Fixed point to fixed circle and activation function in partial metric space

Abstract

We familiarize a notion of a fixed circle in a partial metric space, which is not necessarily the same as a circle in a Euclidean space. Next, we establish novel fixed circle theorems and verify these by illustrative examples with geometric interpretation to demonstrate the authenticity of the postulates. Also, we study the geometric properties of the set of non-unique fixed points of a discontinuous self-map in reference to fixed circle problems and responded to an open problem regarding the existence of a maximum number of points for which there exist circles. This paper is concluded by giving an application to activation function to exhibit the feasibility of results, thereby providing a better insight into the analogous explorations.

Fixed point, its geometry, and application via ω‐interpolative contraction of Suzuki type mapping

“…In this section, following Tomar et al 38 and Joshi et al, 40 we define a fixed circle and a fixed disc to investigate the geometry of nonunique fixed points in a partial b‐metric space. The motivation behind this is the observation that a self mapping satisfying an interpolative contraction or any of its variants may not have a unique fixed point and the set of nonunique fixed points may include some geometric shapes.…”

“…However, the reverse implication may not hold. Following the pattern of Theorem 3.27, we can also establish the uniqueness of fixed circle using $$$ \omega $$$‐ $$$ \psi $$$‐interpolative Suzuki‐ Kannan type contraction, $$$ \omega $$$‐ $$$ \psi $$$‐interpolative Suzuki–Ćirić–Reich–Rus type contraction and other interpolative contractions. For details on the work on the set of non‐unique fixed points forming some fixed figure one may refer to literature, 6,21,38,41–45 and references therein.…”

On unique and non-unique fixed point in parametric N b −metric spaces with application

“…Define a self mapping S : X −→ X as Sx = Cx + d, x, d ∈ R m and C = [c ij ] m×m . First, we show that the self-mapping S satisfies Theorem 1.Then, the unique fixed point of the operator S is the unique solution of a system of linear equations(25). For x, y ∈ R nN (Sx, Sx, Sy, t) = t 3 Σ m i=1 |Sx i − Sx i | + Σ m i=1 |Sx i − Sy i | + Σ m i=1 |Sy i − Sx i | 4t 3 Σ m i=1 |Sx i − Sy i | 4t 3 (Σ m i=1 |Σ m j=1 c ij (x j − y j )|) 2 ≤ 4t 3 Σ m i=1 (Σ m j=1 |c ij | 2 |x j − y j | 2 ) ≤ 4t 3 m max j=1 Σ m i=1 |c ij | 2 Σ m j=1 |x j − y j | 2 < 4t…”

“…For more work on geometry, we may refer to [6]- [10], [25]- [26]. In the following, (X , N ) denotes the parametric N b −metric space.…”

Fixed Circle and Fixed Disc Results for New Types of Θc-Contractive Mappings in Metric Spaces

“…Natural and interesting problems were found through the investigations of the geometric properties of fixed points. In this context, the fixed circle and fixed disc problem have been studied in metric and generalized metric spaces via different contractive conditions (see [15][16][17][18][19][20][21][22][23][24]). For example, in [15], some fixed circle results were proved using the Caristi type contraction on a metric space.…”