Skip to main content
SpringerLink
Log in

Nonlinear Fredholm Inclusions and Applications

  • Chapter
  • First Online:
Method of Guiding Functions in Problems of Nonlinear Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2076))

Abstract

The necessity of studying coincidence points of nonlinear Fredholm operators and nonlinear (compact and condensing) maps of various classes arises in the investigation of many problems in the theory of partial differential equations and optimal control theory.

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 34.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.95
Price excludes VAT (USA)
  • Compact, lightweight 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. P. Benevieri, M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree theory. Ann. Sci. Math. Qué. 22, 131–148 (1998)

    MathSciNet  MATH  Google Scholar 

  2. P. Benevieri, M. Furi, On the concept of orientability for Fredholm maps between real Banach manifolds. Topol. Meth. Nonlinear Anal. 16, 279–306 (2000)

    MathSciNet  MATH  Google Scholar 

  3. Yu.G. Borisovich, Topological characteristics and investigation of the solvability of nonlinear problems. Izv. Vyssh. Uchebn. Zaved. Mat. (Russian) (2), 3–23 (1997); English translation: Russ. Math. (Iz. VUZ) 41(2), 1–21 (1997)

    Google Scholar 

  4. Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis, V.V. Obukhovskii, Topological methods in the theory of fixed points of multivalued mappings. Uspekhi Mat. Nauk (Russian) 35(1)(211), 59–126 (1980); English translation: Russ. Math. Surv. 35, 65–143 (1980)

    Google Scholar 

  5. Yu.G. Borisovich, V.G. Zvyagin, Yu.I. Sapronov, Nonlinear Fredholm mappings, and Leray-Schauder theory. Uspehi Mat. Nauk (in Russian) 32(4)(196), 3–54 (1977). English translation: Russ. Math. Surv. 32(4), 1–54 (1977)

    Google Scholar 

  6. Yu.G. Borisovich, V.G. Zvyagin, V.V. Shabunin, On the solvability in \(W_{p}^{2m+1}\) of the nonlinear Dirichlet problem in a narrow strip. Dokl. Akad. Nauk (in Russian) 334(6), 683–685 (1994). English translation: Russ. Acad. Sci. Dokl. Math. 49(1), 179–182 (1994)

    Google Scholar 

  7. K.D. Elworthy, A.J. Tromba, Differential structures and Fredholm maps on Banach manifolds, in Global Analysis, 1970. Proceedings of the Symposium on Pure Mathematics, vol. XV, Berkeley, CA (American Mathematical Society, Providence, 1968), pp. 45–94

    Google Scholar 

  8. P.M. Fitzpatrick, J. Pejsachowicz, P.J. Rabier, Orientability of Fredholm families and topological degree for orientable non-linear Fredholm mappings. J. Funct. Anal. 124, 1–39 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Gabor, The coincidence index for fundamentally contractible multivalued maps with nonconvex values. Ann. Polon. Math. 75(2), 143–166 (2000)

    MathSciNet  MATH  Google Scholar 

  10. D. Gabor, W. Kryszewski, A coincidence theory involving Fredholm operators of nonnegative index. Topol. Meth. Nonlinear Anal. 15(1), 43–59 (2000)

    MathSciNet  MATH  Google Scholar 

  11. J. Ize, Bifurcation theory for Fredholm operators. Mem. Am. Math. Soc. 7(174), viii + 128 pp (1976)

    Google Scholar 

  12. J. Ize, Topological bifurcation, in Topological Nonlinear Analysis: Degree, Singularity and Variations, ed. by M. Matzeu, A. Vignoli. Progress in Nonlinear Differential Equations and Their Applications, vol. 15 (Birkhäuser, Boston, 1995), pp. 341–463

    Google Scholar 

  13. M. Kamenskii, V. Obukhovskii, P. Zecca, in Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. de Gruyter Series in Nonlinear Analysis and Applications, vol. 7 (Walter de Gruyter, Berlin, 2001)

