非线性方程边值问题解及多解的存在性

【作者】 高岩；

【导师】 张克梅；

【作者基本信息】 曲阜师范大学 ， 基础数学， 2005， 硕士

【摘要】 非线性泛函分析是近代发展起来的一个新的数学分支,它的许多问题来自于化学反应,人口生态,传染病,经济及其它系统的模型.我们需要讨论某些具有特征方程解的存在性.本文利用锥理论,不动点理论,Leggett-Williams不动点定理等研究了几类微分方程边值问题解的情况,得到了一些新的成果.本文内容分为下列四节: 在第一节中,利用拓扑度与上下解相结合的方法研究了二阶积分-微分方程周期边值问题（PBVP）:的正解的存在性,其中,f（t,u,v）∈C[I×R×R,R],I=[0,2π],Tu（t）=integral from n=0 to t（k（t,s）u（s）ds）,k∈C[I×I,R⁺=[0,+∞),这里下解α与上解β满足β≤α. 定理1.2.1 设（1.1）存在下解α,上解β满足β≤α.如下列条件成立:（A₁）f∶I×R×R→R满足Caratheodory条件,且对任意A>0,存在函数h_A∈L¹（I）使得对a.e t∈I及满足（?）u（?）≤A,（?）v（?）≤A的（u,v）∈R²有（?）f（t,u,v）（?）≤h_A（t）; （A₂）（?）g∈L¹（I）使得k（t,s）≤g（s）,（t,s）∈I×I; （A₃）（?）M>0,使得对任意固定的u∈R,都有 f（t,u,v₂）-f（t,u,v₁）≥M（v₂-v₁）, 0≤v₁≤v₂;则周期边值问题（1.1）存在解u（t）满足β（t）≤u（t）≤α（t）. 下面列出进一步假设: （A₄）若u,（?）,v,（?）满足β（t）≤（?）≤u≤α（t）,Tβ（t）≤（?）≤v≤Tα（t）,使得 f（t,u,v）-f（t,（?）,（?））≤M（u-（?））+N（v-（?））a.e t∈I更多还原

