In this article, we establish the concept of intuitionistic fuzzy double-controlled metric-like spaces by “assuming that the self-distance may not be zero”; if the value of the metric is zero, then it has to be “a self-distance”. We derive numerous fixed-point results for contraction mappings. In addition, we provide several non-trivial examples with their graphical views and an application of integral equations to show the validity of the proposed results.

In this paper, the authors introduce the notion of neutrosophic double controlled metric spaces as a generalization of neutrosophic metric spaces. For this purpose, two non-comparable functions, ξ and Γ, are used in triangle inequalities. The authors prove several interesting results for contraction mappings with non-trivial examples. At the end of the paper, the authors prove the existence, and the uniqueness, of the integral equation to support the main result.

In this manuscript, we give sufficient conditions for a sequence to be Cauchy in the context of controlled fuzzy metric space. Furthermore, we generalize the concept of Banach’s contraction principle by utilizing several new contraction conditions and prove several fixed point results. Furthermore, we provide a number of non-trivial examples to validate the superiority of main results in the existing literature. At the end, we discuss an important application to the transformation of solar energy to electric power by utilizing differential equations.

In order to solve issues that arise in various branches of mathematical analysis, such as split feasibility problems, variational inequality problems, nonlinear optimization issues, equilibrium problems, complementarity issues, selection and matching problems, and issues proving the existence of solutions to integral and differential equations, fixed point theory provides vital tools. In this study, we discuss topological structure and several fixed-point theorems in the context of generalized neutrosophic cone metric spaces. In these spaces, the symmetric properties play an important role. We examine the existence and a uniqueness of a solution by utilizing new types of contraction mappings under some circumstances. We provide an example in which we show the existence and a uniqueness of a solution by utilizing our main result. These results are more generalized in the existing literature.

The emission of greenhouse gases from fossil fuels creates devastating effects on Earth’s atmosphere. Therefore, a clean energy source is required to fulfill the energy demand. Hydrogen is considered an energy vector, and the production of green hydrogen is a promising approach. Photoelectrochemical (PEC) water splitting is the best approach to produced green hydrogen, but the efficiency is low. To produce hydrogen by PEC splitting water, semiconductor photocatalysts have received an enormous amount of academic research in recent years. A new class of co-catalysts based on transition metals has emerged as a powerful tool for reducing charge transfer barriers and enhancing photoelectrochemical (PEC) efficiency. In this study, copper oxide (CuO) and tin oxide ($Sn{O}_2$) multilayer thin films were prepared by thermal evaporation to create an energy gradient between $Sn{O}_2$ and CuO semiconductors for better charge transfer. To improve the crystallinity and reduce the defects, the prepared films were annealed in a tube furnace at 400 °C, 500 °C, and 600 °C in an argon inert gas environment. XRD results showed that $Sn{O}_2$/CuO-600 °C exhibited strong peaks, indicating the transformation from amorphous to polycrystalline. SEM images showed the transformation of smooth dense film to a granular structure by annealing, which is better for charge transfer from electrode to electrolyte. Optical properties showed that the bandgap was decreased by annealing, which might be diffusion of Cu and Sn atoms at the interface. PEC results showed that the $Sn{O}_2$/CuO-600 °C heterostructure exhibits the solar light-to-hydrogen (STH%) conversion efficiency of 0.25%.

This manuscript contains several new notions including intuitionistic fuzzy $N_b$ metric space, intuitionistic fuzzy quasi-$S_b$-metric space, intuitionistic fuzzy pseudo-$S_b$-metric space, intuitionistic fuzzy quasi-$N$-metric space and intuitionistic fuzzy pseudo $N_b$ fuzzy metric space. We prove decomposition theorem and fixed-point results in the setting of intuitionistic fuzzy pseudo $N_b$ fuzzy metric space. Further, we provide several non-trivial examples to show the validity of introduced notions and results. At the end, we solve an integral equation, system of linear equations and nonlinear fractional differential equations as applications.