Linear Equations in Banach Spaces

Description

This slim volume contains a general theory of linear equations in Banach spaces with applications to differential and integral equations. It contains a detailed analysis of the uniqueness and correctness problem of over-determined and under-determined equations, connections between the equation and its conjugate. Different transformations of linear equations are considered. Some sections deal with problems of stability and apriori estimates. Different classes of equations, such as Fredholm equations, Noether equations and others are considered. Applications are given to Fredholm integral equations, Volterra integral equations, boundary problems for linear differential equations of hiaher orders and boundary problems for elliptic partial differential equations. To understand the book it is enouh to have a limited knowledqe of the basics of functional analysis. An appendix at the end of the book contains the statements of the theorems of functional analysis which are used in the main text, and this will be found useful to the reader. This book differs from others in its conciseness. It keeps its aim clearly in sight and does not stray from that aim; it is systematic and not difficult to read. It is uncanny how much material the author was able to include ln this slim volume. The author is an outstanding expert in differential equations, the theory of linear operators and applied mathematics. This fact influences the choice of material, the presentation and terminology. One clearly feels the applied motives in the book. The book may be recommended for senior underqraduate students and graduates in mathematics and related fields. In my opinion it is of interest to anybody dealinq with linear equations. Contents:PREFACE . . . . . . INTRODUCTION . . . . . . 1. LINEAR EQUATONS. BASIC NOTIONS. 2. EQUATIONS WITH A CLOSED OPERATOR 3. THE ADJOINT EQUATION . . . . . . . . . . . 4. THE EQUATION ADJOINT TO THE FACTORED EQUATION . .... 5. AN EQUATION WITH A CLOSED OPERATOR WHICH HAS A DENSE DOMAIN 6. NORMALLY SOLVABLE EQUATIONS WITH FINITE DIMENSIONAL KERNEL. 7. A PRIORI ESTIMATES . . . . . . . . . . . . . . . . . 8. EQUATIONS WITH FINITE DEFECT . . . . . . . 9. SOME DIFFERENT ADJOINT EQUATIONS . . . 10. LINEAR TRANSFORMATIONS OF EQUATIONS. . . . . . . 11. TRANSFORMATIONS OF d-NORMAL EQUATIONS. . . . . . . . . 12. NOETHERIAN EQUATIONS. INDEX. . . . . . . . . . . 13. EQUATIONS WITH OPERATORS WHICH ACT IN A SINGLE SPACE 14. FREDHOLM EQUATIONS. REGULARIZATION OF EQUATIONS. . . . . . 15. LINEAR CHANGES OF VARIABLE . . . . . . . . . . 16. STABILITY OF THE PROPERTIES OF AN EQUATION 17. OVERDETERMINED EQUATIONS . . . . 18. UNDETERMINED EQUATIONS 19. INTEGRAL EQUATIONS. . . . 20. DIFFERENTIAL EQUATIONS . APPENDIX. BASIC RESULTS FROM FUNCTIONAL ANALYSIS USED IN THE TEXT LITERATURE CITED . . . . . . . . . . . . . . . . . . . . . . . . .