Stevens Institute of Technology
Crypto Market Analysis & Real-Estate Business Protocol Proposal
Application of Ethereum Blockchain
Sid Bhatia, Samuel Gedal, Himaya Jeyakumar
Grace Lee, Ravinder Chopra, Daniel Roman, Shrijani Chakroborty
26 April 2024

###### Contents

  1. 1 Introduction
  2. 2 Part I: Crypto Market Analysis
    1. 2.1 Overview
    2. 2.2 Detailed Overview and Crypto Protocol
      1. 2.2.1 Bitcoin (BTC)
      2. 2.2.2 Ethereum (ETH)
      3. 2.2.3 XRP (Ripple)
      4. 2.2.4 Dogecoin (DOGE)
      5. 2.2.5 Tether (USDT)
    3. 2.3 Crypto Selection Rationale
    4. 2.4 Market Analysis
      1. 2.4.1 Price Trend Analysis
      2. 2.4.2 Volatility Analysis
      3. 2.4.3 Correlation Analysis
    5. 2.5 FTX Delta Analysis
      1. 2.5.1 Event Overview
      2. 2.5.2 FTX Impact on 11/11/22
      3. 2.5.3 Immediate Impact
      4. 2.5.4 FTX Impact in November 2022
      5. 2.5.5 Long-Term Impact
      6. 2.5.6 Benchmark and Market Performance Comparison
      7. 2.5.7 FTX Conclusion
    6. 2.6 Crypto Market Analysis Synthesis
      1. 2.6.1 Volatility and Stability
      2. 2.6.2 Market Correlations
      3. 2.6.3 FTX Crisis Impact
      4. 2.6.4 Long-Run Market Behavior
    7. 2.7 Part I Key Takeaways
    8. 2.8 Future Implications
  3. 3 Part II: Real-Estate Business Protocol Proposal
    1. 3.1 Business Proposal
      1. 3.1.1 Overview of the Blockchain Protocol
      2. 3.1.2 Transactional Process
      3. 3.1.3 Advantages of Blockchain in Real Estate
    2. 3.2 Target Customers
      1. 3.2.1 Market Strategy and Consumer Segmentation
      2. 3.2.2 Integration of Market Sides
    3. 3.3 Competitive Analysis
      1. 3.3.1 Propy: A Comparative Study
    4. 3.4 Competitive Advantage Analysis
    5. 3.5 Implementation Strategy
    6. 3.6 Economic Viability for Ethereum
    7. 3.7 Comparative Analysis of Blockchain Platforms
    8. 3.8 Synthesis of the Blockchain Real-Estate Protocol
  4. 4 Conclusion
    1. 4.1 Synthesis of Findings
    2. 4.2 Prospects and Implications
    3. 4.3 Summary and Forward Outlook

##  1 Introduction

In the dynamic realm of financial technology, blockchain and cryptocurrencies
represent two of the most significant innovations that have reshaped how
transactions are conducted and assets are managed globally
([futurecryptocurrencies2022]). This paper delves into a dual-focused analysis
and proposal. Firstly, we conduct a thorough market analysis of a select group
of cryptocurrencies, each chosen for its unique role and impact within the
broader digital currency landscape. The cryptocurrencies under review include
Bitcoin, often regarded as the progenitor of all digital currencies; Ethereum,
notable for its robust smart contract capabilities; XRP, designed primarily
for rapid financial transactions; Dogecoin, which began as a meme but has
since gained substantial practical application; and Tether, a stablecoin tied
to the US dollar, offering a less volatile refuge within the highly fluctuant
crypto market ([evolutioncryptomarket2017]).

This study not only examines the price trends, volatilities, and inter-
cryptocurrency correlations but also assesses the impact of significant market
events, such as the FTX bankruptcy, on these digital assets
([ftxresponse2023]). The insights garnered from this analysis aim to provide a
granular understanding of how various cryptocurrencies react to internal and
external pressures, influencing investor sentiment and market dynamics.

Following the market analysis, the second focus of this paper introduces an
innovative business proposal leveraging blockchain technology. This proposal
outlines a new protocol for real estate transactions, allowing property deeds
to be securely managed and transferred without the need for traditional
intermediaries such as lawyers and brokers. By employing blockchain
technology, this protocol seeks to revolutionize the real estate market by
enhancing transparency, reducing transaction costs, and simplifying the
transaction process for buyers and sellers across the globe
([blockchainrealestate2021]).

Through comprehensive analysis and forward-thinking proposals, this paper
contributes to the ongoing discussions surrounding the application of
blockchain technology in traditional sectors, proposing not only a new way to
understand cryptocurrencies in relation to the traditional financial markets
but also offering a practical application that addresses real-world challenges
in real estate transactions.

##  2 Part I: Crypto Market Analysis

###  2.1 Overview

This analysis encompasses a selection of five distinct cryptocurrencies, each
representing a unique facet of the current digital currency ecosystem. Our
selected cryptocurrencies include: Bitcoin (BTC), recognized as the original
and most well-known cryptocurrency; Ethereum (ETH), noted for its advanced
smart contract capabilities; XRP, developed by Ripple Labs with a focus on
rapid digital payments; Dogecoin (DOGE), which has evolved from a meme into a
cryptocurrency with practical uses in tipping and donations; and Tether
(USDT), a stablecoin that introduces a measure of stability in the otherwise
volatile cryptocurrency market. This diverse selection aims to cover a broad
spectrum of functionalities, market positions, and technological innovations
within the crypto space, providing a comprehensive overview of its varied
applications and implications.

###  2.2 Detailed Overview and Crypto Protocol

####  2.2.1 Bitcoin (BTC)

Overview: Introduced in 2009 by an entity under the pseudonym Satoshi
Nakamoto, Bitcoin stands as the inaugural cryptocurrency, designed to operate
as a decentralized digital currency without the oversight of a central
authority. Transactions are conducted directly between users through the peer-
to-peer Bitcoin network.

Protocol: Bitcoin’s network is underpinned by a proof-of-work (PoW) protocol,
wherein miners employ significant computational resources to solve intricate
mathematical problems, thus validating transactions and securing the network,
with new bitcoins awarded as a mining reward.

For more details see [nakamoto2009bitcoin].

####  2.2.2 Ethereum (ETH)

Overview: Launched in 2015, Ethereum transcends the conventional definition of
a cryptocurrency. It serves as a platform for the development of decentralized
applications (DApps) through smart contracts, aiming to democratize access to
a decentralized financial system.

Protocol: Initially based on a proof-of-work mechanism similar to that of
Bitcoin, Ethereum is transitioning to a proof-of-stake (PoS) model with its
Ethereum 2.0 update, which promises enhanced scalability and reduced energy
consumption.

Refer to [buterin2015ethereum] for additional insights.

####  2.2.3 XRP (Ripple)

Overview: Created by Ripple Labs in 2012, XRP is central to a digital payment
protocol that surpasses its identity as a mere cryptocurrency. It facilitates
rapid payment settlements across the network.

Protocol: The XRP Ledger utilizes a consensus protocol that does not rely on
the traditional blockchain mining process; instead, it achieves consensus
through a network of independent validating servers that constantly compare
transaction records.

See [ripple2012xrp] for further information.

####  2.2.4 Dogecoin (DOGE)

Overview: Originating as a humorous take on the cryptocurrency phenomenon in
2013, Dogecoin was inspired by the ”Doge” meme featuring a Shiba Inu. It has
since cultivated a community focused on using the cryptocurrency for
charitable contributions and tipping online content creators.

Protocol: Dogecoin operates on a less energy-intensive proof-of-work algorithm
derived from Litecoin, known as Scrypt, facilitating faster transaction
processing.

Detailed information available at [dogecoin2013meme].

####  2.2.5 Tether (USDT)

Overview: Introduced in 2014, Tether represents a stablecoin that is tethered
to the US dollar, aiming to meld the flexibility of cryptocurrencies with the
stability of fiat currency.

Protocol: Tether supports a hybrid use of protocols, operating on the Omni
Layer of the Bitcoin blockchain and as an ERC-20 token on the Ethereum
blockchain, among other blockchain platforms.

Further details can be found in [tether2014usdt].

These cryptocurrencies were chosen to provide a diverse perspective on the
various applications, market usage, and technological advancements within the
broader cryptocurrency environment. From January 1, 2022, to December 31,
2022, our study observed no missing data, ensuring the completeness and
reliability of the analysis conducted during this period.

###  2.3 Crypto Selection Rationale

The selection of cryptocurrencies for this study was informed by a
multifaceted rationale emphasizing diversity, technological innovation,
community engagement, and market stability. Each cryptocurrency was chosen not
only for its unique position within the market but also for its contribution
to advancing the blockchain technology landscape.

