16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 829 16.4. Linear Integral Equations of the Second Kind with Constant Limits of Integration 16.4.1. Fredholm Integral Equations of the Second Kind. Resolvent 16.4.1-1. Some definitions. The eigenfunctions of a Fredholm integral equation. Linear integral equations of the second kind with constant limits of integration have the general form y(x)–λ b a K(x, t)y(t) dt = f(x), (16.4.1.1) where y(x) is the unknown function (a ≤ x ≤ b), K(x, t)isthekernel of the integral equation, and f(x) is a given function, which is called the right-hand side of equation (16.4.1.1). For convenience of analysis, a number λ is traditionally singled out in equation (16.4.1.1), which is called the parameter of integral equation. The classes of functions and kernels under consideration were defined above in Paragraphs 16.3.1-1 and 16.3.1-2. Note that equations of the form (16.4.1.1) with constant limits of integration and with Fredholm kernels or kernels with weak singularity are called Fredholm equations of the second kind and equations with weak singularity of the second kind, respectively. Equation (16.4.1.1) is said to be homogeneous if f(x) ≡ 0 and nonhomogeneous other- wise. A number λ is called a characteristic value of the integral equation (16.4.1.1) if there exist nontrivial solutions of the corresponding homogeneous equation. The nontrivial solutions themselves are called the eigenfunctions of the integral equation corresponding to the characteristic value λ.Ifλ is a characteristic value, the number 1/λ is called an eigenvalue of the integral equation (16.4.1.1). A value of the parameter λ is said to be regular if for this value the above homogeneous equation has only the trivial solution. Sometimes the characteristic values and the eigenfunctions of a Fredholm integral equation are called the characteristic values and the eigenfunctions of the kernel K(x, t). The kernel K(x, t) of the integral equation (16.4.1.1) is called a degenerate kernel if it has the form K(x, t)=g 1 (x)h 1 (t)+···+ g n (x)h n (t), a difference kernel if it depends on the difference of the arguments (K(x, t)=K(x – t)), and a symmetric kernel if it satisfies the condition K(x, t)=K(t, x). The transposed integral equation is obtained from (16.4.1.1) by replacing the kernel K(x, t)byK(t, x). The integral equation of the second kind with difference kernel on the entire axis (a =–∞, b = ∞)andsemiaxis(a = 0, b = ∞) are referred to as an equation of convolution type of the second kind and a Wiener–Hopf integral equation of the second kind, respectively. 16.4.1-2. Structure of the solution. The resolvent. The solution of equation (16.4.1.1) can be presented in the form y(x)=f(x)+λ b a R(x, t; λ)f (t) dt, where the resolvent R(x, t; λ) is independent of f(x) and is determined by the kernel of the integral equation. 830 INTEGRAL EQUATIONS 16.4.2. Fredholm Equations of the Second Kind with Degenerate Kernel 16.4.2-1. Simplest degenerate kernel. Consider Fredholm integral equations of the second kind with the simplest degenerate kernel: y(x)–λ b a g(x)h(t)y(t) dt = f(x), a ≤ x ≤ b.(16.4.2.1) We seek a solution of equation (16.4.2.1) in the form y(x)=f(x)+λAg(x). (16.4.2.2) On substituting the expressions (16.4.2.2) into equation (16.4.2.1), after simple algebraic manipulations we obtain A 1 – λ b a h(t)g(t) dt = b a f(t)h(t) dt.