Measures of Noncompactness and Condensing Operators

Measures of Noncompactness and Condensing Operators PDF Author: Akhmerov
Publisher: Birkhäuser
ISBN: 3034857276
Category : Science
Languages : en
Pages : 260

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Book Description
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. The degree of noncompactness of a set is measured by means of functions called measures of noncompactness. The contractive maps and the compact maps [i.e., in this Introduction, the maps that send any bounded set into a relatively compact one; in the main text the term "compact" will be reserved for the operators that, in addition to having this property, are continuous, i.e., in the authors' terminology, for the completely continuous operators] are condensing. 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 non-relatively com pact sets. 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 natural measures of noncompactness. The linear condensing operators are characterized by the fact that almost all of their spectrum is included in a disc of radius smaller than one. The examples given above show that condensing operators are a sufficiently typical phenomenon in various applications of functional analysis, for example, in the theory of differential and integral equations. As is turns out, the condensing operators have properties similar to the compact ones.

Measures of Noncompactness and Condensing Operators

Measures of Noncompactness and Condensing Operators PDF Author: Akhmerov
Publisher: Birkhäuser
ISBN: 3034857276
Category : Science
Languages : en
Pages : 260

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Book Description
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. The degree of noncompactness of a set is measured by means of functions called measures of noncompactness. The contractive maps and the compact maps [i.e., in this Introduction, the maps that send any bounded set into a relatively compact one; in the main text the term "compact" will be reserved for the operators that, in addition to having this property, are continuous, i.e., in the authors' terminology, for the completely continuous operators] are condensing. 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 non-relatively com pact sets. 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 natural measures of noncompactness. The linear condensing operators are characterized by the fact that almost all of their spectrum is included in a disc of radius smaller than one. The examples given above show that condensing operators are a sufficiently typical phenomenon in various applications of functional analysis, for example, in the theory of differential and integral equations. As is turns out, the condensing operators have properties similar to the compact ones.

Measures of Noncompactness in Metric Fixed Point Theory

Measures of Noncompactness in Metric Fixed Point Theory PDF Author: J.M. Ayerbe Toledano
Publisher: Birkhäuser
ISBN: 3034889208
Category : Mathematics
Languages : en
Pages : 222

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Book Description
What is clear and easy to grasp attracts us; complications deter David Hilbert The material presented in this volume is based on discussions conducted in peri odically held seminars by the Nonlinear Functional Analysis research group of the University of Seville. This book is mainly addressed to those working or aspiring to work in the field of measures of noncompactness and metric fixed point theory. Special em phasis is made on the results in metric fixed point theory which were derived from geometric coefficients defined by means of measures of noncompactness and on the relationships between nonlinear operators which are contractive for different measures. Several topics in these notes can be found either in texts on measures of noncompactness (see [AKPRSj, [BG]) or in books on metric fixed point theory (see [GK1], [Sm], [Z]). Many other topics have come from papers where the authors of this volume have published the results of their research over the last ten years. However, as in any work of this type, an effort has been made to revise many proofs and to place many others in a correct setting. Our research was made possible by partial support of the D.G.I.C.y'T. and the Junta de Andalucia.

Measures of Noncompactness and Condensing Operators

Measures of Noncompactness and Condensing Operators PDF Author: R. R. Akhmerov
Publisher: Birkhauser
ISBN: 9780817627164
Category : Mathematics
Languages : en
Pages : 249

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Book Description


Measure of Noncompactness, Fixed Point Theorems, and Applications

Measure of Noncompactness, Fixed Point Theorems, and Applications PDF Author: S. A. Mohiuddine
Publisher: CRC Press
ISBN: 1040013368
Category : Mathematics
Languages : en
Pages : 222

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Book Description
The theory of the measure of noncompactness has proved its significance in various contexts, particularly in the study of fixed point theory, differential equations, functional equations, integral and integrodifferential equations, optimization, and others. This edited volume presents the recent developments in the theory of the measure of noncompactness and its applications in pure and applied mathematics. It discusses important topics such as measures of noncompactness in the space of regulated functions, application in nonlinear infinite systems of fractional differential equations, and coupled fixed point theorem. Key Highlights: • Explains numerical solution of functional integral equation through coupled fixed point theorem, measure of noncompactness and iterative algorithm • Showcases applications of the measure of noncompactness and Petryshyn’s fixed point theorem functional integral equations in Banach algebra • Explores the existence of solutions of the implicit fractional integral equation via extension of the Darbo’s fixed point theorem • Discusses best proximity point results using measure of noncompactness and its applications • Includes solvability of some fractional differential equations in the holder space and their numerical treatment via measures of noncompactness This reference work is for scholars and academic researchers in pure and applied mathematics.

Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations

Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations PDF Author: Józef Banaś
Publisher: Springer
ISBN: 8132218868
Category : Mathematics
Languages : en
Pages : 323

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Book Description
This book deals with the study of sequence spaces, matrix transformations, measures of noncompactness and their various applications. The notion of measure of noncompactness is one of the most useful ones available and has many applications. The book discusses some of the existence results for various types of differential and integral equations with the help of measures of noncompactness; in particular, the Hausdorff measure of noncompactness has been applied to obtain necessary and sufficient conditions for matrix operators between BK spaces to be compact operators. The book consists of eight self-contained chapters. Chapter 1 discusses the theory of FK spaces and Chapter 2 various duals of sequence spaces, which are used to characterize the matrix classes between these sequence spaces (FK and BK spaces) in Chapters 3 and 4. Chapter 5 studies the notion of a measure of noncompactness and its properties. The techniques associated with measures of noncompactness are applied to characterize the compact matrix operators in Chapters 6. In Chapters 7 and 8, some of the existence results are discussed for various types of differential and integral equations, which are obtained with the help of argumentations based on compactness conditions.

