Numerical dynamics of integrodifference equations

Abstract

Integrodifference equations are versatile models in theoretical ecology for the spatial dispersal of species evolving in non-overlapping generations. The dynamics of these infinite-dimensional discrete dynamical systems is often illustrated using computational simulations. This paper studies the effect of Nyström discretization to the local dynamics of periodic integrodifference equations having Hölder continuous functions over a compact domain as state space. We prove persistence and convergence for hyperbolic periodic solutions and their associated stable and unstable manifolds respecting the convergence order of the quadrature/cubature method.

1 Introduction

1.1 Growth and dispersal in discrete time

Difference equations of the form

$$\begin{aligned} u_{t+1}=g_t(u_t) \end{aligned}$$

are commonly used to model the temporal evolution of single species which evolve in nonoverlapping generations, reproduce at specific time intervals or are censused at intervals (metered modells). The growth functions $g_t:\mathbb {R}_+\rightarrow \mathbb {R}_+$ capture typical features of a particular population and are of Beverton–Holt, Ricker or Allee type, as well as related forms (see e.g. [22, pp. 11ff, Sect. 2.2]). On larger or inhomogeneous habitats also spatial effects such as dispersal have to be taken into account. This is achieved in terms of a dispersal kernel $k_t(x,y)\ge 0$ indicating the probability of the species to move from position x to position y in the habitat $\varOmega \subset \mathbb {R}^\kappa$. Standard kernels are of Laplace- or Gauß-type, among others, and are discussed in for instance [22, pp. 17ff, Sect. 2.3]. This yields a recursion of the form

$$\begin{aligned} u_{t+1}(x)=\int _\varOmega k_t(x,y)g_t(u_t(y))\,{\mathrm d}y\quad \text {for all }x\in \varOmega , \end{aligned}$$

(1.1)

which is denoted as integrodifference equation (IDE for short, cf. [18, 22]). The state space variable $u_t(x)$ describes the population density in the t-th generation located at $x\in \varOmega$. The right-hand side of (1.1) maps the density in generation t to the density in the next generation $t+1$ in two distinct stages: During the initial sedentary stage individuals grow, reproduce or die, and the local population density $u_t(x)$ evolves into $g_t(u_t(x))$. During the subsequent dispersal stage, the individuals move according to the dispersal kernel $k_t(x,y)$. Of course the above considerations extend to vector-valued growth functions $g_t$ and matrix-valued dispersal kernels $k_t$ in order to model several interacting or single but structured populations. In conclusion, difference equations such as (1.1) describe growth and dispersal, therefore, can be considered as a discrete time counterpart to reaction-diffusion equations, but with greater flexibility in the choice of the kernel.

From a mathematical perspective, IDEs are infinite-dimensional discrete-time dynamical systems. Besides being popular tools in theoretical ecology over the recent years, they canonically arise as time discretizations of integrodifferential equations, as time-1-map of evolutionary partial differential equations or in the iterative solution of (nonlinear) boundary value problems [23, p. 190]. It is understood that IDEs involve an integral operator which in (1.1) is of Hammerstein-, but more general of Urysohn-type. Indeed, for our purposes a sufficiently flexible class are the recursions

whose natural state spaces consists of continuous or integrable functions over a compact subset $\varOmega$ (the habitat in applications from ecology).

1.2 Numerical dynamics

In applied sciences the long-term behavior of IDEs is willingly illustrated using numerical simulations. For this purpose, [22, pp. 112–113] suggests to replace the integral in ($I_{0}$) by the trapezoidal or the Simpson rule. Both are special cases of general Nyström methods

based on convergent quadrature or cubature rules with weights $w_\eta \ge 0$ and nodes $\eta \in \varOmega _n$ over a finite grid $\varOmega _n\subset \varOmega$. Here, $n\in \mathbb {N}$ is related to the number of nodes in $\varOmega _n$ and therefore the accuracy of the approximation, see [8]. We point out that Nyström methods yield full discretizations of ($I_{0}$) and can be evaluated immediately.

While the numerical analysis of integral equations is a well-established field, e.g. [4, 16], this paper enriches it by a dynamical aspect: We study and relate the long-term behavior of the iterates $u_t$ generated by an IDE ($I_{0}$) to those of a Nyström discretization ($I_n$). This brings us to the area of numerical dynamics [21, 31, 32] addressing the following questions:

• Which dynamical or asymptotic properties of an IDE ($I_{0}$) as $t\rightarrow \infty$ are preserved when passing to its Nyström discretizations ($I_n$)?

