# RyanSpeech: A Corpus for Conversational Text-to-Speech Synthesis

###### Abstract

This paper introduces RyanSpeech, a new speech corpus for research on
automated text-to-speech (TTS) systems. Publicly available TTS corpora are
often noisy, recorded with multiple speakers, or lack quality male speech
data. In order to meet the need for a high quality, publicly available male
speech corpus within the field of speech recognition, we have designed and
created RyanSpeech which contains textual materials from real-world
conversational settings. These materials contain over 10 hours of a
professional male voice actor’s speech recorded at 44.1 kHz. This corpus’s
design and pipeline make RyanSpeech ideal for developing TTS systems in real-
world applications. To provide a baseline for future research, protocols, and
benchmarks, we trained 4 state-of-the-art speech models and a vocoder on
RyanSpeech. The results show 3.36 in mean opinion scores (MOS) in our best
model. We have made both the corpus and trained models for public use.

Index Terms: text to speech, speech corpus, speech recognition

##  1 Introduction

The advent of end-to-end deep neural networks (DNN) in the fields of automatic
speech recognition (ASR) and text-to-speech (TTS) has shifted the research
paradigm in these fields. Traditional methods are complex and time-consuming,
requiring pre-defined linguistic features which are typically language
specific. However, new TTS architectures are composed of two parts: first,
they generate a Mel-spectrogram autoregressively or non-autoregressively from
the input text’s phonemes, and then output audio from a separately trained
vocoder. The quality of the final outputs depends heavily on the quality of
speech corpora. Even though researchers [1, 2, 3] in recent years have paid
more attention to create larger speech and language corpora, it is evident
that more research is necessary to fill the gaps of speech corpora.

This paper addresses shortcomings in recent research and scholarship within
the fields of corpus development by introducing RyanSpeech, the first publicly
available male voice TTS corpus in the conversational setting. Using state-of-
the-art deep neural networks, we have found that it is easier to train a model
capable of generating natural speech synthesizers than traditional methods
[2]. However, this is only possible when we have an available corpus at hand.
Most of the publicly available corpora are in the domains of reading or
audiobooks, and most of these corpora are not single-speakers, which is
unsuitable for training the TTS (especially the vocoder models). To have a
high-fidelity TTS, the corpus also needs to be recorded at a high sampling
rate without any background noise. LJSpeech is the only female single-speaker
corpus with low noise and a nonrestrictive license [1]. RyanSpeech includes
features that make it ideal for a TTS system in real-world applications. These
features are:

  * •

RyanSpeech is the only corpus in the domain of conversation. Other speech
corpora in the domain of conversation are multi-speaker and recorded in high-
noise environments that are not appropriate for training TTS systems. In
RyanSpeech, we extracted the most commonly used conversations with prosodic
variations specific to conversations that cannot be replaced by audiobooks or
reading settings.

  * •

All the audio files are recorded by a single professional male speaker with
studio quality at a sampling rate of 44100 Hz. We double-checked all the
recordings and rerecorded sample files that were too slow, too fast, or
contained unwanted speech variations. This makes RyanSpeech the first male
speaker corpus for building high-quality TTS systems.

  * •

We chose the sentences to reflect numerous and diverse real-life
conversational situations, including dialogues on movies, sports, music,
television, restaurants, and nature as well as common questions and general
discussions.

  * •

We open-sourced the corpus including all the original FLAC (Free Lossless
Audio Codec) files with transcripts under the CC BY-NC-ND license to help
rapid development in TTS research.

This paper is organized as follows. Section 2 summarizes speech corpora,
especially those most relevant to our own. Section 3 details the corpus
creation pipeline from collecting raw text to the final audio files. Section 4
provides the corpus’s overall statistics, including both audio and textual
transcripts. We present the details of training different models based on
RyanSpeech as well as the human evaluation and the results in Section 5.
Finally, we conclude the paper in Section 6.

##  2 Related Work

Speech corpora can be categorized into multi- and single-speakers. Multi-
speaker corpora are designed to capture the diversity in spoken language with
different voices, genders, ages, and accents. These corpora are well-suited to
automatic speech recognition systems that require variations in input data
[4]. However, creating a high-quality TTS system necessitates single-speaker
corpora [2], which is especially important for training the vocoder.

Table 1 demonstrates different corpora with public licenses widely used for
research on speech recognition and synthesis. The CMU ARCTIC corpus is a
dataset that has been used for years as a baseline for speech recognition
tasks [5]. The VCTK corpus contains 110 English speakers with various accents
[6]. The main sources for the passages are newspapers selected using a greedy
algorithm to increase phonetic diversity. Common Voice is a project by Mozilla
which collects the biggest speech corpus by crowdsourcing from their community
[7]. This corpus has the greatest diversity, and is also very noisy. VoxForge
is most similar to Common Voice because it is also a community-based project
[8]. Unlike Common Voice, VoxForge does not have a verification process in the
data collection process. LibriSpeech [3] and LibriTTS [2] derive their text
source from the LibriVox project which is based on audiobooks [9]. LibriTTS is
the successor of LibriSpeech and has more robust design choices including
high-quality sampling rate, the removal of noisy subsets, and split speech at
sentence breaks. In the domain of conversational speech corpora, we have
CHiME-5 [10] which consists of 20 parties each recorded in different houses.
Also, the Linguistic Data Consortium (LDC) has developed CALLHOME American
English Speech, which consists of 120 unscripted 30-minute telephone
conversations between native English speakers [11]. Both of these corpora are
noisy and contain multiple speakers, which renders them not particularly
suitable for TTS training.

As we mentioned above, single-speaker corpora are the best candidates for
high-quality TTS systems. The BC2013 corpus comes from the blizzard challenge
and contains a large amount of speech by a single speaker with good quality
[12]. While this corpus is based on Audiobooks, it is not ideal for many
applications that require user interaction. M-AILABS was recorded with a male
and female voice, but it is too noisy and unsuitable for training the speech
synthesizer [13]. The only acceptable quality speech TTS corpus is LJSpeech,
which is all recorded by a female voice. RyanSpeech is the first high-quality
male TTS corpus with a non-restrictive license in the conversational domain
that can be used for training both the vocoder and speech model.

Table 1: List of the publicly available multi and single speaker speech dataset Corpus | Domain | Licence | Duration (hours) | Sampling rate (kHz) | Number of Speakers
---|---|---|---|---|---
CMU ARCTIC [5] | Reading | BSD | 7 | 16 | 7
VCTK [6] | Reading | ODC-By v1.0 | 44 | 48 | 109
Common Voice [7] | Reading | CC-0 1.0 | 1,118 | 48 | 51,072
VoxForge [8] | Reading | GPL | 120 | 16 | 2966
LibriSpeech [3] | Audiobook | CC-BY 4.0 | 982 | 16 | 2,484
LibriTTS [2] | Audiobook | CC-BY 4.0 | 586 | 24 | 2,456
CHiME-5 [10] | Conversation | Commercial | 50 | 16 | 48
CALLHOME [11] | Conversation | LDC | 60 | 8 | 120
BC2013 [12] | Audiobook | Non-commercial | 300 | 44.1 | 1
M-AILABS [13] | Audiobook | BSD | 75 | 16 | 2
LJSpeech [1] | Audiobook | CC-0 1.0 | 25 | 22.05 | 1
RyanSpeech | Conversation | CC BY-NC-ND | 10 | 44.1 | 1

##  3 Corpus creation pipeline

This section describes the data processing which we developed to produce the
RyanSpeech corpus.

###  3.1 Data collection

Since RyanSpeech is a single-speaker speech corpus designed and developed for
conversational systems, we used three different text resources that are most
relevant to this task.

  1. 1.

Ryan Chatbot dataset: This is a specifically designed dataset developed for
Ryan Robot [14, 15] in different conversational settings. It contains more
than 56,000 sentences covering a wide variety of topics, including television,
sports, movies, music, science, food, museums, and history. We randomly
selected 5778 sample sentences from this dataset, and what follows is a short
list of example sentences that appear in our “food” dialogues:

     * •

“Do you want to see another recipe that you could easily prepare at home?”

     * •

“I’m coming over to your house, and I’m coming hungry!”

     * •

“Maybe you just need to have a nice, home-cooked meal.”

  2. 2.

Taskmaster-2: This dataset is also designed for use in goal-oriented dialogue
systems [16]. It includes both user and assistant roles in the conversations.
Taskmaster-2 consists of 17289 dialogues in the seven domains of restaurants,
food ordering, movies, hotels, flights, music, and sports. For RyanSpeech, we
randomly selected 3000 samples in all categories of Taskmaster-2.

  3. 3.

LibriTTS: To balance the dataset, we also included 2501 random samples from
the LibriTTS text corpus. LibriTTS is a multi-speaker speech dataset with
116500 sentences in their train-clean-360 subset. We randomly selected 2501
sentences from this subset.

###  3.2 Text pre-processing

After collecting texts from different sources, we underwent sentence
segmentation and normalization.

  1. 1.

Sentence segmentation: We used Spacy [17] for sentence segmentation on text
from the Ryan Chatbot dataset and Taskmaster-2. It uses a trainable pipeline
component for sentence segmentation.

  2. 2.

Text normalization: we detected non-standard words and normalized them to read
text manually. In this step, we address numbers (cardinal numbers, signed
integers, real numbers, ordinal numbers, roman numerals, fractions, and
sequence of digits like phone numbers), Currency, Time, Date, Abbreviations
and Street addresses.

###  3.3 Trim silence

After recording the audio files, we needed to trim the beginning and end
silences in each file. For this purpose, we wrote a simple script to trim the
audio based on thresholding the sound amplitude. To make it even more
accurate, we manually double-checked all the trimmings with Audacity audio
software to ensure that we correctly removed the silences.

###  3.4 Post processing

Even though all the audios had been recorded in a studio with high-quality, we
normalized the sound amplitude of recordings to ensure that they all had the
same volume level.

