**1. Introduction**

With the introduction of Banach's contraction principle (BCP), the fixed point theory advanced in various directions. Nadler [1] obtained the fundamental fixed point result for set-valued mappings using the notion of Pompeiu–Hausdorff metric which is an extension of the BCP. Later on, many fixed point theorists followed the findings of Nadler and contributed significantly to the development of theory (cf. S. Reich [2,3]).

On the other hand, in order to investigate the semantics of data flow networks; Matthews [4] coined the concept called as partial metric spaces which are used efficiently while building models in computation theory. On the inclusion of partial metric spaces into literature, many fixed point theorems were established in this setting, see [5–16]. Recently, Asadi et al. [17] brought the notion of an *M*-metric as a real generalization of a partial metric into the literature. They also obtained the *M*-metric version of the fixed point results of Banach and Kannan. Also, some fixed point theorems have been established in *M*-metric spaces endowed with a graph, see [18].

In this work, we introduce the *M*-Pompeiu–Hausdorff type metric. Furthermore, we extend the fixed point theorems of Nadler and Kannan to *M*-metric spaces for set-valued mappings. Finally, homotopy results for *M*-metric spaces are discussed.

## **2. Preliminaries**

The symbols N, R and R<sup>+</sup> represent respectively set of all natural numbers, real numbers and nonnegative real numbers. Let us recall some of the concepts for simplicity in understanding.

**Definition 1** ([4])**.** *Let X be a nonempty set. Then a partial metric is a function p* : *X* × *X* → R<sup>+</sup> *satisfying following conditions:*

*(p*1*) a* = *b* ⇐⇒ *p*(*<sup>a</sup>*, *a*) = *p*(*<sup>a</sup>*, *b*) = *p*(*b*, *b*)*; (p*2*) p*(*<sup>a</sup>*, *a*) ≤ *p*(*<sup>a</sup>*, *b*)*; (p*3*) p*(*<sup>a</sup>*, *b*) = *p*(*b*, *a*)*; (p*4*) p*(*<sup>a</sup>*, *b*) ≤ *p*(*<sup>a</sup>*, *c*) + *p*(*<sup>c</sup>*, *b*) − *p*(*<sup>c</sup>*, *c*)*;*

*for all a*, *b*, *c* ∈ *X. The pair* (*<sup>X</sup>*, *p*) *is called a partial metric space.*

The concept of an *M*-metric [17] defined in following definition extends and generalize the notion of partial metric.

**Definition 2** ([17])**.** *Let X be a non empty set. Then an M-metric is a function m* : *X* × *X* → R<sup>+</sup> *satisfying the following conditions:*

*(m*1*) <sup>m</sup>*(*<sup>a</sup>*, *a*) = *<sup>m</sup>*(*b*, *b*) = *<sup>m</sup>*(*<sup>a</sup>*, *b*) ⇔ *a* = *b;*


*for all a*, *b*, *c* ∈ *X. The pair* (*<sup>X</sup>*, *m*) *is called an M-metric space.*

**Remark 1** ([17])**.** *Let us denote Mab* := max{*m*(*<sup>a</sup>*, *<sup>a</sup>*), *<sup>m</sup>*(*b*, *b*)}*, where m is an M-metric on X. Then for every a*, *b* ∈ *X, we have*

*(1)* 0 ≤ *Mab* + *mab* = *<sup>m</sup>*(*<sup>a</sup>*, *a*) + *<sup>m</sup>*(*b*, *b*)*, (2)* 0 ≤ *Mab* − *mab* = |*m*(*<sup>a</sup>*, *a*) − *<sup>m</sup>*(*b*, *b*)|*, (3) Mab*− *mab*≤ (*Mac* − *mac*)+(*Mcb*− *mcb*).

