In this part of the article, we present the convergence theorem of the presented method for PDDEs. Suppose \(V^{n,q}\), \(n\ge 2 \) is a Sobolev space containing all functions \(\rho : [0,R ]\to \mathbb{R}^{n} \) such that \(\rho ^{(j)} \), \(0\le j\le n\), is in \(L^{q} \) equipped with the norm

$$ \Vert \rho \Vert _{v^{n,q} } =\sum_{j=0}^{n} \biggl( \int _{0}^{R} \bigl\Vert \rho ^{(j)} (r) \bigr\Vert _{q} ^{q} \biggr) ^{\frac{1}{q} }. $$

### Lemma 1

([7])

*For any given function* \(\rho (\cdot )\in V^{n,\infty }\), *there is a polynomial* \(p (\cdot ) \in P_{M}\) *such that*

$$ \bigl\Vert \rho (r)-p (r) \bigr\Vert _{\infty }\le CC_{0} M^{-n} , \quad 0\le r\le R, $$

*where* *C* *is a fixed constant independent of* *M* *and* \(C_{0} = \Vert \rho \Vert _{V^{n,\infty } } \).

To guarantee the feasibility of problem (7), we convert it into the following one:

$$ \begin{aligned} &\text{Minimize} \quad I= \Vert \bar{w}_{0} -\gamma \Vert ^{2} \\ &\text{subject to} \quad \Biggl\Vert \sum_{j=0}^{M} \bar{w}_{j} H_{kj} -\alpha (r_{k}) \bar{w}_{k}- \beta (r_{k}) \sum _{j=0}^{M}\bar{w}_{j} L_{j}(qr_{k})- \chi (r_{k}) \Biggr\Vert _{\infty }\\ &\hphantom{\text{subject to} \quad}\quad \le (M-1)^{\frac{3}{2} -n}, \quad k=1,2,\dots ,M. \end{aligned} $$

(11)

### Theorem 1

*Suppose that* \(w(\cdot )\in V^{n,\infty } \), \(n\ge 2\) *is a possible solution to the problem* (2). *There is a positive integer* \({ M}_{1} \) *such that for any integer* \(M>{ M}_{1} \), *problem* (11) *has a feasible solution* \(\bar{w}= (\bar{w}_{0} ,\bar{w}_{1} ,\dots ,\bar{w}_{M} )\) *satisfying*

$$ \bigl\Vert w(r_{k} )-\bar{w}_{k} \bigr\Vert _{\infty } \le L(M-1)^{1-n} ,\quad k=0,1,\dots ,M, $$

*where* \(\{ r_{k} \} _{k=0}^{M} \) *are the collocation points and* *L* *is a positive constant*, *independent of* *M*.

### Proof

There is a polynomial \(p (\cdot )\in P_{M-1}\) and fixed \(C_{1} \) independent of *M*, such that

$$ \bigl\Vert \dot{w}(r)-p(r) \bigr\Vert _{\infty } \le C_{1} (M-1 )^{1-n}. $$

Define

$$ w^{M} (r)= \int _{0}^{r}p(\zeta )\,d\zeta +w(0), \quad r \ge 0. $$

So we have

$$ \dot{w}^{M} (r)=p(r) , \qquad w^{M} (0)=w(0). $$

Hence,

$$ \begin{aligned} \bigl\Vert w(r)-w^{M} (r) \bigr\Vert _{\infty }&= \biggl\Vert \int _{0}^{r}\bigl(\dot{w}(z)-p(z)\bigr)\,dz \biggr\Vert _{\infty }\le \int _{0}^{r} \bigl\Vert \dot{w}(z)-p(z) \bigr\Vert _{\infty }\,dz \\ &\le C_{1} (M-1 )^{1-n} \int _{0}^{r}\,df \le C_{1} R (M-1 )^{1-n}. \end{aligned} $$

(12)

By (12), \(w(r_{k} )\) and \(\bar{w}_{k} \) for \(k=0,1,\dots ,M\) are in a dense set as \(\Psi \subseteq {\mathbb{R}}^{n} \). On the other hand, \(w^{M} (\cdot ) \in P_{M}\) is a polynomial. For any polynomial \(w(\cdot ) \in P_{M}\), its derivative at the transferred LGL nodes \(r_{0} ,r_{1 } ,\dots ,r_{{M} } \) can be computed accurately with differential matrix *H*. Therefore we get

$$ \sum_{j=0}^{M} \bar{w}_{j} H_{kj} =\dot{w}^{M} (r_{k} ). $$

(13)

