Related Fixed Points of Multivalued Mappings of Two Related Orbitally Complete Metric Spaces

Abstract

In the present paper, the concept of the related orbital completeness of two metric spaces for multivalued mappings is introduced. A new related fixed point theorem for multivalued mappings is proven, and some important results are obtained as corollaries of the main theorem. A single-valued version of the main theorem is also derived as a simple corollary, and two illustrative examples are provided.

1. Introduction

Following the famous theorem known as the Contraction Mapping Principle, established by Banach in 1922, fixed point theory has become a fundamental area of research in modern mathematical analysis due to its wide range of applications in nonlinear analysis, differential equations, topology, optimization, computer science, etc. This foundational result has served as a basis for a wide range of extensions and generalizations in fixed point theory. In particular, various authors have pursued different research directions by relaxing the contraction conditions, modifying the completeness assumptions, or by introducing more general structures, such as partial metric spaces, fuzzy metric spaces, and spaces endowed with graph structures. These efforts have broadened the applicability of the original result to more abstract or complex mathematical settings.

Fixed point theorems, particularly generalizations developed for multivalued mappings defined on two different metric spaces, are powerful tools in modeling real-world systems across various disciplines. Such results can serve as a cornerstone in many applications, ranging from differential equations to artificial intelligence, game theory, and network analysis.

Fisher [1,2] established a remarkable result concerning the fixed points of compositions of two mappings on two complete metric spaces, providing a relationship between the fixed points of these mappings. Since then, numerous researchers have extended this result in various directions, exploring different types of theorem for two or more mappings. Examples of such works include [2,3,4,5,6,7,8,9] and others.

In 2000, Fisher and Türkoğlu [10], by considering the multivalued version of the related fixed point theorem for single-valued mappings in [2], established several related fixed point theorems for multivalued mappings on two complete and compact metric spaces. In addition, Chourasia and Fisher [11], Jain and Fisher [12], Popa [13], Rohen and Murthy [14], and Biçer et al. [15] have also proved some related fixed point theorems for multivalued mappings under certain contractive conditions.

In this paper, by considering the multivalued version of the related fixed point theorem for single-valued mappings, we present a new type of related fixed point theorem for multivalued mappings in two related complete metric spaces.

The following are some characteristics of the present work.

• The concept of the related orbital completeness of two metric spaces for multivalued mappings is introduced.
• A new related fixed point theorem for multivalued mappings is established.
• While in existing related fixed point theorems for multivalued mappings, at least two or more contraction conditions are used, in the present main theorem, only one contraction condition is used.
• Unlike other existing theorems for multivalued mappings, only the classical metric is used in the contraction conditions in the main results presented.
• A multivalued version of the Bollenbacher and Hicks’s result [16] is obtained as a corollary of the present main theorem.
• A single-valued version of the present main theorem is obtained like a simple corollary.
• Two illustrative examples are given.

2. Preliminaries

Let $(Z,\rho )$ be a metric space. Throughout the paper we denote by $P\left(Z\right)$ the family of all nonempty subsets of Z, by $CL\left(Z\right)$ the family of all nonempty closed subsets of Z, and by $CB\left(Z\right)$ the family of all nonempty closed bounded subsets of Z.

Let $z_0\in Z$ and let F be a mapping of Z into $P\left(Z\right)$. We shall use the following definitions.

Definition 1 

([17]). An orbit of the multivalued mapping F at the point of $z_0\in Z$ is a sequence

$$O\left(z_0\right)=\{z_n:z_n\in F{z}_{n-1}\}.$$

Definition 2 

([17]). A metric space $(Z,\rho )$ is said to be F-orbitally complete if every Cauchy sequence of the form $\{z_{n_i}:z_{n_i}\in F{z}_{n_i-1}\}$ converges in Z.

Definition 3. 

Let $(Z,\rho )$ and $(Y,\varrho )$ be two metric spaces. Let F be a mapping of Z into $P\left(Y\right)$ and G be a mapping of Y into $P\left(Z\right)$. Let $z_0\in Z$ and $y_0\in Y.$

Consider the following sets,

$$\begin{array}{cc}\hfill O_Z(z_0,y_0)& =\{z_n:z_n\in G{y}_{n-1},n=1,2,\dots \},\hfill \\ \hfill O_Y(z_0,y_0)& =\{y_n:y_n\in F{z}_{n-1},n=1,2,\dots \}.\hfill \end{array}$$

(1)

Then, the metric spaces $(Z,\rho )$ and $(Y,\varrho )$ are called related $FG$-orbitally complete for $(z_0,y_0)\in Z\times Y$ if every Cauchy sequence in $O_Z(z_0,y_0)$ and $O_Y(z_0,y_0)$ converges to a point in Z and converges to a point in Y, respectively.

