The authors show that any operator T in a Banach space satisfying \[ \| Tx-Ty\| \leq a\| x-y\| +b\| x-Tx\| +b\| y- Ty\| \] (a,b\(\geq 0)\) has a unique fixed point. For similar discussions, see the second author’s paper in Pac. J. Math. 123, 189-196 (1986; Zbl 0549.47028).