Abstract
In the paper, the topological degree for a compact continuous operator defined on an open subset of a Menger PN-space is generalized. The new concept of fixed-point index in Menger PN-spaces is introduced, themost important properties of the fixed-point index are established, and some other results are given.
Similar content being viewed by others
References
H. Amman, Lectures on Some Fixed-Point Theorems, in Monografias de Matematica (Rio de Janeiro, 1974).
M. A. Krasnosel’skii, TopologicalMethods in the Theory ofNonlinear Integral Equations (Gostekhizdat, Moscow, 1956; Pergamon Press, Oxford, 1964).
L. Nirenberg, Topics in Nonlinear Functional Analysis (Courant Inst. of Math. Sci., New York, 1974).
J. T. Schwartz, Nonlinear Functional Analysis, in Notes in Mathematics and its Applications (Gordon and Breach Sci. Publ., New York, 1969).
H. Amman, “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,” SIAM Rev. 18(4), 620–709 (1976).
S. S. Chang, J. C. Yeol, and M. K. Shi, Probabilistic Metric Spaces and Nonlinear Operator Theory (Sichuan Univ. Press, Chengdu, 1994).
S. S. Zhang and Y. Q. Chen, “Topological degree theory and fixed point theorems in probabilistic metric spaces,” Appl. Math.Mech. (English Ed.) 10(6), 495–505 (1989).
S. S. Chang, B. S. Lee, Y. J. Cho, Y.Q. Chen, S.M. Kang, and J. S. Jung, “Generalized contraction mapping principle and differential equations in probabilistic metric spaces,” Proc. Amer. Math. Soc. 124, 2367–2376 (1996).
Q.Y. Li and C. X. Zhu, “Some new fixed points in Z-P-S space,” J. Systems Sci.Math. Sci. 28(4), 400–405 (2008).
C. X. Zhu, “Several nonlinear operator problems in the Menger PN space,” Nonlinear Anal. 65(7), 1281–1284 (2006).
C. X. Zhu, “Generalizations of Krasnoselskii’s theorem and Petryshyn’s theorem,” Appl.Math. Lett. 19(7), 628–632 (2006).
C. X. Zhu, “A class of random operator equations in the Hilbert space,” Acta Math. Sinica (Chin. Ser.) 47(4), 641–646 (2004).
C. X. Zhu, “Some theorems in the X-M-PN Space,” Appl. Math. Mech. (English Ed.) 21(2), 181–184 (2000).
C. X. Zhu and C. F. Chen, “Calculations of random fixed-point index,” J. Math. Anal. Appl. 339(2), 839–844 (2008).
C. X. Zhu and Z. B. Xu, “Inequalities and solution of an operator equation,” Appl. Math. Lett. 21(6), 607–611 (2008).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Russian in Matematicheskie Zametki, 2011, Vol. 90, No. 1, pp. 104–112.
The text was submitted by the authors in English.
Rights and permissions
About this article
Cite this article
Li, Q., Zhu, C. The fixed-point index of nonlinear operators in Menger PN-spaces. Math Notes 90, 94 (2011). https://doi.org/10.1134/S0001434611070108
Received:
Published:
DOI: https://doi.org/10.1134/S0001434611070108