The Boundary Problems Of Nonlinear Differential Equation

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 0 and f(t, 1) < h(t) + p(l), (t, 1) E [0,1] x [c, +00);3)?7 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<￡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/...