## Abstract

We obtain a contractive condition for the existence and uniqueness of fixed points for a generalized contraction-type mapping. The present study focuses on providing a method for the existence of fixed points for nonlinear mappings. Sufficient conditions for the existence and uniqueness of such points are obtained using Ishikawa iteration process. Moreover, an example is given.

## Authors

**C. D. Alecsa**

Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

## Keywords

Fixed point; Generalized contraction; Ishikawa; Convergence; Convex metric space;

## Paper coordinates

Cristian-Daniel Alecsa, *Approximating fixed points for nonlinear generalized mappings using Ishikawa iteration, *Rendiconti Del Circolo Matematico Di Palermo, 68 (2019) no. 1, pp. 163-191.

https://doi.org/10.1007/s12215-018-0349-7

## About this paper

##### Journal

Rendiconti del Circolo Matematico di Palermo Series 2

##### Publisher Name

Springer Milan

##### DOI

https://doi.org/10.1007/s12215-018-0349-7

##### Print ISSN

##### Online ISSN

google scholar link

[1] Abbas, M., Khan, S.H., Rhoades, B.E., *Simpler is also better in approximating fixed points*. Appl. Math. Comput. 205(1), 428–431 (2008)

[2] Agarwal, R.P., O’Regan, D., Sahu, D.R., *Fixed Point Theory for Lipschitzian-Type Mappings with Applications*. Springer, Heidelberg (2003)

[3] Asadi, M., *Some results of fixed point theorems in convex metric spaces*. Nonlinear Funct. Anal. Appl. 19(2), 171–175 (2014)

[4] Fukhar-ud-dina, H., Berinde, V.,* Iterative methods for the class of quasi-contractive type operators and comparison of their rate of convergence in convex metric spaces*. Filomat 30(1), 223–230 (2016)

[5] Karapinar, E., *Fixed point theorems in cone banach spaces*. Fixed point theory and applications. Article ID 609281, pp. 1–9 (2009). https://doi.org/10.1155/2009/60928

[6] Moosaei, M., *Fixed point theorems in convex metric spaces*. Fixed Point Theory Appl. 2012, 164 (2012). https://doi.org/10.1186/1687-1812-2012-164

[7] Moosaei, M., *Common fixed points for some generalized contraction pairs in convex metric spaces*. Fixed Point Theory Appl. 2014:98 (2014). http://www.fixedpointtheoryandapplications.com/content/2014/1/

[8] Moosaei, M., Azizi, A., *On coincidence points of generalized contractive pair mappings in convex metric spaces*. J. Hyperstruct. 2(4), 136–141 (2015)

[9] Shimizu, T., Takahashi, W., *Fixed point theorems in certain convex metric spaces*. Math. Jpn. 37, 855–859 (1992)

[10] Takahashi, W., *A convexity in metric spaces and nonexpansive mappings I*. Kodai Math. Semin. Rep. 22, 142–149 (1970)

[11] Wang, C., Zhang, T., *Approximating common fixed points for a pair of generalized nonlinear mappings in convex metric spaces*. J. Nonlinear Sci. Appl. 9, 1–7 (2016)