Eigenvalue Characterization for a Class of Boundary Value Problems

We consider the nth order ordinary dieren tial equation ( 1) n k y (n) = a (t)f(y), t 2 [0; 1], n 3 together with boundary condition y (i) (0) = 0, 0 i k 1, and y (l) (1) = 0, j l j +n k 1, for 1 j k 1 xed. Values of are characterized so that the boundary value problem has a positive solution.


Introduction
Let n ≥ 3, 2 ≤ k ≤ n − 1, and 1 ≤ j ≤ k − 1 be given.In this paper we shall consider the nth order differential equation satisfying the boundary conditions (2) Throughout, we assume the following hypotheses : (H 1 ) a(t) is a continuous nonnegative function on [0, 1] and is not identically equal to zero on any subinterval of [0, 1].
(H 3 ) The limits f 0 = lim u→0 + f (u) u and f ∞ = lim u→∞ f (u) We shall determine values of λ for which the boundary value problem (1), (2) has a positive solution.
The motivation for the present work originates from many recent investigations.In the case n = 2 the boundary value problem ( 1), (2) describes a vast spectrum of scientific phenomena; we refer the reader to [1,3,5,6,14,16].It is noted that only positive solutions are meaningful in those models.
By defining an appropriate Banach space and cone, in Section 3, we characterize the set Sp(a).

Background Notation and Definitions
We first present the definition of a cone in a Banach space and the Krasnosel'skii Fixed Point Theorem.Definition 2.1.Let B be a Banach space over R. A nonempty closed convex set P ⊂ B is said to be a cone provided the following are satisfied: (a) If y ∈ P and α ≥ 0 , then αy ∈ P; (b) If y ∈ P and −y ∈ P , then y = 0. Then T has a fixed point in P ∩ (Ω 2 \ Ω 1 ).
To obtain a solution for (1) and ( 2), we require a mapping whose kernel G(t, s) is the Green's function of the boundary value problem Wong and Agarwal [20] have found that if y satisfies then, for δ ∈ (0, where the functions b and c are defined as Aided by this, we have the following lemma. Let the boundary condition (2) be partitioned into two parts: and Now u satisfies (7), so u (j) satisfies (k − j, n − k) homogeneous conjugate boundary conditions.The conclusion then follows from inequality (6).
It is noted that Eloe [7] proved that G (j) (t, s) = ∂ j ∂t j G(t, s) is the Green's function of y (n−j) = 0 subject to the boundary conditions The proof follows from the four properties of the Green's function.Consequently we have the following result, whose conclusion follows from Lemma 2.2.

Main Results
We are now in a position to give some chacterization of Sp(a).Define a Banach space, B, by . Define a cone, P σ ⊂ B, by To obtain a solution of (1), (2), we shall seek a fixed point of the operator λT in the cone P σ .In order to apply the Krasnosel'skii Fixed Point Theorem, for λ > 0, we need the following.
Furthermore, for any 0 Also, the standard arguments yield that λT is completely continuous.
Finally, we apply part (i) of Krasnosel'skii's Fixed Point Theorem and obtain a fixed point u 1 EJQTDE, 1999 No. 12, p. 7 Hence, u 1 is a nontrivial solution of (1), (2).Successive applications of Rolle's theorem imply that u 1 does not vanish on (0, 1) and so u 1 is a positive solution.
This completes the proof.
Corollary 3.3 Assume all the conditions for Theorem 3.2 hold.Then Theorem 3.4 Assume (H 1 ), (H 2 ) ,and (H 3 ) with f ∞ < f 0 < ∞.Assume there exists a value of λ such that In addition, if f is not bounded, assume also that Then the BVP (1),( 2) has a positive solution in the cone P σ .
On the other hand, consider f ∞ ∈ R + .Given f 0 > f ∞ , there are two subcases for us to consider: Case 1: f is bounded.Let λ > 0 satisfying condition (14) be given throughout this case.Let N > 0 be large enough so that f (u) ≤ N, f or all u ≥ 0, and and Then, for all u ∈ ∂Ω 2 ∩ P σ , EJQTDE, 1999 No. 12, p. 9 Coupled with condition ( 14), we apply part (ii) of Krasnosel'skii's Fixed Point Theorem and obtain a fixed point of λT in P σ ∩ (Ω 2 \Ω 1 ).
Case 2: f is not bounded.Assume now that λ > 0 also satisfies the condition (15).Without loss of generality, we let the preceding also satisfy There exists H2 > 0 such that for all u ≥ H2 , f (u) ≤ (f ∞ + )u.Since f is continuous at u = 0, it is unbounded on (0, ∞) as u approaches +∞.Let and so, Hence, (17) implies that λT u ≤ u .
Finally, we apply part (ii) of Krasnosel'skii's Fixed Point Theorem and obtain a fixed point u 1 of λT in P σ ∩ Ω 2 \Ω 1 .
By an argument similar to that in the proof of Theorem 3.2 there is a positive solution, u 1 , of (1), (2).