下载文档原格式
2m阶差分方程边值问题解的存在性周展;徐菲【摘要】讨论一类2m阶非线性差分方程边值问题.通过建立相应的变分框架,将边值问题的解转换为对应的非线性泛函的临界点.利用环绕定理,获得变分泛函临界点的存在性,进而得到所求边值问题解的存在性.最后给出例子说明本文的结论.【期刊名称】《广州大学学报（自然科学版）》【年(卷),期】2016(015)004【总页数】5页(P13-17)【关键词】2m阶差分方程;环绕定理;边值问题【作者】周展;徐菲【作者单位】广州大学数学与信息科学学院,广东广州510006;广州大学数学与交叉科学广东普通高校重点实验室,广东广州510006【正文语种】中文【中图分类】O175.8差分方程在诸如物理、生态、金融等领域有着广泛的应用.众所周知，差分方程是微分方程离散化, 它与相应的微分方程有很多共同的性质，但很多差分方程与其对应的微分方程有本质不同.因此，在过去几十年里，许多学者把注意力放在差分方程周期解的存在性、振动性、边值问题等方面，获得了丰富的结果，主要方法包括上下解方法、拓扑度理论、不动点理论等经典方法[1-4].2003 年开始，GUO等开始利用临界点理论研究二阶超线性差分方程的周期解和次调和解[3]，后来，这一方法被用来研究差分方程的边值问题.设R, Z分别表示实数集和整数集.对任给的a, b∈Z且a≤b,定义Z(a,b)={a,a+1,…,b}，Z(a)={a,a+1,…}.Δ为向前的差分算子，定义为Δun=un+1-un,Δkun=Δ(Δk-1un), k∈Z(2).设T∈Z(2), 在参考文献[5]中，ATICI 等讨论了如下差分方程的周期边值问题：这里f∈C(Z(1,T)×R,R).方程(1)作为一个二阶微分方程的离散模型，被应用于很多领域，如空气动力学、核物理等.运用上下解方法，ATICI等建立了边值问题(1)存在唯一解的条件.2014年, LIU等在参考文献[6]中利用临界点理论研究了四阶差分边值问题的解的存在性与不存在性条件.其中δ表示正奇数的比，f∈C(Z(1,T)×R, R).在物理学中方程(2)经常被用来模拟弹性梁的弯曲程度.2009年，ZOU等在参考文献[7] 中利用临界点理论讨论了以下2m阶差分方程：Δm(pn-mΔmun-m)+(-1)m+1f(n,un)=0,n∈Z(1,T)在边值条件下的解的存在情况.其中 T和m是任给的正整数,且T>m.然而，可以看到大部分参考文献[5-6,8-12]都是研究二阶或者四阶差分方程的, 对一般高阶差分方程的研究相对来说较少.受文献[4-7,13]的启发，本文讨论更一般的2m阶差分方程Δm(pn-mΔmun-m)+(-1)m+1Δ(g(Δun-1))+(-1)m+1f(n,un)=0, n∈Z(1,T)在边值条件下的解的存在性.其中g∈C(R, R), f(n,·)∈C(R,R)对任意n∈Z(1,T).设m, T∈Z(1)且T>m, 定义向量空间Ω={u={un}|un∈R,n∈Z(1-m,T+m)}，对任意的u,v∈Ω,a,b∈R有au+bv={aun+bvn}.E={u={un}∈Ω|u1-i=u1,uT+i=uT,i∈Z(1,m)}是Ω的一个线性子空间.易知E与RT是同构的，因此，在空间E上可以定义内积如下：由E上的内积可以诱导空间E上的范数：对任意的r≥1, 可以定义空间E上的另一种范数：因为E是有限维空间，所以存在2个常数c2(r)≥c1(r)>0使得c1(r)‖u‖2≤‖u‖r≤c2(r)‖u‖2,∀u∈E这里u.显然J ∈C1(E,R)，其中C1(E, R)表示Hilbert空间E上Fréchet可微且其Fréchet导数是连续的泛函集合.根据E的定义, 有f(n,un),∀n∈Z(1,T).因此，u是泛函J的一个临界点当且仅当u满足边值问题(5)～(6).记u={un}∈E，由于E与RT同构，所以u可写成u=(u1,u2,…,uT)*∈RT.那么存在T×T阶矩阵A 使得显然, A是一个半正定矩阵.令σ+(A)为A的所有正特征值构成的集合.定义}.设W, Y分别为A的0特征值和所有正特征值对应的特征向量空间,则W={(u1,u2,…,uT)*∈RT|ui=w,w∈R,i∈Z(1,T)},且下面介绍一些临界点理论的基本概念和基本结果.定义1 设S是一个实Banach空间, J∈C1(S,R)满足Palais-Smale条件 (简称P.S.条件)，如果对任给的{un}⊂S,{J(un)}有界，当n→∞时J′(un)→0蕴含{un}有收敛的子列.记Bρ={y∈S: ‖y‖<ρ}是以0为中心，半径为ρ的开球，‖y‖=ρ}为Bρ的边界.引理1(环绕定理[14]) 设S=S1⨁S2是一个Hilbert空间, 其中，S1是S的一个有限维的子空间. 若J∈C1(S,R)满足P.S.条件且满足：(1)存在常数σ>0和ρ>0使得J|∂Bρ∩S2≥σ;(2)存在e∈∂B1∩S2和常数R1>ρ使得J|∂Q≤0, 其中⨁{re|0<r<R1}.那么J存在临界值c≥σ, 这里表示∂Q上的恒等算子.定理1 如果以下假设都满足:(A1)f(n,v),g(v)是关于v连续, 且g(0)=0, G(v)≥0对v∈R成立，其中n∈Z(1,T); (A2)对任给的n∈Z(1-m,T),pn>0;(A3)当n∈Z(1,T),v∈R时F(n,v)≥0且(A4)存在正常数R2和β>2使得0<βF(n,v)≤vf(n,v), n∈Z(1,T),|v|≥R2；(A5)存在正常数R3和α<β使得0<sg(s)≤αG(s),n∈Z(1,T),|s|≥R3.那么边值问题(5)～(6)至少存在2个非平凡解.注1 由(A4)知，存在正常数使得,∀(n,v)∈Z(1,T)×R.注2 由(A5)、(A1)知，存在正常数得,∀s∈R.记p*=max{pn, n∈Z(1-m,T)},p*=min{pn, n∈Z(1-m, T)}.则p*≥p*>0.为了方便定理1的证明, 需要验证下面的引理.引理2 假设(A1)～(A5)都满足, 那么泛函J满足P.S.条件.证明设{u(l)}l∈Z(1)⊂E是一个P.S.序列，则存在常数C使得|J(u(l))|≤C,∀l∈Z(1).根据式(11)，注1和注2有‖a1c1β(β)‖‖‖u(l)‖α-‖注意到J(u(l))≥-C, 则由式(13)得‖‖u(l)‖2-‖C.因为β>max{2,α}, 所以存在常数N0>0使得‖u(l)‖≤N0,∀l∈N. 因此, {u(l)}是E 上的有界序列.因为E是有限维的, 所以 {u(l)}存在收敛的子列.即J满足P.S.条件. 定理1的证明由(A3)知f(n,0)=0, n∈Z(1,T), 结合(A1)中g(0)=0知0是 J的一个临界点, 且J(0)=0.式 (13)蕴含lim‖u‖→+∞J(u)=-∞, 因此，J在E上有上界，-J是强制的.记cmax为{J(u)}的上确界,对任给的c0>|cmax|, 存在一个常数t>0,使得|J(u)|>c0>|cmax|,‖u‖>t.根据J在E上的连续性, 存在使得即是J的一个临界点. 可断定cmax>0. 事实上, 由(A3)知存在和η>0使得F(n,u)≤ε|u|2,|u|≤η.对任给的u=(u1,u2,…,uT)*∈Y，‖u‖≤η有|un|≤η,n∈Z(1,T). 因此,‖u‖2-ε‖u‖2=‖u‖2令,∀u∈Y∩∂Bη有J(u)≥σ>0,所以cmax=supu∈EJ(u)≥σ>0, 故cmax对应的临界点是边值问题(5)～(6)的一个非平凡解. 要得到另一个非平凡解可以利用引理1.由引理2 知J满足P.S.条件.其次，令S2=Y,S1=W，则E=S1+S2.由式(14)知J|Y∩∂Bη≥σ，因此J满足引理1的第一个条件.为了验证J满足引理1 的第二个条件，设e∈∂B1∩Y，对任给的w∈W,r∈R,令u=re+w,有‖‖w‖β.定义,‖w‖β.可以得到,因此k1(r)，k2(w)有上界.注意到，则存在一个正常数R4>η使得J(u)≤0,∀u∈∂Q成立, 其中⨁{re|0<r<R4}.由引理1知J存在一个临界值c≥σ>0,其中}.令使得c. 如果,那么定理1的结论成立.不然,有,也即).令h=id,有.与上述方法类似，可以将e换成-e∈∂B1∩Y，同样存在一个常数R5>η使得∀u∈∂Q1,J(u)≤0成立，其中⨁{-re|0<r<R5}，再次利用引理1可以得到J存在一个临界值c′≥σ>0,其中}.同理，存在u′∈E使得J(u′)=c′，如果定理1的结论成立，否则有,即,也即).令h=id,有因为J|∂Q≤0与J|∂Q1≤0,所以u′一定是Q和Q1的内点，然而Q∩Q1⊂W且对任给的u∈W都有J(u)≤0成立，即c′≤0与c′>0矛盾，因此结论成立，定理1 得证.例1 设T为一正整数, 考虑四阶差分方程边值问题Δu-1=Δu0=0,ΔuT=ΔuT+1=0对照式(5), 有因此易知边值问题(15)～(16)满足条件(A1)～(A5), 其中α=4,β=6, 由定理1 知至少存在2个非平凡解.【相关文献】[1] AGARWAL R P, O′REGAN D. Singular discrete (n,p) boundary value problems[J]. Appl Math Lett, 1999, 12(8): 113-119.[2] AGARWAL R P, WONG F H. Upper and lower solutions method for higher-orderdiscrete boundary values problems[J]. Math Ineq Appl, 1998, 1(4): 551-557.[3] GUO Z M, YU J S. Existence of periodic and subharmonic solutions for second-order superlinear difference equations[J]. Sci China Ser A, 2003, 46(4): 506-515.[4] ZHOU Z, YU J S, CHEN Y M. Periodic solutions of a 2nth-order nonliner difference equation[J]. Sci China Math, 2010, 53(1): 41-50.[5] ATICI F M, CABADA A. Existence and uniqueness results for discrete second-order periodic boundary value problems[J]. Comput Math Appl, 2003, 45(6/9): 1417-1427. [6] LIU X, ZHANG Y B, SHI H P. Nonexistence and existence results for a class of fourth-order difference Neumann boundary value problems[J]. Indag Math, 2015, 26(1): 293-305.[7] ZOU Q R, WENG P X. Solutions of 2nth-order boundary value problem for difference equation via variational method[J]. Adv Differ Equ, 2009, Art. ID 730484,10pp.[8] 李龙图, 翁佩萱. 二阶泛函差分方程边值问题[J]. 华南师范大学学报:自然科学版,2003(3): 20-24.LI L T, WENG P X. Boundary value problems of second order functional difference equation[J]. J South China Normal Univ: Nat Sci Edi, 2003(3): 20-24.[9] 梁海华, 翁佩萱. 一类四阶差分边值问题解的存在性与临界点方法[J]. 高校应用数学学报, 2008, 23(1): 67-72.LIANG H H, WENG P X. Existence of solutions for a fourth-order difference boundary value problem and critical point method[J]. Appl Math J Chin Univ Ser A, 2008, 23(1): 67-72.[10]ZHENG B, ZHANG Q Q. Existence and multiplicity of solutions of second-order difference boundary value problems[J]. Act Appl Math, 2010, 110(1): 131-152.[11]LIU X, ZHANG Y B, SHI H P. Periodic solutions for fourth-order nonlinear functional difference equations[J]. Math Meth Appl Sci, 2015, 38(1): 1-10.[12]LIU X, ZHANG Y B, SHI H P. Nonexistence and existence results for a class of fourth-order difference Dirichlet boundary value problems[J]. Math Meth Appl Sci, 2015, 38(4): 691-700.[13]BONANNO G, CANDITO P, D′AUGI G. Variational methods on finite dimentional banach spaces and discrete problems[J]. Adv Nonlin Stud, 2014, 14(4): 915-939.[14]RABINOWITZ P H. Minimax methods in critical point theory with applications to differential equations[M]. USA: CBMS, American Mathematical Society, 1986.。
