This paper develops a comprehensive asymptotic framework for analyzing the long-term behavior of non-oscillatory solutions in high-order forced and disturbed fractional difference systems of the Caputo type. We consider a general nonlinear model in which memory effects, external forcing, and nonlinear disturbance terms interact within a higher-order nabla fractional operator. Despite the growing interest in discrete fractional calculus, existing results primarily address first-order or unforced systems, leaving a significant gap in understanding the asymptotic dynamics of complex high-order models with multiple nonlinearities. To address this gap, we derive new sufficient conditions ensuring that all eventually non-oscillatory solutions remain bounded within an explicit asymptotic envelope of the form
[
|Ψ(ι)| = O!\left((ι^{n-1})^{1/υ} R(ι,c)\right),
]
where (R(ι,c)) is a summable fractional kernel depending on the system’s coefficients. The established criteria incorporate delicate growth restrictions on the forcing term, the nonlinear damping functions, and their relative exponents, thereby generalizing earlier theorems and offering sharper bounds.

The analysis further yields refined exponential-type bounds under additional summability conditions, highlighting how fractional memory and nonlinear perturbations shape the qualitative behavior of solutions. Two detailed numerical examples validate the theoretical findings and illustrate the precision of the derived envelopes. The results significantly extend the current theory of fractional difference equations, providing new analytical tools for models arising in discrete population dynamics, engineering, and other applications where memory and external disturbances play essential roles.

Fractional differential equations with sequential derivatives have become increasingly important because they can capture memory effects and complex behaviours that classical derivatives fail to describe. These tools are widely used today to model real processes in areas such as biology, chemistry, and physics, where systems often evolve in ways that depend not only on their current state but also on their history. Motivated by these applications, we investigate a new multi-term fractional differential equation involving sequential Caputo derivatives, motivated by recent research on fractional models with multiple memory effects. Unlike classical single-term models, the presence of sequential derivatives introduces additional analytical challenges, particularly in constructing explicit solution formulas.

Using techniques inspired by earlier works on sequential fractional operators, we derive the exact analytic solution of the problem in terms of the bivariate Mittag-Leffler function. The obtained representation provides a clear description of how the interaction between the two fractional orders influences the system's temporal dynamics. Additionally, several useful properties of the bivariate Mittag-Leffler function are formulated to support the construction of the solution.

The results demonstrate that multi-term sequential fractional equations admit explicit closed-form solutions under suitable boundary and initial conditions. These findings contribute to the theoretical development of sequential fractional models and offer a practical framework for analysing complex diffusion-type processes with layered memory effects.

The theory of complex Dirichlet spaces is very important in complex analysis. These spaces consist of holomorphic functions whose derivatives are Lebesgue square-integrable in norm. Similarly, the theory of quaternionic Dirichlet modules of slice regular functions is widely known as a natural extension of the theory of complex Dirichlet spaces via quaternionic analysis.

On the other hand, fractional calculus is a theory that allows us to consider integrals and derivatives of any real or complex order, where the fundamental theorem of usual calculus is also extended. Then, by extending the concept of derivative through the fractional proportional derivative with respect to the truncated exponential function in the complex Dirichlet spaces and quaternionic Dirichlet modules, we obtain many families of function spaces.

Some families of complex Banach spaces and quaternionic Banach modules of functions associated with generalized fractional derivatives with respect to a truncated exponential function are presented. The aim of this study is to present an interesting extension of the usual function theory of complex Dirichlet spaces and quaternionic Dirichlet modules in terms of fractal– fractional calculus, finding a large variety of families of complex Banach spaces and quaternionic Banach modules that contain the already known spaces.

In this talk, we consider a class of local Iterated Function Systems (IFSs) whose attractors are the graphs of local fractal functions both in a real Orlicz space and in a real Orlicz–Sobolev space over a domain of $\mathbb{R}^N$. We prove that these maps are in correspondence with the fixed points of the Read–Bajraktarević operator and justify a step in the proof of a theorem recently published in the literature. This result states that local fractal function of an Orlicz–Sobolev class of order $m>1$ ($m$ is an integer) appear naturally as fixed points of the restriction of the Read–Bajraktarević functional. This has been well-known since the works of Massopust et al. in the context of Lebesgue and Sobolev spaces. Our method extends a number of known theorems on the existence of local fractal functions to more general function spaces (where the role of the norm is now played by a Young function, also known as an N function) and to higher orders and dimensions. In this talk, we will relax a condition (inequality) that ensures contractivity of the Read–Bajraktarević operator, and we will prove that an asymptotic quantity appearing in the proof is not required to be greater than 1. The existence of local fractal functions of the Orlicz and of the Orlicz–Sobolev classes is demonstrated through an intermediary result. The realization of a contractive IFS in the (previously untreated) multidimensional case is obtained via a stronger version of the mean-value theorem. Our results demonstrate that it is natural to extend the Read–Bajraktarević operator to other function spaces on subdomains of differentiable and real analytic manifolds. Other questions, such as the existence of fixed points in higher orders, remain open, as well.

