Fixed Point Theorem And Its Application

In this paper, we use Banach's fixed point theorem to prove the existence of solution to the following nonlinear evolution equation: whereΩC RN (N>4) is a smooth bounded domain, the time T＞0 is fixed, g∈H1(Ω),f:Q×R×RN→R and satisfies some other structure conditions.In addition, we also use Schaefer's fixed point theorem to discuss the exsitence of solution to the following biharmonic equation with Dirichlet boundary condition:ΩC RN (N＞4) is a smooth bounded domain, c:RN→R, b:R→R, b, c are Lipschitz continuous functions,|b(p)|≤C1(|p|+1),|c(q)|≤C2(|q|+1), C1, C2 are constants, p∈R, q∈RN.Finally, by using Schaefer's fixed point theorem, we prove the exsitence of solu-tion to the following biharmonic problem with Navier boundary condition: is a smooth bounded domain, c:RN→R, b:R→R, b, c are Lipschitz continuous functions,|b(p)|≤C1(|p|+1),|c(q)|≤C2(|q|+1), C1, C2 are constants, p∈R, q∈RN.