existence and multiplicity results Existence and Multiplicity ResultsExistence and multiplicity results are fundamental concepts in mathematics that pertain to the existence and number of solutions of certain mathematical problems. These results are often studied in various branches of mathematics, including analysis, differential equations, and algebraic topology, to name a few. In this article, we will explore the significance and applications of existence and multiplicity results, step by step.1. Introduction to Existence and Multiplicity Results:Existence results deal with proving the existence of solutions to mathematical problems. They establish the fact that a solution to a given problem does indeed exist. On the other hand, multiplicity results focus on determining the number of distinct solutions to such problems.These results are crucial in mathematics as they provide a foundation for further analysis and allow us to gain a deeper understanding of the structure and behavior of mathematicalsystems. They are used to address a wide range of problems, from solving differential equations and boundary value problems, to studying the existence of critical points in optimization and variational problems.2. Theoretical Framework and Methods:To prove the existence and multiplicity of solutions, various mathematical techniques have been developed over the years. These techniques include fixed point theorems, variational methods, topological methods, and bifurcation theory, among others.Fixed point theorems, such as the Banach fixed point theorem, play a significant role in proving the existence of solutions. These theorems guarantee the existence of a point that remains invariant under a given map or transformation. By establishing a suitable framework and constructing appropriate mappings, one can apply fixed point theorems to demonstrate the existence of solutions for a wide range of problems.Variational methods involve optimizing functionals subjected tocertain constraints. By considering the critical points of these functionals, one can identify solutions to the original problem. Variational methods provide a powerful tool in the study of existence and multiplicity of solutions, particularly in the field of partial differential equations.Topological methods, such as degree theory and index theory, are used to establish the number of solutions to a given problem. These methods utilize topological concepts to assign a numerical index or degree to a map, representing the number of times the map wraps around a particular point. By analyzing the index or degree of a mapping, one can determine the number of solutions that exist.Bifurcation theory studies the qualitative changes in the behavior of solutions as certain parameters of a problem are varied. It explores how the number and types of solutions change as the parameters cross critical values. Bifurcation theory offers valuable insights into the multiplicity and existence of solutions, particularly when studying dynamic systems.3. Applications and Importance:The importance of existence and multiplicity results can be seen in various fields of mathematics as well as in applications to science and engineering. For example, in the study of differential equations, existence and multiplicity results enable us to understand the behavior and stability of solutions. They are essential in modeling physical phenomena, such as heat transfer, fluid dynamics, and population dynamics.In algebraic topology, existence and multiplicity results help us understand the structure and properties of topological spaces. They provide insights into the connectivity and dimensionality of these spaces, aiding in the classification and characterization of geometric objects.Existence and multiplicity results are also crucial in optimization problems. By determining the existence and number of critical points, one can identify the optimal solution to a given problem, whether it is maximizing or minimizing a certain objective function.4. Conclusion:In conclusion, existence and multiplicity results are fundamental concepts in mathematics with significant applications in various fields. They provide a theoretical framework and techniques for proving the existence and determining the number of solutions of mathematical problems. These results have far-reaching implications, enabling us to understand the behavior of solutions in differential equations, analyze the structure of topological spaces, and find optimal solutions in optimization problems. By exploring and applying existence and multiplicity results, mathematicians continue to advance our understanding of the mathematical world and its connections to real-world phenomena.。
Advances in Dynamical Systems and ApplicationsISSN0973-5321,V olume5,Number1,pp.75–85(2010)/adsaExistence of Homoclinic Solutions for a Class of Second-Order Differential Equations with Multiple LagsChengjun GuoGuangdong University of TechnologySchool of Applied Mathematics,510006,P.R.Chinaguochj817@Donal O’ReganNational University of IrelandDepartment of Mathematics,Galway,Irelanddonal.oregan@nuigalway.ieRavi P.AgarwalFlorida Institute of TechnologyDepartment of Mathematical SciencesMelbourne,Florida32901,U.S.A.agarwal@AbstractThis paper is concerned with the existence of homoclinic orbits for second-order differential equations with multiple lags.By using Mawhin’s continuationtheorem,a nontrivial homoclinic orbit is obtained as a limit of a certain sequenceof periodic solutions of the equation.AMS Subject Classifications:34K15,34C25.Keywords:Homoclinic orbit,multiple lags,Mawhin’s continuation theorem.1IntroductionIn recent years several authors studied homoclinic orbits for Hamiltonian systems via critical point theory.In particular second-order systems were considered in[1,2,4–6, Received December4,2009;Accepted December11,2009Communicated by Martin Bohner76Chengjun Guo,Donal O’Regan and Ravi P.Agarwal 12–16,19]andfirst-order systems in[3,7–9,11,17,18].In this paper we consider the existence of homoclinic orbits for FDE by using Mawhin’s continuation theorem.In particular we discuss the existence of homoclinic orbits for the equationx (t)+a1(t)x (t)−a2(t)x(t)=g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f(t),(1.1) whereτi(i=1,2,···,n)are constants,a1(t)and a2(t)are real continuous functions defined on R with positive period T,f:R→R is a continuous and bounded function, g(t,x1,x2,···,x n)∈C(R×R×R×···×R,R),g(t,0,0,···,0)=0,and is T-periodic in t.A solution x of(1.1)is said to be homoclinic(to0)if x(t)→0as t→±∞.In addition,if x≡0then x is called a nontrivial homoclinic solution.This paper is largely motivated by the work of Rabinowitz[15]in which the exis-tence of nontrivial homoclinic solutions for the second-order Hamiltonian system¨q+V q(t,q)=0was proved.For