Other measures were introduced by goldenstein, gohberg and markus. Pdf measure of noncompactness in the study of solutions for. 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. In this paper we calculate the hausdorff and the kuratowski mea sure of noncompactness for any bounded subsets of the space r2 with the river metric or with the radial metric. The basic properties of the hausdorff measure of noncompactness can be found in 2, 79 and for recent developments, see 1018. On the applications of measure of noncompactness to the existence of solutions for fractional difference equations, results math.
A condensing or densifying operator is a mapping under which the image of any set is in a certain sense more compact than the set itself. Pdf measures of noncompactness in orlicz modular spaces. We mention the following result which is used in the subsequent sections. Pdf the objective of this paper is to study the existence of solution of functional integral equations using a measure of noncompactness. Application of measure of noncompactness to infinite systems of differential equations in \. Darbo 2 used the function to prove his xed point theorem which is a very important generalisation of schauders xed point theorem.
Use the measuring tools to measure distances and areas of objects in pdf documents. Measures of noncompactness or weak noncompactness have been success fully applied in operator theory, differential equations and integral equations, see for instance 1, 3, 4, 9, 14, 15 and 16. In this paper, we derive some identities for the hausdorff measures of noncompactness of certain matrix operators on the sequence spaces x r, s of generalized means. However, formatting rules can vary widely between applications and fields of interest or study.
Applications of measure of noncompactness in matrix. We investigate the solutions of functionalintegral equation of fractional order in the setting of a measure of noncompactness on realvalued bounded and continuous banach space. Previous results on interpolation of the measure of noncompactness are due to edmunds and teixeira 11. Measure of noncompactness for compact matrix operators. 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. Measure of noncompactness and fractional integrodifferential. Advances in nonlinear analysis via the concept of measure of noncompactness by jozef banas and publisher springer. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. 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.
How to measure distances and areas in pdf documents. The aim of this section is to discuss conditions on a map f. Danes, on densifying and related mappings and their applications in nonlinear functional analysis, theory of nonlinear operators, akademieverlag, berlin 1974, 1556. Measures of noncompactness and condensing operators. Kuratowski introduced a measure ca of noncompactness for subsets a of a complete metric space x. Measure of weak noncompactness, some new properties in fredholm theory, characterization of the schechter essential spectrum and application to transport. Measures of noncompactness in banach spaces lecture notes. Kuratowskis measure of noncompactness revisited the. Measure of noncompactness, surjectivity of gradient operators. Results are also given for contractive and nonexpansive mappings, and for mappings of an iterative type. Because of the important role of the measure of noncompactness in fixed point. Measure of noncompactness and spectral theory duc 1984. N and by applying this family of measures of noncompactness, we discuss the. Measure of weak noncompactness, some new properties in fredholm theory, characterization of the schechter essential spectrum and application to transport operators.
On the measures of noncompactness in some metric spaces dariusz bugajewski and ewa grzelaczvk received march 1997 abstract. Finally, an example is presented to show the usefulness of the outcome. Axioms free fulltext measure of noncompactness for. Application of measure of noncompactness for the system of. It discusses various applications of the theory of measures of noncompactness, in particular, by addressing the results and methods of fixedpoint theory. Our fixed point results are obtained under a weak topology and measure of weak noncompactness. Application of measures of noncompactness to the system of.
Applications of measure of noncompactness in operators on the. These tools are especially useful when youre creating manuals, instructional guides, forms, and computeraided design cad drawingsor when you want to measure areas of any document before sending to a professional printer. Nonlocal conformablefractional differential equations. Further, we apply the hausdorff measure of noncompactness to obtain the. Axioms free fulltext measure of weak noncompactness and. Eudml on measures of noncompactness in banach spaces.
