- Research Article
- Open Access
© The Author(s). 2009
- Received: 28 March 2009
- Accepted: 19 October 2009
- Published: 28 October 2009
- Positive Integer
- Continuous Function
- Differential Geometry
- Normed Space
- Normal Cone
Definition 1.1 (see ).
for all , where max , with . For normed spaces see .
Lemma 1.3 (see ).
Definition 2.1 (see ).
In practice, such a sequence could arise in the following way. Let be a point in . Set . Let . Now . Because of rounding or discretization in the function , a new value approximately equal to might be obtained instead of the true value of . Then to approximate , the value is computed to yield , an approximation of . This computation is continued to obtain an approximate sequence of .
One of the most popular iteration procedures for approximating a fixed point of is Picard's iteration defined by . If the conditions of Definition 2.1 hold for then we will say that Picard's iteration is -stable.
Recently Qing and Rhoades  established a result for the -stability of Picard's iteration in metric spaces. Here we are going to generalize their result to cone metric spaces and present an application.
Definition 2.5 (see ).
Let and Therefore and satisfy the hypothesis of Theorem 3.1 where has property (i) and has property (ii). So the self maps of defined by and have unique fixed points but Picard's iteration is -stable for but not -stable for
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