On the nonlocal initial value problem for first order differential equations. (English) Zbl 1050.34001
Here, the existence of solutions to the following nontrivial initial value problem is established:
\[
x'(t) = f(t,x(t)) \text{ a.e. } t \in [0,1], \quad x(0) + \sum_{k=1}^m a_kx(t_k) = 0,
\]
with \(\sum_{k=1}^m a_k \neq 1\). The novelty in this paper is the fact that the growth conditions imposed on \(f\) are split in two: one on the interval \([0,t_m]\), and a second one on \([t_m,1]\). The cases where \(f\) is continuous and where \(f\) satisfies a Lipschitz condition with respect to the second variable are treated. The proofs rely on the Leray-Schauder alternative and on the Banach contraction principle according to the cases considered.
Reviewer: Marlène Frigon (Montréal)
MSC:
34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
45G10 | Other nonlinear integral equations |
47H10 | Fixed-point theorems |