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 this article, we find the fixed point, fixed ellipse, and fixed elliptic disc theorems, we introduce the concepts of an orthogonal -contraction, an orthogonal -weak contraction, an orthogonal -weak JS-contraction, an integral-type orthogonal -weak contraction, and an integral-type orthogonal -weak JS-contraction. To further prove the validity of the postulates, we validate these using illustrated cases with geometric meanings. The reason for this work is the possibility of a geometric shape like a circle, ellipse, disc, or elliptic disc being among the set of non-unique fixed points. We provide an orthogonal -contraction application to chemical sciences.

Sets, probability, and neutrosophic logic are all topics covered by neutrosophy. Moreover, the classical set, fuzzy set, and intuitionistic fuzzy set are generalized using the neutrosophic set. A neutrosophic set is a mathematical concept used to solve problems with inconsistent, ambiguous, and inaccurate data. In this article, we demonstrate some basic fixed-point theorems for any even number of compatible mappings in complete neutrosophic metric spaces. Our primary findings expand and generalize the findings previously established in the literature.