On Linear and Nonlinear Fourth-Order Eigenvalue Problems with Nonlocal Boundary Condition

0 q (s) u 󸀠󸀠 (s) ds. (2) By using the Krasnoselskii fixed point theorem, the sufficient conditions for the existence of positive solutions of (2) are obtained. We can find that, however, those conditions do not involve the eigenvalues with respect to the relevant linear operator, and those sufficient conditions are not optimal. The likely reason is that the spectrum structure of the linear eigenvalue problem

Recently, Bai [3] investigated the existence of positive solutions for more general fourth-order nonlocal boundary value problem  (4) () +   () =  (,  () ,   ()) ,  ∈ (0, 1) ,  (0) =  (1) = ∫ (2) By using the Krasnoselskii fixed point theorem, the sufficient conditions for the existence of positive solutions of (2) are obtained.We can find that, however, those conditions do not involve the eigenvalues with respect to the relevant linear operator, and those sufficient conditions are not optimal.
The likely reason is that the spectrum structure of the linear eigenvalue problem  (4) () +   () =  [ () −   ()] ,  ∈ (0, 1) ,  (0) =  (1) = ∫ It is the purpose of this paper to investigate the first eigenvalue of (3) by using the Krein-Rutman theorem [17,18], and then we use this spectrum result to establish the existence of positive solutions of nonlinear nonlocal problem (2).The existence of positive solution is obtained by means of fixed point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operator.The obtained sufficient conditions in this paper are optimal.For the concepts and properties of fixed point index theory, we refer the reader to [19].
In fact, from (ii) of Lemma 2, we can obtain where we can take and also we obtain where for a certain  > 0, and let  be the Banach space [0, 1] with the norm ‖‖ ∞ = max 0≤≤1 |()|.For  ∈ , we have Combining ( 16) and ( 9) with the fact that −() ≤ −  () ≤ (), we conclude that So, we may define the norm of  ∈  by This norm is so called  0 -norm.
It is easy to get the following lemmas.
From ( 18) and ( 19), for all  > 0, there exists  > 0 such that Hence, Let  → ∞ in (20).Then, we have From ( 26), for all  > 0, there exists  > 0 such that combining this fact with (27) we have Then, the cone  is normal and has nonempty interior int .
To prove Theorems 8 and 9, we need the following preliminary results.Lemma 11.For every  ∈ , there exist  0 ,  1 > 0 such that where ) By (0) = (1), there is a  ∈ (0, 1) such that   () = 0, and so, for  ∈ where It is easy to show that  :  →  is a completely continuous operator.In addition, we can verify that the nonzero fixed points of the operator  are positive solutions of the problem (2).
Lemma 13 (see [19]).Let  be Banach space,  a cone in (105) Then,  has at least one fixed point on (  ∩ ) \ (  1 ∩ ).This means that the boundary value problem (2) has at least one positive solution.