Existence of Three Positive Solutions of Semipositone Boundary Value Problems on Time Scales

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Existence of Three Positive Solutions of Semipositone

Boundary Value Problems on Time Scales

Arzu

Denk

∗

,

S. Gulsan

Topal

Department of Mathematics, Ege University, 35100 Bornova, Izmir, Turkey

∗_{Corresponding Author: [email protected]}

Abstract

In this paper, we consider the existence of triple positive solutions for the second order semipositone m-point boundary value problem on time scales. We emphasize that the nonlinear termf may take a negative value.

Keywords

Positive Solutions, Fixed Point Theorems, Semipositone Problems, Time Scales

AMS (MOS) Subject Classification

34N05, 39A10

1 Introduction

LetTbe any time scale (a nonempty closed subset ofR), with[a, b]⊂T. In this paper, we are concerned with the existence of positive solutions for the following second order semipositone m-point eigenvalue problems on time scales:

−[p(t)u△(t)]∇+q(t)u(t) =λf(t, u(t)), t∈[a, b]κ_κ, (1.1)

αu(ρ(a))−βu[△](ρ(a)) =

m_∑−2

i=1

αiu(ξi), (1.2)

γu(b) +δu[△](b) =

m_∑−2

i=1

βiu(ξi), (1.3)

where u[△] := p(t)u△(t), λ > 0 and α, β, γ, δ, ξi, αi, βi (for i ∈ {1,2, ..., m−2}) are complex constants such that

|α|+|β| ̸= 0,|γ|+|δ| ̸= 0andξi ∈ (a, b),q : T → C is a continuous function, p : T → C is∇−differentiable on

Tk,p(t) ̸= 0for allt ∈ T,p∇ : Tk → C is continuous and also the continuous functionf : [ρ(a), b]×[0,∞) → Ris

semipositone, i.e.,f(t, u)needn’t be positive for allt∈[ρ(a), b]and allu≥0. By an interval[a, b], we mean the intersection of the real interval[a, b]with the given time scalesT. Other types of intervals are defined similarly. For the details of basic notions connected to time scales we refer to [4, 5].

In [6, 7], Ma studied the following m-point nonlinear boundary value problem

[p(t)x′(t)]′−q(t)x(t) +f(t, x(t)) = 0, t∈(0,1), αx(0)−βp(0)x′(0) =

m

∑

i=1

aix(ξi),

γx(1) +δp(1)x′(1) =

m

∑

i=1

bix(ξi),

where0< ξ1 < ... < ξm <1, α, β, γ, δ≥0, ai, bi ≥0withρ=γβ+αγ+αδ >0. By using Guo-Krasnoselskii fixed

point theorem the existence and multiplicity of positive solutions were given. In recent paper [10], the following BVP was studied:

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ax(0)−bx′(0) =

m

∑

i=1

aix(ξi),

cx(1) +dx′(1) =

m

∑

i=1

bix(ξi).

Using the cone and the fixed point index theory, the authors showed the existence of multiple positive solutions for the above Sturm-Liouville boundary value problems for second order differential equations. These papers did not discuss the existence of three positive solutions. Besides this, the above papers did not give the results of the existence of positive solutions when the nonlinearity can take negative value. The study of semipositone problems for solutions, as was pointed out by Aris in chemical reactor theory [3], is more interesting. However, there is a little work that has referred to the existence of positive solutions of semipositone problems on time scales. In [2], Anderson and Zhai established the existence of at least two positive solutions for the nonlinear semipositone three-point boundary value problem on time scales. The other works of semipositone problems on time scales were also done by Yang, Meng [8] and Anderson, Wong [1].

This paper is organized as follows: In Section 2, we present some lemmas that will be used later. In Section 3, we will give the main result which state the sufficient conditions for the m-point BVP(1.1)−(1.3)to have at least three positive solutions.

2 The Preliminary Lemmas

Throughout the paper we will assume that the following conditions are satisfied.

(H1) p(t)>0,q(t)≥0,

(H2) α, γ≥0,β, δ >0,αi, βi≥0fori∈ {1,2, ..., m−2},

(H3) Ifq(t)≡0, thenα+γ >0,

(H4) f : [ρ(a), b]×[0,∞] → R is continuous and there exists a constant M > 0 such that f(t, u) ≥ −M for all

(t, u)∈[ρ(a), b]×[0,∞).

