Notes on the paper "Fixed point theory for set-valued quasi-contraction maps in metric spaces" by A. Amini-Harandi

Theorem 1. Let (X, d) be a complete metric space, T → CB(X) a k-set-valued quasi-contraction with k < 1/2. Then T has a fixed point. The author defines a set-valued quasi contraction as one which satisfies H(Tx, Ty) ≤ kmax{d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx)}. (1) Note that kd(x, Ty) ≤ k(d(x, y) + d(y, Ty)) ≤ 2kmax{d(x, y), d(y, Ty)}, and that kd(y, Tx) ≤ k(d(x, y) + d(x, Tx)) ≤ 2kmax{d(x, y), d(x, Tx)}. Since 2k < 1, (1) implies the inequality H(Tx, Ty) ≤ hmax{d(x, y), d(x, Tx), d(y, Ty)}, (2) where h = 2k, and (2) is a special case of inequality (6) of [2], which is H(Tx, Ty) ≤ hmax{d(x, y), d(x, Tx), d(y, Ty), [d(x, Ty) + d(y, Tx)]/2}. Therefore the Theorem of [1] is a special case of Theorem 2 of [2].