The Cerebellum and Premenstrual Dysphoric Disorder

Andrea J. Rapkin; Steven M. Berman; Edythe D. London; Andrea J. Rapkin; Steven M. Berman; Edythe D. London

doi:10.3934/Neuroscience.2014.2.120

AIMS Neuroscience

2014, Volume 1, Issue 2: 120-141. doi: 10.3934/Neuroscience.2014.2.120

Previous Article Next Article

Review Special Issues

The Cerebellum and Premenstrual Dysphoric Disorder

1.
USA David Geffen School of Medicine at UCLA, Box 951740, 27-139 CHS, Los Angeles, CA 90095, USA;
2.
Center for Addictive Behaviors, Department of Psychiatry and Biobehavioral Sciences UCLA, 90095 USA;
3.
Department of Psychiatry and Biobehavioral Sciences, Department of Molecular and Medical Pharmacology, and Brain Research Institute, UCLA, 90095 USA

Received: 30 April 2014 Accepted: 07 July 2014 Published: 30 July 2014

The cerebellum constitutes ten percent of brain volume and contains the majority of brain neurons. Although it was historically viewed primarily as processing motoric computations, current evidence supports a more comprehensive role, where cerebro-cerebellar feedback loops also modulate various forms of cognitive and affective processing. Here we present evidence for a role of the cerebellum in premenstrual dysphoric disorder (PMDD), which is characterized by severe negative mood symptoms during the luteal phase of the menstrual cycle. Although a link between menstruation and cyclical dysphoria has long been recognized, neuroscientific investigations of this common disorder have only recently been explored. This article reviews functional and structural brain imaging studies of PMDD and the similar but less well defined condition of premenstrual syndrome (PMS). The most consistent findings are that women with premenstrual dysphoria exhibit greater relative activity than other women in the dorsolateral prefrontal cortex and posterior lobules VI and VII of the neocerebellum. Since both brain areas have been implicated in emotional processing and mood disorders, working memory and executive functions, this greater activity probably represents coactivation within a cerebro-cerebellar feedback loop regulating emotional and cognitive processing. Some of the evidence suggests that increased activity within this circuit may preserve cerebellar structure during aging, and possible mechanisms and implications of this finding are discussed.

Keywords:

Citation: Andrea J. Rapkin, Steven M. Berman, Edythe D. London. The Cerebellum and Premenstrual Dysphoric Disorder[J]. AIMS Neuroscience, 2014, 1(2): 120-141. doi: 10.3934/Neuroscience.2014.2.120

Related Papers:

[1]	Abdullah Shoaib, Tahair Rasham, Giuseppe Marino, Jung Rye Lee, Choonkil Park . Fixed point results for dominated mappings in rectangular b-metric spaces with applications. AIMS Mathematics, 2020, 5(5): 5221-5229. doi: 10.3934/math.2020335
[2]	Reena Jain, Hemant Kumar Nashine, Jung Rye Lee, Choonkil Park . Unified relational-theoretic approach in metric-like spaces with an application. AIMS Mathematics, 2021, 6(8): 8959-8977. doi: 10.3934/math.2021520
[3]	Sourav Shil, Hemant Kumar Nashine . Positive definite solution of non-linear matrix equations through fixed point technique. AIMS Mathematics, 2022, 7(4): 6259-6281. doi: 10.3934/math.2022348
[4]	Chen Lang, Hongyan Guan . Common fixed point and coincidence point results for generalized $ \alpha $-$ \varphi_{E} $-Geraghty contraction mappings in $ b $-metric spaces. AIMS Mathematics, 2022, 7(8): 14513-14531. doi: 10.3934/math.2022800
[5]	Md Hasanuzzaman, Mohammad Imdad . Relation theoretic metrical fixed point results for Suzuki type $\mathcal{Z_\mathcal{R}}$-contraction with an application. AIMS Mathematics, 2020, 5(3): 2071-2087. doi: 10.3934/math.2020137
[6]	Awais Asif, Sami Ullah Khan, Thabet Abdeljawad, Muhammad Arshad, Ekrem Savas . 3D analysis of modified F-contractions in convex b-metric spaces with application to Fredholm integral equations. AIMS Mathematics, 2020, 5(6): 6929-6948. doi: 10.3934/math.2020444
[7]	Qing Yang, Chuanzhi Bai . Fixed point theorem for orthogonal contraction of Hardy-Rogers-type mapping on $O$-complete metric spaces. AIMS Mathematics, 2020, 5(6): 5734-5742. doi: 10.3934/math.2020368
[8]	Changzhou Li, Chao Yuan, Shiliang Chen . On positive definite solutions of the matrix equation $ X-\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = Q $. AIMS Mathematics, 2024, 9(9): 25532-25544. doi: 10.3934/math.20241247
[9]	Seher Sultan Yeşilkaya, Cafer Aydın, Adem Eroǧlu . Fixed point results on ordered Prešić type mappings. AIMS Mathematics, 2020, 5(5): 5140-5156. doi: 10.3934/math.2020330
[10]	Ahmed Alamer, Faizan Ahmad Khan . Boyd-Wong type functional contractions under locally transitive binary relation with applications to boundary value problems. AIMS Mathematics, 2024, 9(3): 6266-6280. doi: 10.3934/math.2024305

Abstract

1. Introduction and preliminaries

The most papular Banach contraction mapping principle (BCMP) ^[1] is the largest powerful fundamental fixed point result. This principle has a lot of applications in pure and applied mathematics (see ^[2,3,4]). In the past few decades, many authors extended and generalized the (BCMP) in several ways (see ^{[5,6,7,8,9,10]}). Ran and Reurings ^[11] obtained positive definite solutions of matrix equations using the aid of the Banach contraction principle in partially ordered sets. Nieto and Rodriguez-Lopez ^[12] also used partially ordered spaces and fixed point theorems to find solutions of some differential equations. Very recently, Wardowski ^[13] furnished the idea of an $ F $-contraction, which is an extension of the (BCMP). Furthermore, common fixed point theorems for rational $ F_{R} $ -contractive pairs of mappings with applications are announced in ^[14] as an extension of $ F $-contractions in relation theoretic metric spaces. On the other hand, Matthews ^[15] introduced the notion of a partial metric space as a part of the study of semantics of dataflow network, and for more results in this direction see (^{[16,17,18,19,20]}). One of the latest extensions of a metric space and a partial metric space is initiated through the concept of a $ m $-metric space ^[21], and some researchers work in this direction (see more ^{[22,23,24,25,26,27,28,29,30]}). In our article, we utilize two last notions to give an interesting type of generalized $ F_{R}^{m} $-contractions in the frame of relation theoretic $ m $-metric spaces and to prove some fixed point results.

Generally saying that, we generalize and extend some recent results in ^[31]$. $ We also extend the earlier mentioned results in the setting of relation theoretic $ m $-metric spaces, that contain only the last two conditions imposed on the Wardowski function $ F $ in the first section. Furthermore, the consequences of our main results improve and generalize some corresponding theorems appearing in the literature.

Our article consists four sections. In the first section, we recall some fundamental definitions and theorems concerning $ m $-metric spaces and different types of $ F $-contractions. In the second section, we define the notion of generalized $ F_{R}^{m} $-contractions of rational type and generalized $ F_{R}^{m} $-contractions of cyclic type. In the third section, we use the whole Wardowski function in the setting of $ F_{R}^{m} $-contractions of rational type as consequences of main results in section Ⅱ. Using these ideas, we prove some new fixed point results in the frame of relation theoretic $ m $-metric spaces and we present some examples to show that our obtained results are meaningful. In section Ⅳ, we present an application and we ensure the existence of a solution of a class of nonlinear matrix equations.

Throughout this article, $ \mathbb{N} $ indicates a set of all natural numbers, $ \mathbb{R} $ indicates set of real numbers and $ \mathbb{R} ^{+} $ indicates set of positive real numbers, respectively. We also denote $ \mathbb{N} _{0} = \mathbb{N} \cup \left\{ 0\right\}. $ Henceforth, $ U $ will denote a non-empty set. Given a self mapping $ \gamma :U\rightarrow U $. A Picard sequence based on an arbitrary $ \zeta _{0} $ in $ U $ is given by $ \zeta _{\mu } = \gamma \left(\zeta _{\mu -1}\right) = \gamma ^{\mu }\left(\zeta _{0}\right) $ for all $ \mu $ in $ \mathbb{N}, $ where $ \gamma ^{\mu } $ denotes the $ \mu ^{th} $-iteration of $ \gamma. $

In 2013, the notion of a $ m $-metric space was introduced by Asadi et al. ^[21]. They also extended the well known Banach contraction fixed point theorem from partial metric spaces to $ m $-metric spaces. We start recalling some definitions and properties:

Definition 1.1. ^[21] Let $ U\neq \emptyset $. The function $ m:U\times U\rightarrow \mathbb{R} ^{+} $ is a $ m $-metric on the set $ U $ if for all $ \zeta, \Im, \aleph \in U, $

$ (i) $ $ \zeta = \Im \Longleftrightarrow m\left(\zeta, \zeta \right) = m\left(\Im, \Im \right) = m\left(\zeta, \Im \right) \left(T_{0}{\text{ -separation axiom}}\right); $

$ (ii) $ $ m_{\zeta \Im }\leq m(\zeta, \Im) $ $ \left({\text{minimum self distance axiom}}\right); $

$ (iii) $ $ m\left(\zeta, \Im \right) = m\left(\Im, \zeta \right) $ $ \left({\text{symmetry}}\right); $

$ (iv) $ $ m\left(\zeta, \Im \right) -m_{\zeta \Im }\leq (m\left(\zeta, \aleph \right) -m_{\zeta \aleph })+(m\left(\aleph, \Im \right) -m_{\aleph \Im }) $ $ \left({\text{modified triangle inequality}}\right), $

where

$\begin{eqnarray*} m_{\zeta \Im } & = &\min \left\{ m\left( \zeta ,\zeta \right) ,m\left( \Im ,\Im \right) \right\} , \\ M_{\zeta \Im } & = &\max \left\{ m\left( \zeta ,\zeta \right) ,m\left( \Im ,\Im \right) \right\} . \end{eqnarray*}$ $

Here, the pair $ \left(U, m\right) $ is called a $ m $-metric space.

On among the classical examples of $ m $-metric spaces is the pair $ \left(\zeta, m\right) $ where $ U = \left\{ \zeta, \Im, \aleph \right\} $ and $ m\left(\zeta, \zeta \right) = 1 $, $ m\left(\Im, \Im \right) = 9, $ $ m\left(\aleph, \aleph \right) = 5. $ Other examples of $ m $-metric spaces may be found, for instance in ^[21]. Clearly, each partial metric is a $ m $ -metric space, but the converse does not hold (see ^[32,33,34]).

Every $ m $-metric $ m $ on $ U $ generates a $ T_{0} $ topology $ \tau _{m}\left({\text{say}}\right) $ on $ U $ which has a base of collection of $ m $-open balls

$\begin{equation*} \left\{ B_{m}\left( \zeta ,\epsilon \right) :\zeta \in U,\epsilon > 0\right\} , \end{equation*}$ $

where

$\begin{equation*} B_{m}\left( \zeta ,\epsilon \right) = \{\Im \in U:m\left( \zeta ,\Im \right) < m_{\zeta \Im }+\epsilon \}{\text{for all }}\zeta \in U,\varepsilon > 0. \end{equation*}$ $

If $ m $ is a $ m $-metric space on $ U $, then the functions $ m^{w}, $ $ m^{s}:U\times U\rightarrow \mathbb{R} ^{+} $ given by:

$\begin{equation*} m^{w}\left( \zeta ,\Im \right) = m\left( \zeta ,\Im \right) -2m_{\zeta \Im }+M_{\zeta \Im }, \end{equation*}$ $

$\begin{equation*} m^{s} = \left\{ \begin{array}{l} m\left( \zeta ,\Im \right) -m_{\zeta \Im }{\text{, if }}\zeta \neq \Im \\ 0{\text{, if }}\zeta = \Im . \end{array} \right. \end{equation*}$ $

define ordinary metrics on $ U $. It is easy to see that $ m^{w} $ and $ m^{s} $ are equivalent metrics on $ U. $

Definition 1.2. According to ^[21],

$ \left(i\right) $ a sequence $ \{\zeta _{\mu }\} $ in a $ m $-metric space $ \left(U, m\right) $ converges with respect to $ \tau _{m} $ to $ \zeta $ if and only if

$\begin{equation*} \lim\limits_{\mu \rightarrow \infty }\left( m\left( \zeta _{\mu },\zeta \right) -m_{\zeta _{\mu }\zeta }\right) = 0, \end{equation*}$ $

$ (ii) $ a sequence $ \{\zeta _{\mu }\} $ in a $ m $-metric space $ \left(U, m\right) $ is called $ m $-Cauchy if $ \lim_{\mu, \nu \rightarrow \infty }\left(m\left(\zeta _{\mu }, \zeta _{\nu }\right) -m_{\zeta _{\mu }\zeta _{\nu }}\right) $ and $ \lim_{\mu, \nu \rightarrow \infty }\left(M_{\zeta _{\mu }, \zeta _{\nu }}-m_{\zeta _{\mu }\zeta _{\nu }}\right) $ exist and are finite,

$ (iii) $ $ \left(U, m\right) $ is said to be complete if every $ m $ -Cauchy sequence $ \left\{ \zeta _{\mu }\right\} $ in $ U $ is $ m $-convergent to $ \zeta $ with respect to $ \tau _{m} $ in $ U $ such that

$\begin{equation*} \lim\limits_{\mu \rightarrow \infty }\left(m\left( \zeta _{\mu },\zeta \right) -m_{\zeta _{\mu }\zeta }\right) = 0,{\text{ and }}\lim\limits_{\mu \rightarrow \infty }\left( M_{\zeta _{\mu },\zeta }-m_{\zeta _{\mu }\zeta }\right) = 0, \end{equation*}$ $

$ \left(iv\right) $ $ \{\zeta _{\mu }\} $ is a Cauchy sequence in $ \left(U, m\right) $ if and only if it is a Cauchy sequence in the metric space $ \left(U, m^{w}\right), $

$ \left(v\right) $ $ \left(U, m\right) $ is complete if and only if $ \left(U, m^{w}\right) $ is complete.