Letϕ1andϕ2be the solutions of the linear problems

[p(t)ϕ△₁(t)]∇−q(t)ϕ1(t) = 0, t∈[a, b]κκ,

ϕ1(ρ(a)) =β,ϕ [△]

1 (ρ(a)) =α

and

[p(t)ϕ△₂(t)]∇−q(t)ϕ2(t) = 0, t∈[a, b]κκ,

ϕ2(b) =δ,ϕ [△]

2 (b) =−γ

respectively. Let usd=−Wt(ϕ1, ϕ2) =p(t)[ϕ△1 (t)ϕ2(t)−ϕ1(t)ϕ△2(t)].

To prove the main results, we will employ following lemmas.

Lemma 2.1 [4]Under the conditions(H1)and(H2), the solutionsϕ1(t)andϕ2(t)posses the following properties:

ϕ1(t)≥0 t∈[ρ(a), σ(b)], ϕ2(t)≥0 t∈[ρ(a), b],

ϕ[₁△](t)≥0 t∈[ρ(a), b)], ϕ[₂△](t)≤0 t∈[ρ(a), b].

LetG(t, s)be the Green’s function for the boundary value problem

−[p(t)u△(t)]∇+q(t)u(t) = 0, t∈[a, b]κ_κ,

αu(ρ(a))−βu[△](ρ(a)) = 0,

γu(b) +δu[△](b) = 0, is given by

G(t, s) = 1 d

{

ϕ1(s)ϕ2(t), ρ(a)≤s≤t≤b,

ϕ1(t)ϕ2(s) ρ(a)≤t≤s≤b.

(2.1)

Lemma 2.2 [4]Let conditions(H1)−(H3)hold. Then

(i) G(t, s)≥0fort, s∈[ρ(a), b], (ii) G(t, s)>0fort, s∈[a, b].

Set

Ω := −

∑m−2

i=1 αiϕ1(ξi) d−

∑m−2

i=1 αiϕ2(ξi)

d−∑m_i₌₁−2βiϕ1(ξi) −

∑m−2

i=1 βiϕ2(ξi)

.

The lemmas in this section are based on the boundary value problem

−[p(t)u△(t)]∇+q(t)u(t) =v(t), t∈[a, b]κ

κ (2.2)

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Lemma 2.3 [9]Let the conditions(H1)−(H3)be hold. Assume thatΩ̸= 0andv∈C([ρ(a), b]). Then the boundary value

problem(2.2)−(1.2)−(1.3)has a unique solution u(t) =

∫ b

ρ(a)

G(t, s)v(s)∇s+A(v)ϕ1(t) +B(v)ϕ2(t),

whereG(t, s)is given in(2.1),

A(v) := 1 Ω

∑m−2

i=1 αi

∫b

ρ(a)G(ξi, s)v(s)∇s d−

∑m−2

i=1 αiϕ2(ξi)

∑m−2

i=1 βi

∫b

ρ(a)G(ξi, s)v(s)∇s −

∑m−2

i=1 βiϕ2(ξi)

,

B(v) := 1 Ω

−∑m₋2

i=1 αiϕ1(ξi)

∑m₋2

i=1 αi

∫b

ρ(a)G(ξi, s)v(s)∇s

d−∑m_i₌₁−2βiϕ1(ξi)

∑m−2

i=1 βi

∫b

ρ(a)G(ξi, s)v(s)∇s

.

Lemma 2.4 Let(H1)−(H3)hold. Assume that

(H5)Ω<0,d−

∑m₋2

i=1 αiϕ2(ξi)>0,d−

∑m₋2

i=1 βiϕ1(ξi)>0.

Then forv∈C([ρ(a), b])withv≥0, the solutionuof the boundary value problem(2.2)−(1.2)−(1.3)satisfiesu(t)≥0, fort∈[ρ(a), b].

Proof.It is an immediate subsequence of the facts thatG≥0on[ρ(a), b]×[ρ(a), b]andA(v)≥0, B(v)≥0.

Lemma 2.5 Assume that(H1)−(H3)and(H5)hold. Then

g(t)G(s, s)≤G(t, s)≤G(s, s), t, s∈[ρ(a), b], wheregis given by

g(t) = min

t_∈[ρ(a),b]{

ϕ1(t)

ϕ1(b)

, ϕ2(t) ϕ2(ρ(a))}

. (2.3)

Proof.Sinceϕ1(t)is increasing andϕ2(t)is decreasing ont∈[ρ(a), b],G(t, s)≤G(s, s). Now

G(t, s) G(s, s) =

{ _ϕ

2(t)

ϕ2(s), s≤t

ϕ1(t)

ϕ1(s), t≤s

≥

{ _ϕ

2(t)

ϕ2(ρ(a)), s≤t

ϕ1(t)

ϕ1(b), t≤s

≥g(t)

for allt, s∈[ρ(a), b]. This completes the proof.