##  4 Statistics of the corpus

RyanSpeech contains 9.84 hours of high-quality audio recorded in a studio by a
professional voice actor. The speaker has the US English dialect. The sampling
rate of all audio files is 44,100 Hz. We used the audio coding format of FLAC,
which is lossless compression.

Figure 1 shows violin plots of the number of characters per sentence in
LibriTTS, LJSpeech, LibriSpeech, M-AILABS, and RyanSpeech. It can be seen from
the figure that RyanSpeech has relatively shorter sentences compared to other
corpora. The mean sentence length is 58.07 characters with a standard
deviation of 26.09, closest to LibriTTS. The distribution of sentence length
from RyanSpeech to M-AILABS is also significantly different. Our corpus
sentence lengths follow a peaked distribution, whereas in M-AILABS, it has a
larger standard deviation (std=50.80).

In Figure 2, the Box Plot of audio duration for different corpora are
demonstrated. Because our corpus is in the domain of conversation, the audio
files are shorter. The main reason for this observation is that the properties
of conversational recordings are usually shorter and faster than audiobooks
and other reading corpora. The diversity of audio length in terms of standard
deviation in RyanSpeech is less than other corpora, making it suitable for a
stable training corpus for assistant and chat dialogue systems.

Figure 1: The violin plots of the number of characters per sentence for
RyanSpeech, LibriTTS, LibriSpeech, LJSpeech and, M-AILABS. The thick line
shows the interquartile range (from 25% to 75%), and the white dot is the
median value. The width of the violin plot in any point indicates the
frequency. Figure 2: The box plots of audio length for RyanSpeech, LibriTTS,
LibriSpeech, LJSpeech and, M-AILABS. The box shows the interquartile range
(from 25% to 75%); the middle line shows the median value. The whole range is
from the minimum to maximum value. RyanSpeech has the shortest audio length
which is a characteristic of conversational dialogue.

##  5 Experiments

The training pipeline consists of two steps: first, the character level input
text is fed into a TTS system that outputs Mel spectrogram frames, then we use
a trained vocoder to convert the Mel spectrograms to waveforms.

###  5.1 Training neural vocoder

The speech synthesizer gives us the time-domain waveform samples from the Mel
spectrograms feature representations produced by the TTS system. In the first
step, we trained ParallelWaveGAN [18], which is a fast waveform generation
method that uses a non-autoregressive WaveNet model. It is based on Generative
Adversarial Network (GAN) and does not need a two-stage teacher-student
framework for training. Basically, the model is trained by optimizing multi-
resolution Short-time Fourier transform (STFT) on the spectrogram and an
adversarial loss function at the same time. For training the feature
extractor, the sampling rate is set to 22,050 Hz with the FFT window size of
1024. For the generator, we set the number of residual block layers to 30, the
number of dilation cycles to 3, and the number of residual channels to 64. The
number of layers for the discriminator network is set to 10. The loss
balancing coefficient is also set to 4. The model was trained for 400k steps.
Figure 3 shows one example of the generated sample after 400k steps versus the
ground truth. It illustrates that the generative network is capable of
creating a very similar waveform to the ground truth. The training was done on
a NVIDIA TITAN-Xp GPU with 12 GB of memory and a batch size of 6. The training
takes 95.74 hours.

Figure 3: The plot for the audio generated by the ParallelWaveGAN vocoder after training at 400k steps versus the ground truth. Table 2: The comparison between standard deviation ($\sigma$), skewness($\gamma$) and kurtosis ($\kappa$) of pitch in ground-truth and synthesized audio Model | $\sigma$ | $\gamma$ | $\kappa$
---|---|---|---
Ground Truth | 65.64 | -0.630 | 0.256
Tacotron | 63.97 | -0.677 | 0.302
FastSpeech | 68.61 | -0.532 | 0.165
FastSpeech2 | 67.71 | -0.565 | -0.023
Conformer | 65.77 | -0.703 | 0.119

###  5.2 Training text to speech model

We used different architecture for training the TTS system. Tacotron is a RNN-
based model that uses CBHG (1-D convolution bank + highway network +
bidirectional GRU) module [19]. We trained Tacotron for 100k steps.
Transformer-based models have been adopted in TTS systems due to their success
in modeling long-range dependencies. We trained FastSpeech [20] and
FastSpeech2 [21] which are based on transformer. FastSpeech is based on
attention mechanism and 1D convolution. These models have a module for length
regulation that addresses the problem of mismatch between the phoneme and
spectrogram sequence. The main difference between FastSpeech and FastSpeech2
is that the duration predictor for length regulation is trained using a
teacher model in FastSpeech, while in FastSpeech2 it is trained end to end,
which is faster and more accurate. Finally, we used Conformer [22] which
brings the ideas of convolutional networks to the transformer-based model. The
main difference between conformer and transformer models is that conformer has
two feedforward layers which sandwiches not only a multi-head self-attention
module but also a convolution module. The convolution module itself has a
pointwise convolution projecting into a glue activation layer, followed by 1-D
depthwise convolution. Finally, there is swish activation, another pointwise
convolution, and the batch norm. We trained FastSpeech, FastSpeech2, and
Conformer for 500k steps.

We used ESPNet-TTS [23], which is an open-source end-to-end TTS toolkit that
has implemented various state-of-the-art models. It provides recipes that are
inspired by the Kaldi ASR toolkit. The training for all models was done on one
NVIDIA TITAN Xp GPU with a batch size of 20. Table 3 depicts the comparison
between different models in terms of the number of epochs, training time, and
inference speed. Note that training time is only for the acoustic model, not
including vocoder training. Except for Tacotron, the other models have the
same inference speed. Conformer takes a longer time to train, but the quality
is superior to other models. All of our pretrained models are free to download
along with the original dataset.111code:https://github.com/roholazandie/ryan-
tts222corpus:http://mohammadmahoor.com/ryanspeech/ We refer the readers to
[23] for details of hyper-parameters and training configurations for each
model.

Table 3: The comparison of training epochs, training time and inference latency synthesis in different models. RTF (the real-time factor) denotes the time (in seconds) required for the system to synthesize one second waveform. The inference is done on NVIDIA GeForce GTX 1080 with 8GB and training on NVIDIA TITAN Xp GPU with 12GB |  Num epochs |  Training time (h) |  Inference speed (RTF)
---|---|---|---
Tacotron |  200 |  49.58 |  0.12331
FastSpeech |  1000 |  26.19 |  0.04349
FastSpeech2 |  1000 |  26.20 |  0.04355
Conformer |  1000 |  66.71 |  0.04516
Table 4: Comparison of MOS (mean opinion score) among our trained models with 95% confidence intervals. model |  MOS
---|---
Tacotron |  3.00 $\pm$ 0.18
FastSpeech |  3.27 $\pm$ 0.18
FastSpeech2 |  3.27 $\pm$ 0.18
Conformer |  3.36 $\pm$ 0.17

###  5.3 Results

We randomly selected 30 fixed text samples with various lengths from the test
corpus for evaluation. Different models all used the same text so that testers
only reviewed the audio quality without interference factors. We then used a
total of 120 audio samples generated by all four models for human evaluation.
For a fair comparison, all the systems used Parallel WaveGAN as a vocoder.
Twelve human subjects (native American English speakers, 18 years of age or
older) participated in this study. Each participant independently scored all
the models’ outputs that were randomly shuffled. The evaluation was based on
Likert scale score (1: Very Poor, 2:Poor, 3: Fair, 4: Good, 5: Excellent).

Quantitative subjective evaluation, known as mean opinion score (MOS), has
been used for results analysis [24]. MOS is simply the mean of the scores from
all evaluators. Table 4 demonstrates the difference between the evaluations on
each model. The best subjective result on our models is the conformer model.
FastSpeech and FastSpeech2 have the same MOS score in our evaluations. Due to
the effectiveness of variance information such as frequency and energy, as
well as increased accuracy for predictions of duration, FastSpeech models are
superior to Tacotron. However, the conformer model achieved the best score
(closer to good on Likert scale) because of the ability of the model to
harness the power of CNNs in order to extract local features in the creation
of the Mel-spectrogram. Our results are comparable to results from the same
models trained on LJSpeech [20].

To analyze the variance information, we calculated the first three moments of
pitch distribution for the ground truth and synthesized speech [25, 26]. Table
2 shows the results. Among different models, Conformer is closer to the ground
truth in terms of standard deviation, though Tacotron is closest with respect
to skewness and kurtosis. This demonstrates that Conformer can produce pitch
distribution close to natural voices, which results in better prosody.

##  6 Conclusions and Outlook

In this paper, we introduced the RyanSpeech corpus, which we designed
primarily for TTS systems. We collected the text dataset from a variety of
sources with a particular focus on conversational settings, and designed the
corpus to be high-quality in its studio settings. As we have demonstrated,
this is the first large, high-quality male speech corpus that is open-source.
RyanSpeech can open up a number of possibilities for future research in the
areas of ASR and TTS. With the trend in speech synthesis to find models that
replicate human speech with small datasets, RyanSpeech proves an excellent
candidate, especially for TTS evaluation and development in research as well
as applications that require natural sources of speech, such as movies and
podcasts.

##  7 Acknowledgements

Research reported in this paper was supported by the National Institute on
Aging of the National Institutes of Health under award number R44AG059483 to
DreamFace Tech, LLC.

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# Data-Driven Radio Environment Map Estimation Using Graph Neural Networks

Ali Shibli2, Tahar Zanouda {ali.a.shibli<EMAIL_ADDRESS>
Global AI Accelerator, Ericsson, Sweden  2KTH Royal Institute of Technology
<EMAIL_ADDRESS>
###### Abstract

Radio Environment Maps (REMs) are crucial for numerous applications in
Telecom. The construction of accurate Radio Environment Maps (REMs) has become
an important and challenging topic in recent decades. In this paper, we
present a method to estimate REMs using Graph Neural Networks. This approach
utilizes both physical cell information and sparse geo-located signal strength
measurements to estimate REMs. The method first divides and encodes mobile
network coverage areas into a graph. Then, it inputs sparse geo-located signal
strength measurements, characterized by Reference Signal Received Power (RSRP)
and Reference Signal Received Quality (RSRQ) metrics, into a Graph Neural
Network Model to estimate REMs. The proposed architecture inherits the
advantages of a Graph Neural Network to capture the spatial dependencies of
network-wide coverage in contrast with network Radio Access Network node
locations and spatial proximity of known measurements.

