This paper develops an extension of the research work by Proinov to the product spaces X|I| (I is representing an indexing set). This paper also introduces a novel class of (L;Y)-contractions defined on a supremum metric (X|I|,d′). In supremum metric (X|I|,d′), several new fixed point theorems for (L;Y)-contractions have been established that generalize well-known ideas like the Banach, Geraghty, Boyd–Wong, and Wardowski principles. This paper also contributes a new iterated function system (IFS) built on the family of (L;Y)-contractions and demonstrates the existence and uniqueness of fractals (in other words, compact attractors) in a complete supremum metric (X|I|,d′). Theoretical work is illustrated with examples and graphs.

This study aims to analyze the implementation of musyarakah contracts in the mertelu farming system at the Sido Makmur Farmer Group in Sabrang Village, as well as to evaluate the level of farmer welfare through the perspective of maqashid syariah. This study uses a qualitative case study approach, collecting data through observation, documentation, and in-depth interviews with landowners and cultivators. The results show that the mertelu practice (1:3 profit sharing) has substantially fulfilled the principles of muzaraah and mukhabarah contracts. From the perspective of maqashid syariah, the welfare of tenant farmers has been partially achieved, especially in terms of religious preservation (hifz ad-din) and property preservation (hifz al-mal). However, this welfare is still vulnerable in terms of preserving life (hifz an-nafs) and the sustainability of wealth due to dependence on verbal agreements, the absence of written risk mitigation for crop failure, and market price fluctuations. This study concludes that although the mertelu system provides benefits for farmers, formal contracts are needed to ensure legal certainty and more stable economic protection for tenant farmers in the future.

Abstract Fractional iterative differential equations with boundary conditions (FIDEs-wBC) have been mainly studied using the classical Caputo fractional derivative or within the framework of initial value problems. In this paper, we investigate the existence, uniqueness, and Hyers–Ulam stability of solutions for FIDEs-wBC defined by the $\mu$-Caputo fractional derivative. This derivative provides a unified framework that includes several classical fractional derivatives as special cases through suitable choices of the function $\mu$. As a result, this approach allows us to study a broader class of fractional iterative boundary value problems that has not been fully considered in the existing literature. By using the Banach contraction principle and Schauder’s fixed point theorem, we establish results on existence and uniqueness in appropriate function spaces. Several examples are also provided to illustrate the theoretical results.

ABSTRACT This paper is devoted to the study of a coupled system of fractional Langevin equations involving the ‐Hilfer fractional derivative and nonlocal Riemann–Stieltjes integral boundary conditions. By transforming the problem into an equivalent system of integral equations, sufficient conditions for the existence of solutions are established via Schauder's fixed point theorem and the Leray–Schauder nonlinear alternative. In addition, uniqueness results are obtained by applying the Banach contraction principle. Several illustrative examples are provided to demonstrate the applicability of the theoretical results and to validate the proposed approach.

The occurrence of oscillation and non-oscillation is common across various models in real-world applications. For instance, impulsive partial neutral differential equations in mathematical biology and biomathematics often display both oscillatory and non-oscillatory solutions. In our investigation, we establish specific criteria that guarantee the presence of non-oscillatory solutions for variable coefficient nonlinear second-order neutral differential equations with distributed deviating arguments and a forcing term. In this work, we obtained an extension of some existing results in the literature. Using the Banach contraction principle, we obtained new sufficient conditions of existence conditions. The proof of positive solutions was provided by showing the existence of a fixed point. At the end of the paper, we give an example showing how we apply one of the theorems we have learnt.

The e-sports industry in Indonesia is fostering professional relationships between athletes and team management through work agreements. However, the lack of specific legal frameworks has led to concerning contractual clauses, notably the freeze contract clause, which allows team management to indefinitely suspend an athlete's contract while preventing them from pursuing professional opportunities. This study examines the implications of such clauses on athletes' rights and welfare, utilizing normative legal research with legislative and conceptual methodologies. The findings indicate that while the freeze contract is rooted in the freedom of contract principle, it often lacks good faith and balance, resulting in potential bargaining inequalities, loss of wages, and career sustainability issues for athletes. Consequently, there is a pressing need for more precise regulations and contractual guidelines from e-sports institutions to enhance legal protection and equitable welfare for athletes.

