Mathematical and Computer Modelling 49 (2009) 1386–1393

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Mathematical and Computer Modelling

journal homepage: www.elsevier.com/locate/mcm

The existence of three positive solutions ofm-point boundary valueproblems for some dynamic equations on time scalesI

Sihua Liang a,b, Jihui Zhang a,∗, Zhiyong Wang ca Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University, 210097, Jiangsu, PR Chinab College of Mathematics, Changchun Normal University, Changchun 130032, Jilin, PR Chinac College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing, 210044, Jiangsu, PR China

a r t i c l e i n f o

Article history:Received 9 December 2007Received in revised form 12 December 2008Accepted 7 January 2009

Keywords:Time scalesBoundary-value problemsPositive solutionsFixed-point theoremCone

a b s t r a c t

In this paper, we define a new operator which improves and generalizes a p-Laplacianoperator for some p > 1. By using this operator, we consider the existence of triple positivesolutions ofm-point boundary value problems for some dynamic equations on time scales

[ϕ(p(t)u∆(t))]∇ + a(t)f (u(t)) = 0, t ∈ [0, T ]Tκ∩Tk ,

u(0)− B0

(m−2∑i=1

αiu∆(ξi)

)= 0, u∆(T ) = 0,

where ϕ : R → R is an increasing homeomorphism and positive homomorphism andϕ(0) = 0. We show the existence of at least three positive solutions with suitable growthconditions imposed on the nonlinear term by using a new fixed-point theorem.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The theory of dynamic equations on time scales was introduced by Stefan Hilger in his PhD thesis in 1988 [1]. In the pastfew years, it has found a considerable amount of interest and attracted the attention of many researchers. It is still a newsubject, and research in this area is rapidly growing. On the one hand, the time scales approach not only unifies calculus anddifference equations, but also solves other problems that have a mix of stop–start and continuous behavior. On the otherhand, the time scales calculus has tremendous potential for application; for example, it can model insect populations thatare continuous while in season and may follow a difference scheme with variable step size, die out in winter while theireggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population [2]. By usingthe theory of time scales we can also study biological, heat transfer, stock market and epidemic models [3–5], etc.In this paper we deal with the followingm-point boundary value problem for some dynamic equations on time scales

[ϕ(p(t)u∆(t))]∇ + a(t)f (u(t)) = 0, t ∈ [0, T ]Tκ∩Tk , (1.1)

with the boundary value conditions

u(0)− B0

(m−2∑i=1

αiu∆(ξi)

)= 0, u∆(T ) = 0, (1.2)

I This work was supported by NSFC(10871096), Foundation of Major Project of Science and Technology of Chinese Education Ministry, SRFDP of HigherEducation, the project of Nanjing Normal University (181200000214) and Project of Graduate Education Innovation of Jiangsu Province.∗ Corresponding author.E-mail addresses: liangsihua@163.com (S. Liang), jihuiz@jlonline.com (J. Zhang).

0895-7177/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2009.01.001

S. Liang et al. / Mathematical and Computer Modelling 49 (2009) 1386–1393 1387

where ϕ : R → R is an increasing homeomorphism and positive homomorphism and ϕ(0) = 0, p ∈ C([0, T ]T, (0,+∞))and ξi ∈ [0, T ]T with 0 < ξ1 < ξ2 < · · · < ξm−2 < T , 0 <

∑m−2i=1 αi < 1, a : T → [0,+∞) is ld-continuous and not

identical zero on any closed subinterval of [0, T ]T.A projection ϕ : R→ R is called an increasing homeomorphism and positive homomorphism, if the following conditions

are satisfied:

(1) if x ≤ y, then ϕ(x) ≤ ϕ(y), for all x, y ∈ R;(2) ϕ is a continuous bijection and its inverse mapping is also continuous;(3) ϕ(xy) = ϕ(x)ϕ(y), for all x, y ∈ [0,+∞).

In the above definition, condition (3) can be replaced by the following stronger condition:(4) ϕ(xy) = ϕ(x)ϕ(y), for all x, y ∈ R,where R = (−∞,+∞).

Remark 1.1. If conditions (1), (2) and (4) hold, then it implies that ϕ is homogeneous and generates a p-Laplacian operator,i.e., ϕ(x) = |x|p−2x, for some p > 1.

