1 Introduction

Let V be the space of the continuous functions \([ a,b]\rightarrow \mathbb {R}^{d}\) and let U be a Banach space of integrable functions \([a,b] \rightarrow \mathbb {R}^{d}\). We deal with the numerical solution of the functional differential equation boundary value problem (BVP)

$$\begin{aligned} \left\{ \begin{array}{l} y^{\prime }(t) =F\left( t,y,y^{\prime },p \right) ,\quad t\in [a,b] , \\ B\left( y,y^{\prime },p \right) =0, \end{array} \right. \end{aligned}$$
(1)

where the functionals \(F:[a,b] \times V\times U \times \mathbb {R} ^{d_{0}}\rightarrow \mathbb {R}^{d}\) and \(B:V\times U\times \mathbb {R} ^{d_{0}}\rightarrow \mathbb {R}^{d}\times \mathbb {R}^{d_0}\) are given and the pair \((y,p)\in V\times \mathbb {R}^{d_0}\) is unknown.

The reason to include \(p\in \mathbb {R}^{d_0}\) as an unknown of the problem (1) is that, in many real applications, there are parameters to be determined along with the solution y. For example, the determination of periodic solutions for an autonomous functional differential equation reduces to a BVP, where the unknown period of the periodic solution appears as a parameter.

The general functional differential equation

$$\begin{aligned} y^\prime (t)=F(t,y,y^\prime ,p),\quad t\in [a,b], \end{aligned}$$
(2)

in (1) includes the two particular and important cases of differential equations with deviating arguments

$$\begin{aligned} y^{\prime }(t) =f( t,y(t),y(\theta _{1}(t)) ,\ldots ,y(\theta _{k}(t)),y^{\prime }(\vartheta _{1}(t)),\ldots ,y^{\prime }(\vartheta _{l}(t)),p ),\quad t\in [a,b], \end{aligned}$$
(3)

and integro-differential equations

$$\begin{aligned} y^{\prime }(t) =f\left( t,y(t) ,\int \limits _{\alpha (t)}^{\beta (t)}k\left( t,s,y(s) ,y^{\prime }(s) \right) ds,p\right) ,\quad t\in [a,b]. \end{aligned}$$
(4)

In order to restate (3) and (4) in the form (2), it is necessary to have

$$\begin{aligned} \theta _{r}(t),\vartheta _{s}(t)\in [a,b],\quad t\in [a,b]\quad \text {and}\quad r=1,\ldots ,k\quad \text {and}\quad s=1,\ldots ,l, \end{aligned}$$
(5)

for (3) and

$$\begin{aligned} \alpha (t),\beta (t)\in [a,b],\quad t\in [a,b]. \end{aligned}$$
(6)

for (4). However, one often encounters Eq. (3), or Eq. (4), where the condition (5), or the condition (6), is not fulfilled. For example, this happens when some \(\theta _r(t)\) or \(\vartheta _s(t)\) in (3), or one of \(\alpha (t)\) and \(\beta (t)\) in (4), has the form \(t\pm \tau \), where \(\tau >0\). For such equations, we need to specify the solution y and its derivative \(y^{\prime }\) outside the interval [a, b] by the side condition

$$\begin{aligned} y(t) =\phi (t)\quad \text {and}\quad y^{\prime }(t)=\varphi (t),\quad t<a\ \text {or }t>b, \end{aligned}$$
(7)

where \(\phi ,\varphi :(-\infty ,a) \cup ( b,+\infty )\rightarrow \mathbb {R}^d \) are given functions (of course, it makes sense take \(\varphi =\phi ^\prime \)). Then, the equation can be still restated in the form (2) by incorporating the side condition in the functional F: we write the Eq. (3) as

$$\begin{aligned} y^{\prime }(t) =f\left( t,y(t),\varTheta \left( y,t;\phi \right) ,\varTheta \left( y^{\prime },t;\varphi \right) ,p\right) ,\quad t\in [a,b] , \end{aligned}$$

where

$$\begin{aligned} \varTheta (y,t;\phi )&:=\left( \varTheta \left( y,t;\phi \right) _1 ,\ldots ,\varTheta \left( y,t;\phi \right) _k\right) \\ \varTheta \left( y,t;\phi \right) _r&:=\left\{ \begin{array}{ll} y\left( \theta _{r}(t) \right) &{} \text {if}\theta _{r}(t) \in [a,b] \\ \phi \left( \theta _{r}(t) \right) &{}\text {if}\theta _{r}(t) <a\text {or}\theta _{r}(t) >b \end{array} \right. ,\quad r=1,\ldots ,k,\\ \varTheta (y^\prime ,t;\varphi )&:=\left( \varTheta \left( y^{\prime },t;\varphi \right) _1 ,\ldots ,\varTheta \left( y^{\prime },t;\varphi \right) _l\right) \\ \varTheta \left( y^{\prime },t;\varphi \right) _s&:=\left\{ \begin{array}{ll} y^{\prime }\left( \vartheta _{s}(t) \right) &{} \text {if } \vartheta _s(t) \in [a,b] \\ \varphi \left( \vartheta _{s}(t) \right) &{} \text {if}\vartheta _{s}(t) <a \text {or }\vartheta _{s}(t) >b \end{array} \right. ,\quad s=1,\ldots ,l, \end{aligned}$$

and the Eq. (4) as

$$\begin{aligned} y^{\prime }(t) =f\left( t,y(t) ,\int \limits _{\theta _{1}(t)}^{\theta _{2}(t)}K\left( t,s,y,y^{\prime };\phi ,\varphi \right) ds,p\right) ,\quad t\in [a,b] , \end{aligned}$$

where

$$\begin{aligned} K(t,s,y,y^{\prime };\phi ,\varphi ):=\left\{ \begin{array}{ll} k(t,s,y(s),y^{\prime }(s))&{} \text {if}s\in [a,b] \\ k(t,s,\phi (s),\varphi (s))&{} \text {if}s<a\text {or} s>b. \end{array} \right. \end{aligned}$$

Observe that the side condition (7) is considered as a part of the functional differential equation (2), not as a boundary condition.

We recall here that a particular and important case of the Eq. (3) is given by delay differential equations, where

$$\begin{aligned} \theta _{r}(t),\vartheta _{s}(t)\le t,\quad t\in [a,b]\quad \text {and}\quad r=1,\ldots ,k\quad \text {and}\quad s=1,\ldots ,l, \end{aligned}$$

and two particular and important cases of the Eq. (4) are given by Fredholm integro-differential equations, where

$$\begin{aligned} \alpha (t)=a\quad \text {and}\quad \beta (t)=b,\quad t\in [a,b], \end{aligned}$$

and Volterra integro-differential equations, where

$$\begin{aligned} \alpha (t)=a\quad \text {and}\quad \beta (t)=t,\quad t\in [a,b]. \end{aligned}$$

We also remark that the general form (2) includes integro-differential equations

$$\begin{aligned} y^{\prime }(t) =f\left( t,y(t) ,y(\theta (t)),\int \limits _{\alpha (t)}^{\beta (t)}k\left( t,s,y(s) ,y^{\prime }\left( \vartheta (s)\right) \right) ds,p\right) ,\quad t\in [a,b], \end{aligned}$$

which cannot be seen as Eq. (4).

The general boundary condition

$$\begin{aligned} B(y,y^\prime ,p)=0 \end{aligned}$$
(8)

in (1) includes the classical boundary condition

$$\begin{aligned} g(y(a),y(b),p)=0 \end{aligned}$$

and the more general multipoint boundary condition

$$\begin{aligned} g(y(a),y(b),y(t_{1}),\ldots ,y(t_{q}),p)=0, \end{aligned}$$
(9)

where \(t_{i}\in (a,b),\,i=1,\ldots ,q\), and integral boundary condition

$$\begin{aligned} g\left( y(a),y(b),\int \limits _a^b w(t,y(t))dt,p\right) =0. \end{aligned}$$
(10)

Note that, in general, boundary conditions for first order differential equation do not involve the derivative \(y^\prime \). In this paper, we consider boundary conditions involving \(y^\prime \) since our theory can deal with this situation without any further complication.

By following the usual terminology, the functional differential equation (2) can be called neutral since, in general, the values \( F(t,y,y^{\prime },p)\) depend on \(y^\prime \).

Of course, the neutral equation (2) also includes the case where the values of F do not depend on \(y^\prime \). In this case, we say that the Eq. (2) is non-neutral. Similarly, we say that the boundary condition (8) is non-neutral if the values of B do not depend on \(y^\prime \) and that the BVP (1) is non-neutral if both (2) and (8) are non-neutral.

1.1 Numerical literature on BVPs for functional differential equations and aim of the paper

The papers dealing with the numerical solution of functional differential equation BVPs, apart from [13, 30] described below, address special cases of the problem (1). Table 1 collects such papers according to the special case considered.

Table 1 Numerical literature on functional differential equations BVPs
Full size table

Instead, the papers [13, 30] deal with BVPs for general non-neutral second order functional differential equations. The paper [13] deals with BVPs

$$\begin{aligned} \left\{ \begin{array}{l} y^{\prime \prime }(t) =g\left( t,y(t) \right) + \mathcal {F}\left( y\right) (t) ,\quad t\in [0,1] , \\ y\left( 0\right) =y(1) =0, \end{array} \right. \end{aligned}$$

where \(\mathcal {F}\) is an operator acting on y, and considers a discretization of the second derivative by a central difference. As a consequence, a method of order two is obtained. The paper [30] deals with BVPs

$$\begin{aligned} \left\{ \begin{array}{l} y^{\prime \prime }(t) =\mathcal {F}\left( y,y^{\prime }\right) (t) ,\quad t\in [a,b] , \\ y\left( a\right) =\alpha ,\ y\left( b\right) =\beta , \end{array} \right. \end{aligned}$$

where \(\mathcal {F}\) is an operator acting on y and \(y^\prime \), and uses special continuous (dense output) methods for second order differential equations. Such methods can reach an arbitrarily high order, if \(\mathcal {F}\left( y,y^{\prime }\right) \) is independent of \(y^\prime \), and have order two at most, otherwise.

Regarding the theoretical (non-numerical) literature on BVPs for functional differential equations, among many papers, we mention here the monograph [26], which contains a collection of articles dealing with many aspects of the theory of such problems, and the book [2], which considers only the non-neutral case.

Aim of the present paper is to study the numerical solution of problems (1). We restate such problems in an abstract form and, then, we introduce a general type of discretization of the abstract form and develop a convergence analysis of this discretization.

The general discretization studied in this paper includes the two particular discretizations of the problem (1) given by collocation method and the Fourier series method. However, in order to avoid having a very long paper, here we do not deal with these two discretizations. They are the subject of the papers [41, 42] and the forthcoming papers [43, 44]. The present paper contains the theoretical bases for the numerical solution of functional differential equations BVPs.

When compared to the current literature, the research started in this paper and continued in [41–44] contains the following advances.

  1. 1.

    The general form (1) of BVP has not ever been studied in literature, even confining to the non-neutral case. Also the abstract form and the general type of discretization considered in this paper are a novelty.

  2. 2.

