Multiple Positive Solutions for a Class of -Point Boundary Value Problems on Time Scales

Recently, there have been many papers working on the existence of positive solutions to boundary value problems for differential equations on time scales; see, for example, [1–20]. This has been mainly due to its unification of the theory of differential and difference equations. An introduction to this unification is given in [11, 12, 18, 19]. Now, this study is still a new area of fairly theoretical exploration in mathematics. However, it has led to several important applications, for example, in the study of insect population models, neural networks, heat transfer, and epidemic models; see, for example, [10, 11]. For some other excellent results and applications of the case that boundary value problems on time scales to a variety of problems from Khan et al. [21], Agarose et al. [22], Wang [23], Sun [24], Feng et al. [25], Feng et al. [26] and Feng et al. [27].

Motivated by the works mentioned above, we intend in this paper to study the existence of multiple positive solutions for the second-order m-point nonlinear dynamic equation on time scales with polynomial nonlinearity:

(1.1)

where is a time scale,

(1.2)

the points for with ;

(1.3)

(1.4)

Recently, Xu [28] considered the following second-order two-point impulsive singular differential equations boundary value problem:

(1.5)

By means of fixed point index theory in a cone, the author established the existence of two nonnegative solutions for problem (1.5).

More recently, by applying Guo-Krasnosel'skii fixed point theorem in a cone, Anderson and Ma [6] established the existence of at least one positive solution to the multipoint time-scale eigenvalue problem:

(1.6)

where is continuous.

As far as we know, there is no paper to study the existence of multiple positive solutions to problem (1.1) on time scales with polynomial nonlinearity. The objective of the present paper is to fill this gap. On the other hand, many difficulties occur when we study BVPs on time scales. For example, basic tools from calculus such as Fermat's theorem, Rolle's theorem and the intermediate value theorem may not necessarily hold. So it is interesting and important to discuss the problem (1.1). The purpose of this paper is to prove that the problem (1.1) possesses at least two positive solutions. Moreover, the methods used in this paper are different from [6, 28] and the results obtained in this paper generalize some results in [6, 28] to some degree.

The time scale related notations adopted in this paper can be found, if not explained specifically, in almost all literature related to time scales. The readers who are unfamiliar with this area can consult for example [11, 12, 18, 19] for details.

For convenience, we list the following well-known definitions.

Definition 1.1.

A time scale is a nonempty closed subset of .

Definition 1.2.

Define the forward (backward) jump operator at for at for by for all .

We assume throughout that has the topology that it inherits from the standard topology on and say is right-scattered, left-scattered, right-dense and left-dense if and , respectively. Finally, we introduce the sets and . which are derived from the time scale as follows. If has a left-scattered maximum , then , otherwise . If has a right-scattered minimum , then , .

Definition 1.3.

Fix and let . Define to be the number (if it exists) with the property that given there is a neighborhood of with

(1.7)

for all , where denotes the (delta) derivative of with respect to the first variable, then

(1.8)

implies

(1.9)

Definition 1.4.

. Define to be the number (if it exists) with the property that given there is a neighborhood of with

(1.10)

for all . Call the nabla derivative of at the point .

If then . If then is the forward difference operator while is the backward difference operator.

Definition 1.5.

A function is called rd-continuous provided it is continuous at all right dense points of and its left sided limit exists (finite) at left dense . We let denote the set of rd-continuous functions .

Definition 1.6.

is called ld-continuous provided it is continuous at all left and its right sided limit exists (finite) at right dense . We let denote the set of ld-continuous .

Definition 1.7.

A function is called a delta-antiderivative of provided holds for all . In this case we define the delta integral of by

(1.11)

for all .

Definition 1.8.

is called a nabla-antiderivative of provided holds for all . In this case we define the nabla integral of f by

(1.12)

for all .

In this section, we provide some necessary background. In particular, we state some properties of Green's function associated with problem (1.1), and we then state a fixed-point theorem which is crucial to prove our main results.

The basic space used in this paper is . It is well known that is a Banach space with the norm defined by . Let be a cone of , , where .

In this paper, the Green's function of the corresponding homogeneous BVP is defined by

(2.1)

where

(2.2)

and and satisfy

(2.3)

respectively.

Lemma 2.1 (see [6]).

Assume that (1.2) and (1.3) hold. Then and the functions and satisfy

(2.4)

From Lemma 2.1 and the definition of , we can prove that has the following properties.

Proposition 2.2.

For , one has

(2.5)

where

In fact, from Lemma 2.1, we have for Therefore (2.5) holds.

