Solutions of Some Differential Equations 

Author  QianAiXia 
Tutor  ZhaoZengQin 
School  Qufu Normal University 
Course  Basic mathematics 
Keywords  Differential Equations Nonlinear Functional Analysis Nonlinear operator equation Theory of partial differential equations Iterative sequence Monotone iterative technique Hammerstein Normal cone Integral equation theory Cone theory 
CLC  O175 
Type  Master's thesis 
Year  2002 
Downloads  126 
Quotes  0 
Nonlinear Functional Analysis is a subject .old but fashionable.Its abundant theories and advanced methods are providing powerful and fruitful tools in solving ever increasing nonlinear problems in the fields of science and technology. Though the theories of integral and differential equations in Banach spaces, as new branches of Nonlinear Functional Analysis.have developed for no more than thirty years, they are finding extensive applications in such domains as the critical point theory,the theory of partial differential equations,eigenvalue problems.and so on,are attracting much more attentions from both pure and applied mathematicians.Professor Guo Dajun has summarized in his work [7] .such several important tasks and theirs application of Nonlinear Functional Analysis as typical nonlinear operators,Hammerstein integral operations,ordinarily and partially differential equations.the cone theory,the positive solutions of nonlinear operator equations,the number and the branch of solutions,and so on.Reference [1] includes all levels of results of the domain such as Nonlinear Functional Analysis.The present thesis employs the cone theory,monotone iterative technique, the conical expansion and compression principle,the method of upper and lower solutions,the Monch theory of fixed point,and so on,to investigate the existence of solutions of sevral differential equations .The obtained results are either newor intrinsically generalize and improve the previous relevant ones under weaker conditions.The paper is divided into four sections.In section one we investigate the following nonlinear integrodifferential equation in Banach spaces.where.We list below the conditions:(H1) There exists as the lower solution of IVP(1).i.e. satisfy where, M. .V > 0 are constants satisfying such conditions as (i)(ii) of Lemma 1.2.1.(H3) There exist Lebesgue integrable functions L(t), P(t),Q(t) > 0,such that make tt, v € D, u <v satisfyTheorem 1.3.1 Let E be a Banach space and P be a regular cone in E . Assume that (H1)(H2)(H3) are satisfied. Then /KP(1.1.1) possesses only one10solution .and the iterative sequence(1.3.1)converges uniformly on I to w(t). Moreover. there is error estimateTheorem 1.3.2 Let E be a Banach space and P be a regular cone in E. Assume that / satisfys the following assumptions:(H1)t There exists TO 6 C1[I. E] as a upper solution of Then /YP(l.l.l) possesses only one solution w, and D, the iterative sequenceconverges uniformly on I to w(t). Moreover,there is error estimateRemark 1.3.1 The key condition of Theorem 4 in [5] is (H4):There exist constants R, r > 0,such thatthe theorems in the present paper changes B.r into only integarable and nonnegative functions L(t).P(1). broadening this formula and extending this theorem.Remark 1.3.4 Those1 results presented in the given refrenccs cann’t induce the Theorem 1.3.1.1.3.2.These two Theorems in the present paper impove and extend those corresponding theorems .Moreover.its proof methods arc different from theirs.In section two.we (’insider the nonlinear Frdholm integral equation in Banach spacesis a real parameter.For the sake of convenience, we list the follwing conditions: ,satisfies the Caratheodory condition,i.e. for E , H(t, s, x) is measurable in s and continous in x for almost s J.(H2) For Vr > 0, H is boundary on J x J x Tr,so is Ik(k = 1. 2. ...... m) on Tr;(H3) ds is continous in t,and is uniformly continous in here. M = sup {h(t.s)}.771Theroem 2.3.1 Assume the conditions (H1)  (H5) are satisfied.the equation (2.1.1) has solutions in PC{J.E].Remark 2.3.2 When cone P is regular, any ordered sector is boundary.Thus the BH of (III) in the Theorem 2.3.1 is satisfied automatically.when (H5) can be omitted.Let H(t.s.x(s)) admit a decomposition of the formH(t.s.x(s)) = Hi(t.s.x(s)) + H2(t.s.jr(s)) (2.3.24)Definition 2.3.1 If there exist