The Existence For Even Order Nonlinear Differential Equations

It is of great importance to study existence and uniqueness for nonlinear differential equations. There's many methods to study it, such as algebraic method, variational method, fixed point method, topological degree homotopy method, monotone iterated method, homeomorphism method. In this thesis, we discuss the existence for two kinds of nonlinear differential equationsu²ⁿ+(?)G(u) =M(u) and ∑|n α_ju^2j + (?)G(u) = M(u) . First,the existence foreven order Duffing differential equations is discussed, with Lu is self-adjoint operator and M is a bounded compact operator, The result of existence is discussed by virtue of homeomorphism, extension and fixed point method. Secondly, we expand the Duffing differential equations to universal kinds, with virtue of homeomorphism, extension and fixed point method to get the result of existence.