一类非线性分数阶薛定谔-泊松系统非零解的存在性

本文研究了分数阶薛定谔-泊松系统$$\left\{\begin{array}{l}(-\Delta)^su+u+\phi u=\lambda f(u)\ \text {in} \ \mathbb {R}^3, \\ (-\Delta)^{\alpha}\phi =u^2\ \text {in} \ \mathbb {R}^3\emph{},\end{array}\right. $$ 非零解的存在性, 其中$s\in (\frac{3}{4},1), \alpha\in(0,1),\lambda$ 是正参数, $(-\Delta)^s,(-\Delta)^{\alpha}$是分数阶拉普拉斯算子. 在一定的假设条件下, 利用扰动法和Morse迭代法, 得到了系统至少一个非平凡解.     

SECOND INITIAL-BOUNDARY VALUE PROBLEMS FOR QUASI-LINEAR HYPERBOLIC-PARABOLIC COUPLED SYSTEMS

On a rectangular domain \[R(\delta ) = \{ 0 \leqslant t \leqslant \delta ,0 \leqslant x \leqslant 1\} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1)\] We consider the second initial-boundary value problem for the quasi-linear hyperbolic- parabolic coupled system \[{\begin{array}{*{20}{c}} {\sum\limits_{j = 1}^n {{\zeta _{ij}}(t,x,u,v)(\frac{{\partial {u_j}}}{{\partial t}} + {\lambda _l}(t,x,u,v,{v_x})\frac{{\partial {u_j}}}{{\partial x}})} } \\ { = {\zeta _l}(t,x,u,v)(\frac{{\partial v}}{{\partial t}} + {\lambda _l}(t,x,u,v,{v_x})\frac{{\partial v}}{{\partial x}})} \\ { + {\mu _l}(t,x,u,v,{v_x}),(l = 1,...,n)} \\ {\frac{{\partial v}}{{\partial t}} - a(t,x,u,v,{v_x})\frac{{{\partial ^2}v}}{{\partial {x^2}}} = b(t,x,u,v,{v_x})} \end{array}}\] without loss of generatity,the initial conditions may be written as \[t = 0,{u_j} = 0,(j = 1,...,n),v = 0\] and we can suppose that \[\left\{ {\begin{array}{*{20}{c}} {a(0,x,0,0,0) \equiv 1} \\ {b(0,x,0,0,0) \equiv 0} \\ {{\zeta _{ij}}(0,x,0,0) \equiv {\delta _{lj}} = \left\{ {\begin{array}{*{20}{c}} {1,if{\kern 1pt} {\kern 1pt} {\kern 1pt} l = j} \\ {0,if{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l \ne j} \end{array}} \right.} \end{array}} \right.\] The boundary conditions are as follows: \[\begin{gathered} on{\kern 1pt} {\kern 1pt} {\kern 1pt} x = 1,\left\{ {\begin{array}{*{20}{c}} {{u_{\bar r}} = {G_{\bar r}}(t,u,v),(\bar r = 1,...,h;h \leqslant n)} \\ {\frac{{\partial v}}{{\partial x}} = {F_ + }(t,u,v);} \end{array}} \right. \hfill \ on{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{c}} {{u_{\hat s}} = {{\hat G}_{\hat s}}(t,u,v),(\hat s = m + 1,...,n;m \geqslant 0)} \\ {\frac{{\partial v}}{{\partial x}} = {F_ - }(t,u,v){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } \end{array}} \right. \hfill \\ \end{gathered} \] Uf = Q-f(t> u, x), (r = 1> k^n), We assume that the following conditions are satisfied: (1) the orientability condition \[\begin{gathered} {\lambda _{\bar r}}(0,1,0,0,0) < 0,{\lambda _s}(0,1,0,0,0) > 0,\left( {\begin{array}{*{20}{c}} {\bar r = 1,...,h} \\ {s = h + 1,...,n} \end{array}} \right) \hfill \ {\lambda _{\bar r}}(0,0,0,0,0) < 0,{\lambda _{\hat s}}(0,0,0,0,0) > 0,\left( {\begin{array}{*{20}{c}} {\hat r = 1,...,m} \\ {\hat s = m + 1,...,n} \end{array}} \right) \hfill \\ \end{gathered} \] (2) the compatibility condition \[\begin{gathered} \frac{{\partial {G_{\bar r}}}}{{\partial t}}(0,0,0) + \sum\limits_{j = 1}^n {\frac{{\partial {G_{\bar r}}}}{{\partial {u_j}}}} (0,0,0){\mu _j}(0,1,0,0,0) = {\mu _{\bar r}}(0,1,0,0,0) \hfill \ \frac{{\partial {{\hat G}_{\hat s}}}}{{\partial t}}(0,0,0) + \sum\limits_{j = 1}^n {\frac{{\partial {{\hat G}_{\hat s}}}}{{\partial {u_j}}}} (0,0,0){\mu _j}(0,0,0,0,0) = {\mu _{\hat s}}(0,0,0,0,0) \hfill \ (\bar r = 1,...,h;\hat s = m + 1,...,n);{F_ \pm }(0,0,0) = 0 \hfill \\ \end{gathered} \] (3) the condition of characterizing number \[\begin{gathered} \sum\limits_{j = 1}^n {\left| {\frac{{\partial {G_{\bar r}}}}{{\partial {u_j}}}(0,0,0)} \right|} < 1 \hfill \ \sum\limits_{j = 1}^n {\left| {\frac{{\partial {{\hat G}_{\hat s}}}}{{\partial {u_j}}}(0,0,0)} \right|} < 1(\bar r = 1,...,h,\hat s = m + 1,...,n \hfill \\ \end{gathered} \] (4)The smoothness condition: the coefficients of the system and the boundary conditions are suitably smooth. By means of certain a priori estimations for the solution of the heat equation and the linear hyperbolic system, using an iteration method and Leray-Schauder fixed point theorem, we have proved Theorem 1. Under the preceding hypotheses, for the second initial-boundary value problem (2)—(4), (6), (7), there exists uniquely a classical solution on R(8) where \[\delta \]>0 is suitably small. Theorem 2. In theorem the 1,condition of characterizing number (13) may be ameliorated as the following solvable condition; \[\left\{ {\begin{array}{*{20}{c}} {\det |({\delta _{\bar rr'}} - \frac{{\partial {G_{\bar r}}}}{{\partial {u_{r'}}}}(0,0,0)| \ne 0,(\bar r,r' = 1,...,h)} \\ {\det |({\delta _{\hat s\hat s'}} - \frac{{\partial {G_{\hat s}}}}{{\partial {u_{\hat s'}}}}(0,0,0)| \ne 0,(\hat s,\hat s' = m + 1,...,n)} \end{array}} \right.\] i.e,the boundary condition (6),(7)may be written as \[\begin{gathered} on{\kern 1pt} {\kern 1pt} {\kern 1pt} x = 1,\left\{ {\begin{array}{*{20}{c}} {{u_{\bar r}} = {H_{\bar r}}(t,{u_s},v),} \\ {\frac{{\partial v}}{{\partial x}} = {F_ + }(t,u,v);} \end{array}} \right.\left( {\begin{array}{*{20}{c}} {\bar r = 1,...,h} \\ {s = h + 1,...,n} \end{array}} \right) \hfill \ on{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{c}} {{u_{\hat s}} = {H_{\hat s}}(t,{u_{\hat r}},v){\kern 1pt} ,} \\ {\frac{{\partial v}}{{\partial x}} = {F_ - }(t,u,v){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } \end{array}} \right.\left( {\begin{array}{*{20}{c}} {\hat r = 1,...,m} \\ {\hat s = m + 1,...,n} \end{array}} \right) \hfill \\ \end{gathered} \]     

