Second-order Nonlinear Equations And The Equations Of The Existence And Applications

Nonlinear functional analysis is an important branch of mathmatics, and it can explain several kinds of natural phenomena. The boundary value problems (BVPs) for nonlinear differential equations arise in a variety of areas of applied mathematics, physics and variational problems of control theory, it is at present one of the most active fields in analyse mathematics. Among them, mulitipoint BVPs come from a lot of branches of applied mathematics and physics, and it is very meaningful in both practical and theoretical aspects. The present paper employs the cone theory, fixed point theory, topological degree theory and upper-lower solutions method and so on, to investigate the existence of positove solutions to some kinds of nonlinear differential equations, equation systems, and impulsive-integro equations. The results obtained are either new or essentially generalize and improve the previous relevant ones under weaker conditions.The thesis is divided into three chapters according to contents.In chapter 1, By using the fixed point index in a cone, we establish the existence of at least one positive solution of a class of m-point boundary value problems for systems of second-order differential equationswhere a(t),b(t)∈C((0,1),R⁺)∩L¹(0,1)；f,g∈C(J×R⁺×R⁺,R⁺)inwhich J=[0,1],f(t,0,0)=g(t,0,0)=0；α,β>-π²,0<ζ₁<…ζ_m-2<1,a_i,b_i,c_i,d_i∈[0,+∞)(i=1,2,…,m-2)；a(t),b(t) may be singular at t = 0,1.(T_iω)(t)=∫₀^t k_i(t,s)ω(s)ds,in which k_i(t,s)∈C[D,R⁺](i = 1,2) where D = {(t,s) | 0≤s≤t}. Underthe conditions of f, g are suplinear or sublinear, we obtain at least one positive solution. The problem we investigated is more general than that is considered in [9], and our results generalize and extend previous results in the field.In Chapter 2, by using the cone theorem and momotone iterative technique, we study the existence of positive solution of following second-order impulsive differential equation:where T is Volteral opetator and S is Fredholm. With the existence of supersolution or lower solution, the maximal or minimal solutution can be obtained.In Chapter 3, following second-order three-point boundary problemis considered by means of iterative technique and fixed index where 0 <β< 1, 0 <η< 1, /may be singular at u - 0 and t = 0,1. Under some conditions concerning nonlinear term f, one positive solution can be obtained.