Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces
Volume 2016, Article ID 3970621, 15 pages
http://dx.doi.org/10.1155/2016/3970621
Research Article

A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

Received 30 June 2016; Accepted 18 October 2016

Academic Editor: Adrian Petrusel

Copyright © 2016 Teffera M. Asfaw. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. F. E. Browder, “Degree of mapping for nonlinear mappings of monotone type,” Proceedings of the National Academy of Sciences, vol. 80, no. 6, pp. 1771–1773, 1983. View at Publisher · View at Google Scholar
  2. F. E. Browder, “Degree of mapping for nonlinear mappings of monotone type: densely defined mapping,” Proceedings of the National Academy of Sciences of the United States of America, vol. 80, no. 8, pp. 2405–2407, 1983. View at Publisher · View at Google Scholar
  3. J. Kobayashi and M. Ôtani, “Topological degree for (S)+-mappings with maximal monotone perturbations and its applications to variational inequalities,” Nonlinear Analysis, Theory, Methods and Applications, vol. 59, no. 1-2, pp. 147–172, 2004. View at Publisher · View at Google Scholar · View at Scopus
  4. S. C. Hu and N. S. Papageorgiou, “Generalizations of Browder's degree theory,” Transactions of the American Mathematical Society, vol. 347, no. 1, pp. 233–259, 1995. View at Publisher · View at Google Scholar
  5. J. Berkovits and V. Mustonen, “Topological degree for perturbations of maximal monotone mappings and applications to a class of parabolic problems,” Rendiconti di Matematica, vol. 12, pp. 597–621, 1992. View at Google Scholar
  6. J. Berkovits and V. Mustonen, “On the topological degree for mappings of monotone type,” Nonlinear Analysis, vol. 10, no. 12, pp. 1373–1383, 1986. View at Publisher · View at Google Scholar · View at Scopus
  7. J. Berkovits, On the degree theory for nonlinear mapping of monotone type [Dissertations], Annales Academiæ Scientiarum Fennicæ Series A1, 1986.
  8. A. G. Kartsatos and I. V. Skrypnik, “A new topological degree theory for densely defined quasibounded S~+-perturbations of multivalued maximal monotone operators in reflexive Banach spaces,” Abstract and Applied Analysis, vol. 2005, no. 2, pp. 121–158, 2005. View at Publisher · View at Google Scholar
  9. B. T. Kien, M.-M. Wong, and N.-C. Wong, “On the degree theory for general mappings of monotone type,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 707–720, 2008. View at Publisher · View at Google Scholar · View at Scopus
  10. T. M. Asfaw and A. G. Kartsatos, “A Browder topological degree theory for multivalued pseudomonotone perturbations of maximal monotone operators in reflexive Banach spaces,” Advances in Applied Mathematics, vol. 22, pp. 91–148, 2012. View at Google Scholar
  11. T. M. Asfaw, “A new topological degree theory for pseudomonotone perturbations of the sum of two maximal monotone operators and applications,” Journal of Mathematical Analysis and Applications, vol. 434, no. 1, Article ID 19800, pp. 967–1006, 2016. View at Publisher · View at Google Scholar · View at Scopus
  12. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishers, Leyden, The Netherlands, 1975.
  13. V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2010.
  14. D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Sijthoff and Noordhoof, Bucharest, Romania, 1978.
  15. E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B, Springer, New York, NY, USA, 1990.
  16. F. E. Browder and P. Hess, “Nonlinear mappings of monotone type in Banach spaces,” Journal of Functional Analysis, vol. 11, no. 3, pp. 251–294, 1972. View at Publisher · View at Google Scholar · View at Scopus
  17. R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, no. 1, pp. 75–88, 1970. View at Publisher · View at Google Scholar