So by (12) and (13), for \(k=1,2,\dots ,M \), we have

$$\begin{aligned} & \Biggl\Vert \sum_{j=0}^{M} \bar{w}_{j} H_{kj}-\alpha (r_{k})w(r_{k})- \beta (r_{k})w(qr_{k})-\chi (r_{k}) \Biggr\Vert _{\infty } \\ &\quad \leq \bigl\Vert \dot{w}^{M}(r_{k})-\dot{w}(r_{k}) \bigr\Vert _{\infty }+ \bigl\Vert \alpha (r_{k})w^{M}(r_{k})- \alpha (r_{k})w(r_{k}) \bigr\Vert _{\infty }\\ &\qquad {}+ \bigl\Vert \beta (r_{k})w^{M}(qr_{k})-\beta (r_{k})w(qr_{k}) \bigr\Vert _{\infty } \\ &\quad \leq \bigl\Vert \dot{w}^{M}(r_{k})-\dot{w}(r_{k}) \bigr\Vert _{\infty }+ \bigl\Vert \alpha (r_{k}) \bigr\Vert _{\infty } \bigl\Vert w^{M}(r_{k})-w(r_{k}) \bigr\Vert _{\infty }\\ &\qquad {}+ \bigl\Vert \beta (r_{k}) \bigr\Vert _{\infty } \bigl\Vert w^{M}(qr_{k})-w(qr_{k}) \bigr\Vert _{\infty } \\ &\quad \leq C_{1}(M-1)^{1-n}+N_{1}C_{1}R(M-1)^{1-n}+N_{2}C_{1}R(M-1)^{1-n} \\ &\quad =C_{1}(M-1)^{1-n}(1+N_{1}R+N_{2}R), \end{aligned}$$

where \(N_{1}\) and \(N_{2}\) are upper bounds for continuous functions \(a(\cdot )\) and \(b(\cdot )\) on the interval \([0,R]\). Thus by selecting \(M_{1} \in \mathbb{N}\) such that \(C_{1} (1+N_{1}R+N_{2}R)\le (M_{1}-1)^{\frac{1}{2} } \), we get

$$ \Biggl\Vert \sum_{j=0}^{M} \bar{w}_{j} H_{kj}-\alpha (r_{k})w(r_{k})- \beta (r_{k})w(qr_{k})-\chi (r_{k}) \Biggr\Vert _{\infty }\leq (M-1)^{ \frac{3}{2}-n}, \quad k=1,2,\dots , M, $$

(14)

for all integers \(M\ge M_{1} \). □

Let \((\bar{w}_{0}^{*} ,\bar{w}_{1}^{*} ,\dots ,\bar{w}_{M}^{*} )\) be an optimal solution to problem (11) defined by

$$ w_{M}^{*} (r)=\sum_{k=0}^{M} \bar{w}_{k}^{*} L_{k} (r), \quad r \in [0, R ], $$

(15)

where \(L_{k} (\cdot )\), \(k=0,1,\dots ,M\) are the Lagrange interpolating polynomials. We have a sequence of direct solutions \(\{ \bar{w}_{0}^{*} ,\bar{w}_{1}^{*} ,\ldots ,\bar{w}_{M}^{*} \} _{M=M_{1} }^{\infty } \) and corresponding sequences of interpolating functions \(\{ w_{M}^{*} (\cdot ) \} _{M=M_{1} }^{\infty }\).

### Assumption 1

It is supposed that the sequence \(\{ \bar{w}_{0}^{*} ,\dot{w}^{*}_{M} (\cdot ) \} _{M=M_{1} }^{ \infty } \) has a subsequence that uniformly converges to \(\{ {w}_{0}^{\infty } ,q(\cdot ) \} \) where \(q(\cdot )\) is a continuous function and \({w}_{0}^{\infty } \in \mathbb{R}\).

### Theorem 2

*Let* \(\{ \bar{w}_{0}^{*} ,\bar{w}_{1}^{*} ,\ldots ,\bar{w}_{M}^{*} \} _{M=M_{1} }^{\infty } \) *be a sequence for optimal solutions of problem* (11) *and* \(\{ w_{M}^{*} (\cdot ) \} _{M=M_{1} }^{\infty } \) *be their interpolating sequence satisfying Assumption *1. *Then*,

$$ w^{*} (r)= \int _{0}^{r}q(\zeta )\,d\zeta + {w}_{0}^{\infty }, \quad 0 \le r\le R, $$

(16)

*is an optimal solution to the problem* (2).

