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Existence of solutions for fourth-order nonlinear boundary value problems
Advances in Difference Equations volume 2021, Article number: 196 (2021)
Abstract
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 \(\alpha \geq \beta \), we show the existence of (extreme) solution.
1 Introduction
In this paper, we are concerned with the existence and approximation of solutions for the fourth-order nonlinear boundary value problem
where \(I=[0,1]\), \(f:I\times \mathbb{R}\rightarrow \mathbb{R}\) is continuous, and \(A,B,C,D\in \mathbb{R}\).
The quasilinearization method is one of important tools to deal with nonlinear boundary value problems, see [1–5] and the references therein. In [6], Khan studied the second order nonlinear Neumann problem
where \(f:I\times \mathbb{R}\rightarrow \mathbb{R}\) is continuous and \(A,B\in \mathbb{R}\). 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 \(\alpha \geq \beta \). For the case that a lower solution α is not greater than an upper solution β, we also refer the reader to the papers [7–10].
There are a few papers which studied fourth-order boundary value problems with the help of the quasilinearization technique, see [11–14]. In [13], Ma, Zhang, and Fu discussed a fourth-order boundary value problem
where \(g:I\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}\) 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 \(\alpha \leq \beta \).
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 \(\alpha \geq \beta \).
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.
2 Comparison principle
Consider the linear problems
and
where \(M,\lambda ,a,b,A,B,C,D\in R\), \(\sigma ,h \in C(I)\).
From [6], if \(M\neq -n^{2}\pi ^{2}\), (2.1) has a unique solution of the form
where
Lemma 2.1
([6])
(i) Let \(-\frac{\pi ^{2}}{4}\leq M<0\). If \(a\leq 0\leq b\) and \(\sigma (t)\geq 0\), then \(\overline{x}(t)\leq 0\) for all \(t\in I\). If \(b\leq 0\leq a\) and \(\sigma (t)\leq 0\), then \(\overline{x}(t)\geq 0\) for all \(t\in I\).
(ii) Let \(M>0\). If \(a\leq 0\leq b\) and \(\sigma (t)\geq 0\), then \(\overline{x}(t)\geq 0\) for all \(t\in I\). If \(b\leq 0\leq a\) and \(\sigma (t)\leq 0\), then \(\overline{x}(t)\leq 0\) for all \(t\in I\).
Lemma 2.2
Let \(0<\lambda \leq \frac{\pi }{2}\). Then (2.2) has a unique solution \(\widetilde{x}(t)\). Moreover, \(\widetilde{x}(t)\geq 0\) for all \(t\in I\) if \(B\leq 0\leq A\), \(C+\lambda ^{2}A\leq 0\leq D+\lambda ^{2}B\) and \(h(t)\leq 0\); \(\widetilde{x}(t)\leq 0\) for all \(t\in I\) if \(A\leq 0\leq B\), \(D+\lambda ^{2}B\leq 0\leq C+\lambda ^{2}A\), \(h(t)\geq 0\).
Proof
Letting \(y(t)=-x''(t)-\lambda ^{2} x(t)\), we get
and
Hence
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.
If \(C+\lambda ^{2}A\leq 0\leq D+\lambda ^{2}B\) and \(h(t)\leq 0\), using (ii) of Lemma 2.1, we get that \(y(t)\leq 0\). Together with \(B\leq 0\leq A\), we obtain that \(\widetilde{x}(t)\geq 0\) by (i) of Lemma 2.1. Similarly, \(\widetilde{x}(t)\leq 0\) if \(A\leq 0\leq B\), \(D+\lambda ^{2}B\leq 0\leq C+\lambda ^{2}A\), \(h(t)\geq 0\). This completes the proof. □
3 Lower and upper solutions
Definition 3.1
Function \(\alpha \in C^{4}(I)\) is called a lower solution of (1.1) if
An upper solution \(\beta \in C^{4}(I)\) of (1.1) is defined similarly by reversing the inequalities.
