Three-Order Multipoint Boundary Value Problems for -Laplacian Operator on Time Scales

Abstract

In this paper, the existence of positive solutions for the nonlinear four-point singular BVP for there-order with -Laplacian operator on time scales will be studied. By using the fixed-point theory, the existence of positive solutions for nonlinear singular boundary value problem with -Laplacian operator on time scales is obtained.

1. Introduction

In recent years, the nonlinear boundary value problems have been extensively studied. Recently, for the existence of positive solutions of multipoint boundary value problems, some authors have obtained the existence results. The differential equations offer wonderful tools for describing various natural phenomena arising from natural sciences and engineering, many numerical and analytical results, for example [1–20]. However, the multipoint boundary value problems treated in the above mentioned references do not discuss the problems with singularities and the there-order -Laplacian operator. For the singular case of multipoint boundary value problems for higher-order -Laplacian operator, with the author’s acknowledge, no one has studied the existence of positive solutions in this case.

In this paper, we study the following equation with -Laplacian on time scale: with the following boundary value conditions: where , , . is prescribed and , , are both nondecreasing continuous odd functions defined on .

A time scale is a nonempty subset and closed subset of . By an internal , we always mean the intersection of the real internal with the given time scale, that is . The operators and from to which defined by [21], are called the forward jump operator and the backward jump operator, respectively.

The point is left-dense, left-scattered, right-dense, right-scattered if , respectively. If has a right scattered minimum , define ; otherwise set . If has a left scattered maximum , define ; otherwise set .

In this paper, by constructing an integral equation which is equivalent to the problem (1), (2), we research the existence of positive solutions for nonlinear singular boundary value problem (1), (2) when and satisfy some suitable conditions.

2. Preliminaries and Lemmas

For convenience, in the rest of this article, is a closed subset of with .

Letting

Then, is a Banach space with the norm . Suppose

Obviously, is a cone in and on . Set .

In the rest of the paper, we make the following assumptions:

() ;

() and there exists which satisfy

() are both increasing, continuous, odd functions, and at least one of them satisfies the condition that there exists one which satisfy

By direct account, From paper [22], we can easy to obtain the following results.

Lemma 1. Suppose condition holds. Then, there exists a constant satisfies Furthermore, the function is positive ld-continuous function on , therefore, has minimum on . Hence, we suppose that there exists constant which satisfy on .

Lemma 2. Suppose that conditions hold, is a solution of boundary value problems (1), (2) if and only if is a solution of the following integral equation where

Here, is a unique solution of the equation , where

Equation has a unique solution in . Because is strictly monotone increasing on , and is strictly monotone decreasing on , and .

Proof. Necessity. By the equation of the boundary condition and , we have Then, there exist a constant which satisfy . Firstly, by integrating the equation of the problems (1) on , we have then By and condition (2), let on (15), we have Then, we have Then Therefore, by integrating the above equation (19) on , we can east to have Similarly, for , by integrating the equation of problems (1) on , we have Therefore, for any , can be expressed as equation , where is expressed as Lemma 3.
Sufficiency. Suppose . Then we have So, . These imply that the equation (1) holds. Furthermore, we can easily obtain the boundary value equations of (2). This completes the proof of Lemma 3.
Now, we define an operator given by where is given by (15). And we can easily obtain the following Lemma.

Lemma 3. Let and in Lemma 1. Then

Lemma 4. Suppose that conditions hold, then for in Lemma 1, we have

Proof. If is the solution of problem (1), (2), then is a concave function, and .
Thus for , we have . Then by , we have The proof is complete.

Remark. : Obviously, we can obtain the following results, Furthermore, by Arzela-Ascoli Theorem, it is easy to obtain the following Lemma.

Lemma 5. is completely continuous.
For convenience, we set where and are given as Lemma 1.

3. The Existence of Single and Many Positive Solution

In this section, we present the following five main results.

Theorem 6. Suppose that condition (), (), () hold. Assume that also satisfy.
() For , we have
() For , we have where . Then, the boundary value problem (1), (2) has at last one solution such that lies between and .

