Some Fixed Point Results on Relational Quasi Partial Metric Spaces and Application to Non-Linear Matrix Equations

Abstract

We introduce a $q_{\varrho }$-implicit contractive condition by an implicit relation on relational quasi partial metric spaces and establish new (unique) fixed point results and periodic point results based on it. We justify the results by two suitable examples and compare with them related work. We discuss sufficient conditions ensuring the existence of a unique positive definite solution of the non-linear matrix equation $\mathcal{U}=\mathcal{B}+{\sum }_{i=1}^m{\mathcal{A}}_i^{*}\mathcal{G}\left(\mathcal{U}\right){\mathcal{A}}_i$, where $\mathcal{B}$ is an $n\times n$ Hermitian positive definite matrix, ${\mathcal{A}}_1$, ${\mathcal{A}}_2$, … ${\mathcal{A}}_m$ are $n\times n$ matrices, and $\mathcal{G}$ is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Two examples (with randomly generated matrices and complex matrices, respectively) are given, together with convergence and error analysis, as well as average CPU time analysis and visualization of solution in surface plot.

1. Introduction

It is well known that Banach Contraction Principle (BCP) from 1922 was a starting point for the development of an important area called metric fixed point theory, which has an enormous field of application. In particular, in recent decades, a number of mathematicians have obtained fixed point results for contraction type mappings in metric spaces equipped with partial order. The first results in this direction were obtained by Turinici in [1,2] who worked on some results of Matkowski [3,4]. Ran and Reurings [5,6], as well as Nieto and Ródríguez-López [7,8] reinvestigated results of this type. Samet and Turinici [9] established fixed point theorems for non-linear contraction under symmetric closure of an arbitrary relation. Recently, Ahmadullah et al. [10,11,12] and Alam and Imdad [13] employed an amorphous relation to prove a relation-theoretic analogue of BCP which in turn unifies a lot of well known relevant order-theoretic fixed point theorems.

The distance notion in the metric fixed point theory has been introduced and generalized in several different ways by many authors. In particular, Matthews [14] introduced the notion of a partial metric space (PMS, for short) and showed that BCP can be generalized to the partial metric context for applications in program verification. Subsequently, several authors studied fixed point theorems in PMSs, as well as ordered PMSs (see, e.g., [15,16,17]). In [18], Karapinar et al. introduced the notion of quasi partial metric space (QPMS) and proved some fixed point theorems. Popa and Patriciu [19] proved a fixed point result in implicit relation setting. Quite recently, Altun et al. [20] worked on relational PMSs and established some (unique) fixed point and property (P) results under F-contractive conditions with some examples.

Motivated by the concepts of quasi partial metric space and binary relation, in Section 3, we introduce a modified implicit relation of [19] with some illustrations. Further, in Section 4, we define the notion of $q_{\varrho }$-quasi implicit contractive condition for a self-mapping $\mathcal{T}$ on a relational quasi partial metric space and prove respective (unique) fixed point results and a property (P) result for $\mathfrak{R}$-complete space, using $\mathcal{T}$-closedness and $\mathcal{T}$-continuity of the relation $\mathfrak{R}$. Several special cases are considered in Section 5, together with suitable examples, illustrating the obtained results. While doing this, the symmetry condition on the space structure and full completeness of space structure are not required. We show by an illustration that there are situations when the related results of Popa and Patriciu [19] cannot be used in order to obtain the desired conclusions. In fact, the Ćirić type contraction condition is not strong enough in complete quasi partial metric spaces. In Section 6, the obtained fixed point results are used in order to obtain a sufficient condition ensuring the existence of a unique positive definite solution of a non-linear matrix equation. This important application is illustrated by two non-trivial examples in the final Section 7. We visualize this procedure through convergence analysis using three different initializations and a solution graph.

2. Preliminaries

In this article, the notations $\mathbb{Z},\mathbb{N},\mathbb{R},{\mathbb{R}}^{+}$ have their usual meanings, and ${\mathbb{N}}^{*}=\mathbb{N}\cup \left\{0\right\}$.

2.1. Quasi Partial Metric Spaces

Definition 1.

([14]). Let Ξ be a nonempty set. A partial metric on Ξ is a mapping $\varrho :\mathsf{\Xi }\times \mathsf{\Xi }\to {\mathbb{R}}^{+}$ such that for all $\nu ,\vartheta ,\mu \in \mathsf{\Xi }$:

(p${}_1$)

$\nu =\vartheta ⟺\varrho (\nu ,\nu )=\varrho (\nu ,\vartheta )=\varrho (\vartheta ,\vartheta )$,

(p${}_2$)

$\varrho (\nu ,\nu )\le \varrho (\nu ,\vartheta )$,

(p${}_3$)

$\varrho (\nu ,\vartheta )=\varrho (\vartheta ,\nu )$,

(p${}_4$)

$\varrho (\nu ,\vartheta )\le \varrho (\nu ,\mu )+\varrho (\mu ,\vartheta )-\varrho (\mu ,\mu )$.

In this case, the pair $(\mathsf{\Xi },\varrho )$ is called a partial metric space (PMS).

If $\varrho (\nu ,\vartheta )=0$, then from (p${}_1$) and (p${}_2$) it clearly follows that $\nu =\vartheta$. $\varrho (\nu ,\nu )$ may not be 0.

Definition 2.

([18]). Let Ξ be a nonempty set. A quasi partial metric on Ξ is a mapping $q_{\varrho }:\mathsf{\Xi }\times \mathsf{\Xi }\to {\mathbb{R}}^{+}$ such that for all $\nu ,\vartheta ,\mu \in \mathsf{\Xi }$:

(q${}_1$)

$q_{\varrho }(\nu ,\nu )=q_{\varrho }(\nu ,\vartheta )=q_{\varrho }(\vartheta ,\vartheta )⟺\nu =\vartheta$;

(q${}_2$)

$q_{\varrho }(\nu ,\nu )\le q_{\varrho }(\nu ,\vartheta )$,

(q${}_3$)

$q_{\varrho }(\nu ,\nu )\le q_{\varrho }(\vartheta ,\nu )$,

(q${}_4$)

$q_{\varrho }(\nu ,\vartheta )\le q_{\varrho }(\nu ,\mu )+q_{\varrho }(\mu ,\vartheta )-q_{\varrho }(\mu ,\mu )$.

In this case, the pair $(\mathsf{\Xi },q_{\varrho })$ is a quasi partial metric space (QPMS).

Lemma 1.

([18]). Let $(\mathsf{\Xi },q_{\varrho })$ be a QPMS. Then the following hold:

(i)

If $q_{\varrho }(\nu ,\vartheta )=0$, then $\nu =\vartheta$.

(ii)

If $\nu \ne \vartheta$, then $q_{\varrho }(\nu ,\vartheta )>0$ and $q_{\varrho }(\vartheta ,\nu )>0$.

Lemma 2.

([18]). Let $(\mathsf{\Xi },q_{\varrho })$ be a QPMS. Then:

(i)

The function ${\varrho }^{*}$ given by ${\varrho }^{*}(\nu ,\vartheta )=\frac{1}{2}[q_{\varrho }(\nu ,\vartheta )+q_{\varrho }(\vartheta ,\nu )]$ ($\nu ,\vartheta \in \mathsf{\Xi }$) is a partial metric on Ξ.

(ii)

The function $q_{\varrho }^s$ given by

$$q_{\varrho }^s(\nu ,\vartheta )=q_{\varrho }(\nu ,\vartheta )+q_{\varrho }(\vartheta ,\nu )-q_{\varrho }(\nu ,\nu )-q_{\varrho }(\vartheta ,\vartheta )$$

(1)

is a metric on Ξ.

Definition 3.

([18]). Let $(\mathsf{\Xi },q_{\varrho })$ be a QPMS. Then:

• A sequence $\left\{{\nu }_n\right\}$ in $(\mathsf{\Xi },q_{\varrho })$ converges to $\nu \in \mathsf{\Xi }$ if ${lim}_{n\to \infty }{q}_{\varrho }(\nu ,{\nu }_n)={lim}_{n\to \infty }{q}_{\varrho }({\nu }_n,\nu )=q_{\varrho }(\nu ,\nu )$.
• A sequence $\left\{{\nu }_n\right\}$ in $(\mathsf{\Xi },q_{\varrho })$ is called a Cauchy sequence if ${lim}_{n,m\to \infty }{q}_{\varrho }({\nu }_n,{\nu }_m)$ and ${lim}_{n,m\to \infty }{q}_{\varrho }({\nu }_m,{\nu }_n)$ exist (and are finite).
• The space $(\mathsf{\Xi },q_{\varrho })$ is said to be complete if every Cauchy sequence $\left\{{\nu }_n\right\}$ in Ξ converges to a point $\nu \in \mathsf{\Xi }$ such that $q_{\varrho }(\nu ,\nu )={lim}_{n,m\to \infty }{q}_{\varrho }({\nu }_n,{\nu }_m)={lim}_{n,m\to \infty }{q}_{\varrho }({\nu }_m,{\nu }_n)$.

Lemma 3.

([18]). Let $(\mathsf{\Xi },q_{\varrho })$ be a QPMS. Then a sequence $\left\{{\nu }_n\right\}$ is a Cauchy sequence in $(\mathsf{\Xi },q_{\varrho })$ if and only if it is a Cauchy sequence in $(\mathsf{\Xi },q_{\varrho }^s)$. In particular, $(\mathsf{\Xi },q_{\varrho })$ is complete if and only if $(\mathsf{\Xi },q_{\varrho }^s)$ is complete. Moreover,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}{q}_{\varrho }^s(\nu ,{\nu }_n)=0& \iff q_{\varrho }(\nu ,\nu )=\underset{n\to \infty }{lim}{q}_{\varrho }({\nu }_n,\nu )=\underset{n\to \infty }{lim}{q}_{\varrho }(\nu ,{\nu }_n)\\ \hfill & =\underset{n,m\to \infty }{lim}{q}_{\varrho }({\nu }_n,{\nu }_m)=\underset{n,m\to \infty }{lim}{q}_{\varrho }({\nu }_m,{\nu }_n).\hfill \end{array}$$

2.2. Relational Quasi Partial Metric Spaces

We call the pair $(\mathsf{\Xi },\mathfrak{R})$ a relational set if (i) $\mathsf{\Xi }\ne \varnothing$ is a set and (ii) $\mathfrak{R}$ is a binary relation on $\mathsf{\Xi }$.

The following are some standard terms used in the theory of relational sets and spaces (see, e.g., [9,13,21,22,23]).

