Set-Valued Kurzweil–Henstock–Pettis Integral

It is shown that the obvious generalization of the Pettis integral of a multifunction obtained by replacing the Lebesgue integrability of the support functions by the Kurzweil–Henstock integrability, produces an integral which can be described – in case of multifunctions with (weakly) compact convex values – in terms of the Pettis set-valued integral.Mathematics Subject Classifications (2000) Primary: 28B20; secondary: 26A39, 28B05, 46G10, 54C60.     

实值非负函数关于集值序增函数的集值Riemann-Stieltjes积分

本文首先建立了实值非负函数关于集值序增函数的集值Riemann-Stieltjes积分,并讨论了集值Riemann-Stieltjes积分的性质,给出了集值Riemann-Stieltjes可积的充要条件,最后引入了集值Riemann-Stieltjes随机积分．     

Existence results for Urysohn integral inclusions involving the Henstock integral

This paper presents two existence results for Urysohn integral inclusions in Banach spaces, the set-valued integral involved being the Henstock integral. In particular, the existence of continuous solutions for integral inclusions of Volterra and of Hammerstein type is obtained. Our results extend the existence theorems given in literature in the single- or set-valued case, under Bochner or Pettis integrability hypothesis. The compactness of solutions set is also studied.     

A Komlós-type theorem for the set-valued Henstock-Kurzweil-Pettis integral and applications

This paper presents a Komlós theorem that extends to the case of the set-valued Henstock-Kurzweil-Pettis integral a result obtained by Balder and Hess (in the integrably bounded case) and also a result of Hess and Ziat (in the Pettis integrability setting). As applications, a solution to a best approximation problem is given, weak compactness results are deduced and, finally, an existence theorem for an integral inclusion involving the Henstock-Kurzweil-Pettis set-valued integral is obtained.     

On the Pettis Integral of Closed Valued Multifunctions

In this work, the study of Pettis integrability for multifunctions (alias set-valued maps), whose values are allowed to be unbounded, is initiated. For this purpose, two notions of Pettis integrability, and of Pettis integral, are considered and compared. The first notion is similar to that of the weak integral, already known for vector-valued functions, and is defined via support functions. The second notion resembles the classical Aumann definition using integrable selections, but it involves the Pettis integrable selections rather than the Bochner integrable ones. The above two integrals are shown to coincide in a quite general setting. Several criteria for a multifunction to be Pettis integrable (in one sense or the other) are proved. On the other hand, due to the possibility of infinite values for the support functions, we are led to introduce a more general notion of scalar integrability involving the negative part of these functions. We compare the scalar integrability of a multifunction with that of its measurable selections. We also provide some new results concerning multifunctions with bounded values and/or new proofs of already existing ones. Examples are included to illustrate the results and to introduce open problems.     

On Birkhoff integrability for scalar functions and vector measures

The Bartle–Dunford–Schwartz integral for scalar functions with respect to vector measures is characterized by means of Riemann-type sums based on partitions of the domain into countably many measurable sets. In this setting, two natural notions of integrability (Birkhoff integrability and Kolmogoroff integrability) turn out to be equivalent to Bartle–Dunford–Schwartz integrability. A. Fernández, F. Mayoral and F. Naranjo were supported by MEC and FEDER (project MTM2006–11690–C02–02) and La Junta de Andalucía. J. Rodríguez was supported by MEC and FEDER (project MTM2005-08379), Fundación Séneca (project 00690/PI/04) and the Juan de la Cierva Programme (MEC and FSE).     

一类Riccati方程的推广

把 Riccati方程 y′=Py2 + Qy+ R推广成 Riccati型方程 :f′( y) dydx=Pf 2 ( y) + Qf ( y) + R.并给出其可积的条件及其对应的通积分 .     

A Blow-up method to prove formal integrability for some planar differential systems

In this work we provide an effective method to prove the formal integrability of the resonant saddles. The method is based on the use of a blow-up and the resolution of a recurrence differential equation using induction. Using the method some open integrability problems for certain resonant saddles are solved.     

模糊集值映射(K)可积性的一个等价条件

In this paper,we first discuss the relationship between the McShane integral and Pettis integral for vector-valued functions.Then by using the embedding theorems for the fuzzy number space E~1,we give a new equivalent condition for(K) integrability of a fuzzy set-valued mapping F:[a,b]→E~1.     

一道Lebesgue积分题目的注记

设E为n维欧氏空间Rn的可测子集,m E+∞,f(x)为E上的非负可测函数,并记Ek=E{x|k≤f(x)k+1}(k=0,1,2,…),利用Lebesgue积分的性质,通过构造反例,指出"级数∑k km Ek收敛"并不是f(x)在E上Lebesgue可积的充分条件.进一步,通过增加条件"f在E上a.e.有限",得到相应的充分必要条件.     