Note that the two metric spaces $(Z,\rho )$ and $(Y,\varrho )$ are related $FG$-orbitally complete. However, the related $FG$-orbitally complete $(Z,\rho )$ and $(Y,\varrho )$ metric spaces are not necessarily complete, as is shown by the following example.

Example 1. 

Let $Z=Y=[0,1)$ with the Euclidean metric ρ. Let the mappings $F,G:Z\to P\left(Z\right)$ be defined by $F\left(z\right)=[0,z/2], G\left(z\right)=[0,z/4]$ for all $z\in Z$. Then, for $z_0=0$ and $y_0=0$, we have

$$O_Z(0,0)=\{0,0,0,\dots \},O_Y(0,0)=\{0,0,0,\dots \}.$$

Therefore, $(Z,\rho )$ and $(Y,d)$ are related $FG$-orbitally complete for $(0,0)\in Z\times Y$, but $(Z,\rho )$ is not complete.

In a recent paper [18], Romaguera introduced the definition of 0-lower semicontinuity as a generalization of lower semicontinuity.

Similarly, we use the following definition.

Definition 4. 

Let $(Z,\rho )$ and $(Y,\varrho )$ be two metric spaces. We shall say that a real-valued function $\mu :Z\times Y\to [0,\infty )$ is $FG$-orbitally 0-lower semicontinuous (briefly 0-lsc) at $(z,y)\in Z\times Y$ with respect to $(z_0,y_0)$ if ${\lim }_{n\to \infty }{z}_n=z, {\lim }_{n\to \infty }{y}_n=y$ and ${\lim }_{n\to \infty }\mu (z_n,y_n)=0$, then $\mu (z,y)=0$, where $\left\{z_n\right\}$ and $\left\{y_n\right\}$ are sequences in $O_Z(z_0,y_0)$ and $O_Y(z_0,y_0)$, respectively.

Definition 5. 

Let F be a mapping of Z into $P\left(Y\right)$ and G be a mapping of Y into $P\left(Z\right)$. Then, the composition of the mappings F and G is defined by

$$\left(GF\right)\left(z\right)=\bigcup _{v\in F\left(z\right)}G\left(v\right)\mathit{for}z\in Z,\left(FG\right)\left(y\right)=\bigcup _{u\in G\left(y\right)}F\left(u\right)\mathit{for}y\in Y.$$

3. Main Results

In this section, firstly we give the following related fixed point theorem in two related orbitally complete metric spaces, $(Z,\rho )$ and $(Y,\varrho )$.

Theorem 1. 

Let $(Z,\rho )$ and $(Y,\varrho )$ be two metric spaces and let F be a mapping of Z into $P\left(Y\right)$ and G be a mapping of Y into $P\left(Z\right)$. Suppose there exist $z^{\prime }\in G\left(y\right)$ and $y^{\prime }\in F\left(z\right)$ such that

$$\max \{\rho (z,z^{\prime }),\varrho (y,y^{\prime })\}\le \gamma (z,y)-\gamma (z^{\prime },y^{\prime })$$

(2)

for all $z\in Z$ and $y\in Y$, where $\gamma :Z\times Y\to [0,\infty )$. If $(Z,\rho )$ and $(Y,\varrho )$ are related $FG$-orbitally complete for some $(z_0,y_0)\in Z\times Y$, then we have

(a) 

There exist two sequences $\left\{z_n\right\}$ in $O_Z(z_0,y_0)$ and $\left\{y_n\right\}$ in $O_Y(z_0,y_0)$ such that

$$\underset{n\to \infty }{\lim }{z}_n=u\in Zand\underset{n\to \infty }{\lim }{y}_n=v\in Y,$$

(b) 

$\max \{\rho (z_n,u),\varrho (y_n,v)\}\le \gamma (z_0,y_0)$ for all $n=1,2,\dots$,

(c) 

If F is a mapping of Z into $CL\left(Y\right)$ and G is a mapping of Y into $CL\left(Z\right)$, then the following statements are equivalent:

(i) 

$u\in G\left(v\right)$ and $v\in F\left(u\right)$.

(ii) 

$\mu :Z\times Y⟶[0,\infty )$, $\mu (z,y)=\rho (z,G(y\left)\right)$ and $\eta :Z\times Y⟶[0,\infty )$, $\eta (z,y)=\varrho (y,F(z\left)\right)$ are $FG$-orbitally 0-lsc at $(u,v)$ with respect to $(z_0,y_0)$, where $\rho (z,G(y\left)\right)=\inf \left\{\rho \right(z,w):w\in G(y\left)\right\}$ and $\varrho (y,F(z\left)\right)=\inf \left\{\varrho \right(y,x):x\in F(z\left)\right\}$.

(iii) 

$\rho (u,G(v\left)\right)=0$ and $\varrho (v,F(u\left)\right)=0$.

