Multiple Positive Symmetric Solutions to

Discrete Dynamics in Nature and Society Volume 2009, Article ID 141929,27pages doi:10.1155/2009/141929

Research Article

Multiple Positive Symmetric Solutions to

p-Laplacian Dynamic Equations on Time Scales

You-Hui Su

and Can-Yun Huang

1School of Mathematics and Physical Sciences, Xuzhou Institute of Technology, Xuzhou, Jiangsu 221008, China

2Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China

Correspondence should be addressed to Can-Yun Huang,canyun h@sina.com Received 1 July 2009; Accepted 18 November 2009

Recommended by Leonid Shaikhet

This paper makes a study on the existence of positive solution top-Laplacian dynamic equations on time scalesT. Some new sufficient conditions are obtained for the existence of at least single or twin positive solutions by using Krasnosel’skii’s fixed point theorem and new sufficient conditions are also obtained for the existence of at least triple or arbitrary odd number positive solutions by using generalized Avery-Henderson fixed point theorem and Avery-Peterson fixed point theorem.

As applications, two examples are given to illustrate the main results and their differences. These results are even new for the special cases of continuous and discrete equations, as well as in the general time-scale setting.

Copyrightq2009 Y.-H. Su and C.-Y. Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Initiated by Hilger in his Ph.D. thesis 1 in 1988, the theory of time scales has been improved ever since, especially in the unification of the theory of differential equations in the continuous case and the theory of difference equations in the discrete case. For the time being, it remains a field of vitality and attracts attention of many distinguished scholars worldwide.

In particular, the theory is also widely applied to biology, heat transfer, stock market, wound healing, epidemic models2–5, and so forth.

Recent research results indicate that considerable achievement has been made in the existence problems of positive solutions to dynamic equations on time scales. For details, please see6–13and the references therein. Symmetry and pseudosymmetry have been widely used in science and engineering 14. The reason is that symmetry and pseudosymmetry are not only of its theoretical value in studying the metric manifolds15 and symmetric graph16,17, and so forth, but also of its practical value, for example, we can

apply this characteristic to study graph structure18,19and chemistry structure20. Yet, few literature resource21,22is available concerning the characteristics of positive solutions top-Laplacian dynamic equations on time scales.

Throughout this paper, we denote thep-Laplacian operator byϕpu, that is,ϕpu

|u|^p−2uforp >1 withϕp⁻¹ϕqand 1/p 1/q1.

For convenience, we think of the blanket as an assumption thata, bare points inT,for an intervala, b_Twe always meana, b∩T.Other type of intervals is defined similarly.

We would like to mention the results of Sun and Li 11, 12. In 12, Sun and Li considered the two-point BVP

ϕp

u^Δt_Δ

htfu^σt 0, t∈a, b_T, ua−B0

u^Δa

0, u^Δσb 0,

1.1

and established the existence theory for positive solutions of the above problem. They11 also considered them-point boundary value problem withp-Laplacian

ϕp

u^Δt_∇

htft, ut 0, t∈0, T_T,

u^Δ0 0, uT ^m−2

i1

ciuξi,

1.2

and gave the existence of single or multiple positive solutions to the above problem. The main tools used in these two papers are some fixed-point theorems23–25.

It is also noted that the researchers mentioned above 11, 12 only considered the existence of positive solutions. As a results, they failed to further provide characteristics of solutions, such as, symmetry. Naturally, it is quite necessary to consider the characteristics of solutions top-Laplacian dynamic equations on time scales.

Let T be a symmetric time scale such that 0, T ∈ T. we consider the following p- Laplacian boundary value problem on time scalesTof the form:

ϕp

u^Δt_∇

htfut 0, t∈0, T_T, u0 uT 0, u^Δ0 −u^ΔT.

1.3

By using symmetric technique, the Krasnosel’skii’s fixed point theorem24, the generalized Avery-Henderson fixed point theorem26, and Avery-Peterson fixed point theorem 27, we obtain the existence of at least single, twin, triple, or arbitrary odd positive symmetric solutions of problem 1.3. As applications, two examples are given to illustrate the main results and their differences. These results are even new for the special cases of continuous and discrete equations as well as in the general time-scale setting.

The rest of the paper is organized as follows. InSection 2, we present several fixed point results. In Section 3, by using Krasnosel’skii’s fixed point theorem, we obtain the existence of at least single or twin positive symmetric solutions to problem1.3. InSection 4, the existence criteria for at least triple positive or arbitrary odd positive symmetric solutions to

problem1.3are established. InSection 5, we present two simple examples to illustrate our results.

For convenience, we now give some symmetric definitions.

Definition 1.1. The interval0, T_T is said to be symmetric if any givent ∈ 0, T_T,we have T−t∈0, T_T.

We note that such a symmetric time scaleTexists. For example, let

T{0,0.05,0.1,0.15} ∪0.22,0.44∪ {0.5,0.85,0.9,0.95,1} ∪0.56,0.78. 1.4 It is obvious thatTis a symmetric time scale.