Theorem 3.1
Let \(0<\lambda \leq \frac{\pi }{2}\). Suppose that α and β are respectively lower and upper solutions of (1.1) such that \(\alpha (t)\geq \beta (t)\), \(t\in I\). If \(f(t,x)-\lambda ^{4}x\) is nonincreasing in x, then there exists a solution \(x\in C^{4}(I)\) of (1.1) such that
Proof
Define \(p(\alpha (t),x,\beta (t))=\min \{\alpha (t),\max \{x,\beta (t)\}\}\). Then \(p(\alpha (t),x,\beta (t))\) is continuous and satisfies \(\beta (t)\leq p(\alpha (t),x,\beta (t))\leq \alpha (t)\) for all \(x\in \mathbb{R}\), \(t\in I\). Consider the boundary value problem
where \(\psi (t,x)=f(t,p(\alpha (t),x,\beta (t)))-\lambda ^{4}p(\alpha (t),x, \beta (t))\). Problem (3.1) is equivalent to the integral equation
Since α, β, ψ, \(P^{-\lambda ^{2}}_{(A,B)}\), \(P^{\lambda ^{2}}_{(-C-\lambda ^{2}A,-D-\lambda ^{2}B)}\), \(G_{-\lambda ^{2}}\) and \(G_{\lambda ^{2}}\) are continuous, there exist constants \(c_{1},c_{2},c_{3}>0\) such that
Let \(\Vert u \Vert =\max_{t\in I} \vert u(t) \vert \) and \(\Omega =\{u\in C(I): \Vert u \Vert \leq c_{2}+c_{3}(c_{2}+c_{3}c_{1})\}\). It is easy to show that \(T:\Omega \rightarrow \Omega \) is continuous and compact. Hence, T has a fixed point \(x\in \Omega \) by the Schauder fixed point theorem. Moreover, \(x\in C^{4}(I)\) is a solution of (3.1).
Let \(v(t)=\alpha (t)-x(t)\), \(t\in I\). Then \(v'(0)=0=v'(1)\) and \(v'''(0)+\lambda ^{2}v'(0)\leq 0\leq v'''(1)+\lambda ^{2} v'(1)\). Since \(f(t,x)-\lambda ^{4}x\) is nonincreasing in x and \(p(\alpha (t),x,\beta (t))\leq \alpha (t)\), \(t\in I\), we can see that
which implies \(\alpha (t)\geq x(t)\), \(t\in I\). Similarly, \(x(t)\geq \beta (t)\), \(t\in I\). Hence,
that is, x is a solution of (1.1). This completes the proof. □
Theorem 3.2
Assume that α and β are respectively lower and upper solutions of problem (1.1). If \(f:I\times \mathbb{R}\rightarrow \mathbb{R}\) is continuous and for some \(0<\lambda \leq \frac{\pi }{2}\),
then \(\alpha (t)\geq \beta (t)\), \(t\in I\).
Proof
Let \(m(t)=\alpha (t)-\beta (t)\), \(t\in I\). Then \(m(t)\in C^{4}(I)\), \(m'(0)=0=m'(1)\), \(m'''(0)\leq 0 \leq m'''(1)\), and \(m'''(0)+\lambda ^{2}m'(0)\leq 0\leq m'''(1)+\lambda ^{2} m'(1)\). Using the definition of lower and upper solutions, we have
which implies that \(m(t)\geq 0\), \(t\in I\) by Lemma 2.2. This completes the proof. □
4 Main results
To prove the main theorem, we need the following assumptions:
\((H_{1})\) The functions \(\alpha ,\beta \in C^{4}(I)\) are respectively lower and upper solutions of (1.1), and \(\alpha (t)\geq \beta (t)\), \(t\in I\).
\((H_{2})\) \(f\in C^{2}(I\times \mathbb{R},\mathbb{R})\) and \(0< f_{x}(t,x)\leq (\frac{\pi }{2})^{4}\), \(f_{xx}(t,x)\leq 0\) for \((t,x)\in I\times [\min \beta (t),\max \alpha (t)]\).
\((H_{3})\) There exists a constant \(k\in (0,\frac{\pi }{2}]\) such that
for \(\beta (t)\leq x_{2}\leq x_{1}\leq \alpha (t)\), \(t\in I\).
Theorem 4.1
Let \((H_{1})\) and \((H_{2})\) hold. Then there exists a monotone sequence \(\{\omega _{n}\}\) converging uniformly and quadratically to a solution of (1.1).
Proof
Taylor’s theorem and condition \((H_{2})\) imply that
for \((t,x), (t,y)\in I\times [\min \beta (t),\max \alpha (t)]\), where \(\zeta \in (\min \{x,y\},\max \{x,y\})\). Define
then \(F(t,x,y)\) is continuous and satisfies the relations
for \((t,x), (t,y)\in I\times [\min \beta (t),\max \alpha (t)]\).
