Applications of Mathematics

, Volume 58, Issue 6, pp 689–702 | Cite as

Computation of topological degree in ordered Banach spaces with lattice structure and applications

  • Yujun CuiEmail author


Using the cone theory and the lattice structure, we establish some methods of computation of the topological degree for the nonlinear operators which are not assumed to be cone mappings. As applications, existence results of nontrivial solutions for singular Sturm-Liouville problems are given. The nonlinearity in the equations can take negative values and may be unbounded from below.


cone lattice topological degree 

MSC 2010

47H11 34B15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    K. Deimling: Nonlinear Functional Analysis. Springer, Berlin, 1985.CrossRefzbMATHGoogle Scholar
  2. [2]
    D. Guo, V. Lakshmikantham: Nonlinear Problems in Abstract Cones. Notes and Reports in Mathematics in Science and Engineering 5. Academic Press, Boston, 1988.zbMATHGoogle Scholar
  3. [3]
    X. Liu, J. Sun: Computation of topological degree of unilaterally asymptotically linear operators and its applications. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 96–106.CrossRefzbMATHGoogle Scholar
  4. [4]
    W. A. J. Luxemburg, A. C. Zaanen: Riesz Spaces. Vol. I. North-Holland Mathematical Library. North-Holland Publishing Company, Amsterdam, 1971.zbMATHGoogle Scholar
  5. [5]
    M. A. Krasnosel’skij: Positive Solutions of Operator Equations. Translated from the Russian by Richard E. Flaherty (L. F. Boron, ed.). P. Noordhoff Ltd., Groningen, 1964.Google Scholar
  6. [6]
    M. G. Kreĭin, M. A. Rutman: Linear operators leaving invariant a cone in a Banach space. Usp. Mat. Nauk 3 (1948), 3–95. (In Russian.)MathSciNetGoogle Scholar
  7. [7]
    J. Sun: Nontrivial solutions of superlinear Hammerstein integral equations and applications. Chin. Ann. Math., Ser. A 7 (1986), 528–535. (In Chinese.)zbMATHGoogle Scholar
  8. [8]
    J. Sun, X. Liu: Computation for topological degree and its applications. J. Math. Anal. Appl. 202 (1996), 785–796.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    J. Sun, X. Liu: Computation of topological degree for nonlinear operators and applications. Nonlinear Anal., Theory Methods Appl. 69 (2008), 4121–4130.CrossRefzbMATHGoogle Scholar
  10. [10]
    J. Sun, X. Liu: Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations. J. Math. Anal. Appl. 348 (2008), 927–937.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    J. Sun, G. Zhang: Nontrivial solutions of singular sublinear Sturm-Liouville problems. J. Math. Anal. Appl. 326 (2007), 242–251.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    W. Walter: Ordinary Differential Equations. Transl. from the German by Russell Thompson. Graduate Texts in Mathematics. Readings in Mathematics 182. Springer, New York, 1998.Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2013

Authors and Affiliations

  1. 1.Department of MathematicsShandong University of Science and TechnologyQingdaoP. R. China

Personalised recommendations