Positive solutions of discrete φ-Laplacianproblems

2017-05-18,

中山大学学报(自然科学版)(中英文) 2017年1期

( 1. Department of Information Processing and Control Engineering, Lanzhou Petrochemical College of Vocational Technology, Lanzhou 730060, China; 2. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China)

Positive solutions of discreteφ-Laplacianproblems

LIYanming1,BAIDingyong2

discreteφ-Laplacianproblem;positivesolution;semipositoneproblem

Fora,b∈Ζwitha

L[u](·):=Δ(φ(Δu(·)))

whereΔdenotesforwarddifferenceoperator,i.e., Δu(k)=u(k+1)-u(k),k∈[a,b-1]Z.Weconsiderthepositivesolutionsofthefollowingdiscreteφ-Laplacianboundaryvalueproblem

(1)

whereT>5isagivenpositiveinteger,λisanonnegativeparameterandp(k)>0on[1,T]Z.

Recently, the positive solutions ofφ-Laplacianboundaryvalueproblems(i.e.,generalizedp-Laplacian boundary value problems) have been widely studied. For differentialφ-Laplacianproblems,wereferreaderto[1-8]forsomereferences.Forthediscretecase,therearelessstudyresultsthancontinuouscase[9-10].WealsorefertoCabada[11]andCabadaandEspinar[12]fortheresultsofupperandlowersolutionsmethod,andBondar[13]fortheexistenceanduniquenessresultsbythefixedpointtheoryofcontractionmapping.In[9-10],theexistenceandmultiplicityofpositivesolutionsofdiscreteφ-Laplacianboundaryvalueproblemswerestudiedandfwasassumedtobenonnegative.Inthispaper,weconsiderthesemipositonecase(i.e.thecasef(0)<0).Ourinterestistheexplicitopenintervalsofλsuchthat(1)hasatleastonepositivesolution.

(C1)φ-1isconcaveonR+, and for eachδ>0,thereexistsAδ>0suchthatφ-1(δu)≥Aδφ-1(u),u∈R+andlimδ→∞Aδ=∞.

Inourdiscussionstoproblem(1),theconcavityofφ-1isreservedandthefollowingconditionisneeded.

(C2)Thereexistincreasinghomeomorphismsψ,φ:(0,∞)→(0,∞),suchthatforallμ>1,

∀x,y∈R:x≠y

Notethatcondition(C2)implies

ψ(μ)φ(x)≤φ(μx)≤φ(μ)φ(x)
forallx>0andμ>1

(2)

whichwasusedin[3,7]forallx>0,μ>0.Byusingtheinequality(2),theφ-superlinearconditionimposedonfin[2]canbeweakenedbyamoregeneralcondition(see(A2)below).Now,westateourassumptionsasfollows.

(A1)f:R→R is continuous and there existsM>0suchthatf(u)≥-Mholdsforallu≥0;

(A3)φ-1isconcaveonR+.

The following fixed point theorem in cones will be used to prove our main results.

Denote

and

1 Preliminaries

Lemma 2 Leta,b∈Ζwitha

(3)

thenu>von[a,b]Z.

φ(Δ(u-q)(s-1))-φ(Δ(v-q)(s-1))=

φ[Δ(v-q)(k*)]>0

fors∈[a,k*]ZsinceΔ(u-v)(k*)≥0.Thus, Δ(u-v)(s-1)>0,s∈[a,k*]Z.Itfollowsbysummingfromatok*that(u-v)(k*)>0,acontradiction.

Lemma 3 Ifwsatisfies

(4)

Proof By Lemma 2, we knoww>0on[1,T]Z.Itiseasytoseethattheuniquesolutionof(4)canbewrittenas

k∈[0,T+1]Z

and

Lemma 4[2]Let (A3) hold. The following inequalities hold.

φ-1(x+y)≤φ-1(x)+φ-1(y)+φ-1(μ)

forx≥-μ,y≥0,μ≥0,

φ-1(x-y)≥φ-1(x)-2φ-1(y)

forx≥0,y≥0.

(5)

Itiseasytoseethatproblem(5)isequivalentto

k∈[0,T+1]Z

(6)

(7)

(8)

and

(9)

Infact,ontheonehand,by

LetK={u∈E:u(k)≥0,k∈[0,T+1]Z},aconeinE.Withthehelpof(6)and(7),defineF:K→Kby

Fu(k)=

and

respectively.ThenF:K→Kiscontinuousandproblem(5)hasapositivesolutionifandonlyifFhasafixedpointinKforλ>0.

2 Main Results

Foranygivenr>0,set

(10)

Now,westateourmainresult.

Theorem 1 Let (A1)-(A3) and (C2) hold. Assume thatλ0<λrholdsforsomer>0,thenproblem(1)hasatleastonepositivesolutionforλ∈(λ0,λr).Especially,iff∞=∞,thenforeachr>0problem(1)hasonepositivesolutionforλ∈(0,λr).

Proof We divide the proof into two steps.

u(k)=θFu(k)≤

R.Suchusatisfiesthefollowingboundaryvalueproblem

(11)

(12)

Thenwehave

ItfollowsbyLemma2thatu(k)>v(k),k∈[1,k0-1]Z.Notethatthesolutionvof(12)canbeexpressedas

φ-1[Cv-φ(Δw(l-1))]+

Δw(l-1)},k∈[0,k0]Z

(13)

whereCvsatisfiesv(0)=θβ.Thus,

φ-1[Cv-φ(Δw(l-1))]+

Δw(l-1)}

(14)

ItfollowsbyLemma3and4thatfork∈[1,k0]Z,

φ-1(Cv)=φ-1[Cv-φ(Δw(k-1))+

φ(Δw(k-1))]≤

φ-1(Cv-φ(Δw(k-1)))+

Thus,by(10),wehavethatfork∈[1,k0]Z,

φ-1(Cv-φ(Δw(k-1)))+Δw(k-1)≥

Itfollowsthatfork∈[k1,k0]Z,

u(k)-w(k)≥v(k)-w(k)>

sincefork∈[0,T+1]Z,

(15)

Notethat

(16)

By(2),weknow

and

Thusweagainhave

(17)

φ(Δw(l-1))]-

Δw(l-1)},k∈[k0,T+1]Z

(18)

φ(Δw(l-1))]-Δw(l-1)}

(19)

Notethat

(i)If1≤k≤k0,weusetheexpression(6),i.e.,

(20)

Fork∈[0,k0]Z,let

Itiseasytoseethat

isnonincreasingon[1,k0]Z.ThatistosaythatGisadiscreteconcavefunctionon[0,k0].Thenwehave

(21)

From(20)andLemma4wehavethat

φ(Δw(s-1)))+w(k0)≤

Itfollowsby(10)that

(22)

k∈[1,k0]Z

(ii)Thesecondcasek0≤k≤Tcanbesimilarlydealtwith.Weusetheexpression(7),i.e.,

k∈[0,T+1]Z

(23)

Fork∈[k0,T+1]Z,let

k∈[k0,T+1]Z

Thatistosay

(24)

Thuswehaveby(23)andLemma4that

Itfollowsby(10)that

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离散φ-Laplace问题的正解*

2016-05-18 基金项目：国家自然科学基金(11371107)

李延明(1973年生)，男；研究方向：离散动力系统；E-mail:liym@lzpcc.edu.cn

白定勇(1972年生)，男；研究方向：泛函微分方程;E-mail:baidy@gzhu.edu.cn

李延明1，白定勇2

(1. 兰州石化职业技术学院信息处理与控制工程系, 甘肃兰州 730060; 2. 广州大学数学与信息科学学院, 广东广州 510006)