An abstract framework in the numerical solution of boundary value problems for neutral functional differential equations

Abstract

We consider the numerical solution of boundary value problems for general neutral functional differential equations. The problems are restated in an abstract form and, then, a general discretization of the abstract form is introduced and a convergence analysis of this discretization is developed.

Appendix

Lemma 1

Let Y be a Banach space with norm $\left\| \ \cdot \ \right\| _{Y}$, let $A:\Omega \subseteq Y\rightarrow Y$, where $\Omega$ is open, be a Fréchet-differentiable operator and let $y^{*}\in \Omega$ such that $DA\left( y^{*}\right)$ is invertible. For any $r>0$ such that $\overline{B}\left( y^{*},r\right) \subseteq \Omega$, define

$$\begin{aligned} q\left( r\right) :=\sup _{y\in \overline{B}\left( y^{*},r\right) }\left\| DA\left( y^{*}\right) ^{-1}\left( DA(y) -DA\left( y^{*}\right) \right) \right\| . \end{aligned}$$

Now, let $r>0$ be such that $\overline{B}\left( y^{*},r\right) \subseteq \Omega$ . If

$$\begin{aligned} q\left( r\right) <1\quad \text {and}\quad \left\| DA\left( y^{*}\right) ^{-1}Ay^{*} \right\| _{Y}\le \left( 1-q\left( r\right) \right) r, \end{aligned}$$

(71)

then A has a unique zero $\overline{y}^{*}$ in $\overline{B}\left( y^{*},r\right)$ and

$$\begin{aligned} \left\| \overline{y}^{*}-y^{*}\right\| _{Y}\le \frac{ \left\| DA\left( y^{*}\right) ^{-1}A y^{*} \right\| _{Y}}{1-q\left( r\right) }. \end{aligned}$$

(72)

Moreover, we have

$$\begin{aligned} \overline{y}^{*}-y^{*}=-DA\left( y^{*}\right) ^{-1}Ay^{*} +\delta , \end{aligned}$$

(73)

where

$$\begin{aligned} \left\| \delta \right\| _{Y}\le \frac{q\left( r\right) \left\| DA\left( y^{*}\right) ^{-1}Ay^{*} \right\| _{Y}}{ 1-q\left( r\right) }. \end{aligned}$$

(74)

The proof of the first part is more or less similar to the proof of [36, Lemma 19.1, page 293]. The proof of the second part (73) is clear once the proof of first part is understood.

Lemma 2

Let Y be a Banach space with norm $\left\| \ \cdot \ \right\| _{Y}$. Let $A,B,C:Y\rightarrow Y$ be linear bounded operators such that $A=B+C$ and B is invertible. Let $\left\{ A_{K}\right\}$, $\left\{ B_{K}\right\}$ and $\left\{ C_{K}\right\}$ be sequences of linear bounded operators $Y\rightarrow Y$ such that, for any positive integer K, $A_{K}=B_{K}+C_{K}$ and $B_{K}$ is invertible.

If A is invertible,

$$\begin{aligned} \lim _{K\rightarrow \infty }\Vert B_{K}^{-1}\Vert \cdot \Vert \left( B_{K}-B\right) B^{-1}C\Vert =0 \end{aligned}$$

(75)

and

$$\begin{aligned} \lim _{K\rightarrow \infty }\Vert B_{K}^{-1}\Vert \cdot \Vert C_{K}-C\Vert =0, \end{aligned}$$

(76)

then there exists a positive integer $K_{2}$ such that, for any positive integer $K\ge K_{2}$, $A_{K}$ is invertible and

$$\begin{aligned} \Vert A_{K}^{-1}\Vert \le 2\Vert A^{-1}B\Vert \cdot \Vert B_{K}^{-1}\Vert . \end{aligned}$$

Proof

Assume that A is invertible and (75) and (76) hold. For any positive integer K, we have

$$\begin{aligned} A_{K}=B_{K}+C_{K}=B_{K}\left( I_{Y}+B_{K}^{-1}C_{K}\right) \end{aligned}$$

and then $A_{K}$ is invertible if $I_{Y}+B_{K}^{-1}C_{K}$ is invertible. In this case, we have

$$\begin{aligned} A_{K}^{-1}=\left( I_{Y}+B_{K}^{-1}C_{K}\right) ^{-1}B_{K}^{-1}. \end{aligned}$$

Now, since

$$\begin{aligned} I_{Y}+B_{K}^{-1}C_{K}= & {} I_{Y}+B^{-1}C+B_{K}^{-1}C_{K}-B^{-1}C \\= & {} I_{Y}+B^{-1}C+B_{K}^{-1}\left( C_{K}-C\right) +\left( B_{K}^{-1}-B^{-1}\right) C \\= & {} I_{Y}+B^{-1}C+B_{K}^{-1}\left( C_{K}-C\right) -B_{K}^{-1}\left( B_{K}-B\right) B^{-1}C \end{aligned}$$

and

$$\begin{aligned} I_{Y}+B^{-1}C=B^{-1}\left( B+C\right) =B^{-1}A \end{aligned}$$

is invertible with inverse

$$\begin{aligned} \left( I_{Y}+B^{-1}C\right) ^{-1}=A^{-1}B, \end{aligned}$$

the thesis follows by the Banach perturbation Lemma. $\square$