In this talk, we prove the existence of complex local fractal functions of the Hardy-Orlicz class. A local fractal function is the fixed point of the Read-Bajraktarevic (RB) functional. The graph of the fixed point is the attractor of an appropriate iterated function system (IFS), whose construction is fairly standard. When the underlying function space is a Hardy-Orlicz space, the local fractal function of the Hardy-Orlicz class (fixed point of the RB operator) displays peculiar properties stemming from the complex conjugation. For example, it is known that both the fixed point and its graph are intrinsically real. The latter reflects as an embedding of the realified attractor of the induced iterated function system into the product [–1, 1] × [–1, 1]. The induced IFS in this case is denominated a Complex Iterated Function System (CIFS). In this talk, we provide a characterization of this type of IFSs via a quasi-integral representation of the Read-Bajraktarevic (RB) functional. The realization of a contractive IFS in the (previously untreated) analytic case is obtained via a finite-increments theorem for holomorphic functions, dating back to 1965. We prove that both the fractal function and the induced CIFS are complex extensions of their real counterparts. In a broad sense, our results generalize some of the results obtained by Massopust et al. on Lebesgue and Sobolev spaces to higher orders, dimensions, and function spaces (where a Young function now plays the role of the norm). These results somewhat show that it would be natural to extend the Read-Bajraktarević operator to other function spaces on subdomains of differentiable and real analytic manifolds. Other questions, such as the existence of fixed points of higher order, or in more sophisticated function spaces, remain open as well. Our generalizations may be useful in applications.

Determining the order of fractional derivatives in partial differential equations (PDEs) presents a significant challenge in applied mathematics. Fractional derivatives play a crucial role in accurately modeling complex physical phenomena, including anomalous diffusion, wave propagation, viscoelasticity, and various fluid dynamic processes. Unlike classical derivatives, fractional-order models account for memory and nonlocal effects, making them highly relevant for real-world systems. However, the direct measurement of the fractional order is typically impractical due to the lack of appropriate instruments or experimental techniques. As a result, this leads to an inverse problem—finding the unknown fractional order from available, often indirect, observational data of the solution to the governing PDE.

Recent studies have made substantial progress in analyzing such inverse problems, particularly for time-fractional subdiffusion equations, where uniqueness results have been rigorously established. Nevertheless, many physical processes cannot be fully captured by subdiffusion models alone. This presentation explores a newly developed inverse problem formulation that extends the analysis to a broader class of equations, including fractional-wave equations, Rayleigh-Stokes-type models, and mixed-type fractional PDEs.

Our approach not only ensures the uniqueness of the estimated fractional order but also proves the existence of solutions under suitable conditions. This provides a more complete and versatile theoretical framework with strong potential for applications in engineering, physics, and beyond.

The Atangana-Baleanu fractional integral operator is of particular importance due to its nonsingular Mittag-Leffler kernel, which allows it to be used in many branches of applied mathematics for the development and study of mathematical models that involve it. The interest in the study of hypergeometric functions in their connection to the theory of univalent functions has reappeared, as L. de Branges used hypergeometric functions in the proof of the famous Bieberbach conjecture. The confluent (Kummer) hypergeometric function was studied recently from many points of view. Applying the differential subordinations and differential superordinations theories introduced by Miller and Mocanu, interesting inequalities can be established for complex-valued functions, yielding sandwich results. In this paper, we introduce a new operator defined by the Atangana-Baleanu fractional integral applied tothe confluent hypergeometric function
which takes the following form

and we establish several subordination and superordination results, which, taken together, constitute sandwich-type theorems. Interesting corollaries are stated using particular functions as best subordinant and best dominant of the subordinations and superordinations studied in theorems stated in this paper.

