【作者】 冯强；

【导师】 赵增勤；

【作者基本信息】 曲阜师范大学 ， 应用数学， 2006， 硕士

【摘要】 分析学研究对象和方法的发展表明泛函分析的地位日益重要,它在物理工程,化学,生物等方面有着广泛的应用,以泛函分析为工具来解决一些数学问题已成为分析学的一个重要内容,在解决这些领域中的非线性问题的同时逐渐形成了现代分析学中的一个非常重要的分支一非线性泛函分析。它主要包括半序方法,拓扑度方法和变分方法等内容,为当今科技领域中层出不穷的非线性问题提供了富有成效的理论工具,尤其在处理应用科学中提出的各种非线性问题中发挥着不可替代的作用。到上个世纪中叶,非线性泛函分析已初步形成了理论体系。在无穷维空间中,用泛函分析的理论来处理非线性问题也有着巨大的潜力。1921年,L.E.J.Brouwer对有限维空间建立了拓扑度的概念,1934年,J.Leray和J.Schauder将这一概念推广到Banach空间的全连续场。后来,E.Rothe,M.A.Krasnosel’skii,P.H.Rabinowitz,H.Amann,K.Deiming等对拓扑度理论,锥理论及其应用进行了深入的研究。国内张恭庆教授,马如云教授,郭大钧教授,陈文源教授,定光桂教授,孙经先教授,姚庆六教授,赵增勤教授,刘立山教授,张克梅教授等在非线性泛函分析中的众多领域都得到了大量的成就。 对微分方程边值问题的研究已经有大量的结果出现,但近年来奇异边值问题的研究尤为活跃。奇异边值问题在气体动力学,流体力学,核物理,边界层理论等实际问题中有着广泛的应用。爱尔兰著名数学家Donal O’Regan曾对此问题作了系统而详细的论述。一方面实际问题中不断的涌现出大量的微分方程非线性边值问题需要人们去深入研究。另一方面,近几十年来非线性分析有了长足的发展,其应用的理论和先进的方法日渐成熟。所以,运用这几十年来成果的基础上来研究微分方程奇异边值问题是一个富有兴趣和创新性的研究课题。更多还原