    By confining to the non-neutral case, we consider a more general situation than that dealt in the papers [13, 30]. Moreover, the methods that we introduce have arbitrarily high order of convergence, unlike the methods in [13, 30] which have order two only.

  3. 3.

    The study of the numerical solution of BVPs for neutral differential equations with deviating arguments is at a seminal stage. On this subject, there are only the three papers [5, 14, 17], where [17] introduces and proves the convergence of a method of order one and [5, 14] are experimental works without convergence proofs. Moreover, regarding BVPs for neutral integro-differential equations, the literature is confined to Volterra and Fredholm equations. In our research, we introduce methods for general neutral functional differential equations BVPs of arbitrarily high order of convergence.

The plan of the paper is the following. In Sect. 2, we introduce the abstract form of the problem (1). In Sect. 3, we introduce the general type of discretization used for the abstract form. In Sect. 4, we analyze the convergence of this general discretization. In Sect. 5, we specialize the results obtained in Sect. 4, in preparation for their application in [41–44] to the problem (1) discretized by the collocation method and the Fourier series method.

1.2 Notations

We finish this section giving a list of conventions and notations used throughout the paper.

  • The norm of a space Y is denoted by \(\Vert \ \cdot \ \Vert _Y\).

  • Cartesian product spaces are equipped with the norm given by the sum of the norms of the factor spaces.

  • In the space Y, the closed ball of center \(y\in Y\) and radius \(r\ge 0\) is denoted by \(\overline{B}(y,r)\).

  • The identity operator of a space Y is denoted by \(I_{Y}\).

  • The norm of a bounded linear operator L from the space Y to the space Z is denoted by \(\Vert L \Vert \), without any reference to the domain Y and the codomain Z.

  • The Fréchet-derivative of the operator A at the point y is denoted by DA(y) .

  • For an operator \(A:Y\rightarrow Z_1\times \cdots \times Z_k\), we define the operators \( A_{Z_i}:Y\rightarrow Z_i\), \(i=1,\ldots ,k\), by

    $$\begin{aligned} A(y)=(A_{Z_1}(y),\ldots ,A_{Z_k}(y)),\quad y\in Y, \end{aligned}$$

    and call them the components of A.

2 The abstract form

We assume that the functional F in (1) is such that \(F(\cdot ,v,u,\beta ) \in U\), for any \((v,u,\beta )\in V\times U\times \mathbb {R}^{d_{0}}\). This assumption permits to introduce the operator \(\mathcal {F}:V\times U\times \mathbb {R}^{d_{0}}\rightarrow U\) given by

$$\begin{aligned} \mathcal {F}\left( v,u,\beta \right) =F(\cdot ,v,u,\beta ),\quad (v,u,\beta )\in V\times U\times \mathbb {R}^{d_{0}}, \end{aligned}$$

and write the functional differential equation (2) as

$$\begin{aligned} y^{\prime }=\mathcal {F}(y,y^{\prime },p), \end{aligned}$$

which is an equation in the space U.

The BVP (1) is now restated in abstract form, where we use the derivative \(y^{\prime }\), instead of y, as the actual unknown of (1).

Consider the very simple linear differential equation

$$\begin{aligned} v^\prime (t)=u(t),\quad t\in [a,b], \end{aligned}$$
(11)

where \(u\in U\) is given and \(v\in V\) is the unknown. Each solution of this equation is determined by a parameter \(\alpha \in \mathbb {R}^d\). Thus, we introduce a linear operator \(\mathcal {G}:U\times \mathbb {R} ^{d}\rightarrow V\) such that, for any \(u\in U\),

$$\begin{aligned} \{v\in V:v\text { is a solution of (11)}\}=\{ \mathcal {G}(u,\alpha ) :\alpha \in \mathbb {R}^{d}\}. \end{aligned}$$

By following the usual terminology of the differential equations, the linear operator \(\mathcal {G}\) can be called a Green operator for the Eq. (11). Of course, examples of a Green operator for (11) are

$$\begin{aligned} \mathcal {G}(u,\alpha )(t)=\int \limits _c^t u(s)ds+\alpha ,\quad t\in [a,b]\quad \text {and}\quad (u,\alpha )\in U\times \mathbb {R}^d, \end{aligned}$$

where \(c\in [a,b]\).

The abstract form of the BVP (1) is based on the interpretation of the Eq. (11) as

$$\begin{aligned} v=\mathcal {G}(u,\alpha )\quad \text { for some }\alpha \in \mathbb {R}^{d}. \end{aligned}$$

In other words, we replace the derivative operator with the Green operator.

Once a Green operator for (11) is given, the abstract form is introduced by defining what we mean for a solution of (1). Let \((y,p) \in V\times \mathbb {R}^{d_{0}}\). We say that (y, p) is a solution of (1) if \(y=\mathcal {G}(u,\alpha ) \) for some \(u\in U\) and \(\alpha \in \mathbb {R}^{d}\) such that

$$\begin{aligned} \left\{ \begin{array}{l} u=\mathcal {F}(\mathcal {G}(u,\alpha ) ,u,p) \\ B\left( \mathcal {G}(u,\alpha ) ,u,p \right) =0. \end{array} \right. \end{aligned}$$

Hence, we reach the following abstract form of the problem (1).

PAF (Problem in Abstract Form). Given:

  • a normed space \(\mathbb {V}\) and Banach spaces \(\mathbb {U}\), \(\mathbb {A}\) and \(\mathbb {B}\);

  • operators \(\mathfrak {F}:\mathbb {V}\times \mathbb {U}\times \mathbb {B}\rightarrow \mathbb {U}\) and \(\mathfrak {B}:\mathbb {V}\times \mathbb {U}\times \mathbb {B}\rightarrow \mathbb {A}\times \mathbb {B}\);

  • a linear operator \(\mathfrak {G}:\mathbb {U}\times \mathbb {A}\rightarrow \mathbb {V}\);

find a pair \((v,\beta )\in \mathbb {V}\times \mathbb {B}\) such that \(v=\mathfrak {G}(u,\alpha )\) for some \(u\in \mathbb {U}\) and \(\alpha \in \mathbb {A}\) satisfying

$$\begin{aligned} \left\{ \begin{array}{l} u=\mathfrak {F}(\mathfrak {G}(u,\alpha ),u,\beta )\\ \mathfrak {B}\left( \mathfrak {G}(u,\alpha ) ,u,\beta \right) =0. \end{array} \right. \end{aligned}$$

Clearly, the BVPs (1) are in the form PAF with \(\mathbb {V}=V\), \(\mathbb {U}=U\), \(\mathbb {A}=\mathbb {R}^d\), \(\mathbb {B}=\mathbb {R}^{d_0}\), \(\mathfrak {F}=\mathcal {F}\), \(\mathfrak {B}=B\) and \(\mathfrak {G}=\mathcal {G}\).

2.1 Other instances of the abstract form

Besides the BVPs (1), PAF includes other types of BVPs.

For example, consider a second order problem (not restated as a first order problem)

$$\begin{aligned} \left\{ \begin{array}{l} y ^{\prime \prime }(t)=F(t,y,p),\quad t\in [a,b],\\ B(y,p)=0 \end{array} \right. \end{aligned}$$
(12)

where \(F:[a,b]\times V\times \mathbb {R}^{d_0}\rightarrow \mathbb {R}^d\) and \(B:V\times \mathbb {R}^{d_0}\rightarrow \mathbb {R}^d\times \mathbb {R}^d\times \mathbb {R}^{d_0}\), V being, as above, the space of the continuous function \([a,b]\rightarrow \mathbb {R}^d\).

Let U be a Banach space of integrable function \([a,b]\rightarrow \mathbb {R}^d\) and consider the differential equation

$$\begin{aligned} v^{\prime \prime }(t)=u(t),\quad t\in [a,b], \end{aligned}$$
(13)

where \(u\in U\) is given and \(v\in V\) is the unknown. A Green operator for (13) is the linear operator \(\mathcal {G}:U\times \mathbb {R}^d\times \mathbb {R}^d\rightarrow V\) given by

$$\begin{aligned}&\mathcal {G}(u,\alpha _1,\alpha _2)=\text {solution of }\left\{ \begin{array}{l} v^{\prime \prime }(t)=u(t),\quad t \in [a,b],\\ v(a)=\alpha _1, v(b)=\alpha _2, \end{array} \right. \\&(u,\alpha _1,\alpha _2)\in U\times \mathbb {R}^d\times \mathbb {R}^d, \end{aligned}$$

i.e.

$$\begin{aligned}&\mathcal {G}(u,\alpha _1,\alpha _2)(t)=\int \limits _a^t\int \limits _a^s u(\sigma )d\sigma ds-\frac{t-a}{b-a}\int \limits _a^b\int \limits _a^s u(\sigma )d\sigma ds+\frac{b-t}{b-a}\alpha _1+\frac{t-a}{b-a}\alpha _2\\&\quad t\in [a,b]. \end{aligned}$$

Under the assumption that \(F(\cdot ,v,\beta )\in U\) for any \((v,\beta )\in V\times \mathbb {R}^{d_0}\), the problem (12) can be restated in the form PAF by introducing the operator \(\mathcal {F}:V\times \mathbb {R}^{d_0}\rightarrow U\) given by

$$\begin{aligned} \mathcal {F}(v,\beta )=F(\cdot ,v,\beta ),\quad (v,\beta )\in V\times \mathbb {R}^{d_0}. \end{aligned}$$

The BVPs (12) are in the form PAF with \(\mathbb {V}=V\), \(\mathbb {U}=U\), \(\mathbb {A}=\mathbb {R}^d\times \mathbb {R}^d\), \(\mathbb {B}=\mathbb {R}^{d_0}\), \(\mathfrak {F}=\mathcal {F}\), \(\mathfrak {B}=B\) and \(\mathfrak {G}=\mathcal {G}\) (of course, here U, \(\mathcal {F}\), B and \(\mathcal {G}\) are those defined for the problems (12)).

Clearly, PAF also includes BVPs for general differential equations obtained by replacing the second derivative \(y^{\prime \prime }(t)\) in (12) with a general linear differentiation operator.

PAF even includes BVPs for partial functional differential equations. In fact, consider the problem

$$\begin{aligned} \left\{ \begin{array} {l} \Delta v(x)=F(x,v),\quad x\in \Omega ,\\ B(v)=0 \end{array} \right. \end{aligned}$$
(14)

where \(\Omega \) is an open set of \(\mathbb {R}^d\) with boundary \(\partial \Omega \), \(\Delta \) is the Laplacian operator, \(F:\Omega \times V\rightarrow \mathbb {R}\), with V the space of the continuous function \(\overline{\Omega }=\Omega \cup \partial \Omega \rightarrow \mathbb {R}\), and \(B:V\rightarrow A\), with A the space of the continuous function \(\partial \Omega \rightarrow \mathbb {R}\).

