This paper investigates asymptotic bounds for the length density of factorizations in numerical and more general atomic semigroups, introducing a unified algorithmic and analytic treatment through pruning dynamics. Building on the pruning paradigm previously developed for factorization trees, we establish finiteness, correctness, and length-preservation properties under flexible chain conditions such as Finite Factorization (FF)-monoid and prefix-Ascending Chain Condition on Prefixes (ACCP) hypotheses. We then extend these results through a series of major analytical developments that reveal deeper structural behaviour: asymptotic growth laws for normalized length density, categorical representations of pruning as functorial morphisms, entropy-based bounds on factorization complexity, probabilistic models for random pruning trees, spectral analysis of pruning operators, and topological compactification yielding continuum limits. Collectively, these results define an Asymptotic-Analytic Framework for Pruning Dynamics that unifies combinatorial, categorical, and analytic viewpoints of semigroup factorization. The framework establishes new links between algebraic finiteness, entropy growth, spectral stability, and topological convergence, thus extending classical length-density theory toward a continuous and dynamical formulation of semigroup complexity.

In this research, we introduce an improved pseudomonotone subgradient extragradient algorithm for finding common solutions to equilibrium and fixed-point problems in real Hilbert spaces. Under mild and suitable assumptions on the control parameters, we establish strong convergence results for the proposed method. Unlike many existing approaches that depend on contraction mappings or Mann-type techniques requiring heavy computations, our method employs a standard Mann iteration scheme without additional complexity. Moreover, the algorithm integrates a relaxed two-inertial technique, which enhances the convergence speed. We further demonstrate the applicability of our results to variational inequality problems and image recovery tasks. Finally, numerical experiments are presented to validate the theoretical findings and to illustrate the superiority of the proposed method compared with several well-known algorithms in the literature. The results obtained in this paper improve, extend, and unify numerous existing contributions in this research direction.

In this paper, we use the Schuader fixed point and the Banach contraction principle to study sufficient conditions for existence, uniqueness, and Lipschitz stability of solutions for fractional delay differential equations with Mittag-Leffler Kernel (without impulses) and impulsive fractional differential equations (with delay). As a result, with practical applications, the theoretical understanding of fractional differential equations containing memory effects, impulse perturbations, and delay factors is expanded. The developed concept is finally illustrated with examples and applied to impulsive fractional delay biological systems involving a single species population growth model known as, the Hutchinson's equations using the fractional order derivative with Mittag-Leffler kernel.

This paper develops an algebraic framework for modeling multiple-state optimal design problems through semigroup theory. By introducing a natural semigroup structure on the space of feasible design states, we analyze fundamental algebraic properties such as idempotents, Green’s relations, and ideals, linking them to stability, robustness, and subsystem hierarchies in complex designs. We further investigate deeper structural features including regularity, homomorphisms, Rees and Krohn-Rhodes decompositions, and rank, thereby providing new insights into reducibility and minimal generative complexity. From a computational standpoint, we present efficient procedures for detecting idempotents, computing minimal generating sets, and analyzing congruences, alongside algorithmic approaches and flowchart-based representations for decomposition and rank computation in large-scale semigroups. Applications to convex optimization, network flows, and sequential decision processes illustrate the versatility of the framework, while extensions to probabilistic, categorical, and dynamic settings highlight its broad applicability. Overall, the semigroup-theoretic perspective unifies structural insight and computational methods, opening new pathways for theory and practice in optimization and system design.

We investigate the strong convergence of the Halpern iteration for monotone alpha-nonexpansive mappings in uniformly convex Banach spaces endowed with a partial order induced by a normal cone. By establishing new demiclosedness principles, analyzing boundedness and asymptotic regularity, and exploiting order-preserving properties, we prove that the Halpern sequence converges strongly to an extremal fixed point of the operator. The framework is further extended to hybrid iterative schemes, modular and Orlicz function spaces, and applications to variational inequalities, monotone operators in partial differential equations, and equilibrium problems in optimization. Illustrative examples in classical and modular Banach spaces are provided, and convergence is shown to hold under relaxed geometric conditions, such as strict convexity or smoothness, thereby unifying and generalizing existing theories for nonexpansive and alpha-nonexpansive mappings.

This paper presents a comprehensive cone-theoretic comparison principle for Caputo fractional differential equations, thereby addressing a key gap in the qualitative theory of fractional systems. Although classical comparison techniques based on scalar and vector Lyapunov functions have been extended to fractional settings, the more general and powerful framework of cone-valued Lyapunov functions has received little attention. In this work, we develop a complete theoretical foundation for this extension. We establish core results on cone-preserving fractional differential inequalities, prove the existence and characterization of maximal solutions with respect to arbitrary closed convex cones, and derive a general comparison principle for Caputo fractional systems. The central result is a unified comparison theorem that enables stability analysis of fractional systems using cone-valued Lyapunov functions. This approach incorporates both scalar and vector Lyapunov methods as special cases, and can be naturally extended to systems whose dynamics follow non-standard partial orderings. In all, our results provide a robust theoretical basis for studying complex fractional-order systems that lie beyond the reach of traditional comparison techniques.

