Cone expansion and cone compression fixed point theorems for sum of two operators and their applications

Abstract

In this article, we first establish a series of user-friendly versions of fixed point theorems in cones for sum of two operators with one being either contractive or expansive and the other being compact, one of which covers the classical Krasnoselskii’s fixed point theorem concerning cone expansion and compression of norm type. Along the way, we also offer some sufficient conditions which slightly relax the compactness requirement on the one summand operator. Second, as applications to some of our main results, we consider the eigenvalue problems of Krasnoselskii type in critical case and the existence of one positive solution to one parameter operator equations. Finally, to illustrate the usefulness and the applicability of our fixed point results, we study the existence of one nontrivial positive solution to certain integral equations of Hammerstein type and of perturbed Volterra type.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Agarwal, R., O’Regan, D.: Cone compression and expansion fixed point theorems in Fréchet spaces with applications. J. Differ. Equ. 171, 412–429 (2001)

    Article  Google Scholar 

  2. 2.

    Anderson, D.R., Avery, R.I.: Fixed point theorem of cone expansion and compression of functional type. J. Differ. Equ. Appl. 8, 1073–1083 (2002)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Arab, R., Allahyari, R., Haghighi, A.: Construction of a measure of noncompactness on \(BC(\Omega )\) and its application to Volterra integral equations. Mediterr. J. Math. 13, 1197–1210 (2016)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Avery, R., Henderson, J., O’Regan, D.: Functional compression-expansion fixed point theorem, Electron. J. Differ. Equ., No. 22, 12 pp (2008)

  5. 5.

    Avery, R., Anderson, D., Henderson, J.: An extension of the compression-expansion fixed point theorem of functional type, Electron. J. Differ. Equ., Paper No. 253, 12 pp (2016)

  6. 6.

    Avramescu, C., Vladimirescu, C.: Some remarks on Krasnoselskiis fixed point theorem. Fixed Point Theory 4, 3–13 (2003)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Banaś, J., Chlebowicz, A.: On integrable solutions of nonlinear Volterra integral equations under Carathéodory conditions. Bull. Lond. Math. Soc. 41, 1073–1084 (2009)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Barroso, C., Teixeira, E.: A topological and geometric approach to fixed points results for sum of operators and applications. Nonlinear Anal. 60, 625–650 (2005)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Burton, T.: A fixed-point theorem of Krasnolel’skii. Appl. Math. Lett. 1, 85–88 (1998)

    Article  Google Scholar 

  10. 10.

    Burton, T., Purnaras, I.: A unification theory of Krasnosel’skii for differential equations. Nonlinear Anal. 89, 121–133 (2013)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Budisan, S.: Generalizations of Krasnoselskii’s fixed point theorem in cones and applications. Topol. Methods Nonlinear Anal. 43, 23–52 (2014)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    Google Scholar 

  13. 13.

    Frigon, M., O’Regan, D.: Fixed points of cone-compressing and cone-extending operators in Fréchet spaces. Bull. London Math. Soc. 35, 672–680 (2003)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Gripenberg, G., Londen, S.O., Staffans, O.: Volterra Integral and Functional Equations. Cambridge Univ Press, Cambridge (1990)

    Google Scholar 

  15. 15.

    Guo, D.: Some fixed point theorems on cone maps. Kexue Tongbao 29, 575–578 (1984)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988)

    Google Scholar 

  17. 17.

    Krasnosel’skii, M.A.: Two remarks on the method of successive approximations. Uspehi Mat. Nauk 10, 123–127 (1955)

    MathSciNet  Google Scholar 

  18. 18.

    Krasnosel’skii, M.A.: Fixed points of cone-compressing or cone-extending operators. Soviet Math. Dokl. 1, 1285–1288 (1960)

    MathSciNet  Google Scholar 

  19. 19.

    Krasnosel’skii, M.A.: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964)

    Google Scholar 

  20. 20.

    Lan, K.Q.: Multiple positive solutions of semilinear differential equations with singularities. J. London Math. Soc. 63, 690–704 (2001)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Lan, K.Q.: Multiple positive solutions of Hammerstein integral equations and applications to periodic boundary value problems. Appl. Math. Comput. 154, 531–542 (2004)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Latrach, K., Taoudi, M., Zeghal, A.: Some fixed point theorems of the Schauder and the Krasnosel’skii type and application to nonlinear transport equations. J. Differ. Equ. 221, 256–271 (2006)

    Article  Google Scholar 

  23. 23.

    Latrach, K.: An existence result for a class of nonlinear functional integral equations. J. Integral Equ. Appl. 27, 199–218 (2015)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Leggett, R., Williams, L.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28, 673–688 (1979)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Liu, Y., Li, Z.: Krasnoselskii type fixed point theorems and applications. Proc. Am. Math. Soc. 136, 1213–1220 (2008)

    MathSciNet  Article  Google Scholar 

  26. 26.

    O’Regan, D.: Fixed-point theory for the sum of two operators. Appl. Math. Lett. 9, 1–8 (1996)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Park, S.: Generalizations of the Krasnoselskii fixed point theorem. Nonlinear Anal. 67, 3401–3410 (2007)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Petryshyn, W.: Existence of fixed points of positive k-set-contractive maps as consequences of suitable boundary conditions. J. London Math. Soc. 38, 503–512 (1988)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Reich, S.: Characteristic vectors of nonlinear operators. Atti Accad. Naz. Lincei 50, 682–685 (1971)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Reich, S.: Fixed points of condensing functions. J. Math. Anal. Appl. 41, 460–467 (1973)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Smart, D.R.: Fixed Point Theorems. Cambridge University Press, Cambridge (1980)

    Google Scholar 

  32. 32.

    Torres, P.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 190, 643–662 (2003)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Vladimirescu, C.: Remark on Krasnoselskii’s fixed point theorem. Nonlinear Anal. 71, 876–880 (2009)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Webb, J.: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear Anal. 47, 4319–4332 (2001)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Xiang, T., Yuan, R.: A class of expansive-type Krasnosel’skii fixed point theorem. Nonlinear Anal. 71, 3229–3239 (2009)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Xiang, T., Yuan, R.: Critical type of Krasnosel’skii fixed point theorem. Proc. Am. Math. Soc. 139, 1033–1044 (2011)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Xiang, T.: Notes on expansive mappings and a partial answer to Nirenberg’s problem, Electron. J. Differential Equations, No. 02, 16 pp (2013)

  38. 38.

    Xiang, T., Yuan, R.: A note on Krasnosel’skii fixed point theorem, Fixed Point Theory Appl., 99, 8 pp (2015)

  39. 39.

    Xiang, T., Georgiev, S.: Noncompact-type Krasnoselskii fixed-point theorems and their applications. Math. Methods Appl. Sci. 39, 833–863 (2016)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Zhang, G., Sun, J.: A generalization of the cone expansion and compression fixed theorem and applications. Nonlinear Anal. 67, 579–586 (2007)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

We greatly appreciate the four anonymous referees for giving positive and valuable comments from different perspectives, which further helped us to improve the exposition of this work. TX is funded by NSF of China (No. 11601516 and 11871226) and the Research Funds of Renmin University of China (No. 2018030199).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Tian Xiang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Xiang, T., Zhu, D. Cone expansion and cone compression fixed point theorems for sum of two operators and their applications. J. Fixed Point Theory Appl. 22, 49 (2020). https://doi.org/10.1007/s11784-020-00786-5

Download citation

Keywords

  • Cone
  • contraction
  • expansion
  • cone fixed point theorems
  • positive solution
  • integral equation

Mathematics subject classification

  • 47H10
  • 31B10
  • 58J20