Proinov contractions and discontinuity at fixed point

Abstract

In this paper, we show that the contractive definition considered by Proinov [Fixed point theorems in metric spaces, Nonlinear Analysis 64 (2006) 546 -557] is strong enough to generate a fixed point but does not force the mapping to be continuous at the fixed point. Thus we provide several answers to the open question posed by B.E. Rhoades in Contractive definitions and continuity, Contemporary Mathematics 72(1988), 233-245.

A general fixed point theorem

“…Recently some more solutions to the problem of continuity at fixed point and applications of such results in the study of discontinuous activation functions of neural networks have been reported (e.g. Bisht and Pant [2,3], Bisht et al [4], Bisht and Rakočević [5,6], Celik and Özg ür [8], Özg ür and Tas [25,26], Pant and Pant [31], Pant et al [32], Pant et al [33], Pant et al [34], Pant et al [35,36], Pant et al [37], Rashid et al [39], Tas and Özg ür [44], Tas et al [45], Zheng and Wang [48]). Fixed point theorems for discontinuous mappings have found wide applications, for example application of such theorems in the study neural networks with discontinuous activation functions is presently a very active area of research (e. g. Cromme and Diener [13], Cromme [14], Ding et al [15], Forti and Nistri [16], Nie and Zheng [22][23][24], Wu and Shan [47]).…”