BibTex RIS Cite

Common fixed point results for a banach operator pair in cat(0) spaces with applications

Year 2017, Volume: 66 Issue: 2, 195 - 204, 01.08.2017
https://doi.org/10.1501/Commua1_0000000811

Abstract

In this paper, sufficient conditions for the existence of a common fixed point for a Banach operator pair of mappings satisfying generalized contractive conditions in the frame work of CAT(0) spaces are obtained. As an application, related results on best approximation are derived. Our results generalize various known results in contemporary literature

References

  • Akbar, F. and N. Sultana, Dotson’s convexity, Banach operator pair and weak contractions, Bull. Inst. Math. Acad.Sinica, 6(1)(2011), 85-95.
  • Al-Thaga…, M. A., Common …xed points and best approximation, J. Approx. Theory, 85 (1996), 318-320.
  • Beg, I., D. R. Sahu and S. D. Diwan, Approximation of …xed points of uniformly R- subweakly commuting mappings, J. Math. Anal. Appl., 324(2006), 1105-1114.
  • Bridson, M. and A. Hae*iger, Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, Heidelberg, 1999.
  • Bruhat, F. and J. Tits, Groupes réductifs sur un corps local. I. Données radicielles valuées, Inst. Hautes Études Sci. Publ. Math., 41 (1972) 5–251.
  • Burago, D., Y. Burago and S. Ivanov, A Course in Metric Geometry, in: Graduate Studies in Math., vol. 33, Amer. Math. Soc., Providence, RI, 2001.
  • Chen, J. and Z. Li, Common …xed points for Banach operator pairs in best approximation, J. Math. Anal. Appl., (2007) (in press). Dhompongsa, S.
  • W.A. Kirk, B. Sims, Fixed points of uniformly lipschitzian mappings, Nonlinear Anal. 65 (2006) 762–772.
  • Dhompongsa, S. and B.Panyanak, On 4 put. Math.Appl.56 (2008) 2572–2579.
  • Convergence Theorems in CAT(0) Spaces, Com- Dotson, W. J. Jr., Fixed point theorems for nonexpansive mappings on star-shaped subsets of Banach spaces, J. London Math. Soc., 4(1972), 408-410.
  • Goebel, K. and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Map- pings, Marcel Dekker, Inc., New York, 1984.
  • Goebel, K. and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge 1990.
  • Gromov, M., Metric structure for Riemannian and Non-Riemannian spaces, Progr. Math., vol. 152, Birkhauser, Boston,1984.
  • Habiniak, L., Fixed point theorems and invariant approximation, J. Approx. Theory 56(1989), 244.
  • Jungck, G. and S. Sessa, Fixed point theorems in best approximation theory, Math. Japon., (1995), 249-252.
  • Jungck, G. and N. Hussain, Compatible maps and invariant approximations, J. Math. Anal. Appl., 325(2007), 1003-1012.
  • Khan, A. R., N. Hussain and A. B. Thaheem, Applications of …xed point theorems to invariant approximation, Approx. Theory and Appl. 16(2000), 48-55.
  • Khan L. A., and A. R. Khan, An extension of Brosowski-Meinardus theorem on invariant approximations, Approx. Theory and Appl., 11(1995), 1-5.
  • Kirk, W. A., Geodesic geometry and …xed point theory, in Seminar of Mathematical Analysis (Malaga/ Seville, 2002/2003), in: Colecc. Abierta, vol., 64, Univ. Sevilla Secr. Publ., Sevilla, , 195-225.
  • Kirk, W.A., B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008) 3689–3696.
  • Lim, T. C., Remarks on some …xed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179
  • Meinardus, G., Invarianz bei linearn Approximation, Arch. Rat. Mech. Anal., 14 (1963), 303.
  • Reich S., and I. Shafrir, Nonexpansive iterations in hyperbolic space, Nonlinear Anal., 15 (1990) 537–558.
  • Sahab, S. A., M. S. Khan and S. Sessa, A result in best approximation theory, J. Approx. Theory 55(1988), 349-351.
  • Singh, S. P., Application of a …xed point theorem to approximation theory, J. Approx. Theory, (1979), 88-90.
  • Subrahmanyam, P. V., An application of a …xed point theorem to best approximation, J. Approx. Theory 20(1977), 165-172.
  • Hussain, N., Asymptotically Pseudocontractions, Banach Operator Pairs and Best Simultane- ous Approximations, Fixed Point Theory and Applications Volume 2011, Article ID 812813, pages doi:10.1155/2011/812813
  • Hussain, N., M.A Khamsi and A Latif, Banach operator pairs and common …xed points in modular function spaces, Fixed Point Theory and Applications 2011, 2011:75
  • Current address : Safeer Hussain Khan: Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar.
  • E-mail address : safeer@qu.edu.qa; safeerhussain5@yahoo.com Current address : Mujahid Abbas: Department of Mathematics, University of Management and Technology, C-II, Johar Town, Lahore, Pakistan. and Department of Mathematics, University of Jeddah, P. O. Box 80327, Jeddah 21589, Saudi Arabia.
  • E-mail address : mujahid@lums.edu.pk
Year 2017, Volume: 66 Issue: 2, 195 - 204, 01.08.2017
https://doi.org/10.1501/Commua1_0000000811