Lemma 2.6 Letwbe the unique positive solution of the boundary value problem

−[p(t)u△(t)]∇+q(t)u(t) = 1, t∈[a, b]κ

κ (2.4)

with the boundary condition(1.2)−(1.3). Then, w(t)≤Cg(t), t∈[ρ(a), b],

wheregis given in(2.3)and

C= (b−ρ(a)

d +

A(1) ϕ2(b)

+ B(1) ϕ1(ρ(a))

)ϕ1(b)ϕ2(ρ(a)). (2.5)

Proof. Using the expression of the Green function, the definition of the functiongand the properties ofϕ1andϕ2, we have

for allt∈[ρ(a), b],

w(t) =

∫ b

ρ(a)

G(t, s)∇s+A(1)ϕ1(t) +B(1)ϕ2(t)

= 1 d

∫ t

ρ(a)

ϕ1(s)ϕ2(t)∇s+

1 d

∫ b

t

ϕ1(t)ϕ2(s)∇s+A(1)ϕ1(t) +B(1)ϕ2(t)

≤1

d

∫ t

ρ(a)

ϕ1(t)ϕ2(t)∇s+

1 d

∫ b

t

ϕ1(t)ϕ2(t)∇s+A(1)ϕ1(t) +B(1)ϕ2(t)

=1 d

∫ b

ρ(a)

ϕ1(t)ϕ2(t)∇s+A(1)ϕ1(t) +B(1)ϕ2(t)

≤1

dϕ1(b)ϕ2(ρ(a))g(t)(

∫ b

ρ(a)

∇s) + 1 ϕ2(t)

A(1)ϕ1(b)ϕ2(ρ(a))g(t) +

1 ϕ1(t)

B(1)ϕ1(b)ϕ2(ρ(a))g(t)

≤1

dϕ1(b)ϕ2(ρ(a))g(t)(b−ρ(a)) + 1 ϕ2(b)

A(1)ϕ1(b)ϕ2(ρ(a))g(t) +

1 ϕ1(ρ(a))

B(1)ϕ1(b)ϕ2(ρ(a))g(t)

= (b−ρ(a)

d +

A(1) ϕ2(b)

+ B(1) ϕ1(ρ(a))

)ϕ1(b)ϕ2(ρ(a))g(t).

This completes the proof.

LetEdenote the Banach spaceC([ρ(a), b])with the norm∥u∥= max

t_∈[ρ(a),b]|u(t)|. Define the coneP ⊂Eby

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Remark 2.1 Under the condition(H5), the operatorsA, Bwhich are defined in Lemma2.3are increasing and linear.

The following fixed point theorem is fundamental and important to the proof of our main result.

Theorem 2.1 Let P be a cone in a Banach space E. Letα,β andγbe three increasing, nonnegative and continuous

func-tionals onP, satisfying for somec >0andM >0such that γ(x)≤β(x)≤α(x)and∥x∥ ≤M γ(x),

for allx∈P(γ, c). Suppose there exists a completely continuous operatorT:P(γ, c)→Pand0< a < b < csuch that (i)γ(T x)< c, for allx∈∂P(γ, c);

(ii)β(T x)> b, for allx∈∂P(β, b);

(iii)P(α, a)̸=∅, andα(T x)< a, for allx∈∂P(α, a). ThenT has at least three fixed pointsx1, x2, x3∈P(γ, c)such that

0≤α(x1)< a < α(x2), β(x2)< b < β(x3), γ(x3)< c.

Remark 2.2 If the restrictionT θ̸=θis imposed in Theorem2.1, thenT has at least three fixed pointsx1, x2, x3belonging

toP(γ, c)such that

0< α(x1)< a < α(x2), β(x2)< b < β(x3), γ(x3)< c.

For notational convenience, we denoteξ, kandKby

ξ= inf

t∈[ρ(a),b]

g(t),

k=ξ

∫ b

ρ(a)

G(s, s)∇s+A(1)ϕ1(ρ(a)) +B(1)ϕ2(b),

K=

∫ b

ρ(a)

G(s, s)∇s+A(1)ϕ1(b) +B(1)ϕ2(ρ(a)).