Further, if $u\in G\left(v\right)$ and $v\in F\left(u\right)$, then $u\in \left(GF\right)\left(u\right)$ and $v\in \left(FG\right)\left(v\right)$.

Proof. 

Suppose that $(Z,\rho )$ and $(Y,\varrho )$ are related $FG$-orbitally complete for $z_0\in Z$ and $y_0\in Y$. Then, from Inequality (2), there exist $z_1\in G\left(y_0\right)$ and $y_1\in F\left(z_0\right)$ such that

$$\max \{\rho (z_0,z_1),\varrho (y_0,y_1)\}\le \gamma (z_0,y_0)-\gamma (z_1,y_1).$$

Similarly, there exist $z_2\in G\left(y_1\right)$ and $y_2\in F\left(z_1\right)$ such that

$$\max \{\rho (z_1,z_2),\varrho (y_1,y_2)\}\le \gamma (z_1,y_1)-\gamma (z_2,y_2),$$

and continuing in this way, we obtain two sequences $\left\{z_n\right\}$ in $O_Z(z_0,y_0)$ and $\left\{y_n\right\}$ in $O_Y(z_0,y_0)$ such that $z_n\in G\left(y_{n-1}\right)$ and $y_n\in F\left(z_{n-1}\right)$, and

$$\max \{\rho (z_{n-1},z_n),\varrho (y_{n-1},y_n)\}\le \gamma (z_{n-1},y_{n-1})-\gamma (z_n,y_n)$$

(3)

for all $n=1,2,\dots$.

Now we shall show that the sequences $\left\{z_n\right\}$ and $\left\{y_n\right\}$ are the Cauchy sequences.

Using Inequality (3), we get

$$\begin{array}{cc}\hfill r_n& =\sum _{k=1}^n\max \{\rho (z_{k-1},z_k),\varrho (y_{k-1},y_k)\}\hfill \\ \hfill & \le \sum _{k=1}^n[\gamma (z_{k-1},y_{k-1})-\gamma (z_k,y_k)]\hfill \\ \hfill & =\gamma (z_0,y_0)-\gamma (z_n,y_n)\le \gamma (z_0,y_0).\hfill \end{array}$$

Therefore, $\left\{r_n\right\}$ is bounded and also non-decreasing. Thus, $\left\{r_n\right\}$ is convergent. Let $m,n$ be any two positive integers with $m>n$. From the triangle inequality property of the metrics $\rho$ and $\varrho$, we have

$$\begin{array}{cc}\hfill \max \{\rho (z_n,z_m),\varrho (y_n,y_m)\}& \le \max \left\{\sum _{i=n}^{m-1}\rho (z_i,z_{i+1}),\sum _{i=n}^{m-1}\varrho (y_i,y_{i+1})\right\}\hfill \\ \hfill & \le \sum _{i=n}^{m-1}\max \{\rho (z_i,z_{i+1}),\varrho (y_i,y_{i+1})\}.\hfill \end{array}$$

(4)

Since $\left\{r_n\right\}$ is convergent, for any $\epsilon >0$, we can choose a positive integer $n_0$ such that

$$\sum _{i=n}^{\infty }\max \{\rho (z_i,z_{i+1}),\varrho (y_i,y_{i+1})\}<\epsilon$$

for all $n\ge n_0$. Thus, based on Inequality (4), we get

$$\max \{\rho (z_n,z_m),\varrho (y_n,y_m)\}<\epsilon$$

for all $m,n\ge n_0$, and thus $\left\{z_n\right\}$ and $\left\{y_n\right\}$ are two Cauchy sequences in $O_Z(z_0,y_0)$ and $O_Y(z_0,y_0)$, respectively. Since $(Z,\rho )$ and $(Y,\varrho )$ are related $FG$-orbitally complete, the sequence $\left\{z_n\right\}$ has a limit u in Z and the sequence $\left\{y_n\right\}$ has a limit v in Y. Thus, the proof of (a) is complete.

To prove (b), let $m>n$. Then, from Inequalities (2) and (3), we get

$$\begin{array}{cc}\hfill \max \{\rho (z_n,z_m),\varrho & (y_n,y_m)\}\le \sum _{i=n}^{m-1}\max \{\rho (z_i,z_{i+1}),\varrho (y_i,y_{i+1})\}\hfill \\ \hfill & \le \sum _{i=0}^{m-1}[\gamma (z_i,y_i)-\gamma (z_{i+1},z_{i+1})]\hfill \\ \hfill & =\gamma (z_0,y_0)-\gamma (z_m,y_m)\le \gamma (z_0,y_0).\hfill \end{array}$$

Letting m tend to infinity, it follows that

$$\max \{\rho (z_n,u),\varrho (y_n,v)\}\le \gamma (z_0,y_0)$$

Thus the proof of (b) is complete.