Consider a function $ F:\left(0, \infty \right) \rightarrow R $ so that:

$ \left(F_{1}\right) $ $ F\left(\zeta \right) < F\left(\Im \right) $ for all $ \zeta < \Im, $

$ \left(F_{2}\right) $ for each sequence $ \left\{ \varpi _{\mu }\right\} \subseteq \left(0, \infty \right) $, $ \lim_{\mu \rightarrow \infty }\varpi _{\mu } = 0 $ iff $ \lim_{\mu \rightarrow \infty }F\left(\varpi _{\mu }\right) = -\infty, $

$ \left(F_{3}\right) $ there exists $ p\in \left(0, 1\right) $ such that $ \lim_{\varpi _{\mu } \rightarrow 0^{+}}\varpi ^{p}F\left(\varpi \right) = 0. $

According to ^[13], denote by $ \nabla \left(F\right) $ the collection of functions $ F:\left(0, \infty \right) \rightarrow R $ satisfying $ \left(F_{2}\right) $ and $ \left(F_{3}\right) $. Take also

$\begin{equation*} \Pi \left( F\right) = \left\{ F\in \nabla _{F}:F\,\,,\,{\text{verifies}}\left( F_{1}\right) \right\}. \end{equation*}$ $

Example 1.1. ^[13] The following below functions belong to $ \Pi \left(F\right) $:

$ (1) $ $ F\left(s\right) = \ln s $,

$ (2) $ $ F\left(s\right) = s+\ln s $,

$ (3) $ $ F\left(s\right) = \ln \left(s^{2}+s\right) $,

$ (4)$ $ F\left(s\right) = -\frac{1}{\sqrt{s}} $,

for all $ s > 0. $

Example 1.2. The following functions are not strictly increasing and belong to $ \nabla \left(F\right) : $

$ (1) $ $ F\left(s\right) = 100\ln \left(\frac{s}{2}+\sin s\right) $,

$ (2)$ $ F\left(s\right) = \sin s+\ln s $,

$ (3) $ $ F\left(s\right) = \sin s-\frac{1}{\sqrt{s}} $,

for all $ s > 0 $.

Let $ \gamma $ be a self-mapping on a $ mm $-space $ U $. The following are some valuable notations that are useful for the rest.

$ \left(i\right) $ $ \left(\gamma \right) _{Fix} $ is the set of all fixed points of $ \gamma $,

$ \left(ii\right) $ $ \Theta \left(\Psi, {S}\right) = \left\{ \zeta \in U:\left(\zeta, \gamma \left(\zeta \right) \right) \in {R}\right\}, $

$ \left(iii\right) $ $ \digamma \left(\zeta, \Im, \nabla \right) $ is thefashion of all paths in $ \nabla $ from $ \zeta $ to $ \Im. $

Altun et al. ^[35] gave two fixed point results for multivalued $ F $ -contractions on $ mm $-spaces. We ensure the existence of fixed point results for generalized $ F_{R}^{m} $-contractions by using the concept given in ^[35] to the metric space setup. The motivation of this study is to solve nonlinear matrix equations. First, inspired by Altun et al. ^[31] and Wardowski ^[13], we give the following concepts.

Theoretic relations have been used in many research articles, for examples see ^[36]. A non-empty subset $ R $ of $ U^{2} $ is said to be a relation on the $ m $-metric space $ \left(U, m\right) $ if $ {R = }\left\{ \left(\zeta, \Im \right) \in U^{2}:\zeta, \Im \in U\right\}. $ If $ \left(\zeta, \Im \right) \in R $, then we say that $ \zeta \preceq \Im $ $ \left(\zeta {\text{ precede }}\Im \right) $ under $ R $ denoted by $ \left(\zeta, \Im \right) \in { R, } $ and the inverse of $ R $ is denoted as $ R^{-1} = \left\{ \left(\zeta, \Im \right) \in U^{2}:\left(\Im, \zeta \right) \in R\right\}. $ Set $ {S = R\cup R} ^{-1}\subseteq U^{2} $. Consequently, we illustrate another relation on $ U $ denoted $ S^{\ast } $ and is given as $ \left(\zeta, \Im \right) \in S^{\ast }\Leftrightarrow \left(\Im, \zeta \right) \in S $ and $ \Omega \neq \Im. $

Definition 1.3. ^[36] Let $ U\neq \emptyset $ and $ R $ be a binary relation on $ U $. Then $ R $ is transitive if $ \left(\zeta, \xi \right) \in R $ and $ \left(\xi, \Im \right) \in R $ $ \Rightarrow $ $ \left(\zeta, \Im \right) \in R $, for all $ \zeta, \Im, \xi \in U $.

Definition 1.4. ^[36] Let $ U\neq \emptyset $. A sequence $ \zeta _{\mu }\in U $ is called $ R $-preserving, if $ \left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in R. $

Definition 1.5. ^[36] Let $ U\neq \emptyset $ and $ \gamma :U\rightarrow U $. A binary relation $ R $ on $ U $ is called $ \gamma $-closed if for any $ \zeta, \Im $ in $ U $, we deduce $ \left(\zeta, \Im \right) \in R\Rightarrow \left(\gamma \left(\zeta \right), \gamma \left(\Im \right) \right) \in R. $

2. Main results

We begin with the following definitions.

Definition 2.1. We say that $ (U, m, R) $ is regular if for each sequence $ \left\{ \zeta _{\mu }\right\} $ in $ U $,

$\begin{equation*} \left. \begin{array}{c} \left( \zeta _{\mu },\zeta _{\mu +1}\right) \in R\ {\text{ for all }}\mu \in \mathbb{N} \\ \lim\limits_{\mu \rightarrow \infty }\left( m\left( \zeta _{\mu },\zeta \right) -m_{\zeta _{\mu }\zeta }\right) = 0,{\text{ i.e., }}\zeta _{\mu }\overset{t_{m}} {\rightarrow }\zeta \in R, \end{array} \right\} \Rightarrow \left( \zeta _{\mu },\zeta \right) \in R,\ {\text{ for all }}\mu \in \mathbb{N} . \end{equation*}$ $

Definition 2.2. A relation theoretic $ m $-metric space $ \left(U, m, R\right) $ is said to be $ R $-complete if for an $ R $-preserving $ m $-Cauchy sequence $ \left\{ \zeta _{\mu }\right\} $ in $ U $, there exists some $ \zeta $ in $ U $ such that

$\begin{equation*} \lim\limits_{\mu \rightarrow \infty }m\left( \zeta _{\mu },\zeta \right) -m_{\zeta _{\mu }\zeta } = 0,{\text{ and }}\lim\limits_{\mu \rightarrow \infty }\left( M_{\zeta _{\mu },\zeta }-m_{\zeta _{\mu }\zeta }\right) = 0. \end{equation*}$ $

Definition 2.3. Let $ \left(U, m\right) $ be a $ m $-metric space endowed with a binary relation $ R $ on $ U $ and $ \gamma $ be a self-mapping on $ U $. Then, $ \gamma $ is said to be a $ F_{R}^{m} $-contractions, if there exist $ F_{R}^{m}\in \Pi \left(\nabla \right) $ and $ \xi > 0, $ such that

$\begin{equation} \xi +F_{R}^{m}\left( m\left( \gamma \left( \zeta \right) ,\gamma \left( \Im \right) \right) \right) \leq F_{R}^{m}\left( m\left( \zeta ,\Im \right) \right) \end{equation}$ $

(2.1)

for all $ \zeta, \Im \in U $ with $ \left(\zeta, \Im \right) \in S^{\ast }. $

Now, we introduce the concept of a generalized rational type $ F_{R}^{m} $-contraction.

Definition 2.4. Let $ \left(U, m\right) $ be a $ m $-metric space endowed with a binary relation $ R $ on $ U $. Let $ \gamma :U\rightarrow U $ be a self-mapping on $ U $. It is called a generalized rational type $ F_{R}^{m} $-contraction if there are $ F_{R}^{m}\in \nabla \left(F\right) $ and $ \xi > 0 $ such that

$\begin{equation} \xi +F_{R}^{m}\left( m(\gamma \left( \zeta \right) ,\gamma \left( \Im \right) )\right) \leq F_{R}^{m}\left( \max \left\{ \begin{array}{c} m\left( \zeta ,\Im \right) ,m\left( \zeta ,\gamma \left( \zeta \right) \right) ,m\left( \Im ,\gamma \left( \Im \right) \right) , \\ \frac{m\left( \zeta ,\gamma \left( \zeta \right) \right) \left[ 1+m\left( \Im ,\gamma \left( \Im \right) \right) \right] }{1+m\left( \zeta ,\Im \right) } \end{array} \right\} \right) , \end{equation}$ $

(2.2)

for all $ \zeta, \Im \in U $ with $ \left(\zeta, \Im \right) \in S^{\ast }. $

Theorem 2.1. Let $ \left(U, m\right) $ be a complete $ m $-metric space with a binary relation $ R $ on $ U $ and $ \gamma $ be a self-mapping on $ U $ such that:

$ \left(i\right) $ the class $ \Theta \left(\gamma, R\right) $ is nonempty;

$ \left(ii\right) $ $ R $ is $ \gamma $-closed;

$ \left(iii\right) $ the mapping $ \gamma $ is $ R $-continuous;

$ \left(iv\right) $ $ \gamma $ is a generalized rational type $ F_{R}^{m} $-contraction mapping.

Then $ \gamma $ possesses a fixed point in $ U $.

Proof. Let $ \zeta _{0}\in \Theta \left(\left[ \gamma, R\right] \right) $. We define a sequence $ \left\{ \zeta _{\mu}\right\} $ by $ \zeta _{\mu +1} = \gamma \left(\zeta _{\mu }\right) = \gamma ^{\mu }\left(\zeta _{0}\right) $ for each $ \mu \in \mathbb{N}. $ If there is $ \mu _{0} $ in $ \mathbb{N} $ so that $ \gamma \left(\zeta _{\mu _{0}}\right) = \zeta _{\mu _{0}} $, then $ \gamma $ has a fixed point $ \zeta _{\mu _{0}} $ and the proof is complete. Let $ \zeta _{\mu +1}\neq \zeta _{\mu } $ for all $ \mu $ in $ \mathbb{N} $, so $ m\left(\zeta _{\mu +1}, \zeta _{\mu }\right) > 0. $ Since $ \left(\gamma \left(\zeta _{0}\right), \zeta _{0}\right) \in S^{\ast } $, using $ \gamma $-closedness of $ R $, we get $ \left(\gamma \left(\zeta _{\mu +1}\right), \zeta _{\mu}\right) \in S^{\ast } $. Then using the fact that $ \gamma $ is a generalized rational type $ F_{R}^{m} $-contraction mapping, one writes

$\begin{equation*} F_{R}^{m}m\left( \zeta _{\mu +1},\zeta _{\mu }\right) = F_{R}^{m}\left( m\left( \zeta _{\mu +1},\zeta _{\mu }\right) \right) . \end{equation*}$ $

$\begin{equation} \leq F_{R}^{m}\left( \max \left\{ \begin{array}{c} m\left( \zeta _{\mu },\zeta _{\mu -1}\right) ,m\left( \zeta _{\mu },\gamma \left( \zeta _{\mu }\right) \right) ,m\left( \zeta _{\mu -1},\gamma \left( _{\mu -1}\right) \right) , \\ \frac{m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \left[ 1+m\left( \zeta _{\mu -1},\zeta _{\mu }\right) \right] }{1+m\left( \zeta _{\mu -1},\zeta _{\mu }\right) } \end{array} \right\} \right) -\xi \end{equation}$ $

(2.3)

$\begin{eqnarray*} &\leq &F_{R}^{m}\left( \max \left\{ \begin{array}{c} m\left( \zeta _{\mu },\zeta _{\mu -1}\right) ,m\left( \zeta _{\mu },\zeta _{\mu +1}\right) ,m\left( \zeta _{\mu -1},\zeta _{\mu }\right) , \\ \frac{m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \left[ 1+m\left( \zeta _{\mu -1},\zeta _{\mu }\right) \right] }{1+m\left( \zeta _{\mu -1},\zeta _{\mu }\right) } \end{array} \right\} \right) -\xi \\ &\leq &F_{R}^{m}\left( \max \left\{ m\left( \zeta _{\mu },\zeta _{\mu -1}\right) ,m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right\} \right) -\xi . \end{eqnarray*}$ $

If $ \max \left\{ m\left(\zeta _{\mu }, \zeta _{\mu -1}\right), m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right\} = m\left(\zeta _{\mu }, \zeta _{\mu +1}\right), $ then from $ \left(2.3\right), $ we have

$\begin{equation*} F_{R}^{m}\left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) \leq F_{R}^{m}\left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) -\xi < F_{R}^{m}\left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) , \end{equation*}$ $

which is a contradiction. Thus, $ \max \left\{ m\left(\zeta _{\mu }, \zeta _{\mu -1}\right), m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right\} = m\left(\zeta _{\mu }, \zeta _{\mu -1}\right) $ and so from $ \left(2.3\right), $ we have

$\begin{equation} F_{R}^{m}\left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) \leq F_{R}^{m}\left( m\left( \zeta _{\mu -1},\zeta _{\mu }\right) \right) {\text{ for all }}\mu \in \mathbb{N}. \end{equation}$ $

(2.4)

Denote $ \delta _{\mu } = m\left(\zeta _{\mu }, \zeta _{\mu +1}\right). $ We have $ \delta _{\mu } > 0 $ for all $ \mu \in \mathbb{N} $ and using $ \left(2.4\right) $ we deduce that

$\begin{equation} F_{R}^{m}\left( \delta _{\mu }\right) \leq F_{R}^{m}\left( \delta _{\mu -1}\right) -\xi \leq F_{R}^{m}\left( \delta _{\mu -1}\right) -2\xi \leq ...F_{R}^{m}\left( \delta _{\mu -1}\right) -\mu \xi . \end{equation}$ $

(2.5)

It implies that $ \lim_{\mu \rightarrow \infty }F_{R}^{m}\left(\delta _{\mu }\right) = -\infty, $ then by $ \left(F_{2}\right), $ we have $ \lim_{\mu \rightarrow \infty }\delta _{\mu } = 0. $ Due to $ \left(F_{3}\right) $, there exists $ k\in \left(0, 1\right) $ such that $ \lim_{\mu \rightarrow \infty }\delta _{\mu }^{k}F_{R}^{m}\left(\delta _{\mu }\right) = 0. $

From $ \left(2.4\right) $ the following is true for all $ \mu \in \mathbb{N}, $

$\begin{equation} \delta _{\mu }^{k}\left( F_{R}^{m}\left( \delta _{\mu }\right) -F_{R}^{m}\left( \delta _{0}\right) \right) \leq -\delta _{\mu }^{k}\mu \tau \leq 0. \end{equation}$ $

