Degenerate differential operators in weighted Hölder spaces

Abstract

A differential operator ℒ, arising from the differential expression

$$lv(t) \equiv ( - 1)^r v^{[n]} (t) + \sum\nolimits_{k = 0}^{n - 1} {p_k } (t)v^{[k]} (t) + Av(t),0 \leqslant t \leqslant 1,$$

, and system of boundary value conditions

$$P_v [v] = \sum\nolimits_{k = 0}^{n_v } {\alpha _{vk} } r^{[k]} (1) = 0.v - 1, \ldots ,\mu ,0 \leqslant \mu< n$$

is considered in a Banach space E. Herev ^[k](t)=(a(t) d/dt) ^k v(t)a(t) being continuous fort⩾0, α(t) >0 for t > 0 and$\int_0^1 {\frac{{dz}}{{a(z)}} = + \infty ;}$ the operator A is strongly positive in E. The estimates

, are obtained for ℒ: n even, λ varying over a half plane.