On contractive mappings and discontinuity at fixed points

Abstract

This paper deals with an interesting open problem of B.E. Rhoades (Contemporary Math. (Amer. Math. Soc.) 72(1988), 233-245) on the existence of general contractive conditions which have fixed points, but are not necessarily continuous at the fixed points. We propose some more solutions to this problem by introducing two new types of contractive mappings, that is, A-contractive and A -contractive, which are, in some sense, more appropriate than those of the important previous attempts. We establish some new fixed point results involving these two contractive mappings in compact metric spaces and also in complete metric spaces and show that these contractive mappings are not necessarily continuous at their fixed points. Finally, we suggest an applicable area, where our main results may be employed.

On Common fixed point results via implicit relations

“…For examples of particular mappings 'g' belonging to the above two collections, the readers are referred to see [1,7,10,11]. Now, we introduce the notion of A-contraction in case of a couple and a family of mappings.…”

“…Afterwards, this contractive condition have been extended in many ways. Among all such extensions, A-contractive and A -contractive conditions due to Garai et al [7] are one of the most generalized ones, since these two contain a handful number of contractive conditions as particular cases.…”

Discontinuity at fixed point and metric completeness

“…In 1999, Pant [27] proved two fixed point theorems in which the considered mappings were discontinuous at the fixed points, hence gave affirmative solutions to the Rhoades problem for both the single and a pair of self-mappings. Some new solutions to this problem with applications to neural networks have been reported in [2,3,4,5,6,12,23,24,25,26,29,30,31,32,37,39]. Fixed point theorems for discontinuous mappings have found a variety of applications, e.g., neural networks are generally used in character recognition, image compression, stock market prediction and to solve non-negative sparse approximation problems ( [10,11,20,21,22,38]).…”

On proximal contractions via implicit relations and best proximity points

“…(A ′ 1 ) there exists k ∈ [0, 1) such that if r ≤ f (s, 0, r + s), then r ≤ ks for all r, s ∈ R + ; (A ′ 2 ) if t ≤ t 1 , then f (r, s, t) ≤ f (r, s, t 1 ) for all r, s, t, t 1 ∈ R + ; (A ′ 3 ) if r ≤ f (r, r, r), then r = 0. For examples and properties of such collection of mappings, we refer the readers to [1,8,15].…”