Measures of Noncompactness and Condensing Operators

Measures of Noncompactness and Condensing Operators PDF Author: Akhmerov
Publisher: Birkhäuser
ISBN: 9783034857284
Category : Science
Languages : en
Pages : 252

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Book Description
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. The degree of noncompactness of a set is measured by means of functions called measures of noncompactness. The contractive maps and the compact maps [i.e., in this Introduction, the maps that send any bounded set into a relatively compact one; in the main text the term "compact" will be reserved for the operators that, in addition to having this property, are continuous, i.e., in the authors' terminology, for the completely continuous operators] are condensing. 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 non-relatively com pact sets. 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 natural measures of noncompactness. The linear condensing operators are characterized by the fact that almost all of their spectrum is included in a disc of radius smaller than one. The examples given above show that condensing operators are a sufficiently typical phenomenon in various applications of functional analysis, for example, in the theory of differential and integral equations. As is turns out, the condensing operators have properties similar to the compact ones.

Mathematical Analysis of Infectious Diseases

Mathematical Analysis of Infectious Diseases PDF Author: Praveen Agarwal
Publisher: Academic Press
ISBN: 0323904580
Category : Science
Languages : en
Pages : 346

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Book Description
Mathematical Analysis of Infectious Diseases updates on the mathematical and epidemiological analysis of infectious diseases. Epidemic mathematical modeling and analysis is important, not only to understand disease progression, but also to provide predictions about the evolution of disease. One of the main focuses of the book is the transmission dynamics of the infectious diseases like COVID-19 and the intervention strategies. It also discusses optimal control strategies like vaccination and plasma transfusion and their potential effectiveness on infections using compartmental and mathematical models in epidemiology like SI, SIR, SICA, and SEIR. The book also covers topics like: biodynamic hypothesis and its application for the mathematical modeling of biological growth and the analysis of infectious diseases, mathematical modeling and analysis of diagnosis rate effects and prediction of viruses, data-driven graphical analysis of epidemic trends, dynamic simulation and scenario analysis of the spread of diseases, and the systematic review of the mathematical modeling of infectious disease like coronaviruses. - Offers analytical and numerical techniques for virus models - Discusses mathematical modeling and its applications in treating infectious diseases or analyzing their spreading rates - Covers the application of differential equations for analyzing disease problems - Examines probability distribution and bio-mathematical applications

Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces

Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces PDF Author: Mikhail I. Kamenskii
Publisher: Walter de Gruyter
ISBN: 3110870894
Category : Mathematics
Languages : en
Pages : 245

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Book Description
The theory of set-valued maps and of differential inclusion is developed in recent years both as a field of his own and as an approach to control theory. The book deals with the theory of semilinear differential inclusions in infinite dimensional spaces. In this setting, problems of interest to applications do not suppose neither convexity of the map or compactness of the multi-operators. These assumption implies the development of the theory of measure of noncompactness and the construction of a degree theory for condensing mapping. Of particular interest is the approach to the case when the linear part is a generator of a condensing, strongly continuous semigroup. In this context, the existence of solutions for the Cauchy and periodic problems are proved as well as the topological properties of the solution sets. Examples of applications to the control of transmission line and to hybrid systems are presented.

Advances in Differential and Difference Equations with Applications 2020

Advances in Differential and Difference Equations with Applications 2020 PDF Author: Dumitru Baleanu
Publisher: MDPI
ISBN: 3039368702
Category : Computers
Languages : en
Pages : 348

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Book Description
It is very well known that differential equations are related with the rise of physical science in the last several decades and they are used successfully for models of real-world problems in a variety of fields from several disciplines. Additionally, difference equations represent the discrete analogues of differential equations. These types of equations started to be used intensively during the last several years for their multiple applications, particularly in complex chaotic behavior. A certain class of differential and related difference equations is represented by their respective fractional forms, which have been utilized to better describe non-local phenomena appearing in all branches of science and engineering. The purpose of this book is to present some common results given by mathematicians together with physicists, engineers, as well as other scientists, for whom differential and difference equations are valuable research tools. The reported results can be used by researchers and academics working in both pure and applied differential equations.

Sequence Space Theory with Applications

Sequence Space Theory with Applications PDF Author: S. A. Mohiuddine
Publisher: CRC Press
ISBN: 1000610047
Category : Mathematics
Languages : en
Pages : 307

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Book Description
The book features original chapters on sequence spaces involving the idea of ideal convergence, modulus function, multiplier sequences, Riesz mean, Fibonacci difference matrix etc., and illustrate their involvement in various applications. The preliminaries have been presented in the beginning of each chapter and then the advanced discussion takes place, so it is useful for both expert and nonexpert on aforesaid topics. The book consists of original thirteen research chapters contributed by the well-recognized researchers in the field of sequence spaces with associated applications. Features Discusses the Fibonacci and vector valued difference sequence spaces Presents the solution of Volterra integral equation in Banach algebra Discusses some sequence spaces involving invariant mean and related to the domain of Jordan totient matrix Presents the Tauberian theorems of double sequences Discusses the paranormed Riesz difference sequence space of fractional order Includes a technique for studying the existence of solutions of infinite system of functional integro-differential equations in Banach sequence spaces The subject of book is an active area of research of present time internationally and would serve as a good source for researcher and educators involved with the topic of sequence spaces.