ed graph where V denote the set of vertices and E is the set of edges. For all, let denote the set of edges from u to v with elements where denotes the number of elements of. An iterated function system realizing the graph G is given by a collection of metric spaces with contraction mappings corresponding to the edge in the opposite direction of. An attractor (or invariant list) for such an iterated function system is a list of nonempty compact sets such that for all,

Then, is the graph directed iterated function system (GDIFS) realizing the graph G [14] [15] .

Example 1. An example of GDIFS may be seen in [13] [16] .

3. Graph Directed Coalescence FIF

In this section, for a finite number of data sets, generalized graph-directed iterated function system (GDIFS) is defined so that projection of each attractor on is the graph of a CHFIF which interpolates the corre- sponding data set and calls it as graph-directed coalescence hidden-variable fractal interpolation function (GDCHFIF). For simplicity, only two sets of data are considered. Let the two data sets be

where with

(5)

for all and. By introducing two sets of real parameters for and, consider the two generalized data sets

corresponding to and respectively. Also consider the directed graph with such that

To construct a generalized GDIFS associated with the data and realize the graph G, consider the functions defined as

(6)

such that

From each of the above conditions, the following can be derived respectively.

(7)

(8)

(9)

(10)

From the linear system of Equations (7)-(10) the constants, , , , and for, are determined as follows:

The following theorem shows that each map is contraction with respect to metric equivalent to the Euclidean metric and ensures the existence of attractors of generalized GDIFS.

Theorem 2. Let be the generalized GDIFS defined in (6) realizing the graph and associated with the data sets which satisfy (5). If, and are chosen such that for all and. Then there exists a metric on equivalent to the Euclidean metric such that the GDIFS is hyperbolic with respect to. In particular, there exist non empty compact sets such that

Proof. Proof follows in the similar lines of Theorem 2.1.1 of [17] and using the above condition (5). □

Following is the main result regarding existence of coalescence Hidden-variable FIFs for generalized GDIFS.

Theorem 3. Let be the attractors of the generalized GDIFS as in Theorem 2. Then is the graph of a vector valued continuous function such that for, for all. If then the projection of the attractors on is the graph of the continuous function known as CHFIF such that for,. That is

.

Proof. Consider the vector valued function spaces

with metrics

respectively, where denotes a norm on. Since and are complete metric spaces, is also a complete metric space where

Following are the affine maps,

Now define the mapping

where for,

and for,

Now using Equations (7)-(10) it is clear that,

Similarly, ,. It proves that T maps into itself. Since for each

, is continuous and therefore, is continuous on each subintervals.

For, using (7) it follows that.

For, using (8) it follows that.

For, using (7) and (8) it follows that since and.

Hence is continuous on I. Similarly it can be shown that is continuous on J. Consequently T is continuous.

To show that T is a contraction map on, let and. Now,

where and. Therefore

Similarly, it follows that

where and. Then

where and hence T is a contraction mapping. By Banach fixed point theorem, T possesses a unique fixed point, say.

Now, for,

For,

This shows that is the function which interpolates the data. Similarly, it can

be shown that is the function which interpolates the data. For and,

and

If F and H are the graphs of and respectively, then

The uniqueness of the attractor implies that and. That is and. Denoting and, result follows.

Example 4. Consider the data sets as

realizing the graph with, , , as in Figure 1. Take the first set of generalized data

and

corresponding to and respectively. Here for both the generalized data sets. Choose, , for all and. Then Figure 2 is the attractors of the corresponding generalized GDIFS.

Keeping the free variables and constrained variables same, Figure 3 is the attractors of the generalized GDIFS associated with the second set of generalized data

Figure 1. Directed graph for Example 4.

Figure 2. Attractors for the first set of generalized data.

Figure 3. Attractors for the second set of generalized data.

Figure 4. Attractors for the third set of generalized data.

Table 1. The generalized GDIFS with the free variables and constraints variables.

Take the third set of generalized data

and

corresponding to and respectively. For the generalized GDIFS with the free variables and constraints variables given in following Table 1, the attractors are given in Figure 4.

Cite this paper
Akhtar, M. and Prasad, M. (2016) Graph-Directed Coalescence Hidden Variable Fractal Interpolation Functions. Applied Mathematics, 7, 335-345. doi: 10.4236/am.2016.74031.
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