We give a Ćirić-Berinde type contractive condition for multivalued mappings and analyze the existence of fixed point for these mappings.

In 2012, Samet et al. [

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Very recently, Ali et al. [

The purpose of this paper is to introduce the notion of Ćirić-Berinde type contractive multivalued mappings and to generalize and extend the notion of

Let

For

Then, we have

From now on, we denote by

We denote by

Also, we denote by

Note that if

Let

We consider the following conditions:

for any sequence

for any sequence

for any sequence

(1) implies (2) and (2) implies (3).

Note that if

Let

where

Let

Suppose that

Let

Let

Since

Let

If

Let

Since

Let

A function

A function

For a multivalued map

In this section, we establish fixed point theorems for Ćirić-Berinde type contractive multivalued mappings.

Let

Assume that, for all

Also, suppose that the following are satisfied:

there exists

either

Then

Let

If

Let

If

From (

If

Thus,

Hence, there exists

Since

If

Then

From (

If

Thus,

Hence, there exists

Since

By induction, we obtain a sequence

Let

Since

For all

It follows from the completeness of

Suppose that

We have

By letting

Assume that

Then,

Let

Assume that, for all

Also, suppose that conditions (1) and (2) of Theorem

Then

If we have

Let

Let

Assume that, for each

Also, suppose that the following are satisfied:

there exists

either

Then

If we have

From Theorem

Let

Assume that, for all

Also, suppose that conditions (1) and (2) of Theorem

Then

If we have

Let

Assume that, for all

Also, suppose that conditions (1) and (2) of Theorem

Then

In Corollary

Let

Assume that, for all

there exists

for a sequence

Then

Following the proof of Theorem

From (2) there exists a subsequence

Thus, we have

We have

Suppose that

Since

Letting

The following example shows that upper semicontinuity of

Let

Define a mapping

Let

Then,

Let

Obviously, condition (2) of Theorem

We show that (

Let

Then,

If

Let

If

Let

Then, we have

Thus, (

We now show that

Let

Then,

Obviously,

If

Hence,

Note that

Let

Assume that, for all

Then

Let

Assume that, for each

Also, suppose that the following are satisfied:

there exists

for a sequence

Then

Corollary

Let

Assume that, for all

Also, suppose that conditions (1) and (2) of Theorem

Then

By taking

Let

Assume that, for all

Also, suppose that conditions (1) and (2) of Theorem

Then

The author declares that there is no conflict of interests regarding the publication of this paper.

The author would like to thank the anonymous reviewers for their valuable comments.