Given a Banach space U of integrable functions \(\overline{\Omega }\rightarrow \mathbb {R}\), we consider the differential equation

$$\begin{aligned} \Delta v(x)=u(x),\quad x\in \Omega , \end{aligned}$$
(15)

where \(u\in U\) is given and \(v\in V\) is unknown. A Green operator for (15) is the linear operator \(\mathcal {G}:U\times A\rightarrow V\) given by

$$\begin{aligned} \mathcal {G}(u,\alpha )=\text { solution of }\left\{ \begin{array}{ll} \Delta v(x)=u(x),&{}x \in \Omega ,\\ v(x)=\alpha (x),&{}x\in \partial \Omega , \end{array} \right. , (u,\alpha )\in U\times A. \end{aligned}$$

Under the assumption that \(F(\cdot ,v)\in U\) for any \(v\in V\), the problem (14) can be restated in the form PAF (without the space \(\mathbb {B}\)), by introducing the operator \(\mathcal {F}:V\rightarrow U\) given by

$$\begin{aligned} \mathcal {F}(v)=F(\cdot ,v),\quad v\in V. \end{aligned}$$

The BVPs (14) are in the form PAF with \(\mathbb {V}=V\), \(\mathbb {U}=U\), \(\mathbb {A}=A\), \(\mathfrak {F}=\mathcal {F}\), \(\mathfrak {B}=B\) and \(\mathfrak {G}=\mathcal {G}\).

Note that BVPs (14) have the space \(\mathbb {A}=A\) of infinite dimension. However, in the particular and important case of Dirichlet boundary conditions

$$\begin{aligned} B(v)=v|_{\partial \Omega }-g,\quad v\in V, \end{aligned}$$

where \(g\in A\), we can consider \(\mathbb {V}=\left\{ v\in V:v|_{\partial \Omega }\in \mathrm {span}(g)\right\} \), instead of \(\mathbb {V}=V\), and \(\mathbb {A}=\mathrm {span}(g)\), instead of \(\mathbb {A}=A\), where \(\mathrm {span}(g)=\{kg:k\in \mathbb {R}\}\), so to have the space \(\mathbb {A}\) of finite dimension.

2.2 The abstract form as a fixed point problem

From now on we consider the problem PAF with \(\mathbb {A}\) and \(\mathbb {B}\) of finite dimension, rather than its particular instance (1). By introducing the product Banach space

$$\begin{aligned} X:=\mathbb {U}\times \mathbb {A}\times \mathbb {B}, \end{aligned}$$

PAF can be seen as the search for fixed points of the operator \(\varPhi :X\rightarrow X\) given by

$$\begin{aligned} \varPhi \left( x\right) =\left( \mathfrak {F}\left( \mathfrak {G}(u,\alpha ) ,u,\beta \right) ,(\alpha ,\beta )-\mathfrak {B}\left( \mathfrak {G}(u,\alpha ) ,u,\beta \right) \right) ,\quad x=(u,\alpha ,\beta ) \in X. \end{aligned}$$

We have that \((v,\beta ) \in \mathbb {V}\times \mathbb {B}\) is a solution of PAF if and only if \(v=\mathcal {G}(u,\alpha ) \) for some fixed point \((u,\alpha ,\beta ) \in X\) of \(\varPhi \).

Regarding the operators \(\mathfrak {F}\) and \(\mathfrak {B}\) and the linear operator \(\mathfrak {G}\), we do the following assumptions.

  • A \(\varvec{\mathfrak {F}}\varvec{\mathfrak {B}}\) (Assumption \(\varvec{\mathfrak {F}}\varvec{\mathfrak {B}}\)). The operators \(\mathfrak {F}\) and \(\mathfrak {B}\) are Fréchet-differentiable at any point \((v_{0},u_{0},\beta _{0})\in \mathbb {V}\times \mathbb {U}\times \mathbb {B}\).

  • A \(\varvec{\mathfrak {G}}\). The linear operator \(\mathfrak {G}\) is bounded.

Since A \(\varvec{\mathfrak {F}\mathfrak {B}}\) and A \(\varvec{\mathfrak {G}}\) hold, \(\varPhi \) is Fréchet-differentiable at any point \(x_{0}=(u_{0},\alpha _{0},\beta _{0}) \in X\) and the Fréchet-derivative \(D\varPhi \left( x_{0}\right) \) is given by

$$\begin{aligned} D\varPhi \left( x_{0}\right) x&=\left( D\mathfrak {F}(v_0,u_0,\beta _0)\left( v,u,\beta \right) ,(\alpha ,\beta )-D\mathfrak {B}(v_0,u_0,\beta _0)\left( v,u,\beta \right) \right) \\ x&=(u,\alpha ,\beta )\in X, \end{aligned}$$

where \(v_{0}=\mathfrak {G}\left( u_{0},\alpha _{0}\right) \) and \(v=\mathfrak {G} (u,\alpha ) \).

3 Discretization of the abstract form

Our aim is to numerically solve PAF and, in this section, we describe an its quite general discretization. In the following, the positive integer K denotes the level of the discretization: the larger K, the higher is the “quality” of the discretization.

There are two types of discretizations involved in the numerical solution of PAF, that we call secondary discretization and primary discretization.

3.1 The secondary discretization

Consider the BVP (1). In some cases, the values of the functional F cannot be exactly computed. For example, in case of integro-differential equation (4), F involves an integral which has to be replaced with a quadrature rule. Therefore, for any positive integer K, we have to replace F with a suitable functional \(F_{K}:[a,b] \times V\times U\times \mathbb {R}^{d_0}\rightarrow \mathbb {R}^d\), whose values can be exactly computed. If the values of F can be exactly computed, as in case of differential equations with deviating arguments (3), we consider \(F_{K}=F\).

As done for F, we require \(F_K(\cdot ,v,u,\beta ) \in U\), for any positive integer K and \((v,u,\beta )\in V\times U\times \mathbb {R}^{d_{0}}\). Hence, for any positive integer K, we can replace the operator \(\mathcal {F}\) with the operator \(\mathcal {F} _{K}:V\times U\times \mathbb {R} ^{d_0}\rightarrow U\) given by

$$\begin{aligned} \mathcal {F}_K\left( v,u,\beta \right) =F_K\left( \cdot ,v,u,\beta \right) ,\quad (v,u,\beta )\in V\times U\times \mathbb {R}^{d_0}. \end{aligned}$$

Analogously, for any positive integer K, we replace the functional B with a suitable functional \(B_{K}:V\times U\times \mathbb {R}^{d_0}\rightarrow \mathbb {R}^{d}\times \mathbb {R}^{d_0}\), whose values can be exactly computed. If the values of B can be exactly computed, as in case of multipoint boundary conditions (9), we consider \(B_K=B\).

The secondary discretization of PAF consists in replacing, for any positive integer K, the operators \(\mathfrak {F}\) and \(\mathfrak {B}\) with operators \(\mathfrak {F}_K:\mathbb {V}\times \mathbb {U}\times \mathbb {B}\rightarrow \mathbb {U}\) and \(\mathfrak {B}_K:\mathbb {V}\times \mathbb {U}\times \mathbb {B}\rightarrow \mathbb {A}\times \mathbb {B}\), respectively, whose values can be exactly computed. As done for \(\mathfrak {F}\) and \(\mathfrak {B}\), we assume what follows.

  • A \(\varvec{\mathfrak {F}_K}\varvec{\mathfrak {B}_K}\). For any positive integer K, the operators \(\mathfrak {F}_K\) and \(\mathfrak {B}_K\) are Fréchet-differentiable at any point \((v_{0},u_{0},\beta _{0})\in \mathbb {V}\times \mathbb {U}\times \mathbb {B}\).

For any positive integer K, the operator \(\varPhi \) is then replaced with the operator \(\varPhi _K:X\rightarrow X\) given by

$$\begin{aligned} \varPhi _K \left( x\right)&=\left( \mathfrak {F}_K\left( \mathfrak {G}(u,\alpha ) ,u,\beta \right) ,(\alpha ,\beta )-\mathfrak {B}_K\left( \mathfrak {G} (u,\alpha ) ,u,\beta \right) \right) \\ x&=(u,\alpha ,\beta ) \in X. \end{aligned}$$

Since A \(\varvec{\mathfrak {F}_K\mathfrak {B}_K}\) and A \(\varvec{\mathfrak {G}}\) hold, \(\varPhi _K\) is Fréchet-differentiable at any point \( x_{0}=(u_{0},\alpha _{0},\beta _{0})\in X\) and \(D\varPhi _K \left( x_{0}\right) \) is given by

$$\begin{aligned} D\varPhi _K \left( x_{0}\right) x&=\left( D\mathfrak {F}_K(v_{0},u_{0},\beta _{0})\left( v,u,\beta \right) ,(\alpha ,\beta )-D\mathfrak {B}_K(v_{0},u_{0},\beta _{0}\right) \left( v,u,\beta \right) ) \nonumber \\ x&=(u,\alpha ,\beta )\in X, \end{aligned}$$
(16)

where \(v_{0}=\mathfrak {G}\left( u_{0},\alpha _{0}\right) \) and \(v=\mathfrak {G} (u,\alpha ) \).

3.2 The primary discretization

The primary discretization consists in the discretization of the space X into a finite dimensional space and of the operator \(\varPhi \), actually replaced with \(\varPhi _K\) by the secondary discretization, into an operator acting on this finite dimensional space.

Let K be a positive integer (level of discretization). Given a finite dimensional space \(\widehat{\mathbb {U}}_{K}\) and linear bounded operators \(\pi _{K}:\widehat{\mathbb {U}}_{K}\rightarrow \mathbb {U}\) and \(\rho _{K}:\mathbb {U}\rightarrow \widehat{\mathbb {U}}_{K}\), called prolongation to \(\mathbb {U}\) and restriction to \(\widehat{\mathbb {U}}_{K}\), respectively, we consider the finite-dimensional product space

$$\begin{aligned} \widehat{X}_K:=\widehat{\mathbb {U}}_K\times \mathbb {A}\times \mathbb {B} \end{aligned}$$

and the linear bounded operators \(P_K:\widehat{X}_K\rightarrow X\) and \( R_K:X\rightarrow \widehat{X}_K\) defined by

$$\begin{aligned} P_K\widehat{x}=(\pi _K\widehat{u},\alpha ,\beta ),\quad \widehat{x}=(\widehat{u} ,\alpha ,\beta )\in \widehat{X}_K, \end{aligned}$$

and

$$\begin{aligned} R_Kx=(\rho _K u,\alpha ,\beta ),\quad x=(u,\alpha ,\beta )\in X. \end{aligned}$$

Note that if the spaces \(\mathbb {A}\) and \(\mathbb {B}\) were not finite-dimensional, restrictions and prolongations also for these spaces had to be introduced.