Let $(\mathsf{\Xi },\mathfrak{R})$ be a relational set, $(\mathsf{\Xi },q_{\varrho },\mathfrak{R})$ be a relational QPMS (RQPMS, for short), and let $\mathcal{T}$ be a self-mapping on $\mathsf{\Xi }$. Then:

• $\nu \in \mathsf{\Xi }$ is $\mathfrak{R}$-related to $\vartheta \in \mathsf{\Xi }$ if $(\nu ,\vartheta )\in \mathfrak{R}$.
• The set $(\mathsf{\Xi },\mathfrak{R})$ is said to be complete if for all $\nu ,\vartheta \in \mathsf{\Xi },[\nu ,\vartheta ]\in \mathfrak{R}$, where $[\nu ,\vartheta ]\in \mathfrak{R}$ means that $(\vartheta ,\nu )\in \mathfrak{R}$ or $(\nu ,\vartheta )\in \mathfrak{R}$.
• A sequence $\left({\nu }_n\right)$ in $\mathsf{\Xi }$ is said to be $\mathfrak{R}$-preserving if $({\nu }_n,{\nu }_{n+1})\in \mathfrak{R}$, $\forall n\in {\mathbb{N}}^{*}$.
• $\mathfrak{R}$ is said to be $\mathcal{T}$-closed if $(\nu ,\vartheta )\in \mathfrak{R}\Rightarrow (\mathcal{T}\nu ,\mathcal{T}\vartheta )\in \mathfrak{R}$.
• $(\mathsf{\Xi },q_{\varrho },\mathfrak{R})$ is said to be $\mathfrak{R}$-complete if, given an $\mathfrak{R}$-preserving Cauchy sequence $\left\{{\nu }_n\right\}$ in $\mathsf{\Xi }$, there exists some $\nu \in \mathsf{\Xi }$ such that

  $$\begin{array}{cc}\hfill \underset{n,m\to \infty }{lim}{q}_{\varrho }({\nu }_n,{\nu }_m)& =\underset{n,m\to \infty }{lim}{q}_{\varrho }({\nu }_m,{\nu }_n)=q_{\varrho }(\nu ,\nu )\hfill \\ \hfill & =\underset{n\to \infty }{lim}{q}_{\varrho }({\nu }_n,\nu )=\underset{n\to \infty }{lim}{q}_{\varrho }(\nu ,{\nu }_n).\hfill \end{array}$$
• $\mathcal{T}$ is said to be $\mathfrak{R}$-continuous at $\nu$ if, given an $\mathfrak{R}$-preserving sequence $\left\{{\nu }_n\right\}$ with ${\nu }_n\to \nu$, $\mathcal{T}{\nu }_n\to \mathcal{T}\nu$ holds as $n\to \infty$. It is said to be $\mathfrak{R}$-continuous if it is $\mathfrak{R}$-continuous at every point of $\mathsf{\Xi }$.
• $\mathfrak{R}$ is said to be $q_{\varrho }$-self-closed if, given an $\mathfrak{R}$-preserving sequence $\left\{{\nu }_n\right\}$ with ${\nu }_n\to \nu$, there exists a subsequence $\left\{{\nu }_{n_k}\right\}$ of $\left\{{\nu }_n\right\}$, such that $[{\nu }_{n_k},\nu ]\in \mathfrak{R}$, for all $k\in {\mathbb{N}}^{*}$.
• A subset $\mathfrak{Z}$ of $\mathsf{\Xi }$ is called $\mathfrak{R}$-directed if for all $\nu ,\vartheta \in \mathfrak{Z}$, there exists $\mu \in \mathsf{\Xi }$ such that $(\nu ,\mu )\in \mathfrak{R}$ and $(\vartheta ,\mu )\in \mathfrak{R}$. It is called $(\mathcal{T},\mathfrak{R})$-directed if for all $\nu ,\vartheta \in \mathfrak{Z}$, there exists $\mu \in \mathsf{\Xi }$ such that $(\nu ,\mathcal{T}\mu )\in \mathfrak{R}$ and $(\vartheta ,\mathcal{T}\mu )\in \mathfrak{R}$.
• $(\mathsf{\Xi },q_{\varrho },\mathfrak{R})$ is said to be regular if the following condition holds:

  $$\begin{array}{cc}\hfill [\left\{{\mu }_n\right\}\subseteq \mathsf{\Xi }\mathrm{converges}\mathrm{to}\mu \in \mathsf{\Xi }\mathrm{and}{\mu }_n\mathfrak{R}{\mu }_{n+1}& \mathrm{for}\mathrm{all}n\in \mathbb{N}]\hfill \\ \hfill & \Rightarrow [{\mu }_n\mathfrak{R}\mu \mathrm{for}\mathrm{all}n\in \mathbb{N}],\hfill \end{array}$$

  for any sequence $\left\{{\mu }_n\right\}$ in $\mathsf{\Xi }$.
• For $\nu ,\vartheta \in \mathsf{\Xi }$, a path of length k (where k is a natural number) in $\mathfrak{R}$ from $\nu$ to $\vartheta$ is a finite sequence $\{{\mu }_0,{\mu }_1,{\mu }_2,\dots ,{\mu }_k\}\subset \mathsf{\Xi }$ satisfying the following conditions:

  (i)

  ${\mu }_0=\nu$ and ${\mu }_k=\vartheta$,

  (ii)

  $({\mu }_i,{\mu }_{i+1})\in \mathfrak{R}$ for each i$(0\le i\le k-1)$.

Notice that a path of length k involves $k+1$ elements of $\mathsf{\Xi }$, although they are not necessarily distinct.

Remark 1.

Every complete RQPMS is $\mathfrak{R}$-complete but not conversely. On an RQPMS, every continuous mapping is $\mathfrak{R}$-continuous but not conversely.

We fix the following notation for a relational space $(\mathsf{\Xi },\mathfrak{R})$ and a self-mapping $\mathcal{T}$ on $\mathsf{\Xi }$:

(i)

$Fix\left(\mathcal{T}\right):=$ the set of all fixed points of $\mathcal{T}$,

(ii)

$\mathfrak{X}(\mathcal{T},\mathfrak{R}):=\{\nu \in \mathsf{\Xi }:[\nu ,\mathcal{T}\nu ]\in \mathfrak{R}\}$,

(iii)

$\mathfrak{P}(\nu ,\vartheta ,\mathfrak{R}):=$ the class of all paths in $\mathfrak{R}$ from $\nu$ to $\vartheta$, where $\nu ,\vartheta \in \mathsf{\Xi }$.

3. Modified Implicit Relation

In this section, we first modify the notion of implicit relation and examples discussed in [19].

Let $\mathsf{\Phi }$ denote the set of all functions $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ that satisfy the following conditions:

(i)

$\phi$ is increasing and $\phi \left(0\right)=0$;

(ii)

${\sum }_{n=1}^{\infty }{\phi }^n\left(\zeta \right)<\infty$, for all $\zeta >0$, where ${\phi }^n$ is the nth iterate of $\phi$.

It should be noted that $\phi \left(\zeta \right)<\zeta$ always holds and the family $\mathsf{\Phi }\ne \varnothing$.

Example 1.

Consider the mapping $\phi \left(\zeta \right)=\frac{\lambda \zeta }{8}$, where $0<\lambda <1$. Then we have ${\phi }^n\left(\zeta \right)\le \frac{{\lambda }^n\zeta }{8^n}$. Therefore, ${\sum }_{n=1}^{\infty }{\phi }^n\left(\zeta \right)={\sum }_{n=1}^{\infty }\frac{{\lambda }^n\zeta }{8^n}<\infty$ and hence $\phi \in \mathsf{\Phi }$.

Let $\mathfrak{G}$ be the set of all functions $\mathcal{Q}:{\mathbb{R}}_{+}^5\to \mathbb{R}$ which are continuous and satisfy the following conditions:

($\mathcal{Q}$1)

$\mathcal{Q}$ is nonincreasing in the fifth variable;

($\mathcal{Q}$2)

for all $\zeta ,\xi \ge 0$, there exists $\phi \in \mathsf{\Phi }$ such that $\mathcal{Q}(\zeta ,\xi ,\xi ,\zeta ,\zeta +\xi )\le 0$ implies $\zeta \le \phi \left(\xi \right)$.

Let ${\mathfrak{G}}^{\prime }$ be the set of functions $\mathcal{Q}\in \mathfrak{G}$ such that

($\mathcal{Q}$3)

for all $\zeta ,\xi \ge 0$, there exists $\phi \in \mathsf{\Phi }$ such that $\mathcal{Q}(\zeta ,0,0,\xi ,\zeta )\le 0$ implies $\zeta \le \phi \left(\xi \right)$.

Let ${\mathfrak{G}}^{″}$ be the set of functions $\mathcal{Q}\in {\mathfrak{G}}^{\prime }$, such that

($\mathcal{Q}$4)

for all $\zeta ,{\zeta }^{\prime }>0$, there exists $\phi \in \mathsf{\Phi }$ such that $\mathcal{Q}(\zeta ,\zeta ,0,0,\zeta +{\zeta }^{\prime })>0$ implies $\zeta \le \phi \left({\zeta }^{\prime }\right)$.

Example 2.

Let $\mathcal{Q}(s_1,\dots ,s_5)=s_1-\alpha s_2-\beta s_3-\gamma s_4-\delta s_5$, where $\alpha >0$, $\beta ,\gamma ,\delta \ge 0$ and $\alpha +\beta +\gamma +2\delta <1$.

($\mathcal{Q}$2):

Let $\zeta ,\xi \ge 0$ and $\mathcal{Q}(\zeta ,\xi ,\xi ,\zeta ,\zeta +\xi )=\zeta -\alpha \xi -\beta \xi -\gamma \zeta -\delta (\zeta +\xi )\le 0$. Then, $\zeta \le \phi \left(\xi \right)$, where $\phi \left(\xi \right)=h\xi$, $0<h=\frac{\alpha +\beta +\delta }{1-(\gamma +\delta )}<1$.

($\mathcal{Q}$3):

Let $\zeta ,\xi \ge 0$ and $\mathcal{Q}(\zeta ,0,0,\xi ,\zeta )=\zeta -\gamma \xi -\delta \zeta .$ Then, $\zeta \le \left(\frac{\gamma }{1-\alpha }\right)\xi$. Hence, $\zeta \le \phi \left(\xi \right)$, where $\phi \left(\xi \right)=h\xi$, $0<h=\frac{\gamma }{1-\alpha }<1$.

($\mathcal{Q}$4):

Let $\zeta ,{\zeta }^{\prime }>0$, $\mathcal{Q}(\zeta ,\zeta ,0,0,\zeta +{\zeta }^{\prime })=(1-\alpha -\delta )\zeta -\delta {\zeta }^{\prime }\le 0$. Then, $\zeta \le \phi \left({\zeta }^{\prime }\right)$, where $\phi \left({\zeta }^{\prime }\right)=h{\zeta }^{\prime }$, $0<h=\frac{\delta }{1-\alpha -\delta }<1$.