Now suppose that F is a mapping of Z into $CL\left(Y\right)$ and G is a mapping of Y into $CL\left(Z\right)$.

(i) ⇒ (ii): Assume that $u\in G\left(v\right)$ and $v\in F\left(u\right)$. Clearly,

$$\rho (u,G(v\left)\right)=0\mathrm{and}\varrho (v,F(u\left)\right)=0.$$

Let $\left\{z_n\right\}$, $\left\{y_n\right\}$ be two sequences in $O_Z(z_0,y_0)$ and $O_Y(z_0,y_0)$, respectively, with $z_n\to u$, $y_n\to v$. Then, we get

$$\underset{n\to \infty }{\lim }\mu (z_n,y_n)=\underset{n\to \infty }{\lim }\rho (z_n,G\left(y_n\right))\le \underset{n\to \infty }{\lim }\rho (z_n,z_{n+1})=0$$

and

$$\underset{n\to \infty }{\lim }\eta (z_n,y_n)=\underset{n\to \infty }{\lim }\varrho (y_n,F\left(z_n\right))\le \underset{n\to \infty }{\lim }\varrho (y_n,y_{n+1})=0$$

and so $\mu$ and $\eta$ be $FG$-orbitally 0-lsc at $(u,v)$ with respect to $(z_0,y_0)$ since

$$\mu (u,v)=\rho (u,G(v\left)\right)=0\mathrm{and}\eta (u,v)=\varrho (v,F(u\left)\right)=0.$$

(ii) ⇒ (iii): Let $\mu$ and $\eta$ are $FG$-orbitally 0-lsc at $(u,v)$ with respect to $(z_0,y_0)$. From (a), there exist two sequences $\left\{z_n\right\}$ in $O_Z(z_0,y_0)$ and $\left\{y_n\right\}$ in $O_Y(z_0,y_0)$ such that $z_n\to u,y_n\to v$. We also have

$$\underset{n\to \infty }{\lim }\mu (z_n,y_n)=\underset{n\to \infty }{\lim }\rho (z_n,G\left(y_n\right))\le \underset{n\to \infty }{\lim }\rho (z_n,z_{n+1})=0$$

and

$$\underset{n\to \infty }{\lim }\eta (z_n,y_n)=\underset{n\to \infty }{\lim }\varrho (y_n,F\left(z_n\right))\le \underset{n\to \infty }{\lim }\varrho (y_n,y_{n+1})=0.$$

Since $\mu$ and $\eta$ are $FG$-orbitally 0-lsc at $(u,v)$ with respect to $(z_0,y_0)$,

$$\begin{array}{cc}\hfill \rho (u,G(v\left)\right)& =\inf \left\{\rho \right(u,w):w\in G(v\left)\right\}=\mu (u,v)=0\mathrm{and}\hfill \\ \hfill \varrho (v,F(u\left)\right)& =\inf \left\{\varrho \right(v,x):x\in F(u\left)\right\}=\eta (u,v)=0.\hfill \end{array}$$

(iii) ⇒ (i): Now let $\rho (u,G(v\left)\right)=\inf \left\{\rho \right(u,w):w\in G(v\left)\right\}=0$. Then we have $u\in \overline{G\left(v\right)}$. Since $G\left(v\right)$ is a closed subset of Z, $\overline{G\left(v\right)}=G\left(v\right)$, and so $u\in G\left(v\right)$. Similarly, if $\varrho (v,F(u\left)\right)=\inf \left\{\varrho \right(v,x):x\in F(u\left)\right\}=0$, then $v\in F\left(u\right)$.

We now assume that $u\in G\left(v\right)$ and $v\in F\left(u\right)$. Then, $G\left(v\right)\subset \left(GF\right)\left(u\right)$, since $\left(GF\right)\left(u\right)={\bigcup }_{a\in F\left(u\right)}G\left(a\right)$. Therefore, $u\in \left(GF\right)\left(u\right)$. Similarly, $F\left(u\right)\subset \left(FG\right)\left(v\right)$, since $\left(FG\right)\left(v\right)={\bigcup }_{b\in G\left(v\right)}F\left(b\right)$ and so $v\in \left(FG\right)\left(v\right)$, which completes the proof. □

Note that since the inequality

$$\max \{\rho (z,z^{\prime }),\varrho (y,y^{\prime })\}\le \rho (z,z^{\prime })+\varrho (y,y^{\prime })$$

holds for all $z,z^{\prime }\in Z$ and $y,y^{\prime }\in Y$, we obtain the following result.

Corollary 1. 

