Scientific articles
On the dependence of a fixed point on a parameter in $(q_1, q_2)$-quasimetric spaces
R. Senguptaab a Derzhavin Tambov State University
b Artificial Intelligence Research Institute
Abstract:
In the paper, we investigate the problem of continuous dependence of fixed points of contractive mappings in
$(q_{1}, q_{2})$-quasimetric spaces on a parameter. We study equations of the form
$ x = F(x, p)$ where
$x \in X$ is the unknown variable in a complete
$(q_{1}, q_{2})$-quasimetric space
$X,$ the parameter
$p$ lies in a given topological space
$P,$ and
$F : X \times P \to X$ is a prescribed mapping. It is assumed that
$F$ is contractive in the variable
$x.$
Using the classical existence and uniqueness results for fixed points of contractive mappings in complete
$(q_{1}, q_{2})$-quasimetric spaces, we derive sufficient conditions ensuring that the mapping assigning to each parameter
$p\! \in\! P$ the corresponding solution
$x(p)$ of the equation is continuous. As a corollary, we establish continuity of
$x(p)$ in the case where
$X$ is a complete metric space.
We further consider the situation where the parameter space
$P$ itself carries the structure of a
$(q_{1}, q_{2})$-quasimetric space. In this context, sufficient conditions are obtained guaranteeing that the solution map
$x(p)$ is Lipschitz continuous, together with an explicit estimate for its Lipschitz constant. As a consequence, we present a corollary for the case when
$X$ is a complete metric space and
$P$ is a metric space.
Keywords:
quasimetric spaces, continuous dependence on a parameter, solution of implicit equations
UDC:
515.126.4
MSC: 54A05,
54H25 Received: 30.05.2025
Accepted: 12.09.2025
DOI:
10.20310/2686-9667-2025-30-151-267-274
Fulltext:
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