共查询到20条相似文献,搜索用时 140 毫秒
1.
P. Shvartsman 《Geometric And Functional Analysis》2001,11(4):840-868
We prove a Helly-type theorem for the family of all k-dimensional affine subsets of a Hilbert space H. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M,r) ({\cal M},\rho) into this family.¶Let F be such a mapping satisfying the following condition: for every subset M¢ ì M {\cal M'} \subset {\cal M} consisting of at most 2k+1 points, the restriction F|M¢ F|_{\cal M'} of F to M¢ {\cal M'} has a selection fM¢ (i.e. fM¢(x) ? F(x) for all x ? M¢) f_{\cal M'}\,({\rm i.e.}\,f_{\cal M'}(x) \in F(x)\,{\rm for\,all}\,x\,\in {\cal M'}) satisfying the Lipschitz condition ||fM¢(x) - fM¢(y)|| £ r(x,y ), x,y ? M¢ \parallel f_{\cal M'}(x) - f_{\cal M'}(y)\parallel\,\le \rho(x,y ),\,x,y \in {\cal M'} . Then F has a Lipschitz selection f : M ? H f : {\cal M} \to H such that ||f(x) - f(y) || £ gr(x,y ), x,y ? M \parallel f(x) - f(y) \parallel\,\le \gamma \rho (x,y ),\,x,y \in {\cal M} where g = g(k) \gamma = \gamma(k) is a constant depending only on k. (The upper bound of the number of points in M¢ {\cal M'} , 2k+1, is sharp.)¶The proof is based on a geometrical construction which allows us to reduce the problem to an extension property of Lipschitz mappings defined on subsets of metric trees. 相似文献
2.
Zhiting Xu 《Monatshefte für Mathematik》2007,57(5):157-171
Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation
of Emden-Fowler type
(a(t)x¢(t))¢+q1(t)| y(t-s1)|a sgn y(t-s1) +q2(t)| y(t-s2)|b sgn y(t-s2)=0, t 3 t0,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0,
where x(t) = y(t) + p(t)y(t − τ), τ, σ1 and σ2 are nonnegative constants, α > 0, β > 0, and a, p, q
1,
q2 ? C([t0, ¥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R})
. The results of this paper extend and improve some known results. In particular, two interesting examples that point out
the importance of our theorems are also included. 相似文献
3.
Fukun Zhao Leiga Zhao Yanheng Ding 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,15(6):495-511
This paper is concerned with the following periodic Hamiltonian elliptic system
{l-Du+V(x)u=g(x,v) in \mathbbRN,-Dv+V(x)v=f(x,u) in \mathbbRN,u(x)? 0 and v(x)?0 as |x|?¥,\left \{\begin{array}{l}-\Delta u+V(x)u=g(x,v)\, {\rm in }\,\mathbb{R}^N,\\-\Delta v+V(x)v=f(x,u)\, {\rm in }\, \mathbb{R}^N,\\ u(x)\to 0\, {\rm and}\,v(x)\to0\, {\rm as }\,|x|\to\infty,\end{array}\right. 相似文献
4.
P. Sinopoulos 《Aequationes Mathematicae》2000,59(3):255-261
Summary. We determine the general solution g:S? F g:S\to F of the d'Alembert equation¶¶g(x+y)+g(x+sy)=2g(x)g(y) (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)g(y)\qquad (x,y\in S) ,¶the general solution g:S? G g:S\to G of the Jensen equation¶¶g(x+y)+g(x+sy)=2g(x) (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)\qquad (x,y\in S) ,¶and the general solution g:S? H g:S\to H of the quadratic equation¶¶g(x+y)+g(x+sy)=2g(x)+2g(y) (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)+2g(y)\qquad (x,y\in S) ,¶ where S is a commutative semigroup, F is a quadratically closed commutative field of characteristic different from 2, G is a 2-cancellative abelian group, H is an abelian group uniquely divisible by 2, and s \sigma is an endomorphism of S with s(s(x)) = x \sigma(\sigma(x)) = x . 相似文献
5.
Zhiting Xu 《Monatshefte für Mathematik》2009,118(4):187-199
Some necessary and sufficient conditions for nonoscillation are established for the second order nonlinear differential equation
(r(t)y(x(t))|x¢(t)|p-1x¢(t))¢+c(t)f(x(t))=0, t 3 t0,(r(t)\psi(x(t))\vert x^{\prime}(t)\vert^{p-1}x^{\prime}(t))^{\prime}+c(t)f(x(t))=0,\quad t\ge t_0, 相似文献
6.
It is well known that in every inverse semigroup the binary operation and the unary operation of inversion satisfy the following
three identities:
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