In this work, we extend the theory of fractional calculus to hybrid systems that combine continuous and discrete dynamics by establishing the existence, uniqueness, and generalized Ulam–Hyers–Rassias stability of solutions to Caputo fractional delta-differential equations on time scales. Using the Schauder fixed-point theorem, we derive sufficient conditions for well-posedness and stability, building on core concepts from fractional calculus and time-scale theory. To demonstrate the applicability and practicality of the proposed framework, we apply the theoretical results to epidemiological modelling. In particular, SIS and SIR models are employed to illustrate how the approach captures memory effects and hybrid dynamics inherent in real-world systems. Thus, the framework not only advances the theoretical foundations of fractional dynamic equations but also highlights their practical relevance.

This paper concentrates on the analysis of a category of coupled Langevin‐type pantograph differential equations involving the generalized Caputo fractional derivative with nonlocal conditions. We conduct this analysis in two cases for the second member in the nonlinear function; in other words, for the real space and an abstract Banach space Θ. To prove the existence and uniqueness of solutions to our problems, we used different types of the fixed point theorems attributed to Banach, Schaefer and Darbo, along with the measure of noncompactness. Lastly, we provide examples to illustrate our findings.

In this paper, we prove an existence smd uniqueness theorem for a solution of a system of nonlinear equations in the product of b-metric spaces. We obtain generalization of the classical results of Banach, Kannan and Reich in the product of b-metric spaces. Also, some variants of the results of Czerwik, Bhaskar and Lakshmikantham are obtained. Finally, we give an application in support of our result.

The eardrum is one of the most important organs in the body, and disorders such as infection or injury may affect the proper functioning of the eardrum and lead to hearing problems. In this paper, based on a real-world phenomena, we study some mathematical aspects of an abstract fractional $ [\mathtt{p},\mathtt{q}] $-difference equation with initial conditions. Our initial value problem tries to model a vibrating eardrum by using the newly defined fractional Caputo-type $ [\mathtt{p},\mathtt{q}] $-derivatives in two nonlinear single-valued and set-valued structures. We obtain a general form of the solutions in the framework of a $ [\mathtt{p},\mathtt{q}] $-integral equation, and then we investigate the existence and uniqueness properties with the help of fixed points and the end-points of some special $ \mathtt{β} $-$ \mathtt{α} $-contractions and compact mappings. Finally, we simulate this version of the vibrating eardrum model by giving two numerical examples to validate the established theorems.

In this paper, we discuss the existence and uniqueness of solution of Atangana-Baleanu-Caputo impulsive fractional delay differential equations with caratheo-dory function. We further introduce modified Ulam-Hyers-Rassias stability criteria by considering a real-valued function that is Lebesque integrable. This new concept makes the theory more realistic, flexible, and mathematically consistent with modern analysis (fractional calculus, impulsive system and delay equations). It covers unbounded but integrable disturbances, accommodates Caratheordory conditions, and extends applicability to a much larger class of dynamical systems. Extending Ulam-Hyers-Rassias stability to Lebesque integrable perturbations makes it compatible with stronger existence and uniqueness theorems using Banach and Schauder fixed point theorems which often require mappings to be continuous and bounded in $L^{\frac{1}{\theta}}([t_{0}, T])$ L 1 θ ( [ t 0 , T ] ) type norms. The stability of the solution of Atangana-Baleanu-Caputo impulsive fractional delay differential equations with caratheo-dory function is also investigated by using the modified Ulam-Hyers-Rassias stability concept. The outcome will aid in the theoretical development of fractional differential equations with memory effects, impulse perturbations, and delay factors

In this manuscript, we use the concept of multidimensional fixed point in a generalized space, namely, C -distance space with some nonlinear contraction conditions, such as Jaggi- and Dass-Gupta-type contractions. We provide results with a Jaggi-type hybrid contraction for the mentioned space. Moreover, we use control functions to get the desired results. After each theorem, we compare our results with previous ones to show that they are generalized. We provide examples to support our results. An application is also performed to solve the system of integral equations.

Modeling of different processes and phenomena in real-world is one of the most important fields of the mathematics in which qualitative dynamics of such systems are studied from mathematical point of view. In this paper, we discuss the qualitative properties of solutions of a temperature control system in the context of a mathematical model in fractional discrete calculus. We discretize our supposed control system with the help of two delta sum and difference operators in the sense of the Caputo and Riemann–Liouville. By the existing properties of the falling functions, we obtain the equivalent difference formula corresponding to the given discrete delta difference boundary value problems of temperature control system. To conduct an analysis on solutions of this fractional system, the existence results are investigated via fixed points and the stability bahaviors are proved from the Ulam–Hyers point of view. In two applied examples, we use numerical data to simulate solutions of such discrete fractional delta boundary value problems of temperature control system.

