# Common fixed point theorems for complex-valued mappings with applications

## Main Article Content

## Abstract

The aim of this paper is to obtain some results which belong to fixed point theory such as strong convergence, rate of convergence, stability, and data dependence by using the new Jungck-type iteration method for a mapping defined in complex-valued Banach spaces. In addition, some of these results are supported by nontrivial numerical examples. Finally, it is shown that the sequence obtained from the new iteration method converges to the solution of the functional integral equation in complex-valued Banach spaces. The results obtained in this paper may be interpreted as a generalization and improvement of the previously known results.

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## References

[1] W. M. Alfaqih, M. Imdad, and F. Rouzkard, Unified common fixed point theorems in complex valued metric spaces via an implicit relation with applications, Bol. da Soc. Parana. de Mat., 38 (2020), 9–29. Google Scholar

[2] Y. Atalan and V. Karakaya, Investigation of some fixed point theorems in hyperbolic spaces for a three step iteration process, Korean J. Math., 27 (2019), 929–947. Google Scholar

[3] Y. Atalan and V. Karakaya, Obtaining new fixed point theorems using generalized Banach- contraction principle Erciyes Uni. Fen Bil. Enst. Fen Bil. Dergisi., 35 (3) (2020), 34-45. Google Scholar

[4] Y. Atalan and V. Karakaya, Iterative solution of functional Volterra-Fredholm integral equation with deviating argument, J. Non. Con. Anal., 18 (4) (2017), 675–684. Google Scholar

[5] A. Azam, B. Fisher, and M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Funct. Anal. Optim., 32 (2011), 243–253. Google Scholar

[6] V. Berinde, Iterative approximation of fixed points, Springer, 2007. Google Scholar

[7] T. Cardinali and P. Rubbioni, A generalization of the Caristi fixed point theorem in metric spaces, Fixed Point Theory Appl., 11 (2010), 3–10. Google Scholar

[8] R. Chugh and V. Kumar, Strong convergence and stability results for Jungck-SP iterative scheme. Int. J. Comput. Appl., 36 (2011), 40–46. Google Scholar

[9] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Univ. Ostrav., 23 (1993), 5–11 . Google Scholar

[10] D. Dhiman, L. N. Mishra, V. N. Mishra, Solvability of some non-linear functional integral equations via measure of noncompactness, Adv. Stud. Contemp. Math., 32 2 (2022), 157-171. Google Scholar

[11] K. Dogan, F. Gursoy, and V. Karakaya Some fixed point results in the generalized convex metric spaces, TWMS J. of Apl. and Eng. Math., 10 (2020), 11–23. Google Scholar

[12] E. Hacioglu and V. Karakaya, :Existence and convergence for a new multivalued hybrid mapping in cat (k) spaces, Carpathian J. Math., 33 (2017), 319–326. Google Scholar

[13] E. Hacioglu, F. Gursoy, S. Maldar, Y. Atalan, and G. V. Milovanovi ́c, Iterative approximation of fixed points and applications to two-point second-order boundary value problems and to machine learning, Applied Numerical Mathematics., 167 (2021), 143–172. Google Scholar

[14] N. Hussain, V. Kumar, and M. A. Kutbi,On rate of convergence of Jungck-type iterative schemes, Abstr. Appl. Anal., 2013 (2013). Google Scholar

[15] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly, 83 (1976), 261–263. Google Scholar

[16] R. Chugh, S. Kumar, On the stability and strong convergence for Jungck-Agarwal et al. iteration procedure, Int. J. Comput. Appl., 64 (2013), 39-44. Google Scholar

[17] A. R. Khan, F. Gursoy, and V. Karakaya, Jungck-Khan iterative scheme and higher convergence rate, Int. J. Comput. Math., 93 (2016), 2092–2105. Google Scholar

[18] M. Kumar, P. Kumar, and S. Kumar, Common fixed point theorems in complex valued metric spaces, J. Ana. Num. Theor., 2 (2014), 103–109. Google Scholar

[19] S. Maldar, Y. Atalan, and K. Dogan, Comparison rate of convergence and data dependence for a new iteration method, Tbil. Math. J.,13 (2020), 65–79. Google Scholar

[20] S. Maldar, An examination of data dependence for Jungck-type iteration method, Erciyes Uni. Fen Bil. Enst. Fen Bil. Dergisi., 36 (2020), 374–384. Google Scholar