Diversity and Relevance: Bitcoin and Ethereum are selected as foundational
pillars within the cryptocurrency domain, illustrating the broad spectrum of
blockchain applications. Bitcoin, often hailed as the original cryptocurrency,
has pioneered the concept of a decentralized digital currency and enjoys
widespread adoption and recognition. Ethereum, on the other hand, extends the
utility of blockchain beyond mere financial transactions through its support
for smart contracts, thereby catalyzing a plethora of decentralized
applications (DApps). This diversity underscores the significant role these
currencies play in the ongoing development and maturation of the
cryptocurrency market.

Technological Diversity: XRP and Tether were chosen to highlight the
technological diversity within blockchain implementations. XRP, developed by
Ripple, is notable for its rapid transaction capabilities and minimal energy
consumption, diverging from the traditional mining-based consensus used by
currencies like Bitcoin. Similarly, Tether introduces a model of stability in
the highly volatile cryptocurrency market by being pegged to the US dollar,
showcasing a unique application of blockchain technology in creating
stablecoins that mitigate the price volatility typically associated with
cryptocurrencies.

Community and Innovation: Dogecoin exemplifies the impact of community on the
value and adoption of a cryptocurrency. Originating as a meme, Dogecoin has
transcended its initial novelty to foster a robust community that actively
engages in tipping and charitable activities through the currency. This aspect
highlights the role of societal and cultural dynamics in shaping the
cryptocurrency landscape, emphasizing the importance of community-driven
development and innovation.

Market Stability and Innovations: Finally, the inclusion of Tether also
addresses the critical challenge of market stability. By anchoring its value
to a stable fiat currency, Tether offers a pragmatic solution to the issue of
volatility, which is a pervasive concern for investors in cryptocurrencies
like Bitcoin and Ethereum. This approach not only facilitates greater market
stability but also enhances the practicality of cryptocurrencies for everyday
transactions and financial applications.

Collectively, these selections provide a comprehensive overview of the current
state and potential future directions of blockchain technology, illustrating a
spectrum of use cases from foundational cryptocurrencies to innovative
adaptations addressing specific market needs.

###  2.4 Market Analysis

Figure 1: Standardized Daily Prices of Cryptocurrencies for 2022

####  2.4.1 Price Trend Analysis

The analysis of standardized price trends of Bitcoin (BTC), Ethereum (ETH),
XRP, Dogecoin (DOGE), and Tether (USDT) throughout 2022 reveals several key
insights into the dynamics of the cryptocurrency market:

##### Correlated Movements:

The data illustrates that most cryptocurrencies exhibited closely correlated
movements over the course of the year. Such correlation is indicative of the
substantial influence exerted by broader market forces and global economic
events on the cryptocurrency market as a whole, driving collective swings in
investor sentiment—whether bullish or bearish.

##### Volatility Across Assets:

The degree of volatility varied significantly among the analyzed
cryptocurrencies. Bitcoin and Ethereum experienced relatively moderate
fluctuations, maintaining tighter price bands, while Dogecoin displayed higher
volatility, characterized by more pronounced peaks and troughs. This disparity
in volatility underscores the differential market perceptions and investor
bases of these assets.

##### Stablecoin Anomaly:

An unexpected anomaly was observed in the price trend of Tether (USDT),
particularly in May 2022, where it deviated markedly from its expected stable
trajectory. Such a divergence, given the design of stablecoins to maintain
parity with a peg (e.g., USD), suggests potential extraordinary events, data
reporting inaccuracies, or underlying issues with the stability mechanisms
during that period.

##### Market Recovery Ability:

Following significant market dips, the cryptocurrencies demonstrated varying
degrees of recovery. Bitcoin and Ethereum showed robust resilience and
recovery capabilities compared to Dogecoin. This variation could reflect
differing levels of market confidence and inherent stability within these
digital assets.

##### Stablecoin’s Peculiar Trend:

Assuming the accuracy of the observed sharp decline in USDT’s value, this
could represent a period of intense market stress or a temporary disruption in
the stablecoin’s dollar peg. However, such incidents are generally ephemeral,
as corrective mechanisms typically restore stability swiftly, aligning with
the observed rapid return to normalcy.

From the analysis of these price trends, it is evident that while
cryptocurrencies are interconnected and respond collectively to market shifts,
individual assets exhibit distinct behaviors influenced by their specific
market dynamics, investor sentiment, and technological foundations. The
peculiar movement observed in Tether’s price trend during the analyzed period
merits further investigation to ascertain the causes and implications of such
an anomaly.

####  2.4.2 Volatility Analysis

The study of volatility in cryptocurrency markets provides crucial insights
into the risks and stability of digital assets. By calculating daily returns
and examining their standard deviations, we can gauge the unpredictability
associated with each cryptocurrency and identify the factors contributing to
these dynamics.

##### Dogecoin (DOGE-USD):

Dogecoin exhibits the highest volatility among the cryptocurrencies analyzed,
with a standard deviation of approximately 5.64%. This elevated volatility can
primarily be attributed to its relatively low price per unit, which renders it
more susceptible to significant percentage changes on a per-unit basis.
Moreover, Dogecoin’s price is notably influenced by social media trends and
possesses comparatively less market liquidity than more established
cryptocurrencies. These elements combine to increase its price volatility,
reflecting the substantial impact of retail investor sentiment and speculative
trading on its market behavior.

##### Tether (USDT-USD):

In stark contrast, Tether shows the lowest volatility, with a standard
deviation near 0.03%. As a stablecoin, Tether is explicitly designed to be
pegged to a fiat currency, specifically the US dollar, and maintains a stable
value through various regulatory and technological mechanisms. This stability
is critical for its role in providing a safe haven during market turbulence
and for facilitating transactions where volatility can be a deterrent.

##### Bitcoin (BTC-USD) and Ethereum (ETH-USD):

Both Bitcoin and Ethereum exhibit moderate levels of volatility, reflecting
their established presence in the market and larger capitalizations. These
factors typically confer higher liquidity and result in less drastic
percentage changes in daily prices.

##### Benchmark Volatility Analysis:

Comparing the volatility of cryptocurrencies with traditional financial
markets, such as the S&P 500, highlights the unique risk profiles inherent to
digital assets. The S&P 500, with a volatility of 1.00%, offers a contrast to
the higher volatility levels seen in cryptocurrencies, underscoring the
potential for greater price stability in traditional equity markets.

##### Market Implications:

This variability in volatility, especially when benchmarked against
traditional indices like the S&P 500, illustrates the diverse nature of
cryptocurrency markets. While stablecoins like Tether aim to minimize price
fluctuations, other cryptocurrencies such as Dogecoin and Bitcoin exhibit a
range of volatilities, heavily influenced by investor sentiment, liquidity,
and their roles within the digital economy.

The higher volatility of cryptocurrencies compared to traditional markets like
the S&P 500 underscores their speculative nature and the heightened risks they
pose, which investors must navigate carefully. This analysis emphasizes the
importance of strategic risk assessment and portfolio diversification to
mitigate the inherent volatility of cryptocurrencies.

####  2.4.3 Correlation Analysis

Figure 2: Correlation Matrix Heatmap of Cryptocurrencies for 2022

A comprehensive examination of the correlation matrix for daily returns of
Bitcoin (BTC), Ethereum (ETH), XRP, Dogecoin (DOGE), and Tether (USDT) in
tandem with the S&P benchmark (GSPC) elucidates the interrelationships among
these prominent cryptocurrencies:

##### High Correlations:

  * •

Bitcoin and Ethereum: Exhibiting a correlation coefficient of 0.90, BTC and
ETH demonstrate a very strong positive correlation, indicating that these
cryptocurrencies often move in tandem. This strong linkage is primarily due to
their predominant positions in the market, where both are frequently
influenced by similar economic factors, investor sentiments, and regulatory
developments.

  * •

Ethereum and XRP: With a correlation of 0.75, movements in Ethereum frequently
correlate closely with those in XRP, suggesting overlapping functionalities
and investor bases that react similarly to market stimuli in these two
platforms.

##### Moderate Correlations:

  * •

XRP with Bitcoin and Dogecoin: XRP displays moderate correlations of 0.74 with
BTC and 0.61 with DOGE. These correlations suggest a level of synchronicity,
albeit influenced by distinct market dynamics and external factors specific to
each cryptocurrency.

