Interpolative contractions and discontinuity at fixed point

Abstract

In this paper, we investigate new solutions to the Rhoades’ discontinuity problem on the existence of a self-mapping which has a fixed point but is not continuous at the fixed point on metric spaces. To do this, we use the number defined as (FORMULA PRESENTED), where α, β, γ ∈ (0, 1) with α + β + γ < 1 and some interpolative type contractive conditions. Also, we investigate some geometric properties of F ix(T ) under some interpolative type contractions and prove some fixed-disc (resp. fixed-circle) results. Finally, we present a new application to the discontinuous activation functions.