### Abstract

In this paper, we first present a class of first-order nonlinear impulsive integral boundary value problems on time scales. Then, using the well-known Guo-Krasnoselskii fixed point theorem and Legget-Williams fixed point theorem, some criteria for the existence of at least one, two, and three positive solutions are established for the problem under consideration, respectively. Finally, examples are presented to illustrate the main results.

**MSC: **34B10; 34B37; 34N05.

##### Keywords:

integral boundary value problem; fixed point; multiple solutions; time scale### 1 Introduction

In fact, continuous and discrete systems are very important in implementing and applications. It is well known that the theory of time scales has received a lot of attention, which was introduced by Stefan Hilger in order to unify continuous and discrete analyses. Therefore, it is meaningful to study dynamic systems on time scales, which can unify differential and difference systems.

In recent years, a great deal of work has been done in the study of the existence of solutions for boundary value problems on time scales. For the background and results, we refer the reader to some recent contributions [1-5] and references therein. At the same time, boundary value problems for impulsive differential equations and impulsive difference equations have received much attention [6-12], since such equations may exhibit several real-world phenomena in physics, biology, engineering, etc. see [13-15] and the references therein.

In paper [16], Sun studied the first-order boundary value problem on time scales

where 0 *< β < *1. By means of the twin fixed point theorem due to Avery and Henderson, some existence
criteria for at least two positive solutions were established.

Tian and Ge [17] studied the first-order three-point boundary value problem on time scales

Using several fixed point theorems, the existence of at least one positive solution and multiple positive solutions is obtained.

However, except BVP of differential and difference equations, that is, for particular
time scales (
*n*-point boundary value problems, such boundary value problems for continuous systems
have received more and more attention and many results have worked out during the
past ten years, see Refs. [21-27] for more details. To the best of authors' knowledge, up to the present, there is
no paper concerning the boundary value problem with integral boundary conditions on
time scales. This paper is to fill the gap in the literature.

In this paper, we are concerned with the following first-order nonlinear impulsive integral boundary value problem on time scales:

where
*T *are points in
*p *is regressive, ℝ^{+}), *I _{i}*(1 ≤

*i*≤

*m*) ∈

*C*([0, +∞), [0, +∞)),

*g*is a nonnegative integrable function on

*e*

_{p}(0,

*σ*(

*T*)) is the exponential function on time scale

*< t*

_{1 }< · · · <

*t*, and for each

_{m }< T*x*(

*t*) at

**Remark 1.1**. *Let *
*θ*_{1 }*< *· · · *< θ _{q }*≤

*T*,

*σ*(

*θ*

_{0}) = 0,

*θ*

_{q+1 }=

*T. By some basic concepts and time scale calculus formulae in the book by Bohner and Peterson*[28],

*we have*

The main purpose of this paper is to establish some sufficient conditions for the existence of at least one, two, or three positive solutions for BVP (1.3) using Guo-Krasnoselskii and Legget-Williams fixed point theorem, respectively.

For convenience, we introduce the following notation:

where *i *= 1, 2,..., *m*.

This paper is organized as follows. In Section 2, some basic definitions and lemmas on time scales are introduced without proofs. In Section 3, some useful lemmas are established. In particular, Green's function for BVP (1.3) is established. We prove the main results in Sections 4-6.

### 2 Preliminaries

In this section, we shall first recall some basic definitions, lemmas that are used in what follows. For the details of the calculus on time scales, we refer to books by Bohner and Peterson [28,29].

**Definition 2.1**. [28]*A time scale *
*is an arbitrary nonempty closed subset of the real set *ℝ *with the topology and ordering inherited from *ℝ. *The forward and backward jump operators *
*and the graininess *
*are defined, respectively, by*

*In this definition, we put *
*i.e*., *σ*(*t*) = *t if *
*has a maximum t*) *and *
*i.e*., *ρ*(*t*) = *t if *
*has a minimum t*). *The point *
*is called left-dense, left-scattered, right-dense, or right-scattered if ρ*(*t*) = *t, ρ*(*t*) *< t, σ*(*t*) = *t, or σ*(*t*) *> t, respectively. Points that are right-dense and left-dense at the same time are
called dense. If *
*has a left-scattered maximum m*_{1}, *defined *
*otherwise, set *
*If *
*has a right-scattered minimum m*_{2}, *defined *
*otherwise, set *

