This article primarily focuses on the approximate controllability of fractional semilinear integrodifferential equations using resolvent operators. Two alternative sets of necessary requirements have been studied. In the first set, we use theories from functional analysis, the compactness of an associated resolvent operator, for our discussion. The primary discussion is proved in the second set by employing Gronwall’s inequality, which prevents the need for compactness of the resolvent operator and the standard fixed point theorems. Then, we extend the discussions to the fractional Sobolev-type semilinear integrodifferential systems. Finally, some theoretical and practical examples are provided to illustrate the obtained theoretical results.

In this paper, we deal with the existence of integrable solutions of Gripenberg-type equations with m-product of fractional operators on a half-line ${\mathbb{R}}^{+}=[0,\infty )$. We prove the existence of solutions in some weighted spaces of integrable functions, i.e., the so-called $L_1^N$-solutions. Because such a space is not a Banach algebra with respect to the pointwise product, we cannot follow the idea of the proof for continuous solutions, and we prefer a fixed point approach concerning the measure of noncompactness to obtain our results. Appropriate measures for this space and some of its subspaces are introduced. We also study the problem of uniqueness of solutions. To achieve our goal, we utilize a generalized Hölder inequality on the noted spaces. Finally, to validate our results, we study the solvability problem for some particularly interesting cases and initial value problems.

Models of fractional‐order epidemics more accurately represent memory and hereditary effects compared to traditional integer‐order models. However, many Caputo–Fabrizio (CF) SIR models utilize a non‐normalized kernel, which can distort the scaling and understanding of memory. This research introduces a thorough formulation and analysis of an SIR model that uses the normalized CF (NCF) derivative, ensuring both dimensional consistency and meaningful contributions from memory. We identify important theoretical properties, such as existence, uniqueness, positivity, boundedness, and conservation, and we create an efficient time‐stepping method specifically designed for the normalized exponential kernel. Next, we examine the dynamics of the model when influenced by seasonal factors. The numerical results show that normalization significantly impacts outbreak patterns, leading to delayed and reduced infection peaks compared to both classical and unnormalized CF models, as well as altering bifurcation thresholds. Across the parameter ranges we examined, the long‐term behavior aligns into periodic patterns: Poincaré sets are finite, the dispersion of stroboscopic samples diminishes with increasing forcing amplitude, and the leading Lyapunov exponent remains nonpositive, suggesting no sensitive dependence on initial conditions. Therefore, in our investigations, the normalized kernel alters the periodic structure while avoiding the onset of chaos.

This study introduces a fractional‐order mathematical model for alcoholism dynamics using the Hilfer derivative to capture memory effects and hereditary properties within a unified framework. The model incorporates hypothetical social influence through sentiment‐based variables to represent positive and negative social interactions. No real social media data is analyzed in this work; instead, a conceptual framework for future integration of Twitter sentiment analysis using tools such as VADER, TextBlob, or BERT is proposed. Existence, uniqueness, and Ulam–Hyers stability of the model are rigorously established using fixed‐point theory. Numerical simulations are performed via the fractional Adams–Bashforth method, and artificial neural networks (ANNs) are employed solely as a surrogate approximation tool for fractional dynamics, reducing computational cost for future large‐scale simulations. Sensitivity analysis reveals that parameters ζ and δ strongly influence alcoholism prevalence, while the fractional order β governs the persistence of behavioral patterns. Comparative results demonstrate the qualitative advantages of the Hilfer derivative over integer‐order and Caputo models in capturing long‐term dependencies. This work provides a flexible theoretical framework for future empirical validation and real‐time predictive modeling.

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.

In this article, we prove fixed point results for a Meir–Keeler type contraction due to orthogonal M-metric spaces. The results of the paper improve and extend some recent developments in fixed point theory. The extension is assured by the concluding remarks and followed by the main theorem. Finally, an application of the main theorem is established in proving theorems for some integral equations and integral-type contractive conditions.