The finite-dimensional space \(\widehat{X}_K\) is considered as the discretization of level K of X and the operator

$$\begin{aligned} \widehat{\varPhi }_K:=R_K\varPhi _KP_K:\widehat{X}_K\rightarrow \widehat{X}_K \end{aligned}$$

is considered as the discretization of level K of \(\varPhi \). We have

$$\begin{aligned} \widehat{\varPhi }_K \left( \widehat{x}\right)&=\left( \rho _K\mathfrak {F}_K\left( \mathfrak {G}\left( \pi _K \widehat{u},\alpha \right) ,\pi _K \widehat{u},\beta \right) ,(\alpha ,\beta )-\mathfrak {B}_K\left( \mathfrak {G}\left( \pi _K \widehat{u},\alpha \right) ,\pi _K\widehat{u},\beta \right) \right) \nonumber \\ \widehat{x}&=\left( \widehat{u},\alpha ,\beta \right) \in \widehat{X}. \end{aligned}$$
(17)

Given a fixed point \(\widehat{x}_K^*=\left( \widehat{u}_{K}^{*},\alpha _K^{*},\beta _K^{*}\right) \in \widehat{X}_K\) of \(\widehat{\varPhi }_K\) , which can be found by a standard numerical method for solving nonlinear systems of algebraic equations, we consider

$$\begin{aligned} P_{K}\widehat{x}^*_K=\left( \pi _{K}\widehat{u}_{K}^{*},\alpha _K^*,\beta _{K}^{*}\right) \in X \end{aligned}$$
(18)

as an approximation of a fixed point of \(\varPhi \) and

$$\begin{aligned} (v_K^*,\beta _K^*),\quad \text {where}\;v_{K}^*=\mathfrak {G}(\pi _K \widehat{u}_{K}^{*},\alpha _{K}^{*}), \end{aligned}$$
(19)

as an approximation of a solution of PAF.

The papers [41–44], for the particular instance of PAF given by a BVP (1), deal with two types of primary discretization falling in the previous abstract general description, namely the collocation method and the Fourier series method.

Remark 1

Note that, unlike the operators \(\mathfrak {F}\) and \(\mathfrak {B}\), we do not replace the linear operator \(\mathfrak {G}\) with an approximation \(\mathfrak {G}_K\), whose values can be exactly computed. The reason for this is that we assume, as it happens for the primary discretizations dealt in [41–44], the possibility to compute exactly \(\mathfrak {G}(u,\alpha )\) for any \(u\in \pi _K(\widehat{\mathbb {U}}_K)\) and \(\alpha \in \mathbb {A}\) (see (17)).

4 Convergence analysis

Let \(x^{*}=(u^{*},\alpha ^{*},\beta ^{*})\) be a fixed point of \(\varPhi \) and let \((v^*,\beta ^*)\), where \(v^*=\mathfrak {G} (u^*,\alpha ^*)\), be the relevant solution of PAF.

We set \(D^*\varPhi :=D\varPhi (x^*)\) and we make the following two assumptions regarding \(x^*\).

  • A \(\varvec{x^*}\) 1. There exist \(r_0>0\) and \(L\ge 0\) such that

    $$\begin{aligned} \Vert D\varPhi (x)-D^*\varPhi \Vert \le L\Vert x-x^*\Vert _X,\quad x\in \overline{B} (x^*,r_0). \end{aligned}$$

  • A \(\varvec{x^*}\) 2. The linear bounded operator \(I_X-D^*\varPhi \) is invertible, i.e. for any \((u_0,\alpha _0,\beta _0)\) \(\in X\) the linear problem

    $$\begin{aligned} \left\{ \begin{array}{l} u=D^*\mathfrak {F}(\mathfrak {G}(u,\alpha ),u,\beta )+u_0 \\ D^*\mathfrak {B}(\mathfrak {G}(u,\alpha ),u,\beta )=(\alpha _0,\beta _0), \end{array} \right. \end{aligned}$$

    where \(D^*\mathfrak {F}:=D\mathfrak {F}(v^*,u^*,\beta ^*)\) and \( D^*\mathfrak {B}:=D\mathfrak {B}(v^*,u^*,\beta ^*)\), has a unique solution \( (u,\alpha ,\beta )\in X\).

Observe that A \(\varvec{x^*}\) 2 says that \(x^*\) is a simple zero of \(I_X-\varPhi \) and implies that \(x^*\) is an isolated fixed point of \(\varPhi \).

In this section, we study how \(x^*\) and \((v^*,\beta ^*)\) can be approximated by the approximations (18) and (19), respectively, obtained by some fixed point \(\widehat{x}_K^*\) of \(\widehat{\varPhi }_K\).

Our analysis is based on studying of how \(x^{*}\) is approximated by fixed points of the operator

$$\begin{aligned} P_KR_K\varPhi _K:X\rightarrow X. \end{aligned}$$

Unlike \(\widehat{\varPhi }_K=R_K\varPhi _KP_K\), this operator has the advantage to be defined on the space X as \(\varPhi \).

Clearly, the operator \(P_KR_K\varPhi _K\) is Fréchet-differentiable at any point \(x_{0}\in X\) and its Fréchet-derivative at \(x_0\) is \( P_KR_KD\varPhi _K(x_0)\), where \(D\varPhi _K(x_0)\) is given in (16). We set \( D^*\varPhi _K:=D\varPhi _K(x^*)\).

For notational convenience, we also introduce the operator

$$\begin{aligned} \varPsi _K:=I_X-P_KR_K\varPhi _K, \end{aligned}$$

whose zeros are the fixed points of \(P_KR_K\varPhi _K\). Note that \(\varPsi _K\) is Fréchet-differentiable at any point \(x_0\in X\) and

$$\begin{aligned} D\varPsi _K(x_0)=I_X-P_KR_KD\varPhi _K(x_0). \end{aligned}$$

We set

$$\begin{aligned} D^*\varPsi _K:=D\varPsi _K(x^*)=I_X-P_KR_KD^*\varPhi _K. \end{aligned}$$
(20)

Since we consider the operator \(P_KR_K\varPhi _K\) as an approximation of the operator \(\varPhi \), it is expected that \(\varPsi _Kx^*\) has a small norm. We call \(\varPsi _Kx^*\) the consistency error.

Now, we introduce the following two stability conditions.

CS1 (Condition Stability 1) There exist \(r_1>0\) and, for any positive integer K, \( L_K\ge 0\) such that

$$\begin{aligned}&\Vert D\varPsi _K(x)-D^*\varPsi _K\Vert =\Vert P_KR_K\left( D\varPhi _K(x)-D^*\varPhi _K\right) \Vert \le L_K\Vert x-x^*\Vert _X \nonumber \\&x\in \overline{B}(x^*,r_1) \end{aligned}$$
(21)

(compare with A \(\varvec{x^*}\) 1).

CS2. There exists a positive integer \(K_{2}\) such that, for any positive integer \(K\ge K_{2}\), \(D^*\varPsi _{K}\) is invertible and

$$\begin{aligned} \lim _{K\rightarrow \infty } \frac{1}{r_2(K)} \cdot \Vert (D^*\varPsi _{K})^{-1}\Vert \cdot \Vert \varPsi _K x^*\Vert _{X}=0, \end{aligned}$$
(22)

where

$$\begin{aligned} r_{2}(K):=\min \left\{ r_{1},\frac{1}{2\Vert (D^*\varPsi _K)^{-1}\Vert \cdot L_K}\right\} \end{aligned}$$

with \(r_1\) and \(L_K\) given in CS1.

By using the Lemma 1 on the zeros of Fréchet-differentiable operators given in Appendix, we obtain the next theorem.

Theorem 1

Let CS1 and CS2 hold. Then, there exists a positive integer \(\overline{K}\) such that, for any positive integer \( K\ge \overline{K}\), \(P_KR_K\varPhi _{K}\) has a unique fixed point \(x_K^{*}\) in \(\overline{B}\left( x^{*},r_{2}(K)\right) \) and

$$\begin{aligned} \left\| x_K^{*}-x^{*}\right\| _{X}\le 2\Vert (D^*\varPsi _{K}) ^{-1}\Vert \cdot \Vert \varPsi _{K}x^{*}\Vert _{X} \end{aligned}$$
(23)

holds. Moreover, we have the expansion

$$\begin{aligned} x^{*}_K-x^{*}=-(D^*\varPsi _{K} )^{-1}\varPsi _{K}x^{*} +\delta _{K}, \end{aligned}$$
(24)

where

$$\begin{aligned} \Vert \delta _K\Vert _X\le 4L_K\cdot \Vert (D^*\varPsi _K)^{-1}\Vert ^3 \cdot \Vert \varPsi _{K}x^{*}\Vert _{X}^2. \end{aligned}$$
(25)

Here, \(L_K\) is defined in CS1 and \(r_2(K)\) is defined in CS2.

Proof

The proof is an application of the Lemma 1 with \(Y=X\), \( A=\varPsi _K \) and \(y^*=x^*\). Note that \(Y=X\) is a Banach space since \(\mathbb {U}\), \(\mathbb {A}\) and \(\mathbb {B}\) are Banach spaces. For \(K\ge K_2\), where \(K_2\) is defined in CS2, it is immediate to verify that

$$\begin{aligned} q(r_2(K))\le \frac{1}{2}, \end{aligned}$$

where the quantity q(r) is defined in Lemma 1. Now, let \( \overline{K}\ge K_2\) be such that, for \(K\ge \overline{K}\),

$$\begin{aligned} \frac{1}{r_2(K)} \cdot \Vert (D^*\varPsi _{K})^{-1}\Vert \cdot \Vert \varPsi _K x^*\Vert _{X}\le \frac{1}{2} \end{aligned}$$
(26)

(recall (22)). For \(K\ge \overline{K}\), we have

$$\begin{aligned} \Vert (D^*\varPsi _K)^{-1} \varPsi _K x^*\Vert _X \le \frac{1}{2}\cdot r_2(K)\le (1-q(r_2(K)))\cdot r_2(K) \end{aligned}$$

and so, since (71) in Lemma 1 is fulfilled for \( r=r_2(K) \), \(\varPsi _K\) has a unique zero \(x_K^{*}\) in \( \overline{B}\left( x^{*},r_{2}(K)\right) \) and (23) holds by (72).

As for the second part of the theorem, for \(K\ge \overline{K}\), take

$$\begin{aligned} r= 2\Vert (D^*\varPsi _{K}) ^{-1}\Vert \cdot \Vert \varPsi _{K}x^{*}\Vert _{X} \end{aligned}$$

in the second part of Lemma 1. Since \(r\le r_2(K)\) holds (recall (26)), we have

$$\begin{aligned} q(r)\le q(r_2(K))\le \frac{1}{2} \end{aligned}$$

and

$$\begin{aligned} \Vert (D^*\varPsi _K)^{-1}\varPsi _Kx^*\Vert _X \le \frac{1}{2}r\le (1-q(r))r \end{aligned}$$

and so the condition (71) is fulfilled. Then, we obtain (24) by (73) and

$$\begin{aligned} \Vert \delta _K\Vert _X\le 2q(r)\cdot \Vert (D^*\varPsi _{K}) ^{-1}\Vert \cdot \Vert \varPsi _{K}x^{*}\Vert _{X} \end{aligned}$$

by (74). Now, since

$$\begin{aligned} q(r)\le \Vert (D^*\varPsi _K)^{-1}\Vert \cdot L_K \cdot r=2L_K\cdot \Vert (D^*\varPsi _K)^{-1}\Vert ^2\cdot \Vert \varPsi _K x^*\Vert _X \end{aligned}$$

holds, we have (25). \(\square \)

Next result is a consequence of the Theorem 1 and says how \( x^*\) and \((v^*,\beta ^*)\) can be approximated by (18) and (19), respectively.

