利用锥映射的不动点指数证明了含参数λ＞０的Ｓｔｕｒｍ Ｌｉｏｕｖｉｌｅ边值问题至少存在两个正解大庆石油学院自然科学基

中新, 张

令彬, 孔

Xiamen University Institutional Repository

大 庆 石 油 学 院 学 报 第 22 卷　第 3 期 　1998 年 9 月JOURNAL OF DAQING PETROLEUM INSTITUTE Vol.22　　No.3　　Sept.　1998Sturm-Liouville边值问题的多解结果①②孔令彬1)　张中新2)1)　大庆石油学院数学系 ,安达 , 151400　　2)　长春邮电学院基础部 ,长春 , 130012　　摘　要　利用锥映射的不动点指数证明了含参数λ>0的 Sturm-Liouville边值问题至少存在两个正解.主题词　Sturm-Liouville 边值问题;正解;多解;锥;不动点指数分类号　O 175.14第一作者简介　孔令彬 ,男 , 1956年生.硕士.副教授.1982年毕业于大庆石油学院数学专业.现从事微分方程方面的研究.　　Sturm-Liouville 边值问题[ p(t)u′]′+λf(t , u)=0 , 　0 < t <1 (1)au(0)-b lim　t ※0p(t)u′(t)=0 , 　cu(1)+d lim　t ※1p(t)u′(t)=0 (2)来源于边界层理论与应用数学领域之中[ 1 , 2],此处参数 λ>0 ,实常数 a , b , c , d ≥0 ,并满足 ad +bc+ac >0 , f(t , u)是[ 0 ,1] ×[ 0 , ∞)上的非负连续函数.关于边值问题(1),(2)的研究成果 ,有变号解的存在性[ 3 ,4]和正解的存在性[ 1 , 2 , 5～ 7].Erbe 等研究了当λ=1且 p(t)=1的一个特殊情形 ,给出了多解存在性结果[ 8].文中则对 λ>0且 p(t) 1的情形 ,证明了问题(1),(2)至少存在两个正解.假设(H1)　p(t)是[ 0 ,1]上的非负可测函数 ,当 t ∈(0 ,1)时 , p(t)>0并满足∫10dtp(t)<+∞;　　(H2)　limu※0maxt ∈ [ 0 , 1]f(t , u)u=0 , 　 limu ※+∞maxt ∈ [ 0 , 1]f(t , u)u=0;　　(H2)′　limu※0mint ∈ [ 0 , 1]f(t , u)u=+∞, 　 limu※+∞mint ∈ [ 0 , 1]f(t , u)u=+∞;　　(H3)　存在常数 l 1>0和M1>0 ,使当 u≥l1 时 ,有 f(t , u)≥M1;(H3)′　存在常数 l 2>0和M2>0 ,使当 0≤u≤l 2 时 ,有 f(t , u)≤M2.称函数 u(t)是边值问题(1),(2)的一个正解 ,如果它满足:(i)u(t)∈C[ 0 , 1] ∩ C1(0 ,1)并在(0 ,1)内u(t)>0;(ii)p(t)u′(t)在[ 0 ,1]上是绝对连续函数;(iii)u(t)于(0 , 1)内几乎处处满足方程(1)并满足条件(2).由边值问题(1),(2)可导出等价的积分方程为u(t)=λ∫10G(t , s)f(s , u(s))ds式中 G(t , s)=1ρb +a∫s0dτp(τ)yed +c∫1tdτp(τ)　 22 u, 　0 ≤ s ≤ t ≤11ρb +a∫t0dτp(τ)M d +c∫1sdτp(τ)L , 　0 ≤ t ≤ s ≤in 1(3)ρ=ad +bc +ac∫10dτp(τ)>0 (4)　　引理　(1)对任何的 s , t ∈[ 0 ,1]都有 G(t , s)≤G(s , s);·71·①② 大庆石油学院自然科学基金资助项目收稿日期:1997-09-24　　　　审稿人:王守田(2)设 δ∈(0 ,14),则存在 q>0 ,使对 t ∈[ δ,1-δ]和 s ∈[ 0 ,1] ,有 G(t , s)≥qG(s , s);(3)对上述的 δ和 q ,当 s ∈[ δ,1-δ]和 t ∈[ δ,1-δ]时 ,有G(t , s)≥1ρb +a∫1-δδdτp(τ)17 d +c∫1-δδdτp(τ)ur　　证明　由 G(s , s)=1ρb +a∫s0dτp(τ)u0 d +c∫1sdτp(τ)() ,知 　　　　　　 　G(t , s)G(s , s)=[ d +c∫1tdτp(τ)] [ d +c∫1sdτp(τ)] , 　0 ≤ s ≤ t ≤1[ b +a∫t0dτp(τ)] [ b +a∫s0dτp(τ)] , 　0 ≤ t ≤ s ≤1 了当因此 ,(1)式成立.令q =min [ d +c∫11-δdτp(τ)] [ d +c∫10dτp(τ)] , 　[ b +a∫δ0dτp(τ)] [ b +a∫10dτp(τ)]ma则知(2)式成立.(3)式显然.定义映射 Tλ∶K ※K ,(Tλu)(t)=λ∫10G(t , s)f(s , u(s))ds (5)此处 K:={u ∈ C[ 0 , 1] ;u(t)≥0 ,∫1-δδu(t)dt ≥q(1-2δ)‖u ‖ ,0 <δ<14} (6)是 C[ 0 ,1]中的锥.由引理的(1),(2)知道(Tλu)(t)=λ∫10G(t , s)f(s , u(s))ds ≤λq∫10G(s , s)f(s , u(s))ds ,∫1-δδ(Tλu)(t)dt =∫1-δδ[ λ∫10G(t , s)f(s , u(s))ds] dt ≥λq(1-2δ)∫10G(s , s)f(s , u(s))ds ,再由以上两个不等式 ,有∫1-δδ(Tλu)(t)dt ≥q(1-2δ)‖Tλu ‖ , 　δ∈ (0 ,14)因此 ,有 Tλ(K) K.