Let $(Z,\rho )$ and $(Y,\varrho )$ be two metric spaces and let F be a mapping of Z into $P\left(Y\right)$ and G be a mapping of Y into $P\left(Z\right)$. Suppose there exist $z^{\prime }\in G\left(y\right)$ and $y^{\prime }\in F\left(z\right)$ such that

$$\rho (z,z^{\prime })+\varrho (y,y^{\prime })\le \gamma (z,y)-\gamma (z^{\prime },y^{\prime })$$

(5)

for all $z\in Z$ and $y\in Y$, where $\gamma :Z\times Y\to [0,\infty )$. If $(Z,\rho )$ and $(Y,\varrho )$ are related $FG$-orbitally complete for some $(z_0,y_0)\in Z\times Y$, then

(a) 

There exist two sequences $\left\{z_n\right\}$ in $O_Z(z_0,y_0)$ and $\left\{y_n\right\}$ in $O_Y(z_0,y_0)$ such that

$$\underset{n\to \infty }{\lim }{z}_n=u\in Xand\underset{n\to \infty }{\lim }{y}_n=v\in Y,$$

(b) 

$\rho (z_n,u)+\varrho (y_n,v)\le \gamma (z_0,y_0)$ for all $n=1,2,\dots$,

(c) 

If F is a mapping of Z into $CL\left(Y\right)$ and G is a mapping of Y into $CL\left(Z\right)$, then the following statements are equivalent:

(i) 

$u\in G\left(v\right)$ and $v\in F\left(u\right)$.

(ii) 

$\mu :Z\times Y⟶[0,\infty )$, $\mu (z,y)=\rho (z,G(y\left)\right)$ and $\eta :Z\times Y⟶[0,\infty )$, $\eta (z,y)=\varrho (y,F(z\left)\right)$ are $FG$-orbitally 0-lsc at $(u,v)$ with respect to $(z_0,y_0)$, where $\rho (z,G(y\left)\right)=\inf \left\{\rho \right(z,w):w\in G(y\left)\right\}$ and $\varrho (y,F(z\left)\right)=\inf \left\{\varrho \right(y,x):x\in F(z\left)\right\}$.

(iii) 

$\rho (u,G(v\left)\right)=0$ and $\varrho (v,F(u\left)\right)=0$.

Further, if $u\in G\left(v\right)$ and $v\in F\left(u\right)$, then $u\in \left(GF\right)\left(u\right)$ and $v\in \left(FG\right)\left(v\right)$.

Proof. 

We have

$$\begin{array}{cc}\hfill \max \{\rho (z,z^{\prime }),\varrho (y,y^{\prime })\}& \le \rho (z,z^{\prime })+\varrho (y,y^{\prime })\hfill \\ \hfill & \le \gamma (z,y)-\gamma (z^{\prime },y^{\prime }).\hfill \end{array}$$

Then the results (a) and (c) follow immediately from Theorem 1.

To prove (b), let $m>n$. Similarly to in the proof of (b) in Theorem 1, using Inequality (5), we get

$$\begin{array}{cc}\hfill \rho (z_n,z_m)+& \varrho (y_n,y_m)\le \sum _{i=n}^{m-1}[\rho (z_i,z_{i+1})+\varrho (y_i,y_{i+1})]\hfill \\ \hfill & \le \sum _{i=0}^{m-1}[\gamma (z_i,y_i)-\gamma (z_{i+1},y_{i+1})]\le \gamma (z_0,y_0).\hfill \end{array}$$

Letting m tend to infinity, it follows that

$$\rho (z_n,u)+\varrho (y_n,v)\le \gamma (z_0,y_0).$$

□

If we let

$$(Z,\rho )=(Y,\varrho ),T=F=G\mathrm{and}\gamma (z,y)=\vartheta \left(z\right),$$

where $\vartheta :Z\to [0,\infty )$, then from Corollary 1, we have the following multivalued version of Bollenbacher and Hicks’s result [16], which is a version of Caristi’s famous fixed point theorem [19].

Corollary 2. 

Let $(Z,\rho )$ be a metric space and let T be a mapping of Z into $P\left(Z\right)$. Suppose there exists $z^{\prime }\in T\left(z\right)$ such that

$$\rho (z,z^{\prime })\le \vartheta \left(z\right)-\vartheta \left(z^{\prime }\right)$$

(6)

for each $z\in Z$, where $\vartheta :Z\to [0,\infty )$. If $(Z,\rho )$ is T-orbitally complete for some $z_0\in Z$, then

(a) 

There exists a sequence $\left\{z_n\right\}$ in $O_Z\left(z_0\right)$ such that ${\lim }_{n\to \infty }{z}_n=u\in Z$,

(b) 

$\rho (z_n,u)\le \vartheta \left(z_0\right)$ for all $n=1,2,\dots$,

(c) 

If T is a mapping of Z into $CL\left(Z\right)$, then the following statements are equivalent:

(i) 

$u\in T\left(u\right)$.

