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.

I General Introduction.- §1 Fredholm and Volterra equations.- 1.1 Fredholm equations.- 1.2 Equations with a weak singularity.- 1.3 Volterra equations.- §2 Other classes of integral equations.- 2.1 Equations with convolution kernels.- 2.2 The Wiener-Hopf equations.- 2.3 Dual equations.- 2.4 Integral transforms.- 2.5 Singular integral equations.- 2.6 Non-linear integral equations.- §3 Some inversion formulas.- 3.1 The inversion of integral transforms.- 3.2 Inversion formulas for equations with convolution kernels.- 3.3 Volterra’s equations with one independent variable and a convolution kernel.- 3.4 The Abel equation.- 3.5 Integral equations with kernels defined by hypergeometric functions.- II The Fredholm Theory.- §1 Basic concepts and the Fredholm theorems.- 1.1 Basic concepts.- 1.2 The basic theorems.- §2 The solution of Fredholm equations: The method of successive approximation.- 2.1 The construction of approximations: The Neumann series.- 2.2 The resolvent kernel.- §3 The solution of Fredholm equations: Degenerate equations and the general case.- 3.1 Equations with degenerate kernels.- 3.2 The general case.- §4 The Fredholm resolvent.- 4.1 The Fredholm resolvent.- 4.2 Properties of the resolvent.- §5 The solution of Fredholm equations: The Fredholm series.- 5.1 The Fredholm series. Fredholm determinants and minors.- 5.2 The representation of the eigenfunctions of a kernel in terms of the minors of Fredholm.- §6 Equations with a weak singularity.- 6.1 Boundedness of the integral operator with a weak singularity.- 6.2 Iteration of a kernel with a weak singularity.- 6.3 The method of successive approximation.- §7 Systems of integral equations.- 7.1 The vector form for systems of integral equations.- 7.2 Methods of solution for Fredholm kernels.- 7.3 Methods of solution for kernels with a weak singularity.- §8 The structure of the resolvent in the neighbourhood of a characteristic value.- 8.1 Orthogonal kernels.- 8.2 The principal kernels.- 8.3 The canonical kernels.- §9 The rate of growth of eigenvalues.- III Symmetric Equations.- §1 Basic properties.- 1.1 Symmetric kernels.- 1.2 Basic theorems connected with symmetric kernels.- 1.3 Systems of characteristic values and eigenfunctions.- 1.4 Orthogonalization.- §2 The Hilbert-Schmidt series and its properties.- 2.1 Hilbert-Schmidt theorem.- 2.2 The solution of symmetric integral equations.- 2.3 The resolvent of a symmetric kernel.- 2.4 The bilinear series of a kernel and its iterations.- §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.- 6.1 Symmetric equations.- 6.2 Non-symmetric equations.- IV Integral Equations with Non-Negative Kernels.- §1 Positive eigenvalues.- 1.1 The formulation of the problem.- 1.2 The kernels to be examined.- 1.3 Existence of a positive eigenfunction.- 1.4 The comparison of positive eigenvalues with other eigenvalues.- 1.5 The multiplicity of the positive eigenvalue.- 1.6 Stochastic kernels.- 1.7 Notes.- §2 Positive solutions of the non-homogeneous equation.- 2.1 Existence of a positive solution.- 2.2 Convergence of successive approximations.- 2.3 Note.- §3 Estimates for the spectral radius.- 3.1 Formulation of the problem.- 3.2 Upper estimates.- 3.3 The general method.- 3.4 The block method.- 3.5 Supplementary notes.- §4 Oscillating kernels.- 4.1 The formulation of the problem.- 4.2 Oscillating matrices.- 4.3 Vibrations of an elastic continuum with a discrete distribution of mass.- 4.4 Oscillating kernels.- 4.5 Small vibrations of systems with infinite degrees of freedom.- V Continuous and Compact Linear Operators.- §1 Continuity and compactness for linear integral operators.- 1.1 The formulation of the problem.- 1.2 Linear integral operators with their range in C.- 1.3 General properties of integral operators in the Lp-spaces.