Existence Of Solutions For Several Kinds Of Differential Equations

Before the nineteen thirties, the research on functional differential equation was limited to the special properties of some special type equation. In1982and1931, Volterra discussed more general functional differential equations. He defined energy function to observe asymptotic behavior in the short time by using connections between functional differential equations and some physical system, which was a milepost in the development of functional differential equation. After that, a lot of dynamical system problems were proposed in natural science and social science, such as physics, circuit of signal system, ecological system, genetic, epidemiological and so on. With further research on these problems, the theories of functional differential equations are gradually perfect and many creative results on functional differential equations have been obtained. In the qualitative theory of differential equations, periodic solutions are always an important research topic and have been highly concerned by the scholars for decades. So many results on periodic solutions of differential equations are obtained in recent years.There are three chapters in this paper. Chapter1discusses the following variable coefficient nonlinear Duffing equation with impulses and delayswhere0<t1<t2<...<tk<T,xt(s)=x(t+s),s∈[-Ï„,0],Ï„>0,p:Râ†'R+,q: Râ†'R-,tm=tl(m=nk+l,0<l＜k),h:Râ†'R+,R=(-∞,+∞), R+= [0,+∞).This chapter exploits the Krasnoselskii fixed point theorem to obtain the existence of periodic solutionsChapter2applies the fixed point theorem in cone,and investigates the periodic boundary value problems as follows:where I=[0,2Ï€],f1,f2:I×R+×R+â†'R is continuous. By constructing a suitable cone,the existence of positive solutions of(2.1.1)is obtained.Chapter3discusses the following boundary value problems:(φ(u"))’+λf(t,u(t))=0,t∈(0,1)ï¼›(3.1.1)φ(u"(0))=f01β(s)φ(u"(s))ds,u’(0)=0,u(1)=f01α(s)u(s)ds,(3.1.2) whereφ:Râ†'R,saisfying1.if x≤y,then(x)≤φ(y),x,y∈Rï¼›2.φ(xy)=φ(x)φ(y),x,y∈R,φ(0)=0,φ,φ-1are continuous.The boundary value problema with integral boundary conditions studied in this chapter are generalizaions of multi-point boundary value problems,and the sign of nonlinear terms are variable.By constructing a special cone,we exploit the traslation transformation and the fixed point theorem to obtain the existence of positive solutions of the problems.We also discuss the value range of the parameters λ to guarantee the existence of positive solutions.