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In this article, some concepts of locally ( δ , ϕ ) -dominated mappings, locally ( δ , ϕ ) -continuous mappings and L R − ( δ , ϕ ) complete dislocated quasi-metric spaces have been introduced. Fixed point results for such mappings under new contractive conditions in these spaces have been established. As an application, fixed point results endowed with order and graph in such spaces have been examined. Furthermore, the results are substantiated with specific examples. Our empirically validated results represent a novel contribution to the existing body of literature.

The ambition of this work is to introduce the notion of left (right) K-sequentially complete ordered dislocated fuzzy quasimetric spaces and to define a relevant Hausdorff metric on compact sets. A new approach, given in (Shoaib et al. in Filomat 34(2):323–338, 2020) has been used to obtain fixed-point results for multivalued mappings fulfilling generalized contraction in the latest framework. For the authenticity of our result, an example is formulated.

Diabetes is a serious health issue that can be a great risk factor related to numerous physical problems. A class of drugs “Gliflozin” especially Sodium Glucose Co. Transporter 2 was inhibited by a novel drug, which is known as “empagliflozin”. While ZnO nanoparticles (NPs) had considerable promise for combating diabetes, it was employed in the treatment and management of type-2 diabetes mellitus. The new drug empagliflozin was initially incorporated into Zinc Oxide NPs in this study using the surface physio-sorption technique, and the degree of drug adsorption was assessed using the HPLC method. The tailored product was characterized by using the FTIR, EDX, Ultraviolet–Visible, XRD and SEM techniques. With an average particle size of 17 nm, SEM revealed mono-dispersion of NPs and sphere-like form. The Freundlich isotherm model best fits and explains the data for the physio-sorption investigation, which examined adsorption capabilities using adsorption isotherms. The enzymes α-amylase and α-glucosidase, which are involved in the human metabolism of carbohydrates, were used in the in-vitro anti-diabetic assays. It was discovered that the composite showed the highest levels of 81.72 and 92.77% inhibition of -α-amylase and -glucosidase at an absolute concentration of 1000 μg per ml with IC 50 values of 30.6 μg per ml and 72 μg per ml.

The purpose of this paper is to find common fixed point results for two families of multivalued mappings fulfilling generalized rational type –dominated contractive conditions on a closed ball in complete dislocated -metric spaces. Some new fixed point results with graphic contractions on a closed ball for two families of multi-graph dominated mappings on dislocated -metric space have been established. An application to the unique common solution of two families of nonlinear integral equations is presented to show the novelty of our results.

This paper is devoted to finding out some realization of the concept of b-metric like space. First, we attain a fixed point for two fuzzy mappings satisfying a suitable requirement of contractiveness. Subsequently, we apply such a result to graphic contractions. Also, we attain a unique solution for a system of integral equations, and lastly we give an application to ensure that there exists a common bounded solution of a suitable functional equation in dynamic programming.

In this article, common fixed point theorems for a pair of multivalued mappings satisfying a new Ciric-type rational F-contraction condition in complete dislocated metric spaces are established. An example is constructed to illustrate our results. An application to the system of integral equations is presented to support the usability of proved results. Our results combine, extend and infer several comparable results in the existing literature.

Diabetes mellitus (DM) is a rapidly prevailing disease throughout the world that poses boundless risk factors linked to several health problems. Vildagliptin is the standard dipeptidyl peptidase-4 (DPP-4) inhibitor type of medication that is used for the treatment of diabetes anti-hyperglycemic agent (anti-diabetic drug). The current study aimed to synthesize vildagliptin-loaded ZnO NPs for enhanced efficacy in terms of increased retention time minimizing side effects and increased hypoglycemic effects. Herein, Zinc Oxide (ZnO) nanoparticles (NPs) were constructed by precipitation method then the drug vildagliptin was loaded and drug loading efficiency was estimated by the HPLC method. X-ray diffraction analysis (XRD), UV–vis spectroscopy, FT-IR, scanning electron microscope (SEM), and EDX analysis were performed for the characterization of synthesized vildagliptin-loaded ZnO NPs. The UV–visible spectrum shows a distinct peak at 363 nm which confirms the creation of ZnO NPs and SEM showed mono-dispersed sphere-shaped NPs. EDX analysis shows the presence of desired elements along with the elemental composition. The physio-sorption studies, which used adsorption isotherms to assess adsorption capabilities, found that the Freundlich isotherm model explains the data very well and fits best. The maximum adsorption efficiency of 58.83% was obtained. Further, In vitro, anti-diabetic activity was evaluated by determining the α-amylase and DPP IV inhibition activity of the product formed. The formulation gave maximum inhibition of 82.06% and 94.73% of α-amylase and DPP IV respectively. While at 1000 µg/ml concentration with IC 50 values of 24.11 μg/per ml and 42.94 μg/ml. The inhibition of α-amylase can be ascribed to the interactive effect of ZnO NPs and vildagliptin.

The aim of this work is to establish results in fixed point theory for a pair of fuzzy dominated mappings which forms a rational fuzzy dominated V-contraction in modular-like metric spaces. Some results via a partial order and using the graph concept are also developed. We apply our results to ensure the existence of a solution of nonlinear Volterra-type integral equations.

In this paper, we establish some fixed point results for fuzzy mappings in a complete dislocated b-metric space. Our results generalize and extend the results of Joseph et al. (SpringerPlus 5:Article ID 217, 2016). We also give examples to support our results, and applications relating the results to a fixed point for multivalued mappings and fuzzy mappings are studied.