Theorem 2

Let CS1 and CS2 hold. Then, there exists a positive integer \(\widehat{K}\) such that, for any positive integer \(K\ge \widehat{K},\) the operator \(\widehat{\varPhi }_K\) has a fixed point \(\widehat{x}_K^*\) and

$$\begin{aligned} \left\| P_K\widehat{x}_K^{*}-x^{*}\right\| _{X}\le 2\Vert (D^*\varPsi _{K}) ^{-1}\Vert \cdot \Vert \varPsi _{K}x^{*}\Vert _{X} \end{aligned}$$
(27)

and

$$\begin{aligned} P_K\widehat{x}^{*}_K-x^{*}=-(D^*\varPsi _{K} )^{-1}\varPsi _{K}x^{*} +\delta _{K}, \end{aligned}$$
(28)

where \(\delta _K\) is defined in (24) and satisfies (25), hold. Moreover, if \(\widehat{x}_K\) is a fixed point of \(\widehat{\varPhi }_K\) different from \(\widehat{x}^*_K\), then

$$\begin{aligned} \Vert P_K\widehat{x}_K-x^*\Vert _X >r_2(K) \end{aligned}$$
(29)

and

$$\begin{aligned} \Vert \widehat{x}_K-\widehat{x}_K^*\Vert _{\widehat{X}_K}>\frac{r_2(K)}{ 2\max \{\Vert \pi _K \Vert ,1\}}. \end{aligned}$$
(30)

Here, \(r_2(K)\) is defined in CS2. Finally, regarding the approximation \((v_K^*,\beta _K^*)\) of \((v^*,\beta ^*)\), we have

$$\begin{aligned} \Vert (v_K^*,\beta _K^*)-(v^*,\beta ^*)\Vert _{\mathbb {V}\times \mathbb {B}} \le 2\max \left\{ \left\| \mathfrak {G}\right\| ,1\right\} \cdot \Vert (D^*\varPsi _{K}) ^{-1}\Vert \cdot \Vert \varPsi _{K}x^{*}\Vert _{X}. \end{aligned}$$
(31)

Proof

By recalling Theorem 1, for \(K\ge \overline{K}\), let \(x_K^*\) be the unique fixed point of \(P_KR_K\varPhi _K\) in \(\overline{B}(x^*,r_2(K))\) . It is immediate to verify that \(\widehat{x}_K^*=R_K\varPhi _Kx_K^{*}\) is a fixed point of \(\widehat{\varPhi }_K\). Moreover, we have

$$\begin{aligned} P_K\widehat{x}_K^*=P_KR_K\varPhi _Kx_K^*=x^{*}_K. \end{aligned}$$

Therefore, (27) and (28) follow by (23) and (24) in Theorem 1, respectively.

Now, we prove the second part. Let \(\widehat{x}_K\) be a fixed point of \( \widehat{\varPhi }_K\) different from \(\widehat{x}^*_K\). It is immediate to verify that \(P_K\widehat{x}_K\) is a fixed point of \(P_KR_K\varPhi _K\). Since \(x^*_K\) is the unique fixed point of \(P_KR_K\varPhi _K\) in \(\overline{B} \left( x^*,r_2(K)\right) \), we have (29). As for the inequality (30), observe that

$$\begin{aligned} \Vert P_K\widehat{x}_K-P_K\widehat{x}_K^*\Vert _X \ge \Vert P_K\widehat{x} _K-x^*\Vert _X - \Vert x^*_K - x^*\Vert _X > r_2(K)-\Vert x^*_K - x^*\Vert _X. \end{aligned}$$

Since (22) and (23) hold, we have

$$\begin{aligned} \underset{K\rightarrow \infty }{\lim }\frac{1}{r_2(K)} \cdot \Vert x^*_K-x^*\Vert _X=0. \end{aligned}$$

Hence, there exists \(\widehat{K}\ge \overline{K}\) such that, for \(K \ge \widehat{K}\), we have

$$\begin{aligned} \Vert x_K^*-x^*\Vert _X\le \frac{r_2(K)}{2} \end{aligned}$$

and then

$$\begin{aligned} \Vert P_K\widehat{x}_K-P_K\widehat{x}_K^*\Vert _X>\frac{r_2(K)}{2}. \end{aligned}$$
(32)

Now, (30) follows by (32),

$$\begin{aligned} \Vert P_K\widehat{x}_K-P_K\widehat{x}_K^*\Vert _X \le \Vert P_K\Vert \cdot \Vert \widehat{x}-\widehat{x}_K^*\Vert _{\widehat{X}_K} \end{aligned}$$

and

$$\begin{aligned} \Vert P_K \Vert =\max \{\Vert \pi _K\Vert ,1\}. \end{aligned}$$

Finally, the estimate (31) is obtained by

$$\begin{aligned}&\left\| \left( v_{K}^*,\beta _{K}^*\right) -\left( v^*,\beta ^*\right) \right\| _{\mathbb {V}\times \mathbb {B}}\\&\quad =\left\| \mathfrak {G}\left( \pi _K\widehat{u}_{K}^{*},\alpha _{K}^{*}\right) -\mathfrak {G}\left( u^{*},\alpha ^{*}\right) \right\| _{\mathbb {V}} +\left\| \beta _{K}^{*}-\beta ^{*}\right\| _{\mathbb {B}} \\&\quad \le \left\| \mathfrak {G}\right\| \left( \left\| \pi _K \widehat{u} _{K}^{*}-u^{*}\right\| _{\mathbb {U}}+\left\| \alpha _{K}^{*}-\alpha ^{*}\right\| _{\mathbb {A}} \right) +\left\| \beta _{K}^{*}-\beta ^{*}\right\| _{\mathbb {B}} \\&\quad \le \max \left\{ \left\| \mathfrak {G}\right\| ,1\right\} \cdot \left\| P_K\widehat{x}_{K}^{*}-x^*\right\| _{X}. \end{aligned}$$

\(\square \)

In the next subsection, we will give an estimate of the error of the approximation \(( v^*_K,\beta ^*_K)\), better than (31) in some situations.

Remark 2

Regarding the consistency error \(\varPsi _K x^*\), which appears in (27) and (31), we have

$$\begin{aligned} \Vert \varPsi _K x^*\Vert _X \le \Vert (P_KR_K-I_X)x^*\Vert _X +\Vert P_KR_K(\varPhi _K-\varPhi )x^*\Vert _X, \end{aligned}$$

where we have separated the contributions of the primary and secondary discretizations. If only a primary discretization is used, i.e. \(\varPhi _K=\varPhi \), then

$$\begin{aligned} -\varPsi _K x^*=(P_KR_K-I_X)x^*. \end{aligned}$$

Remark 3

Suppose there exists a sequence \(\{\widehat{x}_K\}\) of fixed points of \( \widehat{\varPhi }_K\) such that \(\widehat{x}_K\) is eventually different from \(\widehat{x}_K^*\). By (29) and (30), we obtain

$$\begin{aligned} \frac{1}{\Vert P_K\widehat{x}_K-x^*\Vert _X}=O\left( \frac{1}{r_2(K)} \right) ,\quad K\rightarrow \infty , \end{aligned}$$
(33)

and

$$\begin{aligned} \frac{1}{\Vert \widehat{x}_K-\widehat{x}_K^*\Vert _{\widehat{X}_K}} =O\left( \max \{\Vert \pi _K \Vert ,1\}\cdot \frac{1}{r_2(K)}\right) ,\quad K\rightarrow \infty , \end{aligned}$$
(34)

respectively. Note that, by (33), (27) and (22), we have

$$\begin{aligned} \Vert P_K\widehat{x}_K^*-x^*\Vert _X =o\left( \Vert P_K\widehat{x} _K-x^*\Vert _X\right) ,\ K\rightarrow \infty . \end{aligned}$$
(35)

The estimates (33)–(35) give informations on how much the fixed point \(\widehat{x}^*_K\) is isolated from other fixed points of \(\widehat{\varPhi }_K\).

4.1 The simple case

Let us introduce the space

$$\begin{aligned} Z:=\mathbb {V}\times \mathbb {A}\times \mathbb {B} \end{aligned}$$

and the linear operator \(\varLambda :X\rightarrow Z\) given by

$$\begin{aligned}&\varLambda x =\left( \mathfrak {G}\left( u,0\right) ,\alpha ,\beta \right) ,\quad x=(u,\alpha ,\beta ) \in X. \end{aligned}$$

Clearly, the linear operator \(\varLambda \) is bounded and

$$\begin{aligned} \left\| \varLambda \right\| = \max \left\{ \left\| \mathfrak {G}\left( \cdot ,0\right) \right\| ,1\right\} \end{aligned}$$

holds.

In this subsection, we consider the situation where, for any \( x\in X\), we can factorize \(D\varPhi (x) \) as

$$\begin{aligned} D\varPhi (x)=\varSigma (x)\varLambda , \end{aligned}$$
(36)

and, for any positive integer K, \(D\varPhi _K(x)\) as

$$\begin{aligned} D\varPhi _K(x)=\varSigma _K(x)\varLambda , \end{aligned}$$
(37)

where \(\varSigma (x),\varSigma _K(x):Z\rightarrow X\) are linear bounded operators. We call this situation the simple case.

In the following, we set \(\varSigma ^*:=\varSigma (x^*)\) and \(\varSigma _K^*:=\varSigma _K(x^*)\).

Note that the simple case holds if \(\mathfrak {F}(v,u,\beta )=\mathfrak {F}(v,\beta )\), \(\mathfrak {F}_K(v,u,\beta )=\mathfrak {F}_K(v,\beta )\), \(\mathfrak {B}(v,u,\beta )=\mathfrak {B}(v,\beta )\) and \(\mathfrak {B}_K(v,u,\beta )=\mathfrak {B}_K(v,\beta )\). In fact, for \(x_0=(u_0,\alpha _0,\beta _0)\in X\), factorizations (36) and (37) hold with \(\varSigma (x_0),\varSigma _K(x_0):Z\rightarrow X\) given by

$$\begin{aligned} \varSigma (x_0) z&=(D^0\mathfrak {F}(v+\mathfrak {G}(0,\alpha ),\beta ),(\alpha , \beta )-D^0 \mathfrak {B}( v+\mathfrak {G}(0,\alpha ),\beta ))\\ \varSigma _K(x_0) z&=(D^0\mathfrak {F}_K(v+\mathfrak {G}(0,\alpha ),\beta ) ,(\alpha ,\beta )-D^0 \mathfrak {B}_K( v+\mathfrak {G}(0,\alpha ),\beta ))\\ z&=(v,\alpha ,\beta )\in Z \end{aligned}$$

where \(D^0\mathfrak {F}:=D\mathfrak {F}(v_0,\beta _0)\), \(D^0\mathfrak {F}_K:=D \mathfrak {F}_K(v_0,\beta _0)\), \(D^0 \mathfrak {B}:=D\mathfrak {B}(v_0,\beta _0)\) and \(D^0 \mathfrak {B}_K\) \(:=D\mathfrak {B}_K(v_0,\beta _0)\), with \(v_0=\mathfrak {G}(u_0,\alpha _0)\).

