Positivity Of Green Functions For Linear Fourth-order Boundary Value Problems

In this thesis,by using the the oscillation theory of two-term differential equations,we study the positivity of Green functions of the linear fourth-order boundary value problems whose both end-points being clamped and whose both end-points being fixed,respectively.We describe them in detail as follows.1.We study the positivity of Green function of the linear fourth-order boundary value problem which correspond to both ends being clamped,and obtain the value range of the parameters ?,??R for positive of Green function.When ?? 0,the main re-sults partly extend the ones results of Ulm[Proc.Amer.Math.Soc.,1999],Caba-da,Enguica[Nonlinear Anal.,2011]and Ruyun Ma,Haiyan Wang,Elsanosi[Math.Nachr.,2013].Finally,we obtain existence of positive solutions of the corresponding nonlinear boundary value problems via the method of lower and upper solutions.2.By using the oscillation theory of two-term differential equations.we in-vestigate the positivity of Green function of the linear fourth-order boundary value problem which describes the deformations of an elastic beam whose both end-points are fixed,where ?,? ? R are parametres,and give the sufficient conditions for Pos-itivity of Green Functions.The main results not only directly improve the only results of Ulm[Proc.Amer.Math.Soc.,1999]and Cabada,Cid,Sanchez[Non-linear Anal.,2007],but also partly improve the results for the disconjugacy and equivalence of Ruyun Ma[J.Math.Anal.Appl.,2006].Based on the above result-s,we obtain existence of positive solutions of the corresponding nonlinear boundary value problems by using the method of lower and upper solutions.