Multiple Positive Solutions of Nonlinear Singular M-point Boundary Value Problem for Second-order Dynamic Equations with Sign Changing Coefficients on Time Scales

Let T be a time scale. In this paper, we study the existence of multiple positive solutions for the following nonlinear singular m-point boundary value problem dynamic equations with sign changing coefficients on time scales          u △∇ (t) + a(t)f (u(t)) = 0, (0, T) T , u △ (0) = m−2 i=1 a i u △ (ξ i), u(T) = k i=1 b i u(ξ i) − s i=k+1 b i u(ξ i) − m−2 i=s+1 b i u △ (ξ i), where 1 ≤ k ≤ s ≤ m − 2, a i , b i ∈ (0, +∞) with 0 < k i=1 b i − s i=k+1 b i < 1, 0 < m−2 i=1 a i < 1, 0 < ξ 1 < ξ 2 < · · · < ξ m−2 < ρ(T), f ∈ C([0, +∞), [0, +∞)), a(t) may be singular at t = 0. We show that there exist two positive solutions by using two different fixed point theorems respectively. As an application, some examples are included to illustrate the main results. In particular, our criteria extend and improve some known results.


Introduction
A time scale T is a nonempty closed subset of R. We make the blanket assumption that 0, T are points in T. By an interval (0, T ) T , we always mean the intersection of the real interval (0, T ) T with the given time scale, that is (0, T ) ∩ T. The theory of dynamical systems on time scales is undergoing rapid development as it provides a unifying structure for the study of differential equations in the continuous case and the study of finite difference equations in the discrete case.Here, two-point boundary value problems have been extensively studied; for details, see [3,4,5,7] and the references therein.Since then, more general nonlinear multi-point boundary value problems have been studied by several authors.We refer the reader to [6,8,9,14,16,[18][19][20][21] for some references along this line.Multi-point boundary value problems describe many phenomena in the applied mathematical sciences.
The main tools are the Krasnoselskii's fixed point theorem and Leggett-Williams fixed point theorem.
In 2001, Ma [6] studied m-point boundary value problem (BVP) where and Author established the existence of positive solutions theorems under the condition that f is either superlinear or sublinear.
In [8], Ma and Castaneda considered the following m-point boundary value problem (BVP) where They showed the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones.Motivated by the results mentioned above, in this paper we study the existence of positive solutions for the following nonlinear singular m-point boundary value problem dynamic equation with sign changing coefficients on time scales where For convenience, we list here the following definitions which are needed later.A time scale [0, T ] T is an arbitrary nonempty closed subset of real numbers R. The operators σ and ρ from [0, T ] T to [0, T ] T which defined by [1][2][3], are called the forward jump operator and the backward jump operator, respectively.
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.If T has a right scattered minimum m, define Let f : T → R and t ∈ T k (assume t is not left-scattered if t = sup T), then the delta derivative of f at the point t is defined by Similarly, for t ∈ T (assume t is not right-scattered if t = inf T), the nabla derivative of f at the point t is defined by A function f is left-dense continuous (i.e., ld-continuous), if f is continuous at each left-dense point in T and its right-sided limit exists at each right-dense point in T. It is well-known that if f is ld-continuous.
If F ∇ (t) = f (t), then we define the nabla integral by Throughout this article, T is closed subset of R with 0 ∈ T k , T ∈ T k .By a positive solution of BVP (1.1), we understand a function u which is positive on (0, T ) T and satisfies the differential equations as well as the boundary conditions in BVP (1.1).

Preliminaries and lemmas
In this section, we give some definitions and preliminaries that are important to our main results.Definition 2.1.Let E be a real Banach space over R .A nonempty closed set P ⊂ E is said to be a cone provided that (i) u ∈ P , a ≥ 0 implies au ∈ P ; and EJQTDE, 2010 No. 20, p. 3 (ii) u, −u ∈ P implies u = 0. Definition 2.2.Given a cone P in a real Banach space E, a functional ψ : P → P is said to be increasing on P , provided ψ(x) ≤ ψ(y), for all x, y ∈ P with x ≤ y.Definition 2.3.Given a nonnegative continuous functional γ on P of a real Banach space, we define for each d > 0 the set Theorem 2. 1[See11,12].Let E be a real Banach space, P ⊂ E be a cone.Assume there exist positive numbers c and M, nonnegative increasing continuous functionals α, γ on P , and nonnegative continuous functional θ on P with θ(0) = 0 such that for all x ∈ P (γ, c).Suppose A : P (γ, c) → P is a completely continuous operator and there exist positive numbers a < b < c such that Then A has at least two fixed points x 1 , x 2 ∈ P (γ, c) satisfying a < α(x 1 ), with θ(x 1 ) < b, . Let E be a real Banach space, P ⊂ E be a cone.Assume there exist positive numbers c and M, nonnegative increasing continuous functionals α, γ on P , and nonnegative continuous functional θ on P with θ(0) = 0 such that for all x ∈ P (γ, c).Then A has at least two fixed points x 1 , x 2 ∈ P (γ, c) satisfying a < α(x 1 ), with θ(x 1 ) < b, ) where Proof.Firstly, by integrating the equation of the problems (2.1) on (0, t), we have Integrating (2.3) from 0 to t, we get 3), we have (2.5) The boundary condition u △ (0) = m−2 i=1 a i u △ (ξ i ) and (2.5) yield (2.7) and Using the boundary condition (2.9) Substituting (2.6), (2.9) into (2.4),we have know that u(t) satisfies (2.2).The proof of Lemma 2.2 is completed.
, then E is Banach space, with respect to the norm u = sup t∈[0,T ] |u(t)|.Now we define P = {u ∈ E| u is a concave, nonincreasing and nonnegative function}.Obviously, P is a cone in E.
Define an operator A : P → E by setting It is clear that the existence of a positive solution for the boundary value problems ( EJQTDE, 2010 No. 20, p. 7 So, we get where Hence, we obtain The proof of Lemma 2.4 is completed.

Main results
In this section, let h = max{t ∈ T| 0 ≤ t ≤ T 2 } and fix r ∈ T such that 0 < r < h, and define three nonnegative, increasing and continuous functionals γ, θ and α on P , by It is easy to see that γ(u) = θ(u) ≤ α(u) for u ∈ P .Moreover, by Lemma 2.4 we can On the other hand, we have For notational convenience, we define the following constants Theorem 3.1.Suppose conditions (H 1 ) and (H 2 ) hold, and assume that there are positive numbers a < b < c such that and f satisfies the following conditions (A 1 ): λr for 0 ≤ u ≤ a.Then, the boundary value problem (1.1) has at least two positive solutions u 1 and u 2 such that a < α(u 1 ), with θ(u 1 ) < b, and b < θ(u 2 ) with γ(u 2 ) < c.
Proof.By Lemma 2.3, we can know that A : P (γ, c) → P is completely continuous.We firstly show that if u ∈ ∂P (γ, c), then γ(Au) > c.
From condition(A 3 ), f (u) > a λr for r ≤ t ≤ T .Since Au ∈ P , we have from Lemma 2.4, It is said that f is not superlinear or sublinear, so the conclusion of [6,7,8] is invalid to the example.