  * •

Dogecoin with Ethereum and Bitcoin: Correlation coefficients of 0.67 with ETH
and 0.65 with BTC for Dogecoin indicate a moderate degree of correlation,
influenced by broader market trends that impact all cryptocurrencies, though
each responds according to its unique market niche and investor behavior.

##### Lower Correlations with Tether:

  * •

All Cryptocurrencies with Tether: Tether, being a stablecoin tied closely to
the US dollar, shows significantly lower correlation coefficients with BTC
(0.26), ETH (0.25), XRP (0.28), and DOGE (0.27). This fundamental difference
in design and purpose—aimed at providing stability—results in less
synchronized movements with the more speculative cryptocurrency assets.

##### Benchmark Correlation Comparison:

Comparison with traditional financial markets, specifically through a
correlation study with the S&P 500, reveals additional insights. While
cryptocurrencies such as BTC and ETH show the highest correlation with each
other, they exhibit only moderate correlation levels with the S&P 500, with
BTC showing the highest correlation at 0.52. This suggests that while
cryptocurrencies do move somewhat in sync with traditional financial markets,
they retain distinct market dynamics that set them apart.

##### Market Implications:

These findings highlight the diverse correlation landscapes within the
cryptocurrency markets, where strong intra-crypto correlations contrast with
more moderate interactions with traditional financial indices. This divergence
underscores the necessity for investors to consider the unique correlation
patterns when diversifying portfolios or implementing hedging strategies. The
mixed correlation profiles suggest both opportunities and risks, as
cryptocurrencies can offer portfolio diversification benefits due to their
partial independence from traditional market movements.

###  2.5 FTX Delta Analysis

####  2.5.1 Event Overview

In mid-November 2022, the cryptocurrency exchange FTX filed for bankruptcy,
triggering significant disturbances across the cryptocurrency markets. This
event was exacerbated by the resignation of its CEO, Sam Bankman-Fried,
further destabilizing the market’s confidence.

####  2.5.2 FTX Impact on 11/11/22

Ticker | Impact on Nov 11
---|---
BTC-USD | -3.14%
DOGE-USD | -5.46%
ETH-USD | -0.94%
USDT-USD | 0.04%
XRP-USD | -2.92%
GSPC | 1.00%
Table 1: FTX Impact on Cryptocurrency Prices on November 11, 2022

####  2.5.3 Immediate Impact

The immediate repercussions of the bankruptcy announcement on November 11,
2022, were starkly evident across various cryptocurrencies:

  * •

Bitcoin (BTC) and Ripple (XRP) each faced notable declines, with Bitcoin
falling by -3.14% and XRP by -2.92%.

  * •

Ethereum (ETH) exhibited relative resilience, with a modest decline of -0.94%,
reflecting its robust market presence and investor confidence.

  * •

Dogecoin (DOGE) experienced the most significant drop of -5.46%, illustrating
its susceptibility to market shocks.

  * •

Tether (USDT), maintaining its stability, changed insignificantly by +0.04%,
underscoring its role as a stabilizing force within the volatile
cryptocurrency environment.

####  2.5.4 FTX Impact in November 2022

Ticker | Change in Nov 2022
---|---
BTC-USD | -16.19%
DOGE-USD | -25.05%
ETH-USD | -17.98%
USDT-USD | 0.01%
XRP-USD | -12.02%
GSPC | 5.81%
Table 2: Monthly Impact of FTX Bankruptcy on Cryptocurrency Prices in November
2022

####  2.5.5 Long-Term Impact

The extended impact throughout November painted a grim picture of recovery
challenges:

  * •

Major cryptocurrencies like BTC, ETH, and XRP recorded substantial declines of
-16.19%, -17.98%, and -12.02% respectively.

  * •

DOGE was particularly hard hit, plummeting by -25.05%, marking the highest
vulnerability among the group.

  * •

Conversely, USDT showed remarkable stability with only a 0.01% change,
reinforcing its value proposition as a hedge against volatility.

####  2.5.6 Benchmark and Market Performance Comparison

The correlation and impact studies reveal that while the cryptocurrency market
suffered significant losses in response to the FTX crisis, the traditional
financial markets, as represented by the S&P 500, exhibited contrasting
behavior:

  * •

On November 11, 2022, while cryptocurrencies faced sharp declines, the S&P 500
(GSPC) experienced a rise of 1.00%, demonstrating a decoupling from
cryptocurrency market dynamics.

  * •

Over the entire month of November, the S&P 500 gained 5.81%, further
highlighting the resilience and differing risk profiles of traditional equity
markets compared to the high-risk cryptocurrency sector.

####  2.5.7 FTX Conclusion

The FTX bankruptcy served as a critical stress test, revealing the inherent
volatility and risk exposure of speculative cryptocurrencies compared to the
stability offered by stablecoins like Tether and traditional financial indices
like the S&P 500. This event underscores the need for robust risk management
strategies and diversified investment approaches to navigate the complexities
of cryptocurrency investments effectively.

###  2.6 Crypto Market Analysis Synthesis

Part I of this project delved into a comprehensive analysis of the
cryptocurrency ecosystem, with an emphasis on five key cryptocurrencies
(Bitcoin, Ethereum, Ripple, Dogecoin, and Tether) and comparisons against the
S&P 500 index. Our study covered daily price behavior, volatility,
correlations, and market responses to major events like the FTX bankruptcy.
Let’s synthesize our key findings:

####  2.6.1 Volatility and Stability

  * •

Cryptocurrencies demonstrate substantially higher volatility than traditional
markets like the S&P 500. Dogecoin exhibited the highest volatility due to its
smaller size and speculative nature.

  * •

Tether’s negligible volatility confirms its role as a stablecoin, offering
refuge within the cryptocurrency market.

####  2.6.2 Market Correlations

  * •

Bitcoin, Ethereum, and other major cryptocurrencies are highly correlated,
driven by similar market forces.

  * •

Cryptocurrencies show low-to-moderate correlation with the S&P 500, suggesting
some independence and potential diversification benefits.

####  2.6.3 FTX Crisis Impact

  * •

The FTX bankruptcy severely impacted cryptocurrency prices, while the S&P 500
remained largely unaffected, highlighting sector-specific risks within crypto.

  * •

November 2022’s broader market picture reinforced this divergence.
Cryptocurrencies declined significantly (excluding Tether), while the S&P 500
grew, emphasizing a decoupling during cryptocurrency-specific crises.

####  2.6.4 Long-Run Market Behavior

  * •

2022 data illustrates that while offering potential for growth,
cryptocurrencies also carry substantial risks of sharp declines.

  * •

The S&P 500’s lower volatility and positive November performance underscore
the importance of traditional equity investments for risk mitigation in
diversified portfolios.

###  2.7 Part I Key Takeaways

##### Diversification:

Cryptocurrencies offer diversification potential, but investors must carefully
manage their high-risk profile.

##### Investment Strategy:

Balancing crypto holdings with safer assets like the S&P 500 can mitigate
losses during downturns.

##### Regulatory and Market Sensitivity:

Staying informed about regulatory developments and sector-specific events is
crucial for navigating the dynamic cryptocurrency market.

###  2.8 Future Implications

These insights are vital for developing robust investment strategies
maximizing the potential of cryptocurrencies while safeguarding against their
risks. Monitoring evolving correlations between cryptocurrencies and
traditional markets will aid in understanding market dynamics and adapting
investment strategies accordingly.

##  3 Part II: Real-Estate Business Protocol Proposal

###  3.1 Business Proposal

####  3.1.1 Overview of the Blockchain Protocol

Our business proposal introduces an innovative blockchain protocol designed to
revolutionize the real estate sector. This protocol allows homeowners to store
the deeds of their houses on the blockchain and facilitates the sale of
properties without traditional intermediaries such as lawyers, brokers, or
other third parties. This system not only simplifies transactions but also
enhances security, reduces costs, and increases transparency.

####  3.1.2 Transactional Process

The transactional process under this protocol is streamlined to ensure
efficiency and security:

  1. 1.

Initiation of Sale: Homeowners list their properties on the blockchain
platform directly, bypassing the need for intermediaries. This step
significantly reduces the complexity and duration of property transactions.

  2. 2.

Proof of Ownership: The blockchain technology inherently provides a clear,
immutable record of ownership. This proof is publicly accessible and
verifiable, ensuring that the current owner has indisputable ownership of the
property before proceeding with the sale.

  3. 3.

Payment and Transfer: The buyer pays the seller in cryptocurrency, such as
Bitcoin. Following payment confirmation, the property deed is automatically
transferred to the buyer’s blockchain address via a smart contract, which also
handles the transaction fee, typically associated with platforms like
Ethereum.

  4. 4.