Extensive experiments on real-world data demonstrate that the proposed model
outperforms four other machine learning models, and shows its ability to
generalize across different geographical regions.

###### Index Terms:

Radio Environment Maps, Graph Neural Networks, AI for Telecom Networks.

##  I Introduction

Telecom networks are becoming increasingly complex and heterogeneous. Mobile
networks are constantly evolving, requiring network operators to continuously
introduce changes to network parameters and expand the network infrastructure.
As the complexity of the operating network increases, mobile operators face
new challenges to continuously assess the network and handle upgrade plans.
Traditionally, mobile operators used Radio Environment Maps (REMs) to estimate
network service in various coverage areas. REM refers to multi-domain radio
signal quality information (typically the received signal quality) in various
geographical areas. REMs are estimated using measurements taken from the
network. The data that characterizes the interaction between user equipment
(UE) and the network can be captured in several ways. For instance, Drive Test
is a method of measuring and assessing the coverage, capacity, and Quality of
Service (QoS) of a mobile radio network by collecting minimization of drive
test measurements, which are standardized by 3GPP [1]. These measurements
consist of UE information, field measurements, radio measurements, and
location information. Signal strength measurements are used to interpolate and
estimate radio maps. While the spatial preciseness and the high observation
frequency of signal strength measurements are advantageous, the sparsity of
sampling process of collecting radio measurements remains a challenge.

The accurate REM estimation is an important task for numerous applications [2]
in the telecom domain such as network configuration and parameters
optimization, spectrum estimation, coverage optimization, finding optimal
locations for new cells, and proactive resource management.

In this work, we refer to REMs of received Reference Signal Received Power
(RSRP) in 4G/5G networks. These REMs are constructed using sparse signal
strength measurements. Radio signal quality is estimated for areas and
locations that have not been measured. The aim of this paper is to construct
REMs using sparse measurements that were obtained while sampling the network
signal quality. Previous methods propose various techniques to address these
challenges. These methods, although powerful, require domain knowledge (e.g.,
calculate path loss) and are often computationally costly.

In light of these limitations, we propose a new method using Graph Neural
Networks. The main contributions of this work are summarized as follows:

  1. 1.

A suitable model architecture for REM estimation using Graph Convolution
Networks (GCNs) that is competitive with alternative solutions.

  2. 2.

Comparative analysis with state-of-the-art models: Providing a thorough
evaluation of the proposed GNN model against current machine learning models
such as XGBoost algorithm, known for its efficiency in structured/tabular data
[3], deep learning model based on Fully Connected Network (FCN), and TabNet
[4], a recent innovation in deep learning for tabular data, renowned for its
interpretability and performance.

  3. 3.

Comprehensive data integration and data fusion combining diverse data types,
including physical cell information, geo-located signal strength measurements,
and geographic information.

The paper is organized as follows. Section II reviews the state-of-the-art.
Data is discussed in section III. Section IV mathematically formulates the
problem, and outlines the proposed method. Section V provides details on
evaluation metrics, experimental results, and analysis are introduced in
Section VI and a conclusion to conclude the paper.

##  II Previous work

In recent years, there has been a growing body of research [5] [2] that
focuses on estimating Radio Environment Maps. Radio Environment Map estimation
techniques either relied on physical and statistical propagation models to
describe signal quality propagation properties (i.e., statistical approaches)
or employ a variety of deep learning algorithms (i.e., data-driven
approaches). Traditional and statistical approaches are criticized for relying
on domain knowledge and assumptions which can limit their applications to
real-world scenarios. Methods such [6] are based on ray tracing and stochastic
radio propagation model. Such models require a detailed mapping system that
emulates the target real-world environment. However, the built-up
infrastructure evolves over time and weather conditions change frequently
which makes the process of modeling the environment a time-consuming process.

Recently, machine learning approaches have become increasingly popular due to
the prevalence of data. Alimpertis et al. [7] used Random Forest to generate
predictive signal strength maps based on measurements and other information
such as device location.

Rufaida et al. [8] conducted a comparative analysis of several models: KNN,
SVM, and two decision tree-based models (XGBoost and LightGBM), and showed
that tree-based models outperform other models. The work was only applied to
wireless local area networks.

Anjinappa et al [9] proposed a data-driven unsupervised learning algorithm
named uniform manifold approximation and projection (UMAP) with a focus on
millimeter-wave networks. Millimeter waves have distinctive properties such
its sensitivity to physical obstacles such as buildings and trees. The
planning of millimeter-wave (mmWave) bands in 5G networks usually undergo
different processes.

Levie et al. [10] proposed a Deep Learning method to estimate path loss from a
transmitter location to any point in a flat domain. The results of the paper
showed that the path loss function was estimated in an urban environment with
high accuracy and low computational complexity.

Teganya et al. [11] proposed a Deep Learning architecture based on a fully
convolutional deep completion autoencoder to estimate radio maps. The
numerical experiments with two datasets showed that the method can attain good
RMSE.

Thrane et al. [12] presented a model-aided deep learning approach for path
loss prediction. In their work, they augmented radio data with rich and
unconventional information about the site, e.g. satellite photos, to provide
more accurate and flexible models. Similarly, authors [13] used top-view
geographical images for Radio propagation modeling.

Chaves-Villota et al. [14] proposed Deep Learning method, coined as DeepREM,
to estimate REMs in urban scenarios using combined U-Net and Conditional
Generative Adversarial Network (CGAN) DL architectures.

The approaches, although powerful, struggle to handle data sparsity and do not
incorporate the spatial proximity of coverage areas.

##  III Data

A telecom operational network consists of interconnected Radio nodes (e.g.,
gNB, eNB, etc) that provide coverage to end-users. In each coverage area, User
equipment (UE) communicates via Radio nodes. The coverage area can be divided
into tiles (hexagons) as shown in 1 where UEs receive a signal from the cell.
The interaction between UE and the Radio node can be captured in several ways.
With the rising ubiquity of network data, mobile operators can sample geo-
signal strength measurements to estimate signal quality. Geo-located signal
strength measurements can be sampled from different devices connecting with
the network to sense signal strength and network coverage. Sparse signal
strength measurements contains information about signal quality in some
regions in the city.

In our work, we use a dataset that spans 8 weeks and is collected from 2
cities. Each data point in the dataset encompasses a set of attributes such as
global cell identifier, signal strength measurement values, timestamp of
observation, and the longitude and latitude information of signal
measurements. The dataset does not contain any information about the devices
or users. The dataset covers two cities and includes +1000 deployed Radio
Nodes. Such datasets can be collected using Minimization of Drive Tests (MDT)
feature to assess performance of the network.

Figure 1: A figure showing a coverage area, divided in hexagons, where UEs
receive signal from cells

##  IV Method

This section details the model architectures and the method for generating
REM.

###  IV-A Problem Definition

Given a city $C$, with scattered signal measurements. This work aims to
utilize the collected measurements to predict signal quality in data-
insufficient tiles to facilitate the task of generating coverage maps.

###  IV-B Method Overview

We propose a Graph Neural Network (GNN)-based approach to estimate REMs. The
main steps are described in 1.

1: for each time interval $t=1,\ldots,T$ do

2: Obtain live network inventory data, for each Radio Node $\mathcal{C}_{t}$.

3: Obtain plurality of geo-located signal strength measurements $M$.

4: Map each geo-located signal strength measurements $M_{t}$ to a Radio Nodes
$\mathcal{C}_{t}$ using Global Cell ID.

5: Generate tiles $\mathcal{R}_{t}$ for a geographic area of interest using
hierarchical geospatial indexing system, H3 [15].

6: Determine adjacency matrix, by finding neighboring tiles for each tile.

7: Construct the graph. The city is represented as a temporal graph,
consisting of tiles $\mathcal{R}_{t}$ and edges $\mathcal{E}_{t}$ connecting
the tile

8: Prepare ML model features:

9: for each tile $\mathcal{R}_{t}$ in the city, do

10: \- Calculate distances and bearing values between geo-located signal
strength measurements $M_{t}$, that belong to the same tile, and the
associated Radio Node $\mathcal{C}_{t}$.

11: \- Find Radio Nodes $\mathcal{C}_{t}$ associated with each tile and
extract frequency configuration parameters.

12: \- Extract geospatial features for each tile $\mathcal{R}_{t}$.

13: \- Extract historical measurements characterized by Reference Signal
Received Power (RSRP) and Reference Signal Received Quality (RSRQ) metrics
associated with each tile.

14: end for

15: Preprocess ML model features: \- Apply one-hot encoding on categorical
variables. \- Normalize continuous values.

16: Train GNN model using the model features prepared in previous steps, and
the adjacency matrix extracted from the city graph.

17: Estimate RSRP and RSRQ measurements for each tile $\mathcal{R}_{t}$ in the
city using a trained GNN model, where the sparse measurements are missing.

18: end for

Algorithm 1 Main steps of the method.

###  IV-C Overview of the process

The pipeline consists of the following steps:

####  IV-C1 Data pre-processing

In the first step of the pipeline, during the pre-processing of the data, the
data is processed according to the required input format of the network
architecture. The process begins with the collection of diverse datasets:

  1. 1.

Sampled signal strength measurements.

  2. 2.

Network infrastructure data including Radio node location.

  3. 3.

Spatial data encompasses the geographical locations of network sites.

