Contraction Principle Research Articles

New Results on the Ulam–Hyers–Mittag–Leffler Stability of Caputo Fractional-Order Delay Differential Equations

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability of the same equation in a closed interval using the Picard operator, Chebyshev norm, Bielecki norm and the Banach contraction principle. Finally, we present three examples to show the applications of our results. Although there is an extensive literature on the Lyapunov, Ulam and Mittag–Leffler stability of fractional differential equations (FrDEs) with and without delays, to the best of our knowledge, there are very few works on the UHML stability of FrDEs containing a delay. Thereby, considering a CFrDDE containing multiple variable delays and obtaining new results on the existence and uniqueness of the solutions and UHML stability of this kind of CFrDDE are the important aims of this work.

Investigating the existence, uniqueness, and stability of solutions in boundary value problem of fractional differential equations

This study uses fixed point theory and the Banach contraction principle to prove the existence, uniqueness, and stability of solutions to boundary value problems involving a Ψ-Caputo-type fractional differential equation. The conclusions are supported by illustrative cases, which raise the theoretical framework’s legitimacy. Fractional calculus is widely used in scientific fields, as seen by its applications in beam deflection analysis, groundwater pollution, and biomedical signal processing.

A Comprehensive Study of the Langevin Boundary Value Problems with Variable Order Fractional Derivatives

The authors investigate Langevin boundary value problems containing a variable order Caputo fractional derivative. After presenting the background for the study, the authors provide the definitions, theorems, and lemmas that are required for comprehending the manuscript. The existence of solutions is proved using Schauder’s fixed point theorem; the uniqueness of solutions is obtained by adding an additional hypothesis and applying Banach’s contraction principle. An example is provided to demonstrate the results.

Uniqueness and Hyers–Ulam’s stability for a fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition

Abstract Introducing a pair-parameter matrix Mittag–Leffler function, we study the uniqueness and Hyers–Ulam stability to a new fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition using Banach’s contractive principle as well as Babenko’s approach in a Banach space. These investigations have serious applications since uniqueness and stability analysis are essential topics in various research fields. The techniques used also work for different types of differential equations with initial or boundary conditions, as well as integral equations. Moreover, we present a Python code to compute approximate values of our newly established pair-parameter matrix Mittag–Leffler functions, which extend the multivariate Mittag–Leffler function. A few examples are given to show applications of the key results obtained.

New investigation on controllability of sobolev-type Volterra-Fredholm functional integro-differential equation with non-local condition

In this article, we analyse the controllability criteria on sobolev-type Volterra-Fredholm functional Integro-differential Equations (SVFIDE) via Atangana–Baleanu Caputo (ABC) derivative. The primary outcomes was established by utlising the concepts on the measure of noncompactness, semigroup theory, the contraction principle and Darbo fixed point technique with nonlocal condition. An illustrative example to demonstrate the analytical result is provided at the end.

The Unidroit Principles and the inflexibility of code as law: reconciling smart contracts with general clauses

Abstract Smart contracts are a set of digital technologies enabling the automation of certain aspects of contractual performance and/or remedies. These technologies follow a deterministic, if/then logic that is often difficult to reconcile with the inherent malleability of contract law. This article scrutinizes this tension, with specific reference to the International Institute for the Unification of Private Law’s (Unidroit) Principles of International Commercial Contracts (PICC). The article argues that some recurring PICC general clauses (such as good faith and reasonableness) play a crucial role in ensuring fairness. Therefore, their operation should not be constrained by smart contracts. To the contrary, the parties to an international commercial contract should use technology to empower arbitrators to make full use of the flexibility inherent to the PICC and issue (partially) self-enforcing awards.

A brief survey on the development and applications of Goebel’s coincidence point theorem in differential and integral equations

Abstract Goebel’s coincidence theorem, a remarkably simple extension of Banach’s contraction principle widely applied in analysis, has found significant utility in the theory of differential and integral equations. Over the past fifty years, researchers have endeavored to generalize the definition of a metric space, thereby extending the scope of Goebel’s coincidence theorem to diverse settings. This survey paper overviews valuable insights into Goebel’s coincidence theorem’s historical background, its relevance in the field of fixed point theory, and its practical implications in solving problems related to differential and integral equations.