the sake of completeness,wefirst state Mawhin’s continuation theorem [10].Assume X and Y are two Banach spaces,L:Dom L⊂X→Y is a linear mapping and N:X→Y is a continuous mapping.The mapping L will be called a Fredholm mapping of index zero if dim Ker L=codim Im L<+∞and Im L is closed in Y.If L is a Fredholm mapping of index zero,then there exist continuous projections P:X→X and Q:Y→Y such that Im P=Ker L and Im L=Ker Q=Im(I−Q). It follows that L|Dom L∩Ker P:(I−P)X→Im L has an inverse which will be denoted by K P.IfΩis an open and bounded subset of X,the mapping N will be called L-compact onΩif QN(Ω)is bounded and K P(I−Q)N(Ω)is compact.Since Im Q is isomorphic to Ker L,there exists an isomorphism J:Im Q→Ker L.Theorem1.1(Mawhin’s continuation theorem[10]).Let L be a Fredholm mapping of index zero,and let N be L-compact onΩ.Suppose(1)for eachλ∈(0,1)and x∈∂Ω,Lx=λNx;(2)for each x∈∂Ω∩Ker(L),QNx=0and deg(QN,Ω∩Ker(L),0)=0.Then the equation Lx=Nx has at least one solution inΩ∩D(L).2Main ResultNow we make the following assumptions on a1(t),a2(t)and f(t):(H1)0≤m1≤|a1(t)|≤M1;(H2)M2=maxt∈[0,T]a2(t)≥a2(t)≥m2=mint∈[0,T]a2(t)>0;Existence of Homoclinic Solutions77(H3)f:R→R is continuous and bounded,f≡0andR |f(t)|2dt12≤η,whereη>0is a positive constant.Our main result is the following theorem. Theorem2.1.Suppose(H1)–(H3)and assume(H4)|g(t,x1,x2,···,x n)|≤rni=1|x i|and m2−M214−2rn>0.Then system(1.1)possesses a nontrivial homoclinic solution x∈C2(R,R)such that x (t)→0as t→±∞.In order to prove the main theorem we need some preliminaries.For each k∈N, setX k:={x|x∈C1(R,R),x(t+2kT)=x(t),∀t∈R}and x(0)(t)=x(t),define the norm on X k byx =maxmaxt∈[−kT,kT]|x(t)|,maxt∈[−kT,kT]|x (t)|,and setY k:={y|y∈C(R,R),y(t+2kT)=y(t),∀t∈R}.We define the norm on Y k as y 0=maxt∈[−kT,kT]|y(t)|.Thus both(X k, · )and(Y k, · 0) are Banach spaces.Remark2.2.If x∈X k,then it follows that x(i)(0)=x(i)(2kT)(i=0,1).In the works of Izydorek and Janczewska[12]and Tanaka[19],a homoclinic so-lution of(1.1)is obtained as a limit,as k→±∞,of a certain sequence of functions x k∈X k.So here we will consider a sequence of systems of functional differential equationsx (t)+a1(t)x (t)−a2(t)x(t)=g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t),(2.1) where for each k∈N,f k:R→R is a2kT-periodic extension of the restriction of f to the interval[−kT,kT]and x k is a2kT-periodic solution of(2.1)obtained via Mawhin’s continuation theorem.Define the operators L k:X k→Y k and N k:X k→Y k byL k x(t)=x (t),t∈R,(2.2) andN k x(t)=−a1(t)x (t)+a2(t)x(t)+g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t),t∈R.(2.3)78Chengjun Guo,Donal O’Regan and Ravi P.Agarwal Clearly,Ker L k={x∈X k:x(t)=c∈R}(2.4)andIm L k=y∈Y k:kT−kTy(t)dt=0(2.5)is closed in Y k.Thus L k is a Fredholm mapping of index zero.Let us define P k:X k→X k and Q k:Y k→Y k/Im(L k)byP k x(t)=12kTkT−kTx(t)dt=x(0),t∈R,(2.6)for x=x(t)∈X andQ k y(t)=12kTkT−kTy(t)dt,t∈R(2.7)for y=y(t)∈Y k.It is easy to see that Im P k=Ker L k and Im L k=Ker Q k=Im(I k−Q k).It follows that L k|Dom Lk∩Ker P k :(I k−P k)X k→Im L k has an inversewhich will be denoted by K Pk.LetΩk be an open and bounded subset of X k.We can easily see that Q k N k(Ωk)isbounded and K Pk (I k−Q k)N k(Ωk)is compact.Thus the mapping N k is L-compact onΩk.That is,we have the following result.Lemma2.3.Let L k,N k,P k and Q k be defined by(2.2),(2.3),(2.6)and(2.7)respec-tively.Then L k is a Fredholm mapping of index zero and N k is L-compact onΩk,where Ωk is any open and bounded subset of X k.In order to prove our main result,we need the following lemma[15].Lemma2.4(See Remark2.2and[15]).There is a positive constant such that for each k∈N and x∈X k the following inequality holds:max t∈[−kT,kT]|x(t)|≤kT−kT(|x(t)|2+|x (t)|2)dt12.Now,we consider the auxiliary equationx (t)+λ[a1(t)x (t)−a2(t)x(t)]=λ[g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t)],(2.8) where0<λ<1.Existence of Homoclinic Solutions79 Lemma2.5.Suppose that the conditions of Theorem2.1are satisfied.If x k(t)is a2kT-periodic solution of Eq.(2.8),then there are positive constants D i,i=0,1,which are independent ofλ,such thatx(i)k0≤D i,t∈[−kT,kT],i=0,1.(2.9) Proof.Suppose that x k is a2kT-periodic solution of Eq.(2.8).We have from(2.8)thatkT −kT [xk(t)+λa1(t)xk(t)−λa2(t)x k(t)]x k(t)dt=λkT−kT[g(t,x k(t−τ1),x k(t−τ2),···,x k(t−τn))+f k(t)]x k(t)dt.(2.10)From(2.10),we havekT −kT {|xk(t)|2+λa2(t)|x k(t)|2}dt=−λkT−kT[g(t,x k(t−τ1),x k(t−τ2),···,x k(t−τn))+f k(t)−a1(t)xk(t)]x k(t)dt≤λkT−kT |g(t,x k(t−τ1),x k(t−τ2),···,x k(t−τn))|2dt12×kT−kT |x k(t)|2dt12+λkT−kT|f k(t)|2dt12+M1kT−kT|xk(t)|2dt12×kT−kT |x k(t)|2dt12≤2rnλkT−kT |x k(t)|2dt+λη+M1kT−kT|xk(t)|2dt12 kT−kT|x k(t)|2dt12,soλm2−M214kT−kT|x k(t)|2dt≤λkT−kT|xk(t)|2dt12−M12kT−kT|x k(t)|2dt12 2+λa2(t)−M214kT−kT|x k(t)|2dt+1λ−λkT−kT|xk(t)|2dt ≤2rnλkT−kT|x k(t)|2dt+ηλkT−kT|x k(t)|2dt12,80Chengjun Guo,Donal O’Regan and Ravi P .Agarwalwhich givesm 2−M 214−2rnkT−kT |x k (t )|2dt ≤ηkT−kT|x k (t )|2dt 12.(2.11)From (H 4)and (2.11),there exists a positive constant C 1such thatkT−kT |x k (t )|2dt ≤η2(m 2−M 214−2rn )2=C 1.(2.12)From (2.8),we havekT−kT[x k (t )+λa 1(t )xk (t )−λa 2(t )x k (t )]x k (t )dt=λkT−kT[g (t,x k (t −τ1),x k (t −τ2),···,x k (t −τn ))+f k (t )]x k (t )dt,(2.13)som 1kT−kT|x k (t )|2dt≤kT−kT |a 1(t )||x k (t )|2dt ≤(M 2+2rn )kT−kT |x k (t )|2dt 12+ηkT−kT|x k (t )|2dt12≤[(M 2+2rn )C 1+η]kT−kT|x k (t )|2dt 12,and as a result there exists a positive constant C 2such thatkT−kT|x k (t )|2dt ≤C 2.(2.14)Moreover,for x ∈X k and t,τ∈[−kT,kT ],we have|x (t )|≤x (τ)+ t τx (s )ds .(2.15)Integration of (2.15)over t −12,t +12shows|x (t )|≤ t +12t −12|x (τ)|dτ+t +12t −12t τx (s )dsdτ≤2t +12t −12(|x (τ)|2+|x (t )|2)dτ12.(2.16)Existence of Homoclinic Solutions 81Hence (2.15)and (2.16)implymax t ∈[−kT,kT ]|x (t )|≤kT−kT(|x (t )|2+|x (t )|2)dt 12,x ∈X k ,(2.17)where is given in Lemma 2.4.From Lemma 2.4,(2.12)and (2.14),we havemax t ∈[−kT,kT ]|x k (t )|≤kT−kT(|x k (t )|2+|x k (t ))|2dt 12≤ (C 1+C 2)12=D 0.(2.18)On the other hand,we have from (2.8)thatkT−kT[x k (t )+λa 1(t )x k (t )−λa 2(t )x k (t )]xk (t )dt=λkT−kT[g (t,x k (t −τ1),x k (t −τ2),···,x k (t −τn ))+f k (t )]x k (t )dt,(2.19)so we have kT−kT|x k (t )|2dt≤kT−kT|x k (t )|2dt12M 1 kT−kT|x k (t )|2dt 12+M 2kT−kT|x k (t )|2dt12+ kT−kT |g (t,x k (t −τ1),x k (t −τ2),···,x k (t −τn ))|2dt 12×kT−kT |x k (t )|2dt12+ kT−kT|f k (t )dt12kT−kT|x k (t )|2dt12≤[(2rn +M 2)C 1+M 1 C 2+η]kT−kT|x k (t )|2dt12,and as a result there exists a positive constant C 4such thatkT−kT|x k (t )|2dt ≤[(2rn +M 2) C 1+M 1C 2+η]2=C 4.