(2.6)

Letting $ \mu \rightarrow \infty $ in $ \left(2.6\right) $, we get

$\begin{equation} \lim\limits_{\mu \rightarrow \infty }\mu \delta _{\mu }^{k} = 0. \end{equation}$ $

(2.7)

From $ \left(2.7\right) $, there exists $ \mu _{1}\in \mathbb{N} $ so that $ \mu \delta _{n}^{k}\leq 1 $ for all $ \mu \geq \mu _{1}, $ then we deduce

$\begin{equation*} \delta _{\mu }\leq \frac{1}{\mu ^{\frac{1}{k}}}{\text{ for all }}\mu \geq \mu _{1}. \end{equation*}$ $

We claim that $ \left\{ \zeta _{\mu }\right\} $ is a $ m $-Cauchy sequence in the $ m $-metric space. Let $ \nu, \mu \in \mathbb{N} $ such that $ \nu > \mu \geq \mu _{1}. $ Using the triangle inequality of a $ m $-metric space, one writes

$\begin{eqnarray*} m\left( \zeta _{\mu },\zeta _{\nu }\right) -m_{\zeta _{\mu },\zeta _{\nu }} &\leq &m\left( \zeta _{\mu },\zeta _{\mu +1}\right) -m_{\zeta _{\mu },\zeta _{\mu +1}}+m\left( \zeta _{\mu +1},\zeta _{\mu +2}\right) -m_{\zeta _{\mu +1},\zeta _{\mu +2}}+ \\ &&...+m\left( \zeta _{\nu -1},\zeta _{\nu }\right) -m_{\zeta _{\nu -1},\zeta _{\nu }} \\ &\leq &m\left( \zeta _{\mu },\zeta _{\mu +1}\right) +m\left( \zeta _{\mu +1},\zeta _{\mu +2}\right) +\cdots +m\left( \zeta _{\nu -1},\zeta _{\nu }\right) \\ &\leq &\delta _{\mu }+\delta _{\mu +1}+\cdots +\delta _{v-1} \\ & = &\sum\limits_{i = \mu }^{\nu -1}\delta _{i}\leq \sum\limits_{i = \mu }^{\infty }\delta _{i}\leq \sum\limits_{i = \mu }^{\infty }\frac{1}{i^{\frac{1}{k}}}. \end{eqnarray*}$ $

The convergence of the series $ \sum_{i = \mu }^{\infty }\frac{1}{i^{\frac{1}{k} }} $ yields that $ m\left(\zeta _{\mu }, \zeta _{\nu }\right) -m_{\zeta _{\mu }, \zeta _{\nu }}\rightarrow 0. $ Thus, $ \left\{ \zeta _{\mu }\right\} $ is a $ M $-Cauchy sequence in $ \left(U, m\right) $. Since $ \left(U, m, R\right) $ is $ R $-complete, there exists $ \zeta \in U $ such that $ \left\{ \zeta _{\mu }\right\} $ converges to $ \zeta $ with respect to $ t_{\kappa } $, that is, $ m\left(\zeta _{\mu }, \zeta \right) -m_{\zeta _{\mu }, \zeta }\rightarrow 0 $ as $ \mu \rightarrow \infty. $ Now, the $ R $-continuity of $ \gamma $ implies that

$\begin{equation*} \zeta = \lim\limits_{\mu \rightarrow \infty }\zeta _{\mu +1} = \lim\limits_{\mu \rightarrow \infty }\gamma \left( \zeta _{\mu }\right) = \gamma \left( \zeta \right) . \end{equation*}$ $

Hence, $ \zeta $ is a fixed point of $ \gamma. $

Example 2.1. Let $ U = \left[ 0, \infty \right) $ and $ m $ be defined by $ m\left(\zeta, \Im \right) = \min \left\{ \zeta, \Im \right\} $ for all $ \zeta, \Im \in U $. $ \left(U, m\right) $ is a complete $ m $-metric space. Consider the sequence $ \left\{ z_{\mu }\right\} \subseteq U $ given by $ z_{\mu } = \frac{\mu \left(\mu +1\right) \left(2\mu +1\right) }{6} $ for all $ \mu \geq 2. $ Set an binary relation on $ U $ denoted by $ R $ given by $ R = \left\{ \left(z_{1}, z_{1}\right), \left(z_{\mu -1}, z_{\mu }\right) :\mu = 2, 3, ...100\right\}. $ Now, give $ \gamma :U\rightarrow U $ as

$\begin{equation*} \gamma \left( \zeta \right) = \left\{ \begin{array}{c} \zeta ,{\rm{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if \ }}0\leq \zeta \leq z_{1}, \\ z_{1},{\rm{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if \ }}z_{1}\leq \zeta \leq z_{2}, \\ z_{\mu }+\left( \frac{z_{\mu }-z_{\mu -1}}{z_{\mu +1}-z_{\mu }}\right) \left( \zeta -z_{\mu }\right) ,{\rm{ \ \ \ \ \ \ \ \ \ \ \ if\ \ }}z_{\mu +1}\leq \zeta \leq z_{\mu }\;{\rm{ for\; all }}\;\mu = 2,3,\cdots ,100. \end{array} {\rm{ }}\right. \end{equation*}$ $

Obviously, $ R $ is $ \gamma $-closed and $ \gamma $ is continuous. Choosing $ \zeta = z_{\mu } $ and $ \Im = z_{\mu +1} $ (for $ \mu = 1, 2, 3, \cdots, 100) $, for first condition of $ F $ (which is $ (F_{1}) $), we have

$\begin{equation*} F_{R}^{m}\left( m\left( \gamma \left( z_{\mu }\right) ,\gamma \left( z_{\mu +1}\right) \right) \right) = F_{R}^{m}\left( m\left( z_{\mu -1},z_{\mu }\right) \right) = F_{R}^{m}\left( z_{\mu -1}\right) = 100\ln \left( \frac{ z_{\mu -1}}{2}-\sin z_{\mu -1}\right) , \end{equation*}$ $

and

$\begin{eqnarray*} &&F_{R}^{m}\left( \max \left\{ \begin{array}{c} m\left( z_{\mu },z_{\mu +1}\right) ,m\left( z_{\mu },\gamma \left( z_{\mu }\right) \right) ,m\left( z_{\mu +1},\gamma \left( z_{\mu +1}\right) \right) , \\ \frac{m\left( z_{\mu },\gamma \left( z_{\mu }\right) \right) \left( 1+m\left( z_{\mu +1},\gamma \left( z_{\mu +1}\right) \right) \right) }{ 1+m\left( z_{\mu },z_{\mu +1}\right) } \end{array} \right\} \right) \\ & = &F_{R}^{m}\left( \max \left\{ \begin{array}{c} m\left( z_{\mu },z_{\mu +1}\right) ,m\left( z_{\mu },z_{\mu -1}\right) ,m\left( z_{\mu +1},z_{\mu }\right) , \\ \frac{m\left( z_{\mu },z_{\mu -1}\right) \left( 1+m\left( z_{\mu +1},z_{\mu }\right) \right) }{1+m\left( z_{\mu },z_{\mu +1}\right) } \end{array} \right\} \right) . \\ & = &F_{R}^{m}\left( z_{\mu }\right) . \end{eqnarray*}$ $

Now, for $ \mu = 2, 3, 4, \cdots, 100, $ we have

$\begin{equation} \xi +100\ln \left( \frac{z_{\mu -1}}{2}+\sin z_{\mu -1}\right) \leq 100\ln \left( \frac{z_{\mu }}{2}+\sin z_{\mu }\right) , \end{equation}$ $

(2.9)

implies that

$\begin{equation} \xi \leq 100\ln \left( \frac{\frac{z_{\mu }}{2}+\sin z_{\mu }}{\frac{z_{\mu -1}}{2}+\sin z_{\mu -1}}\right) . \end{equation}$ $

(2.10)

Let

$\begin{equation} f\left( \mu \right) = 100\ln \left( \frac{\frac{z_{\mu }}{2}+\sin z_{\mu }}{ \frac{z_{\mu -1}}{2}+\sin z_{\mu -1}}\right) . \end{equation}$ $

(2.11)

In view of Table 1 and Figure 1, since the function $ \left\{ f\left(\mu \right) \right\} _{\mu \geq 2} $ is decreasing and discontinuous, the smallest value in $ \left(2.11\right) $ is $ 5.02. $ Therefore, the Eq $ \left(2.10\right) $ holds for $ 0 < \xi < 5. $ So

$\begin{equation*} \xi +F_{R}^{m}\left( m(\gamma \left( \zeta \right) ,\gamma \left( \Im \right) )\right) \leq F_{R}^{m}\left( \max \left\{ \begin{array}{c} m\left( \zeta ,\Im \right) ,m\left( \zeta ,\gamma \left( \zeta \right) \right) ,m\left( \Im ,\gamma \left( \Im \right) \right) , \\ \frac{m\left( \zeta ,\gamma \left( \zeta \right) \right) \left[ 1+m\left( \Im ,\gamma \left( \Im \right) \right) \right] }{1+m\left( \zeta ,\Im \right) } \end{array} \right\} \right) , \end{equation*}$ $

Table 1. Iterations and $ f\left(\mu \right) $.

Iter	$ f\left(\mu \right) $	Iter	$ f\left(\mu \right) $
$ \mu=2 $	13.87	$ \mu=13 $	$ 25.21 $
$ \mu=3 $	$ 164.6 $	$ \mu=14 $	$ 23.21 $
$ \mu=4 $	$ 56.16 $	$ \mu=15 $	$ 21.20 $
$ \mu=5 $	$ 63.72 $	$ \mu=16 $	$ 20.20 $
$ \mu=6 $	$ 54.28 $	$ \mu=17 $	$ 18.71 $
$ \mu=7 $	$ 44.23 $	$ \mu=18 $	$ 17.64 $
$ \mu=8 $	$ 36.45 $	$ \mu=19 $	$ 16.53 $
$ \mu=9 $	$ 33.78 $	$... $	$... $
$ \mu=10 $	$ 33.78 $	$ \mu=60 $	$ 5.02 $

| Show Table

DownLoad: CSV

Figure 1. Behaviour of $ f\left(\mu \right), $ for $ \mu \in \left[ 2, 60\right] $.

DownLoad: Full-Size Img PowerPoint

for all $ \zeta, \Im \in U $ such that $ \left(\zeta, \Im \right) \in {\bf{S }}^{\ast } $ with $ mm $-space. Hence, $ \gamma $ is a generalized rational type $ F_{R}^{m} $-contraction mapping with $ 0 < \xi < 5. $ Generally, we can say that $ \gamma $ has infinite $ \left(F.Ps\right) $.

Theorem 2.2. Theorem $ 2.1 $ remains true if the condition $ \left(ii\right) $ is replaced by the following:

$ \left(i\right) $ $ \left(ii\right) ^{^{\prime }}, $

$ \left(ii\right) $ $ \left(X, \kappa, \nabla \right) $ is regular.

Proof. It is a same argument as Theorem $ 2.1 $. Here, the sequence $ \left\{ \zeta _{\mu }\right\} $ is $ m $-Cauchy and converges to some $ \Omega $ in $ U $ such that $ m\left(\zeta _{\mu }, \zeta \right) -m_{\zeta _{\mu }, \zeta } $ as limit $ \mu \rightarrow \infty $ which implies that

$\begin{eqnarray*} \lim\limits_{\mu \rightarrow \infty }m\left( \zeta _{\mu },\zeta \right) & = &\lim\limits_{\mu \rightarrow \infty }m_{\zeta _{\mu },\zeta } = \lim\limits_{\mu \rightarrow \infty }\min \left\{ m\left( \zeta _{\mu }\zeta _{\mu }\right) ,m\left( \zeta ,\zeta \right) \right\} = m\left( \zeta ,\zeta \right) \\ & = &\lim\limits_{\mu ,\nu \rightarrow \infty }m\left( \zeta _{\mu },\zeta _{\nu }\right) = 0{\text{ and }}\lim\limits_{\mu ,\nu \rightarrow \infty }m_{\zeta _{\mu },\zeta _{\nu }} = 0. \end{eqnarray*}$ $

As $ \left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in {R, } $ then $ \left(\zeta _{\mu }, \zeta \right) \in R $ for all $ \mu \in \mathbb{N} $. Set $ \mu = \left\{ \mu \in N:\gamma \left(\zeta _{\mu }\right) = \gamma \left(\zeta \right) \right\}. $ We will take two cases depending on $ \mu $

C-1. If $ \mu $ is a finite set, then there exists $ \mu _{0} $ in $ \mathbb{N}, $ so that $ \gamma \left(\zeta _{\mu }\right) \neq \gamma \left(\zeta \right) $ for every $ \mu \geq \mu _{0}. $ In particular, $ \left(\zeta _{\mu }, \zeta \right) \in S^{\ast } $ and $ \left(\gamma \left(\zeta _{\mu }\right), \gamma \left(\zeta \right) \right) \in S^{\ast }, $ then for all $ \mu \geq \mu _{0} $,

$\begin{equation*} \xi +F_{R}^{m}\left( m\left( \gamma \left( \zeta _{\mu }\right) ,\gamma \left( \zeta \right) \right) \right) \leq F\left( m\left( \zeta _{\mu },\zeta \right) \right). \end{equation*}$ $

Since $ \lim_{\mu \rightarrow \infty }m\left(\zeta _{\mu }, \zeta \right) = 0 $ implies that $ \lim_{\mu \rightarrow \infty }F_{R}^{m}\left(m\left(\zeta _{\mu }, \zeta \right) \right) = -\infty, $ one writes $ \lim_{\mu \rightarrow \infty }F_{R}^{m}\left(m\left(\gamma \left(\zeta _{\mu }\right), \gamma \left(\zeta \right) \right) \right) = -\infty. $ Therefore, $ \lim_{\mu \rightarrow \infty }m\left(\gamma \left(\zeta _{\mu }\right), \gamma \left(\zeta \right) \right) = 0, $ which yields that $ \gamma \left(\zeta \right) = \zeta $, that is, $ \zeta $ is a fixed point of $ \gamma. $

C-2. If $ \mu $ is an infinite set, then there exists a subsequence $ \left\{ \zeta _{\mu _{k}}\right\} $ of $ \left\{ \zeta _{\mu }\right\} $ so that $ \zeta _{\mu _{k}+1} = \gamma \left(\zeta _{\mu _{k}}\right) = \gamma \left(\zeta \right) $ for $ k\in \mathbb{N}, $ so $ \gamma \left(\zeta _{\mu _{k}}\right) \rightarrow \gamma \left(\zeta \right) $ with respect to $ t_{m} $ as $ \zeta _{\mu } $ converges $ \zeta, $ then $ \gamma \left(\zeta \right) = \zeta $, i.e., $ \gamma $ has a fixed point. Hence, the proof is complete.

