,
,
subject to the right focal type two-point boundary conditions
, ![](https://www.scirp.org/html/9-5300273\20c90c8e-aa60-4961-b5bb-e3e5f2e6bb44.jpg)
,
.
We establish a criterion for the existence of at least one positive solution by utilizing Krasnosel’skii fixed point theorem. And then, we establish the existence of at least three positive solutions by utilizing Leggett-Williams fixed point theorem.
1. Introduction
The study of the existence of positive solutions of boundary value problems (BVPs) for higher order differential equations on time scales has gained prominence and it is a rapidly growing field, since it arises, especially for higher order differential equations on time scales arise naturally in technical applications. Meyer [1], strictly speaking, boundary value problems for higher order differential equation on time scales are a particular class of interface problems. One example in which this is exhibited is given by Keener [2] in determining the speed of a flagellate protozoan in a viscous fluid. Another particular case of a boundary value problem for a higher order differential equation on time scales arising as an interface problem is given by Wayner, et al. [3] in dealing with a study of perfectly wetting liquids. In these applied settings, only positive solutions are meaningful. By a time scale we mean a nonempty closed subset of
. For the time scale calculus and notation for delta differentiation, integration, as well as concepts for dynamic equation on time scales we refer to the introductory book on time scales by Bohner and Peterson [4], and denote the time scales by the symbol
.
By an interval we mean the intersection of the real interval with a given time scale. The existence of positive solutions for BVPs has been studied by many authors, first for differential equations, then finite difference equations, and recently, unifying results for dynamic equations. We list some papers, Erbe and Wang [5], and Eloe and Henderson [6,7], Atici and Guseinor [8], and Anderson and Avery [9], and Avery and Peterson [10], Agarwal, Regan and Wang [11], Deimling [12], Gregus [13] Guo and Lakshmikantham [14], Henderson and Ntouyas [15], Hopkins [16] and Li [17]. Recently, in 2008, Moustafa Shehed [18] obtained at least one positive solution to the boundary value problem
![](https://www.scirp.org/html/9-5300273\73ff44f4-c6b8-4bbb-aff6-2054abe76391.jpg)
This paper considers the existence of positive solutions to
order nonlinear differential equation on time scales
(1)
subject to the right focal type boundary conditions
(2)
(3)
These boundary conditions include different types of right focal boundary conditions.
We make the following assumptions throughout:
(A1)
is continuous with respect to
where
is nonnegative real numbers(A2) The point t in
is not left dense and right scattered at the same time.
Define the nonnegative extended real numbers
,
,
and
by
![](https://www.scirp.org/html/9-5300273\d8db2296-6b93-4189-9403-68bf4aef82fc.jpg)
![](https://www.scirp.org/html/9-5300273\68166f80-10e4-485f-a3a4-f58a0da1f685.jpg)
![](https://www.scirp.org/html/9-5300273\866974dc-e86f-4d8c-92e3-e35c4e35ce03.jpg)
and
![](https://www.scirp.org/html/9-5300273\a583594d-77c8-4987-8bef-793189317807.jpg)
This paper is organized as follows; In Section 2, we estimate the bounds for the Greens function which are needed for later discussions. In Section 3, we establish a criteria for the existence of at least one positive solution for the BVP by using Krasnosel’skii fixed point theorem. In Section 4, we establish the existence of at least three positive solutions for the BVP by using Leggett-Williams fixed point theorem. Finally, as an application, we give some examples to demonstrate our result.
2. Green’s Function and Bounds
In this section, first we state a Lemma to compute delta derivatives for
, next, construct a Green’s function for homogeneous two point BVP
with (2), (3) and estimate the bounds to the Green’s function.
Lemma 2.1. Let
, define a function
by
, if we assume that the conditions (A2) and (A3) are satisfied, then
(4)
![](https://www.scirp.org/html/9-5300273\541a22dd-7422-41ad-9480-0ce1a2b77b5e.jpg)
holds for all
where
is the set of all distinct combinations of
such that the sum is equal to given
.
Proof see [19].
