Nontrivial Solutions for the 2 n th Lidstone Boundary Value Problem

In this paper, we study the existence of nontrivial solutions for the 2 n th Lidstone boundary value problem with a sign-changing nonlinearity. Under some conditions involving the eigenvalues of a linear operator, we use the topological degree theory to obtain our main results.

(2) e Lidstone boundary value problem arises in many different areas of applied mathematics and physics. When n � 2, problem (1) describes the deformation of an elastic beam in which both ends are simply supported. Recently, this problem has been extensively studied, and the authors refer the reader to [1][2][3][4][5][6][7][8][9][10][11] and references cited therein. For example, in [1], the authors used a cone-theoretic fixed point theorem to study the existence of nontrivial solutions for the nonlinear Lidstone boundary value problem: y (2m) (t) � λa(t)f y(t), . . . , y (2j) (t), . . . y (2(m− 1)) (t) , 0 < t < 1, where (− 1) m f > 0 is continuous and a is nonnegative. In [2], the authors investigated the existence and uniqueness of positive solutions for the following generalized Lidstone boundary value problem: where α j ≥ 0, β j ≥ 0(j � 0, 1) and α 0 α 1 + α 0 β 1 + α 1 β 0 > 0. In view of symmetry, these results demonstrate that problem (4) is essentially identical with Dirichlet boundary condition (1). Meanwhile, we also note that there are a large number of papers in the literature devoted to sign-changing nonlinearities, and some results can be found in a series of papers  and the references cited therein. For example, in [12], the authors studied the following higher-order nonlinear fractional boundary value problem involving Riemann-Liouville fractional derivatives: where f is a sign-changing nonlinearity. Under some appropriate conditions involving the eigenvalues of the relevant linear operators, they utilized the topological degree to obtain a nontrivial solution for (5). In [13], the authors adopted the similar method in [12] to study the existence of nontrivial solutions for the following system of fractional q-difference equations with q-integral boundary conditions: where α ∈ (2, 3), v ∈ (1, 2), and D α q is the α-order Riemann-Liouville's fractional q-derivative.
Inspired by the aforementioned works, in this paper, we study the existence of nontrivial solutions for (1) where the nonlinearity f is sign-changing. Under some conditions involving the eigenvalues of the revelent linear operators, we use the topological degree to obtain our results.

Preliminaries
is a real Banach space and P is a solid cone in E. In (1), let (− 1) n− 1 u (2n− 2) (t) � v(t) and from [2, P224], we can obtain that (1) is equivalent to the following integral equation: where Next, we provide a lemma, which expresses some vital properties of the functions G i (i � 1, 2, . . .).

Lemma 1 (i) G i are nonnegative continuous functions on
Proof. We only prove (iii) and (iv). For (iii), i � 1 holds obviously. From the definition of G i , we have Noting that Using the symmetry of G i , we easily have is completes the proof. Let α i ≥ 0 with n i�1 α 2 i ≠ 0. en, we have the following equations: 1 0 G α 1 ,...,α n (t, s)sin πtdt � α 1 1 π 2 n + · · · + α n 1 π 2 1 sin πs, where is is a direct result from Lemma 1 (ii), so we omit its proof.
is completes the proof.

Conclusion
In this paper, we use the topological degree to study the nontrivial solutions for the 2n th Lidstone boundary value problem (1). To the best of our knowledge, there are few works that deal with the problem where the nonlinear terms may be unbounded and sign-changing. Moreover, it is remarked that the main result is discussed under some conditions concerning the first eigenvalues corresponding to the relevant linear operators. ese mean that our main result is an improvement in some related works.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors' Contributions
e study was carried out in collaboration among all authors. All authors read and approved the final manuscript.