Error bounds via a new extremum variational principle,mean square residual and weighted mean square residual

Error bounds for a wide class of nonlinear one-dimensional boundary value problems are derived from a new extremum variational principle. A new least-squares approximate technique, based on a weighted mean square residual, is established. Also, the value of the weighted mean square residual and value of the classical mean square residual are used for error estimate. The results are illustrated by four examples.