    Google Scholar 

  14. M.A. Krasnosel’skii, P.P. Zabreiko, Geometrical Methods of Nonlinear Analysis (Nauka, Moscow, 1975); English translation: Grundlehren der Mathematischen Wissenschaften, vol. 263 (Springer, Berlin, 1984)

    Google Scholar 

  15. Y.C. Liou, V. Obukhovskii, J.C. Yao, Application of a coincidence index to some classes of impulsive control systems. Nonlinear Anal. 69(12), 4392–4411 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Nirengerg, in Topics in Nonlinear Functional Analysis. Revised Reprint of the 1974 Original. Courant Lecture Notes in Mathematics, vol. 6, New York University, Courant Institute of Mathematical Sciences, New York (American Mathematical Society, Providence, 2001)

    Google Scholar 

  17. V.V. Obukhovskii, On some fixed point principles for multivalued condensing operators. Trudy Mat. Fac. Voronezh Univ. 4, 70–79 (1971) (in Russian)

    Google Scholar 

  18. V. Obukhovskii, P. Zecca, V. Zvyagin, On coincidence index for multivalued perturbations of nonlinear Fredholm maps and some applications. Abstr. Appl. Anal. 7(6), 295–322 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. V. Obukhovskii, P. Zecca, V. Zvyagin, An oriented coincidence index for nonlinear Fredholm inclusions with nonconvex-valued perturbations. Abstr. Appl. Anal. Art. ID 51794, 21 p. (2006)

    Google Scholar 

  20. M. Väth, in Topological Analysis: From the Basics to the Triple Degree for Nonlinear Fredholm Inclusions. De Gruyter Series in Nonlinear Analysis and Applications, vol. 16 (Walter de Gruyter, Berlin, 2012)

    Google Scholar 

  21. S. Wang, On orientability and degree of Fredholm maps. Mich. Math. J. 53, 419–428 (2005)

    Article  MATH  Google Scholar 

  22. P. Zecca, V.G. Zvyagin, V.V. Obukhovskii, On the oriented coincidence index for nonlinear Fredholm inclusions. Dokl. Akad. Nauk 406(4), 443–446 (2006) (Russian). English translation: [J] Dokl. Math. 73(1), 63–66 (2006)

    Google Scholar 

  23. V.G. Zvyagin, The existence of a continuous branch for the eigenfunctions of a nonlinear elliptic boundary value problem. Differencial’nye Uravnenija 13(8), 1524–1527 (1977) (in Russian)

    MATH  Google Scholar 

  24. V.G. Zvyagin, The oriented degree of a class of perturbations of Fredholm mappings and the bifurcation of the solutions of a nonlinear boundary value problem with noncompact perturbations. Mat. Sb. 182(12), 1740–1768 (1991) (Russian); English translation: Math. USSR-Sb. 74(2), 487–512 (1993)

    Google Scholar 

  25. V.G. Zvyagin, N.M. Ratiner, Oriented degree of Fredholm maps of nonnegative index and its application to global bifurcation of solutions, in Global Analysis – Studies and Applications, V. Lecture Notes in Mathematics, vol. 1520 (Springer, Berlin, 1992), pp. 111–137

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics and Mathematics, Voronezh State Pedagogical University, Voronezh, Russia

    Valeri Obukhovskii & Sergei Kornev

  2. Dipartimento di Matematica e Informatica “U Dini”, Università di Firenze, Firenze, Italy

    Pietro Zecca

  3. Faculty of Fundamental Science, PetroVietNam University, Ba Ria, Vietnam

    Nguyen Van Loi

Authors
  1. Valeri Obukhovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pietro Zecca
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nguyen Van Loi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sergei Kornev
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Obukhovskii, V., Zecca, P., Van Loi, N., Kornev, S. (2013). Nonlinear Fredholm Inclusions and Applications. In: Method of Guiding Functions in Problems of Nonlinear Analysis. Lecture Notes in Mathematics, vol 2076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37070-0_5

Download citation

Publish with us

Policies and ethics

Search

Navigation