On the generalized modulus

The generalized modulus has numerous applications in geometric function theory and analytic number theory. In this paper, we study the monotonicity and convexity of the generalized modulus and obtain sharp functional inequalities for this function. Our results are the extensions of some well-known inequalities of the modulus of the Grötzsch ring.     

Perturbations of functional inequalities using growth conditions

Perturbations of functional inequalities are studied by using merely growth conditions in terms of a distance-like reference function. As a result, optimal sufficient conditions are obtained for perturbations to reach a class of functional inequalities interpolating between the Poincaré inequality and the logarithmic Sobolev inequality.     

Existence of solutions to Hamilton-Jacobi functional differential equations

This paper deals with the Cauchy problem for nonlinear first order partial functional differential equations. The unknown function is the functional variable in the equation and the partial derivatives appear in a classical sense. A theorem on the local existence of a generalized solution is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions of this system is proved by using a method of successive approximations. A method of bicharacteristics and integral inequalities are applied.     

关于一个非齐次核的Hilbert型积分算子

应用权函数的方法及实分析与泛函分析的思想技巧,定义了一个非齐次核的Hilbert型积分算子,并求出其联系范数的两个等价不等式．作为应用,还考虑了其逆形式及一些特殊核的例子．     

Functional inequalities on manifolds with non-convex boundary

In this article, new curvature conditions are introduced to establish functional inequalities including gradient estimates, Harnack inequalities and transportation-cost inequalities on manifolds with non-convex boundary.     

带比例功能反应函数食物链交错扩散模型的整体解

本文研究了带有比例功能反应函数食物链交错扩散模型整体解的存在性和正平衡点的稳定性.利用能量方法和Gagliardo-Nirenberg型不等式,获得了该模型整体解的存在性和一致有界性,同时通过构造Lyapunov函数给出了该模型正平衡点全局渐近稳定的充分条件.     

Banach空间的型与独立Banach空间值随机变量序列的不等式

本文建立了独立Ｂ值随机变量序列部分和的极大值函数Ｓ＊与ｐ方根函数Ｓ（ｐ，Ｘ）的分布函数不等式与矩不等式，讨论了这些不等式的成立与Ｂａｎａｃｈ空间的ｐ型与ｑ余型的关系，同时给出了与Ｈｉｌｂｅｒｔ空间同构的Ｂａｎａｃｈ空间的特征。     

Quermassintegrals of quasi-concave functions and generalized Prékopa–Leindler inequalities

We extend to a functional setting the concept of quermassintegrals, well-known within the Minkowski theory of convex bodies. We work in the class of quasi-concave functions defined on the Euclidean space, and with the hierarchy of their subclasses given by α-concave functions. In this setting, we investigate the most relevant features of functional quermassintegrals, and we show they inherit the basic properties of their classical geometric counterpart. As a first main result, we prove a Steiner-type formula which holds true by choosing a suitable functional equivalent of the unit ball. Then we establish concavity inequalities for quermassintegrals and for other general hyperbolic functionals, which generalize the celebrated Prékopa–Leindler and Brascamp–Lieb inequalities. Further issues that we transpose to this functional setting are integral-geometric formulae of Cauchy–Kubota type, valuation property and isoperimetric/Urysohn-like inequalities.     

Characterization of Pinched Ricci Curvature by Functional Inequalities

In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient estimates, \(L^p\)-inequalities and log-Sobolev inequalities. These results are further extended to differential manifolds carrying geometric flows. As application, it is shown that they can be used in particular to characterize general geometric flow and Ricci flow by functional inequalities.     

Optimality conditions of the maximum principle type in bilinear control problems

The optimization of a bilinear functional related to a linear state system with a modular control constraint is considered. Exact formulas for the functional increment are used to obtain sufficient conditions for the optimality of extremal controls that supplement the maximum principle. These conditions are represented in the form of inequalities and equalities for one-variable functions on a time interval. The optimization of a quadratic functional with the help of a matrix conjugate function is reduced to the bilinear case.     

Functional inequalities in non-Archimedean normed spaces

In this paper, we introduce an additive functional inequality and a quadratic functional inequality in normed spaces, and prove the Hyers–Ulam stability of the functional inequalities in Banach spaces. Furthermore, we introduce an additive functional inequality and a quadratic functional inequality in non-Archimedean normed spaces, and prove the Hyers–Ulam stability of the functional inequalities in non-Archimedean Banach spaces.     

