SOME FIXED POINT RESULTS IN THE GENERALIZED CONVEX METRIC SPACES

Year 2020, Volume: 10 Issue: 1, 11 - 23, 01.01.2020

K. Dogan F. Gursoy V. Karakaya

Abstract

In this study, we introduce a new three step iteration process and show that the iteration process converges to the unique xed point by two theorems under different conditions of contractive mappings on the generalized G- convex metric spaces. Also, we investigate data dependence result for this iterative process in the generalized G- convex metric spaces.

Keywords

Convex G-metric spaces, G-convergence, Fixed point iteration process.

References

• Takahashi, W., A convexity in metric space and nonexpansive mappings I, Kodai Math. Sem. Rep. 22 (1970), 142–149.
• Tian, Y-X., Convergence of an Ishikawa type iterative scheme for asymptotically quasi-nonexpansive mappings, Comput. Math. Appl. 49 (2005) 1905–1912.
• Mustafa, Z., Sims, B., “Some remarks concerning D-metric spaces,” in International Conference on Fixed Point Theory and Applications, pp. 189–198, Yokohama, Yokohama, Japan, 2004.9.
• Mustafa, Z., Obiedat, H., Awawdeh, F., Some of fixed point theorem for mapping on complete G- metric spaces, Fixed Point Theory Appl., 2008(2008), Article ID 189870,page 12.
• Mustafa, Z., A new structure for generalized metric spaces with applications to FIxed point theory, Ph.D. thesis, University of Newcastle, Newcastle, UK, 2005.
• Mustafa, Z., Shatanawi, W. and Bataineh, M., Fixed point theorems on uncomplete G-metric spaces, J. Math. Stat. 4(4)(2008), 196-201.
• Mustafa, Z., Shatanawi, W., Bataineh, M., Existence of fixed point result in G-metric spaces, Int. J. Math. Math. Sci. 2009(2009), page 10, Article ID 283028.
• Mustafa, Z. and Sims, B., Fixed point theorems for contractive mappings in complete G-metric space, Fixed Point Theory Appl. 2009(2009), page 10, Article ID 917175.
• Mustafa, Z. and Sims, B., “A new approach to generalized metric spaces,” Journal of Nonlinear and Convex Analysis, vol. 7, no. 2, pp. 289–297, 2006.
• Rafik, A., Fixed Points of Ciric Quasi-contractive Operators in Generalized Convex Metric Spaces,General Mathematics Vol. 14, No. 3 (2006), 79–90.
• Dogan, K., Karakaya, V., On the Convergence and Stability Results for a New General Iterative Process. The Scientific World Journal, 2014.
• Karakaya, V., Dogan, K., Gursoy, F., Erturk, M., Fixed point of a new three-step iteration algorithm under contractive-like operators over normed spaces. In Abstract and Applied Analysis (Vol. 2013), 2013.
• Gursoy, F. and Karakaya, V., A Picard-S hybrid type iteration method for solving a differential equation with retarded argument. arXiv preprint arXiv:1403.2546, 2014.
• Mann, W. R., Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506-510.
• Ishikawa, S., Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974) 147-150.
• Noor, M. A., New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000) 217-229.
• Weng, X., Fixed point iteration for local strictly pseudo-contractivemapping, Proceedings of the Amer- ican Mathematical Society, 113(1991), 727-731.
• Gahler, S., “2-metrische Raume und ihre topologische Struktur,” Mathematische Nachrichten, vol. 26, pp. 115–148, 1963.
• Gahler, S., “Zur geometric 2-metriche raume,” Revue Roumaine de Math´ematiques Pures et Ap- pliquees, vol. 40, pp. 664–669, 1966.
• Ha, K.S., Cho, Y.J., White, A., “Strictly convex and strictly 2-convex 2-normed spaces,”Mathematica Japonica, vol. 33, no. 3, pp. 375–384, 1988.
• Dhage, B. C., “Generalized metric space and mapping with fixed point,” Bulletin of the Calcutta Mathematical Society, vol. 84, pp. 329–336, 1992. 5.
• Dhage, B. C., “Generalized metric spaces and topological structure.I,”Analele Stiintificeale Universi- tatii Al. I. Cuza din Iasi. Serie Noua. Matematica, vol. 46, no. 1, pp. 3–24, 2000. 6.
• Dhage, B. C., “On generalized metric spaces and topological structure. II,” Pure and Applied Math- ematika Sciences, vol. 40, no. 1-2, pp. 37–41, 1994. 7.
• Dhage, B. C., “On continuity of mappings in D-metric spaces,” Bulletin of the Calcutta Mathematical Society, vol. 86, no. 6, pp. 503–508, 1994. 8.
• Reich, S., “Some remarks concerning contraction mappings,” Canadian Mathematical Bulletin, vol. 14, pp. 121–124, 1971. 11
• Bianchini, R. M. T., “Su un problema di S. Reich riguardante la teoria dei punti fissi,” Bollettino dell’Unione Matematica Italiana, vol. 5, no. 4, pp. 103–108, 1972. 12.
• ´Ciric, Lj. B., “A generalization of Banach’s contraction principle,” Proceedings of the American Mathematical Society, vol. 45, pp. 267–273, 1974. 13.
• Chatterjea, S. K., “Fixed-point theorems,” Doklady Bolgarsko Akademii Nauk. Comptes Rendus de l Academie Bulgare des Sciences, vol. 25, pp. 727–730, 1972.
• Thangavelu P., Shyamala Malini S., Jeyanthi, P., Convexity in D-Metric Spaces and its applications to fixed point theorems, International Journal of Statistika and Mathematika, Vol. 2, Issue 3, (2012), 05-12.
• Modi, G., Bhatt, B., ”Fixed Point Results for Weakly Compatible Mappings in Convex G-Metric Space”, International Journal Of Mathematics And Statistics Invention, Volume 2 Issue 11, 2014, pp 34-38.
• S oltuz SM, Grosan T., Data dependence for Ishikawa iteration when dealing with contractive like operators., Fixed Point Theory A., 2008, 2008, 1-7.