Search Results (381)

This paper considers a class of fractional stochastic differential systems driven jointly by the Rosenblatt process and a compensated Poisson random measure. The Rosenblatt process captures non-Gaussian fluctuations with memory effects; the Poisson jumps model sudden random shocks. By employing the mild solution formulation associated with fractional resolvent operators, we establish the existence of solutions via Krasnoselskii’s fixed point theorem (KFPT) and prove uniqueness through the Banach contraction under suitable Lipschitz conditions on the drift, diffusion, and jump coefficients together with contraction assumptions. Furthermore, Ulam–Hyers stability is established, ensuring that approximate solutions remain close to exact solutions in the mean-square sense. We also establish Mittag–Leffler-type continuous dependence on initial data, demonstrating that solutions depend continuously on their initial histories in the mean-square sense. In addition, open-loop approximate trajectory realization is established through the construction of an explicit open-loop control law that steers the stochastic system along any prescribed admissible trajectory under suitable invertibility and regularity assumptions. An example validates the theoretical results, demonstrating fractional memory, Rosenblatt noise, and Poisson jumps. Full article

We establish a metrization result for some $b_v\left(s\right)$-metric spaces, extending a result recently established for rectangular b-metric spaces. Furthermore, we introduce the notion of an extended polygonal b-metric space (or $b_v\left(\theta \right)$-metric space), which unifies and generalizes several classes of spaces, including metric spaces, rectangular metric spaces, b-metric spaces, rectangular b-metric spaces, polygonal metric spaces, and $b_v\left(s\right)$-metric spaces. Some fixed-point results in $b_v\left(\theta \right)$-metric spaces are established under the weak orbital completeness condition in the framework of the Banach contraction principle and for generalized expansive Hardy-Rogers-type mappings. An a priori error estimate for the iterative process is obtained in both $b_v\left(\theta \right)$-metric and $b_v\left(s\right)$-metric spaces. We also establish the Ulam-Hyers stability of fixed-point equations in both $b_v\left(\theta \right)$-metric and $b_v\left(s\right)$-metric spaces. Several examples are provided, and applications to certain types of integral equations and initial value problems are presented, illustrating the applicability and effectiveness of the obtained results. Full article

In this paper, we introduce the following quadratic functional equation: ${\displaystyle \sum _{i=1}^{m-1}}{\displaystyle \frac{i}{i+1}}f\left({\displaystyle \frac{{\sum }_{j=1}^i{x}_j}{i}}-x_{i+1}\right)+{\displaystyle \frac{1}{m}}f\left({\displaystyle \sum _{i=1}^m}{x}_i\right)={\displaystyle \sum _{i=1}^m}f\left(x_i\right)$ which is derived from a geometric median identity in the Euclidean plane. Then, we find a general solution to the quadratic functional equation and prove the stability results of the quadratic ∗-derivations in $\rho$-complete convex n-modular ∗-algebras. Full article

This paper investigates a class of nonlinear fractional differential equations with boundary value problems involving Caputo–Hadamard-type derivatives. By relaxing the monotonicity constraints on the nonlinear terms and considering more general nonlinear structures, the paper extends the theoretical framework and application scope of the relevant fractional models. Using the upper-lower solution method in conjunction with Schauder’s fixed-point theorem, we establish the existence of exact solutions; furthermore, by applying the Banach contraction mapping principle, we prove the uniqueness of the solutions. Concurrently, we construct a convergent iterative approximation scheme and provide a priori and a posteriori error estimates for numerical solution. Furthermore, the robustness of the solutions to perturbations is characterised via Ulam–Hyers stability analysis, ensuring the reliability of the approximate solutions. Finally, numerical examples are employed to validate all theoretical results. The qualitative theory and numerical analysis methods for Caputo–Hadamard-type fractional boundary value problems have been improved. Full article

In this paper, we introduce $\vartheta$-fuzzy quasi-k-norms with respect to a continuous t-norm $\vartheta$. Also, we investigate fuzzy operators defined on the product of a ∗-fuzzy measure space and an algebraic group, taking values in a $\vartheta$-fuzzy quasi-k-normed space. Furthermore, by employing a fuzzy controller associated with the ct-norm $\vartheta$, we give sufficient conditions for the approximation of a given operator by a homomorphic fuzzy operator. Full article

