Two solutions to Kirchhoff-type fourth-order implusive elastic beam equations

*Correspondence: sxkj163@163.com 1School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, 250014, China Full list of author information is available at the end of the article Abstract In this paper, the existence of two solutions for superlinear fourth-order impulsive elastic beam equations is obtained. We get two theorems via variational methods and corresponding two-critical-point theorems. Combining with the Newton-iterative method, an example is presented to illustrate the value of the obtained theorems.


Introduction
In recent ten years, research on the existence numbers of solutions to nonlinear differential equations has been widely performed via corresponding critical point theorems, for example, [2,6,[9][10][11][14][15][16][17] and the references therein, and the aim of this paper is to obtain the existence of two solutions to Kirchhoff-type fourth-order impulsive elastic beam equations.
In [5], Bonanno   In [13], we obtained the existence of triple solutions of the following second-order Hamiltonian systems with impulsive effects: via variational methods and three-critical-point theorems without adding superlinear or sublinear assumptions to the nonlinearity at zero nor infinity.
In [12], we obtained the existence of at least three solutions to the impulsive equations with small non-autonomous perturbations when the nonlinearity satisfies some superlinear conditions. In [7], D' Aguì et al. considered the fourth-order differential equations with impulsive effects where A, B are real constants, f : [0, 1] × R → R is a L 1 -Carathéodory function. They gave some criteria to guarantee the differential equations have at least two non-trivial solutions.
Motivated by the above-mentioned work, in this article, we consider the following Kirchhoff-type fourth-order differential equations with impulsive effects: where K : [0, +∞) → R is a continuous function such that there exist two constants m 0 and m 1 satisfying 0 < m 0 ≤ K(x) ≤ m 1 , ∀x ≥ 0, α ≤ 0, β ≥ 0 are real constants, λ is a positive real parameter, f : , and I ji (j = 1, 2; i = 1, 2, . . . , l) ∈ C(R, R). We aim to get the existence of at least two solutions. We find the existence of at least two solutions without assuming any asymptotic conditions neither at zero or at infinity on nonlinear items. Our main tools are variational methods and two-critical-point theorems by Bonanno and Marano.

Preliminaries
We consider the spaces and the induced norm By direct calculation one finds that the norm u is equivalent to the following norm: , for more details, see [4]. It is well known that the embedding X → C 1 ([0, 1]) is compact and there exists a positive constant k = 1 + 1 π such that for all u ∈ X (see [18]). We call that u ∈ X is a weak solution of problem (1.6) if Let the functional I λ : X → R be defined by (2.2) Lemma 2.1 , are well defined and Gâteaux differentiable at any u ∈ X and the functionalλ satisfies the (P.S.)-condition and it is unbounded from below. Then, for each λ ∈ (0, r sup (u)≤r (u) ), the functionalλ admits at least two distinct critical points.

Main results
Our main results are the following two theorems about the existence of at least two distinct solutions to the Kirchhoff-type system (1.6).

Conclusion
The main novelty of our paper is that we apply a recent obtained critical-point theorem to the study of the superlinear fourth-order impulsive elastic beam equations, and the existence of at least two solutions of this kind of equations has been studied. The assumptions made and the related considerations are needed to set up the problem in a way that makes it suitable for the abstract framework, and we also improve many previous results.

Funding
This work is supported by National Natural Science Foundation of China (11571197) and the Taishan Scholars Program of Shandong Province (tsqn20161041).