The idea of enriched mappings in normed spaces is relatively a newer idea. In this paper, we initiate the study of enriched mappings in modular function spaces. We first introduce the concepts of enriched $\rho$-contractions and enriched $\rho$-Kannan mappings in modular function spaces. We then establish some Banach Contraction Principle type theorems for the existence of fixed points of such mappings in this setting. Our results for enriched $\rho$-contractions are generalizations of the corresponding results from Banach spaces to modular function spaces and those from contractions to enriched $\rho$-contractions. We make a first ever attempt to prove existence results for enriched $\rho$-Kannan mappings and deduce the result for $\rho$-Kannan mappings. Note that even $\rho$-Kannan mappings in modular function spaces have not been considered yet. We validate our main results by examples.

In this article, mathematical modeling and stability analysis of Lyme disease and its transmission dynamics using Caputo fractional-order derivatives is presented. The compartmental model has been formulated to analyze the spread of Borrelia burgdorferi virus through tick vectors and mammalian hosts. The feasible region is established, and the boundedness of the model is verified. Analytically, the disease-free equilibrium and the basic reproduction number $\left({\Re }_0\right)$ has been determined to assess outbreak potential. By virtue of the fixed-point theory, the existence and uniqueness of solutions has been established. The numerical simulations are obtained via the Runge–Kutta 4 method, demonstrating the model’s ability to capture realistic disease progression. Finally, sensitivity analysis and control strategies (tick population reduction, host vaccination, public awareness, and early treatment) are evaluated, revealing that integrated control measures significantly reduce infection rates and enhance recovery.

Quantitative infrared spectroscopy of bone reads chemistry from ratios of band intensities, but attenuated total reflectance distorts those intensities before they are measured. We treat the distortion as a model to be inverted rather than a nuisance to be filtered. A causal Lorentz parameterisation of the dielectric function, an effective-medium mixture, the Fresnel relations for a finite contact crystal and the instrument line shape are composed into one differentiable operator, and its Jacobian is derived in closed form. Three consequences are quantified. Anomalous dispersion displaces the apparent phosphate maximum by 7 to 36 wavenumbers depending on packing, so band position thresholds do not transfer between laboratories. The displacement varies with composition, so the $I_{1030}/I_{1020}$ crystallinity ratio responds $1.93$ times more strongly to mineral content at fixed crystallinity than to the entire plausible crystallinity range. Absorbance is not affine in composition, which invalidates bilinear unmixing. We prove that closure and structural anticorrelation force the sufficiently scattered condition to fail for bone, and replace it with spectral separability, a checkable condition on band spacing that yields a local identifiability theorem with an explicit lower bound on the smallest singular value. Inference uses a Levenberg–Marquardt iteration unrolled into a network with five learned scalars per layer, for which we prove local linear convergence under a per-layer contraction condition, for constant and for layer-dependent schedules, and a Rademacher bound of order $\sqrt{TlogT/n}$ that is conditional on non-expansive layers; on the trained networks the median contraction is below one at every layer, but non-expansiveness does not hold. On a generator whose band ratios all lie inside published ranges, the estimator reaches the Cramér–Rao bound to within 3 to 6 per cent across a sixteenfold range of noise, recovers volume fractions 143 times more accurately than multivariate curve resolution, and outperforms a network with $1.1\times {10}^5$ parameters by three orders of magnitude at twenty-four training spectra. These figures hold when the data follow the operator. Under deliberate mismatch in band positions, line shape, baseline, effective-medium law, contact physics and noise, the advantage over curve resolution persists at factors of 2 to 108, but the loss rises to many times the bound and the nominal intervals cover the truth in as few as 5 per cent of cases, so sampling uncertainty is reported separately from systematic model uncertainty. The same information matrix prices the experiment: contact variability inflates the standard error on composition by $3.6$ while leaving the band parameters untouched, and an optimal choice of crystal and geometry recovers $23.6$ nats under signal-independent noise, falling to between $0.6$ and $17.9$ nats once throughput loss and detector-limited noise are included.