(ii) 

$\mu :Z⟶[0,\infty )$, $\mu \left(z\right)=\rho (z,T(z\left)\right)$ is T-orbitally 0-lsc at u with respect to $z_0$, where $\rho (z,T(z\left)\right)=\inf \left\{\rho \right(z,x):x\in T(z\left)\right\}$.

(iii) 

$\rho (u,T(u\left)\right)=0$.

Corollary 3. 

Let $(Z,\rho )$ be a metric space and let T be a mapping of Z into $CB\left(Z\right)$. Suppose there exists $z^{\prime }\in T\left(z\right)$ such that

$$\rho (z^{\prime },z^{″})\le k\rho (z,z^{\prime })$$

(7)

for each $z\in Z$ and for all $z^{″}\in T\left(z^{\prime }\right)$, where $0\le k<1$. If $(Z,\rho )$ is T-orbitally complete for some $z_0\in Z$, then

(a) 

There exists a sequence $\left\{z_n\right\}$ in $O_Z\left(z_0\right)$ such that ${\lim }_{n\to \infty }{z}_n=u\in Z$,

(b) 

$\rho (z_n,u)\le {\displaystyle \frac{1}{1-k}}{\sup }_{z\in T\left(z_0\right)}\rho (z_0,z)$ for all $n=1,2,\dots$,

(c) 

The following statements are equivalent:

(i) 

$u\in T\left(u\right)$.

(ii) 

$\mu :Z⟶[0,\infty )$, $\mu \left(z\right)=\rho (z,T(z\left)\right)$ is T-orbitally 0-lsc at u with respect to $z_0$, where $\rho (z,T(z\left)\right)=\inf \left\{\rho \right(z,x):x\in T(z\left)\right\}$.

(iii) 

$\rho (u,T(u\left)\right)=0$.

Proof. 

Define the function $\vartheta$ on Z by $\vartheta \left(z\right)={\displaystyle \frac{1}{1-k}}\left({\sup }_{x\in T\left(z\right)}\rho (z,x)\right)$. Since $T\left(z\right)$ is bounded, $\vartheta$ is a mapping of Z into $[0,\infty )$. Then, from Inequality (7), we get

$$\underset{z^{″}\in T\left(z^{\prime }\right)}{\sup }\rho (z^{\prime },z^{″})\le k\left(\underset{z^{\prime }\in T\left(z\right)}{\sup }\rho (z,z^{\prime })\right).$$

(8)

Thus, from Inequality (8), we have

$$\begin{array}{cc}\hfill \underset{z^{\prime }\in T\left(z\right)}{\sup }\rho (z,z^{\prime })& -k\left(\underset{z^{\prime }\in T\left(z\right)}{\sup }\rho (z,z^{\prime })\right)\hfill \\ \hfill & \le \underset{z^{\prime }\in T\left(z\right)}{\sup }\rho (z,z^{\prime })-\underset{z^{″}\in T\left(z^{\prime }\right)}{\sup }\rho (z^{\prime },z^{″})\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{z^{\prime }\in T\left(z\right)}{\sup }\rho (z,z^{\prime })& \le {\displaystyle \frac{1}{1-k}}\left(\underset{z^{\prime }\in T\left(z\right)}{\sup }\rho (z,z^{\prime })\right)-{\displaystyle \frac{1}{1-k}}\left(\underset{z^{″}\in T\left(z^{\prime }\right)}{\sup }\rho (z^{\prime },z^{″})\right)\hfill \\ \hfill & =\vartheta \left(z\right)-\vartheta \left(z^{\prime }\right)\hfill \end{array}$$

and so

$$\rho (z,z^{\prime })\le \vartheta \left(z\right)-\vartheta \left(z^{\prime }\right).$$

Hence, the results follow, since all the conditions of Corollary 2 are satisfied. □

We need the following definition for the next corollary.

Definition 6. 

If we let F be a single-valued mapping f of Z into Y and G be a single-valued mapping g of Y into Z, then from (1) we get

$$\begin{array}{cc}\hfill O_Z(z_0,y_0)& =\{z_n:z_n=g\left(y_{n-1}\right),n=1,2,\dots \},\hfill \\ \hfill O_Y(z_0,y_0)& =\{y_n:y_n=f\left(z_{n-1}\right),n=1,2,\dots \},\hfill \end{array}$$

where $z_0\in Z$ and $y_0\in Y$.

Then, the metric spaces $(Z,\rho )$ and $(Y,\varrho )$ are called related $fg$-orbitally complete for $(z_0,y_0)\in Z\times Y$ if every Cauchy sequence in $O_Z(z_0,y_0)$ and $O_Y(z_0,y_0)$ converges to a point in Z and converges to a point in Y, respectively.

We finally give the following corollary for single-valued mappings.

Corollary 4. 