(2.20)From Lemma 2.4and (2.20),we havemaxt ∈[−kT,kT ]|x k (t )|≤kT−kT(|x k (t )|2+|x k (t )|2)dt 12≤ (C 2+C 4)12=D 1.(2.21)The proof iscomplete.82Chengjun Guo,Donal O’Regan and Ravi P.Agarwal Lemma2.6.Let k∈N.If(H1)–(H4)hold,then the system(2.1)possesses a2kT-periodic solution.Proof.Suppose that x is a2kT-periodic solution of Eq.(2.8).By Lemma2.5,there exist positive constants D i(i=0,1)which are independent ofλsuch that(2.9)is true.Consider any positive constantαk>max0≤i≤1{D i}+ξ,whereξ=maxt∈R|f(t)|.SetΩk:={x∈X k: x <αk}.We know that L k is a Fredholm mapping of index zero and N k is L-compact onΩk (see[2]).RecallKer(L k)={x∈X k:x(t)=c∈R}and the norm on X k isx =maxmaxt∈[−kT,kT]|x(t)|,maxt∈[−kT,kT]|x (t)|.Then we havex=αk or x=−αk for x∈∂Ωk∩Ker(L k).(2.22) From(H4),we have(ifαk is chosen large enough)a2(t)αk+g(t,αk,αk,···,αk)− f k 0>0t∈[−kT,kT](2.23) andx (t)=0,∀x∈∂Ωk∩Ker(L k).(2.24) Finally from(2.3),(2.7)and(2.22)–(2.24),we have(Q k N k x)=12kTkT−kT[−a1(t)x (t)+a2(t)x(t)+g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t)]dt=12kTkT−kT[a2(t)x(t)+g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t)]dt=0,∀x∈∂Ωk∩Ker(L k).Then,for any x∈Ker L k∩∂Ωk andη∈[0,1],we havexH(x,η)=−ηx2−x2kT(1−η)kT−kT[−a1(t)x (t)+a2(t)x(t)+f k(t)+g(t,x(t−τ1),x(t−τ2),···,x(t−τn))]dt =0.Existence of Homoclinic Solutions 83Thusdeg {Q k N k ,Ωk ∩Ker(L k ),0}=deg −12kT kT−kT[−a 1(t )x (t )+a 2(t )x (t )+f k (t )+g (t,x (t −τ1),x (t −τ2),···,x (t −τn ))]dt,Ωk ∩Ker(L k ),0}=deg {−x,Ωk ∩Ker(L k ),0}=0.From Lemma 2.5,for any x ∈∂Ωk ∩Dom(L k )and λ∈(0,1)we have L k x =λN k x .By Theorem 1.1,the equation L k x =N k x has at least one solution in Dom(L )∩Ωk .So there exists a 2kT -periodic solution x k of the system (2.1).The proof iscomplete.Lemma 2.7.Let {x k }k ∈N be the sequence given by Lemma 2.6.Then there exists x 0anda subsequence of {x n }n ∈N (again we call it {x n }n ∈N )such that x k →x 0in C 1loc (R ,R )as k →+∞.Proof.By (2.18),(2.21)and the Arzel`a –Ascoli theorem,we obtain that a subsequenceof {x k }k ∈N converges in C 1loc (R ,R )to a solution x 0of (1.1)satisfying∞−∞(|x 0(t )|2+|x 0(t )|2)dt <∞.(2.25)To see this note from (2.1)thatlim k →∞[x k (t )+a 1(t )xk (t )−a 2(t )x k (t )−g (t,x k (t −τ1),x k (t −τ2),···,x k (t −τn ))]=x 0(t )+a 1(t )x0(t )−a 2(t )x 0(t )−g (t,x 0(t −τ1),x 0(t −τ2),···,x 0(t −τn ))=lim k →∞f k (t )=f (t ),so x 0is a solution of (1.1).Also,we have∞−∞[|x 0(t )|2+|x 0(t )|2]dt =limk →∞kT−kT[|x k (t )|2+|x k (t )|2]dt <∞.This shows that (2.25)holds.Lemma 2.8.The function x 0determined by Lemma 2.7is the desired homoclinic solu-tion of (1.1).Proof.The proof will be divided into two steps.Step 1:We prove that x 0(t )→0,as t →±∞.By (2.25),we havelim j →∞|t |≥j[|x 0(t )|2+|x 0(t )|2]dt =0.(2.26)84Chengjun Guo,Donal O’Regan and Ravi P.Agarwal Hence(2.18)and(2.26)shows that our claim holds.Step2:We now show that x 0(t)→0as t→±∞.By(2.16),(2.18)and(2.26),it suffices to prove thatj+1 j |x(t)|2dt→0,as j→+∞.(2.27)On the other hand,we obtain from(1.1)thatj+1 j |x(t)|2dt=j+1j|−a1(t)x(t)+a2(t)x0(t)+f(t)+g(t,x0(t−τ1),x0(t−τ2),···,x0(t−τn))|2dt.Since g(t,0,0,···,0)=0for all t∈R,x0(t)→0as t→±∞,j+1j |x(t)|2dt→0and j+1j|f(t)|2dt→0as j→±∞,so(2.27)follows.Proof of Theorem2.1.The result follows now from Lemma2.8.AcknowledgementSupport by grant10871213from NNSF of China and by grant093051from Guangdong University of Technology of China is acknowledged.References[1]A.Ambrosetti,V.Coti Zelati,Multiple homoclinic orbits for a class of conserva-tive system,Rend.Sem.Mat.Univ.Padova.89(1993),177–194.[2]P.C.Carri˜a o,O.H.Miyagaki,Existence of homoclinic solutions for a class oftime-dependent Hamiltonian systems,J.Math.Anal.Appl.230(1999),157–172.[3]V.Coti Zelati,I.Ekeland,E.S´e r´e,A variational approach to homoclinic orbits inHamiltonian systems,Math.Ann.228(1990),133–160.[4]V.Coti Zelati,P.H.Rabinowitz,Homoclinic orbits for second order Hamiltoniansystems possessing superquadratic potentials,J.Amer.Math.Soc.4(1991),693–727.[5]Y.H.Ding,Existence and multiplicity results for homoclinic solutions to a classof Hamiltonian systems,Nonlinear Anal.25(1995),1095–1113.Existence of Homoclinic Solutions85 [6]Y.H.Ding,M.Girardi,Periodic and homoclinic solutions to a class of Hamilto-nian systems with the potentials changing sign,J.Math.Anal.Appl.189(1995), 585–601.[7]Y.H.Ding,L.Jeanjean,Homoclinic orbits for a nonperiodic Hamiltonian system,J.Diff.Eqns.237(2007),473–490.[8]Y.H.Ding,S.J.Li,Homoclinic orbits forfirst order Hamiltonian systems,J.Math.Anal.Appl.189(1995),585–601.[9]Y.H.Ding,M.Willem,Homoclinic orbits of a Hamiltonian system,Z.Angew.Math.Phys.50(1999),759–778.[10]R.E.Gaines,J.L.Mawhin,Coincidence degree and nonlinear differential equa-tion,Lecture Notes in Math,V ol.568,Springer-Verlag,1977.[11]H.Hofer,K.Wysocki,First order elliptic systems and the existence of homoclinicorbits in Hamiltonian systems,Math.Ann.228(1990),483–503.[12]M.Izydorek,J.Janczewska,Homoclinic solutions for a class of the second orderHamiltonian,J.Diff.Eqns.219(2005),375–389.[13]M.J.Ma,Z.M.Guo,Homoclinic orbits and subharmonics for nonlinear secondorder difference equations,Nonlinear.Anal.67(2007),1737–1745.[14]W.Omana,M.Willem,Homoclinic orbits for a class of Hamiltonian systems,Differential Integral Equations.5(1992),1115–1120.[15]P.H.Rabinowitz,Homoclinic orbits for a class of Hamiltonian systems,Proc.Roy.Soc.Edinburgh.114A(1990),33–38.[16]P.H.Rabinowitz,K.Tanaka,Some results on connecting orbits for a class ofHamiltonian systems,Math.Z.206(1991),472–499.[17]E.S´e r´e,Existence of infinitely many homoclinic orbits in Hamiltonian systems,Math.Z.209(1993),561–590.[18]A.Szulkin,W.M.Zou,Homoclinic orbits for asymptotically linear Hamiltoniansystems,J.Funct.Anal.187(2001),25–41.[19]K.Tanaka,Homoclinic orbits for a singular second order Hamiltonian system,Ann.Inst.H.Poincar´e.7(5)(1990),427–438.。
