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Existence and continuity of positive solutions on a parameter for secondorder impulsive differential equations
 Yafang Tian^{1} and
 XueMei Zhang^{1}Email author
 Received: 14 July 2016
 Accepted: 1 September 2016
 Published: 9 September 2016
Abstract
Applying the eigenvalue theory and theory of αconcave operator, we establish some new sufficient conditions to guarantee the existence and continuity of positive solutions on a parameter for a secondorder impulsive differential equation. Furthermore, two nonexistence results of positive solutions are also given. In particular, we prove that the unique solution \(u_{\lambda}(t)\) of the problem is strongly increasing and depends continuously on the parameter λ.
Keywords
 continuity on a parameter
 impulsive differential equations
 transformation technique
 \(L^{p}\)integrable
 eigenvalue
1 Introduction
 (H_{1}):

\(\omega\in L^{p}[0,1]\) for some \(1\leq p\leq+\infty\), and there exists \(\xi>0\) such that \(\omega(t)\geq\xi\) a.e. on J;
 (H_{2}):

\(f\in C([0,+\infty), [0,+\infty))\) with \(f(0)=0\) and \(f(u)>0\) for \(u>0\), \(\{c_{k}\}\) is a real sequence with \(c_{k}>1\), \(k=1, 2, \dots, n\), and \(c(t):=\Pi_{0< t_{k}< t}(1+c_{k})\);
 (H_{3}):

\(g\in C[0,1]\) is nonnegative with$$ \mu:= \int_{0}^{1}g(t)c(t)\,dt\in\bigl[0,ac(1)\bigr). $$(1.2)
Remark 1.1
Remark 1.2
Such problems were first studied by Zhang and Feng [1]. By using transformation technique to deal with impulse term of secondorder impulsive differential equations, the authors obtained existence results of positive solutions by using fixed point theorems in a cone. However, they only considered the case \(\omega(t)\equiv1\) on \(t\in [0,1]\). The other related results can be found in [2–14]. However, there are almost no papers on secondorder boundary value problems, especially secondorder boundary value problems with impulsive effects, using the eigenvalue theory. In this paper, we solve this problem.
The first goal of this paper is to establish several criteria for the optimal intervals of the parameter λ so as to ensure the existence of positive solutions for problem (1.1). Our method is based on transformation technique, Hölder’s inequality, and the eigenvalue theory and is completely different from those used in [1–14].
Another contribution of this paper is to study the expression and properties of Green’s function associated with problem (1.1). It is interesting to point out that the Green’s function associated with problem (1.1) is positive, which is different from that of [15].
Moreover, we give two nonexistence results. The arguments that we present here are based on geometric properties of the supersublinearity of f at zero and infinity, which was first used by Sánchez in [16] (see Properties 1.11.2).
The following geometric Properties 1.11.2 will be very important in our arguments.
Property 1.1
Property 1.2
Finally, we are able to obtain the uniqueness results of problem (1.1) by using theory of αconcave operators. We also obtain the following analytical properties: the unique solution \(u_{\lambda}(t)\) of the above problem is strongly increasing and depends continuously on the parameter λ.
The rest of this paper is organized as follows. In Section 2, we provide some necessary background. In particular, we introduce some lemmas and definitions associated with the eigenvalue theory and theory of αconcave (or −αconvex) operators. Several technical lemmas are given in Section 3. In Section 4, we establish the existence and nonexistence of positive solutions for problem (1.1). In Section 5, we prove the uniqueness of a positive solution for problem (1.1) and its continuity on a parameter . In Section 6, we offer some remarks and comments on the associated problem (1.1). Finally, in Section 7, two examples are also included to illustrate the main results.
2 Preliminaries
In this section, we collect some known results, which can be found in the book by Guo and Lakshmikantham [17].
Definition 2.1
 (i)
\(au+bv \in P\) for all \(u, v \in P\) and all \(a\geq0, b\geq0\), and
 (ii)
\(u, u \in P \) implies \(u=0\).
