Nonlinear Functional Analysis Research Articles

A study for analysing predator–prey ecological system using Legendre polynomial

This paper presents a mathematical model to examine the effects of the coexistence of predators on single prey. Based on Caputo operators, we present a newly developed system of differential equations for the predator–prey system using wavelet method. It is well known that a system of nonlinear singular models cannot operate smoothly since they are singular and nonlinear. Therefore, with the help of this numerical approach, we have converted the system into a nonlinear system of algebraic equations by extending it through operational matrix of Legendre wavelets. Using the wavelet collocation scheme, we have calculated these unknown coefficients. It has been demonstrated in tables and graphs that the developed approach is consistent and proficient. Further bifurcation diagrams, as well as phase portraits, have been used to study the proposed system numerically and to analyse its behaviour. In addition, a nonlinear functional analysis have used to establish uniformly boundedness for the proposed model. Also we have discussed residual error analysis and Lyapunov exponent. The applicability and efficacy of this methodology have been demonstrated through this nonlinear system. Additionally, a comparison with existing results highlights the advantages of our numerical approach. All calculations have been done using MATLAB.

On a Symmetry-Based Structural Deterministic Fractal Fractional Order Mathematical Model to Investigate Conjunctivitis Adenovirus Disease

In the last few years, the conjunctivitis adenovirus disease has been investigated by using the concept of mathematical models. Hence, researchers have presented some mathematical models of the mentioned disease by using classical and fractional order derivatives. A complementary method involves analyzing the system of fractal fractional order equations by considering the set of symmetries of its solutions. By characterizing structures that relate to the fundamental dynamics of biological systems, symmetries offer a potent notion for the creation of mechanistic models. This study investigates a novel mathematical model for conjunctivitis adenovirus disease. Conjunctivitis is an infection in the eye that is caused by adenovirus, also known as pink eye disease. Adenovirus is a common virus that affects the eye’s mucosa. Infectious conjunctivitis is most common eye disease on the planet, impacting individuals across all age groups and demographics. We have formulated a model to investigate the transmission of the aforesaid disease and the impact of vaccination on its dynamics. Also, using mathematical analysis, the percentage of a population which needs vaccination to prevent the spreading of the mentioned disease can be investigated. Fractal fractional derivatives have been widely used in the last few years to study different infectious disease models. Hence, being inspired by the importance of fractal fractional theory to investigate the mentioned human eye-related disease, we derived some adequate results for the above model, including equilibrium points, reproductive number, and sensitivity analysis. Furthermore, by utilizing fixed point theory and numerical techniques, adequate requirements were established for the existence theory, Ulam–Hyers stability, and approximate solutions. We used nonlinear functional analysis and fixed point theory for the qualitative theory. We have graphically simulated the outcomes for several fractal fractional order levels using the numerical method.

Controllability results of discrete boundary control system in Banach spaces

In this paper, we study the boundary controllability of the discrete-time control system in Banach spaces. The aim is to establish sufficient conditions for the existence and uniqueness of solutions, as well as for the approximate controllability of a discrete-time boundary control system. The form of the solution to the considered system and its existence and uniqueness are obtained using non-linear functional analysis, theory of difference equations, semigroup theory and contraction mapping principle. Furthermore, we achieve the approximate controllability results through a sequential approach. In the end, we present some examples to illustrate the application of these analytical results.

Analysis of a class of fractal hybrid fractional differential equation with application to a biological model

Recently, the area devoted to fractional calculus has given much attention by researchers. The reason behind such huge attention is the significant applications of the mentioned area in various disciplines. Different problems of real world processes have been investigated by using the concepts of fractional calculus and important and applicable outcomes were obtained. Because, there has been a lot of interest in fractional differential equations. It is brought on by both the extensive development of fractional calculus theory and its applications. The use of linear and quadratic perturbations of nonlinear differential equations in mathematical models of a variety of real-world problems has received a lot of interest. Therefore, motivated by the mentioned importance, this research work is devoted to analyze in detailed, a class of fractal hybrid fractional differential equation under Atangana- Baleanu- Caputo ABC derivative. The qualitative theory of the problem is examined by using tools of non-linear functional analysis. The Ulam-Hyer’s (U-H) type stability criteria is also applied to the consider problem. Further, the numerical solution of the model is developed by using powerful numerical technique. Lastly, the Wazewska-Czyzewska and Lasota Model, a well-known biological model, verifies the results. Several graphical representations by using different fractals fractional orders values are presented. The detailed discussion and explanations are given at the end.

Existence, Uniqueness, and Stability of a Nonlinear Tripled Fractional Order Differential System

This manuscript investigates the existence, uniqueness, and different forms of Ulam stability for a system of three coupled differential equations involving the Riemann–Liouville (RL) fractional operator. The Leray–Schauder alternative is employed to confirm the existence of solutions, while the Banach contraction principle is used to establish their uniqueness. Stability conditions are derived utilizing classical nonlinear functional analysis techniques. Theoretical findings are illustrated with an example. The proposed system generalizes third-order ordinary differential equations (ODEs) with different boundary conditions (BCs).