Example 3.

$\mathcal{Q}(s_1,\dots ,s_5)=s_1-kmax\{s_2,s_3,s_4,s_5\}$, where $k\in (0,\frac{1}{2})$.

($\mathcal{Q}$2):

Let $\zeta ,\xi \ge 0$ and $\mathcal{Q}(\zeta ,\xi ,\xi ,\zeta ,\zeta +\xi )=\zeta -k(\zeta +\xi )\le 0$. Then, $\zeta \le \phi \left(\xi \right)$, where $\phi \left(\xi \right)=h\xi$, $0<h=\frac{k}{1-k}<1$.

($\mathcal{Q}$3):

Let $\zeta ,\xi \ge 0$ and $\mathcal{Q}(\zeta ,0,0,\xi ,\zeta )=\zeta -kmax\{\zeta ,\xi \}.$ If $\zeta >\nu$, then $(1-k)\zeta \le 0$, a contradiction. Hence $\zeta \le \phi \left(\xi \right)$, where $\phi \left(\xi \right)=h\xi$, $0<h=k<1$.

($\mathcal{Q}$4):

Let $\zeta ,{\zeta }^{\prime }>0$, $\mathcal{Q}(\zeta ,\zeta ,0,0,\zeta +{\zeta }^{\prime })=(1-k)\zeta -k{\zeta }^{\prime }\le 0$. Then, $\zeta \le \phi \left({\zeta }^{\prime }\right)$, where $\phi \left({\zeta }^{\prime }\right)=h{\zeta }^{\prime }$, $h=\frac{k}{1-k}<1$

Example 4.

$\mathcal{Q}(s_1,\dots ,s_5)=s_1^2-\alpha s_2{s}_3-\beta s_4^2-\gamma s_5^2$, where $\alpha >0$, $\beta ,\gamma \ge 0$ and $\alpha +\beta +4\gamma <1$.

($\mathcal{Q}$2):

Let $\zeta ,\xi \ge 0$ and $\mathcal{Q}(\zeta ,\xi ,\xi ,\zeta ,\zeta +\xi )={\zeta }^2-\alpha {\xi }^2-\beta {\zeta }^2-\gamma {(\zeta +\xi )}^2\le 0$. If $\zeta >\xi$, then ${\zeta }^2[1-(\alpha +\beta +4\gamma )]\le 0$, a contradiction. Hence, $\zeta \le \xi$ which implies $\zeta \le \phi \left(\xi \right)$, where $\phi \left(\xi \right)=h\xi$, $0<h=\sqrt{\alpha +\beta +4\gamma }<1$.

($\mathcal{Q}$3):

Let $\zeta ,\xi \ge 0$ and $\mathcal{Q}(\zeta ,0,0,\xi ,\zeta )={\zeta }^2-\beta {\xi }^2-\gamma {\zeta }^2.$ Then, $\zeta \le \sqrt{\frac{b}{1-c}}\xi .$ Hence, $\zeta \le \phi \left(\xi \right)$, where $\phi \left(\xi \right)=h\xi$, $0<h=\sqrt{\frac{\beta }{1-\gamma }}<1$.

($\mathcal{Q}$4):

Let $\zeta ,{\zeta }^{\prime }>0$, $\mathcal{Q}(\zeta ,\zeta ,0,0,\zeta +{\zeta }^{\prime })=(1-\gamma ){\zeta }^2-\gamma {\zeta }^{\prime 2}\le 0$. Then, $\zeta \le \phi \left({\zeta }^{\prime }\right)$, where $\phi \left({\zeta }^{\prime }\right)=h{\zeta }^{\prime }$, $h=\sqrt{\frac{\gamma }{1-\gamma }}<1$

Example 5.

$\mathcal{Q}(s_1,\dots ,s_5)=s_1-\alpha s_2-\beta \frac{(1+s_3)s_4}{1+s_2}-\gamma s_5$, where $\alpha >0$, $\beta ,\gamma \ge 0$ and $\alpha +\beta +2\gamma <1$.

($\mathcal{Q}$2):

Let $\zeta ,\xi \ge 0$ and $\mathcal{Q}(\zeta ,\xi ,\xi ,\zeta ,\zeta +\xi )=\zeta -\alpha \xi -\beta \zeta -\gamma (\zeta +\xi )\le 0$, which implies $\zeta \le \phi \left(\xi \right)$, where $\phi \left(\xi \right)=h\xi$, $0<h=\frac{\alpha +\gamma }{1-(\beta +\gamma )}<1$.

($\mathcal{Q}$3):

Let $\zeta ,\xi \ge 0$ and $\mathcal{Q}(\zeta ,0,0,\xi ,\zeta )=\zeta -\beta \xi -\gamma \zeta .$ Then $\zeta \le \left(\frac{\beta }{1-\gamma }\right)\xi .$ Hence, $\zeta \le \phi \left(\xi \right)$, where $\phi \left(\xi \right)=h\xi$, $0<h=\frac{\beta }{1-\gamma }<1$.

($\mathcal{Q}$4):

Let $\zeta ,{\zeta }^{\prime }>0$, $\mathcal{Q}(\zeta ,\zeta ,0,0,\zeta +{\zeta }^{\prime })=(1-\beta -\gamma )\zeta -\gamma {\zeta }^{\prime }\le 0$. Then, $\zeta \le \phi \left({\zeta }^{\prime }\right)$, where $\phi \left({\zeta }^{\prime }\right)=h{\zeta }^{\prime }$, $h=\frac{\gamma }{1-\beta -\gamma }<1$.

4. ${\mathit{q}}_{\mathit{\varrho }}$-Implicit Contractive Mappings in Relational Quasi Partial Metric Spaces

First we define $q_{\varrho }$-implicit contractive mappings on an RQPMS.

Definition 4.

Let $(\mathsf{\Xi },q_{\varrho },\mathfrak{R})$ be an RQPMS. A mapping $\mathcal{T}:\mathsf{\Xi }\to \mathsf{\Xi }$ is said to be a $q_{\varrho }$-implicit contractive mapping, if there exists $\mathcal{Q}\in \mathfrak{G}$, such that for $\nu ,\vartheta \in \mathsf{\Xi }$ with $(\nu ,\vartheta )\in {\mathfrak{R}}^{*}=\{(\nu ,\vartheta )\in \mathfrak{R}\mid \nu \ne \vartheta \}$,

$$\mathcal{Q}\left(\begin{array}{c}{q}_{\varrho }(\mathcal{T}\nu ,\mathcal{T}\vartheta ),q_{\varrho }(\nu ,\vartheta ),q_{\varrho }(\nu ,\mathcal{T}\nu ),q_{\varrho }(\vartheta ,\mathcal{T}\vartheta ),q_{\varrho }(\nu ,\mathcal{T}\vartheta )+q_{\varrho }(\vartheta ,\mathcal{T}\nu )\end{array}\right)\le 0$$

(2)

holds true.

4.1. Fixed Point Results

Theorem 1.

Let $(\mathsf{\Xi },q_{\varrho },\mathfrak{R})$ be an $\mathit{RQPMS}$ and $\mathcal{T}:\mathsf{\Xi }\to \mathsf{\Xi }$. Suppose that the following conditions hold:

(C₁)

$\mathfrak{X}(\mathcal{T},\mathfrak{R})\ne \varnothing$;

(C₂)

$\mathfrak{R}$ is $\mathcal{T}$-closed;

(C₃)

Ξ is $\mathfrak{R}$-complete;

(C₄)

$\mathcal{T}$ is a $q_{\varrho }$-implicit contractive mapping;

(C₅)

$\mathcal{T}$ is $\mathfrak{R}$-continuous.

Then there exists ${\xi }^{*}\in \mathsf{\Xi }$ such that ${\xi }^{*}\in Fix\left(\mathcal{T}\right)$.

Proof. 

Let ${\xi }_0\in \mathfrak{X}(\mathcal{T},\mathfrak{R})$. Define ${\xi }_{n+1}=\mathcal{T}{\xi }_n={\mathcal{T}}^{n+1}{\xi }_0$ for all $n\in {\mathbb{N}}^{*}$. By ($C_2$), ${\xi }_n\in \mathfrak{X}(\mathcal{T},\mathfrak{R})$, for all $n\in {\mathbb{N}}^{*}$. If there exists $n_0\in {\mathbb{N}}^{*}$ such that ${\xi }_{n_0+1}={\xi }_{n_0}$ then ${\xi }_{n_0}\in Fix\left(\mathcal{T}\right)$ and the statement holds true.

Assume that ${\xi }_{n+1}\ne {\xi }_n$ for all $n\in {\mathbb{N}}^{*}$ so that $q_{\varrho }(\mathcal{T}{\xi }_{n+1},\mathcal{T}{\xi }_n)>0$. As $\mathcal{T}{\xi }_{n+1}\ne \mathcal{T}{\xi }_n$, $({\xi }_n,{\xi }_{n+1})\in {\mathfrak{R}}^{*}$ for all $n\in {\mathbb{N}}^{*}$. As $\mathcal{T}$ is $q_{\varrho }$-implicit contractive mapping, we have, for some $\mathcal{Q}\in \mathfrak{G}$,

$$\mathcal{Q}\left(\begin{array}{c}{q}_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }_{n+1}),q_{\varrho }({\xi }_n,{\xi }_{n+1}),q_{\varrho }({\xi }_n,\mathcal{T}{\xi }_n),\\ q_{\varrho }({\xi }_{n+1},\mathcal{T}{\xi }_{n+1}),q_{\varrho }({\xi }_n,\mathcal{T}{\xi }_{n+1})+q_{\varrho }({\xi }_{n+1},\mathcal{T}{\xi }_n)\end{array}\right)\le 0,$$

that is,

$$\mathcal{Q}\left(\begin{array}{c}{q}_{\varrho }({\xi }_{n+1},{\xi }_{n+2}),q_{\varrho }({\xi }_n,{\xi }_{n+1}),q_{\varrho }({\xi }_n,{\xi }_{n+1}),\\ q_{\varrho }({\xi }_{n+1},{\xi }_{n+2}),q_{\varrho }({\xi }_n,{\xi }_{n+2})+q_{\varrho }({\xi }_{n+1},{\xi }_{n+1})\end{array}\right)\le 0.$$

From the property $\left(q_4\right)$,

$$q_{\varrho }({\xi }_n,{\xi }_{n+2})\le q_{\varrho }({\xi }_n,{\xi }_{n+1})+q_{\varrho }({\xi }_{n+1},{\xi }_{n+2})-q_{\varrho }({\xi }_{n+1},{\xi }_{n+1}).$$