In this paper, we investigate the generalized Hyers–Ulam stability of a multi-variable additive-quadratic functional equation in Banach spaces. Unlike traditional mixed-type equations that segregate variables, the proposed equation embeds a dense combinatorial cross-variable interaction. To overcome the complex analytical challenges arising from asymmetric scaling weights, we prove a complete structural equivalence theorem between this multivariate framework and the classical additive-quadratic system of equations. We establish comprehensive stability results utilizing two distinct methodological paradigms: the direct constructive aggregation method and the generalized metric fixed-point alternative. Furthermore, we provide a rigorous mathematical justification of an analytical singularity by constructing a counterexample that explicitly proves the inherent instability of the system at the critical exponent threshold $p=3$. Full article

This paper investigates a class of Hadamard variable-order fractional differential equations in which the fractional order is determined by the cumulative distribution function (CDF) of a continuous random variable. The proposed framework establishes a novel connection between probability theory and variable-order fractional calculus by allowing the memory index of the fractional operator to evolve according to a prescribed distribution law. To facilitate the analysis, the CDF-based variable order is considered through a piecewise-constant representation on a finite partition of the interval, which transforms the original problem into a family of Hadamard fractional differential equations of constant order on successive subintervals. Existence and uniqueness results are established by converting the differential problem into an equivalent fractional integral equation and applying the Banach contraction principle in suitable Banach spaces. Sufficient conditions ensuring the well-posedness of the problem are derived in terms of explicit bounds involving the fractional order and the nonlinear term. In addition, the Ulam–Hyers stability of the proposed model is investigated, and stability criteria are obtained under the same analytical framework. To illustrate the applicability of the theoretical results, a numerical example involving a CDF-generated variable-order function is presented. The example verifies the assumptions of the existence, uniqueness, and stability theorems and demonstrates the effect of piecewise-constant approximations of the cumulative distribution function on the resulting numerical solutions. The obtained results show that the proposed CDF-based Hadamard variable-order framework provides a mathematically consistent setting for studying fractional differential equations whose memory characteristics depend on probabilistic distributions. Full article

This paper introduces a fractal–fractional SEIR epidemic model based on the Atangana–Baleanu derivative in the Caputo sense augmented by fractal scaling. The fractional order $\alpha \in (0,1]$ captures memory effects while the fractal dimension $\beta \in (0,1]$ accounts for irregular contact networks. We prove global existence, uniqueness, positivity, and boundedness of solutions via fixed-point arguments and establish global Ulam–Hyers stability. An adapted second-order Adams–Bashforth–Moulton predictor-corrector scheme with explicit weights is derived and verified. Numerical simulations across representative $(\alpha ,\beta )$ pairs reveal that decreasing either parameter delays epidemic peaks, reduces peak intensity (with $\beta$ exerting a stronger damping effect), prolongs tails, and induces irregular oscillations—features absent from classical or pure-fractional SEIR models. These results provide a rigorous and reproducible framework for forecasting emerging infections in heterogeneous populations and carry direct implications for targeted public health interventions. Full article

In this paper, we investigate a class of fractional functional differential systems with feedback control, in which the fractional derivatives are interpreted in the Caputo sense. By applying Schauder’s fixed point theorem with the Arzelà–Ascoli compactness criterion, we derive existence results for the solutions. Furthermore, specific hypotheses are introduced to establish uniqueness, continuous dependence of the unique solution, and Hyers–Ulam stability of the considered problem. The applicability and relevance of the theoretical results are further demonstrated through several special cases and illustrative examples. The main contribution of this study is to provide a qualitative analysis of a coupled Caputo fractional functional system involving nonlocal integral conditions and feedback control. Full article

In this paper, we study some existence and stability results related to the boundary value problem involving the Atangana–Baleanu–Caputo hybrid fractional derivative. More precisely, we transform the studied problem to an equivalent integral equation, and after that, by applying appropriate fixed-point theorems and using suitable conditions, we prove the existence of solutions. Furthermore, we derive sufficient conditions that guarantee the stability in the sense of Hyers–Ulam. To support the theoretical findings, we present illustrative examples along with numerical simulations. Full article