Justice for the Parties to the Standard Agreement Relating to the Freedom Contract Principle

This research investigates the considerations necessary for creating a fair standard contract and implementing the principle of freedom of contract. A standard contract, typically drafted by one party with predetermined terms, is crucial in commercial transactions. Despite the principle of freedom of contract, ensuring fairness for all parties involved remains a challenge. The method employed is normative juridical research, utilizing secondary data sources. The findings showed that the standard contracts may not violate the principle of freedom of contract outlined in Article 1320 and Article 1338 of the Civil Code, conflicts may arise, particularly due to restrictive clauses like exoneration clauses. Parties have the right to review the contract and accept or reject its terms. However, disparities persist in standard contract formation, highlighting the need for further examination of their fairness and compliance with legal principles

On the existence, uniqueness, stability, and numerical aspects for a novel mathematical model of HIV/AIDS transmission by a fractal fractional order derivative

In this study, we explore a mathematical model of the transmission of HIV/AIDS. The model incorporates a fractal fractional order derivative with a power-law type kernel. We prove the existence and uniqueness of a solution for the model and establish the stability conditions by employing Banach’s contraction principle and a generalized α-ψ-Geraghty type contraction. We perform stability analysis based on the Ulam–Hyers concept. To calculate the approximate solution, we utilize Gegenbauer polynomials via the spectral collocation method. The presented model includes two fractal and fractional order derivatives. The influence of the fractional and fractal derivatives on the outbreak of HIV is investigated by utilizing real data from the Cape Verde Islands in 1987–2014.

The right to avoid the international sale of goods contract: Comparison of Viet Nam to CISG and PICC

Foreign trade is an indispensable activity in the globalization era. Researching the liabilities of parties in performing the sale contracts and the avoidance right of parties in the contracts is a vital issue. Therefore, this article compared Vietnamese Commercial Law 2005 with the United Nations Convention for the International Sale of Goods (CISG) and Principles of International Commercial Contracts (PICC) regarding the avoidance of sale contracts, including the three aspects, fundamental breach, failure in performance obligations of contract at the end of adding time, and anticipatory breach. The findings not only enable Vietnamese merchants to master international rules in establishing foreign trading but also show differences and similarities between Vietnamese law and international rules. By doing that, this article puts forward some implications for rebuilding the definition of “fundamental breach” in Vietnamese commercial law. Also, this paper suggested that the avoidance clause of the Vietnamese law should include the claim to terminate the agreement of the innocent party due to the non-performance party’s failure to fulfill contractual duties at the end of adding time, and the other party will be constituted an “anticipatory breach”.

Modeling and dynamics of measles via fractional differential operator of singular and non-singular kernels

Young children frequently die from measles, which is a major global health concern. Despite being more prevalent in infants, pregnant women, and people with compromised immune systems, it can infect anyone. Novel fractional operators, the constant-proportional Caputo operator, and the constant-proportional Atangana–Baleanu operator are used to create a hybrid fractional order model that helps analyze the dynamic transmission of the measles virus. We assess the measles-free and endemic equilibrium, reproductive number, biological viability, boundedness, well-posedness, and positivity of the model. We apply the Banach contraction principle to verify the uniqueness of the system’s solutions. The proposed system is confirmed to be Ulam–Hyres stable by using fixed point theory results. The aforementioned operators are further analyzed, and the Laplace-Adomian decomposition method is used to numerically simulate the system of fractional differential equations. To support our findings, the outcomes are graphically displayed. The efficacy and memory impact of fractional operators are illustrated through comparisons. Based on fractional parameter values, the study determines important disease-control strategies and shows that vaccinations greatly reduce the spread of measles. By reducing the number of infected people, increasing vaccination coverage lowers the burden of disease on the general population.

On the ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document}-Caputo Impulsive p-Laplacian Boundary Problem: An Existence Analysis

Due to the importance of some physical systems, in this paper, we aim to investigate a generalized impulsive ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document}-Caputo differential equation equipped with a p-Laplacian operator. In fact, our problem is a generalization of fractional differential equations equipped with the integral boundary conditions, impulsive forms and p-Laplacian operators under the Nemytskii operators. In this direction, we prove some theorems on the existence property along with the uniqueness of solutions under the Nemytskii operator. More precisely, we use the Schauder’s and Schaefer’s fixed point theorems, along with the Banach contraction principle. In the sequel, two examples are provided to show the validity of the obtained results in practical.

Utilitarianism of Claude Adrian Helvetius as one of the philosophical origins of urbanism

The article reveals the connection between the ideas of the Enlightenment and the modern direction of scientific knowledge, urban studies. Urban studies as a concept of the social development of the city has, among other things, the understanding of interest in the interpretation of Helvetius. He thinks that interest is the driving force of society, the nature of which he considers from the standpoint of sensationalism and utilitarianism. Urban studies solve the problems of interaction between a person and a city, which are based on the philosophical problem of interaction between the individual and the general. The purpose of the work is to trace in the ethics of Helvetius the combination of reasonable egoism and the principles of public benefit; the irremovability of natural prerequisites in man preserves the individual in society, and the irremovability of contractual principles makes it necessary to observe the public interest. The article determines that the recognition of the right to personal interest is a great merit of Helvetius and one of the philosophical foundations of urban studies.

