Theory and Modern Applications

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

## 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.

## 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 ). At the same time, the boundary value problems for impulsive differential equations and impulsive difference equations have received much attention . On the other hand, recently, the theory of dynamic equations on time scales has become a new important branch (see, for example, ). Naturally, some authors have focused their attention on the boundary value problems of impulsive dynamic equations on time scales . 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

$\left\{\begin{array}{c}{x}^{\Delta }\left(t\right)+f\left(t,x\left(\sigma \left(t\right)\right)\right)=0,\phantom{\rule{1em}{0ex}}t\in J:={\left[0,T\right]}_{\mathbf{T}},\phantom{\rule{1em}{0ex}}t\ne {t}_{k},\phantom{\rule{1em}{0ex}}k=1,2,\dots ,m,\hfill \\ x\left({t}_{k}^{+}\right)-x\left({t}_{k}^{-}\right)={I}_{k}\left(x\left({t}_{k}^{-}\right)\right),\phantom{\rule{1em}{0ex}}k=1,2,\dots ,m,\hfill \\ x\left(0\right)=x\left(\sigma \left(T\right)\right),\hfill \end{array}\right\$
(1.1)

where T is an arbitrary time scale, T > 0 is fixed, 0, T T, f C (J × [0, ∞), (-∞, ∞)), I k C([0, ∞), [0, ∞)), t k (0, T) T , 0 < t1 < < t m < T, and for each k = 1, 2,..., m, $x\left({t}_{k}^{+}\right)=\underset{h\to {0}^{+}}{\text{lim}}x\left({t}_{k}+h\right)$ and $x\left({t}_{k}^{-}\right)=\underset{h\to {0}^{-}}{\text{lim}}x\left({t}_{k}+h\right)$ represent the right and left limits of x(t) at t = t k . We always assume the following hypothesis holds (semi-position condition):

1. (H)

There exists a positive number M such that

$Mx-f\left(t,x\right)\ge 0\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}x\in \left[0,\infty \right),\phantom{\rule{1em}{0ex}}t\in {\left[0,T\right]}_{\mathbf{T}}.$

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

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

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 $0\in {\Omega }_{1}\subset {\stackrel{̄}{\Omega }}_{1}\subset {\Omega }_{2}$ and Φ: $K\cap \left({\stackrel{̄}{\Omega }}_{2}\{\Omega }_{1}\right)\to K$ is a completely continuous operator. If

1. (i)

There exists u 0 K\{0} such that u - Φu ≠ λu 0, u K ∂ Ω2, λ ≥ 0; Φuτu, u K ∂Ω1, τ ≥ 1, or

2. (ii)

There exists u 0 K\{0} such that u - Φu ≠ λu 0, u K ∂Ω1, λ ≥ 0; Φuτu, u K ∂Ω2, τ ≥ 1.

Then Φ has at least one fixed point in $K\cap \left({\stackrel{̄}{\Omega }}_{2}\{\Omega }_{1}\right)$.

## 2 Preliminaries

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

We define

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

Let

$X=\left\{x:x\in PC,\phantom{\rule{1em}{0ex}}x\left(0\right)=x\left(\sigma \left(T\right)\right)\right\}$

with the norm $∥x∥=\underset{t\in {\left[0,\sigma \left(T\right)\right]}_{\mathbf{T}}}{\text{sup}}\left|x\left(t\right)\right|$, 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

$x\left(t\right)=\underset{0}{\overset{\sigma \left(T\right)}{\int }}G\left(t,s\right)h\left(s\right)\Delta s+\sum _{k=1}^{m}G\left(t,{t}_{k}\right){I}_{k}\left(x\left({t}_{k}\right)\right),\phantom{\rule{1em}{0ex}}t\in {\left[0,\sigma \left(T\right)\right]}_{\mathbf{T}},$

where $G\left(t,s\right)=\left\{\begin{array}{cc}\frac{{e}_{M}\left(s,t\right){e}_{M}\left(\sigma \left(T\right),0\right)}{{e}_{M}\left(\sigma \left(T\right),0\right)-1},\hfill & \hfill 0\le s\le t\le \sigma \left(T\right),\hfill \\ \frac{{e}_{M}\left(s,t\right)}{{e}_{M}\left(\sigma \left(T\right),0\right)-1},\hfill & \hfill 0\le t

