In this paper some theorems on continuous dependence of the fixed point on a parameter are proved. As applications, the following result was obtained. \emph{Theorem 5. Let $\wedge$ be a topological spave, $f(x,t,\lambda)$ be a continuous mapping $E\times[t_0-a,t_0+a]\times\wedge$ in $E$, and $\lambda\to x_0(\lambda)$ continuous mapping $\wedge$ in $E$. Let for every $\lambda\in\wedge$ the mapping $f(x,t,\lambda)$ satisfy the following conditions:} \emph{1. For every $\alpha\in I$ there exists $k_\alpha(t,\lambda)$, integrable over $[t_0-a,\;t_0\neq a]$, so that:} \[ |f(x,t,ambda)-f(y,t,ambda)|lphaeq k_lpha(t,ambda)|x-y|ǎrphi_ambda(lpha)\quad x,yı E,\;tı I. \] \emph{2. There exists $h'>0$, natural number $n_\alpha$, and positive constants $A(\alpha)$ and $Q(\alpha)$ independents of $\alpha$, so that:} \[ a)\quad\max\Big(ıt_{t_0-h'}^{t_0}kǎrphi^n_ambda(lpha)(t,ambda)dt,\;ıt_{t_0}^{t_0+h'}kǎrphi^n_ambda(lpha)(t,ambda)dt\Big)eq A(lpha),\quad n<n_lpha \] \[ a)\quad\max\Big(ıt_{t_0-h'}^{t_0}kǎrphi^n_ambda(lpha)(t,ambda)dt,\;{ıt'}_{t_0}^{t_0+h}kǎrphi^n_ambda(lpha)(t,ambda)dt\Big)eq Q(lpha)<1,\quad n\geq n_lpha \] \emph{3. For every $\alpha\in I$, there exists $\beta(\alpha)\in I$ and $m_\alpha>0$ independent of $\lambda$, so that:} \[ |x|ǎrphi^n_ambda(lpha)eq m_lpha|x|\beta(lpha)\quad xı E \] \emph{Then: $(t,\lambda)\to x(t,\lambda)$ is a continuous mapping $[t_0-h,\;t_0+h]\times\wedge$ in $E$, where $x(t,\lambda)$ is the solution of the differential equation \[ \frac{dx}{dt}=f(x,t,ambda)\quad x(t_0)=x_0(ambda) \] defined in $I'=[t_0-h,\;t_0+h],\quad h=\min(a,h')$.}