Now, we prove a result of uniqueness.

Theorem 2.3. Following Theorems 2.1 and 2.2 $ \gamma $ possesses a unique fixed point if $ \digamma \left(\zeta, \Im, \nabla \right) \neq \emptyset, $ for all $ \zeta, \Im \in \left(\gamma \right) _{Fix}. $

Proof. Let $ \zeta, \Im \in \left(\gamma \right) _{Fix} $ such that $ \zeta \neq \Im. $ Since $ \digamma \left(\zeta, \Im, \nabla \right) \neq \emptyset, $ there exists a path $ \left(\left\{ a_{0}, a_{1}, ...a_{\mu }\right\} \right) $ of some finite length $ \mu $ in $ \nabla $ from $ \Omega $ to $ \Im $ $ \left({\text{with }}a_{s}\neq a_{s+1}{\text{ for all }}s\in \left[ 0, p-1\right] \right) $. Then $ a_{0} = \zeta $, $ a_{k} = \Im, $ $ \left(a_{s}, a_{s+1}\right) \in S^{\ast } $ for every $ s\in \left[ 0, p-1\right]. $ As $ a_{s}\in \gamma \left(U\right), $ $ \gamma \left(a_{s}\right) = a_{s} $ for all $ s\in \left[ 0, p-1\right] $ we deduce that

$\begin{eqnarray*} F_{R}^{m}\left( m\left( a_{s},a_{s+1}\right) \right) & = &F_{R}^{m}\left( m\left( \gamma \left( a_{s}\right) ,\gamma \left( a_{s+1}\right) \right) \right) \\ &\leq &F_{R}^{m}\left\{ \max \left\{ \begin{array}{c} m\left( a_{s},a_{s+1}\right) ,m\left( a_{s},\gamma \left( a_{s}\right) \right) ,m\left( a_{s+1},\gamma \left( a_{s+1}\right) \right) , \\ \frac{m\left( a_{s},\gamma \left( a_{s}\right) \right) ,\left[ 1+m\left( a_{s},\gamma \left( a_{s+1}\right) \right) \right] }{1+m\left( a_{s},a_{s+1}\right) } \end{array} \right\} \right\} -\xi \\ & = &F_{R}^{m}\left\{ \max \left\{ \begin{array}{c} m\left( a_{s},a_{s+1}\right) ,m\left( a_{s},a_{s}\right) ,m\left( a_{s+1},a_{s+1}\right) , \\ \frac{m\left( a_{s},a_{s}\right) \left[ 1+m\left( a_{s},a_{s+1}\right) \right] }{1+m\left( a_{s},a_{s+1}\right) } \end{array} \right\} \right\} -\xi \\ & < &F_{R}^{m}\left\{ \left( m\left( a_{s},a_{s+1}\right) \right) \right\} . \end{eqnarray*}$ $

It is a contradiction. Hence, $ \gamma $ possesses a unique fixed point.

Now, we say that $ \gamma :U\rightarrow U $ has the property $ P $ if

$\begin{equation*} \left( \gamma ^{\mu }\right) _{{\text{Fix}}} = \left( \gamma \right) _{{\text{Fix}} }{\text{ for each }}\mu {\text{ is member of }} \mathbb{N} . \end{equation*}$ $

In this theorem, we use above condition having property $ P $.

Theorem 2.4. Let $ \left(U, m\right) $ be a complete $ m $-metric space with a binary relation $ R $ on $ U $ and $ \gamma $ be a self-mapping such that:

$ \left(i\right) $ the class $ \Theta \left(\gamma, R\right) $ is nonempty,

$ \left(ii\right) $ the binary relation $ R $ is $ \gamma $-closed,

$ \left(iii\right) $ $ \gamma $ is $ R $-continuous,

$ \left(iv\right) $ there are $ F_{R}^{m}\in \nabla \left(F\right) $ and $ \xi > 0 $ so that

$\begin{equation*} \xi +F_{R}^{m}\left( m\left( \gamma \left( \zeta \right) ,\gamma ^{2}\left( \zeta \right) \right) \right) \leq F_{R}^{m}\left( \max \left\{ \begin{array}{c} m\left( \zeta ,\gamma \left( \zeta \right) \right) ,m\left( \gamma \left( \zeta \right) ,\gamma ^{2}\left( \zeta \right) \right) , \\ \frac{m\left( \zeta ,\gamma \left( \zeta \right) \right) \left[ 1+m\left( \gamma \left( \zeta \right) ,\gamma ^{2}\left( \zeta \right) \right) \right] }{1+m\left( \gamma \left( \zeta \right) ,\gamma ^{2}\left( \zeta \right) \right) } \end{array} \right\} \right) \end{equation*}$ $

for all $ \zeta \in U, $ with $ \left(\gamma \left(\zeta \right), \gamma ^{2}\left(\zeta \right) \right) \in S^{\ast }. $

Then $ \gamma $ has a fixed point. Furthermore, if

$ \left(v\right) $ $ \left(iv\right) ^{^{\prime }} $;

$ \left(vi\right) $ $ \zeta \in \left(\gamma ^{\mu }\right) _{Fix} $ $ \left(for \;some\;\mu \in \mathbb{N} \right) $ which implies that $ \left(\zeta, \gamma \left(\zeta \right) \right) \in R, $

then $ \gamma $ has a property $ P. $

Proof. Let $ \zeta _{0}\in \Theta \left(\left[ \gamma, R\right] \right) $, i.e., $ \left(\zeta _{0}, \gamma \left(\zeta _{0}\right) \right) \in {\bf{ R}} $ $, $therefore using assumption $ \left(ii\right) $, we get $ \left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in R $ for each $ \mu \in \mathbb{N} $. Denote $ \zeta _{\mu +1} = \gamma \left(\zeta _{\mu }\right) = $ $ \gamma ^{\mu +1}\left(\zeta _{0}\right), $ for all $ \mu \in \mathbb{N}. $ If there exists $ \mu _{0}\in \mathbb{N} $ so that $ \gamma \left(\zeta _{\mu _{0}}\right) = \zeta _{\mu _{0}}, $ then $ \gamma $ has a fixed point $ \zeta _{\mu _{0}} $ and it completes the proof. Otherwise, assume that $ \zeta _{\mu +1}\neq \zeta _{\mu } $ for every $ \mu \in \mathbb{N} $. Then $ \left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in R $ $ \left({\rm{for \;all}}\;\mu \in \mathbb{N} \right) $. Continuing this process and using the assumption $ \left(iv\right) $, we deduce $ \left({\rm{for \;all}}\;\mu \in \mathbb{N} \right) $

$\begin{eqnarray*} F_{R}^{m}\left( m\left( \gamma \left( \zeta _{\mu -1}\right) ,\gamma ^{2}\left( \zeta _{\mu -1}\right) \right) \right) &\leq &F_{R}^{m}\left( \max \left\{ \begin{array}{c} m\left( \zeta _{\mu -1},\gamma \left( \zeta _{\mu -1}\right) \right) ,m\left( \gamma \left( \zeta _{\mu -1}\right) ,\gamma ^{2}\left( \zeta _{\mu -1}\right) \right) , \\ \frac{m\left( \zeta _{\mu -1},\gamma \left( \zeta _{\mu -1}\right) \right) \left[ 1+m\left( \gamma \left( \zeta _{\mu -1}\right) ,\gamma ^{2}\left( \zeta _{\mu -1}\right) \right) \right] }{1+m\left( \gamma \left( \zeta _{\mu -1}\right) ,\gamma ^{2}\left( \zeta _{\mu -1}\right) \right) } \end{array} \right\} \right) \\ &&-\xi \\ & = &F_{R}^{m}\left( \max \left\{ \begin{array}{c} m\left( \zeta _{\mu -1},\zeta _{\mu }\right) ,m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \\ \frac{m\left( \zeta _{\mu -1},\zeta _{\mu }\right) \left[ 1+m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right] }{1+m\left( \zeta _{\mu },\zeta _{\mu +1}\right) } \end{array} \right\} \right) \\ &&-\xi \\ &\leq &F_{R}^{m}\left( \max \left\{ m\left( \zeta _{\mu -1},\zeta _{\mu }\right) ,m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right\} \right) -\xi . \end{eqnarray*}$ $

Assume that $ \max \left\{ m\left(\zeta _{\mu -1}, \zeta _{\mu }\right), m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right\} = m\left(\zeta _{\mu }, \zeta _{\mu +1}\right), $ then we get

It is a contradiction. Hence, $ \max \left\{ m\left(\zeta _{\mu -1}, \zeta _{\mu }\right), m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right\} = m\left(\zeta _{\mu -1}, \zeta _{\mu }\right), $ and so

$\begin{equation*} F_{R}^{m}\left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) \leq F_{R}^{m}\left( m\left( \zeta _{\mu -1},\zeta _{\mu }\right) \right) -\xi\;{\rm{ for\; all}}\;\mu \in \mathbb{N} . \end{equation*}$ $

This yields that $ \left({\rm{ for\; all}}\;\mu \in \mathbb{N} \right) $

$\begin{eqnarray} F_{R}^{m}\left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) &\leq &F_{R}^{m}\left( m\left( \zeta _{\mu -1},\zeta _{\mu }\right) \right) -\xi \\ &\leq &F_{R}^{m}\left( m\left( \zeta _{\mu -2},\zeta _{\mu -1}\right) \right) -2\xi \\ &\leq &... \\ &\leq &F_{R}^{m}\left( m\left( \zeta _{0},\zeta _{1}\right) \right) -\mu \xi . \end{eqnarray}$ $

(2.1)

By applying limit as $ \mu $ goes to $ \infty $ in above equation, we deduce $ \lim_{\mu \rightarrow \infty }F_{R}^{m}\left(m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right) = -\infty. $ Since $ F_{R}^{m}\in \nabla \left(F\right), $ we deduce that $ \lim_{\mu \rightarrow \infty }m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) = 0. $ Using $ (F_{3}) $, there is $ k\in \left(0, 1\right) $ so that

$\begin{equation*} \lim\limits_{\mu \rightarrow \infty }\left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) ^{k}F_{R}^{m}\left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) = 0. \end{equation*}$ $

Now, from $ \left(2.9\right) $, we have

$\begin{eqnarray} &&\left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) ^{k}F_{R}^{m}\left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) -\left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) ^{k}F_{R}^{m}\left( m\left( \zeta _{0},\zeta _{1}\right) \right) \\ &\leq &-\left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) ^{k}\mu \leq 0. \end{eqnarray}$ $

(2.2)

Letting $ \mu \rightarrow \infty $ in $ \left(2.10\right) $, we get $ \lim_{\mu \rightarrow \infty }\left(m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right) ^{k} = 0. $ There is $ \mu _{1} $ in $ \mathbb{N} $ so that

$\begin{equation*} \mu \left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) ^{k}\leq 1 \;{\rm{for \;all}}\;\mu \geq \mu _{1}. \end{equation*}$ $

That is,

$\begin{equation*} m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \leq \frac{1}{\mu ^{\frac{1}{k}} }\;{\rm{for \;all}}\;\mu \geq \mu _{1}. \end{equation*}$ $

Now, for $ \nu > \mu > \mu _{1} $, we have

$\begin{equation*} m\left( \zeta _{\mu },\zeta _{\nu }\right) -m_{\zeta _{\mu },\zeta _{\nu }}\leq \sum\limits_{i = \mu }^{\nu -1}m\left( \zeta _{\mu },\zeta _{\nu }\right) -m_{\zeta _{\mu },\zeta _{\nu }}\leq \sum\limits_{i = \mu }^{\nu -1}m\left( \zeta _{\mu },\zeta _{\nu }\right) \leq \sum\limits_{i = \mu }^{\nu -1}\frac{1}{i^{\frac{1}{ k}}}. \end{equation*}$ $

Since the series $ \sum_{i = \mu }^{\nu -1}\frac{1}{i^{\frac{1}{k}}} $ is convergent, i.e., $ m\left(\zeta _{\mu }, \zeta _{\nu }\right) -m_{\zeta _{\mu }, \zeta _{\nu }} $ converges to $ 0 $, the sequence $ \left\{ \zeta _{\mu }\right\} $ is a $ m $-Cauchy sequence. Since $ \left(U, m, R\right) $ is $ R $-complete and $ \left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in R $ for all $ \mu \in \mathbb{N}, $ $ \left\{ \zeta _{\mu }\right\} $ converges to $ \zeta \in U. $ Now, using the $ R $-continuity of $ \gamma $, we deduce that

Finally, we will prove that $ \left(\gamma ^{\mu }\right) _{Fix} = \left(\gamma \right) _{Fix} $ where $ \mu \in \mathbb{N}. $ Assume on contrary that $ \zeta \in \left(\gamma ^{\mu }\right) _{Fix} $ and $ \zeta \notin \left(\gamma \right) _{Fix} $ for some $ \mu \in \mathbb{N} $. Then $ m\left(\zeta, \gamma \left(\zeta \right) \right) > 0 $, $ \left(\zeta, \gamma \left(\zeta \right) \right) \in R\left({\rm{from \;condition}}\;\left(iv\right) ^{^{\prime }}\right). $ From assumption $ \left(ii\right) $ we obtain $ \left(\gamma ^{\mu }\left(\zeta \right), \gamma ^{\mu +1}\left(\zeta \right) \right) \in R $ for all $ \mu \in \mathbb{N} $. Assumption $ \left(iv\right) $ implies that

$\begin{eqnarray} F_{R}^{m}\left( m\left( \zeta ,\gamma \left( \zeta \right) \right) \right) & = &F_{R}^{m}\left( m\left( \gamma \left( \gamma ^{\mu -1}\left( \zeta \right) \right) ,\gamma ^{2}\left( \gamma ^{\mu -1}\left( \zeta \right) \right) \right) \right) \\ &\leq &F_{R}^{m}\left( m\left( \gamma ^{\mu -1}\left( \zeta \right) \right) ,\gamma ^{\mu }\left( \zeta \right) \right) -\xi \\ &\leq &F_{R}^{m}\left( m\left( \gamma ^{\mu -2}\left( \zeta \right) \right) ,\gamma ^{\mu -1}\left( \zeta \right) \right) -2\xi \\ &&... \\ &\leq &F_{R}^{m}\left( m\left( \zeta ,\gamma \left( \zeta \right) \right) \right) -\mu \xi . \end{eqnarray}$ $

Taking $ \mu \rightarrow \infty $ in above inequality, we obtain $ F_{R}^{m}\left(m\left(\zeta, \gamma \left(\zeta \right) \right) \right) = -\infty, $ a contradiction. So, $ \left(\gamma ^{\mu }\right) _{Fix} = \left(\gamma \right) _{Fix} $ for any $ \mu \in \mathbb{N}. $

Corollary 2.1. Let $ \left(U, m\right) $ be a complete $ m $-metric space with a binary relation $ R $ on $ U $ and $ \gamma $ be a self-mapping such that:

$ \left(i\right) $ the class $ \Theta \left(\gamma, R\right) $ is nonempty;

$ \left(ii\right) $ the binary relation $ R $ is $ \gamma $-closed;

$ \left(iii\right) $ $ \gamma $ is $ R $-continuous;

$ \left(iv\right) $ $ \gamma $ is a $ F_{R}^{m} $-contraction mapping.

