# Modified random errors S-iterative process for stochastic fixed point theorems in a generalized convex metric space

### Abstract

In this paper, we suggest the modified random S-iterative process and prove the common random fixed point theorems of a finite family of random uniformly quasi-Lipschitzian operators in a generalized convex metric space. Our results improves and extends various results in the literature.### References

A. Spacek, Zufallige Gleichungen, Czechoslov. Math. J. vol. 5, no. 80, pp. 462–466 1955.

O. Hans, Random operator equations In: Proceedings of 4th Berkeley Sympos. Math. Statist. Prob., vol. II, part I, pp. 185–202, University of California Press, Berkeley, 1961.

O. Hans, Reduzierende zufallige transformationen, Czechoslov. Math. J., vol. 7, no. 82, pp. 154–158, 1957.

A. Mukherjee, Transformation aleatoires separable theorem all point fixed aleatoire, C.R. Acad. Sci. Paris, Ser. A-B, vol. 263, pp. 393–395, 1966.

A.T., Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc., vol. 82, pp. 641–657, 1976.

S. Itoh, Random fixed-point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl., vol. 67, pp. 261–273, 1979.

V.M. Sehgal, C.Waters, Some random fixed point theorems for condensing operators, Proc. Amer. Math. Soc., vol. 90, pp. 425–429, 1984.

E. Rothe, Zur theorie der topologischen ordnung und der Vektorfelder in Banachschen Raumen, Compos. Math., vol. 5, pp. 177–197, 1938.

I. Beg, N. Shahzad, Random fixed points of random multivalued operator on Polish spaces, to appear in Nonlinear Anal.

P. Kumam, W. Kumam, Random fixed points of multivalued random operators with property (D), Random Oper.Stoch. Equat., vol. 15, pp. 127–136, 2007.

P. Kumam, S. Plubtieng, Random fixed point theorems for asymptotically regular random operators, Demonst. Math., vol. XLII, pp. 131–141, 2009.

P. Kumam, S. Plubtieng, The characteristic of noncompact convexity and random fixed point theorem for set-valued operators, Czechoslov. Math. J., vol. 57, no. 132, pp. 269–279, 2007.

P. Kumam, Random common fixed points of single-valued and multivalued random operators in a uniformly convex Banach space, J. Comput. Anal. Appl., vol. 13, pp. 368–375, 2011.

W. Kumam, P. Kumam, Random fixed point theorems for multivalued subsequentially limit-contractive maps satisfying inwardness conditions, J. Comput. Anal. Appl., vol. 14, pp. 239–251, 2012.

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 Mathematical Journal, Vol. 50, Issue 2, pp. 379–396, 2000.

M. Saha, On some random fixed point of mappings over a Banach space with a probability measure, Proc. Natl. Acad. Sci. India, Sect. A, vol. 76, pp. 219–224, 2006.

M. Saha, L. Debnath, Random fixed point of mappings over a Hilbert space with a probability measure, Adv. Stud. Contemp. Math., vol. 1, pp. 79–84, 2007.

W.J. Padgett, On a nonlinear stochastic integral equation of the hammerstein type, Proc. Amer. Soc., vol. 38, pp. 625–631, 1973.

J. Achari, On a pair of random generalized nonlinear contractions, Internat. J. Math. Math. Sci., vol. 6, pp. 467–475, 1983

M. Saha, D. Dey, Some random fixed point theorems for (;L)−weak contractions to appear in Hacet. J. Math. Stat.

M. Patriche, Random fixed point theorems under mild continuity assumptions, Fixed Point Theory and Applications, pp. 1–14, 2014.

V.N. Mishra, Some Problems on Approximations of Functions in Banach Spaces, Ph.D.Thesis (2007), Indian Institute of Technology, Roorkee 247 667, Uttarakhand, India.

L.N. Mishra, S.K. Tiwari and V.N. Mishra, Fixed point theorems for generalized weakly S-contractive mappings in partial metric spaces, Journal of Applied Analysis and Computation, vol. 5, pp. 600–612, 2015, doi:10.11948/2015047.