DMk OPERATORS AND SPECTRAL OPERATORS

1谱位于平面上的有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子 记号与[1,2]相同，不再一一赘述.设序列 {Mk}满足（M.1)，(M.2)，(M.3)即.对数凸性、非拟解析性、可微性[1]. 由{M（k）}我们可以 定义二元相关函数\[M({t_1},{t_2})\](详见[7])以及二元\[{\mathcal{D}_{ < {M_k} > }}\]空间 \[{\mathcal{D}_{ < {M_k} > }} = \{ \varphi |\varphi \in \mathcal{D};\exists \nu ,st{\left\| \varphi \right\|_\nu } = \mathop {\sup }\limits_\begin{subarray}{l} s \in {R^2} \\ {k_i} \geqslant 0 \\ (i = 1,2) \end{subarray} |\frac{{{\partial ^{{k_1} + {k_2}}}}}{{{\partial ^{{k_1}}}{s_1}\partial _{{s_2}}^{{k_2}}}}\varphi (s)|/{\nu ^k}{M_k} < + \infty \} \] 其中\[s = ({s_1},{s_2})k = {k_1} + {k_2}\].关于谱位于复平面上的有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子的定义及性质可 参看[3,4].设X为Banach空间，B(X)为X上有界线性算子的全体组成的环.当 \[T \in B(X)\]为\[{\mathcal{D}_{ < {M_k} > }}\]型算子时，有\[T = {T_1} + i{T_2};{T_1} = {U_{Ret}}{T_2}{\text{ = }}{U_{\operatorname{Im} {\kern 1pt} t}}\] ,此处U为T的谱超广义函数，t为复变量.由于supp(U)为紧集，故可将U延拓到\[{\varepsilon _{ < {M_k} > }}\]上且保持连续性. 经过简单的计算，若\[T \in B(X)\]为谱位于平面上的一个\[{\mathcal{D}_{ < {M_k} > }}\]型算子，则T的一个谱 超广义函数（1）U可表成 \[{U_\varphi } = \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{e^{i({t_1}{T_1} + {t_2}{T_2})}}\hat \varphi } } ({t_1},{t_2})d{t_1}d{t_2}\] 设\[T \in B(X)\]为谱算子，S、N、E（.)分别为T的标量部分、根部、谱测度.下面的定理给出了谱算子成为\[{\mathcal{D}_{ < {M_k} > }}\]型算子的一个充分条件： 定理1设T为谱算子适合下面的条件 \[\mathop {\sup }\limits_{k > 0} \mathop {\sup }\limits_\begin{subarray}{l} |{\mu _j}| < 1 \\ {\delta _j} \in \mathcal{B} \\ j = 1,2,...,k \end{subarray} {(\left\| {\frac{{{N^n}}}{{n!}}\sum\limits_{j = 1}^k {{\mu _j}E({\delta _j})} } \right\|{M_n})^{\frac{1}{n}}} \to 0(n \to \infty )\] 其中\[\mathcal{B}\]为平面本的Borel集类.则T为\[{\mathcal{D}_{ < {M_k} > }}\]型算子且它的一个谱广义函数可表为 \[{U_\varphi } = \sum\limits_{n = 0}^\infty {\frac{{{N^n}}}{{n!}}} \int {{\partial ^n}} \varphi (s)dE(s)\] 推论1设E（?），N满足 \[{(\frac{{{M_n}}}{{n!}} \vee ({N^n}E))^{\frac{1}{n}}} \to 0\] 则T为\[{\mathcal{D}_{ < {M_k} > }}\]型算子. 推论2设N为广义幂零算子，则对于任何与N可换的标量算子S,S+N为\[{\mathcal{D}_{ < {M_k} > }}\]型算子的充分必要条件是 \[{(\frac{{\left\| {{N^n}} \right\|}}{{n!}}{M_n})^{\frac{1}{n}}} \to 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (n \to \infty )\] 在[4]中称满足上式的算子为\[\{ {M_k}\} \]广义幂零算子.显然\[\{ {M_k}\} \]广义幂零算子必为通 常的广义幂零算子.下面的命题给出了\[\{ {M_k}\} \] 广义幂零算子的一些性质. 命题 设N为广义幂零算子，则下列事实等价： (i ) N为\[\{ {M_k}\} \]广义幂零算子； (ii)对于任给的\[\lambda > 0\],存在\[{B_\lambda } > 0\]使(1) \[\left\| {R(\xi ,N)} \right\| \leqslant {B_\lambda }{e^{{M^*}(\frac{\lambda }{{|\xi |}})}}\](\[{|\xi |}\]充分小)； (iii)对于任给的\[\mu > 0\],存在\[{A_\mu } > 0\]使 \[\left\| {{e^{izN}}} \right\| \leqslant {A_\mu }{e^{M(\mu |z|)}}\] 2谱位于实轴上的有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子本节讨论有界\[{\mathcal{D}_{ < {M_k} > }}\]型算子T成为谱算子 的条件，这里假定\[{\mathcal{D}_{ < {M_k} > }}\]中的函数是一元的，于是Т的谱位于实轴上.X^*表示X的共轭 空间. 设\[f \in {\mathcal{D}^'}_{ < {M_k} > }\]，由[8, 9]，存在测度\[{\mu _n}(n \geqslant 0)\]使得对任何h>0,存在A>0适合 \[\sum\limits_{n = 0}^\infty {\frac{{{h^n}}}{{n!}}} {M_n}\int {|d{\mu _n}| \leqslant A} \]且 \[ < f,\varphi > = \sum\limits_{n = 0}^\infty {\frac{1}{{n!}}} \int {{\varphi ^{(n)}}} (t)d{\mu _n}(t)\] 一般说，上述\[{\mu _n}(n \geqslant 0)\]不是唯一的，为此我们引入 定义设\[{n_0}\]为正整，如果对一切\[n \geqslant {n_0}\]，存在测度\[{{\mu _n}}\]，它们的支集均包含在某一L 零测度闭集内，则称f是\[{n_0}\]奇异的，若\[{n_0}\] = 1，则称f是奇异的.设\[T \in B(X)\]为\[{\mathcal{D}_{ < {M_k} > }}\]型 算子，U为其谱超广义函数，如果对于任何\[x \in X{x^*} \in {X^*},{x^*}U\].x是\[{n_0}\]奇异的（奇异 的)，则称T是\[{n_0}\]奇异的(奇异的）\[{\mathcal{D}_{ < {M_k} > }}\]型算子. 经过若干准备，可以证明下面的 定理2 设X为自反的Banach空间，则\[T \in B(X)\]为奇异\[{\mathcal{D}_{ < {M_k} > }}\]型算子的充分必要 条件是T为满足下列条件的谱算子： (i)对每个\[x \in X\]及\[{x^*} \in X\]，\[\sup p({x^*}{N^n}E()x)\]包含在一个与\[n \geqslant 1\]无关的L零测 度闭集F内(F可以依赖于\[x{x^*}\])，此处E(?)、N分别是T的谱测度与根部； (ii)算子N是\[\{ {M_k}\} \]广义幂零算子. 推论 设X为自反的banach空间，\[T \in B(X)\]为奇异\[{\mathcal{D}_{ < {M_k} > }}\]型算子且\[\sigma (T)\]的测度 为零的充分必要条件是T为满足下列条件的谱算子： (i) E(?）的支集为L零测度集； (ii) 算子N是\[\{ {M_k}\} \]广义幂零算子.；     

QUSAI-PRIMITIVE RINFS(II)

