Continuous dependence on a parameter of the countable fractal interpolation function (Q2919610)

The author considers the continuous dependence with respect to a parameter of the countable fractal interpolation function.NEWLINENEWLINE For each \(t\in P\), a function \(\Delta(t)=(x_n(t),y_n(t))_{n\geq 0}\subset [a,b]\times Y\) is a countable system of data (CSD) called a countable system of data with parameter \(t\), where \((P,d_P)\) and \((Y,d_Y)\) are two metric spaces. Let \(c,s:P\to\mathbb R\) be two mappings satisfy: (1) \(c(t)>0,\;s(t)\geq 0\), for \(t\in P\); and \(\sup c(t)\in\mathbb R\), \(\sup s(t)<1\); (2) a mapping \(\varphi_n:P\times [a,b]\times Y\to Y\) satisfies NEWLINE\[NEWLINEd_Y(\varphi_n(t,x,y),\varphi_n(t,x',y))\leq c(t)| x-x'|,\qquad {\text{ for }}\forall t\in P,x,x'\in [a,b], y\in Y;NEWLINE\]NEWLINE NEWLINE\[NEWLINEd_Y(\varphi_n(t,x,y),\varphi_n(t,x,y'))\leq s(t)d_Y(y,y'),\qquad {\text{ for }}\forall t\in P,x,x'\in [a,b], y\in Y.NEWLINE\]NEWLINE Moreover, the functions \(l_n:P\times [a,b]\to [a,b]\) defined by \(l_n(t,x)=a_n(t)+e_n(t)\), with \(a_n(t)=\frac{x_n(t)-x_{n-1}(t)}{b-a}\) and \(e_n(t)=\frac{bx_{n-1}(t)-ax_n(t)}{b-a}\). Then the main results are:NEWLINENEWLINE 1. in the metric space \(X=[a,b]\times Y\), \(d((x_1,y_1),(x_2,y_2))=| x_1-x_2| +\theta d_Y(y_1,y_2)\) with \(\theta=\frac{\inf_{n,t} (1-a_n(t))}{2\sup_{t\in P} c(t)}>0\), the sequence of mappings \(\omega_n:P\times X\to X\) defined by \(\omega_n(t,x,y)=(l_n(t,x),\varphi_n(t,x,y))\) is a Countable Iterated Function System (CIFS) on the space \((x,d_x)\).NEWLINENEWLINE 2. under certain conditions, there exists a countable interpolation function \(x\to f(t,x)\) corresponding to the CSD \(\Delta(t)\) such that \(A(t)=\{(x,f(t,x)):x\in [a,b]\}\) is the attractor of CIFS \((\omega_n(t,\cdot,\cdot))_n\).NEWLINENEWLINE 3. by certain hypothesis, the attractor \(A(t)\) of CIFS associated to CSD \(\Delta(t)\) depends continuously of \(t\in P\), and the mapping \(t\to A(t)\) is Lipschitz.NEWLINENEWLINE An example in \(\mathbb R^2\) is given.