§ 1. Statement of the Problem and Formulation of the Existence and Uniqueness Theorems

1991 Mathematics Subject Classification. 34K10, 34K15 Key words and phrases. Nonlinear two-point boundary value problem, two-dimensional differential system with advanced arguments, nonnegative solution § 1. Statement of the Problem and Formulation of the Existence and Uniqueness Theorems Consider the differential system u i (t) = f i t, u 1 (τ i1 (t)), u 2 (τ i2 (t)) (i = 1, 2) (1.1) with the boundary conditions ϕ u 1 (0), u 2 (0) = 0, u 1 (t) = u 1 (a), u 2 (t) = 0 for t ≥ a, and the function ϕ satisfies one of the following two conditions:

If (1.3) holds, then it is obvious that each component of a nonnegative solution of the problem (1.1), (1.2) is a nonincreasing function.
Theorems 1.1-1.4proven below and also their corollaries make the previous wellknown results [1][2][3][9][10][11][12] on the solvability and unique solvability of the boundary value problems for the differential systems with deviated arguments more complete.
Theorem 1.1 below reduces the question of the solvability of the problem (1.1), (1.2) to obtaining uniform a priori estimates of second components of solutions of the problem (1.1 ε ), (1.2) with respect to the parameter ε.Such estimates can be derived in rather general situations and therefore from Theorem 1.1 the effective and optimal in a sense conditions are obtained for the solvability of the problem (1.1), (1.2) (see Corollaries 1.1-1.5 and Theorem 1.2).
Remark 1.1.The condition (1.9) in Corollary 1.1 is essential and it cannot be replaced by the condition (1.7).To convince ourselves that this is so, consider the boundary value problem ) where It is seen that for that problem all the conditions of Corollary 1.1, except (1.9), are fulfilled.Instead of (1.9) there takes place the condition (1.7).Nevertheless, the problem (1.10), (1.11) has no solution.Indeed, should this problem have a solution (u 1 , u 2 ), the function u 2 would be positive on [0, a[ and But the latter inequality contradicts (1.12).
EJQTDE, 1999 No. 5, p. 3 Corollary 1.2.Let the conditions (1.3), (1.4), and (1.7) be fulfilled and let there exist y 0 > 0 such that no matter how small ε > 0 is.As an example verifying this fact, consider the differential system with the boundary conditions (1.11), where ε ∈ ]0, a[ and It is seen that for this problem the conditions (1.
In contrast to Corollary 1.3, Corollaries 1.4 and 1.5 below catch the effect of an advanced argument τ 22 .
Remark 1.5.The condition (1.25) in Corollary 1.5 cannot be replaced by the condition no matter how small ε > 0 is.As an example, consider the differential system with the boundary conditions (1.11), where 0 and η = min{a, 1}.For the system (1.27) the conditions (1.22)-(1.24)hold and instead of (1.25) there takes place the condition (1.26), where Show that the problem (1.27), (1.11) has no solution.Assume the contrary that this problem has a solution (u 1 , u 2 ).Then The integration of the latter inequality from 0 to t yields and hence EJQTDE, 1999 No. 5, p. 6 Integrating this inequality from 0 to η, we obtain However, since 0 < u 1 (η) < 1, we have ηu which contradicts (1.28).
The uniqueness of a solution of the problem (1.1), (1.2) is closely connected with the uniqueness of a solution of the system (1.1) with the Cauchy conditions (1.29) The following theorem is valid.
Theorem 1.3.Let the condition (1.7) be fulfilled and for any c ∈ R the Cauchy problem (1.1), (1.29) have no more than one solution.Let, moreover, the functions f i (i = 1, 2) not increase in the last two arguments, while the function ϕ increase in the first argument and not decrease in the second argument.Then the problem (1.1), (1.2) has no more than one solution.
Theorem 1.4.Let (1.7) be fulfilled and for any c ∈ R the Cauchy problem (1.1), (1.29) have no more than one solution.Let, moreover, the function f 1 not increase in the last two arguments, f 2 decrease in the second argument and not increase in the third argument, while the function ϕ be such that ϕ(x, y) < ϕ(x, y) for x < x, y < y. (1.30) Then the problem (1.1), (1.2) has no more than one solution.
Remark 1.6.For the uniqueness of a solution of the problem (1.1), (1.29) it is sufficient that either the functions f i (i = 1, 2) satisfy in the last two arguments the local Lipschitz condition or the functions τ ik (i, k = 1, 2) satisfy the inequalities As an example, consider the boundary value problem where   Proof.In view of (2.1) there exists a natural number m such that Put Then Suppose that for some c ≥ 0 the problem (1.1), (1.29) has a solution (u 1 (•,c), u 2 (•,c)).Then on account of where ) we conclude that if the problem (1.1), (1.29) has a solution, then this solution is unique since the functions u ij (i = 1, 2; j = m, . . ., 0) are defined uniquely.

