Skip to main content
SpringerLink
Log in

Common fixed point results for generalized \(\alpha _{*}\)-\(\psi \)-contraction multivalued mappings in b-metric spaces

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

Abstract

The purpose of this paper was to introduce the notion of generalized \( \alpha _{*}\)-\(\psi \)-Geraghty contraction type for multivalued mappings and establish common fixed point theorems for such multivalued mappings in an \(\alpha \)-complete b-metric spaces. We give an example for supporting these results. Moreover, an application of our main result to the existence of solutions to systems of ordinary differential equations is also presented.

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. Felhi, A., Sahmim, S., Aydi, H.: Ulam–Hyers stability and well-posedness of fixed point problems for \(\alpha \)-\(\lambda \)-contractions on quasi b-metric spaces. Fixed Point Theory Appl. 2016, 1 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  2. Reich, S.: Some remarks concerning contraction mappings. Can. Math. Bull. 14, 121–124 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  3. Reich, S.: Fixed points of contractive functions. Boll. Un. Mat. Ital. 5, 26–42 (1972)

    MATH  MathSciNet  Google Scholar 

  4. Cho, S.H., Bae, J.S., Karapinar, E.: Fixed point theorems for \( \alpha \)-Geraghty contraction type maps in metric spaces. Fixed Point Theory Appl., 11 (2013)

  5. Chuadchawna, P., Kaewcharoen, A., Plubtieng, S.: Fixed point theorems for generalized \(\alpha \)-\(\eta \)-\(\psi \)-Geraghty contraction type mappings in \(\alpha \)-\(\eta \)-complete metric spaces. J. Nonlinear Sci. Appl. 9, 471–485 (2016)

    MATH  MathSciNet  Google Scholar 

  6. Karapinar, E.: \(\alpha \)-\(\psi \)-Geraghty contraction type mappings and some related fixed point results. Filomat 28, 37–48 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gupta, V., Shatanawi, W., Mani, N.: Fixed point theorems for \( (\Psi,\beta )\)-Geraghty contraction type maps in ordered metric spaces and some applications to integral and ordinary differential equations. J. fixed point theory appl. (2016). doi:10.1007/s11784-016-0303-2

    MATH  Google Scholar 

  8. Aydi, H., Bota, M.F., Karapinar, E., Moradi, S.: A common fixed point for weak \(\phi \)-contractions on b-metric spaces. Fixed Point Theory 13(2), 337–346 (2012)

    MATH  MathSciNet  Google Scholar 

  9. Aydi, H., Karapinar, E., Bota, M.F., Mitrović, S.: A fixed point theorem for set-valued quasi-contractions in b-metric spaces. Fixed Point Theory Appl. 2012, 88 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Aydi, H., Felhi, A., Sahmim, S.: Common fixed points in rectangular b-metric spaces using (E:A) property. J. Adv. Math. Stud. 8(2), 159–169 (2015)

    MATH  MathSciNet  Google Scholar 

  11. Arshad, M., Ameer, E., Karapinar, E.: Generalized contractions with triangular \(\alpha \)-orbital admissible mapping on Branciari metric spaces. J Inequal. Appl. 2016, 63 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jleli, M., Samet, B., Vetro, C., Vetro, F.: Fixed points for multivalued mappings in b-metric spaces. Abstr. Appl. Anal., 7 (Article ID 718074) (2015)

  13. Jovanović, M., Kadelburg, Z., Radenović, S.: Common XED point results in metric-type spaces. Fixed Point Theory Appl. 15 (Article ID 978121) (2010)

  14. Huang, H., Xu, S.: Fixed point theorems of contractive mappings in cone b-metric spaces and applications. Fixed Point Theory Appl. 2013, 112 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hussian, N., Shah, M.H.: KKM mappings in cone b-metric spaces. Comput. Math. Appl. 62, 1677–1684 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hussain, N., Kutbi, M.A., Salimi, P.: Fixed point theory in \( \alpha \)-complete metric space with applications. Abstr. Appl. Anal., 11 1,1, 1.11, 1.13 (2014)

  17. Roshan, J.R., Parvaneh, V., Sedghi, Sh, Shobkolaei, N., Shatanawi, W.: Common xed points of almost generalized \(\left( \psi -\varphi \right) _{s}\)-contraction mappings in ordered b-metric spaces. Fixed Point Theory Appl. 2013, 159 (2013)