**Example 1** ([17])**.** *Let m be an M-metric on X. Then*

$$\begin{aligned} (1) \quad &m^{\text{uv}}(a,b) = m(a,b) - 2m\_{ab} + M\_{ab,\prime} \\ (2) \quad &m^{\text{s}}(a,b) = \begin{cases} m(a,b) - m\_{ab} & \text{if } a \neq b, \\\\ 0 & \text{if } a = b, \end{cases} \end{aligned}$$

*are ordinary metrics on X.*

Two new examples of *M*-metrics are as follows:

**Example 2.** *Let X* = [0, <sup>∞</sup>)*. Then*

$$\begin{array}{ll}(a) & m\_1(a,b) = |a-b| + \frac{a+b}{2}, \\ (b) & m\_2(a,b) = |a-b| + \frac{a+b}{3} \end{array}$$

*are M-metrics on X.*

Let *Bm*(*<sup>a</sup>*, *η*) = {*b* ∈ *X* : *<sup>m</sup>*(*<sup>a</sup>*, *b*) < *mab* + *η*} be the open ball with center *a* and radius *η* > 0 in *M*-metric space (*<sup>X</sup>*, *<sup>m</sup>*). The collection {*Bm*(*<sup>a</sup>*, *η*) : *a* ∈ *X*, *η* > <sup>0</sup>}, acts as a basis for the topology *τm* (say) on *M*-metric *X*.

**Remark 2** ([17])**.** *τm is T*0 *but not Hausdorff.*

**Definition 3** ([17])**.** *Let* {*ak*} *be a sequence in M-metric spaces* (*<sup>X</sup>*, *<sup>m</sup>*)*.*

*(1)* {*ak*} *is called M-convergent to a* ∈ *X if and only if*

$$\lim\_{k \to \infty} (m(a\_{k'}a) - m\_{a\_k a}) = 0.$$


**Lemma 1** ([17])**.** *Let* {*ak*} *be a sequence in M-metric spaces* (*<sup>X</sup>*, *<sup>m</sup>*)*. Then*


**Example 3.** *Let X and m*1, *m*2 : *X* × *X* → [0, ∞) *be as defined in Example 2 for all a*, *b* ∈ *X. Then* (*<sup>X</sup>*, *<sup>m</sup>*1) *and* (*<sup>X</sup>*, *<sup>m</sup>*2) *are M-complete. Indeed,* (*<sup>X</sup>*, *m<sup>w</sup>*) = ([0, <sup>∞</sup>), *k*|*x* − *y*|) *is a complete metric space, where k* = 52 *for m*1 *and k* = 2 *for m*2*.*

**Lemma 2** ([17])**.** *Let ak* → *a and bk* → *b as k* → ∞ *in* (*<sup>X</sup>*, *<sup>m</sup>*)*. Then as k* → <sup>∞</sup>*,* (*m*(*ak*, *bk*) − *makbk* ) → (*m*(*<sup>a</sup>*, *b*) − *mab*).

**Lemma 3** ([17])**.** *Let ak* → *a as k* → ∞ *in* (*<sup>X</sup>*, *<sup>m</sup>*)*. Then* (*m*(*ak*, *b*) − *makb*) → (*m*(*<sup>a</sup>*, *b*) − *mab*), *k* → <sup>∞</sup>*, for all b* ∈ *X.*

**Lemma 4** ([17])**.** *Let ak* → *a and ak* → *b as k* → ∞ *in* (*<sup>X</sup>*, *<sup>m</sup>*)*. Then <sup>m</sup>*(*<sup>a</sup>*, *b*) = *mab. Further, if <sup>m</sup>*(*<sup>a</sup>*, *a*) = *<sup>m</sup>*(*b*, *b*)*, then a* = *b.*

**Lemma 5** ([17])**.** *Let* {*ak*} *be a sequence in* (*<sup>X</sup>*, *m*) *such that for some r* ∈ [0, <sup>1</sup>)*, <sup>m</sup>*(*ak*+1, *ak*) ≤ *rm*(*ak*, *ak*−<sup>1</sup>)*, k* ∈ N *then*

*(a)* lim *<sup>m</sup>*(*ak*, *ak*−<sup>1</sup>) = 0*;*


*k*,*j*<sup>→</sup><sup>∞</sup> *(d)*{*ak*}*is M-Cauchy.*

#### **3.** *M***-Pompeiu–Hausdorff Type Metric**

The concept of a partial Hausdorff metric is defined in [19,20]. Following them we initiate the notion of an *M*-Pompeiu–Hausdorff type metric induced by an *M*-metric in this section. Let us begin with the following definition.