Definition 2.2
A cone P of a real Banach space E is a solid cone if \(P^{\circ}\) is not empty, where \(P^{\circ}\) is the interior of P.
Every cone \(P \subset E\) induces a semiorder in E given by “≤”. That is, \(x\leq y\) if and only if \(yx \in P\). If a cone P is solid and \(yx\in P^{\circ}\), then we write \(x\ll y\).
Definition 2.3
Geometrically, normality means that the angle between two positive unit vectors is bounded away from π. In other words, a normal cone cannot be too large.
Lemma 2.1
 (i)
P is normal;
 (ii)There exists a constant \(\gamma>0\) such that$$\x+y\\geq\gamma\max\bigl\{ \x\,\y\\bigr\} ,\quad \forall x, y\in P; $$
 (iii)
There exists a constant \(\eta>0\) such that \(0\leq x\leq y\) implies that \(\x\\leq\eta\y\\), that is, the norm \(\\cdot\\) is semimonotone;
 (iv)
There exists an equivalent norm \(\\cdot\_{1}\) on E such that \(0\leq x\leq y\) implies that \(\x\_{1}\leq\y\_{1}\), that is, the norm \(\\cdot\_{1}\) is semimonotone;
 (v)
\(x_{n}\leq z_{n}\leq y_{n}\) (\(n=1,2,3,\ldots\)) and \(\ x_{n}x\\rightarrow0, \y_{n}x\\rightarrow0\) imply that \(\ z_{n}x\\rightarrow0\);
 (vi)The set \((B+P)\cap(BP)\) is bounded, where$$B=\bigl\{ x\in E:\x\\leq1\bigr\} ; $$
 (vii)
Every order interval \([x,y]=\{z\in E:x\leq z\leq y\}\) is bounded.
Remark 2.1
Some authors use assertion (iii) as the definition of normality of a cone P and call the smallest number η the normal constant of P.
Definition 2.4
Definition 2.5
An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.
Lemma 2.2
(ArzelàAscoli)
 (i)
All the functions in the set M are uniformly bounded, which means that there exists a constant \(r>0\) such that \(u(t)\leq r, \forall t\in J, u\in M\);
 (ii)All the functions in the set M are equicontinuous, which means that for every \(\varepsilon>0\), there is \(\delta=\delta (\varepsilon)>0\), which is independent of the function \(u\in M\), such that$$\biglu(t_{1})u(t_{2})\bigr< \varepsilon $$
Lemma 2.3
Lemma 2.4
Suppose that P is a normal cone of a real Banach space and \(A: P^{\circ}\rightarrow P^{\circ}\) is an αconcave increasing (or −αconvex decreasing) operator. Then A has exactly one fixed point in \(P^{\circ}\).
3 Some lemmas
 (i)
\(u(t)\) is absolutely continuous on each interval \((0,t_{1}]\) and \((t_{k},t_{k+1}]\), \(k=1, 2, \dots, n\);
 (ii)
for any \(k=1, 2, \dots, n\), \(u(t_{k}^{+})\) and \(u(t_{k}^{})\) exist, and \(u(t_{k}^{})=u(t_{k})\);
 (iii)
\(u(t)\) satisfies (1.1).
Lemma 3.1
Proof
It is obvious that \(u(t)\) satisfies the boundary conditions.
Then \(u(t)\) is a solution of problem (1.1) on J.
Then \(y(t)\) is a solution of problem (3.2) on J. □
Lemma 3.2
Proof
Lemma 3.3
Proof
This gives the proof of (3.14).
To obtain some of the norm inequalities in our main results, we employ Hölder’s inequality.
Lemma 3.4
(Hölder)
Lemma 3.5
Assume that (H_{1})(H_{3}) hold. Then \(T(K)\subset K\), and \(T: K\rightarrow K\) is completely continuous.
Proof
Next, we prove that the operator \(T: K\rightarrow K\) is completely continuous by standard methods and the ArzelàAscoli theorem.