Final Ownership: The new owner receives the property deed securely stored on
the blockchain, ensuring both safety and accessibility. This digital deed is
resistant to tampering, loss, or theft, providing a permanent record of
ownership.

####  3.1.3 Advantages of Blockchain in Real Estate

The integration of blockchain into real estate transactions offers several
improvements over traditional methods:

Transparency:

The blockchain’s immutable ledger ensures that all transactions, including
historical ownership data and property details (e.g., square footage, number
of bedrooms, date of last renovation), are permanently recorded and openly
verifiable. This level of transparency significantly reduces the potential for
fraud and disputes.

Cost Efficiency:

By eliminating the need for various intermediaries and reducing paperwork and
manual verification processes, the blockchain protocol cuts down on
significant transactional costs. These savings make real estate transactions
more economical for both buyers and sellers.

Global Accessibility:

The blockchain protocol enables international transactions without the
complexities of cross-border legalities and financial transactions, opening up
the property market to global participants and investors.

Market Liquidity:

The use of blockchain can enhance market liquidity. Buyers who may not have
immediate access to traditional financing options can leverage decentralized
finance (DeFi) solutions, such as Aave, for quicker funding solutions, thereby
accelerating the buying process.

###  3.2 Target Customers

####  3.2.1 Market Strategy and Consumer Segmentation

Our blockchain protocol adopts a dual-sided market strategy designed to
address distinct needs within the real estate transaction process. This
approach targets two main customer segments: property sellers (and deed
holders) and property buyers, along with a third, indirect segment involving
financial lenders.

##### Property Sellers and Deed Holders:

The primary market segment consists of current property owners who stand to
benefit substantially from blockchain integration. Traditional methods of deed
storage involve physical documentation, which not only increases the risk of
loss and damage but also complicates the verification and transfer processes.
Our protocol offers a secure and immutable storage solution on the blockchain,
eliminating the need for physical safekeeping.

In tandem with disproportionately high the current real estate market
structure necessitates multiple intermediaries, including real estate agents,
brokers, and legal advisors, each adding significant transaction costs in the
form of commissions and fees. Historical data indicates that commission rates
have remained relatively stable over the past three decades, despite
substantial increases in property values, leading to disproportionately high
costs for sellers (refer to Figure 3, [commratetrends2013]). Moreover, median
housing prices have soared, as vividly depicted in the provided data from the
Federal Reserve Economic Data (FRED, [fredmspus2024]) (see Figure 4).

Our protocol simplifies this process, allowing sellers to initiate and
complete sales directly on the blockchain, thereby reducing or eliminating
traditional commission fees.

Figure 3: Historical Analysis of Average Commission Rates in Real Estate
Transactions Figure 4: Federal Reserve Economic Data (FRED) on Median House
Sale Prices

##### Property Buyers:

The second primary target segment includes potential property buyers who
benefit from the streamlined purchase process. Through our protocol, buyers
can directly engage with sellers, conduct swift and secure transactions, and
gain immediate access to verified property deeds, significantly speeding up
the acquisition process. The use of smart contracts ensures that all
conditions of the sale are met before the transaction is finalized, offering
additional security and efficiency.

##### Financial Lenders:

An emerging market segment within our protocol includes financial lenders,
particularly those operating in the decentralized finance (DeFi) space. With
the rise of blockchain technology, platforms like Aave have demonstrated
significant demand for more dynamic lending solutions that offer higher yields
compared to traditional financial products. Our protocol can connect these
lenders directly with real estate buyers, providing a new avenue for secured
lending at competitive interest rates, reflective of the increased risk
profiles associated with cryptocurrency-based transactions (refer to Figure
5).

Figure 5: Top Total Value Locked (TVL) in DeFi; Growth in DeFi Loan Market
Size

####  3.2.2 Integration of Market Sides

By effectively integrating these two sides of the market—sellers and buyers
with the financial backing of lenders—our blockchain protocol facilitates a
comprehensive ecosystem that enhances liquidity, reduces transaction latency,
and improves overall market efficiency. This integrated approach not only
serves the immediate participants but also introduces a scalable model for
future expansions in global real estate markets.

###  3.3 Competitive Analysis

####  3.3.1 Propy: A Comparative Study

Propy emerges as a significant player within the blockchain-based real estate
marketplace, providing a global platform that aligns closely with the
decentralized ethos of the blockchain revolution. As a competitor, Propy’s
operational model is built upon eliminating traditional intermediaries from
the property transaction process. By leveraging smart contracts on the
blockchain, Propy ensures secure and efficient property transactions.

##### Operational Model:

The core of Propy’s proposition is its decentralized marketplace, which
facilitates the buying and selling of properties. This innovative approach
circumvents the need for brokers and agents, potentially reducing
transactional friction and cost. Furthermore, Propy offers digital deeds
alongside automated escrow services, thus simplifying real estate transactions
and enhancing user experience.

##### Economic Structure:

Propy’s economic framework incorporates the use of its native cryptocurrency,
PRO, alongside a designated fee for smart contract execution, termed PGas. The
integration of PRO within their platform ecosystem not only facilitates
transactional activities but also extends utility to users engaging with
Propy’s services. The current market valuation of PRO stands at $2.99, which
plays a pivotal role in the cost structure of property sales on Propy’s
platform.

Figure 6: Analysis of Propy’s Transactional Fees for Property Sales

This competitive analysis provides a deeper understanding of Propy’s strategic
positioning within the blockchain-based real estate sector. By dissecting
their transactional fee structure and operational model, we can assess the
potential impact on our blockchain protocol’s market penetration and user
adoption.

###  3.4 Competitive Advantage Analysis

Our protocol presents a unique value proposition in the blockchain real estate
marketplace, offering comprehensive solutions that address various stages of
the real estate transaction process. It capitalizes on the inherent advantages
of blockchain technology to deliver an end-to-end service that simplifies the
complexities traditionally associated with real estate transactions.

##### Comprehensive Transactional Solutions:

At the heart of our protocol is the capability to facilitate complete real
estate transactions on the blockchain. This ranges from secure deed storage to
the actual execution of property sales via cryptocurrency. By providing a
single, unified platform, we significantly reduce the dependency on multiple
services, thereby streamlining the transaction process for all stakeholders
involved.

##### Transparency and Security:

The blockchain’s immutable ledger is a cornerstone feature that enhances our
protocol’s appeal. It serves as an unalterable record of transactions,
ensuring complete transparency and security for the transaction history. This
transparency is a critical factor for buyers and sellers who prioritize trust
and verifiable transparency in their transactions, eliminating the traditional
concerns of fraud and ambiguity in property ownership and history.

##### Cost Efficiency:

By obviating the need for intermediaries such as agents, brokers, and legal
consultants, our protocol minimizes the associated transaction costs. The
conventional commission-based model, which significantly increases transaction
expenses, is replaced by a more cost-effective structure that aligns with the
economic preferences of a market leaning towards efficiency and reduced
overhead.

##### Global Market Accessibility:

Our protocol removes the barriers to entry for international buyers and
sellers, thereby facilitating global transactions. Without the constraints
imposed by legal and regulatory compliance typical of centralized systems, our
platform paves the way for a more inclusive and expansive real estate market,
appealing to a broader investor base and contributing to the diversity of real
estate offerings.

In summary, our competitive advantage stems from a holistic approach that not
only provides practical transactional capabilities but also fosters trust,
reduces costs, and embraces global inclusivity. This strategic positioning is
poised to disrupt the traditional real estate market, leveraging blockchain
technology to its fullest potential.

###  3.5 Implementation Strategy

The implementation of our business model onto the blockchain comprises a
systematic approach, focused on smart contract formulation, asset
tokenization, oracle integration, privacy considerations, and user interface
development.

##### Defining Smart Contracts:

The foundation of our blockchain protocol is the design of smart contracts,
which are digital representations of real estate assets, mortgages, and deed
transfers. The contracts will encapsulate the logic for buying, selling,
transferring ownership, and managing mortgage payments. This will involve:

  1. 1.

Structuring smart contracts to encapsulate real estate transaction
requirements.

  2. 2.

Integrating functions for various transaction processes within these
contracts.

##### Tokenizing Real Estate Assets:

We will transform real estate properties into digital tokens on the Ethereum
blockchain, where each token signifies property ownership. This process will:

  1. 1.

Utilize established token standards, such as ERC-20, for the representation of
real assets in the digital domain.

  2. 2.

Deploy contracts to issue and regulate these tokens, assigning unique
identifiers to each property ([erc20whitepaper]).

##### Integration of Oracles:

Oracles will be employed to incorporate off-chain data, like property
specifications and legal documents, into the blockchain. These oracles will:

  1. 1.