Measures of noncompactness are very useful tools in banach spaces. Measures of noncompactness in the space of continuous and. For a bounded set b in a banach space x, the hausdorff measure of noncompactness. The function is called the hausdorff measure of noncompactness. By scott nguyen, integrated sales and support foxit phantompdf makes it easy to measure distances and areas of objects in pdf documents with our measure tools. We use topological tools to study measures of weak noncompactness in banach spaces. Mursaleen, sequence spaces and measures of noncompactness with applica. The weak measure of noncompactness of d the measure of noncompactness of d in the weak topology wd, is defined as od w,d. Our main result is more general than the main result of 2. Let x, d be a metric space and d be a bounded subset of x.
The university of hochiminh city faculty of mathematics hochiminh city box 923, s. The formula for computing measure of noncompactness for a general mnc in a given metric or normed space is a rigorous task, however in some normed spaces the exact formula is available for hausdor mnc. Measure of open sets approximate from within by polygons measure of compact sets approximate from outside by opens outer and inner measures. The basic properties of the hausdor measure of noncompactness can be found in, 12. The purpose of this thesis is to set forth some fixed point theorems in metric and banach spaces for densifying mappings and ksetcontractions. A cone measure of noncompactness and some generalizations of darbo s theorem with applications to functional integral equations mohamedjleli, 1 mohammadmursaleen, 2 kishinsadarangani, 3 andbessemsamet 1 department of mathematics, college of science, king sau d university, p. How to measure distances and areas in pdf documents foxit.
The degree of noncompactness of a set is measured by means of functions called measures of noncompactness. Applied mathematics letters an application of a measure of. Application of measure of noncompactness 3 now, we mention some results and facts about measure of noncompactness. In this research article, we introduce a new class of hybrid langevin equation involving two distinct fractional order derivatives in the caputo sense and riemannliouville fractional integral. Advances in nonlinear analysis via the concept of measure. Measure of noncompactness in banach algebra and application. The existence theorems for nonlinear integral equations have been studied in many papers with the help of the technique of measures of non compactness which. Supported by threepoint boundary conditions, we discuss the existence of a solution to this boundary value problem. The kuratowski measure of noncompactness of d ad is defined as the infimum of positive numbers e such that d can be covered by a finite number of sets of diameter not greater than e. Kuratovskiy in 1930 in connection with problems of general typology. Now we present an example of a measure of noncompactness in banach algebra which satis. Advances in nonlinear analysis via the concept of measure of. The technique used is a generalization of the classical darbo fixed point theorem for frechet spaces associated with the concept of measures of. The measuring tools are useful for showing the distances and areas associated with objects in a form or computeraided design cad drawing.
Measures of noncompactness in what follows we will give a brief description of measures of noncompactness and condensing oper. A family of measures of noncompactness in the locally sobolev. An application of measure of noncompactness in study of. Applications of measure of noncompactness in matrix operators. Measures of noncompactness in the space of regulated functions. They are widely used in fixed point theory, differential equations, functional.
Solvability of functionalintegral equations fractional. Research article measure of noncompactness for compact. You can also use these tools to measure certain areas of a document before sending it to a professional printer. In functional analysis, two measures of noncompactness are commonly used. Axioms free fulltext measure of weak noncompactness. In this paper, we introduce a new family of measures of noncompactness in the frechet space lp loc. Consider the following measure of noncompactness e a. This book offers a comprehensive treatment of the theory of measures of noncompactness.
Measure of noncompactness in the study of solutions for a system of integral equations article pdf available march 2019 with 115 reads how we measure reads. Pdf application of measure of noncompactness for the system of. Application of measure of noncompactness for the system of functional integral equations reza araba adepartment of mathematics, sari branch, islamic azad university, sari, iran. Wc prove some fixed point theorems for wcontractive, tocondensing and cononexpansive mappings. Compactness by the hausdorff measure of noncompactness.
Pdf applications of measure of noncompactness and operator. We do this using the definition and properties of the measure of noncompactness of sets and operators see, for instance, 1, chapter 2, or. Kuratowskis measure of noncompactness revisited, the quarterly journal of mathematics, volume 39, issue 2, 1 june 1988, pages 235254. Measure of noncompactness, densifying mappings and fixed. For contractive maps one can take as measure of noncompactness the diameter of a set, while for compact maps can take the indicator function of a family of nonrelatively com pact sets. We also illustrate our results through some concrete examples to the memory of professor lj. Moreover, the choice of suitable measure of noncompactness allows us to. Measures of noncompactness in banach spaces book, 1980. Pdf in this paper we introduce the notion of the generalized darbo fixed point theorem and prove some fixed and coupled fixed point. Applications of measure of noncompactness in operators on.
An illustration is given in support of the solution of a. Measure of noncompactness, surjectivity of gradient. Definition of lebesgue measurable for sets with finite outer measure remove restriction of finite outer measure rn, l, lambda is a measure space, i. The operators of the form f x g x, x, where g is contractive in the first argument and compact in the second, are also condensing with respect to some. Measures of noncompactness are numerical characteristics of bounded subsets of metric space, equal to zero on relatively compact subsets. The quantitative characteristic of measure of noncompactness of metric space subset was intro duced by k. In this paper, by applying a measure of noncompactness in the space l. Bx,ythe hausdorff measure of noncompactness of a, denoted by a. Numerous and frequentlyupdated resource results are available from this search.
Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Pdf compactness by the hausdorff measure of noncompactness. Mar 22, 2019 instant access to the full article pdf. In chapter i, we discuss the banach contraction principle and give an extensive coverage of its generalizations. If, then the hausdorff measure of noncompactness of the set, denoted by, is defined by. Measure of noncompactness for compact matrix operators on. Save up to 80% by choosing the etextbook option for isbn. Grids, guides, and measurements in pdfs, adobe acrobat.
Pdf measure of noncompactness in the study of solutions. Real interpolation and measure of weak noncompactness. Measure of noncompactness, darbo property, fixed point, ordinary differential equation in banach space. In this paper we introduce the notion of the generalized darbo. Pure mathematics research article a family of measures of. The first result on fixed points, in which a measure of noncompactness is used, was published, as far as i know, in an article by g darbo 1955. In this paper, the existence of solutions for an initial value problem of a fractional differential equation is obtained by means of monchs fixed point theorem and the technique of measures of noncompactness. Measure of weak noncompactness and real interpolation of.
Research article measure of noncompactness for compact matrix operators on some bk spaces e. Research article a cone measure of noncompactness and. Measures of noncompactness in orlicz modular spaces 5 t o show. In this chapter we consider the basic notions connected with measures of noncompactness mncs for brevity and condensing or densifying operators.
Further, we denote by m e the family of all nonempty bounded subsets of eand r e its subfamily consisting of all relatively compact sets. Goebel, measures of noncompactness in banach spaces. Weighted estimates of a measure of noncompactness for. We define and study in detail the three main and most frequently used mncs. Measures of noncompactness in banach spaces lecture notes in. Buy measures of noncompactness in banach spaces lecture notes in pure and applied mathematics on free shipping on qualified orders. The results of 11 referto the special case when a0 a1, or b0 b1, or the couple b0. Measures of noncompactness and condensing operators book.
Request pdf an application of measure of noncompactness in study of asymptotic stability using the technique of fixedpoint theorem of darbo type associated with measures of noncompactness, we. Lecture notes measure and integration mathematics mit. View the article pdf and any associated supplements and figures for a period of 48 hours. Measures of noncompactness in banach spaces lloyd 1981. Foxit phantompdf makes it easy to measure distances and areas of objects in pdf documents with our measure tools these tools are especially useful when youre creating manuals, instructional guides, forms, and computeraided design drawingsor when you want to measure areas of any document before sending to a professional printer. Appell,measures of noncompactness and condensing operators, vol. Goebel, measures of noncompactness in banach spaces, preprint. We denote by xand conv xthe closure and the convex closure of xas a subset of e, respectively.