**Connection 5**

**Connection 3**

where *x*<sup>0</sup>

**Connection 4**

*d*∗

we can see that for some *Ldn* > 0,

Let *dn* : *I* ! be a continuously differentiable function such that

*Mathematical Theorems - Boundary Value Problems and Approximations*

max

ing max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣ ≤ 1, but the converse is not true in general.

*<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup>*dn*ð Þ *<sup>x</sup>* ∣ ≤ <sup>1</sup>*:*

Then, by the Differential Mean Value Theorem and the extreme value theorem,

∣*dn x*<sup>0</sup> ð Þ� *dn x*<sup>00</sup> ð Þ∣ ≤*Ldn* ∣*x*<sup>0</sup> � *x*00∣,

1.The function *dn*ð Þ *x sn*ð Þ*y* is a generalisation of the bivariable function *dn*ð Þ *x y* with vertical scaling factors as (continuous) 'contraction functions'. In fact, in

the case when 0< max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣<1 (see [20], p. 3), obviously,

max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣

*<sup>n</sup>* ð Þ¼ *x*

2.The functional condition max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣ ≤1 is essential in order to show the difference between Banach contractibility of *Fn*ð Þ �, *y* and Rakotch contractibility

of *Fn*ð Þ �, *y* ; compare with [20]. In fact, since *φ*ð Þ*t* <*t* for any *t* >0,

∣*Fn x*, *y*<sup>0</sup> ð Þ� *Fn x*, *y*<sup>00</sup> ð Þ∣ ¼ ∣*dn*ð Þk *x sn y*<sup>0</sup> ð Þ� *sn y*<sup>00</sup> ð Þ∣

≤ max

≤ max

≤ max

standing each *sn* is a Rakotch contraction that is not a Banach contraction, each

On the other hand, if max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣ ¼ 1, then we conclude that each *Fn* is Rakotch contraction with respect to the second variable whenever each *sn* is a

*Fn* is Banach contraction with respect to the second variable because

∣*Fn x*, *y*<sup>0</sup> ð Þ� *Fn x*, *y*<sup>00</sup> ð Þ∣ ≤ max

∣*Fn x*, *y*<sup>0</sup> ð Þ� *Fn x*, *y*<sup>00</sup> ð Þ∣ ≤ max

*dn*ð Þ *<sup>x</sup> <sup>y</sup>* <sup>¼</sup> *dn*ð Þ *<sup>x</sup>*

Let *sn*ð Þ¼ *<sup>y</sup>* max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *<sup>x</sup>* <sup>∣</sup>*<sup>y</sup>* and *<sup>d</sup>*<sup>∗</sup>

*<sup>n</sup>* ð Þ *<sup>x</sup> sn*ð Þ*<sup>y</sup>* , max *<sup>x</sup>*∈*<sup>I</sup>*∣*d*<sup>∗</sup>

where *<sup>x</sup>*, *<sup>y</sup>*<sup>0</sup> ð Þ, *<sup>x</sup>*, *<sup>y</sup>*<sup>00</sup> ð Þ<sup>∈</sup> <sup>2</sup>

Rakotch contraction because

**126**

, *x*<sup>00</sup> ∈*I*. Hence, *dn* is Lipschitz continuous function defined on *I* satisfy-

max *<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup>*dn*ð Þ *<sup>x</sup>* <sup>∣</sup>*y:*

*dn*ð Þ *x* max *<sup>x</sup>*∈*I*∣*dn*ð Þ *x* ∣

*<sup>n</sup>* ð Þ *x* ∣ ¼ 1 and *sn* is a Banach (or Rakotch) contraction.

*<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup>*dn*ð Þk *<sup>x</sup> sn <sup>y</sup>*<sup>0</sup> ð Þ� *sn <sup>y</sup>*<sup>00</sup> ð Þ<sup>∣</sup>

. Hence, if max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣<1, as can be seen, notwith-

*<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup>*dn*ð Þk *<sup>x</sup> <sup>y</sup>*<sup>0</sup> � *<sup>y</sup>*00∣*:*

*<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup>*dn*ð Þ *<sup>x</sup>* <sup>∣</sup>*<sup>φ</sup>* <sup>j</sup>*y*<sup>0</sup> � *<sup>y</sup>*<sup>00</sup> ð Þj *:*

*<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup>*dn*ð Þ *<sup>x</sup>* <sup>∣</sup>*<sup>φ</sup>* <sup>j</sup>*y*<sup>0</sup> � *<sup>y</sup>*<sup>00</sup> ð Þj

*<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup>*dn*ð Þk *<sup>x</sup> <sup>y</sup>*<sup>0</sup> � *<sup>y</sup>*00∣,

. Then *dn*ð Þ *x y* ¼

In the case where the vertical scaling factors are constants, in [1], the existence of affine FIFs by using the Banach fixed point theorem was investigated, whereas in [20], a generalisation of affine FIFs by using vertical scaling factors as (continuous) 'contraction functions' and Banach's fixed point theorem was introduced. Theorem 2.2 gives the existence of fractal interpolation curves by using the Rakotch fixed point theorem and vertical scaling factors as (continuous) 'contraction functions'.