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

$\left\{\begin{array}{c}{x}^{\Delta }\left(t\right)+Mx\left(\sigma \left(t\right)\right)=h\left(t\right),\phantom{\rule{1em}{0ex}}t\in J:={\left[0,T\right]}_{\mathbf{T}},\phantom{\rule{1em}{0ex}}t\ne {t}_{k},\phantom{\rule{1em}{0ex}}k=1,2,\dots ,m,\hfill \\ x\left({t}_{k}^{+}\right)-x\left({t}_{k}^{-}\right)={I}_{k}\left(x\left({t}_{k}^{-}\right)\right),\phantom{\rule{1em}{0ex}}k=1,2,\dots ,m,\hfill \\ x\left(0\right)=x\left(\sigma \left(T\right)\right).\hfill \end{array}\right\$

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

$\frac{1}{{e}_{M}\left(\sigma \left(T\right),0\right)-1}\le G\left(t,s\right)\le \frac{{e}_{M}\left(\sigma \left(T\right),0\right)}{{e}_{M}\left(\sigma \left(T\right),0\right)-1}\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{all}}\phantom{\rule{2.77695pt}{0ex}}t,s\in {\left[0,\sigma \left(T\right)\right]}_{\mathbf{T}}.$

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

Remark 2.1. Let G(t, s) be defined as in Lemma 2.1, then ${\int }_{0}^{\sigma \left(T\right)}G\left(t,s\right)\Delta s=\frac{1}{M}$.

For u X, we consider the following problem:

$\left\{\begin{array}{l}{x}^{\Delta }\left(t\right)+Mx\left(\sigma \left(t\right)\right)=Mu\left(\sigma \left(t\right)\right)-f\left(t,u\left(\sigma \left(t\right)\right),\phantom{\rule{0.1em}{0ex}}t\in {\left[0,T\right]}_{T},\phantom{\rule{0.1em}{0ex}}t\ne {t}_{k},\phantom{\rule{0.1em}{0ex}}k=1,2,\dots ,m,\hfill \\ x\left({t}_{k}^{+}\right)-x\left({t}_{k}^{-}\right)={I}_{k}\left(x\left({t}_{k}^{-}\right)\right),\phantom{\rule{0.1em}{0ex}}k=1,2,\dots ,m,\hfill \\ x\left(0\right)=x\left(\sigma \left(T\right)\right).\hfill \end{array}$
(2.1)

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

$x\left(t\right)=\underset{0}{\overset{\sigma \left(T\right)}{\int }}G\left(t,s\right){h}_{u}\left(s\right)\Delta s+\sum _{k=1}^{m}G\left(t,{t}_{k}\right){I}_{k}\left(x\left({t}_{k}\right)\right),\phantom{\rule{1em}{0ex}}t\in {\left[0,\sigma \left(T\right)\right]}_{\mathbf{T}},$

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

We define an operator Φ: XX by

$\Phi \left(u\right)\left(t\right)=\underset{0}{\overset{\sigma \left(T\right)}{\int }}G\left(t,s\right){h}_{u}\left(s\right)\Delta s+\sum _{k=1}^{m}G\left(t,{t}_{k}\right){I}_{k}\left(u\left({t}_{k}\right)\right),\phantom{\rule{1em}{0ex}}t\in {\left[0,\sigma \left(T\right)\right]}_{\mathbf{T}}.$

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

Lemma 2.3. Φ: XX is completely continuous.

Proof. The proof is divided into three steps.

