Research

# Periodic boundary value problems for nonlinear first-order impulsive dynamic equations on time scales

Da-Bin Wang

Author Affiliations

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, People’s Republic of China

Advances in Difference Equations 2012, 2012:12  doi:10.1186/1687-1847-2012-12

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2012/1/12

 Received: 23 August 2011 Accepted: 15 February 2012 Published: 15 February 2012

© 2012 Wang; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

By using the classical fixed point theorem for operators on cone, in this article, some results of one and two positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales are obtained. Two examples are given to illustrate the main results in this article.

Mathematics Subject Classification: 39A10; 34B15.

##### Keywords:
time scale; periodic boundary value problem; positive solution; fixed point; impulsive dynamic equation

### 1 Introduction

Let T be a time scale, i.e., T is a nonempty closed subset of R. Let 0, T be points in T, an interval (0, T)T denoting time scales interval, that is, (0, T)T: = (0, T) ⋂ T. Other types of intervals are defined similarly.

The theory of impulsive differential equations is emerging as an important area of investigation, since it is a lot richer than the corresponding theory of differential equations without impulse effects. Moreover, such equations may exhibit several real world phenomena in physics, biology, engineering, etc. (see [1-3]). At the same time, the boundary value problems for impulsive differential equations and impulsive difference equations have received much attention [4-18]. On the other hand, recently, the theory of dynamic equations on time scales has become a new important branch (see, for example, [19-21]). Naturally, some authors have focused their attention on the boundary value problems of impulsive dynamic equations on time scales [22-36]. However, to the best of our knowledge, few papers concerning PBVPs of impulsive dynamic equations on time scales with semi-position condition.

In this article, we are concerned with the existence of positive solutions for the following PBVPs of impulsive dynamic equations on time scales with semi-position condition

x Δ ( t ) + f ( t , x ( σ ( t ) ) ) = 0 , t J : = [ 0 , T ] T , t t k , k = 1 , 2 , , m , x ( t k + ) - x ( t k - ) = I k ( x ( t k - ) ) , k = 1 , 2 , , m , x ( 0 ) = x ( σ ( T ) ) , (1.1)

where T is an arbitrary time scale, T > 0 is fixed, 0, T T, f C (J × [0, ∞), (-∞, ∞)), Ik C([0, ∞), [0, ∞)), tk ∈ (0, T)T, 0 < t1 < ⋯ < tm < T, and for each k = 1, 2,..., m, x ( t k + ) = lim h 0 + x ( t k + h ) and x ( t k - ) = lim h 0 - x ( t k + h ) represent the right and left limits of x(t) at t = tk. We always assume the following hypothesis holds (semi-position condition):

(H) There exists a positive number M such that

M x - f ( t , x ) 0 for x [ 0 , ) , t [ 0 , T ] T .

By using a fixed point theorem for operators on cone [37], some existence criteria of positive solution to the problem (1.1) are established. We note that for the case T = R and Ik(x) ≡ 0, k = 1, 2,..., m, the problem (1.1) reduces to the problem studied by [38] and for the case Ik(x) ≡ 0, k = 1, 2,..., m, the problem (1.1) reduces to the problem (in the one-dimension case) studied by [39].

In the remainder of this section, we state the following fixed point theorem [37].

Theorem 1.1. Let X be a Banach space and K X be a cone in X. Assume Ω1, Ω2 are bounded open subsets of X with 0 Ω 1 Ω ̄ 1 Ω 2 and Φ: K ( Ω ̄ 2 \ Ω 1 ) K is a completely continuous operator. If

(i) There exists u0 K\{0} such that u - Φu ≠ λu0, u K ⋂ ∂ Ω2, λ≥ 0; Φu τu, u K ⋂ ∂Ω1, τ ≥ 1, or

(ii) There exists u0 K\{0} such that u - Φu ≠ λu0, u K ⋂ ∂Ω1, λ≥ 0; Φu τu, u K ⋂ ∂Ω2, τ ≥ 1.

Then Φ has at least one fixed point in K ( Ω ̄ 2 \ Ω 1 ) .

### 2 Preliminaries

Throughout the rest of this article, we always assume that the points of impulse tk are right-dense for each k = 1, 2,...,m.