• What can be said about convergence as $n\rightarrow \infty$ when the approximations become increasingly more accurate? In particular, are convergence rates of the integration rules preserved?

For the classical qualitative behavior of autonomous ODEs, such problems originate in [6] and are surveyed in [32]. In between various contributions to continuous-time infinite-dimensional dynamical systems generated by functional differential equations [15] or evolutionary (e.g. parabolic) partial differential equations [21, 31] arose, both for spatial, as well as for full discretizations. IDEs merely require spatial discretization, but have in common with these problems that conventional error estimates are unsuitable to describe asymptotic behavior. In fact, bounds for the global discretization error typically grow exponentially in time and therefore establish convergence as $n\rightarrow \infty$ only over compact time intervals [25, Thm. 4.1]. Thus, techniques extending those of standard numerical analysis are required to tackle the above problems.

Previous contributions to the numerical dynamics of IDEs address basics and error estimates [25], as well as the persistence/convergence of globally asymptotically stable solutions [26]. This paper focusses on an another important aspect, namely the local saddle-point structure near periodic solutions to ($I_{0}$). Related work, but for autonomous evolutionary differential equations near equilibria, is due to [6, 14] (ODEs), [1] (parabolic PDEs) and [12] (retarded FDEs).

In contrast, we study time-periodic IDEs ($I_{0}$) in the vicinity of periodic solutions. We stress that periodic time-dependence is strongly motivated by applications to incorporate seasonal influences. While [25, 26] apply to semi-discretizations of ($I_{0}$) of collocation- or degenerate kernel-type [4, 16], which act between finite-dimensional function spaces, but still contain integrals, we tackle Nyström discretizations ($I_n$), because they can be evaluated immediately. At this point the question for an ambient state space of ($I_{0}$) arises. A natural choice are the continuous functions $C(\varOmega )$ over a compact $\varOmega \subset \mathbb {R}^\kappa$. Here however, already for linear integral operators, the discretization error under Nyström methods converges only in the strong, but not in the uniform topology as $n\rightarrow \infty$, see [16, pp. 130–131, Lemma 4.7.6]. Using the theory of collectively compact operators [2] one can still establish that fixed-points of ($I_{0}$) (and their stability properties) persist [3, 33]. Nonetheless, it is not clear how to establish convergence of the associated stable and unstable manifolds of ($I_n$) to those of the original problem ($I_{0}$). For this reason we retreat to the Hölder continuous functions $C^\alpha (\varOmega )$ as state space. This set-up is sufficiently general to capture most relevant applied problems [18, 22] and has the advantage that a more conventional perturbation theory (see App. A) applies to realize our goals. It should not be concealed, though, that the prize for this endeavor are more involved assumptions and technical preliminaries on Urysohn operators (well-definedness, complete continuity, differentiability). For the sake of a brief presentation they are outsourced to [27, 28].

The structure of our presentation is as follows: In Sect. 2 we introduce a flexible framework for general periodic difference equations in Banach spaces and their linearization. Perturbation results for the Floquet spectrum of linear periodic equations are given in Theorem 2.1, while Theorem 2.2 addresses persistence and convergence of hyperbolic solutions and Theorem 2.3 the associated stable and unstable manifolds – when dealing with periodic equations we speak of fiber bundles. Although tailor-made for Nyström discretizations of IDEs, these results also apply to collocation- or degenerate kernel-discretizations, as well as when studying time-periodic evolutionary differential equations via their time-h-maps. The concrete case of Urysohn IDEs ($I_{0}$) is saved for Sect. 3 and illustrates how Thms. 2.1–2.3 apply. One obtains convergence of both hyperbolic solutions, and of the functions parametrizing their invariant fiber bundles with a rate given by the Hölder exponent $\alpha \in (0,1]$ of the kernel functions $f_t$ in the first variable. Nonetheless, for smooth $f_t$ the higher-order convergence rates inherited from the particular quadrature/cubature rules are established. Section 4 contains numerical simulations confirming our theoretical results. An example with separable kernel (logistic IDE) allows explicit results and a direct comparison between exact with numerically obtained periodic solutions in the Hölder norm. The closing Sect. 5 comments on related approaches and simulation techniques. Finally, an “Appendix” summarizes the technical ingredients required in our analysis.