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

Let ${\left\{{u}_{n}\right\}}_{n=1}^{\infty }$ be a sequence such that u n u (n → ∞) in X. Since f(t, u) and I k (u) are continuous in x, we have

$\begin{array}{c}\left|{h}_{un}\left(t\right)-{h}_{u}\left(t\right)\right|=\left|M\left({u}_{n}-u\right)-\left(f\left(t,{u}_{n}\right)-f\left(t,u\right)\right)\right|\to 0\left(n\to \infty \right),\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left|{I}_{k}\left({u}_{n}\left({t}_{k}\right)\right)-{I}_{k}\left(u\left({t}_{k}\right)\right)\right|\to 0\left(n\to \infty \right).\end{array}$

So

$\begin{array}{c}\phantom{\rule{1em}{0ex}}\left|\Phi \left({u}_{n}\right)\left(t\right)-\Phi \left(u\right)\left(t\right)\right|\\ =\left|\underset{0}{\overset{\sigma \left(T\right)}{\int }}G\left(t,s\right)\left[{h}_{{u}_{n}}\left(s\right)-{h}_{u}\left(s\right)\right]\Delta s+\sum _{k=1}^{m}G\left(t,{t}_{k}\right)\left[{I}_{k}\left({u}_{n}\left({t}_{k}\right)\right)-{I}_{k}\left(u\left({t}_{k}\right)\right)\right]\right|\\ \le \frac{{e}_{M}\left(\sigma \left(T\right),0\right)}{{e}_{M}\left(\sigma \left(T\right),0\right)-1}\left[\underset{0}{\overset{\sigma \left(T\right)}{\int }}\left|{h}_{{u}_{n}}\left(s\right)-{h}_{u}\left(s\right)\right|\Delta s+\sum _{k=1}^{m}\left|{I}_{k}\left({u}_{n}\left({t}_{k}\right)\right)-{I}_{k}\left(u\left({t}_{k}\right)\right)\right|\right]\to 0\left(n\to \infty \right),\end{array}$

which leads to ||Φu n - Φu|| → 0 (n → ∞). That is, Φ: XX 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 I k (u), there exist c > 0, c k > 0 such that

$\left|f\left(t,u\right)\right|\le c,\phantom{\rule{1em}{0ex}}\left|{I}_{k}\left(u\right)\right|\le {c}_{k},\phantom{\rule{1em}{0ex}}k=1,2,\dots ,m.$

We get

$\begin{array}{ll}\hfill \left|\Phi \left(u\right)\left(t\right)\right|& =\left|\underset{0}{\overset{\sigma \left(T\right)}{\int }}G\left(t,s\right){h}_{u}\left(s\right)\Delta s+\sum _{k=1}^{m}G\left(t,{t}_{k}\right){I}_{k}\left(u\left({t}_{k}\right)\right)\right|\phantom{\rule{2em}{0ex}}\\ \le \underset{0}{\overset{\sigma \left(T\right)}{\int }}G\left(t,s\right)\left|{h}_{u}\left(s\right)\right|\Delta s+\sum _{k=1}^{m}G\left(t,{t}_{k}\right)\left|{I}_{k}\left(u\left({t}_{k}\right)\right)\right|\phantom{\rule{2em}{0ex}}\\ \le \frac{{e}_{M}\left(\sigma \left(T\right),0\right)}{{e}_{M}\left(\sigma \left(T\right),0\right)-1}\left[\sigma \left(T\right)\left(Mr+c\right)+\sum _{k=1}^{m}{c}_{k}\right].\phantom{\rule{2em}{0ex}}\end{array}$

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 (t k , tk+1] T [0, σ(T)] T , u B, then

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\left|\Phi \left(u\right)\left({t}_{1}\right)-\Phi \left(u\right)\left({t}_{2}\right)\right|\phantom{\rule{2em}{0ex}}\\ \le \underset{0}{\overset{\sigma \left(T\right)}{\int }}\left|G\left({t}_{1},s\right)-G\left({t}_{2},s\right)\right|\left|{h}_{u}\left(s\right)\right|\Delta s+\sum _{k=1}^{m}\left|G\left({t}_{1},{t}_{k}\right)-G\left({t}_{2},{t}_{k}\right)\right|\left|{I}_{k}\left(u\left({t}_{k}\right)\right)\right|.\phantom{\rule{2em}{0ex}}\end{array}$

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: XX is completely continuous.