(16.4.2.3) Both integrals occurring in equation (16.4.2.3) are supposed to exist. On the basis of (16.4.2.1) and (16.4.2.3) and taking into account the fact that the unique characteristic value λ 1 of equation (16.4.2.1) is given by the expression λ 1 = b a h(t)g(t) dt –1 ,(16.4.2.4) we obtain the following results: 1 ◦ .Ifλ ≠ λ 1 , then for an arbitrary right-hand side there exists a unique solution of equa- tion (16.4.2.1), which can be written in the form y(x)=f(x)+ λλ 1 f 1 λ 1 – λ g(x), f 1 = b a f(t)h(t) dt.(16.4.2.5) 2 ◦ .Ifλ = λ 1 and f 1 = 0, then any solution of equation (16.4.2.1) can be represented in the form y = f(x)+Cy 1 (x), y 1 (x)=g(x), (16.4.2.6) where C is an arbitrary constant and y 1 (x) is an eigenfunction that corresponds to the characteristic value λ 1 . 3 ◦ .Ifλ = λ 1 and f 1 ≠ 0, then there are no solutions. 16.4.2-2. Degenerate kernel in the general case. In the general case, a Fredholm integral equation of the second kind with degenerate kernel has the form y(x)–λ b a n k=1 g k (x)h k (t) y(t) dt = f (x), n = 2, 3, (16.4.2.7) 16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 831 Let us rewrite equation (16.4.2.7) in the form y(x)=f(x)+λ n k=1 g k (x) b a h k (t)y(t) dt, n = 2, 3, (16.4.2.8) We assume that equation (16.4.2.8) has a solution and introduce the notation A k = b a h k (t)y(t) dt.(16.4.2.9) In this case we have y(x)=f(x)+λ n k=1 A k g k (x), (16.4.2.10) and hence the solution of the integral equation with degenerate kernel is reduced to the definition of the constants A k . Let us multiply equation (16.4.2.10) by h m (x) and integrate with respect to x from a to b. We obtain the following system of linear algebraic equations for the coefficients A k : A m – λ n k=1 s mk A k = f m , m = 1, , n,(16.4.2.11) where s mk = b a h m (x)g k (x) dx, f m = b a f(x)h m (x) dx; m, k = 1, , n.(16.4.2.12) Once we construct a solution of system (16.4.2.11), we obtain a solution of the integral equation with degenerate kernel (16.4.2.7) as well. The values of the parameter λ at which the determinant of system (16.4.2.11) vanishes are characteristic values of the integral equation (16.4.2.7), and it is clear that there are just n such values counted according to their multiplicities. Example. Let us solve the integral equation y(x)–λ π –π (x cos t + t 2 sin x +cosx sin t)y(t) dt = x,–π ≤ x ≤ π. (16.4.2.13) Let us denote A 1 = π –π y(t)costdt, A 2 = π –π t 2 y(t) dt, A 3 = π –π y(t)sintdt, (16.4.2.14) where A 1 , A 2 ,andA 3 are unknown constants. Then equation (16.4.2.13) can be rewritten in the form y(x)=A 1 λx + A 2 λ sin x + A 3 λ cos x + x. (16.4.2.15) On substituting the expression (16.4.2.15) into relations (16.4.2.14), we obtain A 1 = π –π (A 1 λt + A 2 λ sin t + A 3 λ cos t + t)costdt, A 2 = π –π (A 1 λt + A 2 λ sin t + A 3 λ cos t + t)t 2 dt, A 3 = π –π (A 1 λt + A 2 λ sin t + A 3 λ cos t + t)sintdt. 832 INTEGRAL EQUATIONS On calculating the integrals occurring in these equations, we obtain the following system of algebraic equations for the unknowns A 1 , A 2 ,andA 3 : A 1 – λπA 3 = 0, A 2 + 4λπA 3 = 0, –2λπA 1 – λπA 2 + A 3 = 2π. (16.4.2.16) System (16.4.2.16) has the unique solution A 1 = 2λπ 2 1 + 2λ 2 π 2 , A 2 =– 8λπ 2 1 + 2λ 2 π 2 , A 3 = 2π 1 + 2λ 2 π 2 . On substituting the above values of A 1 , A 2 ,andA 3 into (16.4.2.15), we obtain the solution of the original integral equation: y(x)= 2λπ 1 + 2λ 2 π 2 (λπx – 4λπ sin x +cosx)+x. 16.4.3. Solution as a Power Series in the Parameter. Method of Successive Approximations 16.4.3-1. Iterated kernels. Consider the Fredholm integral equation of the second kind: y(x)–λ b a K(x, t)y(t) dt = f(x), a ≤ x ≤ b.