Then $ \gamma $ possesses a fixed point in $ U $.

Here, we use the definition of $ F $-contractions with the standard conditions $ \left(i-iii\right). $

Definition 2.5. Given a $ mm $-space $ \left(U, m\right) $ and a binary relation $ R $ on $ U. $ Suppose that

$\begin{equation*} \varpi = \left\{ \left( \zeta ,\Im \right) \in S^{\ast }:\kappa \left( \zeta ,\Im \right) > 0\right\} . \end{equation*}$ $

We say that a self-mapping $ \gamma :U\rightarrow U $ is a rational type $ F_{R}^{m} $-contraction if there exists $ F_{R}^{m}\in \Pi \left(\nabla \right) $ such that

(3.1)

for all $ \left(\zeta, \Im \right) \in \Xi. $

Theorem 2.5. Let $ \left(U, m\right) $ be a complete $ m $-metric space, $ R $ be a binary relation on $ U $ and $ \gamma $ be a self-mapping on $ U. $ Assume that:

$ \left(i\right) $ the class $ \Theta \left(\gamma, R\right) $ is non-empty;

$ \left(ii\right) $ the binary relation $ R $ is $ \gamma $-closed;

$ \left(iii\right) $ $ \gamma $ is $ R $-continuous;

$ \left(iv\right) $ $ \gamma $ is a rational type $ F_{R}^{m} $ -contraction mapping.

Then $ \gamma $ possesses a fixed point in $ U $.

Proof. Let $ \zeta _{0}\in \Theta \left(\left[ \gamma, R\right] \right) $, i.e., $ \left(\left[ \zeta _{0}, \gamma \left(\zeta _{0}\right) \right] \right) \in R $ $ \ $. We define a sequence $ \left\{ \zeta _{\mu +1}\right\} $ given as $ \zeta _{\mu +1} = \gamma \left(\zeta _{\mu }\right) = $ $ \gamma ^{\mu +1}\left(\zeta _{0}\right) $. We have $ \left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in R $ for all $ \mu $ in $ \mathbb{N} $. If there exists $ \mu _{0}\ $ in $ \mathbb{N} $ such that $ \gamma \left(\zeta _{\mu _{0}}\right) = \zeta _{\mu _{0}}, $ then $ \zeta _{\mu _{0}} $ is a fixed point of $ \gamma $ and the proof is finished. Now, assume that $ \zeta _{\mu +1}\neq \zeta _{\mu } $ for all $ \mu \in \mathbb{N} $. Then $ \left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in R\left({\text{for all }}\mu \in \mathbb{N} \right) $. Using the condition $ \left(iv\right) $, we deduce $ \left({\text{for all }}\mu \in \mathbb{N} \right) $

$\begin{eqnarray} F\left( m\left( \gamma \left( \zeta _{\mu -1}\right) ,\gamma ^{2}\left( \zeta _{\mu -1}\right) \right) \right) &\leq &F_{R}^{m}\left( \max \left\{ \begin{array}{c} m\left( \zeta _{\mu -1},\gamma \left( \zeta _{\mu -1}\right) \right) ,m\left( \gamma \left( \zeta _{\mu -1}\right) ,\gamma ^{2}\left( \zeta _{\mu -1}\right) \right) , \\ \frac{m\left( \zeta _{\mu -1},\gamma \left( \zeta _{\mu -1}\right) \right) \left[ 1+m\left( \gamma \left( \zeta _{\mu -1}\right) ,\gamma ^{2}\left( \zeta _{\mu -1}\right) \right) \right] }{1+m\left( \gamma \left( \zeta _{\mu -1}\right) ,\gamma ^{2}\left( \zeta _{\mu -1}\right) \right) } \end{array} \right\} \right) -\xi \\ & = &F_{R}^{m}\left( \max \left\{ \begin{array}{c} m\left( \zeta _{\mu -1},\zeta _{\mu }\right) ,m\left( \zeta _{\mu },\zeta _{\mu +1}\right) , \\ \frac{m\left( \zeta _{\mu -1},\zeta _{\mu }\right) \left[ 1+m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right] }{1+m\left( \zeta _{\mu },\zeta _{\mu +1}\right) } \end{array} \right\} \right) -\xi \\ &\leq &F_{R}^{m}\left( \max \left\{ m\left( \zeta _{\mu -1},\zeta _{\mu }\right) ,m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right\} \right) -\xi. \end{eqnarray}$ $

(2.3)

By $ (F_{1}), $ we have $ \max \left\{ m\left(\zeta _{\mu -1}, \zeta _{\mu }\right), m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right\} = m\left(\zeta _{\mu }, \zeta _{\mu +1}\right), $ then we get a contradiction. Thus, $ \max \left\{ m\left(\zeta _{\mu }, \zeta _{\mu -1}\right), m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right\} = m\left(\zeta _{\mu }, \zeta _{\mu -1}\right) $ and so from $ \left(2.3\right) $ we have

$\begin{equation} F_{R}^{m}\left( m\left( \zeta _{\mu },\zeta _{\mu +1}\right) \right) \leq F_{R}^{m}\left( m\left( \zeta _{\mu },\zeta _{\mu -1}\right) \right) -\xi {\text{ for all }}\mu \in \mathbb{N} . \end{equation}$ $

(3.3)

The proof of Theorem 2.1 is complete.

Corollary 2.2. Let $ \left(U, m\right) $ be a complete $ m $-metric space, $ R $ be a binary relation on $ U $ and $ \gamma $ be a self-mapping on $ U. $ Assume that:

$ \left(i\right) $ the class $ \Theta \left(\gamma, R\right) $ is non-empty;

$ \left(ii\right) $ $ R $ is $ \gamma $-closed;

$ \left(iii\right) $ $ \gamma $ is $ R $-continuous;

$ \left(iv\right) $ $ \gamma $ is a $ F_{R}^{m} $-contraction mapping.

Then $ \gamma $ possesses a fixed point in $ U $.

Example 2.2. Let $ U = \left[ 0, 1\right] $ and $ m $ be a relation theoretic $ m $-metric defined by

$\begin{equation*} m\left( \zeta ,\Im \right) = \frac{\zeta +\Im }{2}{\text{ for all }}\zeta ,\Im \in U. \end{equation*}$ $

We define the binary relation

$\begin{equation*} \left( \zeta ,\Im \right) \in S^{\ast }\Leftrightarrow m\left( \zeta ,\zeta \right) = m\left( \zeta ,\Im \right) \Leftrightarrow \frac{\zeta +\Im }{2}. \end{equation*}$ $

$ \left(U, m\right) $ is a complete $ m $-metric space with a binary relation. Define a mapping $ \gamma :U\rightarrow U $ by

$\begin{equation*} \gamma \left( \zeta \right) = \left\{ \begin{array}{c} \frac{\zeta }{5}{\rm{ \ \ \ \ if \ }}\zeta \in \left[ 0,1\right) \\ 0{\rm{ \ \ \ \ \ \ \ \ if \ \ }}\zeta = 1. \end{array} \right. \end{equation*}$ $

Obviously, $ \Re $ is $ \gamma $-closed, also and $ \gamma $ is $ \Re $ -continuous. Define $ F_{\Re }^{m}:\left(0, \infty \right) \rightarrow R $ by

$\begin{equation*} F_{\Re }^{m}\left( a\right) = \ln \left( a+a^{2}\right) {{\rm{ \ \ {\ for\; all }}\ \ }}\xi \in \left( 0,\infty \right) .{{\rm{ \ \ \ \ \ \ \ }}} \end{equation*}$ $

Assume that $ \left(\zeta, \Im \right) \in \Xi = \left\{ \left(\zeta, \Im \right) \in S^{\ast }:m\left(\gamma \left(\zeta \right), \gamma \left(\Im \right) \right) > 0\right\}. $ Therefore, for all $ \zeta, \Im \in U, $ with $ 0 < \zeta < 1, \Im = 1, $ we have

$\begin{eqnarray*} F_{\Re }^{m}\left( m\left( \gamma \left( \zeta \right) ,\gamma \left( \Im \right) \right) \right) & = &F_{\Re }^{m}\left( m\left( \frac{\zeta }{5} ,0\right) \right) = F_{\Re }^{m}\left( \frac{\zeta }{10}\right) \\ & = &\ln \left( \frac{\zeta }{100}^{2}+\frac{\zeta }{10}\right) . \end{eqnarray*}$ $

Now, consider $ Z_{A} = \max \left\{

$\begin{array}{c} m\left(\zeta, 1\right), m\left(\zeta, \gamma \left(\zeta \right) \right), m\left(\Im, \gamma \left(\Im \right) \right), \\ \frac{m\left(\zeta, \gamma \left(\zeta \right) \right) \left[ 1+m\left(\Im, \gamma \left(\Im \right) \right) \right] }{1+m\left(\Omega, 1\right) } \end{array}$ \right\} $

$\begin{eqnarray*} &&F_{R}^{m}\left( \max \left\{ \begin{array}{c} m\left( \zeta ,1\right) ,m\left( \zeta ,\gamma \left( \zeta \right) \right) ,m\left( \Im ,\gamma \left( \Im \right) \right) , \\ \frac{m\left( \zeta ,\gamma \left( \zeta \right) \right) \left[ 1+m\left( \Im ,\gamma \left( \Im \right) \right) \right] }{1+m\left( \zeta ,1\right) } \end{array} \right\} \right) \\ & = &F_{R}^{m}\left( \max \left\{ \begin{array}{c} \frac{\zeta +1}{2},\frac{\zeta +\frac{\zeta }{5}}{2},\frac{1+0}{2}, \\ \frac{\frac{\zeta +\frac{\zeta }{5}}{2}\left[ 1+\frac{1+0}{2}\right] }{1+ \frac{\zeta +1}{2}} \end{array} \right\} \right) \\ & = &F_{R}^{m}\left( \frac{\zeta +1}{2}\right) \\ & = &F_{R}^{m}\left( Z_{A}\right) \\ & = &\ln \left( \left( \frac{\zeta +1}{2}\right) ^{2}+\frac{\zeta +1}{2} \right) . \end{eqnarray*}$ $

From Table Table.2, $ \gamma $ is a rational type $ F_{R}^{m} $ -contraction mapping with $ \xi = 2. $ Moreover, there is $ \zeta _{0} = 0.1 $ in $ U $ so that$ \left(\xi _{0, }\gamma \left(\xi _{0}\right) \right) \in S^{\ast } $ and the class $ \Theta \left(\gamma, {R} \right) $ is nonempty. Hence, all conditions of Theorem 2.5 hold, and therefore $ \gamma $ has a fixed point.

Table 2. $ \xi +F_{\Re }^{m}\left( m(\gamma \left( \zeta \right),\gamma \left( \Im \right) )\right) $ and $ F_{R}^{m}\left( Z_{A}\right) $.

$ \zeta $	$ \Im $	$ \xi +F_{\Re }^{m}\left( m(\gamma \left( \zeta \right) ,\gamma \left( \Im \right) )\right) $	$ F_{R}^{m}\left( Z_{A}\right) $
$ 0.1 $	$ 1 $	$ -4.595 $	$ -0.159 $
$ 0.2 $	$ 1 $	$ -3.892 $	$ -0.040 $
$ 0.3 $	$ 1 $	$ -3.479 $	$ 0.069 $
$ 0.4 $	$ 1 $	$ -3.179 $	$ 0.174 $
$ 0.5 $	$ 1 $	$ -2.947 $	$ 0.272 $
$ 0.6 $	$ 1 $	$ -2.755 $	$ 0.364 $
$ 0.7 $	$ 1 $	$ -2.591 $	$ 0.452 $
$ 0.8 $	$ 1 $	$ -2.448 $	$ 0.536 $
$ 0.9 $	$ 1 $	$ -2.321 $	$ 0.616 $

| Show Table

DownLoad: CSV

3. Some fixed point results for cyclic contractions

In ^[37], Kirk et al. gave the concept of a cyclic contraction, which is the extension of the Banach contraction. It is utilized in the following theorem.

Theorem 3.1. Suppose that $ \left(U, m\right) $ is a compete $ m $-metric space, $ G $, $ H $ are two nonempty closed subsets of $ U\ $ and $ \gamma :U\rightarrow U $ verifies the following conditions:

$ \left(i\right) $ $ \gamma \left(B\right) \subseteq D $ and $ \gamma \left(D\right) \subseteq B; $

$ \left(ii\right) $ there exists a constant $ k\in \left(0, 1\right) $ such that

$\begin{equation*} m\left( \gamma \left( \zeta \right) ,\gamma \left( \Im \right) \right) \leq km\left( \zeta ,\Im \right) for\; all\;\zeta \in B, \Im \in D. \end{equation*}$ $

Then $ B\cap D $ is nonempty and there is $ \zeta \in B\cap D $ a fixed point of $ \gamma $.

By Theorems 2.1 and 3.1, we obtain successive fixed point results for cyclic rational type $ F_{R}^{m} $- generalized contraction mappings.

Theorem 3.2. Let $ \left(U, m\right) $ be a complete $ m $-metric space, $ G $ and $ H $ be two nonempty closed subsets of $ U $ and $ \gamma :U\rightarrow U $ be an operator. Assume that the successive axioms hold:

$ \left(i\right) $ $ \gamma \left(G\right) \subseteq H $ and $ \gamma \left(H\right) \subseteq G; $

$ \left(ii\right) $ there exist $ F_{R}^{m}\in \nabla \left(F\right) $ and $ \xi > 0 $ such that

(2.10)

for all $ \zeta $ in $ G $ and $ \Im $ in $ H $.