We define

P C = { x [ 0 , σ ( T ) ] T R : x k C ( J k , R ) , k = 0 , 1 , 2 , , m  and there exist x ( t k + ) and x ( t k ) with x ( t k ) = x ( t k ) , k = 1 , 2 , , m } ,

where xk is the restriction of x to Jk = (tk, tk+1]T ⊂ (0, σ(T)]T, k = 1, 2,..., m and J0 = [0, t1]T, tm +1 = σ(T).

Let

X = { x : x P C , x ( 0 ) = x ( σ ( T ) ) }

with the norm x = sup t [ 0 , σ ( T ) ] T x ( t ) , then X is a Banach space.

Lemma 2.1. Suppose M > 0 and h: [0, T]T R is rd-continuous, then x is a solution of

x ( t ) = 0 σ ( T ) G ( t , s ) h ( s ) Δ s + k = 1 m G ( t , t k ) I k ( x ( t k ) ) , t [ 0 , σ ( T ) ] T ,

where G ( t , s ) = e M ( s , t ) e M ( σ ( T ) , 0 ) e M ( σ ( T ) , 0 ) - 1 , 0 s t σ ( T ) , e M ( s , t ) e M ( σ ( T ) , 0 ) - 1 , 0 t < s σ ( T ) ,

if and only if x is a solution of the boundary value problem

x Δ ( t ) + M x ( σ ( t ) ) = h ( t ) , t J : = [ 0 , T ] T , t t k , k = 1 , 2 , , m , x ( t k + ) - x ( t k - ) = I k ( x ( t k - ) ) , k = 1 , 2 , , m , x ( 0 ) = x ( σ ( T ) ) .

Proof. Since the proof similar to that of [34, Lemma 3.1], we omit it here.

Lemma 2.2. Let G(t, s) be defined as in Lemma 2.1, then

1 e M ( σ ( T ) , 0 ) - 1 G ( t , s ) e M ( σ ( T ) , 0 ) e M ( σ ( T ) , 0 ) - 1 for all t , s [ 0 , σ ( T ) ] T .

Proof. It is obviously, so we omit it here.

Remark 2.1. Let G(t, s) be defined as in Lemma 2.1, then 0 σ ( T ) G ( t , s ) Δ s = 1 M .

For u X, we consider the following problem:

{ x Δ ( t ) + M x ( σ ( t ) ) = M u ( σ ( t ) ) f ( t , u ( σ ( t ) ) , t [ 0 , T ] T , t t k , k = 1 , 2 , , m , x ( t k + ) x ( t k ) = I k ( x ( t k ) ) , k = 1 , 2 , , m , x ( 0 ) = x ( σ ( T ) ) . (2.1)

It follows from Lemma 2.1 that the problem (2.1) has a unique solution:

x ( t ) = 0 σ ( T ) G ( t , s ) h u ( s ) Δ s + k = 1 m G ( t , t k ) I k ( x ( t k ) ) , t [ 0 , σ ( T ) ] T ,

where hu(s) = Mu(σ(s)) - f(s, u(σ(s))), s ∈ [0, T]T.

We define an operator Φ: X X by

Φ ( u ) ( t ) = 0 σ ( T ) G ( t , s ) h u ( s ) Δ s + k = 1 m G ( t , t k ) I k ( u ( t k ) ) , t [ 0 , σ ( T ) ] T .

It is obvious that fixed points of Φ are solutions of the problem (1.1).

Lemma 2.3. Φ: X X is completely continuous.

Proof. The proof is divided into three steps.

Step 1: To show that Φ: X X is continuous.

Let { u n } n = 1 be a sequence such that un u (n → ∞) in X. Since f(t, u) and Ik(u) are continuous in x, we have

h u n ( t ) - h u ( t ) = M ( u n - u ) - ( f ( t , u n ) - f ( t , u ) ) 0 ( n ) , I k ( u n ( t k ) ) - I k ( u ( t k ) ) 0 ( n ) .

So

Φ ( u n ) ( t ) - Φ ( u ) ( t ) = 0 σ ( T ) G ( t , s ) [ h u n ( s ) - h u ( s ) ] Δ s + k = 1 m G ( t , t k ) [ I k ( u n ( t k ) ) - I k ( u ( t k ) ) ] e M ( σ ( T ) , 0 ) e M ( σ ( T ) , 0 ) - 1 0 σ ( T ) h u n ( s ) - h u ( s ) Δ s + k = 1 m I k ( u n ( t k ) ) - I k ( u ( t k ) ) 0 ( n ) ,

which leads to ||Φun - Φu|| → 0 (n → ∞). That is, Φ: X X is continuous.

