On $$\Gamma $$ Γ -convergence of vector-valued mappings

On $$\Gamma $$ Γ -convergence of vector-valued mappings This paper deals with a new concept of limit for sequences of vector-valued mappings in normed spaces. We generalize the well-known concept of $$\Gamma $$ Γ -convergence to the case of vector-valued mappings and specify notion of $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -convergence similar to the one previously introduced in Dovzhenko et al. (Far East J Appl Math 60:1–39, 2011). In particular, we show that $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -convergence concept introduced in this paper possesses a compactness property whereas this property was failed in Dovzhenko et al. (Far East J Appl Math 60:1–39, 2011). In spite of the fact this paper contains another definition of $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -limits for vector-valued mapping we prove that the $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -lower limit in the new version coincides with the previous one, whereas the $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -upper limit leads to a different mapping in general. Using the link between the lower semicontinuity property of vector-valued mappings and the topological properties of their coepigraphs, we establish the connection between $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -convergence of the sequences of mappings and $$K$$ K -convergence of their epigraphs and coepigraphs in the sense of Kuratowski and study the main topological properties of $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -limits. The main results are illustrated by numerous examples. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

On $$\Gamma $$ Γ -convergence of vector-valued mappings

Positivity , Volume 18 (4) – Jan 14, 2014
Loading next page...
 
/lp/springer_journal/on-gamma-convergence-of-vector-valued-mappings-zWCG3IHxwq
Publisher
Springer Basel
Copyright
Copyright © 2014 by Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-013-0272-2
Publisher site
See Article on Publisher Site

Abstract

This paper deals with a new concept of limit for sequences of vector-valued mappings in normed spaces. We generalize the well-known concept of $$\Gamma $$ Γ -convergence to the case of vector-valued mappings and specify notion of $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -convergence similar to the one previously introduced in Dovzhenko et al. (Far East J Appl Math 60:1–39, 2011). In particular, we show that $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -convergence concept introduced in this paper possesses a compactness property whereas this property was failed in Dovzhenko et al. (Far East J Appl Math 60:1–39, 2011). In spite of the fact this paper contains another definition of $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -limits for vector-valued mapping we prove that the $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -lower limit in the new version coincides with the previous one, whereas the $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -upper limit leads to a different mapping in general. Using the link between the lower semicontinuity property of vector-valued mappings and the topological properties of their coepigraphs, we establish the connection between $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -convergence of the sequences of mappings and $$K$$ K -convergence of their epigraphs and coepigraphs in the sense of Kuratowski and study the main topological properties of $$\Gamma ^{\Lambda ,\mu }$$ Γ Λ , μ -limits. The main results are illustrated by numerous examples.

Journal

PositivitySpringer Journals

Published: Jan 14, 2014

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off