We denote
![](https://www.scirp.org/html/9-5300273\80fdeb0e-d5c1-4a6d-8814-b64c60527c12.jpg)
![](https://www.scirp.org/html/9-5300273\4c3437cb-a80e-41de-b362-ea9a9db5732f.jpg)
Theorem 2.2. Green’s function for the homogeneous BVP
![](https://www.scirp.org/html/9-5300273\4fecf3a4-ee6f-48cd-942c-540f1374bcb9.jpg)
with the boundary conditions (2), (3) is given by
![](https://www.scirp.org/html/9-5300273\78462cb7-a3b0-491e-9221-b77e7eec0fc0.jpg)
where
![](https://www.scirp.org/html/9-5300273\381ae5b2-4e0d-4a67-9145-d132327272c1.jpg)
for all ![](https://www.scirp.org/html/9-5300273\ca0765aa-77cb-459a-b8ba-eb7834e47283.jpg)
Proof: It is easy to check that the BVP
with the boundary conditions (2) and (3) has only trivial solution. Let
be the Cauchy function for
, and is given by
![](https://www.scirp.org/html/9-5300273\c63729da-0d22-43de-947d-c46e017d904d.jpg)
For each fixed
let
be the unique solution of the BVP
![](https://www.scirp.org/html/9-5300273\48e14a2f-51a1-4cd5-9706-d822aecff205.jpg)
and
![](https://www.scirp.org/html/9-5300273\3eec03d7-ac41-4113-9778-147e8beb6d3d.jpg)
![](https://www.scirp.org/html/9-5300273\e63ecee6-d611-44ac-9fac-0da2382d9449.jpg)
Since
![](https://www.scirp.org/html/9-5300273\7ae41970-3507-476c-b619-5c168936e944.jpg)
are the solutions of ![](https://www.scirp.org/html/9-5300273\81a86f3a-e16c-46ed-9ec3-50825fc3603e.jpg)
![](https://www.scirp.org/html/9-5300273\84f09d3e-820a-4718-843e-e590c032f824.jpg)
By using boundary conditions,
,
, we have
. Therefore
![](https://www.scirp.org/html/9-5300273\77eb0584-f70b-44c2-9d5e-c9531d80c7b2.jpg)
Since,
![](https://www.scirp.org/html/9-5300273\8b5e1c66-6aa9-4dcc-815f-4399e667dfce.jpg)
It follows that
![](https://www.scirp.org/html/9-5300273\b09b948f-6df0-4bbd-9e45-4e56cfb4ca60.jpg)
Hence
has the form for ![](https://www.scirp.org/html/9-5300273\eaa40fa2-ad1e-48be-9e61-d4b28d97fd80.jpg)
![](https://www.scirp.org/html/9-5300273\13890685-5af8-41ab-b62b-3c1f5eba5594.jpg)
And for
,
. It follows that
![](https://www.scirp.org/html/9-5300273\a8debc3b-23a7-46a1-804c-ca71ade5c266.jpg)
where
![](https://www.scirp.org/html/9-5300273\e4b3a1bf-87a7-4d8e-ae9d-5231d0adc06f.jpg)
Lemma 2.3. For
, we have
(5)
Proof: For
, we have
![](https://www.scirp.org/html/9-5300273\630b852f-0521-4d69-9e81-dc48d287d426.jpg)
Similarly, for
we have
Thus, we have
![](https://www.scirp.org/html/9-5300273\73143d88-a867-4e29-afdf-6ba8c21d7668.jpg)
for all
![](https://www.scirp.org/html/9-5300273\ca8dfd5c-61f4-424e-885d-5d5385fce8b9.jpg)
Lemma 2.4. Let
. For
, we have
(6)
Proof: The Green’s function
for the homogeneous BVP corresponding to (1)-(3) is positive on ![](https://www.scirp.org/html/9-5300273\9a98f0c6-c779-4b47-993d-a409e9f6860b.jpg)
For
and
, we have
![](https://www.scirp.org/html/9-5300273\68c8a8b3-9f6a-4b97-b601-7c80f6373772.jpg)
Similarly, for
and
we have
![](https://www.scirp.org/html/9-5300273\0361ea51-3d37-4316-a33f-a78463c294bc.jpg)
![](https://www.scirp.org/html/9-5300273\19a4a6be-3acc-45bc-9a70-6a38ba5ae4bb.jpg)
3. Existence of at Least One Positive Solution
In this section, we establish a criteria for the existence of at least one positive solution of the BVP (1)-(3). Let
be the solution of the BVP (1)-(3), and is given by
(7)
for all ![](https://www.scirp.org/html/9-5300273\efb8a22c-08aa-41a1-a0a2-bf5c7c029028.jpg)
Define
with the norm
![](https://www.scirp.org/html/9-5300273\d0838847-efd2-4951-a5dd-73a8c7fa211d.jpg)
Then
is a Banach space. Define a set
by
(8)
We define the operator
by
(9)
for all ![](https://www.scirp.org/html/9-5300273\e2f44845-6210-446f-befd-3800bb751ba4.jpg)
Theorem 3.1. (Krasnosel’skii) Let
be a Banach space,
be a cone, and suppose that
,
are open subsets of
with
and
. Suppose further that
is completely continuous operator such that either 1)
,
and
,
, or 2)
,
and
,
holds. Then T has a fixed point in ![](https://www.scirp.org/html/9-5300273\bb4bffe1-fe15-4ff6-a798-92911269d725.jpg)
Theorem 3.2. If
and
, then the BVP (1)-(3) has at least one positive solution that lies in
.
