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具积分边值条件的二阶、三阶和四阶微分方程的正解

作　者: 卡达（Khidir Shaib）
导　师: 高文杰
学　校: 吉林大学
专　业: 应用数学
关键词: 四阶微分方程边值条件三阶二阶 existence integral solutions boundary fixed 硕士学位论文 nonnegative there subinterval operator ordinary uniqueness following conditions defined semilinear
分类号: O175.8
类　型: 硕士论文
年　份: 2011年
下　载: 23次
引　用: 0次
阅　读: 论文下载

内容摘要

This thesis is a survey of the recent results in studying the boundary value problems of ordinary differential equations with integral boundary conditions. We briefly overviewthe resent situation for studying the problems and also survey the results with concentration on some important equations of second, third and fourth order, some systems, some nonlocal differential equations, some nonlocal boundary conditions and some boundary value problems of partial differential equations.We introduce the existence of solutions; introduce the existence of positive solutions; introduce the existence and uniqueness of a solutions; introduce the existence of positive solutions to a system of some partial differential equations of second order. Introduce existence of multiple positive solutions of nonlocal boundary conditions; introduce existence of solutions of third order. We also introduce existence and multiplicity of positive solutions for a class of nonlinear boundary value problems; introduce the symmetric positive solutions for nonlocal boundary value problems; introduce symmetric positive solutions for p-laplacian and introduce the positive solutions of some nonlocal to some partial equations of fourth order.This thesis consists of 4 Chapters. In Chapter 1, contain the introductions of the second, third and fourth order boundary value problem with integral boundary condition.In Chapter 2, we introduce several important theorems are used Leary-schauder type and the Banach contraction principle, Fixed point in a cone, fixed point index and also methods of upper and lower solutions in proving the existence of solutions our main results to the boundary value problems of second order equations. We first list some necessary and sufficient conditions to the existence of solutions to the boundary value problemBy using Leary-Schauder type and the Banach contraction principle theorems, the following hypothesisf:[0,1]×R→R is a given function and g:[0,1]→R is an integrable function Theorem 1. Assume that f is a an L1-Carathe’odary function and the following hypothesis (A1) there exists l∈L1([0,1],R+) such that |f(t,x)-f(t,x]≤l(t)[x-x| (?)x,-x∈R,l∈[0,1] holds, if‖l‖L1+(‖g‖L1‖l‖L1)/g*<4, then the BVP(1) has a unique solution.Next, we introduce the results for studying the existence of positive solutions to the following problem with the use of fixed point in a cone to completely continuous integral boundary condition, where f: [0,1]×R→R is continuous, g0,g1:[0,1]→[0,+∞) are continuous and positive, a, b are nonnegative real parameters. Theorem 2. Assume (H0), (H1), (H2) and (H3) are satisfied. Then (2) has at least one positive solution.At the end of this chapter, we introduce existence of positive solutions to a system of second order nonlocal boundary value problems by using fixed point index theory in a cone. we consider the following problem whereαandβare increasing nonconstant functions defined on [0,1]withα(0)=0=β(0); f∈C([0,1]×R+×R+,R+), g∈C([0,1]×R+xR+,R+) and Hi∈C(R+,R+)(i=1,2). here (?) u(τ)dα(τ) and (?) v(τ)dβ(τ) denote the Riemann-Stieltjes integrals. (H1) lim supu→0+(Hi(u))/u<1/λ1(i=1,2) (H2) lim supu→+∞(Hi(u))/u<1/λ1(i=1, 2). (H3) there exist p∈C(R+, R+) and q∈C(R+,R+) such that(1) p is concave and strictly increa sin g on R+;(2) lim infv→+∞(f(t,u,v))/(p(v)>0 unifrmly with respect to (t, u)∈[0,1]×R+ and (t,v)∈[0,1]×R+ respectively;(3) limu→+∞(p(Cq(u)))/u=+∞, for all C>0. (H4) lim supu+v→+0 (f(t,u,v))/(u+v)=0, lim supu+v→+0 (g(t.u,v))/(uv)=0,unifrmly with respect t燵 0,1] (H5) there exist a∈C(R+,R+) and b∈C(R+, R+)such that(1) a is concave and strictly increasing on R+ with a(0)=0;(2) lim infv→0+(f(t,u,v))/(a(v))>0 lim inf (g(t,u,v))/(b(u)>0 uniformly with respect to (t,u)∈[0,1]×R+ and (t, v)∈[0,1]×R+ respectively;(3) limu→0+ (a(Cb(u)))/u=+∞, for all C>0. (H6) lim supu+v→+∞(f(t,u,v))/(u+v)=0, lim supu+v→+∞(g(t,u,v))/(u+v)=0,uniformly with respect to燵 0,1] (H7) There exist L1∈(0,1/λ1) and L2∈(,1/λ2) such that Hi(x)≤Lix(i=1,2) for all x∈R+ (H8)f(t, u, v) and g(t, u, v) are increasing in u and v and there exists N>0 such that for all t∈[0,1], whereγdenotes the constantWe obtain the following theorem: Theorem 3. Suppose (H1), (H3) and (H4) hold. Then (3) has at least one positive solution (u,v)∈(p×p)\{0}.Chapter 3, we use fixed point of the corresponding Hammerstein integral operator to introduce the existence of multiple positive solutions a third order of the following problem where G is the Green function.The main conclusion is the following theorem. Theorem 4. supposeλ[1]≠1. then the unique solution of the BVP um(t)=y(t), u(0)=0, u′(p)=0, u″(1)=λ[u″] is given by Also, we introduce the existence solutions a third order to the following problem with the use of a priori bounds and fixed point theorems, where f:[0,1]×R3→R, and h1,h2:R2→R are continuous and satisfy some other conditions that will be specified later, a,b are positive real parameters. The main result of the following.Theorem 5. ([12,theorem 2.1]). Let U be an open set in a closed, convex set C of a Banach space E. assume that 0∈U, F(U) bounded and F:U→C is given by F=F1+F2 where F1: U→E is continuous and Completely continuous and F2:U→E is a nonlinear contraction, then either,(A1) F has a fixed point in U ; or(A2) there is a point u∈(?)U andλ∈(0,1) with u=λF(u). At the end of this chapter, we introduce a generalization of Mawhin’s coincidence degree theory to proving the existence of a solution to the following problem where 0<p<1, the nonlinear termfsatisfies Carathe’odory conditions with respect to L1[0, T],λ[v]=(?) v(t)dA(t), and the functionalλsatisfies the resonance condition.The main results in the following theorem. Theorem 6. LetΩ(?) X be open and bounded. Let L be a freholm mapping of index zero and let Nbe L-compact onΩ. Assume that the following conditions are satisfied:(ⅰ) Lu≠μNu for every (u, v)∈((dom L\ker L)×(0,1);(ⅱ) Nu (?)Im L for every u∈ker L (?) drΩ;(ⅲ)degB(JQN|ker L(?)ΩΩ(?)ker L,0)≠0, with Q:Z→Z a continuous projector, such that ker Q=Im L and J:Im Q→ker L is an isomorphism. Then the equation Lu=Nu has at least one solution in dom L(?)Ω.we say that the map f:[0,T]×Rn→R satisfies Carathe’odory conditions with respect to L1[0, T] if the following conditions holds.(ⅰ) for each z∈Rn the mapping t→f(t, z) is Lebesgue measurable.(ⅱ) For a.e. t∈[0, T], the mapping z→f(t, z) is continuous on Rn.(ⅲ) For each r>0, there existsαr∈([0,T], R) such that for a.e. t∈[0,T] and for all z such that |z|<r, we have |f(t, z)|≥αr(t).Chapter 4, we study the following problem where w may be singular at t=0, and(or)t=1, f∈C([0,1]×[0,+∞)×(-∞,0], [0,+∞)), and g,h∈L1[0,1] are nonnegative. Assume that(H1)w∈C((0,1), [0,+∞)), 0<(?)w(s)ds <+∞and w does not vanish on any subinterval of (0,1)(H2) f∈C([0,1]×[0,+∞)×(-∞,0],[0,+∞));(H3) g, h∈L1[0,1] are nonnegative, andμ∈[0,1), v∈[0,1), whereThe main theorem isTheorem 7. Assume that (H1)-(H3) hold. If▽f0<1 <△f∞, then problem (7) has at least onepositive solution.At the end of this Chapter, By the use of the Krasnosel’skii’s fixed point theorem, we introduce the existence positive solution for the nonlocal fourth-order boundary-value problem of following problem where p,q∈L[0,1],λ>0 and f∈C[0,1]×[0,∞)×(-∞,0],[0,∞)).Assume that(A1)λ>0 and 0<β.<π2.(A2) f∈C[0,1]×[0,∞)×(-∞,0], [0,∞)), p,q∈L[0,1], p(s)≥0, q(s)≥0, Theorem 8. Assume (A1)and (A2) hold. (ⅰ)f0<1/λη0,f(?)>1/λη1；(ⅰ)f0>1/λη0,f0<1/λη0.Then problem(8)has at least one positive solution,if one of the following cases holds.