Let $(Z,\rho )$ and $(Y,\varrho )$ be two metric spaces and let f be a mapping of Z into Y and g be a mapping of Y into Z satisfying the inequality

$$\max \left\{\rho \right(z,g\left(y\right)),\varrho (y,f\left(z\right)\left)\right\}\le \gamma (z,y)-\gamma \left(g\right(y),f(z\left)\right)$$

(9)

for all $z\in Z$ and $y\in Y$, where $\gamma :Z\times Y\to [0,\infty )$. If $(Z,\rho )$ and $(Y,\varrho )$ are related $fg$-orbitally complete for some $(z_0,y_0)\in Z\times Y$, then

(a) 

${\lim }_{n\to \infty }{z}_n=g\left(y_{n-1}\right)=u\in Zand{\lim }_{n\to \infty }{y}_n=f\left(z_{n-1}\right)=v\in Y,$ for $z_0\in Z$ and $y_0\in Y$, exist.

(b) 

$\max \{\rho (z_n,u),\varrho (y_n,v)\}\le \gamma (z_0,y_0)$ for all $n=1,2,\dots$,

(c) 

$g\left(v\right)=u$ and $f\left(u\right)=v$ if and only if $\mu :Z\times Y⟶[0,\infty )$, $\mu (z,y)=\rho (z,g(y\left)\right)$ and $\eta :Z\times Y⟶[0,\infty )$, $\eta (z,y)=\varrho (y,f(z\left)\right)$ are $fg$-orbitally 0-lsc at $(u,v)$ with respect to $(z_0,y_0)$.

Further, if $u=g\left(v\right)$ and $v=f\left(u\right)$, then $u=\left(gf\right)\left(u\right)$ and $v=\left(fg\right)\left(v\right)$.

Proof. 

Define two mappings f of Z into $P\left(Y\right)$ and g of Y into $P\left(Z\right)$ by putting $F\left(z\right)=f\left(z\right)$ for all $z\in Z$ and $G\left(y\right)=g\left(y\right)$ for all $y\in Y$, respectively. It follows that F and G satisfy Inequality (2). Then, the results (a) and (b) follow, since all the conditions of Theorem 1 are satisfied.

Now we prove (c). Suppose that $f\left(u\right)=v$, $g\left(v\right)=u$ and $\left\{z_n\right\}$ and $\left\{y_n\right\}$ are sequences in $O_Z(z_0,y_0)$ and $O_Y(z_0,y_0)$, respectively, with $z_n\to u,y_n\to v$. Then, we get

$$\underset{n\to \infty }{\lim }\mu (z_n,y_n)=\underset{n\to \infty }{\lim }\rho (z_n,g\left(y_n\right))=\underset{n\to \infty }{\lim }\rho (z_n,z_{n+1})=0$$

and also $\mu (u,v)=\rho (u,g(v\left)\right)=\rho (u,u)=0$.

Similarly we have ${\lim }_{n\to \infty }\eta (z_n,y_n)=0$ and $\eta (u,v)=0$. Thus, $\mu$ and $\eta$ are 0-lsc at $(u,v)$.

Now $\mu ,\eta$ are 0-lsc at $(u,v)$ and let $z_n=g\left(y_{n-1}\right),y_n=f\left(z_{n-1}\right)$. It follows from (a) that ${\lim }_{n\to \infty }\rho (z_n,z_{n+1})=0$ and ${\lim }_{n\to \infty }\varrho (y_n,y_{n+1})=0$. Then,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{\lim }\mu (z_n,y_n)& =\underset{n\to \infty }{\lim }\rho (z_n,g\left(y_n\right))=\underset{n\to \infty }{\lim }\rho (z_n,z_{n+1})=0\mathrm{and}\hfill \\ \hfill \underset{n\to \infty }{\lim }\eta (z_n,y_n)& =\underset{n\to \infty }{\lim }\varrho (y_n,f\left(z_n\right))=\underset{n\to \infty }{\lim }\varrho (y_n,y_{n+1})=0.\hfill \end{array}$$

Since $\mu ,\eta$ are 0-lsc at $(u,v)$, we have $d(u,g(v\left)\right)=\mu (u,v)=0$ and $\varrho (v,f(u\left)\right)=\eta (u,v)=0$ and so $f\left(u\right)=v$ and $g\left(v\right)=u$. □

4. Examples

We finally give two examples which support our main result.

Example 2. 

Let $Z=[0,1)$ and $Y=[0,1/2)$ with the Euclidean metrics ρ and ϱ, respectively. Define the mappings $F:Z\to CL\left(Y\right)$ and $G:Y\to CL\left(Z\right)$ by

$$F\left(z\right)=[0,z/4],G\left(y\right)=[0,y/2]$$

for all $z\in Z$ and for all $y\in Y$. Then, for $z_0=0\in Z$ and $y_0=0\in Y$, we have

$$O_Z(0,0)=\{0,0,0,\dots \},O_Y(0,0)=\{0,0,0,\dots \}.$$

Therefore, $(Z,\rho )$ and $(Y,\varrho )$ are related $FG$-orbitally complete.

