Nonlocal boundary value problem for the inhomogeneous equation with the differentiation operator $z\frac{\partial}{\partial z}$ and homogeneous boundary conditions

Abstract

The paper investigates a nonlocal problem with homogeneous boundary conditions for a nonhomogeneous partial differential equation involving the generalized differentiation operator $z\frac{\partial}{\partial z}$, acting on a function of the complex variable $z$ with scalar coefficients. The uniqueness and existence theorems for the solution of the problem in the space $\mathbf{H}_{q}^n(\mathcal{D})$ are proved. The Hadamard well-posedness of the problem is established. This distinguishes the problem from Hadamard ill-posed nonlocal problems with a vector complex variable $z$.