The shooting method and positive solutions of fourth-order impulsive differential equations with multi-strip integral boundary conditions
- Yuke Zhu1 and
- Huihui Pang1Email author
https://doi.org/10.1186/s13662-017-1453-2
© The Author(s) 2018
Received: 17 August 2017
Accepted: 15 December 2017
Published: 10 January 2018
Abstract
In this paper, we investigate the existence results of a fourth-order differential equation with multi-strip integral boundary conditions. Our analysis relies on the shooting method and the Sturm comparison theorem. Finally, an example is discussed for the illustration of the main work.
Keywords
MSC
1 Introduction
In fact, multi-strip conditions correspond to a linear combination of integrals of unknown function on the sub-intervals of J. The multi-strip conditions appear in the mathematical modelings of the physical phenomena, for instance, see [1, 2]. They have various applications in realistic fields such as blood flow problems, semiconductor problems, hydrodynamic problems, thermoelectric flexibility, underground water flow and so on. For a detailed description of multi-strip integral boundary conditions, we refer the reader to the papers [3–6].
Impulsive differential equations, which provide a natural description of observed evolution processes, are regarded as important mathematical tools for a better understanding of several real world problems in the applied sciences. For an overview of existing results and of recent research areas of impulsive differential equations, see [7–10].
The existing literature indicates that research of fourth-order nonlocal integral boundary value problems is excellent, and the relevant methods are developed to be various. Generally, the fixed point theorems in cones, the method of upper and lower solutions, the monotone iterative technique, the critical point theory and variational methods play extremely important roles in establishing the existence of solutions to boundary value problems. It is well known that the classical shooting method could be effectively used to prove the existence results for differential equation boundary value problems. To some extent, this approach has an advantage over the traditional methods. Readers can see [11–16] and the references therein for details.
To the best of our knowledge, no paper has considered the existence of positive solutions for a fourth-order impulsive differential equation multi-strip integral boundary value problem with the shooting method till now. Motivated by the excellent work mentioned above, in this paper, we try to employ the shooting method to establish the criteria for the existence of positive solutions to BVP (1.1).
The rest of the paper is organized as follows. In Section 2, we provide some necessary lemmas. In particular, we transform fourth-order impulsive problem (1.1) into a second-order differential integral equation BVP (2.10), and by using the shooting method, we convert BVP (2.10) into a corresponding IVP (initial value problem). In Section 3, the main theorem is stated and proved. Finally, an example is discussed for the illustration of the main work.
- (H1):
-
\(f\in C(\mathbb{R^{+}}\times\mathbb{R^{-}}\times\mathbb{R^{-}} , \mathbb{R}^{+})\), \(I_{k}\in C(\mathbb{R^{+}},\mathbb{R^{-}})\), \(J_{k}\in C(\mathbb{R^{+}}\times\mathbb{R},\mathbb{R^{-}})\) for \(1\leqslant k\leqslant p \), here \(\mathbb{R}^{+}=[0,\infty)\), \(\mathbb{R}^{-}=(-\infty ,0]\);
- (H2):
-
h, \(g_{1}\) and \(g_{2}\in C(J,\mathbb{R}^{+}) \);
- (H3):
-
\(\lambda,\mu,\gamma_{i} ,\rho_{i} >0\) for \(i=1,2,\ldots,n\) and \(0<\Gamma=\sum_{i=1}^{n} \gamma_{i}\int_{\alpha_{i}}^{\beta _{i}}g_{1}(t)\,dt<1\).
2 Preliminaries
Lemma 2.1
Proof
Lemma 2.2
Assume that conditions (H2)-(H3) hold. \(G(t,s)\) and \(H(t,s)\) are given as in the statement of Lemma 2.1. Then \(G(t,s)\geqslant0\), \(H(t,s)\geqslant0\) for any \(t,s\in[0,1]\).
Proof
Under assumptions (H1)-(H3), denote by \(y(t,m)\) the solution of IVP (2.11). We assume that f is strong continuous enough to guarantee that \(y(t,m)\) is uniquely defined and that \(y(t,m)\) depends continuously on both t and m. The studies of this kind of problem can be available in [17]. Consequently, the solution of IVP (2.11) exists.
