ABSTRACT
Preliminary remarks. A number λ is called a characteristic value of the integral equation
y(x) – λ ∫ b a
K(x, t)y(t) dt = f (x)
if there exist nontrivial solutions of the corresponding homogeneous equation (with f (x) ≡ 0). The nontrivial solutions themselves are called the eigenfunctionsof the integral equation corresponding to the characteristic value λ. If λ is a characteristic value, the number 1/λ is called an eigenvalue of the integral equation. A value of the parameter λ is said to be regular if for this value the 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). In the above equation, it is usually assumed that a ≤ x ≤ b.