Definition 1.2. A functionu:0, T_T → Ris said to be symmetric ifuis symmetric over the interval0, T_T.That is,ut uT−t, for any givent∈0, T_T.

Definition 1.3. We sayuis a symmetric solution to problem1.3on0, T_Tprovided thatuis a solution to boundary value problem1.3and is symmetric over the interval0, T_T.

Basic definitions on time scale can be found in6,7,28. Another excellent sources on dynamical systems on measure chains are the book in29.

Throughout this paper, it is assumed that

H1f :0,∞ → 0,∞is continuous, and does not vanish identically;

H2h ∈ Cld0, T_T,0,∞is symmetric over the interval0, T_T and does not vanish identically on any closed subinterval of0, T_T,whereCld0, T_T,0,∞denotes the set of all left dense continuous functions from0, T_Tto0,∞.

2. Preliminaries

LetECld0, T_T,Rand equip norm

u sup

t∈0,TT

|ut|, 2.1

thenEis a Banach space. Define a coneP ⊂Eby

P

u∈E|u0uT0, uis symmetric,nonnegative,and concave on the interval0, T_T 2.2.

Assume that r, η ∈ 0, T/2_T with η < r. By using the symmetric and concave characters ofu∈Pandu0 uT 0, it is easy to obtain the following results.

Lemma 2.1. Assume thatr, η∈0, T/2_Twithη < r. Ifu∈P,then iuη≥η/rur;

ii T/2ur≥ruT/2.

From the previous lemma we know thatuuT/2foru∈P.

The operatorA:P → Eis defined by

Aut

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩ _t

0

ϕq

_T/2

s

hrfur∇r

Δs, t∈

0,T 2

T, _T

t

ϕq

_s

T/2

hrfur∇r

Δs, t∈ T

2, T

T.

2.3

It is obvious thatAis completely continuous operator and all the fixed points ofAare the solutions to the boundary value problem1.3.

In addition, it is easy to see that the operatorAis symmetric. In fact, fort∈0, T/2_T, we haveT−t∈T/2, T_T, by using the integral transform, we have

AuT−t _T

T−tϕq

_s

T/2

hrfur∇r

Δs

s→T−s

_t

0

ϕq

_T−s

T/2

hrfur∇r

Δs

r→T−r _t

0

ϕq

T/2 s

hT−rfuT−r∇r

Δs

_t

0

ϕq

T/2 s

hrfur∇r

ΔsAut.

2.4

Hence,Ais symmetric.

Now, we provide some background material from the theory of cones in Banach spaces 24,26,27,30, and then state several fixed point theorems needed later.

Firstly, we list the Krasnosel’skii’s fixed point theorem24.

Lemma 2.2see24. LetPbe a cone in a Banach spaceE.Assume thatΩ1,Ω2are open subsets of Ewith 0∈Ω1,Ω1 ⊂ Ω2.IfA:P∩Ω2\Ω1 → Pis a completely continuous operator such that either

iAx ≤ x, for allx∈P∩∂Ω1andAx ≥ x, for allx∈P∩∂Ω2or iiAx ≥ x, for allx∈P∩∂Ω1andAx ≤ x, for allx∈P∩∂Ω2, thenAhas a fixed point inP∩Ω2\Ω1.

Given a nonnegative continuous functionalγon a conePof a real Banach spaceE,we define, for eachd >0,the setPγ, d {x∈P:γx< d}.

Secondly, we state the generalized Avery-Henderson fixed point theorem26.

Lemma 2.3see 26. Let P be a cone in a real Banach spaceE. Letα,β, andγ be increasing, nonnegative continuous functional onP such that for somec >0 andH > 0, γx≤ βx ≤αx andx ≤Hγxfor allx∈Pγ, c.Suppose that there exist positive numbersaandbwitha < b < c andA:Pγ, c → Pis a completely continuous operator such that

iγAx< cfor allx∈∂Pγ, c;

iiβAx> bfor allx∈∂Pβ, b;

iiiPα, a/∅andαAx< aforx∈∂Pα, a,

thenAhas at least three fixed pointsx1,x2, andx3belonging toPγ, csuch that

0≤αx1< a < αx2 with βx2< b < βx3, γx3< c. 2.5

The following lemma can be found in21.

Lemma 2.4see 21. Let P be a cone in a real Banach spaceE. Letα,β, andγ be increasing, nonnegative continuous functional onP such that for somec >0 andH > 0, γx≤ βx ≤αx andx ≤Hγxfor allx∈Pγ, c.Suppose that there exist positive numbersaandbwitha < b < c andA:Pγ, c → Pis a completely continuous operator such that:

iγAx> cfor allx∈∂Pγ, c;

iiβAx< bfor allx∈∂Pβ, b;

iiiPα, a/∅andαAx> aforx∈∂Pα, a,

thenAhas at least three fixed pointsx1, x2, andx3belonging toPγ, csuch that

0≤αx1< a < αx2 with βx2< b < βx3, γx3< c. 2.6

Letβandφ be nonnegative continuous convex functionals onP,λ is a nonnegative continuous concave functional on P, and ϕ is a nonnegative continuous functional, respectively onP.We define the following convex sets:

P

φ, λ, b, d

x∈P :b≤λx, φx≤d , P

φ, β, λ, b, c, d

x∈P :b≤λx, βx≤c, φx≤d

, 2.7

and a closed setRφ, ϕ, a, d {x∈P:a≤ϕx, φx≤d}.