Then there is $ \zeta \in G\cap H $ a fixed point of $ \gamma $.

Proof. $ Z = G\cup H $ is closed, so $ Z $ is a closed subspace of $ U $. Therefore, $ \left(U, m\right) $ is a complete $ m $-metric space. Set a binary relation on $ Z $ denoted by $ R $ given as

$\begin{equation*} R = G\times H. \end{equation*}$ $

It means that

$\begin{equation*} \left( \zeta ,\Im \right) \in {R\Leftrightarrow }\left( \zeta ,\Im \right) \in B\times D{\text{ for all }}\zeta ,\Im \in Z{\text{.}} \end{equation*}$ $

Set $ {S = R\cup R}^{-1} $ an asymmetric relation. Directly, $ \left(U, m, S\right) $ is regular. Assume $ \left\{ \zeta _{\mu }\right\} \in Z $ is any sequence and $ \zeta \in Z $ so that

$\begin{equation*} \left( \zeta _{\mu },\zeta _{\mu +1}\right) \in S{\text{ for all }} \mu \in \mathbb{N}, \end{equation*}$ $

and

$\begin{equation*} \lim\limits_{\mu \rightarrow \infty }m\left( \zeta _{\mu },\zeta \right) = \lim\limits_{\mu \rightarrow \infty }\min \left\{ m\left( \zeta _{\mu },\zeta _{\mu }\right) ,m\left( \zeta ,\zeta \right) \right\} = m\left( \zeta ,\zeta \right) . \end{equation*}$ $

Using the definition of $ {\bf{S, }} $ we obtain

$\begin{equation} \left( \zeta _{\mu },\zeta _{\mu +1}\right) \in \left( B\times D\right) \cup \left( D\times B\right) {\text{ for all }}\mu \in \mathbb{N} {\text{.}} \end{equation}$ $

(3.2)

Immediately, the product fashion $ Z\times Z $ involves a $ mm $-space $ m $ given as

$\begin{equation*} m\left( \left( \zeta _{1},\Im _{1}\right) ,\left( \zeta _{2},\Im _{2}\right) \right) = \frac{m\left( \zeta _{1},\Im _{1}\right) +m\left( \zeta _{2},\Im _{2}\right) }{2}. \end{equation*}$ $

Since $ \left(U, m\right) $ is a complete $ m $-metric space, we obtain $ \left(Z\times Z, m\right) $ is complete. Furthermore, $ G\times H $ and $ H\times G $ are closed in $ \left(Z\times Z, m\right), $ because $ G $ and $ H $ are closed in $ \left(U, m\right). $ Letting $ \mu \rightarrow \infty $ in $ \left(2.11\right) $, we have $ \left(\zeta, \zeta \right) \in \left(B\times D\right) \cup \left(D\times B\right). $ This implies that $ \zeta \in B\cap D. $ Furthermore, from Eq $ \left(2.11\right) $, we have $ \zeta _{\mu }\in B\cup D. $ Thus, we get $ \left(\zeta _{\mu }, \zeta \right) \in S\left({\text{for all }}\mu \in \mathbb{N} \right). $ Therefore, our assertions hold. Furthermore, since $ \gamma $ is a self-mapping and from condition $ \left(i\right) $, we obtain for all $ \zeta, \Im \in U, $

$\begin{eqnarray*} \left( \zeta ,\Im \right) {\text{ in }}G\times H{\text{ which implies }}\left( \gamma \left( \zeta \right) ,\gamma \left( \Im \right) \right) &\in &H\times G, \\ \left( \zeta ,\Im \right) {\text{ in }}H\times G{\text{ which implies }}\left( \gamma \left( \zeta \right) ,\gamma \left( \Im \right) \right) &\in &G\times H. \end{eqnarray*}$ $

The binary relation $ R $ is $ \gamma $-closed. As $ B\neq \emptyset $, there exists $ \zeta _{0}\in B, $ such that $ \gamma \left(\zeta _{0}\right) \in D $ that is $ \left(\zeta _{0}, \gamma \left(\zeta _{0}\right) \right) \in R. $ Therefore, all the hypotheses of Theorem $ 2.2 $ are satisfied. Hence, $ \left(\gamma \right) _{Fix} $ $ \neq \emptyset $ and also$ \left(\gamma \right) _{Fix}\subseteq B\cap D $. Finally, as $ \left(\zeta, \Im \right) \in R $ for all $ \zeta, \Im \in G\cap H $, $ G\cap H $ is $ \nabla $-directed. Hence, the main conditions of Theorem $ \ 2.2 $ are satisfied, so $ \gamma $ has a unique fixed point. It finishes the proof.

4. Application

In this section, we illustrate how to guarantee existence of a solution of a matrix type equation. We shall use the following notations. Let $ A\left(\mu \right) $ be the set of all $ \mu \times \mu $ complex matrices, let $ H\left(\mu \right) \subseteq A\left(\mu \right) $ be the family of all $ \mu \times \mu $ Hermitian matrices, let $ G\left(\mu \right) \subseteq A\left(\mu \right) $ be the set of all $ \mu \times \mu $ positive definite matrices, $ H^{+}\left(\mu \right) \subseteq F\left(\mu \right) $ be the set of all $ \mu \times \mu $ positive semidefinite matrices. For $ \Lambda $ in $ G\left(\mu \right), $ we will also denote $ \Lambda \succ 0. $ Furthermore, $ \Lambda \succeq 0 $ means that $ \Lambda $ in $ H^{+}\left(\mu \right) $ $. $ As a different notation for $ \Lambda -\Delta \succeq 0 $ and $ \Lambda -\Delta \succ 0, $ we will denote $ \Lambda \succeq \Delta $ and$ \Lambda \succ \Delta $, respectively. Also, for each $ \ \Lambda, \Delta $ in $ A\left(\mu \right) $ there is a greatest lower bound and least upper bound, see ^[38]$. $ In addition, take

$\begin{equation*} \left\Vert .\right\Vert = {\text{ denote the spectral norm of matrix }}Q{\text{ i.e }}\left\Vert Q\right\Vert = \left( \lambda ^{+}\left( Q^{\ast }Q\right) \right) ^{\frac{1}{2}}, \end{equation*}$ $

such that

$\begin{equation*} \lambda ^{+}\left( Q^{\ast }Q\right) = {\text{ is the largest eigenvalue of }} Q^{\ast }Q,{\text{ where }}Q^{\ast }{\text{ is the conjugate transport of }}Q. \end{equation*}$ $

We use the $ m $-metric induced by the trace norm $ \left\Vert.\right\Vert _{tr} $ given as $ \left\Vert Q\right\Vert _{tr} = \sum_{i = 1}^{\mu }\Xi _{i}\left(Q\right), $ where $ \Xi _{i}\left(Q\right), $ $ i = 1, 2, ..., \mu $ are the singular values of $ Q $ in $ A\left(\mu \right). $ The set $ H\left(\mu \right) $ endowed with this norm is a complete $ m $-metric space. Moreover, we see that

$\begin{equation*} H\left( \mu \right) = {\text{is a partial ordered set with partial order}} \preceq ,{\text{ where }}\Lambda \preceq \Delta \Leftrightarrow \Lambda \succeq \Delta . \end{equation*}$ $

Consider the following nonlinear matrix equation

$\begin{equation} \Lambda = S+\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left( \Lambda \right) Q_{i} , \end{equation}$ $

(4.1)

where $ \vartheta $ is a positive definite matrix, $ Q_{1}, Q_{2}, \cdots, Q_{m} $ are $ \mu \times \mu $ matrices and $ \Xi $ is an order persevering continuous map from $ H\left(\mu \right) $ to $ G\left(\mu \right) $. Then, $ F_{R}^{m}\in \nabla \left(F\right) $ and $ \left(A\left(\mu \right), m\right) $ is a complete $ mm $-space, where

$\begin{equation} m\left( \Lambda ,\Delta \right) = \left\Vert \frac{\Lambda +\Delta }{2} \right\Vert _{tr} = \frac{1}{2}\left( tr\left( \Lambda +\Delta \right) \right) . \end{equation}$ $

(4.2)

In this section, we prove the existence of the positive definite solution to the nonlinear matrix Eq $ \left(4.1\right). $

Theorem 4.1. Assume that there are positive real numbers $ C $ and $ \xi $ such that:

$ \left(i\right) $ for each $ \Lambda, \Delta $ in $ H\left(\mu \right) $ such that $ \left(\Lambda, \Delta \right) $ in $ \preceq $ with $ \sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\Lambda \right) Q_{i}\neq \sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\Delta \right) Q_{i} $,

$\begin{equation*} \left\vert \frac{tr\left( \Xi \left( \Lambda \right) +\Xi \left( \Delta \right) \right) }{2}\right\vert \leq \frac{\left\vert \frac{tr\left( \Lambda +\Delta \right) }{2}\right\vert }{C\left( 1+\xi \sqrt{\frac{tr\left( \Lambda +\Delta \right) }{2}}\right) ^{2}}, \end{equation*}$ $

$ \left(ii\right) $ there exists a positive number $ N $ for which $ \sum\limits_{i = 1}^{\mu }Q_{i}Q_{i}^{\ast } < CI_{\mu } $ and $ \sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\Lambda \right) Q_{i} > 0. $

Then the matrix Eq $ (4.1) $ has a solution$. $ Furthermore, the iteration

$\begin{equation} \Lambda _{\mu } = S+\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left( \Lambda _{\mu -1}\right) Q_{i} , \end{equation}$ $

(4.2)

where $ \Lambda _{0} $ in $ F\left(\mu \right) $ satisfies $ \Lambda _{0}\preceq \vartheta +\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(S_{\mu -1}\right) Q_{i} $, converges in the sense of trace norm $ \left\Vert.\right\Vert _{tr} $ to the solution of the matrix Eq $ \left(4.1\right). $

Proof. We define the mapping $ \gamma :H\left(\mu \right) \rightarrow H\left(\mu \right) $ and $ F_{R}^{m}:R^{+}\rightarrow R $ by

$\begin{equation*} \gamma \left( \Lambda \right) = S+\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left( \Lambda \right) Q_{i},{\text{ for all }}\Lambda \in F\left( \mu \right), \end{equation*}$ $

and set

$\begin{equation*} H^{+}\left( \mu \right) \left( \gamma ,\preceq \right) = \left\{ Q\in F\left( \mu \right) :Q\preceq \gamma \left( Q\right) {\text{ or }}\gamma \left( Q\right) -Q{\text{ }}\succeq 0\right\}. \end{equation*}$ $

Then, $ \gamma $ is well defined and $ \preceq $ is a relation under $ R $, and $ \preceq $ on $ F\left(\mu \right) $ is $ \gamma $-closed. $ F_{R}^{m}\left(a\right) = -\frac{1}{\sqrt{a}} $ for all $ a\in R^{+} $. Furthermore, a fixed point of $ \gamma $ is a positive solution of $ \left(4.1\right). $ Now, we want to prove that $ \gamma $ is a $ F_{R}^{m} $-contraction mapping with $ \xi $ Let $ \left(\Lambda, \Delta \right) \in \varpi = \left\{ \left(\left(\Lambda, \Delta \right) \in R:\Xi \left(\Lambda \right) \neq \Xi \left(\Delta \right) \right) \right\} $ which implies that $ \Lambda \prec \Delta. $ Since $ \Xi $ is an order preserving mapping, we deduce that $ \Xi \left(\Lambda \right) \prec \Xi \left(\Delta \right). $ We have

$\begin{eqnarray*} \left\Vert \frac{\gamma \left( \Lambda \right) +\gamma \left( \left( \Delta \right) \right) }{2}\right\Vert _{tr} & = &\frac{1}{2}\left( tr\left( \gamma \left( \Lambda \right) +\gamma \left( \Delta \right) \right) \right) \\ & = &\sum\limits_{i = 1}^{m}\frac{1}{2}\left( tr\left( Q_{i}Q_{i}^{\ast }\left( \gamma \left( \Lambda \right) +\gamma \left( \Delta \right) \right) \right) \right) \\ & = &\frac{1}{2}tr\left( \left( \sum\limits_{i = 1}^{m}Q_{i}Q_{i}^{\ast }\right) \gamma \left( \Lambda \right) +\gamma \left( \Delta \right) \right) \\ &\leq &\left\Vert \sum\limits_{i = 1}^{m}E_{i}E_{i}^{\ast }\right\Vert \frac{1 }{2}\left\Vert \gamma \left( \Lambda \right) +\gamma \left( \Delta \right) \right\Vert _{1} \\ &\leq &\frac{\left\Vert \sum\limits_{i = 1}^{m}E_{i}E_{i}^{\ast }\right\Vert }{C}\left( \frac{\left\Vert \frac{\Lambda +\Delta }{2}\right\Vert }{\left( 1+\xi \sqrt{\left\Vert \frac{\Lambda +\Delta }{2}\right\Vert }\right) ^{2}} \right) \\ & < &\left( \frac{\left\Vert \frac{\Lambda +\Delta }{2}\right\Vert }{\left( 1+\xi \sqrt{\left\Vert \frac{\Lambda +\Delta }{2}\right\Vert }\right) ^{2}} \right), \end{eqnarray*}$ $

and so

$\begin{equation*} \frac{\left( 1+\xi \sqrt{\left\Vert \frac{\Lambda +\Delta }{2}\right\Vert _{tr}}\right) ^{2}}{\left\Vert \frac{\Lambda +\Delta }{2}\right\Vert _{tr}} \leq \frac{1}{\left\Vert \frac{\gamma \left( \Lambda \right) +\gamma \left( \Lambda \right) }{2}\right\Vert _{tr}}. \end{equation*}$ $

This implies that

$\begin{equation*} \left( \xi +\frac{1}{\sqrt{\left\Vert \frac{\Lambda +\Delta }{2}\right\Vert _{tr}}}\right) ^{2}\leq \frac{1}{\left\Vert \frac{\gamma \left( \Lambda \right) +\gamma \left( \Delta \right) }{2}\right\Vert _{tr}}, \end{equation*}$ $

and then

$\begin{equation*} \xi +\frac{1}{\sqrt{\left\Vert \frac{\Lambda +\Delta }{2}\right\Vert _{tr}}} \leq \frac{1}{\sqrt{\left\Vert \frac{\gamma \left( \Lambda \right) +\gamma \left( \Delta \right) }{2}\right\Vert _{tr}}}. \end{equation*}$ $

Consequently,

$\begin{equation*} \xi -\frac{1}{\sqrt{\left\Vert \frac{\gamma \left( \Lambda \right) +\gamma \left( \Delta \right) }{2}\right\Vert _{tr}}}\leq -\frac{1}{\sqrt{\left\Vert \frac{\Lambda +\Delta }{2}\right\Vert _{tr}}}. \end{equation*}$ $

