact solutions of this equation are obtained by using various methods such as direct integration, homogeneous balance method, Exp-function method, and Kudryashov method.     

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.