Finally, we list the fixed point theorem due to Avery-Peterson27.

Lemma 2.5see27. LetP be a cone in a real Banach spaceE and β, φ, λ, ϕdefined as above, moreover,ϕsatisfiesϕλx≤λϕxfor 0≤λ≤1 such that, for some positive numbershandd,

λx≤ϕx, x ≤hφx 2.8

for allx∈ Pφ, d.Suppose thatA : Pφ, d → Pφ, dis completely continuous and there exist positive real numbersa,b,c, witha < bsuch that

i{x∈Pφ, β, λ, b, c, d:λx> b}/∅andλAx> bforx∈Pφ, β, λ, b, c, d;

iiλAx> bforx∈Pφ, λ, b, dwithβAx> c;

iii0/∈Rφ, ϕ, a, dandλAx< afor allx∈Rφ, ϕ, a, dwithϕx a, thenAhas at least three fixed pointsx1, x2, x3∈Pφ, dsuch that

φxi≤d fori1,2,3, b < λx1, a < ϕx2, λx2< b withϕx3< a.

2.9

3. Single or Twin Solutions

Let

f0 lim

u→0

fu

ϕpu, f∞ lim

u→ ∞

fu

ϕpu. 3.1

We definei0 number of zeros in the set {f0, f∞}andi∞ number of infinities in the set {f0, f_∞}. Clearly,i0,i_∞0,1,or 2 and there exist six possible cases:ii0 1 andi_∞ 1;ii i0 0 andi∞ 0;iiii0 0 andi∞ 1;ivi0 0 andi∞ 2;vi0 1 andi∞ 0;vi i0 2 andi∞ 0. In the following, by using Krasnosel’skii’s fixed point theorem in a cone, we study the existence of positive symmetric solutions to problem1.3under the above six possible cases.

3.1. For the Casei01andi∞1

In this subsection, we discuss the existence of single positive symmetric solution of the problem1.3underi01 andi∞1.

Theorem 3.1. Problem 1.3 has at least one positive symmetric solution in the case i0 1 and i∞1.

Proof. We divide the proof into two cases.

Case 1 f0 0 and f_∞ ∞. In view of f0 0, there exists an H1 > 0 such that fu ≤ ϕpεϕpu ϕpεu for u ∈ 0, H1, where ε arbitrary small and satisfies 0 <

εT/2ϕq_T/2

0 hr∇r≤1.

Ifu∈P withuH1,then

Au sup

t∈0,TT

|Au|

Au T

2

_T/2

0

ϕq

_T/2

s

hrfur∇r

Δs

≤εu _T/2

0

ϕq

_T/2

0

hr∇r

Δs

≤ uεT 2 ϕq

_T/2

0

hr∇r

≤ u.

3.2

We letΩH1 {u∈E:u< H1},thenAu ≤ uforu∈P∩∂ΩH1.

From f∞ ∞, there exists an H₂ > 0 such that fu ≥ ϕpkϕpu ϕpku for u∈H₂,∞, wherek >0,and satisfies the following inequality:

2kη² T ϕq

_T/2

η

hr∇r

≥1. 3.3

Set

H2max

2H1, H₂ T 2η

, ΩH2 {u∈E:u< H2}. 3.4

Ifu∈PwithuH2,then, byLemma 2.1, one has

t∈η,T/2min _Tut≥ 2η T u

T 2

≥H₂. 3.5

Foru∈P∩∂ΩH2,in terms of3.3and3.5, we get Au sup

t∈0,T_T|Au|

≥Au η

_η

0

ϕq

_T/2

s

hrfur∇r

Δs

≥ _η

0

ϕq

_T/2

η

hrfur∇r

Δs

≥ _η

0

ϕq

_T/2

η

hrϕpkur∇r

Δs

≥ 2kη² T uϕq

_T/2

η

hr∇r

≥ u.

3.6

Thus, byiofLemma 2.2, problem1.3has at least single positive symmetric solutionuin P∩ΩH2\ΩH1withH1≤ u ≤H2.

Case 2 f0 ∞ and f∞ 0. Since f0 ∞, there exists an H3 > 0 such that fu ≥ ϕpmϕpu ϕpmuforu∈0, H3,wheremis such that

2mη² T ϕq

_T/2

η

hr∇r

≥1. 3.7

Ifu∈PwithuH3,then, by3.7, one has Au sup

t∈0,TT

|Au|

≥ _η

0

ϕq

_T/2

η

hrfur∇r

Δs

≥ηϕq

_T/2

η

hrϕpmur∇r

≥ηm2η T uϕq

_T/2

η

hr∇r

≥ u.

