# Existence and continuity of positive solutions on a parameter for second-order impulsive differential equations

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## Abstract

Applying the eigenvalue theory and theory of *α*-concave operator, we establish some new sufficient conditions to guarantee the existence and continuity of positive solutions on a parameter for a second-order 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

*g*satisfy

- (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

_{2}) and the definition of \(c(t)\), we know that \(c(t)\) is a step function, which is bounded on

*J*, and

Such problems were first studied by Zhang and Feng [1]. By using transformation technique to deal with impulse term of second-order 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, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. However, there are almost no papers on second-order boundary value problems, especially second-order 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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 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 super-sublinearity of *f* at zero and infinity, which was first used by Sánchez in [16] (see Properties 1.1-1.2).

The following geometric Properties 1.1-1.2 will be very important in our arguments.

### Property 1.1

*R̄*be a point where

*f*attains its maximum on the interval \((0,R]\).

### Property 1.2

Finally, we are able to obtain the uniqueness results of problem (1.1) by using theory of *α*-concave 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 *α*-concave (or −*α*-convex) 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

*E*be a real Banach space over

**R**. A nonempty closed set \(P \subset E\) is said to be a cone if

- (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 \(y-x \in P\). If a cone *P* is solid and \(y-x\in P^{\circ}\), then we write \(x\ll y\).

### Definition 2.3

*P*is said to be normal if there exists a positive constant

*δ*such that

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

*Let*

*P*

*be a cone in*

*E*.

*Then the following assertions are equivalent*:

- (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*\((B+P)\cap(B-P)\)*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

*P*be a solid cone of a real Banach space

*E*. An operator \(A: P^{\circ}\rightarrow P^{\circ}\) is called an

*α*-concave operator (−

*α*-convex operator) if

*A*is increasing (decreasing) if \(x_{1}, x_{2}\in P^{\circ}\) and \(x_{1}\leq x_{2}\) imply \(Ax_{1}\leq Ax_{2} \) (\(Ax_{1}\geq Ax_{2}\)), and further, the operator

*A*is strongly increasing (decreasing) if \(x_{1}, x_{2}\in P^{\circ}\) and \(x_{1}< x_{2}\) imply \(Ax_{2}- Ax_{1}\in P^{\circ}\) (\(Ax_{1}- Ax_{2}\in P^{\circ}\)). Let \(x_{\lambda}\) be a proper element of an eigenvalue

*λ*of

*A*, that is, \(Ax_{\lambda}=\lambda x_{\lambda}\). Then \(x_{\lambda}\) is called strongly increasing (decreasing) if \(\lambda _{1}>\lambda_{2}\) implies that \(x_{\lambda_{1}}-x_{\lambda_{2}}\in P^{\circ}\) (\(x_{\lambda_{2}}-x_{\lambda_{1}}\in P^{\circ}\)), which is denoted by \(x_{\lambda_{1}}\gg x_{\lambda_{2}}\) (\(x_{\lambda_{2}}\gg x_{\lambda _{1}}\)).

### Definition 2.5

An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.

### Lemma 2.2

(Arzelà-Ascoli)

*A set*\(M\subset C(J,R)\)

*is said to be a precompact set if the following two conditions are satisfied*:

- (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*$$\bigl|u(t_{1})-u(t_{2})\bigr|< \varepsilon $$

*whenever*\(|t_{1}-t_{2}|<\delta, t_{1}, t_{2}\in J\).

### Lemma 2.3

*Suppose*

*D*

*is an open subset of an infinite*-

*dimensional real Banach space E*, \(\theta\in D\),

*and*

*P*

*is a cone of*

*E*.

*If the operator*\(\Gamma: P\cap D\rightarrow P\)

*is completely continuous with*\(\Gamma\theta=\theta\)

*and satisfies*

*then*Γ

*has a proper element on*\(P\cap\partial D\)

*associated with a positive eigenvalue*.