- 1.4 L-characteristics of linear integral operators.- 1.5 Linear U-bounded operators.- 1.6 Linear U-cobounded operators.- 1.7 Theorems with two conditions.- 1.8 A subtle continuity and compactness condition.- 1.9 Some particular classes of integral operators.- 1.10 Additional notes.- §2 Equations of the second kind. The resolvent of an integral operator.- 2.1 The formulation of the problem.- 2.2 The resolvent of a linear operator arid the spectrum.- 2.3 The space of kernels.- 2.4 The resolvent of a linear integral operator.- 2.5 The resolvent of ‘improving’ operators.- 2.6 Conditions for unique solvability.- 2.7 Equations with iterated kernels.- 2.8 The conjugate equation.- 2.9 Supplementary notes.- §3 Equations of the second kind with compact operators in a Banach space.- 3.1 The formulation of the problem.- 3.2 The spectrum of a compact operator.- 3.3 The splitting of compact operators.- 3.4 The spectrum of a compact integral operator.- 3.5 Fredholm theorems.- 3.6 The resolvent of a compact operator.- 3.7 Equations with improving operators.- 3.8 Supplementary notes.- §4 Equations of the second kind with compact operators in a Hilbert space.- 4.1 Preliminary notes.- 4.2 Equations with self-adjoint operators.- 4.3 The resolvent and the spectrum of a self-adjoint integral operator.- 4.4 Hilbert-Schmidt operators.- 4.5 Mercer operators.- 4.6 Self-adjoint operators with their range in a Banach space.- 4.7 Positive definite self-adjoint operators.- 4.8 Hilbert-Schmidt decomposition for compact operators.- §5 Positive operators.- 5.1 Semi-ordered spaces.- 5.2 General theorems relating to positive operators.- 5.3 Estimates for the spectral radius.- 5.4 The non-homogeneous equation.- 5.5 Existence of an eigenvector.- 5.6 Properties of the eigenvalue ?(K).- 5.7 Eigenvalues of the conjugate operator.- 5.8 Operators which leave a miniedral cone invariant.- §6 Volterra equations of the second kind.- 6.1 The formulation of the problem.- 6.2 Basic theorems.- 6.3 Supplementary notes.- §7 Equations of the first kind.- 7.1 The formulation of the problem.- 7.2 Equations in Hilbert space.- 7.3 The regularization method.- VI One-Dimensional Singular Equations.- §1 Basic notions.- 1.1 The singular integral.- 1.2 The singular Cauchy integral.- 1.3 The singular Hilbert integral.- §2 Some properties of singular integrals.- 2.1 Assumptions about the contour.- 2.2 On the existence of singular Cauchy integrals.- 2.3 Limit formulas.- 2.4 Integration by parts.- 2.5 The distortion of the removed arc.- 2.6 Change of variable.- §3 Singular operators in functional spaces.- 3.1 The singular operator.- 3.2 Invariant spaces.- 3.3 Non-Liapunov contours.- 3.4 Further theorems about invariant spaces.- §4 Differentiation and integration formulas involving singular integrals.- 4.1 Differentiation formulas.- 4.2 Integration formulas.- §5 Regularization.- 5.1 Right and left regularization.- 5.2 The index of an operator.- §6 Closed contours; symbols; Nöther theorems.- 6.1 The general singular operator.- 6.2 The symbol of a singular operator.- 6.3 Nöther theorems.- 6.4 Singular equations with Hilbert kernels.- 6.5 Supplementary notes.- §7 The Carleman method for a closed contour.- 7.1 Reduction of a singular equation to a boundary value problem.- 7.2 The solution of the boundary value problem.- §8 Systems of singular equations defined on a closed contour.- 8.1 The singular operator of a system.- 8.2 The symbol.- 8.3 Nöther theorems.- §9 The open contour case.- 9.1 Example.- 9.2 The general case.- 9.3 Supplementary notes.- §10 Tricomi and Gellerstedt equations.- 10.1 The formulation of the problem.- 10.2 Reduction to a boundary value problem.- 10.3 Solution of the homogeneous problem.- 10.4 Solution of the non-homogeneous problem.- §11 Equations with degenerate symbol.