Remark 1.2. It is well known that the p-Laplacian operator is odd. However, the operator which we defined above is notnecessary odd, see Example 4.1.

Recently, for p(t) ≡ 1 and ϕ(u) = |u|p−2u, p > 1, the existence problems of positive solutions of boundary valueproblems on time scales, some authors have obtained many results; for details, see [6–18] and the references therein. Fromthe process of proving above references, one can see that the oddness of a p-Laplacian operator is key to the proof. Howeverin this paper we define a new operator which improves and generalizes a p-Laplacian operator for some p > 1 and ϕ isnot necessary odd. Moreover, for the increasing homeomorphism and positive homomorphism operator, the research hasproceeded very slowly (see [19,20]) and the existence of positive solutions for dynamic equations on time scales still remainunknown.In [20], Liu and Zhang studied the existence of positive solutions of the quasilinear differential equation{

(ϕ(x′))′ + a(t)f (x(t)) = 0, 0 < t < 1,x(0)− βx′(0) = 0, x(1)+ δx′(1) = 0,

where ϕ : R→ R is an increasing homeomorphism and positive homomorphism and ϕ(0) = 0. They obtained the existenceof one or two positive solutions by using a fixed-point index theorem in cones.In 2005, He [11] considered the following boundary value problem[ϕp(u

∆(t))]∇ + a(t)f (u(t)) = 0, t ∈ [0, T ]T,u(0)− B0(u∆(η)) = 0, u∆(T ) = 0, oru∆(0) = 0, u(T )+ B1(u∆(η)) = 0,

where T is a time scale, ϕp(s) = |s|p−2s, p > 1, η ∈ (0, ρ(t))T. The authors showed the existence of at least two positivesolutions by way of a new double fixed-point theorem.Anderson et al. [7] studied the delta-nabla dynamic equation as follows{[g(u∆(t))]∇ + c(t)f (u(t)) = 0, t ∈ (a, b),u(a)− B0(u∆(v)) = 0, u∆(b) = 0,

where g(z) = |z|p−2z, p > 1 and v ∈ (a, b) ⊂ T. By virtue of the fixed point theorem of cone expansion and compressionof functional type, the authors proved the existence of at least one solution for the corresponding boundary value problem.In [10], Sang and Xi studied the existence of positive solutions of the p-Laplacian dynamic equation on time scales by

using fixed point theorems in cones of the form(φp(u∆(t)))∇ + a(t)f (t, u(t)) = 0, t ∈ [0, T ]T,

φp(u∆(0)) =m−2∑i=1

aiφp(u∆(ξi)), u(T ) =m−2∑i=1

biu(ξi),

where T is a time scale, φp(s) = |s|p−2s, p > 1. We emphasize that the results of the paper [10] are not replaced by ϕ whichwe defined above.Throughout the paper, the following conditions hold:

(C1) f : R+ → R+ is continuous and 0 <∑m−2i=1 αi < 1;

(C2) p ∈ C([0, T ]T, (0,+∞)) and nondecreasing on [0, T ]T;(C3) a : T→ [0,+∞) is ld-continuous and not identical zero on any closed subinterval of [0, T ]T;(C4) B0(v) is a continuous function on R and satisfies that there exist A ≥ B > 0 such that

Bv ≤ B0(v) ≤ Av, for all v ≥ 0.

1388 S. Liang et al. / Mathematical and Computer Modelling 49 (2009) 1386–1393

2. Some definitions and fixed point theorems

For convenience, we list the following definitions which can be found in [7,21–23,11,12,17,18].By a time scale, we mean an arbitrary nonempty closed subset of R and we denote the time scale by T throughout.

Examples of time scales are N, Z, R; Cantor set and so on.The handicap that time scales are not necessarily connected is eliminated by utilizing the notion of jump operators which

we shall next define.

Definition 2.1. For t < sup T and r > inf T, the mappings σ , ρ : T→ T,

σ(t) = inf{s ∈ T : s > t}, ρ(r) = sup{s ∈ T : s < r}

are called the forward jump operator and the backward jump operator, respectively.In the case T = R, σ(t) = ρ(t) = t , and T = Z, σ (t) = t + 1, ρ(t) = t − 1.

Definition 2.2. For t ∈ T, t is said to be right-scattered if σ(t) > t , right-dense if σ(t) = t , left-scattered if ρ(t) < t ,left-dense if ρ(t) = t .

