COUNTABLY INFINITELY MANY POSITIVE SOLUTIONS FOR EVEN ORDER BOUNDARY VALUE PROBLEMS WITH STURM-LIOUVILLE TYPE INTEGRAL BOUNDARY CONDITIONS ON TIME SCALES

In this paper, we establish the existence of countably infinitely many positive solutions for a certain even order two-point boundary value problem with integral boundary conditions on time scales by using Hölder’s inequality and Krasnoselskii’s fixed point theorem for operators on a cone.


Introduction
The study of dynamic equations on time scales unifies existing results in differential and finite difference equations, and provides powerful new tools for exploring connections between the traditionally separated fields. For details refer to the books by Bohner and Peterson [6], [7], Lakshmikantham et al. [23] and the papers [1], [3], [19].
However, to the best of our knowledge, little work has been done on the existence of positive solutions for higher order boundary value problems with integral boundary conditions on time scales. We would like to mention some results of Karasa and Tokmak [20], Li and Wang [25], Li and Sun [24], Cetin and Topal [8], and Sreedhar et al [31] which motivate us to consider the problem (1.6)-(1.7). In 2013, Karasa and Tokmak [20] established the existence of a positive solution of the following third order boundary value problem with integral boundary conditions, φ(−u ∆∆ (t)) ∆ + q(t)f (t, u(t), u ∆ (t)) = 0, t ∈ [0, 1] T , au(0) − bu ∆ (0) = by applying a generalization of the Leggett-Williams fixed point theorem. In [24], Li and Sun studied the following boundary value problem on time scales, by using Schauder fixed point theorem in a cone and by the method of upper and lower solutions. In 2017, Sreedhar et al [31] considered the 2n th order boundary value problem with integral boundary conditions on time scales, where n act as positive. By using Avery-Henderson fixed point theorem, the authors established the existence of even number of positive solutions for (1.5). Motivated by the work mentioned above, in this paper we investigate the existence of infinitely many positive solutions for the even order boundary value problem on time scales given by satisfying the Sturm-Liouville type integral boundary conditions where n ≥ 1, T is a time scale, f ∈ C [0, +∞), [0, +∞) , ω(t) ∈ L p ∇ [0, 1] for some p ≥ 1 and has countably many singularities in (0, 1 2 ) T . We show that the boundary value problem (1.6)-(1.7) has countably infinitely many positive solutions by imposing suitable conditions on ω and f . The key tool in our approach is the Hölder's inequality and Krasnoselskii's fixed point theorem for operators on a cone.
Definition 2.1. A time scale T is a nonempty closed subset of the real numbers R. T has the topology that it inherits from the real numbers with the standard topology. It follows that the jump operators σ, ρ : T → T, σ(t) = inf{r ∈ T : r > t}, ρ(t) = sup{r ∈ T : r < t} (supplemented by inf ∅ := sup T and sup ∅ := inf T) are well defined. The point t ∈ T is left-dense, left-scattered, right-dense, right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t, σ(t) < t, respectively.
By an interval time scale, we mean the intersection of a real interval with a given time scale. i.e., Definition 2.2. Let µ ∆ and µ ∇ be the Lebesgue ∆− measure and the Lebesgue ∇−measure on T, respectively. If A ⊂ T satisfies µ ∆ (A) = µ ∇ (A), then we call A is measurable on T, denoted µ(A) and this value is called the Lebesgue measure of A. Let P denote a proposition with respect to t ∈ T.
(i) If there exists E 1 ⊂ A with µ ∆ (E 1 ) = 0 such that P holds on A\E 1 , then P is said to hold ∆-a.e. on A. (ii) If there exists E 2 ⊂ A with µ ∇ (E 2 ) = 0 such that P holds on A\E 2 , then P is said to hold ∇-a.e. on A.
We make the following assumptions throughout the paper: The rest of the paper is organized in the following fashion. In Section 2, we provide some definitions and lemmas that provide us with some useful information concerning the behavior of solution of the boundary value problem (1.6)-(1.7). In Section 3, we construct the Green's function for the homogeneous problem corresponding to (1.6)-(1.7), estimate bounds for the Green's function, and some lemmas which are needed in establishing our main results are provided. In Section 4, we establish a criteria for the existence of countable number of positive solutions for the boundary value problem (1.6)-(1.7) by applying Krasnoselskii's fixed point theorem in cones. Finally, we provide an example of a family of functions ω(t) that satisfy conditions (H1) − (H3).