Denote R an associative ring,\[\mathcal{M}\] a right modular idea of R,i,e,there exists an \[a \in R\] such that for all \[r \in R\],\[r + ar \in \mathcal{M}\], Let \[\{ {\mathcal{M}_i}\} \] be a given set of modular right ideals of R.Then introduce the following definition: Definition 1.Let \[\mathcal{M}\] be a modular right ideals of R. An element a of \[\mathcal{M}\] is called an \[\mathcal{M}\]-right quasi-regular element,if{i+ai}=\[\mathcal{M}\] for all \[i \in \mathcal{M}\].A right ideal L of R is called \[i \in \mathcal{M}\]-regular right ideal if every element of L is an \[i \in \mathcal{M}\]-right quasiregular element. Definition 2. Let \[i \in \mathcal{M}\] and \[{\mathcal{M}^'}\] be two right ideals of R,\[{\mathcal{M}^'}\] is called \[{\mathcal{M}^'}\]-modular if \[{\mathcal{M}^'} \subset \mathcal{M}\] and if there exist an element \[a \in \mathcal{M}\] such that for all \[i \in \mathcal{M}\],\[i + ai \in {\mathcal{M}^'}\]. Now we introduce the symbol \[{\hat \mathcal{M}}\].Let \[\mathcal{M} \in \sum \].Then if \[\mathcal{M}\] is an \[\mathcal{M}\]-regular right ideal,we put \[\hat \mathcal{M} = \mathcal{M}\];if \[\mathcal{M}\] is not an \[\mathcal{M}\]-regular ideal,we put \[{\hat \mathcal{M}}\] to be an \[\mathcal{M}\]-maximal modular right ideal in \[\mathcal{M}\].Let \[\mathcal{M} \in \sum \].Then if \[\mathcal{M}\] is not an \[\mathcal{M}\]-regular right ideal,we put \[\hat \mathcal{M} = \mathcal{M} \in {{\hat \sum }_\mathcal{M}} = \{ \hat \mathcal{M}|\hat \mathcal{M} is \mathcal{M}\} \]-maximal modular right ideal};if \[\mathcal{M}\] is an \[\mathcal{M}\]-right regular right idal,we put \[{{\hat \sum }_\mathcal{M}} = \mathcal{M}\]. Now we put \[\hat \sum = \{ \hat \mathcal{M}|\hat \mathcal{M} \in {{\hat \sum }_\mathcal{M}},\mathcal{M} \in \sum \} \] and \[\hat J = \cup {L_i}\] (1) for an element \[\mathcal{M} \in \sum \],where \[{L_i}\] are \[\mathcal{M}\]-regular right ideal,and U is set theoretical sum.Furthermore we put \[\hat J = \mathop \cap \limits_{\mathcal{M} \in \sum } {{\hat J}_\mathcal{M}}\] (2) and \[{J_1} = \{ b|b \in \mathop \cap \limits_{\mathcal{M} \in \sum } \mathcal{M},\],b satisfying the following condition}, (3) i,e,if |b)+\[{\mathcal{M}^{{\text{1}}}}{\text{ = }}\mathcal{M} \in \sum \] for an \[\mathcal{M}\]-modular right ideal \[{\mathcal{M}^{{\text{1}}}}\],then it must be \[{\mathcal{M}^{{\text{1}}}}{\text{ = }}\mathcal{M}\],where |b) is the intersection of all right ideals including b. Definition 3.an element \[\mathcal{M}\] of \[\sum \] is called satisfying J1-left idealizer condition,if \[x \in {J_1},y \in \mathcal{M}\],then \[rx + ryx \in \mathcal{M}\] for all \[r \in R\].The \[\sum \] is called satisfying J1-left idealizer condition(briefly,J1-l,i,c) if every \[\mathcal{M}\] \[\mathcal{M}\] of \[\sum \] is satisfying J1-l,i.c. Theorem 1. Suppose that \[\sum = \{ \mathcal{M}\} \] is satisfying J1-l.i.c.and put \[\beta = \hat \mathcal{M}\];\[R = \{ x \in R|Rx \subset \hat \mathcal{M}\} ,\hat \mathcal{M} \in \hat \sum \],then J1 is an ideal and \[{J_1} = \hat J = \sum\limits_{\hat \mathcal{M} \in \hat \sum } {\hat \mathcal{M} = \mathop \cap \limits_{\hat \mathcal{M} \in \hat \sum } } \beta \] Definition 4. Let \[\sum = \{ \mathcal{M}\} \] be satisfying J1-l.i.c.