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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
Online
ISSN
1898-9934
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Volume 21, Issue 4 (Dec 2013)

Issues

Differential Equations on Functions from R into Real Banach Space

Keiko Narita / Noboru Endou / Yasunari Shidama
Published Online: 2013-12-27 | DOI: https://doi.org/10.2478/forma-2013-0028

Abstract

In this article, we describe the differential equations on functions from R into real Banach space. The descriptions are based on the article [20]. As preliminary to the proof of these theorems, we proved some properties of differentiable functions on real normed space. For the proof we referred to descriptions and theorems in the article [21] and the article [32]. And applying the theorems of Riemann integral introduced in the article [22], we proved the ordinary differential equations on real Banach space. We referred to the methods of proof in [30].

Keywords: formalization of differential equations

References

  • [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Google Scholar

  • [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Google Scholar

  • [3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Google Scholar

  • [4] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Google Scholar

  • [5] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Google Scholar

  • [6] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Google Scholar

  • [7] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Google Scholar

  • [8] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Google Scholar

  • [9] Czesław Bylinski. Introduction to real linear topological spaces. Formalized Mathematics, 13(1):99-107, 2005.Google Scholar

  • [10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Google Scholar

  • [11] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Google Scholar

  • [12] Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Google Scholar

  • [13] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001.Google Scholar

  • [14] Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007. doi:10.2478/v10037-007-0008-5.CrossrefGoogle Scholar

  • [15] Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321-327, 2004.Google Scholar

  • [16] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Google Scholar

  • [17] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Google Scholar

  • [18] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Google Scholar

  • [19] Keiichi Miyajima, Takahiro Kato, and Yasunari Shidama. Riemann integral of functions from R into real normed space. Formalized Mathematics, 19(1):17-22, 2011. doi:10.2478/v10037-011-0003-8.CrossrefGoogle Scholar

  • [20] Keiichi Miyajima, Artur Korniłowicz, and Yasunari Shidama. Contracting mapping on normed linear space. Formalized Mathematics, 20(4):291-301, 2012. doi:10.2478/v10037-012-0035-8.CrossrefGoogle Scholar

  • [21] Keiko Narita, Artur Korniłowicz, and Yasunari Shidama. The differentiable functions from R into Rn. Formalized Mathematics, 20(1):65-71, 2012. doi:10.2478/v10037-012-0009-x.CrossrefGoogle Scholar

  • [22] Keiko Narita, Noboru Endou, and Yasunari Shidama. The linearity of Riemann integral on functions from R into real Banach space. Formalized Mathematics, 21(3):185-191, 2013. doi:10.2478/forma-2013-0020. CrossrefGoogle Scholar

  • [23] Takaya Nishiyama, Artur Korniłowicz, and Yasunari Shidama. The uniform continuity of functions on normed linear spaces. Formalized Mathematics, 12(3):277-279, 2004.Google Scholar

  • [24] Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Google Scholar

  • [25] Hiroyuki Okazaki, Noboru Endou, Keiko Narita, and Yasunari Shidama. Differentiable functions into real normed spaces. Formalized Mathematics, 19(2):69-72, 2011. doi:10.2478/v10037-011-0012-7.CrossrefGoogle Scholar

  • [26] Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. More on continuous functions on normed linear spaces. Formalized Mathematics, 19(1):45-49, 2011. doi:10.2478/v10037-011-0008-3.CrossrefGoogle Scholar

  • [27] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Google Scholar

  • [28] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Google Scholar

  • [29] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Google Scholar

  • [30] Laurent Schwartz. Cours d’analyse, vol. 1. Hermann Paris, 1967.Web of ScienceGoogle Scholar

  • [31] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Google Scholar

  • [32] Yasunari Shidama. Differentiable functions on normed linear spaces. Formalized Mathematics, 20(1):31-40, 2012. doi:10.2478/v10037-012-0005-1.CrossrefGoogle Scholar

  • [33] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Google Scholar

  • [34] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Google Scholar

  • [35] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Google Scholar

  • [36] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Google Scholar

  • [37] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Google Scholar

  • [38] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Google Scholar

About the article

Received: 2013-12-31

Published Online: 2013-12-27

Published in Print: 2013-12-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/forma-2013-0028.

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© by Keiko Narita . This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. (CC BY-SA 3.0) BY-SA 3.0

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