### Abstract

In this article, we investigate the multiplicity of positive solutions for a fourth-order system of integral boundary value problem on time scales. The existence of multiple positive solutions for the system is obtained by using the fixed point theorem of cone expansion and compression type due to Krasnosel'skill. To demonstrate the applications of our results, an example is also given in the article.

##### Keywords:

positive solutions; fixed points; integral boundary conditions; time scales### 1 Introduction

Boundary value problem (BVP) for ordinary differential equations arise in different areas of applied mathematics and physics and so on, the existence and multiplicity of positive solutions for such problems have become an important area of investigation in recent years, lots of significant results have been established by using upper and lower solution arguments, fixed point indexes, fixed point theorems and so on (see [1-8] and the references therein). Especially, the existence of positive solutions of nonlinear BVP with integral boundary conditions has been extensively studied by many authors (see [9-18] and the references therein).

However, the corresponding results for BVP with integral boundary conditions on time scales are still very few [19-21]. In this article, we discuss the multiple positive solutions for the following fourth-order system of integral BVP with a parameter on time scales

where *a _{i}*,

*b*,

_{i}*c*,

_{i}*d*≥ 0, and

_{i }*ρ*=

_{i }*a*(

_{i}c_{i}σ*T*) +

*a*+

_{i}d_{i }*b*0(

_{i}c_{i }>*i*= 1, 2), 0

*< λ*,

*μ <*+∞,

*f*,

^{+ }= [0, +∞),

*A*and

_{i }*B*are nonnegative and

_{i }*rd*-continuous on

The main purpose of this article is to establish some sufficient conditions for the existence of at least two positive solutions for system (1.1) by using the fixed point theorem of cone expansion and compression type. This article is organized as follows. In Section 2, some useful lemmas are established. In Section 3, by using the fixed point theorem of cone expansion and compression type, we establish sufficient conditions for the existence of at least two positive solutions for system (1.1). An illustrative example is given in Section 4.

### 2 Preliminaries

In this section, we will provide several foundational definitions and results from the calculus on time scales and give some lemmas which are used in the proof of our main results.

A time scale

**Definition 2.1**. [22]*For
*,

*while the backward jump operator*

*by*

In this definition, we put
*σ*(*t*) *> t*, we say that *t *is right-scattered, while if *ρ*(*t*) *< t*, we say that *t *is left-scattered. Also, if
*σ*(*t*) = *t*, then *t *is called right-dense, and if
*ρ*(*t*) = *t*, then *t *is called left-dense. We also need, below, the set
*m*, then

**Definition 2.2**. [22]*Assume that *
*is a function and let *
*Then x is called differentiable at *
*if there exists a θ *∈ ℝ *such that for any given ε > *0*, there is an open neighborhood U of t such that*

*In this case, x*^{Δ}(*t*) *is called the delta derivative of x at t. The second derivative of x*(*t*) *is defined by x*^{ΔΔ}(*t*) = (*x*^{Δ})^{Δ}(*t*).

In a similar way, we can obtain the fourth-order derivative of *x*(*t*) is defined by *x*^{(4Δ)}(*t*) = (((*x*^{Δ})^{Δ})^{Δ})^{Δ}(*t*).

**Definition 2.3**. [22]*A function *
*is called rd-continuous provided it is continuous at right-dense points in *
*and its left-sided limits exist at left-dense points in *
*The set of rd-continuous functions *
*will be denoted by *

**Definition 2.4**. [22]*A function *
*is called a delta-antiderivative of *
*provide F*^{Δ}(*t*) = *f*(*t*) *holds for all *
*In this case we define the integral of f by*

For convenience, we denote
*i *= 1, 2, we set

where

To establish the existence of multiple positive solutions of system (1.1), let us list the following assumptions:

In order to overcome the difficulty due to the dependence of *f*, *g *on derivatives, we first consider the following second-order nonlinear system

where *A*_{0 }is the identity operator, and

For the proof of our main results, we will make use of the following lemmas.

**Lemma 2.1**. *The fourth-order system *(1.1) *has a solution *(*x*, *y*) *if and only if the nonlinear system *(2.1) *has a solution *(*u*, *v*).

*Proof*. If (*x*, *y*) is a solution of the fourth-order system (1.1), let *u*(*t*) = *x*^{ΔΔ}(*t*), *v*(*t*) = *y*^{ΔΔ}(*t*), then it follows from the boundary conditions of system (1.1) that

Thus (*u*, *v*) = (*x*^{ΔΔ}(*t*), *y*^{ΔΔ}(*t*)) is a solution of the nonlinear system (2.1).