Let

$K=\left\{u\in X:u\left(t\right)\ge \delta ∥u∥,\phantom{\rule{1em}{0ex}}t\in {\left[0,\sigma \left(T\right)\right]}_{\mathbf{T}}\right\},$

where $\delta =\frac{1}{{e}_{M}\left(\sigma \left(T\right),0\right)}\in \left(0,1\right)$. 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

$\begin{array}{ll}\hfill {f}^{0}& =\underset{u\to {0}^{+}}{\text{lim}}\text{sup}\underset{t\in {\left[0,T\right]}_{\mathbf{T}}}{\text{max}}\frac{f\left(t,u\right)}{u},\phantom{\rule{1em}{0ex}}{f}^{\infty }=\underset{u\to \infty }{\text{lim}}\text{sup}\underset{t\in {\left[0,T\right]}_{\mathbf{T}}}{\text{max}}\frac{f\left(t,u\right)}{u},\phantom{\rule{2em}{0ex}}\\ \hfill {f}_{0}& =\underset{u\to {0}^{+}}{\text{lim}}\text{inf}\underset{t\in {\left[0,T\right]}_{\mathbf{T}}}{\text{min}}\frac{f\left(t,u\right)}{u},\phantom{\rule{1em}{0ex}}{f}_{\infty }=\underset{u\to \infty }{\text{lim}}\text{inf}\underset{t\in {\left[0,T\right]}_{\mathbf{T}}}{\text{min}}\frac{f\left(t,u\right)}{u}.\phantom{\rule{2em}{0ex}}\end{array}$

and

${I}_{0}=\underset{u\to {0}^{+}}{\text{lim}}\frac{{I}_{k}\left(u\right)}{u},\phantom{\rule{1em}{0ex}}{I}_{\infty }=\underset{u\to \infty }{\text{lim}}\frac{{I}_{k}\left(u\right)}{u}.$

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

$f\left(t,u\right)\ge \epsilon u,\phantom{\rule{1em}{0ex}}t\in {\left[0,T\right]}_{\mathbf{T}},\phantom{\rule{1em}{0ex}}u\in \left(0,\alpha \right],$
(3.1)
${I}_{k}\left(u\right)\le \frac{\left[{e}_{m}\left(\sigma \left(T\right),0\right)-1\right]\epsilon }{2Mm{e}_{M}\left(\sigma \left(T\right),0\right)}u,u\in \left(0,\alpha \right],\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{any}}\phantom{\rule{2.77695pt}{0ex}}k,$
(3.2)

and

$f\left(t,u\right)\le -\epsilon u,\phantom{\rule{1em}{0ex}}t\in {\left[0,T\right]}_{\mathbf{T}},\phantom{\rule{1em}{0ex}}u\in \left[\beta ,\infty \right).$
(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

$\begin{array}{ll}\hfill \Phi \left(u\right)\left(t\right)& =\underset{0}{\overset{\sigma \left(T\right)}{\int }}G\left(t,s\right){h}_{u}\left(s\right)\Delta s+\sum _{k=1}^{m}G\left(t,{t}_{k}\right){I}_{k}\left(u\left({t}_{k}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \underset{0}{\overset{\sigma \left(T\right)}{\int }}G\left(t,s\right)\left(M-\epsilon \right)u\left(\sigma \left(s\right)\right)\Delta s+\sum _{k=1}^{m}G\left(t,{t}_{k}\right)\frac{\left[{e}_{M}\left(\sigma \left(T\right),0\right)-1\right]\epsilon }{2Mm{e}_{M}\left(\sigma \left(T\right),0\right)}u\left({t}_{k}\right)\phantom{\rule{2em}{0ex}}\\ \le \frac{\left(M-\epsilon \right)}{M}∥u∥+\frac{{e}_{M}\left(\sigma \left(T\right),0\right)}{{e}_{M}\left(\sigma \left(T\right),0\right)-1}\sum _{k=1}^{m}\frac{\left[{e}_{M}\left(\sigma \left(T\right),0\right)-1\right]\epsilon }{2Mm{e}_{M}\left(\sigma \left(T\right),0\right)}∥u∥\phantom{\rule{2em}{0ex}}\\ =\frac{\left(M-\frac{\epsilon }{2}\right)}{M}∥u∥\phantom{\rule{2em}{0ex}}\\ <∥u∥,t\in {\left[0,\sigma \left(T\right)\right]}_{\mathbf{T}},\phantom{\rule{2em}{0ex}}\end{array}$

which yields ||Φ(u)|| < ||u||.

