We begin this section by proving the following crucial lemmas.
Lemma 2.1 Let C be a closed interval on the real line (can be unbounded) and let be a continuous function. Let , , , and be sequences in with and . Let be a sequence generated iteratively by and
(2.1)
where , , and .
If , then a is a fixed point of f.
Proof Let and suppose . Then is bounded. So, is bounded by the continuity of f. So are , , and . Moreover, since and . We also have since and .
From (2.1) we obtain
(2.2)
Let , and . Then we observe
So, from (2.2) we obtain
It is easy to see that since and . Similarly, we have since and . This shows that is a divergent sequence since and . This contradicts the convergence of . Hence and a is a fixed point of f. □
Lemma 2.2 Let C be a closed interval on the real line (can be unbounded) and let be a continuous function. Let , , , and be sequences in with and . Let be a sequence generated iteratively by and
where , , and .
If is bounded, then is convergent.
Proof Suppose is not convergent. Let and . Then . We first show that if , then . Suppose . Without loss of generality, we suppose . Since f is continuous, there exists δ with such that for ,
Since is bounded and f is continuous, is bounded. Hence , , and are all bounded. Using
we can easily show that , and . Thus there exists a positive integer N such that for all ,
(2.3)
Since , there exists such that . Let , then . For , there exist two cases as follows.
-
(i)
, then using (2.3). So, we have .
-
(ii)
, then and by (2.3). So, we obtain , , . Hence
(2.4)
We observe that
(2.5)
From (2.2), (2.4) and (2.5), we have
Thus . From (i) and (ii), we have . Similarly, we get that , , …. Thus we have for all . So, , which is a contradiction with . Thus .
We next consider the following two cases.
-
(i)
There exists such that . Then . It follows that
and
Hence, we obtain
Similarly, we obtain . So, we conclude that . Since there exists , . This shows that , which is a contradiction.
-
(ii)
For all or . Since and , there exists such that for . So, it is always that for ; or it is always that for . If for , then , which is a contradiction with . If for , then , which is a contradiction with . Thus we conclude that . This completes the proof. □
We are now ready to prove the main results of this paper.
Theorem 2.3 Let C be a closed interval on the real line (can be unbounded) and let be a continuous function. Let , , , and be sequences in with and . Let be a sequence generated iteratively by and
where , , and .
If is bounded, then converges to a fixed point of f.
Proof Let be a bounded sequence. Then, by Lemma (2.2), is a convergent sequence. Hence, by Lemma (2.1), it converges to a fixed point of f. □
As a direct consequence of Theorem 2.3, we obtain the following result.
Theorem 2.4 Let C be a closed interval on the real line (can be unbounded) and let be a continuous function. Let , , , and be sequences in with and . Let be a sequence generated iteratively by and
where , , and .
Then converges to a fixed point of f if and only if is bounded.
Corollary 2.5 Let be a continuous function. Let , , , and be sequences in with and . Let be a sequence generated iteratively by and
where , , and .
Then converges to a fixed point of f.
If we take , then we obtain the following result.
Corollary 2.6 Let C be a closed interval on the real line (can be unbounded) and let be a continuous function. Let , and be sequences in . Let be a sequence generated iteratively by and
where , , and .
Then converges to a fixed point of f if and only if is bounded.
If we take and , then we obtain the following result.
Corollary 2.7 Let C be a closed interval on the real line (can be unbounded) and let be a continuous function. Let , and be sequences in . Let be a sequence generated iteratively by and
where , , and .
Then converges to a fixed point of f if and only if is bounded.
Remark 2.8 Corollary 2.7 extends the main result obtained in [8] from the modified Ishikawa iteration to the modified Noor iteration.