Skip to main content
SpringerLink
Log in

Fixed point theorems in partially ordered Banach spaces with applications to nonlinear fractional evolution equations

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, a notion of partially Hausdorff measure of noncompactness in partially ordered Banach spaces is introduced, and some Krasnoselskii-type fixed point theorems under certain mixed conditions are proved. Some applications of the obtained fixed point theorems are given to a class of fractional hybrid evolution equations for proving the existence of mild solutions under certain monotonicity conditions. At the end, an example of the fractional parabolic equation is given to illustrate the abstract results.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Appell, J.: Measure of noncompactness, condensing operators and fixed points: an application-oriented survey. Fixed Point Theory 6, 157–229 (2005)

    MathSciNet  MATH  Google Scholar 

  2. Bashiri, T., Vaezpour, S., Park, C.: A coupled fixed point theorem and application to fractional hybrid differential problems. Fixed Point Theory Appl. 2016, 23 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berinde, V.: Coupled fixed point theorems for \(\phi \)-contractive mixed monotone mapping in partially ordered metric spaces. Nonliear Anal. 75, 3218–3228 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dhage, B.: On extension of Tarski’s fixed point theorem and applications. Pure Appl. Math. Sci. 25, 37–42 (1987)

    MathSciNet  MATH  Google Scholar 

  5. Dhage, B.: Some fixed point theorems for in ordered Banach spaces and applications. Math. Stud. 61, 81–88 (1992)

    MathSciNet  MATH  Google Scholar 

  6. Dhage, B.: Fixed point theorems in ordered Banach algebras and applications. J. Panam. Math. 9, 93–102 (1999)

    MathSciNet  MATH  Google Scholar 

  7. Dhage, B.: A nonlinear alternative with applications to nonlinear pertrubed differential equations. Nonlinar Stud. 13, 343–354 (2006)

    MATH  Google Scholar 

  8. Dhage, B.: Hybrid fixed point theory in partially ordered normed linear spaces and asplications to fractional integral equations. J. Differ. Equ. Appl. 2, 155–184 (2013)

    MATH  Google Scholar 

  9. Dhage, B.: Partially continuous mappings in partially ordered normed linear spaces and applications to functional integral equations. Tamkang J. Math. 45, 397–426 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dhage, B., Dhage, S., Ntouyas, S.: Approximating solutions of nonlinear hybrid differential equations. Appl. Math. Lett. 34, 76–80 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guo, D., Lakshmikantham, V., Liu, X.: Nonlinear Integral Equations in Abstract Spaces. Mathematics and Its Applications, vol. 373. Kluwer Academic Publishers, Dordrecht (1996)

  12. Harjani, J., Sadarangani, K.: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 72, 1188–1197 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Heinz, H.: On the behavior of measure of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlinear Anal. 7, 1351–1371 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kamenskii, M., Obukhovskii, V., Zecca, P.: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. De Gruyter, Berlin (2001)

    Book  MATH  Google Scholar 

  15. Krasnoselskii, M.: Topological Methods in the Theory of Nonlinear Integral Equations. Macmillan, New York (1964)

  16. Li, Y.X.: Existence and asymptotic stability of periodic solution for evolution equations with delays. J. Funct. Anal. 261, 1309–1324 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liang, J., Yang, H.: Controllability of fractional integro-differential evolution equations with nonlocal conditions. Appl. Math. Comput. 254, 20–29 (2015)

    MathSciNet  Google Scholar 

  18. Nashine, H., Samet, B.: Fixed point results for mappings satisfying \((\psi, \varphi )\)-weakly contractive conditions in partially orderen metric spaces. Nonlinear Anal. 74, 2201–2209 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nieto, J., Rodríguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223–239 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nieto, J., Rodríguez-López, R.: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Eng. Ser. 23, 2205–2212 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pathak, H., Rodríguez-López, R.: Existence and approximation of solutions to nonlinear hybrid ordinary differential equations. Appl. Math. Lett. 39, 101–106 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  23. Ran, A., Reurings, M.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Reich, S.: Fixed points in locally convex spaces. Math. Z. 125, 17–31 (1972)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. 11, 4465–4475 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The first author is thankful to the NSFC (No. 11261053) and the third author is thankful to the United State-India Education Foundation, New Delhi, India and IIE/CIES, Washington, DC, USA on selection for Fulbright-Nehru PDF Award (No. 2052/FNPDR/2015).

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    He Yang

  2. Department of Mathematics, Texas A& M University-Kingsville, Kingsville, TX 78363, USA

    Ravi P. Agarwal & Hemant K. Nashine

  3. College of Science, Gansu Agricultural University, Lanzhou, 730070, People’s Republic of China

    Yue Liang

Authors
  1. He Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ravi P. Agarwal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hemant K. Nashine
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yue Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to He Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, H., Agarwal, R.P., Nashine, H.K. et al. Fixed point theorems in partially ordered Banach spaces with applications to nonlinear fractional evolution equations. J. Fixed Point Theory Appl. 19, 1661–1678 (2017). https://doi.org/10.1007/s11784-016-0316-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11784-016-0316-x

Keywords

Mathematics Subject Classification

Search

Navigation