On discontinuity problem with an application to threshold activation function

Abstract

In this paper, some discontinuity results are obtained using the number M-C(t, t*) defined as

M-C(t, t*) = max {(d(t, t*), ad(t, Tt) + (1-a)d(t*, St*))((1 -a)d(t, Tt) + ad(t*, St*),b/2 [d(t, St*) + d(t*, Tt))},

at the common fixed point. Our results provide a new and distinct solution to an open problem "What are the contractive conditions which are strong enough to generate a fixed point but which do not force the map to be continuous at fixed point?" given by Rhoades [33]. To do this, we investigate a new discontinuity theorem at the common fixed point on a complete metric space. Also an application to threshold activation function is given.