Definition 2.3. If T has a right-scattered minimum m, define Tκ = T − {m}; otherwise set Tk = T. If T has left-scatteredmaximumM , define Tκ = T− {M}; otherwise let Tκ = T.

Definition 2.4. For x : T → R and t ∈ Tκ , we define the delta derivative of x(t), x∆(t), to be the number (when it exists)with the property that, for any ε > 0, there is a neighborhood U of t such that

|[x(σ (t))− x(s)] − x∆(t)[σ(t)− s]| < ε|σ(t)− s| for all s ∈ U .

For x : T→ R and t ∈ Tκ , we define the nabla derivative of x(t), x∇(t), to be the number (when it exists) with the propertythat, for any ε > 0, there is a neighborhood V of t such that

|[x(ρ(t))− x(s)] − x∇(t)[ρ(t)− s]| < ε|ρ(t)− s| for all s ∈ V .

Definition 2.5. If F∆(t) = f (t), then the delta integral is defined by∫ t

af (s)∆s = F(t)− F(a).

If G∇(t) = g(t), then the nabla integral is defined by∫ t

ag(s)∇s = G(t)− G(a).

Definition 2.6. The mapping f : T→ R is called ld-continuous provided it is continuous at each left-dense point and has aright-side limit at each right-dense point in T, and we write f ∈ Cld(T,R).

Definition 2.7. We say that u is a solution of (1.1) and (1.2) if u : T→ R is delta differentiable, ϕ(pu∆) : Tκ → R is nabladifferentiable on Tκ ∩ Tκ , [ϕ(pu∆)]∇ : Tκ ∩ Tκ → R is continuous, u satisfies (1.1) and (1.2).

Lemma 2.1. [ [8]] The following formulas hold:

(i)(∫ ta f (s)∆s

)∆= f (t),

(ii)(∫ ta f (s)∆s

)∇= f (ρ(t)),

(iii)(∫ ta f (s)∇s

)∆= f (σ (t)),

(iv)(∫ ta f (s)∇s

)∇= f (t).

In this paper, we suppose that T is a closed subset of R with 0 ∈ Tκ , T ∈ Tκ .Next we list some concepts of cones and a new fixed point theorem.

Definition 2.8. Let (E, ‖.‖) be a real Banach space. A nonempty, closed, convex set P ⊂ E is said to be a cone provided thefollowing are satisfied:

(a) if y ∈ P and λ ≥ 0, then λy ∈ P;

S. Liang et al. / Mathematical and Computer Modelling 49 (2009) 1386–1393 1389

(b) if y ∈ P and−y ∈ P , then y = 0.

If P ⊂ E is a cone, we denote the order induced by P on E by≤, that is, x ≤ y if and only if y− x ∈ P .

Definition 2.9. A map α is said to be a nonnegative, continuous, concave functional on a cone P of a real Banach space E, if

α : P → [0,∞)

is continuous, and

α(tx+ (1− t)y) ≥ tα(x)+ (1− t)α(y)

for all x, y ∈ P and t ∈ [0, 1].

Definition 2.10. Given a nonnegative continuous functional γ on a cone P of E, for each d > 0 we define the set

P(γ , d) = {x ∈ P : γ (x) < d}.

The following fixed point theorem is fundamental and important to the proofs of our main results.

Theorem 2.1 ([24]). Let P be a cone in a Banach space E. Let α, β and γ be three increasing, nonnegative and continuousfunctionals on P, satisfying for some c > 0 and M > 0 such that

γ (x) ≤ β(x) ≤ α(x), ‖x‖ ≤ Mγ (x)

for all x ∈ P(γ , c). Suppose there exists a completely continuous operator T : P(γ , c)→ P and 0 < a < b < c such that

(i) γ (Tx) < c, for all x ∈ ∂P(γ , c);(ii) β(Tx) > b, for all x ∈ ∂P(β, b);(iii) P(α, a) 6= ∅, and α(Tx) < a, for all x ∈ ∂P(α, a).