Abstract - The Banach contraction principle serves as a foundational result in nonlinear analysis and is extensively applied to establish the existence and uniqueness of solutions to mathematical problems, including differential equations and dynamic programming. The classical metric space framework, however, presumes ideal precision in distance measurements. To address the impact of experimental errors, Jleli and Samet recently introduced perturbed metric spaces, in which a perturbation mapping modifies the distance function to account for inherent measurement inaccuracies. Concurrently, the geometric generalization of functional analysis has advanced through the study of 2-Banach spaces, a concept introduced by Gähler and subsequently formalized by White. In these spaces, the traditional notion of distance between two points is replaced by the area determined by three points. This framework provides a multidimensional perspective on fixed-point theory, as recently examined by Ettayb. This paper unifies these research directions by introducing the concept of Perturbed 2-Banach Spaces. This framework facilitates rigorous analysis of two-dimensional geometric structures subject to non-zero perturbation errors. The study extends contraction mapping theory in this context by examining Hardy-Rogers-type contractions, which unify and generalize the contraction conditions of Banach, Kannan, and Reich. Sufficient conditions are established for the existence and uniqueness of fixed points for such mappings in complete perturbed 2-Banach spaces. To illustrate the significance and applicability of these results, examples are provided that differentiate the findings from classical 2-normed space theory, along with a concrete application to the solvability of nonlinear integral equations. Key Words: Perturbed metric space, 2-Banach space, Hardy-Rogers contraction, Fixed point theory, Error analysis, Integral equations.

In the present paper, time-fractional order linear and nonlinear heat type Emden-Fowler equations are reformulated from existing classical equations by applying Caputo-Fabrizio time-fractional derivative. Then, a semi-analytical scheme, that is an amalgamation of Laplace transformation and Picard’s iterative technique, is exploited to simulate singular initial value problems for corresponding time-fractional order heat type Emden-Fowler equations. Further, the stability of developed scheme is also assessed by exploiting R-stable mapping and Banach contraction principle. Numerical results, error estimation, and comparison of obtained results with exact solutions are presented through graphs and tables to exhibit the efficiency of time-fractional order derivative and implemented semi-analytical scheme.

This paper introduces polynomial F-contractions, a novel category of contractive mappings within metric spaces. This concept synthesizes two powerful generalizations of the Banach contraction principle: the F-contractions originally developed by Wardowski and the polynomial-type contractions studied very recently by Jleli et al. We formulate fixed point theorems for this new class of mappings in complete metric spaces, which extends and unifies several established theorems in fixed point theory. We first prove our main result for continuous mappings and then extend it to a broader class of mappings that are not necessarily continuous but satisfy the Picard continuity condition. The significance and novelty of our results are highlighted through illustrative examples and further supported by applications to a fractional boundary value problem.

This paper introduces and develops the framework of complex-valued extended fuzzy b-metric spaces, thereby broadening the scope of classical fuzzy metric and fuzzy b-metric spaces into a richer and more versatile setting. We first establish the fundamental structural properties of these spaces and then investigate the existence of fixed points within this generalized framework, thus advancing the theoretical underpinnings of fuzzy analysis. In particular, we derive fixed point theorems tailored to complex-valued extended fuzzy b-metric spaces, laying a rigorous foundation for subsequent research. An illustrative example is provided to elucidate the applicability of the proposed results, while a concrete application to integral equations demonstrates the practical relevance of the theory. By extending classical principles, most notably the Banach Contraction Principle, our findings encompass more intricate spaces that integrate both real and imaginary components. This generalization not only enriches the theory of fixed points but also situates it within a broader perspective, offering new avenues for exploration in fuzzy spaces and related mathematical disciplines.

We introduce a new class of mappings, referred to as large triangle–perimeter contractions, which simultaneously extend Petrov’s triangle–perimeter contractions and Burton’s large contraction principle. The proposed approach combines a strict local reduction of triangle perimeters with a nonuniform contractive mechanism that becomes effective whenever the underlying triangle is sufficiently nondegenerate. Within this two-scale setting, we establish a fixed-point theorem showing that every such mapping defined on a complete metric space admits a unique fixed point, provided that one orbit is bounded. The proof follows the spirit of Burton’s decay technique, adapted here to control the behavior of triangle perimeters rather than pairwise distances. Several illustrative examples, including both continuous and discrete cases, demonstrate that this class strictly contains mappings that fail to satisfy Petrov’s uniform perimeter contraction condition.