(16.4.3.1) We seek the solution in the form of a series in powers of the parameter λ: y(x)=f(x)+ ∞ n=1 λ n ψ n (x). (16.4.3.2) Substitute series (16.4.3.2) into equation (16.4.3.1). On matching the coefficients of like powers of λ, we obtain a recurrent system of equations for the functions ψ n (x). The solution of this system yields ψ 1 (x)= b a K(x, t)f(t) dt, ψ 2 (x)= b a K(x, t)ψ 1 (t) dt = b a K 2 (x, t)f(t) dt, ψ 3 (x)= b a K(x, t)ψ 2 (t) dt = b a K 3 (x, t)f(t) dt,etc. Here K n (x, t)= b a K(x, z)K n–1 (z, t) dz,(16.4.3.3) where n = 2, 3, , and we have K 1 (x, t) ≡ K(x, t). The functions K n (x, t)defined by formulas (16.4.3.3) are called iterated kernels. These kernels satisfy the relation K n (x, t)= b a K m (x, s)K n–m (s, t) ds,(16.4.3.4) where m is an arbitrary positive integer less than n. 16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 833 The iterated kernels K n (x, t) can be directly expressed via K(x, t) by the formula K n (x, t)= b a b a ··· b a n–1 K(x, s 1 )K(s 1 , s 2 ) K(s n–1 , t) ds 1 ds 2 ds n–1 . All iterated kernels K n (x, t), beginning with K 2 (x, t), are continuous functions on the square S = {a ≤ x ≤ b, a ≤ t ≤ b} if the original kernel K(x, t) is square integrable on S. If K(x, t) is symmetric, then all iterated kernels K n (x, t) are also symmetric. 16.4.3-2. Method of successive approximations. The results of Subsection 16.4.3-1 can also be obtained by means of the method of successive approximations. To this end, one should use the recurrent formula y n (x)=f(x)+λ b a K(x, t)y n–1 (t) dt, n = 1, 2, , with the zeroth approximation y 0 (x)=f(x). 16.4.3-3. Construction of the resolvent. The resolvent of the integral equation (16.4.3.1) is defined via the iterated kernels by the formula R(x, t; λ)= ∞ n=1 λ n–1 K n (x, t), (16.4.3.5) where the series on the right-hand side is called the Neumann series of the kernel K(x, t). It converges to a unique square integrable solution of equation (16.4.3.1) provided that |λ| < 1 B , B = b a b a K 2 (x, t) dx dt.(16.4.3.6) If, in addition, we have b a K 2 (x, t) dt ≤ A, a ≤ x ≤ b, where A is a constant, then the Neumann series converges absolutely and uniformly on [a, b]. A solution of a Fredholm equation of the second kind of the form (16.4.3.1) is expressed by the formula y(x)=f(x)+λ b a R(x, t; λ)f (t) dt, a ≤ x ≤ b.(16.4.3.7) Inequality (16.4.3.6) is essential for the convergence of the series (16.4.3.5). However, a solution of equation (16.4.3.1) can exist for values |λ| > 1/B as well. 834 INTEGRAL EQUATIONS Example. Let us solve the integral equation y(x)–λ 1 0 xty(t) dt = f(x), 0 ≤ x ≤ 1, by the method of successive approximations. Here we have K(x, t)=xt, a = 0,andb = 1. We successively define K 1 (x, t)=xt, K 2 (x, t)= 1 0 (xz)(zt)dz = xt 3 , K 3 (x, t)= 1 3 1 0 (xz)(zt)dz = xt 3 2 , , K n (x, t)= xt 3 n–1 . According to formula (16.4.3.5) for the resolvent, we obtain R(x, t; λ)= ∞ n=1 λ n–1 K n (x, t)=xt ∞ n=1 λ 3 n–1 = 3xt 3 – λ , where |λ| < 3, and it follows from formula (16.4.3.7) that the solution of the integral equation can be rewritten in the form y(x)=f(x)+λ 1 0 3xt 3 – λ f(t) dt, 0 ≤ x ≤ 1, λ ≠ 3. 16.4.4. Fredholm Theorems and the Fredholm Alternative 16.4.4-1. Fredholm theorems. THEOREM 1. If λ is a regular value, then both the Fredholm integral equation of the second kind and the transposed equation are solvable for any right-hand side, and both the equations have unique solutions. The corresponding homogeneous equations have only the trivial solutions. THEOREM 2. For the nonhomogeneous integral equation to be solvable, it is necessary and sufficient that the right-hand side f(x) satisfies the conditions b a f(x)ψ k (x) dx = 0, k = 1, , n, where ψ k (x) is a complete set of linearly independent solutions of the corresponding transposed homogeneous equation. THEOREM 3. If λ is a characteristic value, then both the homogeneous integral equation and the transposed homogeneous equation have nontrivial solutions. The number of linearly independent solutions of the homogeneous integral equation is finite and is equal to the number of linearly independent solutions of the transposed homogeneous equation. THEOREM 4. A Fredholm equation of the second kind has at most countably many characteristic values, whose only possible accumulation point is the point at infinity. 