scholarly journals Differential Equations on Functions from R into Real Banach Space

2013 ◽  
Vol 21 (4) ◽  
pp. 261-272
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Normed Space ◽  
Real Banach Space ◽  
Riemann Integral ◽  
Differentiable Functions ◽  
Real Normed Space

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].

scholarly journals Riemann Integral of Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (2) ◽  
pp. 145-152
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Normed Space ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Uniform Continuity ◽  
Riemann Integral ◽  
Real Normed Space ◽  
Finite Sequences ◽  
Convergence Of Sequences

Summary In this article we deal with the Riemann integral of functions from R into a real Banach space. The last theorem establishes the integrability of continuous functions on the closed interval of reals. To prove the integrability we defined uniform continuity for functions from R into a real normed space, and proved related theorems. We also stated some properties of finite sequences of elements of a real normed space and finite sequences of real numbers. In addition we proved some theorems about the convergence of sequences. We applied definitions introduced in the previous article [21] to the proof of integrability.

scholarly journals Contracting Mapping on Normed Linear Space

2012 ◽  
Vol 20 (4) ◽  
pp. 291-301
Author(s):  
Keiichi Miyajima ◽  
Artur Korniłowicz ◽  
Yasunari Shidama
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Space ◽  
Normed Space ◽  
Normed Linear Space ◽  
Contracting Mapping ◽  
Real Normed Space ◽  
The One

Summary In this article, we described the contracting mapping on normed linear space. Furthermore, we applied that mapping to ordinary differential equations on real normed space. Our method is based on the one presented by Schwarz [29].

scholarly journals The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (3) ◽  
pp. 185-191
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Riemann Integral ◽  
Basic Properties ◽  
Bounded Functions ◽  
Definition Of

Summary In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10], we also proved some theorems on bounded functions.

scholarly journals A reinterpretation of set differential equations as differential equations in a Banach space

2017 ◽  
Vol 148 (2) ◽  
pp. 429-446
Author(s):  
Martin Rasmussen ◽  
Janosch Rieger ◽  
Kevin Webster
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Existence And Uniqueness ◽  
Geometric Interpretation ◽  
Uniqueness Theorems ◽  
Support Functions ◽  
Existence And Uniqueness Theorems ◽  
Compact Subsets

Set differential equations are usually formulated in terms of the Hukuhara differential. As a consequence, the theory of set differential equations is perceived as an independent subject, in which all results are proved within the framework of the Hukuhara calculus. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of ℝd with their support functions. Using this representation, standard existence and uniqueness theorems for ordinary differential equations can be applied to set differential equations. We provide a geometric interpretation of the main result, and demonstrate that our approach overcomes the heavy restrictions that the use of the Hukuhara differential implies for the nature of a solution.

scholarly journals Riemann Integral of Functions from R into Real Normed Space

2011 ◽  
Vol 19 (1) ◽  
pp. 17-22 ◽  
Author(s):  
Keiichi Miyajima ◽  
Takahiro Kato ◽  
Yasunari Shidama
Keyword(s):  
Normed Space ◽  
Riemann Integral ◽  
Real Normed Space ◽  
Range Of Functions ◽  
Riemann Integration

Riemann Integral of Functions from R into Real Normed Space In this article, we define the Riemann integral on functions from R into real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to a wider range of functions. The proof method follows the [16].

scholarly journals The sufficient conditions for polystability of solutions of~nonlinear systems of ordinary differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 304-317
Author(s):  
Pavel A. Shamanaev ◽  
Olga S. Yazovtseva
Keyword(s):  
Banach Space ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Approximation ◽  
Critical Case ◽  
Sufficient Conditions ◽  
Asymptotic Equivalence ◽  
Theorem Proof

The article states the sufficient polystability conditions for part of variables for nonlinear systems of ordinary differential equations with a sufficiently smooth right-hand side. The obtained theorem proof is based on the establishment of a local componentwise Brauer asymptotic equivalence. An operator in the Banach space that connects the solutions of the nonlinear system and its linear approximation is constructed. This operator satisfies the conditions of the Schauder principle, therefore, it has at least one fixed point. Further, using the estimates of the non-zero elements of the fundamental matrix, conditions that ensure the transition of the properties of polystability are obtained, if the trivial solution of the linear approximation system to solutions of a nonlinear system that is locally componentwise asymptotically equivalent to its linear approximation. There are given examples, that illustrate the application of proven sufficient conditions to the study of polystability of zero solutions of nonlinear systems of ordinary differential equations, including in the critical case, and also in the presence of positive eigenvalues.

Qualitative properties of nonlinear ordinary differential equations

1977 ◽  
Vol 79 (1-2) ◽  
pp. 79-85 ◽  
Author(s):  
Nelson Onuchic ◽  
Plácido Z. Táboas
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Ordinary Differential Equation ◽  
Point Of View ◽  
Nonlinear Ordinary Differential Equations ◽  
Qualitative Properties ◽  
Image Position

SynopsisThe perturbed linear ordinary differential equationis considered. Adopting the same approach of Massera and Schäffer [6], Corduneanu states in [2] the existence of a set of solutions of (1) contained in a given Banach space. In this paper we investigate some topological aspects of the set and analyze some of the implications from a point of view ofstability theory.

scholarly journals An existence theorem for ordinary differential equations in Banach spaces

1984 ◽  
Vol 30 (3) ◽  
pp. 449-456 ◽  
Author(s):  
Bogdan Rzepecki
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Regularity Condition ◽  
Measure Of Noncompactness ◽  
Bounded Solution ◽  
Carathéodory Conditions

We prove the existence of bounded solution of the differential equation y′ = A(t)y + f(t, y) in a Banach space. The method used here is based on the concept of “admissibility” due to Massera and Schäffer when f satisfies the Caratheodory conditions and some regularity condition expressed in terms of the measure of noncompactness α.

A three-dimensional Boltzmann-like model of outgassing and contamination

1997 ◽  
Vol 8 (2) ◽  
pp. 229-249 ◽  
Author(s):  
A. BELLENI-MORANTE ◽  
G. BUSONI
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Three Dimensional ◽  
Inert Gas ◽  
Strict Solution

We consider a Boltzmann-like model of outgassing and contamination in a three-dimensional region V= V1∪V2∪V3. V1 is the region where the contaminant particles are produced, and V2 is the region where such particles migrate and interact with some inert gas. V3 is where contamination takes place because of the particles emanating from V2. In each of the three regions, the behaviour of the contaminant particles is represented by means of a Boltzmann-like equation. We show that such a problem has a unique positive strict solution, belonging to a suitable L1 Banach space X. Finally, a system of ordinary differential equations is derived which gives the evolution of the total number of contaminant particles in each of the three regions.

Sign in / Sign up

Export Citation Format

Share Document