Let \(\lambda >0\) and \(\lambda ^{4}=\max \{f_{x}(t,x):(t,x)\in I\times [\min \beta (t), \max \alpha (t)]\}\). Then \(0<\lambda \leq \frac{\pi }{2}\). Put \(\varphi \in [\beta ,\alpha ]=\{x\in C^{4}(I):\beta (t)\leq x(t)\leq \alpha (t),t\in I\}\) and consider the problem
Clearly,
that is, α and β are respectively lower and upper solutions of (4.1).
On the other hand, the function
is nonincreasing in x. From Theorem 3.1, (4.1) has a solution \(\omega _{1}\in [\beta ,\alpha ]\). Moreover, \(\omega _{1}\) is an upper solution of (4.1), which implies that
has a solution \(\omega _{2}\in [\omega _{1},\alpha ]\). Repeating the process, we obtain a sequence \(\{\omega _{n}\}\) satisfying
and \(\{\omega _{n}\}\) is uniformly convergent. Let \(\lim_{n\rightarrow \infty }\omega _{n}(t)=x\). 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 \(\{\omega _{n}\}\) is quadratic, we begin by writing \(e_{n}(t)=x(t)-\omega _{n}(t)\), \(t\in I\), \(n\in \mathbb{N}^{+}\), where x is a solution of (1.1). It is clear that \(e_{n}\geq 0\) on I and \(e_{n}'(0)=e_{n}'(1)=e_{n}'''(0)=e_{n}'''(1)=0\). Let \(\rho ^{4}=\min \{f_{x}(t,x):(t,x)\in I\times [\min \beta (t),\max \alpha (t)]\}\). Then \(0<\rho \leq \frac{\pi }{2}\). In view of Taylor’s theorem, we obtain
where \(\omega _{n-1}(t)<\xi (t)<x(t)\), \(t\in I\). Let \(\gamma (t)\) be the unique solution of the boundary value problem
Similar to (3.1), γ satisfies
where
Setting \(K_{n}(t)=e_{n}(t)-\gamma (t)\), \(t\in I\), we get \(K_{n}'(0)=K_{n}'(1)=K_{n}'''(1)=K_{n}'''(0)=0\) and \(K_{n}^{(4)}-\rho ^{4}K_{n}\geq 0\) on I. By Lemma 2.2, we easily obtain \(e_{n}(t)\leq \gamma (t)\), \(t\in I\). Thus \(\Vert e_{n} \Vert \leq \delta \Vert e_{n-1} \Vert ^{2}\) and we conclude that the convergence of the sequence \(\{\omega _{n}\}\) is quadratic. This completes the proof. □
Remark 4.1
In \((H_{2})\), if the assumption \(f_{xx}(t,x)\leq 0\) is replaced by \(f_{xx}(t,x)\geq 0\) for \((t,x)\in I\times [\min \beta (t),\max \alpha (t)]\), and we let the other assumptions in Theorem 4.1 hold, then
for \(x,y\in [\min \beta (t),\max \alpha (t)]\), \(t\in I\). One can obtain a monotonically nonincreasing sequence \(\{\omega _{n}\}\) of solutions of (4.1) with
which converges uniformly and quadratically to a solution of (1.1).
Theorem 4.2
Let \((H_{1})\) and \((H_{3})\) hold. Then there exist monotone sequences \(\{\alpha _{n}\}\), \(\{\beta _{n}\}\) with \(\alpha _{0}=\alpha \), \(\beta _{0}=\beta \) such that \(\lim_{n\rightarrow \infty }\alpha _{n}(t)=u(t)\), \(\lim_{n\rightarrow \infty }\beta _{n}(t)=r(t)\) uniformly on I, and r, u are the minimal and maximal solutions of (1.1), respectively, such that
on I, where x is a solution of (1.1) such that \(\beta (t)\leq x(t)\leq \alpha (t)\), \(t\in I\).
Proof
For any \(\phi \in [\beta ,\alpha ]\), we consider the equation
From \((H_{3})\), we have
Hence, α, β are respectively lower and upper solution of (4.3). By Lemma 2.2 and Theorem 3.1, (4.3) has a unique solution \(x\in [\beta ,\alpha ]\). We can define an operator Q by \(x=Q\phi \) and Q is an operator from \([\beta ,\alpha ]\) to \([\beta ,\alpha ]\). Thus \(Q\alpha \leq \alpha \), \(\beta \leq Q\beta \).
