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Discrete Dynamics in Nature and Society
Volume 2015, Article ID 143510, 10 pages
http://dx.doi.org/10.1155/2015/143510
Research Article

Nonlinear Dynamics, Fixed Points and Coupled Fixed Points in Generalized Gauge Spaces with Applications to a System of Integral Equations

Faculty of Mathematics and Computer Science and Faculty of Business, Babeş-Bolyai University Cluj-Napoca, Kogălniceanu Street No. 1, 400084 Cluj-Napoca, Romania

Received 2 June 2015; Accepted 31 July 2015

Academic Editor: Gabriele Bonanno

Copyright © 2015 Adrian Petruşel and Gabriela Petruşel. 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. A. I. Perov, “On the Cauchy problem for a system of ordinary differential equations,” Pviblizhen. Met. Reshen. Differ. Uravn, vol. 2, pp. 115–134, 1964. View at Google Scholar
  2. A. I. Perov and A. V. Kibenko, “On a certain general method for investigation of boundary value problems,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 30, pp. 249–264, 1966 (Russian). View at Google Scholar · View at MathSciNet
  3. R. P. Agarwal, “Contraction and approximate contraction with an application to multipoint boundary value problems,” Journal of Computational and Applied Mathematics, vol. 9, no. 4, pp. 315–325, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. A.-D. Filip and A. Petruşel, “Fixed point theorems on spaces endowed with vector-valued metrics,” Fixed Point Theory and Applications, vol. 2010, Article ID 281381, 15 pages, 2010. View at Google Scholar · View at MathSciNet
  5. D. O'Regan, N. Shahzad, and R. P. Agarwal, “Fixed point theory for generalized contractive maps on spaces with vector-valued metrics,” in Fixed Point Theory and Applications, pp. 143–149, Nova Science Publishers, New York, NY, USA, 2007. View at Google Scholar · View at MathSciNet
  6. R. Precup and A. Viorel, “Existence results for systems of nonlinear evolution equations,” International Journal of Pure and Applied Mathematics, vol. 47, no. 2, pp. 199–206, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. R. Precup and A. Viorel, “Existence results for systems of nonlinear evolution inclusions,” Fixed Point Theory, vol. 11, no. 2, pp. 337–346, 2010. View at Google Scholar · View at MathSciNet
  8. R. Precup, “The role of matrices that are convergent to zero in the study of semilinear operator systems,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 703–708, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. A. Petruşel, “Multivalued weakly Picard operators and applications,” Scientiae Mathematicae Japonicae, vol. 59, no. 1, pp. 169–202, 2004. View at Google Scholar · View at MathSciNet
  10. I.-R. Petre and A. Petruşel, “Krasnoselskii's theorem in generalized Banach spaces and applications,” Electronic Journal of Qualitative Theory of Differential Equations, no. 85, pp. 1–20, 2012. View at Google Scholar · View at MathSciNet
  11. Sh. Rezapour and P. Amiri, “Some fixed point results for multivalued operators in generalized metric spaces,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2661–2666, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. A. Petruşel, C. Urs, and O. Mleşniţe, “Vector-valued metrics in fixed point theory,” Contemporary Mathematics, vol. 636, pp. 149–164, 2015. View at Publisher · View at Google Scholar
  13. A. Petruşel, G. Petruşel, and C. Urs, “Vector-valued metrics, fixed points and coupled fixed points for nonlinear operators,” Fixed Point Theory and Applications, vol. 2013, article 218, 2013. View at Publisher · View at Google Scholar · View at Scopus
  14. P. P. Zabrejko, “K-metric and K-normed linear spaces: survey,” Collectanea Mathematica, vol. 48, no. 4–6, pp. 825–859, 1997. View at Google Scholar
  15. L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1467–1475, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. G. Marinescu, Spaţii Vectoriale Topologice şi Pseudotopologice, vol. 4, Biblioteca Matematică, Editura Academiei Republicii Populare Romîne, Bucharest, Romania, 1959.