**Definition 2.2**. [28]*A function *
*is rd continuous provided it is continuous at each right-dense point in *
*and has a left-sided limit at each left-dense point in *
*The set of rd-continuous functions *
*will be denoted by *

**Definition 2.3**. [28]*If *
*is a function and *
*then the delta derivative of f at the point t is defined to be the number f*^{Δ}(*t*) *(provided it exists) with the property that for each ε > *0 *there is a neighborhood U of t such that*

**Definition 2.4**. [28]*For a function *
*the range *ℝ *of f may be actually replaced by Banach space*), *the *(*delta*) *derivative is defined by*

*if f is continuous at t and t is right-scattered. If t is not right-scattered, then
the derivative is defined by*

*provided this limit exists*.

**Definition 2.5**. [28]*If F*^{Δ}(*t*) = *f*(*t*), *then we define the delta integral by*

**Definition 2.6**. [28]*A function *
*is said to be regressive provided *1 + *μ*(*t*)*p*(*t*) ≠ 0 *for all *
*where μ*(*t*) = *σ*(*t*) - *t is the graininess function. The set of all regressive rd-continuous functions *
*is denoted by *
*while the set *
*is given by *
*for all*
*Let *
*The exponential function is defined by*

*where ξ*_{h(z) }*is the so-called cylinder transformation*.

**Lemma 2.1**. [28]*Let p,
*

(1) *e*_{0}(*t*, *s*) ≡ 1 *and e _{p}*(

*t*,

*t*) ≡ 1;

(2) *e _{p}*(

*σ*(

*t*),

*s*) = (1 +

*μ*(

*t*)

*p*(

*t*))

*e*(

_{p}*t*,

*s*);

(3)
*where *

(4) *e _{p}*(

*t*,

*s*)

*e*(

_{p}*s*,

*r*) =

*e*(

_{p}*t*,

*r*),

(5)

**Lemma 2.2**. [28]*Assume that *
*are delta differentiable at *
*Then*

**Lemma 2.3**. [28]*Let *
*and assume that *
*is continuous at *(*t*, *t*), *where *
*with t > a. Also, assume that f*^{Δ}(*t*, ·) *is rd-continuous on *[*a*, *σ*(*t*)]. *Suppose that for each ε > *0 *there exists a neighborhood U of t, independent of τ *∈ [*a*, *σ*(*t*)], *such that*

*where f*^{Δ }*denotes the derivative of f with respect to the first variable. Then*

(1)

(2)

### 3 Foundational lemmas

In this section, we first introduce some background definitions, fixed point theorems in Banach space, then present basic lemmas that are very crucial in the proof of the main results.

We define
*PC *is a Banach Space.

**Definition 3.1**. *A function x is said to be a positive solution of problem *(1.3) *if x *∈ *PC satisfying problem *(1.3) *and x*(*t*) *> *0 *for all *

**Definition 3.2**. *Let X be a real Banach space, the nonempty set K *⊂ *X is called a cone of X, if it satisfies the following conditions*.

(1) *x *∈ *K and λ *≥ 0 *implies λx *∈ *K;*

(2) *x *∈ *K and *-*x *∈ *K implies x *= 0.

Every cone *K *⊂ *X *induces an ordering in *X*, which is given by *x *≤ *y *if and only if *y *- *x *∈ *K*.

**Definition 3.3**. *An operator is called completely continuous if it is continuous and maps bounded sets
into precompact sets*.

**Lemma 3.1**. *(Guo-Krasnoselskii *[30]) *Let X be a Banach space and K *⊂ *X be a cone in X. Assume that *Ω_{1}, Ω_{2 }*are bounded open subsets of X with *
*and *
*is a completely continuous operator such that, either*

(1) ||Φ*x*|| ≤ ||*x*||, *x *∈ *K *∩ ∂Ω_{1}, *and *||Φ*x*|| ≥ ||*x*||, *x *∈ *K *∩ ∂Ω_{2}; *or*

(2) ||Φ*x*|| ≥ ||*x*||, *x *∈ *K *∩ ∂Ω_{1}, *and *||Φ*x*|| ≤ ||*x*||, *x *∈ *K *∩ ∂Ω_{2}.

