FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

46 T. S. Aleroev, H. T. Aleroeva, Ning-Ming Nie, and Yi-Fa TangThenIn fact, from (2.22) we haveλ(χ, ε) − λ = tr [ T (χ, ε)P (χ, ε) ] = − 1 ∫2πi tr (ζ − λ)R(ζ, χ, ε) dζ.|λ(χ, ε) − λ| = 12π tr ∫Sincewe haveΓ(ζ − λ)R(ζ, χ, ε) dζ ≤≤ ρ2π tr ∫ρtrP (χ, ε) = dim P (χ, ε) = 1,2πΓΓR(ζ, χ, ε) dζ = ρ trP (χ, ε).2π|λ(χ, ε) − λ| < ρ(2π) −1 . (2.24)A simple analysis of the formula (2.24) shows that the less is ρ, the closergets λ(χ, ε) to λ as expected. Moreover, the less are χ, ε, the less should be|χ|, |ε| in general, for λ(χ, ε) to get within Γ.Assertion 2.6. Let T ′ R(ζ, T ) ∈ G F :∥∥(χT ′ + εT ′′ )R(ζ, T ) ∥ ∥ < 1, (2.25){T ′ R(ζ, T ) + T ′′ R(ζ, T ) } v = Σ [ a k (v i ϕ i )ϕ i + b k (v i ϕ i )ψ i], (2.26)where (ϕ i ϕ j ) = 0, (ϕ i ϕ i ) = 1, (ϕ i ψ j ) = 0. Then(T ′ R + T ′′ R) 2n = (T ′ R) 2n−1 + (T ′′ R) 2n−1 ,(T ′ R + T ′′ R) 2n+1 = (T ′ R) 2n+1 + T ′ (T ′′ ) 2n R 2n+1 .Proof. Consider the double power series1+ ( −χT ′ R(ζ, T ) ) + ( −χT ′ R(ζ, T ) ) + ( −χT ′ R(ζ, T ) ) 2+(−χT ′ R(ζ, T ) ) 3++ · · · + ( − εT ′′ R(ζ, T ) ) + ( − χT ′ R(ζ, T ) )( − εT ′′ R(ζ, T ) ) ++ ( − εT ′′ R(ζ, T ) )( − χT ′ R(ζ, T ) ) ++ · · · + ( − χT ′′ R(ζ, T ) ) + ( − εT ′′ R(ζ, T ) ) 2+ · · · .The general member of this series can be rewritten as[(χT ′ + εT ′′ )(R(ζ, T )) ] p.Rewrite the formula (2.26):[χT ′ R(ζ, T ) + εT ′′ R(ζ, T ) ] v 2 ={ [χT= ′ R(ζ, T ) ] }2+ χεT ′ R(ζ, T )T ′′ R(ζ, T ) + χεT ′′ R(ζ, T )T ′ R(ζ, T )}={χ 2[ T ′ R(ζ, T ) ] 2χε[T ′′ R(ζ, T )T ′ R(ζ, T ) ] v,v =