I need the proof of (19) and (20) by the difference operators Λ(1)y = ỹ(xx) + byx for b > 0 or Λ(2)y = y(x̄x) + byx for b < 0 with y in hat(Ω)h. Let widetilde(Λ)y be an operator from Ω(h) into Ω(h) coinciding with Λy for y in Ωdeg(h). The operators A(1) = -widetilde(Λ)(1) and A(2) = -widetilde(Λ)(2), acting from Ω(h) into Ω(h), are positive definite for any h. Indeed, {(A(1)y, y) = (-y(x̄x), y) - b(yx, y) = (1 + (1/2)hb)||y(x̄)|||^(2), (A(2)y, y) = (1 - (1/2)hb)||y(x̄)|||^(2) = (1 + (1/2)h|b|)||y(x̄)||. We conclude from here that (A(α)y, y) >= 8(1 + (1/2)h|b|)||y||^(2) for α = 1, 2. By re-ordering A(1) = A + bΛ(+) and A(2) = A + |b|Λ(-), where A = -Λ, Λy = y(x̄x) and ||Λ(+-)|| <= (2)/h, ||A|| <= (4)/h^(2), we are led due to the triangle inequality for the norms to ||A(1)|| <= ||A|| + b||Λ(+)|| < (4)/h^(2)(1 + (1/2)hb), b > 0, ||A(2)|| <= ||A|| + b||Λ(-)|| < (4)/h^(2)(1 + (1/2)h|b|), b < 0. We note in passing that if we approximate the operator Lu = u'' + bu' by the expression Λy = y(x̄x) + by(x̄) for b > 0, then the member 1 - (1/2)hb arises in place of 1 + (1/2)hb in (19) and thereby the operator -Λ will be positive definite only for h < (2)/b. The operator Lu = u''(x) + bu'(), x ∈ [0,1], b = const, is approximated by the difference operators Ay = y- + byx for b > 0 or Ay = yx + by. for b < 0 with y ∈ Lh. Let Ay be an operator from S into Sp coinciding with Ay for y ∈ Sh. The operators A = -A and A = -A, acting from S into Sh, are positive definite for any h. Indeed, Ayy = (-yxy) - byxy) = (1 + hb)[|y,]|^(2) (19) Ayy = 1 - hb)||y] = 1 + h|b|1y]^(2) We conclude from here that (Aay, y)8(I + zh|b1)||y|I2 for a = 1, 2 Be re-ordering A, == A + b + and A = A + |b|-, where A = -- Ay = Y, and [|| 2/h, |A|4/h2, we are led due to the triangle inequality for the norms to IAl<Aj+b[+< b2I+hb b > 0, (20) HA|<HA{}+b||X< h2(I+h|b1), '0>9 We note in passing that if we approximate the operator L u = u" + bu' by the expression Ay = Ye- + b y, for b > 0, then the member 1 - , h b arises in place of 1 + h b in (19) and thereby the operator -- will be positive definite only for h < 2/b.
