On a Second Derivative Test due to Qi

Qi [12] has given a theorem which guarantees the existence and uniqueness of a zero x* of a function f : Rn → Rn in a bounded closed rectangular convex set [x] ⊂ Rn under more general sufficient conditions than those described by Moore [9] and has defined an operator (the so-called second-derivative operator) together with a test, involving the second-derivative operator, for the existence but not the uniqueness of a zero x* of f in [x].The present paper has three purposes: (i) to establish sufficient conditions for the uniqueness of x* involving the second-derivative operator; (ii) to show that under the hypotheses for the convergence of sequences generated from Newton's method given in [15] a set [x] exists which satisfies the sufficient conditions in (i); (iii) to show how the second-derivative operator can be used in a manner similar to that which has been done with the Krawczyk operator in [4].