Then T has at least three fixed points x1, x2, x3 ∈ P(γ , c) such that

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

3. Solutions of (1.1) and (1.2) in a cone

In this section, by defining an appropriate Banach space and cone, we impose the growth conditions on f which allowus to apply the new fixed-point theorem in establishing the existence of three positive solutions of (1.1) and (1.2). We notethat, from the nonnegativity of a, f , a solution of (1.1) and (1.2) is nonnegative and concave on [0, T ]T.Let the Banach space E = Cld([0, T ]T,R)with norm ‖u‖ = supt∈[0,T ]T |u(t)| and define the cone P ⊂ E by

P ={u ∈ E : u(t) is a concave and nonnegative nondecreasing function on [0, T ]Tκ∩Tk

}.

Lemma 3.1 ([11]). If u ∈ P, then

u(t) ≥tT‖u‖, t ∈ [0, T ]T,

where ‖u‖ = supt∈[0,T ]T |u(t)|.Define an operator F : P → E by

(Fu)(t) = B0

(m−2∑i=1

αi1p(ξi)

ϕ−1(∫ T

ξi

a(r)f (u(r))∇r))+

∫ t

0

1p(s)

ϕ−1(∫ T

s(a(r)f (u(r))∇r

)∆s. (3.1)

Lemma 3.2. F : P → P is completely continuous.

Proof. ∀ u ∈ P , (Fu)(t) ≥ 0,∀ t ∈ [0, T ]T. It follows from Lemma 2.1 that we have

(Fu)∆(t) =1p(t)

ϕ−1(∫ T

ta(r)f (u(r))∇r

).

Obviously, (Fu)∆(t) is a continuous function and (Fu)∆(t) ≥ 0, that is (Fu)(t) is decreasing on [0, T ]T. The following proofis similar to that of [2, Theorem 8.39, p. 331]. For convenience, we list it here.

1390 S. Liang et al. / Mathematical and Computer Modelling 49 (2009) 1386–1393

If t ∈ [0, T ]Tκ∩Tκ is left scattered, we have

(Fu)∆∇(t) =(Fu)∆(t)− (Fu)∆(ρ(t))

t − ρ(t)≤ 0.

If t ∈ [0, T ]Tκ∩Tκ is left dense, we have

(Fu)∆∇(t) = lims→t

(Fu)∆(t)− (Fu)∆(s)t − s

≤ 0.

So we obtain (Fu)∆∇(t) ≤ 0, i.e., (Fu) is concave on [0, T ]T. This implies that Fu ∈ P and F : P → P . With standard argumentone may show that F : P → P is completely continuous. �

Notice that[ϕ(p(t)(Fu)∆(t))

]∇= −a(t)f (u(t)),

B0

(m−2∑i=1

αi(Fu)∆(ξi)

)= B0

(m−2∑i=1

αi1p(ξi)

ϕ−1(∫ T

ξi

a(r)f (u(r))∇r))= (Fu)(0),

(Fu)∆(T ) = 0.

So finding solutions to (1.1) and (1.2) are equivalent to seek fixed points of F in the cone P .Let us define the increasing, nonnegative, continuous functionals γ , β and α on P by

γ (u) = maxt∈[0,ξ1]T

u(t) = u(ξ1),

β(u) = mint∈[ξ1,ξm−2]T

u(t) = u(ξ1),

α(u) = maxt∈[0,ξm−2]T

u(t) = u(ξm−2).

Obviously for every u ∈ P

γ (u) ≤ β(u) ≤ α(u).

In addition, for each u ∈ P , Lemma 3.1 implies γ (u) = u(ξ1) ≥ξ1T ‖u‖. That is ‖u‖ ≤

Tξ1γ (u), for all u ∈ P .

For notational convenience, we denotem,M and λ by

m =1p(T )

(Bm−2∑i=1

αi + ξ1

)ϕ−1

(∫ T

ξm−2

a(r)∇r

),

M =1p(0)

(Am−2∑i=1

αi + ξ1

)ϕ−1

(∫ T

0a(r)∇r

),

λ =1p(0)

(Am−2∑i=1

αi + ξm−2

)ϕ−1

(∫ T

0a(r)∇r

).

Theorem 3.1. Assume that conditions (C1)–(C4) are satisfied. Let

0 < a <ξ1

Tb < b <

mMc,

and suppose that f satisfies the following conditions

(C5) f (u) < ϕ( cM

), for all u ∈ [0, T

ξ1c];

(C6) f (u) > ϕ( bm

), for all u ∈ [b, T

ξ1b];

(C7) f (u) < ϕ( aλ

), for all u ∈ [0, T

ξ1a].

