International Journal of Mathematics and Mathematical Sciences
Volume 12 (1989), Issue 2, Pages 247-256
doi:10.1155/S0161171289000281
Coincidence and fixed points of nonlinear hybrid contractions
Shynam Lal Singh1
, Ki Sik Ha2
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
Abstract
In this Der, we show the existence of solutions of functional equations fx∈sx∩tx and x=fx∈sx∩Tx 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.