CARPATHIAN J. MATH. Volume 36 (2020), No. 1, Pages 127 - 139 Online version at https://www.carpathian.cunbm.utcluj.ro/ Print Edition: ISSN 1584 - 2851; Online Edition: ISSN 1843 - 4401 DOI: https://doi.org/10.37193/CJM.2020.01.12 Dedicated to Prof. Hong-Kun Xu on the occasion of his 60 th anniversary Modified inertial double Mann type iterative algorithm for a bivariate weakly nonexpansive operator ANANTACHAI PADCHAROEN 1 and KAMONRAT SOMBUT 2,* ABSTRACT. We introduce a modified inertial double Mann type iterative method to approximate coupled solutions of a bivariate nonexpansive operator T : C × C → C, where C is a nonempty closed and convex subset of a Hilbert space. The one theorem and complement important old and recent results in coupled fixed point theory. Some appropriate examples to illustrate our results and their generalization are also given. Acknowledgments. The first author thanks for the support of Rambhai Barni Rajabhat University. The authors thank you very much Prof. Vasile Berinde for his suggestions and comments. Finally, Kamonrat Sombut was financial supported by RMUTT annual government statement of expendture in 2019 and RMUTT Research Grant for New Scholar for fiscal year of 2019 (Grant no.NSF62D0603) was gratefully acknowledged. REFERENCES [1] Aghajani, A., Abbas, M. and Kallehbasti, E. P., Coupled fixed point theorems in partially ordered metric spaces and application, Math. Commun., 17 (2002), No. 2, 497–509 [2] Aghajani, A. and Arab, R., Fixed points of (ψ,ϕ,θ)-contractive mappings in partially ordered b-metric spaces and application to quadratic integral equations, Fixed Point Theory Appl., 2013, 245 (2013), doi:10.1186/1687-1812-2013-245 [3] Amini-Harandi, A., Coupled and tripled fixed point theory in partially ordered metric spaces with application to initial value problem, Math. Comput. Model., 57 (2013), No. (9-10), 2343–2348 [4] Auslender, A., Teboulle, M. and Ben-Tiba, S., A logarithmic-quadratic proximal method for variational in- equalities, Comput. Optim. Appl., 12 (1999), 31–40 [5] Bauschke, H. H. and Combettes, P. L., Convex Analysis and Monotone Operator Theory, in Hilbert Spaces (Springer, Berlin, 2011) [6] Berinde, V., Khan, A. R. and P˘ acurar, M., Coupled solutions for a bivariate weakly nonexpansive operator by iterations, Fixed Point Theory Appl., 2014 (2014), Article ID 149 [7] Berinde, V., Coupled fixed point theorems for ϕ-contractive mixed monotone mappings in partially ordered metric spaces, Nonlinear Anal., 75 (2012), No. 6, 3218–3228 [8] Berinde, V. and Borcut, M., Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal., 74 (2011), 4889–4897 [9] Berinde, V. and P˘ acurar, M., Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces, Fixed Point Theory Appl., 2012, 2012:115, 11 pp. [10] Berzig, M. and Samet, B., An extension of coupled fixed point’s concept in higher dimension and applications, Comput. Math. Appl., 63 (2012), No. 8, 1319–1334 Received: 29.05.2019; In revised form: 30.06.2019; Accepted: 14.07.2019 2010 Mathematics Subject Classification. 47H09, 47H10, 49M05. Key words and phrases. Inertial Mann method, coupled solutions, bivariate nonexpansive and weekly nonexpansive. Corresponding author: Kamonrat Sombut; kamonrat s@rmutt.ac.th 127