Unveiling the Complexity of HIV Transmission: Integrating Multi-Level Infections via Fractal-Fractional Analysis

This article presents a non-linear deterministic mathematical model that captures the evolving dynamics of HIV disease spread, considering three levels of infection in a population. The model integrates fractal-fractional order derivatives using the Caputo operator and undergoes qualitative analysis to establish the existence and uniqueness of solutions via fixed-point theory. Ulam-Hyer stability is confirmed through nonlinear functional analysis, accounting for small perturbations. Numerical solutions are obtained using the fractional Adam-Bashforth iterative scheme and corroborated through MATLAB simulations. The results, plotted across various fractional orders and fractal dimensions, are compared with integer orders, revealing trends towards HIV disease-free equilibrium points for infective and recovered populations. Meanwhile, susceptible individuals decrease towards this equilibrium state, indicating stability in HIV exposure. The study emphasizes the critical role of controlling transmission rates to mitigate fatalities, curb HIV transmission, and enhance recovery rates. This proposed strategy offers a competitive advantage, enhancing comprehension of the model’s intricate dynamics.

Dynamical behavior of fractal-fractional order monkeypox virus model

A virus that infects both humans and animals, monkeypox is common in many West African countries and has sporadically spread to other parts of the world. There are serious public health concerns around the globe as a result of the recent spike in cases of monkeypox among both endemic and non-endemic populations. This paper proposes to use the Caputo sense of fractal fractional-order derivatives to examine the dynamics of monkeypox transmission. The study uses Schauder’s fixed point theorem to evaluate the solutions qualitatively and establish their uniqueness inside the model. The next generation matrix technique is used to compute the essential reproduction number. The stability of equilibrium points is further investigated in the research, and a sensitivity analysis of model parameters is carried out. When the basic reproduction number R0 is less than 1, the equilibrium without infections is locally stable. Also, when R0 exceeds 1, this equilibrium becomes unstable. The proposed model incorporates Ulam–Hyers stability through nonlinear functional analysis. To estimate solutions for the fractal-fractional order monkeypox model, the Lagrange’s interpolation method is utilized. In addition, numerical simulations are carried out to examine the influence of some parameters on the overall dynamics of the model. Numerical simulations are performed using MATLAB software to exemplify the model behavior in the context of the Nigerian case study. The graphical representations suggest that the fractal fractional order affects the dynamics of the monkeypox. The findings indicate that isolation of infected individuals in the human population helps to reduce disease transmission.

Equilibrium analysis in game theory using nonlinear functional analysis techniques

Game theory provides a mathematical framework for analyzing the tactical choices of rational decision makers. The idea of equilibrium, which states that no player has a motivation to change their existing approach given the strategies of other players, is one of the core ideas of game theory. This study examines a type of undirected graph-based distributed quadratic game. The problem of communication topology constraints is presented and nonlinear dynamics with uncertain time-dependent perturbations are present in the participant's dynamics. Based on a high-gain observer approach, a distributed Nash equilibrium (NE) finding technique is given, and the Lyapunov stability theory is used to study the convergence. It represents that every player approximates the positions of their rival players and that there are differences between the NE and the placements of the minor limitation that finally restricts the players. In addition, chattering problems are eliminated since the offered theory's formulation employs the hyperbolic tangent function to control the perturbation rather than the signum function. In an imitation of the oligopoly match, five enterprises manufacture identical materials in a duopoly market framework; this is done to confirm the effectiveness of the recommended strategy. Our results provide novel perspectives and methods for understanding complicated strategic situations, helping to close the gap between mathematical theory and real-world applications.

Mathematical insights of social media addiction: fractal-fractional perspectives

The excessive use of social media has become a growing concern in the current century, with dependence on these platforms developing into a complex behavioral addiction. Addressing this issue requires the employment of well-directed and inclusive efforts. In pursuit of continuous development in existent strategies, this article presents a non-linear deterministic mathematical model that encapsulates the dynamics of social media addiction within a population. The proposed model incorporates the fractal-fractional order derivative in the sense of the Caputo operator. The objectives of this research are attained by groping the dynamics of the social media addiction model through the stratification of the population into five compartments: susceptible individuals, exposed individuals, addicted individuals, recovery individuals, and those who have quit using social media. The validity of the devised model is established by proving the existence and uniqueness of the solution within the framework of the fixed-point theory. The Ulam-Hyer’s stability is established through nonlinear functional analysis, perturbing the problem with a small factor. Utilizing the Adam Bashforth numerical scheme, we obtain numerical solutions, which we validate through MATLAB simulations. Additionally, we explore the application of artificial neural networks (ANNs) to approximate solutions, presenting a significant innovation in this domain. We propose the adoption of this novel method for solving integral equations that elucidate the dynamics of social media addiction, surpassing traditional numerical methods. Numerical results are illustrated across various fractional orders and fractal dimensions, with comparisons made against integer orders. Our study indicates that ANN outperforms the Adams-Bashforth algorithm, offering a dependable approach to problem-solving. Throughout the article, we underscore the competitive advantage of our proposed strategy, providing a more nuanced understanding of the complex dynamics outlined in the model.