These datasets undergo feature engineering to extract and structure the data
into a format amenable for processing by the GNN. Two GNN models are
evaluated: (1) node regression model, where the GNN predicts a continuous RSRP
score, and (2) node classification, where it categorizes RSRP values into four
distinct classes— Very weak, Weak, Average, and Good according to Table I.
This dual approach allows for a comprehensive understanding of signal quality
across different dimensions, enhancing the accuracy and utility of the
coverage maps generated.

The datasets are then preprocessed as described in the following steps:

  1. 1.

Min-max scale for continuous values in each set to the range [0,1].

  2. 2.

One-hot encoding for categorical values.

RSRP (in dBm) | Classification Label
---|---
Less than -120 | Very weak
Between -120 and -106 | Weak
Between -105 and -90 | Average
Greater than -90 | Good
TABLE I: Classification of RSRP values.

####  IV-C2 Coverage Map Representation using a Temporal Graph

In this work, we partition the city into $K$ equally-sized regions (or tiles),
each represented in a hexagon shape. Each region is represented by a unique
geographic identifier corresponding to a region on the Earth. In our work, we
use the H3 index from Sahr et al. [15]. H3 is a framework comprising of a
global grid system that is suitable for analyzing large spatial datasets, by
partitioning areas of the Earth into identifiable grid tiles such as hexagons.
The resolution/hexagon size reflects the size of homogeneous hexagons used to
divide the earth. The choice of the hexagon size can be fine-tuned during the
training process. H3 divides regions into approximately equal areas regardless
of location on the planet. It is important to consider equal areas when
developing a solution at a global scale. Dividing geographical regions into
squares can result in biased results between northern or southern parts of the
globe and regions close to the equator. With a hierarchical hexagon grid, we
can control the granularity of the region for each use case and easily find
the right resolution for each use case. Each sample in the dataset (the signal
strength measurements) is mapped into the corresponding tile, represented by a
geographic identifier corresponding to a region on the Earth. The H3 method is
used to map (latitude, longitude) to spatial hexagons and generate the geo
index (geo hash). It is worth mentioning that there are other indexing systems
(e.g., S2) that divide geographic regions into rectangular squares. However,
H3 keeps both hexagon area and shape distortion to a minimum together,
compared to square and rectangular grids.

We treat each hexagon as a Region $R$. Each Region $R$ is characterized by a
set of features associated with a vector $X_{r}$. We structure the city as a
temporal graph, consisting of regions $\mathcal{R}$ and edges $\mathcal{E}$.
It is represented as follows:
$(\mathcal{G}_{\text{C}}^{t})_{t=1}^{T}=(\\{\mathcal{R}^{t},\mathcal{E}\\})_{t=1}^{T}$,
where $\mathcal{R}^{t}=\\{x_{1}^{t},\ldots,x_{k}^{t}\\}$ represents the vertex
features corresponding to the multivariate time series $(\mathbf{x}^{t})_{t}$
at time step $t$, and $(\mathcal{E})_{t}=\mathcal{E}$ is the static edge set
of the region graph at time step $t$. For each hexagon tile, we determine its
neighbors in the H3 grid system. The number of neighboring levels considered
can be a hyper-parameter. For instance, we benchmarked k-level graphs (kth
outer level circle of neighbors) as well.

Figure 2: Representing Coverage Areas using H3 Hexagons

####  IV-C3 Feature Engineering

Once the measurements are grouped by each cell in the network, we perform the
following data processing tasks to prepare a set of features for the model:

  * •

(1) Cell Parameters encompassing attributes like Channelbandwidth (data
transmission capacity) and Earfcndl (central downlink frequency channel
number), Radio Access Technology (4G/5G).

  * •

(2) Historical geo-located signal strength measurements including RSRQ
(Reference Signal Received Quality), RSRP (Reference Signal Received Power),
RSSI (Received Signal Strength Indicator), and SINR/SNR (Signal-to-noise ratio
of the given signal).

    * –

(2.1) Index measurement location: we use the H3 method [15] to index
measurements (measurement latitude, measurement longitude) to spatial hexagon
and generate geo index (geo hash).

    * –

(2.2) Calculate the Bearing azimuth between the measurement location and cell
location. Bearing azimuth, in this context, refers to the angular direction of
the signal measurement’s location corresponding to the cell location. The
values can vary between 0 and 360.

    * –

(2.3) Calculate the geographic distance between the sample location and cell
location.

    * –

(2.4) Extract historical signal strength values such as RSRQ, RSRP, RSSI, and
SINR/SNR.

    * –

(2.5) Extract statistics on device brand usage.

  * •

(3) Spatial context values displaying geographical attributes like terrain
type, and functional area type.

####  IV-C4 Coverage Map Estimation Model using Graph Neural Networks

GNNs [16] are a class of machine learning models designed for learning and
reasoning over graph-structured data. The network architecture for this task
is Graph Conventional Networks (GCNs) [17]. The idea behind such models is to
aggregate neighboring nodes’ content features while taking graph structures
(edges/topology) into consideration. In our work, we represent a geographical
region (e.g., city) using a set of tiles. Each tile is seen as a graph node.
Figure 3 depicts how coverage areas are represented as a graph. Links between
coverage areas exist when the network nodes are geographically located in
close proximity. Then, we use Graph Conventional Networks [17] to predict
signal strength values characterized by RSRQ and RSRP.

The model was trained in an unsupervised manner. The labeled samples from the
dataset were used in the evaluation phase to quantify the performance of the
prediction.

Figure 3: REMs estimation using GCN model.

Regression Task: Given a temporal graph
$(\mathcal{G}_{\text{C}}^{t})_{t}$=$(\mathcal{V}^{t},\mathcal{E}^{t})_{t}$ as
input with a time length of $T$, we aim to predict:

$(\mathbf{y}^{t})_{t=1}^{T}=\\{y_{1}^{t},\ldots,y_{K}^{t}\\}_{t=1}^{T}$

where $y_{k}^{t}$ denotes the predicted continuous value of signal quality.

Classification Task: Given a temporal graph
$(\mathcal{G}_{\text{C}}^{t})_{t}$=$(\mathcal{V}^{t},\mathcal{E}^{t})_{t}$ as
input with a time length of $T$, we aim to predict:

$(\mathbf{y}^{t})_{t=1}^{T}=\\{y_{1}^{t},\ldots,y_{K}^{t}\\}_{t=1}^{T}$

where $y_{k}^{t}\in\\{0,1,3,4\\}$ denotes the quality of signal based on Table
I.

##  V Experiments

We compare our proposed method against several state-of-the-art models for
multivariate time-series prediction and classification. The models are trained
on the same features described in section IV:

  * •

Model 1: XGBoost [3]: an implementation of gradient-boosted decision trees
designed for speed and performance, XGBoost is a powerful approach for
regression and classification problems on tabular data.

  * •

Model 2: FCN: fully connected neural network with 3 hidden layers of
dimensions 256-512-128.

  * •

Model 3: TabNet [4]: Attention-based neural network approach for tabular data,
TabNet uses sequential attention to choose which features to process at each
decision step, making it highly interpretable and capable of handling both
sparse and dense features, as in our case of RSRP signal quality.

  * •

Model 4: GCN [17]: Graph Convolutional Network. Since the sites in the network
acquire topological attributes and are interconnected, the application of
graph theory via GCNs becomes critical. These networks extend the
convolutional concept to graph-structured data, allowing the model to learn a
representation of each site by considering not just its own attributes but
also those of its neighbors. Such information is aggregated via the signal
measurements as well as the geospatial data. This is particularly useful for
RSRP KPI prediction, as the signal quality at one site may be influenced by
the signal qualities of neighboring sites. This leads to more accurate and
robust models for network coverage estimation.

###  V-A Evaluation Metrics

To assess the performance of our models, we divide our dataset into 20% for
testing and 80% for training. We use well-established metrics: the $R^{2}$
score for regression tasks, and accuracy for classification tasks. The
accuracy metric is an indicator of the model’s overall classification
performance, representing the ratio of correctly predicted observations to the
total observations.

$Accuracy=\frac{TP+TN}{TP+TN+FP+FN}$ (1)

where $tp$ stands for true positive, $fp$ stands for false positive, and $fn$
stands for false negative.

Our choice of loss functions is aligned with these metrics: cross-entropy for
classification models and root-mean-squared error for regression models.

To ensure a comprehensive evaluation, the results, including accuracy and
$R^{2}$ scores, are averaged across all H3 index areas. This averaging process
provides a more robust and generalizable measure of model performance, taking
into account the variability and specific characteristics of different
geographical regions covered by each H3 index area.

###  V-B Regression Task

In the regression task, the performance of the models was evaluated using the
R2 score. As presented in Table II, the GCN model outperformed the other
models with an R2 score of 0.83, indicating a high level of predictive
accuracy and suggesting that the GCN’s structure is well-suited to capturing
the topological and relational data inherent in the network’s dataset. The FCN
also showed strong performance with a score of 0.77, followed closely by
TabNet with 0.74, which highlights the efficiency of attention mechanisms in
tabular data prediction tasks. XGBoost, with a score of 0.62, lagged behind
the neural network-based models, potentially due to its comparatively lower
capacity to model complex non-linear interactions in the data. These results
demonstrate the varying levels of efficacy of state-of-the-art models in
handling the intricacies of regression tasks within multivariate time-series
data.

Model | R2
---|---
XGBoost | 0.62
FCN | 0.77
TabNet | 0.74
GCN | 0.83
TABLE II: The numerical results of the evaluated models (XGBoost, FCN, TabNet,
GCN) for the regression task using the R2 score.

###  V-C Classification Task

For the classification task, we assess the accuracy of the same four models.
Table III shows the accuracy scores, where the GCN again demonstrates superior
performance with an accuracy of 92%. This suggests that the GCN’s ability to
leverage the relational structure of the data is particularly advantageous in
classification tasks, allowing it to achieve a high degree of precision. Both
the FCN and TabNet report a commendable accuracy of 74%, indicating that these
models are also effective at capturing the relevant patterns within the data
for classification purposes. XGBoost, while still achieving a moderate level
of accuracy at 63%, appears less adept at this task compared to the more
complex neural network-based approaches. These results underscore the
importance of choosing a model that aligns well with the nature of the data
and the specific requirements of the classification task at hand.

