Positive solutions for systems of nth order threepoint nonlocal boundary value problems, Electron

Intervals of the parameterare determined for which there exist positive solutions for the system of nonlinear differential equations, u (n) + �a(t)f(v) = 0, v (n) +�b(t)g(u) = 0, for 0 < t < 1, and satisfying three-point nonlocal bound- ary conditions, u(0) = 0,u 0 (0) = 0,...,u (n 2) (0) = 0, u(1) = �u(�),v(0) = 0,v 0 (0) = 0,... ,v (n 2) (0) = 0, v(1) = �v(�). A Guo-Krasnosel'skii fixed point

There is currently a great deal of interest in positive solutions for several types of boundary value problems.While some of the interest has focused on theoretical questions [5,9,13,26], an equal amount of interest has been devoted to applications for which only positive solutions have meaning [1,8,17,18].While most of the above studies have dealt with scalar problems, some recent work has addressed questions of positive solutions for systems of boundary value problems [3,12,14,15,16,19,22,25,27,30].In addition, some studies have been directed toward positive solutions for nonlocal boundary value problems; see, for example, [4,6,10,17,18,19,21,22,20,24,26,28,29,30].
Additional attention has been directed toward extensions to higher order problems, such as in [2,4,7,8,11,23,29].Recently Benchohra et al. [3] and Henderson and Ntouyas [12] studied the existence of positive solutions of systems of nonlinear eigenvalue problems.Here we extend these results to eigenvalue problems for systems of higher order three-point nonlocal boundary value problems.
The main tool in this paper is an application of the Guo-Krasnosel'skii fixed point theorem for operators leaving a Banach space cone invariant [9].A Green's function plays a fundamental role in defining an appropriate operator on a suitable cone.

Some preliminaries
In this section, we state some preliminary lemmas and the well-known Guo-Krasnosel'skii fixed point theorem.
has a unique solution where k(t, s) and By simple calculation we have (see [11]) We note that a pair (u(t), v(t)) is a solution of eigenvalue problem (1), (2) if, and only if, where Values of λ for which there are positive solutions (positive with respect to a cone) of ( 1), (2) will be determined via applications of the following fixed point theorem.
be a completely continuous operator such that, either Then T has a fixed point in P ∩ (Ω 2 \ Ω 1 ).

Positive solutions in a cone
In this section, we apply Theorem 2.1 to obtain solutions in a cone (that is, positive solutions) of ( 1), (2).For our construction, let B = C[0, 1] with supremum norm, • , and define a cone P ⊂ B by For our first result, define positive numbers L 1 and L 2 by there exists a pair (u, v) satisfying (1), ( 2) such that u(x) > 0 and v(x) > 0 on (0, 1).
Proof.Let λ be as in (9).And let > 0 be chosen such that We seek suitable fixed points of T in the cone P. By Lemma 2.2, T P ⊂ P. In addition, standard arguments show that T is completely continuous.Now, from the definitions of f 0 and g 0 , there exists an H 1 > 0 such that f (x) ≤ (f 0 + )x and g(x) ≤ (g 0 + )x, 0 < x ≤ H 1 .
Let u ∈ P with u = H 1 .We first have from ( 7) and choice of , As a consequence, we next have from (7), and choice of , EJQTDE, 2007 No. 18, p. 5 So, T u ≤ u .If we set Next, from the definitions of f ∞ and g ∞ , there exists H 2 > 0 such that Let Let u ∈ P and u = H 2 .Then, from ( 8) and choice of , And so, we have from ( 8) and choice of , Hence, T u ≥ u .So, if we set Applying Theorem 2.1 to ( 11) and ( 12), we obtain that T has a fixed point u ∈ P ∩ (Ω 2 \ Ω 1 ).As such, and with v defined by the pair (u, v) is a desired solution of (1), (2) for the given λ.The proof is complete.
Prior to our next result, we introduce another hypothesis.
Proof.Let λ be as in (13).And let > 0 be chosen such that Let T be the cone preserving, completely continuous operator that was defined by (10).
From the definitions of f 0 and g 0 , there exists H 1 > 0 such that Now g(0) = 0 and so there exists 0 < H 2 < H 1 such that Choose u ∈ P with u = H 2 .Then Then, by ( 8) and (D) Next, by definitions of f ∞ and g ∞ , there exists H 1 such that There are two cases, (a) g is bounded, and (b) g is unbounded.
For case (a), suppose N > 0 is such that g(x) ≤ N for all 0 < x < ∞.Then, for u For case (b), there exists H 3 > max{2H 2 , H 1 } such that g(x) ≤ g(H 3 ), for 0 < x ≤ In either of the cases, application of part (ii) of Theorem 2.1 yields a fixed point u of T belonging to P ∩ (Ω 2 \ Ω 1 ), which in turn yields a pair (u, v) satisfying ( 1), (2) for the chosen value of λ.The proof is complete.
Theorem 2.1 Let B be a Banach space, and let P ⊂ B be a cone in B. Assume Ω 1 and Ω 2 are open subsets of B with 0 ∈ Ω 1 ⊂ Ω 1 ⊂ Ω 2 , and let EJQTDE, 2007 No. 18, p. 3