另外知 Tλ是全连续的.定理 1　假设条件(H1),(H2),(H3)成立 ,则存在常数 l>0和 λ1>0 ,使当 λ>λ1 时 ,边值问题(1),(2)都至少存在两个正解 u1(t)和 u2(t),并满足 0<‖ u1 ‖<l<‖u2 ‖ .证明　取 0<δ<1 4 , l=(l 1+1) q ,令 Ψl ={u ∈K ;‖ u ‖ ≤l},再令 Eδ={t ∈ [ δ, 1-δ] ;u(t)≥l1},则对 u ∈ Ψl ,有∫1-δδu(t)dt ≥q(1-2δ)‖u ‖ =ql(1-2δ)∫1-δδu(t)dt =∫Eδu(t)dt +∫[ δ, 1-δ]  Eδu(t)d t ≤ lmesEδ+l1(1 -2δ-mesEδ)由以上两个不等式推出 mesEδ=[ (ql-l 1)(1-2δ)] (l-l 1)>1 (2l)>0再由(H3),有 ∫Eδf(s , u(s))ds ≥M1mesEδ>M1 (2l) (7)　　记 γ=1ρ(1-2δ) b +a∫δ0dτp(τ)in d +c∫11-δdτp(τ),利用引理的(3)和(7)式 ,对 u ∈ Ψl ,有‖Tλu ‖ =∫10[ maxt ∈ [ 0 ,1]λ∫10G(t , s)f(s , u(s))ds] dt ≥λ∫1-δδdt∫1-δδG(t , s)f(s , u(s))ds ≥λγ∫Ψδf(s , u(s))ds ≥λγM1 (2l2)‖ u ‖因此 ,对任何 λ>λ1=2l2 (γM1),都有‖Tλu ‖>‖u ‖ .由文献[ 8]知 , i(Tλ, Ψl ,K)=0. (8)·72·大　庆　石　油　学　院　学　报　　　　　　　　　　　　　　第 22卷　1998年记M =∫10G(s , s)ds , 由假设(H2),存在 0<r<l ,使当 0≤u ≤r时 ,有 f(t , u)<εu ,此处 ε>0 ,满足ελM<1.令 Ψr={u ∈K ;‖ u ‖ ≤r},则对 u ∈ Ψr ,有(Tλu)(t)≤ελ∫10G(s , s)u(s)ds < ‖ u ‖ , 即有‖Tλu ‖ <‖ u ‖.再由文献[ 8] ,有 i(Tλ, Ψr ,K)=1. (9)根据(H2),对上述 ε,存在 R 1>0 ,使 f(t , u)≤R1+εu ,对一切 t∈[ 0 ,1]和 u≥0成立.取 R>max{l ,λR1M (1-ελM)},并令 ΨR ={u ∈K;‖u ‖ <R},则对 u ∈ ΨR ,有(Tλu)(t)≤λ∫10G(t , s)(R1 +εu(s))ds ≤λ(R1 +εR)M <‖u ‖即有 ‖Tλu ‖ <‖ u ‖.由文献[ 8] ,有 i(Tλ, ΨR ,K)=1. (10)根据(8),(9),(10)式 ,得 i(Tλ, ΨR  Ψ。l ,K)=1 , i(Tλ, Ψl  Ψ。r , K)=1 ,因此 Tλ在 Ψl  Ψ。r 和 ΨR  Ψ。l中各有一个不动点u1(t)和 u2(t),并有 0<‖u1 ‖<l<‖ u2 ‖ .定理 2　假设条件(H1),(H2)′,(H3)′成立 ,则存在常数 λ2>0和 l2>0 ,使对任何 0<λ<λ2 ,边值问题(1),(2)都至少存在两个正解 u1(t)和 u2(t),并且有 0<‖ u1 ‖<l2<‖u2 ‖.证明　令 Ψl2={u ∈K;‖ u ‖≤l2}.对 u ∈ Ψl2,由(H3)′,有(Tλu)(t)=λ∫10G(t , s)f(s , u(s))ds ≤λMM2故对任何 0<λ<λ2=l2 (MM2),都有‖ Tλu ‖ <‖u ‖ , Tλ见式(5).由文献[ 8]知i(Tλ, Ψl2,K)=1. (11)　　由(H2)′,存在 0<r<l2 和 R 2>l 2 ,使当0≤u≤r和 u≥R 2 时都有 f(t , u)≥Nu ,此处 N>0使 λNqγ(1-2δ)>1 ,0<δ<(1 4).选择 R >R2 ,则对 u ∈ Ψr 和u ∈ ΨR ,都有‖Tλu ‖ =∫10[ maxt ∈ [ 0 ,1]λ∫10G(t , s)f(s , u(s))ds] dt ≥λ∫1-δδdt∫1-δδG(t , s)f(s , u(s))ds ≥λγN∫1-δδu(s)ds ≥λγqN(1-2δ)‖u ‖ >‖ u ‖即有 ‖Tλu ‖ >‖ u ‖.于是 i(Tλ, Ψr ,K)=0 , 　i(Tλ, ΨR ,K)=0 (12)　　　从(11),(12)式知 , i(Tλ, Ψl2 Ψ。r ,K)=1 , i(Tλ, ΨR  Ψ。l2,K)=-1.因此 , Tλ在Ψl2 Ψ。r 和ΨR  Ψ。l2中各有一个不动点 u1(t)和 u2(t),满足 0<‖u1 ‖<l2<‖u2 ‖ .