Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Delta-exponential mappings in Banach algebras

  • Conference paper
General Inequalities 7

Part of the book series: ISNM International Series of Numerical Mathematics ((ISNM,volume 123))

  • 613 Accesses

Abstract

An intriguing interplay between the theory of delta-convex mappings (in the sense of Veselý and Zajiček) and the Hyers-Ulam stability problems is developed by studying a functional inequality

$$ \left\| {F\left( {x + y} \right) - F(x)F(y)} \right\| \leqslant f(x)f(y) - f\left( {x + y} \right). $$
(*)

This is an “exponential version” of the inequality

$$ \left\| {F\left( {x + y} \right) - F(x) - F(y)} \right\| \leqslant \left\| x \right\| + \left\| y \right\| - \left\| {x + y} \right\|, $$

proposed first by D. Yost and then generalized to

$$ \left\| {F\left( {x + y} \right) - F(x) - F(y)} \right\| \leqslant f(x)f(y) - f\left( {x + y} \right). $$

Superstability phenomenon in connection with (*) is examined.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J.A. Baker, The stability of the cosine equation. Proc. Amer. Math. Soc. 80 (1980), 411–416.

    Article  MathSciNet  MATH  Google Scholar 

  2. J.A. Baker, J. Lawrence and F. Zorzitto, The stability of the equation f(x + y) = f(x)f(y). Proc. Amer. Math. Soc. 74 (1979), 242–246.

    MathSciNet  MATH  Google Scholar 

  3. Z. Gajda, On the stability of additive mappings. Internat. J. Math. & Math. Sci. 14 (1991), 431–434.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Ger, On functional inequalities stemming from stability questions. International Series of Numerical Analysis, Birkhäuser Verlag Basel, vol. 103 (1992), 227–240.

    MathSciNet  Google Scholar 

  5. R. Ger, The singular case in the stability behaviour of linear mappings. Grazer Mathematische Berichte 316 (1992), 59–70.

    MathSciNet  MATH  Google Scholar 

  6. R. Ger, Superstability is not natural. Rocznik Naukowo-Dydaktyczny WSP w Krakowie 159 (1993), 109–123.

    MathSciNet  Google Scholar 

  7. R. Ger, Stability aspects of delta-convexity. In “Stability of Hyers-Ulam type”, A Volume dedicated to D.H. Hyers & S. Ulam, (ed. Th.M. Rassias & J. Tabor), Hadronic Press, Inc., Palm Harbor (1994), 99–109.

    Google Scholar 

  8. R. Ger & P. Šemrl, The stability of the exponential equation. Proc. Amer. Math. Soc. (in print).

    Google Scholar 

  9. D.H. Hyers, On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.

    Article  MathSciNet  Google Scholar 

  10. J. Lawrence, The stability of multiplicative semi-group homomorphisms to real normed algebras, I. Aequationes Math. 28 (1985), 94–101.

    Article  MathSciNet  MATH  Google Scholar 

  11. Th.M. Rassias, On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72 (1978), 297–300.

    Article  MathSciNet  MATH  Google Scholar 

  12. Th.M. Rassias & P. Šemrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proc. Amer. Math. Soc. To appear.

    Google Scholar 

  13. L. Veselý & L. Zajiček, Delta-convex mappings between Banach spaces and applications. Dissert at iones Math. 289, Polish Scientific Publishers, Warszawa, 1989.

    Google Scholar 

  14. D. Yost, Talk at the 17-th Winter School on Abstract Analysis (Section of Analysis), January, 1989, Šrni, Czech Republic.

    Google Scholar 

  15. D. Yost, Talk at the 18-th Winter School on Abstract Analysis (Section of Analysis), January, 1990, Šrni, Czech Republic.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Silesian University, Bankowa 14, 40-007, Katowice, Poland

    Roman Ger

Authors
  1. Roman Ger
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Mathematics, University of Basel, Rheinsprung 21, 4051, Basel, Switzerland

    Catherine Bandle

  2. School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, England, UK

    William N. Everitt

  3. Department of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, 13060, Safat, Kuwait

    Laszlo Losonczi

  4. Institue of Mathematics I, University of Karlsruhe, Kaiserstr. 12, 76128, Karlsruhe, Germany

    Wolfgang Walter

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer Basel AG

About this paper

Cite this paper

Ger, R. (1997). Delta-exponential mappings in Banach algebras. In: Bandle, C., Everitt, W.N., Losonczi, L., Walter, W. (eds) General Inequalities 7. ISNM International Series of Numerical Mathematics, vol 123. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8942-1_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8942-1_23

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9837-9

  • Online ISBN: 978-3-0348-8942-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation