非线性方程组边值问题的解及其应用

【作者】 刘炳妹；

【导师】 刘立山；

【作者基本信息】 曲阜师范大学 ， 应用数学， 2006， 硕士

【摘要】 非线性泛函分析是现代分析数学的一个重要分支,因其能很好的解释自然界中的各种各样的自然现象受到了越来越多的数学工作者的关注。其中,非线性边值问题来源于应用数学和物理的多个分支,是目前分析数学中研究最为活跃的领域之一。本文利用锥理论,不动点理论,Krasnoselskii不动点定理等研究了几类微分方程奇异边值问题解的情况,得到了一些新成果。 根据内容本文分为以下四章: 第一章 在Banach空间中,证明了不连续非线性二阶常微分方程组边值问题的唯一解,且唯一解可以由迭代序列的一致极限得到,并给出了迭代序列的误差估计式。 我们得到了如下结果: 定理1.3.1 设E是实Banach空间,P是E中的一个正规锥,正规常数是N,u₀∈C[I,E],D={u∈C[I,E]|u≥u₀},若f（t,v）,g（t,u）满足下列条件: （H₁）:f（t,v）,g（t,u）分别把连续函数u,v∈D映为强可测函数; （H₂）:v₀（s）∈D,且对t∈I有 （H₃）:存在非负常数L,M,且LM<64,使得当u,v∈D,u≤v时,有 θ≤f（t,v）-f（t,u）≤L（v-u）,θ≤g（t,v）-g（t,u）≤M（v-u）;更多还原

【Abstract】 Nonlinear functional analysis is an important branch of morderm analysis mathmatics, because it can explain all kinds of natural phenomena, more and more mathematicans are devoting their time to it. Among them, the nonlinear boundary value problem comes from a lot of branches of applied mathematics and physics, it is at present one of the most active fields that is studied in analyse mathematics. The present paper employs the cone theory, fixed point index theory, and Krasnoselskii fixed point theorem and so on, to investigate the existence of solutions to boundary value problem of several kinds of nonlinear systems of differential equations. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions.The thesis is divided into four sections according to contents.In chapter 1, we consider the systems for boundary value problem of discontinue nonlinear second order ordinary differential equationsWe obtain that the systems has a unique solution and the unique solution can be obtained by the uniformly limit of the iterative sequences. And the error estimate of the iterative sequences of approximation solution is given.The main results are as follows:Theorem 1.3.1 Let P be a normal cone of real Banach space E. The normal number of P is N. Let u₀ ∈C[I,E],D = {u ∈ C[I,E] | u ≥ u₀}. Suppose that f（t,v）,g（t,u） satisfy the following conditions:（H₁） : f（t,v）,g（t,u） change repectively continuous function u,v ∈ D into strongly mesurable functions.（H2） : vo（s） e D, for t e /k（t,s）f（s,vo（s））ds, f{s,vo{s))eL[I,E},vo{t) < I k（t,s）g{s,uo{s))ds, g（s,uo（s）） G L[I, E\. Jo（H3） : There exists nonnegative constants L, M with LM < 64 such that u,v G D,u < v implies that0 < f（t, v） - /（*, u）<L（v-u）, 6 < g（t, v） - g（t, u） < M（v - u）.Then Problem （1.1.1） has a unique solution （u*,v*） € DxD, and u*（t） = ui*（t）, v*（t） = Jo k（t, s）g（s, cu*（s））ds, where uj* is the unique solution of equation （1.3.3）. For any u>o € D, we have that the sequenceun{t)= I k{t,s)f （s, I k{s,z）g{z,ojn^{z))dz\ds, n = l,2,--- （1.3.4）converges uniformly to ui*{t) on / and there exists n0 € ^V such that（LM）nN （LMYNIK tj’llc < 64n iko - nolle + -63 g4n ||mi - uo\\c, n>n0. （1.3.5）Theorem 1.3.2 Let P be a normal cone of real Banach space E. The normal number of P is TV. Let v0 e C[I,E],D = {v e C[I,E] | v > v0}-Suppose that f（t,v）,g（t,u） satisfy the following conditions:-（Hi）, （H3） and （H2）* : uc（s） € D, for t e Iuo{t）< [ k（t,s）f（s,vQ（s））ds, f（s,vo（s））eL[I,E}, Jovo（t）< k（t,s）g{s,uo（s）)ds, g（s,uo（s）） e L[I,E]. JoThen Problem （1.1.1） has a unique solution （u*,i>*） G DxD and u\（t） = w*（i）, Vj（i） = Jo k（t, s）g（s, cjI（s））ds, where u>\ is the unique solution of equation （1.3.3）. For any u>q G D, we have that the sequenceujn（t）= I k（t,s）gis, I k{s,z)f（z,Ljn-l{z）)dzjds, n = l,2,--- （1.3.14）converges uniformly to cj*（i） on / and there exists n0 € N such that（LM）nN （LM）nNIK teller < 64n 11^0 "ollc + gg ^ IK U0||c, n > n0. （1.3.15）Theorem 1.3.3 Let F be a normal cone of real Banach space E. The normal const number of P is N. Let u0 e C[I, E], D — {u € C[7, iS1] | u < u0}, Suppose that f（t,v）,g（t,u） satisfy the following conditions:（#1） and {H2y* : uo（s） G D,, for t € I/ k（t,s）f（s,vo{s）)ds, f（s,vo（s））eL[I,E], ovo（t） > / k（t,s）g（s,uo（s））ds, g（s,uo（s）） e L[I, E}. Jo（H3） : There exist nonnegative constants L, M with LM < 64, such that u,v e D,u < v imply that9<f（t,v）-f（t,u）<L（v-u）, 0<g（t,v）-g（t,u）<M（v-u）.Then Problem （1.1.1） has a unique solution （u\, v%） € DxD and u*2（t） = ^（t）, w2（0 = Jo k（t, s）g（s,u>2（s））ds. where W2 is the unique solution of equation （1.3.3）. For any uio € D, we have that the sequencewn{t）= [ k{t,s)f（s,f k（s,z）g{z,u>n-i{z）)dz\ds, n = l,2,--- （1.3.16）converges uniformly to u]%（t） on / and there is no E N such that（LMYN （LM）nNlK-^Hc<V ^ IN-nolle + 63 ,g4n IK-^ollc, n>n0. （1.3.17）Theorem 1.3.4 Let P be a normal cone of real Banach space E. The normal const number of P is TV. Let v0 ￡ C[I, E], D = {v e C[I, E] \ v < v0}. Suppose that f{t,v）,g（t,u） satisfy the following conditions:（Hi）, （H3） and （#2）*** : uQ（s） G D, for t € /k（t,s）f（s,vo（s））ds, f（s,vo（s））￡L{I,E},vo（t）> / k（t,s）g（s,uo{s）)ds, g（s,uo（s）） ￡ L[I, E}. JoThen the problem （1.1.1） has a unique solution （u^v^） € D x D and u^（t） = ^W) u3i（0 — /o k（t,s）g（s,cul（s））ds. where W3 is the unique solution of equation （1.3.3）. For any u>q G D, we have thatun（t）= k（t,s）g（s, k{s,z）f{z,ion₁{z）)dz)ds, ra = l,2, ■?■ （1.3. converges uniformly to ^（i） on / and there is n0 € N such that18）（LM）nN K 64;\\u0（LM）nNV63 ’un \\ux - uolU n>n0. （1.3.19）? Remark 1.3.1 We ont only obtain the unique solution, but also give the iterative sequences. And the error estimate of the iterative sequences of approximation solution is given. Theorem 1.3.1, Theorem 1.3.2 just use lower solution.Remark 1.3.2 The above results can be generalized the following mixed systems of second-order and fourth-order boundary value problem-?（4） = /（?,?）, 0 < t < 1, -v" = g（t,u）, 0 < t < 1, u（0） = u（l） = ?"（0） = u"（l） = 0, u（0） = u（l） =0.or <-u" = f（t,v）, 0<t<l, -vW=g（t,u）, 0 < t < 1, 1,（0） = v（l） = v"（0） = v"（l） = 0,In Chapter 2, by constructing a special cone and applying fixed point theory, we investigate the existence of positive solutions for the following boundaryvalue problem’ -u’" = fi（t,u,v,u’,v’）, 0 < t < 1,-u’" = /2（i,u, *;,?