*That is*,

*there exist*\(x_{0}\in P\cap\partial D\)

*and*\(\mu_{0}\)

*such that*\(\Gamma x_{0}=\mu _{0}x_{0}\).

### Lemma 2.4

*Suppose that* *P* *is a normal cone of a real Banach space and* \(A: P^{\circ}\rightarrow P^{\circ}\) *is an* *α*-*concave increasing* (*or* −*α*-*convex decreasing*) *operator*. *Then* *A* *has exactly one fixed point in* \(P^{\circ}\).

## 3 Some lemmas

*J*if:

- (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

*J*. It is easy to see that \(u(t)=c(t)y(t)\) is absolutely continuous on each interval \((t_{k},t_{k+1}]\), \(k=1,2,\dots,n\). By the definition of \(c(t)\) we have \(c^{\prime}(t)=0\) for \(t\neq t_{k}\). Then, for \(t\neq t_{k}\), we have

It is obvious that \(u(t)\) satisfies the boundary conditions.

Then \(u(t)\) is a solution of problem (1.1) on *J*.

*J*. It is easy to prove that \(y(t)\) is absolutely continuous on

*J*and satisfies the boundary conditions.

Then \(y(t)\) is a solution of problem (3.2) on *J*. □

### Lemma 3.2

*If*(H

_{1})-(H

_{3})

*hold*,

*then problem*(3.2)

*has a solution*

*y*,

*and*

*y*

*can be expressed in the form*

*where*

### Proof

*y*is a solution of problem (3.2). Integrating problem (3.2) from 0 to

*t*, by the boundary conditions we obtain that

*t*, we have

### Lemma 3.3

*Let*\(\mu\in[0,ac(1))\),

*G*,

*and*

*H*

*be given as in Lemma*3.2.

*Then we have the following results*:

*where*\(0\leq e(t)=1-t\leq1\),

*and*

*where*

### 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)

*Let*\(e\in L^{p}[a,b]\)

*with*\(p>1\), \(h\in L^{q}[a,b]\)

*with*\(q>1\),

*and*\(\frac{1}{p}+\frac{1}{q}=1\).

*Then*\(eh\in L^{1}[a,b]\)

*and*

*Let*\(e\in L^{1}[a,b]\), \(h\in L^{\infty}[a,b]\).

*Then*\(eh\in L^{1}[a,b]\),

*and*

*E*is a real Banach space with the norm \(\|\cdot\| \) defined by

*K*and \(K_{1}\) in

*E*by

*K*and \(K_{1}\) are two solid normal cones and

*T*: \(K\rightarrow K\) by

### 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 Arzelà-Ascoli theorem.

Thus, the set \(\{T: y\in B_{r} \}\) is equicontinuous. The Arzelà-Ascoli 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

*Assume that*(H

_{1})-(H

_{3})

*hold*.

*If*\(0< f_{\infty }<+\infty\),

*then there exists*\(R_{0}>0\)

*such that for any*\(r>R_{0}\),

*problem*(1.1)

*has a positive solution*\(u_{r}(t)\)

*satisfying*\(\| u_{r}(t)\|=c_{M}r\)

*for any*

*where*\(\lambda_{1}\)

*and*\(\lambda_{2}\)

*are two positive finite numbers*.

### 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

*Assume that*(H

_{1})-(H

_{3})

*hold*.

*If*\(0< f_{\infty}<+\infty\),

*then there exists*\(R_{1}>0\)

*such that for any*\(r>R_{1}\),

*problem*(1.1)

*has a positive solution*\(u_{r}(t)\)

*satisfying*\(\|u_{r}(t)\|=c_{M}r\)

*for any*

*where*

### 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

*Assume that*(H

_{1})-(H

_{3})

*hold*.

*If*\(0< f_{\infty}<+\infty\),

*then there exists*\(R_{2}>0\)

*such that for any*\(r>R_{2}\),

*problem*(1.1)

*has a positive solution*\(u_{r}(t)\)

*satisfying*\(\|u_{r}(t)\|=c_{M}r\)

*for any*

*where*

### 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

*Assume that*(H

_{1})-(H

_{3})

*hold*.

