# Stochastic fixed point theorems for a random Z-contraction in a complete probability measure space with application to non-linear stochastic integral equations

Volume 1, Issue 1, pp 40--48
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### Authors

Plern Saipara - KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand Poom Kumam - KMUTT-Fixed Point Theory and Applications Research Group (KMUTT-FPTA), Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand Apirak Sombat - KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand Anantachai Padcharoen - KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand Wiyada Kumam - Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi, Thanyaburi (RMUTT), Pathumthani 12110, Thailand

### Abstract

In this paper, we propose the random $\mathcal{Z}$-contraction, prove a stochastic fixed point theorem for this contraction, and show that a solution for a non-linear stochastic integral equations exists in Banach spaces.

### Keywords

• $\mathcal{Z}$-contraction
• stochastic fixed point theorem
• complete probability measure spaces

### References

[1] J. Achari, On a pair of random generalized nonlinear contractions, Internat. J. Math. Math. Sci., 6 (1983), 467–475.
[2] R. F. Arens, A topology for spaces of transformations, Ann. of Math., 47 (1946), 480–495.
[3] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181.
[4] I. Beg, N. Shahzad, Random fixed points of random multivalued operators on Polish spaces, Nonlinear Anal., 20 (1993), 835–847 .
[5] A. T. Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc., 82 (1976), 641–657.
[6] Y. J. Cho, J. Li, N. J. Huang, Random Ishikawa iterative sequence with errors for approximating random fixed points, Taiwanese J. Math., 12 (2008), 51–61.
[7] O. Hanš, Reduzierende Zufällige Transformationen, (German) Czechoslovak Math. J., 7 (1957), 154–158.
[8] O. Hanš, Random operator equations, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Univ. California Press, Berkeley, Calif., II (1961), 185–202.
[9] S. Itoh, Random fixed-point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl., 67 (1979), 261–273.
[10] J. S. Jung, Y. J. Cho, S. M. Kang, B. S. Lee, B. S. Thakur, Random fixed point theorems for a certain class of mappings in Banach spaces, Czechoslovak Math. J., 50 (2000), 379–396.
[11] F. Khojasteh, S. Shukla, S. Radenovi´c, A new approach to the study of fixed point theory for simulation functions, Filomat, 29 (2015), 1188–1194.
[12] P. Kumam, Random common fixed points of single-valued and multivalued random operators in a uniformly convex Banach space, J. Comput. Anal. Appl., 13 (2011), 368–375.
[13] P. Kumam, W. Kumam, Random fixed points of multivalued random operators with property (D), Random Oper. Stoch. Equ., 15 (2007), 127–136.
[14] W. Kumam, P. Kumam, Random fixed point theorems for multivalued subsequentially limit-contractive maps satisfying inwardness conditions, J. Comput. Anal. Appl., 14 (2012), 239–251.
[15] P. Kumam, S. Plubtieng, Random coincidence and random common fixed points of nonlinear multivalued random operators, Thai J. Math., 5 (2007), 155–163.
[16] P. Kumam, S. Plubtieng, The characteristic of noncompact convexity and random fixed point theorem for set-valued operators, Czechoslovak Math. J., 57 (2007), 269–279.
[17] P. Kumam, S. Plubtieng, Some random fixed point theorems for random asymptotically regular operators, Demonstratio Math., 42 (2009), 131–141.
[18] A. C. H. Lee, W. J. Padgett, On random nonlinear contractions, Math. Systems Theory, 11 (1977/78), 77–84.
[19] A. Mukherjee, Transformation aleatoires separable theorem all point xed aleatoire, C. R. Acad. Sci. Paris, Ser. A-B, 263 (1966), 393–395.
[20] W. J. Padgett, On a nonlinear stochastic integral equation of the Hammerstein type, Proc. Amer. Math. Soc., 38 (1973), 625–631.
[21] S. Plubtieng, P. Kumam, Random fixed point theorems for multivalued nonexpansive non-self-random operators, J. Appl. Math. Stoch. Anal., 2006 (2006), 9 pages.
[22] E. Rothe, Zur Theorie der topologischen Ordnung und der Vektorfelder in Banachschen Räumen, (German) Compositio Math., 5 (1938), 177–197.
[23] M. Saha, On some random fixed points of mappings over a Banach space with a probability measure, Proc. Natl. Acad. Sci. India Sect. A Phys. Sci., 76 (2006), 219–224.
[24] M. Saha, L. Debnath, Random fixed point of mappings over a Hilbert space with a probability measure, Adv. Stud. Contemp. Math. (Kyungshang), 18 (2009), 97–104.
[25] M. Saha, D. Dey, Some random fixed point theorems for ($\theta,L$)-weak contractions, Hacet. J. Math. Stat., 41 (2012), 795– 812.
[26] M. Saha, A. Ganguly, Random fixed point theorem on a iri-type contractive mapping and its consequence, Fixed Point Theory Appl., 2012 (2012), 18 pages.
[27] V. M. Sehgal, C. Waters, Some random fixed point theorems for condensing operators, Proc. Amer. Math. Soc., 90 (1984), 425–429.
[28] A. Špaček, Zufällige Gleichungen, (German) Czechoslovak Math. J., 5 (1955), 462–466.
[29] K. Yosida, Functional analysis, Die Grundlehren der Mathematischen Wissenschaften, Band 123 Academic Press, Inc., New York; Springer-Verlag, Berlin, (1965).