3.8

If we letΩH3{u∈E:u< H3},thenAu ≥ uforu∈P∩∂ΩH3.

Now, we considerf∞0.By definition, there existsH₄ >0 such that

fu≤ϕpδϕpu ϕpδu foru∈ H₄,∞

, 3.9

whereδ >0 satisfies

δT 2 ϕq

_T/2

0

hr∇r

≤1. 3.10

Suppose thatf is bounded, thenfu ≤ ϕpKfor allu ∈ 0,∞and some constant K >0.Pick

H4max

2H3,KT 2 ϕq

_T/2

0

hsΔs

. 3.11

Ifu∈PwithuH4,then

AuAu T

2

≤ _T/2

0

ϕq

_T/2

0

hrfur∇r

Δs

≤KT 2ϕq

_T/2

0

hsΔs

≤H4

u.

3.12

Suppose thatf is unbounded. Fromf ∈ C0, ∞,0, ∞,we havefu ≤ C3 for arbitraryu∈0, C4,hereC3andC4are arbitrary positive constants. This implies thatfu → ∞ifu → ∞. Hence, it is easy to know that there existsH4 ≥ max{2H3,T/2ηH₄}such

thatfu≤fH4foru∈0, H4.Ifu∈P withuH4,then by using3.9and3.10, we have

AuAu T

2

_T/2

0

ϕq

_T/2

s

hrfur∇r

Δs

≤ _T/2

0

ϕq

_T/2

0

hrfH4∇r

Δs

≤δH4T 2ϕq

_T/2

0

hr∇r

≤ u.

3.13

Consequently, in either case, if we takeΩH4 {u∈E:u< H4},then, foru∈P∩∂ΩH4,we haveAu ≤ u. Thus, conditioniiofLemma 2.2is satisfied. Consequently, problem1.3 has at least single positive symmetric solutionuinP∩ΩH4\ΩH3withH3≤ u ≤H4.The proof is complete.

3.2. For the Casei00andi∞0

In this subsection, we discuss the existence of positive symmetric solutions to problems1.3 underi00 andi∞0 .

First, we will state and prove the following main result of problem1.3.

Theorem 3.2. Suppose that the following conditions hold:

ithere exists constant p > 0 such that fu ≤ ϕppΛ1 for u ∈ 0, p, where Λ1 {T/2ϕq_T/2

0 hr∇r}⁻¹;

iithere exists constant q > 0 such that fu ≥ ϕpqΛ2foru ∈ 2η/Tq, q,where Λ2{ηϕq_T/2

η hr∇r}⁻¹,furthermore,p/q,

then problem1.3has at least one positive symmetric solutionusuch thatulies betweenpandq.

Proof. Without loss of generality, we may assume thatp< q.

LetΩp{u∈E:u< p}.For anyu∈P∩∂Ωp,in view of conditioni, we have AuAu

T 2

_T/2

0

ϕq

_T/2

s

hrfur∇r

Δs

≤pΛ1T 2ϕq

_T/2

0

hr∇r

p,

3.14

which yields

Au ≤ u foru∈P∩∂Ωp. 3.15 Now, setΩq{u∈E:u< q}.Foru∈P∩∂Ωq,Lemma 2.1implies that

2η

T q≤ut≤q fort∈

η,T 2

T. 3.16

Hence, by conditioniiwe get AuAu

T 2

≥ _η

0

ϕq

_T/2

η

hrfur∇r

Δs

≥qΛ2ηϕq

_T/2

η

hr∇r

q.

3.17

So, if we takeΩq {u∈E:u< q},then

Au ≥ u, u∈P∩∂Ωq. 3.18 Consequently, in view ofp < q,3.15and3.18, it follows fromLemma 2.2that problem 1.3has a positive symmetric solutionuinP∩Ωq\Ωp.The proof is complete.

3.3. For the Casei01andi∞0ori00andi∞1

In this subsection, under the conditionsi0 1 andi∞0 ori00 andi∞1,we discuss the existence of positive symmetric solutions to problem1.3.

Theorem 3.3. Suppose thatf0 ∈ 0, ϕpΛ1and f∞ ∈ ϕpT/2ηΛ2,∞ hold. Then problem 1.3has at least one positive symmetric solution.

Proof. It is easy to see that under the assumptions, conditionsiandiiinTheorem 3.2are satisfied. So the proof is easy and we omit it here.

Theorem 3.4. Suppose thatf0 ∈ϕpT/2ηΛ2,∞andf_∞∈0, ϕpΛ1hold, then problem1.3 has at least one positive symmetric solution.

Proof. Firstly, letε1f0−ϕpT/2ηΛ2>0,there exists a sufficiently smallq>0 that satisfies fu

ϕpu ≥f0−ε1ϕp

T 2ηΛ2

foru∈ 0, q

. 3.19

Thus,u∈2η/Tq, q,we have

fu≥ϕp

T 2ηΛ2

ϕpu≥ϕp

Λ2q

, 3.20

which implies that conditioniiinTheorem 3.2holds.