Step 2: To show that Φ maps bounded sets into bounded sets in X.

Let B X be a bounded set, that is, ∃ r > 0 such that ∀ u B we have ||u|| ≤ r. Then, for any u B, in virtue of the continuities of f(t, u) and Ik(u), there exist c > 0, ck > 0 such that

f ( t , u ) c , I k ( u ) c k , k = 1 , 2 , , m .

We get

Φ ( u ) ( t ) = 0 σ ( T ) G ( t , s ) h u ( s ) Δ s + k = 1 m G ( t , t k ) I k ( u ( t k ) ) 0 σ ( T ) G ( t , s ) h u ( s ) Δ s + k = 1 m G ( t , t k ) I k ( u ( t k ) ) e M ( σ ( T ) , 0 ) e M ( σ ( T ) , 0 ) - 1 σ ( T ) ( M r + c ) + k = 1 m c k .

Then we can conclude that Φu is bounded uniformly, and so Φ(B) is a bounded set.

Step 3: To show that Φ maps bounded sets into equicontinuous sets of X.

Let t1, t2 ∈ (tk, tk+1]T ⋂ [0, σ(T)]T, u B, then

Φ ( u ) ( t 1 ) - Φ ( u ) ( t 2 ) 0 σ ( T ) G ( t 1 , s ) - G ( t 2 , s ) h u ( s ) Δ s + k = 1 m G ( t 1 , t k ) - G ( t 2 , t k ) I k ( u ( t k ) ) .

The right-hand side tends to uniformly zero as |t1 - t2| → 0.

Consequently, Steps 1-3 together with the Arzela-Ascoli Theorem shows that Φ: X X is completely continuous.

Let

K = { u X : u ( t ) δ u , t [ 0 , σ ( T ) ] T } ,

where δ = 1 e M ( σ ( T ) , 0 ) ( 0 , 1 ) . It is not difficult to verify that K is a cone in X.

From condition (H) and Lemma 2.2, it is easy to obtain following result:

Lemma 2.4. Φ maps K into K.

### 3 Main results

For convenience, we denote

f 0 = lim u 0 + sup max t [ 0 , T ] T f ( t , u ) u , f = lim u sup max t [ 0 , T ] T f ( t , u ) u , f 0 = lim u 0 + inf min t [ 0 , T ] T f ( t , u ) u , f = lim u inf min t [ 0 , T ] T f ( t , u ) u .

and

I 0 = lim u 0 + I k ( u ) u , I = lim u I k ( u ) u .

Now we state our main results.

Theorem 3.1. Suppose that

(H1) f0 > 0, f< 0, I0 = 0 for any k; or

(H2) f> 0, f0 < 0, I= 0 for any k.

Then the problem (1.1) has at least one positive solutions.

Proof. Firstly, we assume (H1) holds. Then there exist ε > 0 and β > α > 0 such that

f ( t , u ) ε u , t [ 0 , T ] T , u ( 0 , α ] , (3.1)

I k ( u ) [ e m ( σ ( T ) , 0 ) - 1 ] ε 2 M m e M ( σ ( T ) , 0 ) u , u ( 0 , α ] , for any k , (3.2)

and

f ( t , u ) - ε u , t [ 0 , T ] T , u [ β , ) . (3.3)

Let Ω1 = {u X: ||u|| < r1}, where r1 = α. Then u K ⋂ ∂Ω1, 0 < δα = δ ||u|| ≤ u(t) ≤ α, in view of (3.1) and (3.2) we have

Φ ( u ) ( t ) = 0 σ ( T ) G ( t , s ) h u ( s ) Δ s + k = 1 m G ( t , t k ) I k ( u ( t k ) ) 0 σ ( T ) G ( t , s ) ( M - ε ) u ( σ ( s ) ) Δ s + k = 1 m G ( t , t k ) [ e M ( σ ( T ) , 0 ) - 1 ] ε 2 M m e M ( σ ( T ) , 0 ) u ( t k ) ( M - ε ) M u + e M ( σ ( T ) , 0 ) e M ( σ ( T ) , 0 ) - 1 k = 1 m [ e M ( σ ( T ) , 0 ) - 1 ] ε 2 M m e M ( σ ( T ) , 0 ) u = M - ε 2 M u < u , t [ 0 , σ ( T ) ] T ,

which yields ||Φ(u)|| < ||u||.

