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.
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| Giovanni Dore |
Department of Mathematics
University of Bologna
Piazza di Porta San Donato 5, I-40126, Bologna, Italy
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