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

φ ( u1(0), u2(0) ) = 0, u1(t) = u1(a), u2(t) = 0 for t ≥ a, (1.2) where fi : [0, a] × R 2 → R (i = 1, 2) satisfy the local Carathéodory conditions, while φ : R → R and τik : [0, a] → [0,+∞[ (i, k = 1, 2) are continuous functions. We are interested in the case, where fi(t, 0, 0) = 0, fi(t, x, y) ≤ 0 for 0 ≤ t ≤ a, x ≥ 0, y ≥ 0 (i = 1, 2) (1.3) and the function φ satisfies one of the following two conditions: φ(0, 0) < 0, φ(x, y) > 0 for x > r, y ≥ 0 (1.4) and φ(0, 0) < 0, φ(x, y) > 0 for x ≥ 0, y ≥ 0, x+ y > r, (1.5)