Then there exist at least three positive solutions u1, u2, u3 of (1.1) and (1.2) such that

0 ≤ α(u1) < a < α(u2), β(u2) < b < β(u3), γ (u3) < c.

S. Liang et al. / Mathematical and Computer Modelling 49 (2009) 1386–1393 1391

Proof. We define the completely continuous operator F by (3.1). Let u ∈ ∂P(γ , c), then (Fu)(t) ≥ 0 for t ∈ [0, T ]T. ByLemma 3.2 we know that F : P(γ , c)→ P .We now show that all the conditions of Theorem 2.1 are satisfied. To make use of property (i) of Theorem 2.1, we choose

u ∈ ∂P(γ , c). Then γ (u) = maxt∈[0,ξ1]T u(t) = u(ξ1) = c. If we recalling that ‖u‖ ≤Tξ1γ (u) = T

ξ1c. So we have

0 ≤ u(t) ≤Tηc, for all t ∈ [0, T ]T.

Then assumption (C5) implies

f (u(s)) < ϕ( cM

), for s ∈ [0, T ]T.

Since Fu ∈ P , we get

γ (Fu) = (Fu)(ξ1)

= B0

(m−2∑i=1

αi1p(ξi)

ϕ−1(∫ T

ξi

a(r)f (u(r))∇r))+

∫ ξ1

0

1p(s)

ϕ−1(∫ T

s(a(r)f (u(r))∇r

)∆s.

≤Ap(0)

m−2∑i=1

αiϕ−1(∫ T

0a(r)f (u(r))∇r

)+

1p(0)

∫ ξ1

0ϕ−1

(∫ T

0(a(r)f (u(r))∇r

)∆s

<1p(0)

(Am−2∑i=1

αi + ξ1

)ϕ−1

(∫ T

0a(r)∇r

)cM

= c.

Hence condition (i) of Theorem 2.1 is satisfied.Secondly, we show that (ii) of Theorem2.1 is fulfilled. For this, we select u ∈ ∂P(β, b). Thenβ(u) = mint∈[ξ1,ξm−2]T u(t) =

u(ξ1) = b. This means u(t) ≥ b, t ∈ [ξ1, T ]T and since u ∈ P , we have b ≤ u(t) ≤ ‖u‖ = u(T ), for t ∈ [ξ1, T ]T. Note that‖u‖ ≤ T

ξ1γ (u) = T

ξ1β(u) = T

ξ1b, for all u ∈ P . So we have

b ≤ u(t) ≤Tξ1b, for all t ∈ [ξ1, T ]T.

Then assumption (C6) implies

f (u(s)) > ϕ

(bm

), for s ∈ [ξ1, T ]T.

Therefore

β(Fu) = (Fu)(ξ1)

= B0

(m−2∑i=1

αi1p(ξi)

ϕ−1(∫ T

ξi

a(r)f (u(r))∇r))+

∫ ξ1

0

1p(s)

ϕ−1(∫ T

s(a(r)f (u(r))∇r

)∆s.

>1p(T )

(Bm−2∑i=1

αi + ξ1

)ϕ−1

(∫ T

ξm−2

a(r)∇r

)bm

= b.

Then condition (ii) of Theorem 2.1 holds.Finally, we verify that (iii) of Theorem 2.1 is also satisfied. We note that u(t) ≡ a

2 , t ∈ [0, T ]T is a member of P(α, a) andα(u) = a

2 < a. So P(α, a) 6= ∅. Now let u ∈ ∂P(α, a). Then α(u) = maxt∈[0,ξm−2]T u(t) = u(ξm−2) = a. This implies that0 ≤ u(t) ≤ a, for t ∈ [0, ξm−2]T. Note that ‖u‖ ≤ T

ξm−2γ (u) ≤ T

ξm−2α(u) ≤ T

ξ1a, for all u ∈ P . So

0 ≤ u(t) ≤Tξ1a, for all t ∈ [0, T ]T.

From (C7), we have

f (u(s)) < ϕ

(aλl

), for s ∈ [0, T ]T.

