Integral Equations

Thus the common term of the series (87) is estimated by the quantity√D C 1 |λ| m B m−1 , (96)so that the series converges faster than the progression with the denominator |λ|B, which provesthe theorem.If we take the first n terms in the series (87) then the resulting error will be not greaterthan|λ|D√C n+1 B n11 − |λ|B . (97)5.5 ResolventIntroduce the notation for the resolvent∞∑∫ bΓ(x, y, λ) = λ m−1 K m (x, y) (98)m=1aChanging the order of summation and integration in (87) we obtain the solution in the compactform∫ bφ(x) = f(x) + λ Γ(x, y, λ)f(y)dy. (99)aOne can show that the resolvent satisfies the IE∫ bΓ(x, y, λ) = K(x, y) + λ K(x, t)Γ(t, y, λ)dt. (100)a6 IEs as linear operator equations. Fundamental propertiesof completely continuous operators6.1 Banach spacesDefinition 3 A complete normed space is said to be a Banach space, B.Example 9 The space C[a, b] of continuous functions defined in a closed interval a ≤ x ≤ bwith the (uniform) norm||f|| C = maxa≤x≤b |f(x)|[and the metricρ(φ 1 , φ 2 ) = ||φ 1 − φ 2 || C = maxa≤x≤b |φ 1(x)| − φ 2 (x)|]is a normed and complete space. Indeed, every fundamental (convergent) functional sequence{φ n (x)} of continuous functions converges to a continuous function in the C[a, b]-metric.Definition 4 A set M in the metric space R is said to be compact if every sequence of elementsin M contains a subsequence that converges to some x ∈ R.17