Model | Accuracy
---|---
XGBoost | 0.63
FCN | 0.74
TabNet | 0.74
GCN | 0.92
TABLE III: The numerical results of the evaluated models (XGBoost, FCN,
TabNet, GCN) for the classification task using the Accuracy score.

##  VI Conclusion

In this paper, we have presented an approach to impute and predict signal
quality in geographical areas, which is fundamental to the creation of
accurate and reliable Radio Environment Maps. The method provides an approach
based on Graph Neural Networks for predicting signal quality maps in telecom
networks using sparse signal strength measurements. The performance of GNN-
based models showed better results (in both classification and regression
tasks) as compared to models presented in other works (XGBoost, MLP, and
TabNet). GNNs are capable of capturing the complex spatial relationships
inherent in network topologies, thereby providing a more accurate
representation of the inter-dependencies between network nodes. This approach
not only enhances the accuracy of signal quality predictions but also offers
insights into the spatial dynamics of network coverage.

To further improve the capabilities and applicability of GNN models for REMs,
future work should prioritize the following tasks:

\- Investigating multimodal deep learning architectures. Our initial
experiments using satellite imagery did not show any improvement. Many recent
works showed promising results when incorporating environmental data. This may
involve exploring the correlation between physical infrastructure and signal
quality, enhancing the model’s ability to predict network performance in
varying geographic and urban landscapes.

\- Focusing on the refinement and optimization of the GCN model, enhancing its
predictive accuracy and efficiency. This may involve exploring various neural
network architectures, advanced training techniques, and hyperparameter
optimization.

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# Nonlinear contraction in $b$-suprametric spaces

Maher Berzig  Maher Berzig, Université de Tunis, École Nationale Supérieure
d’Ingénieurs de Tunis, Département de Mathématiques, 5 avenue Taha Hussein
Montfleury, 1008 Tunis, Tunisie<EMAIL_ADDRESS>
###### Abstract.

We introduce the concept of $b$-suprametric spaces and establish a fixed point
result for mappings satisfying a nonlinear contraction in such spaces. The
obtained result generalizes a fixed point theorem of Czerwik and a recent
result of the author.

###### Key words and phrases:

$b$-suprametric space, fixed point theorem, contraction-type mapping.

###### 2010 Mathematics Subject Classification:

54A05, 47H10, 47H09.

## 1\. Introduction

The triangle inequality is one of the most important and useful inequalities
in analysis. However, the full power of the triangle inequality is not
necessarily essential for certain applications, such as in the measures used
for the distance between shapes [12], the problems of traveling salesman or in
the nearest neighbor search [15], and even for the study of denotational
semantics of data in program verification [23]. There are currently several
generalizations of the triangle inequality satisfied by various distance
functions, such as those of the partial metric [23], the $b$-metric [11] or
the rectangular metric [8].
The partial metric was introduced by Matthews [23] by modifying the triangular
inequality to handle non-zero self-distance, which allows him to extend the
Banach contraction principle and to develop some applications in programming
language theory. Further investigations have since been carried out to explore
the usefulness of this distance function, see for instance [10, 16, 17, 20].
A relaxed triangular inequality, via a real coefficient $b$ greater or equal
to one, was used by Czerwik [11] and the space equipped with a distance
satisfying such an inequality called it $b$-metric space. Furthermore, he
established a fixed point theorem in the context of $b$-metric spaces. Due to
the importance of this distance function, it has been extended in various ways
and adapted to a wide range of applications, see e.g. [2, 6, 7, 21].
In [8], Branciari replaced the triangle inequality of the distance function
with a rectangular one, which is also called a quadrilateral inequality. The
rectangular distance between two points is less than or equal to the sum of
their distances to any two distinct points and the distance between them.
Currently, there are many fixed point results developed in spaces equipped
with distance functions satisfying rectangular type inequalities, we refer to
the non-exhaustive list of references [3, 9, 29, 26].
In addition, and in order to cope with the increasing complexity of real-world
applications, many new distance functions appear by merging, relaxing or
extending some of the axioms of the existing distance functions. Moreover,
several fixed point results have been developed in sets equipped with these
distances, see for instance the results obtained in the context of the partial
rectangular metric spaces [28], $b$-rectangular metric spaces [27], extended-
Branciari $b$-distance spaces [1], quasi-partial $b$-metric space [13] and
extended $b$-metric spaces [32].

Recently, a new relaxed triangular inequality is used in the so-called
suprametric, which was introduced by the author in [5]. In particular, the
Banach contraction principle is proven in suprametric spaces, along with other
results in ordered vector spaces. Some nonlinear integral and matrix equations
have been studied in this context. The suprametric was used by Panda et al.
[24] to analyze the stability and the existence of equilibrium points of some
fractional-order complex-valued neural networks. Additionally, the suprametric
space was extended by Panda et al. [25] and a study of the existence of a
solution of Ito-Doob type stochastic integral equations was proven via this
extension. However, it is well known that the space of integrable functions
$L_{1}[0,1]$ is a $b$-metric space (see for instance [4]), but it is not known
whether the $b$-metric is a suprametric or an extended suprametric. In order
to study the existence of positive solutions to certain nonlinear equations in
this space, it will be useful to develop a context including the suprametric
and the $b$-metric spaces.
In this paper, we explore an important step toward this task by unifying these
two concepts and introduce the $b$-suprametric spaces, which merge the
advantages of both the $b$-metric and the suprametric spaces. Then, we
establish a fixed point theorem for mappings satisfying nonlinear contractions
via Matkowski functions.

## 2\. Preliminaries

Throughout this paper, we respectively denote by $\mathbb{R}$,
$\mathbb{R}_{+}$ and $\mathbb{N}$ the sets of all real numbers, all
nonnegative real numbers and all nonnegative natural numbers.

######  Definition 2.1.

Let $X$ be a nonempty set. A function $d\colon X\times X\to\mathbb{R}_{+}$ is
called semimetric if for all $x,y\in X$ the following properties hold:

  1. (d1)

$d(x,y)=0$ if and only if $x=y$,

  2. (d2)

$d(x,y)=d(y,x)$.

A semimetric space is a pair $(X,d)$ where $X$ is a nonempty set and $d$ is a
semimetric.

Recall now the concept of $b$-metric spaces [11].

######  Definition 2.2.

Let $(X,d)$ be a semimetric space and $b\geq 1$. The function $d$ is called
$b$-metric if it satisfies the following additional property:

  1. (d3)

$d(x,y)\leq b\,(d(x,z)+d(z,y))$ for all $x,y,z\in X$.

A $b$-metric space is a pair $(X,d)$ where $X$ is a nonempty set and $d$ is a
$b$-metric.

For more information on the topology of the $b$-metric spaces, we refer the
reader to [30]. Let us recall a few examples of such spaces.

######  Example 2.1.

Let $(X,d)=(\ell_{p}(\mathbb{R}),d_{\ell_{p}})$ with $0<p<1$ where

$\ell_{p}(\mathbb{R})\coloneqq\big{\\{}\\{x_{n}\\}\subset\mathbb{R}\text{ such
that }\textstyle\sum\limits_{n=1}^{\infty}|x_{n}|^{p}<\infty\big{\\}},$

and $d_{\ell_{p}}\colon X\times X\to\mathbb{R}_{+}$ is given by

$d_{\ell_{p}}(\\{x_{n}\\},\\{y_{n}\\})=\left(\textstyle\sum\limits_{n=1}^{\infty}|x_{n}-y_{n}|^{p}\right)^{\textstyle\frac{1}{p}}\text{
for all }\\{x_{n}\\},\\{y_{n}\\}\in X.$

Then $(X,d_{\ell_{p}})$ is a $b$-metric space with $b=2^{\frac{1}{p}}$.

######  Example 2.2.

Let $(X,d)=(L_{p}[0,1],d_{L_{p}})$ with $0<p<1$ where

$L_{p}[0,1]\coloneqq\left\\{x\colon[0,1]\to\mathbb{R}\text{ such that
}\int_{0}^{1}|x(t)|^{p}<\infty\right\\},$

and $d_{L_{p}}\colon X\times X\to\mathbb{R}_{+}$ is given by

$d_{L_{p}}(x,y)=\left(\int_{0}^{1}|x(t)-y(t)|^{p}\right)^{\textstyle\frac{1}{p}}\text{
for all }x,y\in X.$

Then $(X,d_{L_{p}})$ is a $b$-metric space with $b=2^{\frac{1}{p}}$.

In the sequel, we denote by $\mathbb{M}$ the set of Matkowski functions [22],
$\psi\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ that satisfy

  1. (i)

$\psi$ is increasing on $\mathbb{R}_{+}$,

  2. (ii)

$\lim\limits_{n\to\infty}\psi^{n}(t)=0$ for all $t>0$.

Denote by $\mathbb{M}_{b}$ $(b\geq 1)$ the set of functions of $\mathbb{M}$
that satisfy

$\limsup_{n\to\infty}\textstyle\frac{\psi^{n+1}(t)}{\psi^{n}(t)}<\frac{1}{b},\;\text{
for all }t>0,$ (1)

######  Remark 2.1.

It is not difficult to see that if $\psi\in\mathbb{M}$, then $\psi(t)<t$ for
all $t>0$, $\psi$ is continuous at $0$ and $\psi(0)=0$.

Recall next the result of Czerwik [11], see also Kajántó and Lukács [18].

######  Theorem 2.1.

Let $(X,d)$ be a complete $b$-metric space and $f\colon X\to X$ be a mapping.
Assume there exists $\psi\in\mathbb{M}$ such that $f$ is a $\psi$-contraction,
that is,

$d(fx,fy)\leq\psi(d(x,y))\;\text{ for all }x,y\in X.$

Then, $f$ has a unique fixed point and $\\{f^{n}x_{0}\\}_{n\in\mathbb{N}}$
converges to it for every $x_{0}\in X$.