Now, we prove that Q is nondecreasing in \([\beta ,\alpha ]\). Let \(\beta \leq \mu _{1}\leq \mu _{2}\leq \alpha \) and \(\eta =Q\mu _{1}-Q\mu _{2}\). Then, by \((H_{3})\), we have
By Lemma 2.2, \(\eta (t)\leq 0\), which implies \(Q\mu _{1}\leq Q\mu _{2}\).
Define the sequences \(\{ \alpha _{n}\}\), \(\{ \beta _{n}\}\) with \(\alpha _{0}=\alpha \), \(\beta _{0}=\beta \) such that \(\alpha _{n+1}=Q\alpha _{n}\), \(\beta _{n+1}=Q\beta _{n}\) for \(n=0,1,2,\dots \) From the fact that \(Q\alpha \leq \alpha \), \(\beta \leq Q\beta \) and the monotonicity of Q, we have
on I and
Therefore \(\lim_{n\rightarrow \infty }\alpha _{n}(t)=u(t)\), \(\lim_{n\rightarrow \infty }\beta _{n}(t)=r(t)\) uniformly on I. Clearly, u, r are solutions of (1.1).
Finally, we prove that if \(x\in [\beta ,\alpha ]\) is a solution of (1.1), then \(r\leq x\leq u\) on I. To this end, we assume that, without loss of generality, \(\beta _{n}(t)\leq x(t)\leq \alpha _{n}(t)\) for some n. We also know that \(\beta _{n+1}\leq x\leq \alpha _{n+1}\) on I from the monotonicity of Q. Since \(\beta _{0}\leq x\leq \alpha _{0}\) on I, we can conclude that
Passing to the limit as \(n\rightarrow \infty \), we obtain that \(r\leq x\leq u\). This completes the proof. □
Example 4.1
Consider the following problem:
Let \(\alpha =\sqrt{2}\) and \(\beta =\frac{1}{2}\). It is easy to check that α, β are respectively lower and upper solutions of (4.4). Moreover, for
we can easily check that \(0< f_{x}(t,u)\leq (\frac{\pi }{2})^{4}\) and \(f_{xx}(t,u)\geq 0\) for \((t,u)\in I\times [\frac{1}{2},\sqrt{2}]\). By Remark 4.1, there exists a monotonically nonincreasing sequence \(\{\omega _{n}\}\) which converges uniformly and quadratically to a solution of (4.4).
Example 4.2
Consider the following problem:
where \(c>0\) is sufficiently small.
Let \(\alpha (t)=A+t\) and \(\beta (t)=t-c\frac{(t-t^{2})^{2}}{24}\), \(t\in I\), here \(A>\sqrt{(\tan 0.25)^{-1}}\). We have \(\alpha '(0)=\alpha '(1)=1\), \(\alpha '''(0)=\alpha '''(1)=0\),
which means that α is a lower solution of (4.5).
Moreover, \(\beta '(0)=\beta '(1)=1\), \(\beta '''(0)=\frac{c}{2}>0\), \(\beta '''(1)=- \frac{c}{2}<0\), and
There exist a constant \(c^{*}>0\) such that \(\arctan \frac{1}{c}>\frac{\pi }{4}\) if \(c\in (0,c^{*})\). For such \(c\in (0,c^{*})\),
hence β is an upper solution of (4.5) and \((H_{1})\) is satisfied.
On the other hand, for \(\forall (t,u)\in I\times [\min \beta (t),\max \alpha (t)]\), we have
Hence \((H_{3})\) is satisfied for \(c>0\) sufficiently small. By Theorem 4.2, (4.5) has the maximal and minimal solutions on \([\beta ,\alpha ]\).
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Acknowledgements
The author would like to express her sincere thanks to the referees for the careful reading and their important comments which helped improve the original paper.
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Mingzhu Huang received B.S. degree at the Department of Mathematics, Hunan University of Science and Technology. Now she is a Master’s student and her research direction is differential equations.
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Huang, M. Existence of solutions for fourth-order nonlinear boundary value problems. Adv Differ Equ 2021, 196 (2021). https://doi.org/10.1186/s13662-021-03354-4
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DOI: https://doi.org/10.1186/s13662-021-03354-4
Keywords
- Boundary value problem
- Quasilinearization method
- Upper solution and lower solution
- Extreme solution