Therefore

$\Phi u\ne \tau u,\phantom{\rule{1em}{0ex}}u\in K\cap \partial {\Omega }_{1},\phantom{\rule{1em}{0ex}}\tau \ge 1.$
(3.4)

On the other hand, let Ω2 = {u X: ||u|| < r2}, where ${r}_{2}=\frac{\beta }{\delta }$.

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

$u-\Phi u\ne \lambda {u}_{0},\phantom{\rule{1em}{0ex}}u\in K\cap \partial {\Omega }_{2},\phantom{\rule{1em}{0ex}}\lambda \ge 0.$
(3.5)

Suppose on the contrary that there exist $ū\in K\cap \partial {\Omega }_{2}$ and $\stackrel{̄}{\lambda }\ge 0$ such that

$ū-\Phi ū=\stackrel{̄}{\lambda }{u}_{0}.$

Let $\varsigma =\underset{t\in {\left[0,\sigma \left(T\right)\right]}_{\mathbf{T}}}{\text{min}}ū\left(t\right)$, then $\varsigma \ge \delta ∥ū∥=\delta {r}_{2}=\beta$, we have from (3.3) that

$\begin{array}{ll}\hfill ū\left(t\right)& =\Phi \left(ū\right)\left(t\right)+\stackrel{̄}{\lambda }\phantom{\rule{2em}{0ex}}\\ =\underset{0}{\overset{\sigma \left(T\right)}{\int }}G\left(t,s\right){h}_{ū}\left(s\right)\Delta s+\sum _{k=1}^{m}G\left(t,{t}_{k}\right){I}_{k}\left(ū\left({t}_{k}\right)\right)+\stackrel{̄}{\lambda }\phantom{\rule{2em}{0ex}}\\ \ge \underset{0}{\overset{\sigma \left(T\right)}{\int }}G\left(t,s\right){h}_{ū}\left(s\right)\Delta s+\stackrel{̄}{\lambda }\phantom{\rule{2em}{0ex}}\\ \ge \frac{\left(M+\epsilon \right)}{M}\varsigma +\stackrel{̄}{\lambda },\phantom{\rule{1em}{0ex}}t\in {\left[0,\sigma \left(T\right)\right]}_{\mathbf{T}}.\phantom{\rule{2em}{0ex}}\end{array}$

Therefore,

$\varsigma =\underset{t\in {\left[0,\sigma \left(T\right)\right]}_{\mathbf{T}}}{\text{min}}ū\left(t\right)\ge \frac{\left(M+\epsilon \right)}{M}\varsigma +\stackrel{̄}{\lambda }>\varsigma ,$

It follows from (3.4), (3.5) and Theorem 1.1 that Φ has a fixed point ${u}^{*}\in K\cap \left({\stackrel{̄}{\Omega }}_{2}\{\Omega }_{1}\right)$, 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

$f\left(t,u\right)\ge {\epsilon }^{\prime }u,\phantom{\rule{1em}{0ex}}t\in {\left[0,T\right]}_{\mathbf{T}},\phantom{\rule{1em}{0ex}}u\in \left[{\beta }^{\prime },\infty \right),$
(3.6)
${I}_{k}\left(u\right)\le \frac{\left[{e}_{M}\left(\sigma \left(T\right),0\right)-1\right]{\epsilon }^{\prime }}{2Mm{e}_{M}\left(\sigma \left(T\right),0\right)}u,\phantom{\rule{1em}{0ex}}u\in \left[{\beta }^{\prime },\infty \right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{any}}\phantom{\rule{2.77695pt}{0ex}}k,$
(3.7)

and

$f\left(t,u\right)\le -{\epsilon }^{\prime }u,\phantom{\rule{1em}{0ex}}t\in {\left[0,T\right]}_{\mathbf{T}},\phantom{\rule{1em}{0ex}}u\in \left(0,{\alpha }^{\prime }\right].$
(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