*If*\(f_{\infty }=+\infty\),

*then there exists*\(R_{3}>0\)

*such that for any*\(r>R_{3}\),

*problem*(1.1)

*has a positive solution*\(u_{r}(t)\)

*satisfying*\(\|u_{r}\| =c_{M}r\)

*for any*

*where*\(\lambda_{3}\)

*is a positive finite number*.

### 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

*Assume that*(H

_{1})-(H

_{3})

*hold*.

*If*\(0< f_{0}<+\infty\),

*then there exists*\(r_{0}>0\)

*such that for any*\(0< r< r_{0}\),

*problem*(1.1)

*has a positive solution*\(u_{r}(t)\)

*satisfying*\(\|u_{r}(t)\|=c_{M}r\)

*for any*

*where*\(\hat{\lambda}_{1}\)

*and*\(\hat{\lambda}_{2}\)

*are two positive finite numbers*.

### 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

*Assume that*(H

_{1})-(H

_{3})

*hold*.

*If*\(f_{0}=+\infty\),

*then there exists*\(r_{1}>0\)

*such that for any*\(0< r< r_{1}\),

*problem*(1.1)

*has a positive solution*\(u_{r}(t)\)

*satisfying*\(\|u_{r}\|=c_{M}r\)

*for any*

*where*\(\hat{\lambda}_{3}\)

*is a positive finite number*.

### 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

*n*, problem (3.2) has a positive solution \(y_{n}\in K\). Let \(\mu_{n}=\frac {1}{\lambda_{n}}\). Since \((Ty_{n})(t)=\mu_{n} y_{n}(t)\) for \(t\in J\) and \(f(u)\geq Nu\) for all \(u>0\), where \(N=\frac{f(R)}{R}\), we have

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

*Suppose that*\(f(u): [0,+\infty)\rightarrow[0,+\infty )\)

*is a nondecreasing function with*\(f(u)>0\)

*for*\(u>0\)

*and satisfies*\(f(\rho u)\geq\rho^{\alpha}f(u)\)

*for any*\(0<\rho<1\),

*where*\(0\leq\alpha <1\).

*Then*,

*for any*\(\lambda\in(0,\infty)\),

*problem*(1.1)

*has a unique positive solution*\(u_{\lambda}(t)\).

*Furthermore*,

*such a solution*\(u_{\lambda}(t)\)

*satisfies the following properties*:

- (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

*T*is the same as in (3.18). Similarly to Lemma 3.5, the operator Ψ maps \(K_{1}\) into \(K_{1}\). In view of \(H(t,s)>0, \omega(s)>0, c^{-1}(s)>0\), and \(f(u)>0\) for \(u>0\), it is easy to see that \(\Psi: K_{1}^{0}\rightarrow K_{1}^{0}\). We assert that \(\Psi: K_{1}^{0}\rightarrow K_{1}^{0}\) is an

*α*-concave increasing operator. Indeed,

*γ*. Namely, \(y_{\lambda}(t)\) is strongly increasing in

*λ*. By Lemma 3.1, (i) is proved.

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 second-order 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

*If*(H

_{1})-(H

_{3})

*hold*,

*then problem*(6.2)

