FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

48 T. S. Aleroev, H. T. Aleroeva, Ning-Ming Nie, and Yi-Fa TangIf T coincides with a projector, the statement of Corollary 2.1 is obviouswithout planimetric integration. Asλ n (x) − λ n = − 1 ∫tr [ (ζ − λ)R(ζ, χ, ε) ] dζ,2iπwe haveλ n (x) − λ n = 1 ∫ ( ∑ [tr (χT ′ + εT ′′ )R(ζ, T ) ] P 1)dζ =2iπP !Γ n= 1 ∫ [tr χT ′ R(ζ, T ) + εT ′′ R(ζ, T ) + χ 2 (T ′ ) 2 (R(ζ, T )) 2 +2iπΓ nΓ n+χεT ′ R(ζ, T )T ′′ R(ζ, T )+χεT ′′ R(ζ, T )T ′ R(ζ, T )+ε 2 (T ′′ R(ζ, T )) 2 +· · ·]dζ.Theorem 2.1. Let the operator T ′ be permutable with the operatorT , and the operator T ′′ R(ζ, T ) be a nilpotent operator with parameter ofnilpotence equal to 2, T ′ R, T ′′ R ∈ G F . Thenλ n (x) − λ n = χspT ′ P n + εspT ′′ P n ,where P n is a Riesz’s projector of the operator T corresponding to λ n :P n = 1 ∫R(ζ, T ) dζ2iπΓ nProof. From (2.27),wherêλ nm = (−1)m+n+12iπλ n (χ, ε) − λ n =∫Γ n∞∑m=0 n=0∞∑χ m ε n̂λnm ,a nm tr(T ′ R(ζ, T )) m (T ′′ R(ζ, T )) n dζ.As the operator T ′′ R(ζ, T ) is nilpotent, we have∫̂λ 0m = (−1)m+1 Cmkk (T ′ ) m (R(ζ, T )) m ,2iπΓ n∫̂λ 1m = (−1)m+2 Cmkm (T ′′ ) m R(ζ, T )(T ′ )R m ,2iπΓ n∫̂λ jm = (−1)m+1 a jm (T ′′ R(ζ, T )) m T ′ R.2iπΓ n