A fractal-fractional order Susceptible-Exposed-Infected-Recovered (SEIR) model with Caputo sense

This study explores the intricacies of the COVID-19 pandemic by employing a four-compartment model with a fractal-fractional derivative based on Caputo concept. The analysis hinges on Schauder fixed point theorem, used to qualitatively examine the solutions and ascertain their existence and uniqueness within the model. The fundamental reproduction number is determined through the next-generation matrix approach. This study delves into the stability of equilibrium points and conducts a sensitivity analysis of model parameters. The equilibrium without infections is locally and globally stable when the basic reproduction number is less than 1. Also, this equilibrium becomes unstable when the basic reproduction number exceeds 1. Applying Lyapunov principles and the Routh–Hurwitz criteria, it is established that the endemic equilibrium point is globally stable for the basic reproduction number values greater than 1. The proposed model incorporates Ulam-Hyers stability through nonlinear functional analysis. Lagrange interpolation method estimates solutions for the fractal-fractional order COVID-19 model. Numerical simulations are performed using MATLAB software to exemplify the model behavior in the context of the Italian case study. Furthermore, fractal-fractional calculus techniques hold significant promise for comprehending and predicting the pandemic’s global dynamics in other countries.

Fixed point theorems for continuous single-valued and upper semicontinuous set-valued mappings in $p$-vector and locally $p$-convex spaces

The goal of this paper is to establish a general fixed point theorem for compact single-valued continuous mappings in Hausdorff $p$-vector spaces, and a fixed point theorem for upper semicontinuous set-valued mappings in locally $p$-convex spaces for $p\in (0, 1]$. These results not only provide a solution to Schauder conjecture in the affirmative under the setting of $p$-vector spaces for compact single-valued continuous mappings, but also show the existence of fixed points for upper semicontinuous set-valued mappings defined on $s$-convex subsets in Hausdorff locally $p$-convex spaces, which would be fundamental for nonlinear functional analysis, where $s, p \in (0, 1]$.

On Nonlinear Analysis for Multi-term Delay Fractional Differential Equations Under Hilfer Derivative

AbstractIn this manuscript, a class of multi-term delay fractional differential equations (FDEs) under the Hilfer derivative is considered. Some newly updated results are established under boundary conditions. For the required results, we utilize the fixed point theory and tools of the nonlinear functional analysis. Further keeping in mind the importance of stability results, we develop some adequate results about the said aspect. The Hyers-Ulam (H-U)-type concept is used to derive the required stability for the solution of the considered problem. Finally, by appropriate test problems, we justify our findings.

AN ALGORITHMIC APPROACH TO LOCAL SOLUTION OF THE NONLINEAR HIGHER ORDER ORDINARY HYBRID DIFFERENTIAL EQUATIONS

In this paper, we introduce a new notion of local or neighborhood solutions and establish a couple of approximation results for local existence and uniqueness of the solution of an IVP of nonlinear higher order ordinary hybrid differential equations by using the Dhage monotone iteration method based on the recent hybrid fixed point theorems of Dhage (2023). An approximation result for Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Our main abstract results of this paper are also illustrated with a couple of numerical examples. Finally, we compare our existence and uniqueness results with those existing in the literature via other operator theoretic methods from nonlinear functional analysis. We claim that the method and the results of this paper are new to the literature.

Unveiling the dynamics of drug transmission: A fractal-fractional approach integrating criminal law perspectives

<abstract><p>The excessive use of drugs has become a growing concern in the current century, with the global toll of drug-related deaths and disabilities posing a significant public health challenge in both developed and developing countries. In pursuit of continuous improvement in existing strategies, this article presented a nonlinear deterministic mathematical model that encapsulates the dynamics of drug addiction transmission while considering the legal implications imposed by criminal law within a population. The proposed model incorporated the fractal-fractional order derivative using the Atangana-Baleanu-Caputo ($ \mathbb{ABC} $) operator. The objectives of this research were achieved by examining the dynamics of the drug transmission model, which stratifies the population into six compartments: The susceptible class to drug addicts, the number of individuals receiving drug misuse education, the count of mild drug addicts, the population of heavy-level drug addicts, individuals subjected to criminal law, and those who have ceased drug use. The qualitative analysis of the devised model established the existence and uniqueness of solutions within the framework of fixed-point theory. Furthermore, Ulam-Hyer's stability was established through nonlinear functional analysis. To obtain numerical solutions, the fractional Adam-Bashforth iterative scheme was employed, and the results were validated through simulations conducted using MATLAB. Additionally, numerical results were plotted for various fractional orders and fractal dimensions, with comparisons made against integer orders. The findings underscored the necessity of controlling the effective transmission rate to halt drug transmission effectively. The newly proposed strategy demonstrated a competitive advantage, providing a more nuanced understanding of the complex dynamics outlined in the model.</p></abstract>

