Narrow Results

This paper introduces a novel nonlinear fractal fractional model to comprehensively analyze influenza epidemics. By integrating fractional calculus and considering nonlinear interactions among individuals, the model utilizes the Fractal-Fractional (FF) operator. This operator, combining fractal and fractional calculus, establishes a unique framework for investigating influenza virus propagation dynamics and potential vaccination strategies. Amid growing concerns over influenza outbreaks, fractional derivatives are employed to address intricate challenges. The proposed model sheds light on virus spread dynamics and countermeasures. The integration of the FF operator enriches analysis, while the application of the fixed point theory of Schauder and Banach demonstrates solution existence and uniqueness. Model validation employs numerical simulations with MATLAB12̂ and the Adams–Bashforth method, confirming its ability to capture influenza propagation dynamics accurately. Ulam–Hyers stability techniques ensure the model’s reliability. Beyond its scientific contributions, the model underscores the significance of studying influenza epidemics via mathematical modeling in understanding disease dynamics and guiding effective intervention strategies. Through a synergy of mathematical innovation and epidemiological insights, this study establishes a robust foundation for more effective strategies against influenza epidemics.

The primary objective of this study is to analyze the Hyers–Ulam stability of fractional derivatives for the Black–Scholes model, involving two underlying asset systems and utilizing Caputo fractional derivatives. We employ a fixed-point approach to examine the existence and uniqueness of solutions and to investigate the Hyers–Ulam stability of the given problem. Additionally, we analyze the graphical behavior of the obtained results, demonstrating that the analytical method is highly efficient and delivers precise results for determining approximate numerical solutions. The findings highlight the significant role of fractional approaches in studying nonlinear systems of scientific and physical importance. Furthermore, the graphical analysis, considering various fractional orders and parameter values, unveils new insights and intriguing phenomena associated with the Black–Scholes model.

Wireless sensor networks have been extensively studied for their potential applications in both civil and military domains. However, due to their inherent resource constraints, sensor nodes are highly susceptible to cyber threats, particularly worm attacks, which pose significant security challenges. This study explores the dynamics of potential worm attacks within wireless sensor networks using a compartmental epidemic model. The model analyzes the temporal evolution of worm propagation while effectively capturing both spatial and temporal aspects. We determine the reproduction number and equilibrium of the system, with the local stability assessed using the Jacobian matrix. The linear growth and Lipschitz conditions are used to establish the existence and uniqueness of the solution. Furthermore, the Hyers–Ulam stability of the proposed model is also evaluated within the context of the fractal-fractional operator. Finally, a numerical method is developed to investigate the dynamic behavior of the wireless sensor network model under fractal-fractional orders, providing valuable insights into its robustness and security against potential worm attacks.

In this paper, we study the stability of specific local functional inequalities within quasi-$\beta$-Banach spaces, focusing on mappings that satisfy modular concavity and bounded perturbations. The analysis establishes the existence and uniqueness of $ℂ$-linear mappings that approximate these functions. Furthermore, we apply these results to investigate the stability of Lie homomorphisms and Lie derivations in quasi-$\beta$-Banach spaces.

In the present paper, we consider the generalized Hyers–Ulam stability for fractional differential equations of the form:

In this paper, our main objective is to develop the conditions that assure the existence of solution to a system of boundary value problems (BVPs) of sequential hybrid fractional differential equations (SHFDEs). The problem is considered under the nonlinear boundary conditions. Nonlinear functions involved in the considered system of SHFDEs are continuous and satisfy the growth conditions. We convert the system of SHFDEs to the system of fixed points problem by using the technique of the topological degree theory also called prior estimate method. We establish sufficient conditions that guarantee the existence and uniqueness of positive solution to the system under consideration. Moreover, suitable results are also developed for the Hyers–Ulam stability analysis for the solution of the considered problem. An example is also included to reveal our main result.

We investigate the appropriate and sufficient conditions for the existence and uniqueness of a solution for a coupled system of Atangana–Baleanu fractional equations with a p-Laplacian operator. We also study the HU-stability of the solution by using the Atangana–Baleanu–Caputo (ABC) derivative. To achieve these goals, we convert the coupled system of Atangana–Baleanu fractional equations into an integral equation form with the help of Green functions. The existence of the solution is proven by using topological degree theory and Banach’s fixed point theorem, with which we analyze the solution’s continuity, equicontinuity and boundedness. Then, we use Arzela–Ascolli theory to ensure that the solution is completely continuous. Uniqueness is established using the Banach contraction principle. We also investigate several adequate conditions for HU-stability and generalized HU-stability of the solution. An illustrative example is presented to verify our results.

This paper investigates the necessary conditions relating to the existence and uniqueness of solution to impulsive system fractional differential equation with a nonlinear p-Laplacian operator. Our problem is based on two kinds of fractional order derivatives. That is, Atangana–Baleanu–Caputo (ABC) fractional derivative and the Caputo–Fabrizio derivative. To achieve our main aims, we will first convert the proposed impulse system into an integral equation form. Next, we prove the existence and uniqueness of solutions with the help of Leray–Schauder’s theory and the Banach contraction principle. We analyze the operator for continuity, boundedness, and equicontinuity. Further, we investigate the stability solution to the proposed impulsive system by using stability techniques. In the last part, we demonstrate the results via an illustrative example for the application of the results.