一类具有交叉扩散的捕食-食饵模型正解的存在性吕杨; 郭改慧; 袁海龙; 李书选【期刊名称】《《工程数学学报》》【年(卷),期】2019(036)006【总页数】9页(P658-666)【关键词】捕食-食饵模型; 交叉扩散; 正解; 存在性【作者】吕杨; 郭改慧; 袁海龙; 李书选【作者单位】陕西科技大学文理学院西安 710021; 西安交通大学数学与统计学院西安 710049【正文语种】中文【中图分类】O175.261 引言本文考虑如下具有交叉扩散的捕食-食饵模型其中Ω是RN 上的有界区域，∂Ω 是光滑边界；u, v 分别代表食饵与捕食者的密度.a, c, d,µ是正常数；α, β 和m 是非负常数；b 可能变号.当α = β = 0 且m > 0 时，系统(1)已被许多学者研究[1-10].特别地，文[2]利用分歧理论证明了正解的存在性.文[7]讨论了当某些参数充分大时，建立了系统正解的存在性、多解性、唯一性和稳定性.进一步，Du 和Lou[8]说明了当m 充分大时，奇异扰动系统存在Hopf 分歧，进而说明原系统存在Hopf 分歧.当α,β > 0 且m = 0 时，则系统(1)变成具有Lotka-Volterra 捕食-食饵模型，已经被许多生物数学家所探究[11-15].特别地，文[11]用分歧理论研究了正解的存在性，并证明共存区域随着β 增大而增大，随着α 增大而减小.进一步，文[14]讨论了当空间维数小于5 时，他们寻找到了一个与α, β 无关的先验估计.在文[16]，Wang 和Li 利用分歧理论说明当b ∈(b∗,b∗)，系统(1)至少有一个正解.特别地，当α 充分大时，他们以b 为分歧参数证明了全局分歧解在b = b∗从半平解(θa,0)出发连接到在b = (µ+1)λ1 的半平凡解(0,sϕ1)，从而形成一个有界解，其中s>0. 本文主要讨论当交叉扩散系数α 充分大时正解的极限行为.特别地，我们以a 为分歧参数，利用全局分歧理论说明极限系统的正解将随着分歧参数a 在正锥内到达无穷.令λ1(p)<λ2(p)≤λ3(p)≤···是下列问题的特征值其中我们知道λ1(p)是简单的、实的，λ1(p)关于p 是严格单调递增的.当p ≡ 0 时，我们记λ1(0)为λ1.进一步，我们记ϕ1 是λ1 对应的特征函数且满足ϕ1 >0,||ϕ1||2 =1.我们知道当a>λ1 时，下列问题存在唯一的正解θa，且a →θa 在a ∈(λ1,+∞)是连续的，θa 关于a 是单调递增的.进一步，θa 非退化，线性稳定.2 预备知识则我们有系统(1)可以写成引理1 令(u,v)是系统(1)的任意正解，则存在一个与α 无关的常数C1，使得证明令(u,v)是系统(1)的任意正解，则根据比较原理，我们知道其中U∗是方程在Ω 上且满足U∗|∂Ω =0 的解.从而说明 u, U 存在与α 无关的先验估计.对于V 的方程，我们有同理，我们说明v, V 存在与α 无关的先验估计.定义下面的引理说明a∗(b)的一些性质.由于证明过程比较简单，我们在此略去其证明而仅陈述其结果.引理2[11] 集合S 形成一个有界的曲线其中a=a∗(b)是b ∈(0,(µ+1)λ1]的正的连续函数，且满足下面的性质：(i) a=a∗(b)关于b ∈ (0,(µ +1)λ1)是严格单调递减的；(ii) a∗((µ +1)λ1)= λ1.在此，我们定义ϕ∗>0 满足下面的方程对p>N，我们定义Banach 空间X 和Y 如下通过Sobolev 嵌入定理，我们知道3 主要结论在本节中，我们主要考虑当α 充分大时，极限系统正解的存在性.通过引理1 我们知道，系统(1)或(2)存在与α 无关的先验估计.下面的引理表明当α 充分大时，系统(1)的任意正解收敛到下述系统(4)的正解. 引理3[16] 存在充分大的Λ，使得当α ≥Λ，且α=αi 时，(ui,vi)是系统(1)的任意正解，则上成立，其中是下面系统的正解我们通过分歧理论建立系统(4)正解的存在性，而这只需建立下面系统(5)正解的存在性.我们通过分歧理论说明当a ∈(a∗,∞)时，系统(5)至少存在一个正解.令则系统(4)可以写成当a>λ1 时，系统(5)存在半平凡解我们以a 为分歧参数，利用局部和全局分歧理论来建立系统(5)正解的存在性.令其中是的函数.通过在处Taylor 展开，令则显然f(θa,0)=−∆θa, g(θa,0)=0.令则其中令K 是−∆在齐次Dirichlet 边界条件下的逆算子，则定义T :R×X →X则是X 空间上的紧算子.令显然H(a;0,0)=0，且记的Frchet 导数为下面，我们利用局部分歧理论说明系统(5)在(a∗;θa∗,0)附近存在局部分歧解.定理1 设a > λ1，则(a∗;θa∗,0) ∈ R × X 是系统 (5)的分歧点，且存在δ > 0，在分歧点(θa,0)的邻域内存在如下形式的正解其中是光滑函数且满足证明令则令则如果矛盾.因此a=a∗, ker L1(a∗;0,0)=span{ψ∗,ϕ∗}，其中我们定义算子L1(a∗;0,0)的伴随算子为令则根据Fredholm 选择公理，我们知算子L1(a∗;0,0)的值域为因此R(L1(a∗;0,0))的余维数是1.为了在处应用局部分歧理论[17]，我们断言显然我们采用反证法.假设存在满足也就是在上面的等式两边同乘以ϕ∗，再积分，我们有矛盾.根据全局分歧理论，我们断言局部分歧解可以延拓为全局分歧解，并且该全局分歧解随着分歧参数a 在正锥内延伸到无穷.令假设σ ≥ 1 是算子T′(a)的特征值且其对应的特征函数为(ξ,η)，则则σ ≥ 1 是算子T′(a)的特征值的充要条件是：存在一些i = 1,2,···，使得a = ai(µ).进而，若a<a∗，则i(T(a,·),0)=1，如果a∗ <a<a2(1)，则i(T(a,·),0)= −1.根据全局分歧理论[18]，我们知道在R+×X 内，存在从(a∗;0,0)出发的连通分支C0，满足H(a;ξ,η) = 0，且在(a∗;0,0)附近，H(a;ξ,η)的所有零点都在定理1 中得到的那条分歧曲线令则C 是系统(5)从(a∗;0,0)分歧的解曲线，令在(a∗;θa∗,0)的小邻域内，解曲线C ⊂ P0.定理2 C −{(a∗;θa∗,0)}在正锥内随着分歧参数a 延伸到无穷.证明根据Rabinowitz 全局分歧理论[18]和更加一般的分歧理论[19,20]，我们断言全局分歧解C −{(a∗;θa∗,0)}必满足下列三个选择之一：(i) 全局分歧解C 连接半平凡解{a;θa,0}，其中 aa∗且I − T′(a)是不可逆的；(ii) 全局分歧解C 在R×X 是无界的；(iii) 存在a = 和W ∈\{0}，满足(;W) ∈C，其中是空间{(0,−∆ϕ∗)}的补空间，{(0,−∆ϕ∗)}由定理1 给出.如果则存在使得在Ω 的极限.由于由最大值原理我们知道，在Ω.假设令则根据二阶椭圆型方程的正则化理论，我们假设在C1 上成立且满足和在Lp 弱收敛.在上面系统的两边同时取极限，我们有根据最大值原理我们知道，>0 在Ω 上成立.因此由a∗的定义我们知道，有=a∗，矛盾.假设类似地，我们可以得到矛盾.因此，我们说明C −{(a∗;θa∗,0)} ⊂ P0.进而(i)不可能发生.由于在Ω 上ϕ∗>0，补空间不可能包含不变号的元素，从而(iii)也不可能发生.由Lp 估计和Sobolev 嵌入定理，我们知道存在正常数，使得因此，全局分歧解C 只能随着分歧参数a 在正锥内到达无穷.参考文献：【相关文献】[1]袁海龙，李艳玲.一类捕食-食饵模型共存解的存在性与稳定性[J].陕西师范大学学报(自然科学版)，2014, 42(1):15-18 Yuan H L, Li Y L.Coexistence of existence and stability of a predator-prey model[J].Journal of Shaanxi Normal University (Natural Science Edition), 2014, 42(1): 15-18[2]Blat J, Brown K J.Global bifurcation of positive solutions in some systems of elliptic equations[J].SIAM Journal on Mathematical Analysis, 1986, 17(6): 1339-1353[3]Brown K J.Nontrivial solutions of predator-prey systems with smalldiffusion[J].Nonlinear Analysis:Theory Methods & Applications, 1987, 11(6): 685-689[4]Casal A, Eilbeck J C, Lpez-Gmez J.Existence and uniqueness of coexistence states for a predator-prey model with diffusion[J].Differential & Integral Equations, 1994, 7(2): 411-439[5]Conway E D, Gardner R, Smoller J.Stability and bifurcation of steady-state solutions for predator-prey equations[J].Advances in Applied Mathematics, 1982, 3(3): 288-334[6]Dancer E N.On positive solutions of some pairs of differential equations[J].Transactions of the American Mathematical Society, 1984, 284(2): 729-743[7]Du Y, Lou Y.Some uniqueness and exact multiplicity results for a predator-preymodel[J].Transactions of the American Mathematical Society, 1997, 346(6): 2443-2475 [8]Du Y,Lou Y.S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model[J].Journal of Differential Equations, 1998, 144(2): 390-440[9]Du Y, Shi J.A diffusive predator-prey model with a protection zone[J].Journal of Differential Equations,2006, 229(1): 63-91[10]Du Y, Peng R, Wang M.Effect of a protection zone in the diffusive Leslie predator-prey model[J].Journal of Differential Equations, 2009, 246(10): 3932-3956[11]Kadota T,Kuto K.Positive steady states for a prey-predator model with some nonlinear diffusion terms[J].Journal of Mathematical Analysis & Applications, 2006, 323(2): 1387-1401[12]Kuto K, Yamada Y.Multiple coexistence states for a prey-predator system with cross-diffusion[J].Journal of Differential Equations, 2004, 197(2): 315-348[13]Kuto K.A strongly coupled diffusion effect on the stationary solution set of a prey-predator model[J].Advances in Differential Equations, 2007, 12(2): 145-172[14]Kuto K, Yamada Y.Coexistence problem for a prey-predator model with density-dependent diffusion[J].Nonlinear Analysis, 2009, 71(12): e2223-e2232[15]Nakashima K, Yamada Y.Positive steady states for prey-predator models with cross-diffusion[J].Advances in Differential Equations, 1996, 1(6): 1099-1122[16]Wang Y X, Li W T.Stationary problem of a predator-prey system with nonlinear diffusion effects[J].Computers & Mathematics with Applications, 2015, 70(8): 2102-2124 [17]Crandall M G, Rabinowitz P H.Bifurcation from simple eigenvalues[J].Journal of Functional Analysis,1971, 8(2): 321-340[18]Rabinowitz P H.Some global results for nonlinear eigenvalue problems[J].Journal of Functional Analysis,1971, 7(3): 487-513[19]Lpez-Gmez J.Spectral Theory and Nonlinear Function Analysis[M].Boca Raton: Chapman and Hall/CRC, 2001[20]Dancer E N.Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one[J].Bulletin of the London Mathematical Society, 2002, 34(5): 533-538。