$u-\Phi u\ne \lambda {u}_{0},\phantom{\rule{1em}{0ex}}u\in K\cap \partial {\Omega }_{3},\phantom{\rule{1em}{0ex}}\lambda \ge 0.$
(3.9)

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

$\Phi u\ne \tau u,\phantom{\rule{1em}{0ex}}u\in K\cap \partial {\Omega }_{4},\phantom{\rule{1em}{0ex}}\tau \ge 1.$
(3.10)

It follows from (3.9), (3.10) and Theorem 1.1 that Φ has a fixed point ${u}^{*}\in K\cap \left({\stackrel{̄}{\Omega }}_{4}\{\Omega }_{3}\right)$, 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

$\text{min}\left\{f\left(t,u\right)-u|t\in {\left[0,T\right]}_{\mathbf{T}},\phantom{\rule{1em}{0ex}}\delta \rho \le u\le \rho \right\}>0;$
(3.11)
${I}_{k}\left(u\right)\le \frac{\left[{e}_{M}\left(\sigma \left(T\right),0\right)-1\right]}{Mm{e}_{M}\left(\sigma \left(T\right),0\right)}u,\phantom{\rule{1em}{0ex}}\delta \rho \le u\le \rho ,\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{any}}\phantom{\rule{2.77695pt}{0ex}}k.$
(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

$u-\Phi u\ne \lambda {u}_{0},\phantom{\rule{1em}{0ex}}u\in K\cap \partial {\Omega }_{5},\phantom{\rule{1em}{0ex}}\lambda \ge 0,$
(3.13)
$u-\Phi u\ne \lambda {u}_{0},\phantom{\rule{1em}{0ex}}u\in K\cap \partial {\Omega }_{6},\phantom{\rule{1em}{0ex}}\lambda \ge 0,$
(3.14)

where Ω5 = {u X: ||u|| < r5}, Ω6 = {u X: ||u|| < r6}, ${r}_{5}={\alpha }^{″},{r}_{6}=\frac{{\beta }^{″}}{\delta }.$

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

$f\left(t,u\right)\ge \left(1+\epsilon \right)u,\phantom{\rule{1em}{0ex}}t\in {\left[0,T\right]}_{\mathbf{T}},\phantom{\rule{1em}{0ex}}\delta \rho \le u\le \rho .$
(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

$\Phi u\ne \tau u,\phantom{\rule{1em}{0ex}}u\in K\cap \partial {\Omega }_{7},\phantom{\rule{1em}{0ex}}\tau \ge 1.$
(3.16)

By Theorem 1.1, we conclude that Φ has two fixed points ${u}^{**}\in K\cap \left({\stackrel{̄}{\Omega }}_{6}\{\Omega }_{7}\right)$ and ${u}^{***}\in K\cap \left({\stackrel{̄}{\Omega }}_{7}\{\Omega }_{5}\right)$, 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

$\text{max}\left\{f\left(t,u\right)|t\in {\left[0,T\right]}_{\mathbf{T}},\phantom{\rule{1em}{0ex}}\delta \rho \le u\le \rho \right\}<0.$

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

$\left\{\begin{array}{c}{x}^{\Delta }\left(t\right)+f\left(t,x\left(\sigma \left(t\right)\right)\right)=0,\phantom{\rule{1em}{0ex}}t\in {\left[0,3\right]}_{\mathbf{T}},\phantom{\rule{1em}{0ex}}t\ne \frac{1}{2},\hfill \\ x\left({\frac{1}{2}}^{+}\right)-x\left({\frac{1}{2}}^{-}\right)=I\left(x\left(\frac{1}{2}\right)\right),\hfill \\ x\left(0\right)=x\left(3\right),\hfill \end{array}\right\$
(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

