Study On The Periodic Solution To The Variation Problem Of Semilinear Differential Equation

As everyone knows, it is of great importance to study existence and uniquenessfor differential equations. There are many methods to study this, such as variationalmethod, homeomorphism method, semi group of operators etc. Hilbert space methodattracts a great many scholar's interest as an important furthermore creative method.In this thesis, we discuss the existence and uniqueness for the weak solution of twokinds of semilinear differential equations with Hilbert space method. First, bySobolev embedding theorem and Schauder fixed point theorem, we establish Hilbertspace method for the systems having meshed spectral conditions and obtain existenceof solution to the semilinear operator equation. Then we apply this result tospecifically ordinary equation. Second, with Galerkin approximation procedure, wediscuss a type of second-order semilinear hyperbolic systems without resonance. Ateach finite step, we prove the existence of an approximate solution by applying aminimax principle. Then we give an estimate for the approximate solutions. In theend, by virtue of Schauder fixed point theorem, the existence of periodic weaksolution to hyperbolic equation is proved as well.