全文目录

Abstract  4-11
List of contents  11-13
Chapter 1. Introduction  13-17
Chapter 2. Second-order boundary value problem with integral boundary conditions  17-32
  2.1 The existence of solution for the case wthen f is independent of y'  17-19
  2.2 The existence of positive solutions  19-23
  2.3 The positive solutions to the case when the nonlinear term depends on y'  23-24
  2.4 The existence and uniqueness of solutions to quasilinear equation with integral boundary conditions  24-28
  2.5 The existence of positive solutions to a system of second order nonlocal boundary value problems  28-32
Chapter 3. Third-order boundary value problems with nonlocal boundary Conditions  32-48
  3.1 The existence of multiple positive solutions for a third order boundary value problem with nonlocal boundary conditions  32-37
  3.2 The existence of solutions to a semilinear third order differential equations with integral boundary conditions  37-41
  3.3 The existence of solutions for a third order quasilinear differential equation with integral boundary conditions  41-45
  3.4 The third order nonlocal boundary value problem at resonance  45-48
Chapter 4. Fourth order boundary problem with integral boundary conditions  48-66
  4.1 The existence and nonexistence of positive solutions for a class of boundary value problems with integral boundary conditions  48-54
  4.2 symmetric positive solution  54-57
  4.3 The symmetric positive solutions for p-Laplacian fourth-order differential equations with integral boundary condition  57-62
  4.4 The positive solutions of some nonlocal fourth-order boundary value problem  62-66
Reference  66-70
Acknowledgement  70

具积分边值条件的二阶、三阶和四阶微分方程的正解

内容摘要

全文目录

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