Abstract differential operators generating fractional resolvent families

In the paper under review,we consider the generation of fractional resolvent families by abstract differential operators.Our results can be simply incorporated in the study of corresponding abstract time-fractional equations with Caputo fractional derivatives.     

A generalized Gronwall-Bellman-Ou-Iang type inequality for discontinuous functions and applications to BVP

In this paper we investigate a generalized Gronwall-Bellman-Ou-Iang type inequality for piecewise continuous functions. By studying the unknown function in different continuous regions Ω_kj of Ω, we give the complete estimate for the unknown function. We apply our result to a boundary value problem of a partial impulsive differential equation for estimation of its solution.     

The complex inversion formula in UMD spaces for families of bounded operators

In this article, we prove that in UMD Banach spaces the complex inversion formula of the Laplace transform is valid, in the strong sense, for wide classes of families of bounded linear operators. Our approach allows us to recover (in a unified way) known results about C ₀-semigroups, cosine functions and resolvent families as well as to prove new results for k-convoluted semigroups and integrated semigroups, among others.     

Lyapunov-type inequality for a class of odd-order differential equations

In this paper, we give a generalization of the well-known Lyapunov-type inequality for a class of odd-order differential equations, the result of this paper is new and generalizes some early results on this topic.     

The sharp quantitative Sobolev inequality for functions of bounded variation

The classical Sobolev embedding theorem of the space of functions of bounded variation BV(Rn) into Ln^′(Rn) is proved in a sharp quantitative form.     

A weak-type inequality for differentially subordinate harmonic functions

Assuming an extra condition, we decrease the constant in the sharp inequality of Burkholder for two harmonic functions and . That is, we prove the sharp weak-type inequality under the assumptions that , and the extra assumption that . Here is the harmonic measure with respect to and the constant is the one found by Davis to be the best constant in Kolmogorov's weak-type inequality for conjugate functions.

     

Perturbations of existence families for abstract Cauchy problems

In this paper, we establish Desch-Schappacher type multiplicative and additive perturbation theorems for existence families for arbitrary order abstract Cauchy problems in a Banach space: ; . As a consequence, we obtain such perturbation results for regularized semigroups and regularized cosine operator functions. An example is also given to illustrate possible applications.

     

An order preserving inequality for three operators via furuta inequality

Furuta showed that if A≥B≥0,then for each r≥0,f（p）=（A^r/2 B^p A^r/2）^t＋r/p＋r is decreasing for p≥t≥0.Using this result,the following inequality（C^r/2（AB^2A）^δC^ r/2）^ p-1＋r/4δ＋r ≤C^p-1＋r is obtained for 0〈p ≤1,r≥1,1/4≤δ≤1 and three positive operators A, B, C satisfy（A^1/2BA^1/2）^p/2≤A^p,（B^1/2AB^1/2）^p/2≥B^p,（C^1/2AC^1/2）^p/2≤C^p,（A^1/2CA^1/2）^p/2≥A^p.     

Lyapunov-type inequality for a class of even-order differential equations

In this paper, we generalize the well-known Lyapunov-type inequality for second-order linear differential equations to a class of even-order linear differential equations, the results of this paper are new and generalize some known inequalities in the literatures.     

Strict inequality of Robin eigenvalues for elliptic differential operators on Lipschitz domains

On a bounded Lipschitz domain we consider two selfadjoint operator realizations of the same second order elliptic differential expression subject to Robin boundary conditions, where the coefficients in the boundary conditions are functions. We prove that inequality between these functions on the boundary implies strict inequality between the eigenvalues of the two operators, provided that the inequality of the functions in the boundary conditions is strict on an arbitrarily small nonempty, open set.     

The Turán-Kubilius inequality for additive arithmetical semigroups

We show that the Turán-Kubilius inequality holds for additive arithmetical semigroups satisfying the following conditions: G(n) = q ⁿ (A+O(1/ln n)) (where A > 0 and q > 1) for the number of elements of degree n and P(n) = O(q ⁿ /n) for the number of prime elements of degree n. This is an improvement of a result of Zhang. We also give some variants of the inequality under some stronger or weaker assumptions and applications for the prime divisor function ω and related functions. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 457–471, July–September, 2006.     

A sharp exponential inequality for Lorentz-Sobolev spaces on bounded domains

This paper generalizes an inequality of Moser from the case that is in the Lebesgue space to certain subspaces, namely the Lorentz spaces , where . The conclusion is that is integrable, where . This is a higher degree of integrability than in the Moser inequality when . A formula for is given and it is also shown that no larger value of works.

     

A submartingale inequality

This paper extends Burkholder's inequality between a nonnegative submartingale and a process strongly differentially subordinate to it.

     

A Cauchy-Schwarz type inequality for fuzzy integrals

In this paper we prove a Cauchy-Schwarz type inequality for fuzzy integrals.     

Generalizations of Bohr inequality for Hilbert space operators

Let B(H) be the space of all bounded linear operators on a complex separable Hilbert space H. Bohr inequality for Hilbert space operators asserts that for A,B∈B(H) and p,q>1 real numbers such that 1/p+1/q=1,

²|A+B|?p²|A|+q²|B|     

解一类抽象积—微分方程的积分半群方法

在这篇文章内，研究一类含参数线性一算子族Ａ（ｔ）和线性算子Ｂ的积－微分方程，应用积分半群方法证明了该方程存在指数有界解，和含Ａ（ｔ），Ｂ的二阶微分方程存在指数有界和分解。这里Ａ（ｔ），Ｂ不必满足Ｈｉｌｌｅ－Ｙｏｓｉｄａ条件。     

On the Bohr inequality of operators

This paper is focused on the operator inequalities of the Bohr type. We will give a new and transparent proof for the operator Bohr inequality through an absolute value operator identity, show some related operator inequalities by means of 2×2 (block) operator matrices, and finally we will present a generalization of the operator Bohr inequality for multiple operators.     

Almost automorphic solutions for semilinear boundary differential equations

In this work, we use the extrapolation methods to study the existence and uniqueness of almost automorphic solutions to the semilinear boundary differential equation

where generates a hyperbolic -semigroup on a Banach space and are almost automorphic functions which take values in and a ``boundary space' , respectively. These equations are an abstract formulation of partial differential equations with semilinear terms at the boundary, such as population equations, retarded differential equations and boundary control systems. An application to retarded differential equations is given.

     

关于Pan—Yang不等式的新证明

曲率积分不等式是研究平面曲线的演化问题的重要组成部分,Pan—Yang在研究一类缩短流时得到一个关于曲率积分的不等式．本文主要利用傅里叶分析的方法给出了此不等式的一种简单的证明方法．