The suprametric space introduced in [5] as follows.

######  Definition 2.3.

Let $(X,d)$ be a semimetric space and $\rho\in\mathbb{R}_{+}$. The function
$d$ is called suprametric if it satisfies the following additional property:

  1. (d3)

$d(x,y)\leq d(x,z)+d(z,y)+\rho\,d(x,z)d(z,y)$ for all $x,y,z\in X$.

A suprametric space is a pair $(X,d)$ where $X$ is a nonempty set and $d$ is a
suprametric.

######  Example 2.3.

Let $a>0$, $b\geq 1$, $d_{m}$ be a metric on $X$ and $d^{(a,b)}\colon X\times
X\to\mathbb{R}_{+}$ be a function given by

$d^{(a,b)}(x,y)=d_{m}(x,y)(a\,d_{m}(x,y)+b)\text{ for all }x,y\in X.$

Then $(X,d^{(a,b)})$ is a suprametric with $\rho=\frac{2a}{b}$.

######  Example 2.4.

Let $\beta\geq 1$, $d_{m}$ be a metric on $X$ and $d^{(\beta)}_{1}\colon
X\times X\to\mathbb{R}_{+}$ be a function given by

$d^{(\beta)}_{1}(x,y)=(e^{-\beta d_{m}(x,y)^{2}}-1)\text{ for all }x,y\in X.$

Then $(X,d^{(\beta)}_{1})$ is a suprametric with $\rho=1$.

######  Example 2.5.

Let $\gamma>0$, $d_{m}$ be a metric on $X$ and $d^{(\gamma)}_{2}\colon X\times
X\to\mathbb{R}_{+}$ be a function given by

$d^{(\gamma)}_{2}(x,y)=\gamma\,(e^{d_{m}(x,y)}-1)\text{ for all }x,y\in X.$

Then $(X,d^{(\gamma)}_{2})$ is a suprametric with $\rho=\frac{1}{\gamma}$.

######  Remark 2.2.

Let $X=\mathbb{R}$ and $d_{m}(x,y)=|x-y|$ for all $x,y\in X$. If $d$ is equal
to $d^{(1,1)}$, $d^{(1)}_{1}$ or $d^{(1)}_{2}$, we get $d(0,1)+d(1,2)<d(0,2)$,
which proves that there are not classical metrics.

Recall now a fixed point result from [5].

######  Theorem 2.2.

Let $(X,d)$ be a complete suprametric space and $f\colon X\to X$ be a mapping.
Assume there exists $c\in[0,1)$ such that

$d(fx,fy)\leq c\,d(x,y)\;\text{ for all }x,y\in X.$

Then, $f$ has a unique fixed point and $\\{f^{n}x_{0}\\}_{n\in\mathbb{N}}$
converges to it for every $x_{0}\in X$.

We next introduce the concept of $b$-suprametric spaces.

######  Definition 2.4.

Let $(X,d)$ be a semimetric space, $b\geq 1$ and $\rho\geq 0$. The function
$d$ is called $b$-suprametric if it satisfies the following additional
property:

  1. (d3)

$d(x,y)\leq b\,(d(x,z)+d(z,y))+\rho d(x,z)d(z,y)$ for all $x,y,z\in X$.

A $b$-suprametric space is a pair $(X,d)$ where $X$ is a nonempty set and $d$
is a $b$-suprametric.

######  Example 2.6.

Let $d\colon X\times X\to\mathbb{R}_{+}$ be given by

$d(x,y)=d^{(1,2)}(x,y)(d^{(1,2)}(x,y)+1)\text{ for all }x,y\in X.$

Then $(X,d)$ is a $b$-suprametric with $(b,\rho)=(1,1)$.

######  Example 2.7.

Let $d\colon X\times X\to\mathbb{R}_{+}$ be given by

$d(x,y)=e^{-d^{(1)}_{1}(x,y)^{2}}-1\text{ for all }x,y\in X.$

Then $(X,d)$ is a $b$-suprametric with $(b,\rho)=(1,1)$.

######  Example 2.8.

Let $d\colon X\times X\to\mathbb{R}_{+}$ be given by

$d(x,y)=d_{\ell_{p}}(x,y)(d_{\ell_{p}}(x,y)+1)\text{ for all }x,y\in X.$

Then $(X,d)$ is a $b$-suprametric with
$(b,\rho)=(4^{\frac{1}{p}},8^{\frac{1}{p}})$.

######  Example 2.9.

Let $d\colon X\times X\to\mathbb{R}_{+}$ be given by

$d(x,y)=d_{L_{p}}(x,y)(d_{L_{p}}(x,y)+1)\text{ for all }x,y\in X.$

Then $(X,d)$ is a $b$-suprametric with
$(b,\rho)=(4^{\frac{1}{p}},8^{\frac{1}{p}})$.

Inspired from [30, Example 3.9], we consider a non-trivial structure and endow
it with a $b$-suprametric.

######  Example 2.10.

Let $X=\\{0,1,\frac{1}{2},\ldots,\frac{1}{n},\ldots\\}$ and

$d(x,y)=\left\\{\begin{array}[]{ll}0&\text{ if }x=y\\\ \frac{1}{5}&\text{ if
}x\neq y\text{ and }x,y\in\\{0,1\\}\\\ 1-e^{-|x-y|}&\text{ if }x\neq y\text{
and }x,y\in\\{0\\}\cup\\{\frac{1}{2n},n=1,2,\ldots\\}\\\ \frac{1}{4}&\text{
otherwise}.\end{array}\right.$

Then

  1. (i)

$d$ is a $b$-suprametric space on $X$ with $b=\frac{3}{2}$ and $\rho=7$.

  2. (ii)

$d$ is not continuous in each variable.

  3. (iii)

$B(1,\frac{1}{4})\coloneqq\big{\\{}x\in X:d(1,x)<\frac{9}{40}\big{\\}}$ is not
open.

For the proof of the example, we need the following lemma.

######  Lemma 2.1.

We have:

  1. (i)

$e^{-|s-t|}-1\leq e^{-s}-e^{-t}$ for all $s,t>0$.

  2. (ii)

$e^{-s}+e^{-(t+s)}\leq e^{-t}+1$ for all $s,t>0$.

  3. (iii)

$e^{-s}+e^{-t}\leq e^{-|s-t|}+1$ for all $s,t>0$.

In particular, for all $b\geq 1$ we have

  1. (i’)

$b(2-e^{-t}-e^{-|s-t|})\geq 1-e^{-s}$ for all $s,t>0$.

  2. (ii’)

$b(2-e^{-s}-e^{-(t+s)})\geq 1-e^{-t}$ for all $s,t>0$.

  3. (iii’)

$b(2-e^{-t}-e^{-s})\geq 1-e^{-|s-t|}$ for all $s,t>0$.

###### Proof.

(i): The case $s=t$ is trivial. Assume now that $t>s$, then by using that
$t\mapsto\frac{1+e^{-t}}{1+e^{t}}$ is strictly decreasing for $t>0$, we obtain
$\frac{1+e^{-s}}{1+e^{s}}>\frac{1+e^{-t}}{1+e^{t}}=e^{-t}$, thus
$e^{-|s-t|}-1=e^{s-t}-1<e^{-s}-e^{-t}$. Assume next that $s>t$, so by using
that $t\mapsto\frac{1-e^{-t}}{e^{t}-1}$ is strictly deceasing for $t>0$, we
deduce that $\frac{1-e^{-t}}{e^{t}-1}>\frac{1-e^{-s}}{e^{s}-1}=e^{-s}$, so
$e^{-|s-t|}-1=e^{t-s}-1<e^{-s}-e^{-t}$.

(ii): Obvious.

(iii): The case $s=t$ is trivial. Assume that $t>s$. Then from
$-e^{-s}=\frac{e^{-s}-1}{e^{s}-1}<e^{-t}$, we obtain
$e^{-|s-t|}+1=e^{s-t}+1>e^{-s}+e^{-t}$. Similarly, if we assume that $s>t$, we
deduce from $-e^{-t}=\frac{e^{-t}-1}{e^{t}-1}<e^{-s}$ that
$e^{-|s-t|}+1=e^{t-s}+1>e^{-s}+e^{-t}$. ∎

###### Proof of Example 2.10.

(i): Note that for all $x,y\in X$, $d(x,y)\geq 0$, $d(x,y)=0$ if and only if
$x=y$. We next discuss several cases according to different values of
$x,y,z(z\notin\\{x,y\\})$. Denote by $\delta(x,z,y)\coloneqq
b(d(x,z)+d(z,y))+\rho\,d(x,z)d(z,y)$, where $b=\frac{3}{2}$ and $\rho=7$.

• If $d(x,y)=d(0,1)=\frac{1}{5}$, then
$\delta(x,z,y)=\frac{29}{8}-\frac{13}{4}e^{-\frac{1}{2n}}$ if $z=\frac{1}{2n}$
and $\frac{19}{16}$ otherwise.

