On nonnegative solutions of nonlinear two-point boundary value problems for two-dimensional differential systems with advanced arguments

I. Kiguradze, Georgian Academy Of Sciences, Georgia
N. Partsvania, A. Razmadze Mathematical Institute, Tbilisi, Georgia

E. J. Qualitative Theory of Diff. Equ., No. 5. (1999), pp. 1-22.

Abstract: In this paper we consider the differential system (1.1)
$u_i'(t)=f_i\big(t,u_1(\tau_{i1}(t)),u_2(\tau_{i2}(t))\big) (i=1,2)$
with the boundary conditions (1.2)
$\varphi\big(u_1(0),u_2(0)\big)=0, u_1(t)=u_1(a), u_2(t)=0 for t\geq a,$
where $f_i: [0,a]\times \Bbb{R}^2\to \Bbb{R}$ $(i=1,2)$ satisfy the local Carath\'{e}odory conditions, while $\varphi: \Bbb{R}^2\to \Bbb{R}$ and $\tau_{ik}: [0,a]\to [0,+\infty[$ $(i,k=1,2)$ are continuous functions. The optimal, in a certain sense, sufficient conditions are obtained for the existence and uniqueness of a nonnegative solution of the problem (1.1), (1.2).

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