Abstract

References

  • Akbar, F. and N. Sultana, Dotson’s convexity, Banach operator pair and weak contractions, Bull. Inst. Math. Acad.Sinica, 6(1)(2011), 85-95.
  • Al-Thaga…, M. A., Common …xed points and best approximation, J. Approx. Theory, 85 (1996), 318-320.
  • Beg, I., D. R. Sahu and S. D. Diwan, Approximation of …xed points of uniformly R- subweakly commuting mappings, J. Math. Anal. Appl., 324(2006), 1105-1114.
  • Bridson, M. and A. Hae*iger, Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, Heidelberg, 1999.
  • Bruhat, F. and J. Tits, Groupes réductifs sur un corps local. I. Données radicielles valuées, Inst. Hautes Études Sci. Publ. Math., 41 (1972) 5–251.
  • Burago, D., Y. Burago and S. Ivanov, A Course in Metric Geometry, in: Graduate Studies in Math., vol. 33, Amer. Math. Soc., Providence, RI, 2001.
  • Chen, J. and Z. Li, Common …xed points for Banach operator pairs in best approximation, J. Math. Anal. Appl., (2007) (in press). Dhompongsa, S.
  • W.A. Kirk, B. Sims, Fixed points of uniformly lipschitzian mappings, Nonlinear Anal. 65 (2006) 762–772.
  • Dhompongsa, S. and B.Panyanak, On 4 put. Math.Appl.56 (2008) 2572–2579.
  • Convergence Theorems in CAT(0) Spaces, Com- Dotson, W. J. Jr., Fixed point theorems for nonexpansive mappings on star-shaped subsets of Banach spaces, J. London Math. Soc., 4(1972), 408-410.
  • Goebel, K. and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Map- pings, Marcel Dekker, Inc., New York, 1984.
  • Goebel, K. and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge 1990.
  • Gromov, M., Metric structure for Riemannian and Non-Riemannian spaces, Progr. Math., vol. 152, Birkhauser, Boston,1984.
  • Habiniak, L., Fixed point theorems and invariant approximation, J. Approx. Theory 56(1989), 244.
  • Jungck, G. and S. Sessa, Fixed point theorems in best approximation theory, Math. Japon., (1995), 249-252.
  • Jungck, G. and N. Hussain, Compatible maps and invariant approximations, J. Math. Anal. Appl., 325(2007), 1003-1012.
  • Khan, A. R., N. Hussain and A. B. Thaheem, Applications of …xed point theorems to invariant approximation, Approx. Theory and Appl. 16(2000), 48-55.
  • Khan L. A., and A. R. Khan, An extension of Brosowski-Meinardus theorem on invariant approximations, Approx. Theory and Appl., 11(1995), 1-5.
  • Kirk, W. A., Geodesic geometry and …xed point theory, in Seminar of Mathematical Analysis (Malaga/ Seville, 2002/2003), in: Colecc. Abierta, vol., 64, Univ. Sevilla Secr. Publ., Sevilla, , 195-225.
  • Kirk, W.A., B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008) 3689–3696.
  • Lim, T. C., Remarks on some …xed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179
  • Meinardus, G., Invarianz bei linearn Approximation, Arch. Rat. Mech. Anal., 14 (1963), 303.
  • Reich S., and I. Shafrir, Nonexpansive iterations in hyperbolic space, Nonlinear Anal., 15 (1990) 537–558.
  • Sahab, S. A., M. S. Khan and S. Sessa, A result in best approximation theory, J. Approx. Theory 55(1988), 349-351.
  • Singh, S. P., Application of a …xed point theorem to approximation theory, J. Approx. Theory, (1979), 88-90.
  • Subrahmanyam, P. V., An application of a …xed point theorem to best approximation, J. Approx. Theory 20(1977), 165-172.
  • Hussain, N., Asymptotically Pseudocontractions, Banach Operator Pairs and Best Simultane- ous Approximations, Fixed Point Theory and Applications Volume 2011, Article ID 812813, pages doi:10.1155/2011/812813
  • Hussain, N., M.A Khamsi and A Latif, Banach operator pairs and common …xed points in modular function spaces, Fixed Point Theory and Applications 2011, 2011:75
  • Current address : Safeer Hussain Khan: Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar.
  • E-mail address : safeer@qu.edu.qa; safeerhussain5@yahoo.com Current address : Mujahid Abbas: Department of Mathematics, University of Management and Technology, C-II, Johar Town, Lahore, Pakistan. and Department of Mathematics, University of Jeddah, P. O. Box 80327, Jeddah 21589, Saudi Arabia.
  • E-mail address : mujahid@lums.edu.pk
There are 31 citations in total.

