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

Neumann boundary value problems are an important class of boundary value problems in ordinary differential equations.In particular,the fourth-order Neumann differential equation boundary value problem describes the deformation of elastic beams sliding at both ends,and it has a wide range of applications in elastic mechanics and engineering physics.This paper mainly studies the existence and uniqueness of solutions for two classes of fourth-order Neumann boundary value problems.For the first class of problem,under the premise that the lower solution is not less than the upper solution,the existence of the solution is proved by using the quasilinearization technique,and the quadratic convergence of the approximation sequence is discussed;the monotonic iterative method is used to prove the existence of monotonic sequences that converge to the maximum and minimum solutions respectively.For the second class of problem,the expression of Green function is given,and the existence and uniqueness of the positive solution is discussed by using the fixed point theorem in cones on Banach space and the fixed point theorem of concave operator.