$Mx-f\left(t,x\right)=\left(t+1\right){x}^{2}\ge 0\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}x\in \left[0,\infty \right),\phantom{\rule{1em}{0ex}}t\in {\left[0,3\right]}_{\mathbf{T}},$

and

${f}_{0}\ge 1,\phantom{\rule{1em}{0ex}}{f}^{\infty }=-\infty ,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}{I}_{0}=0.$

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

$\left\{\begin{array}{c}{x}^{\Delta }\left(t\right)+f\left(t,x\left(\sigma \left(t\right)\right)\right)=0,\phantom{\rule{1em}{0ex}}t\in {\left[0,3\right]}_{\mathbf{T}},\phantom{\rule{1em}{0ex}}t\ne \frac{1}{2},\hfill \\ x\left({\frac{1}{2}}^{+}\right)-x\left({\frac{1}{2}}^{-}\right)=I\left(x\left(\frac{1}{2}\right)\right),\hfill \\ x\left(0\right)=x\left(3\right),\hfill \end{array}\right\$
(4.2)

where $T=3,f\left(t,x\right)=4{e}^{1-4{e}^{2}}x-\left(t+1\right){x}^{2}{e}^{-x}$, and I(x) = x2e-x.

Choose M = 1, ρ = 4e2, then $\delta =\frac{1}{2{e}^{2}}$, it is easy to see that

$\begin{array}{ll}\hfill Mx-f\left(t,x\right)& =x\left(1-4{e}^{1-4{e}^{2}}\right)+\left(t+1\right){x}^{2}{e}^{-x}\ge 0\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}x\in \left[0,\infty \right),\phantom{\rule{1em}{0ex}}t\in {\left[0,3\right]}_{\mathbf{T}},\phantom{\rule{2em}{0ex}}\\ \hfill {f}_{0}& \ge 4{e}^{1-4{e}^{2}}>0,\phantom{\rule{1em}{0ex}}{f}_{\infty }\ge 4{e}^{1-4{e}^{2}}>0,\phantom{\rule{1em}{0ex}}{I}_{0}=0\phantom{\rule{1em}{0ex}},{I}_{\infty }=0,\phantom{\rule{2em}{0ex}}\end{array}$

and

$\mathrm{max}\left(f\left(t,u\right)|t\in {\left[0,T\right]}_{T},\delta \rho \le u\le \rho \right\}=\mathrm{max}\left\{f\left(t,u\right)|t\in {\left[0,3\right]}_{T},2\le u\le 4{e}^{2}\right\}=16{e}^{3-4{e}^{2}}\left(1-e\right)<0.$

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

## 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. Volume 2. 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 2000, 114: 51–59. 10.1016/S0096-3003(99)00074-0

5. Feng M, Du B, Ge W: Impulsive boundary value problems with integral boundary conditions and one-dimensional p -Laplacian. Nonlinear Anal 2009, 70: 3119–3126. 10.1016/j.na.2008.04.015

6. Feng M, Xie D: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. J Comput Appl Math 2009, 223: 438–448. 10.1016/j.cam.2008.01.024

7. He Z, Zhang X: Monotone iteative technique for first order impulsive differential equations with peroidic boundary conditions. Appl Math Comput 2004, 156: 605–620. 10.1016/j.amc.2003.08.013

8. Li JL, Nieto JJ, Shen J: Impulsive periodic boundary value problems of first-order differential equastions. J Math Anal Appl 2007, 325: 226–236. 10.1016/j.jmaa.2005.04.005

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

10. Nieto JJ: Periodic boundary value problems for first-order impulsive ordinary diffeer-ential equations. Nonlinear Anal 2002, 51: 1223–1232. 10.1016/S0362-546X(01)00889-6

11. Nieto JJ, O'Regan D: Variational approach to impulsive differential equations. Nonlinear Anal Real World Appl 2009, 10: 680–690. 10.1016/j.nonrwa.2007.10.022

12. Nieto JJ, Rodriguez-Lopez R: Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations. J Math Anal Appl 2006, 318: 593–610. 10.1016/j.jmaa.2005.06.014

