On the Existence of Positive Solutions for a Fourth-Order Boundary Value Problem

where f : [0, 1] ×R3+ ×R− → R+ is continuous. BVP (1) is used tomodel such phenomena as the deformations of an elastic beam in equilibrium state, whose one endpoint is fixed and the other is freed. In mechanics, BVP (1) is called cantilever beam equation. Owing to its importance in mechanics, the existence of solutions to this problemhas been studied by many authors; see [1–7] and references therein. Very recently, Li [3] used the theory of the fixed point index to discuss the existence of solutions of BVP (1) when f(t, x0, x1, x2, x3) is superlinear or sublinear growth on x0, x1, x2, x3. It should be remarked that two constants 1 and λ1, where λ1 is the first eigenvalue of the linear eigenvalue problem corresponding to BVP (1), play an important role in the discussion. The first eigenvalue principles were also used in [8–24]. However, none of these known results use the first eigenvalues of the corresponding linear differential equation that contains the derivative of the unknown function. This is because the presence of derivatives in linear eigenvalue problemwill make the study extremely difficult. To overcome this difficulty, we employ the method of order reduction to develop spectral properties of associated linear differential equation that contains the derivative of the unknown function. Then, we use the fixed point index theory to investigate the existence results of positive solutions for BVP (1) under some conditions concerning the first eigenvalue corresponding to the relevant linear eigenvalue problem that contains the derivative of the unknown function. It should be noted that the method of order reduction was also used in [7, 25] to transform higher order boundary value problem to a lower order integrodifferential equation.

However, none of these known results use the first eigenvalues of the corresponding linear differential equation that contains the derivative of the unknown function.This is because the presence of derivatives in linear eigenvalue problem will make the study extremely difficult.To overcome this difficulty, we employ the method of order reduction to develop spectral properties of associated linear differential equation that contains the derivative of the unknown function.Then, we use the fixed point index theory to investigate the existence results of positive solutions for BVP (1) under some conditions concerning the first eigenvalue corresponding to the relevant linear eigenvalue problem that contains the derivative of the unknown function.It should be noted that the method of order reduction was also used in [7,25] to transform higher order boundary value problem to a lower order integrodifferential equation.

Let
We assume the following hypothesis holds throughout this paper.
Firstly, we consider the existence of solutions to BVP (1) when the nonlinearity  satisfies the sublinear growth property.To this end, we use the method of order reduction to transform BVP (1).For V ∈ [0, 1], we set where Then we have From the above expressions, we easily see that   : [0, 1] → [0, 1] ( = 1, 2, 3) are three completely continuous linear operators, and  1 and  2 have the following properties: Hence we conclude that Using the above transformations   ( = 1, 2, 3), BVP (1) can be converted into the equivalent nonlinear integral equation: Define an operator  by Then the existence of a positive solution of BVP ( 1) is equivalent to the existence of a nontrivial fixed point of  on .Now condition ( 1 ) implies that  :  →  is a completely continuous operator. For Clearly,  a :  →  is a completely continuous linear operator.
Lemma 1.For the operator  a defined by (8), the spectral radius ( a ) ̸ = 0 and  a had unique positive unit eigenfunction  a corresponding to its first eigenvalue Proof.Take V() = 1 − ; clearly V ∈ ; by (3) we have So we have Thus there exists a constant  > 0 such that ( a V)() ≥ V(),  ∈ [0, 1].From Lemma 2 in [22], we know that the spectral radius ( a ) ̸ = 0 and  a had a positive eigenfunction corresponding to its first eigenvalue  1 = (( a )) −1 .Hence Lemma 1 holds.
It follows from the definition of operator  a that the function  a , corresponding to the first eigenvalue  1 = (( a )) −1 of the operator  a , belongs to  4 [0, 1] and satisfies the equation In particular, for a = ( 1 ,  1 , 0, 0) ∈ R 4 + , the positive eigenfunction  a of the operator  a corresponding to the first eigenvalue  1 = (( a )) −1 belongs to  4 [0, 1] and satisfies the equation Theorem 2. If  satisfies ( 1 ) and the following conditions, then BVP ( 1) has at least one positive solution.
The condition ( a ) < 1 ensures that  −  a has a bounded inverse operator ( −  a ) −1 which is given by where Let  b be the positive eigenvalue function of  b (with replacement of a by b in Lemma 1).We may suppose that  has no fixed point on Ω  ∩  (otherwise, the proof is finished).Now we shall show that where Ω  = { ∈ [0, 1] : ‖‖  < }.In fact, if (17) is not valid, there exist V 0 ∈ Ω  ∩  and  0 ≥ 0 such that V 0 − V 0 =  0  b .Hence we have  0 > 0. Since V 0 ∈ Ω  ∩ , by the definitions of  and   ( = 1, 2, 3), we have Hence from ( 3 ), we obtain that from which it follows that Set Therefore, by (20), which contradicts the definition of  * .Hence ( 17) is true and we have from Corollary 2.3.1 in [26] that By ( 16) and ( 22), we have that Then  has at least one fixed point in (Ω  ∩ ) \ (Ω  ∩ ) which means that BVP (1) has a positive solution.The proof of Theorem 2 is completed.
Based on the definition of operator  c , (12) can be rewritten in the following form: which means that the operator  c has an eigenvalue (( c )) −1 , and its related eigenfunction can be taken by  1  c .In fact, we have ( c ) = ( c ) from equivalent differential equation (12).
Theorem 5.If  satisfies ( 1 ) and the following conditions, then BVP (24) has at least one positive solution.