Then by $\left({\mathcal{Q}}_1\right)$, we obtain

$$\mathcal{Q}\left(\begin{array}{c}{q}_{\varrho }({\xi }_{n+1},{\xi }_{n+2}),q_{\varrho }({\xi }_n,{\xi }_{n+1}),q_{\varrho }({\xi }_n,{\xi }_{n+1}),\\ q_{\varrho }({\xi }_{n+1},{\xi }_{n+2}),q_{\varrho }({\xi }_n,{\xi }_{n+1})+q_{\varrho }({\xi }_{n+1},{\xi }_{n+2})\end{array}\right)\le 0.$$

It follows from $\left({\mathcal{Q}}_2\right)$ that there is $\phi \in \mathsf{\Phi }$ such that

$$q_{\varrho }({\xi }_{n+1},{\xi }_{n+2})\le \phi \left(q_{\varrho }({\xi }_n,{\xi }_{n+1})\right).$$

For $m>n$, using $\left(q_4\right)$, we obtain

$$\begin{array}{cc}\hfill q_{\varrho }({\xi }_n,{\xi }_m)& \le q_{\varrho }({\xi }_n,{\xi }_{n+1})+\dots +q_{\varrho }({\xi }_{n+m-1},{\xi }_{n+m})\hfill \\ \hfill & -q_{\varrho }({\xi }_{n+1},{\xi }_{n+1})-q_{\varrho }({\xi }_{n+2},{\xi }_{n+2})-\dots \hfill \\ \hfill & \le q_{\varrho }({\xi }_n,{\xi }_{n+1})+\dots +q_{\varrho }({\xi }_{n+m-1},{\xi }_{n+m})\hfill \\ \hfill & \le \sum _{k=n}^m{\phi }^k\left(q_{\varrho }({\xi }_0,{\xi }_1)\right)\hfill \\ \hfill & \le \sum _{k\ge n}{\phi }^k\left(q_{\varrho }({\xi }_0,{\xi }_1)\right)\to 0\mathrm{as}n\to \infty .\hfill \end{array}$$

Thus, ${lim}_{n,m\to \infty }{q}_{\varrho }({\xi }_n,{\xi }_m)=0$. Again, since $({\xi }_{n+1},{\xi }_n)\in {\mathfrak{R}}^{*}$ for all $n\in {\mathbb{N}}^{*}$, with the same arguments, we can show that ${lim}_{n,m\to \infty }{q}_{\varrho }({\xi }_m,{\xi }_n)=0$. Therefore, by Lemma 3, $\left\{{\xi }_n\right\}$ is a Cauchy sequence, both in $(\mathsf{\Xi },q_{\varrho })$ and in $(\mathsf{\Xi },q_{\varrho }^s)$. Since $(\mathsf{\Xi },q_{\varrho })$ is $\mathcal{R}$-complete, so is $(\mathsf{\Xi },q_{\varrho }^s)$ and the sequence $\left\{{\xi }_n\right\}$ converges to some ${\xi }^{*}$ in $(\mathsf{\Xi },q_{\varrho }^s)$. Now, $\mathcal{R}$-continuity of $\mathcal{T}$ implies that

$${\xi }^{*}=\underset{n\to \infty }{lim}{\xi }_{n+1}=\underset{n\to \infty }{lim}\mathcal{T}{\xi }_n=\mathcal{T}{\xi }^{*}.$$

Hence, ${\xi }^{*}\in Fix\left(\mathcal{T}\right)$.  □

Theorem 2.

The conclusion of Theorem 1 remains true for $\mathcal{Q}\in {\mathfrak{G}}^{\prime }$, if the condition $\left(C_5\right)$ is replaced by the following one:

($C_5^{\prime }$)

$(\mathsf{\Xi },q_{\varrho },\mathfrak{R})$ is regular.

Proof. 

Following the proof of Theorem 1, we obtain that the sequence $\left\{{\mathcal{T}}^n{\xi }_0\right\}$ is a Cauchy sequence, and so there exists a point ${\xi }^{*}\in \mathsf{\Xi }$, such that

$$\begin{array}{cc}\hfill q_{\varrho }({\xi }^{*},{\xi }^{*})& =\underset{n\to \infty }{lim}{q}_{\varrho }({\xi }_n,{\xi }^{*})=\underset{n\to \infty }{lim}{q}_{\varrho }({\xi }^{*},{\xi }_n)\hfill \\ \hfill & =\underset{n,m\to \infty }{lim}{q}_{\varrho }({\xi }_n,{\xi }_m)=\underset{n,m\to \infty }{lim}{q}_{\varrho }({\xi }_m,{\xi }_n)=0.\hfill \end{array}$$

As $({\xi }_n,{\xi }_{n+1})\in {\mathfrak{R}}^{*}$, then $({\xi }_n,{\xi }^{*})\in {\mathfrak{R}}^{*}$ for all $n\in \mathbb{N}$.

Let $\mathsf{\Lambda }=\{n\in \mathbb{N}:\mathcal{T}{\xi }_n=\mathcal{T}x\}$ consider the following two cases.

• If $\mathsf{\Lambda }$ is finite, then there exists $n_0\in \mathbb{N}$ such that $\mathcal{T}{\xi }_n\ne \mathcal{T}{\xi }^{*}$ for all $n\ge n_0$. In particular, $({\xi }_n,{\xi }^{*})\in {\mathfrak{R}}^{*}$ and $(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }^{*})\in {\mathfrak{R}}^{*}$, and then, for all $n\ge n_0$,

  $$\begin{array}{c}\hfill \mathcal{Q}\left(\begin{array}{c}{q}_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }^{*}),q_{\varrho }({\xi }_n,{\xi }^{*}),q_{\varrho }({\xi }_n,\mathcal{T}{\xi }_n),\\ q_{\varrho }({\xi }^{*},\mathcal{T}{\xi }^{*}),q_{\varrho }({\xi }^{*},\mathcal{T}{\xi }_n)+q_{\varrho }({\xi }_n,\mathcal{T}{\xi }^{*})\end{array}\right)\le 0.\end{array}$$

  Since $\mathcal{Q}$ is continuous, applying the limit as $n\to \infty$, we get

  $$\begin{array}{c}\hfill \mathcal{Q}(\underset{n\to \infty }{lim}{q}_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }^{*}),0,0,q_{\varrho }({\xi }^{*},\mathcal{T}{\xi }^{*}),\underset{n\to \infty }{lim}{q}_{\varrho }({\xi }_n,\mathcal{T}{\xi }^{*}))\le 0.\end{array}$$

  By $\left(q_4\right)$, we have

  $$\begin{array}{cc}\hfill q_{\varrho }({\xi }_n,\mathcal{T}{\xi }^{*})& \le q_{\varrho }({\xi }_n,\mathcal{T}{\xi }_n)+q_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }^{*})-q_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }_n)\hfill \\ \hfill & \le q_{\varrho }({\xi }_n,\mathcal{T}{\xi }_n)+q_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }^{*})\hfill \end{array}$$

  so that

  $$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}{q}_{\varrho }({\xi }_n,\mathcal{T}{\xi }^{*})& \le \underset{n\to \infty }{lim}{q}_{\varrho }({\xi }_n,\mathcal{T}{\xi }_n)+\underset{n\to \infty }{lim}{q}_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }^{*})\hfill \\ \hfill & \le \underset{n\to \infty }{lim}{q}_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }^{*}).\hfill \end{array}$$

  Then by $\left({\mathcal{Q}}_1\right)$ we obtain

  $$\begin{array}{c}\hfill \mathcal{Q}(\underset{n\to \infty }{lim}{q}_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }^{*}),0,0,q_{\varrho }(\xi ,\mathcal{T}{\xi }^{*}),\underset{n\to \infty }{lim}{q}_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }^{*}))\le 0.\end{array}$$

  It follows from $\left({\mathcal{Q}}_3\right)$ that there is $\phi \in \mathsf{\Phi }$ such that

  $$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{q}_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }^{*})\le \phi \left(q_{\varrho }({\xi }^{*},\mathcal{T}{\xi }^{*})\right).\end{array}$$

  (3)

  On the other hand, by $\left(q_4\right)$ we have

  $$\begin{array}{cc}\hfill q_{\varrho }({\xi }^{*},\mathcal{T}{\xi }^{*}))& \le q_{\varrho }({\xi }^{*},\mathcal{T}{\xi }_n)+q_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }^{*})-q_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }_n)\hfill \\ \hfill & \le q_{\varrho }({\xi }_n,\mathcal{T}{\xi }_n)+q_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }^{*})\hfill \end{array}$$

  so that

  $$\begin{array}{cc}\hfill q_{\varrho }({\xi }^{*},\mathcal{T}{\xi }^{*})& \le \underset{n\to \infty }{lim}{q}_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }^{*}).\hfill \end{array}$$

  (4)

  Combining (3) and (4), we get $q_{\varrho }({\xi }^{*},\mathcal{T}{\xi }^{*})\le \phi \left(q_{\varrho }({\xi }^{*},\mathcal{T}\xi )\right)<q_{\varrho }({\xi }^{*},\mathcal{T}{\xi }^{*})$, a contradiction. This implies that $\mathcal{T}{\xi }^{*}={\xi }^{*}$, i.e., ${\xi }^{*}$ is a fixed point of $\mathcal{T}$.
• If $\mathsf{\Lambda }$ is not finite, then there is a subsequence $\left\{{\xi }_{n\left(\varsigma \right)}\right\}$ of $\left\{{\xi }_n\right\}$ such that

  $${\xi }_{n\left(\varsigma \right)+1}=\mathcal{T}{\xi }_{n\left(\varsigma \right)}=\mathcal{T}{\xi }^{*},\forall \varsigma \in \mathbb{N}.$$

  As ${\xi }_n\to {\xi }^{*}$, therefore $\mathcal{T}{\xi }^{*}={\xi }^{*}$.

 □

Further, we present a sufficient condition for the uniqueness of fixed point in Theorems 1 and 2.

Theorem 3.

In addition to the assumptions of Theorem 1 (or Theorem 2), let $\mathfrak{P}(\xi ,\vartheta ;\mathfrak{R})\ne \varnothing$ for all $\xi ,\vartheta \in Fix\left(\mathcal{T}\right)$ and let $\mathcal{Q}\in {\mathfrak{G}}^{″}$. Then the fixed point of $\mathcal{T}$ is unique.

Proof. 