Notation We write $\mathbb {R}_+:=[0,\infty )$ for the nonnegative reals, $\mathbb {S}^1:=\left\{ z\in \mathbb {C}:\,\left| z\right| =1\right\}$ for the unit circle in $\mathbb {C}$, $[\cdot ]:\mathbb {R}\rightarrow \mathbb {Z}$ is the integer function and $\left| \cdot \right|$ denotes norms on finite-dimensional spaces. On the Cartesian product $X\times Y$ of normed spaces X, Y,

$$\begin{aligned} \left\| (x,y)\right\| :=\max \bigl \{\left\| x\right\| _X,\left\| y\right\| _Y\bigr \} \end{aligned}$$

(1.2)

is the product norm and we proceed accordingly on products of more than two spaces. The open resp. closed balls in X with radius $r\ge 0$ and center $x\in X$ are

$$\begin{aligned} B_r(x,X)&:=\left\{ y\in X:\,\left\| y-x\right\| <r\right\} ,&{\bar{B}}_r(x,X)&:=\left\{ y\in X:\,\left\| y-x\right\| \le r\right\} ; \end{aligned}$$

on a finite-dimensional X we write $B_r(x)$ and ${\bar{B}}_r(x)$. For nonempty $A\subseteq X$, ${{\,\textrm{diam}\,}}A$ denotes the diameter of A, ${{\,\textrm{dist}\,}}_A(x):=\inf _{a\in A}\left\| x-a\right\|$ the distance of a point $x\in X$ from A, ${{\,\textrm{dist}\,}}(B,A):=\sup _{b\in B}{{\,\textrm{dist}\,}}_A(b)$ the Hausdorff semidistance of $B\subseteq X$ from A and we set $B_r(A):=\{x\in X:\,{{\,\textrm{dist}\,}}_A(x)<r\}$. We denote a subset $\mathcal {A}\subseteq \mathbb {Z}\times X$ as nonautonomous set having the fibers $\mathcal {A}(t):=\{x\in X:\,(t,x)\in \mathcal {A}\}$, $t\in \mathbb {Z}$ and write $\mathcal {B}_r(\phi ):=\left\{ (t,u)\in \mathbb {Z}\times X:\,\left\| u-\phi _t\right\| <r\right\}$ for the r-neighborhood of a sequence $\phi =(\phi _t)_{t\in \mathbb {Z}}$ in X.

The bounded k-linear maps from the Cartesian product $X^k$ to Y are denoted by $L_k(X,Y)$, $L(X,Y):=L_1(X,Y)$ and $L_0(X,Y):=Y$. Moreover, we abbreviate $L_k(X):=L_k(X,X)$, $L(X):=L(X,X)$, GL(X) are the invertible maps in L(X) and $I_X$ is the identity on X. Furthermore, N(T) is the kernel and R(T) the range of $T\in L(X,Y)$; $\sigma (S)$ is the spectrum and $\sigma _p(S)$ the point spectrum of $S\in L(X)$.

2 Difference equations and perturbation

Let $(X,\left\| \cdot \right\| )$ denote a Banach space.

2.1 Periodic difference equations

Abstractly, we are interested in a family of nonautonomous difference equations

with right-hand sides on open sets $U_t\subseteq X$, $t\in \mathbb {Z}$, parametrized by $n\in \mathbb {N}_0$. In the following, $n\in \mathbb {N}$ is a discretization parameter such that are understood as approximations converging to the original problem as $n\rightarrow \infty$ in a sense to be defined below. A nonautonomous set $\mathcal {A}\subseteq \mathbb {Z}\times X$ with fibers $\mathcal {A}(t)\subseteq U_t$ for all $t\in \mathbb {Z}$ is called forward invariant or invariant (w.r.t. ($\varDelta _n$)), provided

resp., holds. Given an initial time $\tau \in \mathbb {Z}$, a forward solution to ($\varDelta _n$) is a sequence $\phi =(\phi _t)_{\tau \le t}$ satisfying $\phi _t\in U_t$ and the solution identity for all $\tau \le t$, a backward solution fulfills the solution identity for $t<\tau$ and for an entire solution $(\phi _t)_{t\in \mathbb {Z}}$ one has on $\mathbb {Z}$. The forward solution starting at $\tau$ in the initial state $u_\tau \in U_\tau$ is uniquely determined as composition

