The Research On The Two-point Boundary Value Problem For Ordinary Differential Equations With Nonlinearity Dependencing On Derivative

The boundary value problems for ordinary diferential equations are very im-portant. With the development of science and technology, all kinds of natural andmarginal problems in the felds of engineering, mechanics, astronomy, economics,cybernetics, biology etc can be described into the ordinary diferential equationsboundary value problems. As we all know, it is quite difcult to fnd out thesolutions of the diferential equations. So, the theoretical study of the existenceof the solutions and their character from the theory attracts a great attention.Two-point boundary value problems for ordinary diferential equations alsohave broad actual signifcance. It has broad application in the physics, life scienceand cybernetics.For example,the ploblem of Sturm-Lioville,the string vibration the-ory,the fluid mechanics problem and so on. In recent years,the theory of two-pointboundary value with ordinary diferential equation is an important developing. Atpresent, the existence of solutions for boundary value problems have been widelystudied. For instance, Ma ruyun~[20-23]and Sun jingxian~[12,26,33]at home, B.P.Rynne~[1-3]and Jacek Gulgowsk~[9,10]abroad have obtained many results under diferentconditions. These results are mostly on nonlinearity f~[1-3],[7-12],[19-23,26,28,31-34]without derivative dependence, but the case f with derivative dependence is stud-ied very little. Now in the thesis we allow the existence of a derivative sub-term inthe nonlinear term.This greatly expands the scope of the nonlinear term.The thesiscontains two chapters.In the frst chapter, we consider the second-order two-point boundary value problemswhere f (ξ,η) =βξ+αη+ F (ξ,η), F (ξ,η) : R×Râ†'R is continuous, F (ξ,η) = ((ξ~2+η~2)21 )(when (ξ,η)â†'(0, 0)),β∈C [0, 1] andβ> 0,α= 0.Jingxian Sun[20 23]and so on obtain a global bifurcation result for a relatedbifurcation problem,and then use this to obstain multiple (at least eight) solutionsof the Sturm-Liouville problem having specifed nodal properties.This paper usingthe same method and in the similar conditions investigates the spectrum structureof corresponding guide operators and a global bifurcation problem with Rabinowitzbifurcation theorem.We then use this to obtain multiple solutions of the two-pointboundary value problems.In the second chapter,we consider the second-order two-point boundary valueproblemswheref0= f (s,Yulian An[1 3]investigate the global structure of second-order m-point bound-ary value problems with superlinear nonlinear. In the thesis we allow the existenceof a derivative sub-term in the nonlinear term, and then investigate the globalstructure of second-order two-point boundary value problems and the number ofthe corresponding solutions.The thesis not only prove each existence theory,but also give an example behindthat.