Nextly, forε2ϕpΛ1−f∞>0,there exists a sufficiently largep > qsuch that fu

ϕpu ≤f∞ ε2 ϕpΛ1 foru∈ p,∞

. 3.21

We consider two cases.

Case 1. Assume thatfis bounded, that is,

fu≤ϕpK1 foru∈0,∞, 3.22

hereK1 > 0 some constant. If we take sufficiently large psuch thatp ≥ max{K1/Λ1, p}, then

fu≤ϕpK1≤ϕp

Λ1p

foru∈ 0, p

. 3.23

Consequently, from the above inequality, conditioniofTheorem 3.2is true.

Case 2. Assume thatfis unbounded.

Fromf∈C0,∞,0,∞,there existsp> psuch that fu≤f

p

foru∈ 0, p

. 3.24

Sincep> p, by3.21, we getfp≤ϕpΛ1p,hence

fu≤f p

≤ϕp

Λ1p

foru∈ 0, p

. 3.25

Thus, conditioniofTheorem 3.2is fulfilled.

Consequently,Theorem 3.2implies that the conclusion of this theorem holds.

From the proof of Theorems 3.1 and 3.2, respectively, we have the following two results.

Corollary 3.5. Suppose thatf00 and condition (ii) inTheorem 3.2hold, then problem1.3has at least one positive symmetric solution.

Corollary 3.6. Suppose thatf∞ 0 and condition (ii) inTheorem 3.2hold, then problem1.3has at least one positive symmetric solution.

Theorem 3.7. Suppose thatf0 ∈0, ϕpΛ1andf_∞ ∞hold, then problem1.3has at least one positive symmetric solution.

Proof. First, in view off∞∞,then by inequality3.7, we haveAu ≥ uforu∈P∩∂ΩH2. Next, byf0∈0, ϕpΛ1,forε3ϕpΛ1−f0>0,there exists a sufficiently smallp∈0, H2 such that

fu≤

f0 ε3

ϕpu ϕpΛ1u≤ϕp

Λ1p

foru∈ 0, p

, 3.26

which implies thatiofTheorem 3.2holds, that is,3.14is true. Hence, we obtainAu ≤ uforu∈P∩∂Ωp.The result is obtained and the proof is complete.

Theorem 3.8. Suppose thatf0 ∞andf_∞ ∈0, ϕpΛ1hold, then problem1.3has at least one positive symmetric solution.

Proof. On one hand, sincef0∞,by inequality3.9, one getsAu ≥ u,u∈P∩∂ΩH3.On the other hand, sincef_∞ ∈ 0, ϕpΛ1,from the technique similar to the second part proof inTheorem 3.4, one obtains that conditioniofTheorem 3.2is satisfied, that is, inequality 3.14holds, one hasAu ≤ u,u∈P ∩∂Ωp,wherep> H3.Hence, problem1.3has at least one positive symmetric solution. The proof is complete.

From Theorems 3.7 and 3.8, respectively, it is easy to obtain the following two corollaries.

Corollary 3.9. Assume thatf∞ ∞and condition (i) inTheorem 3.2hold, then problem1.3has at least one positive symmetric solution.

Corollary 3.10. Assume thatf0 ∞and condition (i) inTheorem 3.2hold, then problem1.3has at least one positive symmetric solution.

3.4. For the Casei00andi∞2ori02andi∞0

In this subsection, underi0 0 andi_∞ 2 or i0 2 andi_∞ 0,we study the existence of multiple positive solutions to problems1.3.

Combining the proofs of Theorems3.1and3.2, it is easy to prove the following two theorems.

Theorem 3.11. Suppose thati0 0 andi∞2 and condition (i) ofTheorem 3.2hold, then problem 1.3has at least two positive solutionsu1, u2∈Psuch that 0<u1< p<u2.

Theorem 3.12. Suppose thati0 2 andi_∞0 and condition (ii) ofTheorem 3.2hold, then problem 1.3has at least two positive solutionsu1, u2∈Psuch that 0<u1< q<u2.

4. Triple Solutions

In the previous section, we have obtained some results on the existence of at least single or twin positive symmetric solutions to problem1.3. In this section, we will further discuss the existence of positive symmetric solutions to problem1.3by using two different methods.

And the conclusions we will arrive at are different with their own distinctive advantages.

Based on the obtained symmetric solution position and local properties, we can only get some local properties of solutions by using method one; however, the position of solutions is not determined. In contrast, by means of method two, we cannot only get some local properties of solutions but also give the position of all solutions, with regard to some subsets of the cone, which has to meet some conditions which are stronger than those of method one.

Obviously, the local properties of obtained solutions are different by using the two different methods. Hence, it is convenient for us to comprehensively comprehend the solutions of the models by using the two different techniques.

