Abstract. In this paper, we prove random common fixed point theorem for two pairs of random self mappings under a generalized contractive condition using subsequential continuity with compatibility of type (E). An example is given to justify our theorem. Our results in randomness extend and improve the results of S. Beloul [4]. Finally, we give an application to discuss the existence of a solution of random Hammerstein integral equations.

Received: July 14, 2016

Revised: August 18, 2016

Published: October 9, 2016

AMS Subject Classification: 47H10, 24H25

Key Words and Phrases: polish spaces, random operators, compatible of type (E), weakly subsequentially continuous
Download paper from here.

Bibliography

1

I. Beg, M. Abbas, Common random fixed point of compatible random operators, Hindawi Pub. Corp., 2006 (2006), 1-15.

2

I. Beg, M. Abbas, Iterative procedures for solution of random equations in Banach spaces, J. Math. Anal. Appl., 315 (2006), 181-201., 369-396.

3

I. Beg, N. Shahzad, Random fixed point theorems for nonexpansive and contractive-type random operators on Banach spaces, J. Appl. Math. Stoch. Anal., 7 (1994), 569-580.

4

S. Beloul, Common fixed point theorems for weakly subsequentially continuous generalized contractions with applications, J. App. Math. E-notes, 15 (2015), 173-186.

5

A. T. Bharucha-Reid, fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc., 82 (1976), 641-657.

6

H. Bouhadjera, C. G. Thobie, Common fixed point theorems for pairs of subcompatible maps, J. App.Math. E-notes, arXiv0906.3159v1 [math.FA],(2009)

7

B. S. Choudhury, Convergence of a random iteration scheme to a random fixed point, J. Appl. Math. Stochastic Anal., 8 (1995), 139-142.

8

B. S. Choudhury, M. Ray, Convergence of an iteration leading to a soulation of a random operator equation, J. Appl. Math. Stoch. Anal., 12 (1999), 161-168.

9

B. S. Choudhury, A. Upadhyay, An iteration leading to random solutions and fixed points of operators, Soochow J.Math., 25 (1999), 395-400.

10

V. Gupta, A. Saini, R. Kumar, Common fixed points for weakly compatible Maps in 2-metric space, Int. J. Math. Archive, 3 (2012), 3670-3675.

11

O. Hans, Random operator equations, Proc. 4th Berkeley Symp. Math. Stat. Prob. Vol. II, Part I niversity of California Press, Berkeley, (1961), 185-202.

12

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

13

G. Jungck, ommuting mappings and fixed points, Amer. Math. Monthly, 83 (1976), 261-263.

14

G. Jungck, Compatible mappings and common fixed points, Int. J. Math. Math. Sci., 9 (1986), 771-779.

15

G. Jungck, B. E Rhoades, ixed point for set valued functions without continuity, JIndian J. Pure Appl. Math., 29 1998), 227-238.

16

T. C. Lin, Random approximations and random fixed point theorems for non-self maps, Proc. Am. Math. Soc., 103 ((1988), 1129-1135.

17

D. Oregan, Fixed points and random fixed points for weakly inward approximable maps, Proc. Ameri. Math. Soc., 126 (1998), 3045-3053.

18

V. M. Sehgal, C. Waters, Some random fixed point theorems for condensing operators, Proc. Amer. Math. Soc., 90 (1984), 425-429.

19

S. Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math. (Beograd) (N.S.), 46 (1982) 149-153.

20

M. R. Singh, Y. M. Singh, Compatible mappings of type (E) and common fixed point theorems of Meir-Keeler type, Int. J. Math. Sci. Engg. Appl., 1 (2007), 299-315.

21

A. Spacek, zufăllige Gleichungen, Czechoslovak, Math. J., 80 (1955) 462-466.

20

J. Yin, Z. Liu, Random fixed point theorems of random comparable operators and an application, Dis. Dynam. in Nat. and Soc., 2015 (2015), 1-6.

.

---

DOI: 10.12732/ijpam.v109i4.5 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 109
Issue: 4
Pages: 813 - 826

Google Scholar; DOI (International DOI Foundation); WorldCAT.

This work is licensed under the Creative Commons Attribution International License (CC BY).

---