Let the increasing, nonnegative and continuous functionalsα,βandγbe defined on the coneP by,

α(u) = max

t∈[ρ(a),b]

u(t) =u(t0)

β(u) = min

t_∈[ρ(a),b]u(t) =u(t1)

γ(u) = min

t∈[ρ(a),b]

u(t) =u(t1).

We see that, for eachu∈P,γ(u) =β(u)≤α(u). In addition, for eachu∈P, we know that∥u∥ ≤ u(t1) g(t) ≤

u(t1)

ξ . That

is∥u∥ ≤ 1

ξγ(u), for allu∈P.

3 Main Results

First we shall show that the following boundary value problem

−[p(t)y△(t)]∇+q(t)y(t) =λF(t, y∗(t)) (3.1) αy(ρ(a))−βy[△]₍_ρ₍_a_{)) =}∑m−2

i=1 αiy(ξi), (3.2)

γy(b) +δy[△](b) =∑m_i₌₁−2βiy(ξi), (3.3)

has at least three positive solutions where

F(t, z) =

{

f(t, z) +M, z≥0, f(t,0) +M, z≤0,

y∗(t) = max{y(t)−x(t),0} andx(t) = λM w(t) such thatw is the unique solution of the boundary value problem

(2.4)−(1.2)−(1.3). Thereafter we shall obtain at least three positive solutions for the boundary value problem(1.1)−(1.3). We give the following assumption:

(H6)f(t, u(t))̸≡0for(t, u)∈[ρ(a), b]×[0,∞).

Theorem 3.1 Assume that conditions(H1)−(H6)are satisfied. Let

0< λ < b

M∥w∥, 0< a < b−λM∥w∥< b ξ <

k Kc, and suppose thatFsatisfies the following conditions:

(C1)F(t, y)<

c Kλ

−1

, for(t, y)∈[ρ(a), b]×[0,c ξ],

(C2)F(t, y)>

b kλ

−1_{, for}₍_{t, y}₎_∈_[_ρ₍_a₎_{, b}_]_×_[_b₋_λM_∥_w_∥_,b

ξ],

(C3)F(t, y)<

a Kλ

−1_{, for}₍_{t, y}₎_∈_[_ρ₍_a₎_{, b}_]_×_[0_{, a}_]_.

Then the boundary value problem(3.1)−(3.3)has at least three positive solutionsy1, y2, y3and there existsd >0such that

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Proof.It is well known that the existence of positive solution to the boundary value problem(3.1)−(3.3)is equivalent to the existence of fixed point of the operatorT. So we shall seek a fixed point ofTin our conePwhere the operatorT :E →E

is defined by

T y(t) =λ

∫ b

ρ(a)

G(t, s)F(s, y∗(s))∇s+λA(F)ϕ1(t) +λB(F)ϕ2(t), t∈[ρ(a), b].

First, it is obvious thatTis completely continuous.

Now we shall prove thatT(P)⊆ P. Lety ∈ P. Then, using Lemma2.5and the properties ofϕ1 andϕ2, we get for

t∈[ρ(a), b],

T y(t) =λ

∫ b

ρ(a)

G(t, s)F(s, y∗(s))∇s+λA(F)ϕ1(t) +λB(F)ϕ2(t)

≤λ

∫ b

ρ(a)

G(s, s)F(s, y∗(s))∇s+λA(F)ϕ1(b) +λB(F)ϕ2(ρ(a)),

and so

∥T y∥ ≤λ

∫ b

ρ(a)

G(s, s)F(s, y∗(s))∇s+λA(F)ϕ1(b) +λB(F)ϕ2(ρ(a)). (3.4)

Now, using Lemma2.5again and(3.4), we obtain fort∈[ρ(a), b],

T y(t) =λ

∫ b

ρ(a)

G(t, s)F(s, y∗(s))∇s+λA(F)ϕ1(t) +λB(F)ϕ2(t)

≥λg(t)

∫ b

ρ(a)

G(s, s)F(s, y∗(s))∇s+λA(F)ϕ1(t) ϕ1(b)

ϕ1(b) +λB(F)

ϕ2(t)

ϕ2(ρ(a))

≥λg(t)

∫ b

ρ(a)

G(s, s)F(s, y∗(s))∇s+λg(t)A(F)ϕ1(b) +λg(t)B(F)ϕ2(ρ(a))

≥g(t)∥T y∥. This shows thatT(P)⊆P.

