| A General Version of the Retract Method for Discrete Equations |
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| 作者姓名： | Josef DIBLIK Irena RICKOVA Miroslava ROZICKOVA |
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| 作者单位： | [1]Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Teehniekd 8, 61600 Brno, Czech Republic [2]Department of Applied Mathematics, Faculty of Science, Zilina University, Hurbanova 15, 010 26 Zilina, Slovak Republic |
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| 基金项目： | This investigation is supported by the Grant 201/04/0580 of the Czech Grant Agency （Prague） and by the Grant No 1/0026/03 and No 1/3238/06 of the Grant Agency of Slovak Republic （VEGA） |
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| 摘 要： | We study a problem concerning the compulsory behavior of the solutions of systems of discrete equations u（k ＋ 1） = F（k, u（k））, k ∈ N（a） = {a, a ＋ 1, a ＋ 2 }, a ∈ N,N= {0, 1,... } and F ： N（a） × R^n→R^n. A general principle for the existence of at least one solution with graph staying for every k ∈ N（a） in a previously prescribed domain is formulated. Such solutions are defined by means of the corresponding initial data and their existence is proved by means of retract type approach. For the development of this approach a notion of egress type points lying on the defined boundary of a given domain and with respect to the system considered is utilized. Unlike previous investigations, the boundary can contain points which are not points of egress type, too. Examples are inserted to illustrate the obtained result.
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| 关 键 词： | 离散方程 收缩核法 非线性常微分方程 相邻点 |
| 收稿时间： | 8 March 2005 |
| 修稿时间： | 2005-03-082005-06-15 |
A General Version of the Retract Method for Discrete Equations |
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| Josef DIBLIK Irena RICKOVA Miroslava ROZICKOVA.A General Version of the Retract Method for Discrete Equations[J].Acta Mathematica Sinica,2007,23(2):341-348. |
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| Authors: | Josef Diblík Irena Růžičková Miroslava Růžičková |
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| Institution: | (1) Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, Czech Republic;(2) Department of Applied Mathematics, Faculty of Science, Žilina University, Hurbanova 15, 010 26 Žilina, Slovak Republic |
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| Abstract: | In this paper we study a problem concerning the compulsory behavior of the solutions of systems of discrete equations u(k + 1) = F(k, u(k)), k ∈ N(a) = {a, a + 1, a + 2, . . . }, a ∈ ℕ, ℕ = {0, 1, . . . } and F : N(a) × ℝ
n
→ ℝ
n
. A general principle for the existence of at least one solution with graph staying for every k ∈ N(a) in a previously prescribed domain is formulated. Such solutions are defined by means of the corresponding initial data and
their existence is proved by means of retract type approach. For the development of this approach a notion of egress type
points lying on the defined boundary of a given domain and with respect to the system considered is utilized. Unlike previous
investigations, the boundary can contain points which are not points of egress type, too. Examples are inserted to illustrate
the obtained result.
This investigation is supported by the Grant 201/04/0580 of the Czech Grant Agency (Prague) and by the Grant No 1/0026/03
and No 1/3238/06 of the Grant Agency of Slovak Republic (VEGA) |
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| Keywords: | discrete equation consequent point retract |
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