\[\hat \sum = \{ \hat \mathcal{M}|\hat \mathcal{M} \in {{\hat \sum }_\mathcal{M}},\mathcal{M} \in \sum \} \] as stated in (1), then we call ideal \[{J_1} = \mathop \cup \limits_{\hat \mathcal{M} \in \hat \sum } \hat \mathcal{M}\] the \[\sum \]-radioal of R. If J1=0, then R is called \[\sum \]-semisimple ring. Theorem 2. Let \[\sum = \{ \mathcal{M}\} \] be satisfying J1-'.i.c,where J1 is \[\sum \]-radical of R}, and \[\bar \sum = \{ \bar \mathcal{M}\} ,\bar \mathcal{M} = \mathcal{M}/{J_1},\mathcal{M} \in \sum ,\bar \hat \sum = \{ \bar \hat \mathcal{M}\} ,\hat \mathcal{M} \in \hat \sum ,\bar \hat \mathcal{M} = \hat \mathcal{M}/{J_1}\] then the \[{\bar \sum }\]-radical of \[\bar R = R/{J_1}\] is \[{\bar 0}\]. Definition 5. Let \[\sum = \{ \mathcal{M}\} \] be satisfying J1-l.i.c. and \[\hat \sum = \{ \hat \mathcal{M}\} \], then R is called a basic ring if and only if there exists an element \[{\hat \mathcal{M}}\] of such that \[\hat \mathcal{M}:R = 0\]. Let \[\beta \] be an ideal of R, if \[\beta = \hat \mathcal{M}\]\[:R\], \[\hat \mathcal{M} \in \hat \sum \],then \[\beta \] is called a basic ideal of R. Theorem 3. The \[\sum \]-rdical of R is the intersection of all basic ideals of R. Theorem 4. Any \[\sum \]-semisimple ring is isomorphic to a subdirect sum of basic rings. Theorem 5. Let R be an associative ring. Suppose that the set \[\sum \] includes only one element R, then the \[\sum \]-radieal of R, the \[\sum \]-semisimfple and the basic rings become the Jacobson radical, the Jacobson semisimple and the primitive rings respectively. Definition 6. An element \[m \in \mathfrak{M}\] is called strictly cyclic if \[m \in mR\]. \[\mathfrak{M}\] is called special if there exists a subset M of \[\mathfrak{M}\] such that every element \[m \in M\] is strictly cyclic and 0：\[\mathfrak{M} = \mathop \cap \limits_{m \in M} 0:m\] Definition 7. A module \[\mathfrak{M}\] is called a special dense module if and only if (i)\[\mathfrak{M}\] is special, (ii) \[\mathfrak{M}\] is a F-space as stated in [1] ,(\[\mathfrak{M}\]) suppose that\[{u_{{i_1}}},{u_{{i_2}}},...,{u_{{i_n}}}\] be arbitrary finite F-independent elements and \[{u_{{i_1}}}r \ne 0,{u_{{i_j}}} = 0,j \ne 1\] for an element \[r \in R\], then there exists an element \[t \in R\] such that .\[{u_{{i_1}}}tR = \mathfrak{M},{u_{{i_j}}} = 0,j \ne 1\]. Let S be the set of all free elements of \[\mathfrak{M}\] as stated in [1]. It is clear that S is a strictly cyclic set and \[\mathfrak{M}\] is a special module. Now put I to be the class of all speciall dense modules with M = S, Denote \[{\Lambda _s} = \{ {\mathcal{M}_m}\} \] where =\[{\mathcal{M}_m} = 0:m,m \in S\], and \[\sum = \{ \mathcal{M}|\mathcal{M} \in {\Lambda _s},s \subset \mathfrak{M} \in I\} \]； \[{\hat \sum }\] as stated before. Then we can show that \[{J^*} = \mathop \cap \limits_{\mathcal{M} \in \sum } \mathcal{M} = \mathop \cap \limits_{\hat \mathcal{M} \in \hat \sum } \hat \mathcal{M}\] is a \[\sum \] -radical and \[{J^*} \subset J\], where J is Jacobson radical. Definition 8. The above stated \[\sum \]-radical \[{J^*}\] will be called the quasi Jacobson radical. A ring R is Called quasi Jacobson semisimple ring if and only if the quasi Jacobson radical \[{J^*}\] = 0. Theorem 6. Let R be a quasi Jacobson semisimple ring, then R is isomorphic to a subdirect sum of quasi primitive rings.     