InSection 5, two examples are given to illustrate the differences of the results obtained by the two different methods.

For the notational convenience, we denote

Mξηϕq

_T/2

0

hr∇r

, Nξ ηϕq

_T/2

η

hr∇r

,

Lξrϕq

_T/2

0

hr∇r

, Lθrϕq

_T/2

r

hr∇r

, Wξ T

2ϕq

_T/2

0

hr∇r

. 4.1

4.1. Result 1

In this subsection, in view of the generalized Avery-Henderson fixed-point theorem26, the existence criteria for at least triple and arbitrary odd positive symmetric solutions to problems 1.3are established.

Foru∈P,we define the nonnegative, increasing, continuous functionalsγ, β, andαby

γu max

t∈0,η_Tut u η

, βu min

t∈η,T/2_Tut u η

,

αu max

t∈0,r_Tut ur. 4.2

It is obvious thatγu≤βu≤αufor eachu∈P.ByLemma 2.1, one obtainsu ≤C^∗γu for allu∈P, hereC^∗T/2η.

We now present the results in this subsection.

Theorem 4.1. If there are positive numbersa,b,csuch thata<2r/Tb<2r/TcNξ/Mξ. In addition,fusatisfies the following conditions:

ifu< ϕpc/Mξforu∈0,T/2ηc; iifu> ϕpb/Nξforu∈b,T/2ηb; iiifu< ϕpa/Lξforu∈0,T/2ra.

Then problem1.3has at least three positive symmetric solutionsu1,u2, andu3such that

0< max

t∈0,rT

u1t< a< max

t∈0,rT

u2t,

t∈η,T/2min _Tu2t< b< min

t∈η,T/2_Tu3t, max

t∈0,η_Tu3t< c. 4.3

Proof. By the definition of completely continuous operatorAand its properties, it has to be demonstrated that all the conditions ofLemma 2.3hold with respect toA.It is easy to obtain thatA:Pγ, c → P.

Firstly, we verify that ifu∈∂Pγ, c, thenγAu< c. Ifu∈∂Pγ, c,then

γu max

t∈0,η_Tut u η

c. 4.4

Lemma 2.1implies that

u ≤ T 2ηu

η T

2ηc, 4.5

we have

0≤ut≤ T

2ηc, t∈

0,T 2

T. 4.6

Thus, by conditioni, one has

γAu max

t∈0,ηT

Aut Au

η

_η

0

ϕq

_T/2

s

hrfur∇r

Δs

≤ _η

0

ϕq

_T/2

0

hrfur∇r

Δs

< ηϕq

_T/2

0

hrϕp

c Mξ

∇r

c Mξηϕq

_T/2

0

hr∇r

c.

4.7

Secondly, we show thatβAu> bforu∈∂Pβ, b.

If we chooseu∈∂Pβ, b,thenβu min_t∈η,T/2Tut b.In view ofLemma 2.1, we have

u ≤ T 2ηu

η T

2ηb. 4.8

So

b≤ut≤ T

2ηb, t∈

η,T 2

T. 4.9

Using conditionii, we get

βAu Au

η

≥ _η

0

ϕq

_T/2

η

hrfur∇r

Δs

> ηϕq

_T/2

η

hrϕp

b Nξ

∇r

> b Nξηϕq

_T/2

η

hr∇r

b.

4.10

Finally, we prove thatPα, a/∅andαAu< afor allu∈∂Pα, a.

In fact, the constant function a/2 ∈ Pα, a.Moreover, for u ∈ ∂Pα, a,we have αu max_t∈0,rTut a,which implies 0≤ut≤afort∈0, r_T.Hence,u ≤T/2rur.

Therefore

0≤ut≤ T

2ra, t∈

0,T 2

T. 4.11

By using assumptioniii, one has

αAu Aur

_r

0

ϕq

_T/2

s

hrfur

∇rΔs

≤ _r

0

ϕq

_T/2

0

hrfur∇r

Δs

≤ _r

0

ϕq

_T/2

0

hrϕp

a Lξ

∇r

Δs

a Lξrϕq

_T/2

0

hr∇r

a.

4.12

Thus, all the conditions inLemma 2.3are satisfied. FromH1and H2, we have that the solutions to problem 1.3 do not vanish identically on any closed subinterval of 0, T_T. Consequently, problem1.3 has at least three positive symmetric solutionsu1,u2, and u3

belonging toPγ, c,and satisfying4.3. The proof is complete.

FromTheorem 4.1, we see that, when assumptions asi,ii, and iiiare imposed appropriately onf, we can establish the existence of an arbitrary odd number of positive symmetric solutions to problem1.3.

Theorem 4.2. Letl1,2, . . . , n.Suppose that there exist positive numbersa_sl,b_sl,c_sl such that

a_s1< 2r

T b_s1 < 2r T

Nξ

Mξc_s1< a_s2< 2T

r b_s2< 2r T

Nξ

Mξc_s2 < a_s3 <· · ·< a_sl. 4.13 In addition,fusatisfies the following conditions:

ifu< ϕpc_sl/Mξforu∈0,T/2ηc_sl; iifu> ϕpb_sl/Nξforu∈b_sl,T/2ηb_sl; iiifu< ϕpa_sl/Lξforu∈0,T/2ra_sl.