E X IST E NC E AN D M U LT IP LIC IT Y OF S OLU T IO N S T O A C LASS OF D IR I C HLE T P R OB LE M S W IT H I M P ULSI VE E F F E C T SYingliang Song(Sc hool of Ma th.and C omputer Sc ience,Fujia n Nor ma l Univer sity,F uzhou 350007,E-m a il:wyxsx17@)Ann.of Di .Eqs.29:2(2013),195-202Ab st ra ctI n this pa per ,we st udy the exist ence a nd mult iplicity of solut ions t o a cla ss of Di-r ic h let p roble ms with imp ulsive e ect s via va r ia tional met hods.Under an a ssum ption t ha t t he nonlinea rity f is super linea r bu t does not necessa r ily sa tisfy th e Am br oset ti-R a binowit z condit ion,we ext end a nd im prove some recent r esult s.K e yw or d s im pulsive di er ential equat ion;va riat iona l m ethod;Dir ic h let b ou n-d ar y con dit ion2000M a t h e m a t ic s S u b je c t C la ss i c a t io n 34A371In t r o d u ct io nIn t his paper,we st udy t he following problemu ′′(t )+g(t )u(t )=f (t,u(t )),a.e.t ∈[0,T],(u ′(t j ))=I j (u(t j )),j =1,2,,p,u(0)=u(T)=0,(1.1)where t j ,j =1,2,,p are the inst ant s where t he impulses occur and 0=t 0<t 1<t 2<<t p <t p+1=T,(u ′(t j ))=u ′(t +j )u ′(t j )=lim s →t +ju ′(s)lim s →t ju ′(s ).Impulsive e ect s exist widely in many evolut ion processes whose st at es are subject ed t o discont inuous changes at t he cert ain moment of t ime.Impulsive di erent ial equat ions can mat hemat ically describe t hose phenomena.Some processes of t his eld mot ivat ed by cont rol t heory,opt imizat ion t heory,populat ion dynamics,biology m edical and engineering can be seen in [1-6].For t he general aspect s of impulsive di erent ial equat ions,we refer t he readers t o t he classical monograph [7].Some classical t echniques and tools,such as xed point t heory,t opological degree t heory (including cont inuat ion method and coincidence degree theory)and comparison met hod (including upper and lower solut ions m et hods and monot one iterat ive met hod)(see [8-15]and references t herein)have been used t o obt ain t he existence of solut ions t o im pulsive di erent ial equat ion.For some general and recent works on t he impulsive di erent ial equat ion via variat ional met hods,we refer t he reader t o [16-29].Also,variat ional met hods,recent ly,have been proved t o be very crucial t o deal with impulsive di erent ial equat ion.For som e general and recent work on the t heory of crit ical point t heory and variat ional m et hods,we refer t he reader t o [30-33].Suppor ted by t he key pr oject of NNSF of C hina (10831005).Ma nusc ript r eceived Sept emb er 29,2012195196ANN.OF DIF F.E QS.Vol.29More precisely,in [16],t he aut hors used t he variat ional met hod t o obt ain the mult iplicit y of solut ions to problem (1.1).Brie y,Zhang [17],Liu [18]and Chen [19]ext ended t he result s of [16]using some new crit ical point pared wit h [16],t hey did not assume t hat t he nonlinearit y f sat is es t he Ambroset t i-Rabinowitz condit ion,t hat is,t here exist >2and R >0such t hat0<F (t ,u)≤uf (t,u),u ∈R \{0}and for all t ∈[0,T],or0<F (t,u)≤uf (t ,u),|u|>R and for all t ∈[0,T].Mot ivat ed by t he above fact ,in t his paper,our aim is also t o st udy problem (1.1)under t he case t hat t he nonlinearit y f is superlinear which was not considered in [16-29].In t his paper,we do not require t hat F (t ,u)>0,for u ∈R \{0}and f (t,u)=o(|u|),as u →0,or 0≤F (t,u)=o(|u|2),as u →0.We use variat ional met hods t o get t he exist ence and mult iplicit y of solut ions t o problem (1.1),which ext ends t he exist ing result s.In t his paper,we always assume that λ=π2/T 2is t he rst eigenvalue of t he problem {u ′′(t)=λt ,in [0,T],u(0)=u(T)=0,(f)f :[0,T]×R →R is cont inuous,f (t ,s )=f 1(t ,s )f 2(t ,s );(g)g ∈L ∞[0,T],and essinf t ∈[0,T ]g(t)=m >λ1;(i)I j :R →R,j =1,2,,p,are cont inuous,F (t,u)=∫uf (t ,s )ds,F 1(t ,u)=∫uf 1(t ,s )ds ,F 2(t ,u)=∫uf 2(t ,s )ds.Now we st at e t he assum pt ions:(H 1)f 1(t ,u)and f 2(t,u)are cont inuous;(H 2)t here exist s a >2,such t hat 0<F 1(t,u)≤uf 1(t,u),u ∈R \{0};(H 3)t here exist s an 1<α<2,such t hat 0<uf 2(t ,u)≤αF 2(t ,u),u ∈R \{0};(H 4)t here exist s a 0<β<,such t hat 0<uI j (u)≤β∫u0I j (s )ds ,u ∈R \{0},j =1,2,,p;(H 5)I j (u)are odd about u,j =1,2,,p;(H 6)f (t,u)is odd about u.Now we st at e our m ain result s.T he or em 1.1Suppose that (H 1),(H 2),(H 3)and (H 4)hold,t hen problem (1.1)has a t least one solution.T he or em 1.2Suppose that (H 1),(H 2),(H 3),(H 4),(H 5)and (H 6)hold,then problem (1.1)has in nitely many solutions.R em a r k (H 2)and (H 3)can guarantee t hat f is superlinear,but does not necessarily sat isfy F (t,u)>0,for u ∈R \{0}and f (t ,u)=o(|u|),as u →0,or 0≤F (t,u)=o(|u|2),as u →0.So f does not sat isfy t he Ambroset t i-Rabinowitz condit ion,which is considered in t he rst t im e.2P r elim in ar iesIn t his sect ion,we rst int roduce som e t heorems which are necessary.Suppose that ENo.2Y.L.Song,SOLUT IONS TO IMPULSIVE E QUATION 197is a Banach space and φ∈C 1(E ,R ).We say t hat φsat is es t he P.S.condit ion,t hat is,every sequence {u j }E sat isfying t hat {φ(u j )}is bounded and lim j →∞φ′(u j )=0cont ains aconvergent subsequence.T he or em 2.1[30](Mountain P ass Theorem)Let E be a real Ba nach space and I ∈C 1(E ,R )sa tis es the P.S.condit ion with I (0)=0.If I sat is es the following condit ions :(1)T here exist consta nts ρ,α>0such that I |B ρ≥α;(2)t here exis ts an e ∈E \B ρ,such that I (e)≤0.Then I possesses a crit ical value c ≥α.Moreover,c is given by c =inf g ∈Γmax s ∈[0,1]I (g(s)),whereΓ={g ∈C ([0,1],E )|g(0)=0,g(1)=e}.T he or em 2.2[30](Symmetric Mount ain P ass T heorem)L et E be a real Banach space and I ∈C 1(E,R )be even,sat isfying t he P.S.condit ion w it h I (0)=0.Let E =V ⊕Y,where Vis nit e dimensional,and φsatis es the follow ing conditions:(1)T here exist consta nts ρ,α>0such that I |B ρ∩Y ≥α;(2)for each nite dimensional subspace W E ,there is an R =R (W )such t hat φ(u)≤0for all u ∈W with ∥u ∥>R.Then φhas a unbounded crit ical values.The space H 10(0,T)is equipped wit h t he inner product u,v =∫Tu ′(t)v ′(t)dt,and with t he associat ed norm∥u ∥H 10(0,T )=(∫T(u ′(t ))2dt)12.We also use the inner productu,v =∫Tu ′(t )v ′(t )dt +∫Tg(t )u(t )v(t)dt,and the norm∥u ∥=(∫T(u ′(t))2dt +∫Tg(t)(u(t))2dt)12.Taking v ∈H 10(0,T)and mult iplying (1.1)by v,we have∫T 0u ′′v +∫T 0guv =∫Tf (t ,u)v.The rst t erm in t he left of the above equality is ∫T 0u ′′v =p ∑j =0∫t j +1t ju ′′v,and∫t j +1t ju ′′v =u ′(t j +1)v(t j +1)u ′(t +j )v(t +j )∫t j +1t j u ′v ′.Hence∫Tu ′′v =p∑j =0u ′(t j )v(t j )+u ′(0)v(0)u ′(T)v(T )+∫Tu ′v ′=p ∑j =1I j (u(t j ))v(t j )+∫Tu ′v ′.198ANN.OF DIF F.E QS.Vol.29A weak solut ion t o (1.1)is a funct ion u ∈H 10(0,T )such t hat ∫T 0u ′(t)v ′(t)dt +∫T 0g(t)u(t)v(t )dt =∫Tf (t,u(t ))v(t)dtp∑j =1I j (u(t j ))v(t j ),for every v ∈H 10(0,T).For each u ∈H 10(0,T),we de ne φon H 10(0,T)byφ(u)=12∥u ∥2∫T 0F (t,u(t ))dt +p ∑j =1∫u (t j )I j (s )ds ,(2.1)where F (t ,u)=∫u0f (t,s)ds.Since f 1,f 2and I j ,j =1,2,,p are cont inuous,one can easily verify φ∈C 1(H 10,R ).For any v ∈H 10(0,T),we haveφ′(u),v =∫T 0u ′(t)v ′(t )dt +∫T 0g(t)u(t)v(t )dt ∫T 0f (t,u(t ))v(t)dt +p∑j =1I j (u(t j ))v(t j ).(2.2)Thus,t he weak solut ions t o problem (1.1)are t he corresponding crit ical point s of φ.3P r o ofs of T h eor em sLem m a 3.1[21]Suppose t hat (H 2)holds,then for every t ∈[0,T],t he following inequa-lities hold F (t,u)≤F (t ,u|u|)|u|,0<|u|≤1,(3.1)F (t,x)≥F (t ,u |u|)|u|,|u|≥1.