一致L-Lipschitz的渐近伪压缩映象不动点的Ishikawa迭代逼近问题

在任意实的Banach空间中研究了用具误差的修正的Ishikawa与Mann迭代程序来逼近一致L-Lipschitz的渐近伪压缩映象不动点的强收敛性问题,在去掉条件$$\sum\limits_{n=0}^{\infty}\alpha_{n}^{2}<\infty, \q \sum\limits_{n=0}^{\infty }\gamma_{n}<\infty,\q \sum\limits_{n=0}^{\infty }\alpha_{n}(\beta_{n}+\delta_{n})<\infty,\q \sum\limits_{n=0}^{\infty}\alpha_{n}(k_{n}-1)<\infty$$之下,证明了相关文献的结果仍然成立.所得结果不但改进和推广了最近一些人的最新结果,而且也从根本上改进了定理的证明方法.     

The Generalized Riemann^Haseman Problem

In this paper we study the generalized Riemann—Haseman problem which was given by Vekua, I. N. Problem (R-H). Find a sectionally generalized holomorphic function w(z) = {w^+(z), w^-(z)} such that $\frac {\partial w}{\partial \bar z}+B(z)\bar w=0,z\in E$ Here $B(z)\in C_\alpha ^n-1(D^++L),B(z)\in C_\alpha ^n-1(D^- +L),L\in C_\alpha ^n-1,0<\alpha \leq 1,|B(z)|\leq \frac {K}{|z|^1+s}(z\rightarrow \infty),K>0,\varepsilon >0;w(z)$ Satisfies the boundary condition $\sum\limits_{k=0}^n {a_k(t)\frac {\partial ^k w^+}{\partial t^k}|b_k \frac{\bar \partial ^kw^+}{\partial t^k}}_{t=\alpha(z)}-\sum\limits_{k=0}^n{c_k(t)\frac{\partial ^k w^-}{\partial t^k}+d_k(t)\frac{\bar \partial ^kw^-}{\partial t^k}}=f(t),t\in L$, Where a_k(t)、 b_k (t) 、 c_k (t) 、d_k(t)、f(t)\in H;\alpha(t) is a mapping of L into itself, $\alpha'(t)\ne 0$and \alpha [\alpha(t)] \equal t. We study the conditions of the solubility and the number of linearly independent solutions.     

