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RETRACTED ARTICLE: New Riesz representations of linear maps associated with certain boundary value problems and their applications
- Wei Yang^{1},
- Jiannan Duan^{1},
- Wenmin Hu^{2} and
- Jing Zhang^{3}Email author
- Received: 26 May 2017
- Accepted: 27 October 2017
- Published: 9 November 2017
The Retraction Note to this article has been published in Boundary Value Problems 2019 2019:93
Abstract
In this paper, we obtain new Riesz representations of continuous linear maps associated with certain boundary value problems in the set of all closed bounded convex non-empty subsets of any Banach space. As applications, the Riesz integral representation results are also given.
Keywords
- Riesz decomposition method
- representation
- vector-valued map
1 Introduction
By using augmented Riesz decomposition methods developed by Wang, Huang and Yamini [17], the purpose of this paper is to study the product \(G^{l}\cdot\delta ^{(k)}(G)\) and then study a more general product of \(f(G)\cdot\delta^{(k)}(G)\), where f is a \(C^{\infty}\)-function on \(\mathbb{R}\) and \(\delta^{(k)}(G)\) is the Dirac delta function with k-derivatives. Meanwhile, we shall show that we can control the \(L^{\infty}\) norm by the \(H^{1}\) norm and a stronger norm with a logarithmic growth or double logarithmic growth. The inequality is sharp for the double logarithmic growth. The result there is used earlier in our paper to obtain a boundary limit theorem for sturdy harmonic functions and continuous linear maps. Before proceeding to our main results, the following definitions and concepts are required.
2 Preliminaries
Definition 2.1
Definition 2.2
The proof of the following lemma is given in [17].
Lemma 2.1
In particular, for \(m = 1\), \(\delta_{1}^{*(k)}(G)\) is reduced to \(\delta _{1}^{(k)}(G)\), and \(\delta_{2}^{*(k)}(G)\) is reduced to \(\delta _{2}^{(k)}(G)\) (see [4, p.250]).
3 Main results
Theorem 3.1
Proof
Example 3.1
Theorem 3.2
Proof
Example 3.2
4 Numerical simulations
In this section, we give the bifurcation diagrams, phase portraits of model (2.1) to confirm the above theoretic analysis and show the new interesting complex dynamical behaviors by using numerical simulations. The bifurcation parameters are considered in the following two cases:
5 Conclusions
In this paper, we firstly obtained the representation of continuous linear maps in the set of all closed bounded convex non-empty subsets of any Banach space. As applications, we secondly deduced the Riesz integral representation results for set-valued maps, for vector-valued maps of Diestel-Uhl and for scalar-valued maps of Dunford-Schwartz. Finally, we gave the bifurcation diagrams, phase portraits of related models to confirm the above theoretic analysis and showed the new interesting complex dynamical behaviors by using numerical simulations.
Notes
Declarations
Acknowledgements
The authors thank the referees for their valuable comments, which greatly improved their paper. This work was supported by the National Natural Science Foundation of China (Grant No. 41171176) and Postgraduate Technology Innovation Project of Hunan Province (Grant No. CX2015B243).
Authors’ contributions
JZ drafted the manuscript. WY helped to prepare the revised manuscript and JD carried out the transformation process according to the referee reports. WH corrected typos and grammatical errors throughout the manuscript, making it more readable. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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