- 11.1 Unbounded regularization.- 11.2 The general singular equation.- 11.3 Systems of singular equations.- §12 Singular equations in generalized function spaces.- 12.1 Equations with a non-degenerate symbol.- 12.2 Equations with a degenerate symbol.- VII The Integral Equations of Mathematical Physics.- §1 The integral equations of potential theory.- 1.1 The integral equations of the Dirichlet and Neumann problems, for simply connected boundaries.- 1.2 The Robin problem.- 1.3 The external Dirichlet problem.- 1.4 The case of a disconnected boundary.- 1.5 Mixed problems in potential theory.- 1.6 The distribution of the characteristic values of the integral equations of potential theory.- 1.7 Extension to multi-dimensional spaces.- §2 The application of complex variable to the problems of potential theory in plane regions.- 2.1 Dirichlet’s problem for a simply connected plane region.- 2.2 Dirichlet’s problem for multi-connected regions.- 2.3 The Neumann problem.- 2.4 The conformal mapping of multi-connected regions.- 2.5 The Green’s function and the Schwarz kernel.- §3 The biharmonic equation and the plane problem in the theory of elasticity.- 3.1 The application of Green’s function. The formulation and the study of the plane problem of the theory of elasticity.- 3.2 Simply connected regions.- 3.3 Application of Cauchy-type integrals. Muskhelishvili’s equations.- 3.4 Lauricella-Sherman equations.- 3.5 Periodic problem of the theory of elasticity.- 3.6 The characteristic values of the integral equations of the theory of elasticity.- §4 Potentials for the heat conduction equation.- 4.1 Integral equations for heat conduction.- 4.2 An examination of the integral equations of heat potentials and the convergence of the method of successive approximations.- §5 The generalized Schwarz algorithm.- 5.1 The general formulation and the convergence of the algorithm for the plane problem of potential theory.- 5.2 Application to three-dimensional problems.- 5.3 Applications to the theory of elasticity.- §6 Application of the theory of symmetric integral equations.- 6.1 The Sturm-Liouville problem.- 6.2 The fundamental vibrations of a string.- 6.3 The stability of a rod under compression.- 6.4 The fundamental vibrations of a membrane.- 6.5 The pressure of a rigid stamp on an elastic half-space.- §7 Certain applications of singular integral equations.- 7.1 Mixed problem of potential theory.- 7.2 Mixed problems in the half-plane.- 7.3 The problem of two elastic half-planes in contact.- 7.4 The pressure of a rigid stamp on an elastic half-plane.- 7.5 The mixed problem of the theory of elasticity.- 7.6 The problem of flow past an arc of given shape.- VIII Integral Equations with Convolution Kernels.- §1 General introduction.- 1.1 The basic equation and special cases.- 1.2 The symbol. Conditions for normal solvability.- 1.3 Equations with elementary solutions.- 1.4 Equations which reduce to the form (8.1).- 1.5 Function spaces.- §2 Examples.- 2.1 The basic problem of the theory of radiation.- 2.2 The problem of linear smoothing and forecasting.- 2.3 The electromagnetic coastal effect.- 2.4 A problem in the theory of hereditary elasticity.- 2.5 The potential of a conducting disk.- §3 Equations defined on a semi-infinite interval with summable kernels.- 3.1 The solvability conditions.- 3.2 Factorization.- 3.3 Solution of the non-homogeneous equation.- 3.4 Solution of the homogeneous equation.- §4 Dual equations with summable kernels and their adjoints.- 4.1 Reduction of the dual equation to an equivalent equation defined on a semi-infinite interval.- 4.2 The formula for the index and properties of the basis for the solutions of the homogeneous equation.- 4.3 The equation adjoint to the dual.- §5 Examples.- §6 Dual equations with kernels of exponential type.- 6.1 Reduction to a boundary value problem.