Random Coincidence Point Theorem in Fréchet Spaces with Applications

Abstract

We proved a random coincidence point theorem for a pair of commuting random operators in the setup of Fréchet spaces. As applications, we obtained random fixed point and best approximation results for *-nonexpansive multivalued maps. Our results are generalizations or stochastic versions of the corresponding results of Shahzad and Latif [Shahzad, N.; Latif, A. A random coincidence point theorem. J. Math. Anal. Appl. 2000, 245, 633–638], Khan and Hussain [Khan, A.R.; Hussain, N. Best approximation and fixed point results. Indian J. Pure Appl. Math. 2000, 31 (8), 983–987], Tan and Yaun [Tan, K.K.; Yaun, X.Z. Random fixed point theorems and approximation. Stoch. Anal. Appl. 1997, 15 (1), 103–123] and Xu [Xu, H.K. On weakly nonexpansive and *-nonexpansive multivalued mappings. Math. Japon. 1991, 36 (3), 441–445].

Key Words:

• Random fixed point
• Random coincidence point
• *-Nonexpansive map
• Opial's condition
• Fréchet space

2000 Subject Classification:

• Primary 28B20
• 41A65
• 47H10
• 60H25
• Secondary 46A04
• 54H25

Acknowledgment

The author A.R. Khan gratefully acknowledges support provided by the King Fahd University of Petroleum and Minerals during this research.