Functional differential inequalities and estimation of the Cauchy function of an equation with aftereffect

We consider scalar functional differential inequalities that are used to estimate solutions to differential equations with deviating argument. A theorem on positiveness of the Cauchy function of a differential equation with aftereffect is derived from a theorem on a functional differential inequality with nonlinearmonotone operator, which is a direct generalization of the simplest classical theorem on a differential inequality. The suggested proofs rely on local properties of continuous functions only.     

Infinite Systems of Quasilinear Differential Difference Inequalities and Applications

The article deals with the initial boundary value problem for an infinite system of first order quasilinear functional differential equations. A comparison result concerning infinite systems of differential difference inequalities is proved. A function satisfying such inequalities is estimated by a solution of a suitable Cauchy problem for an ordinary functional differential system. The comparison result is used in an existence theorem and in the investigation of the stability of the numerical method of lines for the original problem. A theorem on the error estimate of the method is given. The infinite system of first order functional differential equations contains, as particular cases, equations with a deviated argument and integral differential equations of the Volterra type.     

Optimality conditions for extremal controls in bilinear and quadratic problems

We consider the optimization problem for a bilinear functional with respect to a linear phase system with a modularly constrained control. On the base of exact formulas for the functional increment we establish sufficient optimality conditions for extremal controls. These conditions are stated as inequalities for one-dimensional functions on a time interval. They supplement the maximum principle, keeping the implementation complexity at the same level. The optimization problem for a quadratic functional is reduced to the bilinear case with the help of the matrix conjugate function.     

Affine isoperimetric inequalities in the functional Orlicz–Brunn–Minkowski theory

In this paper, we develop a basic theory of Orlicz affine and geominimal surface areas for convex and s-concave functions. We prove some basic properties for these newly introduced functional affine invariants and establish related functional affine isoperimetric inequalities as well as functional Santaló type inequalities.     

On a class of weighted anisotropic Sobolev inequalities

In this article, motivated by a work of Caffarelli and Cordoba in phase transitions analysis, we prove new weighted anisotropic Sobolev type inequalities where different derivatives have different weight functions. These inequalities are also intimately connected to weighted Sobolev inequalities for Grushin type operators, the weights being not necessarily Muckenhoupt. For example we consider Sobolev inequalities on finite cylinders, the weight being a power of the distance function from the top or the bottom of the cylinder. We also prove similar inequalities in the more general case in which the weight is a power of the distance function from a higher codimension part of the boundary.     

Quadratic variation functionals and dilation equations

Here we present some distribution function inequalities between certain functionals defined relative to a convolution approximation procedure. Such inequalities are best known when the approximation is made using dilations of the Gaussian or Cauchy kernels. In these cases, classical differential equations, the heat equation or Laplace's equation, provide the basis for comparisons; in the latter case, the quadratic functional is known as the Lusin area integral. The kernels we consider are compactly supported, and satisfy a dilation equation, rather than a differential equation. For these kernels, there is an intrinsic quadratic variation, defined from the dilation structure. We obtain good lambda distribution function inequalities between a maximal function and the quadratic variation functional.     

Comparing Nonsmooth Nonconvex Bundle Methods in Solving Hemivariational Inequalities

Hemivariational inequalities can be considered as a generalization of variational inequalities. Their origin is in nonsmooth mechanics of solid, especially in nonmonotone contact problems. The solution of a hemivariational inequality proves to be a substationary point of some functional, and thus can be found by the nonsmooth and nonconvex optimization methods. We consider two type of bundle methods in order to solve hemivariational inequalities numerically: proximal bundle and bundle-Newton methods. Proximal bundle method is based on first order polyhedral approximation of the locally Lipschitz continuous objective function. To obtain better convergence rate bundle-Newton method contains also some second order information of the objective function in the form of approximate Hessian. Since the optimization problem arising in the hemivariational inequalities has a dominated quadratic part the second order method should be a good choice. The main question in the functioning of the methods is how remarkable is the advantage of the possible better convergence rate of bundle-Newton method when compared to the increased calculation demand.     

FORCED OSCILLATION AND APPLICATION OF FUNCTIONAL DIFFERENTIAL INEQUALITIES

In this paper we study the oscillations for a class of functional differential inequalities. By using these properties some forced oscillations to the boundary value problems of functional partial differential equations are established.     

Некоторые точные нер авенства в теории при ближения функций

Exact inequalities are obtained that illuminate the interrelation between best polynomial approximations of functions, analytic in the disk and the modulus of continuity of the derivatives of the boundary values of these functions.For various classes of functions exact estimates are given for the derivative of a function by means of the modulus of continuity of this function and the modulus of continuity of its second derivative.As application, exact inequalities are deduced, analogous to the well-known Bernstein and Hardy inequalities.