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
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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

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