Therefore

Φ u τ u , u K Ω 1 , τ 1 . (3.4)

On the other hand, let Ω2 = {u X: ||u|| < r2}, where r 2 = β δ .

Choose u0 = 1, then u0 K\{0}. We assert that

u - Φ u λ u 0 , u K Ω 2 , λ 0 . (3.5)

Suppose on the contrary that there exist ū K Ω 2 and λ ̄ 0 such that

ū - Φ ū = λ ̄ u 0 .

Let ς = min t [ 0 , σ ( T ) ] T ū ( t ) , then ς δ ū = δ r 2 = β , we have from (3.3) that

ū ( t ) = Φ ( ū ) ( t ) + λ ̄ = 0 σ ( T ) G ( t , s ) h ū ( s ) Δ s + k = 1 m G ( t , t k ) I k ( ū ( t k ) ) + λ ̄ 0 σ ( T ) G ( t , s ) h ū ( s ) Δ s + λ ̄ ( M + ε ) M ς + λ ̄ , t [ 0 , σ ( T ) ] T .

Therefore,

ς = min t [ 0 , σ ( T ) ] T ū ( t ) ( M + ε ) M ς + λ ̄ > ς ,

which is a contradiction.

It follows from (3.4), (3.5) and Theorem 1.1 that Φ has a fixed point u * K ( Ω ̄ 2 \ Ω 1 ) , and u* is a desired positive solution of the problem (1.1).

Next, suppose that (H2) holds. Then we can choose ε' > 0 and β' > α' > 0 such that

f ( t , u ) ε u , t [ 0 , T ] T , u [ β , ) , (3.6)

I k ( u ) [ e M ( σ ( T ) , 0 ) - 1 ] ε 2 M m e M ( σ ( T ) , 0 ) u , u [ β , ) for any k , (3.7)

and

f ( t , u ) - ε u , t [ 0 , T ] T , u ( 0 , α ] . (3.8)

Let Ω3 = {u X: ||u|| < r3}, where r3 = α'. Then for any u K ⋂ ∂Ω3, 0 < δ ||u|| ≤ u(t) ≤ ||u|| = α'.

It is similar to the proof of (3.5), we have

u - Φ u λ u 0 , u K Ω 3 , λ 0 . (3.9)

Let Ω4 = {u X: ||u|| < r4}, where r 4 = β δ . Then for any u K ⋂ ∂Ω4, u(t) ≥ δ ||u|| = δr4 = β', by (3.6) and (3.7), it is easy to obtain

Φ u τ u , u K Ω 4 , τ 1 . (3.10)

It follows from (3.9), (3.10) and Theorem 1.1 that Φ has a fixed point u * K ( Ω ̄ 4 \ Ω 3 ) , and u* is a desired positive solution of the problem (1.1).

Theorem 3.2. Suppose that

(H3) f0 < 0, f< 0;

(H4) there exists ρ > 0 such that

min { f ( t , u ) - u | t [ 0 , T ] T , δ ρ u ρ } > 0 ; (3.11)

I k ( u ) [ e M ( σ ( T ) , 0 ) - 1 ] M m e M ( σ ( T ) , 0 ) u , δ ρ u ρ , for any k . (3.12)

Then the problem (1.1) has at least two positive solutions.

Proof. By (H3), from the proof of Theorem 3.1, we should know that there exist β" > ρ > α" > 0 such that

u - Φ u λ u 0 , u K Ω 5 , λ 0 , (3.13)

u - Φ u λ u 0 , u K Ω 6 , λ 0 , (3.14)

where Ω5 = {u X: ||u|| < r5}, Ω6 = {u X: ||u|| < r6}, r 5 = α , r 6 = β δ .

By (3.11) of (H4), we can choose ε > 0 such that

f ( t , u ) ( 1 + ε ) u , t [ 0 , T ] T , δ ρ u ρ . (3.15)

Let Ω7 = {u X: ||u|| < ρ}, for any u K ⋂ ∂Ω7, δρ = δ ||u|| ≤ u(t) ≤ ||u|| = ρ, from (3.12) and (3.15), it is similar to the proof of (3.4), we have

Φ u τ u , u K Ω 7 , τ 1 . (3.16)

By Theorem 1.1, we conclude that Φ has two fixed points u * * K ( Ω ̄ 6 \ Ω 7 ) and u * * * K ( Ω ̄ 7 \ Ω 5 ) , and u** and u*** are two positive solution of the problem (1.1).

