Set-Valued Interpolative Hardy–Rogers and Set-Valued Reich–Rus–Ćirić-Type Contractions in b-Metric Spaces

Abstract

In this paper, using an interpolative approach, we investigate two fixed point theorems in the framework of a b-metric space whose all closed and bounded subsets are compact. One of the theorems is for set-valued Hardy–Rogers-type and the other one is for set-valued Reich–Rus–Ćirić-type contractions. Examples are provided to validate the results.

1. Introduction

After Banach proved the celebrated contraction principle [1] in 1922, numerous researchers have tried to improve or generalize it. The generalizations were mainly done in two directions—either the contractive condition was replaced by some more general ones, or new metric spaces were defined by incorporating additional conditions. In the former direction, many significant and improved results appeared, like Kannan’s [2], Chatterjea’s [3], Ćirić’s [4], Meir-Keeler’s [5], Boyd-Wong’s [6], etc. The later direction produced introduction of semimetric, quasimetric, symmetric, partial metric, b-metric, fuzzy metric, and many other generalized classes of metric spaces. However, not all of such attempts of generalizations were useful in applications, which was the main motive of such investigations. Some of them were not even true generalizations as they appeared to be equivalent to already existing ones. For some recent relevant work we refer to the works by the authors of [7,8,9].

Among the many generalized versions of metric spaces, the one due to Bakhtin [10] and Czerwik [11], called b-metric space, has drawn the attention of researchers worldwide, due to its importance and ease of applicability in many fields. The initial motivation for the introduction of b-metric space by both Bakhtin and Czerwik was to study the issue of convergence of measurable functions in regard to measure. Since then, immense development of fixed point theory in the framework of b-metric space has taken place [12,13,14,15,16,17,18,19,20,21].

In the current paper, our aim is to study two significant set valued interpolative contractions in the setting of a b-metric space whose all closed and bounded subsets are compact.

2. Preliminaries

First, we list some important definitions and relevant theorems, which are useful in our main results. Throughout the paper $\mathbb{N}$, $\mathbb{R}$, and ${\mathbb{R}}_{+}$ denote the set of natural numbers, set of real numbers, and set of non-negative real numbers, respectively.

Definition 1.

[10,11] Let Ω be a nonempty set and the mapping $\delta :\mathsf{\Omega }\times \mathsf{\Omega }\to [0,\infty )$ satisfies:

(1) 

$\delta (u,v)=0$ if and only if $u=v$;

(2) 

$\delta (u,v)=\delta (v,u)$ for all $u,v\in \mathsf{\Omega }$;

(3) 

there exists a real number $s\ge 1$ such that $\delta (u,z)\le s\left[\delta \right(u,v)+\delta (v,z\left)\right]$ for all $u,v,z\in \mathsf{\Omega }$.

Then δ is called a b-metric on Ω and $(\mathsf{\Omega },\delta ,s)$ is called a b-metric space with coefficient s.

The definitions of convergent sequence, Cauchy sequence, and complete b-metric space are exactly same as that of usual metric spaces.

Recently, some interesting fixed point results for interpolative contractions for single valued maps have been studied by Debnath et al. [22] and Karapinar et al. [23,24]. Our current results are generalized versions of those results for set-valued case.

Fixed point results for set-valued mappings play a fundamental role in nonlinear analysis. Fixed point theory for multivalued operators is also an important topic for set-valued analysis. The theory of set-valued mappings has applications in control theory, convex optimization, differential inclusions, and economics. The study of fixed point theorems for set-valued mappings was initiated by Markin [25] and Nadler [26]. In 1969, via Hausdorff concept of a distance between two arbitrary sets, Nadler proved some breakthrough fixed point results for set-valued maps.

The Pompeiu–Hausdorff metric $\mathcal{PH}$ is defined in the following manner in a complete metric space $(\mathsf{\Omega },\delta )$.

