arXiv subjects

Hassan Khandani

Publications and source records attributed to Hassan Khandani.

7 recordsLinked to original sources

arXiv2025

A Study of Kirk's Asymptotic Contractions via Leader Contractions

This paper investigates asymptotic fixed point results for nonlinear contractions, with emphasis on Kirk-type theorems and their generalizations. A central difficulty in the literature has been the requirement that the mapping possesses a bounded orbit, a condition that is often hard to verify and traditionally viewed as essential for guaranteeing the existence of fixed points. We eliminate this boundedness assumption by proving that every asymptotic Kirk contraction is a Leader contraction, which inherently guarantees orbit boundedness. This observation simplifies fixed point arguments and broadens the scope of applicable mappings. We also resolve an open question by showing that the standard upper semicontinuity condition on the control function phi can be weakened to right-upper semicontinuity, addressing a conjecture posed by Jachymski et al. These contributions unify and generalize several foundational results, including those of Boyd-Wong, Kirk, Chen, Arav et al., and Reich and Zaslavski, under the more flexible framework of Leader contractions. The results offer streamlined and more practical conditions for convergence and fixed point existence in generalized metric spaces.

math.FA↗

arXiv2025

Interpolative Metrics Are Not New: A Study of Generalized Contractions in b-Suprametric Spaces

{Researchers recently introduced interpolative metric spaces and established fixed-point theorems in this setting. We demonstrate that these metrics are a special case of b-metrics. On the other hand, suprametrics and b-suprametrics have also been introduced, and we show that b-suprametric spaces generalize b-metric spaces. We establish corresponding results previously presented in interpolative metric spaces in the framework of b-suprametric spaces with weaker conditions.

math.MG↗

arXiv2025

Fixed Points of Meir-Keeler and Leader Contractions with bounded orbits in b-Metric Spaces

We establish fixed-point theorems for Meir-Keeler-type contractions in b-metric spaces. While Lu et al. demonstrated via an explicit counterexample that classical Meir-Keeler contractions may fail to admit fixed points in this setting, we prove that a natural strengthening of the conditions yields existence results. Specifically, we show that every non-expansive Leader contraction with bounded orbits in a b-metric space possesses a fixed point. To contextualize our findings, we present a hierarchical diagram illustrating that the fixed-point theory of non-expansive Leader contractions subsumes earlier results, including Meir-Keeler contractions, the primary focus of this work. Our proofs hold in arbitrary b-metric spaces, without relying on the triangle inequality, requiring instead only the assumption of unique limits. This work not only resolves the limitation exposed by Lu et al.'s counterexample but also establishes a unifying framework for future research in the literature.

math.MG↗

arXiv2022

Some extensions of Krasnoselskii's fixed point result for real functions

We extend Krasnoselskii's fixed point result to non-self-real functions. We find a new and simple proof for Hillam's result. In our approach, we don't assume the image of the related mapping to be compact or bounded. In this way, we extend Hillam's result to self-mappings on $\mathbb R$. Finally, we present a new proof for the global convergence of the Newton-Raphson method.

math.OC↗

arXiv2022

A new method for estimating the real roots of real differentiable functions

We introduce a new type of Krasnoselskii's result. Using a simple differentiability condition, we relax the nonexpansive condition in Krasnoselskii's theorem. More clearly, we analyze the convergence of the sequence $x_{n+1}=\frac{x_n+g(x_n)}{2}$ based on some differentiability condition of $g$ and present some fixed point results. We introduce some iterative sequences that for any real differentiable function $g$ and any starting point $x_0\in \mathbb [a,b]$ converge monotonically to the nearest root of $g$ in $[a,b]$ that lay to the right or left side of $x_0$. Based on this approach, we present an efficient and novel method for finding the real roots of real functions. We prove that no root will be missed in our method. It is worth mentioning that our iterative method is free from the derivative evaluation which can be regarded as an advantage of this method in comparison with many other methods. Finally, we illustrate our results with some numerical examples.

math.FA↗

arXiv2021

A convergence condition for Newton-Raphson method

In this paper we study the convergence of Newton-Raphson method. For this method there exists some convergence results which are practically not very useful and just guarantee the convergence of this method when the first term of this sequence is very close to the guessed root \cite{sulimayer}. Khandani et al. introduced a new iterative method to estimate the roots of real-valued functions \cite{khandani}. Using this method we introduce some simple and easy-to-test conditions under which Newton-Raphson sequence converges to its guessed root even when the initial point is chosen very far from this root. More clearly, for a real-valued second differentiable function $f:[a,c]\to \mathbb R$ with $f^{''}f\ge 0$ on $(a,c)$ where $c$ is the unique root of $f$ in $[a,c]$, the Newton-Raphson sequence $f$ converges to $c$ for each $x_0\in[a,c]$ provided $f$ satisfies some other simple conditions on this interval. A similar result holds if $[a,c]$ be replaced with $[c,b]$. Our study will enable us to predict accurately where Newton-Raphson sequence converges.

math.GM↗

arXiv2015

Some consequences of Caristi's fixed point theorem, partial answers to some known open problems and its applications

In this paper, we show that several extension of Banach contraction principle, can be easily derived from the Caristi's theorem is one of the useful generalization of Banach contraction principle in the setting of the complete metric spaces. Moreover, some partial answers to some known open problems are given via Caristi's corollaries. Finally, existence of bounded solutions of a functional equation is studied to support our results.

math.MG↗