Measures of noncompactness in banach spaces lecture notes in. It is worthwhile mentioning that the important measures of noncompactness in notable spaces satisfy condition m see 8,9,10,12,18. Measure of noncompactness in banach algebra and application. Pisier, similarity problems and completely bounded maps, p 27 for a banach. A new measure of noncompactness on banach spaces is defined from the hausdorff measure of noncompactness, giving a quantitative version of a classical result by r. View the article pdf and any associated supplements and figures for a period of 48 hours. On the other hand, in certain classical spaces the two approaches are equivalent 17. Several quasibanach sequence spaces are thus characterized as a particular case of s. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. Pdf note on measures of noncompactness in banach sequence. Several properties of the defined measures are given. The first measure of noncompactness, the function a, was defined and studied by.
Advances in nonlinear analysis via the concept of measure of. Measure of noncompactness, darbo property, fixed point, ordinary differential equation in banach space. Entropy, volume growth and srb measures for banach space. Measures of weak noncompactness and fixed point theory for 1set weakly contractive operators on unbounded domains. Request pdf measures of weak noncompactness in banach spaces. Measures of noncompactness are however useful in the study of infinitedimensional banach spaces, for. In mathematics, more specifically in functional analysis, a banach space pronounced is a complete normed vector space. The authors thank the referees for their constructive comments, which suggest that besides the hausdorff and kuratowski measures of noncompactness, many other measures of noncompactness are also widely used in many aspects of nonlinear analysis.
This thesis studies various measures of noncompactness and some geometrical coefficients in metric or bauach spaces. Appell,measures of noncompactness and condensing operators, vol. Buy measures of noncompactness in banach spaces lecture notes in pure and applied mathematics on free shipping on qualified orders. Note on measures of noncompactness in banach sequence spaces article pdf available in mathematica slovaca january 1990 with 20 reads how we measure reads. Banach spaces rather fragmented, maybe you could say it is underdeveloped, but one can argue that linear approximations are often used for considering nonlinear problems. The degree of noncompactness of a set is measured by means of functions called measures of noncompactness. Measures of noncompactness in banach spaces, lecture notes in pure and applied mathematics 60, marcel dekker, new york and basel, 1980 with k. On existence theorems for differential equations in banach spaces. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. They are widely used in fixed point theory, differential equations, functional equations, integral and integrodifferential equations, optimization, and so on 28. Caratheodory solutions of sturmliouville dynamic equation. On the measure of noncompactness in banach spaces and. In the paper we formulate a criterion for relative compactness in the space r j, e consisting of all regulated functions, defined on the compact interval j and with values in the banach space e. Goebel, measures of noncompactness in banach spaces, preprint.
In functional analysis, two measures of noncompactness are commonly used. Moreover for bounded sets h of a banach space e we consider the. On measures of noncompactness in banach spaces dmlcz. In this article measures of noncompactness are used to study inequalities, more exactly the extension of an equality, studied in many works and having wide application. This work was supported by national natural science foundation of china grant nos. Entropy, volume growth and srb measures for banach space mappings alex blumenthal. They are widely used in fixed point theory, differential equations, functional equations, integral and integrodifferential equations, optimization, and so on 2 8.
To generalize the hausdorff measure of noncompactness to other classes of bounded sets like e. Advances in nonlinear analysis via the concept of measure. A family of measures of noncompactness in the locally sobolev. On the measures of noncompactness in some metric spaces dariusz bugajewski and ewa grzelaczvk received march 1997 abstract. View the article pdf and any associated supplements and figures. This chapter deals with measure of noncompactness briefly mnc in banach spaces and application to the theory of differential and integral equations. Essential spectra of some matrix operators by means of. Measures of weak noncompactness are formulae that quantify different characterizations of weak compactness in banach spaces. Measures of noncompactness or weak noncompactness have been successfully applied in operator theory, differential equations and integral equations, see for instance 1, 3, 4, 9, 14, 15 and 16.
Danes, on densifying and related mappings and their applications in nonlinear functional analysis, theory of nonlinear operators, akademieverlag, berlin 1974, 1556. Generalized quasibanach sequence spaces and measures of. Schauder 20 let c be a closed and convex subset of a banach space e. On measures of noncompactness in banach spaces by jozef banas topics. Measures of noncompactness are very useful tools in banach spaces. However, formatting rules can vary widely between applications and fields of interest or study. We use topological tools to study measures of weak noncompactness in banach spaces. In this paper an axiomatic approach to the notion of a measure of weak noncompactness is presented. Measures of noncompactness in what follows we will give a brief description of measures of noncompactness and condensing operators which will be used in subsequent sections. Numerous and frequentlyupdated resource results are available from this search.