*Then *Φ *has at least one fixed point in *

**Lemma 3.2**. *Suppose *
*ν _{i }*∈ ℝ,

*then x is a solution of*

*where*

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

*Proof*. Assume that *x*(*t*) is a solution of (3.2). By the first equation in (3.2), we have

If *t *∈ [0, *t*_{1}], integrating (3.3) from 0 to *t*, we get

while *t *→ *t*_{1}, we have

then

Now, let *t *∈ (*t*_{1}, *t*_{2}], integrating (3.3) from *t*_{1 }to *t*, we obtain

For *t *∈ (*t _{k}*,

*t*

_{k+1}], repeating the above process, we can get

that is

It follows from

where

This means that if *x *is a solution of (3.2) then *x *satisfies (3.1).

On the other hand, if *x *satisfies (3.1), we have

Then

where

Notice that

Similarly,

Hence, we get from (3.5) that

that is

Finally, we can obtain from (3.1) that

and

So the proof of this lemma is completed.

**Lemma 3.3**. *Let G*(*t*, *s*) *be defined the same as that in Lemma *3.2, *then the following properties hold*.

(1) *G*(*t*, *s*) *> *0 *for all *

(2) *A *≤ *G*(*t*, *s*) ≤ *B for all *
*where*

*Proof*. Since

Hence, the left-hand side of (2) holds. And it is easy to show that the right-hand side of (2) also holds. The proof is complete. ■

Define an operator Φ : *PC *→ *PC *by

By Lemma 3.2, the fixed points of Φ are solutions of problem (1.3).

**Lemma 3.4**. *The operator *Φ : *PC *→ *PC is completely continuous*.

*Proof*. The first step we will show that Φ : *PC *→ *PC *is continuous. Let
*PC*. Then

Since *f*(*t*, *x*) and *I _{i}*(

*x*)(1 ≤

*i*≤

*m*) are continuous in

*x*, we have |(Φ

*x*)(

_{n}*t*) - (Φ

*x*)(

*t*)| → 0, which leads to ||Φ

*x*- Φ

_{n }*x*||

*→ 0, as*

_{PC }*n*→ ∞. That is, Φ :

*PC*→

*PC*is continuous.

Next, we will show that Φ : *PC *→ *PC *is a compact operator by two steps.

Let *U *⊂ *PC *be a bounded set.

Firstly, we will show that {Φ*x *: *x *∈ *U*}is bounded. For any *x *∈ *U*, we have

In virtue of the continuity of *f*(*t*, *x*) and *I _{i}*(

*x*)(1 ≤

*i*≤

*m*), we can conclude that {Φ

*x*:

*x*∈

*U*} is bounded from above inequality.

Secondly, we will show that {Φ*x *: *x *∈ *U*} is the set of equicontinuous functions. For any *x*, *y *∈ *U*, then

In virtue of the continuity of *f*(*t*, *x*) and *I _{i}*(

*x*)(1 ≤

*i*≤

*m*), the right-hand side tends to zero uniformly as |

*x*-

*y*| → 0. Consequently, {Φ

*x*:

*x*∈

*U*} is the set of equicontinuous functions.

By Arzela-Ascoli theorem on time scales [31], {Φ*x *: *x *∈ *U*} is a relatively compact set. So Φ maps a bounded set into a relatively compact set,
and Φ is a compact operator.

From above three steps, it is easy to see that Φ : *PC *→ *PC *is completely continuous. The proof is complete. ■

Let
*PC*.

**Lemma 3.5**. Φ *maps K into K*.

*Proof*. Obviously, Φ(*K*) ⊂ *PC*. ∀*x *∈ *K*, we have

which implies

Therefore,

Hence, Φ(*K*) ⊂ *K*. The proof is complete. ■

### 4 Existence of at least one positive solution

In this section, we will state and prove our main result about the existence of at least one positive solution of problem (1.3).

**Theorem 4.1**. *Assume that one of the following conditions is satisfied:*

(*H*_{1}) max *f*_{0 }= 0, min *f*_{∞ }= ∞, *and I*_{i0 }= 0, *i *= 1, 2,..., *m*; *or*

(*H*_{2}) max *f*_{∞ }= 0, min *f*_{0 }= ∞, *and I*_{i∞ }= 0, *i *= 1, 2,..., *m*.

*Then, problem *(1.3) *has at least one positive solution*.

