As the generalization of the fixed-point theory, the fixed-circle problems are interesting and notable geometric constructions. In this paper, we prove that some new necessary conditions are investigated for the existence of a fixed circle of a given self-mapping in -metric spaces. The well-known Braincari and Chatterjea contractive conditions are generalized for proving the uniqueness of obtained theorems. Finally, an application to parametric rectified linear unit activation functions are given to show the importance of studying the fixed-circle problem.

In this paper, we modify various contractive conditions (C.C.)s such as Ciric type (C.C.), Rhoades type (C.C.), Seghal type (C.C.), Bianchini type (C.C.), and Berinde type (C.C.) for two self-mappings, considering that the contractive property plays a major role in establishing a fixed circle (F.C.) on both metric spaces (M-s) and $\mathbb{S}$-(M-s) where the symmetry condition is satisfied, and we utilize them to establish a common (F.C.). We prove new (F.C.) results on both (M-s) and $\mathbb{S}$-(M-s) with illustrative examples. Finally, we provide an application to activation functions such as rectified linear unit activation functions and parametric rectified linear unit activation functions.

In this study, the fixed point theory on fuzzy metric spaces has been generalized to the fixed-circle theory by making a geometric interpretation. The necessary conditions to exist the fixed circles of a self-mapping have been investigated and the uniqueness of the circle is examined under suitable conditions. We present some illustrative examples of obtained results and also offer an application to confirm the utility of our established result for finding the unique solution of an integral equation appearing in the dynamic market equilibrium aspects of economics.

Recently, the fixed-circle problems have been studied with different approaches as an interesting and geometric generalization. In this paper, we present some solutions to an open problem $CC$: what is (are) the condition(s) to make any circle $C_{{\varpi }_0,\sigma }$ as the common fixed circle for two (or more than two) self-mappings? To do this, we modify some known contractions which are used in fixed-point theorems such as the Hardy–Rogers-type contraction, Kannan-type contraction, etc.

This paper investigates the fixed‐circle problem in metric spaces within the framework of multivalued mappings. We introduce three novel classes of bilateral contractions, namely, the Jaggi‐type bilateral, Dass‐Gupta type I bilateral, and Dass‐Gupta type II bilateral multivalued contractions, each specifically formulated to extend the fixed‐circle theory to the setting of multivalued mappings. By employing these generalized contractive conditions, we establish several fixed‐circle and fixed‐disc theorems and further extend our results to encompass integral‐type contractions. The theoretical findings are supported by illustrative examples that confirm the applicability and robustness of the proposed approach.