Similar to Theorem 3.2, we have:

Theorem 3.3. Suppose that

(H4) f0 > 0, f> 0, I0 = 0, I= 0;

(H5) there exists ρ > 0 such that

max { f ( t , u ) | t [ 0 , T ] T , δ ρ u ρ } < 0 .

Then the problem (1.1) has at least two positive solutions.

### 4 Examples

Example 4.1. Let T = [0, 1] ∪ [2,3]. We consider the following problem on T

x Δ ( t ) + f ( t , x ( σ ( t ) ) ) = 0 , t [ 0 , 3 ] T , t 1 2 , x 1 2 + - x 1 2 - = I x 1 2 , x ( 0 ) = x ( 3 ) , (4.1)

where T = 3, f(t, x) = x - (t + 1)x2, and I(x) = x2

Let M = 1, then, it is easy to see that

M x - f ( t , x ) = ( t + 1 ) x 2 0 for x [ 0 , ) , t [ 0 , 3 ] T ,

and

f 0 1 , f = - , and I 0 = 0 .

Therefore, by Theorem 3.1, it follows that the problem (4.1) has at least one positive solution.

Example 4.2. Let T = [0, 1] ∪ [2,3]. We consider the following problem on T

x Δ ( t ) + f ( t , x ( σ ( t ) ) ) = 0 , t [ 0 , 3 ] T , t 1 2 , x 1 2 + - x 1 2 - = I x 1 2 , x ( 0 ) = x ( 3 ) , (4.2)

where T = 3 , f ( t , x ) = 4 e 1 - 4 e 2 x - ( t + 1 ) x 2 e - x , and I(x) = x2e-x.

Choose M = 1, ρ = 4e2, then δ = 1 2 e 2 , it is easy to see that

M x - f ( t , x ) = x ( 1 - 4 e 1 - 4 e 2 ) + ( t + 1 ) x 2 e - x 0 for x [ 0 , ) , t [ 0 , 3 ] T , f 0 4 e 1 - 4 e 2 > 0 , f 4 e 1 - 4 e 2 > 0 , I 0 = 0 , I = 0 ,

and

max ( f ( t , u ) | t [ 0 , T ] T , δ ρ u ρ } = max { f ( t , u ) | t [ 0 , 3 ] T , 2 u 4 e 2 } = 16 e 3 4 e 2 ( 1 e ) < 0.

Therefore, together with Theorem 3.3, it follows that the problem (4.2) has at least two positive solutions.

### Competing interests

The author declares that they have no competing interests.

### Acknowledgements

The author thankful to the anonymous referee for his/her helpful suggestions for the improvement of this article. This work is supported by the Excellent Young Teacher Training Program of Lanzhou University of Technology (Q200907)

### References

1. Bainov, DD, Simeonov, PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical, Harlow (1993)

2. Benchohra, M, Henderson, J, Ntouyas, SK: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)

3. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995)

4. Agarwal, RP, O'Regan, D: Multiple nonnegative solutions for second order impulsive differential equations. Appl Math Comput. 114, 51–59 (2000). Publisher Full Text

5. Feng, M, Du, B, Ge, W: Impulsive boundary value problems with integral boundary conditions and one-dimensional p-Laplacian. Nonlinear Anal. 70, 3119–3126 (2009). Publisher Full Text

6. 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). Publisher Full Text

7. He, Z, Zhang, X: Monotone iteative technique for first order impulsive differential equations with peroidic boundary conditions. Appl Math Comput. 156, 605–620 (2004). Publisher Full Text

8. Li, JL, Nieto, JJ, Shen, J: Impulsive periodic boundary value problems of first-order differential equastions. J Math Anal Appl. 325, 226–236 (2007). Publisher Full Text

9. Li, JL, Shen, JH: Positive solutions for first-order difference equation with impulses. Int J Diff Equ. 2, 225–239 (2006)

10. Nieto, JJ: Periodic boundary value problems for first-order impulsive ordinary diffeer-ential equations. Nonlinear Anal. 51, 1223–1232 (2002). Publisher Full Text

11. Nieto, JJ, O'Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal Real World Appl. 10, 680–690 (2009). Publisher Full Text

12. Nieto, JJ, Rodriguez-Lopez, R: Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations. J Math Anal Appl. 318, 593–610 (2006). Publisher Full Text

