Necessary and Sufficient Conditions for the Existence of Positive Solution for Singular Boundary Value Problems on Time Scales

Abstract

By constructing available upper and lower solutions and combining the Schauder's fixed point theorem with maximum principle, this paper establishes sufficient and necessary conditions to guarantee the existence of as well as positive solutions for a class of singular boundary value problems on time scales. The results significantly extend and improve many known results for both the continuous case and more general time scales. We illustrate our results by one example.

1. Introduction

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, [1–22]. This has been mainly due to its unification of the theory of differential and difference equations. An introduction to this unification is given in [10, 14, 23, 24]. 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, [9, 10].

Motivated by works mentioned previously, we intend in this paper to establish sufficient and necessary conditions to guarantee the existence of positive solutions for the singular dynamic equation on time scales:

(11)

subject to one of the following boundary conditions:

(12)

or

(13)

where is a time scale, , where is right dense and is left dense. and is continuous. Suppose further that is nonincreasing with respect to , and there exists a function such that

(14)

A necessary and sufficient condition for the existence of as well as positive solutions is given by constructing upper and lower solutions and with the maximum principle. The nonlinearity may be singular at and/or . By singularity we mean that the functions in (1.1) is allowed to be unbounded at the points and/or A function is called a (positive) solution of (1.1) if it satisfies (1.1) (); if even exist, we call it is a solution.

To the best of our knowledge, there is very few literature giving sufficient and necessary conditions to guarantee the existence of positive solutions for singular boundary value problem on time scales. So it is interesting and important to discuss these problems. Many difficulties occur when we deal with them. For example, basic tools from calculus such as Fermat's theorem, Rolle's theorem, and the intermediate value theorem may not necessarily hold. So we need to introduce some new tools and methods to investigate the existence of positive solutions for problem (1.1) with one of the above boundary conditions.

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, [6, 11–13, 25, 26] for details.

The organization of this paper is as follows. In Section 2, we provide some necessary background. In Section 3, the main results of problem (1.1)-(1.2) will be stated and proved. In Section 4, the main results of problem (1.1)–(1.3) will be investigated. Finally, in Section 5, one example is also included to illustrate the main results.

2. Preliminaries

In this section we will introduce several definitions on time scales and give some lemmas which are useful in proving our main results.

Definition 2.1.

A time scale is a nonempty closed subset of .

Definition 2.2.

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

(21)

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 , otherwise .

Definition 2.3.

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

(22)

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

(23)

implies

(24)

Definition 2.4.

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

(25)

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 2.5.

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

Definition 2.6.

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

Definition 2.7.

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

(26)

for all .

Definition 2.8.

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

(27)

for all .

Throughout this paper, we assume that is a closed subset of with .

Let , equipped with the norm

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It is clear that is a real Banach space with the norm.

Lemma 2.9 (Maximum Principle).

Let and . If , , and Then

3. Existence of Positive Solution to (1.1)-(1.2)

In this section, by constructing upper and lower solutions and with the maximum principle Lemma 2.9, we impose the growth conditions on which allow us to establish necessary and sufficient condition for the existence of (1.1)-(1.2).

We know that

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is the Green's function of corresponding homogeneous BVP of (1.1)-(1.2).

We can prove that has the following properties.

Proposition 3.1.

For , one has

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To obtain positive solutions of problem (1.1)-(1.2), the following results of Lemma 3.2 are fundamental.

Lemma 3.2.

Assume that holds. If and exist and are finite, then one has

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Proof.

Without loss of generality, we suppose that there is only one right-scattered point . Then we have

(34)

that is,

(35)

Similarly, we can prove

(36)

The proof is complete.

Theorem 3.3.

Suppose that holds. Then problem (1.1)-(1.2) has a positive solution if and only if the following integral condition holds:

(37)

Proof.

• (1) Necessity

By (H), there exists such that . Without loss of generality, we assume that is nonincreasing on with .

Suppose that is a positive solution of problem (1.1)-(1.2), then

(38)

which implies that is concave on . Combining this with the boundary conditions, we have Therefore . So by [10, Theorem  1.115], there exists satisfying or . And for , , for . Denote , then .

First we prove .

By (), for any fixed , we have

(39)

It follows that

(310)

If , then we have by (3.10)

(311)

This means , then , which is a contradiction with being positive solution. Thus , then .

Second, we prove .

If then

(312)

If , then , , and

(313)

It follows that

(314)

By (3.14) we have

(315)

(316)

Combining this with (3.10) we obtain

(317)

Similarly

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Then we can obtain

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• (2) Sufficiency

Let

(320)

Then

(321)

Let

(322)

then

Let then

(323)

So, we have

(324)

and . Hence are lower and upper solutions of problem (1.1)-(1.2), respectively. Obviously for .

Now we prove that problem (1.1)-(1.2) has a positive solution with

Define a function

(325)

Then is continuous. Consider BVP

(326)

Define mapping by

(327)

Then problem (1.1)-(1.2) has a positive solution if and only if has a fixed point with

Obviously is continuous. Let By (3.7) and (3.16), for all , we have

(328)

Then is bounded. By the continuity of we can easily found that are equicontinuous. Thus is completely continuous. By Schauder fixed point theorem we found that has at least one fixed point .

We prove . If there exists such that

(329)

Let then for Thus . By (3.24) we know that And . By Lemma 2.9 we have , which is a contradiction. Then . Similarly we can prove . The proof is complete.

Theorem 3.4.

Suppose that () holds. Then problem (1.1)-(1.2) has a positive solution if and only if the following integral condition holds:

(330)

Proof.

• (1) Necessity

Let be a positive solution of problem (1.1)-(1.2). Then is decreasing on . Hence is integrable and

(331)

By simple computation and using [10, Theorem  1.119], we obtain . So there exist such that . By we obtain

(332)

By

(333)

we have

• (2) Sufficiency

Let , then

(334)

Similar to Theorem 3.3, let , there exists satisfying , and

(335)

then is integral and exist, hence is a positive solution in . The proof is complete.

4. Existence of Positive Solution to (1.1)–(1.3)

Now we deal with problem (1.1)–(1.3). The method is just similar to what we have done in Section 3, so we omit the proof of main result of this section.

Let

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be the Green's function of corresponding homogeneous BVP of (1.1)–(1.3).

We can prove that has the following properties.

Similar to (3.2), we have

(42)

Theorem 4.1.

Suppose that holds, then problem (1.1)–(1.3) has a positive solution if and only if the following integral condition holds:

(43)

Theorem 4.2.

Suppose that holds, then problem (1.1)–(1.3) has a positive solution if and only if the following integral condition holds:

(44)

5. Example

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

Example 5.1.

We have

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where Select , then we have Moreover, we have

(52)

By Theorem 3.3, problem (5.1) has a positive solution in .

Remark 5.2.

Example 5.1 implies that there is a large number of functions that satisfy the conditions of Theorem 3.3. In addition, the conditions of Theorem 3.3 are also easy to check.