    Article  MATH  Google Scholar 

  18. Shahkoohi, R.J., Razani, A.: Some fixed point theorems for rational Geraghty contractive mappings in ordered b-metric spaces. J. Ineq. Appl. 2014, 373 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  19. Shi, L., Xu, S.: Common fixed point theorems for two weakly compatible self-mappings in cone b-metric spaces. Fixed Point Theory Appl. 2013, 120 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  20. Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux equations itegrales. Fund. Math. 3, 133–181 (1922)

    Article  MATH  Google Scholar 

  21. Geraghty, M.: On contractive mappings. Proc. Am. Math. Soc. 40, 604–608 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  22. Czerwik, S.: Nonlinear set-valued contraction mappings in b-metric spaces. Atti Semin. Mat. Fis. Univ. Modena. 46(2), 263–276 (1998)

    MATH  MathSciNet  Google Scholar 

  23. Czerwik, S.: Contraction mappings in b-metric spaces. Acta Math. Inf. Univ. Ostrav. 1, 5–11 (1993)

    MATH  MathSciNet  Google Scholar 

  24. Salimi, P., Latif, A., Hussain, N.: Modified \(\alpha -\psi -\) contractive mappings with applications. Fixed Point Theory Appl. 2013, 151 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  25. Karapınar, E., Kumam, P., Salimi, P.: On \(\alpha \)-\(\psi \)-Meir–Keeler contractive mappings. Fixed Point Theory Appl. 2013, 94 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  26. Berzig, M., Karapınar, E.: On modified \(\alpha \)–\(\psi \)-contractive mappings with application. Thai. J. Math. 13(1), 147–152 (2015)

    MATH  MathSciNet  Google Scholar 

  27. Mohammadi, B., Rezapour, Sh, Shahzad, N.: Some results of fixed point of \(\alpha \)–\(\psi \)-quasi-contractive multifunctions. Fixed Point Theory Appl. 2013, 112 (2013)

    Article  Google Scholar 

  28. Nadler, S.B.: Multivalued contraction mappings. Pac. J. Math. 30, 475–488 (1969)

    Article  MATH  Google Scholar 

  29. Popescu, O.: Some new fixed point theorems for \(\alpha \)-Geraghty contraction type maps in metric spaces. Fixed Point Theory Appl. 2014, 190 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  30. Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for \( \alpha \)–\(\psi \)-contractive type mappings. Nonlinear Anal. 75, 2154–2165 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Shatanawi, W., Hani, M.B.: A fixed point theorem in b-metric spaces with nonlinear contractive condition. Far East J. Math. Sci. 100, 1901–1908 (2016)

    MATH  Google Scholar 

  32. Shatanawi, W., Hani, M.B.: A coupled fixed point theorem in b-metric spaces. Int. J. Pure Appl. Math. 109, 889–897 (2016)

    Article  MATH  Google Scholar 

  33. Shatanawi, W., Pitea, B., Lazovic, R.: Contraction conditions using comparison functions on b-metric spaces. Fixed Point Theory Appl. 2014(135), 1–11 (2014)

    MATH  MathSciNet  Google Scholar 

  34. Shatanawi, W.: Fixed and common fixed point for mappings satisfying nonlinear contractive in \(b\)-metric spaces. J. Math. Anal. 7(4), 1–12 (2016)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the reviewer for his/her valuable remarks which improved the paper in nice form.

Author information

Authors and Affiliations

  1. Department of Mathematics, Taiz University, Taiz, Yemen

    Eskandar Ameer

  2. Department of Mathematics, International Islamic University, H-10, 44000, Islamabad, Pakistan

    Eskandar Ameer & Muhammad Arshad

  3. Department of General Sciences, Prince Sultan University, Riyadh, Saudi Arabia

    Wasfi Shatanawi

  4. Department of Mathematics, Hashemite University, Zarqa, Jordan

    Wasfi Shatanawi

Authors
  1. Eskandar Ameer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Muhammad Arshad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wasfi Shatanawi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eskandar Ameer.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ameer, E., Arshad, M. & Shatanawi, W. Common fixed point results for generalized \(\alpha _{*}\)-\(\psi \)-contraction multivalued mappings in b-metric spaces. J. Fixed Point Theory Appl. 19, 3069–3086 (2017). https://doi.org/10.1007/s11784-017-0477-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11784-017-0477-2

Keywords

Mathematics Subject Classification

Search

Navigation