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Mappings of Generalized Condensing Type in Metric Spaces with Busemann Convex Structure

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Abstract

Related to earlier work on existence results of best proximity points (pairs) for cyclic (noncyclic) condensing operators of integral type in the setting of reflexive Busemann convex spaces, in this paper, we introduce another class of cyclic (noncyclic) condensing operators and study the existence of best proximity points (pairs) as well as coupled best proximity points (pairs) in such spaces. Then, we present an application of our main existence result to study the existence of an optimal solution for a system of differential equations.

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References

  1. Abkar, A., Gabeleh, M.: Best proximity points for asymptotic cyclic contraction mappings. Nonlinear Anal. 74, 7261–7268 (2011)

    Article  MathSciNet  Google Scholar 

  2. Akhmerov, R.R., Kamenskii, M.I., Potapov, A.S., Rodkina, A.E., Sadovskii, B.N.: Measures of Noncompactness and Condensing Operators, vol. 55. Birkhauser, Basel (1992)

    Book  Google Scholar 

  3. Al-Thagafi, M.A., Shahzad, N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70, 3665–3671 (2009)

    Article  MathSciNet  Google Scholar 

  4. Amini-Harandi, A.: Fixed and coupled fixed points of a new type set-valued contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012(215), 1–7 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Ahuja, G.C., Narang, T.D., Trehan, S.: Best approximation on convex sets in a metric space. J. Approx. Theory 12, 94–97 (1974)

    Article  MathSciNet  Google Scholar 

  6. Anuradha, J., Veeramani, P.: Proximal pointwise contraction. Topol. Appl. 156, 2942–2948 (2009)

    Article  MathSciNet  Google Scholar 

  7. Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Springer, Berlin, Heidelberg (1999)

    Book  Google Scholar 

  8. Busemann, H.: Geometry of Geodesics. Academic Press, New York (1955)

    MATH  Google Scholar 

  9. Eldred, A.A., Kirk, W.A., Veeramani, P.: Proximal normal structure and relatively nonexpansive mappings. Studia Math. 171, 283–293 (2005)

    Article  MathSciNet  Google Scholar 

  10. Eldred, A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323, 1001–1006 (2006)

    Article  MathSciNet  Google Scholar 

  11. Espínola, R.: A new approach to relatively nonexpansive mappings. Proc. Am. Math. Soc. 136, 1987–1996 (2008)

    Article  MathSciNet  Google Scholar 

  12. Espínola, R., Nicolae, A.: Mutually nearest and farthest points of sets and the drop theorem in geodesic spaces. Monatsh. Math. 165, 173–197 (2012)

    Article  MathSciNet  Google Scholar 

  13. Espínola, R., Pia̧tek, B.: Fixed point property and unbounded sets in CAT(0) spaces. J. Math. Anal. Appl. 408, 638–654 (2013)

    Article  MathSciNet  Google Scholar 

  14. Espínola, R., Madiedo, O., Nicolae, A.: Borsuk-Dugundji type extensions theorems with Busemann convex target spaces. Ann. Acad. Sci. Fennicae Math. 43, 225–238 (2018)

    Article  MathSciNet  Google Scholar 

  15. Fan, K.: Extensions of two fixed point theorems of F.E. Browder. Math. Z. 122, 234–240 (1969)

    Article  MathSciNet  Google Scholar 

  16. Gabeleh, M., Markin, J.: Optimum solutions for a system of differential equations via measure of noncompactness. Indag. Math. 29, 895–906 (2018)

    Article  MathSciNet  Google Scholar 

  17. Gabeleh, M., Künzi, H.P.: Condensing operators of integral type in Busemann reflexive convex spaces. Bull. Malays. Math. Sci. Soc. (2019). https://doi.org/10.1007/s40840-019-00785-x

  18. Gabeleh, M.: Best proximity points for cyclic mappings, Ph.D Thesis (2012)

  19. Gabeleh, M., Moshomoa, S.P., Vetro, C.: Solvability of a system of integral equations and coupled best proximity points, Preprint (2019)

  20. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)

    MATH  Google Scholar 

  21. Samadi, A.: Applications of measure of noncompactness to coupled fixed points and system of integral equations. Miskolc Math. Notes 19, 537–553 (2018)

    Article  MathSciNet  Google Scholar 

  22. Istratescu, V.I.: Fixed Point Theory. Reidel Publishing Company, Boston (1981)

    Book  Google Scholar 

  23. Leon, A.Fernandez, Nicolae, A.: Best proximity pair results relatively nonexpansive mappings in geodesic spaces. Numer. Funct. Anal. Optim. 35, 1399–1418 (2014)

    Article  MathSciNet  Google Scholar 

  24. Toledano, M.A., Benavides, T.D., Acedo, G.L.: Measures of noncompactness in metric fixed point theory. In: Operator Theory: Advances and Applications, vol. 99, Birkhäuser, Basel (1997)

  25. Wardowski, D.: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, 1–6 (2006)

    MathSciNet  Google Scholar 

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Acknowledgements

The second author acknowledges partial support from the National Research Foundation of South Africa under Grant 114773.

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Authors and Affiliations

  1. Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran

    Moosa Gabeleh

  2. Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701, South Africa

    Hans-Peter A. Künzi

Authors
  1. Moosa Gabeleh
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  2. Hans-Peter A. Künzi
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Correspondence to Hans-Peter A. Künzi.

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Communicated by Ali Abkar.

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Gabeleh, M., Künzi, HP.A. Mappings of Generalized Condensing Type in Metric Spaces with Busemann Convex Structure. Bull. Iran. Math. Soc. 46, 1465–1483 (2020). https://doi.org/10.1007/s41980-019-00336-x

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  • DOI: https://doi.org/10.1007/s41980-019-00336-x

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