• If $d(x,y)=d(0,\frac{1}{2n})=1-e^{-\frac{1}{2n}}$, then

$\delta(x,z,y)=\left\\{\begin{array}[]{ll}\frac{3}{2}(2-e^{-\frac{1}{2m}}-e^{-|\frac{1}{2m}-\frac{1}{2n}|})+4(1-e^{-\frac{1}{2m}})(1-e^{-|\frac{1}{2m}-\frac{1}{2n}|})&\text{
if }z=\frac{1}{2m}\\\ \frac{41}{40}&\text{ if }z=1\\\ \frac{19}{16}&\text{
otherwise}.\end{array}\right.$

• If $d(x,y)=d(0,\frac{1}{2n+1})=\frac{1}{4}$ $(n\neq 0)$, then

$\delta(x,z,y)=\left\\{\begin{array}[]{ll}\frac{29}{8}-\frac{13}{4}e^{-\frac{1}{2m}}&\text{
if }z=\frac{1}{2m}\\\ \frac{41}{40}&\text{ if }z=1\\\ \frac{19}{16}&\text{
otherwise}.\end{array}\right.$

• If $d(x,y)=d(\frac{1}{2n},1)=\frac{1}{4}$, then

$\delta(x,z,y)=\left\\{\begin{array}[]{ll}\frac{29}{8}-\frac{13}{4}e^{-|\frac{1}{2n}-\frac{1}{2m}|}&\text{
if }z=\frac{1}{2m}\\\ \frac{16}{5}-\frac{29}{10}e^{-\frac{1}{2n}}&\text{ if
}z=0\\\ \frac{19}{16}&\text{ otherwise}.\end{array}\right.$

• If $d(x,y)=d(\frac{1}{2n+1},1)=\frac{1}{4}$ $(n\neq 0)$, then
$\delta(x,z,y)=\frac{41}{40}$ if $z=0$ and $\frac{19}{16}$ otherwise.

• If $d(x,y)=d(\frac{1}{2m},\frac{1}{2n})=1-e^{-|\frac{1}{2m}-\frac{1}{2n}|}$,
then

$\delta(x,z,y)=\left\\{\begin{array}[]{ll}\frac{3}{2}(2-e^{-|\frac{1}{2m}-\frac{1}{2k}|}-e^{-|\frac{1}{2k}-\frac{1}{2n}|})+4(1-e^{-|\frac{1}{2m}-\frac{1}{2k}|})(1-e^{-|\frac{1}{2k}-\frac{1}{2n}|})&\text{
if }z=\frac{1}{2k}\\\
\frac{3}{2}(2-e^{-\frac{1}{2m}}-e^{-\frac{1}{2n}})+4(1-e^{-\frac{1}{2m}})(1-e^{-\frac{1}{2n}})&\text{
if }z=0\\\ \frac{19}{16}&\text{ otherwise}.\end{array}\right.$

• If $d(x,y)=d(\frac{1}{2m},\frac{1}{2n+1})=\frac{1}{4}$ $(n\neq 0)$, then

$\delta(x,z,y)=\left\\{\begin{array}[]{ll}\frac{29}{8}-\frac{13}{4}e^{-|\frac{1}{2m}-\frac{1}{2k}|}&\text{
if }z=\frac{1}{2k}\\\ \frac{29}{8}-\frac{13}{4}e^{-\frac{1}{2m}}&\text{ if
}z=0\\\ \frac{19}{16}&\text{ otherwise}.\end{array}\right.$

• If $d(x,y)=d(\frac{1}{2m+1},\frac{1}{2n+1})=\frac{1}{4}$, then
$\delta(x,z,y)=\frac{19}{16}$ for all $z\in X$.

We conclude by using Lemma 2.1 that

$d(x,y)\leq b(d(x,z)+d(z,y))+\rho d(x,z)+d(z,y),$

where $b=\frac{3}{2}$ and $\rho=7$.

(ii): We have

$\lim_{n\to\infty}d(0,\textstyle\frac{1}{2n})=\lim\limits_{n\to\infty}(1-e^{-\frac{1}{2n}})=0.$

However, we also have

$\lim_{n\to\infty}d(1,\textstyle\frac{1}{2n})=\textstyle\frac{1}{4}\neq\frac{1}{5}=d(1,0).$

This proves that $d$ is not continuous in each variable.

(iii): Observe that $B(1,\frac{9}{40})=\\{0,1\\}$. Since
$d(0,\textstyle\frac{1}{2n})=1-e^{-\frac{1}{2n}}$, then for each $r>0$,
$\frac{1}{2n}\in B(0,r)$ for a sufficiently large $n$ and since
$\frac{1}{2n}\not\in B(1,\frac{9}{40})$, $B(0,r)\not\subset
B(1,\frac{9}{40})$, which proves that $B(1,\frac{9}{40})$ is not open. ∎

The following definition will be important in what follows.

######  Definition 2.5.

Let $(X,d)$ be a $b$-suprametric space.

  1. (i)

The sequence $\\{x_{n}\\}_{n\in\mathbb{N}}$ converges to $x\in X$ if and only
if $\lim\limits_{n\to\infty}d(x_{n},x)=0$.

  2. (ii)

The sequence $\\{x_{n}\\}_{n\in\mathbb{N}}$ is Cauchy if and only if
$\lim\limits_{n,m\to\infty}d(x_{n},x_{m})=0$.

  3. (iii)

$(X,d)$ is complete if and only if any Cauchy sequence in $X$ is convergent.

## 3\. The main result

The main result is the following theorem.

######  Theorem 3.1.

Let $(X,d)$ be a complete $b$-suprametric space and $f\colon X\to X$ be a
mapping. Assume there exists $\psi\in\mathbb{M}$ such that

$d(fx,fy)\leq\psi(d(x,y))\;\text{ for all }x,y\in X.$ (2)

Then, $f$ has a unique fixed point and $\\{f^{n}x_{0}\\}_{n\in\mathbb{N}}$
converges to it for every $x_{0}\in X$.

###### Proof.

Define $x_{n}=f^{n}x_{0}$ for all $n\in\mathbb{N}$, where $x_{0}$ is an
arbitrary point of $X$. For the sake of simplicity, we will use the following
notations:

$d_{i,j}\coloneqq d(x_{i},x_{j})\text{ and }d_{i}\coloneqq d_{i,i+1}\;\text{
with }i,j\in\mathbb{N}.$

We now apply (2), $mn$ times, where $m,n\in\mathbb{N}$, we obtain

$d_{(m+1)n,mn}\leq\psi^{mn}(d_{n,0}),$

which implies

$\lim_{m\to\infty}d_{(m+1)n,mn}=0\,\text{ for all }n\in\mathbb{N}.$ (3)

Next, we shall show that $\\{x_{n}\\}$ is a Cauchy sequence. Let
$\varepsilon>0$ and take $q\in\mathbb{N}$ ($q>1$) such that
$\psi^{q}(\varepsilon)<\frac{\varepsilon}{b+\sqrt{b^{2}+\rho\varepsilon}}$. By
using (3), one can choose $p\in\mathbb{N}$ such that

$d_{(m+1)q,mq}<\textstyle\frac{\varepsilon}{b+\sqrt{b^{2}+\rho\varepsilon}}\;\text{
for all }m\geq p.$ (4)

Hence, for all $z\in B(x_{pq},\varepsilon)\coloneqq\\{x\in
X:d(x_{pq},x)<\varepsilon\\}$, we have

$d(f^{q}z,f^{q}x_{pq})\leq\psi^{q}(d(z,x_{pq}))\leq\psi^{q}(\varepsilon)<\textstyle\frac{\varepsilon}{b+\sqrt{b^{2}+\rho\varepsilon}},$
(5)

and since from (4), we have

$d_{(p+1)q,pq}<\textstyle\frac{\varepsilon}{b+\sqrt{b^{2}+\rho\varepsilon}},$
(6)

then by Definition (d3), we obtain

$\displaystyle d(f^{q}z,x_{pq})$ $\displaystyle\leq
b\left(d(f^{q}z,f^{q}x_{pq})+d_{(p+1)q,pq}\right)+\rho
d(f^{q}z,f^{q}x_{pq})\,d_{(p+1)q,pq}$
$\displaystyle<2b\textstyle\frac{\varepsilon}{{b+\sqrt{b^{2}+\rho\varepsilon}}^{\phantom{2}}}+\rho\textstyle\frac{\varepsilon^{2}}{\left(b+\sqrt{b^{2}+\rho\varepsilon}\right)^{2}}=\varepsilon.$

We conclude that for all $\varepsilon>0$ there exist $p,q\in\mathbb{N}$ such
that

$f^{q}(B(x_{pq},\varepsilon))\subseteq B(x_{pq},\varepsilon).$ (7)

This implies that for all $m\geq p$, we have

$d_{mq,pq}<\varepsilon.$ (8)

Next, we shall show that there exists $m_{0}\in\mathbb{N}$ such that

$d_{mq,mq+k}<\varepsilon\;\text{ for all }m\geq m_{0}\text{ and all
}k\in\\{1,\ldots,q-1\\}.$ (9)

Now, if $n=1$ in (3), we observe that there exists $i_{0}\in\mathbb{N}$ such
that for all $i\geq i_{0}$, we have

$d_{i}<\frac{\varepsilon}{\varepsilon+qc_{q}},$

where

$c_{q}=\max\left\\{(\begin{smallmatrix}q\\\
i\end{smallmatrix})b^{q-i}\rho^{i-1}:i=1,\ldots,q-1\right\\}.$

Let $m_{0}\in\mathbb{N}$ such that $m_{0}q>i_{0}$. We use Definition (d3) and
deduce that for all $m>m_{0}$ and all $k\in\\{1,\ldots,q-1\\}$, we have

$\displaystyle d_{mq,mq+k}$
$\displaystyle\leq\sum_{i=1}^{k}b^{k-i}\rho^{i-1}\textbf{e}_{i}(d_{mq},\ldots,d_{mq+k})$
$\displaystyle\leq\sum_{i=1}^{k}\left(\begin{smallmatrix}k\\\
i\end{smallmatrix}\right)b^{k-i}\rho^{i-1}\left(\frac{\varepsilon}{\varepsilon+qc_{q}}\right)^{i}$
$\displaystyle\leq\sum_{i=1}^{q-1}\left(\begin{smallmatrix}q\\\
i\end{smallmatrix}\right)b^{q-i}\rho^{i-1}\left(\frac{\varepsilon}{\varepsilon+qc_{q}}\right)^{i}$
$\displaystyle\leq\sum_{i=1}^{q-1}\left(\begin{smallmatrix}q\\\
i\end{smallmatrix}\right)b^{q-i}\rho^{i-1}\frac{\varepsilon}{\varepsilon+qc_{q}}$
$\displaystyle\leq\sum_{i=1}^{q-1}c_{q}\frac{\varepsilon}{qc_{q}}=\varepsilon,$

where e is the elementary symmetric polynomial in $k$ variables, that is,

$\textbf{e}_{i}(x_{1},\ldots,x_{k})=\sum_{1\leq j_{1}<j_{2}<\cdots<j_{i}\leq
k}\;\prod_{t=1}^{i}x_{j_{t}}\quad\text{for }i=1,\ldots,q-1.$