Conversely, if (*u*, *v*) is a solution of the nonlinear system (2.1), let *x*(*t*) = *A*_{2}*u*(*t*), *y*(*t*) = *A*_{2}*v*(*t*), then we have

which imply that

Consequently, (*x*, *y*) = (*A*_{2}*u*(*t*), *A*_{2}*v*(*t*)) is a solution of the fourth-order system (1.1). This completes the proof.

**Lemma 2.2**. *Assume that D*_{11}*D*_{21 }≠ 1 *holds. Then for any h*_{1 }∈ *C*(*I*', ℝ^{+})*, the following BVP*

*has a solution*

*where*

*Proof*. First suppose that *u *is a solution of system (2.3). It is easy to see by integration of BVP(2.3) that

Integrating again, we can obtain

Let *t *= *σ*(*T*) in (2.4) and (2.5), we obtain

Substituting (2.6) and (2.7) into the second boundary value condition of system (2.3), we obtain

From (2.8) and the first boundary value condition of system (2.3), we have

Substituting (2.9) and (2.10) into (2.5), we have

By (2.11), we get

By (2.12) and (2.13), we get

Substituting (2.14) and (2.15) into (2.11), we have

Conversely, suppose

Direct differentiation of (2.17) implies

and

and it is easy to verify that

This completes the proof.

**Lemma 2.3**. *Assume that D*_{12}*D*_{22 }≠ 1 *holds. Then for any h*_{2 }∈ *C*(*I*', ℝ^{+})*, the following BVP*

*has a solution*

*where*

*Proof*. The proof is similar to that of Lemma 2.2 and will omit it here.

**Lemma 2.4**. *Suppose that *(*H*_{1}) *is satisfied, for all t*, *s *∈ *I and i *= 1, 2*, we have*

(i) *G _{i}*(

*t*,

*s*)

*>*0

*, H*(

_{i}*t*,

*s*)

*>*0,

(ii) *Lim _{i}G_{i}*(

*σ*(

*s*),

*s*) ≤

*H*(

_{i}*t*,

*s*) ≤

*M*(

_{i}G_{i}*σ*(

*s*),

*s*),

(iii) *mG _{i}*(

*σ*(

*s*),

*s*) ≤

*H*(

_{i}*t*,

*s*) ≤

*MG*(

_{i}*σ*(

*s*),

*s*),

*where*

*Proof*. It is easy to verify that *G _{i}*(

*t*,

*s*)

*>*0,

*H*(

_{i}*t*,

*s*)

*>*0 and

*G*(

_{i}*t*,

*s*) ≤

*G*(

_{i}*σ*(

*s*),

*s*), for all

*t*,

*s*∈

*I*. Since

Thus *G _{i}*(

*t*,

*s*)/

*G*(

_{i}*σ*(

*s*),

*s*) ≥

*L*and we have

_{i }

On the one hand, from the definition of *L _{i }*and

*m*, for all

_{i}*t*,

*s*∈

*I*, we have

and on the other hand, we obtain easily that from the definition of *M _{i}*, for all

*t*,

*s*∈

*I*,

Finally, it is easy to verify that *mG _{i}*(

*σ*(

*s*),

*s*) ≤

*H*(

_{i}*t*,

*s*) ≤

*MG*(

_{i}*σ*(

*s*),

*s*). This completes the proof.

**Lemma 2.5**. [23]*Let E be a Banach space and P be a cone in E. Assume that *Ω_{1 }*and *Ω_{2 }*are bounded open subsets of E, such that *0 ∈ Ω_{1},
*and let *
*be a completely continuous operator such that either*

(*i*) ||*Tu*|| ≤ ||*u*||, ∀*u *∈ *P *∩ ∂Ω_{1 }*and *||*Tu*|| ≥ ||*u*||, ∀*u *∈ *P *∩ ∂Ω_{2}*, or*

(*ii*) ||*Tu*|| ≥ ||*u*||, ∀*u *∈ *P *∩ ∂Ω_{1 }*and *||*Tu*|| ≤ ||*u*||, ∀*u *∈ *P *∩ ∂Ω_{2}

*holds. Then T has a fixed point in *

To obtain the existence of positive solutions for system (2.1), we construct a cone
*P *in the Banach space *Q *= *C*(*I*, ℝ^{+}) × *C*(*I*, ℝ^{+}) equipped with the norm

It is easy to see that *P *is a cone in *Q*.