13. Sun, J, Chen, H, Nieto, JJ, Otero-Novoa, M: The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. 72, 4575–4586 (2010). Publisher Full Text

14. Tian, Y, Ge, W: Applications of variational methods to boundary-value problem for impulsive differential equations. Proceedings of the Edinburgh Mathematical Society. 51, 509–527 (2008)

15. Xiao, J, Nieto, JJ: Variational approach to some damped Dirichlet nonlinear impulsive differential equations. J Frankl Inst. 348, 369–377 (2011). Publisher Full Text

16. Zhou, J, Li, Y: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Anal. 71, 2856–2865 (2009). Publisher Full Text

17. Zhang, H, Li, Z: Variational approach to impulsive differential equations with periodic boundary conditions. Nonlinear Anal Real World Appl. 11, 67–78 (2010). Publisher Full Text

18. Zhang, Z, Yuan, R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal Real World Appl. 11, 155–162 (2010). Publisher Full Text

19. Bohner, M, Peterson, A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhauser, Boston (2001)

20. Bohner, M, Peterson, A: Advances in Dynamic Equations on Time Scales. Birkhauser, Boston (2003)

21. Hilger, S: Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)

22. Benchohra, M, Henderson, J, Ntouyas, SK, Ouahab, A: On first order impulsive dynamic equations on time scales. J Diff Equ Appl. 6, 541–548 (2004)

23. Benchohra, M, Ntouyas, SK, Ouahab, A: Existence results for second-order bounary value problem of impulsive dynamic equations on time scales. J Math Anal Appl. 296, 65–73 (2004). Publisher Full Text

24. Benchohra, M, Ntouyas, SK, Ouahab, A: Extremal solutions of second order impulsive dynamic equations on time scales. J Math Anal Appl. 324, 425–434 (2006). Publisher Full Text

25. Chen, HB, Wang, HH: Triple positive solutions of boundary value problems for p-Laplacian impulsive dynamic equations on time scales. Math Comput Model. 47, 917–924 (2008). Publisher Full Text

26. Geng, F, Zhu, D, Lu, Q: A new existence result for impulsive dynamic equations on time scales. Appl Math Lett. 20, 206–212 (2007). Publisher Full Text

27. Geng, F, Xu, Y, Zhu, D: Periodic boundary value problems for first-order impulsive dynamic equations on time scales. Nonlinear Anal. 69, 4074–4087 (2008). Publisher Full Text

28. Graef, JR, Ouahab, A: Extremal solutions for nonresonance impulsive functional dynamic equations on time scales. Appl Math Comput. 196, 333–339 (2008). Publisher Full Text

29. Henderson, J: Double solutions of impulsive dynamic boundary value problems on time scale. J Diff Equ Appl. 8, 345–356 (2002). Publisher Full Text

30. Li, JL, Shen, JH: Existence results for second-order impulsive boundary value problems on time scales. Nonlinear Anal. 70, 1648–1655 (2009). Publisher Full Text

31. Li, YK, Shu, JY: Multiple positive solutions for first-order impulsive integral boundary value problems on time scales. Boundary Value Probl. 2011, 12 (2011). BioMed Central Full Text

32. Liu, HB, Xiang, X: A class of the first order impulsive dynamic equations on time scales. Nonlinear Anal. 69, 2803–2811 (2008). Publisher Full Text

33. Wang, C, Li, YK, Fei, Y: Three positive periodic solutions to nonlinear neutral functional differential equations with impulses and parameters on time scales. Math Com-put Model. 52, 1451–1462 (2010). Publisher Full Text

34. Wang, DB: Positive solutions for nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales. Comput Math Appl. 56, 1496–1504 (2008). Publisher Full Text

35. Wang, ZY, Weng, PX: Existence of solutions for first order PBVPs with impulses on time scales. Comput Math Appl. 56, 2010–2018 (2008). Publisher Full Text

36. Zhang, HT, Li, YK: Existence of positive periodic solutions for functional differential equations with impulse effects on time scales. Commun Nonlinear Sci Numer Simul. 14, 19–26 (2009). Publisher Full Text

37. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)

38. Peng, S: Positive solutions for first order periodic boundary value problem. Appl Math Comput. 158, 345–351 (2004). Publisher Full Text

39. Sun, JP, Li, WT: Positive solution for system of nonlinear first-order PBVPs on time scales. Nonlinear Anal. 62, 131–139 (2005). Publisher Full Text