Existence of solutions for fourth-order nonlinear boundary value problems

In this paper, we discuss the existence and approximation of solutions for a fourth-order nonlinear boundary value problem by using a quasilinearization technique. In the presence of a lower solution α and an upper solution β in the reverse order α≥β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \geq \beta $\end{document}, we show the existence of (extreme) solution.


Introduction
In this paper, we are concerned with the existence and approximation of solutions for the fourth-order nonlinear boundary value problem ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ The quasilinearization method is one of important tools to deal with nonlinear boundary value problems, see [1][2][3][4][5] and the references therein. In [6], Khan studied the second order nonlinear Neumann problem ⎧ ⎨ ⎩ -x (t) = f (t, x(t)), t ∈ I, where f : I × R → R is continuous and A, B ∈ R. By using the quasilinearization technique, the author obtained the existence and approximation of solutions of (1.2) in the presence of a lower solution α and an upper solution β in the reverse order α ≥ β. For the case that a lower solution α is not greater than an upper solution β, we also refer the reader to the papers [7][8][9][10].
There are a few papers which studied fourth-order boundary value problems with the help of the quasilinearization technique, see [11][12][13][14]. In [13], Ma, Zhang, and Fu discussed a fourth-order boundary value problem where g : I × R × R → R is continuous. They showed the existence of solutions between a lower solution α and an upper solution β without any growth restriction on g by means of the monotonicity method. Li [14] obtained the existence and uniqueness result for (1.3) by the method of lower and upper solutions in the presence of a lower solution α and an upper solution β with α ≤ β.
Inspired by [6,14], in this paper, we study the existence of solution for (1.1) in the presence of a lower solution α and an upper solution β in the reverse order α ≥ β.
The paper is organized as follows. In Sect. 2, we establish a comparison principle related to problem (1.1). In Sect. 3, the concept of a lower and upper solution of (1.1) is introduced and the method of a lower and upper solution is mentioned. In Sect. 4, using the approach of quasilinearization, we obtain the existence result of (extreme) solution for (1.1), and we also discuss the quadratic convergence of the approximate sequence.

Comparison principle
Consider the linear problems where M, λ, a, b, A, B, C, D ∈ R, σ , h ∈ C(I).
is a solution of (2.2), where Clearly, the solution of (2.2) is unique since the solution of (2.4) or (2.5) is unique.
This completes the proof.

Lower and upper solutions
An upper solution β ∈ C 4 (I) of (1.1) is defined similarly by reversing the inequalities.
Suppose that α and β are respectively lower and upper solutions of (1.1) such that α(t) ≥ β(t), t ∈ I. If f (t, x)λ 4 x is nonincreasing in x, then there exists a solution x ∈ C 4 (I) of (1.1) such that Let u = max t∈I |u(t)| and = {u ∈ C(I) : u ≤ c 2 + c 3 (c 2 + c 3 c 1 )}. It is easy to show that T : → is continuous and compact. Hence, T has a fixed point x ∈ by the Schauder fixed point theorem. Moreover, x ∈ C 4 (I) is a solution of (3.1). Let that is, x is a solution of (1.1). This completes the proof.
which implies that m(t) ≥ 0, t ∈ I by Lemma 2.2. This completes the proof.

Main results
To prove the main theorem, we need the following assumptions: (H 1 ) The functions α, β ∈ C 4 (I) are respectively lower and upper solutions of (1.1), and α(t) ≥ β(t), t ∈ I.
On the other hand, the function is nonincreasing in x. From Theorem 3.1, (4.1) has a solution ω 1 ∈ [β, α]. Moreover, ω 1 is an upper solution of (4.1), which implies that has a solution ω 2 ∈ [ω 1 , α]. Repeating the process, we obtain a sequence {ω n } satisfying and {ω n } is uniformly convergent. Let lim n→∞ ω n (t) = x. Since F is continuous, we have which implies that x is a solution of problem (1.1).
To show that the convergence of the sequence {ω n } is quadratic, we begin by writing Then 0 < ρ ≤ π 2 . In view of Taylor's theorem, we obtain where ω n-1 (t) < ξ (t) < x(t), t ∈ I. Let γ (t) be the unique solution of the boundary value problem Setting K n (t) = e n (t)γ (t), t ∈ I, we get K n (0) = K n (1) = K n (1) = K n (0) = 0 and K (4) nρ 4 K n ≥ 0 on I. By Lemma 2.2, we easily obtain e n (t) ≤ γ (t), t ∈ I. Thus e n ≤ δ e n-1 2 and we conclude that the convergence of the sequence {ω n } is quadratic. This completes the proof.
, and we let the other assumptions in Theorem 4.1 hold, then , t ∈ I. One can obtain a monotonically nonincreasing sequence {ω n } of solutions of (4.1) with which converges uniformly and quadratically to a solution of (1.1).