ON THE CHARACTERISTIC FUNCTION OF SEMI-HYPONORMAL OPERATOR

本文继[3]之后，研究拟亚正常算子和半亚正常算子的特征函数.设\[A = U|A{|_r}\]是\[H{\kern 1pt} {\kern 1pt} \] 上拟亚正常算子，\[U\]是酉算子，\[B = |A{|_ + } - |A{|_ - }\],作算子\[A\]的特征函数\[W(\lambda ,A) = I - {B^{\frac{1}{2}}}{(\lambda I - {A_ - })^{ - 1}}U{B^{\frac{1}{2}}}\] 定理1 设\[A = U|A{|_r}\]及\[{A^'} = {U^'}|{A^'}{|_r}\]为\[\varphi - \]拟亚正常算子而且都是简单的.又设 \[U\]与\[{U^'}|\]是酉算子.如果有酉算\[T\]将\[H\]映照成\[{H^'}\]而且\[|{A^'}{|_ \pm } = T|A{|_ \pm }{T^{ - 1}}\]，\[{U^'} = TU{T^{ - 1}}\]那末必有\[{\cal B}(A)\]到\[{\cal B}({A^'}){\kern 1pt} \]上的酉算子\[S{\kern 1pt} {\kern 1pt} \]使当\[\lambda \notin \sigma ({A_ - }) = \sigma (A_ - ^')\]时\[W(\lambda ,{A^'}) = SW(\lambda ,A){S^{ - 1}}\]反之亦真. 下面设\[A\]是半亚正常的.又设\[{\cal D}\]为一辅助的希尔伯特空间，\[K\]为\[{\cal D}\]到\[{\kern 1pt} H\]中的线 性算子使\[Q = |A{|_{\rm{r}}} - |A{|_l} = K{K^*}{\kern 1pt} {\kern 1pt} \]，当\[\lambda \in \rho (A)\]，\[|Z| \ne 1\]时作 \[Y(z,\lambda ) = I - {\kern 1pt} {\kern 1pt} z{K^*}{(I - z{U^*})^{ - 1}}{(A - \lambda I)^{ - 1}}K\] 定理2设\[A = U|A{|_r}{\kern 1pt} {\kern 1pt} {\kern 1pt} \]及\[{A^'} = {U^'}|{A^'}{|_r}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \]分别是\[H\]与\[{H^'}{\kern 1pt} {\kern 1pt} {\kern 1pt} \]中的半亚正常算子，\[U\]与 \[{U^'}\]是酉算子而且\[A\]与\[{A^'}\]都是简单的.如果存在\[{\cal D} \to {{\cal D}^'}{\kern 1pt} \]上的酉算子\[S\]使 \[{Y^'}(z,\lambda ) = SY(z,\lambda ){S^{ - 1}}\] 那末必有由\[H\]到\[{H^'}{\kern 1pt} {\kern 1pt} {\kern 1pt} \]上的酉算子\[T\]使(1)成立，反之亦真. 定理3 若\[K\]是希尔伯特-许密特算子则\[Y(z,\lambda )\]的行列式（当\[|Z| \ne 1\]时）存在， 且\[\det (Y(z,\lambda )) = \det ((I - z{U^*})(A - \lambda I){(I - z{U^*})^{ - 1}}{(A - \lambda I)^{ - 1}})\] 下面只考虑奇型积分模型这时\[W(\lambda ,A)\]成为乘法算子，\[(W(\lambda ,A)f)({e^{i\theta }}) = W({e^{i\theta }},\lambda )f({e^{i\theta }})\]其中\[W({e^{i\theta }},\lambda ) = I - \alpha ({e^{i\theta }}){(\lambda {e^{i\theta }}I - \beta ({e^{i\theta }}))^{ - 1}}\alpha ({e^{i\theta }})\] 我们又假设\[A\]是完全非正常的.记\[{Y_ \pm }({e^{i\theta }},\lambda )a = \mathop {\lim }\limits_{r \to 1 \pm 0} Y({e^{i\theta }},\lambda )a\] 定理4设\[\lambda \in \rho (A){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \]，\[a \in {\cal D}{\kern 1pt} {\kern 1pt} {\kern 1pt} \]为固定的，那末\[{Y_ \pm }({e^{i\theta }},\lambda )a\]为黎曼-希尔伯特问题 \[{Y_ - }({e^{i\theta }},\lambda )a = W({e^{i\theta }},\lambda ){Y_ + }({e^{i\theta }},\lambda )a\] 的解. 设\[{\cal L}({\cal D}{\kern 1pt} {\kern 1pt} {\kern 1pt} ){\kern 1pt} {\kern 1pt} {\kern 1pt} \]为\[{\cal D}{\kern 1pt} {\kern 1pt} {\kern 1pt} \]上线性有界算子全体所成的Banach空间，\[H_ \pm ^p({\cal L}{\kern 1pt} ({\cal D}{\kern 1pt} {\kern 1pt} ){\kern 1pt} {\kern 1pt} ){\kern 1pt} {\kern 1pt} \]为单位圆 外，内取值于\[{\cal L}({\cal D}{\kern 1pt} {\kern 1pt} {\kern 1pt} ){\kern 1pt} \]的某些解析函数所成的Hardy空间.设\[f({e^{i\theta }})\]是单位圆周上的函 数，如果有\[{u_ \pm } \in H_ \pm ^p({\cal L}{\kern 1pt} ({\cal D}{\kern 1pt} {\kern 1pt} ){\kern 1pt} {\kern 1pt} ){\kern 1pt} {\kern 1pt} (p > 2)\]使\[u_ - ^{ - 1}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \]存在\[{u_ - }{\kern 1pt} {\kern 1pt} {\kern 1pt} {({e^{i\theta }})^{ - 1}}{u_ + }{\kern 1pt} ({e^{i\theta }}) = f({e^{i\theta }})\]则称\[f\]是可分解的.     

强不可约算子的K0群及其近似相似不变量

令 $\mathcal H$ 表示复可分的Hilbert空间, ${\mathcal L}({\mathcal H})$ 表示 $\mathcal H$上全体有界线性算子的集合. 算子 $T \in{\mathcal L}{(\mathcal H)}$称为是强不可约的, 如果不存在非平凡的幂等元与 T 可交换. 对强不可约算子的近似不变量给出比以往文献更精细的刻画. 主要结果如下: 对任意具有连通谱的有界线性算子 T 及 ε>0, 存在强不可约算子A, 使得 $\|A-T\|<\varepsilon$, $V({\mathcal A}^{\prime}(A))\cong{\mathbb{N}}$, $K_{0}({\mathcal A}^{\prime}(A))\cong{\mathbb{Z}}$, 且 ${{\mathcal A}^{\prime}(A)}/{\rm rad}{{\mathcal A}^{\prime}(A)}$ 可交换, 这里${\mathcal A}^{\prime}(A)$ 表示A 的换位代数, 且 ${\rm rad}{\mathcal A}^{\prime}(A)$ 表示${\mathcal A}^{\prime}(A)$的Jacobson根.     