and denoted as the general solution to ($\varDelta _n$); it is defined as long as the compositions stay in $U_t$. A difference equation ($\varDelta _n$) is called $\theta _0$-periodic, if both and $U_{t+\theta _0}=U_t$ hold for all $t\in \mathbb {Z}$ with some basic period $\theta _0\in \mathbb {N}$; an autonomous equation is 1-periodic, i.e. there exists a with , $U_t\equiv U$ on $\mathbb {Z}$. A $\theta _1$-periodic solution to ($\varDelta _n$) is an entire solution satisfying $\phi _t\equiv \phi _{t+\theta _1}$ on $\mathbb {Z}$.

Given a fixed $\theta \in \mathbb {N}$ and a sequence $u=(u_t)_{t\in \mathbb {Z}}$ with $u_t\in U_t$, $t\in \mathbb {Z}$, let us introduce the open product ${\hat{U}}:=U_0\times \ldots \times U_{\theta -1}$ and

$$\begin{aligned} {\hat{u}}&:=(u_0,\ldots ,u_{\theta -1})\in {\hat{U}},&(\overline{u_0,\ldots ,u_{\theta -1}})&:=(u_{t\text { mod }\theta })_{t\in \mathbb {Z}}. \end{aligned}$$

In order to characterize and compute periodic solutions to ($\varDelta _n$), $n\in \mathbb {N}_0$, we introduce the nonlinear operators

(2.1)

and use the norm induced via (1.2) on the Cartesian product $X^\theta$.

The next two results are immediate:

Lemma 2.1

Let $n\in \mathbb {N}_0$, ($\varDelta _n$) be $\theta _0$-periodic and $\theta$ be a multiple of $\theta _0$:

• (a) If $(\phi _t)_{t\in \mathbb {Z}}$ is a $\theta$-periodic solution to ($\varDelta _n$), then ${\hat{\phi }}\in {\hat{U}}$ is a fixed point of .

• (b) Conversely, if ${\hat{\phi }}\in {\hat{U}}$ is a fixed point of , then $(\overline{\phi _0,\ldots ,\phi _{\theta -1}})$ is a $\theta$-periodic solution to ($\varDelta _n$).

This characterization of periodic solutions to ($\varDelta _n$) via the mapping has the numerical advantage to avoid the computation of compositions $\varphi ^n(\theta +\tau ;\tau ,\cdot ):U_\tau \rightarrow X$, $\tau \in \mathbb {Z}$, and therefore preserves (numerical) backward stability (see [13]).

Lemma 2.2

Let $n\in \mathbb {N}_0$, $m\in \mathbb {N}$, ($\varDelta _n$) be $\theta _0$-periodic and $\theta$ be a multiple of $\theta _0$. If every , $0\le t<\theta _0$, is m-times continuously (Fréchet) differentiable, then is of class $C^m$ and for every ${\hat{u}}\in {\hat{U}}$ one has

2.2 Linear periodic difference equations

Suppose that , $t\in \mathbb {Z}$, and consider a family of linear difference equations

in X parametrized by $n\in \mathbb {N}_0$. As above we understand ($L_n$), $n\in \mathbb {N}$, as perturbations of an initial problem $(L_0)$. The transition operator $\varPhi ^n(t,\tau )\in L(X)$ of ($L_n$) is

(2.2)

We are interested in $\theta$-periodic equations ($L_n$), that is

(2.3)

allowing us to introduce the period operator $\varXi _\theta ^n:=\varPhi ^n(\theta ,0)\in L(X)$ of ($L_n$). Its eigenvalues are the Floquet multipliers and $\sigma _p(\varXi _\theta ^n)$ is the Floquet spectrum of ($L_n$).