Then problem1.3has at least 2l 1 positive symmetric solutions.

Proof. Whenl1,it is clear thatTheorem 4.1holds. Then we can obtain at least three positive symmetric solutionsu1, u2, andu3satisfying

0< max

t∈0,rT

u1t< as1< max

t∈0,rT

u2t,

ti∈η,T/2min _Tu2t< bs1< min

ti∈η,T/2_Tu3t, max

t∈0,η_Tu3t< cs1. 4.14 Following this way, we finish the proof by induction. The proof is complete.

UsingLemma 2.4, it is easy to have the following results.

Theorem 4.3. Suppose that there are positive numbers a, b, c such that a < Lθ/Mξb <

2η/TLθ/Mξc.In addition,fusatisfies the following conditions:

ifu> ϕpc/Nξforu∈c,T/2ηc; iifu< ϕpb/Mξforu∈0,T/2ηb; iiifu> ϕpa/Lθforu∈a,T/2ra.

Then problem1.3has at least three positive symmetric solutionsu1,u2, andu3such that 0< max

t∈0,r_Tu1t< a< max

t∈0,r_Tu2t,

t∈η,T/2min Tu2t< b< min

t∈η,T/2Tu3t, max

t∈0,ηTu3t< c. 4.15 FromTheorem 4.3, we can obtainTheorem 4.4andCorollary 4.5.

Theorem 4.4. Letl1,2, . . . , n.Suppose that there existence positive numbersa_λ

l,b_λ

l,c_λ

lsuch that

a_λ1< Lθ

Mξb_λ1 < 2η T

Lθc_λ

1

Nξ < a_λ2< Lθ

Mξb_λ2< 2η T

Lθc_λ

2

Nξ < a_λ3 <· · ·< a_λl. 4.16 In addition,fusatisfies the following conditions:

ifu> ϕpc_λ

l/Nξforu∈c_λ

l,T/2ηc_λ

l;

iifu< ϕpb_λl/Mξforu∈0,T/2ηb_λl; iiifu> ϕpa_λ

l/Lθforu∈a_λ

l,T/2ra_λ

l.

Then problem1.3has at least 2l 1 positive symmetric solutions.

Corollary 4.5. Assume thatfsatisfies the following conditions:

if0∞, f_∞∞,

iithere existsc0>0 such thatfu< ϕp2η/Tc0/Mξforu∈0, c0, then problem1.3has at least three positive symmetric solutions.

Proof. First, by conditionii, letb 2η/Tc0,one gets fu< ϕp

b Mξ

foru∈

0, T 2ηb

, 4.17

which implies thatiiofTheorem 4.3holds.

Second, chooseK3sufficiently large to satisfy

K3LθK3rϕq

_T/2

r

hr∇r

>1. 4.18

Sincef0∞,there existsr₁ >0 sufficiently small such that fu≥ϕpK3ϕpu ϕpK3u foru∈

0, r₁

. 4.19

Without loss of generality, supposer₁ ≤LθT/2rMξb.Choosea>0 such thata<2r/Tr₁. Fora≤u≤T/2ra,we haveu≤r₁ anda<Lθ/Mξb.Thus, by4.18and4.19, we have

fu≥ϕpK3u≥ϕpK3a> ϕp

a Lθ

foru∈

a, T 2ra

, 4.20

this implies thatiiiofTheorem 4.3is true.

Third, chooseK2sufficiently large such that

K2NξK2ηϕq

_T/2

η

hr∇r

>1. 4.21

Sincef_∞∞,there existsr₂ >0 sufficiently large such that

fu≥ϕpK2ϕpu ϕpK2u foru≥r₂. 4.22 Without loss of generality, supposer₂ >T/2ηb.Choosecr₂.Then

fu≥ϕpK2u≥ϕpK2c> ϕp

c Nξ

foru∈

c, T 2ηc

, 4.23

which means thatiofTheorem 4.3holds.

From above analysis, we get

0< a< Lθ

Mξb< 2η T

Lθc

Mξ, 4.24

then, all conditions in Theorem 4.3 are satisfied. Hence, problem 1.3 has at least three positive symmetric solutions.

In terms ofTheorem 4.1, we also have the following corollary.

Corollary 4.6. Assume thatfsatisfies conditions if00, f∞0;

iithere existsc0>0 such thatfu> ϕp2η/Tc0/Nξforu∈2η/Tc0, c0, then problem1.3has at least three positive symmetric solutions.

4.2. Result 2

In this subsection, the existence criteria for at least triple positive or arbitrary odd positive symmetric solutions to problems1.3are established by using the Avery-Peterson fixed point theorem27.

