Integral Equations--A Reference Text

I General Introduction.- x1 Fredholm and Volterra equations.- 1.1 Fredholm equations.- 1.2 Equations with a weak singularity.- 1.3 Volterra equations.- x2 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.- x3 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.- x1 Basic concepts and the Fredholm theorems.- 1.1 Basic concepts.- 1.2 The basic theorems.- x2 The solution of Fredholm equations: The method of successive approximation.- 2.1 The construction of approximations: The Neumann series.- 2.2 The resolvent kernel.- x3 The solution of Fredholm equations: Degenerate equations and the general case.- 3.1 Equations with degenerate kernels.- 3.2 The general case.- x4 The Fredholm resolvent.- 4.1 The Fredholm resolvent.- 4.2 Properties of the resolvent.- x5 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.- x6 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.- x7 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.- x8 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.- x9 The rate of growth of eigenvalues.- III Symmetric Equations.- x1 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.- x2 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.- x3 The classification of symmetric kernels.- x4 Extremal properties of characteristic values and eigenfunctions.- x5 Schmidt kernels and bilinear series for non-symmetric kernels.- x6 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.- x1 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.- x2 Positive solutions of the non-homogeneous equation.- 2.1 Existence of a positive solution.- 2.2 Convergence of successive approximations.- 2.3 Note.- x3 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.- x4 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.- x1 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.- x2 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.- x3 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.- x4 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.- x5 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.- x6 Volterra equations of the second kind.- 6.1 The formulation of the problem.- 6.2 Basic theorems.- 6.3 Supplementary notes.- x7 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.- x1 Basic notions.- 1.1 The singular integral.- 1.2 The singular Cauchy integral.- 1.3 The singular Hilbert integral.- x2 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.- x3 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.- x4 Differentiation and integration formulas involving singular integrals.- 4.1 Differentiation formulas.- 4.2 Integration formulas.- x5 Regularization.- 5.1 Right and left regularization.- 5.2 The index of an operator.- x6 Closed contours; symbols; Nother theorems.- 6.1 The general singular operator.- 6.2 The symbol of a singular operator.- 6.3 Nother theorems.- 6.4 Singular equations with Hilbert kernels.- 6.5 Supplementary notes.- x7 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.- x8 Systems of singular equations defined on a closed contour.- 8.1 The singular operator of a system.- 8.2 The symbol.- 8.3 Nother theorems.- x9 The open contour case.- 9.1 Example.- 9.2 The general case.- 9.3 Supplementary notes.- x10 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.- x11 Equations with degenerate symbol.- 11.1 Unbounded regularization.- 11.2 The general singular equation.- 11.3 Systems of singular equations.- x12 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.- x1 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.- x2 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.- x3 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.- x4 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.- x5 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.- x6 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.- x7 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.- x1 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.- x2 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.- x3 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.- x4 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.- x5 Examples.- x6 Dual equations with kernels of exponential type.- 6.1 Reduction to a boundary value problem.- 6.2 Case 1.- 6.3 Case 2.- x7 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.- x8 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.- x9 Examples.- x10 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.- x11 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.- x1 Basic concepts and theorems.- 1.1 Notations.- 1.2 The singular integral.- 1.3 Existence conditions for singular integrals.- x2 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.- x3 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.- x4 Singular integrals over a manifold.- 4.1 The definition of a singular operator and its symbol.- 4.2 Another definition for the singular operator.- x5 Regularization and Fredholm theorems.- x6 Systems of singular equations.- 6.1 Matrix singular operators.- 6.2 The symbol matrix.- 6.3 The index.- x7 Singular equations in Lipschitz spaces.- x8 Singular equations on a cylinder.- x9 Singular equations in spaces of generalized functions.- x10 Equations with degenerate symbol.- x11 Singular integro-differential equations.- x12 Singular equations on a manifold with boundary.- X Non-Linear Integral Equations.- x1 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.- x2 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.- x3 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.