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RETRACTED ARTICLE: A sharp Trudinger type inequality for harmonic functions and its applications
- Yili Tan^{1},
- Yongli An^{2},
- Hong Wang^{1} and
- Jing Liu^{3}Email author
https://doi.org/10.1186/s13660-017-1522-9
© The Author(s) 2017
- Received: 29 May 2017
- Accepted: 20 September 2017
- Published: 6 October 2017
The Retraction Note to this article has been published in Journal of Inequalities and Applications 2019 2019:156
Abstract
The present paper introduces a sharp Trudinger type inequality for harmonic functions based on the Cauchy-Riesz kernel function, which includes modified Poisson type kernel in a half plane considered by Xu et al. (Bound. Value Probl. 2013:262, 2013). As applications, we not only obtain Morrey representations of continuous linear maps for harmonic functions in the set of all closed bounded convex nonempty subsets of any Banach space, but also deduce the representation for set-valued maps and for scalar-valued maps of Dunford-Schwartz.
Keywords
- Trudinger type inequality
- Cauchy-Riesz kernel function
- modified Poisson type kernel
- Morrey representation
1 Introduction
By using augmented Riesz decomposition methods developed by Xie and Viouonu [12], the purpose of this paper is to obtain a sharp Trudinger type inequality for harmonic functions based on a Cauchy-Riesz kernel function and study the product \(G^{l}(P)\cdot\delta^{(k+1)}(P)\) and then study a more general product of \(f(P)\cdot\delta^{(k+1)}(P)\), where f is a \(C_{1}^{\infty}\)-function on \(\mathbb{R}\) and \(\delta^{(k+1)}(G)\) is the Dirac delta function with k-derivatives. As applications, we not only obtain Morrey representations of continuous linear maps for harmonic functions in the set of all closed bounded convex nonempty subsets of any Banach space, but also deduce the representation for set-valued maps and for scalar-valued maps of Dunford-Schwartz. 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 [12].
Lemma 2.3
In particular, for \(m = 1\), \(\delta_{1}^{*(k+1)}(G)\) is reduced to \(\delta_{1}^{(k+1)}(G)\), and \(\delta_{2}^{*(k+1)}(G)\) is reduced to \(\delta_{2}^{(k+1)}(G)\) (see [5, 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 obtained the representation of continuous linear maps in the set of all closed bounded convex nonempty subsets of any Banach space. Meanwhile, we 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.
Notes
Declarations
Acknowledgements
We would like to thank the editor, the associate editor and the anonymous referees for their careful reading and constructive comments which have helped us to significantly improve the presentation of the paper. This paper was written during a short stay of the corresponding author at the School of Mathematics of Osaka Kyoyobu University as a visiting professor. He would also like to thank the School of Mathematics and their members for their warm hospitality. This work was supported by the Natural Science Foundation of China (Grant No. 11401160) and the Natural Science Foundation of Hebei Province (No. A2015209040).
Authors’ contributions
YT designed the solution methodology. YT and YA prepared the revised manuscript according to the referee reports. HW participated in the design of the study. JL drafted the manuscript. 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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