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In this paper, we delve into the ideas of Geraghty-type proximal contractions and their relation to multivalued, single-valued, and self mappings. We begin by introducing the notions of (ψω)MCP-proximal Geraghty contraction and rational (ψω)RMCP-proximal Geraghty contraction for multivalued mappings, aimed at establishing coincidence point results. To enhance our understanding and illustrate the concepts, practical examples are provided with each definition. This study extends these contractions to single-valued mappings with the introduction of (ψω)SCP-proximal Geraghty contraction and rational (ψω)RSCP-proximal Geraghty contraction, supported by relevant examples to reinforce the main results. Then, we explore (ψω)SFP Geraghty contraction and rational (ψω)RSFP contraction for self-mappings, obtaining fixed point theorems and clearly illustrating them through examples. Finally, we apply the theoretical framework developed to investigate the existence and uniqueness of solutions to certain two-dimensional Volterra integral equations. Specifically, we consider the transformation of first-kind Volterra integral equations, which play crucial roles in modeling memory in diverse scientific fields like biology, physics, and engineering. This approach provides a powerful tool for solving difficult integral equations and furthering applied mathematics research.

Background: Nanoparticles modulate several physiochemical and biological properties. In this regard, silver nanoparticles (AgNPs) have shown remarkable applications. The present research work comprises green synthesis of AgNPs using aqueous leaves extract of Eranthemum pulchellum and evaluation of its efficacy as antifungal and antioxidant agents. Methods: Synthesized AgNPs have been characterized by Fourier Transforms Infrared (FTIR) Spectroscopy, X-ray diffraction (XRD), transmission electron microscopy (TEM), and ultraviolet-visible spectrophotometry. Qualitative phytochemical analysis of aqueous leaves extract of E. pulchellum and FTIR spectrum of the synthesized AgNPs suggest about the presence of different phytochemicals and functional groups, respectively, which are responsible for reducing silver ions as well as capping and stabilizing synthesized nanoparticles. Results: The wavelength of maximum absorbance of AgNPs solution near to 439 nm indicates the spherical morphology. XRD infers about average crystalline size of the synthesized AgNPs to be ~12 nm. Selected area electron diffraction pattern of the synthesized AgNPs shows four visible diffraction rings corresponding to (111), (200), (220), and (311) set of planes which are attributed to face centered cubic metallic silver. The size and spherical shape of the synthesized AgNPs have been further determined by TEM. The synthesized AgNPs have shown a significant antifungal activity against Aspergillus flavus (AF-LHP-NS7) strain with minimum inhibitory concentration value of 200 μg/mL. The synthesized AgNPs have also shown strong antioxidant efficacy through Azino-bis-3-ethylbenzothiazoline-6-sulfonic acid assay with IC50 value of 462.56 μg/mL. Conclusions: The present study shows a green and facile synthesis of AgNPs. Leaves extract of E. pulchellum has been first time utilized as efficient reductant for the AgNPs synthesis. These AgNPs have shown potent antifungal and antioxidant activity.

Two simple, accurate, novel, safe and precise methods were developed for the simultaneous estimation of poorly water-soluble drugs Metronidazole (MTR) and Furazolidone (FZ) in a tablet dosage form using 2 M sodium acetate and 8 M urea solution (50:50%V/V) as a mixed hydrotropic solution. MTR and FZ show maximum absorbances at 319 and 364 nm, respectively. Sodium acetate and urea solution did not show any absorbance above 240 nm and thus no interference in the estimation of drugs was seen. MTR and FZ follow Beer’s law in the concentration range of 10–50 μg/ml and 5–25 μg/ml (r2 = 0.9992 and 0.9996). Method-A employs the simultaneous equation method using 319 and 364 nm as two analytical wavelengths, method-B employs the absorption ratio method, which uses 339.2 and 364 nm as two analytical wavelengths for estimation of MTR and FZ. The mean percent label claims of tablet dosage were found to be 98.715 ± 1.012 and 98.74 ± 0.912 in method A, 98.99 ± 0.872 and 97.89 ± 0.903 in method B for MTR and FZ, respectively. The developed methods were validated according to ICH guidelines and values of accuracy, precision and other statistical analysis were found to be in good accordance with the prescribed values therefore both methods can be used for routine monitoring of MTR and FZ in industry in the assay of bulk drug and tablets.