Now, let $p_{0}=\max\\{p,m_{0}\\}$ and two integers $k_{1},k_{2}\geq p_{0}q$
and $r_{1},r_{2}\in\\{0,\ldots,q-1\\}$ such that $k_{1}=p_{1}q+r_{1}$,
$k_{2}=p_{2}q+r_{2}$ and $p_{1},p_{2}\geq p_{0}$. We conclude from (8) and
(9), that we have

$\max\\{u_{1},u_{2},u_{3},u_{4}\\}<\varepsilon,$

where

$\displaystyle u_{1}\coloneqq d_{k_{1},p_{1}q},\quad u_{2}\coloneqq
d_{p_{1}q,pq},\quad u_{3}\coloneqq d_{pq,p_{2}q},\quad u_{4}\coloneqq
d_{p_{2}q,k_{2}}.$

Hence, by (d3) it follows

$\displaystyle d_{k_{1},k_{2}}$
$\displaystyle\leq\left(u_{{3}}+u_{{4}}\right){b}^{3}+u_{{2}}{b}^{2}+\left(u_{{3}}u_{{4}}+u_{{2}}\left(u_{{3}}+u_{{4}}\right)+u_{{1}}\left(u_{{3}}+u_{{4}}\right)\right)\rho\,{b}^{2}$
$\displaystyle\quad+\left(u_{{2}}u_{{3}}u_{{4}}+u_{{1}}\left(u_{{3}}u_{{4}}+u_{{2}}\left(u_{{3}}+u_{{4}}\right)\right)\right)b{\rho}^{2}+u_{{1}}u_{{2}}b\rho+bu_{{1}}+{\rho}^{3}u_{{1}}u_{{2}}u_{{3}}u_{{4}}$
$\displaystyle\leq\varepsilon\left(\varepsilon^{3}\rho^{3}+4\varepsilon^{2}{b}\,\rho^{2}+5\varepsilon\rho\,{b}^{2}+\varepsilon{b}\,\rho+2{b}^{3}+b^{2}+{b}\right)$
$\displaystyle\leq\varepsilon\left(\varepsilon^{3}+\varepsilon^{2}+\varepsilon+1\right)\max\\{\rho^{3},4\,{b}\,\rho^{2},6\,{b}^{2}\,\rho,4\,{b}^{3}\\}.$

We conclude that $\\{x_{n}\\}$ is a Cauchy sequence. As $(X,d)$ is complete,
then $\\{x_{n}\\}$ converges to some $x_{\ast}\in X$, say. Using (d3), (2) and
Remark 2.1, we obtain

$\displaystyle d(x_{\ast},fx_{\ast})$ $\displaystyle\leq
b\,d(x_{\ast},x_{k+1})+b\,d(x_{k+1},fx_{\ast})+\rho\,d(x_{\ast},x_{k+1})d(x_{k+1},fx_{\ast})$
$\displaystyle=b\,d(x_{\ast},x_{k+1})+b\,d(fx_{k},fx_{\ast})+\rho\,d(x_{\ast},x_{k+1})d(fx_{k},fx_{\ast})$
$\displaystyle\leq
b\,d(x_{\ast},x_{k+1})+b\,\psi(d(x_{k},x_{\ast}))+\rho\,d(x_{\ast},x_{k+1})\psi(d(x_{k},x_{\ast})),$

thus as $k$ tends to infinity, we conclude that $x_{\ast}=fx_{\ast}$. Finally,
assume that $x_{\ast}$ and $y_{\ast}$ are two distinct fixed points of $f$,
then by (2) we deduce

$d(x_{\ast},y_{\ast})=d(fx_{\ast},fy_{\ast})\leq\psi(d(x_{\ast},y_{\ast}))<d(x_{\ast},y_{\ast}),$

which is a contradiction, so $x_{\ast}=y_{\ast}$ and the fixed point is
unique. ∎

Next, if $\psi\in\mathbb{M}_{b}$, we can use a proof similar to that of [5,
Theorem 2.1].

######  Theorem 3.2.

Let $(X,d)$ be a complete $b$-suprametric space and $f\colon X\to X$ be a
mapping. Assume there exists $\psi\in\mathbb{M}_{b}$ such that (2) holds.
Then, $f$ has a unique fixed point and $\\{f^{n}x_{0}\\}_{n\in\mathbb{N}}$
converges to it for every $x_{0}\in X$.

###### Proof.

We adopt the same notations as the previous proof. Assume that $d_{i}\neq 0$
for all $i$ (otherwise $x_{i}$ becomes a fixed point of $f$). So, from (2), we
deduce that $d_{i+1}\leq\psi(d_{i})<d_{i}$ for all $i$. Next, we show that
$\\{x_{i}\\}$ is a Cauchy sequence. By Definition 2.4, it follow that

$\displaystyle d_{p,q}$ $\displaystyle\leq
bd_{p}+bd_{p+1,q}+\rho\,d_{p}\,d_{p+1,q}$ $\displaystyle\leq
b\psi^{p}(d_{0})+(b+\rho\,\psi^{p}(d_{0}))d_{p+1,q},$

where

$\displaystyle d_{p+1,q}$ $\displaystyle\leq
bd_{p+1}+bd_{p+2,q}+\rho\,d_{p+1}\,d_{p+2,q}$ $\displaystyle\leq
b\psi^{p+1}(d_{0})+(b+\rho\,\psi^{p+1}(d_{0}))d_{p+2,q},$

Then combining the previous inequalities, we obtain

$\displaystyle d_{p,q}\leq$ $\displaystyle
b\psi^{p}(d_{0})+b\psi^{p+1}(d_{0})(b+\rho\,\psi^{p}(d_{0}))$
$\displaystyle+(b+\rho\,\psi^{p}(d_{0}))(b+\rho\,\psi^{p+1}(d_{0}))d_{p+2,q}.$

Continuing this process, we obtain

$d_{p,q}\leq\sum_{i=p}^{q-1}b\psi^{i}(d_{0})\prod_{j=p}^{i-1}(b+\rho\psi^{j}(d_{0})).$

We also have

$d_{p,q}\leq\sum_{i=p}^{q-1}b\psi^{i}(d_{0})\prod_{j=0}^{i-1}(b+\rho\psi^{j}(d_{0})).$

In order to study the behavior of $d_{p,q}$, we need to study the convergence
of the series $\sum\limits_{i=0}^{\infty}u_{i}$, where

$u_{i}=b\psi^{i}(d_{0})\prod_{j=0}^{i-1}(b+\rho\psi^{j}(d_{0})).$

To this end, observe that from $\psi^{i+1}(d_{0})\to 0$ as $i\to\infty$, we
obtain

$\lim_{i\to\infty}\frac{u_{i+1}}{u_{i}}\leq\limsup_{i\to\infty}\frac{\psi^{i+1}(d_{0})}{\psi^{i}(d_{0})}(b+\rho\psi^{i}(d_{0}))<1,$

which implies that the series $\sum_{i=0}^{\infty}u_{i}$ converges, so
$d_{p,q}$ tends to zero as $p,q$ tend to infinity. Hence, the sequence
$\\{x_{n}\\}$ is Cauchy, and by completeness of $(X,d)$ we conclude that
$\\{x_{n}\\}$ converges to some $x\in X$. We next show that $x$ is a fixed
point of $f$. By using Definition 2.4 and (2), we get

$\displaystyle d(x,fx)$ $\displaystyle\leq bd(x,x_{n+1})+bd(fx_{n},fx)+\rho
d(x,x_{n})d(fx_{n},fx)$ $\displaystyle\leq
bd(x,x_{n+1})+b\psi(d(x_{n},x))+\rho d(x,x_{n})\psi(d(x_{n},x))$
$\displaystyle\leq bd(x,x_{n+1})+bd(x_{n},x)+\rho d(x,x_{n})^{2}.$

Thus, as $n$ tends to infinity, we deduce that $x=fx$. Finally, the uniqueness
of the fixed point follows immediately from (2) and Remark 2.1. ∎

We immediately derive the following corollaries.

######  Corollary 3.1.

Let $(X,d)$ be a complete $b$-suprametric space and $f\colon X\to X$ be a
mapping. Assume there exists $c\in[0,1)$ such that

$d(fx,fy)\leq c\,d(x,y)\;\text{ for all }x,y\in X.$

Then, $f$ has a unique fixed point and $\\{f^{n}x_{0}\\}_{n\in\mathbb{N}}$
converges to it for every $x_{0}\in X$.

######  Corollary 3.2.

Let $(X,d)$ be a complete suprametric space and $f\colon X\to X$ be a mapping.
Assume there exists $\psi\in\mathbb{M}$ such that

$d(fx,fy)\leq\psi(d(x,y))\;\text{ for all }x,y\in X.$

Then, $f$ has a unique fixed point and $\\{f^{n}x_{0}\\}_{n\in\mathbb{N}}$
converges to it for every $x_{0}\in X$.

As immediate consequences, we obtain the following propositions.

######  Proposition 3.1.

Theorem 3.1 generalizes Theorem 2.1.

######  Proposition 3.2.

Corollary 3.2 generalizes Theorem 2.2.

Acknowledgment: The author thanks the referees for valuable suggestions.

## Compliance with Ethical Standards

Funding: Not Applicable.
Conflict of Interest: The author declares that he has no conflict of interest.
Ethical approval: This article does not contain any studies with human
participants or animals performed by the author.
Data availability: Not Applicable.

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