We investigate convergence properties in weighted spaces of distributions D′ Lp and their test spaces DLq , 1 p + 1 q = 1. Also we give characterization of weak limits of weakly convergent sequences of L-distributions. AMS Mathematics Subject Classification (2010): 46F05

We prove that RO^fNJ/fin, C*) is isomorphic under p = cf(2No) to the Boolean completion of the comparison ordering of the absolutely divergent series (downwards) and under Nj = cf(2No) to the Boolean completion of the ratio comparison ordering of absolutely convergent series (upwards).

We prove a powerful scaling property for the extremality condition in the recently developed variational perturbation theory which converts divergent perturbation expansions into exponentially fast convergent ones. The proof is given for the energy eigenvalues of an anharmonic oscillator with an arbitrary x p-potential. The scaling property greatly increases the accuracy of the results.

We design a new continuation method for the solution of nonlinear complementarity problems with uniform P−functions. Similar to interior-point methods, we try to follow the central path inexactly. In contrast to interior-point methods, however, our iterates are allowed to stay outside of the positive orthant. The method is shown to be globally and superlinearly (quadratically) convergent.

We consider in dimension four weakly convergent sequences of approximate biharmonic maps into sphere with bi-tension fields bounded in L for some p > 1. We prove an energy identity that accounts for the loss of Hessian energies by the sum of Hessian energies over finitely many nontrivial biharmonic maps on R.