ABSTRACT
Preliminary remarks. For equations of the form∫ x a
K(x, t)y(t) dt = f (x), a ≤ x ≤ b,
where the functions K(x, t) and f (x) are continuous, the right-hand side must satisfy the following conditions: 1◦. If K(a, a) ≠ 0, then we must have f (a) = 0 (for example, the right-hand sides of equations 1.1.1 and 1.2.1 must satisfy this condition). 2◦. If K(a, a) = K ′x(a, a) = · · · = K (n-1)x (a, a) = 0, 0 <
∣∣K (n)x (a, a)∣∣ <∞, then the right-hand side of the equation must satisfy the conditions
f (a) = f ′x(a) = · · · = f (n)x (a) = 0. For example, with n = 1, these are constraints for the right-hand side of equation 1.1.2. 3◦. If K(a, a) = K ′x(a, a) = · · · = K (n-1)x (a, a) = 0, K (n)x (a, a) =∞, then the right-hand side of the equation must satisfy the conditions
f (a) = f ′x(a) = · · · = f (n-1)x (a) = 0. For example, with n = 1, this is a constraint for the right-hand side of equation 1.1.30. 4◦. For unbounded K(x, t) with integrable power-law or logarithmic singularity at x = t and continuous f (x), no additional conditions are imposed on the right-hand side of the integral equation (e.g., see Abel’s equation 1.1.36).
