Overview

The investigation of many mathematical problems is significantly simplified if it is possible to reduce them to equations involving continuous or com­ pletely continuous operators in function spaces. In particular, this is true for non-linear boundary value problems and for integro-differential and integral equations. To effect a transformation to equations with continuous or completely continuous operators, it is usually necessary to reduce the original problem to one involving integral equations. Here, negative and fractional powers of those unbounded differential operators which constitute 'principal parts' of the original problem, are used in an essential way. Next there is chosen or constructed a function space in which the corresponding integral oper­ ator possesses sufficiently good properties. Once such a space is found, the original problem can often be analyzed by applying general theorems (Fredholm theorems in the study of linear equations, fixed point principles in the study of non-linear equations, methods of the theory of cones in the study of positive solutions, etc.). In other words, the investigation of many problems is effectively divided into three independent parts: transformation to an integral equation, investi­ gation of the corresponding integral expression as an operator acting in function spaces, and, finally, application of general methods of functional analysis to the investigation of the linear and non-linear equations.

Full Product Details

Author:   M.A. Krasnosel'skii ,  P.P. Zabreyko ,  E.I. Pustylnik ,  P.E. Sobolevski
Publisher:   Springer
Imprint:   Springer
Edition:   Softcover reprint of the original 1st ed. 1976
Volume:   1
Dimensions:   Width: 15.20cm , Height: 2.70cm , Length: 22.90cm
Weight:   0.782kg
ISBN:  

9789401015448

ISBN 10:   9401015449
Pages:   536
Publication Date:   08 November 2011
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

1. Linear operators in L? spaces.- 1. The space L?.- 2. Continuous linear operators.- 3. Compact linear operators.- 2. Continuity and compactness of linear integral operators.- 4. General theorems on continuity on integral operators.- 5. General theorems on compactness of integral operators.- 6. Linear Uo-bounded operators.- 7. Integral operators with kernels satisfying conditions of kantorovic type.- 8. Operators of potential type.- 3. Fractional powers of selfadjoint operators.- 9. Splitting of linear operators.- 10. Fractional powers of bounded operators.- 11. Unbounded selfadjoint operators.- 12. Properties of fractional powers of unbounded operators.- 4. Fractional powers of operators of positive type.- 13. Semi-groups of operators.- 14. Fractional powers of positive-type operators.- 15. Moment inequalities and L-characteristics of fractional powers.- 16. Fractional powers of elliptic operators.- 5. Non-linear integral operators.- 17. The superposition operators.- 18. Conditions for continuity of integral operators.- 19. Conditions for complete continuity of an Uryson operator.- 20. Differentiation of non-linear operators.- 6. Some applications.- 21. Equations with completely continuous operators.- 22. Convergence of Fouriers’ method.- 23. Translation operators along trajectories of differential equations.- Index of terminologies.- Index of notations.- Author index.

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