Existence Of Positive Solutions For Boundary Value Problems Of Fourth Order Differential Equations

Fourth-order differential equation has a wide application background.It is often used to describe a large number of physical,biological and chemical phenomena.Specially,the solution of fourth-order boundary value problem can be used to describe the bending of elastic beams in equilibrium state.Therefore,many scholars have studied the qualitative properties of solutions to fourth-order boundary value problems.Some fruitful results have also been achieved.In this paper,we mainly study the existence of positive solutions for boundary value problems of fourth-order differential equations.According to the content,there are five chapters in this paper.In chapter one,we introduce the background and development of boundary value problems for fourth-order differential equations.In chapter two,we discuss the existence of positive solutions for a class of fourth order nonlocal boundary value problem u(4)(t)+?u"(t)=f(t,u(t),u'(t),u"(t),u"'(t)),u(0)=u(1)=?01p(s)u(s)ds,u"(0)=u"(1)=?01q(s)u"(s)ds,where 0<t<1,0<?<?2,f?[0,1]×R4?R+ is continuous,p,q?L[0,1],p(t)?0,q(t)?0 and ?01p(s)ds<1,.B(?)y means of a new fixed point theorem on cones and the properties of Green's functions,the existence of positive solutions for boundary value problems is obtained.In chapter three,we study the existence of positive solutions for a class of fourth order implicit differential equation boundary value problems u(4)(t)=f(t,u(t),u'(t),u"(t),u"'(t),u(4)(t)),u(0)=u(1)=u"(0)=u"(1)=0,where0<t<1,f?[0,1]×R4?R+ is continuous.The Green's function is constructed by Laplace transformation,and then the existence of positive solutions is obtained by a numerical iteration method.In chapter four,we study the existence of positive solutions for a class of fourth order implicit differential equation boundary value problems with parameters u(4)(t)-(k1+k2)u"(t)+k1+k2u(t)=f(t,u(t),u'(t),u"(t),u"'(t),u(4)(t)),u(0)=u(1)=u"(0)=u"(1)=0,where 0<t<1,f?[0,1]×R4?R+ is continuous,k1,k2 is zero at different times,and k1,k2 ?(-?2,+?).The Green's function is constructed by Laplace transformation,and then the existence of positive solutions is obtained by a numerical iteration method.The five chapter is the summary and prospect of this paper.