16.4.4-2. Fredholm alternative. The Fredholm theorems imply the so-called Fredholm alternative, which is most frequently used in the investigation of integral equations. T HE FREDHOLM ALTERNATIVE. Either the nonhomogeneous equation is solvable for any right-hand side or the corresponding homogeneous equation has nontrivial solutions. The first part of the alternative holds if the given value of the parameter is regular and the second if it is characteristic. Remark. The Fredholm theory is also valid for integral equations of the second kind with weak singularity. 16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 835 16.4.5. Fredholm Integral Equations of the Second Kind with Symmetric Kernel 16.4.5-1. Characteristic values and eigenfunctions. Integral equations whose kernels are symmetric, that is, satisfy the condition K(x, t)= K(t, x), are called symmetric integral equations. Each symmetric kernel that is not identically zero has at least one characteristic value. For any n, the set of characteristic values of the nth iterated kernel coincides with the set of nth powers of the characteristic values of the first kernel. The eigenfunctions of a symmetric kernel corresponding to distinct characteristic values are orthogonal, i.e., if ϕ 1 (x)=λ 1 b a K(x, t)ϕ 1 (t) dt, ϕ 2 (x)=λ 2 b a K(x, t)ϕ 2 (t) dt, λ 1 ≠ λ 2 , then (ϕ 1 , ϕ 2 )=0,(ϕ, ψ) ≡ b a ϕ(x)ψ(x) dx. The characteristic values of a symmetric kernel are real. The eigenfunctions can be normalized; namely, we can divide each characteristic func- tion by its norm. If several linearly independent eigenfunctions correspond to the same characteristic value, say, ϕ 1 (x), , ϕ n (x), then each linear combination of these func- tions is an eigenfunction as well, and these linear combinations can be chosen so that the corresponding eigenfunctions are orthonormal. Indeed, the function ψ 1 (x)= ϕ 1 (x) ϕ 1 , ϕ 1 = (ϕ 1 , ϕ 1 ), has the norm equal to one, i.e., ψ 1 = 1. Let us form a linear combination αψ 1 + ϕ 2 and choose α so that (αψ 1 + ϕ 2 , ψ 1 )=0, i.e., α =– (ϕ 2 , ψ 1 ) (ψ 1 , ψ 1 ) =–(ϕ 2 , ψ 1 ). The function ψ 2 (x)= αψ 1 + ϕ 2 αψ 1 + ϕ 2 is orthogonal to ψ 1 (x) and has the unit norm. Next, we choose a linear combination αψ 1 + βψ 2 + ϕ 3 , where the constants α and β can be found from the orthogonality relations (αψ 1 + βϕ 2 + ϕ 3 , ψ 1 )=0,(αψ 1 + βψ 2 + ϕ 3 , ψ 2 )=0. For the coefficients α and β thus defined, the function ψ 3 = αψ 1 + βψ 2 + ϕ 2 αψ 1 + βϕ 2 + ϕ 3 is orthogonal to ψ 1 and ψ 2 and has the unit norm, and so on. . Neumann series converges absolutely and uniformly on [a, b]. A solution of a Fredholm equation of the second kind of the form (16.4.3.1) is expressed by the formula y(x)=f(x)+λ b a R(x, t; λ)f. value. For any n, the set of characteristic values of the nth iterated kernel coincides with the set of nth powers of the characteristic values of the first kernel. The eigenfunctions of a symmetric. ≤ x ≤ b), K(x, t)isthekernel of the integral equation, and f(x) is a given function, which is called the right-hand side of equation (16.4.1.1). For convenience of analysis, a number λ is traditionally