1392 S. Liang et al. / Mathematical and Computer Modelling 49 (2009) 1386–1393

As before, we get

α(Fu) = (Fu)(ξm−2)

= B0

(m−2∑i=1

αi1p(ξi)

ϕ−1(∫ T

ξi

a(r)f (u(r))∇r))+

∫ ξm−2

0

1p(s)

ϕ−1(∫ T

s(a(r)f (u(r))∇r

)∆s.

≤Ap(0)

m−2∑i=1

αiϕ−1(∫ T

0a(r)f (u(r))∇r

)+

1p(0)

∫ ξm−2

0ϕ−1

(∫ T

0(a(r)f (u(r))∇r

)∆s

<1p(0)

(Am−2∑i=1

αi + ξm−2

)ϕ−1

(∫ T

0a(r)∇r

)aλ

= a.

Then condition (iii) of Theorem 2.1 is satisfied. Therefore Theorem 2.1 implies that F has at least three fixed points whichare positive solutions u1, u2 and u3 belonging to P(γ , c) of (1.1) and (1.2) such that

0 ≤ α(u1) < a < α(u2), β(u2) < b < β(u3), γ (u3) < c.

The proof of Theorem 3.1 is complete. �

Remark 3.1. If we add the condition of a(t)f (u(t)) 6≡ 0, t ∈ [0, T ]T, to Theorem 3.1, we can get at least three positivesolutions u1u2 and u3 satisfying

0 < α(u1) < a < α(u2), β(u2) < b < β(u3), γ (u3) < c.

4. Examples and remark

Example 4.1. Let T = {( 23 )N0} ∪ {1 − ( 23 )

N0}, where N0 denotes the set of all nonnegative integers. Consider the followingproblem

[ϕ(u∆(t))]∇ + f (u(t)) = 0, t ∈ [0, 1]T, (4.1)

u(0)−(u∆(13

)+ u∆

(23

))= 0, u∆(1) = 0, (4.2)

where a(t) ≡ 1, ξ1 = 13 , ξ2 =

23 , α1 = α2 =

14 , A = B = 4, p(t) ≡ 1 and

ϕ(u) =

u5

1+ u2, u ≤ 0,

u2, u > 0.

and

f (u) =

0.1, 0 ≤ u ≤ 3,

0.1+90

7√3− 3

(u− 3), 3 ≤ u ≤ 7√3,

90.1, 7√3 ≤ u.

Then problem (4.1) and (4.2) have at least three solutions.

Proof. Select a = 1, b = 7√3, c = 105. Then

m =1p(T )

(Bm−2∑i=1

αi + ξ1

)ϕ−1

(∫ T

ξm−2

a(r)∇r

)=

(2+

13

)ϕ−1

(∫ 1

23

1∇r

)=7√39,

M =1p(0)

(Am−2∑i=1

αi + ξ1

)ϕ−1

(∫ T

0a(r)∇r

)=

(2+

13

)ϕ−1

(∫ 1

0a(r)∇r

)=73,

λ =1p(0)

(Am−2∑i=1

αi + ξm−2

)ϕ−1

(∫ T

0a(r)∇r

)=

(2+

23

)ϕ−1

(∫ 1

0a(r)∇r

)=83.

It is easy to see that

0 < a <ξ1

Tb < b <

mMc,

S. Liang et al. / Mathematical and Computer Modelling 49 (2009) 1386–1393 1393

and f satisfies that

f (u) < ϕ( cM

)=

(10573

)2= 2025, for all 0 ≤ u ≤ 315;

f (u) > ϕ

(bm

)=

(7√3

7√39

)2= 81, for all 7

√3 ≤ u ≤ 21

√3;

f (u) < ϕ

(aλl

)=

(183

)2=964≈ 0.1159, for all 0 ≤ u ≤ 3.

Therefore by Theorem 3.1, The problem (4.1) and (4.2) has at least three positive solutions u1, u2, u3 such that

maxt∈[0, 13 ]T

ui(t) ≤ 105, for i = 1, 2, 3,

0 ≤ maxt∈[0, 23 ]T

u1(t) < 1 < maxt∈[0, 23 ]T

u2(t)

with

mint∈[ 13 ,

23 ]T

u2(t) < 7√3 < min

t∈[ 13 ,23 ]T

u3(t), maxt∈[0, 13 ]T

u3(t) ≤ 105. �

Acknowledgments

The authors of this paper wish to thank the referee for the valuable suggestions regarding the original manuscript.

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