On improper integrals and differential equations in ordered Banach spaces. (English) Zbl 1105.34037

Heikkilä, S.; Kumpulainen, M.

J. Math. Anal. Appl. 319, No. 2, 579-603 (2006).

This paper deals with the existence of least and greatest solutions of initial and boundary value problems in ordered Banach spaces with regular order cone. The right-hand sides of the discussed differential equations comprise locally integrable Banach-space-valued functions possessing improper integrals. Therefore, the authors study in a first section improper integrals of functions with values in such Banach spaces. Combining this with a fixed-point theorem (derived from earlier results), they prove their main result in Theorem 3.1. The problem considered in 3.1 is of a general form, so that several applications and examples can be given.

Reviewer: Marianne Reichert (Frankfurt / Main)

MSC:

34G20	Nonlinear differential equations in abstract spaces
34A99	General theory for ordinary differential equations
34B99	Boundary value problems for ordinary differential equations
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations

Keywords:

improper integral; ordered Banach space; regular order cone; initial value problem; boundary value problem; singular; functional; nonlocal; discontinuous; implicit; least solution; greatest solution

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