Details

Primary Language English
Journal Section Research Articles
Authors

Safeer Hussain Khan This is me

Mujahid Abbas This is me

Publication Date August 1, 2017
Published in Issue Year 2017 Volume: 66 Issue: 2

Cite

APA Khan, S. H., & Abbas, M. (2017). Common fixed point results for a banach operator pair in cat(0) spaces with applications. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 66(2), 195-204. https://doi.org/10.1501/Commua1_0000000811
AMA Khan SH, Abbas M. Common fixed point results for a banach operator pair in cat(0) spaces with applications. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. August 2017;66(2):195-204. doi:10.1501/Commua1_0000000811
Chicago Khan, Safeer Hussain, and Mujahid Abbas. “Common Fixed Point Results for a Banach Operator Pair in cat(0) Spaces With Applications”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 66, no. 2 (August 2017): 195-204. https://doi.org/10.1501/Commua1_0000000811.
EndNote Khan SH, Abbas M (August 1, 2017) Common fixed point results for a banach operator pair in cat(0) spaces with applications. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 66 2 195–204.
IEEE S. H. Khan and M. Abbas, “Common fixed point results for a banach operator pair in cat(0) spaces with applications”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 66, no. 2, pp. 195–204, 2017, doi: 10.1501/Commua1_0000000811.
ISNAD Khan, Safeer Hussain - Abbas, Mujahid. “Common Fixed Point Results for a Banach Operator Pair in cat(0) Spaces With Applications”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 66/2 (August 2017), 195-204. https://doi.org/10.1501/Commua1_0000000811.
JAMA Khan SH, Abbas M. Common fixed point results for a banach operator pair in cat(0) spaces with applications. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2017;66:195–204.
MLA Khan, Safeer Hussain and Mujahid Abbas. “Common Fixed Point Results for a Banach Operator Pair in cat(0) Spaces With Applications”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 66, no. 2, 2017, pp. 195-04, doi:10.1501/Commua1_0000000811.
Vancouver Khan SH, Abbas M. Common fixed point results for a banach operator pair in cat(0) spaces with applications. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2017;66(2):195-204.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.