**Connection 1**

*dm*,*n*ð Þ¼ *x*0, *y dm*,*n*ð Þ¼ *xM*, *y dm*,*<sup>n</sup> x*, *y*<sup>0</sup>

*Mathematical Theorems - Boundary Value Problems and Approximations*

∣*dm*,*n*ð Þ� *x*, *y dm*,*<sup>n</sup> x*<sup>0</sup>

*Tf x*ð Þ¼ , *<sup>y</sup> Fm*,*<sup>n</sup> <sup>u</sup>*�<sup>1</sup>

For the next, see [12] for details.

3.*if G* ⊂ *D* � *is a graph of f, then*

**Example 3.** *Let <sup>φ</sup>*ð Þ*<sup>t</sup>* <sup>≔</sup> *<sup>t</sup>*

*Let for z*∈½ Þ 0, þ∞ ,

n o

<sup>¼</sup> *em*,*nu*�<sup>1</sup>

<sup>þ</sup> *dm*,*<sup>n</sup> <sup>u</sup>*�<sup>1</sup>

for ð Þ *x*, *y* ∈ *Dm*,*<sup>n</sup>*, *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*.

*defined above. If each sm*,*<sup>n</sup> be a bounded function, then.*

and for some *L*1, *L*<sup>2</sup> >0,

Let

*<sup>C</sup>*<sup>∗</sup> ð Þ! *<sup>D</sup> B D*ð Þ by

2.*f xi*, *y <sup>j</sup>* � �

*xi*, *y <sup>j</sup>* , *zi*,*<sup>j</sup>* � �

**124**

� � <sup>¼</sup> *dm*,*<sup>n</sup> <sup>x</sup>*, *yN*

∣ þ *L*2∣*y* � *y*<sup>0</sup>

*<sup>m</sup>* ð Þ *<sup>x</sup> <sup>v</sup>*�<sup>1</sup> *<sup>n</sup>* ð Þ*y*

*<sup>m</sup>* ð Þ *<sup>x</sup>* , *<sup>v</sup>*�<sup>1</sup> *<sup>n</sup>* ð Þ*<sup>y</sup>* � � � � <sup>þ</sup> *hm*,*<sup>n</sup>*

*be given, where* 0 ¼ *x*<sup>0</sup> < *x*<sup>1</sup> < *x*<sup>2</sup> ¼ 1*,* 0 ¼

*:*

, *y*<sup>0</sup> ð Þ∣ ≤*L*1∣*x* � *x*<sup>0</sup>

*Fm*,*n*ð Þ¼ *x*, *y*, *z em*,*nx* þ *f <sup>m</sup>*,*<sup>n</sup> y* þ *gm*,*nxy* þ *dm*,*n*ð Þ *x*, *y sm*,*n*ð Þþ *z hm*,*n*,

*<sup>n</sup>* ð Þ*<sup>y</sup>* , *f u*�<sup>1</sup>

*<sup>n</sup>* ð Þ*<sup>y</sup>* � �*sm*,*<sup>n</sup> f u*�<sup>1</sup>

**Corollary 2.2.** *Let D*f g � ; *wm*,*<sup>n</sup>*, *m* ¼ 1, 2, … , *M*; *n* ¼ 1, 2, … , *N denote the IFS*

1.*there is a unique continuous function f* : *D* ! *which is a fixed point of T*;

¼ *zi*,*<sup>j</sup> for all i* ¼ 0, 1, … , *M and j* ¼ 0, 1, … , *N;*

*G* ¼ ⋃ *M m*¼1 ⋃ *N n*¼1

An especially simple example is the following; see [12].

*<sup>s</sup>*1,1ð Þ*<sup>z</sup>* <sup>≔</sup> <sup>1</sup>

*<sup>s</sup>*2,1ð Þ*<sup>z</sup>* <sup>≔</sup> *<sup>z</sup>*

1 þ *z*

1 þ 2*z*

: *i* ¼ 0, 1, 2; *j* ¼ 0, 1, 2

*all i* ¼ 1, 2; *j* ¼ 1, 2 *and x*ð Þ , *y* ∈½ �� 0, 1 ½ � 0, 1 ,

*<sup>n</sup>* ð Þ*<sup>y</sup>* � � � �

*<sup>m</sup>* ð Þ *<sup>x</sup>* , *<sup>v</sup>*�<sup>1</sup>

*<sup>n</sup>* ð Þþ *<sup>y</sup> gm*,*<sup>n</sup>u*�<sup>1</sup>

*wm*,*<sup>n</sup>*ð Þ *G :*

<sup>1</sup>þ*<sup>t</sup> for t*∈ð Þ 0, <sup>þ</sup><sup>∞</sup> *. Let a set of data*

