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The basic motivation of this paper is to extend, generalize, and improve several fundamental results on the existence (and uniqueness) of coincidence points and fixed points for well-known maps in the literature such as Kannan type, Chatterjea type, Mizoguchi-Takahashi type, Berinde-Berinde type, Du type, and other types from the class of self-maps to the class of non-self-maps in the framework of the metric fixed point theory. We establish some fixed/coincidence point theorems for multivalued non-self-maps in the context of complete metric spaces.

During the last few decades, the celebrated Banach contraction principle, also known as the Banach fixed point theorem [

Let

Let

The characterization of the renowned Banach fixed point theorem in the setting of multivalued maps is one of the most outstanding ideas of research in fixed point theory. The remarkable examples in this trend were given by Nadler [

The following attractive result was reported by M. Berinde and V. Berinde [

Let

If we take

Let

Recently, Du [

Let

The basic objective of this paper is to investigate the existence of coincidence and fixed points of multivalued non-self-maps under the certain conditions in the setting of metric spaces. The presented results generalize, improve, and extend several crucial and notable results that examine the existence of the coincidence/fixed point of well-known maps such as Kannan type, Chatterjea type, Mizoguchi-Takahashi type, Berinde-Berinde type, Du type, and other types in the context of complete metric spaces.

Let

Let

Let

A function

It is evident that if

In what follows that, we recall some characterizations of

Let

For each

For each

For each

For each

For any nonincreasing sequence

In this section, we prove the existence of coincidence points and fixed points of multivalued non-self-maps of Kannan type and Chatterjea type. For this purpose, we first established a new intersection theorem of

Let

there exist a function

Then

Since

In Theorem

there exist a function

there exist a function

there exist a function

there exist a function

there exist a function

there exist a function

there exist a function

Then

It is obvious that any of these conditions (K1)–(K7) implies condition (D3) as in Theorem

The following fixed point theorem for multivalued non-self-maps of generalized Kannan type can be established immediately from Theorem

Let

there exist a function

there exist a function

there exist a function

there exist a function

there exist a function

there exist a function

there exist a function

there exist a function

Then

As a consequence of Theorem

Let

(a) If

(b) Theorems

Let

Let

there exists

By (

In Theorem

Applying Theorem

Let

there exist a function

there exist a function

there exist a function

there exist a function

there exist a function

there exist a function

there exist a function

there exist a function

Then

The following result is a generalized Chatterjea’s type fixed point theorem for multivalued maps in complete metric spaces.

Let

(a) If

(b) Theorems

In this section, we prove some coincidence and fixed point theorems for multivalued non-self-maps of Mizoguchi-Takahashi type, Berinde-Berinde type, and Du type.

Recall first the following auxiliary result.

Let

Let

Since

In Theorem

Let

there exist an

Then

Let

there exist an

Then

As a direct consequence of Theorems

Let

there exist an

there exist an

Then

Let

The following fixed point theorems for multivalued non-self-maps of generalized Berinde-Berinde type and generalized Mizoguchi-Takahashi type are established immediately from Theorem

Let

there exist an

Then

Let

there exist an

Then

Let

there exists an

Then

(a) If

(b) Theorems

The first author was supported partially by grant no. NSC 101-2115-M-017-001 of the National Science Council of the Republic of China.