Source trusted data essential for executing real estate transactions.

  2. 2.

Update on-chain records to reflect accurate off-chain information, ensuring
the veracity and reliability of data ([blockchainoracle2020]).

##### Privacy Enhancements:

Our protocol will incorporate privacy measures to safeguard sensitive
transaction data, allowing for public verification while maintaining
confidentiality. We will:

  1. 1.

Develop smart contracts with robust privacy controls.

  2. 2.

Implement encryption techniques to restrict data access to authorized entities
only.

##### User Interface Development:

To facilitate user interaction with our blockchain platform, we will develop
accessible and intuitive interfaces. These interfaces will:

  1. 1.

Offer a seamless user experience for property searches, transaction
initiation, mortgage tracking, and deed management.

  2. 2.

Provide tools that are comprehensible and efficient for users, irrespective of
their familiarity with blockchain technology.

The strategic deployment of these elements will result in a robust blockchain
protocol for real estate, streamlining the transaction process and enhancing
the overall experience for users in the real estate market. The careful
orchestration of smart contracts, tokenization, oracles, privacy
considerations, and user interfaces are essential components of our strategy
to integrate real estate transactions with blockchain technology effectively.

###  3.6 Economic Viability for Ethereum

Evaluating the economic feasibility of our business proposal on the Ethereum
platform involves careful consideration of various factors, including
scalability, transaction fees, and the complexity of smart contracts.

##### Scalability Concerns:

Ethereum’s scalability is a pivotal concern, particularly as transaction
volumes escalate. As the leading blockchain platform, Ethereum faces
challenges in maintaining performance amid rising demand. The introduction of
Ethereum 2.0 promises to alleviate these issues through sharding and a proof-
of-stake consensus mechanism, potentially enhancing throughput and lowering
transaction costs ([blockchaingasfees2021]).

##### Transaction Fees:

Gas fees on Ethereum are known for their volatility and can constitute a
significant portion of transaction costs. These fees tend to surge during
periods of network congestion, impacting the cost-benefit analysis for users
engaging in real estate transactions. Monitoring the historical trends of
Ethereum’s average gas fee is crucial in forecasting and managing the
financial viability of transactions ([ethereumgasprices2024]).

Figure 7: Average Ethereum Gas Fees Over the Last Five Years

##### Smart Contract Complexity:

The smart contracts at the core of our real estate protocol, which will enable
property transfers, loan repayments, and the enforcement of covenants, are
inherently complex. The intricacy of these contracts is directly proportional
to the computational resources required, thus influencing the overall gas fees
incurred. This complexity must be carefully managed to ensure that the
benefits of using the Ethereum platform outweigh the costs for all parties
involved.

The prospective enhancements with Ethereum 2.0 alongside strategic management
of smart contract complexity and monitoring of gas fees are essential for
ensuring the economic viability of deploying our real estate transaction
protocol on the Ethereum blockchain. As we progress, it will be imperative to
remain adaptable to the evolving blockchain landscape to maintain a
competitive and cost-effective platform for real estate transactions.

###  3.7 Comparative Analysis of Blockchain Platforms

Figure 8: Comparative Analysis of Ethereum Versus Alternative Blockchain
Platforms

In exploring the economic viability and technical suitability of our real
estate transaction protocol, we extend our analysis beyond Ethereum to other
leading blockchain platforms, each with distinct attributes and potential
advantages.

##### Binance Smart Chain:

Binance Smart Chain (BSC) emerges as a viable alternative, promising higher
throughput and lower latency compared to Ethereum, which is advantageous for
high-demand scenarios. Despite these benefits, BSC’s more centralized nature
may raise concerns among stakeholders seeking a fully decentralized solution.

##### Solana:

Solana presents a compelling case for applications necessitating rapid
transaction processing, offering superior transaction speeds and scalability
([solana2022]). While Solana provides an efficient alternative to Ethereum,
its developer ecosystem and tooling are less mature, potentially imposing
limitations on development and integration efforts.

##### Polkadot:

The multi-chain framework of Polkadot facilitates cross-chain
interoperability, which can significantly enhance the scope and flexibility of
our protocol. Polkadot’s design allows for seamless integration with a variety
of blockchain networks, potentially expanding the protocol’s reach.
Nevertheless, Polkadot’s infrastructure and tooling are still evolving, which
may introduce challenges during early adoption phases.

This comparative analysis underscores the importance of selecting a blockchain
platform that aligns with our protocol’s requirements for security,
decentralization, transaction speed, and scalability. As we proceed with our
implementation strategy, ongoing evaluation of these platforms’ evolving
capabilities will be imperative to maintain an innovative and user-centric
service in the dynamic real estate market.

###  3.8 Synthesis of the Blockchain Real-Estate Protocol

The business protocol presented in Part II encapsulates an innovative,
blockchain-based approach to real estate transactions. The proposed system
introduces a transformative model that empowers homeowners to engage directly
in the sale and purchase of properties, effectively circumventing the
traditional, intermediary-reliant processes that are often cumbersome and less
secure.

##### Overview and Efficacy of Protocol Application

The cornerstone of the proposed protocol lies in its utilization of
blockchain’s inherent properties such as immutability, transparency, and
distributed consensus. These properties facilitate a seamless transition of
deeds and payments, providing a robust proof of ownership and streamlining
transactions. By leveraging smart contract technology, the protocol ensures
that all prerequisites of a property transaction are automatically met,
heralding a new era of efficiency in property dealings.

##### Enhancing Real Estate Transaction Dynamics

The protocol offers multiple benefits over conventional methods. It provides a
public, transparent ledger for ownership and transaction history, reduces the
costs associated with property transactions by eliminating intermediary fees,
and enables global participation in the real estate market. Additionally, the
protocol introduces greater liquidity to the market by integrating with
decentralized financial platforms, allowing prospective buyers to secure
funding rapidly.

##### Consumer-Centric Market Strategy

This protocol advocates a dual-sided market strategy that aims to reform the
real estate transaction paradigm. Property sellers are afforded the ability to
secure deed storage on the blockchain while benefiting from direct market
access for sales, effectively bypassing intermediary overheads. For property
buyers, the protocol simplifies the purchase process, offering immediate
access to property details and streamlining the transfer of ownership.
Moreover, financial lenders find a new marketplace in which to offer secured
loans, augmented by blockchain’s security features.

##### Competitive Landscape and Advantages

In the competitive landscape, the proposed protocol differentiates itself by
presenting a comprehensive and integrated solution that extends beyond mere
transaction facilitation. It anticipates the current and future needs of the
real estate market, focusing on user experience, transactional integrity, and
market inclusivity. Against competitors like Propy, RealT, and Deedcoin, the
protocol asserts its edge through its amalgamation of transactional
efficiency, cost savings, and market reach.

##### Strategic Implementation and Economic Considerations

The practical implementation of the protocol on the Ethereum blockchain, and
the considerations for its economic viability, are outlined with a foresight
into potential scalability issues and transaction costs. Alternative
blockchains such as Binance Smart Chain, Solana, and Polkadot are appraised
for their suitability, with an emphasis on their comparative advantages in
terms of transaction speed, costs, and infrastructural development.

##### Implications and Future Prospects

In conclusion, the blockchain real-estate protocol promises to not only
revolutionize the manner in which real estate transactions are conducted but
also to serve as a blueprint for future applications of blockchain technology
in other domains. The synthesis of the protocol’s operational model, strategic
implementation, and competitive positioning underscores its potential to offer
a superior alternative to the established real estate transaction processes,
setting a new benchmark in efficiency, security, and global accessibility.

##  4 Conclusion

This paper has presented an in-depth examination of the cryptocurrency market,
followed by a pioneering proposal for a blockchain-based real estate
transaction protocol. Through meticulous analysis, it has provided evidence of
the intricate dynamics governing cryptocurrency price movements, volatility,
and correlations, particularly in the context of the FTX bankruptcy event,
thereby illuminating the vulnerabilities and resilience inherent in the
digital currency landscape. In tandem, it has offered a visionary blueprint
for the utilization of blockchain technology in streamlining real estate
transactions, proposing a model that is poised to redefine the sector.

###  4.1 Synthesis of Findings

The market analysis revealed that cryptocurrencies are not only highly
volatile but are also subject to correlated movements, which can lead to
systemic risks within the digital asset class. Yet, this volatility and
interconnectivity also underscore the potential of cryptocurrencies to
diversify investment portfolios when judiciously balanced with traditional
assets. The FTX bankruptcy served as a litmus test for the market, delineating
the stability offered by stablecoins and the S&P 500 in contrast to the
pronounced volatility of cryptocurrencies like Bitcoin and Dogecoin.

