ON A SYSTEM OF HIGHER-ORDER MULTI-POINT BOUNDARY VALUE PROBLEMS

We investigate the existence and nonexistence of positive solutions for a system of nonlinear higher-order ordinary differential equations subject to some multi-point boundary conditions. AMS Subject Classification: 34B10, 34B18.

By using the Schauder fixed point theorem, we shall prove the existence of positive solutions of problem (S)−(BC). By a positive solution of (S)−(BC) we mean a pair of functions (u, v) ∈ C n ([0, T ]; R + )×C m ([0, T ]; R + ) satisfying (S) and (BC) with u(t) > 0, v(t) > 0 for all t ∈ (0, T ]. We shall also give sufficient conditions for the nonexistence of positive solutions for this problem.
Our results obtained in this paper were inspired by the paper [16], where the authors studied the existence and nonexistence of positive solutions for the m-point boundary value problem on time scales where in this case (0, T ) denotes a time scale interval. Multi-point boundary value problems for ordinary differential equations or finite difference equations have applications in a variety of different areas of applied mathematics and physics. For example the vibrations of a guy wire of a uniform cross-section and composed of N parts of different densities can be set up as a multi-point boundary value problem (see [22]); also many problems in the theory of elastic stability can be handled as multi-point problems (see [24]). The study of multi-point boundary value problems for second order differential equations was initiated by Il'in and Moiseev (see [13], [14]). Since then such multi-point boundary value problems (continuous or discrete cases) have been studied by many authors, by using different methods, such as fixed point theorems in cones, the Leray-Schauder continuation theorem, nonlinear alternatives of Leray-Schauder and coincidence degree theory.
In Section 2, we shall present some auxiliary results which investigate a boundary value problem for a n-th order differential equation (problem (1) − (2) below). In Section 3, we shall prove our main results, and in Section 4, we shall present a simple example which illustrate the obtained results.

Auxiliary results
In this section, we shall present some auxiliary results from [15] and [19] related to the following n-th order differential equation with p-point boundary conditions , then the solution of (1)-(2) is given by [19]) Under the assumptions of Lemma 2.1, Green's function for the boundary value problem (1)-(2) is given by Using the above Green's function the solution of problem (1) We can also formulate similar results as Lemma 2.1 -Lemma 2.5 above for the boundary value problem we denote by G 2 Green's function corresponding to problem (3)-(4), that is Under similar assumptions as those from Lemma 2.5, we have the inequality inf

Main results
We present the assumptions that we shall use in the sequel:

Proof. We consider the problems
The above problems (5) and (6) have the solutions We define the functions x(t) and y(t), t ∈ [0, T ] by where (u, v) is solution of (S) − (BC). Then (S) − (BC) can be equivalently written as with the boundary conditions Using the Green's functions given in Section 2, a pair (x, y) is a solution of the problem (8)- (9) if and only if (x, y) is a solution for the nonlinear integral equations where h(t), w(t), t ∈ [0, T ] are given by (7).
We consider the Banach space X = C([0, T ]) with the supremum norm · and define the set We also define the operator A : K → X by For sufficiently small a 0 > 0 and b 0 > 0, by (H3), we deduce Then, by using Lemma 2.3, we obtain A(x)(t) ≥ 0 for all t ∈ [0, T ] and x ∈ K. By Lemma 2.4, for all x ∈ K, we have Therefore A(K) ⊂ K.
Using standard arguments, we deduce that A is completely continuous (A is compact, that is for any bounded set B ⊂ K, A(B) ⊂ K is relatively compact by Arzèla-Ascoli theorem, and A is continuous). By the Schauder fixed point theorem, we conclude that A has a fixed point x ∈ K. This element together with y given by y(t) = 2 In what follows, we present sufficient conditions for the nonexistence of the positive solutions of (S) − (BC). Proof. We suppose that (u, v) is a positive solution of (S) − (BC). Then x = u − a 0 h, y = v − b 0 w is a solution for (8)- (9), where h and w are the solutions of problems (5) and (6) (given by (7)). By Lemma 2.3, we have x(t) ≥ 0, y(t) ≥ 0 for all t ∈ [0, T ], and by (H2) we deduce that x > 0, y > 0. Using Lemma 2.5, we also have inf y(t) ≥ γ 2 y , where γ 1 , γ 2 are defined in Section 2.
Using now (7), we deduce that inf In a similar manner we obtain inf We now consider By (H4), for R defined above, we deduce that there exists M > 0 such that f (u) > 2Ru, g(u) > 2Ru for all u ≥ M.
We consider a 0 > 0 and b 0 > 0 sufficiently large such that By using Lemma 2.4 and the above considerations, we have Therefore, we obtain x ≤ y(η q−2 )/2 ≤ y /2.
In a similar manner, we deduce By (11) and (12), we obtain x ≤ y /2 ≤ x /4, which is a contradiction, because x > 0. Then, for a 0 and b 0 sufficiently large, our problem (S) − (BC) has no positive solution. 2