Integral Equations

The general relationship has the formwheren∑∫ bφ n (x) = f(x) + λ m K m (x, y)f(y)dy, (84)m=1aK 1 (x, y) = K(x, y), K m (x, y) =∫ baK(x, t)K m−1 (t, y)dt. (85)and K m (x, y) is called the mth iterated kernel. One can easliy prove that the iterated kernelssatisfy a more general relationshipK m (x, y) =∫ baK r (x, t)K m−r (t, y)dt, r = 1, 2, . . . , m − 1 (m = 2, 3, . . .). (86)Assuming that successive approximations (84) converge we obtain the (unique) solution to IE(70) by passing to the limit in (84)∞∑∫ bφ(x) = f(x) + λ m K m (x, y)f(y)dy. (87)m=1aIn order to prove the convergence of this series, write∫ band estimate C m . Setting r = m − 1 in (86) we haveaK m (x, y) =|K m (x, y)| 2 dy ≤ C m ∀x ∈ [a, b], (88)∫ bApplying to (89) the Schwartz inequality we obtaina|K m (x, y)| 2 =Integrating (90) with respect to y yields∫ bawhich gives∫ b|K m (x, y)| 2 dy ≤ B 2 |K m−1 (x, t)| 2 ≤ B 2 C m−1 (B =and finally the required estimateaK m−1 (x, t)K(t, y)dt, (m = 2, 3, . . .). (89)∫ ba∫ b|K m−1 (x, t)| 2 |K(t, y)| 2 dt. (90)a∫ b ∫ baa|K(x, y)| 2 dxdy) (91)C m ≤ B 2 C m−1 , (92)C m ≤ B 2m−2 C 1 . (93)Denoting√ ∫ bD = |f(y)| 2 dy (94)aand applying the Schwartz inequality we obtain∫ b2 ∫ bK∣ m (x, y)f(y)dy≤a∣ a∫ b|K m (x, y)| 2 dy |f(y)| 2 dy ≤ D 2 C 1 B 2m−2 . (95)a16