New results on the positive solutions of nonlinear second-order differential systems

In this paper, we study the three-point boundary value problems for systems of nonlinear second order ordinary differential equations of the form      u = −f (t, v), 0 < t < 1, v = −g(t, u), 0 < t < 1 u(0) = v(0) = 0, ςu(ζ) = u(1), ςv(ζ) = v(1), ςζ < 1, f may be singular at t = 0 and/or t = 1. Under some rather simple conditions, by means of monotone iterative technique, a necessary and sufficient condition for the existence of positive solutions is established, a result on the existence and uniqueness of the positive solution and the iterative sequence of solution is given.


Introduction
Recently an increasing interest has been observed in investigating the existence of positive solutions of boundary value problems for systems of differential equations.Singular differential systems arise in many branches of applied mathematics and physics such as gas dynamics, Newtonian fluid mechanics, nuclear physics.Such systems have been widely studied by many authors (see [1][2][3][4][5][6][7][8] and references therein).The study of positive radial solutions for elliptic problems in annular regions can usually be transformed into that of positive solutions of two-point boundary value problems for ordinary differential equations.The study of multi-point boundary value problems for linear second order ordinary differential equations was initiated by Il'in and Moiseev [5].Since then, nonlinear multi-point boundary value problems have been studied by several authors using the Leray-Schauder continuation theorem, nonlinear alternatives of Leray-Schauder, coincidence degree theory, and fixed point theorem in cones.In particular, Zhou and Xu [6] gave some existence results for positive solutions for the following third-order boundary value problem by applying the fixed point index theory in cones.However, most of the above mentioned work is concerned only with sufficient conditions for solvability of systems.As far as we know, the study about a necessary and sufficient condition of positive solutions to differential system has been received much less attention.But to seek a necessary and sufficient condition of solutions for singular differential systems is also a important and interesting work.
Motivated by the work and the reasoning mentioned above, in this paper, by means of monotone iterative technique, we establish a necessary and sufficient condition of the existence of positive solutions for the above nonlinear singular differential system (1.1),where f : (0, 1) × [0, +∞) → [0, +∞), g : [0, 1] × [0, +∞) → [0, +∞), 0 < ζ < 1, ς > 0, and ςζ < 1, f may be singular at t = 0 and/or t = 1.At the same time, we also give the existence and uniqueness of solutions and the iterative sequence of solutions.To that end, two classes extending boundary value differential systems are discussed and some further results are obtained.
Theorem 1. Suppose that f satisfies (A) and g satisfies (B).Then (1) The system (1.1) has a is unique, and for any initial value u 0 ∈ P l , let where Remark 2. In fact, G(t, s) can be rewritten as where where is a solution of the following system of nonlinear integral equations: Moreover, the system can be written as the nonlinear integral equation

Now let us define a nonlinear operator
Thus the existence of solutions to system (1.1) is equivalent to the existence of fixed points of nonlinear operator F, i.e., if x is a fixed point of F in C[0, 1], then system (1.1) has a solution (u, v), which is given by (2.1) ] positive solution of system (1.1).Clearly x (t) ≤ 0, which implies that x is a convex function on [0,1].Combining with the boundary conditions, we assert that there exist constants 0 < µ 1 < 1 < µ 2 such that (2.2) EJQTDE, 2009 No. 3, p. 5 In the following, we show that (2.2) is right.In fact, there exists a t 0 ∈ (0, 1] such that x = x(t 0 ).Since it is easy to check that Thus, For t ∈ [0, 1], by the concavity of x and the conditions of (1.1), we have that On the other hand, 3 Proof of the Theorem 1 On the other hand, (ii)The sufficiency.
For any u ∈ P l , there exist l u and L u such that The above facts imply that there exist positive numbers l F u , L F u such that l F u t ≤ F u(t) ≤ L F u t, where l F u = sup{δ > 0 : F u(t) ≥ δt} and L F u = inf{φ > 0 : F u(t) ≤ φt}.Thus F : P l → P l .At the same time, in view of the monotonicity of f and g in the second variables, it is easy to see that F is also an increasing operator in E. Taking Then, we have and Since αt = u 0 ≤ z 0 = βt and F is an increasing operator, we have Noticing that Let n → +∞, we have This and (3.1) imply that Let ρ n = sup{ξ > 0 : ξu * ≤ u n , ξz n ≤ z * } and then and In what follows, we shall prove that In fact, From the definition of ρ n , we have which contradicts with (3.4).Hence (3.5) holds.
Notice that F is a increasing operator and F u = u, one has u n (t) ≤ u(t) ≤ z n (t).Let n → +∞, then u * = u, and so v * = v.Hence the C 1 [0, 1]× C 1 [0, 1] positive solution to system (1.1) exists and is unique.
also an equicontinuous set.It follows from (3.1) that {u n } and {z n } are relatively compact sets in E. Since P is normal, there exist u