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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.

##### 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
##### Keywords:
nonlinear differential equation; nonlocal