Integral equations—a reference text

The title 'Integral equations' covers many things which have very little connection with each other. However, they are united by the following important feature. In most cases, the equations involve an unknown function operated on by a bounded and often compact operator defined on some Banach space. The aim of the book is to list the main results concerning integral equations. The classical Fredholm theory and Hilbert-Schmidt theory are presented in Chapters II and III. The preceding Chapter I contains a description of the most important types of integral equations which can be solved in 'closed' form. Chapter IV is an important addition to Chapters II and III, as it contains the theory of integral equations with non-negative kernels. The development of this theory is mainly due to M. G. Krein. The content of the first four chapters is fairly elementary. It is well known that the Fredholm theory has been generalized for equations with compact operators. Chapter V is devoted tothis generalization. In Chapter VI one-dimensional (i.e. with one dependent variable) singular integral equations are considered. The last type of equations differ from that considered in the preceding chapters in that singular integral operators are not compact but only bounded in the usual functional spaces.

Table of Contents:
I General Introduction.- §1 Fredholm and Volterra equations.- §2 Other classes of integral equations.- §3 Some inversion formulas.- II The Fredholm Theory.- §1 Basic concepts and the Fredholm theorems.- §2 The solution of Fredholm equations: The method of successive approximation.- §3 The solution of Fredholm equations: Degenerate equations and the general case.- §4 The Fredholm resolvent.- §5 The solution of Fredholm equations: The Fredholm series.- §6 Equations with a weak singularity.- §7 Systems of integral equations.- §8 The structure of the resolvent in the neighbourhood of a characteristic value.- §9 The rate of growth of eigenvalues.- III Symmetric Equations.- §1 Basic properties.- §2 The Hilbert-Schmidt series and its properties.- §3 The classification of symmetric kernels.- §4 Extremal properties of characteristic values and eigenfunctions.- §5 Schmidt kernels and bilinear series for non-symmetric kernels.- §6 The solution of integral equations of the first kind.- IV Integral Equations with Non-Negative Kernels.- §1 Positive eigenvalues.- §2 Positive solutions of the non-homogeneous equation.- §3 Estimates for the spectral radius.- §4 Oscillating kernels.- V Continuous and Compact Linear Operators.- §1 Continuity and compactness for linear integral operators.- §2 Equations of the second kind. The resolvent of an integral operator.- §3 Equations of the second kind with compact operators in a Banach space.- §4 Equations of the second kind with compact operators in a Hilbert space.- §5 Positive operators.- §6 Volterra equations of the second kind.- §7 Equations of the first kind.- VI One-Dimensional Singular Equations.- §1 Basic notions.- §2 Some properties of singular integrals.- §3 Singular operators in functional spaces.- §4Differentiation and integration formulas involving singular integrals.- §5 Regularization.- §6 Closed contours; symbols; Nöther theorems.- §7 The Carleman method for a closed contour.- §8 Systems of singular equations defined on a closed contour.- §9 The open contour case.- §10 Tricomi and Gellerstedt equations.- §11 Equations with degenerate symbol.- §12 Singular equations in generalized function spaces.- VII The Integral Equations of Mathematical Physics.- §1 The integral equations of potential theory.- §2 The application of complex variable to the problems of potential theory in plane regions.- §3 The biharmonic equation and the plane problem in the theory of elasticity.- §4 Potentials for the heat conduction equation.- §5 The generalized Schwarz algorithm.- §6 Application of the theory of symmetric integral equations.- §7 Certain applications of singular integral equations.- VIII Integral Equations with Convolution Kernels.- §1 General introduction.- §2 Examples.- §3 Equations defined on a semi-infinite interval with summable kernels.- §4 Dual equations with summable kernels and their adjoints.- §5 Examples.- §6 Dual equations with kernels of exponential type.- §7 The Wiener-Hopf method.- §8 Equations with degenerate symbol.- §9 Examples.- §10 Systems of equations on a semi-infinite interval.- §11 Equations defined on a finite interval.- IX Multidimensional Singular Equations.- §1 Basic concepts and theorems.- §2 The symbol.- §3 Singular operators in Lp(Em).- §4 Singular integrals over a manifold.- §5 Regularization and Fredholm theorems.- §6 Systems of singular equations.- §7 Singular equations in Lipschitz spaces.- §8 Singular equations on a cylinder.- §9 Singular equations in spaces of generalized functions.- §10 Equationswith degenerate symbol.- §11 Singular integro-differential equations.- §12 Singular equations on a manifold with boundary.- X Non-Linear Integral Equations.- §1 Non-linear integral operators.- §2 The existence and uniqueness of solutions.- §3 The extension and bifurcation of solutions of non-linear integral equations.- References.