Generalized Meir-Keeler type contractions and discontinuity at fixed point

Abstract

Abstract. In this paper, we show that generalized Meir-Keeler type contractive definitions are strong enough to generate a fixed point but do not force the mapping to be continuous at the fixed point. Thus we provide more answers to the open question posed by B.E. Rhoades in the paper Contractive definitions and continuity, Contemporary Mathematics 72(1988), 233-245.

Nemytzki-Edelstein-Meir-Keeler Type Results in b-Metric Spaces

“…A mapping satisfying (38) is called an eventually contractive mapping. It is obvious that any contractive mapping is eventually contractive (it satisfies (38) with ( , ) = 1), but the implication is not reverse.…”

Caristi type and meir-keeler type fixed point theorems

“…Fixed point theorems for contractive mappings which admit discontinuity at the fixed point and their applications to neural networks with discontinuous activation functions have emerged as a very active area of research (e.g. Bisht and Rakocevic [4,5], Ozgur and Tas [27,28], Rashid et al [33], Tas and Ozgur [38], Tas et al [39], Zheng and Wang [44]). The question of the existence of contractive mappings which admit discontinuity at the fixed point arose with the publication of two papers by Kannan [18,19].…”

“…Recently some more solutions to the problem of continuity at fixed point and applications of such results to neural networks with discontinuous activation functions have been reported (e.g. Bisht and Pant [2,3], Bisht and Rakocevic [4,5], Ozgur and Tas [27,28], Rashid et al [33], Tas and Ozgur [38], Tas et al [39], Zheng and Wang [44]). All the known solutions of the Rhoades' problem (e.g.…”

“…All the known solutions of the Rhoades' problem (e.g. [2,3], [4,5], [27,28], [30], [31], [44]) employ condition (ii) or some generalized form of (ii) together with a condition similar to condition (iii) or some weaker form of continuity, e.g., orbital continuity, continuity of f k or k-continuity for some k > 1.…”

Proinov contractions and discontinuity at fixed point

“…ı/ condition and a -contractive condition to prove a fixed point in which the fixed point may be a point of discontinuity. Recently in [3] Bisht and Rakočević have studied generalized Meir-Keeler type contractions and discontinuity at fixed point.…”