Proposition 2.3.

If (1.2) holds, then for , one has

(2.6)

Proof.

In fact, from Lemma 2.1, we obtain for So .

On the other hand, from Lemma 2.1, we know that for . This together with implies that for . Hence is nondecreasing on , is nonincreasing on . So (2.6) holds.

Proposition 2.4.

For all one has

(2.7)

where

(2.8)

Proof.

In fact, for , we have

(2.9)

Therefore (2.7) holds.

It is easy to see that , for . Thus, there exists such that for , where

(2.10)

We remark that Proposition 2.2 implies that there exists such that for

(2.11)

Set

(2.12)

Lemma 2.5 (see [6]).

Assume that (1.2) and (1.3) hold. If and , then the nonhomogeneous boundary value problem

(2.13)

has a unique solution for which the formula

(2.14)

holds, where

(2.15)

(2.16)

By similar method, one can define

(2.17)

The following lemma is crucial to prove our main results.

Lemma 2.6 (see[29, 30]).

Let and be two bounded open sets in a real Banach space , such that and . Let the operator be completely continuous, where is a cone in . Suppose that one of the two conditions

(2.18)

or

(2.19)

is satisfied. Then has at least one fixed point in .

In this section, we apply Lemma 2.6 to establish the existence of at least two positive solutions for BVP (1.1).

The following assumptions will stand throughout this paper.

(H₁)There exist such that

(3.1)

where and are defined in (1.4), respectively.

(H₂)We have

(3.2)

for and given in (2.2) and (2.12), respectively.

If holds, then we can show that have the following properties.

Proposition 3.1.

If (1.2)–(1.4) and hold, then from (2.15), for , one has

(3.3)

where

Proof.

Let

(3.4)

Then from (1.2)–(1.4) and , we obtain Therefore,

On the other hand, since

(3.5)

we have So one has

(3.6)

This and imply (3.3) holds.

Proposition 3.2.

If (1.2)–(1.4) and hold, then from (2.16), , one has

(3.7)

Proof.

The proof is similar to that of Proposition 3.1. So we omit it.

For the sake of applying fixed point theorem on cone, we construct a cone in by

(3.8)

where is defined in (2.10).

Define by

(3.9)

By (2.14), it is well known that the problem (1.1) has a positive solution if and only if is a fixed point of .

Lemma 3.3.

Suppose that (1.2)–(1.4) and - hold. Then and is completely continuous.

Proof.

For by (2.14), we have and

(3.10)

On the other hand, for , by (3.9),(3.10) and (2.7), we obtain

(3.11)

Therefore , that is, .

Next by standard methods and the Ascoli-Arzela theorem one can prove that is completely continuous. So it is omitted.

Theorem 3.4.

Suppose that (1.2)–(1.4) and - hold. Then problem (1.1) has at least two positive solutions provided

(3.12)

where and are defined in (3.3), (3.7) and in Proposition 3.1, respectively.

Proof.

Let be the cone preserving, completely continuous operator that was defined by (3.9).

Let , where . Choosing and satisfy

(3.13)

Now we prove that

(3.14)

(3.15)

In fact, if there exists such that , then for , we have

(3.16)

where defined by (2.17).

Therefore , that is, , which is a contradiction. Hence (3.14) holds.

Next, turning to (3.15). If there exists such that , then for , we have

(3.17)

where are defined by (2.17).

Therefore , that is, , which is a contradiction. Hence (3.15) holds.

It remains to prove

(3.18)

In fact, if there exists such that , then for , we have

(3.19)

that is,

(3.20)

which is a contradiction, where are defined by (2.17). Hence (3.18) holds. From Lemma 2.6, (3.14), (3.15) and (3.18) yield that the problem (1.1) has at least two solutions and . The proof is complete.

Example 4.1.

To illustrate how our main results can be used in practice we present an example.

Let . Take in (1.1). Now we consider the following three point boundary value problem

(4.1)

where

(4.2)

It is not difficult to see that

(4.3)

On the other hand, by calculating we have ,

(4.4)

and

Let . Then and

(4.5)

It follows that and hold.

Finally, we prove that

(4.6)

In fact, from , we have and

(4.7)

Therefore, the conditions of Theorem 3.4 hold. Hence problem (4.1) has at least two positive solutions.

Remark 4.2.

Example 4.1 implies that there is a large number of functions that satisfy the conditions of Theorem 3.4. In addition, the conditions of Theorem 3.4 are also easy to check.