If CB(Ω) denotes the class of all nonempty closed and bounded subsets of Ω, then for $\mathcal{A},\mathcal{B}\in CB\left(\mathsf{\Omega }\right)$, we define

$$\mathcal{PH}(\mathcal{A},\mathcal{B})=max\{\underset{\xi \in \mathcal{B}}{sup}\Delta (\xi ,\mathcal{A}),\underset{\eta \in \mathcal{A}}{sup}\Delta (\eta ,\mathcal{B})\},$$

where $\Delta (\eta ,\mathcal{B})={inf}_{\xi \in \mathcal{B}}\delta (\eta ,\xi )$. $\left(CB\right(\mathsf{\Omega }),\mathcal{PH})$ is called the Pompeiu–Hausdorff metric space induced by $\delta$.

Definition 2.

[26] The point $x\in \mathsf{\Omega }$ is said to be a fixed point of the set-valued map $T:\mathsf{\Omega }\to CB\left(\mathsf{\Omega }\right)$ if $x\in Tx$.

Theorem 1.

[26] Let $(\mathsf{\Omega },\delta )$ be a complete metric space and $T:\mathsf{\Omega }\to CB\left(\mathsf{\Omega }\right)$ is a mapping such that for all $u,v\in \mathsf{\Omega }$

$$\mathcal{PH}(Tu,Tv)\le \alpha \delta (u,v);$$

where $0\le \alpha <1$. Then T has a unique fixed point.

Some other results that will be used in the sequel are also stated below.

Lemma 1.

[27] Suppose that $(\mathsf{\Omega },\delta )$ is a metric space and $Cl\left(\mathsf{\Omega }\right)$ be the class of all nonempty closed subsets of Ω. Let $\mathcal{B}\in Cl\left(\mathsf{\Omega }\right)$. Then for each $u\in \mathsf{\Omega }$ and $s>1$, there exists an element $\xi \in \mathcal{B}$ such that $\delta (u,\xi )\le s·\Delta (u,\mathcal{B})$.

Remark 1.

[27] If $\mathcal{B}$ is a compact subset of a metric space $(\mathsf{\Omega },\delta )$, then there exists an element $\xi \in \mathcal{B}$ such that $\delta (u,\xi )=\delta (u,\mathcal{B}).$

Lemma 2.

[28] Let $(\mathsf{\Omega },\delta )$ be a metric space. Let $\mathcal{A},\mathcal{B}\subset \mathsf{\Omega }$ and $s>1$. Then, for every $\eta \in \mathcal{A}$, there exists $\xi \in \mathcal{B}$ such that $\delta (\eta ,\xi )\le s·\mathcal{PH}(\mathcal{A},\mathcal{B})$.

Remark 2.

[26] If $\mathcal{A},\mathcal{B}\in CB\left(\mathsf{\Omega }\right)$ and let $\eta \in \mathcal{A}$, then for $\alpha >0$, there exists $\xi \in \mathcal{B}$, such that

$$\delta (\eta ,\xi )\le \mathcal{PH}(\mathcal{A},\mathcal{B})+\alpha .$$

However, there may not be a point $\xi \in \mathcal{B}$ such that

$$\delta (\eta ,\xi )\le \mathcal{PH}(\mathcal{A},\mathcal{B}).$$

If $\mathcal{B}$ is compact, then such a point ξ does exists, i.e., $\delta (\eta ,\xi )\le \mathcal{PH}(\mathcal{A},\mathcal{B}).$

3. Main Results

First, we define set-valued Hardy–Rogers-type contraction in a b-metric space and discuss the corresponding fixed point theorem.

Definition 3.

Let $(\mathsf{\Omega },\delta ,s)$ be a b-metric space. The map $T:\mathsf{\Omega }\to CB\left(\mathsf{\Omega }\right)$ is called a set-valued interpolative Hardy–Rogers-type contraction if there exist $\lambda \in [0,1)$ and $p,q,r\in (0,1)$ with $p+q+r<1$ such that

$$\mathcal{PH}(Tu,Tv)\le \lambda {\left[\delta (u,v)\right]}^q{[\Delta (u,Tu)]}^p{[\Delta (v,Tv)]}^r{\left[\frac{1}{2s}(\Delta (u,Tv)+\Delta (v,Tu))\right]}^{1-p-q-r}$$

(1)

for all $u,v\in \mathsf{\Omega }/Fix\left(T\right)$.