判别函数可积性的一个新的重要命题

在抽象测度空间中,用可测集EK去逼近集E的办法,从函数f在E上的可测性去推f在E上的可积性,是判别函数可积性的一个新的重要命题,但[2]在证明这一命题时有误.本文作了更正,并从距离空间中的积分推广到抽象测度空间中的积分.     

Birkhoff integral for multi-valued functions

The aim of this paper is to study Birkhoff integrability for multi-valued maps , where (Ω,Σ,μ) is a complete finite measure space, X is a Banach space and cwk(X) is the family of all non-empty convex weakly compact subsets of X. It is shown that the Birkhoff integral of F can be computed as the limit for the Hausdorff distance in cwk(X) of a net of Riemann sums ∑_nμ(A_n)F(t_n). We link Birkhoff integrability with Debreu integrability, a notion introduced to replace sums associated to correspondences when studying certain models in Mathematical Economics. We show that each Debreu integrable multi-valued function is Birkhoff integrable and that each Birkhoff integrable multi-valued function is Pettis integrable. The three previous notions coincide for finite dimensional Banach spaces and they are different even for bounded multi-valued functions when X is infinite dimensional and X∗ is assumed to be separable. We show that when F takes values in the family of all non-empty convex norm compact sets of a separable Banach space X, then F is Pettis integrable if, and only if, F is Birkhoff integrable; in particular, these Pettis integrable F's can be seen as single-valued Pettis integrable functions with values in some other adequate Banach space. Incidentally, to handle some of the constructions needed we prove that if X is an Asplund Banach space, then cwk(X) is separable for the Hausdorff distance if, and only if, X is finite dimensional.     

Quantitative analysis of some system of integral equations

In this paper, we study the integrability of the non-negative solutions to the Euler-Lagrange equations associated with Weighted Hardy-Littlewood-Sobolev (HLS) inequality. We obtain the optimal integrability for the solutions. The integrability and the radial symmetry (which we derived in our earlier paper) are the key ingredients to study the growth rate at the center and the decay rate at infinity of the solutions. These are also the essential properties needed to classify all non-negative solutions. Some simple generalizations are also provided here.     

一类具有1:-4共振奇点的复三次Lotka-Volterra系统的可积性条件

对于一类具有1:-4共振奇点的复三次Lotka-Volterra系统,通过前12阶广义奇点量的计算,给出系统可积的充分条件.这些条件通过构造积分因子或形式积分得以证明.     

Darboux-Jouanolou-Ghys integrability for one-dimensional foliations on toric varieties

We use the existence of homogeneous coordinates for simplicial toric varieties to prove a result analogous to the Darboux-Jouanolou-Ghys integrability theorem for the existence of rational first integrals for one-dimensional foliations.     

反函数法求定积分

由定积分的可积条件与分部积分法推出一种利用反函数求解定积分的简捷方法.     

Topology of real algebraic curves

The problem on the existence of an additional first integral of the equations of geodesics on noncompact algebraic surfaces is considered. This problem was discussed as early as by Riemann and Darboux. We indicate coarse obstructions to integrability, which are related to the topology of the real algebraic curve obtained as the line of intersection of such a surface with a sphere of large radius. Some yet unsolved problems are discussed.     

Exponential square integrability for non-tangential limits of certain smooth functions

Under the assumption of the boundedness of certain operator (resembling Lusin's area function) of a smooth function u that satisfies certain interior regularity property, we prove the exponential square integrability of the boundary values of u.     

Local Darboux first integrals of analytic differential systems

We discuss local and formal Darboux first integrals of analytic differential systems, using the theory of Poincaré–Dulac normal forms, and we study the effect of local Darboux integrability on analytic normalization. Moreover we determine local restrictions on classical Darboux integrability of polynomial systems.     

PARTIAL BOCHNER INTEGRABILITY (I)

1991MRSubjectClassification46B1PreliminariesThenotati()11ofpartialBO(thllerintegrals,ajsanextensiollofBochnerintegrals,isintroduce(lill[8]forthestlldyofpartialRadoll-Nikodyrxlderivatives(e.g.[1]).Weshrillg(!ller;llizctilei'-leasplltforthin[4]alld[7]all(lshowhowfarthesegelleralizationsgotowar(lschara(:t(frizillgpartialBochnerilitegrability.I,etusfixsometerrliinologyandnotation.Throughoutthispaper,(fi,I,p)isafinitemeasurespace,thetriple([(),1],Z,A)referstotheLebesguemeasurespace.Xdenotesa…