Weak theta-phi-contraction and discontinuity

Abstract

In this paper, we introduce the notion of weak θ-φ-contraction ensuring a convergence of successive approximations but does not force the mapping to be continuous at the fixed point. Thus, we answer one more solution to the open question raised

Weak θ-contractions and some fixed point results with applications to fractal theory

“…Observe that condition Θ1 can be withdrawn, and still Theorem 3.1 (also, most of the existence results in the literature (e.g., results of[25][26][27][28][29])) survives (in view of Proposition 3.1). Now, Proposition 3.1 and Remark 3.2 led us to define a weaker contraction under the name of weak θ -contraction as follows.…”

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]).…”

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.…”