When the goose stops laying golden eggs: supervening loss of trust in long-term contracts under the PICC

Purpose The purpose of this paper is to examine the supervening loss of inter-organisational trust in long-term commercial contracts. The underlying research question is whether contract law – the legal institution regulating economic exchanges – should intervene and enable a party to a long-term commercial contract to extricate itself from a situation where a relationship of trust has broken down irretrievably. Design/methodology/approach This paper uses doctrinal methodology and theoretical conceptualisation to answer the underlying research question. The legal instrument chosen for analysis purposes is the UNIDROIT Principles of International Commercial Contracts. This paper also draws on extant literature on inter-organisational trust (including conceptual and empirical studies) to support the arguments and propositions. Furthermore, this study proceeds to assess the substantive justifiability of the proposed remedial measure using four normative values: legal certainty and predictability, protection of the performance interest, economic efficiency and the preservation of the relation. Findings The central argument put forward in this paper is the reformulation of draft Article 6.3.1 proposed by the UNIDROIT Working Group on Long-Term Contracts, which confers a novel right to terminate for a compelling reason. This paper presents a multidimensional model of inter-organisational trust that would serve as the conceptual framework for the proposed reformulation of the provision and establishes a coherent juridical basis for the legal solution that would accord with the Principles of International Commercial Contracts’ general remedial scheme. As for the normative assessment, this paper demonstrates that the proposed remedial measure would significantly promote efficient outcomes and positively serve the norms of legal certainty, protection of the performance interest and the preservation of the relation. Originality/value This paper addresses the lacuna in current legal scholarship in relation to the adverse socio-economic effects following trust violation and deterioration in inter-organisational relationships. Additionally, the propositions and findings should contribute to the workings of the UNIDROIT in adopting new rules and principles that would serve the special requirements of cross-border trade.

Global exponential convergence and synchronization for exponential numerical competitive neural networks with different time scales and fuzzy logic

For nonlocal fuzzy delayed competitive neural networks, a novel difference model is first developed in this article using the Mittag-Leffler Euler difference method. Secondly, the existence of a sole globally bounded solution and globally exponential stability to the discrete-time networks are addressed based on Banach contractive mapping principle and the constant variation formula for sequences. Besides, exponential synchronization of the discrete-time networks is investigated and a feedback controller is designed from the viewpoint of the theory of controls. It is the first time to consider nonlocal fuzzy delayed competitive neural networks by using Mittag-Leffler Euler differences. In contrast to nonlocal discrete approaches, such difference competitive neural networks could preferably figure the features of continuous networks. The difference method in this paper extends and improves the traditional difference approaches, for example, Adams-Bashforth Euler differences. Moreover, this article explores some avenue for the research of discrete-time fractional-order systems and develops a set of theories and methodologies for the future investigations.

The effect of curative and preventive optimal control measures on a fractional order plant disease model

In this paper, we present a novel mathematical model of fractional order for epidemic dynamics in plants, this fractional order model (FOM) in the sense of Caputo derivatives governing fractional differential equations (FDEs), introducing modified parameters to coincide both sides dimensions for FOM that means enhancing its accuracy in representing the real-life scenarios to control the spread of the epidemic. Our contributions include a comprehensive qualitative analysis of the proposed FOM, such as proving the existence and uniqueness of the projected solution by transforming our problem into a fixed point problem and applying a Banach contraction principle and Schauder’s fixed-point theorem. The positivity and boundedness of this solution are also demonstrated. We precisely evaluate all equilibrium points (EPs), examine their local and global stability by using Routh–Hurwitz conditions and fractional LaSalle’s invariance principle (LIP), respectively and shed light on the dynamic behavior of the FOM. Furthermore, we discuss the sensitivity analysis for parameters of the control reproduction number (CRN), with insightful plots to illustrate the model’s response to parameter changes (replanting and roguing parameters). From this foundation, we formulate a fractional optimal control problem (FOCP) by incorporating preventive u1 and curative u2 controls to effectively eliminate the propagation of the epidemic in plants. The fractional necessary optimality conditions (FNOCs) are derived attribution to Pontryagin’s maximum principle (PMP). We employ the forward–backward sweep method (FBSM) based on fractional Euler method (FEM) in order to facilitate control implementation and show the different suggested strategies. By enhancing this method, optimal control efforts are projected onto the FOM and yield three various strategies to impact the epidemic’s trajectory. For each strategy, we explain how the presence (i.e. with control cases) and absence (without control cases) of the proposed controls affected the susceptible, protected and infected plants. Some attractive figures with various values of the weight factor and fractional order α for the CRN are also presented, allowing a visual assessment of their impact on the dynamics of the epidemic in plants. In addition, we calculate the objective function for controlled and uncontrolled cases to provide a quantitative measure of its effectiveness in containing the outbreak in the plants. The results gained from our analyses and simulations provide valuable guidance for the management and elimination of epidemics in plants by offering various scenarios to implement the proposed controls.