Theorem 2.

Let $(\mathsf{\Omega },\delta ,s)$ be a complete b-metric space whose all closed and bounded subsets are compact and T be a set-valued interpolative Hardy–Rogers-type contraction. Then, T has a fixed point.

Proof. 

Let $u_0\in \mathsf{\Omega }$ and choose $u_1\in T{u}_0$. By Lemma 2, we can select $u_2\in T{u}_1$ such that $\delta (u_2,u_1)\le \mathcal{PH}(T{u}_1,T{u}_0)$. Similarly we may choose $u_3\in T{u}_2$ such that $\delta (u_3,u_2)\le \mathcal{PH}(T{u}_2,T{u}_1)$. Continuing in this manner we construct a sequence $\left\{u_n\right\}$ satisfying $u_n+1\in T{u}_n$ such that $\delta (u_n+1,u_n)\le \mathcal{PH}(T{u}_n,T{u}_{n-1})$.

Now if there exists $n_0\in \mathbb{N}$, such that $u_{n_0}\in T{u}_{n_0}$, then $u_{n_0}$ becomes a fixed point of T and the proof is complete. Thus, assume that $u_n\notin T{u}_n$ for all $n\ge 0$. We show that the sequence $\left\{u_n\right\}$ above is a Cauchy sequence.

Replacing u by $u_n$ and v by $u_{n-1}$ in (1), we have

$$\begin{array}{c}\delta (u_{n+1},u_n)\le \mathcal{PH}(T{u}_n,T{u}_{n-1})\hfill \\ \le \lambda {\left[\delta (u_n,u_{n-1})\right]}^q{[\Delta (u_n,T{u}_n)]}^p{[\Delta (u_{n-1},T{u}_{n-1})]}^r{\left[\frac{1}{2s}(\Delta (u_n,T{u}_{n-1})+\Delta (u_{n-1},T{u}_n))\right]}^{1-p-q-r}\hfill \\ \le \lambda {\left[\delta (u_n,u_{n-1})\right]}^q{\left[\delta (u_n,u_{n+1})\right]}^p{\left[\delta (u_{n-1},u_n)\right]}^r{[\frac{1}{2s}·\delta (u_{n-1},u_{n+1})]}^{1-p-q-r}\hfill \\ \le \lambda {\left[\delta (u_n,u_{n-1})\right]}^q{\left[\delta (u_n,u_{n+1})\right]}^p{\left[\delta (u_{n-1},u_n)\right]}^r{[\frac{1}{2s}·s(\delta (u_{n-1},u_n)+\delta (u_n,u_{n+1}))]}^{1-p-q-r}.\hfill \end{array}$$

(2)

Assume that $\delta (u_{n-1},u_n)<\delta (u_n,u_{n+1})$ for some $n\ge 1$. Then $\frac{1}{2}[\delta (u_{n-1},u_n)+\delta (u_n,u_{n+1})]\le \delta (u_n,u_{n+1}).$

Thus, from Equation (2), we have

$$\begin{array}{cc}\hfill \delta (u_{n+1},u_n)& \le \lambda {\left[\delta (u_n,u_{n-1})\right]}^q{\left[\delta (u_n,u_{n+1})\right]}^p{\left[\delta (u_{n-1},u_n)\right]}^r{\left[\right(\delta (u_n,u_{n+1})]}^{1-p-q-r}\hfill \\ \hfill & =\lambda {\left[\delta (u_n,u_{n-1})\right]}^{q+r}{\left[\delta (u_n,u_{n+1})\right]}^{1-q-r}.\hfill \end{array}$$

This implies

$${\left[\delta (u_n,u_{n+1})\right]}^{q+r}\le \lambda {\left[\delta (u_{n-1},u_n)\right]}^{q+r}.$$

(3)

