First we extend Zhang and Song's theorem (Theorem 1.4) to the case where one of the mappings is multivalued.

Theorem 4.1.

Let be a complete metric space and let and be two mappings such that for all ,

(i.e., generalized -weak contractive) where is l.s.c. with and for all . Then there exists a unique point such that .

Proof.

Unicity of the common fixed point follows from (4.1).

Obviously if and only if is a common fixed point of and .

Let and . Let . By Lemma 2.3, there exists such that

We let . Inductively, we let , and by Lemma 2.3, we choose such that

We break the argument into four steps.

Step 1.

Proof.

Using (4.1) and (4.3),

where

So . Hence by (4.4),

Also

where

So . Hence by (4.7),

Therefore, by (4.6) and (4.9), we conclude that

for all .

Therefore, the sequence is monotone nonincreasing and bounded below. So there exists such that

Since is l.s.c.,

By (4.4), we conclude that

and so . Hence .

Step 2.

is a bounded sequence.

Proof.

If were unbounded, then by Step 1, and are unbounded. We choose the sequence such that , is even and minimal in the sense that , and , and similarly is odd and minimal in the sense that , and is even and minimal in the sense that and , and is odd and minimal in the sense that and .

Obviously for every . By Step 1, there exists such that for all we have . So for every , we have and

Hence Also

and this shows that

So if is odd, then

where

and this shows that Since is l.s.c. and (4.16) holds, we have . So and this is a contradiction.

Step 3.

is Cauchy.

Proof.

Let . Since is bounded, for all . Obviously is decreasing. So there exists such that . We need to show that .

For every , there exists such that and

By (4.18), we conclude that

From Step 1 and (4.19), we have

So we may assume that for every , is odd and is even. Hence

where

This inequality shows that Since is l.s.c. and (4.21) holds, we have . Hence and so . Therefore, is a Cauchy sequence.

Step 4.

and have a common fixed point.

Proof.

Since is complete and is Cauchy, there exists such that For every

where

and this shows that

Since is l.s.c. and (4.23) holds, letting in (4.23) we get

So and hence . Since , then .Also

where

So from (4.26), we have

Thus , and hence . Therefore, .

Remark 4.2.

In the proof of Theorem in Zhang and Song [8], the boundedness of the sequence is used, but not proved. Also, for the proof that is a Cauchy sequence, the monotonicity of is used, without being explicitly mentioned.

In our proof of Theorem 4.1, which is different from [8, Theorem ], is not assumed to be nondecreasing.

The following theorem extends Rhoades' theorem by assuming to be only l.s.c..

Theorem 4.3.

Let be a complete metric space, and let be a mapping such that

for every (i.e., -weak contractive), where is an l.s.c. function with and for all . Then has a unique fixed point.

Proof.

The proof is similar to the proof of Theorem 4.1, by taking , and replacing with .

Remark 4.4.

With a similar proof as in Theorem 4.1, in Theorem 4.3 we can replace the inequality (4.29) by the following inequality (4.30) for two single valued mappings .