FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Boundary Value Problems for Differential Equations of Fractional Order 45As∫Assertion 2.3 is proved.ΓR(ζ, χ, ε) dζ = P (χ, ε),∫Γdζ = 0,Assertion 2.4. The following equality holdsλ(χ, ε) − λ = − 1 ∫log [ 1 + (χT ′ + εT ′′ ) ] R(ζ, T ) dζ.2iπΓProof. From the formula (2.22) we haveλ(χ, ε) − λ = − 1 ∫ ∞∑2iπ tr 1p (ζ − λ) d [− (χT ′ + εT ′′ )R(ζ, T ) ] pdζ.dζΓp=0Substitution of (2.21) instead of the resolvent R(ζ, χ, ε) gives usλ(χ, ε) − λ = − 1 ∫2iπ tr (ζ − λ)R(ζ, χ, ε) dζ =Γ= 1 ∫2iπΓlog (1 + [ (χT ′ + εT ′′ )R(ζ, T ) ] ) dζ.Until now we have considered various power series of χ and ε and did notspecify explicitly a condition of their convergence. Now we investigate suchconditions.The series (2.21) obviously converges if the equality∥∥(χT ′ + εT ′′ ) ∥ ∥ < 1holds and this condition takes place if|χ| < 1 (‖T ′ R(ζ, T )‖ ) , |ε| < 1 (‖T ′′ ‖R(ζ, T ) ) .22Denote by r 0 (χ) and r 1 (ε) the values of χ and ε, respectively for which wehave(‖T ′ R(ζ, T )‖ ) −1 (= 2|χ|, ‖T ′′ R(ζ, T )‖ ) −1= 2ε.Clearly the series (2.21) converges in regular intervals on ζ ∈ Γ if|χ| ≤ r 0 = minζ∈Γ r 0(ζ),|ε| < r 1 = minζ∈Γ r 1(ζ). (2.23)Hence it is clear that the radia of convergence r 0 and r 1 depend on thecontour Γ.□Assertion 2.5. Let ρ = maxζ∈Γ r 1|ζ − λ|. Then|λ(χ, ε) − λ| < β 2π .