So, we must have $\delta (u_n,u_{n+1})\le \delta (u_{n-1},u_n)$, which is a contradiction to the previous assumption. Thus,

$$\delta (u_n,u_{n+1})\le \delta (u_{n-1},u_n)\mathrm{for}\mathrm{all}n\ge 1.$$

(4)

Now, from (2) we have

$$\begin{array}{cc}\hfill \delta (u_{n+1},u_n)& \le \lambda {\left[\delta (u_n,u_{n-1})\right]}^q{\left[\delta (u_n,u_{n+1})\right]}^p{\left[\delta (u_{n-1},u_n)\right]}^r\left[\delta (u_{n-1},u_n)\right){]}^{1-p-q-r}\hfill \\ & =\lambda {\left[\delta (u_{n-1},u_n)\right]}^{1-p}{\left[\delta (u_n,u_{n+1})\right]}^p.\hfill \end{array}$$

This implies

$${\left[\delta (u_n,u_{n+1})\right]}^{1-p}\le \lambda {\left[\delta (u_{n-1},u_n)\right]}^{1-p}\mathrm{for}\mathrm{all}n\ge 1.$$

(5)

Combining (4) and (5) we conclude that

$$\delta (u_n,u_{n+1})\le \lambda \delta (u_{n-1},u_n)\mathrm{for}\mathrm{all}n\ge 1.$$

(6)

It has been proved in the work by the authors of [29] that every sequence $\left\{u_n\right\}$ in a b-metric space $(\mathsf{\Omega },\delta ,s)$ having the property that there exists $\lambda \in [0,1)$, such that $\delta (u_n,u_{n+1})\le \lambda \delta (u_{n-1},u_n)\mathrm{for}\mathrm{all}n\ge 1$ is Cauchy.

Thus from (6), we conclude that $\left\{u_n\right\}$ is Cauchy. Again $(\mathsf{\Omega },\delta ,s)$ being complete, there exists $l\in \mathsf{\Omega }$ such that ${lim}_{n\to \infty }{u}_n=l$.

Since $u_n\notin T{u}_n$ for all $n\ge 0$, replacing u by $u_n$ and v by l in (1), we have

$$\begin{array}{cc}\hfill \Delta (u_{n+1},Tl)& \le \mathcal{PH}(T{u}_n,Tl)\hfill \\ & \le \lambda {\left[\delta (u_n,l)\right]}^q{[\Delta (u_n,T{u}_n)]}^p{[\Delta (l,Tl)]}^r{[\frac{1}{2s}(\Delta (u_n,Tl)+\Delta (l,T{u}_n))]}^{1-p-q-r}\hfill \end{array}$$

(7)

Taking limit as $n\to \infty$ in (7), we have $\Delta (l,Tl)=0$, which implies that $l\in Tl$.

 □

Example 1.

Let $\mathsf{\Omega }=[0,\infty )$ and $\delta (u,v)={(u-v)}^2$. Then $(\mathsf{\Omega },\delta ,2)$ is a complete b-metric space whose every closed and bounded subset is compact.

Define $T:\mathsf{\Omega }\to CB\left(\mathsf{\Omega }\right)$ by

$$Tu=\left\{\begin{array}{cc}\left\{0\right\},\hfill & ifu\in [0,1)\hfill \\ \{u,u+1\},\hfill & ifu\ge 1.\hfill \end{array}\right.$$

Let $u,v\in \mathsf{\Omega }/Fix\left(T\right)$. Then obviously $u,v\in (0,1)$. Now $\mathcal{PH}(Tu,Tv)=\mathcal{PH}\left(\right\{0\},\{0\left\}\right)=0$, i.e., T is a set valued interpolative Hardy–Rogers-type contraction and the inequality (1) holds. Therefore, all hypotheses of Theorem 2 are true and thus T has a fixed point. Here, it is easy to see that T has infinitely many fixed points.

Next we prove a fixed point theorem for set-valued Reich–Rus–Ćirić-type contraction in b-metric spaces.

Definition 4.

