Asymptotic properties of solutions of equations in Banach spaces.

• Main Author: Schulzer, Michael
• Language: English
• Published: University of British Columbia 2012
• Subjects: Equations; Generalized spaces
• Online Access: http://hdl.handle.net/2429/39917

Certain properties of the solution u of the equation Pu = v in a Banach space will be investigated. It will be assumed that v is a prescribed element of the space, P is a transformation defined on a closed subset in the space and consisting of the sum of a linear transformation and a contraction mapping, and that P and v depend on a real variable λ. which assumes values over the half-open positive interval 0 < λ ≤ λₒ. Then a theorem will be proved, establishing the existence and uniqueness of the solution u(λ) of P(λ)u(λ) = v(λ) . Under the hypothesis that P and v possess asymptotic expansions as λ→0, it will be shown that asymptotic solutions exist, that they are asymptotically unique, and that they possess asymptotic expansions which may be determined by a recursive process from those of P and v. The results obtained will be applied to particular types of Banach spaces, such as finite-dimensional Euclidean spaces, spaces of Lebesgue-square-summable functions and of continuous functions over a closed interval. === Science, Faculty of === Mathematics, Department of === Graduate