USING LEARNING ABILITIES OF COMPUTATION IN COUNTING

In this paper, we introduce a new notion of local or neighborhood solutions and establish a couple of approximation results for local existence and uniqueness of the solution of an IVP of nonlinear higher order ordinary hybrid differential equations by using the Dhage monotone iteration method based on the recent hybrid fixed point theorems of Dhage (2023). An approximation result for Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Our main abstract results of this paper are also illustrated with a couple of numerical examples. Finally, we compare our existence and uniqueness results with those existing in the literature via other operator theoretic methods from nonlinear functional analysis. We claim that the method and the results of this paper are new to the literature.

A FRACTAL-FRACTIONAL ORDER MODEL TO STUDY MULTIPLE SCLEROSIS: A CHRONIC DISEASE

A mathematical model of progressive disease of the nervous system also called multiple sclerosis (MS) is studied in this paper. The proposed model is investigated under the concept of the fractal-fractional order derivative (FFOD) in the Caputo sense. In addition, the tools of nonlinear functional analysis are applied to prove some qualitative results including the existence theory, stability, and numerical analysis. For the recommended results of the existence theory, Banach and Krassnoselski’s fixed point theorems are used. Additionally, Hyers–Ulam (HU) concept is used to derive some results for stability analysis. Additionally, for numerical illustration of approximate solutions of various compartments of the considered model, the modified Euler method is utilized. The aforementioned results are displayed graphically for various values of fractal-fractional orders.

Existence theory and Ulam’s stabilities for switched coupled system of implicit impulsive fractional order Langevin equations

In this work, a system of nonlinear, switched, coupled, implicit, impulsive Langevin equations with two Hilfer fractional derivatives is introduced. The suitable conditions and results are established to discuss existence, uniqueness, and Ulam-type stability results of the mentioned model, with the help of nonlinear functional analysis techniques and Banach’s fixed-point theorem. Furthermore, we examine our results with the help of example.

Functional Data Clustering Via Functional Mahalanobis Distance

As an exploratory data analysis method, functional data clustering aims to identify the underlying features of the observed data. In this context, this paper proposes a functional data clustering method based on functional Mahalanobis distance. As a distance-based non-parametric clustering model, the proposed method can effectively avoid the disadvantages of generative models and has excellent properties of decoupling and dimension standardization. Compared with other functional data clustering models, this method has lower computational complexity. In addition, the proposed method can be applied to any distance-based multivariate clustering method, thus generalizing it to the case of functional data. In practical data analysis, this paper compares the performance of this method with some other functional clustering methods, using k-means clustering as an example, and finds that it has better performance in terms of purity, adjusted Rand index, and computational speed. Finally, the idea of using Mahalanobis distance for functional data distance measurement can also be extended to construct kernel functions for measuring similarity between functional data samples, thus developing non-linear functional data analysis methods based on reproducing kernel theory.

A chaos study of fractal–fractional predator–prey model of mathematical ecology

This paper presents a mathematical model to examine the effects of the coexistence of predators on single prey. Based on fractal–fractional Atangana–Baleanu (AB) and Caputo operators, we present a newly developed system of differential equations for the predator–prey system. Our study utilized the fixed point postulate to investigate the uniqueness and existence of solutions. Additionally, Ulam’s type of stability of the proposed model is established with the help of nonlinear functional analysis. Further bifurcation diagrams, as well as phase portraits, have been used to study the proposed system numerically and to analyze its behavior. The generalized non-linear system with fractal–fractional Atangana–Baleanu (AB) and Caputo non-integer operators have been solved numerically via the Toufik–Atangana (TA) scheme respectively. We have demonstrated the applicability and effectiveness of these methods by analyzing numerical simulations for the fractal–fractional predator–prey ecological model and the numerical simulation has been calculated by MATLAB programming.

Dynamics and Stability of $\Xi$-Hilfer Fractional Fuzzy Differential Equations with Impulses

This paper deals with the existence, uniqueness, and Ulam-stability outcomes for $\Xi$-Hilfer fractional fuzzy differential equations with impulse. Further, by using the techniques of nonlinear functional analysis, we study the Ulam-Hyers-Rassias stability.