Measures of noncompactness in banach spaces lecture notes. Measures of weak noncompactness in banach spaces request pdf. In the paper we construct and investigate two convenient measures of noncompactness in the space of socalled regulated functions see. These geometrical numbers are useful in the study of measures of noncompactness, some of which are interesting quantities in fixed point theory, in chapter 1, we give some definitions and km. Measures of noncompactness in banach spaces, lecture notes in pure and applied mathematics 60 marcel dekker, new york, 1980 with k. Nonlocal conformablefractional differential equations with a. Pdf generalized measures of noncompactness of sets and. Mursaleen, sequence spaces and measures of noncompactness with applica. Operations on measurable functions sums, products, composition realvalued measurable functions. Strongly generated banach spaces and measures of noncompactness. Sequence spaces and measures of noncompactness with. Keywords quadratic integral equation banach space measure of noncompactness fixed point theorem of darbo type asymptotic stability citation banas, jozef. These results are exploited to investigate the essential spectra of some matrix operators on banach spaces. A quantitative version of a classical criterion for compactness of bounded sets in banach spaces by.
Measure of weak noncompactness and real interpolation of. Goebel, measures of noncompactness in banach spaces, lecture notes in pure and applied mathematics 60, marcel dekker, new york, 1980. Moreover, we provide a few concrete realizations of the accepeted axiomatic system in some banach spaces. In section 2, we recall some definitions and facts concerning the general measures of noncompactness in banach spaces and regulated functions. So, it seems to be an interesting problem whether there is some natural classical space in which the two approaches are not equivalent. For a bounded set b in a banach space x, the hausdorff measure of noncompactness. Such a condition will guarantee the solvability of operator equations in banach algebra. Every banach space admits a homogenous measure of non.
This work contains some results which extend some well known ones in the literature. Measures of noncompactness in the space of regulated functions. For y a banach space, the space bx, y is a banach space with respect to this norm if x is a banach space, the space bx bx, x forms a unital banach algebra. Nonlocal conformablefractional differential equations. It is known that the only banach space that satisfies the vonneumann inequality is the hilbert space. Moreover for bounded sets h of a banach space e we consider the worst distance k h of the weak. Dhage, hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, differ. Note that these measures of noncompactness are useless for subsets of euclidean space r n. Measures of weak noncompactness in banach spaces let h be a bounded subset of the banach space e. Commentationes mathematicae universitatis carolinae, vol.
Measures of weak noncompactness in banach spaces request. Measures of noncompactness in banach spaces book, 1980. He used professor viaclovskys handwritten notes in producing them. Measures of noncompactness and asymptotic stability of solutions of a quadratic hammerstein integral equation. Measures of noncompactness in the study of solutions of. So, one relies on the fact that the linear problems are relatively tractable, and on the theory we will consider. New measures of noncompactness for bounded sets and linear operators, in the setting of abstract measures and generalized limits, are constructed. Measures of weak noncompactness in banach spaces sciencedirect. For these spaces, new measures of noncompactness are also defined, related to the hausdorff measure of noncompactness. Furthermore, we will introduce the family of measures of noncompactness. On the basis of the criterion we construct two arithmetically convenient and regular measures of noncompactness and investigate the connections of these measures with the hausdorff measure. Ainterpolation spaces of a quasibanach couple are studied. Measures of noncompactness in banach spaces lloyd 1981. Generalized measures of noncompactness of sets and.
Thus, a banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a cauchy sequence of vectors always converges to a well defined limit that is within the space. On existence theorems for differential equations in banach. Geometrical coefficients and measures of noncompactness. Applied mathematics letters an application of a measure of. The lecture notes were prepared in latex by ethan brown, a former student in the class. In this paper, we develop representation spaces based on the nonstandard hull construction which is equivalent to the ultrapower construction. A family of measures of noncompactness in the locally. On measures of noncompactness in banach spaces core. Goebel, measures of noncompactness in banach spaces. Eudml on measures of noncompactness in banach spaces. Magenes this inequality is proved for embedding operators in banach spaces a particular case of metric spaces. For this reason, measures of weak noncompactness have been axiomatized 12 allowing thus several authors to construct measures of weak noncompactness in several banach spaces 7, 11, 47, 48.