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2 Journal of Applied Analysis

2 2022

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Relation theoretic contractions and their applications in b-metric like spaces

Tomar Anita, Joshi Meena, Padaliya S. K.

Journal of Applied Analysis

2022

Vol. 28, nr 2

295--309

We introduce a relational generalized Meir-Keeler contraction and a relational generalized Meir-Keeler contraction with rational terms in non-complete relational b-metric like spaces in order to establish non-unique fixed point results for a discontinuous single-valued map. Also, we provide an illustrative example to demonstrate that a relational generalized Meir-Keeler contraction with rational terms in a relational b-metric like space admits discontinuity at the fixed point. Thereby,we provide a novel explanation via a binary relation to the question of the existence of a contractive map admitting a fixed point at the point of discontinuity. Finally, we give applications to solve an initial value problem and a non-linear matrix equation which demonstrate the usability and effectiveness of our results.

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

Tomar Anita, Joshi Meena, Padaliya S. K.

Journal of Applied Analysis

2022

Vol. 28, nr 1

57--66

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 forwhich 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.