In this paper, we investigate the existence, uniqueness, and Hyers-Ulam stability of a nonlinear fractional Langevin equation involving a generalized fractional derivative that unifies the ψ-Hilfer and ψ-Caputo types. The problem incorporates mixed and nonlocal boundary conditions. Existence results are derived using Krasnoselskii's fixed-point theorem, while uniqueness is established through Banach's contraction principle. Moreover, Ulam-type stability is analyzed under both Ulam-Hyers and Ulam-Hyers-Rassias criteria. Finally, a numerical example is presented, and the computed results are summarized in a table, showing the influence of different ψ-functions on the stability constants and confirming the consistency between theoretical and numerical findings.

This paper develops a nonlinear fractal-fractional predator-prey model that incorporates logistic prey growth and immigration effects. The predator-prey interaction is characterized by a Holling type II functional response, capturing the saturation phenomenon in the predator's feeding rate. Using the Caputo-Fabrizio (CF) fractional operator, the model integrates memory effects into the population dynamics. Theoretical investigations establish the existence and uniqueness of solutions by applying Krasnoselskii's fixed point theorem and Banach's contraction principle, followed by the stability analysis of equilibrium points. For the numerical approximation, a modified Adams-Bashforth method adapted to the CF operator is employed. Simulation results reveal that small yet positive immigration rates promote asymptotically stable coexistence between prey and predator populations, emphasizing the stabilizing influence of immigration on ecosystem dynamics. The study demonstrates how fractal-fractional calculus can provide deeper insight into ecological stability and long-term behavior of interacting species.

Abstract In this work, we investigate the existence and uniqueness of solutions to Hilfer fractional neutral impulsive stochastic delayed differential equations with nonlocal conditions. These equations, which are characterized by Hilfer fractional derivatives, stochastic perturbations, impulses, delays, and nonlocal components, naturally occur in a number of practical domains, including signal processing, control theory, and systems with memory and abrupt transitions. Krasnoselskii’s fixed point theorem is used to prove the existence of solutions, and the Banach contraction principle guarantees uniqueness. Two examples with particular parameter values are analyzed to show how the theoretical results are applicable. Graphical analysis is used to show how the solutions behave dynamically under different stochastic and impulsive inputs. Furthermore, simulations provide further details about the qualitative behavior of the solutions. By providing new insights into the theory of fractional stochastic differential equations with complex structures, this work broadens the potential applications of these equations in simulating real-world events.

Novel results for Wardowski contraction principle in triple-controlled orthogonal $\mathcal{S} $-metric space with applications to fractional equations

This study examines the juridical implications of penalty clauses in State-Owned Enterprise (SOE) construction contracts through a case study of Housing Project C between Company P and Company M. The research addresses the complex intersection of private law, public audit mechanisms, and judicial review in SOE contract enforcement. Using normative legal research methodology, this study analyzes the construction contract, BPK (Supreme Audit Institution) audit findings, and Supreme Court decisions to understand the legal position and enforceability of penalty clauses in SOE construction contracts. The findings reveal that while penalty clauses in SOE construction contracts are normatively valid under civil law provisions, their enforcement is subject to significant limitations arising from the hybrid legal character of SOEs. The BPK audit findings, which identified potential state losses due to project delays, compelled Company P to enforce contractual penalties as part of accountability and Good Corporate Governance principles. However, the Supreme Court's decision limited the maximum penalty to 2% instead of the contractually stipulated 5%, demonstrating that the principle of pacta sunt servanda is not applied absolutely in SOE construction contracts. This study concludes that SOE construction contracts operate at the intersection of three legal regimes: private law (contractual freedom), public law (state audit oversight), and judicial law (court interpretation). The Supreme Court's emphasis on fairness and proportionality principles over strict contractual interpretation reflects the unique legal challenges facing SOE contracts. These findings indicate that SOEs cannot rely solely on textual contract formulations but must consider judicial interpretation possibilities, fairness principles, and public accountability requirements when drafting construction contracts. The research contributes to legal theory development in contract law, construction law, and SOE business law, while providing practical recommendations for SOEs to draft more robust, audit-resistant construction contracts with greater legal certainty. The study recommends an integrative legal approach in designing SOE construction contracts that balances private contractual principles with public law obligations and anticipates judicial review considerations.