Thus, the set \(\{T: y\in B_{r} \}\) is equicontinuous. The ArzelàAscoli theorem implies that T is completely continuous, and Lemma 3.5 is proved. □
4 Existence and nonexistence of positive solutions on a parameter
In this section, we establish some sufficient conditions for the existence and nonexistence of positive solutions of problem (1.1). We consider the following three cases for \(\omega\in L^{p}[0,1]: p>1, p=1\), and \(p=\infty\). The case \(p>1\) is treated in the following theorem.
Theorem 4.1
Proof
By (3.3) and (3.18) problem (1.1) has a positive solution \(u_{r}(t)\) associated with \(\lambda>0\) if and only if the operator T has a proper element \(y_{r}\) associated with the eigenvalue \(\frac {1}{\lambda}>0\).
By Lemma 2.3, for any \(r>R_{0}\), the operator T has a proper element \(y_{r}\in K\) associated with the eigenvalue \(\gamma>0\); further, \(y_{r}\) satisfies \(\y_{r}\=r\). Let \(\lambda=\frac{1}{\gamma}\). Then problem (3.2) has a positive solution \(y_{r}(t)\) associated with λ.
Hence, it follows from Lemma 3.1 that problem (1.1) has a positive solution \(u_{r}(t)\) associated with λ and satisfying \(\u_{r}\=c_{M}r\).
In conclusion, \(\lambda\in[\lambda_{1},\lambda_{2}]\). The proof is complete. □
The following Corollary 4.1 deals with the case \(p=\infty\).
Corollary 4.1
Proof
Replacing \(\h\_{q}\\omega\_{p}\) by \(\h\_{1}\\omega\ _{\infty}\) and repeating the argument above, we get the corollary. □
Finally, we consider the case of \(p=1\).
Corollary 4.2
Proof
Replacing \(\beta^{\prime}\h\_{q}\\omega\_{p}\) by \(\beta ^{*}\\omega\_{1}\) and repeating the argument above, we get the corollary. □
In the following theorems, we only consider the case \(1< p<+\infty\).
Theorem 4.2
Proof
Similarly to the proof of Theorem 4.1, it is easy to see from (4.2) and (4.3) that Theorem 4.2 is also true. □
Theorem 4.3
Proof
By (3.3) and (3.18) problem (1.1) has a positive solution \(u_{r}(t)\) associated with \(\lambda>0\) if and only if the operator T has a proper element \(y_{r}\) associated with the eigenvalue \(\frac {1}{\lambda}>0\).
Then \(U_{r}\) is a bounded open subset of the Banach space E, and \(\theta\in U_{r}\).
Now, we prove that \(r_{0}=\frac{\eta'}{c_{M}}\) is required.
By Lemma 2.3, for any \(0< r< r_{0}\), the operator T has a proper element \(y_{r}\in K\) associated with the eigenvalue \(\gamma>0\); further, \(y_{r}\) satisfies \(\y_{r}\=r\). Letting \(\lambda=\frac {1}{\gamma}\) and following the proof of Theorem 4.1, we complete the proof of Theorem 4.3. □
Theorem 4.4
Proof
The proof is similar to that of Theorem 4.3, so we omit it here. □
Theorem 4.5
Assume that (H_{1})(H_{3}) hold. If \(f_{0}=f_{\infty}=+\infty\), then there exists \(\bar{\lambda}>0\) such that problem (1.1) has no positive solutions for all \(\lambda\in[\bar {\lambda},+\infty)\).
Proof
Since n may be arbitrarily large, we obtain a contradiction.
Therefore, by Lemma 3.1 problem (1.1) has no positive solutions for all \(\lambda\geq\bar{\lambda}\). This gives the proof of Theorem 4.5. □
Theorem 4.6
Assume that (H_{1})(H_{3}) hold. If \(f_{0}=f_{\infty}=0\), then there exists \(\underline{\lambda}>0\) such that problem (1.1) has no positive solutions for \(\lambda\in (0,\underline{\lambda})\).
