Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales

Abstract

We are concerned with singular three-point boundary value problems for delay higher-order dynamic equations on time scales. Theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is given to illustrate our main result.

1. Introduction

In this paper, we are concerned with the following singular three-point boundary value problem (BVP for short) for delay higher-order dynamic equations on time scales:

(1.1)

where , , , , , and . The functional is continuous and is continuous. Our nonlinearity may have singularity at and/or and may have singularity at .

To understand the notations used in (1.1), we recall the following definitions which can be found in [1, 2].

(a) A time scale is a nonempty closed subset of the real numbers . has the topology that it inherits from the real numbers with the standard topology. It follows that the jump operators ,

(1.2)

(supplemented by and ) are well defined. The point is left-dense, left-scattered, right-dense, right-scattered if , , , , respectively. If has a left-scattered maximum (right-scattered minimum ), define (); otherwise, set (). By an interval we always mean the intersection of the real interval with the given time scale, that is, . Other types of intervals are defined similarly.

(b) For a function and , the -derivative of at , denoted by , is the number (provided it exists) with the property that, given any , there is a neighborhood of such that

(1.3)

(c) For a function and , the -derivative of at , denoted by , is the number (provided it exists) with the property that, given any , there is a neighborhood of such that

(1.4)

(d) If then we define the integral

(1.5)

Theoretically, dynamic equations on time scales can build bridges between continuous and discrete mathematics. Practically, dynamic equations have been proposed as models in the study of insect population models, neural networks, and many physical phenomena which include gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems as well as concentration in chemical of biological problems [2]. Hence, two-point and multipoint boundary value problems for dynamic equations on time scales have attracted many researchers' attention (see, e.g., [1–19] and references therein). Moreover, singular boundary value problems have also been treated in many papers (see, e.g., [4, 5, 12–14, 18] and references therein).

In 2004, J. J. DaCunha et al. [13] considered singular second-order three-point boundary value problems on time scales

(1.6)

and obtained the existence of positive solutions by using a fixed point theorem due to Gatica et al. [14], where is decreasing in for every and may have singularity at .

In 2006, Boey and Wong [11] were concerned with higher-order differential equation on time scales of the form

(1.7)

where are fixed integers satisfying , . They obtained some existence theorems of positive solutions by using Krasnosel'skii fixed point theorem.

Recently, Anderson and Karaca [8] studied higher-order three-point boundary value problems on time scales and obtained criteria for the existence of positive solutions.

The purpose of this paper is to investigate further the singular BVP for delay higher-order dynamic equation (1.1). By the use of the fixed point theorem of cone expansion and compression type, results on the existence of positive solutions to the BVP (1.1) are established.

The paper is organized as follows. In Section 2, we give some lemmas, which will be required in the proof of our main theorem. In Section 3, we prove some theorems on the existence of positive solutions for BVP (1.1). Moreover, we give an example to illustrate our main result.

2. Lemmas

For , let be Green's function of the following three-point boundary value problem:

(2.1)

where and satisfy the following condition:

• (C)
  

  (2.2)

Throughout the paper, we assume that .

From [8], we know that for any and ,

(2.3)

where

(2.4)

The following four lemmas can be found in [8].

Lemma 2.1.

Suppose that the condition (C) holds. Then the Green function of in (2.3) satisfies

(2.5)

Lemma 2.2.

Assume that the condition (C) holds. Then Green's function in (2.3) satisfies

(2.6)

Remark 2.3.

  If , we know that is nonincreasing in and

(2.7)

Therefore, we have

(2.8)

where

(2.9)

If and satisfy the other cases, then we get that is nondecreasing in and

(2.10)

Lemma 2.4.

Assume that (C) holds. Then Green's function in (2.3) verifies the following inequality:

(2.11)

Remark 2.5.

If , then we find

(2.12)

So there exists a misprint on [8, Page 2431, line 23]. From (2.3), it follows that

(2.13)

Consequently, we get

(2.14)

If , , then, from (2.8), we obtain

(2.15)

Remark 2.6.

If we set , then we have

(2.16)

Denote

(2.17)

Thus we have

(2.18)

Lemma 2.7.

Assume that condition (C) is satisfied. For as in (2.3), put and recursively define

(2.19)

for . Then is Green's function for the homogeneous problem

(2.20)

Lemma 2.8.

Assume that (C) holds. Denote

(2.21)

then Green's function in Lemma 2.7 satisfies

(2.22)

where

(2.23)

Proof.

We proceed by induction on . We denote the statement by . From Lemma 2.7, it follows that

(2.24)

and from (2.18), we have

(2.25)

So is true.

We now assume that is true for some positive integer . From Lemma 2.7, it follows that

(2.26)

So holds. Thus is true by induction.

Lemma 2.9 (see [20]).

