On the solvability and the dimension of the solution set of abstract linear boundary value problems and applications to ODE'S

• Published in: Journal of mathematical analysis and applications Vol. 534; no. 1; p. 128029
• Main Author: Milojević, P.S.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-06-2024
• Subjects: Asymptotic hyperbolicity; Covering dimension; Exponential dichotomy; Linear boundary values problems; Normal solvability; Ordinary differential equations; Ordinary differential equations; Linear boundary values problems; Asymptotic hyperbolicity; Normal solvability; Exponential dichotomy; Covering dimension
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2023.128029

A solvability theory for linear abstract boundary value problems, that is of linear operator equations subject to an additional (“boundary”) condition given by a linear operator, is developed using a new approach based on normal solvability of the induced map or a pseudoinverse of the boundary map when restricted to the null space of the homogeneous problem. Uniqueness and dimension of their solution sets is also established. Applications to BVP's for finite-dimensional as well as infinite-dimensional systems of ordinary differential equations defined on the half line or the whole line are given. Systems having exponential dichotomy, or satisfying Riccati type inequalities are studied, as well as systems with asymptotically hyperbolic linear part and systems between pairs of admissible Banach spaces.