【Abstract】 Nonlinear functional analysis is a new branch of modern analysis math-matics. Many problems arise from models of chemical reators, neutron transport, population biology, contagious disease, economics, and other systems. We need to discuss the existence of nonnegative solutions with certain desired qualitative properties. The paper employs the cone theory, fixed point index theory, and Leggett-Williams fixed point theorem and so on, to investigate the existence of positive solutions of several classes of boundary value problem for differential equations. The paper is divided into four sections.In the first section, by using the topological degree theory, we prove the existence of solutions between lower and upper solutions for periodic boundary value problems of second order differential equations, where the lower and upper solution α and p satisfy relations: β < α.whereTheorem 1.2.1 Suppose that (1.1) has the low and upper solutions α, β satisfying β <α, and the following conditions hold:(A1) f: I x Rx R—> R satisfies Caratheodory condition , and for all A > 0, there exists hA G L1(I) such that for all t ∈ I and (u, v) G R2 with |u| ≤ A, |v|≤ A, we have |f(t,u,v)| ≤ hA(t);(A2) (?)g G L1(I) such that k(t, s) ≤ g(s), (t, s) G I xI;(A3) (?)M > 0, for all fixed u G R, we havef(t, u, v2) - f(t, u,v1) > M(v2 — v1), 0 ≤ v1 < v2Then PBVP(l.l) has a solution u(t) satisfying β(t) < u(t) < α(t).(A4) Suppose that u, u, v, v satisfy β(t) ≤u≤u≤ α(t), Tβ(t) ≤v < v < Ta(t), andf(t, u, v) - f(t, % v) < M(u -u) + N(v -v) a.ete I hold, where M, N > 0 satisfy 4tt2(M + 2ttA^A;o) < 1 and2nNk0) < 1.Theorem 1.3.1 Suppose that(Ax) - (A4) hold, there exist the monotone sequences {pn(t)}, {an(t)}, where /?0 = f3(t), a0 = a(t), such thatlim pn(t) = pit), lim an(t) = v{t)uninformly in t G /, and p, v are the maximal and minimal solutions of PBVP(l.l) between a and p.Remark 1.3.1 In this paper, we extend the periodic boundary value problem of first order of [24-26] to PBVP of second order.Remark 1.3.2 Many authors get the maximal and minimal solutions only under the condition: a < f3, but we give the solutions under the condition: (3 < a.In the second section, by the two-point expasion condition of fixed point for the mixed monotone operator, the existence of positive solution to a class of singular second-order boundary value problems:au(0) - (3u’(0) = 0, ju(l) + 6u’(l) = 0is first established.Suppose that the following conditions hold :(Bi) a, b : (0,1) —* [0,+00) is continuous, and a, b is singular at t = 0,1;(￡2) There exists t0 G (0,1), such that a(t0) > 0, b(t0) > 0 and Jo G{s, s)(a(s) + b(s))ds < 00;(B3) /, g : [0, +00) —> [0, +00) is continuous;(￡4) a, 7 > 0, 0, 6>Q, A = 7/? + cry + a5 > 0.Theorem 2.3.1 Suppose that the following conditions hold:(C\) f : [0, +00) —? [0, +00) is continuous, strictly increasing and ^+oo f(u) = +00;(C2) g : [0, +00) —y [0, +00) is continuous and decreasing; (C3) 3m, M > 0, such thatwhere ￡ = sup{6 > 0, ￡~￡G(t,s) > b ￡ G{t,s)ds}, then BVP(2.1) has a positive solution .Remark 2.4.1 We translate BVP(2.1) to a mixed monotone operator. In the present results they all require Uo < vo, so the BVP cannot be proved by the had method under the condition: Uo ^ ^o- But we get it.Remark 2.4.2 Prom theorem 2.2.1 and theorem 2.2.2 and proof of them, we know A(u, v) can only be a strictly increasing about u operator.Remark 2.4.3 If A(u, v) is a only strictly decreasing about v operator, we can get the same result by getting rid of (Ci), and suppose the same conditions hold to g , then theorem 2.3.1 hold .In the third section, by using the fixed point theorems of cone expansion and cone compression, we prove a necessary and sufficient condition for the existence of positive solution to a class of singular fourth-order boundary value problems:), 0 < * < 1,u(0) = u(l) = 0, (3-1)au"(0) - 6u"(0) = 0, cu"(l) - du"{\) = 0,Where p, q e C((0,1), [0,+oo)), and p, q are singular at t = 0, t = 1,/, p € C([0, +oo), [0, +oo)) satisfying:(￡>i) a>0, 6>0, c>0, rf>0, a + b > 0, c + d > 0, p =ac + ad + bc> 0;(D2) There exist contants Ai, /Hi(0 < Ai < /^ < 1), A2, /J2(0 < A2 < H2 < +00), co(0 < c0 < 1), such that for all t G (0,1), u G C(0, +00), we have(D2)’ There exist contants Ai, >Lti(O < Ai < nx < 1), A2, /x2(0 < A2 < Hi < +00), Co(0 < Co < 1), such that for all t G (0,1), u e C(0,+oo), we haved?f{u) < f(cou) < cj?/(u);(D3) There exists Jo = [a, yS] C (0,1), and we mark mint€j0 p(t)f(l) = n > 0, minteJo q{t)g{l) = r2 > 0.Remark 3.1.1 If Co > 1, from (3.2) we know that for all t e (0,1), u G C(0, +00), we haveIn fact, if Co > 1, from (3.2) we get 0 < ￡ < 1.Remark 3.1.2 From {D2), we obtain that / is sublinear and g is super linear.Theorem 3.2.1 Suppose that (Di), (D2), (D3) or (Dx), (D2)’, (D3) hold, and0 < f\p(t)f(l) + q(t)g(l))dt < -i-, (3.4)Jo Mothen the necessary and sufficient condition that the singular boundary value problem (3.1) has a solution in C3[0,1] is0 < / \p(t)f(G(s, s)) + q{t)g(G(s, s))]dt < 00,whereMo = max G(t,s),0<t,s<l v更多还原