On the frontier of innovation, the blockchain real estate proposal detailed in
Part II of this paper is set to disrupt a long-established industry. By
removing intermediaries, reducing transaction costs, and enhancing
transparency, the protocol demonstrates a tangible application of blockchain
beyond speculative trading, embodying the technology’s transformative
potential in real-world asset management and exchange.

###  4.2 Prospects and Implications

The synthesis of the paper’s findings paints a nuanced picture of the
cryptocurrency market’s complexities and introduces a sophisticated approach
to real estate transactions that capitalizes on blockchain technology’s
strengths. As the cryptocurrency market continues to mature, it is anticipated
that investor strategies will adapt to encompass both traditional and digital
assets, ensuring balanced portfolios that mitigate risk while capitalizing on
growth opportunities.

The real estate blockchain protocol, while nascent, holds promise for a
radical shift in property ownership transfer, marking a significant leap
towards a more interconnected, efficient, and accessible global market. Its
implications extend beyond the real estate sector, signaling the advent of a
broader adoption of blockchain in various facets of commerce and governance,
ushering in a new era of decentralized digital solutions.

###  4.3 Summary and Forward Outlook

In summary, the research presented in this paper contributes meaningfully to
the understanding of cryptocurrencies and offers a progressive application of
blockchain technology. As we witness the convergence of traditional financial
methodologies with groundbreaking digital solutions, the potential for
innovation in both markets and technology is boundless. Future research and
development will undoubtedly continue to expand on these foundations, further
integrating the burgeoning possibilities of blockchain technology into the
fabric of societal and economic structures.

In moving forward, continuous monitoring of market trends, regulatory
developments, and technological advancements will be crucial in optimizing the
strategies and applications discussed herein. The cryptocurrency market’s
evolution and the blockchain real estate protocol’s maturation will
undoubtedly serve as critical barometers for the future trajectory of digital
finance and property transactions. The journey ahead promises to be as
challenging as it is exciting, with the potential to redefine the very essence
of investment, ownership, and exchange in an increasingly digital world.


# Remarks on Fixed Point Assertions in Digital Topology, 5

Laurence Boxer Department of Computer and Information Sciences, Niagara
University, NY 14109, USA; and Department of Computer Science and Engineering,
State University of New York at Buffalo
email<EMAIL_ADDRESS>
Paper accepted for publication in Applied General Topology

###### Abstract

As in [6, 3, 4, 5], we discuss published assertions concerning fixed points in
“digital metric spaces” - assertions that are incorrect or incorrectly proven,
or reduce to triviality.

MSC: 54H25

Key words and phrases: digital topology, fixed point, metric space

##  1 Introduction

As stated in [3]:

> The topic of fixed points in digital topology has drawn much attention in
> recent papers. The quality of discussion among these papers is uneven; while
> some assertions have been correct and interesting, others have been
> incorrect, incorrectly proven, or reducible to triviality.

Paraphrasing [3] slightly: in [6, 3, 4, 5], we have discussed many
shortcomings in earlier papers and have offered corrections and improvements.
We continue this work in the current paper.

Authors of many weak papers concerning fixed points in digital topology seek
to obtain results in a “digital metric space” (see section 2.1 for its
definition). This seems to be a bad idea. We quote [5]:

>   * •
>
> Nearly all correct nontrivial published assertions concerning digital metric
> spaces use either the adjacency of the digital image or the metric, but not
> both.
>
>   * •
>
> If $X$ is finite (as in a “real world” digital image) or the metric $d$ is a
> common metric such as any $\ell_{p}$ metric, then $(X,d)$ is uniformly
> discrete as a topological space, hence not very interesting.
>
>   * •
>
> Many of the published assertions concerning digital metric spaces mimic
> analogues for subsets of Euclidean ${\mathbb{R}}^{n}$. Often, the authors
> neglect important differences between the topological space
> ${\mathbb{R}}^{n}$ and digital images, resulting in assertions that are
> incorrect, trivial, or trivial when restricted to conditions that many
> regard as essential. E.g., in many cases, functions that satisfy fixed point
> assertions must be constant or fail to be digitally continuous [6, 3, 4].
>
>

Since the publication of [5], additional papers concerning fixed points in
digital metric spaces have come to our attention. This paper continues the
work of [6, 3, 4, 5] in discussing shortcomings of published assertions
concerning fixed points in digital metric spaces.

Many of the definitions and assertions we discuss were written with
typographical and grammatical errors, and mathematical flaws. We have quoted
these by using images of the originals so that the reader can see these errors
as they appear in their sources (we have removed or replaced with a different
style labels in equations and inequalities in the images to remove confusion
with labels in our text).

##  2 Preliminaries

Much of the material in this section is quoted or paraphrased from [5].

We use ${\mathbb{N}}$ to represent the natural numbers, ${\mathbb{Z}}$ to
represent the integers, and ${\mathbb{R}}$ to represent the reals.

A digital image is a pair $(X,\kappa)$, where $X\subset{\mathbb{Z}}^{n}$ for
some positive integer $n$, and $\kappa$ is an adjacency relation on $X$. Thus,
a digital image is a graph. In order to model the “real world,” we usually
take $X$ to be finite, although there are several papers that consider
infinite digital images. The points of $X$ may be thought of as the “black
points” or foreground of a binary, monochrome “digital picture,” and the
points of ${\mathbb{Z}}^{n}\setminus X$ as the “white points” or background of
the digital picture.

For this paper, we need not specify the details of adjacencies or of digitally
continuous functions.

A fixed point of a function $f:X\to X$ is a point $x\in X$ such that $f(x)=x$.

###  2.1 Digital metric spaces

A digital metric space [9] is a triple $(X,d,\kappa)$, where $(X,\kappa)$ is a
digital image and $d$ is a metric on $X$. The metric is usually taken to be
the Euclidean metric or some other $\ell_{p}$ metric. We are not convinced
that the digital metric space is a notion worth developing. Typically,
assertions in the literature do not make use of both $d$ and $\kappa$, so that
“digital metric space” seems an artificial notion. E.g., for a discrete
topological space $X$, all functions $f:X\to X$ are continuous, although on
digital images, many functions $g:X\to X$ are not digitally continuous
(digital continuity is defined in [2], generalizing an earlier definition
[15]).

##  3 Assertions for contractions in [11]

The paper [11] claims fixed point theorems for several types of digital
contraction functions. Serious errors in the paper are discussed below.

###  3.1 Fixed point for Kannan contraction

Figure 1 shows the definition appearing in [11] of a Kannan digital
contraction.

Figure 1: Definition of Kannan digital contraction in [11] Figure 2: Fixed
point assertion for Kannan digital contractions in [11]

Figure 2 shows a fixed point assertion for Kannan digital contractions in
[11]. The “proof” of this assertion has errors discussed below.

  * •

In the fourth and fifth lines of the “proof” is the claim that

$\lambda[d(x_{n},x_{n-1})+d(x_{n-1},x_{n-2})]\leq 2\lambda d(x_{n},x_{n-1}).$

Since $\lambda>0$, this claim implies

$d(x_{n-1},x_{n-2})\leq d(x_{n},x_{n-1}),$

so if any $d(x_{n},x_{n-1})$ is positive, the sequence $\\{x_{n}\\}$ is not a
Cauchy sequence, contrary to a claim that appears later in the argument.

  * •

Towards the end of the existence argument, the authors claim that a Kannan
digital contraction is digitally continuous. This assumption is contradicted
by Example 4.1 of [6].

In light of these errors, we must conclude that the assertion of Figure 2 is
unproven.

###  3.2 Example of pp. 10769 - 10770

This example is shown in Figure 3. One sees easily the following.

  * •

$d(0,1)=d(0,2)=0$, so $d$, contrary to the claim, is not a metric.

  * •

$T(1)=1/2\not\in X$.

Figure 3: The example of [11], pp. 10769 - 10770

###  3.3 Fixed point for generalization of Kannan contraction

Figure 4 shows an assertion of a fixed point result on p. 10770 of [11].

Figure 4: A “theorem” of [11], p. 10770

Note “And let $f$ satisfies” should be “and let $T$ satisfy”.