The proof of Theorem 6. From Condition , for , we can suppose that , and . By Lemma 3, for any , we can obtain that

We define the following two open subset and of :

For any , by (29), we have

For and , we shall discuss it from three perspectives. (i)If , thus for , by () and Lemma 3, we have

Then, with the case of and , we have (ii)If , thus for , by () and Lemma 3, we have

Then, with the case of and , we have (iii)If , thus for , by () and Lemma 3, we have

Then, with the case of and , we have

Therefore, for any case of , we all easy to obtain that

Then, by fixed point theorem of cone expansion-compression type in [23, 24], we have

Secondly, for , using , from (), we can easily know that

Thus, we have

Then, by fixed point theorem of cone expansion-compression type in [23, 24], we have

Therefore, by (36), (39), we have

Then, operator has at last one fixed point , and . This completes the proof of Theorem 6.

Theorem 7. Suppose that condition (), (), () hold. Assume that also satisfy.
() ;
() .

Then, the boundary value problem (1), (2) has at last one solution .

The proof of Theorem 7.

First, by , letting , we know that there exists an appropriately small positive number which satisfy as , and we have

Then, letting , thus by the above equation, we can have

So condition () in Theorem 6 holds.

Next, by , letting , we know that there exists an adequately big positive number which satisfy as , and we have

Letting , thus by the above equation, we have that () in Theorem 6 holds. Therefore, by Theorem 6, we can easily obtain the results of Theorem 7 holds. The proof of Theorem 3.2 is complete.

Corollary 8. Suppose that condition (), (), () hold. Assume that also satisfy.
() ;
() .
Then, the boundary value problem (1), (2) has at last one solution .

The proof of Corollary 8. Similar to the proof of Theorem 7, we can obtain Corollary 8.

Theorem 9. Suppose that conditions (), (), (), and in Theorem 6 hold. Assume that also satisfy.
() ;
() .
Then, the boundary value problem (1), (2) have at least two solutions which satisfy The proof of Theorem 9.

Firstly, by , for any , there exists a constant which satisfy

Set , similar to the previous proof of Theorem 6, for any , from the above discussion and Lemma 2, we can have from three perspectives

Then, by fixed point theorem of cone expansion-compression type, we can have

Secondly, for any , by , there exists a constant which satisfy

Therefore, we choose a constant , obviously . Set . For any , by Lemma 2, we can easily obtain

Then, by the above discussion and also similar to the previous proof of Theorem 6, we can also have from three perspectives

Then, by fixed point theorem of cone expansion-compression type, we have

Finally, imitating the latter proof of Theorem 6, for any , by , setting , we can also easy to have

Then, by fixed point theorem of cone expansion-compression type, we have

Therefore, by (47), (51), (53), we have

Then, have fixed point , and fixed point .

Obviously, are all positive solutions of problem (1), (2) and . The proof of Theorem 9 is complete.

Theorem 10. Suppose that conditions (), (), () and in Theorem 6 hold. Assume that also satisfy.
() ;
() .
Then, the boundary value problem (1), (2) have at least two solutions which satisfy .
The proof of Theorem 10.

Firstly, by , for , there exists a constant which satisfy

Set , for any , by (23), we have

Then, by fixed point theorem of cone expansion-compression type, we have

Secondly, letting , we can easy to know that is monotone increasing with respect to .

Therefore by , we can easy to have

Therefore, for any , there exists a constant which satisfy

Set , for any , by (4.8), we have

Then, by fixed point theorem of cone expansion-compression type, we have

Finally, imitating the previous proof of Theorem 6, for any , by , setting , For any , we can also easy to have

Then, by fixed point theorem of cone expansion-compression type, we have

Therefore, by (57), (60), (62), , we have

Then, have fixed point , and fixed point .

Obviously, are all positive solutions of problem (1),(2) and . The proof of Theorem 10 is complete.

4. Application

Example. Consider the following three-order BVP with -Laplacian where , Then obviously, , so conditions (), , , (), hold.
Next, , we choose and for , because of the monotone increasing of on , then Therefore, using , we have , we know so conditions holds. Then, by Theorem 9, the Example has at least two positive solutions and .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Authors’ Contributions

The study was carried out in collaboration among all authors. All authors read and approved the final manuscript.

Acknowledgments

The authors really appreciate the anonymous reviewers for their pertinent comments and suggestions, which were helpful to improve the earlier manuscript. The author was supported by the Project of National Social Science Fund of China (NSSF) (18BTY015) and Shandong Province Higher Educational Science and Technology Program (J16LI01).