Let $(\mathsf{\Omega },\delta ,s)$ be a b-metric space. A map $T:\mathsf{\Omega }\to CB\left(\mathsf{\Omega }\right)$ is said to be a set valued interpolative Reich–Rus–Ćirić-type contraction if there are constants $\lambda \in [0,1)$ and $A,B\in (0,1)$ with $A+B<1$, such that

$$\mathcal{PH}(Tu,Tv)\le \lambda {\left[\delta (u,v)\right]}^B{[\Delta (u,Tu)]}^A{[(\Delta (v,Tv)]}^{1-A-B}$$

(8)

for all $u,v\in \mathsf{\Omega }/Fix\left(T\right)$.

Theorem 3.

Let $(\mathsf{\Omega },\delta ,s)$ be a complete b-metric space whose all closed and bounded subsets are compact. If $T:\mathsf{\Omega }\to CB\left(\mathsf{\Omega }\right)$ is an interpolative Reich–Rus–Ćirić-type contraction, then T has a fixed point in Ω.

Proof. 

Let $u_0\in \mathsf{\Omega }$ and choose $u_1\in T{u}_0$. By Lemma 2, we can select $u_2\in T{u}_1$ such that $\delta (u_2,u_1)\le \mathcal{PH}(T{u}_1,T{u}_0)$. Similarly we may choose $u_3\in T{u}_2$ such that $\delta (u_3,u_2)\le \mathcal{PH}(T{u}_2,T{u}_1)$. Continuing in this manner we construct a sequence $\left\{u_n\right\}$ satisfying $u_n+1\in T{u}_n$ $\delta (u_n+1,u_n)\le \mathcal{PH}(T{u}_n,T{u}_{n-1})$.

Now if there exists $n_0\in \mathbb{N}$ such that $u_{n_0}\in T{u}_{n_0}$, then $u_{n_0}$ becomes a fixed point of T and the proof is complete. Hence assume that $u_n\notin T{u}_n$ for all $n\ge 0$. We show that the sequence $\left\{u_n\right\}$ above is a Cauchy sequence.

Substituting u by $u_n$ and v by $u_{n-1}$ in (8), we have

$$\begin{array}{c}\delta (u_{n+1},u_n)\le \mathcal{PH}(T{u}_n,T{u}_{n-1})\hfill \\ \le \lambda {\left[\delta (u_n,u_{n-1})\right]}^B{[\Delta (u_n,T{u}_n)]}^A{[\Delta (u_{n-1},T{u}_{n-1})]}^{1-A-B}\hfill \\ \le \lambda {\left[\delta (u_n,u_{n-1})\right]}^B{\left[\delta (u_n,u_{n+1})\right]}^A{\left[h(u_{n-1},u_n)\right]}^{1-A-B}\hfill \\ =\lambda {\left[\delta (u_n,u_{n-1})\right]}^{1-A}{\left[\delta (u_n,u_{n+1})\right]}^A.\hfill \end{array}$$

(9)

From the above, we obtain

$${\left[\delta (u_n,u_{n+1})\right]}^{1-A}\le \lambda {\left[\delta (u_n,u_{n-1})\right]}^{1-A},$$

(10)

which implies that

$$\delta (u_n,u_{n+1})\le \delta (u_n,u_{n-1})\mathrm{for}\mathrm{all}n\ge 0.$$

(11)

Using (10) and (11), we have

$$\delta (u_n,u_{n+1})\le {\lambda }^{\frac{1}{1-A}}\delta (u_{n-1},u_n)\mathrm{for}\mathrm{all}n\ge 1.$$

(12)

However, we know from the work by the authors of [29] that every sequence $\left\{u_n\right\}$ in a b-metric space $(\mathsf{\Omega },\delta ,s)$ satisfying the property (12) is Cauchy.

Thus, we conclude that as $\left\{u_n\right\}$ is a Cauchy sequence and $(\mathsf{\Omega },\delta ,s)$ is complete, there exists $\xi \in \mathsf{\Omega }$ such that ${lim}_{n\to \infty }{u}_n=\xi$.

Next we show that $\xi$ is a fixed point of T. Suppose that $\xi \notin T\xi$ so that $\Delta (T\xi ,\xi )>0$. Also, our assumption is that $u_n\notin T{u}_n$ for all $n\ge 0$.

