Random fixed point results for generalized asymptotically nonexpansive random operators

Year 2023, Volume: 72 Issue: 3, 570 - 586, 30.09.2023

İsa Yildirim , Muhammed Emin Batuhan

https://doi.org/10.31801/cfsuasmas.1211661

Abstract

In this paper, we define an implicit random iterative process with errors for three finite families of generalized asymptotically nonexpansive random operators. We also prove some convergence theorems using this iteration method in separable Banach spaces.

Keywords

Generalized asymptotically nonexpansive random operators, measurable spaces, Banach spaces, condition (B*), implicit random iteration

Thanks

The second author would like to thank TUBITAK (Scientific and Technological Research Council of Turkey) for financial support with the TUBITAK 2211 Domestic Graduate Scholarship Program during his graduate studies.

References

• Banerjee, S., Choudhury, B. S., Composite implicit random iterations for approximating common random fixed point for a finite family of asymptotically nonexpansive random operators, Communications of the Korean Mathematical Society, 26(1) (2011), 23-35. https://doi.org/10.4134/CKMS.2011.26.1.023
• Beg, I., Abbas, M., Iterative procedures for solutions of random operator equations in Banach spaces, J. Math. Anal. Appl., 315(1) (2006), 181-201. https://doi.org/10.1016/j.jmaa.2005.05.073
• Beg, I., Shahzad, N., Random fixed point theorems for nonexpansive and contractive type random operators on Banach spaces, J. Appl. Math. Stochastic Anal., 7(4) (1994), 569-580. https://doi.org/10.1155/S1048953394000444
• Choudhury, B. S., A common unique fixed point theorem for two random operators in Hilbert spaces, International Journal of Mathematics and Mathematical Sciences, 32(3) (2002), 177-182. https://doi.org/10.1155/S0161171202005616
• Choudhury, B. S., A random fixed point iteration For three random operators on uniformly convex Banach spaces, Analysis in Theory and Application 19(2) (2003), 99-107. https://doi.org/10.1007/BF02835233
• Choudhury, B. S., An iteration for finding a common random fixed point, Journal of Applied Mathematics and Stochastic Analysis, 2004(4) (2004), 385-394. https://doi.org/10.1155/S1048953304208012
• Choudhury, B. S., Random Mann iteration scheme, Appl. Math. Lett. 16(1) (2003), 93-96. https://doi.org/10.1016/S0893-9659(02)00149-0
• Choudhury, B. S., Upadhyay A., An iteration leading to random solutions and fixed points of operators, Soochow J. Math. 25(4) (1999), 395-400.
• Hans, O., Random fixed point theorems, Transactions of the 1st Prague Conf. on Information Theory, Statistics, Decision Functions and Random Processes, Czeschosl. Acad. Sci., Prague (1957), 105-125.
• Itoh, S., Random fixed point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl., 67(2) (1979), 261-273. https://doi.org/10.1016/0022-247X(79)90023-4
• Khan, A. R., Thaheem, A. B., Hussain, N., Random fixed points and random approximations in nonconvex domains, J. Appl. Math. Stochastic Anal., 15(3) (2002), 247-253. https://doi.org/10.1155/S1048953302000217
• Lin, T. C., Random approximations and random fixed point theorems for continuous 1-set-contractive random maps, Proc. Amer. Math. Soc., 123(4) (1995), 1167-1176. https://doi.org/10.2307/2160715
• O’Regan, D., Random fixed point theory for multivalued maps, Stochastic Analysis and Applications, 17(4) (1999), 597-607. https://doi.org/10.1080/07362999908809623
• Plubtieng, S., Kumam, P.,Wangkeeree, R., Approximation of a common random fixed point for a finite family of random operators, Int. J. Math. Math. Sci., (2007). https://doi.org/10.1155/2007/69626
• Rhoades, B. E., Iteration to obtain random solutions and fixed points of operators in uniformly convex Banach spaces, Soochow J. Math., 27(4) (2001), 401-404.
• Schu, J., Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43(1) (1991), 153-159. https://doi.org/10.1017/S0004972700028884
• Spacek, A., Zuffalige gleichungen, Czech. Math. Jour., 80(5) (1955), 462-466. https://doi.org/10.21136/CMJ.1955.100162
• Tan, K. K., Xu, H. K., Approximating fixed points of nonexpansive mapping by the Ishikawa iteration process, J. Math. Anal. Appl., 178(2) (1993), 301-308. https://doi.org/10.1006/jmaa.1993.1309