DEGENERATE FIRST ORDER DIFFERENTIAL-OPERATOR EQUATIONS

Abstract

We consider boundary value problem for degenerate first order differentialoperator equation $Lu \mathclose{\equiv} t^{\alpha} u^{\prime} \mathclose{-} P u \mathclose{=} f $, $ u(0) \mathclose{-} \mu u(b) \mathclose{=} 0 $, where $ t \mathclose{\in} (0,b) $,  $ a \mathclose{\geq} 0 $, $ P: H \mathclose{\rightarrow} H $ is linear operator in separable Hilbert space $ H $, $ f \mathclose{\in} L_{2, \beta} ((0,b),H) $,  $ \mu \mathclose{\in} \mathbb{C} $. We prove that under some conditions on the operator $ P $ and number $ \mu $ the boundary value problem has unique generalized solution $ u \mathclose{\in} L_{2, \beta} ((0,b),H) $ when $ 2 \alpha \mathclose{+} \beta \mathclose{<} 1 $,  $ \beta \mathclose{\geq} 0 $ and for any $ f \mathclose{\in} L_{2, \beta} ((0,b),H) $.