Let $\xi ,\vartheta \in Fix\left(\mathcal{T}\right)$ such that $\xi \ne \vartheta$. Since $\mathfrak{P}(\xi ,\vartheta ;\mathfrak{R})\ne \varnothing$, there exists a path (say $\{{\mu }_0,{\mu }_1,\dots ,{\mu }_k\})$ of some finite length k in $\mathfrak{R}$ from $\xi$ to $\vartheta$ (with ${\mu }_i\ne {\mu }_{i+1}$ for all $0\le i\le k-1)$. Then

$${\mu }_0=\xi ,{\mu }_k=\vartheta ,({\mu }_i,{\mu }_{i+1})\in {\mathfrak{R}}^{*}\mathrm{for} \mathrm{each} 0\le i\le k-1.$$

As ${\mu }_i\in Fix\left(\mathcal{T}\right)$, $\mathcal{T}{\mu }_i={\mu }_i$ for all $0\le i\le k-1$, we have

$$\mathcal{Q}\left(\begin{array}{c}{q}_{\varrho }(\mathcal{T}{\mu }_i,\mathcal{T}{\mu }_{i+1}),q_{\varrho }({\mu }_i,{\mu }_{i+1}),q_{\varrho }({\mu }_i,\mathcal{T}{\mu }_i),\\ q_{\varrho }({\mu }_{i+1},\mathcal{T}{\mu }_{i+1}),q_{\varrho }({\mu }_i,\mathcal{T}{\mu }_{i+1})+q_{\varrho }({\mu }_{i+1},\mathcal{T}{\mu }_i)\end{array}\right)\le 0,$$

i.e.,

$$\mathcal{Q}\left(\begin{array}{c}{q}_{\varrho }({\mu }_i,{\mu }_{i+1}),q_{\varrho }({\mu }_i,{\mu }_{i+1}),q_{\varrho }({\mu }_i,{\mu }_i),\\ q_{\varrho }({\mu }_{i+1},{\mu }_{i+1}),q_{\varrho }({\mu }_i,{\mu }_{i+1})+q_{\varrho }({\mu }_{i+1},{\mu }_i)\end{array}\right)\le 0,$$

hence,

$$\mathcal{Q}\left(\begin{array}{c}{q}_{\varrho }({\mu }_i,{\mu }_{i+1}),q_{\varrho }({\mu }_i,{\mu }_{i+1}),0,0,q_{\varrho }({\mu }_i,{\mu }_{i+1})+q_{\varrho }({\mu }_{i+1},{\mu }_i)\end{array}\right)\le 0.$$

It follows from $\left({\mathcal{Q}}_4\right)$ that there is $\phi \in \mathsf{\Phi }$ such that

$$q_{\varrho }({\mu }_i,{\mu }_{i+1})\le \phi \left(q_{\varrho }({\mu }_{i+1},{\mu }_i)\right).$$

Similarly we can get

$$q_{\varrho }({\mu }_{i+1},{\mu }_i)\le \phi \left(q_{\varrho }({\mu }_i,{\mu }_{i+1})\right).$$

On combining the above conclusions, we have

$$q_{\varrho }({\mu }_{i+1},{\mu }_i)\le {\phi }^2\left(q_{\varrho }({\mu }_{i+1},{\mu }_i)\right).$$

a contradiction. Thus, $\mathcal{T}$ has a unique fixed point.  □

4.2. Periodic Point Results

Definition 5.

([24]). A mapping $\mathcal{T}:\mathsf{\Xi }\to \mathsf{\Xi }$ is said to have the property (P) if $Fix\left({\mathcal{T}}^n\right)=Fix\left(\mathcal{T}\right)$ for every $n\in \mathbb{N}$, i.e., $\mathcal{T}$ has no periodic point.

Theorem 4.

In addition to the assumptions of Theorem 1, if ${\xi }^{*}\in Fix\left({\mathcal{T}}^n\right)$ and ${\xi }^{*}\notin Fix\left(\mathcal{T}\right)$ implies ${\xi }^{*}\mathfrak{R}\mathcal{T}{\xi }^{*}$, then $\mathcal{T}$ has the property (P).

Proof. 

Let ${\xi }_0\in \mathfrak{X}(\mathcal{T},\mathfrak{R})$, i.e., $[{\xi }_0,\mathcal{T}{\xi }_0]\in \mathfrak{R}$, so by condition ($C_2$), we have $({\xi }_n,{\xi }_{n+1})\in \mathfrak{R}$, for all $n\in {\mathbb{N}}_0$. Define ${\xi }_{n+1}=\mathcal{T}{\xi }_n={\mathcal{T}}^{n+1}{\xi }_0$, for all $n\in {\mathbb{N}}^{*}$. If there exists $n_0\in {\mathbb{N}}^{*}$ such that $\mathcal{T}{\xi }_{n_0}={\xi }_{n_0}$, then ${\xi }_{n_0}\in Fix\left(\mathcal{T}\right)$ and there is nothing to prove.

Assume that ${\xi }_{n+1}\ne {\xi }_n$, for all $n\in {\mathbb{N}}^{*}$. Then $({\xi }_n,{\xi }_{n+1})\in {\mathfrak{R}}^{*}$ (for all $n\in {\mathbb{N}}^{*}$). Using the condition ($C_4$) for $\nu ={\xi }_n$ and $\vartheta =\mathcal{T}{\xi }_n$, we have

$$\mathcal{Q}\left(\begin{array}{c}{q}_{\varrho }(\mathcal{T}{\xi }_n,{\mathcal{T}}^2{\xi }_n),q_{\varrho }({\xi }_n,\mathcal{T}{\xi }_n),q_{\varrho }({\xi }_n,\mathcal{T}{\xi }_n),\\ q_{\varrho }(\mathcal{T}{\xi }_n,{\mathcal{T}}^2{\xi }_n),q_{\varrho }({\xi }_n,{\mathcal{T}}^2{\xi }_n)+q_{\varrho }(\mathcal{T}{\xi }_n,\mathcal{T}{\xi }_n)\end{array}\right)\le 0,$$

i.e.,

$$\mathcal{Q}\left(\begin{array}{c}{q}_{\varrho }({\xi }_{n+1},{\xi }_{n+2}),q_{\varrho }({\xi }_n,{\xi }_{n+1}),q_{\varrho }({\xi }_n,{\xi }_{n+1}),\\ q_{\varrho }({\xi }_{n+1},{\xi }_{n+2}),q_{\varrho }({\xi }_n,{\xi }_{n+2})+q_{\varrho }({\xi }_{n+1},{\xi }_{n+1})\end{array}\right)\le 0.$$

Using the properties ($q_4$) and $\left({\mathcal{Q}}_1\right)$, we get

$$\mathcal{Q}\left(\begin{array}{c}{q}_{\varrho }({\xi }_{n+1},{\xi }_{n+2}),q_{\varrho }({\xi }_n,{\xi }_{n+1}),q_{\varrho }({\xi }_n,{\xi }_{n+1}),\\ q_{\varrho }({\xi }_{n+1},{\xi }_{n+2}),q_{\varrho }({\xi }_n,{\xi }_{n+1})+q_{\varrho }({\xi }_{n+1},{\xi }_{n+2})\end{array}\right)\le 0.$$

This yields (from (${\mathcal{Q}}_2$)) that, for all $n\in {\mathbb{N}}^{*}$,

$$q_{\varrho }({\xi }_{n+1},{\xi }_{n+2})\le \phi \left(q_{\varrho }({\xi }_n,{\xi }_{n+1})\right).$$

(5)

Following the rest of the proof of Theorem 1, $\left\{{\xi }_n\right\}$ is a Cauchy sequence in $(\mathsf{\Xi },q_{\varrho })$ and in $(\mathsf{\Xi },q_{\varrho }^s)$. Using the $\mathfrak{R}$-completeness and $\mathfrak{R}$-continuity of $\mathcal{T}$, $\mathcal{T}\xi =\xi$. Finally, we show that $Fix\left({\mathcal{T}}^n\right)=Fix\left(\mathcal{T}\right)$ for any $n\in \mathbb{N}$. Let, on contrary, $\xi \in Fix\left({\mathcal{T}}^n\right)$ and $\xi \notin Fix\left(\mathcal{T}\right)$ hold for some $n\in \mathbb{N}$. Then $q_{\varrho }(\xi ,\mathcal{T}\xi )>0$ and $(\xi ,\mathcal{T}\xi )\in {\mathfrak{R}}^{*}$ ( by assumption ). By condition ($C_2$), we have $({\mathcal{T}}^i\xi ,{\mathcal{T}}^{i+1}\xi )\in {\mathfrak{R}}^{*}$ for all $i\in {\mathbb{N}}^{*}$. From (5), we have

$$\begin{array}{cc}\hfill q_{\varrho }(\xi ,\mathcal{T}\xi )& =q_{\varrho }({\mathcal{T}}^n\xi ,\mathcal{T}{\mathcal{T}}^n\xi )=q_{\varrho }({\xi }_n,{\xi }_{n+1})\hfill \\ \hfill & \le \phi \left(q_{\varrho }({\xi }_{n-1},{\xi }_n)\right)\le \cdots \le {\phi }^n\left(q_{\varrho }(\xi ,\mathcal{T}\xi )\right),\hfill \end{array}$$

a contradiction as $\phi \left(\zeta \right)<\zeta$. Therefore, $Fix\left({\mathcal{T}}^n\right)=Fix\left(\mathcal{T}\right)$ for all $n\in \mathbb{N}.$  □

5. Illustrations and Consequences

Example 6.

Consider the set $\mathsf{\Xi }=[0,1]$ and define a quasi partial metric $q_{\varrho }$ by $q_{\varrho }(\nu ,\vartheta )=|\nu -\vartheta |+\left|\nu \right|$, for all $\nu ,\vartheta \in \mathsf{\Xi }$. Define a binary relation $\mathfrak{R}$ by

$$(\nu ,\vartheta )\in \mathfrak{R}\iff \nu ,\vartheta \in \left\{0\right\}\cup \left\{\frac{1}{9^n}:n\in \mathbb{N}\right\}\mathrm{for}\mathrm{all}\nu ,\vartheta \in \mathsf{\Xi }.$$

Then Ξ is $\mathcal{R}$-complete. Consider the self-mapping $\mathcal{T}$ on Ξ given by $\mathcal{T}\nu =\frac{\nu }{9}$ for all $\nu \in \mathsf{\Xi }$.

It is easy to see that $\mathfrak{X}(\mathcal{T},\mathfrak{R})\ne \varnothing$, that $\mathcal{R}$ is $\mathcal{T}$-closed, and that $\mathcal{T}$ is $\mathcal{R}$-continuous.

Considering Example 2, (2) reduces to

$$q_{\varrho }(\mathcal{T}\nu ,\mathcal{T}\vartheta )\le \alpha q_{\varrho }(\nu ,\vartheta )+\beta q_{\varrho }(\nu ,\mathcal{T}\nu )+\gamma q_{\varrho }(\vartheta ,\mathcal{T}\vartheta )+\delta [q_{\varrho }(\nu ,\mathcal{T}\vartheta )+q_{\varrho }(\vartheta ,\mathcal{T}\nu )].$$

(6)

We show that $\mathcal{T}$ satisfies (6) by dividing the proof in two parts.