Define two operators *T _{λ}*,

*T*:

_{μ }*P*→

*C*(

*I*, ℝ

^{+}) by

Then we can define an operator *T *: *P *→ *C*(*I*, ℝ^{+}) by

**Lemma 2.6**. *Let *(*H*_{1}) *hold. Then T *: *P *→ *P is completely continuous*.

*Proof*. Firstly, we prove that *T *: *P *→ *P*. In fact, for all (*u*, *v*) ∈ *P *and *t *∈ *I*, by Lemma 2.4(i) and (*H*_{1}), it is obvious that *T _{λ}*(

*u*,

*v*)(

*t*)

*>*0,

*T*(

_{μ}*u*,

*v*)(

*t*)

*>*0. In addition, we have

which implies

In a similar way,

Therefore,

This shows that *T *: *P *→ *P*.

Secondly, we prove that *T *is continuous and compact, respectively. Let {(*u _{k}*,

*v*)} ∈

_{k}*P*be any sequence of functions with

from the continuity of *f*, we know that ||*T _{λ}*(

*u*,

_{k}*v*) -

_{k}*T*(

_{λ}*u*,

*v*)|| → 0 as

*k*→ ∞. Hence

*T*is continuous.

_{λ }*T _{λ }*is compact provided that it maps bounded sets into relatively compact sets. Let

*P*, then there exists

*r >*0 such that ||(

*u*,

*v*)|| ≤

*r*for all (

*u*,

*v*) ∈ Ω. Obviously, from (2.16), we know that

so, *T _{λ}*Ω is bounded for all (

*u*,

*v*) ∈ Ω. Moreover, let

We have

Thus, for any (*u*, *v*) ∈ Ω and ∀*ε > *0, let
*t*_{1}, *t*_{2 }∈ *I*, |*t*_{1 }- *t*_{2}| *< δ*, we have

So, for all (*u*, *v*) ∈ Ω, *T _{λ}*Ω is equicontinuous. By Ascoli-Arzela theorem, we obtain that

*T*:

_{λ }*P*→

*P*is completely continuous. In a similar way, we can prove that

*T*:

_{μ }*P*→

*P*is completely continuous. Therefore,

*T*:

*P*→

*P*is completely continuous. This completes the proof.

### 3 Main results

In this section, we will give our main results on multiplicity of positive solutions of system (1.1). In the following, for convenience, we set

where *q _{i}*(

*t*),

*q*(

_{j}*t*) ∈

*C*(

_{rd}*I*', ℝ

^{+}) satisfy

**Theorem 3.1**. *Assume that *(*H*_{1}) *holds. Assume further that*

(*H*_{2}) *there exist a constant R > *0*, and two functions p _{i}*(

*t*) ∈

*C*(

_{rd}*I*,

*R*

_{+})

*satisfying*

*such that*

*and one of the folloeing conditions is satisfied*

(*E*_{1})

(*E*_{2})

(*E*_{3})
*μ *∈ (0, *N*_{4}),

(*E*_{4}) *λ *∈ (0, *M*_{4}),

*where*

*O*_{1}*, O*_{2 }*satisfy *
*Then system (1.1) has at least two positive solutions*.

*Proof*. We only prove the case in which (*H*_{2}) and (*E*_{1}) hold, the other case can be proved similarly. Firstly, from (2.2), we have

Take
_{1 }= {(*u*, *v*) ∈ *Q*; ||(*u*, *v*)|| *< R*_{1}}. For any *t *∈ *I*, (*u*, *v*) ∈ ∂Ω_{1 }∩ *P*, it follows from *λ < M*_{4}, *μ < N*_{4 }and (*H*_{2}) that

and

Consequently, for any (*u*, *v*) ∈ ∂Ω_{1 }∩ *P*, we have

Second, from
*ε*_{1 }*> *0 such that *λf*_{0 }*> M*_{3 }+ *ε*_{1}, then there exists 0 *< l*_{1 }*< NR*_{1 }such that for any
*t *∈ *I*,

And since

Take

For all (*u*, *v*) ∈ Ω_{2 }∩ *P*, where Ω_{2 }= {(*u*, *v*) ∈ *Q*; ||(*u*, *v*)|| *< R*_{2}}, we have

Thus, for all (*u*, *v*) ∈ Ω_{2 }∩ *P*, we have