One says a linear difference equation ($L_n$) is weakly hyperbolic, if $1\not \in \sigma (\varXi _\theta ^n)$, and hyperbolic, if $\mathbb {S}^1\cap \sigma (\varXi _\theta ^n)=\emptyset$ holds. In the hyperbolic situation, the spectrum can be decomposed as $\sigma (\varXi _\tau ^n)=\sigma _+{\dot{\cup }}\sigma _-$ with spectral sets

$$\begin{aligned} \sigma _+&\subseteq B_1(0),&\sigma _-&\subseteq \mathbb {C}\setminus {\bar{B}}_1(0). \end{aligned}$$

With the spectral projections $P_+^n:=\frac{1}{2\pi \i }\int _{\mathbb {S}^1}(zI_{X}-\varXi _\theta ^n)^{-1}\,{\mathrm d}z$, $P_-^n:=I_X-P_+^n$ we introduce the fibers $\mathcal {V}_+^n(t):=\varPhi ^n(t,0)R(P_+^n)$ and $\mathcal {V}_-^n(t):=\varPhi ^n(t,0)R(P_-^n)$, first for $t\ge 0$ and then by $\theta$-periodic continuation on $\mathbb {Z}$. This yields $\theta$-periodic nonautonomous sets $\mathcal {V}_+^n\subseteq \mathbb {Z}\times X$ (stable vector bundle) and $\mathcal {V}_-^n\subseteq \mathbb {Z}\times X$ (unstable vector bundle) of ($L_n$). Then $\mathcal {V}_+^n$ is forward invariant, while $\mathcal {V}_-^n$ is invariant w.r.t. ($L_n$).

For compact operators $\varXi _\theta ^n\in L(X)$ the Riesz-Schauder theory [19, pp. 428ff] applies: Every Floquet multiplier $\lambda \in \sigma _p(\varXi _\theta ^n)$ possesses a minimal $\iota (\lambda )\in \mathbb {N}$ so that $N(\lambda I_X-\varXi _\theta ^n)^j=N(\lambda I_X-\varXi _\theta ^n)^{j+1}$ for all $j\ge \iota (\lambda )$ leading to finite-dimensional generalized eigenspaces $N(\lambda I_X-\varXi _\theta ^n)^{\iota (\lambda )}$. All unstable fibers $\mathcal {V}_-^n(t)$, $t\in \mathbb {Z}$, have a constant finite dimension, which is denoted as the Morse index of ($L_n$) and equals the finite sum $\sum _{\lambda \in \sigma _-}\dim N(\lambda I_X-\varXi _\theta ^n)^{\iota (\lambda )}$ of algebraic multiplicities.

We begin with a perturbation result for hyperbolic linear systems ($L_n$) under uniform convergence:

Theorem 2.1

(Perturbed hyperbolicity) Suppose that the $\theta$-periodic linear difference equations ($L_n$), $n\in \mathbb {N}_0$, fulfill:

• (i) for all $0\le t<\theta$,

• (ii) $\varXi _\theta ^n\in L(X)$ is compact for all $n\in \mathbb {N}$.

Then also the period operator $\varXi _\theta ^0\in L(X)$ of $(L_0)$ is compact and there exists a $N\in \mathbb {N}$ such that the following holds for all $n\ge N$ or $n=0$:

• (a) With $(L_0)$ also the perturbed equation ($L_n$) is weakly hyperbolic,

• (b) with $(L_0)$ also the perturbed equation ($L_n$) is hyperbolic. In particular, for reals $\beta \in \bigl (\max \left\{ 0,1-\tfrac{1}{2}{{\,\textrm{dist}\,}}(\sigma (\varXi _\theta ^0),\mathbb {S}^1)\right\} ,1\bigr )$, there exists a $\theta$-periodic sequence $(P_t^n)_{t\in \mathbb {Z}}$ of invariant projectors in L(X) with for all $t\in \mathbb {Z}$, so that the transition operators $\varPhi ^n(t,s)$ satisfy $\dim \mathcal {V}_-^n=\dim \mathcal {V}_-^0$ and the estimates

  $$\begin{aligned} \begin{aligned} \left\| \varPhi ^n(t,s)P_s^n\right\| _{L(X)}&\le K\beta ^{t-s}\quad \text {for all }s\le t,\\ \left\| \varPhi ^n(t,s)[I_X-P_s^n]\right\| _{L(X)}&\le K\beta ^{s-t}\quad \text {for all }t\le s, \end{aligned} \end{aligned}$$

  (2.4)

• (c) $\lim _{n\rightarrow \infty }\left\| P_t^n-P_t^0\right\| _{L(X)}=0$ for all $t\in \mathbb {Z}$.