By substituting u by $u_n$ and v by $\xi$ in (8), we have

$$\begin{array}{cc}\hfill \Delta (\xi ,T\xi )& \le s[\delta (\xi ,u_{n+1})+\Delta (u_{n+1},T\xi )]\hfill \\ & \le s[\delta (\xi ,u_{n+1})+\mathcal{PH}(T{u}_n,T\xi )]\hfill \\ & \le sh(\xi ,u_{n+1})+\lambda s{\left[\delta (u_n,\xi )\right]}^B{[\Delta (u_n,T{u}_n)]}^A{[\Delta (\xi ,T\xi )]}^{1-A-B}\hfill \\ & =s\delta (\xi ,u_{n+1})+\lambda s{\left[\delta (u_n,\xi )\right]}^B{\left[\delta (u_n,u_{n+1})\right]}^A{[\Delta (\xi ,T\xi )]}^{1-A-B}.\hfill \end{array}$$

(13)

Taking limit as $n\to \infty$ in (13), we have $\Delta (\xi ,T\xi )=0$, which is a contradiction to our last hypothesis. Hence $\xi \in T\xi$.

 □

It is to be noted that in the previous theorem the role of coefficient s from the definition of b-metric space is not so visible except for the last part of the proof due to the fact that the definition of set-valued Reich–Rus–Ćirić-type contraction in b-metric spaces is almost analogous to its single-valued counterpart.

Below we give an example of Theorem 3.

Example 2.

Let $\mathsf{\Omega }=\{0,1,2\}$ and $\delta :\mathsf{\Omega }\times \mathsf{\Omega }\to [0,\infty )$ be defined as $\delta (u,v)=0$, $\delta (u,v)=\delta (v,u)$ for all $u,v\in \mathsf{\Omega }$, $\delta (0,1)=1$, $\delta (0,2)=2.2$ and $\delta (1,2)=1.1$. Then we can verify that $(\mathsf{\Omega },\delta ,\frac{22}{21})$ is a complete b-metric space (but it is not a metric space) whose every closed and bounded subset is compact.

Define the map $T:\mathsf{\Omega }\to CB\left(\mathsf{\Omega }\right)$ on Ω by

$$Tu=\left\{\begin{array}{cc}\left\{0\right\},\hfill & ifu\ne 2\hfill \\ \{0,1\},\hfill & ifu=2.\hfill \end{array}\right.$$

Further we can see that

$$\mathcal{PH}(Tu,Tv)=\left\{\begin{array}{cc}\mathcal{PH}\left(\right\{0\},\{0\left\}\right)=0,\hfill & ifu\ne 2,v\ne 2\hfill \\ \mathcal{PH}\left(\right\{0,1\},\{0\left\}\right)=1,\hfill & ifu=2,v\ne 2\hfill \\ \mathcal{PH}\left(\right\{0\},\{0,1\left\}\right)=1,\hfill & ifu\ne 2,v=2\hfill \\ \mathcal{PH}\left(\right\{0,1\},\{0,1\left\}\right)=1,\hfill & ifu=2,v=2.\hfill \end{array}\right.$$

Let $u,v\in \mathsf{\Omega }/Fix\left(T\right)$. Then clearly the maximum value of $\mathcal{PH}(Tu,Tv)$ is 1, i.e., Inequality (8) and all hypotheses of Theorem 3 hold if we choose $\lambda =\frac{3}{100},A=\frac{1}{3},B=\frac{1}{5}$. Thus, T is a set-valued Reich–Rus–Ćirić-type contraction and has a (unique) fixed point $0\in T0$.

4. Conclusions

In this paper, we considered two types of set valued interpolative contractions in b-metric spaces. We have imposed a strong condition on the b-metric space that all its closed and bounded subsets are compact. It would be an interesting future study to see if this condition can be dropped. These new interpolative approaches helped us to establish the existence of fixed points for those set-valued contractions. Study of the uniqueness of their fixed points is also a suggested future work.