Existence of Multiple Positive Solutions for n th Order Two-Point Boundary Value Problems on Time Scales

We consider the order nonlinear differential equation on time scales th n       , 0 y t f t y t   n   ,  , t a b    , subject to the right focal type two-point boundary conditions 0 y a i   , 0 2 i n        0 n p y b    p  ,   1, but fixed 

, subject to the right focal type two-point boundary conditions

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 subject to the right focal type boundary conditions 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,  is not left dense and right scattered at the same time.
Define the nonnegative extended real numbers , 0 f , 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.

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.

 
, if we assume that the conditions (A2) and (A3) are satisfied, then where is the set of all distinct combinations of such that the sum is equal to given .
Proof see [19].We denote , , , , , , 0 1 let u be the unique solution of the BVP Hence has the form for Proof: For , we have a t Similarly, for we have Thus, we have . For , we have Proof: The Green's function for the homogeneous BVP corresponding to (1)-( 3) is positive on

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 We define the operator by are open subsets of  with and , and 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.: Hence, .Standard argument involving the Arzela-Ascoli theorem shows that T is completely continuous operator.Since , there exist 0 And we have Proof: Let T be the cone preserving, completely continuous operator defined as in (9).Since 0 f   , there exist 1 0 , and It remains for us to consider , in this case, there exist 2 0   and 2 There are two subcases.
Then, for , we have

Ty t G t s f s y s s G b s f s y s s
An application of Theorem 3.1, to ( 12), ( 13) and ( 14) yields a fixed point of T that lies in . This fixed point is a solution of the BVP (1)-(3).

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   a b Let and x y P for all  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 S y y  for all  , and c y P .Suppose that there exist a , , For convenience, we let , and Assume that there exist real numbers , , and c with     , then : Moreover, for t I for all  .Thus, in view of (16) we see that

S Ty G t s f s y s s G b s f s y s s G t s f s y s s
. Thus 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.  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.

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.


be the solution of the BVP (1)-(3), and is given by

3 y
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 solu- 2 hold.It follows from Theorem 3.2, the BVP (18) has at least one positive solution.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.