Proof

Let $0\le s<\theta$. Due to (i) the sequence is bounded and consequently we obtain from and the periodicity condition (2.3) that for all $t\in \mathbb {Z}$.

• (I) Claim: $\lim _{n\rightarrow \infty }\Vert \varPhi ^n(t,0)-\varPhi ^0(t,0)\Vert =0$ for all $0\le t$. We proceed by mathematical induction. Thanks to (2.2), for $t=0$ the assertion is trivial and for $t=1$ it results from (i). In the induction step $t\rightarrow t+1$ we obtain

  

  from the induction hypothesis and the triangle inequality, yielding the claim.

• (II) Claim: $\varXi _\theta ^0\in L(X)$ is compact. If we set $t=\theta$ in claim (I), then the period operators satisfy

  $$\begin{aligned} \lim _{n\rightarrow \infty }\left\| \varXi _\theta ^n-\varXi _\theta ^0\right\| =0. \end{aligned}$$

  (2.5)

  Hence, $\varXi _\theta ^0$ is the uniform limit of by (ii) compact operators $\varXi _\theta ^n$, $n\in \mathbb {N}$, and consequently compact [19, p. 416, Thm. 1.1].

• (III) Claim: For every nonempty closed $S\subseteq \mathbb {C}$ with $\sigma (\varXi _\theta ^0)\cap S=\emptyset$ there exists a $n_1\in \mathbb {N}$ such that $\sigma (\varXi _\theta ^n)\cap S=\emptyset$ for all $n\ge n_1$.

Since the closed S and the compact $\sigma (\varXi _\theta ^0)$ are disjoint they have a positive distance. Therefore, there is an $\varepsilon >0$ so that $S\cap B_\varepsilon (\sigma (\varXi _\theta ^0))=\emptyset$. By the upper semicontinuity of the spectrum [5, p. 80, Lemma 3] and relation (2.5) there is a $n_1\in \mathbb {N}$ with $\sigma (\varXi _\theta ^n)\subset B_\varepsilon (\sigma (\varXi _\theta ^0))$ and consequently $\sigma (\varXi _\theta ^n)$ stays disjoint from S for all $n\ge n_1$.

• (a) If $(L_0)$ is weakly hyperbolic, then $\sigma (\varXi _\theta ^0)\cap \left\{ 1\right\} =\emptyset$ and (III) applied to the singleton $S=\left\{ 1\right\}$ yields the assertion.

• (b) The hyperbolicity of ($L_n$) results as above in (a) with $S=\mathbb {S}^1$. Furthermore, then [30, p. 44, Prop. 3.13] implies that ($L_n$) possess an exponential dichotomy on $\mathbb {Z}$ as claimed with the $\theta$-periodic invariant projectors $P_t^n$ satisfying

  

  (2.6)

  and $I_X-P_0^n=P_-^n$. In particular, by (2.5) and [5, p. 80, Cor. 1] the spectral projections $P_-^n$ associated to the unstable spectral parts of $\varXi _\theta ^n$, $n\in \mathbb {N}_0$, fulfill that $\dim R(P_-^n)=\dim R(P_-^0)$ for large n, say for $n\ge n_2$. Thanks to (2.6) this extends to the dimension of the unstable bundles $\mathcal {V}_-^n$. Finally, we set $N:=\max \left\{ n_1,n_2\right\}$.

• (c) Combining (2.5) with [5, p. 80, Lemma 4] yields that the spectral projections satisfy $\lim _{n\rightarrow \infty }\left\| P_-^n-P_-^0\right\| =0$. Together with claim (I) we obtain for $t\in \mathbb {Z}$ that

  $$\begin{aligned}&\left\| P_t^n-P_t^0\right\| = \left\| [I_X-P_t^n]-[I_X-P_t^0]\right\| \\&\quad {\mathop {\le }\limits ^{(2.6)}} \left\| \varPhi ^n(t,0)\right\| \left\| P_-^n-P_-^0\right\| \left\| \varPhi ^0(0,t)\right\| + \left\| \varPhi ^n(t,0)-\varPhi ^0(t,0)\right\| \left\| P_-^0\varPhi ^0(0,t)\right\| \\&\quad \le \bigg (\prod _{r=0}^{t-1}c_r\bigg )\left\| P_-^n-P_-^0\right\| \left\| \varPhi ^0(0,t)\right\| + \left\| \varPhi ^n(t,0)-\varPhi ^0(t,0)\right\| \left\| P_-^0\varPhi ^0(0,t)\right\| \end{aligned}$$

  from the triangle inequality, whose right-hand side converges to 0 as $n\rightarrow \infty$. $\square$

