The methods of pseudolinear equations and successive approximations for quasilinear two-point boundary-value problems (Q1084840)

Quasilinear vector two-point boundary value problems for equations of the type \((-1)^ kd^ k/dx^ k[p_ 1(d^ ku/dx^ k,u,x)]+p_ 2(d^ ku/dx^ k,u,x)=f(x)\) are considered using two competitive methods: pseudolinear equations (PE) and successive approximations (SA). The strategy of the methods is the same, namely, iterative calculation of approximate solutions using fixed operators (based on fixed symmetric positive definite coefficient matrices Q and \(\hat Q\) of order 2m\(\times 2m)\). The PE method consists in iteratively minimizing the \((L_ 2(a,b))^ m\) error with weights Q and \(\hat Q\) when the exact solutions of the given and conjugate problems respectively are replaced by approximate ones, i.e., the PE method solves the problems in tandem, while the SA method does it independently. Conditions on the coefficient functions used in the two methods that are sufficient for global linear convergence in the energy spaces are developed.