Convergence of CR-iteration procedure for a nonlinear quasi contractive map in convex metric spaces

Document Type : Original Article


Department of Mathematics, Andhra University, Visakhapatnam-530 003, India


We prove that the modified CR-iteration procedure converges strongly to a fixed point
of a generalized quasi contraction map in convex metric spaces which is the main result
of this paper. The convergence of Picard-S iteration procedure follows as a corollary to
our main result.


[1] M. Bridson and A. Hae iger, Metric spaces of non-positive curvature, Springer-Verlag, Berlin, Heidelberg, New-York, 1999.
[2] R. Chugh, V. Kumar, and S. Kumar, Strong convergence of a new three step iterative scheme in Banach spaces, Amer. J. Compu. Math., 2 (2012) 345-357.
[3] L. B. Ciric, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc., 45(2) (1974), 267-273.
[4] L. B. Ciric, Convergence theorems for a sequence of Ishikawa iterations for nonlinear quasi contractive mappings, Indian J. pure appl. Math., 30(4), (1999) 425-433.
[5] X. P. Ding, Iteration process for Nonlinear mappings in Convex metric spaces, J. Math. Anal. Appl., 132, (1988), 114-122. 
[6] F. Gurusoy and V. Karakaya, A Picad-S Hybrid type iteration method for solving a differential equation with retarted argument, arXiv:1403.2546v2.
[7] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147-150.
[8] M. Moosaei, Fixed point theorems in convex metric spaces, Fixed Point Theory and Appl., Vol. 2012, article 164, (2012) 6 pages.
[9] Renu Chugh, Preety Malik, Convergence and fi xed point theorems in convex metric spaces: a survey, International Journal of Applied Mathematical Research, 3(2) (2014)133-160. 
[10] K. P. R. Sastry, G. V. R. Babu and Ch. Srinivasa Rao, Convergence of an Ishikawa iteration scheme for nonlinear quasi-contractive mappings in convex metric spaces, Tamkang J. Math., 32 (2), (2001), 117-126.
[11] W. Takahashi, A convexity in metric space and nonexpansive mappings, Kodai Math. Sem. Rep., 22 (1970),142-149.