Even Number of Positive Solutions for 3n th Order Three-Point Boundary Value Problems on Time Scales

We establish the existence of at least two positive solutions for the 3n th order three-point boundary value problem on time scales by using Avery-Henderson fixed point theorem. We also establish the existence of at least 2m positive solutions for an arbitrary positive integer m.


Introduction
The theory of time scales was introduced and developed by Hilger [13] to unify both continuous and discrete analysis.Time scales theory presents us with the tools necessary to understand and explain the mathematical structure underpinning the theories of discrete and continuous dynamic systems and allows us to connect them.The theory is widely applied to various situations like epidemic models, the stock market and mathematical modeling of physical and biological systems.Certain economically important phenomena contain processes that feature elements of both the continuous and discrete.
In recent years, the existence of positive solutions of the higher order boundary value problems (BVPs) on time scales have been studied extensively due to their striking applications to almost all area of science, engineering and technology.The existence of positive solutions are studied by many authors.A few papers along these lines are Henderson [11], Anderson [1,2], Kaufmann EJQTDE, 2011 No. 98, p. 1 [15], Anderson and Avery [3], DaCunha, Davis and Singh [10], Peterson, Raffoul and Tisdell [18], Sun and Li [19], Luo and Ma [17], Cetin and Topal [8], Karaca [14] and Anderson and Karaca [4].

Green's Function and Bounds
In this section, we construct the Green's function for the homogeneous problem corresponding to (1.1)-(1.2) and estimate bounds for the Green's function.
Let G i (t, s) be the Green's function for the homogeneous BVP, satisfying the general three-point boundary conditions, where Lemma 2.2 Assume that the conditions (A1)-(A4) are satisfied.Then, for 3).We prove the result for G i 1 (t, s).Then, G i 1 (t, s) = g i 1 (t)l i 1 (s), where Using the conditions (A1) and (A4), g i 1 (t) has maximum at t = m i 23 , and hence g i 1 (t) > 0 on [t 1 , σ(t 3 )] by conditions (A2) and (A3).From conditions (A2) and (A4), Similarly, we can establish the positivity of the Green's function in the remaining cases.
(2.4) where In each case, we prove the inequality as in (2.4). .
Lemma 2.5 Assume that the conditions (A1)-(A4) hold.If we define then the Green's function H n (t, s) in Lemma 2.4 satisfies and where m n is given as in Theorem 2.3, EJQTDE, 2011 No. 98, p. 7 In this section, we establish the existence of at least two positive solutions for the BVP (1.1)-(1.2) by using an Avery-Henderson functional fixed point theorem.And then, we establish the existence of at least 2m positive solutions for an arbitrary positive integer m.
Let B be a real Banach space.A nonempty closed convex set P ⊂ B is called a cone, if it satisfies the following two conditions: (i) y ∈ P, λ ≥ 0 implies λy ∈ P , and (ii) y ∈ P and −y ∈ P implies y = 0.
Let ψ be a nonnegative continuous functional on a cone P of the real Banach space B. Then for a positive real number c ′ , we define the sets P (ψ, c ′ ) = {y ∈ P : ψ(y) < c ′ } and P a = {y ∈ P : y < a}.
In obtaining multiple positive solutions of the BVP (1.1)-(1.2), the following Avery-Henderson functional fixed point theorem will be the fundamental tool.
where M is given as in (3.1).Define the nonnegative, increasing, continuous functionals γ, θ and α on the cone P by We observe that for any y ∈ P , , where m n and M are defined in Theorem 2.3 and (3.1) respectively.Then the BVP (1.1)-(1.2) has at least two positive solutions y 1 and y 2 such that It is obvious that a fixed point of T is the solution of the BVP (1.1)-(1.2).We seek two fixed points y 1 , y 2 ∈ P of T .First, we show that T : P → P .Let EJQTDE, 2011 No. 98, p. 9 y ∈ P .From Theorem 2.3 and Lemma 2.5, we have T y(t) ≥ 0 on [t 1 , σ(t 3 )] and also, Next, if y ∈ P , then we have Hence T y ∈ P and so T : P → P .Moreover, T is completely continuous.From (3.2) and (3.3), for each y ∈ P , we have γ(y) ≤ θ(y) ≤ α(y) and y ≤ 1 M γ(y).Also, for any 0 ≤ λ ≤ 1 and y ∈ P , we have θ(λy) = max t∈[t 2 ,σ(t 3 )] (λy)(t) = λ max t∈[t 2 ,σ(t 3 )] y(t) = λθ(y).It is clear that θ(0) = 0. We now show that the remaining conditions of Theorem 3.1 are satisfied.Firstly, we shall verify that condition (B1) of Theorem 3.1 is satisfied.Since y ∈ ∂P (γ, c ′ ), from (3.3) we have that using hypothesis (D1).Now we shall show that condition (B2) of Theorem 3.1 is satisfied.Since EJQTDE, 2011 No. 98, p. 10 Thus by hypothesis (D2).Finally, using hypothesis (D3), we shall show that condition (B3) of Theorem 3.1 is satisfied.Since 0 ∈ P and a ′ > 0, Thus, all the conditions of Theorem 3.1 are satisfied and so there exist at least two positive solutions y 1 , y 2 ∈ P (γ, c ′ ) for the BVP ( . This completes the proof of the theorem. 2 Theorem 3.3 Let m be an arbitrary positive integer.Assume that there exist numbers a r (r = 1, 2,  subject to the boundary conditions,