(3.2)From t he above fact ,we can easily get t he following t hree lemm as.Lem m a 3.2Suppose that (H 2)holds,thenF 1(t,u)≤M 1|u|,0<|u|≤1,(3.3)F 1(t,u)≥m 1|u|,|u|≥1,(3.4)where M 1=maxt ∈[0,T ],|u |=1F 1(t,u),m 1=mint ∈[0,T ],|u |=1F 1(t ,u).Lem m a 3.3Suppose that (H 3)holds,thenF 2(t ,u)≥m 2|u|α,0<|u|≤1,(3.5)F 2(t ,u)≤M 2|u|α,|u|≥1,(3.6)where M 2=maxt ∈[0,T ],|u |=1F 2(t,u),m 2=mint ∈[0,T ],|u |=1F 2(t ,u).Lem m a 3.4Suppose that (H 4)holds,thenI j (u)≥m I j |u|β,0<|u|≤1,(3.7)I j (u)≤M I j |u|β,|u|≥1,(3.8)where M I j =m ax |u |=1I j (u),m I j =min |u |=1I j (u),j =1,2,,p.Lem m a 3.5[16]If essinf t ∈[0,T ]g(t )=m >λ1,then the norms ∥∥and ∥∥H 10(0,T )a re equivalent.No.2Y.L.Song,SOLUT IONS TO IMPULSIVE E QUATION199Lem m a 3.6[33]There exists a const ant C(depending only on|I|≤∞)such that ∥u∥L∞≤C∥u∥W1,p(I),for any u∈W1,p(I),1≤p≤∞.In other words,W1,p(I)L∞(I),w i th continuous injection for all1≤p≤∞.Further, if I is bounded then:t he injection W1,p(I)C(I)is compa ct for all1<p≤∞;and the injection W1,p(I)L q(I)is compact for all1≤p<∞.From Lemm a3.6,t here exists a C0≥0such that∥u∥L∞≤C0∥u∥,for any u∈H1(0,T),(3.9)where∥u∥L∞=ess sup{|u|:t∈[0,T]}.Lem m a 3.7If(H1),(H2),(H3)and(H4)hold,t henφsatis es the P.S.condition.P r oof Let{u n}be a sequence in H10(0,T)such t hat{φ(u n)}is bounded andφ′(u n)→0,as n→∞.F irst,we prove t hat{u n}is bounded.By(2.1)and(2.2),one hasφ(u n)φ′(u n)u n=(21)∥u n∥2+∫Tf1(t,u n)u n dt∫TF1(t,u n)dt∫Tf2(t,u n)u n dt +∫TF2(t,u n)dt+p∑j=1∫u n(t j)I j(s)dsp∑j=1u n(t j)I j(u n(t j)).By(H2),(H3)and(H4),we can deduce t hat∫T0f1(t,u n)u j dt∫TF1(t,u n)dt≥0,(3.10)∫T0F2(t,u n)dt∫Tf2(t,u n)u n dt≥0,(3.11)p ∑j=1∫u n(t j)I j(s)dsp∑j=1u n(t j)I j(u n(t j))≥0.(3.12)By(3.10),(3.11)and(3.12),we conclude t hatφ(u n)φ′(u n)u n≥(21)∥u n∥2.Since>2,it follows t hat{u n}is bounded in H1(0,T).Therefore t here exist s a subsequence also denot ed by u n∈H1(0,T)such t hatu n u,in H1(0,T),u n→u,in C([0,T],R),as n→∞.Henceφ′(u n)φ′(u),u n u→0,∫T 0(f(t,u n)f(t,u))(u n u)dt→0,p ∑j=1(u n(t j)I j(u n(t j)u(t j)I j(u(t j)))(u n(t j)u(t j))→0,as n→∞.Moreover,an easy comput at ion shows t hat200ANN.OF DIF F.E QS.Vol.29φ′(u n)φ′(u),u n u=∥u n u∥2∫T0(f(t,u n)f(t,u))(u n u)dt+p∑j=1(u n(t j)I j(u n(t j))u(t j)I j(u(t j)))(u n(t j)u(t j)).So∥u n u∥→0,as n→∞.That is,{u n}converges st rongly t o u in H10(0,T). Therefore,φsat is es t he P.S.condit ion.P r oof of T he or em 1.1Since f1,f2and I j,j=1,2,,p are cont inuous,we obt ain t hatφ∈C1(H10(0,T),R),φ(0)=0.By Lemm a3.7,φsat is es t he P.S.condit ion.It is clear t hat∥u∥≤1/C0,where C0is de ned in(3.9),implies∥u∥∞≤1.According t o Lemmas3.2and3.3,by(3.9),one has∫T0F1(t,u)dt≤∫TM1|u|dt≤M1TC∥u∥,(3.13)∫T0F2(t,u)dt≥∫Tm2|u|αdt≥m2T Cα0∥u∥α.(3.14)Combining(3.13),(3.14),(H4)and(3.9),one hasφ(u)≥12∥u∥2M1TC∥u∥+m2TCα∥u∥α≥12∥u∥2M1TC∥u∥.Since>2,one can choose aρ>0sm all enough such t hatφ(u)≥0.Finally,it remains t o show t hat t here exist s an e such t hat∥e∥>ρandφ(e)≤0.In fact,by(2.1),(3.4),(3.6)and(3.9),for everyξ∈R\{0},u∈H10(0,T)\{0}wit h ∥ξu∥∞≥1w e have t he following inequalit yφ(ξu)=12∥u∥2ξ2∫TF1(t,ξu(t))dt+∫TF2(t,ξu(t))dt+p∑j=1∫ξu(tj)I j(s)ds≤12∥u∥2ξ2m1TC∥u∥ξ+M2TCα∥u∥αξα+p∑j=1M IjCβ∥u∥βξβ.Since>2,1<α<2and0<β<,(3.6)im plies that t here exist s aξ∈R\{0}such t hat∥ξu∥>ρandφ(ξu)≤0,so we set e:=ξu.By Theorem2.1,φhas a crit ical value. Consequently,we obt ain a weak solut ion.P r oof of T heor em 1.2f1,f2and I j are odd funct ions which implies t hatφis even. Moreover,by t he assumptions of Theorem1.2,φ∈C1(H10(0,T),R)wit hφ(0)=0sat is es t he P.S.condit ion.To apply Sym met ric M ount ain Pass Theorem,it su ces t o prove t hatφsat is es condi-t ions(1)and(2)of Theorem2.2.First ly one can easily verify(1)similar t o t hat in t he proof of Theorem1.1.It rem ains t o prove t hat,for each nit e dimensional subspace W H1(0,T),t here is an R=R(W) such t hatφ(u)≤0for all u∈W wit h∥u∥>R.By(3.8),t here exist s a constant C1for everyξ∈R\{0},u∈W\{0}wit h∥ξu∥∞≥1such t hatp∑j=1∫ξu(tj)I j(s)ds≤C1p∑j=1M Ij∥u∥βξβ.(3.15)No.2Y.L.Song,SOLUT IONS TO IMPULSIVE E QUATION201By(3.4),(3.6),t here exist constant s C2and C3for everyξ∈R\{0},u∈W\{0}with ∥ξu∥∞≥1such t hat∫T0F1(t,ξu(t))dt≥C2m1T∥u∥ξ,(3.16)∫TF2(t,ξu(t))dt≤C3M2T∥u∥αξα.(3.17) Combining(2.1),(3.16),(3.17)and(3.15),one hasφ(ξu)=12∥u∥2ξ2∫TF1(t,ξu(t))dt+∫TF2(t,ξu(t))dt+p∑j=1∫ξu(tj)I j(s)ds≤12∥u∥2ξ2C2m1T∥u∥ξ+C3M2T∥u∥αξα+C1p∑j=1M Ij∥u∥βξβ.Since>2,1<α<2and0<β<,t here is aξ∈R\{0}such t hat∥ξu∥>ρand φ(ξu)≤0,so we set e:=ξu.By T heorem2.2,t he funct ionalφpossesses in nit ely m any crit ical point s,t hat is,t he im pulsive problem(1.1)has in nit ely many weak solut ions.4E xam p leIn t his sect ion,we give an example t o illust rat e our result s.E xam ple4.1Consider t he following problem:u′′(t)+(t2+2)u(t)=f(t,u(t)), a.e.in[0,π],(u′(t j))=I j(u(t j)),j=1,2,,p,u(0)=u(π)=0,(4.1)where f(t,u(t))=u5u1/3,I j(u(t j))=u3.Com paring problem(4.1)wit h(1.1),let f1(t,u(t))=u5and f2(t,u(t))=u1/3.Then,it is easy t o verify t hat all condit ions of Theorem2.2are sat is ed.By T heorem2.2,problem (4.1)has in nit ely many solut ions.R efer en ces[1]R.K.Geor ge, A.K.Na ndakum a ra n, A.Ar ap ost at his,A not e on cont rolla bility of impu lsivesyst ems,J.Ma th.Ana l.Appl.,241(2000),276-283.[2]S.Nen ov,I mpulsive cont r olla bility and opt im iz at ion pr oblem s in popula t ion dynam ics,Non li-nea r An al.,36(1999),881-890.[3]G.Jia ng,Q.Lu,Im pulsive st at e feedba ck cont rol of a pr eda tor-pre y m odel,J.C om put.Appl.Ma th.,200(2007),193-207.[4]M.C hoisy,J.F.Gu ega n,P.R oha ni,Dyna m ics of infect ious d iseases an d pulse vaccina tion:t easin g a pa rt th e emb edded r esona nce e ect s,P hysica D,223(2006),26-35.[5] A.D’Onof r io,On p ulse va ccina t ion st ra t egy in t he SIR e pidem ic m odel with vert ic al t r ans-m ission,Appl.Ma th.Let t.,18(2005).729-732.[6]S.Ga o,L.Che n,J.J.Nieto, A.Torr es,Ana lysis of a delayed epide mic m odel wit h p ulseva ccinat ion a nd sa tur at ion incidence,Va c c ine,24(2006),6037-6045.[7]kshm ikant ham,D.D.B ainov,P.S.Simeonov,Theor y of Im pulsive Di e rent ia l Equa t ions,in:Se ries M oder n Appl.M at h.,Vol.6,World Scienti c,Te aneck,NJ,1989.[8] D.Fr anco,J.J.Niet o,Non linea r b ound ar y va lue problem s for rst or der im pulsive funct ionaldi er ential equa tions, put.Appl.Ma th.,88(1998),149-159.202ANN.OF DIF F.E QS.Vol.29 [9]X.Lin,D.J ian g,Mult iple p osit ive solut ions of Dir ichlet bounda ry value pr oblem s for secondor der im pulsive di er ential equa tion s,J.Ma th.Ana l.Appl.,321(2006),501-514.