, 
,
subject to the right focal type two-point boundary conditions
, 
,
.
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
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
and
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)
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
Theorem 2.2. Green’s function for the homogeneous BVP
with the boundary conditions (2), (3) is given by
where
for all 
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
For each fixed 
let 
be the unique solution of the BVP
and
Since
are the solutions of 
By using boundary conditions, 
, 
, we have
. Therefore
Since,
It follows that
Hence 
has the form for 
And for
,
. It follows that
where
Lemma 2.3. For
, we have
(5)
Proof: For
, we have
Similarly, for 
we have 
Thus, we have
for all 
Lemma 2.4. Let
. For
, we have
(6)
Proof: The Green’s function 
for the homogeneous BVP corresponding to (1)-(3) is positive on 
For 
and
, we have
Similarly, for 
and 
we have
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 
Define 
with the norm
Then 
is a Banach space. Define a set 
by
(8)
We define the operator 
by
(9)
for all 
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 
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
so that
Next, if
, then
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,
Therefore, 
Hence, if we set
Then
(10)
Since
, there exist 
and 
such that
, for 
and 
If we set
and define
If
, so that
, then
And we have
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). 
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
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
Then, for
, we have
and so
(13)
Case (ii) f is unbounded. Let 
be such that 
for
. Let
Choosing
,
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). 
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
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
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 
2) 
for 
3) 
for 
with 
Then 
has at least three fixed points
, 
, 
in 
satisfying
For convenience, we let
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
We denote
Then, it is obvious that P is a cone in E. For
, we define
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
.
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
Moreover, for
, we have
for all
. Thus, in view of (16) we see that
as required. Finally, we show that if 
and
, then
. To see this, we suppose that 
and
, then, by Lemma 2.4, we have
for all
. Thus
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
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
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
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
The Green’s function for the homogeneous boundary value problem is given by
A simple calculation shows that
,
and
. If we choose
, 
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.