Green's Function and Bounds
In this section, we construct the Green's function for the homogeneous problem corresponding to (1.6)-(1.7) and estimate bounds for the Green's function.
Lemma 3.1. Let (H4), (H5) hold. Then for any h(t) ∈ C(J 0 ), the boundary value problem, has a unique solution where, Proof. Suppose that u is a solution of (3.1), then, we have Thus, we have from which, we obtain After certain computations we can determined, PRASAD AND KHUDDUSH From (3.7) and (3.8), (3.6) can be written as The proof is complete.
Lemma 3.2. Assume that (H4), (H5) hold and for τ ∈ (0, 1 Then G i (t, s) for 1 ≤ i ≤ n, satisfies the following properties: Assume that (H4), (H5) holds. Then K i (t, s) for 1 ≤ i ≤ n, have the following properties: Lemma 3.4. Assume that (H4), (H5) hold and K j (t, s) for 1 ≤ j ≤ n, is given in (3.5). Let H 1 (t, s) = K 1 (t, s) and recursively define Then H n (t, s) is the Green's function for the homogeneous boundary value problem then the Green's function H n (t, s) satisfies the following inequalities: Proof. It is clear that Green's function H n (t, s) ≥ 0, for all t, s ∈ J 0 . Now we prove the inequality by induction on n and denote the statement by p(n). From (3.5) we have So, p(m + 1) holds. Thus p(n) is true by induction where ξ τ = ητ gτ gφ . For any u ∈ P τ , define an operator T : P τ → X by (T u)(t) = Lemma 3.6. Assume that (H1)-(H3) hold. Then T (P τ ) ⊂ P τ and T : P τ → P τ is completely continuous for each τ ∈ (0, 1 2 ) T .
Proof. Fix τ ∈ (0, 1 2 ). Since ω(s)f (u(s)) ≥ 0 for all s ∈ J 0 , u ∈ P τ and since H n (t, s) ≥ 0 for all t, s ∈ J 0 , then T (u(t)) ≥ 0 for all t ∈ J 0 , u ∈ P τ . On the other hand, by Lemma 3.5 we obtain (T u)(t) = T H n (t, s)ω(s)f u(s) ∇s for all t ∈ J 0 . Thus min T u(t) ≥ ξ τ T u . So, T u ∈ P τ and then T (P τ ) ⊂ P τ . Next, by standard methods and the Arzela-Ascoli theorem, one can easily prove that the operator T is completely continuous. The proof is complete.

Main Results
In this section, we establish the existence of countably infinitely many positive solutions for the boundary value problem (1.6)-(1.7) by applying Krasnoselskii's fixed point theorem in cones.
Assume that f satisfies Then the boundary value problem (1.6)-(1.7) has countably infinitely many positive solutions {u k } ∞ k=1 .
Furthermore, r k ≤ u k ≤ R k for each k ∈ N.
Now we deal with the case p = 1.
and (A2). Then the boundary value problem (1.6)-(1.7) has countably infinitely many positive solutions Proof. For a fixed k, let Ω 1,k be as in the proof of Theorem 4.3 and let u ∈ P τ k ∩ ∂Ω 2,k . Again for all s ∈ J 0 . By (B1) and Theorem 4.3,
Then, the argument leading to (4.2) carries over to the present case and completes the proof.
Finally we consider the case of p = ∞.
By the Theorem 4.1, completes the proof.