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Nonlocal initial value problems for first order differential systems. (English) Zbl 1286.34034
From the text: The proof for the existence of a solution is based on the Perov, Schauder and Leray-Schauder fixed point principles which are applied to a nonlinear integral operator.
We deal with the nonlocal initial value problem for the first-order differential system $\begin{gathered} x'(t)= f_1(t, x(t), y(t)),\\ y'(t)= f_2(t, x(t), y(t))\quad\text{for a.e. on }[0,1],\\ x(0)= \alpha[x],\qquad y(t)= \beta[y].\end{gathered}$ Here $$f_1,f_2: [0,1]\times \mathbb{R}^2\to\mathbb{R}$$ are Carathéodory functions, $$\alpha,\beta: C[0,1]\to \mathbb{R}$$ are linear and continuous functionals such that $$1-\alpha[1]\neq 0$$ and $$1-\beta[1]\neq 0$$.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 47H10 Fixed-point theorems
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