Now, we get

$\begin{equation*} \xi +F_{R}^{m}\left( \left\Vert \frac{\gamma \left( \Lambda \right) +\gamma \left( \Delta \right) }{2}\right\Vert _{tr}\right) \leq F_{R}^{m}\left( \left\Vert \frac{\Lambda +\Delta }{2}\right\Vert _{tr}\right) . \end{equation*}$ $

This shows that $ \gamma $ is a $ F_{R}^{m} $-contraction. Using $ \sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\vartheta \right) Q_{i}\succ 0 $, we deduce that $ \vartheta \preceq \gamma \left(\vartheta \right). $ This means that $ \vartheta $ in $ H^{+}\left(\gamma, \preceq \right). $ From Corollary $ 2.2 $, there exists $ \Lambda _{0}\in H\left(\mu \right) $ such that $ \gamma \left(\Lambda _{0}\right) = $ $ \Lambda _{0} $. Hence, the matrix Eq $ \left(4.1\right) $ has a solution$. $

Example 4.1. Now, consider the matrix equation

$\begin{equation*} \Lambda = S+\sum\limits_{i = 1}^{2}Q_{i}^{\ast }\Xi \left( \Lambda \right) Q_{i}, \end{equation*}$ $

where

$\begin{equation*} \ \vartheta = \left( \begin{array}{ccc} 0.1 & 0.01 & 0.01 \\ 0.01 & 0.1 & 0.01 \\ 0.01 & 0.01 & 0.1 \end{array} \right), \end{equation*}$ $

$\begin{equation*} Q_{1} = \left( \begin{array}{ccc} 0.2 & 0.01 & 0.01 \\ 0.01 & 0.4 & 0.01 \\ 0.01 & 0.01 & 0.4 \end{array} \right), \end{equation*}$ $

$\begin{equation*} Q_{2} = \left( \begin{array}{ccc} 0.6 & 0.01 & 0.01 \\ 0.01 & 0.6 & 0.01 \\ 0.01 & 0.01 & 0.6 \end{array} \right), \end{equation*}$ $

and Define $ F_{R}^{m}:R^{+}\rightarrow R $ by

$\begin{equation*} F_{R}^{m}\left( a\right) = -\frac{1}{\sqrt{a}}, \end{equation*}$ $

for all $ a\in R^{+}, $ and $ \Xi :H\left(\mu \right) \rightarrow H\left(\mu \right) $ is given by $ \Xi \left(\Lambda \right) = \frac{\Lambda }{3}. $ Then, all conditions of Corollary 2.2 are satisfied for $ N = \frac{6}{10} $ by using the iterative sequence

$\begin{equation*} \Lambda _{\mu +1} = S+\sum\limits_{i = 1}^{2}Q_{i}^{\ast }\Xi \left( \Lambda \right) Q_{i}, \end{equation*}$ $

$\begin{equation*} \Lambda _{0} = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right). \end{equation*}$ $

After some iterations, we get the approximation solution

$\begin{equation*} \vartheta _{15} = \left( \begin{array}{ccc} 0.0233 & 0.0102 & 0.0912 \\ 0.0102 & 0.0466 & 0.0214 \\ 0.0912 & 0.0532 & 0.0326 \end{array} \right). \end{equation*}$ $

Hence, all the conditions of Theorem $ 4.1 $ are satisfied$. $

5. Conclusions

In this paper, a relation theoretic $ M $-metric fixed point algorithm under rational type $ \ F_{R}^{m} $-contractions $ \left({\text{respectively, rational type}}\ {\text{generalized }}F_{R}^{m}{\text{-contractions}}\right) $ is proposed to solve the nonlinear matrix equation $ \Lambda = S+\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\Lambda \right) Q_{i}. $ Some numerical comparison experiments with existing algorithms are presented within given tables and figures. Analogously, this proposed work can be extended to generalized distance spaces, such as symmetric spaces, $ m_{b}m $ -spaces $ rmm $-spaces, $ rm_{b}m $-spaces, $ pm $-spaces, $ p_{b}m $-spaces, etc. Some problems of fixed point results could be studied in near future.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work grant code: 23UQU4331214DSR001.

Conflict of interest

The authors declare that they have no conflict of interest.

