Random fixed point theorems for multi-valued contraction mappings in complete metric space

“In this paper the concept of contraction mapping for multi-valued maps in complete metric space is introduced. The point to set map is also called multi-valued (multifunction) which guarantees the existence of fixed point generalization for some multi-valued contraction mappings such as Banach Contraction principle extended by Nadler (S. B. Nadler Jr., Multi-valued contraction mappings, pacific J. Math.30(1969)475-488), generalization of Banach and Caristi fixed point theorem for multi-valued maps was developed by many authors such as (Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued caristi type mappings, J.Math.Anal.Appl. 317(2006)103-112) and (N. Mizoguchi, W. Takahashi, Fixed point theorems for multi-valued mappings on complete metric space, J.Math.Anal.Appl.141(1989)177-188). Here we generalize N. Mizoguchi, W. Takahashi fixed point theorem and our result improves a latest result by Klim and Wadowski (D. Klim, D. Wardowski, Fixed point theorems for set valued contractions in complete metric space, J.Math.Anal.Appl.334(1)(2007)132-139).
Mathematics Subject Classification: - 54H25, 47H10