Let be a real Banach space and a cone. Assume that is completely continuous operator such that

(i) for and for ,

(ii) for and for .

Then has a fixed point with .

3. Main Results

We assume that and are strictly decreasing and strictly increasing sequences, respectively, with , and . A Banach space is the set of real-valued continuous (in the topology of ) functions defined on with the norm

(3.1)

Define a cone by

(3.2)

Set

(3.3)

Assume that

(C1) is continuous;

(C2) we have

(3.4)

for constants and with ;

(C3) the function is continuous and is continuous satisfying

(3.5)

We seek positive solutions , satisfying (1.1). For this end, we transform (1.1) into an integral equation involving the appropriate Green function and seek fixed points of the following integral operator.

Define an operator by

(3.6)

where .

Proposition 3.1.

Let (C1), (C2), and (C3) hold, and let , be fixed constants with . Then is completely continuous.

Proof.

We separate the proof into four steps.

Step 1.

For each , is bounded.

By condition (C3), there exists some positive integer satisfying

(3.7)

where

(3.8)

here, we used the fact that for each and ,

(3.9)

where

(3.10)

Set

(3.11)

Then we obtain

(3.12)

Consequently, is bounded and well defined.

Step 2.

. For every , we get from (2.22)

(3.13)

Then by the above inequality

(3.14)

This leads to .

Step 3.

We will show that is continuous. Let be any sequence in such that . Notice also that as ,

(3.15)

Now these together with (C2) and the Lebesgue dominated convergence theorem [10] yield that as ,

(3.16)

Step 4.

is compact.

Define

(3.17)

and an operator sequence for a fixed by

(3.18)

Clearly, the operator sequence is compact by using the Arzela-Ascoli theorem [3], for each . We will prove that converges uniformly to on . For any , we obtain

(3.19)

From (C1), (C2), and the Lebesgue dominated convergence theorem [10], we see that the right-hand side (3.19) can be sufficiently small for beingbig enough. Hence the sequence of compact operators converges uniformly to on so that operator is compact. Consequently, is completely continuous by using the Arzela-Ascoli theorem [3].

Proposition 3.2.

It holds that is a solution of (1.1)  if and only if .

Proof.

If and , then we have

(3.20)

and for any ,

(3.21)

From [8, Lemma  3.1], we know that on . So we conclude that is the solution of BVP (1.1).

For convenience, we list the following notations and assumptions:

(3.22)

(3.23)

(3.24)

(3.25)

From condition (C2) and (3.12), we have .

Theorem 3.3.

Assume that there exist positive constants with , and such that

(i) and ;

(ii) , for all and .

If (C1), (C2), and (C3) hold, then the boundary value problem (1.1) has at least one positive solution such that

(3.26)

Proof.

Define the operator by (3.6). From (i) and (3.23), it follows that there exists such that

(3.27)

We claim that

(3.28)

If it is false, then there exists some with , that is, which implies that for .

Set

(3.29)

We know from (2.22) and (3.27) that for ,

(3.30)

the first inequality of (C2) implies that

(3.31)

Clearly, (3.31) contradicts (3.29). This means that (3.28) holds.

Next we will show that

(3.32)

Suppose on the contrary that there exists some with for all .

For , from (i) and (3.24), there exists such that

(3.33)

and for , there exists , from (ii), such that

(3.34)

Put

(3.35)

If , then we take . It is easy to see that for and , , that is, . From (3.33) and (3.34), we find that

(3.36)

yielding a contradiction with for all . This means that (3.32) holds. Therefore, from (3.28), (3.32) and Lemma 2.9, we conclude that the operator has at least one fixed point . From the definition of the cone and (2.18), we see that for all . Thus, Proposition 3.2 implies that is a solution of BVP (1.1). So we obtain the desired result.

Adopting the same argument as in Theorem 3.3, we obtain the following results.

Corollary 3.4.

Let be as in Theorem 3.3.Suppose that (ii) of Theorem 3.3 holds and . If (C1), (C2), and (C3) holds, then boundary value problem (1.1) has at least one positive solution such that

(3.37)

Theorem 3.5.

Assume that there exist positive constants with , and , such that

(iii) and

(iv) , for all and .

If (C1), (C2), and (C3) hold, then boundary value problem (1.1) has at least positive solutions such that for

(3.38)

Example 3.6.

Let . Consider the following singular three-point boundary value problems for delay four-order dynamic equations:

(3.39)

where, for any , , , , , and ,

(3.40)

Clearly, we know that

(3.41)

Simple computations yield

(3.42)

Obviously,

(3.43)

If , then we have

(3.44)

Therefore, we get

(3.45)

From (3.25), it follows that

(3.46)

Thus,

(3.47)

Therefore, by Theorem 3.3, the BVP (3.39) has at least one positive solution such that

(3.48)