[10]X.Yan g,J.Sh en,Nonlinea r bounda ry va lue pr oblem s for r st or der imp ulsive f u nctionaldi er ential equa tions,Appl.M ath.C omput.,189(2007),1943-1952.[11]J.J.Niet o,B asic theor y f or nonresonanc e im pulsive per iodic pr oblem s of rst or der,J.M ath.An al.Appl.,205(1997),423-433.[12]J.J.Niet o,P eriodic bounda ry va lue problems for r st-or der imp ulsive or dina r y di er entialequat ions,Nonline a r Ana l.T MA,51(2002),1223-1232.[13] B.Ahm a d,J.J.Nieto,E xistenc e a nd app roxim at ion of solu tions f or a c la ss of nonlinear im-pulsive func tiona l d i e rent ia l equa t ions with a nti-per bounda ry conditions,Nonlinear Ana l.TM A,69(2008),3291-3298.[14]X.Ya ng,J.Shen,P eriodic bou ndar y va lu e problems for second-or der im pulsive int egro-di er ential equa tions,put.Appl.Ma th.,209(2007),176-186.[15]J.Yan,A.Zha o,J.J.Nieto,Exist ence and global a tt r act ivity of p osit ive per iodic solut ion ofper iodic single-spe cies im pulsive Lotka-Volter ra syste m,Mat h.C om put.M ode l.,40(2004),509-518.[16]J.Zhou,Y.Li,Exist ence a nd m ult iplicity of solut ions for som e D ir ichlet pr oblem s wit h im-pulsive e ect s,Nonlinea r An al.TM A,71(2009),2856-2865.[17]Dan Zha ng,Binxia ng Dai,In nit ely m a ny solut ions for a c la ss of nonlinea r impu lsive di er en-tia l e qua t ions wit h p eriodic bounda ry c ondit ions,C omput.Ma th.Appl.,61(2011),3153-3160.[18]Zh isu Liu,Haibo C hen and T iejun Zhou,Var ia t ional m et hods to t he second-or der im pul-sive di ere nt ial equat ion wit h Dirichlet bounda ry value pr oblem,Ma th.C omput.Model., 61(2011),1154-1161.[19]P.C hen,X.H.Tang,New exist ence a nd m ultiplicity of solut ions for som e Dirich-let pr oblem with im pulsive e ect s,Ma thema tical a nd Com pute r Mode lling,(2011).doi:10.1016/j.m cm.2011.08.046[20] D.Zha ng, B.Da i,Exist ence of solut ions for non linea r impu lsive di er ential equa tions wit hDir ic h let bounda ry conditions,Ma put.M odel.,53(2011),1687-1690.[21]Z.Zha ng,R.Yuan,An a pplica tion of va r ia tional m et hods to Dir ichlet bounda r y va lu e pr ob lemwit h im pulse s,Non linear Ana l.RWA,11(2010),155-162.[22]J.J.Niet o, D.O’R ega n,Va r ia tional appr oach to im pulsive di er ent ia l equa tions,Nonlin e a rAn al.RWA,10(2009),680-690.[23]J.Sun,H.C hen,Multiplicity of solu tions for a class of im pulsive d i erent ia l equat ionswit h Dir ichlet bounda r y condition s via va riant founta in t heore ms,Nonlin e a r An al.RWA, 11(2010),4062-4071.[24]H.Zha ng,Z.Li,Va ria tiona l app roach to impu lsive di er ent ia l e qu at ions wit h pe riodic boun-da ry conditions,Nonlinear Ana l.RWA,11(2010),67-78.[25]Y.Tia n,W.Ge,Var iat iona l met hods t o St urm-Liouville b ound ar y va lue pr oblem for im pulsivedi er ential equa tions,Nonlinear Ana l.T MA,72(2010),277-287.[26]J.Sun,H.C hen,L.Yan g,T he exist ence and m ult ip licit y of solut ions for an imp ulsive di er en-tia l equa tion wit h two pa ra m ete rs via va r ia tion al me thod,Nonlinea r An al.TM A,73(2010),440-449.[27]Y.T ia n,W.G.Ge,Var ia t ional met hods t o St ur m-Liou ville bounda ry va lue pr oblem for im-pulsive di erent ia l equat ions,Nonlin e a r Ana l.,72(2010),277-287.[28]J.Xiao,J.J.Niet o,Va riat iona l appr oach t o som e da mp ed Dirichlet nonlinea r im pulsive di e-re nt ial equat ions,J.Fra nklin I nst.,348(2011),369-377.[29]Y.T ia n,W.G.G e,Applicat ions of va r ia tional me thods to bounda r y value pr oblem for im pul-sive di er ent ial equa tions,P roc.Edinb.Ma th.Soc.,51(2008),509-527.[30]P.H.R a binowitz,Minima x Met hods in Cr itical P oint T heory wit h Applica tions to Di er entialEqua tion s,C.B.M.S.65,Am er.Ma th.Soc.,P rovidenc e,R.I.,1985.[31]M.St ruwe,Va r ia tion al Met hods,Sp ringe r-Ver lag,1990.[32] E.Zeidler,T he Ljust ern ik-Schnirem a n th eor y for in de nite a nd not n ecessa rily odd n onlin earop era t or s and it s a pplica t ions,Nonlinea r Anal.TM A,4(1980),451-489.[33]H.Br ezis,Funct iona l Ana lysis,Sobolev Spa ces a nd P a rt ia l Di er entia l Equa tions,S pringer,2010.(edited by Liangwei Huang)。
（n-1,1）-型分数阶共轭边值问题的正解王勇;杨阳【期刊名称】《数学杂志》【年(卷),期】2015(000)001【摘要】This paper studies the existence and multiplicity of positive solutions for (n−1, 1)–type fractional conjugate boundary value problem. By virtue of Krasnoselskii–Zabreiko fixed point theorem, several results are formulated in terms of some inequalities associated with Green’s func-tion. The results obtained here improve some existing results in the literature.%本文研究了(n−1,1)–型分数阶共轭边值问题正解的存在性与多解性问题。
利用Krasnoselskii–Zabreiko不动点定理,结合与Green函数相关的不等式,获得了几个存在性结果,推广了一些现有的结果。
【总页数】8页(P35-42)【作者】王勇;杨阳【作者单位】江南大学理学院，江苏无锡 214122;江南大学理学院，江苏无锡214122【正文语种】中文【中图分类】O175.8【相关文献】1.一类(n-1,1)共轭型边值问题的正解存在性 [J], 马永梅;谭春晓2.一类分数阶共轭边值问题正解的存在唯一性 [J], 郑凤霞;文武;谢茂森3.具有适型分数阶导数的边值问题的正解 [J], 董晓玉;白占兵;孙苏菁4.一类具有适型分数阶导数的分数阶微分方程m点边值问题的正解 [J], 赵微5.依赖于一阶导数的(n-1,1)共轭m点边值问题正解的存在性 [J], 纪玉德;郭彦平;禹长龙因版权原因，仅展示原文概要，查看原文内容请购买。
In 1887, the German physicist Erwin Schrödinger proposed a radial solution to the Maxwell-Schrödinger equation. This equation describes the behavior of an electron in an atom and is used to calculate its energy levels. The radial solution was found to be valid for all values of angular momentum quantum number l, which means that it can describe any type of atomic orbital.The existence and multiplicity of this radial solution has been studied extensively since then. It has been shown that there are infinitely many solutions for each value of l, with each one corresponding to a different energy level. Furthermore, these solutions can be divided into two categories: bound states and scattering states. Bound states have negative energies and correspond to electrons that are trapped within the atom; scattering states have positive energies and correspond to electrons that escape from the atom after being excited by external radiation or collisions with other particles.The existence and multiplicity of these solutions is important because they provide insight into how atoms interact with their environment through electromagnetic radiation or collisions with other particles. They also help us understand why certain elements form molecules when combined together, as well as why some elements remain stable while others decay over time due to radioactive processes such as alpha decay or beta decay.。