【Abstract】 Nonlinear functional analysis is more and more important and the importance is embodied by the improvement of the subjects it has studied and the development of the method it has used. During the development of solving such problems, nonliear functional analysis has been one of the most important reseach fields in modern mathematics. It mainly includes partial ordering method, topological tool for solving many nonlinear problems in the fields of the science and techology. And what is more, it is an importanr approach for studying nonlinear integral equations, differential equations and partial equations arising from many applied mathematics. L. E. J. Brouwer had established the conception of topological degree for finite dimensional space in 1912. J. Leray and J, Schauder had extended the conception to completely continous field of Banach space in 1943. After E. Rothe, M. A. Krasonsel’skii, P. H. Rabinowitz, H. Ama nn, K. eiming had carried on embeded reseach on topological degree and cone theory. Many well known mathematicians in China. say Zhang Gongqing, Ma Ruyun, Guo Dajun, Chen Wenyuan, Ding guanggui, Sun Jingxian,Yao Qinliu, Zhao Zengqin, Liu Lishan and Zhang Kemei etc., have great works in various fields of nonlinear functional analysis.The reseach of differential equations boundary value problems has a lot of achievements. It is well known that the ordinary differential equations singular boundary value problems arises in the fields of gas dynamics, newtonain fluid mechanics, the theory of boundary layer, epidemic problems and so on, andhas been considered extensively. Donal O’regan,the Ireland mathematician, dealt with the singular theory in detail and systematically. On the other hand, many nonlinear ordinary differential equations singular boundary value problems come forth all sorts of applied subjects. This forces many people to study them. On the other hand, nonlinear functional analysis has made great progress. Its powerful and fruitful theoretical tools and its advanced methods have been ripeness gradually, Thus, by using many advanced analysis of nonlinear analysis in recent years, to study differential equations singular boundary value problem is a subject which is much more interesting and may gain much more important fruitful new results.The paper is divided into five chapters according to contents.The first chapter is introduction.In it ,we narrated that the history and current situation method of boundary value problems this paper studied.In the second chapter ,a singular nonlinear boundary value problem of second order three-pointI u"{t) + f(t,u(t)) = 0, 0 < t < 1, { u(0) = 0 u{l) = au{rj)Where rj G (0,1) is a constant, / € C((0,1), [0,+oo)), is considered by Schauder fixed-point theorem, and we obtain the existence of the positive solution of the boundary problem. We state the main results as follows:Firstly,we assume the four following conditions:(Hi) 7] e (0,1), 0 < ar] < 1;(H2) f(t, u) is a continuous function on (0,1) x [0, +oo) and for all u G [0, +oo), f(t, u) is non-negative measurable about t in(0,1);/Jo(H3) V u > 0, 3 u0 > 0 we have f(t, u) < f(t, uo),t e [0,1];andv ri(t,uo)ds+ / (1 - s)f{s,uo)ds < +oo;(H4) 3f0e (0,1], for evrey u e E we have /(￡0, w) > 0;We get the rusult as follows:Theorem 2.3.1 Suppose (Hi)-(H4)(§2.2) hold, then the BVP (2.1.1) has at least one positive solution.Remark The conditions of theorem 2.3.1 are weaker than the other .relevant papers.In the third chapter, a nonlinear boundary value problem of fouth order three-point?(0) = w(l) = 0, (3.1.1)u"(0) = 0, au"{rf) = u"(l). where 0<?7<l,0<?<i,/€ C([0,1] x [0,+oo), [0,+oo)), and / is not always zero in every subinterval of [0,1] for t about u E [0, +00), is considered by making use of the Krssonsel skii fixed-point theorem of cone expansion type.We get the rusult as follows:Theorom 3.2.1 If the following three conditions hold1) /ieL1([0,l],[0,+oo)),pGC([0,+oo),[0,+oo)) andlim"^ < ^1;2) c > 0 and f(t, 1) < h(t) + p(l), (t, 1) E [0,1] x [c, +00);3)?7<ATheorom 3.3.2 If the following three conditions holdl)he ^([0,1], [0, +oo)),p e C([0, +oo), [0, +oo))and TinT^ < A2;I—too2) f(t,l) < h(t)p(l), (t,l) G [0,1] x [0,+oo);3)?7<AThoerom 4.3.1 Let (Hi)-(H5) hold, and there exsist // > 0, A > 0 where rj < p, < A < 1 and .Urn min{/(t,c) : (t, c) € [fi, A] x [0,i] > B.I—v+oothen BVP (4.1.1 ) has at least one positive solution, where-AP 7?In the fifth chapter, a singular nonlinear boundary value problem of second order three-pointu"(t) + f(t,u(t)) = 0, 0<￡<l, mi(0) - /3u’(l) = 0, u(l) - jfeu(77) = 0.Where rj e (0,1) is a constant, / € C((0,1), [0, +oo)), is considered by simple application of Schauder fixed-piont theorem, and we obtain the existence of the positive solution of the boundary problem.In this paper, we let the foil wing conditions hold firstly:Let E = C[0,1] be a Banach space, ||w|| = max u(t).Note: K(r) = {u e C+[0,1} : ||u|| < r}.We assume the follwing conditions:(Hi) o;>0,/3>0oro;>0,/5>0,0<77< 1,0 < k < g^(< J), andp := a(l - krj) + 0(1 - k) > 0;(H2) h e C((0,l)[0,+oo)),p e C([0,+oo),[0,+oo)) and iiE^ < A where A = [Sf^G{s,s)ds}-1;(H3) r > 0,c> 0 where /(*, I) < h(t)ll+T + p(l), (t, 1) e [0,1] x [0, c];(H4) Jq(I3 + as)h(s)ds < +00, J1 h(s)ds < +00;(H5) f(t, u) is not always zero for every t G [0,1] about every u G [0, +00).We get the result as follows:theorem5.3.1 If (Hi)-(H5) hold and 0 < f(t,O) < Ae, then the BVP(5.1.1) has at least one positive solution.Remark The conditions and proof method in theorem 5.3.1 is different from theorem 4.3.1/更多还原