Discontinuity at the fixed point in suprametric spaces

Technical Summary: Discontinuity at the Fixed Point in Suprametric Spaces

Problem Statement and Motivation
The paper addresses the existence and uniqueness of fixed points for self-mappings on complete suprametric spaces, a generalization of metric spaces introduced by Berzig. A central motivation for this work is the open problem posed by B. E. Rhoades (1988): Is there a contractive definition strong enough to generate a fixed point but does not force the mapping to be continuous at the fixed point?

While classical fixed point theorems (e.g., Banach–Picard–Caccioppoli) often require continuity, and Istrˇatescu's results on convex contractions of order 2 also rely on global continuity, this paper investigates whether weaker conditions—specifically $k$-continuity or orbital lower semi-continuity—suffice to guarantee a fixed point in the context of suprametric spaces. The authors aim to generalize existing results regarding convex contractions and quasi-contractions (Sehgal, ´Ciri´c, and Fisher) to this broader setting while resolving the continuity issue raised by Rhoades.

Methodology and Framework
The authors operate within the framework of suprametric spaces $(X, d)$, defined by a function satisfying symmetry, identity of indiscernibles, and a modified triangle inequality:
$$d(x, y) \leq d(x, z) + d(z, y) + \rho d(x, z)d(z, y)$$
for some constant $\rho \geq 0$.

The methodology involves:

1. Generalizing Contractive Conditions: Extending the definitions of convex contractions of order $m$ and quasi-contractions (including those by ´Ciri´c and Fisher) to the suprametric setting.
2. Relaxing Continuity Assumptions: Replacing the requirement of global continuity with weaker conditions:
   • $k$-continuity: A mapping $T$ is $k$-continuous if $T^{k-1}x_n \to z$ implies $T^k x_n \to Tz$.
   • $T$-orbitally lower semi-continuity: The displacement function $D(x) = d(x, Tx)$ is lower semi-continuous along the orbit of a point.
   • Condition (C): A specific condition relating the convergence of a sequence and its displacement to the fixed point property.
3. Iterative Analysis: Utilizing Picard iterations ($x_n = T^n x_0$) and establishing bounds on successive iterates to prove the Cauchy property of the sequence. The proofs often involve transforming the suprametric inequality into a form compatible with standard metric space techniques (e.g., using logarithmic transformations or equivalent metrics) to establish convergence.

Key Contributions and Results

1. Fixed Point Theorem for Convex Contractions of Order $m$:
   The paper establishes Theorem 2.1, proving that a self-mapping $T$ on a complete suprametric space satisfying a convex contraction condition of order $m$ (where $d(T^m x, T^m y) \leq \sum_{i=0}^{m-1} a_i d(T^i x, T^i y)$ with $\sum a_i < 1$) possesses a unique fixed point. Crucially, the theorem holds if $T$ satisfies any one of the following:

   • $T$ is $k$-continuous.
   • The displacement function $D(x)$ is $T$-orbitally lower semi-continuous.
   • $T$ satisfies condition (C).
     This result generalizes Istrˇatescu's theorem for order 2 and extends it to order $m$ in suprametric spaces.

2. Resolution of Rhoades' Open Problem:
   The authors claim that their results provide a "new and distinct solution" to Rhoades' open problem. By demonstrating that convex contractions of order $m$ generate unique fixed points without requiring continuity at the fixed point (relying instead on the weaker conditions listed above), they affirm the existence of such contractive definitions.

3. Generalization of Quasi-Contractions:

   • ´Ciri´c's Quasi-contraction: Theorem 3.1 generalizes ´Ciri´c's result to suprametric spaces, proving the existence of a unique fixed point for mappings satisfying a generalized quasi-contractive inequality involving iterates $n(x)$.
   • Sehgal's Result: Theorem 3.2 extends Sehgal's 1969 generalization of the contraction principle to this setting.
   • Fisher's Quasi-contraction: Theorem 4.1 extends Fisher's 1979 results involving two fixed integers $n$ and $m$ and a maximum over various iterates, again proving uniqueness under the relaxed continuity conditions.

4. Applications:
   The paper applies the main theoretical results to the existence and uniqueness of solutions for a Fredholm integral equation of the second kind:
   $$f(x) = \int_a^b K(x, t)f(t) dt$$
   By defining a specific suprametric on the space of continuous functions $C[a, b]$, the authors show that the integral operator satisfies the convex contraction condition of order 2, thereby guaranteeing a unique solution without requiring the operator to be continuous at the fixed point in the traditional sense (though it is Lipschitz continuous, satisfying the $k$-continuity condition).

Significance and Claims
The paper asserts that the class of convex contractions of order $m$ is "strong enough" to generate fixed points in complete suprametric spaces while being "weak enough" to allow for discontinuity at the fixed point. This distinction is presented as a significant advancement over previous results that necessitated global continuity. The authors provide specific examples (including finite sets and interval-based functions) to demonstrate that their conditions are satisfied in cases where classical metric space theorems (like those strictly requiring metric properties or continuity) would fail or be inapplicable. The work effectively bridges the gap between Istrˇatescu's convex contractions and Rhoades' open problem within the generalized topology of suprametric spaces.