References

[1]	Association AP (2013) American Psychiatric Association: Diagnositc and Statistical Manual of Mental Disorders, 5 Eds. American Psychiatric Association: Diagnositc and Statistical Manual of Mental Disorders.
[2]	Epperson CN (2013) Premenstrual dysphoric disorder and the brain. Am J Psychiatry 170:248-252. doi: 10.1176/appi.ajp.2012.12121555
[3]	O'Brien PM, Backstrom T, Brown C, et al. (2011) Towards a consensus on diagnostic criteria, measurement and trial design of the premenstrual disorders: the ISPMD Montreal consensus. Arch Womens Ment Health 14: 13-21. doi: 10.1007/s00737-010-0201-3
[4]	Veith I (1965) The History of a Disease. Chicago: Chicago University Press.
[5]	Herculano-Houzel S (2010) Coordinated scaling of cortical and cerebellar numbers of neurons. Front Neuroanat 4: 12.
[6]	Tien RD, Ashdown BC (1992) Crossed cerebellar diaschisis and crossed cerebellar atrophy: correlation of MR findings, clinical symptoms, and supratentorial diseases in 26 patients. AJR Am J Roentgenol 158: 1155-1159. doi: 10.2214/ajr.158.5.1566683
[7]	Ito M (1993) New concepts in cerebellar function. Rev Neurol (Paris) 149: 596-599.
[8]	Harlow HF, Harlow M (1962) Social deprivation in monkeys. Sci Am 207: 136-146. doi: 10.1038/scientificamerican1162-136
[9]	Prescott JW (1970) Early somatosensory deprivation as ontogenic process in the abnormal development of the brain and behavior. Medical Primatology 1970: 356-375.
[10]	Nashold BS, Jr. , Slaughter DG (1969) Effects of stimulating or destroying the deep cerebellar regions in man. J Neurosurg 31: 172-186. doi: 10.3171/jns.1969.31.2.0172
[11]	Heath RG (1977) Modulation of emotion with a brain pacemaker. Treatment for intractable psychiatric illness. J Nerv Ment Dis 165: 300-317.
[12]	Cooper IS, Amin, L. , Gilman, S. , Waltz, J. M. (1974) The Effect of chronic stimulation of cerebellar cortex on epilepsy in Man. The Cerebellum, Epilepsy and Behavior. New York: Plenum Press. pp. 199-172.
[13]	Heath RG, Franklin DE, Shraberg D (1979) Gross pathology of the cerebellum in patients diagnosed and treated as functional psychiatric disorders. J Nerv Ment Dis 167: 585-592. doi: 10.1097/00005053-197910000-00001
[14]	Heath RG, Llewellyn RC, Rouchell AM (1980) The cerebellar pacemaker for intractable behavioral disorders and epilepsy: follow-up report. Biol Psychiatry 15: 243-256.
[15]	Heath RG, Franklin DE, Walker CF, et al. (1982) Cerebellar vermal atrophy in psychiatric patients. Biol Psychiatry 17: 569-583.
[16]	Schmahmann JD, Sherman JC (1998) The cerebellar cognitive affective syndrome. Brain 121 ( Pt4): 561-579.
[17]	Schmahmann JD, Weilburg JB, Sherman JC (2007) The neuropsychiatry of the cerebellum - insights from the clinic. Cerebellum 6: 254-267. doi: 10.1080/14734220701490995
[18]	Schmahmann JD (1991) An emerging concept. The cerebellar contribution to higher function. Arch Neurol 48: 1178-1187.
[19]	Schmahmann JD (1996) From movement to thought: anatomic substrates of the cerebellar contribution to cognitive processing. Hum Brain Mapp 4: 174-198. doi: 10.1002/(SICI)1097-0193(1996)4:3<174::AID-HBM3>3.0.CO;2-0
[20]	Allen G, Buxton RB, Wong EC, et al. (1997) Attentional activation of the cerebellum independent of motor involvement. Science 275: 1940-1943. doi: 10.1126/science.275.5308.1940
[21]	Stoodley CJ, Schmahmann JD (2009) Functional topography in the human cerebellum: a meta-analysis of neuroimaging studies. Neuroimage 44: 489-501. doi: 10.1016/j.neuroimage.2008.08.039
[22]	Koziol LF, Budding D, Andreasen N, et al. (2014) Consensus paper: the cerebellum's role in movement and cognition. Cerebellum 13: 151-177. doi: 10.1007/s12311-013-0511-x
[23]	Schraa-Tam CK, Rietdijk WJ, Verbeke WJ, et al. (2012) fMRI activities in the emotional cerebellum: a preference for negative stimuli and goal-directed behavior. Cerebellum 11: 233-245. doi: 10.1007/s12311-011-0301-2
[24]	Ferrucci R, Giannicola G, Rosa M, et al. (2012) Cerebellum and processing of negative facial emotions: cerebellar transcranial DC stimulation specifically enhances the emotional recognition of facial anger and sadness. Cogn Emot 26: 786-799. doi: 10.1080/02699931.2011.619520
[25]	Grimaldi G, Argyropoulos GP, Boehringer A, et al. (2014) Non-invasive cerebellar stimulation--a consensus paper. Cerebellum 13: 121-138. doi: 10.1007/s12311-013-0514-7
[26]	West RL (1996) An application of prefrontal cortex function theory to cognitive aging. Psych Bull120: 272-292.
[27]	Hogan MJ (2004) The cerebellum in thought and action: a fronto-cerebellar aging hypothesis. New Ideas in Psychology 22: 97-125. doi: 10.1016/j.newideapsych.2004.09.002
[28]	Eckert MA (2011) Slowing down: age-related neurobiological predictors of processing speed. Front Neurosci 5: 25.
[29]	Woodruff-Pak DS, Vogel RW, Ewers M, et al. (2001) MRI-assessed volume of cerebellum correlates with associative learning. Neurobiology of Learning and Memory 76: 342-357. doi: 10.1006/nlme.2001.4026
[30]	MacLullich AMJ, Edmond CL, Ferguson KJ, et al. (2004) Size of the neocerebellar vermis is associated with cognition in healthy elderly men. Brain and Cognition 56: 344-348. doi: 10.1016/j.bandc.2004.08.001
[31]	Paul R, Grieve SM, Chaudary B, et al. (2009) Relative contributions of the cerebellar vermis and prefrontal lobe volumes on cognitive function across the adult lifespan. Neurobiol Aging 30:457-465. doi: 10.1016/j.neurobiolaging.2007.07.017
[32]	Eckert MA, Keren NI, Roberts DR, et al. (2010) Age-related changes in processing speed: unique contributions of cerebellar and prefrontal cortex. Front Hum Neurosci 4: 10.
[33]	Hogan MJ, Staff RT, Bunting BP, et al. (2011) Cerebellar brain volume accounts for variance in cognitive performance in older adults. Cortex 47: 441-450. doi: 10.1016/j.cortex.2010.01.001
[34]	Rasgon N, Serra M, Biggio G, et al. (2001) Neuroactive steroid-serotonergic interaction: responses to an intravenous L-tryptophan challenge in women with premenstrual syndrome. Eur J Endocrinol 145: 25-33. doi: 10.1530/eje.0.1450025
[35]	Hamakawa H, Kato T, Murashita J, et al. (1998) Quantitative proton magnetic resonance spectroscopy of the basal ganglia in patients with affective disorders. Eur Arch Psychiatry Clin Neurosci 248: 53-58. doi: 10.1007/s004060050017
[36]	Renshaw PF, Levin JM, Kaufman MJ, et al. (1997) Dynamic susceptibility contrast magnetic resonance imaging in neuropsychiatry: present utility and future promise. Eur Radiol 7 Suppl 5:216-221.
[37]	Buchpiguel C, Alavi A, Crawford D, et al. (2000) Changes in cerebral blood flow associated with premenstrual syndrome: a preliminary study. J Psychosom Obstet Gynaecol 21: 157-165. doi: 10.3109/01674820009075623
[38]	Rasgon NL, Thomas MA, Guze BH, et al. (2001) Menstrual cycle-related brain metabolite changes using 1H magnetic resonance spectroscopy in premenopausal women: a pilot study. Psychiatry Res 106: 47-57. doi: 10.1016/S0925-4927(00)00085-8
[39]	Epperson CN, Haga K, Mason GF, et al. (2002) Cortical gamma-aminobutyric acid levels across the menstrual cycle in healthy women and those with premenstrual dysphoric disorder: a proton magnetic resonance spectroscopy study. Arch Gen Psychiatry 59: 851-858. doi: 10.1001/archpsyc.59.9.851
[40]	Jovanovic H, Cerin A, Karlsson P, et al. (2006) A PET study of 5-HT1A receptors at different phases of the menstrual cycle in women with premenstrual dysphoria. Psychiatry Res 148:185-193. doi: 10.1016/j.pscychresns.2006.05.002
[41]	Eriksson O, Wall A, Marteinsdottir I, et al. (2006) Mood changes correlate to changes in brain serotonin precursor trapping in women with premenstrual dysphoria. Psychiatry Res 146: 107-116. doi: 10.1016/j.pscychresns.2005.02.012
[42]	Batra NA, Seres-Mailo J, Hanstock C, et al. (2008) Proton magnetic resonance spectroscopy measurement of brain glutamate levels in premenstrual dysphoric disorder. Biol Psychiatry 63:1178-1184. doi: 10.1016/j.biopsych.2007.10.007
[43]	Protopopescu X, Tuescher O, Pan H, et al. (2008) Toward a functional neuroanatomy of premenstrual dysphoric disorder. J Affect Disord 108: 87-94. doi: 10.1016/j.jad.2007.09.015
[44]	Bannbers E, Gingnell M, Engman J, et al. (2012) The effect of premenstrual dysphoric disorder and menstrual cycle phase on brain activity during response inhibition. J Affect Disord 142:347-350. doi: 10.1016/j.jad.2012.04.006
[45]	Gingnell M, Morell A, Bannbers E, et al. (2012) Menstrual cycle effects on amygdala reactivity to emotional stimulation in premenstrual dysphoric disorder. Horm Behav 62: 400-406. doi: 10.1016/j.yhbeh.2012.07.005
[46]	Gingnell M, Bannbers E, Wikstrom J, et al. (2013) Premenstrual dysphoric disorder and prefrontal reactivity during anticipation of emotional stimuli. Eur Neuropsychopharmacol 23: 1474-1483. doi: 10.1016/j.euroneuro.2013.08.002
[47]	Baller EB, Wei SM, Kohn PD, et al. (2013) Abnormalities of dorsolateral prefrontal function in women with premenstrual dysphoric disorder: a multimodal neuroimaging study. Am J Psychiatry170: 305-314.
[48]	Jeong HG, Ham BJ, Yeo HB, et al. (2012) Gray matter abnormalities in patients with premenstrual dysphoric disorder: an optimized voxel-based morphometry. J Affect Disord 140: 260-267. doi: 10.1016/j.jad.2012.02.010
[49]	Berman SM, London ED, Morgan M, et al. (2013) Elevated gray matter volume of the emotional cerebellum in women with premenstrual dysphoric disorder. J Affect Disord 146: 266-271. doi: 10.1016/j.jad.2012.06.038
[50]	Rapkin A (2003) A review of treatment of premenstrual syndrome and premenstrual dysphoric disorder. Psychoneuroendocrinology 28 Suppl 3: 39-53.
[51]	Halbreich U (2008) Selective serotonin reuptake inhibitors and initial oral contraceptives for the treatment of PMDD: effective but not enough. CNS Spectr 13: 566-572. doi: 10.1017/S1092852900016849
[52]	Nevatte T, O'Brien PM, Backstrom T, et al. (2013) ISPMD consensus on the management of premenstrual disorders. Arch Womens Ment Health 16: 279-291. doi: 10.1007/s00737-013-0346-y
[53]	Raichle M (1987) Circulatory and Metabolic Correlates of brain function in normal humans. Handbook of Physiology-The nervous system Bethesda: American Physiological Society V:643-674.
[54]	Rapkin AJ, Berman SM, Mandelkern MA, et al. (2011) Neuroimaging evidence of cerebellar involvement in premenstrual dysphoric disorder. Biol Psychiatry 69: 374-380. doi: 10.1016/j.biopsych.2010.09.029
[55]	Mackenzie G, Maguire J (2014) The role of ovarian hormone-derived neurosteroids on the regulation of GABA receptors in affective disorders. Psychopharmacology (Berl).
[56]	Schmidt PJ, Nieman LK, Danaceau MA, et al. (1998) Differential behavioral effects of gonadal steroids in women with and in those without premenstrual syndrome. New England Journal of Medicine 338: 209-216. doi: 10.1056/NEJM199801223380401
[57]	Hanstock C, Allen, PS (2000) Segmentation of brain from a PRESS localized single volume using double inversion recovery for simultaneous T1 nulling. 8th Annual Meeting of the International Society for Magnetic Resonance in Medicine. Denver, Colorado.
[58]	Diedrichsen J, Balsters JH, Flavell J, et al. (2009) A probabilistic MR atlas of the human cerebellum. Neuroimage 46: 39-46. doi: 10.1016/j.neuroimage.2009.01.045
[59]	Kalpouzos G, Persson J, Nyberg L (2012) Local brain atrophy accounts for functional activity differences in normal aging. Neurobiol Aging 33: 623 e621-623 e613.
[60]	Diedrichsen J, Verstynen T, Schlerf J, et al. (2010) Advances in functional imaging of the human cerebellum. Curr Opin Neurol 23: 382-387.
[61]	Baldacara L, Nery-Fernandes F, Rocha M, et al. (2011) Is cerebellar volume related to bipolar disorder? J Affect Disord 135: 305-309. doi: 10.1016/j.jad.2011.06.059
[62]	De Bellis MD, Kuchibhatla M (2006) Cerebellar volumes in pediatric maltreatment-related posttraumatic stress disorder. Biol Psychiatry 60: 697-703. doi: 10.1016/j.biopsych.2006.04.035
[63]	Frodl TS, Koutsouleris N, Bottlender R, et al. (2008) Depression-related variation in brain morphology over 3 years: effects of stress? Arch Gen Psychiatry 65: 1156-1165. doi: 10.1001/archpsyc.65.10.1156
[64]	Peng J, Liu J, Nie B, et al. (2011) Cerebral and cerebellar gray matter reduction in first-episode patients with major depressive disorder: a voxel-based morphometry study. Eur J Radiol 80:395-399. doi: 10.1016/j.ejrad.2010.04.006
[65]	Kim D, Cho HB, Dager SR, et al. (2013) Posterior cerebellar vermal deficits in bipolar disorder. J Affect Disord 150: 499-506. doi: 10.1016/j.jad.2013.04.050
[66]	Schutter DJ, Koolschijn PC, Peper JS, et al. (2012) The cerebellum link to neuroticism: a volumetric MRI association study in healthy volunteers. PLoS One 7: e37252. doi: 10.1371/journal.pone.0037252
[67]	Adler CM, DelBello MP, Jarvis K, et al. (2007) Voxel-based study of structural changes in first-episode patients with bipolar disorder. Biol Psychiatry 61: 776-781. doi: 10.1016/j.biopsych.2006.05.042
[68]	Spinelli S, Chefer S, Suomi SJ, et al. (2009) Early-life stress induces long-term morphologic changes in primate brain. Arch Gen Psychiatry 66: 658-665. doi: 10.1001/archgenpsychiatry.2009.52
[69]	Draganski B, Gaser C, Busch V, et al. (2004) Neuroplasticity: changes in grey matter induced by training. Nature 427: 311-312. doi: 10.1038/427311a
[70]	Kwok V, Niu Z, Kay P, et al. (2011) Learning new color names produces rapid increase in gray matter in the intact adult human cortex. Proc Natl Acad Sci U S A 108: 6686-6688. doi: 10.1073/pnas.1103217108
[71]	Oral E, Ozcan H, Kirkan TS, et al. (2013) Luteal serum BDNF and HSP70 levels in women with premenstrual dysphoric disorder. Eur Arch Psychiatry Clin Neurosci 263: 685-693. doi: 10.1007/s00406-013-0398-z
[72]	Anim-Nyame N, Domoney C, Panay N, et al. (2000) Plasma leptin concentrations are increased in women with premenstrual syndrome. Hum Reprod 15: 2329-2332. doi: 10.1093/humrep/15.11.2329
[73]	Oldreive CE, Harvey J, Doherty GH (2008) Neurotrophic effects of leptin on cerebellar Purkinje but not granule neurons in vitro. Neurosci Lett 438: 17-21. doi: 10.1016/j.neulet.2008.04.045
[74]	Riad-Gabriel MG, Jinagouda SD, Sharma A, et al. (1998) Changes in plasma leptin during the menstrual cycle. Eur J Endocrinol 139: 528-531. doi: 10.1530/eje.0.1390528
[75]	Narita K, Kosaka H, Okazawa H, et al. (2009) Relationship between plasma leptin level and brain structure in elderly: a voxel-based morphometric study. Biol Psychiatry 65: 992-994. doi: 10.1016/j.biopsych.2008.10.006
[76]	Matochik JA, London ED, Yildiz BO, et al. (2005) Effect of leptin replacement on brain structure in genetically leptin-deficient adults. J Clin Endocrinol Metab 90: 2851-2854. doi: 10.1210/jc.2004-1979
[77]	London ED, Berman SM, Chakrapani S, et al. (2011) Short-term plasticity of gray matter associated with leptin deficiency and replacement. J Clin Endocrinol Metab 96: E1212-1220. doi: 10.1210/jc.2011-0314
[78]	Tommaselli GA, Di Carlo C, Bifulco G, et al. (2003) Serum leptin levels in patients with premenstrual syndrome treated with GnRH analogues alone and in association with tibolone. Clin Endocrinol (Oxf) 59: 716-722. doi: 10.1046/j.1365-2265.2003.01911.x
[79]	Akturk M, Toruner F, Aslan S, et al. (2013) Circulating insulin and leptin in women with and without premenstrual disphoric disorder in the menstrual cycle. Gynecol Endocrinol 29: 465-469. doi: 10.3109/09513590.2013.769512
[80]	Eikelis N, Esler M, Barton D, et al. (2006) Reduced brain leptin in patients with major depressive disorder and in suicide victims. Mol Psychiatry 11: 800-801. doi: 10.1038/sj.mp.4001862
[81]	Westling S, Ahren B, Traskman-Bendz L, et al. (2004) Low CSF leptin in female suicide attempters with major depression. J Affect Disord 81: 41-48. doi: 10.1016/j.jad.2003.07.002
[82]	Yoshida-Komiya H, Takano K, Fujimori K, et al. (2014) Plasma levels of leptin in reproductive-aged women with mild depressive and anxious states. Psychiatry Clin Neurosci.
[83]	Lawson EA, Miller KK, Blum JI, et al. (2012) Leptin levels are associated with decreased depressive symptoms in women across the weight spectrum, independent of body fat. Clin Endocrinol (Oxf) 76: 520-525. doi: 10.1111/j.1365-2265.2011.04182.x
[84]	Chirinos DA, Goldberg R, Gellman M, et al. (2013) Leptin and its association with somatic depressive symptoms in patients with the metabolic syndrome. Ann Behav Med 46: 31-39. doi: 10.1007/s12160-013-9479-5
[85]	Kloiber S, Ripke S, Kohli MA, et al. (2013) Resistance to antidepressant treatment is associated with polymorphisms in the leptin gene, decreased leptin mRNA expression, and decreased leptin serum levels. Eur Neuropsychopharmacol 23: 653-662. doi: 10.1016/j.euroneuro.2012.08.010
[86]	Johnston JM, Greco SJ, Hamzelou A, et al. (2011) Repositioning leptin as a therapy for Alzheimer's disease. Therapy 8: 481-490. doi: 10.2217/thy.11.57
[87]	Johnston J HW, Fardo D, Greco S, Perry G, Montine T, Trojanowski J, Shaw L, Ashford J, Tezapsidis N (2013) For The Alzheimer's Disease Neuroimaging Initiative. Low Plasma Leptin in Cognitively Impaired ADNI Subjects- Gender Differences and Diagnostic and Therapeutic Potential. Curr Alzheimer Res.
[88]	Rapkin AJ, Morgan M, Goldman L, et al. (1997) Progesterone metabolite allopregnanolone in women with premenstrual syndrome. Obstet Gynecol 90: 709-714. doi: 10.1016/S0029-7844(97)00417-1
[89]	Singh M, Su C (2013) Progesterone and neuroprotection. Horm Behav 63: 284-290. doi: 10.1016/j.yhbeh.2012.06.003
[90]	Azcoitia I, Arevalo MA, De Nicola AF, et al. (2011) Neuroprotective actions of estradiol revisited. Trends Endocrinol Metab 22: 467-473. doi: 10.1016/j.tem.2011.08.002
[91]	Gao Q, Horvath TL (2008) Cross-talk between estrogen and leptin signaling in the hypothalamus. Am J Physiol Endocrinol Metab 294: E817-826. doi: 10.1152/ajpendo.00733.2007
[92]	Hedges VL, Ebner TJ, Meisel RL, et al. (2012) The cerebellum as a target for estrogen action. Front Neuroendocrinol 33: 403-411. doi: 10.1016/j.yfrne.2012.08.005
[93]	Ghidoni R, Boccardi M, Benussi L, et al. (2006) Effects of estrogens on cognition and brain morphology: involvement of the cerebellum. Maturitas 54: 222-228. doi: 10.1016/j.maturitas.2005.11.002
[94]	Boccardi M, Ghidoni R, Govoni S, et al. (2006) Effects of hormone therapy on brain morphology of healthy postmenopausal women: a Voxel-based morphometry study. Menopause 13: 584-591. doi: 10.1097/01.gme.0000196811.88505.10
[95]	Robertson D, Craig M, van Amelsvoort T, et al. (2009) Effects of estrogen therapy on age-related differences in gray matter concentration. Climacteric 12: 301-309. doi: 10.1080/13697130902730742
[96]	Kim SG, Ogawa S (2012) Biophysical and physiological origins of blood oxygenation level-dependent fMRI signals. J Cereb Blood Flow Metab 32: 1188-1206. doi: 10.1038/jcbfm.2012.23
[97]	D'Esposito M, Deouell LY, Gazzaley A. (2003) Alterations in the BOLD fMRI signal with ageing and disease: a challenge for neuroimaging. Nat Rev Neurosci 4: 863-872. doi: 10.1038/nrn1246
[98]	Ances BM, Liang CL, Leontiev O, et al. (2009) Effects of aging on cerebral blood flow, oxygen metabolism, and blood oxygenation level dependent responses to visual stimulation. Hum Brain Mapp 30: 1120-1132. doi: 10.1002/hbm.20574
[99]	Gauthier CJ, Madjar C, Desjardins-Crepeau L, et al. (2013) Age dependence of hemodynamic response characteristics in human functional magnetic resonance imaging. Neurobiol Aging 34:1469-1485. doi: 10.1016/j.neurobiolaging.2012.11.002
[100]	Sui R, Zhang L (2012) Cerebellar dysfunction may play an important role in vascular dementia. Med Hypotheses 78: 162-165. doi: 10.1016/j.mehy.2011.10.017
[101]	arrett DD, Kovacevic N, McIntosh AR, et al. (2010) Blood oxygen level-dependent signal variability si more than just noise. J Neurosci 30: 4914-4921.
[102]	Grady CL, Garrett DD (2013) Understanding variability in the BOLD signal and why it matters for aging. Brain Imaging Behav.

This article has been cited by:

1.	Muhammad Tariq, Muhammad Arshad, Eskandar Ameer, Ahmad Aloqaily, Suhad Subhi Aiadi, Nabil Mlaiki, On Relational Weak Fℜm,η-Contractive Mappings and Their Applications, 2023, 15, 2073-8994, 922, 10.3390/sym15040922
2.	Maide Gökșin Taș, Duran Türkoğlu, Ishak Altun, Fixed point results for $ P $-contractive mappings on $ M $-metric space and application, 2024, 9, 2473-6988, 9770, 10.3934/math.2024478

Reader Comments

Your name:*

Email:*
© 2014 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Neuroscience

3.1 4.2

Metrics

Article views(6435) PDF downloads(936) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Tables(1)

AIMS Neuroscience

The Cerebellum and Premenstrual Dysphoric Disorder

Related Papers:

Abstract

1. Introduction and preliminaries

2. Main results

3. Some fixed point results for cyclic contractions

4. Application

5. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Neuroscience

The Cerebellum and Premenstrual Dysphoric Disorder

Related Papers:

Abstract

1. Introduction and preliminaries

2. Main results

3. Some fixed point results for cyclic contractions

4. Application

5. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog