The main results of this paper are:
Theorem 3.1. Let X be a Banach space, and K a nonempty, closed subset of X,T:K a mapping satisfying the condition Tx for every (the boundary of K), and an increasing continuous function satisfying
Furthermore, let b and c be decreasing functions from into [0,1) such that for every . Suppose that T satisfies the condition
. Then T has a unique fixed point.
Theorem 1 of N. A. Assad, Tamkang J. Math. 7, 91-94 (1976; Zbl 0356.47027) is a special case of Theorem 3.1, Theorem 3.1 generaizes Theorem 2 of M. S. Khan, M. Swaleh and S. Sessa, Bull. Aust. Math. Soc. 30, 1-9 (1984; Zbl 0553.54023). Theorem 4.1. Let X be a Banach space and K a nonempty compact subset of X. Let T:K be a continuous mapping satisfying the condition that T(x) for every and an increasing continuous function satisfying property (1) of Theorem 3.1. Furthermore for all distinct x,y in K the inequality
holds, where . Then T has a unique fixed point.
Theorem 4.1 generalizes the fixed point theorem of B. Fisher, Calcutta Math. Soc. 68, 265-266 (1976; Zbl 0378.54035) in case X is a Banach space.