>’）, 0<*<l, w’（0） = u’（l） = u’（0） = i/（l） = 0.where /x, /2 is nonnegative or /i is nonnegative and /2 is allowed to change sign.First let us list the assumptions: {A)fi G C[I xR+ x R+ xR+ x R+,R+],i = 1,2. （B）/i eC[/xi?+xi?xi?+xfl, i?+], /2 g C[/ x R+ x i? x i?+ x R, R).Our main results are as follows:Theorem 2.3.1 E = C[I,R],T = {（x,y） G ^ x ￡, | x > 6,y > 6}, P is a cone in E. Let T : Tuj{i) = Jo cu（s）ds,\/u> <E P. Suppose /i（i = 1,2） satisfy （A） and the following assumptions:（#0 （￡,!/） G P, 0 < limsup fi&X’y’TZ’Ty） < 8 uniformly on < e /.ll（a;y）ll（H2） （x,y） E P,16 < liminf f^x^^Ty) < +oo uniformiy onll（x,v）H-?oo （x,y）te/.Then Problem （2.1.1） has at least one positive solution.Theorem 2.3.2 E,P,T are the same as Theorem 2.3.1, Suppose /,（i = 1, 2） satisfy （^4） and the following assumptions:（H[） （x, y）eP,0< limsup ^x^T^V) < 8 unif0rmly out el.Mxy)\\（Hi） （x, y） G P,16 < liminf fi^^V^^y) < +oo uniformiv on t G J. ll（*）ll>o \\{x,y)\\)\\ Then Problem （2.1.1） has at least one positive solution.Theorem 2.3.3 P,E,T are the same as the above. Suppose fi（i — 1,2） satisfy （B） and the following assumptions:（H"） J2{t,x,\y\,Tx,T\y\) > Q,x G P,y G jB, there exist two constant M,N with 0 < N < M <1, such that -Nfi（t,x,y,Tx,Ty） < f2{t,x,y,Tx,Ty) <Mh{t,x,y,Tx,Ty),V（x,y） € P x E,t G /.（#￡） （x,y） G P x ￡, 0 < limsup ^ n^’^’ < 8 uniformly on||（s,j/）||-+o 11（^2/） II（if’’） （x, y） G P x ￡,16 < liminf /ifo*;^3^’7^ < +oo uniformly onte/.Then Problem （2.1.1） has at least one solution.Theorem 2.3.4 P,E,T are the same as Theorem 2.3.1. Suppose fi（i =1,2） satisfy （A） and the following assumptions: fAt x y Tx Ty)（Ai） （x, y） € P, limsup ——’ ’ ’ ’-----= aQ（t） uniformly on t G / and\\（x,y）\\supie/ aQ（t） = a < 8.（A2）（x,y）EP, limsup fJh^tllllA = ,aoo（t） uniformly on t G /ii（x,3/）n->+oo IKz.yJIIand supJe7 aoo（f） = a < 8.（^4.3） There exist （x0, j/o） G int（P）, 1$ = [ao, bo] C / and a positive number L such that L ff k（t,s）ds > l,foe any t G /o, and V（x,y） > （xo,yo） implying that （Fl（x,y）,F2{x,y）) > L（xo,yo）-Then （2.1.1） has at least two positive solutions {xi,yi),（x2,y2） € P, andRemark 2.3.1 The mathods applied in this paper is different from [3], [9] and the conditions are more weaker than those in [9]. In addition, the existence and multiplicity of solutions are obtained.Remark 2.3.2 In （2.1.1）, if f\ just contains t,v’ and , f2 just contains t, u’, then （2.1.1） is hte same as [3] in form. So this paper contains more affluent content and wider applying domain. Especially, we boarden the assumptions on super limit and lower limit in Theorem 2.3.1, Theorem 2.3.2 and Theorem 2.3.3, which is seldom in this kind of references.In Chapter 3, by constructing a special cone and applying fixed index theory in cone, we investigate the existence and multiplicity of positive solutionsof the following singular boundary value problemswhere / e C[（0,1） x’R+,R+],g G C[（0,1） x R+,R+], f{t,y) and g（t,x） are allowed to be singular at t = 0 and 2 = 1.For convenience, let us list the following assumptions:（Hi）f,g G C[{0,l)xR+,R+] and there exist functions/i2 eC[（0,l）,R+],u2 ￡ C[R+,R+) and a nondecreasing function u\ G C[i?