Take $(\nu ,\vartheta )\in {\mathfrak{R}}^{*}$ and so $0\le \vartheta ,\nu \le \frac{1}{9}$. Consider two cases:

Case I:If $\nu =0$ and $\vartheta =1/9^n$ (similarly for $\vartheta =0$ and $\nu =1/9^n$), $n\in \mathbb{N}$, then (6) reduces to

$$\frac{1}{9^{n+1}}\le \alpha ·\frac{1}{9^n}+\gamma \left[\left|\frac{1}{9^n}-\frac{1}{9^{n+1}}\right|+\frac{1}{9^n}\right]+\delta \left[\frac{1}{9^{n+1}}+\frac{2}{9^n}\right].$$

Case II:Let $\vartheta ,\nu \in \{1/9^n\mid n\in \mathbb{N}\}$ with $0<\vartheta <\nu$, i.e., $\vartheta \le \nu /9$. Then (6) reduces to

$$\begin{array}{cc}\left|\frac{\nu }{9}-\frac{\vartheta }{9}\right|+\frac{\nu }{9}& \le \alpha \left[\right|\nu -\vartheta |+\nu ]+\beta \left[\left|\nu -\frac{\nu }{9}\right|+\nu \right]+\gamma \left[\left|\vartheta -\frac{\vartheta }{9}\right|+\vartheta \right]\\ & +\delta \left[\left|\nu -\frac{\vartheta }{9}\right|+\nu +\left[\left|\vartheta -\frac{\nu }{9}\right|+\vartheta \right]\right].\hfill \end{array}$$

It can be easily checked that the above inequalities hold true for $\alpha =\beta =\gamma =\delta =1/8$ (so that $\alpha +\beta +\gamma +2\delta <1$).

Thus, $\mathcal{T}$ is a $q_{\varrho }$-implicit contractive mapping. Therefore, all the conditions of Theorem 1 are satisfied and ${\xi }^{*}=0$ is the unique fixed point of $\mathcal{T}$ in Ξ.

Example 7.

Consider the set $\mathsf{\Xi }=\{0,\frac{1}{5},1\}$ and define a quasi partial metric $q_{\varrho }$ by

$$q_{\varrho }(\nu ,\vartheta )=\left\{\begin{array}{cc}15\nu +\vartheta +2,\hfill & \nu \ne \vartheta \hfill \\ 1,\hfill & \nu =\vartheta ,\hfill \end{array}\right.$$

and a binary relation $\mathfrak{R}$ by $\mathfrak{R}=\left\{(0,1)\left(\frac{1}{5},1\right),(1,1)\right\}$. Consider a self-mapping $\mathcal{T}$ on Ξ given by $\mathcal{T}0=\frac{1}{5}$, $\mathcal{T}\frac{1}{5}=1$, $\mathcal{T}1=1$.

It is clear that Ξ is $\mathfrak{R}$ complete and $\mathfrak{R}$ is $\mathcal{T}$-closed. In addition, ${\mathfrak{R}}^{*}=\{(0,1),(\frac{1}{5},1)\}$, and $\mathfrak{X}(\mathcal{T};\mathfrak{R})\ne \varnothing$ as $(\frac{1}{5},\mathcal{T}\frac{1}{5})=(\frac{1}{5},1)\in \mathfrak{R}$ and $(1,\mathcal{T}1)=(1,1)\in \mathfrak{R}$.

Considering Example 3, (2) reduces to

$$q_{\varrho }(\mathcal{T}\nu ,\mathcal{T}\vartheta )\le kmax\left\{q_{\varrho }(\nu ,\vartheta ),q_{\varrho }(\nu ,\mathcal{T}\nu ),q_{\varrho }(\vartheta ,\mathcal{T}\vartheta ),q_{\varrho }(\nu ,\mathcal{T}\vartheta )+q_{\varrho }(\vartheta ,\mathcal{T}\nu )\right\}.$$

(7)

We show that $\mathcal{T}$ satisfies (7). We have to check two possible cases when $(\nu ,\vartheta )\in {\mathfrak{R}}^{*}$.

• Let $(\nu ,\vartheta )=(0,1)$. Then $q_{\varrho }(\mathcal{T}\nu ,\mathcal{T}\vartheta )=6$, $q_{\varrho }(\nu ,\vartheta )=3$, $q_{\varrho }(\nu ,\mathcal{T}\nu )=\frac{11}{5}$, $q_{\varrho }(\vartheta ,\mathcal{T}\vartheta )=1$, $q_{\varrho }(\nu ,\mathcal{T}\vartheta )=3$, $q_{\varrho }(\vartheta ,\mathcal{T}\nu )=\frac{86}{5}$. Therefore, the condition (7) reduces to $6\le 86k/5$.
• Let $(\nu ,\vartheta )=(\frac{1}{5},1)$. Then, $q_{\varrho }(\mathcal{T}\nu ,\mathcal{T}\vartheta )=1$, $q_{\varrho }(\nu ,\vartheta )=6$, $q_{\varrho }(\nu ,\mathcal{T}\nu )=6$, $q_{\varrho }(\vartheta ,\mathcal{T}\vartheta )=1$, $q_{\varrho }(\nu ,\mathcal{T}\vartheta )=6$, $q_{\varrho }(\vartheta ,\mathcal{T}\nu )=1$. Therefore, the condition (7) reduces to $1\le 6k$.

Obviously, the above cases hold true for any k with $\frac{15}{43}<k<\frac{1}{2}$. Thus, $\mathcal{T}$ is a $q_{\varrho }$-implicit contractive mapping.

Let $\left({\nu }_n\right)$ be an $\mathfrak{R}$-preserving sequence converging to ν as $n\to \infty$. Then, we must have

$$({\nu }_n,{\nu }_{n+1})\in \left\{(0,1),\left(\frac{1}{5},1\right),(1,1)\right\}$$

implying that ${\nu }_n\in \left\{1,\frac{1}{5}\right\}$. This implies that either ${\nu }_n\to 1$ or ${\nu }_n\to \frac{1}{5}$ as $n\to \infty$ and clearly, we have $[{\nu }_n,\nu ]\in \mathfrak{R}$ for all $n\in \mathbb{N}$, where $\nu =1$ or $\frac{1}{5}$. This shows that $(\mathsf{\Xi },q_{\varrho },\mathfrak{R})$ is regular. Thus, all the conditions of Theorem 2 are satisfied; hence, $\mathcal{T}$ has a fixed point (${\xi }^{*}=1$) which is also a periodic point of $\mathcal{T}$.

It can be noted that the contraction condition considered in Theorem 3.1 or, in particular, Ćirić type contraction condition taken in Theorem 3.2 of [19], do not hold at $(\nu ,\vartheta )=(\frac{1}{5},0)\notin \mathfrak{R}$ as (7) at this point reduces to

$$\frac{86}{5}\nleqq \frac{41k}{5}=kmax\left\{5,6,\frac{11}{5},\frac{41}{5}\right\}$$

for any $k\in (0,1/2)$. Hence, these results from [19] cannot be used for obtaining the desired conclusion.

By choosing particular $\mathcal{Q}\in \mathfrak{G}$ from Examples 2–5, we have the following consequences.

Corollary 1.

Let all the conditions of Theorems 1–3 be satisfied, except that the assumption of $q_{\varrho }$-implicit contractive mapping for $\mathcal{Q}\in {\mathfrak{G}}^{″}$ is replaced by either of the form

(I)

$$\begin{array}{c}\hfill q_{\varrho }(\mathcal{T}\nu ,\mathcal{T}\vartheta )\le \alpha q_{\varrho }(\nu ,\vartheta )+\beta q_{\varrho }(\nu ,\mathcal{T}\nu )+\gamma q_{\varrho }(\vartheta ,\mathcal{T}\vartheta )+\delta [q_{\varrho }(\nu ,\mathcal{T}\vartheta )+q_{\varrho }(\vartheta ,\mathcal{T}\nu )]\end{array}$$

where $\alpha >0$, $\beta ,\gamma ,\delta \ge 0$, and $\alpha +\beta +\gamma +2\delta <1$, or

(II)

$$\begin{array}{c}\hfill q_{\varrho }(\mathcal{T}\nu ,\mathcal{T}\vartheta )\le kmax\left\{q_{\varrho }(\nu ,\vartheta ),q_{\varrho }(\nu ,\mathcal{T}\nu ),q_{\varrho }(\vartheta ,\mathcal{T}\vartheta ),q_{\varrho }(\nu ,\mathcal{T}\vartheta )+q_{\varrho }(\vartheta ,\mathcal{T}\nu )\right\}\end{array}$$

(8)

where $0<k<1/2$, or

(III)

$$\begin{array}{c}\hfill {\left[q_{\varrho }(\mathcal{T}\nu ,\mathcal{T}\vartheta )\right]}^2\le \alpha q_{\varrho }(\nu ,\vartheta )·q_{\varrho }(\nu ,\mathcal{T}\nu )+\beta {\left[q_{\varrho }(\vartheta ,\mathcal{T}\vartheta )\right]}^2+\gamma {[q_{\varrho }(\nu ,\mathcal{T}\vartheta )+q_{\varrho }(\vartheta ,\mathcal{T}\nu )]}^2\end{array}$$

(9)

where $\alpha >0$, $\beta ,\gamma \ge 0$, and $\alpha +\beta +4\gamma <1$, or

(IV)

$$\begin{array}{c}\hfill q_{\varrho }(\mathcal{T}\nu ,\mathcal{T}\vartheta )\le \alpha q_{\varrho }(\nu ,\vartheta )+\beta \frac{1+q_{\varrho }(\nu ,\mathcal{T}\nu )q_{\varrho }(\vartheta ,\mathcal{T}\vartheta )}{1+q_{\varrho }(\nu ,\vartheta )}+\gamma [q_{\varrho }(\nu ,\mathcal{T}\vartheta )+q_{\varrho }(\vartheta ,\mathcal{T}\nu )]\end{array}$$

(10)

where $\alpha >0$, $\beta ,\gamma \ge 0$, and $\alpha +\beta +2\gamma <1$.

Then $Fix\left(\mathcal{T}\right)$ is a singleton.

6. Application

Let $\mathcal{M}\left(n\right)$ denote the set of all $n\times n$ matrices over $\mathbb{C}$ and $\mathcal{H}\left(n\right)$ (resp. $\mathcal{K}\left(n\right)$, $\mathcal{P}\left(n\right)$) denote the set of all Hermitian (resp. positive semi-definite, positive definite) matrices from $\mathcal{M}\left(n\right)$. For a matrix $\mathcal{B}\in \mathcal{H}\left(n\right)$, we will denote by ${\parallel \mathcal{B}\parallel }_{tr}$ its trace norm, i.e., the sum of all of its singular values. For $\mathcal{C},\mathcal{D}\in \mathcal{H}\left(n\right)$, $\mathcal{C}⪰\mathcal{D}$ (resp. $\mathcal{C}\succ \mathcal{D}$) will mean that the matrix $\mathcal{C}-\mathcal{D}$ belongs to $\mathcal{K}\left(n\right)$ (resp. $\mathcal{P}\left(n\right)$).