**Definition 4.** *A subset A of an M-metric space* (*<sup>X</sup>*, *m*) *is called bounded if for all a* ∈ *A, there exist b* ∈ *X and K* ≥ 0 *such that a* ∈ *Bm*(*b*, *<sup>K</sup>*)*, that is, <sup>m</sup>*(*<sup>a</sup>*, *b*) < *mba* + *K.*

Let CB*<sup>m</sup>*(*X*) denotes the family of all nonempty, bounded, and closed subsets in (*<sup>X</sup>*, *<sup>m</sup>*). For *P*, *Q* ∈ CB*<sup>m</sup>*(*X*), define

$$\mathcal{H}\_m(P, \mathbb{Q}) = \max \{ \delta\_m(P, \mathbb{Q}), \delta\_m(\mathbb{Q}, P) \}\_{\prime\prime}$$

where *<sup>δ</sup>m*(*<sup>P</sup>*, *Q*) = sup{*m*(*<sup>a</sup>*, *Q*) : *a* ∈ *P*} and *<sup>m</sup>*(*<sup>a</sup>*, *Q*) = inf{*m*(*<sup>a</sup>*, *b*) : *b* ∈ *Q*}.

Let *P* denote the closure of *P* with respect to *M*-metric *m*. Note that *P* is closed in (*<sup>X</sup>*, *m*) if and only if *P* = *P*.

**Lemma 6.** *Let P be any nonempty set in an M-metric space* (*<sup>X</sup>*, *<sup>m</sup>*)*, then a* ∈ *P if and only if <sup>m</sup>*(*<sup>a</sup>*, *P*) = sup*x*∈*<sup>P</sup> max.*

**Proof.**

$$\begin{aligned} &a \in \overline{P} \Leftrightarrow B\_{\mathfrak{m}}(a, \eta) \cap P \neq \bigotimes\_{\ell} \text{ for all } \eta > 0\\ &\Leftrightarrow m(a, \mathbf{x}) < m\_{\mathfrak{a}\mathbf{x}} + \eta \text{ for some } \mathbf{x} \in P\\ &\Leftrightarrow m(a, \mathbf{x}) - m\_{\mathfrak{a}\mathbf{x}} < \eta\\ &\Leftrightarrow \inf\{m(a, \mathbf{x}) - m\_{\mathfrak{a}\mathbf{x}} : \mathbf{x} \in P\} = 0\\ &\Leftrightarrow \inf\{m(a, \mathbf{x}) : \mathbf{x} \in P\} = \sup\{m\_{\mathfrak{a}\mathbf{x}} : \mathbf{x} \in P\} \\ &\Leftrightarrow m(a, P) = \sup\_{\mathbf{x} \in P} m\_{\mathfrak{a}\mathbf{x}}.\end{aligned}$$

**Proposition 1.** *Let P*, *Q*, *R* ∈ CB*<sup>m</sup>*(*X*)*, then we have*

$$\begin{aligned} (a) \quad \delta\_{\mathfrak{m}}(P, P) &= \sup\_{a \in P} \{ \sup\_{b \in P} m\_{ab} \}; \\ (b) \quad (\delta\_{\mathfrak{m}}(P, Q) - \sup\_{a \in P} \sup\_{b \in Q} m\_{ab}) &\leq (\delta\_{\mathfrak{m}}(P, R) - \inf\_{a \in P} \inf\_{c \in R} m\_{ac}) + (\delta\_{\mathfrak{m}}(R, Q) - \inf\_{c \in R} \inf\_{b \in Q} m\_{cb}). \end{aligned}$$
