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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.

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