Coincidence and fixed points of nonlinear hybrid contractions (Q1118854)

We show the existence of solutions of functional equations fx\(\in Sx\cap Tx\) and \(x=fx\in Sx\cap Tx\) under certain contraction and asymptotic regularity conditions, where f, S and T are single-valued and multi- valued mappings on a metric space, respectively. We also observe that Mukherjee's fixed point theorem for a single-valued mapping commuting with a multi-valued mapping admits of a counterexample and suggest some modifications. While doing so, we also answer an open question raised by \textit{B. E. Rhoades}, \textit{S. L. Singh} and \textit{Ch. Kulshrestha}, in Int. J. Math. Math. Sci. 7, 429-434 (1984; Zbl 0571.54034) and \textit{S. L. Singh} and \textit{Ch. Kulshrestha}, in Indian J. Math. Phys. Nat. Sci. 2B, 19-22 (1982).