The following lemmas are needed in the subsequent discussion.

Lemma 4.

([5]). If $\mathcal{A}⪰O$ and $\mathcal{B}⪰O$ are $n\times n$ matrices, then

$$0\le tr\left(\mathcal{AB}\right)\le {\parallel \mathcal{A}\parallel }_{tr}tr\left(\mathcal{B}\right).$$

Lemma 5.

([5]). If $\mathcal{A}\in \mathcal{H}\left(n\right)$ satisfies $\mathcal{A}\prec I_n$, then ${\parallel \mathcal{A}\parallel }_{tr}<1$.

We establish the existence and uniqueness of the solution of non-linear matrix equation (NME)

$$\mathcal{U}=\mathcal{D}+\sum _{i=1}^m{\mathcal{A}}_i^{*}\mathcal{G}\left(\mathcal{U}\right){\mathcal{A}}_i,$$

(11)

where $\mathcal{D}\in \mathcal{P}\left(n\right)$, ${\mathcal{A}}_i^{*}$ stands for the conjugate transpose of ${\mathcal{A}}_i\in \mathcal{M}\left(n\right)$ and $\mathcal{G}:\mathcal{P}\left(n\right)\to \mathcal{P}\left(n\right)$ is an order-preserving continuous mapping such that $\mathcal{G}\left(O\right)=O$.

Theorem 5.

Consider the NME (11) and assume the following:

(H₁)

there exists $\mathcal{D}\in \mathcal{P}\left(n\right)$ with $\mathcal{D}⪯\mathcal{D}+\sum _{i=1}^m{\mathcal{A}}_i^{*}\mathcal{G}\left(\mathcal{D}\right){\mathcal{A}}_i$;

(H₂)

for all $\mathcal{U},\mathcal{V}\in \mathcal{P}\left(n\right)$, $\mathcal{U}⪯\mathcal{V}$ implies $\sum _{i=1}^m{\mathcal{A}}_i^{*}\mathcal{G}\left(\mathcal{U}\right){\mathcal{A}}_i⪯\sum _{i=1}^m{\mathcal{A}}_i^{*}\mathcal{G}\left(\mathcal{V}\right){\mathcal{A}}_i$.

(H₃)

there exists $k\in (0,1/2)$ such that for all $\mathcal{U},\mathcal{V}\in \mathcal{P}\left(n\right)$ with $\mathcal{U}⪯\mathcal{V}$, the following inequality holds

$$\begin{array}{cc}\hfill & ∥\sum _{i=1}^m{\mathcal{A}}_i^{*}(\mathcal{G}\left(\mathcal{U}\right)-\mathcal{G}\left(\mathcal{V}\right))\mathcal{A}∥+∥\mathcal{D}+\sum _{i=1}^m{\mathcal{A}}_i^{*}\mathcal{G}\left(\mathcal{U}\right)\mathcal{A}∥\le \hfill \\ \hfill & kmax\left\{\begin{array}{c}\parallel \mathcal{U}-\mathcal{V}\parallel +\parallel \mathcal{U}\parallel ,\parallel \mathcal{U}-\mathcal{D}-{\displaystyle \sum _{i=1}^m}{\mathcal{A}}_i^{*}\mathcal{G}\left(\mathcal{U}\right){\mathcal{A}}_i\parallel +\parallel \mathcal{U}\parallel ,\\ \parallel \mathcal{V}-\mathcal{D}-{\displaystyle \sum _{i=1}^m}{\mathcal{A}}_i^{*}\mathcal{G}\left(\mathcal{V}\right){\mathcal{A}}_i\parallel +\parallel \mathcal{V}\parallel ,\\ \parallel \mathcal{U}-\mathcal{D}-{\displaystyle \sum _{i=1}^m}{\mathcal{A}}_i^{*}\mathcal{G}\left(\mathcal{V}\right){\mathcal{A}}_i\parallel +\parallel \mathcal{U}\parallel +\parallel \mathcal{V}-\mathcal{D}-{\displaystyle \sum _{i=1}^m}{\mathcal{A}}_i^{*}\mathcal{G}\left(\mathcal{U}\right){\mathcal{A}}_i\parallel +\parallel \mathcal{V}\parallel \end{array}\right\}.\hfill \end{array}$$

Then the NME (11) has a unique solution $\widehat{\mathcal{U}}$. Moreover, the iteration

$${\mathcal{U}}_n=\mathcal{D}+\sum _{i=1}^m{\mathcal{A}}_i^{*}\mathcal{G}\left({\mathcal{U}}_{n-1}\right){\mathcal{A}}_i,$$

(12)

where ${\mathcal{U}}_0\in \mathcal{P}\left(n\right)$ satisfies ${\mathcal{U}}_0⪯\mathcal{D}+\sum _{i=1}^m{\mathcal{A}}_i^{*}\mathcal{G}\left({\mathcal{U}}_0\right){\mathcal{A}}_i$, converges to $\widehat{\mathcal{U}}$ in the sense of trace norm ${\parallel ·\parallel }_{tr}$.

Proof. 

Define a mapping $\mathcal{T}:\mathcal{P}\left(n\right)\to \mathcal{P}\left(n\right)$ by

$$\mathcal{T}\left(\mathcal{U}\right)=\mathcal{D}+\sum _{i=1}^m{\mathcal{A}}_i^{*}\mathcal{G}\left(\mathcal{U}\right){\mathcal{A}}_i,\mathrm{for}\mathrm{all}\mathcal{U}\in \mathcal{P}\left(n\right),$$

and a binary relation

$$\mathfrak{R}=\left\{\right(\mathcal{U},\mathcal{V})\in \mathcal{P}(n)\times \mathcal{P}(n):\mathcal{U}⪯\mathcal{V}\}.$$

Then a fixed point of the mapping $\mathcal{T}$ is a solution of the matrix Equation (11). Notice that $\mathcal{T}$ is well defined, $\mathfrak{R}$-continuous and $\mathfrak{R}$ is $\mathcal{T}$-closed. By the assumption ($H_1$), we have $(\mathcal{D},\mathcal{T}(\mathcal{D}\left)\right)\in \mathfrak{R}$ and hence $\mathfrak{X}(\mathcal{T};\mathfrak{R})\ne \varnothing$.

Now, let $(\mathcal{U},\mathcal{V})\in {\mathfrak{R}}^{*}=\{(\mathcal{U},\mathcal{V})\in \mathfrak{R}:\mathcal{U}\ne \mathcal{V}\}$. Define $q_{\varrho }:\mathcal{P}\left(n\right)\times \mathcal{P}\left(n\right)\to {\mathbb{R}}_{+}$ by

$$q_{\varrho }(\mathcal{U},\mathcal{V})={\parallel \mathcal{U}-\mathcal{V}\parallel }_{tr}+{\parallel \mathcal{U}\parallel }_{tr}\mathrm{for}\mathrm{all}\mathcal{U},\mathcal{V}\in \mathcal{P}\left(n\right).$$

Then $(\mathcal{P}\left(n\right),q_{\varrho },\mathfrak{R})$ is a complete $\mathfrak{R}$-relational quasi partial metric space.

Consider $\mathcal{Q}\in \mathfrak{G}$ given by

$$\mathcal{Q}(s_1,s_2,s_3,s_4,s_5)=s_1-kmax\{s_2,s_3,s_4,s_5\}$$

where $k\in (0,1/2)$ is given in ($H_3$) (see Corollary 1.(II)). Then the condition

$$\begin{array}{c}\hfill \mathcal{Q}\left(\begin{array}{c}{q}_{\varrho }(\mathcal{T}\left(\mathcal{U}\right),\mathcal{T}\left(\mathcal{V}\right)),q_{\varrho }(\mathcal{U},\mathcal{V}),q_{\varrho }(\mathcal{U},\mathcal{T}\left(\mathcal{U}\right)),\\ q_{\varrho }(\mathcal{V},\mathcal{T}\left(\mathcal{V}\right)),q_{\varrho }(\mathcal{U},\mathcal{T}\left(\mathcal{V}\right))+q_{\varrho }(\mathcal{V},\mathcal{T}\left(\mathcal{U}\right))\end{array}\right)\le 0.\end{array}$$

reduces to

$$\begin{array}{cc}\hfill & q_{\varrho }(\mathcal{T}\left(\mathcal{U}\right),\mathcal{T}\left(\mathcal{V}\right))\le \hfill \\ \hfill & kmax\{q_{\varrho }(\mathcal{U},\mathcal{V}),q_{\varrho }(\mathcal{U},\mathcal{T}\left(\mathcal{U}\right)),q_{\varrho }(\mathcal{V},\mathcal{T}\left(\mathcal{V}\right)),q_{\varrho }(\mathcal{U},\mathcal{T}\left(\mathcal{V}\right))+q_{\varrho }(\mathcal{V},\mathcal{T}\left(\mathcal{U}\right))\}\hfill \end{array}$$

and is fulfilled by the assumption ($H_3$). Thus, all the hypotheses of Theorem 1 are satisfied, and therefore there exists $\widehat{\mathcal{U}}\in \mathcal{P}\left(n\right)$ such that $\mathcal{T}\left(\widehat{\mathcal{U}}\right)=\widehat{\mathcal{U}}$, and hence the matrix Equation (11) has a solution in $\mathcal{P}\left(n\right)$. Furthermore, due to the existence of least upper bound and greatest lower bound for all $\mathcal{U},\mathcal{V}\in \mathcal{P}\left(n\right)$, we have $\mathfrak{P}(\mathcal{U},\mathcal{V},\mathfrak{R})\ne \varnothing$. Thus, on using Theorem 3, $\mathcal{T}$ has a unique fixed point, and hence we conclude that the matrix Equation (11) has a unique solution in $\mathcal{P}\left(n\right)$, obtained as the limit of iterative sequence (12).  □

7. Numerical Experiments

Example 8.

