On the dependence of a fixed point on a parameter\\ in $(q_1, q_2)$-quasimetric spaces

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.