Existence Of Positive Solutions For Two Classes Of Nonlinear Fourth-order Boundary Value Problems

This dissertation obtains the existence and the global structure of positive solutions for two kinds of nonlinear fourth-order ordinary differential equations with two-point boundary value problems by using the global bifurcation theory.The main results are described as follows:1.We use the global bifurcation theory to study the existence and multiplicity of positive solutions for the following nonlinear fourth-order boundary value problem(?)where f:[0,1]× R × R?R is a continuous function which satisfies the sign condition and there exist the function q ? C([0,1],(0,?))such that f(t,u,v)=p(t)u+o(|(u,v)|)near(0,0)and f(t,u,v)=q(t)u+v(|(u,v)|)near ?.Wei,Yongfang et al.[Appl.Math.Lett.,87(2019)]only obtained the existence and uniqueness of the solution.Therefore,the main results in this section are more abundant than those of Wei,Yongfang et al.[Appl.Math.Lett.,2019].2.We use the global bifurcation theory to study the global structure of positive solutions for the following nonlinear fourth-order boundary value problem(?)where M?[-m04,m14),m0?4.73004,m1?5.553,? is a positive parameter,f:[0,1]×[0,?)?[0,?)is a continuous function,f(t,s)>0,t ?[0,1],s ?(0,?)and f is asymptotically linear growth at s=0.Furthermore,when the nonlinear term f satisfies the oscillation condition,we obtain the oscillation of the connected branches of positive solutions and the oscillation amplitude is bounded.When u(1)=k(?)01 u(s)ds,where k is a bounded and positive parameter,Cabada et al.[Discrete Contin.Dyn.Syst.Ser.B.,2020]obtained the existence of infinitely many positive solutions by using the fixed point theorem of extension and compression of cones,however,they haven't studied the existence of infinitely many positive solutions for the case u(1)=0.Therefore,the main results in this section supplement the results of Cabada et al.[Discrete Contin.Dyn.Syst.Ser.B.,2020].