Proof: We seek a fixed point of T in
. We prove this by showing the conditions in Theorem 3.1 hold.
First, if
, then
![](https://www.scirp.org/html/9-5300273\15e44b27-4881-49d9-ab78-12ac16561d4c.jpg)
so that
![](https://www.scirp.org/html/9-5300273\391b0c50-41f8-4507-a484-60f894ad74a0.jpg)
Next, if
, then
![](https://www.scirp.org/html/9-5300273\a46dccf9-4d1f-40f8-b6ac-d5a714c20f4a.jpg)
Hence,
. Standard argument involving the Arzela-Ascoli theorem shows that T is completely continuous operator. Since
, there exist
and
such that
for
, and
Let us choose
with
. Then, we have from Lemma 2.3,
![](https://www.scirp.org/html/9-5300273\772bd5f5-5242-4e56-842b-478e47759a00.jpg)
Therefore,
Hence, if we set
![](https://www.scirp.org/html/9-5300273\2689fddc-585e-4364-834f-b13fd2b5717b.jpg)
Then
(10)
Since
, there exist
and
such that
, for
and
If we set
![](https://www.scirp.org/html/9-5300273\04cb0f2e-ea81-4375-befc-5272e9207f34.jpg)
and define
![](https://www.scirp.org/html/9-5300273\21dc09bc-edb5-4f59-b6a7-4da90ab300e1.jpg)
If
, so that
, then
![](https://www.scirp.org/html/9-5300273\839e902a-9f9c-4a42-80d6-b251560a9da2.jpg)
And we have
![](https://www.scirp.org/html/9-5300273\1b1360a5-065a-4284-be2a-a215eeddb352.jpg)
Thus,
, and so
(11)
An application of Theorem 3.1 to (10) and (11) yields a fixed point of
that lies in
. This fixed point is a solution of the BVP (1)-(3). ![](https://www.scirp.org/html/9-5300273\d9a19902-b60d-4fed-94af-19cf0d153bf9.jpg)
Theorem 3.3. If
and
, then the BVP (1)-(3) has at least one positive solution that lies in
.
Proof: Let T be the cone preserving, completely continuous operator defined as in (9). Since
, there exist
and
such that
for
, and
In this case, define
Then, for
we have
and moreover,
,
. Thus
![](https://www.scirp.org/html/9-5300273\603da86a-ec3b-45e0-b8d2-8980851d280b.jpg)
From which we have
(12)
It remains for us to consider
, in this case, there exist
and
such that
, for
, and
There are two subcases.
Case (i)
is bounded. Suppose
is such that
, for all
.
Let
and let
![](https://www.scirp.org/html/9-5300273\77c997f1-0ad8-4986-9e2d-d42ee8fec6e1.jpg)
Then, for
, we have
![](https://www.scirp.org/html/9-5300273\c795a298-da29-42ef-bb82-0a5e37c2bcea.jpg)
and so
(13)
Case (ii) f is unbounded. Let
be such that
for
. Let
![](https://www.scirp.org/html/9-5300273\6423afaa-4ca2-4564-8f05-05571c434c81.jpg)
Choosing
,
![](https://www.scirp.org/html/9-5300273\8349dec7-c9a3-4865-9134-8c088c0b334c.jpg)
And so
(14)
An application of Theorem 3.1, to (12), (13) and (14) yields a fixed point of
that lies in
. This fixed point is a solution of the BVP (1)-(3). ![](https://www.scirp.org/html/9-5300273\f4b386f6-0078-4dcf-ba95-bb48effd525b.jpg)
4. Existence of Multiple Positive Solutions
In this section, we establish the existence of at least three positive solutions to the BVP (1)-(3).
Let
be a real Banach space with cone
. A map
is said to be a nonnegative continuous concave functional on
, if
is continuous and
![](https://www.scirp.org/html/9-5300273\21f84182-65b0-4300-9763-2e4ec95bd99c.jpg)
for all
and
Let
and
be two real numbers such that
and
be a nonnegative continuous concave functional on
. We define the following convex sets
![](https://www.scirp.org/html/9-5300273\9ef9e1c4-5209-475b-9ba8-dd06f82d51e9.jpg)
![](https://www.scirp.org/html/9-5300273\fff18256-56fc-472d-a9d3-07806454f5ad.jpg)
We now state the famous Leggett-Williams fixed point theorem.