Define the nonnegative continuous convex functionals φ and β, nonnegative continuous concave functionalλ,and nonnegative continuous functionalϕ, respectively, on Pby

φu max

t∈0,T/2_Tut u T

2

, βu max

t∈r,T/2_Tκ

u^Δtu^Δr,

λu ϕu min

t∈η,T/2_Tut u η

.

4.25

Now, we list and prove the results in this subsection.

Theorem 4.7. Suppose that there exist constants a^∗, b^∗, d^∗ such that 0 < a^∗ < 2η/Tb^∗ <

2η/TNξd^∗/Wξ.In addition, suppose thatWξ > ϕq_T/2

η hs∇sholds,fsatisfies the following conditions:

ifu≤ϕpd^∗/Wξforu∈0, d^∗; iifu> ϕpb^∗/Nξforu∈b^∗, d^∗; iiifu< ϕpa^∗/Mξforu∈0,T/2ηa^∗,

then problem1.3has at least three positive symmetric solutionsu1, u2, u3such that

ui ≤d^∗, i1,2,3, b^∗< u1

η

, u2

η

< b^∗, u3

η

< a^∗. 4.26

Proof. By the definition of completely continuous operatorAand its properties, it suffices to show that all the conditions ofLemma 2.5hold with respect toA.

For allu∈P, λu ϕuanduuT/2 φu. Hence, condition2.8is satisfied.

Firstly, we show thatA:Pφ, d^∗ → Pφ, d^∗.

For anyu∈Pφ, d^∗,in view ofφu u ≤d^∗and assumptioni, one has AuAu

T 2

_T/2

0

ϕq

_T/2

s

hrfur∇r

Δs

≤ _T/2

0

ϕq

_T/2

0

hrfur∇r

Δs

≤ d^∗ Wξ

T 2ϕq

_T/2

0

hr∇r

d^∗.

4.27

From the above analysis, it remains to show thati–iiiofLemma 2.5hold.

Secondly, we verify that conditioni of Lemma 2.5holds, let ut ≡ kb^∗ with k Wξ/Nξ>1.From the definitions ofNξ, Wξ, andβu, respectively, it is easy to see thatut kb^∗> b^∗andβu 0< kb^∗. In addition, in view ofb^∗<Nξ/Wξd^∗, we haveφu kb^∗< d^∗. Thus

u∈P

φ, β, λ, b^∗, kb^∗, d^∗

:λx> b^∗

/∅. 4.28

For anyu∈ Pφ, β, λ, b^∗, kb^∗, d^∗,then we getb^∗ ≤ ut≤ d^∗for allt∈ η, T/2_T. Hence, by assumptionii, we have

λAu Au

η

_η

0

ϕq

_T/2

s

hrfur∇r

Δs

≥ _η

0

ϕq

_T/2

η

hrfur∇r

Δs

> b^∗ Nξηϕq

_T/2

η

hr∇r

b^∗.

4.29

Thirdly, we prove that conditioniiofLemma 2.5holds. For anyu ∈ Pφ, λ, b^∗, d^∗ withβAu> kb^∗,that is,

βAu Au^Δrϕq

T/2 r

hsfus∇s

> kb^∗. 4.30

So, in view ofkWξ/Nξ,Wξ> ϕq_T/2

η hs∇sand4.30, one has

λAu Au

η

_η

0

ϕq

T/2 s

hrfur∇r

Δs

≥ _η

0

ϕq

_T/2

η

hrfur∇r

Δs

>

_η

0

ϕq

_T/2

r

hrfur∇r

Δs

> ηkb^∗

> b^∗.

4.31

Finally, we check conditioniiiofLemma 2.5. Clearly, sinceϕ0 0 < a^∗,we have 0/∈Rφ, ϕ, a^∗, d^∗.Ifu ∈ Rφ, ϕ, a^∗, d^∗with ϕu min_t∈η,T/2Tut a^∗,then Lemma 2.1 implies that

u ≤ T 2ηu

η T

2ηa^∗. 4.32

This yields 0≤ut≤T/2ηa^∗for allt∈0, T/2_T.Hence, by assumptioniii, we have

λAu Au

η

≤ _η

0

ϕq

_T/2

0

hrfur∇r

Δs < a^∗ Mξηϕq

_T/2

0

hr∇r

a^∗. 4.33

Consequently, all conditions ofLemma 2.5are satisfied. The proof is completed.

We remark that conditioniinTheorem 4.7can be replaced by the following condition i:

ulim→ ∞

fu ϕpu ≤ϕp

1 Wξ

, 4.34

which is a special case ofi.

Corollary 4.8. If condition (i) inTheorem 4.7is replaced by (i), then the conclusion ofTheorem 4.7 also holds.

Proof. By Theorem 4.7, we only need to prove thati’implies thatiholds, that is, ifi’

holds, then there is a numberd^∗ ≥max{T/2ηa^∗,Wξ/Nξb^∗}such thatfu≤ ϕpd^∗/Wξ foru∈0, d^∗.