Two theorems are proved. They are with principal significance in functional analysis, for they imply some well known theorems, such as the open mapping theorem, the closed graph theorem and the Banach Steinhaus theo...Two theorems are proved. They are with principal significance in functional analysis, for they imply some well known theorems, such as the open mapping theorem, the closed graph theorem and the Banach Steinhaus theorem.展开更多
The main result of this paper is a basic theorem about generalized Galerkin approximations for pseudoinverses and operator equations of the first kind, which is presented as follows ：Let H be a Hilbert space, { Hn } ...The main result of this paper is a basic theorem about generalized Galerkin approximations for pseudoinverses and operator equations of the first kind, which is presented as follows ：Let H be a Hilbert space, { Hn } a sequence of closed subspaces of H, Pn the orthogonal projection of H onto Hn, A∈B（H） and An∈B（Hn）. Suppose s-lim↑n→∞Hn=H, lim↑n→∞｜｜Pn°（A-An） ｜｜n=0,-↑R（An）=R（An）（n∈N）. Then the following four propositions are equivalent ： （a） sup↑n∈Nv∈An^-1 inf ｜｜υ｜｜〈∞ if un∈R（An） and lim↑n→∞un=0; （b） sup↑n∈N｜｜ An ｜｜ 〈∞; （c） if un∈R（An） and lim↑n→∞ un=u, then u∈R（A） and s-lim↑n→∞An^-1（un）=A^-1（u）; （d） if un∈R（An） and lim↑n→∞un=u.then u∈R（A） and lim↑n→∞Au^＋（un）=A^＋（u）. Furtherrnore, if any of the above propositions holds, we have thin N（A）=s-lim↑n→∞N（An ）,R（A） = s-lim↑n→∞R（An ）, -↑R（A） =R（A）.展开更多
A coupled system of three nonlinear oscillators with symmetric couplings is studied.It turns out that there exists a reduced variational equation to the in phase periodic solution of such a coupled system.By the symme...A coupled system of three nonlinear oscillators with symmetric couplings is studied.It turns out that there exists a reduced variational equation to the in phase periodic solution of such a coupled system.By the symmetry one establishes a structural property for the monodromy matrix of the reduced variational equation,which simplifies the computation of multipliers to a great extent.As an application of the above results,a coupled system of three Poincare oscillators is discussed as well.展开更多
文摘Two theorems are proved. They are with principal significance in functional analysis, for they imply some well known theorems, such as the open mapping theorem, the closed graph theorem and the Banach Steinhaus theorem.
基金Supported by the Wuhan University Teaching Re-search Foundation (TS2004030)
文摘The main result of this paper is a basic theorem about generalized Galerkin approximations for pseudoinverses and operator equations of the first kind, which is presented as follows ：Let H be a Hilbert space, { Hn } a sequence of closed subspaces of H, Pn the orthogonal projection of H onto Hn, A∈B（H） and An∈B（Hn）. Suppose s-lim↑n→∞Hn=H, lim↑n→∞｜｜Pn°（A-An） ｜｜n=0,-↑R（An）=R（An）（n∈N）. Then the following four propositions are equivalent ： （a） sup↑n∈Nv∈An^-1 inf ｜｜υ｜｜〈∞ if un∈R（An） and lim↑n→∞un=0; （b） sup↑n∈N｜｜ An ｜｜ 〈∞; （c） if un∈R（An） and lim↑n→∞ un=u, then u∈R（A） and s-lim↑n→∞An^-1（un）=A^-1（u）; （d） if un∈R（An） and lim↑n→∞un=u.then u∈R（A） and lim↑n→∞Au^＋（un）=A^＋（u）. Furtherrnore, if any of the above propositions holds, we have thin N（A）=s-lim↑n→∞N（An ）,R（A） = s-lim↑n→∞R（An ）, -↑R（A） =R（A）.
文摘A coupled system of three nonlinear oscillators with symmetric couplings is studied.It turns out that there exists a reduced variational equation to the in phase periodic solution of such a coupled system.By the symmetry one establishes a structural property for the monodromy matrix of the reduced variational equation,which simplifies the computation of multipliers to a great extent.As an application of the above results,a coupled system of three Poincare oscillators is discussed as well.