ON MILIN-LEBDEV INEQUALITIES

Let \[\varphi (x) = \sum\limits_{k = 1}^\infty {{A_k}} {x^k},\Phi (x) = {e^{\varphi (x)}} = \sum\limits_{k = 1}^\infty {{D_k}} {x^k}\] \[\begin{gathered} \frac{1}{{{{(1 - x)}^\lambda }}} = \sum\limits_{k = 1}^\infty {{d_k}} (\lambda ){x^k} \hfill \ {\overline \Delta _n}(\lambda ) = {\lambda ^{2 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} \mathop {|{A_k}|}\nolimits_{}^p - \sum\limits_{k = 1}^\infty {\frac{1}{k}} \hfill \\ \end{gathered} \] Milin-Lebedey proved that \[\sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^{p - 1}(\lambda )}}} \leqslant \exp \{ {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} |{A_k}{|^p}\} \] where p>l and \[\lambda \]>0. In this paper, we have proved the following theorems； Theorem 1. Let \[p \geqslant 1,\lambda > 0\] and \[F(x) = \sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}} {x^p}\exp \{ - {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}|{A_k}{|^p}{x^k}} \} (2)\] then F(x) is a decreasing function of x on [0, 1]. This theorem is stronger than the result (1). Theorem 2. Let \[p \geqslant 2,\lambda > 0\] and \[{{\bar Q}_n}(\lambda ) = \frac{1}{{n + 1}}\sum\limits_{k = 0}^n {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}\exp } \{ - \frac{1}{{n + 1}}\sum\limits_{v = 1}^n {\overline {{\Delta _p}} } (\lambda )\} \] then \[{{\bar Q}_n}(\lambda )\] is a decreasing fimctLon of n(n=l, 2，...)In the case p=2 this is contained in the Miiin-Lebedev's result.     

A FINITE STRUCTURE THEOREM BETWEEN PRIMITIVE RINGS AND ITS APPLICATION TO GALOIS THEORY

设\[\mathfrak{M} = \sum {F{u_i}} \]是除环F上向量空间，P是F的一个子除环且在F中是Galois，即存 在F的一个自同构群G使\[I(G) = P\].记Ф是F的中心，\[{G_0}\]是属于G的内自同构群， \[{G_0}\]的元素记为\[{I_r},r \in F\]；，记\[{E^'} = \sum\limits_{{I_{{r_j}}} \in {G_0}} {{\Phi _{{r_j}}}} \]是G的代数，\[P' = {C_F}({E^'})\]是\[{E^'}\]在F中的中心化子.记\[\mathfrak{U}(F,\mathfrak{M})\]是\[\mathfrak{M}\]的F-线性变换完全环，\[{T_v}(F,\mathfrak{M})\]是\[\mathfrak{U}(F,\mathfrak{M})\]中所有秩小于\[\mathcal{X}{_v}\]的元素集合，那末我们有如下主要结果： (1)\[{[F:P']_L} = n\]有限当且仅当\[{T_v}(P',\mathfrak{M}) = \sum\limits_{j = 1}^n \oplus {r_{jL}}{T_v}(F,\mathfrak{M})\]，其中\[{r_j} \in {E^'}\]，\[{r_{jL}}\]表示元素\[{r_j}\]的标量左乘. （2）\[{[P':P]_L} = t\]有限当且仅当凡\[{T_v}(P,\mathfrak{M}) = \sum\limits_{j = 1}^t \oplus {S_j}{T_v}(P',\mathfrak{M})\],其中\[{S_j}\]表示\[\mathfrak{M}\]的F-半线变换自同构，它的伴随同构\[{\psi _j} \in G\]. ⑶如有某个序数v使\[{T_v}(P,\mathfrak{M})\]，\[{T_v}(P',\mathfrak{M})\]及\[{T_v}(F,\mathfrak{M})\]满足⑴及(2)中的关系 式，那末对任何\[{T_\mu }(P,\mathfrak{M})\]，\[{T_\mu }(P',\mathfrak{M})\]及\[{T_\mu }(F,\mathfrak{M})\]皆满足(1)及(2)中的关系式.特别 对\[\mathfrak{U}(P,\mathfrak{M})\]，\[\mathfrak{U}(P',\mathfrak{M})\]及\[\mathfrak{U}(F,\mathfrak{M})\]也是如此. ⑷如果\[{[F:P]_L}\]有限，那末必有\[{C_p}({C_F}(E')) = E'\]，\[{[F:P']_L} = \dim E'\]，\[{[P':P']_L} = [G/{G_0}]\]，其中dim E'表示E'在\[\Phi \]上的维数，\[[G/{G_0}]\]表示\[{G_0}\]在G中的指数,特别\[G\]是 Galois 群，则 \[{C_F}(P') = {C_F}(P) = E'\]. (5)若\[{\tilde G}\]是F的另一自同构群且\[I(G) = I(\tilde G)\]，那末必有\[[G/{G_0}] = [\tilde G/{{\tilde G}_0}]\]， \[\dim {\kern 1pt} {\kern 1pt} E' = \dim {\kern 1pt} {\kern 1pt} \tilde E'\]. 其中\[{\kern 1pt} \tilde E'\]表示\[{\tilde G}\]的代数. 如果P取为F的中心时，于是从上述结果(1)就得出熟知的定理：\[[F:\Phi ]\]是有限的当 且仅当\[\mathfrak{U}(\Phi ,\mathfrak{M}) = \mathfrak{U}(F,\mathfrak{M}){ \otimes _\Phi }{F_L}\]. 另方面，运用我们上述的结果，可导出除环F的有限Galois理论.     