This paper establishes a rigorous analysis of a coupled hybrid fractional differential system involving a generalized Hilfer operator under integral and antiperiodic boundary conditions. The existence and uniqueness of solutions are proved using Dhage’s fixed point theorem for existence and the Banach contraction principle for uniqueness. Furthermore, we establish Ulam–Hyers stability by deriving the following explicit and computable bound estimate: $\left(\begin{array}{c}\parallel \widehat{u}-u\parallel \\ \parallel \widehat{v}-v\parallel \end{array}\right)\le {(I-\mathit{\chi })}^{-1}\left(\begin{array}{c}{C}_1{ϵ}_1\\ C_2{ϵ}_2\end{array}\right)$, where $C_1$ and $C_2$ are positive constants depending on the system parameters, ${ϵ}_1,{ϵ}_2$ denote the perturbation bounds, and $\mathit{\chi }$ is the associated Lipschitz matrix. This formulation provides a more detailed stability description than scalar criteria, as it captures the interactions among the system components through the entries of $\mathit{\chi }$, leading to a more informative stability estimate. Two illustrative examples confirm the theoretical results and demonstrate their potential applicability for modeling real-world phenomena where memory effects are present. Full article

We study a perturbed coupled system of generalized hybrid pantograph equations involving the deformable derivative of Zulfeqarr–Ujlayan–Ahuja. A central point of the revision is made explicit: for classically differentiable functions this derivative is local and satisfies $D^{\tau }u=(1-\tau )u+\tau u^{\prime }$. Therefore, in the present differentiable setting the memory or aftereffect is produced by the proportional pantograph delays, while the deformable order $\tau$ supplies an order-dependent local relaxation/drift term. After rewriting the system as an equivalent integral equation on $\mathcal{X}=C(I,{\mathbb{R}}^2)$, we establish invariant-ball conditions, existence and uniqueness within invariant balls, generalized Ulam–Hyers stability, and Lipschitz continuous dependence on the perturbation amplitude $\epsilon$. The assumptions and constants are stated so that the restrictive roles of the Lipschitz bounds, the interval length, and $\left|\epsilon \right|$ are transparent. We then provide numerical parameter sensitivity diagrams for illustrative pantograph systems and include step-size refinement checks and performance indices. The numerical and plasma-inspired sections are deliberately framed as exploratory delay-feedback examples rather than as first-principles plasma models or rigorous bifurcation theory. Full article

This article employs a fractional differential equation model to probe the dynamic mechanism of animal avoidance learning and memory retention. This model encompasses both linear and nonlinear scenarios. We first obtain the series-type analytical solution for the linear scenario and its absolute uniform convergence by Laplace transform and Mittag–Leffler function. Secondly, we establish the existence, uniqueness and Ulam–Hyers stability for the nonlinear scenario via the fixed point theorem and analytical techniques. Eventually, some examples and numerical simulations are provided to examine the effectiveness and availability of the main findings. Full article

Diabetes mellitus is a chronic disease with complex progression dynamics. This study introduces a fractional order compartmental model based on the Caputo derivative, a singular-kernel derivative, to describe disease progression across four compartments: susceptible, insulin-resistant, diabetic without complications, and diabetic with complications. The model is novel for integrating memory effects into disease-stage transitions while maintaining dimensional consistency. Key mathematical properties, including existence, uniqueness, positivity, boundedness, equilibrium analysis, and both local and global stability, are established. Ulam–Hyers stability is also examined to evaluate the robustness of the model solutions. Numerical approximations are obtained using the Adomian Decomposition Method and its Laplace variant. Simulations indicate that lower fractional orders enhance memory effects, slow disease progression, and influence long-term dynamics. These results demonstrate that the proposed approach provides a flexible and robust framework for studying chronic disease progression and makes a meaningful contribution to the literature on fractional diabetes models. Full article

This study formulates closed-form solution expressions for linear discrete matrix equations that involve several time delays, without requiring the coefficient matrices or the non-homogeneous term to commute. Using a generalized multinomial series and exponential matrix functions adapted to multiple delays, we establish fundamental solutions in a setting where matrix multiplication is not assumed to be commutative. These explicit representations are subsequently utilized to analyze the stability properties of the system, specifically establishing Hyers–Ulam stability. The analysis elucidates the influence of both delay structure and noncommutativity on solution behavior and robustness. A representative example is provided to illustrate the practical applicability of the proposed method and to highlight the significant qualitative effects induced by delays and noncommutative matrix interactions. Notably, the results extend classical theories by addressing noncommutative settings and yield novel contributions that remain significant even in the absence of delays. Full article