*has a solution*

*y*,

*and*

*y*

*can be expressed in the form*

*where*

*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

*g*that (H

_{1})-(H

_{3}) hold and

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. □

## Notes

### 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.

### References

- 1.Zhang, X, Feng, M: Transformation techniques and fixed point theories to establish the positive solutions of second-order impulsive differential equations. J. Comput. Appl. Math.
**271**, 117-129 (2014) MathSciNetCrossRefMATHGoogle Scholar - 2.Guo, D: Multiple positive solutions of impulsive nonlinear Fredholm integral equations and applications. J. Math. Anal. Appl.
**173**, 318-324 (1993) MathSciNetCrossRefMATHGoogle Scholar - 3.Agarwal, RP, Franco, D, O’Regan, D: Singular boundary value problems for first and second order impulsive differential equations. Aequ. Math.
**69**, 83-96 (2005) MathSciNetCrossRefMATHGoogle Scholar - 4.Lin, X, Jiang, D: Multiple solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl.
**321**, 501-514 (2006) MathSciNetCrossRefMATHGoogle Scholar - 5.Feng, M, Xie, D: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. J. Comput. Appl. Math.
**223**, 438-448 (2009) MathSciNetCrossRefMATHGoogle Scholar - 6.Li, Q, Cong, F, Jiang, D: Multiplicity of positive solutions to second order Neumann boundary value problems with impulse actions. Appl. Math. Comput.
**206**, 810-817 (2008) MathSciNetMATHGoogle Scholar - 7.Zhou, Q, Jiang, D, Tian, Y: Multiplicity of positive solutions to period boundary value problems for second order impulsive differential equations. Acta Math. Appl. Sinica (Engl. Ser.)
**26**, 113-124 (2010) MathSciNetCrossRefMATHGoogle Scholar - 8.Liu, Y, O’Regan, D: Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations. Commun. Nonlinear Sci. Numer. Simul.
**16**, 1769-1775 (2011) MathSciNetCrossRefMATHGoogle Scholar - 9.Ma, R, Yang, B, Wang, Z: Positive periodic solutions of first-order delay differential equations with impulses. Appl. Math. Comput.
**219**, 6074-6083 (2013) MathSciNetMATHGoogle Scholar - 10.Hao, X, Liu, L, Wu, Y: Positive solutions for second order impulsive differential equations with integral boundary conditions. Commun. Nonlinear Sci. Numer. Simul.
**16**, 101-111 (2011) MathSciNetCrossRefMATHGoogle Scholar - 11.Feng, M: Positive solutions for a second-order
*p*-Laplacian boundary value problem with impulsive effects and two parameters. Abstr. Appl. Anal.**2014**, 4 (2014) MathSciNetGoogle Scholar - 12.Feng, M, Qiu, J: Multi-parameter fourth order impulsive integral boundary value problems with one-dimensional
*m*-Laplacian and deviating arguments. J. Inequal. Appl.**2015**, 64 (2015) MathSciNetCrossRefMATHGoogle Scholar - 13.Zhou, J, Feng, M: Green’s function for Sturm-Liouville-type boundary value problems of fractional order impulsive differential equations and its application. Bound. Value Probl.
**2014**, 69 (2014) MathSciNetCrossRefMATHGoogle Scholar - 14.Liu, X, Guo, D: Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space. Comput. Math. Appl.
**38**, 213-223 (1999) MathSciNetCrossRefMATHGoogle Scholar - 15.Lu, G, Feng, M: Positive Green’s function and triple positive solutions of a second-order impulsive differential equation with integral boundary conditions and a delayed argument. Bound. Value Probl.
**2016**, 88 (2016) CrossRefMATHGoogle Scholar - 16.Sánchez, J: Multiple positive solutions of singular eigenvalue type problems involving the one-dimensional
*p*-Laplacian. J. Math. Anal. Appl.**292**, 401-414 (2004) MathSciNetCrossRefMATHGoogle Scholar - 17.Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988) MATHGoogle Scholar
- 18.Zhang, X, Feng, M: Existence of a positive solution for one-dimensional singular
*p*-Laplacian problems and its parameter dependence. J. Math. Anal. Appl.**413**, 566-582 (2014) MathSciNetCrossRefMATHGoogle Scholar - 19.Liu, X, Li, W: Existence and uniqueness of positive periodic solutions of functional differential equations. J. Math. Anal. Appl.
**293**, 28-39 (2004) MathSciNetCrossRefMATHGoogle Scholar - 20.Zhou, J, Feng, M: Triple positive solutions for a second order
*m*-point boundary value problem with a delayed argument. Bound. Value Probl.**2015**, 178 (2015) MathSciNetCrossRefMATHGoogle Scholar

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