+,i?+] satisfying（H2） There exist two constants Lx > 0, L2 > 0 satisfying 0 < LiL2CiC2 < 1 and0 < limsup^M < Lu 0 < limsup^M < ^y-40+ 1/ x->o+ ywhere 0 < d = /^ ^（1 - OHCl^ < oo, i = 1,2.（i/3） There exist two nonnegative numbers Mi,M2 satisfying MxM2 > 43（/i3/44 s（! - s）ds）-2 such thatliminf ^^ > M1;liminf ^^ > M2,uniformly with respect to t € [1/4,3/4].（H4） There exists ip G ￡[0,1] satisfying liminfx_）.o+ f{t, y) > fp{t) uniformly with respect to t G [1/4,3/4] such that（H5） There exists a constant L3 > 0 such that1- W2（Z） rhm —-— = L3.0+The main results are as follows:Theory 3.3.1 Assume （Hi） - （H3） hold. Then BVP （3.1.1） has at least one positive solution.Theory 3.3.2 Assume （Hi）, （H4） and （H5） hold. Then BVP （3.1.1） has at least one positive solution.Theory 3.3.3 Assume （Hi） - （H5） hold. Then BVP （3.1.1） has at least two positive solutions.Remark 3.3.1 Suppose that （Hi）, （H2） and the following condition hold.（H’） m , — -4-no m , v _<sub>- — -l-no\Ji3l Hilly—?-f-oo — TUU, lliUjj^-j-OQ — TLaJ,uniformly with respect to t e [1/4,3/4].Then BVP （3.1.1） has at least one positive solution.In Chapter 4, we investigates the singular systems of nonlinear second order three-point boundary value problem-u" = f（t,v）, 0<t<l,-v" = g（t,u）, 0 < t < 1, （4.1.1）i（0） = v（0） = 0, tz（l） = au（r]）, v（l） — olv（t\）,where r\ 6 （0, l）,0 < arj < 1 and f,g may be singular at t = 0 and/or 0. Under some weaker conditions the existence of positive solutions is obtained by applying the fixed point theorem of cone expansion and compression. And two examples are worked out to demonstrate our main results.（Hi） / € C(（0,l] x R+,R+）,g e C(（0,1] x R+,R+） and there exist Pi e C(（0,1], R+）, qz e C（R+, R+）, i = 1,2, q2（0） = 0 such thatf（t,v）<pi（t）qi（v）, g（t,u）<P2（t）q2（u）, t￡（0,l},u,veR+anda= / s（l — s）p\（s）ds < +00, 6= / 5（1 — s）p2（s）ds < +00. 7o ■ io（H2） There exist rx,r2 G （0,+00） and rxr2 > 1 such thatqi（v） Q2（u） hmsup------<+00, am sup------ = 0.（H3） There exist lx, l2 G （0, +00） and lil2 > 1 such thatlimini mm —;— > 0, liminr mm —;— = +00.（H4） There exist ax,a2 G （0,+00） and axai < 1 such thatqi{v） Q2M hm sup------< +00, hm sup------= 0.II—> + OO U U—> + OO u（H5） There exist /3X,jS2 G （0, +00） and fixj32 < 1 such thatliminf min —-5— > 0, liminf min —5— = +00.o+ [i] ^1 o+ [] foOur main results are as follows:Theorem 4.1.1. Assume that （Hi）-（H3） hold. Then BVP （4.1.1） has at least one positive solution.Theorem 4.1.2. Assume that （r^）, （H3） and （H5） hold. Then BVP （4.1.1） has at least one positive solution.更多还原