2.3 Perturbation of hyperbolic solutions and invariant bundles

We next address the robustness of $\theta _1$-periodic solutions $\phi ^0$ to general $\theta _0$-periodic difference equations $(\varDelta _0)$, as well as their nearby saddle-point structure consisting of stable and unstable bundles (see [17, pp. 143ff, Chap. 6], [24, pp. 256ff, Sect. 4.6]) under perturbation. By imposing a natural hyperbolicity condition on the solution $\phi ^0$ it is shown that also the perturbations ($\varDelta _n$) have (locally unique) periodic solutions $\phi ^n$ for sufficiently large n, which converge to $\phi ^0$ in the limit $n\rightarrow \infty$.

Let $\theta :={{\,\textrm{lcm}\,}}\left\{ \theta _0,\theta _1\right\}$. We suppose that the right-hand sides of ($\varDelta _n$) are continuously differentiable. Our endeavor is based on the variational equations

associated to $\theta$-periodic solutions $\phi ^n$ of ($\varDelta _n$), $n\in \mathbb {N}_0$. Since the linear equations ($V_n$) are $\theta$-periodic, the terminology and results from Sect. 2.2 apply to ($V_n$) with and the period operator $\varXi _\theta ^n$, $n\in \mathbb {N}_0$. In this context, we understand a solution $\phi ^n$ of ($\varDelta _n$) as (weakly) hyperbolic, if ($V_n$) has the corresponding property.

Lemma 2.3

Let $n\in \mathbb {N}_0$. If is continuously differentiable for all $0\le t<\theta _0$, then the derivatives of the mappings defined in (2.1) satisfy and .

Based on this result, the (Floquet) spectrum of ($V_n$) can be computed from the (point) spectrum of the cyclic block operator given in Lemma 2.2. This has the numerical advantage of avoiding to evaluate the compositions (matrix products) $\varXi _\theta ^n$.

Proof

Keeping $n\in \mathbb {N}_0$ fixed, we abbreviate , $t\in \mathbb {Z}$, and observe that the $\theta$th power of given in Lemma 2.2 becomes a block diagonal operator

Referring to [30, p. 42, Prop. 3.11(a)] one has for all $t\in \mathbb {Z}$ and therefore . Now the Spectral Mapping Theorem [5, p. 65, Thm. 2] yields the assertion for the spectra. Concerning the point spectrum the claim follows directly from the corresponding eigenvalue-eigenvector relations and the solution identity for ($V_n$). $\square$

Our next result establishes persistence of hyperbolic periodic solutions to $(\varDelta _0)$:

Theorem 2.2

(Perturbed periodic solutions) Let $\theta ={{\,\textrm{lcm}\,}}\left\{ \theta _0,\theta _1\right\}$. Suppose that the $\theta _0$-periodic difference equations ($\varDelta _n$), $n\in \mathbb {N}_0$, fulfill:

• (i) are continuously differentiable for all $0\le t<\theta _0$ and $n\in \mathbb {N}_0$,

• (ii) , $n\in \mathbb {N}$, are uniformly continuous on bounded sets uniformly in $n\in \mathbb {N}$, the family is equicontinuous for all $0\le t<\theta _0$ and for every $n\in \mathbb {N}$ there exists a $0\le t<\theta _0$ such that has compact values.

If $\phi ^0$ is a weakly hyperbolic $\theta _1$-periodic solution to $(\varDelta _0)$ and there exists a function $\varGamma _0:\mathbb {R}_+\rightarrow \mathbb {R}_+$ with $\lim _{\varrho \searrow 0}\varGamma _0(\varrho )=0$ satisfying for all $0\le t<\theta$ that

(2.7)

(2.8)

then there exist reals $\rho _0>0$ and $N_0\in \mathbb {N}$ such that the following hold for all $n\ge N_0$:

• (a) There is a unique $\theta$-periodic solution $\phi ^n$ to ($\varDelta _n$) in the neighborhood $\mathcal {B}_{\rho _0}(\phi ^0)$, it is weakly hyperbolic and there exists a constant $K_0\ge 0$ such that

  $$\begin{aligned} \sup _{t\in \mathbb {Z}}\left\| \phi _t^n-\phi _t^0\right\| _X \le K_0\varGamma _0(\tfrac{1}{n}), \end{aligned}$$

  (2.9)

• (b) with $\phi ^0$ also the solution $\phi ^n$ to ($\varDelta _n$) is hyperbolic with the same Morse index.