[21] S. Maldar, Gelecegin dunyasinda bilimsel ve mesleki calismalar: Matematik ve fen bilimleri, Ekin Basım Yayın Da ̆gıtım, 2019. Google Scholar

[22] S. Maldar, Yeni bir iterasyon y ̈ontemi i ̧cin yakınsaklık hızı, Igdır Uni. Fen Bil. Ens. Dergisi, 10 (2) (2020), 1263-1272. Google Scholar

[23] L. N. Mishra, V. Dewangan, V. N. Mishra, S. Karateke, Best proximity points of admissible almost generalized weakly contractive mappings with rational expressions on b-metric spaces, J. Math. Computer Sci., 22 (2) (2021), 97–109. Google Scholar

[24] L. N. Mishra, V. Dewangan, V. N. Mishra, H. Amrulloh, Coupled best proximity point theorems for mixed g-monotone mappings in partially ordered metric spaces, J. Math. Comput. Sci., 11 (5) (2021),6168-6192. Google Scholar

[25] Z. Mustafa and B. Sims, A new approach to generalized metric spaces, J Nonlinear Convex Anal., 7 (2006), 289. Google Scholar

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

[27] G. A. Okeke and K. J. Kim, Fixed point theorems in complex valued Banach spaces with appli- cations to a nonlinear integral equation, Nonlinear Funct. Anal. Appl., 25 (2020), 411–436. Google Scholar

[28] M. Ozturk, I. A. Kosal, and H. H. Kosal, Coincidence and common fixed point theorems via C-class functions in elliptic valued metric spaces, An. S.t. Univ. Ovidius Constanta., 29 (1) (2021), 165–182. Google Scholar

[29] A. Petrusel and I. Rus, A class of functional-integral equations via picard operator technique, Ann. Acad. Rom. Sci.,10 (2018), 15–24. Google Scholar

[30] W. Phuengrattana and S. Suantai, Comparison of the rate of convergence of various iterative methods for the class of weak contractions in Banach spaces Thai J. Math., 11, 217–226 (2012). Google Scholar

[31] D. R. Sahu, Applications of the S-iteration process to constrained minimization problems and split feasibility problems Fixed point theo., 12 (1) (2011), 187–204. Google Scholar

[32] N. Sharma, L. N. Mishra, V. N. Mishra, H. Almusawa, Endpoint approximation of standard three-step multi-valued iteration algorithm for nonexpansive mappings Appl. Math. Inf. Sci., 15 (1) (2021), 73-81. Google Scholar

[33] N. Sharma, L. N. Mishra, V. N. Mishra, S. Pandey,Solution of delay differential equation via N1v iteration algorithm European J. Pure Appl. Math., 13 (5) (2020), 1110-1130. Google Scholar

[34] N. Sharma, L.N. Mishra, S.N. Mishra, V.N. Mishra, Empirical study of new iterative algorithm for generalized nonexpansive operators J. Math. Comput. Sci., 25 (3) (2022), 284-295. Google Scholar

[35] S. L. Singh, C. Bhatnagar, and S. N. Mishra, Stability of Jungck-type iterative procedures, Int. J. Math. Sci., 19 (2005), 3035-3043. Google Scholar

[36] K. Sitthikul and S. Saejung, Some fixed point theorems in complex valued metric spaces, Fixed Point Theory Appl.,(2012), 189. Google Scholar

[37] S. M. Soltuz and T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive-like operators, Fixed Point Theory Appl., 2008 (2008), 1–7. Google Scholar

[38] S. Soursouri, N. Shobkolaei, S. Sedghi, I. Altun, A Common fixed point theorem on ordered partial S-metric spaces and applications, Korean J. Math., 28 (2) (2020), 169-189. Google Scholar

[39] X. Weng, Fixed point iteration for local strictly pseudo-contractive mapping, Proc. Am. Math. Soc.,113 (1991), 727–731. Google Scholar

[40] T. Zamfirescu, Fixed point theorems in metric spaces, Archiv der Mathematik, 23 (1972), 292–298. Google Scholar

[41] F. M. Zeyada, G. H. Hassan, and M. A. Ahmed, A generalization of a fixed point theorem due to hitzler and seda in dislocated quasi-metric spaces, Arab. J. Sci. Eng., 31 (2006), 111. Google Scholar