Convergence of the Sequence of Successive Approximations to a Fixed Point
© Tomonari Suzuki. 2010
Received: 29 September 2009
Accepted: 21 December 2009
Published: 2 February 2010
If is a complete metric space and is a contraction on , then the conclusion of the Banach-Caccioppoli contraction principle is that the sequence of successive approximations of starting from any point converges to a unique fixed point. In this paper, using the concept of -distance, we obtain simple, sufficient, and necessary conditions of the above conclusion.
The following famous theorem is referred to as the Banach-Caccioppoli contraction principle. This theorem is very forceful and simple, and it became a classical tool in nonlinear analysis.
Theorem 1.2 (see ).
We obtained a condition equivalent to (B) in .
Theorem 1.4 (see ).
(ii)The following hold.
We cannot tell that the conditions (ii) of Theorems 1.2 and 1.4 are simple. Motivated by this, we obtain simpler conditions which are equivalent to Conditions (A) and (B).
Definition 2.1 (see ).
Theorem 2.2 (see ).
Theorem 2.3 (see).
The following lemmas are useful in our proofs.
Lemma 2.4 (see ).
Lemma 2.5 (see ).
The following lemmas are easily deduced from Lemmas 2.4 and 2.5.
Lemma 2.8 (see ).
The following is obvious.
Then the following hold.
3. Condition (B)
In this section, we discuss Condition (B).
As a direct consequence of Theorem 3.1, we obtain the following.
Corollary 3.2 characterizes Condition (B).
Then satisfies ( 2) and by Lemma 2.8. In order to show ( 3), we assume that and . Then without loss of generality, we may assume that . Thus for . It is obvious that for with . We consider the following two cases.
Therefore (3.3) holds.
Combining Theorem 6 in , we obtain the following.
4. Condition (A)
In this section, we discuss Condition (A).
which implies a contradiction. Thus the fixed point is unique.
We shall show that Theorems 4.1 and 4.2 characterize Condition (A).
Therefore (4.1) holds.
The proof of Theorem 4.3 works.
Combining Theorem 7 in , we obtain the following.
5. Additional Result
Since Theorem 2.2 deduces Corollary 3.2, we can tell that Theorem 2.2 characterizes Condition (B). However, the following example tells that Theorem 2.3 does not characterize Condition (A).
This is a contradiction.
The author is supported in part by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science. The author wishes to express his gratitude to the referees for careful reading and giving a historical comment.
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