In � 1 of this paper the notion of a multi-valued Lipschitz mapping is defined and, in � 2, some elementary results and examples are given. In � 3 the two fixed point theorems for multi-valued contraction mappings are proved. The first, a generalization of the contraction mapping principle of Banach, states that a multi-valued contraction mapping of a complete metric space X into the nonempty closed and bounded subsets of X has a fixed point. The second, a generalization of a result of Edelstein, is a fixed point theorem for compact setvalued local contractions. A counterexample to a theorem about (ε, λ)-uniformly locally expansive (single-valued) mappings is given and several fixed point theorems concerning such mappings are proved. In � 4 the convergence of a sequence of fixed points of a convergent sequence of multi-valued contraction mappings is investigated. The results obtained extend theorems on the stability of fixed points of single-valued mappings [19].