Theorem 4.1. See ref. [20] Let
be completely continuous and S be a nonnegative continuous concave functional on P such that
for all
. Suppose that there exist
,
,
, and
with
such that 1)
and
for ![](https://www.scirp.org/html/9-5300273\fe6f0e1a-e7e1-4e32-930b-47e4a446f875.jpg)
2)
for ![](https://www.scirp.org/html/9-5300273\bfcd5d75-3abc-4847-ac39-d8ab341eb5ec.jpg)
3)
for
with ![](https://www.scirp.org/html/9-5300273\37ecd739-d3cb-4950-92dd-9c8d0446e50e.jpg)
Then
has at least three fixed points
,
,
in
satisfying
![](https://www.scirp.org/html/9-5300273\6ce35e5a-9855-45aa-b84b-273d384cda9a.jpg)
For convenience, we let
![](https://www.scirp.org/html/9-5300273\5d4c5c94-7dfd-4713-80ac-b29b45a0f78c.jpg)
![](https://www.scirp.org/html/9-5300273\ee688d7e-25ff-4990-a17d-f5b7305a1845.jpg)
Theorem 4.2. Assume that there exist real numbers
,
, and c with
such that
(15)
(16)
(17)
Then the BVP (1)-(3) has at least three positive solutions.
Proof: Let the Banach space
be equipped with the norm
![](https://www.scirp.org/html/9-5300273\250cffc0-c563-4e78-ab41-923f612f917b.jpg)
We denote
![](https://www.scirp.org/html/9-5300273\9eec5906-0e88-4edc-ba04-59409c72aeca.jpg)
Then, it is obvious that P is a cone in E. For
, we define
![](https://www.scirp.org/html/9-5300273\64948ba6-9666-47b3-810e-760cef3a784b.jpg)
It is easy to check that
is a nonnegative continuous concave functional on
with
for
and that
is completely continuous and fixed points of
are solutions of the BVP (1)-(3). First, we prove that if there exists a positive number
such that
for
, then
. Indeed, if
, then for
.
![](https://www.scirp.org/html/9-5300273\1ff15a25-7e1d-482a-9ab7-febf4ce2b346.jpg)
Thus,
, that is,
Hence, we have shown that if (15) and (17) hold, then
maps
into
and
into
. Next, we show that
and
for all
. In fact, the constant function
![](https://www.scirp.org/html/9-5300273\31cbbc9f-c98f-4b0e-b353-6d0868fca605.jpg)
Moreover, for
, we have
![](https://www.scirp.org/html/9-5300273\b2a61349-b407-4b16-b265-12af28d93fb0.jpg)
for all
. Thus, in view of (16) we see that
![](https://www.scirp.org/html/9-5300273\fdc77308-8d9e-48fd-a0ab-c88846dcd266.jpg)
as required. Finally, we show that if
and
, then
. To see this, we suppose that
and
, then, by Lemma 2.4, we have
![](https://www.scirp.org/html/9-5300273\584337a3-a607-471b-8b2e-e3772abfa43b.jpg)
for all
. Thus
![](https://www.scirp.org/html/9-5300273\a6e606da-7582-4598-92a8-9789dc951d33.jpg)
To sum up the above, all the hypotheses of Theorem 4.1 are satisfied. Hence
has at least three fixed points, that is, the BVP (1)-(3) has at least three positive solutions
,
and
such that
![](https://www.scirp.org/html/9-5300273\545ed30a-564f-4334-bed3-2d783660216b.jpg)
5. Examples
Now, we give some examples to illustrate the main result.
Example 1
Consider the following boundary value problem
(18)
The Green’s function for the homogeneous boundary value problem is given by
![](https://www.scirp.org/html/9-5300273\0857da03-3a5f-42bd-a34c-97f8d06b3718.jpg)
It is easy to see that all the conditions of Theorem 3.2 hold. It follows from Theorem 3.2, the BVP (18) has at least one positive solution.
Example 2
Consider the following boundary value problem
(19)
The Green’s function for the homogeneous boundary value problem is given by
![](https://www.scirp.org/html/9-5300273\4a54c5b9-49d6-4962-a181-c18d6b7066f1.jpg)
It is easy to see that all the conditions of Theorem 3.3 hold. It follows from Theorem 3.3, the BVP (19) has at least one positive solution.
Example 3
Consider the following boundary value problem on time scale
(20)
where
![](https://www.scirp.org/html/9-5300273\dd5e9943-3d03-4b6d-a871-d43049e037cc.jpg)
The Green’s function for the homogeneous boundary value problem is given by
![](https://www.scirp.org/html/9-5300273\802047cb-d4cf-4583-ac3c-c4f08fa33a86.jpg)
A simple calculation shows that
,
and
. If we choose
, ![](https://www.scirp.org/html/9-5300273\ae67a557-6186-4be7-976b-b4a227ab774c.jpg)
and
then, we see that all the conditions of Theorem 4.2 hold. It follows from Theorem 4.2, the BVP (20) has at least three positive solutions.
6. Conclusion
In this paper, we have established the existence of positive solutions for higher order boundary value problems on time scales which unifies the results on continuous intervals and discrete intervals, by using Leggett-Williams fixed point theorem. These results are rapidly arising in the field of modelling and determination of flagellate protozoan in a viscous fluid in further research.