*Proof*. Firstly, we assume that (*H*_{1}) holds. In this case, since max *f*_{0 }= 0 and *I*_{i0 }= 0, *i *= 1, 2,..., *m*, for *ε *≤ (*Bσ*(*T*) + *Bm*)^{-1}, there exists a positive constant *r*_{1 }such that

In view of min *f*_{∞ }= ∞, we have that for *M *≥ (*Aσ*(*T*)*δ*)^{-1}, there exists a constant

Let Ω* _{i }*= {

*x*∈

*PC*: ||

*x*||

*< r*},

_{i}*i*= 1, 2.

On the one hand, if *x *∈ *K *∩ ∂Ω_{1}, we have

which yields

On the other hand, if *x *∈ *K *∩ ∂Ω_{2}, we have

which implies

Therefore, by (4.1), (4.2), and Lemma 3.1, it follows that Φ has a fixed point in

Next, we assume that (*H*_{2}) holds. In this case, since max *f*_{∞ }= 0 and *I*_{i∞ }= 0, *i *= 1, 2,..., *m*, for *ε' *≤ (*Bσ*(*T*) + *Bm*)^{-1}, there exists a positive constant *r*_{3 }such that

In view of min *f*_{∞ }= ∞, we have that for *M' *≥ (*Aσ*(*T*)*δ*)^{-1}, there exists a positive constant *r*_{4 }*< δr*_{3 }such that

Let Ω* _{i }*= {

*x*∈

*PC*: ||

*x*||

*< r*},

_{i}*i*= 3, 4.

On the one hand, if *x *∈ *K *∩ ∂Ω_{3}, we have

which yields

On the other hand, if *x *∈ *K *∩ ∂Ω_{4}, we have

which implies

Hence, from (4.3) and (4.4) and Lemma 3.1, we conclude that Φ has a fixed point in

### 5 Existence of at least two positive solutions

In this section, we will state and prove our main results about the existence of at least two positive solutions to problem (1.3).

**Theorem 5.1**. *Assume that the following conditions hold*.

(*H*_{3}) min *f*_{0 }= +∞, min *f*_{∞ }= +∞.

(*H*_{4}) *There exists a positive constant R such that *
*for all *0 < *x *≤ *R*.

(*H*_{5})
*x *∈ (0, ∞), *i *= 1, 2,..., *m*.

*Then, problem *(1.3) *has at least two positive solutions*.

*Proof*. Let Ω* _{R }*= {

*x*∈

*PC*: ||

*x*||

*< R*}. From (

*H*

_{4}) and (

*H*

_{5}), for

*x*∈

*K*∩ ∂Ω

*, we get*

_{R}

So

Since min *f*_{0 }= +∞, for *M *≥ (*Aσ*(*T*)*δ*)^{-1}, there exists a positive constant *R*_{1 }*< δ _{R }*such that

Let

Hence,

Similarly, since min *f*_{∞ }= +∞, for *M' *≥ (*Aσ*(*T*)*δ*)^{-1}, there exists a positive constant

Let

Hence,

Equations 5.1 and 5.2 imply that Φ has at least one fixed point in
*x*_{1 }and *x*_{2 }satisfying 0 *< R*_{1 }≤ ||*x*_{1}|| *< R < *||*x*_{2}|| ≤ *R*_{2}. The proof is complete. ■

**Theorem 5.2**. *Assume that the following conditions hold*.

(*H*_{6}) max *f*_{0 }= 0, max *f*_{∞ }= 0, *I*_{i0 }= 0, *I*_{i∞ }= 0, *i *= 1, 2,..., *m*.

(*H*_{7}) *There exists a positive constant r such that *
*for all *0 < *x *≤ *r*.

*Then problem *(1.3) *has at least two positive solutions*.

*Proof*. Let Ω* _{r }*= {

*x*∈

*PC*: ||

*x*||

*< r*}. From (

*H*

_{7}), for

*x*∈

*K*∩ ∂Ω

*, we get*

_{r}

So

Since max *f*_{0 }= 0 and *I*_{i0 }= 0, *i *= 1, 2,..., *m*, for *ε *≤ (*Bσ*(*T*) + *Bm*)^{-1}, there exists a positive constant *r*_{1 }*< δ _{r }*such that

Let

Hence,

Similarly, since max *f*_{∞ }= 0 and *I*_{i∞ }= 0, *i *= 1, 2,..., *m*, for *ε' *≤ (*Bσ*(*T*) + *Bm*)^{-1}, there exists a positive constant

Let