If $z^{\prime }={\displaystyle \frac{y}{2}}\in G\left(y\right)$ and $y^{\prime }={\displaystyle \frac{z}{4}}\in F\left(z\right)$ are taken for each $z\in Z$ and $y\in Y$, then we get

$$\begin{array}{cc}\hfill \max \{\rho (z,z^{\prime }),\varrho (y,y^{\prime })\}& =\max \left\{|z-{\displaystyle \frac{y}{2}}|,|y-{\displaystyle \frac{z}{4}}|\right\}\hfill \\ \hfill & \le {\displaystyle \frac{3z}{2}}+y=\gamma (z,y)-\gamma (z^{\prime },y^{\prime })\hfill \end{array}$$

where $\gamma :Z\times Y\to [0,\infty ),\gamma (z,y)=2(z+y)$. Thus, Inequality (2) is satisfied.

The sequences

$$\left\{z_n\right\}=\{0,0,0,\dots \},\left\{y_n\right\}=\{0,0,0,\dots \}$$

in $O_Z(0,0)$ and $O_Y(0,0)$ converge to 0. Also, $0\in F\left(0\right)$ and $0\in G\left(0\right)$ and so $0\in GF\left(0\right)$ and $0\in FG\left(0\right)$.

Example 3. 

Let $Z=[-1,1]$ and $Y=[-1,2]$ with the Euclidean metrics ρ and ϱ, respectively. Define the mappings $F:Z\to CL\left(Y\right)$ and $G:Y\to CL\left(Z\right)$ by

$$F\left(z\right)=\{z/2,z\},G\left(y\right)=[-1,y/2]$$

for all $z\in Z$ and for all $y\in Y$.

If $z^{\prime }={\displaystyle \frac{y}{2}}\in G\left(y\right)$ and $y^{\prime }={\displaystyle \frac{z}{2}}\in F\left(z\right)$ are taken for each $z\in Z$ and $y\in Y$, then we get

$$\begin{array}{cc}\hfill \max \{\rho (z,z^{\prime }),\varrho (y,y^{\prime })\}& =\max \left\{|z-{\displaystyle \frac{y}{2}}|,|y-{\displaystyle \frac{z}{2}}|\right\}\hfill \\ \hfill & \le z+y=\gamma (z,y)-\gamma (z^{\prime },y^{\prime })\hfill \end{array}$$

where $\gamma :Z\times Y\to [0,\infty ),\gamma (z,y)=2(z+y)$. Thus, Inequality (2) is satisfied.

Take the points $z_0=1/2\in Z$ and $y_0=0\in Y$. If we choose $z_n\in G\left(y_{n-1}\right)$ as ${\displaystyle \frac{y_{n-1}}{2}}$ and $y_n\in F\left(z_{n-1}\right)$ as ${\displaystyle \frac{z_{n-1}}{2}}$, then we obtain the following sequences in $O_Z(1/2,0)$ and $O_Y(1/2,0)$, respectively.

$$\left\{z_n\right\}=\left\{0,{\displaystyle \frac{1}{2^3}},{\displaystyle \frac{1}{2^5}},{\displaystyle \frac{1}{2^7}},\dots \right\},\left\{y_n\right\}=\{{\displaystyle \frac{1}{2^2}},{\displaystyle \frac{1}{2^4}},{\displaystyle \frac{1}{2^6}},\dots \}$$

and so the sequences converge to 0. Also, $0\in F\left(0\right)$ and $0\in G\left(0\right)$ and so $0\in GF\left(0\right)$ and $0\in FG\left(0\right)$.

Note that the closedness of the set $G\left(z\right)$ and $F\left(y\right)$, for all $z\in Z$ and for all $y\in Y$, is a necessary condition in (c) of Theoremn 1. For example, if we take $G\left(y\right)=[-1,y/2)$ in the example above, then $0\notin G\left(0\right)$.

5. Conclusions

In this research article, by introducing the concept of the related orbital completeness of two metric spaces for multivalued mappings, a theorem concerning the fixed points of the compositions of two multivalued mappings defined on two orbitally complete metric spaces is presented. The relationship between the fixed points of these mappings is also investigated. Some important results are derived as corollaries of the main theorem. Finally, two concrete examples are provided to illustrate the significance of the main result.

We believe that the introduced concept and the corresponding fixed point theorem open up new directions for future research, including potential extensions to more general spaces, other types of contractions, and applications in various mathematical and applied fields. In particular, it would be interesting to investigate how the findings of this paper behave in the setting of b-metric spaces, as considered in [20].