- 6.2 Case 1.- 6.3 Case 2.- §7 The Wiener-Hopf method.- 7.1 A description of the method.- 7.2 The reduction of (8.14) to a boundary value problem.- 7.3 An example.- §8 Equations with degenerate symbol.- 8.1 The Riemann problem.- 8.2 The Wiener-Hopf equation of the second kind.- 8.3 The Wiener-Hopf equation of the first kind.- 8.4 The dual equation of the second kind.- §9 Examples.- §10 Systems of equations on a semi-infinite interval.- 10.1 The basic assumptions.- 10.2 Factorization of matrix-functions.- 10.3 The solvability conditions.- 10.4 Dual equations.- 10.5 Kernels with an exponential decay at infinity.- §11 Equations defined on a finite interval.- 11.1 Reduction to a boundary value problem.- 11.2 Kernels with rational Fourier transforms.- 11.3 Reduction to a differential equation with constant coefficients.- 11.4 Eigenvalues.- IX Multidimensional Singular Equations.- §1 Basic concepts and theorems.- 1.1 Notations.- 1.2 The singular integral.- 1.3 Existence conditions for singular integrals.- §2 The symbol.- 2.1 The singular operator.- 2.2 The symbol of a singular operator.- 2.3 Formulas for the determination of the symbol.- 2.4 Some new concepts.- 2.5 Variation in the symbol with respect to changes in the independent variables.- 2.6 An integral representation for the operator A in terms of its symbol.- §3 Singular operators in Lp(Em).- 3.1 Boundedness conditions for singular operators.- 3.2 The decomposition of the simplest singular operator into a series.- 3.3 The multiplication rule for symbols.- 3.4 The operator conjugate to a singular operator.- §4 Singular integrals over a manifold.- 4.1 The definition of a singular operator and its symbol.- 4.2 Another definition for the singular operator.- §5 Regularization and Fredholm theorems.- §6 Systems of singular equations.- 6.1 Matrix singular operators.- 6.2 The symbol matrix.- 6.3 The index.- §7 Singular equations in Lipschitz spaces.- §8 Singular equations on a cylinder.- §9 Singular equations in spaces of generalized functions.- §10 Equations with 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.- 1.1 The basic concepts of the theory of non-linear operators.- 1.2 Urison operators with their range in the space C.- 1.3 Hammerstein operators with range in Lq.- 1.4 The continuity of Urison operators with range in Lq.- 1.5 The complete continuity of Urison operators with range in Lq.- 1.6 Some special conditions.- 1.7 Operators with range in L?.- 1.8 Continuity and complete continuity for other types of integral operators.- 1.9 Derivatives of non-linear operators.- 1.10 Derivatives of the Hammerstein operator.- 1.11 Auxiliary theorems about superposition operators.- 1.12 The differentiability of Urison operators.- 1.13 Continuous differentiability of Urison operators.- 1.14 Higher order derivatives.- 1.15 Analytic operators.- 1.16 Asymptotically linear operators.- 1.17 Supplementary notes.- §2 The existence and uniqueness of solutions.- 2.1 The formulation of the problem.- 2.2 Equations with operators satisfying a Lipschitz condition.- 2.3 Equations with completely continuous operators.- 2.4 The use of upper estimates.- 2.5 Equations with asymptotically linear operators.- 2.6 Variational methods.- 2.7 The existence of non-zero solutions.- 2.8 The existence of a positive solution.- 2.9 Equations with concave non-linearities.- 2.10 Equations with a parameter.- 2.11 Supplementary notes.- §3 The extension and bifurcation of solutions of non-linear integral equations.- 3.1 The formulation of the problem.- 3.2 The basic theorem for implicit functions.- 3.3 Differential properties of an implicit function.- 3.4 The analyticity of solutions.- 3.5 The bifurcation equation.- 3.6 The Nekrasov-Nazarov method.- 3.7 The bifurcation points.- 3.8 Supplementary notes about bifurcation points.- References.