Existence And Multiplicity Of Positive Solutions For Periodic Boundary Value Problems

In recent years, periodic boundary value problem has been an important area ofinvestigation, the theory of periodic boundary value problem has been applied in alot of practical problem. Thus there has been an increased interest in studying theexistence and multiplicity of periodic boundary value problem. For example, in popu-lation dynamics,nonlinear dispersion, ecology, biological systems, etc. In practice,onlypositive solution is significant.In this paper,we discuss the existence and multiplicity of positive solutions fortwo classes of periodic boundary value problem in Banach. The paper includes twochapters.In Chapter 1,we consider a kind of periodic boundary value problem of second-order functional differential equationwhere I=[0, w],w＞0,f:I×C→R is continuous and bounded function ,C=C([-Ï„,0],R),τ≥0,:x_t∈C, x_t(θ) = x(t+θ),t∈I,-τ≤θ≤0. ForΦ∈C ,let|Φ|=(?)|. By using the Krasnoselskii fixed point theorem,we obtained the existence and multiplicity of positive solutions to periodic boundary value problem(1.1.1).As an application,we also give an example to demonstrate our result.In Chapter 2,we consider a kind of fourth-order periodic boundary value problemwhereα,β∈R,α＞0 and f : [0,2Ï€]×R⁺→R⁺ is continuous.This kind ofPBVP(2.1.1)describes the deformations of an elastic beam in equilibrium state withperiodic boundary condition.In practice,only positive solution is significant.In Section 2.2,some new conditions are obtained,which satisfy a new maximumprinciple of linear differential operator L₄u = u⁴-βu″+αu in periodic condi- tion.Especially sufficient and necessary conditions are obtained in the caseâ–³=β²-4α＜0.That is to say if one of the following cases holdsthen L₄u is strongly inverse positive, wherez=a+bi is a root of p(λ) =λ⁴-βλ″+α.Be based on this new maximum principle and a fixed point theorem in cone,the existence and multiplicity of positive solutions are obtained.Thus we improvedthe corresponding result in paper [22]. As an application,we also give an example todemonstrate our result.In this exampleα,βsatisfy(A₃) and do not satisfy conditionin paper [22].