  18. H. Brézis and L. Nirenberg, “Characterizations of the ranges of some nonlinear operators and applications to boundary value problems,” Annali della Scuola Normale Superiore di Pisa—Classe di Scienze, vol. 5, no. 2, pp. 225–326, 1978. View at Google Scholar
  19. J. Berkovits and V. Mustonen, “An extension of Leray-Schauder degree and applications to nonlinear wave equations,” Differential Integral Equations, vol. 3, no. 5, pp. 945–963, 1990. View at Google Scholar
  20. P. H. Rabinowitz, “Free vibrations for a semilinear wave equation,” Communications on Pure and Applied Mathematics, vol. 31, no. 1, pp. 31–68, 1978. View at Publisher · View at Google Scholar
  21. H. Brézis and L. Nirenberg, “Forced vibrations for a nonlinear wave equation,” Communications on Pure and Applied Mathematics, vol. 31, no. 1, pp. 1–30, 1978. View at Publisher · View at Google Scholar
  22. H. Brezis, “Periodic solutions of nonlinear vibrating strings and duality principles,” Bulletin of the American Mathematical Society, vol. 8, no. 3, pp. 409–427, 1983. View at Publisher · View at Google Scholar
  23. V. Barbu and N. H. Pavel, “Periodic solutions to nonlinear one dimensional wave equation with x-dependent coefficients,” Transactions of the American Mathematical Society, vol. 349, no. 5, pp. 2035–2048, 1997. View at Publisher · View at Google Scholar · View at Scopus
  24. J. M. Coron, “Periodic solutions of a nonlinear wave equation without assumption of monotonicity,” Mathematische Annalen, vol. 262, no. 2, pp. 273–285, 1983. View at Publisher · View at Google Scholar · View at Scopus
  25. H. Hofer, “On the range of a wave operator with nonmonotone nonlinearity,” Mathematische Nachrichten, vol. 106, no. 1, pp. 327–340, 1982. View at Publisher · View at Google Scholar
  26. R. Landes and V. Mustonen, “On pseudo-monotone operators and nonlinear noncoercive variational problems on unbounded domains,” Mathematische Annalen, vol. 248, no. 3, pp. 241–246, 1980. View at Publisher · View at Google Scholar · View at Scopus
  27. V. Mustonen, “On elliptic operators in divergence form; old and new with applications,” in Proceedings of the Conference Function Spaces, Differential Operators and Nonlinear Analysis (FSDONA '04), pp. 188–200, May-June 2004.
  28. V. Mustonen and M. Tienari, “A remark on the Leray-Lions condition,” Nonlinear Analysis, Theory, Methods and Applications, vol. 49, no. 7, pp. 1015–1022, 2002. View at Publisher · View at Google Scholar · View at Scopus
  29. N. Kenmochi, “Monotonicity and compactness methods for nonlinear variational inequalities,” in Handbook of Differential Equations, IV, pp. 203–298, Elsevier/North-Holland, Amsterdam, The Netherlands, 2007. View at Google Scholar
  30. A. G. Kartsatos, “New results in the perturbation theory of maximal monotone and M-accretive operators in banach spaces,” Transactions of the American Mathematical Society, vol. 348, no. 5, pp. 1663–1707, 1996. View at Publisher · View at Google Scholar · View at Scopus
  31. T. M. Asfaw, Topological degree and variational inequality theories for pseudomonotone perturbations of maximal monotone operators [Dissertations], ProQuest, Ann Arbor, Mich, USA, 2013.
  32. T. M. Asfaw, “New surjectivity results for perturbed weakly coercive operators of monotone type in reflexive Banach spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 113, pp. 209–229, 2015. View at Publisher · View at Google Scholar · View at Scopus
  33. T. M. Asfaw, “New variational inequality and surjectivity theories for perturbed noncoercive operators and application to nonlinear problems,” Advances in Mathematical Sciences and Applications, vol. 24, pp. 611–668, 2014. View at Google Scholar
  34. V. K. Le, “A range and existence theorem for pseudomonotone perturbations of maximal monotone operators,” Proceedings of the American Mathematical Society, vol. 139, no. 5, pp. 1645–1658, 2011. View at Publisher · View at Google Scholar · View at Scopus