Consider matrices with randomly generated coefficients by

$$\begin{array}{ccc}\hfill {\mathcal{A}}_1={\mathcal{Q}}_1^{-(1/2)}\tilde{\mathcal{A}}{\mathcal{Q}}_1^{-(1/2)},& & {\mathcal{A}}_2={\mathcal{Q}}_2^{-(1/2)}\tilde{\mathcal{B}}{\mathcal{Q}}_2^{-(1/2)},\hfill \\ \hfill {\mathcal{A}}_3={\mathcal{Q}}_3^{-(1/2)}\tilde{\mathcal{C}}{\mathcal{Q}}_3^{-(1/2)},\end{array}$$

where

$${\mathcal{Q}}_1=\mathcal{I}+{\tilde{\mathcal{A}}}^{*}\tilde{\mathcal{A}},{\mathcal{Q}}_2=\mathcal{I}+{\tilde{\mathcal{B}}}^{*}\tilde{\mathcal{B}},{\mathcal{Q}}_3=\mathcal{I}+{\tilde{\mathcal{C}}}^{*}\tilde{\mathcal{C}},$$

and

$${\left(\tilde{\mathcal{A}}\right)}_{ij}=\frac{180}{i+j-1},\tilde{\mathcal{B}}=\frac{1}{4}\tilde{\mathcal{A}},{\left(\tilde{\mathcal{C}}\right)}_{ij}=\frac{240}{i+j-1}.$$

Take $n=4$ and

$$\mathcal{D}=\left[\begin{array}{cccc}0.002001041665461& 0.000001388855136& 0.000001735883345& 0.000002082504561\\ 0.000001388855136& 0.002001874902367& 0.000002360744740& 0.000002846107903\\ 0.000001735883345& 0.000002360744740& 0.002002985358081& 0.000003609419653\\ 0.000002082504561& 0.000002846107903& 0.000003609419653& 0.002004372108965\end{array}\right].$$

We take $\eta =5$, and $\mathcal{G}\left(\mathcal{U}\right)={\mathcal{U}}^{5/2}$ to test our algorithm. The numerical results are given in Table 1.

Table 1. Three initialization based analysis.

Int. Mat	$\mathcal{G}\left(\mathit{U}\right)$	k	$\mathit{\eta }$	Dim	Iter No.	CPU (Sec)	Error ($1\times$ 10 ${}^{-15}$)	Min (Eig)
$U_0$	$U^{5/2}$	0.2	5	4	6	0.025963	0.303848	0.002
$V_0$	$V^{5/2}$	0.2	5	4	6	0.023645	0.0594	0.002
$W_0$	$W^{5/2}$	0.2	5	4	6	0.021197	0.0104	0.002

We use the initial values

$${\mathcal{U}}_0=\left[\begin{array}{cccc}0.009259103097439& 0.012341315127054& 0.015402771909257& 0.018412458542713\\ 0.012341315127054& 0.016654059129192& 0.020940448808588& 0.025165938208678\\ 0.015402771909257& 0.020940448808588& 0.026446565358956& 0.031881958855667\\ 0.018412458542713& 0.025165938208678& 0.031881958855667& 0.038521230063628\end{array}\right],$$

$${\mathcal{V}}_0=\left[\begin{array}{cccc}0.004629551548720& 0.006170657563527& 0.007701385954628& 0.009206229271357\\ 0.006170657563527& 0.008327029564596& 0.010470224404294& 0.012582969104339\\ 0.007701385954628& 0.010470224404294& 0.013223282679478& 0.015940979427834\\ 0.009206229271357& 0.012582969104339& 0.015940979427834& 0.019260615031814\end{array}\right],$$

$${\mathcal{W}}_0=\left[\begin{array}{cccc}0.002314775774360& 0.003085328781763& 0.003850692977314& 0.004603114635678\\ 0.003085328781763& 0.004163514782298& 0.005235112202147& 0.006291484552169\\ 0.003850692977314& 0.005235112202147& 0.006611641339739& 0.007970489713917\\ 0.004603114635678& 0.006291484552169& 0.007970489713917& 0.009630307515907\end{array}\right],$$

where ${\mathcal{U}}_0,{\mathcal{V}}_0,{\mathcal{W}}_0\in \mathcal{P}\left(4\right)$.

After 6 successive iterations, we obtain the following positive-definite solution

$$\begin{array}{cc}\hfill & \widehat{\mathcal{X}}=\left[\begin{array}{cccc}0.002001043436828& 0.000001391216745& 0.000001738831424& 0.000002086029718\\ 0.000001391216745& 0.002001878089562& 0.000002364752730& 0.000002850925622\\ 0.000001738831424& 0.000002364752730& 0.002002990420241& 0.000003615523118\\ 0.000002086029718& 0.000002850925622& 0.000003615523118& 0.002004379484192\end{array}\right].\hfill \end{array}$$

The graphical view of convergence and $\widehat{\mathcal{X}}$ are shown in Figure 1 and Figure 2 respectively:

Figure 1. Convergence analysis grpah.

Figure 2. Solution graph.

Example 9.

Consider the following matrices ${\mathcal{A}}_1,{\mathcal{A}}_2,{\mathcal{A}}_3,\mathcal{D}\in \mathcal{M}\left(4\right)$:

$${\mathcal{A}}_1=\left[\begin{array}{cccc}0.0241+0.0194i& 0.0220-0.0067i& 0.0461-0.0037i& 0.0578+0.0157i\\ 0.0841-0.0062i& 0.0226+0.0022i& 0.0639+0.0032i& 0.0433-0.0087i\\ 0.0857-0.0030i& 0.0537+0.0165i& 0.0917+0.0187i& 0.0884+0.0108i\\ 0.0964-0.0093i& 0.0762+0.0064i& 0.0162-0.0063i& 0.0393+0.0194i\end{array}\right],$$

$${\mathcal{A}}_2=\left[\begin{array}{cccc}0.1943+0.0020i& -0.0668+0.0019i& -0.0375+0.0013i& 0.1571+0.0087i\\ -0.0619+0.0035i& 0.0222+0.0045i& 0.0323+0.0080i& -0.0870+0.0069i\\ -0.0303+0.0022i& 0.1652+0.0068i& 0.1869+0.0008i& 0.1075+0.0051i\\ -0.0929+0.0000i& 0.0644+0.0047i& -0.0628+0.0012i& 0.1937+0.0062i\end{array}\right],$$

$${\mathcal{A}}_3=\left[\begin{array}{cccc}0.0199+0.0024i& 0.0192+0.0022i& 0.0128+0.0046i& 0.0871+0.0058i\\ 0.0353+0.0084i& 0.0450+0.0023i& 0.0804+0.0064i& 0.0685+0.0043i\\ 0.0218+0.0086i& 0.0682+0.0054i& 0.0076+0.0092i& 0.0512+0.0088i\\ 0.0003+0.0096i& 0.0468+0.0076i& 0.0123+0.0016i& 0.0616+0.0039i\end{array}\right],$$

$$\mathcal{D}=\left[\begin{array}{cccc}0.0020+0.0000i& 0.0000+0.0000i& 0.0000-0.0000i& 0.0000-0.0000i\\ 0.0000-0.0000i& 0.0020+0.0000i& 0.0000-0.0000i& 0.0000-0.0000i\\ 0.0000+0.0000i& 0.0000+0.0000i& 0.0020+0.0000i& 0.0000-0.0000i\\ 0.0000+0.0000i& 0.0000+0.0000i& 0.0000+0.0000i& 0.0020+0.0000i\end{array}\right].$$

In order to show the convergence of the sequence $\left\{{\mathcal{U}}_n\right\}$ defined in (12), we start with the following initializations and discuss the CPU time and errors in Table 2:

$${\mathcal{U}}_0=\left[\begin{array}{cccc}0.0756+0.0000i& -0.0206+0.0004i& 0.0055+0.0013i& 0.0176+0.0021i\\ -0.0206-0.0004i& 0.0268+0.0000i& 0.0204+0.0006i& 0.0008+0.0007i\\ 0.0055-0.0013i& 0.0204-0.0006i& 0.1020+0.0000i& 0.0401+0.0005i\\ 0.0176-0.0021i& 0.0008-0.0007i& 0.0401-0.0005i& 0.0718+0.0000i\end{array}\right],$$

$${\mathcal{V}}_0=\left[\begin{array}{cccc}0.0378+0.0000i& -0.0103+0.0002i& 0.0028+0.0006i& 0.0088+0.0011i\\ -0.0103-0.0002i& 0.0134+0.0000i& 0.0102+0.0003i& 0.0004+0.0003i\\ 0.0028-0.0006i& 0.0102-0.0003i& 0.0510+0.0000i& 0.0201+0.0003i\\ 0.0088-0.0011i& 0.0004-0.0003i& 0.0201-0.0003i& 0.0359+0.0000i\end{array}\right],$$

$${\mathcal{W}}_0=\left[\begin{array}{cccc}0.0189+0.0000i& -0.0052+0.0001i& 0.0014+0.0003i& 0.0044+0.0005i\\ -0.0052-0.0001i& 0.0067+0.0000i& 0.0051+0.0002i& 0.0002+0.0002i\\ 0.0014-0.0003i& 0.0051-0.0002i& 0.0255+0.0000i& 0.0100+0.0001i\\ 0.0044-0.0005i& 0.0002-0.0002i& 0.0100-0.0001i& 0.0179+0.0000i\end{array}\right],$$

Table 2. Three initialization based analysis.

Int. Mat	$\mathcal{G}\left(\mathit{U}\right)$	k	$\mathit{\eta }$	Dim	Iter No.	CPU (Sec)	Error ($1\times$ 10 ${}^{-15}$)	Min (Eig)
$U_0$	$U^{3/2}$	0.3	5	4	9	0.025963	0.012884	0.002005
$V_0$	$V^{3/2}$	0.3	5	4	8	0.023645	0.009381	0.002005
$W_0$	$W^{3/2}$	0.3	5	4	8	0.021197	0.008276	0.002005

where ${\mathcal{U}}_0,{\mathcal{V}}_0,{\mathcal{W}}_0\in \mathcal{P}\left(4\right)$. We obtain

The positive definite solution is given by

$$\widehat{\mathcal{X}}=\left[\begin{array}{cccc}0.0020-0.0000i& 0.0000+0.0000i& 0.0000-0.0000i& 0.0000-0.0000i\\ 0.0000-0.0000i& 0.0020-0.0000i& 0.0000-0.0000i& 0.0000-0.0000i\\ 0.0000+0.0000i& 0.0000+0.0000i& 0.0020-0.0000i& 0.0000-0.0000i\\ 0.0000+0.0000i& 0.0000+0.0000i& 0.0000+0.0000i& 0.0020+0.0000i\end{array}\right].$$

The graphical view of convergence is shown in Figure 3 below:

Figure 3. Convergence analysis graph.

8. Conclusions

In this paper, investigations of fixed point problems in so-called metric fixed point theory have been broadened to problems formulated in terms of $q_{\varrho }$-quasi implicit contractive condition for a self-mapping on a relational quasi partial metric space. In such a way, more general results have been obtained than those existing in literature. It has been shown by explicit examples that these generalizations are proper. The obtained results have been applied to the field of nonlinear matrix equations which is a very important and applicable area by its own.