*y*<sup>0</sup> <*y*<sup>1</sup> <*y*<sup>2</sup> ¼ 1 *and zi*,*<sup>j</sup>* ∈½ � 0, 1 *for all i* ¼ 0, 1, 2; *j* ¼ 0, 1, 2*. Here, a set of data points is not necessarily the case that z*0,*<sup>j</sup>* ¼ *zi*,0 ¼ *z*2,*<sup>j</sup>* ¼ *zi*,2 *for all i* ¼ 0, 1, 2; *j* ¼ 0, 1, 2*. Let for*

*dm*,*<sup>n</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>≔</sup> 22ð Þ *<sup>m</sup>*þ*<sup>n</sup> <sup>x</sup><sup>m</sup>*ð Þ <sup>1</sup> � *<sup>x</sup> myn*ð Þ <sup>1</sup> � *<sup>y</sup> <sup>n</sup>*

, *<sup>s</sup>*1,2ð Þ*<sup>z</sup>* <sup>≔</sup> *<sup>z</sup>*

, *<sup>s</sup>*2,2ð Þ*<sup>z</sup>* <sup>≔</sup> *<sup>z</sup>*

1 þ *z* ,

1 þ 3*z :*

where *sm*,*<sup>n</sup>* is a Rakotch or Geraghty contraction. For *<sup>f</sup>* <sup>∈</sup>*C*<sup>∗</sup> ð Þ *<sup>D</sup>* , we define *<sup>T</sup>* :

*<sup>m</sup>* ð Þ *<sup>x</sup>* , *<sup>v</sup>*�<sup>1</sup>

*<sup>m</sup>* ð Þþ *<sup>x</sup> <sup>f</sup> <sup>m</sup>*,*<sup>n</sup>v*�<sup>1</sup>

*<sup>m</sup>* ð Þ *<sup>x</sup>* , *<sup>v</sup>*�<sup>1</sup>

� � <sup>¼</sup> <sup>0</sup>

∣*:*


### **Connection 2**

1. *C I*ð Þ, *dC I*ð Þ � �, *<sup>C</sup>*<sup>∗</sup> ð Þ*<sup>I</sup>* , *dC I*ð Þ � � and *<sup>C</sup>*∗ ∗ ð Þ*<sup>I</sup>* , *dC I*ð Þ � � are complete metric spaces, where

$$d\_{C(I)}(f, \mathbf{g}) \coloneqq \max\_{\mathbf{x} \in I} |f(\mathbf{x}) - \mathbf{g}(\mathbf{x})|$$

for all *f*, *g* ∈*C I*ð Þ (see [2]).

2. *B D*ð Þ, *dB D*ð Þ � �, *<sup>B</sup>*<sup>∗</sup> ð Þ *<sup>D</sup>* , *dB D*ð Þ � � and *<sup>B</sup>*∗ ∗ ð Þ *<sup>D</sup>* , *dB D*ð Þ � � are complete metric spaces, where

$$d\_{B(D)}(f, \mathbf{g}) \coloneqq \sup\_{(\mathbf{x}, \mathbf{y}) \in D} |f(\mathbf{x}, \mathbf{y}) - \mathbf{g}(\mathbf{x}, \mathbf{y})|$$

for all *f*, *g* ∈ *B D*ð Þ [10].

3.*C*∗ ∗ ð Þ *<sup>D</sup>* , *<sup>C</sup>*<sup>∗</sup> ð Þ *<sup>D</sup>* , *<sup>C</sup>*∗ ∗ ð Þ *<sup>D</sup>* , *<sup>C</sup>*<sup>∗</sup> ð Þ *<sup>D</sup>* and *C D*ð Þ are closed subspaces of *B D*ð Þ with *C*∗ ∗ ð Þ *<sup>D</sup>* <sup>⊂</sup>*C*<sup>∗</sup> ð Þ *<sup>D</sup>* <sup>⊂</sup>*C*<sup>∗</sup> ð Þ *<sup>D</sup>* <sup>⊂</sup>*C D*ð Þ⊂*B D*ð Þ and *<sup>C</sup>*∗ ∗ ð Þ *<sup>D</sup>* <sup>⊂</sup>*C*∗ ∗ ð Þ *<sup>D</sup>* <sup>⊂</sup>*C*<sup>∗</sup> ð Þ *<sup>D</sup>* ⊂*C D*ð Þ⊂ *B D*ð Þ, and so they are complete metric spaces.