### Proof

Under Assumption 1, there exists a subsequence \(\{ \dot{w}^{*}_{M_{i}} (\cdot ) \} _{i=1}^{\infty } \) of sequence \(\{ \dot{w}^{*}_{M} (\cdot ) \} _{M=M_{1}}^{\infty } \) such that \(\lim_{i\to \infty } M_{i} =\infty \) and \(\lim_{i\to \infty } \dot{w}^{*}_{M_{i} }(\cdot ) =q( \cdot )\). From (16) and Assumption 1, we get

$$ \lim_{i\to \infty } \dot{w}^{*}_{M_{i} } (\cdot )= \dot{w}^{*} (\cdot ). $$

In the first step, we demonstrate that \(w^{*} (\cdot )\) is a feasible solution for problem (2). In the second step, we show that \(w^{*} (\cdot )\) is an optimal solution for the problem (2).

Step 1. Suppose that \(w^{*} (\cdot )\) does not satisfy the restriction of problem (2). There is a time \(\bar{r}\in [0,R ]\) such that

$$ \dot{w}^{*}(\overline{r})-\alpha (\overline{r})w^{*}( \overline{r})- \beta (\overline{r})w^{*}(q\overline{r})-\chi ( \overline{r})\ne 0. $$

Since nodes \(\{ r_{k} \} _{k=0}^{\infty } \) are dense in \([0,R ]\) (see [10]), there exists a subsequence \(k_{M_{i} } \) such that \(0< k_{M_{i} } < M_{i} \), \(\lim_{i\to \infty } r_{k_{M_{i} } } =\bar{r}\). Thus,

$$ \begin{aligned} &\dot{w}^{*}(\bar{r})-\alpha (\overline{r})w^{*}( \overline{r})-\beta ( \overline{r})w^{*}(q\overline{r})-\chi ( \overline{r})\\ &\quad =\lim_{i\to \infty } \bigl(\dot{w}_{M_{i}}^{*}(r_{k_{M_{i}}} )- \alpha (r_{k_{M_{i}}} ) w_{M_{i}}^{*}(r_{k_{M_{i}}} )- \beta (r_{k_{M_{i}}} )w_{M_{i}}^{*}(qr_{k_{M_{i}}} )-\chi (r_{k_{M_{i}}} ) \bigr)\ne 0. \end{aligned} $$

(17)

On the other hand, \(\lim_{i\to \infty } (M_{i} -1 )^{{ \frac{3}{2}} -n} =0\), so that for problem (11), we obtain

$$ \lim_{i\to \infty } \bigl(\dot{w}_{M_{i}}^{*}(r_{k_{M_{i} } } )-\alpha (r_{k_{M_{i}}} ) w_{M_{i}}^{*}(r_{k_{M_{i}}} )- \beta (r_{k_{M_{i}}} )w_{M_{i}}^{*}(qr_{k_{M_{i}}} )-\chi (r_{k_{M_{i}}} )\bigr)=0, $$

which is a contradiction to (17). So, \(w^{*}(\cdot )\) is a possible solution to problem (2).

Step 2. Let \(w^{**} (\cdot )\in V^{n,\infty } \), \(n\ge 2\) be an optimal solution to the problem (2). In view of Theorem 1, there exists a sequence of possible solutions \(\{ \tilde{w}=(\tilde{w}_{0}^{*} ,\tilde{w}_{1}^{*} ,\dots , \tilde{w}_{M}^{*}) \} _{M=M_{1} }^{\infty } \) for problem (11) which converges uniformly to \(w^{**} (\cdot )\). With optimality of \(w^{**} (\cdot )\) and \(\bar{w}^{*}= (\bar{w}_{0}^{*}, \bar{w}_{1}^{*}, \dots , \bar{w}_{M}^{*} )\), we obtain

$$\begin{aligned} 0&= \bigl\Vert w^{**} (0)-\gamma \bigr\Vert ^{2} \le \bigl\Vert w^{*} (0)- \gamma \bigr\Vert ^{2} =\lim _{i\to \infty } \bigl\Vert w_{M_{i} }^{*} (0)-\gamma \bigr\Vert ^{2} \\ &= \bigl\Vert \bar{w}_{0}^{*} -\gamma \bigr\Vert ^{2} \le \bigl\Vert \tilde{w}_{0}^{*} -\gamma \bigr\Vert ^{2} = \bigl\Vert w^{**} (0)- \gamma \bigr\Vert ^{2} =0. \end{aligned}$$

So \(\| w^{*}(0)-\gamma \|^{2}=0 \). Therefore, \(w^{*} (\cdot )\) is an optimal solution for problem (2). □