Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 107, pp. 1-16.

Title: Dirichlet problem for second-order abstract differential equations
       
Author: Giovanni Dore (Univ. of Bologna, Italy)

Abstract:
 We study the well-posedness in the space of continuous functions of
 the Dirichlet boundary value problem for a homogeneous linear
 second-order differential equation u''+Au = 0, where A is a linear closed
 densely defined operator in a Banach space.
 We give necessary conditions for the well-posedness, in terms of the
 resolvent operator of A. In particular we obtain an estimate on the norm of
 the resolvent at the points k^2, where k is a positive integer, and we show that
 this estimate is the best possible one, but it is not sufficient for the 
 well-posedness  of the problem.
 Moreover we characterize the bounded operators for which the problem is well-posed.
 
Submitted August 19, 2018. Published October 29, 2020.
Math Subject Classifications: 34G10.
Key Words: Boundary value problem; differential equations in Banach spaces.