Index Formulas for Countably φ-Set Contraction

Article Sidebar

PDF
Cite as:
H. Salahifard, S. M. Vaezpour, Index Formulas for Countably φ-Set Contraction, Int. J. Anal. Appl., 6 (2) (2014), 154-163.

Main Article Content

H. Salahifard, S. M. Vaezpour

Abstract

In this paper, we study the index formulas for a class of bounded linear operators, namely φ-set contractions, acting on a Banach space and we discuss some application of this class of operators to the theory of bifurcation points. In particular our results generalize and improve some recent results mentioned in the literature.

Article Details

References

  1. R.R. Akhmerov, M.I. Kamenskii, A.S. Patapov, A.E. Rodkina, B.N. Sadovskii. Measures of noncompactness and condensing operators, Operator Theory, Advances and Applications.55 (1992), Birkhuser Verlag, Basel.
  2. H. Amann. Fixed point equations and nonlinear eigenvalue problems in ordered Banach space, SIAM Rev.18 (1976), 620-09.
  3. J. Appel. Measures of noncompactness, condensing operators and fixed points: an appicationoriented survey, Fixed Point Theory. 6 (2005), 157-29.
  4. F. E. Browder. On the spectral theory of elliptic differential operators. I, Math. Ann. 142(1961), 22-30.
  5. B. C. Dhage. Condensing mappings and applications to existence theorems for common solution of differential equations, Bull. Korean Math. Soc. 36(1999) No. 3, 565-578.
  6. L.Guozhen. The fixed point index and the fixed point theorems of 1-set contrac- tion mappings, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1163-1170.
  7. A. Granas, J. Dugundji. Fixed Point Theory, Springer-Verlag, New York, Berlin, 2003.
  8. G. Hetzer, V. Stallbohm. Coincidence degree and Rabinowitz's bifurcation theorem, Publications de l'institut mathematique, Nouvelle series, tome 20 (1976), 117-129.
  9. In-Sook Kim, Sunghui Kwon. Global bifurcation for generalized k-set contractions, Nonlinear Analysis. 68 (2008), 3224-231.
  10. In-Sook Kim. Index formulas for countably k-set contractive operators, Nonlinear Analysis.69 (2008), 4182-189.
  11. C. Kuratowski. Sur les espaces complets, Fund. Math.15 (1930), 301-09.
  12. L.S. Liu. Approximation theorems and fixed point theorems for various class of 1-setcontractive mappings in Banach spaces, Acta Math. Sinica.17(2001), 103-12.
  13. R. D. Nussbaum. The radius of the essential spectrum, Duke Math. J.38 (1970), 473-78.
  14. R.D. Nussbaum. The fixed point index and fixed point theorems for k-set contractions, Bull. Amer. Math. Soc. 75(1969), 490-95.
  15. R.D. Nussbaum. Asymptotic fixed point theorems for local condensing maps, Math. Ann. 191 (1971), 181-195.
  16. Martin Schechter. Principles of Functional Analysis, Academic Press. (1971).
  17. Martin Schechter. On the Essential Spectrum of an Arbitrary Operator. I, Journal of Mathematical Analysis and Applications.13 (1966), 205-15.
  18. C.A. Stuart, J.F. Toland. The fixed point index of a linear k-set contraction, J. London Math. Soc.6 (1973), 317-20.
  19. J.M. Ayerbe Toledano, T. Dominguez Benavides, G. Lopez Acedo. Measures of Noncompactness in Metric Fixed Point Theory, Birkhauser, Basel,. (1997).
  20. M. Vath. Fixed point theorems and fixed point index for countably condensing maps, Topol. Methods Nonlinear Anal.13 (1999), 341-63.