We now show that all the conditions of Theorem2.1are satisfied. To make use of property(i) of Theorem2.1, we choose

y∈∂P(γ, c). Thenγ(y) = min

t∈[ρ(a),b]

y(t1) =c. If we recalling that∥y∥ ≤

1 ξγ(y) =

1

ξc. So we have 0≤y∗(t)≤y(t)≤ ∥y∥ ≤ c

ξ

and so

0≤y∗(t)≤ c

ξ, for allt∈[ρ(a), b].

From(C1), we get

γ(T y) = (T y)(t1) =λ

∫ b

ρ(a)

G(t1, s)F(s, y∗(s))∇s+λA(F)ϕ1(t1) +λB(F)ϕ2(t1)

≤λ

∫ b

ρ(a)

G(s, s)F(s, y∗(s))∇s+λA(F)ϕ1(b) +λB(F)ϕ2(ρ(a))

< λλ−1 c K

∫ b

ρ(a)

G(s, s)∇s+λλ−1 c

KA(1)ϕ1(b) +λλ

−1 c

KB(1)ϕ2(ρ(a))

= c K

[ ∫ b

ρ(a)

G(s, s)∇s+A(1)ϕ1(b) +B(1)ϕ2(ρ(a))]

=c.

Then condition(i)of Theorem2.1holds.

Secondly, we show that(ii)of Theorem2.1is fulfilled. For this, we selecty∈∂P(β, b). Thenβ(y) = min

t_∈[ρ(a),b]y(t1) =b.

This meansy(t)≥b, for allt∈[ρ(a), b]and sincey∈P, we have fort∈[ρ(a), b],

b−λM∥w∥ ≤b−λM w(t)≤y(t)−x(t)≤y∗(t)≤y(t)≤ ∥y∥. Note that∥y∥ ≤ 1 ξβ(y) =

1

ξb. So we have b−λM∥w∥ ≤y∗(t)≤1

ξb, for allt∈[ρ(a), b].

Therefore

β(T y) = (T y)(t1) =λ

∫ b

ρ(a)

≥λ

∫ b

ρ(a)

g(t1)G(s, s)F(s, y∗(s))∇s+λA(F)ϕ1(ρ(a)) +λB(F)ϕ2(b)

> λλ−1b

k

∫ b

ρ(a)

g(t1)G(s, s)∇s+λλ−1

b

kA(1)ϕ1(ρ(a)) +λλ

−1b

kB(1)ϕ2(b)

≥ b

k

[ ∫ b

ρ(a)

ξG(s, s)∇s+A(1)ϕ1(ρ(a)) +B(1)ϕ2(b)]

=b.

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Finally, we verify that(iii)of Theorem2.1is also satisfied. We note thaty(t)≡ a

2,t∈[ρ(a), b]is a member ofP(α, a)

sinceα(y) = a

2 < a. ThereforeP(α, a)̸=∅. Now lety ∈ ∂P(α, a). Thenγ(y) = t_∈max[ρ(a),b] =y(t0) = a. This means

0≤y∗(t)≤y(t)≤a. So we have0≤y∗(t)≤a, for allt∈[ρ(a), b]. Then, by the condition(C3)of this theorem, we have

α(T y) = (T y)(t0) =λ

∫ b

ρ(a)

≤λ

∫ b

ρ(a)

G(s, s)F(s, y∗(s))∇s+λA(F)ϕ1(b) +λB(F)ϕ2(ρ(a))

< λλ−1 a K

∫ b

ρ(a)

G(s, s)∇s+λλ−1a

KA(1)ϕ1(b) +λλ

−1a

KB(1)ϕ2(ρ(a))

= a K

[ ∫ b

ρ(a)

G(s, s)∇s+A(1)ϕ1(b) +B(1)ϕ2(ρ(a))]

=a.

Hence condition(iii)of Theorem2.1is satisfied.

Therefore it follows from Remark2.2that T has at least three fixed points which are positive solutionsy1, y2 andy3

belonging toP(γ, c)of(3.1)−(3.3)such that

0< α(y1)< a < α(y2), β(y2)< b < β(y3), γ(y3)< c,

and so there existsd >0such that

d≤ max

t∈[ρ(a),b]y1(t) =α(y1). (3.5)

Lemma 3.1 y(t) is the solution of the boundary value problem(3.1)−(3.3)withy(t)> x(t)for allt∈[ρ(a), b]if and only ifu(t) =y(t)−x(t)is the positive solution of the boundary value problem(1.1)−(1.3).