参 考 文 献1　O , Regan.Singular second order boundary value problem.Nonlinear Analysis , , 1990 , 65:1079～ 11902　O , Regan.Some existence principle and some general results for nonlinear two point boundary value problem.J Math Anal Appl , 1992, 166:24～ 403　Li Yong.Singular two point boundary value problems for second order differential equations.J Math Anal Appl , 1992 , 170:245～ 2544　Dunninger , Kurtz.Existence of solutions for some nonlinear singular boundary value problems.JMath Anal Appl , 1986 , 115:396～ 4055　Wang Junyu.On positive solutions of singular nonlinear two point boundary value problems.J Dif f Eqns , 1994 , 107:163～ 1746　Anuradha V , Hai D D , Shivaji R.Existence results for superlinear semipositone BVP , s.Proc Amer Math Soc , 1996 , 124:757～ 7637　Gatica.Singular nonlinear BVP for second order ordinary differential equations.J Diff Eqns , 1989 ,79:62～ 788　Erbe L H , Hu Shouchuan , Wang Haiyan.Multiple positive solutions of some boundary value problems.J Math Anal Appl , 1994 ,184:640～ 648·73·第 3期　　　　　　　　　　　孔令彬等:Sturm-Liouville 边值问题的多解结果Measurement of Temperature Coefficient of ThermistorWang Hongbo(Dept of Electronics Engineering , Daqing Petroleum Institute , Anda , Heilongjiang , China , 151400)Abstract　In order to keep the measuring system in isolated and balanced state , isothermal field radiation method wasapplied to measure the temperature factor of thermistor .The characteristic and classification of thermistor and the de-sign idea of the measuring instrument were demonstrated.This instrument guarantees the precision of temperature mea-suring and decreases the system error caused by the non-thermal balance .In order to ensure the synchronization of theresistance and temperature of thermistor , two electric-bridge time synchronizous automatic recording instruments wereused.Finally , the measuring results were analyzed and discussed.Subject terms　thermistor , temperature factor , isothermal field radiation method , thermal balance , least squaremethodMultiple Positive Solutions to Sturm-Liouville Boundary Value ProblemKong Lingbin(Dept of Mathematics , Daqing Petroleum Institute , Anda , Heilongjiang , China , 151400)Abstract　It is proved that Sturm-Liouville boundary value problem with λ>0 has at least two positive solutions byusing the fixed point index for the mapping in cones.Subject terms　Sturm-Liouville boundary value problem , positive solution , multiple solution , cone , fixed point indexSedimentary Features of Dujiatai Reservoir of Shu 22 BlocksLiu Jiyu(Dept of Petroleum Prospecting , Daqing Petroleum Institute , Anda , Heilongjiang , China , 151400)Abstract　To gain a clear idea of the sedimentary environment and its sedimentary feature of a sand body is