This paper introduces a novel extension of the classical Banach Contraction Principle, focusing on "perimetric contractions" in n-gon. Unlike traditional contractions that deal with the distances between pairs of points, perimetric contractions are concerned with the contraction of the entire perimeter of an n-gon, considering the distances between consecutive points along the boundary. This new perspective enables the development of fixed-point results in higher-dimensional metric spaces. The core objective is to establish a fixed-point theorem for mappings that contract the perimeters of n-gon, providing a generalization of Banach's original theorem. The paper demonstrates that such mappings are continuous and presents conditions under which fixed points exist and are unique. Additionally, the relationships between perimetric contractions and conventional contraction mappings are examined, thus expanding the applicability of fixed-point theorems in more complex settings.

This paper presents a comprehensive analysis of the existence, uniqueness, and Ulam–Hyers stability of solutions for a class of Cauchy‐type nonlinear fractional differential equations with variable order and finite delay. The motivation for this study lies in the increasing importance of variable‐order fractional calculus in modeling real‐world systems with time‐dependent memory and hereditary properties, where the order of differentiation evolves with time or state. Moreover, incorporating finite delay allows the model to capture the influence of past states on current dynamics, making it applicable to a wider range of physical and biological processes. By employing tools from fixed point theory, we derive new sufficient conditions for the existence and uniqueness of solutions using both the Banach contraction principle and Schauder’s fixed point theorem. Furthermore, we establish the Ulam–Hyers stability of the proposed system under suitable assumptions on the nonlinear term. To illustrate the validity of our theoretical findings, a numerical example is provided, demonstrating how variations in the fractional order affect the system’s behavior. The obtained results enrich the theoretical framework of variable‐order fractional calculus and extend its applicability to delayed systems. These findings may serve as a mathematical foundation for future research on more general models, including neutral or implicit fractional differential equations of variable order, which are of growing relevance in applied sciences and engineering.

Excessive fiduciary fees exacted from retirement savings exacerbate an overlooked crisis in which Americans descend into poverty after decades of work. Though ERISA fiduciary relationships and their associated fees are memorialized through contracts, most analysis by courts and scholars centers on ERISA's foundation in trust law. Examining ERISA fiduciary arrangements as relational contracts exposes structural misalignments in fee compositions, incentive mechanisms, and core fiduciary duties. It also supplies tools for redesigning these relationships to better protect American workers-without the need for congressional or regulatory intervention.  <br><br>Drawing from the robust cross-disciplinary literature on relational contracts, this Article explores how relational contract principles should inform analyses-and restructuring-of the contracts underpinning the ERISA fiduciary arrangement. It contributes by proposing practicable revisions to fiduciary arrangements that rectify suboptimal fiduciary compensation structures and lopsided risk allocation. With the aim of preserving the $13 trillion in assets defined contribution retirement plans hold, this Article proposes incentivization and bonding mechanisms previously unapplied in the ERISA fiduciary context.

This paper establishes a comprehensive analysis of a coupled system of nonlinear Hadamard-type fractional differential equations subject to generalized nonlocal integral boundary conditions. The distinct logarithmic kernel of the Hadamard derivative makes this framework particularly suitable for modeling scale-invariant processes and ultraslow diffusion phenomena. The existence and uniqueness of solutions are rigorously investigated using fixed point theory: Banach’s contraction principle ensures uniqueness, while the Leray-Schauder nonlinear alternative guarantees existence under more general growth conditions. Furthermore, the system is proven to be Ulam-Hyers stable, ensuring that approximate solutions remain close to exact solutions, which is crucial for the robustness of the model in practical applications. The theoretical findings are effectively validated through two detailed numerical examples, demonstrating the applicability of the established results to different classes of nonlinearities.OPEN ACCESS Received: 22/08/2025 Accepted: 03/11/2025 Published: 23/01/2026