This paper presents and examines a newly improved linear technique for solving the equilibrium problem of a pseudomonotone operator and the fixed point problem of a nonexpansive mapping within a real Hilbert space framework. The technique relies two modified mildly inertial methods and the subgradient extragradient approach. In addition, it can be viewed as an advancement over the previously known inertial subgradient extragradient approach. Based on common assumptions, the algorithm's weak convergence has been established. Finally, in order to confirm the efficiency and benefit of the proposed algorithm, we present a few numerical experiments.

In this study, changes in westerly waves and their connections to increased global warming under the influence of greenhouse gases were investigated via a Caputo fractional four-dimensional atmospheric system. The idea of the existence of chaotic behavior in the westerly wind's motion was depicted. It has been noted that westerlies are becoming stronger due to rising air temperatures. An analysis of the existence, uniqueness, boundedness, stability of equilibrium points, and conservative behavior of the solutions was conducted. To prove the existence of chaos in the modified model, the Lyapunov exponents, Poincaré map, and bifurcation were computed. A sliding mode controller to control the chaos in this novel fractional-order system was designed, and conditions for the global stability of the controlled system with and without external disturbances and uncertainties were derived. The finite-time interval for the system to reach the sliding surface was computed. The developed controller's performance was evaluated with respect to both commensurate and non-commensurate fractional derivatives. In each scenario, the impact of fractional orders was investigated. Numerical simulations were used to support theoretical statements about how the controller affects the system.

We introduce the concept of controlled extended Branciari quasi-b-metric spaces, as well as a Gq-implicit type mapping. Under this new space setting, we derive some new fixed points, periodic points, right and left Ulam–Hyers stability, right and left weak well-posed properties, and right and left weak limit shadowing results. Additionally, we use these findings to solve the fractional differential equations of a Riesz–Caputo type with integral anti-periodic boundary values, as well of nonlinear matrix equations. All ideas, results, and applications are properly illustrated with examples.

In this paper, we introduce the notion of orthogonal α–F–convex contraction mapping and prove some fixed-point theorems for self-mapping in orthogonal complete metric spaces. The proven results generalize and extend some of the well-known results in the literature. Following the derivation of these fixed-point results, we propose a solution for the fractional integro-differential equation, utilizing the fixed-point technique within the context of orthogonal complete metric spaces.

This manuscript is devoted to constructing a novel iterative scheme and reckoning of fixed points for generalized contraction mappings in hyperbolic spaces. Also, we establish Δ\Delta and strong convergence results by the considered iteration under the class of mappings satisfying condition (E). Moreover, some qualitative results of the suggested iteration, like weak w2{w}^{2}-stability and data dependence results, are discussed. Furthermore, to test the efficiency and effectiveness of the proposed iteration, practical experiments are given. To support the theoretical results, illustrative examples are presented. Finally, our results improve and generalize several classical results in the literature of fixed point iterations.

In this study, we investigate the stability and asymptotic stability properties of Caputo fractional time-dependent systems with delay by employing vector Lyapunov functions. Utilizing the Caputo fractional Dini derivative on Lyapunov-like functions, along with a new comparison theorem and differential inequalities, we derive and prove sufficient conditions for the stability and asymptotic stability of these complex systems. An example is included to showcase the method's practicality and to specifically illustrate its advantages over scalar Lyapunov functions. Our results improves, extends, and generalizes several existing findings in the literature.

This work explores the existence and uniqueness criteria for the solution of hybrid Caputo–Hadamard fractional sequential differential equations (HCHFSDEs) by employing Darbo’s fixed-point theorem. Fractional differential equations play a pivotal role in modeling complex phenomena in various areas of science and engineering. The hybrid approach considered in this work combines the advantages of both the Caputo and Hadamard fractional derivatives, leading to a more comprehensive and versatile model for describing sequential processes. To address the problem of the existence and uniqueness of solutions for such hybrid fractional sequential differential equations, we turn to Darbo’s fixed-point theorem, a powerful mathematical tool that establishes the existence of fixed points for certain types of mappings. By appropriately transforming the differential equation into an equivalent fixed-point formulation, we can exploit the properties of Darbo’s theorem to analyze the solutions’ existence and uniqueness. The outcomes of this research expand the understanding of HCHFSDEs and contribute to the growing body of knowledge in fractional calculus and fixed-point theory. These findings are expected to have significant implications in various scientific and engineering applications, where sequential processes are prevalent, such as in physics, biology, finance, and control theory.