  17. I. Colojoară, “On a fixed point theorem in complete uniform spaces,” Com. Acad. R.P.R., vol. 11, pp. 281–283, 1961. View at Google Scholar
  18. N. Gheorghiu, “Contraction theoremin uniform spaces,” Studii și Cercetări Matematice, vol. 19, pp. 119–122, 1967 (Romanian). View at Google Scholar
  19. R. J. Knill, “Fixed points of uniform contractions,” Journal of Mathematical Analysis and Applications, vol. 12, pp. 449–455, 1965. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. J. L. Cain Jr. and M. Z. Nashed, “Fixed points and stability for a sum of two operators in locally convex spaces,” Pacific Journal of Mathematics, vol. 39, pp. 581–592, 1971. View at Publisher · View at Google Scholar · View at MathSciNet
  21. M. Frigon, “Fixed point results for generalized contractions in gauge spaces and applications,” Proceedings of the American Mathematical Society, vol. 128, no. 10, pp. 2957–2965, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. M. Frigon, “Fixed point and continuation results for contractions in metric and gauge spaces,” Fixed Point Theory and Its Applications, vol. 77, pp. 89–114, 2007. View at Publisher · View at Google Scholar
  23. J. Dugundji, Topology, Allyn & Bacon, Boston, Mass, USA, 1966.
  24. A. Novac and R. Precup, “Perov type results in gauge spaces and their applications to integral systems on semi-axis,” Mathematica Slovaca, vol. 64, no. 4, pp. 961–972, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  25. V. I. Opoitsev, “Heterogeneous and combined concave operators,” Siberian Mathematical Journal, vol. 16, no. 4, pp. 597–605, 1975 (Russian). View at Publisher · View at Google Scholar · View at Scopus
  26. V. I. Opoitsev, “Dynamics of collective behavior. III. Heterogenic systems,” Avtomatika i Telemekhanika, vol. 36, pp. 124–138, 1975 (Russian). View at Google Scholar
  27. V. I. Opoitsev and T. A. Khurodze, “Nonlinear operators in spaces with a cone,” Tbilisskiü Gosudarstvennyü Universitet, Tbilisi, vol. 271, 1984 (Russian). View at Google Scholar
  28. D. J. Guo and V. Lakshmikantham, “Coupled fixed points of nonlinear operators with applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 11, no. 5, pp. 623–632, 1987. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 7, pp. 1379–1393, 2006. View at Publisher · View at Google Scholar · View at Scopus
  30. V. Lakshmikantham and L. Ćirić, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4341–4349, 2009. View at Publisher · View at Google Scholar · View at Scopus
  31. D. Guo, Y. J. Cho, and J. Zhu, Partial Ordering Methods in Nonlinear Problems, Nova Science Publishers, Hauppauge, NY, USA, 2004. View at MathSciNet
  32. S. Hong, “Fixed points for mixed monotone multivalued operators in Banach spaces with applications,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 333–342, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  33. V. Berinde, “Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces,” Nonlinear Analysis, vol. 74, no. 18, pp. 7347–7355, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. R. S. Varga, Matrix Iterative Analysis, vol. 27 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  35. G. Allaire and S. M. Kaber, Numerical Linear Algebra, Springer, New York, NY, USA, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  36. A. Petruşel and I. A. Rus, “The theory of a metric fixed point theorem for multivalued operators,” in Fixed Point Theory and Its Applications, L. J. Lin, A. Petruşel, and H. K. Xu, Eds., pp. 167–176, Yokohama, 2010. View at Google Scholar
  37. A. Granas and J. Dugundji, Fixed Point Theory, Springer, Berlin, Germany, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  38. M. Frigon, “Fixed point results for multivalued contractions on gauge spaces,” in Set-Valued Mappings with Applications in Nonlinear Analysis, vol. 4 of Mathematical Analysis and Applications, pp. 175–181, Taylor & Francis, London, UK, 2002. View at Google Scholar