Therefore, the simple case holds for the particular instance of PAF given by a non-neutral BVP (1). As it is shown in [43], the simple case can hold also in case of BVP for neutral functional differential equations. In particular, it holds in case of BVPs for neutral integro-differential equation (4) and non-neutral boundary conditions.

Now, we present two theorems for the simple case. The first result is a condition under which the invertibility of the linear bounded operator \(D^*\varPsi _K\), and the uniform boundedness with respect to K of the norm of its inverse, are guaranteed. We recall that \(D^*\varPsi _K\) is defined in (20) and the norm of its inverse appears in CS2 and in the error estimates of Theorem 2.

Theorem 3

Assume the simple case. If

$$\begin{aligned} \lim _{K\rightarrow \infty }\Vert (P_KR_K\varSigma _{K}^*-\varSigma ^*)\varLambda \Vert =0, \end{aligned}$$
(38)

then there exists a positive integer \(K_{2}\) such that, for any positive integer \(K\ge K_{2}\), \(D^*\varPsi _{K}\) is invertible and

$$\begin{aligned} \Vert (D^*\varPsi _{K})^{-1}\Vert \le 2\Vert (I_X-D^*\varPhi )^{-1}\Vert . \end{aligned}$$

Note that the previous theorem implicitly requires the invertibility of \(I_X-D^*\varPhi \), which is assumed in A \(\varvec{x^*}\) 2.

Proof

By recalling (20), we have

$$\begin{aligned} D^*\varPsi _K=I_X-D^*\varPhi -(P_KR_K\varSigma _K^*-\varSigma ^*)\varLambda . \end{aligned}$$

The theorem now follows by an application of the Banach perturbation lemma. \(\square \)

Note that in (38) we can take advantage of the fact that the error operator \( P_KR_K\varSigma _{K}^*-\varSigma ^*\) is applied to elements that have been regularized by the operator \(\varLambda \).

Remark 4

By separating the contributions of the primary and secondary discretizations, we have that (38) holds if

$$\begin{aligned} \lim _{K\rightarrow \infty }\Vert (P_KR_K-I_X)\varSigma ^*\varLambda \Vert =0 \end{aligned}$$

and

$$\begin{aligned} \lim _{K\rightarrow \infty } \Vert P_KR_K(\varSigma _{K}^*-\varSigma ^*)\varLambda \Vert =0. \end{aligned}$$

The second result for the simple case is an estimate of the error of \(\left( v^{*}_K,\beta ^{*}_K\right) \) different from (31).

Theorem 4

Let CS1 and CS2 hold. Assume the simple case. We have

$$\begin{aligned}&\Vert \left( v_{K}^{*},\beta _{K}^{*}\right) -\left( v^{*},\beta ^{*}\right) \Vert _{\mathbb {V}\times \mathbb {B}} \nonumber \\&\quad \le \max \{\Vert \mathfrak {G }(0,\cdot )\Vert ,1\} \cdot \left( \Vert \varXi _K\Vert \cdot \Vert \varLambda \varPsi _{K}x^{*}\Vert _{Z} +\Vert \varLambda \Vert \cdot \Vert \delta _{K}\Vert _{X}\right) , \end{aligned}$$
(39)

where

$$\begin{aligned} \varXi _K:=I_Z+\varLambda (D^{*}\varPsi _{K})^{-1}P_KR_K\varSigma _K^*\end{aligned}$$
(40)

and \(\Vert \delta _K\Vert _X\) satisfies (25).

Proof

Of course, all that is stated in Theorem 2 holds. Now, consider the expansion (28):

$$\begin{aligned} P_K\widehat{x}_K^*-x^*=-w_k+\delta _K, \end{aligned}$$

where we set

$$\begin{aligned} w_K :=(D^{*}\varPsi _{K})^{-1}\varPsi _{K}x^{*}. \end{aligned}$$

We have

$$\begin{aligned}&\left\| \left( v_{K}^{*},\beta _{K}^{*}\right) -\left( v^{*},\beta ^{*}\right) \right\| _{\mathbb {V}\times \mathbb {B}} \nonumber \\&\quad =\left\| \mathfrak {G}\left( \pi _{K}\widehat{u}_{K}^{*},\alpha _{K}^{*}\right) -\mathfrak {G}\left( u^{*},\alpha ^{*}\right) \right\| _{\mathbb {V}} +\left\| \beta _{K}^{*}-\beta ^{*}\right\| _{\mathbb {B}} \nonumber \\&\quad =\left\| \mathfrak {G}\left( \pi _{K}\widehat{u}_{K}^{*}-u^{*},0\right) +\mathfrak {G}\left( 0,\alpha _{K}^{*}-\alpha ^{*}\right) \right\| _{\mathbb {V}}+\left\| \beta _{K}^{*}-\beta ^{*}\right\| _{\mathbb {B}} \nonumber \\&\quad \le \max \{\Vert \mathfrak {G}(0,\cdot )\Vert ,1\}\cdot \left( \left\| \mathfrak {G}\left( \pi _{K}\widehat{u}_{K}^{*}-u^{*},0\right) \right\| _{\mathbb {V}}+\left\| \alpha _{K}^{*}-\alpha ^{*}\right\| _{\mathbb {A}}+\left\| \beta _{K}^{*}-\beta ^{*}\right\| _{ \mathbb {B}}\right) \nonumber \\&\quad =\max \{\Vert \mathfrak {G}(0,\cdot )\Vert ,1\}\cdot \left\| \varLambda \left( \pi _{K}\widehat{u}_{K}^{*}-u^{*},\alpha _{K}^{*}-\alpha ^{*},\beta _{K}^{*}-\beta ^{*}\right) \right\| _{Z} \nonumber \\&\quad =\max \{\Vert \mathfrak {G}(0,\cdot )\Vert ,1\}\cdot \Vert \varLambda (-w_K+\delta _K) \Vert _Z \nonumber \\&\quad \le \max \{\Vert \mathfrak {G}(0,\cdot )\Vert ,1\}\cdot (\left\| \varLambda w_K \right\| _{Z}+\left\| \varLambda \right\| \left\| \delta _{K}\right\| _{X}). \end{aligned}$$
(41)

From (20), we obtain

$$\begin{aligned} \varLambda (D^{*}\varPsi _{K})^{-1}&= \varLambda (D^{*}\varPsi _{K})^{-1}(D^{*}\varPsi _{K}+P_KR_KD^{*}\varPhi _{K}) \\&=\varLambda +\varLambda (D^{*}\varPsi _{K})^{-1}P_KR_KD^{*}\varPhi _{K} \end{aligned}$$

and then

$$\begin{aligned} \varLambda w_K&= \varLambda (D^{*}\varPsi _{K})^{-1}\varPsi _{K}x^{*} \\&= \varLambda \varPsi _{K}x^{*}+\varLambda (D^{*}\varPsi _{K})^{-1}P_KR_K D^*\varPhi _K\varPsi _{K}x^{*}. \end{aligned}$$

Now, since \(D^*\varPhi _K=\varSigma _K^*\varLambda \) holds, we have

$$\begin{aligned} \varLambda w_K=\varXi _K\varLambda \varPsi _{K}x^{*}, \end{aligned}$$

with \(\varXi _K\) defined in (40), and the estimate (39) follows by (41). \(\square \)

This result is indeed useful since it can happen, as it is illustrated in [41] in case of the collocation method (version finite element method), that \(\Vert \varLambda \varPsi _{K}x^{*}\Vert _{Z}\) has an order of convergence to zero, as \(K\rightarrow \infty \), higher than \(\Vert \varPsi _{K}x^{*}\Vert _{X}\).

Remark 5

Regarding the regularized consistency error \(\varLambda \varPsi _Kx^*\), we have

$$\begin{aligned} \Vert \varLambda \varPsi _K x^*\Vert _Z \le \Vert \varLambda (P_KR_K-I_X)x^*\Vert _Z+\Vert \varLambda P_KR_K (\varPhi _K-\varPhi )x^*\Vert _Z, \end{aligned}$$

where we have separated the contributions of the primary and secondary discretizations. If only a primary discretization is used, i.e. \(\varPhi _K=\varPhi \), then

$$\begin{aligned} -\varLambda \varPsi _K x^*=\varLambda (P_KR_K-I_X)x^*. \end{aligned}$$

4.2 Invertibility of \(D^*\varPhi _K\)

In the previous subsection, in Theorem 3, it has been presented a condition under which the invertibility of \(D^*\varPhi \) is guaranteed in the simple case. In this subsection, we study the invertibility of \(D^*\varPhi \) in the general case.

We consider a splitting

$$\begin{aligned} D^*\varPhi =\varGamma ^*+\varSigma ^*\varLambda \end{aligned}$$
(42)

of \(D^*\varPhi \), where \(\varGamma ^*:X\rightarrow X\) and \(\varSigma ^*:Z \rightarrow X\) are linear bounded operators. (Recall that Z and \(\varLambda \) have been introduced at the beginning of the previous subsection). Similarly, for any positive integer K, we consider a splitting

$$\begin{aligned} D^*\varPhi _K=\varGamma _K^*+\varSigma _K^*\varLambda , \end{aligned}$$
(43)

of \(D^*\varPhi _K\), where \(\varGamma _K^*:X\rightarrow X\) and \( \varSigma _K^*:Z\rightarrow X\) are linear bounded operators.

Note that in the simple case, described in the previous subsection, we have splittings (42) and (43) with \(\varGamma ^*=\varGamma _K^*=0\).

In this subsection, by using splittings (42) and (43), we give a theorem concerning the invertibility of \(D^*\varPsi _K\) and the norm of its inverse. This theorem is an extension of the Theorem 3 (which is valid only for the simple case) and it is based on the Lemma 2 in Appendix.

Theorem 5

Assume that there exist a splitting (42) such that \(I_X-\varGamma ^*\) is invertible and, for any positive integer K , a splitting (43) such that \(I_X-P_KR_K\varGamma _K^*\) is invertible.

If

$$\begin{aligned} \lim _{K\rightarrow \infty }\Vert (I_X-P_KR_K\varGamma _{K}^*)^{-1}\Vert \cdot \Vert (P_KR_K \varGamma _{K}^*-\varGamma ^*)(I_X-\varGamma ^*)^{-1}\varSigma ^*\varLambda \Vert =0 \end{aligned}$$
(44)

and

$$\begin{aligned} \lim _{K\rightarrow \infty }\Vert (I_X-P_KR_K\varGamma _{K}^*)^{-1}\Vert \cdot \Vert (P_KR_K\varSigma _{K}^*-\varSigma ^*)\varLambda \Vert =0, \end{aligned}$$
(45)

then there exists a positive integer \(K_{2}\) such that, for any positive integer \(K\ge K_{2}\), \(D^*\varPsi _{K}\) is invertible and

$$\begin{aligned} \Vert (D^*\varPsi _{K})^{-1}\Vert \le 2\Vert (I_X-D^*\varPhi )^{-1}(I_X-\varGamma ^*)\Vert \cdot \Vert (I_X-P_KR_K\varGamma _K^*)^{-1}\Vert . \end{aligned}$$
(46)

Proof

The proof is an application of the Lemma 2 with \(Y=X\), \(A=I_X-D^*\varPhi \) (recall A \(\varvec{x^*}\) 2), \(B=I_X-\varGamma ^*\), \(C=-\varSigma ^*\varLambda \) and, for any positive integer K, \(A_K=D^*\varPsi _K=I_X-P_KR_KD^*\varPhi _K\), \(B_K=I_X-P_KR_K\varGamma _K^*\) and \( C_K=-P_KR_K\varSigma _K^*\varLambda \). \(\square \)

The previous theorem reduces the invertibility of \(D^*\varPsi _{K}=I_X-P_KR_KD^*\varPhi _K\) to the invertibility of \(I_X-P_KR_K\varGamma _K^*\). Note that in (44) and (45), we can take advantage of the fact that the error operators \(P_KR_K\varGamma _{K}^*-\varGamma ^*\) and \( P_KR_K\varSigma _{K}^*-\varSigma ^*\) are applied to elements that have been regularized by \(\varLambda \).

