ABSTRACT
In this chapter we introduce two other methods, namely the monotone iterative method and invariant regions. These two methods were mainly developed in the 1970s, but they are still important and useful nowadays. The basic strategy of the monotone iterative method for the second-order elliptic equations or the second-order parabolic equations relies on the comparison principle, an invariant of the well-known maximum principle. By the comparison principle one can define the upper solution and lower solution or supersolution and subsolution for nonlinear second-order parabolic equations or nonlinear second-order elliptic equations, and construct the corresponding monotone iterative sequence. If a pair of upper solution and lower solution u, u satisfying u ≤ u can be found, then by the comparison principle, it can be proved that the corresponding sequence un obtained from u or u by the iterative method is monotone, and it stays between u and u, i.e., u ≤ un ≤ u. From the monotonicity and uniform bound of un, the pointwise convergence of un immediately follows. Then one could prove that un has a subsequence that converges in a more regular space, and it turns out that the limit function u is indeed a classical solution of the original problem. The general idea of the monotone iterative method, including the
interval iteration method, probably goes back to an early work [34] by German mathematician L. Collatz in 1964. Applications to semilinear second-order elliptic equations were proposed by H. Amann in 1971 [3] where the topological method was also used to conclude existence of multiple solutions. In 1972, D. Sattinger [129] used the monotone iterative method to study the existence of solution to the initial boundary value problem for the semilinear second-order parabolic equation. Since then, this method has been successively extended to semilinear
the tem. We refer the reader to the book [117] by C.V. Pao for the details. A closely related method, namely the method of invariant regions or
invariant sets, was developed in the later 1970s. This method allows one to obtain uniform L∞ norm bounds of solution to some nonlinear evolution equations. Therefore, it also provides a powerful tool to deal with the issue of existence of solutions to these equations. In this aspect, H. Amann [7] and K. Chueh, C. Conley and J. Smoller in [33] made important contributions. In this chapter, we first introduce the monotone iterative method. In the final section of this chapter, the method of invariant regions is also introduced.