More importantly: In the argument offered as proof of this assertion, the
authors let $x_{0}\in X$, and, inductively,

$x_{n+1}=T(x_{n}),~{}~{}~{}~{}~{}a_{n+1}=d(x_{n},x_{n+1}).$

They claim that by using the statements marked (i) and (ii) in Figure 4, it
follows that

$a_{n+1}=d(x_{n},x_{n+1})\leq\Upsilon(d(x_{n},x_{n-1})+d(x_{n-1},x_{n-2}))$
$<2d(x_{n},x_{n-1})=2a_{n},$ (1)

which, despite the authors’ claim, does not show that the sequence
$\\{a_{n}\\}$ is decreasing. However, what correctly follows from the
statements marked (i) and (ii) in Figure 4 is

$a_{n+1}=d(x_{n},x_{n+1})=d(T(x_{n-1}),T(x_{n}))\leq$
$\Upsilon(d(x_{n-1},T(x_{n-1}))+d(x_{n},T(x_{n})))=\Upsilon(d(x_{n-1},x_{n})+d(x_{n},x_{n+1}))$
$=\Upsilon(a_{n}+a_{n+1})<\frac{a_{n}+a_{n+1}}{2}.$ (2)

From this we see that $a_{n+1}<a_{n}$, so the sequence $\\{a_{n}\\}$ is
decreasing and bounded below by 0, hence tends to a limit $a\geq 0$.

The authors then claim that if $a>0$ then $a_{n+1}\leq Y(a_{n})$. However,
what we showed in (2) does not support this conclusion, which is not justified
in any obvious way. Since the authors wish to contradict the hypothesis that
$a>0$ in order to derive that the sequence $\\{x_{n}\\}$ is a Cauchy sequence,
we must regard the assertion shown in Figure 4 as unproven.

###  3.4 Fixed point for Zamfirescu contraction

A Zamfirescu digital contraction is defined in Figure 5. This notion is used
in [11, 14], and will be discussed in the current section and in section 6.

Figure 5: Definition 3.2 of [13], used in [11, 14].

Figure 6 shows an assertion found on p. 10770 of [11].

Figure 6: Another “theorem” of [11], p. 10770

The argument offered as “proof” of this assertion considers cases. For an
arbitrary $x_{0}\in X$, a sequence is inductively defined via
$x_{n+1}=T(x_{n})$. For convenience, let us define

$M(x,y)=\max\left\\{d(x,y),\frac{d(x,Tx)+d(y,Ty)}{2},\frac{d(x,Ty)+d(y,Tx)}{2}\right\\}.$

The argument considers several cases.

  * •

Case 1 says if $d(x_{n+1},x_{n})=M(x_{n+1},x_{n})$ then, by implied induction,

$d(x_{n+1},x_{n})\leq\lambda^{n}d(x_{1},x_{0}).$

But this argument is based on the unproven assumption that this is also the
case for all indices $i<n$; i.e., that $d(x_{i+1},x_{i})=M(x_{i+1},x_{i})$.

  * •

Case 2 says if

$\frac{d(x_{n+1},Tx_{n+1})+d(x_{n},Tx_{n})}{2}=$
$\max\left\\{\begin{array}[]{l}d(x_{n+1},x_{n}),\frac{d(x_{n+1},Tx_{n+1})+d(x_{n},Tx_{n})}{2},\\\
\\\ \frac{d(x_{n+1},Tx_{n})+d(x_{n},Tx_{n_{1}})}{2}\end{array}\right\\},$

then

$d(x_{n+1},x_{n})=d(Tx_{n},Tx_{n-1})\leq\lambda\frac{d(x_{n},x_{n-1})+d(x_{n-1},x_{n-2})}{2}\leq$
$\lambda d(x_{n},x_{n-1}).$

But in order for the second inequality in this chain to be true, we would need
$d(x_{n-1},x_{n-2})\leq d(x_{n},x_{n-1})$, and no reason is given to believe
the latter.

  * •

Case 3 says that if
$\frac{d(x_{n+1},Tx_{n})+d(x_{n},Tx_{n+1})}{2}=M(x_{n+1},x_{n})$ then

$d(x_{n+1},x_{n})=d(Tx_{n},Tx_{n-1})\leq\lambda\frac{d(x_{n},x_{n-2})+d(x_{n-1},x_{n-1})}{2}.$

The correct upper bound, according to the definition shown in Figure 5, is

$\lambda\frac{d(x_{n+1},Tx_{n})+d(x_{n},Tx_{n+1})}{2}=\lambda\frac{d(x_{n+1},x_{n+1})+d(x_{n},x_{n+2})}{2}=$
$\lambda\frac{d(x_{n},x_{n+2})}{2}.$

Further, the conclusion reached by the authors for this case, that the
distances $d(x_{n+1},x_{n})$ are bounded above by an expression that tends to
0 as $n\to\infty$, depends on the unproven hypothesis that an analog of this
case holds for all indices $i<n$.

Thus all three cases considered by the authors are handled incorrectly. We
must conclude that the assertion of Figure 6 is unproven.

###  3.5 Fixed point for Rhoades contraction

Figure 7 shows the definition appearing in [11] of a Rhoades digital
contraction. The paper [11] claims the fixed point result shown in Figure 8
for such functions. The argument offered in “proof” of this assertion has
errors that are discussed below.

Figure 7: Definition of Rhoades digital contraction, [11], p. 10769  Figure 8:
Fixed point assertion for Rhoades digital contraction, [11], p. 10771

For convenience, let

$M(x,y)=\max\left\\{d(x,y),~{}\frac{d(x,Tx)+d(y,Ty)}{2},~{}d(x,Ty),~{}d(y,Tx)\right\\}.$

The authors’ argument considers cases corresponding to which of the embraced
expressions above gives the value of $M(x_{n+1},x_{n})$. In each case, the
authors assume without proof that the same case is valid for
$M(x_{i+1},x_{i})$, for all indices $i<n$.

Additional errors:

  * •

In case 2, the inequality

$d(Tx_{n},Tx_{n-1})\leq\lambda\frac{d(x_{n},x_{n-1})+d(x_{n-1},x_{n-2})}{2}$

should be, according to Figure 7,

$d(x_{n},x_{n+1})=d(Tx_{n},Tx_{n-1})\leq\lambda\frac{d(x_{n},Tx_{n})+d(x_{n+1},Tx_{n-1})}{2}$
$=\lambda\frac{d(x_{n},x_{n+1})+d(x_{n+1},x_{n})}{2}=\lambda
d(x_{n},x_{n+1}).$

Note this implies

$x_{n}=x_{n+1},$ (3)

which would imply the existence of a fixed point. Also, the authors claim that

$\lambda\frac{d(x_{n},x_{n-1})+d(x_{n-1},x_{n-2})}{2}\leq\lambda
d(x_{n},x_{n-1}),$

which is equivalent to

$d(x_{n-1},x_{n-2})\leq d(x_{n},x_{n-1}).$

No reason is given in support of the latter; further, it undermines the later
claim that $\\{x_{n}\\}$ is a Cauchy sequence, since the authors did not
deduce (3).

  * •

In case 3, it is claimed that

$\lambda d(x_{n},Tx_{n-1})\leq\lambda d(x_{n},x_{n-1}).$

This should be corrected to

$\lambda d(x_{n},Tx_{n-1})=\lambda d(x_{n},x_{n})=0,$

which would guarantee a fixed point.

  * •

In case 4, we see the claim

$d(x_{n-1},Tx_{n})=d(x_{n-1},x_{n-1})=0.$

This should be corrected to

$d(x_{n-1},Tx_{n})=d(x_{n-1},x_{n+1}).$

In view of these errors, we must regard the assertion shown in Figure 8 as
unproven.

##  4 Assertions for weakly compatible maps in [1]

In this section, we show that the assertions of [1] are trivial or incorrect.

###  4.1 Theorem 3.1 of [1]

######  Definition 4.1.

[8] Let $S,T:X\to X$. Then $S$ and $T$ are weakly compatible or coincidentally
commuting if, for every $x\in X$ such that $S(x)=T(x)$ we have
$S(T(x))=T(S(x))$.

The assertion stated as Theorem 3.1 in [1] is shown in Figure 9. In this
section, we show the assertion is false except in a trivial case.

Note if $d$ is any $\ell_{p}$ metric (including the usual Euclidean metric)
then the requirements of closed subsets of $X$ are automatically satisfied,
since $(X,d)$ is a discrete space.

######  Theorem 4.2.

If functions $G,H,P,Q$ satisfy the hypotheses of Theorem 3.1 of [1] then
$G=H=P=Q$. Therefore, each of the pairs $(P,G)$ and $(Q,H)$ has a unique
common point of coincidence if and only if $N$ consists of a single point.

###### Proof.

We observe the following.

