# Existence and approximation of fixed point of a nonlinear mapping satisfying rational type contractive inequality condition in complex-valued Banach spaces

### Abstract

We prove the existence of a unique xed point for a mapping satisfying a rational type

contractive inequality condition in complex-valued Banach spaces. We approximate this xed

point via some xed point iterative processes with high rate of convergence. We then prove

that our results are valid in cone metric spaces with Banach algebras. Furthermore, our results

are applied in nding the solutions of delay dierential equations. Our results extend and

generalize several known results in the literature.

### References

Mann, W. R, Mean value methods in iteration, Proc. Amer. Math. Soc. 44 (1953), 506-510.

Ishikawa, S.,~ Fixed point by a new iteration method, Proc. Amer. Math. Soc. 44(1) (1974), 147-150.

Akewe, H., Okeke, G.A., Olayiwola, A.F., Strong convergence and stability of Kirk-multistep-type iterative schemes for contractive-type operators, Fixed Point Theory and Applications 2014, 2014:46, 24 pages.

Akewe, H., Okeke, G.A., Convergence and stability theorems for the Picard-Mann hybrid iterative scheme for a general class of contractive-like operators, Fixed Point Theory and Applications (2015) 2015:66, 8 pages.

Okeke, G.A., Convergence analysis of the Picard-Ishikawa hybrid iterative process with applications, Afrika Matematika, 30 (2019) 817-835.

Okeke, G.A., Abbas, M., A solution of delay differential equations via Picard-Krasnoselskii hybrid iterative process, Arab. J. Math. (2017) 6:21-29.

Olatinwo, M. O., Some stability and strong convergence results for the Jungck-Ishikawa iteration process, Creative Math. & Inf. 17 (2008), 33-42.

Berinde, V., Iterative approximation of fixed points, Lecture Notes in Mathematics, Springer-Verlag, Berlin Heidelberg, 2007.

Olaleru, J.O., Murthy, P.P., Common fixed point theorems of Gregubreves type mappings in a complete linear metric space, JP Journal of Fixed Point Theory and Applications, Vol. 4, No. 3, 2009, 193-208.

Olatinwo, M. O. and Ishola, B. T, Some fixed point theorems involving rational type contractive operators in complete metric spaces, Surveys in Mathematics and its Applications 13 (2018), 107-117.

Huang, L.-G., Zhang, X., Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), 1468-1476.

Azam, A., Fisher, B., Khan, M., Common fixed point theorems in complex valued metric spaces, Numerical Functional Analysis and Optimization, 32(3)(2011), 243-253.

Abbas, M., Raji'c V.'C., Nazir, T., Radenovi'c, S., Common fixed point of mappings satisfying rational inequalities in ordered complex valued generalized metric spaces, Afr. Mat. DOI: 10.1007/s13370-013-0185-z, 2013, 14 pages.

Abbas, M., Arshad, M., Azam, A., Fixed points of asymptotically regular mappings in complex-valued metric spaces, Georgian Math. J. 20 (2013), 213-221.

Abbas, M., Sen M. De la, Nazir, T., Common fixed points of generalized cocyclic mappings in complex valued metric spaces, Discrete Dynamics in Nature and Society, Vol. 2015, Article ID: 147303, 2015, 11 pages.

Ahmad,J., Hussain, N., Azam, A., Arshad, M., Common fixed point results in complex valued metric space with applications to system of integral equations, Journal of Nonlinear and Convex Analysis, 29(5) (2015), 855-871.

Alfaqih, W.M., Imdad, M., Rouzkard, F., Unified common fixed point theorems in complex valued metric spaces via an implicit relation with applications, Bol. Soc. Paran. Mat. (3s.) v. 38 (4) (2020): 9-29.

Rouzkard, F., Imdad, M., Some common fixed point theorems on complex valued metric spaces, Computers and Mathematics with Applications 64(2012) 1866-1874.

Shukla, S., Rodr'iguez-L'opez, R., Abbas, M., Fixed point results for contractive mappings in complex valued fuzzy metric spaces, Fixed Point Theory 19 (2018), No. 2, 751-774.

Singh, N., Singh, D., Badal, A., Joshi, V., Fixed point theorems in complex valued metric spaces, Journal of the Egyptian Math. Soc. 24 (2016), 402-409.

Okeke, G.A., Iterative approximation of fixed points of contraction mappings in complex valued Banach spaces, Arab J. Math. Sci. 25(1) (2019), 83-105.

Okeke, G.A., Abbas, M., Fej'er monotonicity and fixed point theorems with applications to a nonlinear integral equation in complex valued Banach spaces, Appl. Gen. Topol. 21, no. 1 (2020), 135-158.

Noor, M. A., textitNew approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000) 217–229.

Khan, S.H., A Picard-Mann hybrid iterative process, Fixed Point Theory and Applications 2013, 2013:69, 10 pages.

Jaggi, D. S. and Dass, B. K., An extension of Banach’s fixed point theorem through rational expression, Bull. Cal. Math. 72 (1980), 261-266.

Dass, B. K. and Gupta, S., An extension of Banach contraction principle through rational expression, Indian J. Pure and Appl. Math 6 (1975), 1455-1458.

Jaggi, D. S., Some unique fixed point theorems, Indian Journal of Pure and Applied Mathematics 8(2) (1977), 223-230.

Das, G., Debata, J.P., Fixed points of quasi-nonexpansive mappings, Indian J. Pure Appl. Math. 17 (1986) 1263-1269.

Soltuz, S.M., Otrocol, D., Classical results via Mann-Ishikawa iteration, Rev. d'Analyse Numer. de l'Approximation, 36 (2) (2007), 193-197.

Rudin, W., Functional analysis, 2nd edn. McGraw-Hill, New York, 1991.

Ozturk, M., Bacsarir, M., On some common fixed point theorems with rational expressions on cone metric spaces over a Banach algebra, Hacettepe J. Math. and Stat. Vol. 41 (2) (2012), 211-222.

Olaleru, J.O., Okeke, G.A., Akewe, H., Coupled fixed point theorems for generalized varphi-mappings satisfying contractive condition of integral type on cone metric spaces, International Journal of Mathematical Modelling & Computations, Vol. 02, No. 02, 2012, 87-98.

Liu, H., Xu, S., Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory and Appl. 2013, 2013:320, 10 pages.

*International Journal of Mathematical Sciences and Optimization: Theory and Applications*,

*2020*(1), 707 - 717. Retrieved from http://ijmso.unilag.edu.ng/article/view/951