ABSTRACT

We consider the order nonlinear differential equation on time scales

, ,

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.

REFERENCES