Using the transformation
we can rewrite the system (1.1 ) in the form where EJQTDE, 1999 No. 5, p. 9 On the basis of (1.6) and (1.7) we have According to one of the conditions of the lemma, for each j ∈ {1, 2} the system (2.7) under the initial conditions v 1 (0) = c j , v 2 (0) = 0 has the unique solution (v 1 (•, c j ), v 2 (•, c j )), and On the other hand, c 1 < c 2 , the functions f i (i = 1, 2) do not decrease in the last two arguments, and the functions ζ ik (i, k = 1, 2) satisfy (2.8).By virtue of Corollary 1.9 from [13] the above conditions guarantee the validity of the inequalities It is obvious that Consequently, Hence the inequalities (2.6) follow immediately.
The following lemma holds.

Lemmas on a priori estimates.
First of all consider the system of differential inequalities with the initial condition where δ : [0, ) is said to be a nonnegative solution of the problem (2.17), (2.18) if the functions u 1 and u 2 are absolutely continuous, the function u 1 satisfies the inequality (2.18), and the system of differential inequalities (2.17 Then there exists a positive number ρ 0 such that the second component of an arbitrary nonnegative solution (u 1 , u 2 ) of the problem (2.17), (2.18) admits the estimate Proof.By virtue of (2. 19) and (2.20) there exist positive numbers ρ 0 and ρ 1 such that Let (u 1 , u 2 ) be an arbitrary nonnegative solution of the problem (2.17), (2.18).Then which, owing to (2.17), results in Taking into account (2.22), from (2.24) we get EJQTDE, 1999 No. 5, p. 13 On the basis of the last inequality and (2.23), from (2.25) we find the estimate (2.21).
According to the latter estimate and the inequalities (2.26), we have
In view of (1.3) and the fact that u 1m and u 2m are nonnegative, we can conclude that these functions are nonincreasing.Thus it becomes clear from (3.2) and (3.3) that for every natural m the estimate (2.13), where ρ = r + ρ 0 , is valid.It is also obvious that (u 1m , u 2m ) is a solution of the problem (2.9), (2.10), where c m = u 1m (a).Without loss of generality it can be assumed that the sequence (c m ) +∞ m=1 is convergent.Denote by c the limit of that sequence.
Proof of Corollary 1.1.Choose a natural number m so that Introduce the function and the numbers Let ε ∈ ]0, 1] be an arbitrarily fixed number and (u 1 , u 2 ) be a nonnegative solution of the problem (1.1 ε ), (1.2).By Theorem 1.1, to prove Corollary 1.1 it is sufficient to show that u 2 admits the estimate (1.8).
By virtue of (1.3), the functions u 1 and u 2 are nonincreasing.From this fact, on account of (1.4) and (3.5) it follows that If now we suppose ω(y) = ω(y 0 ) for 0 ≤ y ≤ y 0 , then owing to (1.13), we will have Hence by virtue of (3.12) we obtain the estimate (1.8).
Proof of Corollary 1.4.By virtue of (1.18) we can find ρ 1 ≥ y 0 so that +∞[ is a summable in the first and nondecreasing in the second argument function satisfying the condition (1.18).Then the problem (1.1), (1.2) has at least one nonnegative solution.Remark 1.4.It is obvious from the example (1.20), (1.11) that it is impossible in Corollary 1.4 to replace the condition (1.21) by the condition