the basisand key to understanding the heterogeneity , evaluating the exploitation effect and making further adjustment.Dujiataireservoir of Shu 22 blocks is a set of fan delta sediment which developed on the base of complex paleostructure and pal-eotopography.It is necessary to dissect the DuⅡ8 ～ Ⅱ11 and DuⅢ3 ～ Ⅲ5 layer of the research area and make furtherstudy about its sedimentary microfacies by applying electrical assemblage , combining the development extent of thesand body with the condition of its plane and space distribution.The two major reservoirs of the research area are boththe frontal subfacies of fan delta and the subfacies of frontal fan delta.And the frontal subfacies of fan delta can alsobe subdivided into distributary microfacies , interdistributary microfacies channel mouth bar microfacies and frontal thinbedded sand microfacies.Each microfacies developed from different small layers , and each microfacies has its ownsedimentary features.Subject terms　Shu 22 blocks , sedimentary feature , sedimentary microfacies , frontal subfacies of fan delta , subfaciesof frontal fan deltaMicro Heterogeneous Characteristics of Dujiatai Reservoir of Shu 22 BlocksLiu Jiyu(Dept of Petroleum Prospecting , Daqing Petroleum Institute , Anda , Heilongjiang , China , 151400)Abstract　According to the data of core analysis as the major information , we have studied the micro heterogeneous ofDujiatai reservoir of Shu 22 blocks through the analysis of mercury-injection ,SEM , thin section of cast mold , oil bear-ing thin section and pore image.The clay mineral content of the reservoir is high and its main components are mixed-layer of illite and montmorillonite.Intergranular pore and vugular solution pore are the major pore type , and the largeor middle pore with moderate permeability and fine throat is the important type of the pore structure.Subject terms　Shu 22 blocks , micro heterogeneity , clay mineral , type of pore , pore structureMacro Heterogeneous Characteristics of Dujiatai Reservoir of Shu 22 BlocksLiu Jiyu(Dept of Petroleum Prospecting , Daqing Petroleum Institute , Anda , Heilongjiang , China , 151400)·111·Abstracts　　　　　　　Journal of Daqing Petroleum Institute　　　　　　　　Vol.22　No.3　Sept.1998

Sturm-Liouville边值问题的多解结果

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Sturm-Liouville边值问题的多解结果

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