Advances in Fixed Point Theory and Applications

Dear Colleagues,

Kannan (1968 Bull. Calcutta Math. Soc., 1969 Amer. Math. Monthly) obtained a contraction mapping theorem that, compared to the celebrated theorems of Banach, Meir and Keeler, and Boyd and Wong, does not require the mapping to be continuous. Kannan’s theorem was followed by a multitude of contraction theorems that do not require the mapping to be continuous, but none of these results reported discontinuity at the fixed point despite the fact that such mappings could be discontinuous at every other point. This made the search for contraction mappings that admit discontinuity at the fixed point an important area of study. Besides being the genesis of the question of the continuity of contraction mappings at the fixed point, the Kannan fixed point theorem characterizes the completeness of the metric space (Subrahmanyam, Monatsh. Math. 80, 1975). In 1988, Rhoades (Contemporary Math 72, Amer Math. Soc.) examined a large number of the most general contractive definitions for continuity at the fixed point and found that each of these mappings were continuous at the fixed point. Rhoades (1988) concluded this study by stating ”An open question is whether there exists a contractive definition which is strong enough to generate a fixed point, but which does not force the map to be continuous at the fixed point”. This statement came to be known as the Rhoades Problem on continuity at the fixed point. In 1992, Hicks and Rhoades (International J. Math. Math. Sci. 15-1) studied a large number of contractive multivalued mappings for continuity at the fixed point and found that each mapping was continuous at the fixed point. In 1999, Pant (JMAA 240), by employing a new (?, δ) condition that was an extended form of the Kannan contraction condition but was independent of the Meir–Keeler condition, resolved the Rhoades Question in the affirmative. For pairs of mappings, Pant resolved the Rhoades question by employing the notion of reciprocal continuity (Bull. Calcutta Math. Soc. 90, 1998, Indian J. Pure Appl. Math. 30-2, 1999). Since then, and particularly since 2017, various authors have obtained interesting solutions of the Rhoades Problem in a variety of settings and in different types of spaces (e.g., N. Ozgur, Ozgur et al, N. Tas,

Tas et al, Rale Nikoli′c, Sinisa Jeˇsi′c, Bisht and Rakoˇcevi′c, L K Dey, P. Bala-subramaniam, D. Gopal et al, W. Sintunavarat). Applications of contractive functions exhibiting discontinuity at the fixed point in activation functions of neural networks have also been obtained by many authors. Solutions of the Rhoades Problem by using Meir–Keeler-type conditions or in the setting of fixed circles have also been obtained. Rhoades’ statement on the problem of the continuity of contractive maps at the fixed point is three decades old in 2024, while the first solution of this problem (Pant, 1999) is 25 years old in 2024. The Kannan contraction theorem, which is the genesis of the Rhoades question, and the Meir–Keeler theorem are 55 years old in 2024. The Meir–Keeler theorem is also important in the context of the Rhoades Problem, since the majority of the solutions of the Rhoades Problem employ (?, δ)-type conditions.

The topics of this Special Issue include, but are not limited to, the following: Rhoades Problem in fixed point theory; operators in fixed point theory; best proximity points, measures of noncompactness, applications of fixed point theory; Perov-type contractions, etc.

Prof. Dr. Vladimir Rakocevic
Prof. Dr. Rajandra Prasad Pant
Guest Editors

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• Rhoades Problem in fixed point theory
• Operators in fixed point theory
• Applications of fixed point theory
• Fixed point theory