A random arbitrary-order mathematical system is investigated via the global and non-singular kernel of Atangana–Baleanu in the sense of Caputo $({𝒜}{ℬ}{𝒞})$ derivative in this study where the proposed problem is divided into four general compartments for the explanation. To show the existing result, the Krasnosilkii’s theorem from the theory of fixed points is used, whereas the well-known Banach theorem is utilized in order to show that the solution is unique to the proposed problem. Furthermore, by using the idea of Hyers–Ulam (UH) stability, the generalized problem is perturbed little for the purpose of checking its stability. The numerical solution is evaluated by applying the Adams–Bashforth iterative techniques. The numerical examples derived are tested in order to illustrate the established outcomes along with the numerical simulation to demonstrate the verification of the results obtained. The dynamics of every compartment is examined on different non-integer order ${𝒷}$ and by choosing arbitrary time $t$ by the taken approximate solution employing the AB numerical technique. Ultimately, the total continuous spectrum on the dynamics of each quantity in any arbitrary order lying between any of the two natural values, namely 0 and 1, has been achieved based on the investigated analyses.

It is known that hyperbolic nonautonomous linear delay differential equations in a finite dimensional space are Hyers–Ulam stable and hence shadowable. The converse result is available only in the special case of autonomous and periodic linear delay differential equations with a simple spectrum. In this paper, we prove the converse and hence the equivalence of all three notions in the title for a general class of nonautonomous linear delay differential equations with uniformly bounded coefficients. The importance of the boundedness assumption is shown by an example.

In this paper, integro-differential equations are solved by using an efficient numerical technique, namely, Multistage Optimal Homotopy Asymptotic method. The existence and uniqueness of solutions are established by the application of Lipschitz condition. Convergence of approximate solutions along with stability are also carried out. Some examples are solved to highlight the vital characteristics of the applied numerical scheme. Error estimation and comparison of derived results with existing exact solutions and those results which already available in the literature through graphical illustrations and tables reveal that Multistage Optimal Homotopy Asymptotic algorithm is more efficient and fruitful.

Gordji et al. proved the Hyers–Ulam stability and the superstability of J*-derivations in J*-algebras for the generalized Jensen type functional equation

Eshaghi Gordji and Ghobadipour proved the Hyers–Ulam stability of (α, β, γ)-derivations on Lie C*-algebras associated with the following functional equation

In this work, we introduce quadratic Jordan *-derivations on real C*-algebras and real JC*-algebras and prove the Hyers–Ulam stability of Jordan *-derivations and of quadratic Jordan *-derivations on real C*-algebras and real JC*-algebras. We also establish the superstability of such derivations on real C*-algebras and real JC*-algebras by using a fixed point theorem.

Ebadian et al. [Stability of bi-θ-derivations on JB*-triples, Int. J. Geom. Meth. Mod. Phys.9(7) (2012) 1250051, 12 pp.] proved the Hyers–Ulam stability of bi-θ-derivations on JB*-triples. Under the conditions of the definition of bi-θ-derivation given in the above paper, the bi-θ-derivation must be zero. So the results given in the above paper must be trivial. Under the conditions in the main theorem, we can show that the related mapping must be zero. In this paper, we correct the statements of the results and prove the corrected theorems and corollaries.

Shokri et al. [Approximate bihomomorphisms and biderivations in 3-Lie algebras, Int. J. Geom. Methods Mod. Phys.10 (2013) 1220020, 13pp.] proved the Hyers–Ulam stability of bihomomorphisms and biderivations on normed 3-Lie algebras. It is easy to see that the definition of bihomomorphism in normed 3-Lie algebras is meaningless and so the results of [J. Shokri, A. Ebadian and R. Aghalari, Approximate bihomomorphisms and biderivations in 3-Lie algebras, Int. J. Geom. Methods Mod. Phys.10 (2013) 1220020, 13pp., Sec. 3] are meaningless. Moreover, there is a serious problem in the main functional equation (1.2). So, we replace the functional equation (1.2) by a suitable functional equation. In this paper, we correct the definition of bihomomorphism and the statements of the results in [J. Shokri, A. Ebadian and R. Aghalari, Approximate bihomomorphisms and biderivations in 3-Lie algebras, Int. J. Geom. Methods Mod. Phys.10 (2013) 1220020, 13pp., Sec. 3], and prove the corrected theorems.

The primary varicella-zoster virus (VZV) infection that causes chickenpox (also known as varicella), spreads quickly among people and, in severe circumstances, can cause to fever and encephalitis. In this paper, the Mittag-Leffler fractional operator is used to examine the mathematical representation of the VZV. Five fractional-order differential equations are created in terms of the disease’s dynamical analysis such as S: Susceptible, V: Vaccinated, E: Exposed, I: Infectious and R: Recovered. We derive the existence criterion, positive solution, Hyers–Ulam stability, and boundedness of results in order to examine the suggested fractional-order model’s wellposedness. Finally, some numerical examples for the VZV model of various fractional orders are shown with the aid of the generalized Adams–Bashforth–Moulton approach to show the viability of the obtained results.

In this paper, we give the general solution of the Euler–Lagrange–Rassias-type quadratic functional equation

In this paper, we introduce a new form of the quadratic reciprocal functional equation. Then, we study the Hyers–Ulam stability for this quadratic reciprocal functional equation in non-Archimedean fields.

In this paper, we establish some Hyers–Ulam stability and hyperstability results of the following functional equation