Lemma 2.4
([18])
- (i)
the solution \(y(t,m_{1})\) of (2.11) remains positive in \((0,1)\) and \(k(m_{1})\leqslant1\);
- (ii)
the solution \(y(t,m_{2})\) of (2.11) remains positive in \((0,1)\) and \(k(m_{2})\geqslant1\).
Now we introduce the Sturm comparison theorem derived from [19].
Lemma 2.5
Proof
Lemma 2.6
Proof
Lemma 2.7
Let \(\sum_{i=1}^{n}\rho_{i}\int_{\xi_{i}}^{\eta_{i}}g_{2}(t)\,dt>1\), then BVP (2.10) has no positive solution.
Proof
- (H4):
-
\(0<\Lambda=\sum_{i=1}^{n} \rho_{i}\int_{\xi_{i}}^{\eta _{i}}g_{2}(t)\,dt<1\).
3 Existence results
Lemma 3.1
Proof
Since the function \(f_{1}(x)\) is continuous on \((0,\frac{\pi}{2} )\), there exists a real number \(A_{1}\in (0,\frac{\pi}{2} )\) such that \(f_{1}(A_{1})\leqslant1\).
Theorem 3.1
- (i)
\(0\leqslant f^{0} < \frac{\underline{A}^{2}}{h^{L}}\), \(f_{\infty}>\frac{\bar{A}^{2}}{h^{l}}\);
- (ii)
\(0\leqslant f^{\infty}< \frac{\underline{A}^{2}}{h^{L}}\), \(f_{0}>\frac{\bar{A}^{2}}{h^{l}}\).
Proof
In what follows, we need to find a positive number \(m_{2}\) satisfying \(k(m_{2})>1\).
Claim
Since the function \(y(t,m)\) is concave and \(y'(0,m)=0\), the function \(y(t,m)\) and the line \(y=L\) intersects at most one time for the constant L defined in (3.5) and \(t\in(0,1]\). The intersecting point is denoted as \(\bar{\delta} _{m}\) provided it exists. Furthermore, we set \(I_{m}=(0,\bar{\delta} _{m}]\subseteq (0,1]\). If \(y(1,m)\geqslant L\), then \(\bar{\delta} _{m}=1\).
Next, we divide the discussion into three steps.
Step 1. We declare that there exists \(m_{0}\) large enough such that \(0\leqslant y(t,m_{0})\leqslant L\) for \(t\in[\bar{\delta} _{m_{0}},1]\) and \(y(t,m_{0})\geqslant L\) for \(t\in I_{m_{0}}\).
Step 3. Search a suitable \(m_{2}^{*}\) and a positive number σ such that \(0<\max \{\frac{A_{2}}{A_{2}+\varepsilon}, \eta_{n} \}\leqslant\sigma\leqslant1\) and \(y(t,m_{2}^{*})\geqslant L\) for \(t\in(0,\sigma]\).
Next, we show that \(k(m_{2}^{*})\geqslant1\) for the chosen \(m_{2}^{*}\) and σ.
On account of Lemma 2.4, some initial value \(m^{*}\) can be found such that \(y(t,m^{*})\) is the solution of (2.10). Consequently, \(u(t,m^{*})=(Ay)(t,m^{*})\) is the solution of BVP (1.1).
We omit the derivation for (ii) since it is similar to the above proof. □
4 Example
It is easy to check that (H1) and (H2) are satisfied. Simple calculation shows that \(\Gamma=\Lambda=\frac{9}{16}<1\), which implies that (H3) and (H4) are satisfied.
Let \(A_{1}=\frac{1}{2}\) and \(A_{2}=\frac{3}{2}\) such that \(f_{1}(A_{1})=0.9552<1\) and \(f_{2}(A_{2})>4.2304>1\).
Then condition (ii) of Theorem 3.1 is satisfied. Consequently, Theorem 3.1 guarantees that problem (4.1) has at least one positive solution \(u(t)\).
Declarations
Acknowledgements
The work is supported by Chinese Universities Scientific Fund (Project No. 2017LX003) and National Training Program of Innovation (Project No. 201710019252).
Authors’ contributions
All authors contributed equally to the manuscript, read and approved the final draft.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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