International Journal of Mathematics and Mathematical Sciences
Volume 12 (1989), Issue 2, Pages 247-256

Coincidence and fixed points of nonlinear Hybrid contractions

Shyam Lal Singh,1 Ki Sik Ha,2 and Yeol Je Cho3

1Department of Mathematics, Gurukul Kangrl Unlversity, Hardwar 249404, India
2Department of Mathematics, Pusan National University, Pusan 607, Korea
3Department of Mathematics, Gyeongsang National University, JinJu 620, Korea

Received 8 June 1987; Revised 2 November 1987

Copyright © 1989 Shyam Lal Singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In this Der, we show the existence of solutions of functional equations fxsxtx and x=fxsxTx under certain contraction and asymptotic regularity conditions, where f, S and T are single-valued and multl-valued mappings on a metric space, respectively. We also observe that MukherJee's fixed point theorem for a single-valued mapping commuting with a multl-valued mapping admits of a counterexample and suggest some modifications. While doing so, we also answer an open question raised in [I] and [2]. Moreover, our results extend and unify a multitude of fixed point theorems for multi-valued mappings.