Deepmala, Study on Fixed Point Theorems for Nonlinear Contractions and its Applications, Ph.D. Thesis (2014), Pt. Ravishankar Shukla University, Raipur 492 010, Chhatisgarh, India.

H.K. Pathak and Deepmala, Common fixed point theorems for PD-operator pairs under Relaxed conditions with applications, Journal of Computational and Applied Mathematics, vol. 239, pp. 103–113, 2013.

V.N. Mishra, L.N. Mishra, Trigonometric Approximation of Signals (Functions) in Lp (p ≥ 1) norm, International Journal of Contemporary Mathematical Sciences, vol. 7, no. 19, pp. 909–918, 2012.

W. Takahashi, A convexity in metric space and nonexpansive mapping, Kodai.Math.Sem.Rep., vol. 22, pp. 142–149, 1970.

W.A. Kirk, Krasnoselskfi’s iteration process in hyperbohc space, Number, Funet., Anal. Optm, vol. 4, pp. 371-381, 1982.

K. Goebel, W.A. Kirk, Iteration processes for nonexpansive mappings, In Contemporary Math, Amer Math. Soc, Providence, RI, vol. 1, pp. 115–123, 1983.

Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications, vol. 207, no. 1, pp. 96–103, 1997.

Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 18–24, 2001.

Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach space, Journal of Mathematical Analysis and Applications, vol. 266, no. 2, pp. 468–471, 2002.

Y.X. Tian, Convergence of an Ishikawa type iterative scheme for asymptotically quasi-nonexpansive mappings, Comput.Math.Appl., vol. 49, pp. 1905–1912, 2005.

G. Das, J.P. Debata, Fixed points of quasi-onexpansive mappings, Indian J. Pure. Appl. Math., vol. 17, pp. 1263–1269, 1986.

A.R. Khan, M.A. Ahmed, Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces and applications, Comput.Math.Appl., vol. 59, pp. 2990–2995, 2010.

C.Wang, L. Liu, Convergence theorems for fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces, Nonlinear Analysis : TMA., vol. 70, pp. 2067–2071, 2009.

R.P. Agarwal, D. O’Regan and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex. Anal., vol. 8, no. 1, pp. 61–79, 2007.

P. Saipara, P. Chaipunya, Y.J. Cho and P. Kumam, On Strong and Δ-convergence of modified S-iteration for uniformly continuous total asymptotically nonexpansive mappings in CAT() spaces, The Journal of Nonlinear Science and Applications, vol. 8, pp. 965–975, 2015.

R. Suparatulatorn and P. Cholumjiak, The modified S-iteration process for nonexpansive mappings in CAT() spaces, Fixed Point Theory and Applications, vol. 1, pp. 1–12, 2016.

V. Kumar, A. Latif, A. Rafiq and N. Hassain, S-iteration process for quasi-contr active mappings, Journal of Inequalities and Applications, vol. 206, 2013.

S. Kosol, Rate of convergence of S-iteration and Ishikawa iteration for continuous functions on closed intervals, Thai Journal of Mathematics, vol.11, pp. 703–709, 2013.

P. Kumam, G.S. Saluja and H.K. Nashine, Convergence of modified S-iteration process for two asymptotically nonexpansive mappings in the intermediate sense in CAT(0) spaces, Journal of Inequalities and Applications, vol. 368, 2014.

I. Beg, Approximation of random fixed point in normed space, Nonlinear Analysis: TMA., vol. 51, pp. 1363–1372, 2002.

K.K. Tan, X.Z. Yuan, Some random fixed point theorems, in: K.K.Tan (Ed.), Fixed Point Theory and Applications,World Scientific, Singapore, pp.334-345, 1992.

*Statistics, Optimization & Information Computing*,

*5*(1), 65-74. https://doi.org/10.19139/soic.v5i1.250

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).