Existence And Multiplicity Of Positive Solutions For A Class Of Nonlinear Fourth-order Neumann Boundary Value Problem

In this paper,the existence of positive solutions to a class of fourth-order nonlinear Neumann boundary value problemswith two parameters is studied.By introducing the research background and status analysis of the equation of elastic beam,the main problems discussed in this paper are proposed,specifically including:Firstly,we may discussed when k1?k2<0,k1<0<k2??2/4 and 0<k1<k2??2/4 three cases of linear Neumann boundary value problems of expression of Green's function and the corresponding definite properties;Secondly,by using Green's function and fixed point index theory in cones,we obtain the optimal conditions for existence of positive solutions of the fourth-order Neumann boundary value problemwith conditions k1?k2<0,where ??C([0,1]×[0,?),[0,?)).Lastly,by using the fixed point theorem of cone mapping,we obtain the exis-tence and multiplicity of positive solutions for the semipositone fourth-order Neu-mann boundary value problemwith conditions k1<0 k2 ??2/4,where ?>0 is a parameter and f?C([0,1]×[0,?),(-?,?)),with ?(x,y)?-X for some positive constants X.