Stability and Numerical Analysis of a Coupled System of Piecewise Atangana–Baleanu Fractional Differential Equations with Delays

This paper focuses on using piecewise derivatives to simulate the dynamic behavior and investigate the crossover effect within the coupled fractional system with delays by dividing the study interval into two subintervals. We establish and prove significant lemmas concerning piecewise derivatives. Furthermore, we extend and develop the necessary conditions for the existence and uniqueness of solutions, while also investigating the Hyers–Ulam stability results of the proposed system. The results are derived using the Banach contraction principle and the Leary–Schauder alternative fixed-point theorem. Additionally, we employ a numerical method based on Newton’s interpolation polynomials to compute approximate solutions for the considered system. Finally, we provide an illustrative example demonstrating our theoretical conclusions’ practical application.

What does Covid-19 teach us about English contract law?

AbstractThis paper examines how English courts have responded to the contract problems generated by the Covid-19 pandemic and considers what this tells us about future contract law development. In relation to consumers, the case law on pandemic-affected contracts, though limited, indicates that traditional contract doctrine does not necessarily produce beneficial outcomes for consumers. This further diminishes the importance of the common law in the consumer contracting context. In the commercial sector, contracting parties were encouraged by government and other organisations to co-operate with one another and act in good faith during the crisis, but this has not influenced the courts applying contract law in the pandemic aftermath. The emerging case law suggests that contract law has retained its commitment to certainty, freedom of contract and sanctity of contract, notwithstanding the extraordinary circumstances around the outbreak and its unpredictable effects on contracts. The unalloyed application of formal contract law in the post-pandemic case law augments the position of relational norms as extra-contractual in English law, putting the further judicial development of relational contract principles in doubt. The paper concludes that despite the considerable social and economic upheaval caused by the pandemic, its impact on contract law development is likely to be minimal.

Relatively exact controllability of fractional stochastic neutral system with two incommensurate constant delays

This paper is devoted to analyzing a kind of fractional stochastic neutral system (FSNS). Firstly, by introducing the notion of newly defined two‐parameter Mittag–Leffler matrix function, we derive the solution of the corresponding linear stochastic system. Subsequently, for the linear case, by virtue of the Grammian matrix, we give a suffcient and necessary condition to guarantee the relatively exact controllability for the addressed case. Furthermore, for the nonlinear one, the relatively exact controllability is obtained by fixed point and explore it via Banach contraction principle. Finally, two examples are provided to intensify our theoretical conclusions.

The Position of Smart Contracts in the Light of Islamic Contract Theory

Smart contracts are simply programs stored in a blockchain that run under predetermined conditions; however, they are yet to be implemented commercially in the financial industry, including the Islamic financial industry. It has not been entirely implemented in the Islamic financial industry because it is unstable and there are debates regarding its conformity with Shariah principles. Since the development of the smart contract is still in the preliminary stages, its position in an Islamic contract is yet to be determined. Does a smart contract blockchain comply with Islamic contract theory? This qualitative study aims to analyse the smart contract’s position based on Islamic contract theory. Data were obtained using content analysis and interview methods, in which the semi-structured interview involved Islamic financial experts and industryplayers. Data were then analysed using the QDA Miner version 5.0.31 software. Findings indicate that a smart contract differs from other contracts because it records every transaction using hash cryptography and computer codes known as solidity. Besides that, transactions did not adhere to two principles of an Islamic contract, namely the existence of autonomy in the contracting parties and the ability to manipulate the contract. Hence, Shariah-based risks in a smart contract can be decreased by improving the Shariah compliance aspect in the transaction to solve autonomy issues and the manipulation of contracts. The study implies that a smart contract has the potential to become an innovation in the Islamicfinancial industry if it can adhere to the principles of an Islamic contract and it can be monitored by relevant authorities.