Remark 6

By separating the contributions of the primary and secondary discretizations, we have that (44) holds if

$$\begin{aligned} \lim _{K\rightarrow \infty }\Vert (I_X-P_KR_K\varGamma _{K}^*)^{-1}\Vert \cdot \Vert (P_KR_K-I_X)\varGamma ^*(I_X-\varGamma ^*)^{-1}\varSigma ^*\varLambda \Vert =0 \end{aligned}$$

and

$$\begin{aligned} \lim _{K\rightarrow \infty }\Vert (I_X-P_KR_K\varGamma _{K}^*)^{-1}\Vert \cdot \Vert P_KR_K(\varGamma _K^*-\varGamma ^*)(I_X-\varGamma ^*)^{-1}\varSigma ^*\varLambda \Vert =0, \end{aligned}$$

and (45) holds if

$$\begin{aligned} \lim _{K\rightarrow \infty }\Vert (I_X-P_KR_K\varGamma _{K}^*)^{-1}\Vert \cdot \Vert (P_KR_K-I_X)\varSigma ^*\varLambda \Vert =0 \end{aligned}$$

and

$$\begin{aligned} \lim _{K\rightarrow \infty }\Vert (I_X-P_KR_K\varGamma _{K}^*)^{-1}\Vert \cdot \Vert P_KR_K (\varSigma _{K}^*-\varSigma ^*)\varLambda \Vert =0. \end{aligned}$$

4.3 The nilpotency case and the splitting case

In view of Theorem 5, it remains to study the invertibility of \(I_X-P_KR_K\varGamma _K^*\). To this aim, we consider the situation where there exist a positive integer c, a splitting (42) such that \((\varGamma ^*)^c=0\) and, for any positive integer K, a splitting (43) such that \((P_KR_K\varGamma _K^ *)^c=0\). We call this situation the nilpotency case.

Remark 7

If the nilpotency case holds, then \(I_X-\varGamma ^*\) is invertible and, for any positive integer K, \(I_X-P_KR_K\varGamma _K^*\) is invertible and

$$\begin{aligned} \Vert (I_X-P_KR_K\varGamma _K^*)^{-1}\Vert \le \sum _{i=0}^{c-1} \Vert P_KR_K\Vert ^i\cdot \Vert \varGamma _K^*\Vert ^i. \end{aligned}$$

holds.

On the other hand, we call the splitting case the more general situation (including the nilpotency case) described in the premise in Theorem 5, namely there exist a splittings (42) such that \(I_X-\varGamma ^*\) is invertible and, for any positive integer K , a splitting (43) such that \(I_X-P_KR_K\varGamma _K^*\) is invertible.

4.4 The operators \(P_KR_K\) and \(P_KR_K-I_X\)

Regarding the primary discretization, the previous subsections have shown that the role played by the linear operators \(P_{K}R_{K}:X\rightarrow X\) and \(P_{K}R_{K}-I_{X}:X\rightarrow X\) given by

$$\begin{aligned} P_{K}R_{K}x=(\pi _{K}\rho _{K}u,\alpha ,\beta ),\quad x=(u,\alpha ,\beta )\in X, \end{aligned}$$

and

$$\begin{aligned} (P_{K}R_{K}-I_{X})x=((\pi _{K}\rho _{K}-I_{\mathbb {U}})u,0,0),\quad x=(u,\alpha ,\beta )\in X, \end{aligned}$$
(47)

is crucial. In this subsection, we list some simple facts about them to be used in the next section.

We have

$$\begin{aligned} \Vert (P_K R_K -I_X)x \Vert _X=\Vert (\pi _K\rho _K-I_{\mathbb {U}})u\Vert _\mathbb {U},\quad x=(u,\alpha ,\beta )\in X, \end{aligned}$$

and, for a linear bounded operator \(A:X\rightarrow X\),

$$\begin{aligned} \Vert (P_K R_K -I_X)A \Vert =\Vert (\pi _K \rho _K-I_\mathbb {U})A_\mathbb {U}\Vert , \end{aligned}$$
(48)

where \(A_\mathbb {U}\) is the \(\mathbb {U}\)-component of A defined in Sect. 1.2.

Moreover, note that

$$\begin{aligned} \lambda _K:=\Vert P_K R_K \Vert =\max \{\Vert \pi _K \rho _K \Vert , 1\} \end{aligned}$$
(49)

and

$$\begin{aligned} \Vert (P_KR_K-I_X)x^*\Vert _X=\Vert e_K^*\Vert _\mathbb {U}, \end{aligned}$$
(50)

where \(x^*=(u^*,\alpha ^*,\beta ^*)\) is the fixed point of \(\varPhi \) and

$$\begin{aligned} e_K^*:=(\pi _K\rho _K-I_\mathbb {U})u^*\end{aligned}$$
(51)

can be called the consistency error of the primary discretization (see Remark 2).

5 Specialization of the convergence results

In the convergence analysis presented above, we have considered the general situation where, beside a primary discretization, also a secondary discretization is introduced. This means that approximations \(\mathfrak {F}_K\) of \(\mathfrak {F}\) and \(\mathfrak {B}_K\) of \(\mathfrak {B}\) are used and then the operator \(\varPhi \) is actually replaced by \(\varPhi _K\). However, in the papers [41, 44], where the results of this paper are specialized to the problem (1) for two particular primary discretizations, we do not consider a secondary discretization, in order to avoid giving results with too many assumptions and details.

As previously remarked, in case of the problem (1), approximations \(\mathfrak {F}_K=\mathcal {F}_K\) of \(\mathfrak {F}=\mathcal {F}\) and \(\mathfrak {B}_K=B_K\) of \(\mathfrak {B}=B\) are used for integro-differential equation (4) and integral boundary conditions (10), respectively, where the involved integrals are approximated by quadrature rules. Convergence results, when quadrature rules are used in integro-differential equations BVPs, can be deduced from the general theory given above and they are addressed in [43]. We also remark that the choice of considering the exact computation of integrals in integro-differential equations BVPs is adopted in the papers [21, 22, 25, 27].

The case of BVPs (1) for differential equations with deviating arguments (3) and multipoint boundary conditions (9), where a secondary discretization is not necessary, is dealt in [42].

In this section, for the situation where only a primary discretization is used (i.e., for any positive integer K, we have \(\mathfrak {F}_K=\mathfrak {F} \) and \(\mathfrak {B}_K=\mathfrak {B}\) and then \(\varPhi _K=\varPhi \)), we give two convergence theorems for the problem PAF, less abstract than Theorem 2. The first is for the simple case and the second is for the splitting case, which includes the nilpotency case. Such theorems are used in [41, 44], in case of the problem (1), for the two particular primary discretizations given by the collocation method and the Fourier series method.

As already remarked, the simple case holds for non-neutral BVPs (1) and for BVPs given by neutral integro-differential equation (4) and non-neutral boundary conditions. Moreover, for the collocation method and the Fourier series method, the nilpotency case holds for BVPs given by neutral differential equations with deviating arguments (3) and non-neutral boundary conditions, whenever the neutral deviating arguments \(\vartheta _s\), \(s=1,\ldots ,l\), are such that

$$\begin{aligned} \vartheta _s (t) \le t-\tau ,\quad s=1,\ldots ,l\quad \text {and}\quad t\in [a,b], \end{aligned}$$

or

$$\begin{aligned} \vartheta _s (t) \ge t+\tau ,\quad s=1,\ldots ,l\quad \text {and}\quad t\in [a,b], \end{aligned}$$

for some \(\tau >0\). This is shown in [42, 44].

Below, since we are considering only a primary discretization, we have that:

  • the simple case reduces to the sole factorization (36);

  • the splitting case uses the sole splitting (42) and requires the invertibility of \(I_X-\varGamma ^*\) and, for any positive integer K, of \(I_X-P_KR_K\varGamma ^*\);

  • the nilpotency case uses the sole splitting (42) and requires \(\left( \varGamma ^*\right) ^c=0\) and, for any positive integer K, \((P_KR_K\varGamma ^*)^c=0\), for some positive integer c.

Moreover, we use diffusely the notation \(A_\mathbb {U}\) of the \(\mathbb {U}\)-component of an operator A introduced in Sect. 1.2. Finally, we remark that the quantities \(\lambda _K\) and \(\Vert e_K^*\Vert _\mathbb {U}\) (see (49)–(51)) play a crucial role. In particular, we have

$$\begin{aligned} \Vert \varPsi _K x^*\Vert _X =\Vert (P_KR_K-I_X)x^*\Vert _X=\Vert e_K^*\Vert _\mathbb {U} \end{aligned}$$
(52)

(see Remark 2).

5.1 The simple case

Here is the theorem for the simple case. Before to present it, we introduce the following condition, which is formulated only for the simple case.

