Journal of Inequalities and Applications最新文献

For inequality constrained optimization problem, we first propose a new smoothing method to the lower order exact penalty function, and then show that an approximate global solution of the original problem can be obtained by solving a global solution of a smooth lower order exact penalty problem. We propose an algorithm based on the smoothed lower order exact penalty function. The global convergence of the algorithm is proved under some mild conditions. Some numerical experiments show the efficiency of the proposed method.

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 [Formula: see text] and a new criterion for the monotonicity of the quotient of two power series.

The aim of present work is to study some kinds of well-posedness for a class of generalized variational-hemivariational inequality problems involving set-valued operators. Some systematic approaches are presented to establish some equivalence theorems between several classes of well-posedness for the inequality problems and some corresponding metric characterizations, which generalize many known results. Finally, the well-posedness for a class of generalized mixed equilibrium problems is also considered.

This work is devoted to discuss some spectral properties and the scattering function of the impulsive operator generated by the Sturm-Liouville equation. We present a different method to investigate the spectral singularities and eigenvalues of the mentioned operator. We also obtain the finiteness of eigenvalues and spectral singularities with finite multiplicities under some certain conditions. Finally, we illustrate our results by a detailed example.

In this paper, we study the smallest constant α in the anisotropic Sobolev inequality of the form ${\parallel u\parallel }_p^p\le \alpha {\parallel u\parallel }_2^{\frac{2(2N-1)+(3-2N)p}{2}}{\parallel u_x\parallel }_2^{\frac{N(p-2)}{2}}\prod _{k=1}^{N-1}{\parallel D_x^{-1}{\partial }_{y_k}u\parallel }_2^{\frac{p-2}{2}}$ and the smallest constant β in the inequality ${\parallel u\parallel }_{p_{\ast }}^{p_{\ast }}\le \beta {\parallel u_x\parallel }_2^{\frac{2N}{2N-3}}\prod _{k=1}^{N-1}{\parallel D_x^{-1}{\partial }_{y_k}u\parallel }_2^{\frac{2}{2N-3}},$ where $V:=(x,y_1,\dots ,y_{N-1})\in R^N$ with $N\ge 3$ and $2<p<p_{\ast }=\frac{2(2N-1)}{2N-3}$ . These constants are characterized by variational methods and scaling techniques. The techniques used here seem to have independent interests.

In this paper, we present four new Windschitl type approximation formulas for the gamma function. By some unique ideas and techniques, we prove that four functions combined with the gamma function and Windschitl type approximation formulas have good properties, such as monotonicity and convexity. These not only yield some new inequalities for the gamma and factorial functions, but also provide a new proof of known inequalities and strengthen known results.

We consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a smooth metric measure space [Formula: see text]: [Formula: see text] 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 [Formula: see text]. In particular, we find that any bounded positive solution of the above equation must be constant under some suitable assumptions.

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.

In this paper, we introduce and study a generalization of the k-Bessel function of order ν given by $W_{\nu ,c}^k\left(x\right):=\sum _{r=0}^{\infty }\frac{{(-c)}^r}{{\Gamma }_k(rk+\nu +k)r!}{\left(\frac{x}{2}\right)}^{2r+\frac{\nu }{k}}.$ We also indicate some representation formulae for the function introduced. Further, we show that the function $W_{\nu ,c}^k$ is a solution of a second-order differential equation. We investigate monotonicity and log-convexity properties of the generalized k-Bessel function $W_{\nu ,c}^k$ , particularly, in the case $c=-1$ . We establish several inequalities, including a Turán-type inequality. We propose an open problem regarding the pattern of the zeroes of $W_{\nu ,c}^k$ .

The alternating direction method of multipliers (ADMM) is one of the most powerful and successful methods for solving convex composite minimization problem. The generalized ADMM relaxes both the variables and the multipliers with a common relaxation factor in $(0,2)$ , which has the potential of enhancing the performance of the classic ADMM. Very recently, two different variants of semi-proximal generalized ADMM have been proposed. They allow the weighting matrix in the proximal terms to be positive semidefinite, which makes the subproblems relatively easy to evaluate. One of the variants of semi-proximal generalized ADMMs has been analyzed theoretically, but the convergence result of the other is not known so far. This paper aims to remedy this deficiency and establish its convergence result under some mild conditions in the sense that the relaxation factor is also restricted into $(0,2)$ .