Proof
Remark 4.1
The method to study the existence and nonexistence results of positive solutions is completely different from those of Zhang and Feng [18].
5 Uniqueness and continuity of positive solution on a parameter
In the previous section, we have established some existence and nonexistence criteria of positive solutions for problem (1.1). Next, we consider the uniqueness and continuity of positive solutions on a parameter for problem (1.1).
Theorem 5.1
 (i)
\(u_{\lambda}(t)\) is strongly increasing in λ, that is, \(\lambda_{1}>\lambda_{2}>0\) implies \(u_{\lambda_{1}}(t)\gg u_{\lambda _{2}}(t)\) for \(t\in J\).
 (ii)
\(\lim_{\lambda\rightarrow0^{+}}\u_{\lambda}\=0, \lim_{\lambda\rightarrow+\infty}\u_{\lambda}\=+\infty\).
 (iii)
\(u_{\lambda}(t)\) is continuous with respect to λ, that is, \(\lambda\rightarrow\lambda_{0}>0\) implies \(\u_{\lambda}u_{\lambda _{0}}\\rightarrow0\).
Proof
Similarly, letting \(\gamma_{1}=\gamma\) and fixing \(\gamma_{2}\), again by (5.2) and the normality of \(K_{1}\) we have \(\lim_{\lambda \rightarrow+\infty}\y_{\lambda}(t)\=+\infty\). Then, it follows from Lemma 3.1 that \(\lim_{\lambda\rightarrow +\infty}\u_{\lambda}(t)\=+\infty\).
This gives the proof of (ii).
6 Remarks and comments
In this section, we offer some remarks and comments on the associated problem (1.1).
Remark 6.1
Some ideas of the proof of Theorem 5.1 come from Theorem 2.2.7 in [17] and Theorem 6 in [19], but there are almost no papers considering the uniqueness of positive solution for second impulsive differential equations, especially in the case where \(\omega (t)\) is \(L^{p}\)integrable.
Remark 6.2
Generally, it is difficult to study the uniqueness of a positive solution for nonlinear secondorder differential equations with or without impulsive effects (see, e.g., [4, 5, 20] and references therein).
Using a proof similar to that of Lemma 3.2, we can obtain the following results.
Lemma 6.1
It is not difficult to prove that \(H^{*}(t,s)\) and \(G^{*}(t,s)\) have similar properties to those of \(H(t,s)\) and \(G(t,s)\). However, we cannot guarantee that \(H^{*}(t,s)>0\) for any \(t,s\in J\). This implies that we cannot apply Lemma 2.4 to study the uniqueness of a positive solution for problem (6.1).
Remark 6.3
In Theorem 5.1, even though we do not assume that T is completely continuous or even continuous, we can assert that \(u_{\lambda}\) depends continuously on λ.
Remark 6.4
If we replace \(K_{1}, K_{1}^{0}\) by \(K, K^{0}\), respectively, then Theorem 5.1 also holds.
7 Examples
To illustrate how our main results can be used in practice, we present two examples.
Example 7.1
Conclusion
Problem (7.1) has at least one positive solution for any \(\lambda\in[0.0056, 0.09]\).
Proof
Hence, by Theorem 4.1 the conclusion follows, and the proof is complete. □
Example 7.2
Conclusion
Problem (7.3) has at least one positive solution for any \(\lambda\in[\frac{1}{504}, \frac{1}{30}]\).
Proof
Therefore, it follows from the definitions \(\omega(t), f\), and g that (H_{1})(H_{3}) hold.
Hence, by Corollary 4.2 the conclusion follows, and the proof is complete. □
Declarations
Acknowledgements
This work is sponsored by the National Natural Science Foundation of China (11301178, 11371117), the Beijing Natural Science Foundation of China (1163007), and the Scientific Research Project of Construction for Scientific and Technological Innovation Service Capacity (71E1610973). The authors are grateful to anonymous referees for their constructive comments and suggestions, which have greatly improved this paper.
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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