Nica, Octavia Nonlocal initial value problems for first order differential systems. (English) Zbl 1286.34034 Fixed Point Theory 13, No. 2, 603-612 (2012). 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 \(1-\beta[1]\neq 0\). Cited in 8 Documents 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 Keywords:nonlinear differential system; nonlocal initial condition; fixed point PDFBibTeX XMLCite \textit{O. Nica}, Fixed Point Theory 13, No. 2, 603--612 (2012; Zbl 1286.34034) Full Text: Link