  • CSC (Condition Simple Case) There exist \(r_{2}>0\) and, for any positive integer K, \(\sigma _K\ge 0\) such that

    $$\begin{aligned} \Vert (\pi _K\rho _K-I_\mathbb {U})(D\varPhi (x)-D^*\varPhi )_{\mathbb {U}}\Vert&=\Vert (\pi _K\rho _K-I_\mathbb {U})((\varSigma (x)-\varSigma ^*)\varLambda )_\mathbb {U}\Vert \\&\le \sigma _K\Vert x-x^*\Vert _X,\quad x\in \overline{B}(x^*,r_2), \end{aligned}$$

    and

    $$\begin{aligned} \sigma _K=O(1),\quad K\rightarrow \infty . \end{aligned}$$

Theorem 6

Assume that only a primary discretization is used. Moreover, assume the simple case,

$$\begin{aligned} \lim _{K\rightarrow \infty }\Vert (\pi _K\rho _K-I_\mathbb {U})(D^*\varPhi )_\mathbb {U}\Vert =\lim _{K\rightarrow \infty }\Vert (\pi _K\rho _K-I_\mathbb {U})(\varSigma ^*\varLambda )_\mathbb {U}\Vert =0 \end{aligned}$$
(53)

and

$$\begin{aligned} \underset{K\rightarrow \infty }{\lim }\left\{ \begin{array}{ll} \lambda _K\ \cdot \Vert e_K^*\Vert _\mathbb {U}&{} \\ \Vert e_K^*\Vert _\mathbb {U}&{}\quad \text {if}\;\mathbf {CSC}\,\text {holds} \end{array} \right. =0. \end{aligned}$$
(54)

(One has to read the lower row after \(\{,\) instead of the upper one, if CSC holds). Then, there exists a positive integer \(\widehat{K}\) such that, for any positive integer \(K\ge \widehat{K}\), \(\widehat{\varPhi }_K\) has a fixed point \(\widehat{x}_K^*\) such that

$$\begin{aligned} \left\| P_K\widehat{x}_K^{*}-x^{*}\right\| _{X}=O(\Vert e_K^*\Vert _\mathbb {U}),\quad K\rightarrow \infty . \end{aligned}$$
(55)

Moreover, for the approximation \((v_K^*,\beta _K^*)\) of \((v^*,\beta ^*)\), we have the two estimates

$$\begin{aligned} \Vert (v_K^*,\beta _K^*)-(v^*,\beta ^*)\Vert _{\mathbb {V}\times \mathbb {B}}=O(\Vert e_K^*\Vert _\mathbb {U}),\quad K\rightarrow \infty , \end{aligned}$$
(56)

and

$$\begin{aligned}&\Vert (v_K^*,\beta _K^*)-(v^*,\beta ^*)\Vert _{\mathbb {V}\times \mathbb {B}} \nonumber \\&\quad =O\left( \lambda _K\cdot \Vert \mathfrak {G}(e_K^*,0)\Vert _\mathbb {V}\right) +\left\{ \begin{array}{ll} O(\lambda _K\cdot \Vert e_K^*\Vert _\mathbb {U}^2)&{}\\ O(\Vert e_K^*\Vert _\mathbb {U}^2)&{}\quad \text {if}\;\mathbf {CSC}\,\text {holds} \end{array}\right. ,\quad K\rightarrow \infty . \end{aligned}$$
(57)

Finally, suppose there exists a sequence \(\{\widehat{x}_K\}\) of fixed points of \( \widehat{\varPhi }_K\) such that \(\widehat{x}_K\) is eventually different from \(\widehat{x}^*_K\). Then

$$\begin{aligned} \frac{1}{\Vert P_K\widehat{x}_K-x^*\Vert _X}=\left\{ \begin{array}{ll} O(\lambda _K)&{} \\ O(1)&{}\quad \text {if}\;\mathbf {CSC}\,\text {holds} \end{array} \right. ,\quad K\rightarrow \infty , \end{aligned}$$
(58)

and

$$\begin{aligned} \frac{1}{\Vert \widehat{x}_K-\widehat{x}_K^*\Vert _{\widehat{X}_K}} =\left\{ \begin{array}{ll} O(\max \{\Vert \pi _K \Vert ,1\}\cdot \lambda _K)&{} \\ O(\max \{\Vert \pi _K \Vert ,1\})&{}\quad \text {if}\;\mathbf {CSC}\,\text {holds} \end{array} \right. ,\quad K\rightarrow \infty . \end{aligned}$$
(59)

Note that, in CSC and (53), the error operator \(\pi _K\rho _K-I_\mathbb {U}\) is applied to elements regularized by means of \(\varLambda \).

Proof

Since A \(\varvec{x^*}\) 1 holds, CS1 is fulfilled with a constant

$$\begin{aligned} L_K=\left\{ \begin{array}{ll} O(\lambda _K)&{} \\ O(1)&{}\quad \text {if}\;\mathbf {CSC}\,\text {holds} \end{array} \right. ,\quad K\rightarrow \infty . \end{aligned}$$

Now, we show that CS2 holds. By (53), (48), Remark 4 and Theorem 3, we obtain that there exists a positive integer \(K_{2}\) such that, for any positive integer \(K\ge K_{2}\), \( D^*\varPsi _{K}\) is invertible and

$$\begin{aligned} \Vert (D^*\varPsi _{K})^{-1}\Vert =O\left( 1\right) ,\quad K\rightarrow \infty . \end{aligned}$$
(60)

Now, we have

$$\begin{aligned} \frac{1}{r_2(K)}=\left\{ \begin{array}{ll} O(\lambda _K) \\ O(1)&{}\quad \text {if}\,\mathbf {CSC}\,\text { holds} \end{array} \right. ,\quad K\rightarrow \infty . \end{aligned}$$
(61)

By (61), (60), (52) and (54), we conclude that CS2 is fulfilled.

Then, Theorem 2 says that there exists a positive integer \( \widehat{K}\) such that, for any positive integer \(K\ge \widehat{K}\), \( \widehat{\varPhi }_K\) has a fixed point \(\widehat{x}_K^*\) such that (55) and (56) hold: see (60) and (52). Moreover, by Remark 3, we obtain (58) and (59): see (61).

It remains to prove (57). Since

$$\begin{aligned} \Vert \varLambda \varPsi _Kx^*\Vert _Z=\Vert \varLambda (P_KR_K-I_X)x^*\Vert _Z =\Vert \mathfrak {G}(e_K^*,0) \Vert _\mathbb {V} \end{aligned}$$

(see Remark 5, (47) and (51)) and

$$\begin{aligned} \Vert \varXi _K \Vert =O(\lambda _K),\quad K\rightarrow \infty , \end{aligned}$$

(see (40) with \(\varSigma ^*_K=\varSigma ^*\)) and

$$\begin{aligned} \Vert \delta \Vert _X=\left\{ \begin{array}{ll} O(\lambda _K\cdot \Vert e_K^*\Vert _\mathbb {U}^2)&{}\\ O(\Vert e_K^*\Vert _\mathbb {U}^2)&{}\quad \text {if}\;\mathbf {CSC}\,\text {holds} \end{array} \right. ,\quad K\rightarrow \infty , \end{aligned}$$

(see (25)), we obtain (57) by Theorem 4. \(\square \)

5.2 The splitting case

Now, we give the theorem for the splitting case. If the splitting case holds, we set, for any positive integer K,

$$\begin{aligned} \mu _K:=\Vert \left( I_X-P_KR_K\varGamma ^*\right) ^{-1}\Vert \end{aligned}$$

If the nilpotency case holds, then, by Remark 7, we have

$$\begin{aligned} \mu _K=O\left( \lambda _K^{c-1}\right) ,\quad K\rightarrow \infty . \end{aligned}$$

Theorem 7

Assume that only a primary discretization is used. Moreover, assume the splitting case,

$$\begin{aligned}&\lim _{K\rightarrow \infty }\mu _K\cdot \Vert (\pi _K\rho _K-I_\mathbb {U})(\varGamma ^*(I_X-\varGamma ^*)^{-1}\varSigma ^*\varLambda )_\mathbb {U} \Vert =0,\end{aligned}$$
(62)
$$\begin{aligned}&\lim _{K\rightarrow \infty }\mu _K\cdot \Vert (\pi _K\rho _K-I_\mathbb {U})(\varSigma ^*\varLambda )_\mathbb {U}\Vert =0, \end{aligned}$$
(63)

and

$$\begin{aligned} \underset{K\rightarrow \infty }{\lim }\mu _K^{2}\lambda _K\cdot \Vert e_K^*\Vert _\mathbb {U}=0. \end{aligned}$$
(64)

Then, there exists a positive integer \(\widehat{K}\) such that, for any positive integer \(K\ge \widehat{K}\), \(\widehat{\varPhi }_K\) has a fixed point \( \widehat{x}_K^*\) such that

$$\begin{aligned} \left\| P_K\widehat{x}_K^{*}-x^{*}\right\| _{X}=O(\mu _K\cdot \Vert e_K^*\Vert _\mathbb {U}),\quad K\rightarrow \infty . \end{aligned}$$
(65)

Moreover, for the approximation \((v_K^*,\beta _K^*)\) of \((v^*,\beta ^*)\), we have the estimate

$$\begin{aligned} \Vert (v_K^*,\beta _K^*)-(v^*,\beta ^*)\Vert _{\mathbb {V}\times \mathbb {B}}=O(\mu _K\cdot \Vert e_K^*\Vert _\mathbb {U}),\quad K\rightarrow \infty . \end{aligned}$$
(66)

Finally, suppose there exists a sequence \(\{\widehat{x}_K\}\) of fixed points of \(\widehat{\varPhi }_K\) such that \(\widehat{x}_K\) is eventually different from \(\widehat{x}^*_K\). Then

$$\begin{aligned} \frac{1}{\Vert P_K\widehat{x}_K-x^*\Vert _X}=O(\mu _K\lambda _K),\quad K\rightarrow \infty , \end{aligned}$$
(67)

and

$$\begin{aligned} \frac{1}{\Vert \widehat{x}_K-\widehat{x}_K^*\Vert _{\widehat{X}_K}} =O\left( \max \{\Vert \pi _K \Vert ,1\}\cdot \mu _K\lambda _K\right) ,\quad K\rightarrow \infty . \end{aligned}$$
(68)

Proof

Since A \(\varvec{x^*}\) 1 holds, CS1 is fulfilled with

$$\begin{aligned} L_{K}=O(\lambda _K),\quad K\rightarrow \infty . \end{aligned}$$

Now, we show that CS2 holds. By (62), (63), (48), Remark 6 and Theorem 5, we obtain that there exists a positive integer \(K_{2}\) such that, for any positive integer \(K\ge K_{2}\), \(D^*\varPsi _{K}\) is invertible and

$$\begin{aligned} \Vert (D^*\varPsi _{K})^{-1}\Vert =O\left( \mu _K\right) ,\quad K\rightarrow \infty . \end{aligned}$$
(69)

Now, we have

$$\begin{aligned} \frac{1}{r_2(K)}=O(\mu _K\lambda _K),\quad K\rightarrow \infty . \end{aligned}$$
(70)

By (70), (69), (52) and (64), we conclude that \(\mathbf {CS2}\) is fulfilled.

Then, Theorem 2 says that there exists a positive integer \( \widehat{K}\) such that, for any positive integer \(K\ge \widehat{K}\), \( \widehat{\varPhi }_K\) has a fixed point \(\widehat{x}_K^*\) such that (65) and (66) hold: see (69) and (52). Moreover, by Remark 3 we obtain (67) and (68): see (70). \(\square \)

6 Conclusions

In this paper we have studied the numerical solution of PAF, introduced in Sect. 2, in case of spaces \(\mathbb {A}\) and \(\mathbb {B}\) of finite dimension. PAF has been discretized by a primary discretization and a secondary discretization, as explained in Sect. 3. A convergence analysis has been carried out in Sect. 4. In Sect. 4, we have also addressed the two particular situations of the simple case and the splitting case (which includes the nilpotency case). In Sect. 5, under the assumption that only a primary discretization is used, the convergence results have been specialized to these two particular situations.

The functional differential equation BVP (1) is a particular instance of PAF. The results of Sect. 5 are applied in [41, 44] to this particular instance for the two particular primary discretizations given by the collocation method and the Fourier series method. The present paper provides the theoretical basis for the analysis of such methods.

We finish observing that PAF also includes BVPs for partial functional differential equations, as it has been illustrated in Sect. 2.1. Apart from the possible infinite-dimensionality of the space \(\mathbb {A}\), a numerical study of such problems in the context of PAF has to take into account the use of approximations \(\mathfrak {G}_K\) of the linear operator \(\mathfrak {G}\).