We prove a lemma which plays key role to establish strong convergence of the iterative schemes (1.3) and (1.4).

Lemma 2.1.

Let be a uniformly convex metric space. Let be a nonempty closed convex subset of a quasi-nonexpansive map and as in (1.3). If and then

Proof.

For , we consider

This implies that the sequence is nonincreasing and bounded below. Thus exists. We may assume that

For any , we have that

Since exists, so is bounded and hence exists. We show that . Assume that

Then

Hence by Theorem 1.3, there exists such that

That is,

Taking and summing up the terms on the both sides in the above inequality, we have

Let . Then, we have

This is contradiction and hence

In the light of above result, we can construct subsequences and of and , respectively, such that and hence

Now we state and prove Ishikawa type convergence result in uniformly convex metric spaces.

Theorem 2.2.

Let be a uniformly convex complete metric space with continuous convex structure and let be its nonempty closed convex subset. Let be a continuous quasi-nonexpansive map of into itself satisfying Condition 3. If is as in (1.3), where and , then converges to a fixed point of .

Proof.

In Lemma 2.1, we have shown that Therefore . This implies that the sequence is nonincreasing and bounded below. Thus exists. Now by Condition 3, we have

Using the properties of we have . As exists, therefore

Now, we show that is a Cauchy sequence. For there exists a constant such that for all we have In particular, That is, There must exist such that Now, for , we have that

This proves that is a Cauchy sequence in . Since is a closed subset of a complete metric space therefore it must converge to a point in .

Finally, we prove that is a fixed point of

Since

therefore As is closed, so

Choose for all in the above theorem; it reduces to the following Mann type convergence result.

Theorem 2.3.

Let be a uniformly convex complete metric space with continuous convex structure and let be its nonempty closed convex subset. Let be a continuous quasi-nonexpansive map of into itself satisfying Condition 1. If is as in (1.4), where , then converges to a fixed point of .

Next we establish strong convergence of Ishikawa iterates of a generalized nonexpansive map.

Theorem 2.4.

Let and be as in Theorem 2.3. Let be a continuous generalied nonexpansive map of into itself with at least one fixed point. If is as in (1.3), where and then converges to a fixed point of .

Proof.

Let be any fixed point of Then setting in (*), we have

which implies

Thus is quasi-nonexpansive.

For any we also observe that

If where then

Using (2.14) in (2.13), we have

Also it is obvious that

Combining (2.16) and (2.17), we get that

Now inserting (2.15) in (2.18), we derive

That is,

where Thus satisfies Condition 4 (and hence Condition 3). The result now follows from Theorem 2.2.

Remark 2.5.

In the above theorem, we have assumed that the generalied nonexpansive map has a fixed point. It remains an open questions: what conditions on , and in (*) are sufficient to guarantee the existence of a fixed point of even in the setting of a metric space.

Choose for all in Theorem 2.4 to get the following Mann type convergence result.

Theorem 2.6.

Let and be as in Theorem 2.4. If is as in (1.4), where then converges to a fixed point of .

Proof.

For for all , the inequality (2.20) in the proof of Theorem 2.4 becomes

Thus satisfies Condition 2 (and hence Condition 1) and so the result follows from Theorem 2.3.

The analogue of Kannan result in uniformly convex metric space can be deduced from Theorem 2.6 (by taking , and for all ) as follows.

Theorem 2.7.

Let be a uniformly convex complete metric space with continuous convex structure and let be its nonempty closed convex subset. Let be a continuous map of into itself with at least one fixed point such that for all . Then the sequence where and converges to a fixed point of

Next we give sufficient conditions for the existence of fixed point of a -Lipschitz map in terms of the Ishikawa iterates.

Theorem 2.8.

Let be a convex metric space and let be its nonempty convex subset. Let be a -Lipschitz selfmap of Let be the sequence as in (1.3), where and satisfy (i) for all (ii) and (iii) If and then is a fixed point of

Proof.

Let Then

That is,

Since therefore there exists such that for all This implies that

Taking on both the sides in the above inequality and using the condition , we have

Finally, using a generalized nonexpansive map on a metric space , we provide a necessary and sufficient condition for the convergence of an arbitrary sequence in to a fixed point of in terms of the approximating sequence

Theorem 2.9.

Suppose that is a closed subset of a complete metric space and is a continuous map such that for some , the following inequality holds:

for all Then a sequence in converges to a fixed point of if and only if

Proof.

Suppose that First we show that is a Cauchy sequence in To acheive this goal, consider:

That is,

Since and therefore from the above inequality, it follows that is a Cauchy sequence in In view of closedness of this sequence converges to an element of Also gives that Now using the continuity of we have Hence is a fixed point of

Conversely, suppose that converges to a fixed point of Using the continuity of we have that Thus

Remark 2.10.

Theorem 2.8 improves Lemma 2 in [3] from real line to convex metric space setting. Theorem 2.9 is an extension of Theorem 4 in [21] to metric spaces. If we choose in Theorem 2.9, it is still an improvement of [21, Theorem 4].

Remark 2.11.

We have proved our results (2.1)–(2.8) in convex metric space setting. All these results, in particular, hold in Banach spaces if we set