Proof.Lety(t)is the solution of the boundary value problem(3.1)−(3.3). Then

y(t) =λ

∫ b

ρ(a)

G(t, s)(f(s, y∗(s)) +M)∇s+λA(f+M)ϕ1(t) +λB(f+M)ϕ2(t).

SinceAandBare linear , we have fort∈[ρ(a), b],

y(t) =λ

∫ b

ρ(a)

G(t, s)f(s, y(s)−x(s))∇s+λM

∫ b

ρ(a)

G(t, s)∇s+λA(f)ϕ1(t)

+λM A(1)ϕ1(t) +λB(f)ϕ2(t) +λM B(1)ϕ2(t),

Noticing that,

w(t) =

∫ b

ρ(a)

G(t, s)∇s+A(1)ϕ1(t) +B(1)ϕ2(t),

we have fort∈[ρ(a), b],

y(t) =λ

∫ b

ρ(a)

G(t, s)f(s, y(s)−x(s))∇s+λA(f)ϕ1(t) +λB(f)ϕ2(t) +λM w(t),

or

y(t)−x(t) =λ

∫ b

ρ(a)

G(t, s)f(s, y(s)−x(s))∇s+λA(f)ϕ1(t) +λB(f)ϕ2(t),

and hence

u(t) =λ

∫ b

ρ(a)

G(t, s)f(s, u(s))∇s+λA(f)ϕ1(t) +λB(f)ϕ2(t).

Theorem 3.2 Assume that the hypotheses of Theorem3.1are satisfied. If

0< λ < d

M C, (3.6)

wheredis given(3.5)in Theorem3.1, then the boundary value problem(1.1)−(1.3)has at least three positive solutions u1, u2, u3such that

d≤α(u1)< a < α(u2) +λM∥w∥, β(u2)< b, γ(u3)< c.

Proof. Since the hypotheses of Theorem3.1are satisfied, then the boundary value problem(3.1)−(3.3)has at least three positive solutionsy1, y2, y3such that

d≤α(y1)< a < α(y2), β(y2)< b < β(y3), γ(y3)< c. (3.7)

Letu1(t) =y1(t)−x(t),u2(t) =y2(t)−x(t),u3(t) =y3(t)−x(t),we shall show thatu1, u2, u3are positive. From(3.7)

and the definition of the functionalsα, βandγ, we obtain

d≤ ∥y1∥, a <∥y2∥, b <∥y3∥. (3.8)

Moreover, using(3.6),(3.8)and Lemma2.6, we obtain fort∈[ρ(a), b],

y1(t)≥g(t)∥y1∥ ≥dg(t)> λM g(t)≥λM w(t) =x(t),

(7)

y3(t)≥g(t)∥y3∥ ≥bg(t)> dg(t)≥λM g(t)≥λM w(t) =x(t).

Thus, it follows from Lemma3.1 that u1, u2, u3 are positive solutions of the boundary value problem(1.1)−(1.3). In

addition, from(3.7), we get

α(u1)≤α(y1)< a < α(y2)< α(u2) +λM∥w∥, β(u2)≤β(y2)< b, γ(u3)≤γ(y3)< c.

4 Conclusions

There is a little work that gives the results of the existence of three positive solutions of semipositone problems on time scales in the literature.

REFERENCES

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[2] D. R. Anderson, C. Zhai, Positive solutions to semi-positone second-order three-point problems on time scales 215 (2010) 3713-3720.

[3] R. Aris, Introduction to the Analysis of Chemical Reactors, Prentice-Hall, Englewood Cliffs, NJ, 1965.

[4] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhauser, Boston, Cambridge, MA (2001).

[5] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, Cambridge, MA (2003).

[6] R. Ma , Multiple positive solutions for nonlinear m-point boundary value problems, Appl. Math. Comput., 148 (2004) 249-262.

[7] R. Ma and B. Thompson, Positive solutions for nonlinear m-point eigenvalue problems, Journal of Mathematical Analysis and Applications, 297 (2004) 24-37.

[8] Y. Yang, F. Meng, Positive solutions for the singular semipositone bounadary value problem on time scales, Mathematical and Computer Modelling, 52 (2010) 481-489.

[9] I. Yaslan Karaca, Multiple positive solutions for dynamic m-point boundary value problems, Dynamic Systems and Applications 17 (2008) 25-42.