Figure 9: The assertion stated as Theorem 3.1 of [1]

  * •

The inequality in the assertion simplifies as

$\Psi(d(Px,Qy))\leq-\,\frac{1}{2}\Psi(d_{G,H}(x,y)).$

Since the function $\Psi$ is non-negative, we have

$\Psi(d(Px,Qy))=\Psi(d_{G,H}(x,y))=0.$ (4)

Therefore, we have

$P(x)=Q(y)\mbox{ for all }x,y\in N.$ (5)

  * •

In the equation for $d_{G,H}$ in Figure 9, the pairs of points listed on the
right side should be understood as having $d$ applied, i.e.,

$d_{G,H}(x,y)=\max\left\\{\begin{array}[]{c}d(G(x),H(y)),d(G(x),P(x)),d(H(y),Q(y)),\\\
\frac{1}{2}[d(G(x),Q(y))+d(H(y),P(x))]\end{array}\right\\}.$ (6)

Since $\Psi(x)=0$ if and only if $x=0$, we have from (4) that
$d_{G,H}(x,y)=0$, so (5) and (6) imply

$G(x)=P(x)=Q(x)=H(x)\mbox{ for all }x\in N.$

We conclude that $(P,G)$ and $(Q,H)$ are pairs of functions whose respective
common points of coincidence are unique if and only if $N$ consists of a
single point. ∎

###  4.2 Example 3.1 of [1]

In Figure 10, we see the assertion presented as Example 3.1 of [1].

Figure 10: The assertion stated as Example 3.1 of [1]

Note the following.

  * •

If “$X=[4,40]$” is meant to mean the real interval from 4 to 40, then $X$ is
not a digital image, as all coordinates of members of a digital image must be
integers. Perhaps $X$ is meant to be the digital interval
$[4,40]_{{\mathbb{Z}}}=[4,40]\cap{\mathbb{Z}}$.

  * •

The function $G$ is not defined on all of $[4,40]_{{\mathbb{Z}}}$, appears to
be doubly defined for some values of $x$, (notice the incomplete inequality at
the end of the 3rd line) and is not restricted to integer values.

  * •

The function $H$ is not defined on all of $[4,40]_{{\mathbb{Z}}}$ and is
doubly defined for $x\in\\{13,14\\}$.

  * •

The function $Q$ is not defined for $x\in\\{9,10,11,12\\}$, and is doubly
defined for $x\in\\{14,15\\}$.

  * •

The “above theorem” referenced in the last sentence of Figure 10 is the
assertion discredited by our Theorem 4.2, which shows that $P=Q=G=H$. Clearly,
the assertion shown in Figure 10 fails to satisfy the latter.

Thus, Example 3.1 of [1] is not useful.

###  4.3 Corollary 3.2 of [1]

The assertion presented as Corollary 3.2 of [1] is presented in Figure 11.
Notice that “weakly mappings” is an undefined phrase. No proof is given in [1]
for this assertion, and it is not clear how the assertion might follow from
previous assertions of the paper (which, as we have seen above, are also
flawed).

Perhaps “weakly mappings” is intended to be “weakly compatible mappings”. At
any rate, by labeling this assertion as a Corollary, the authors suggest that
it follows from the paper’s flawed “Theorem” 3.1.

The assertion presented as “Corollary” 3.2 of [1] must be regarded as
undefined and unproven.

Figure 11: The assertion stated as Corollary 3.2 of [1]

##  5 Assertion for coincidence and fixed points in [12]

Figure 12: The assertion presented as Theorem 4 of [12]

Figure 12 shows the assertion presented as Theorem 4 of [12]. The assertion as
stated is false. Flaws in this assertion include:

  * •

“$D$” apparently should be “$L$”, and “$x_{x}$” apparently should be “$x$”.

  * •

No restriction is stated for the value of $h$. Therefore, we can take $h=0$,
leaving the inequality in i) as $b(fx,fy)\geq 0$; since $b$ is a metric, this
inequality is not a restriction. Thus $f$ and $g$ are arbitrary; they need not
have a coincidence point or fixed points.

##  6 Assertion for Zamfirescu contractions in [14]

Let $f:X\to X$, where $(X,d,\kappa)$ is a digital metric space. Recall that a
Zamfirescu digital contraction [13] is defined in Figure 5.

We show the assertion presented as Theorem 4.1 of [14] in Figure 13.

Figure 13: The assertion presented as Theorem 4.1 of [14]

Observe:

  * •

The symbol $\theta$ has distinct uses in this assertion. In the first line,
$\theta$ is introduced as both the metric and the adjacency of $X$. Since our
discussion below does not use an adjacency, we will assume $\theta$ is the
metric $d$.

  * •

The symbol $\varphi$ seems intended to be a real number satisfying some
restriction, but no restriction is stated. Alternately, it may be that
$\varphi$ is intended to be a function to be applied to the $Max$ value in the
statement, but no description of such a function appears.

Perhaps most important, we have the following.

######  Theorem 6.1.

If $Y$ has more than one point and $d$ is any $\ell_{p}$ metric, then no
function $T$ satisfies the hypotheses shown in Figure 13.

###### Proof.

Suppose there is such a function $T$. By choice of $d$, there exist
$u_{0},v_{0}\in X$ such that

$d(u_{0},v_{0})=\min\\{d(x,y)~{}|~{}x,y\in X,x\neq y\\}.$

By the inequality stated in item 1 of Figure 13, $d(Tu_{0},Tv_{0})=0$. This
contradicts the hypothesis that $T$ is injective. ∎

##  7 Assertion for expansive map in [7]

The paper [7] claims to have a fixed point theorem for digital metric spaces.
However, it is not clear what the authors intend to assert, as the paper has
many undefined and unreferenced terms and many obscuring typographical errors.

The assertion stated as “Preposition” 2.7 of [7] (and also as an unlabeled
Proposition on p. 10769 of [11]) was shown in Example 4.1 of [6] to be false.

Figure 14: The statement presented as Definition 3.1 of [7]

The definition of what this paper calls a Generalized ($\alpha~{}-~{}\phi$) -
$\psi$ expansive mapping for random variable is shown in Figure 14. We observe
the following.

  * •

This is not the same as a $\beta~{}-~{}\psi~{}-~{}\phi$ expansive mapping
defined in [10].

  * •

Notice the $\rho$ that appears intended to be the adjacency of the digital
metric space. This is significant in our discussion later.

  * •

The set $\Psi$ is not defined anywhere in the paper. Perhaps it is meant to be
the set $\Psi$ of [10].

  * •

The functions $d$, $\alpha$, and $M$ all have a third parameter $a$ that
appears to be extraneous, since each of these functions is used later with
only two parameters.

  * •

One supposes $\mu$ and $\omega$ must be non-negative, but this is not stated.

The assertion presented as Theorem 3.3 of [7] is shown in Figure 15.

Figure 15: The assertion presented as Theorem 3.3 of [7]

In the statement of this assertion:

  * •

“DGMS” appears, although it is not defined anywhere in the paper. Perhaps it
represents “digital metric space”.

  * •

The term “exclusive RFP” is not defined anywhere in the paper. One supposes
the “FP” is for “fixed point”.

In the argument offered as “Verification” of this assumption, we note the
following.

  * •

The second line of the verification contains an undefined operator,  $+$ $n$
which perhaps is meant to be $+$.

  * •

The same line contains part of the phrase “$u_{n}$ is a unique point of $S$.”
What the authors intend by this is unclear.

  * •

At the start of the long statement (3e), it is claimed that $M(\xi u_{n},\xi
u_{n+1})$ is the maximum of three expressions. The second term of the
expression for $M(\xi u_{n},\xi u_{n+1})$ applies $\rho$ to a numeric
expression. This makes no sense, since $\rho$ is the adjacency of $Y$ (see
Figure 14). Notice also that Figure 14 shows no such term in its expression
for the function $M$. The use of $\rho$, as a numeric value that has neither
been defined nor restricted to some range of values, propagates through both
of the cases considered.

  * •

In the expression for $M(\xi u_{n},\xi u_{n+1})$, the third term, $\omega(\xi
u_{n},\xi u_{n-1})$, should be $\omega d(\xi u_{n},\xi u_{n-1})$ according to
Figure 14. This error repeats several times in statement (3e).

Other errors are present, but we have established enough to conclude that
whatever the authors were trying to prove is unproven.

##  8 Further remarks

We have shown that nearly every assertion introduced in the papers [11, 1, 12,
14, 7] is incorrect, unproven due to errors in the “proofs,” or trivial. These
papers are part of a larger body of highly flawed publications devoted to
fixed point assertions in digital metric spaces, and emphasize our contention
that the digital metric space is not a worthy subject of study.

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