Application of a Fractional Fourier Transform with Memory to nonlocal fractional differential
equations is proposed in this work. The new transform is defined as
\[
\mathcal{F}_{\alpha,\beta}[u](\omega) :=
\int_{-\infty}^{+\infty}
u(t)\,
\exp\!\Bigg(
i\, I^{\beta}\, g(\omega)\,
\int_{0}^{t} h(s)\,ds
\Bigg)\, dt ,
\]
where $g(\omega)$ and $h(s)$ characterize memory effects, while $I^{\beta}$ provides an
additional fractional degree of freedom. This formulation allows us to capture nonlocal and
hereditary behaviors commonly observed in fractional-order systems, and reduces to the
classical Fourier transform as a special case when $g(\omega)$ is constant, $h(s)=1$, and
$\beta=0$.

Beyond its formal definition, the proposed framework offers a more flexible mathematical
structure that is particularly well-suited to problems where memory effects play a dominant
role. Many physical and engineering systems—such as anomalous diffusion, viscoelasticity, and
biological processes—cannot be accurately described without explicitly incorporating
hereditary dynamics. The memory-based kernel of the transform provides a natural mechanism
for integrating such effects, thereby extending the applicability of Fourier analysis to a
much broader class of problems.

In addition to its theoretical relevance, the transform establishes new operational relations
with Caputo-type derivatives and fractional integrals. These relations enable the development
of efficient analytical and semi-analytical methods for solving nonlocal fractional
differential equations. Numerical illustrations further confirm that the proposed transform
achieves better adaptability to varying memory characteristics compared with the classical
Fourier and even standard fractional Fourier frameworks.

Overall, this study highlights the potential of the Fractional Fourier Transform with Memory
as a versatile and powerful tool, bridging the gap between traditional Fourier analysis and
the growing need for models that faithfully represent memory-dependent, nonlocal, and
fractional-order dynamics.

Riesz fractional integro-differentiation is recognized as an analytical framework that defines fractional powers of the Laplace operator through potentials and hypersingular integrals. Furthermore, hypersingular integrals are established as natural extensions of partial differential operators to fractional orders, thereby enabling the development of fractional calculus for multivariate functions based on these operator classes. This study is devoted to the generalization of Riesz fractional calculus to abstract metric measure spaces, where classical results are subsumed as particular instances. The central objective is to enhance the theoretical framework with new fundamental results on weighted continuity characterized by specific generalizaitons of the Hölder condition. Three principal contributions are presented. First, the local modulus of continuity is refined through a generalized definition compatible with advanced theoretical constructs, extending its applicability to broader function classes on metric measure spaces. Second, Zygmund-type estimates are derived for power-weighted functions via a novel analytical approach, resolving technical challenges introduced by power-law weights that are absent in unweighted settings. Third, boundedness theorems are established to characterize the behavior of potential-type operators and hypersingular integrals on power-weighted variable generalized Hölder spaces, thereby completing the analytical foundation of Riesz fractional calculus in these generalized settings. The scientific significance of this work lies in its advancement of fractional calculus as a discipline within function analysis and operator theory. Specifically, the developed theory rigorously formalizes continuity properties of functions on abstract metric measure spaces, offering an approach that unifies classical and generalized perspectives. These contributions are anticipated to hold substantial implications for applications in integral equations and mathematical modeling, where the derived continuity estimates and operator boundedness properties may be leveraged to analyze complex systems under non-standard regularity assumptions.

In this work, we constructed the dynamics equilibrium equation for fractal manifolds based on the concept of fractal calculus introduced recently by Balankin, using their fractal continuum derivatives, as an extension of traditional differential calculus to spaces and functions characterized by fractal geometry.

The generalized form of the equation of motion for two-dimensional self-similar frames subjected to forced vibration incorporates both the stiffness and mass matrices, which account for the hierarchical structure and scaling properties of the frame's geometry; allowing the accurate modeling of dynamic responses by considering the influence of fractal continuity and irregular distribution of mass and rigidity, reflecting the distinctive physical behavior of such frames under external influence.

The fundamental interrelation between Balankin fractal derivatives and ordinary derivatives establishes a connection that enables the transformation of vector differential operators defined in the fractal domain R_x³ into their counterparts within the fractal continuum R_ξ³. This relationship behaves as a mathematical bridge, mapping complex fractal structures (characterized by non-integer dimensions and self-similarity) into a generalized fractal continuum framework.

The alpha and beta parameters of order α, β ∈ (0,1] are added to the new fractal matrices via this transformation, preserving essential fractal properties (geometry and topology) and providing new analytical tools and models for physical phenomena occurring within fractal continua. The influence of these parameters on the vibrational characteristics of the frames is analyzed graphically. Some mechanical implications are also discussed.