Skip to main content

Measures of Noncompactness and Well-Posed Minimization Problems

  • Chapter
  • First Online:
Nonlinear Analysis

Part of the book series: Trends in Mathematics ((TM))

Abstract

This chapter presents facts concerning the theory of well-posed minimization problems. We recall some classical results obtained in the framework of the theory but focus mainly on the detailed presentation of the application of the theory of measures of noncompactness to investigations of the well-posedness of minimization problem.

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

Access this chapter

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 EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
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

References

  1. Akhmerov, R.R., Kamenski, M.I., Potapov, A.S., Rodkina, A.E., Sadovskii, B.N.: Measures of Noncompactness and Condensing Operators. Birkháuser, Basel (1992)

    Book  MATH  Google Scholar 

  2. Ansari, Q.H. (ed.): Topics in Nonlinear Analysis and Optimization. World Education, Delhi (2012)

    Google Scholar 

  3. Banaś, J.: Measures of noncompactness in the space of continuous tempered functions. Demonstratio Math. 14, 127–133 (1981)

    MATH  MathSciNet  Google Scholar 

  4. Banaś, J.: On drop property and nearly uniformly smooth Banach spaces. Nonlinear Anal. Theory Meth. Appl. 14, 927–933 (1990)

    Google Scholar 

  5. Banaś, J.: Compactness conditions in the geometric theory of Banach spaces. Nonlinear Anal. Theory Meth. Appl. 16, 669–682 (1991)

    Google Scholar 

  6. Banaś, J.: Measures of noncompactness in the study of soultions of nonlinear differential and integral equations. Cent. Eur. J. Math. 10, 2003–2011 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Banaś, J., Fra̧czek, K.: Conditions involving compactness in geometry of Banach spaces. Nonlinear Anal. Theory Meth. Appl. 20, 1217–1230 (1993)

    Google Scholar 

  8. Banaś, J., Fra̧czek, K.: Locally nearly uniformly smooth Banach spaces. Collect. Math. 44, 13–22 (1993)

    Google Scholar 

  9. Banaś, J., Goebel, K.: Measures of Noncompactness in Banach spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Marcel Dekker, New York (1980)

    Google Scholar 

  10. Banaś, J., Martinon, A.: Some properties of the Hausdorff distance in metric spaces. Bull. Austral. Math. Soc. 42, 511–516 (1990)

    Google Scholar 

  11. Banaś, J., Martinon, A.: Measures of noncompactness in Banach sequence spaces. Math. Slovaca 42, 497–503 (1992)

    Google Scholar 

  12. Banaś, J., Sadarangani, K.: Compactness conditions in the study of functional, differential and integral equations. Abstr. Appl. Anal. 2013, Article ID 819315 (2013)

    Google Scholar 

  13. Bednarczuk, E., Penot, J.P.: Metrically well-set minimization problems. Appl. Math. Optim. 26, 273–285 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Springer, Berlin (1993)

    Google Scholar 

  15. Dunford, N., Schwartz, J.T.: Linear Operators I. International Publications, Leyden (1963)

    Google Scholar 

  16. Furi, M., Vignoli, A.: About well-posed optimization problems for functionals in metric spaces. J. Optim. Theory Appl. 5, 225–229 (1970)

    Article  MATH  Google Scholar 

  17. Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  18. Goebel, K., Sȩkowski, T.: The modulus of noncomapct convexity. Ann. Univ. Mariae Curie-Skłodowska Sect. A 38, 41–48 (1984)

    Google Scholar 

  19. Golden\(\check{\text{ s }}\)tein, L.S., Markus, A.S.: On a measure of noncompactness of bounded sets and linear operators. In: Studies in Algebra and Mathematical Analysis, Kishinev pp. 45–54 (1965)

    Google Scholar 

  20. Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)

    Google Scholar 

  21. Hu, R., Fang, Y., Huang, N., Wong, M.: Well-posedness of systems of equilibrium problems. Taiwan. J. Math. 14, 2435–2446 (2010)

    MATH  MathSciNet  Google Scholar 

  22. Huff, R.: Banach spaces which are nearly uniformly convex. Rocky Mount. J. Math. 10, 743–749 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  23. Knap, Z., Banaś, J.: Characterization of the well-posed minimum problem (in Polish). Tow. Nauk. w Rzeszowie Met. Numer. 6, 51–62 (1980)

    Google Scholar 

  24. Kuratowski, K.: Sur les espaces complets. Fund. Math. 15, 301–309 (1930)

    MATH  Google Scholar 

  25. Kuratowski, K.: Topology. Academic Press, New York (1968)

    Google Scholar 

  26. Levitin, E.S., Polyak, B.T.: Convergence of minimizing sequences in conditional extremum problem. Soviet Math. Dokl. 7, 764–767 (1966)

    MATH  Google Scholar 

  27. Long, X.J., Huang, N.J., Teo, K.L.: Levitin-Polyak well-posedness for equilibrium problems with functional constrains. J. Inequal. Appl. 2008, Article ID 657329 (2008)

    Google Scholar 

  28. Montesinos, V.: Drop property equals reflexivity. Studia Math. 87, 93–100 (1987)

    MATH  MathSciNet  Google Scholar 

  29. Revalski, J.P.: Hadamard and strong well-posedness for convex programs. SIAM J. Optim. 7, 519–526 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  30. Rolewicz, S.: On drop property. Studia Math. 85, 27–35 (1987)

    MathSciNet  Google Scholar 

  31. Rolewicz, S.: On \(\delta \)-uniform convexity and drop property. Studia Math. 87, 181–191 (1987)

    MATH  MathSciNet  Google Scholar 

  32. Sȩkowski, T., Stachura, A.: Noncompact smoothness and noncompact convexity. Atti. Sem. Mat. Fis. Univ. Modena 36, 329–338 (1988)

    MathSciNet  Google Scholar 

  33. Tikhonov, A.N.: On the stability of the functional optimization problem. USSR J. Comput. Math. Math. Phys. 6, 631–634 (1966)

    Google Scholar 

  34. Zolezzi, T.: Extended well-posedness of optimization problem. J. Optim. Theory Appl. 91, 257–266 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Józef Banaś .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer India

About this chapter

Cite this chapter

Banaś, J. (2014). Measures of Noncompactness and Well-Posed Minimization Problems. In: Ansari, Q. (eds) Nonlinear Analysis. Trends in Mathematics. Birkhäuser, New Delhi. https://doi.org/10.1007/978-81-322-1883-8_4

Download citation

Publish with us

Policies and ethics