This paper examines the asymptotic behavior of solutions to linear fractal differential equations within the framework of F^α -calculus. We identify the conditions that determine whether the solutions remain stable, grow, or decay. These dynamics are further explored through comprehensive examples and theoretical findings, emphasizing the self-similar characteristics of solutions, including first- and second-order higher α -order fractal differential equations.

This paper presents the foundational concepts of fractal calculus before generalizing the Dirac Constraint Formalism and the Faddeev-Jackiw Formalism for first α -order Lagrangian systems in fractal spaces with non-integer dimensions. We provide a detailed analysis of the generalization process, highlighting the theoretical framework and key results, including the extended structure of the constraint systems in these Lagrangian formulations. Specific examples are discussed to demonstrate the practical application of the generalized formalism and to validate the consistency of our results. Moreover, graphical visualizations are included to enhance clarity, offering a visual interpretation of the findings and illustrating the relationship between the theory and its real-world implications.

This paper presents a novel low-pass filtering framework based on Fractal First α -order and Second α -order designs, formulated within the framework of fractal calculus. By incorporating the structure of fractal time, the proposed filters can effectively process signals with intricate, non-differentiable characteristics. The fractal second α -order low-pass filter is applied to a simulated noisy ECG signal, demonstrating significant noise suppression while preserving the essential morphological features of the waveform. A comparative study with the classical Bessel low-pass filter further illustrates the advantages of the fractal approach in capturing scale-invariant and self-similar properties of biomedical signals. These results highlight the potential of fractal-order filters for advanced biomedical signal processing.

In this paper, we give a brief summary of fractal calculus. Fractal functional differential equations are formulated as a framework that provides a mathematical model for the phe-nomena with fractal time and fractal structure. Fractal retarded, neutral, and renewal delay differential equations with constant coefficients are solved by the method of steps and us-ing Laplace transform. The graphs of solutions are given to show the details.(c) 2023 Elsevier Inc. All rights reserved.

A non-linear system of Volterra integro-fractional delay differential equations with Caputo fractional derivatives is considered. New sufficient conditions for uniform stability, asymptotic stability, and Mittag-Leffler stability of the zero solution of the unperturbed system, and the boundedness of all solutions of the perturbed system, are presented. The technique of proof involves the Razumikhin method with an appropriate Lyapunov function. For illustrative purposes, two examples are provided. (C) 2021 Published by Elsevier B.V.

Let G be a locally compact abelian group and let M(G) be the convolution measure algebra of G. A measure μ∈M(G) is said to be power bounded if supn≥0⁡‖μn‖1<∞, where μn denotes nth convolution power of μ. We show that if μ∈M(G) is power bounded and A=[an,k]n,k=0∞ is a strongly regular matrix, then the limit limn→∞⁡∑k=0∞an,kμk exists in the weak⁎ topology of M(G) and is equal to the idempotent measure θ, where θˆ=1intFμ. Here, θˆ is the Fourier-Stieltjes transform of θ, Fμ:={γ∈Γ:μˆ(γ)=1}, and 1intFμ is the characteristic function of intFμ. Some applications are also given.

In this paper, we summarize the local fractal calculus, called $$F^{\alpha }$$ -calculus, which defines derivatives and integrals of functions with fractal domains of non-integer dimensions, functions for which ordinary calculus fails. Hyers–Ulam stability provides a method to find approximate solutions for equations where the exact solution cannot be found. Here, we generalize Hyers–Ulam stability to be applied to $$\alpha $$ -order linear fractal differential equations. The nuclear decay law involving fractal time is suggested, and it is proved to be fractally Hyers–Ulam stable.

In 1933, Adams [1] developed Hausdorff transformations for double sequences. H. ?evli and R. Sava? [18] proved some result for the double Endl- Jakimovski (E-J) generalization. In this study, we consider some further results for E-J Hausdorff transformations for double sequences

In 1933, Adams [1] developed Hausdorff transformations for double sequences. H. Şevli and R. Savaş [18] proved some result for the double EndlJakimovski (E-J) generalization. In this study, we consider some further results for E-J Hausdorff transformations for double sequences.

The object of the present paper is to examine the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of a nonlinear Volterra integro-differential equation by using the fixed point method. c ©2016 All rights reserved.

Das (Proc. Camb. Philos. Soc. 67:321-326, 1970) proved that every conservative Hausdorff matrix is absolutely k th power conservative. Savaş and Rhoades (Anal. Math. 35:249-256, 2009) proved the result of Das for double Hausdorff summability. In this paper we will consider the double Endl-Jakimovski (E-J) generalization and we will prove the corresponding result of Savaş and Şevli (J. Comput. Anal. Appl. 11:702-710, 2009) for double E-J generalized Hausdorff matrices. MSC: 40F05, 40G05.

In this paper, we will establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.MSC: 34A05, 39B82, 26D10, 34A40.

In this paper, we prove some inequalities related to the concept of C11(st2)-conservative matrices, C11(st2)−lim sup and C11(st2)−lim inf which are natural analogues of (Cbp,st2∩L∞)-matrices, st2−lim sup and st2−lim inf, respectively.

In this work, we will prove that every solution of a perturbed Volterra integro-differential equation can be approximated by a solution of the Volterra integro-differential equation.

Using the power series method, we solve the inhomogeneous linear rst order dierential equation y 0 (x) + (x )y(x) = 1 X m=0 am(x ) m ; and prove an approximation property of Gaussian functions. an(x)y (n) (x) +an 1(x)y (n 1) (x) + +a1(x)y 0 (x) +a0(x)y(x) +h(x) for all x2 I and for some 0, there exists a solution f0 : I! Y of the dierential equation an(x)y (n) (x) +an 1(x)y (n 1) (x) + +a1(x)y 0 (x) +a0(x)y(x) +h(x) = 0 such that kf(x) f0(x)k K() for any x 2 I, where K() depends on only, then we say that the above dierential equation satises

In this paper using $ \delta $-quasi-monotone sequences a theorem on $ | C, \alpha, \beta; \delta |_k $ summability factors of infinite series, which generalizes a theorem of Mazhar[6] on $ | C, 1 |_k $ summability factors, has been proved.

In this paper, we study the concepts of statistically convergent and statistically Cauchy double sequences in the framework of fuzzy normed spaces which provide better tool to study a more general class of sequences. We also introduce here statistical limit point and statistical cluster point for double sequences in this framework and discuss the relationship between them.

The object of the present paper is to determine the stability of the Hyers–Ulam–Rassias type theorem concerning the Pexiderized quadratic functional equation in intuitionistic fuzzy normed spaces (IFNS).

The object of this paper is to determine Hyers–Ulam–Rassias stability concerning the Jensen functional equation in intuitionistic fuzzy normed space (IFNS) by using the fixed point method. Further, we establish stability of the Cauchy functional equation in IFNS.