Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable solutions across various disciplines. In differential equations, regularity refers to the smoothness of solution behavior. The averaging principle offers an approximation that balances complexity and simplicity. Our research contributes to establishing the well-posedness, regularity, and averaging principle of FPSDE solutions in ${\mathcal{L}}^{\mathrm{p}}$ spaces with $\mathrm{p}\ge 2$ under Caputo derivatives. The main ingredients in the proof include the use of Hölder, Burkholder–Davis–Gundy, Jensen, and Grönwall–Bellman inequalities, along with the interval translation approach. To understand the theoretical results, we provide numerical examples at the end.

In this research work, we use the concepts of contraction mapping to establish the existence and uniqueness results and also study the averaging principle in ${\mathbb{L}}^p$ space by using Jensen’s, Grönwall–Bellman’s, Hölder’s, and Burkholder–Davis–Gundy’s inequalities, and the interval translation technique for a class of fractional neutral stochastic differential equations. We establish the results within the framework of the $\Psi$-Caputo derivative. We generalize the two situations of $p=2$ and the Caputo derivative with the findings that we obtain. To help with the understanding of the theoretical results, we provide two applied examples at the end.

In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear smoking model concerning the Caputo derivative. The outcomes of the proposed technique exhibit good agreement with the Laplace decomposition method, demonstrating that our technique is an excellent alternative to various series solution methods. Our approach utilizes the simple limit principle at zero, making it the easiest way to extract series solutions, while variational iteration, Adomian decomposition, and homotopy perturbation methods require integration. Moreover, our technique is also superior to the residual method by eliminating the need for derivatives, as fractional integration and differentiation are particularly challenging in fractional contexts. Significantly, our technique is simpler than other series solution techniques by not relying on Adomian’s and He’s polynomials, thereby offering a more efficient way of solving nonlinear problems.

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for $\mathrm{p}=2$ to results in the ${\mathbb{L}}^{\mathrm{p}}$ space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish.

We develop and analyze a Zika transmission model that couples mosquito‐borne and sexual pathways with host awareness and vector control interventions, assuming no disease‐induced mortality. The dynamics are formulated in a fractal‐fractional framework with order ℘ and fractal dimension ς , allowing memory and nonlocal effects. Existence and uniqueness of solutions are established via compactness and a Banach fixed‐point argument, and Ulam–Hyers stability is derived for the integral equation representing the system. For computation, we design a fractional Adams–Bashforth scheme and report simulations using baseline parameters from the literature. One at a time, sensitivity experiments identify the dominant amplifiers of infection (mosquito biting b ₂ and transmission probabilities, together with sexual contact c and α ₂ ) and show that awareness a and vector control b suppress prevalence; the fractional parameters modulate persistence, with larger ℘ / ς prolonging transients. We further employ a feedforward artificial neural network as a surrogate to approximate the numerical solution, using a training, validation, testing split, and standard performance diagnostics. Finally, we compare operator choices integer, Caputo, Hilfer, Atangana–Baleanu in Caputo sense, and FF under identical settings; all memory‐bearing models decay more slowly than the classical system, with ABC/FF exhibiting the strongest persistence. Future research will extend this framework to spatially structured and seasonally varying settings, calibrated with surveillance data, to support optimal intervention design and real‐time decision‐making.