FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Boundary Value Problems for Differential Equations of Fractional Order 55Let |λ| ≤ N. ThenFor n ≥ 2 we have≤Hence≤∫ x0∣∣u 1 (x, λ) − u 0 (x, λ) ∣ ≤ x2−α2 − α + N x22 .∣∣u 2 (x, λ) − u 0 (x, λ) ∣ ∣ ≤{}(x − t) 1−α u(t) + λ(x − t)u 1 (t) − (x − t) 1−α t − λ(x − t)t dt.x 3−α(x3+N(2−α)(3−α) 3! +N∣∣u 2 (x, λ) − u 1 (x, λ) ∣ ∣ ≤x 3−α(2−α)(3−α)and in general∣∣u n (x, λ) − u n−1 (x, λ) ∣ ≤ (N + M) n[Hence the seriesu(x, λ) = u 0 (x, λ) +) [=(N +1)x n−αx 3−α(2−α)(3−α)],(2 − α)(3 − α) · · · (n − α) + N xnn!].∞∑ {un (x, λ) − u n−1 (x, λ) } (2.31)n=1converges in regular intervals of λ, for |λ| ≤ N and for 0 ≤ x ≤ 1. As forany n ≥ 2=∫ x0u ′ n(x, λ) − u ′ n−1(x, λ) ={(x − t) −α (u n−1 (t, λ) − u n−2 (t, λ)) + λ ( u n−1 (t, λ) − u n−2 (t, λ) )} dt,u ′′ n(x, λ) − u ′′ n−1(x, λ) = {D α 0x + λ} ( u n−1 (x, λ) − u n−2 (x, λ) ) ,we have that the series obtained by once and twice differentiation of theseries (2.31) also converge in regular intervals of x. Thus∞∑u ′′ {(x, λ) = u′′n(x, λ) − u ′′ n−1(x, λ) } =n=1= u ′′1(x, λ) − u ′′0(x, λ) += {D α 0x + λ}{u 0 (x, λ)} +∞∑n=2{u′′n(x, λ) − u ′′ n−1(x, λ) } =∞∑ {un (x, λ) − u n−1 (x, λ) } =n=2= {D α 0x + λ}{u 0 (x, λ)} + λu 0 (x, λ),and u(x, λ) satisfies the equation (2.29). Clearly u(x, λ) satisfies the conditions(2.30).