每个点都是链回归点的树映射

设f是端点数为n的树T上的连续自映射且T上的每一点都是f的链回归点.本文证明了: (1)如果T的某个端点是f的不动点,那么,T上的每个点都是f的周期为r≤n-1的周期点,或存在自然数r ≤ n-1,使得fr含有湍流; (2)如果f的不动点都在T的内部,那么,T上的每个点都是f的周期为r≤n的周期点,或存在自然数r≤n,使得,fr含有湍流.     

Line-critical point determining and point distinguishing graphs

Sumner defined a graph to be point determining if and only if distinct points have distinct neighborhoods and he has characterized connected line-critical point determining graphs. Here a short alternate proof of his characterization is provided and arbitrary line-critical point determining graphs are then characterized. Next line-critical point distinguishing graphs are considered; a graph is point distinguishing if and only if it is the complement of a point determining graph. Finally, line-critical graphs that are both point determining and point distinguishing are characterized.     

一些连续统的孤立链回归点

对树或图上的连续自映射,本文证明了孤立链回归点是终周期点,如孤立边回归不在临界点的轨道中且它的轨道中也不含有临界点,则它还是周期点,我们还证明了上述性质可以扩张到一类特殊的λ－dendroid上。     

解非线性约束方程的拉格朗日全局投影方法

基于最优化方法求解约束非线性方程组的一个突出困难是计算 得到的仅是该优化问题的稳定点或局部极小点,而非方程组的解点.由此引出的问题是如何从一个稳定点出发得到一个相对于方程组解更好的点. 该文采用投影型算法,推广了Nazareth-Qi$^{[8,9]}$ 求解无约束非线性方程组的拉格朗日全局算法(Lagrangian Global-LG)于约束方程上; 理论上证明了从优化问题的稳定点出发,投影LG方法可寻找到一个更好的点. 数值试验证明了LG方法的有效性.     

On farthest points of sets

For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.     

Convergence Properties of a Smoothing Approach for Mathematical Programs with Second-Order Cone Complementarity Constraints

This paper discusses the convergence properties of a smoothing approach for solving the mathematical programs with second-order cone complementarity constraints (SOCMPCCs). We first introduce B-stationary, C-stationary, M(orduckhovich)-stationary, S-stationary point, SOCMPCC-linear independence constraint qualification (denoted by SOCMPCC-LICQ), second-order cone upper level strict complementarity (denoted by SOC-ULSC) condition at a feasible point of a SOCMPCC problem. With the help of the projection operator over a second-order cone, we construct a smooth optimization problem to approximate the SOCMPCC. We demonstrate that any accumulation point of the sequence of stationary points to the sequence of smoothing problems, when smoothing parameters decrease to zero, is a C-stationary point to the SOCMPCC under SOCMPCC-LICQ at the accumulation point. We also prove that the accumulation point is an M-stationary point if, in addition, the sequence of stationary points satisfy weak second order necessary conditions for the sequence of smoothing problems, and moreover it is a B-stationary point if, in addition, the SOC-ULSC condition holds at the accumulation point.     

Saddle points for vector valued functions: existence, necessary and sufficient theorems

Necessary and sufficient conditions for a point to be a weak saddle point of a vector valued function (i.e. to be a solution of the vector saddle point problem) are given. Also, an existence result for a vector saddle point to have a solution is given.     

Point determination in graphs

A point determining graph is defined to be a graph in which distinct nonadjacent points have distinct neighborhoods. Those graphs which are critical with respect to this property are studied. We show that a graph is complete if and only if it is connected, point determining, but fails to remain point determining upon the removal of any edge. We also show that every connected, point determining graph contains at least two points, the removal of either of which will result again in a point determining graph. Graphs which are point determining and contain exactly two such points are shown to have the property that every point is adjacent to exactly one of these two points.     

Families of paths in the plane that connect all point pairs

A mesh is a family of paths in the plane connecting every pair of points. A crossing point of a mesh is a point which is an interior point to more than one path. A simple crossing point is an interior point of exactly two paths. We give an example of a mesh with only simple crossing points. We characterize subsets of the plane that can be the set of crossing points of a mesh. Our emphasis is on constructive methods.     

超空间与原空间动力系统之间的关系

设(X,d,f)为拓扑动力系统,其中X为局部紧可分的可度量化空间,d为紧型度量,f为完备映射,用2X表示由X的所有非空闭子集构成的集族,(2X,ρ,2f)为由(X,d,f)所诱导的赋予hit-or-miss拓扑的超空间动力系统.本文引入了余紧点传递和弱拓扑传递的定义.特别的,在X满足一定的条件时,给出了点传递,弱拓扑传递和余紧点传递之间的关系,并研究了(X,d,f)的余紧传递点,回复点和几乎周期点分别与(2X,ρ,2f)的传递点,回复点和几乎周期点之间的蕴含关系.这些结论丰富了赋予hit-or-miss拓扑的超空间的研究内容.