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 2010, 72: 4575–4586. 10.1016/j.na.2010.02.034

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

15. Xiao J, Nieto JJ: Variational approach to some damped Dirichlet nonlinear impulsive differential equations. J Frankl Inst 2011, 348: 369–377. 10.1016/j.jfranklin.2010.12.003

16. Zhou J, Li Y: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Anal 2009, 71: 2856–2865. 10.1016/j.na.2009.01.140

17. Zhang H, Li Z: Variational approach to impulsive differential equations with periodic boundary conditions. Nonlinear Anal Real World Appl 2010, 11: 67–78. 10.1016/j.nonrwa.2008.10.016

18. Zhang Z, Yuan R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal Real World Appl 2010, 11: 155–162. 10.1016/j.nonrwa.2008.10.044

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 1990, 18: 18–56.

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

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 2004, 296: 65–73. 10.1016/j.jmaa.2004.02.057

24. Benchohra M, Ntouyas SK, Ouahab A: Extremal solutions of second order impulsive dynamic equations on time scales. J Math Anal Appl 2006, 324: 425–434. 10.1016/j.jmaa.2005.12.028

25. Chen HB, Wang HH: Triple positive solutions of boundary value problems for p -Laplacian impulsive dynamic equations on time scales. Math Comput Model 2008, 47: 917–924. 10.1016/j.mcm.2007.06.012

26. Geng F, Zhu D, Lu Q: A new existence result for impulsive dynamic equations on time scales. Appl Math Lett 2007, 20: 206–212. 10.1016/j.aml.2006.03.013

27. Geng F, Xu Y, Zhu D: Periodic boundary value problems for first-order impulsive dynamic equations on time scales. Nonlinear Anal 2008, 69: 4074–4087. 10.1016/j.na.2007.10.038

28. Graef JR, Ouahab A: Extremal solutions for nonresonance impulsive functional dynamic equations on time scales. Appl Math Comput 2008, 196: 333–339. 10.1016/j.amc.2007.05.056

29. Henderson J: Double solutions of impulsive dynamic boundary value problems on time scale. J Diff Equ Appl 2002, 8: 345–356. 10.1080/1026190290017405

30. Li JL, Shen JH: Existence results for second-order impulsive boundary value problems on time scales. Nonlinear Anal 2009, 70: 1648–1655. 10.1016/j.na.2008.02.047

31. Li YK, Shu JY: Multiple positive solutions for first-order impulsive integral boundary value problems on time scales. Boundary Value Probl 2011, 2011: 12. 10.1186/1687-2770-2011-12

32. Liu HB, Xiang X: A class of the first order impulsive dynamic equations on time scales. Nonlinear Anal 2008, 69: 2803–2811. 10.1016/j.na.2007.08.052

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 2010, 52: 1451–1462. 10.1016/j.mcm.2010.06.009

34. Wang DB: Positive solutions for nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales. Comput Math Appl 2008, 56: 1496–1504. 10.1016/j.camwa.2008.02.038

35. Wang ZY, Weng PX: Existence of solutions for first order PBVPs with impulses on time scales. Comput Math Appl 2008, 56: 2010–2018. 10.1016/j.camwa.2008.04.012

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 2009, 14: 19–26. 10.1016/j.cnsns.2007.08.006

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 2004, 158: 345–351. 10.1016/j.amc.2003.08.090

39. Sun JP, Li WT: Positive solution for system of nonlinear first-order PBVPs on time scales. Nonlinear Anal 2005, 62: 131–139. 10.1016/j.na.2005.03.016

## 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)

## Author information

Authors

### Corresponding author

Correspondence to Da-Bin Wang.

### Competing interests

The author declares that they have no competing interests.

## Rights and permissions

Reprints and Permissions

Wang, DB. Periodic boundary value problems for nonlinear first-order impulsive dynamic equations on time scales. Adv Differ Equ 2012, 12 (2012). https://doi.org/10.1186/1687-1847-2012-12

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1687-1847-2012-12

### Keywords

• time scale
• periodic boundary value problem
• positive solution
• fixed point
• impulsive dynamic equation 