As the subsequent proof and Lemma 2.3 reveal, the constant $K_0\ge 0$ essentially depends on the distance of the Floquet spectrum of $\phi ^0$ to the point $1\in \mathbb {C}$. The value of $K_0$ blows up as this distance shrinks to 0, i.e. when (weak) hyperbolicity is lost.

Proof

Let I denote the identity mapping on the Cartesian product $X^\theta$. Our aim is to apply the quantitative Implicit Function Theorem A.1 with the open set $\varOmega ={\hat{U}}$, Banach spaces , the parameter space $\varLambda :=\left\{ 0\right\} \cup \left\{ \tfrac{1}{n}:\,n\in \mathbb {N}\right\} \subseteq \mathbb {R}$ with metric $d(\lambda _1,\lambda _2):=\left| \lambda _1-\lambda _2\right|$, $\lambda _0:=0$, $x_0:={\hat{\phi }}^0$, $y_0:=0$ and the mapping

with $x={\hat{\phi }}$. Let us first verify the assumptions of Theorem A.1. It follows from (2.1) that the mapping T is well-defined.

ad (i’): Thanks to Lemma 2.1(a), for the $\theta$-periodic solution $\phi ^0$ of $(\varDelta _0)$ the resulting tuple ${\hat{\phi }}^0$ is a fixed point of and therefore .

ad (ii’): Referring to Lemma 2.2 and assumption (i) every mapping is continuously differentiable and so is each $T(\cdot ,\lambda )$, $\lambda \in \varLambda$. Moreover, the partial derivative is invertible, because otherwise and thus by Lemma 2.3. This contradicts the weak hyperbolicity assumption on the solution $\phi ^0$.

ad (iii’): First, we obtain (A.1) from the estimates

and thus $\left\| T(x_0,\lambda )-T(x_0,\lambda _0)\right\| \le \varGamma _0(\lambda )$ (cf. (2.7)) for all $\lambda =\tfrac{1}{n}\in \varLambda$. Second, by assumption (ii) the derivatives are uniformly continuous on bounded sets, uniformly in $n\in \mathbb {N}$, and consequently there exist moduli of continuity $\omega _t:\mathbb {R}_+\rightarrow \mathbb {R}_+$ satisfying $\lim _{\varrho \searrow 0}\omega _t(\varrho )=0$ and

where ${\hat{\phi }}\in {\hat{U}}$. By the triangle inequality this results in

for all $\lambda =\tfrac{1}{n}\in \varLambda$. Now, with satisfying $\lim _{\varrho \searrow 0}\varOmega '(\varrho )=0$ due to (2.8), this gives for all $\lambda =\tfrac{1}{n}\in \varLambda$ that

$$\begin{aligned} \left\| D_1T(x,\lambda )-D_1T(x_0,\lambda _0)\right\| {\mathop {\le }\limits ^{(2.7)}} \max _{t=1}^\theta \omega _t(\left\| \phi _t-\phi _t^0\right\| )+\varOmega '(\tfrac{1}{n}) \le \varGamma \bigl (\bigl \Vert {\hat{\phi }}-{\hat{\phi }}^0\bigr \Vert ,\lambda \bigr ), \end{aligned}$$

with the function $\varGamma (\varrho _1,\varrho _2):=\max _{t=1}^\theta \omega _t(\varrho _1)+\varOmega '(\varrho _2)$, which clearly satisfies the limit relation $\lim _{\varrho _1,\varrho _2\searrow 0}\varGamma (\varrho _1,\varrho _2)=0$, i.e. (A.2) holds.

(a) Because the assumptions (i’–iii’) of Theorem A.1 hold, we can choose $\rho ,\delta >0$ so small that (A.3) holds for e.g. $q:=\tfrac{1}{2}$. Moreover, there exists a unique fixed point function ${\hat{\phi }}:B_\delta (\lambda _0)\rightarrow {\bar{B}}_{\rho _0}({\hat{\phi }}^0,X^\theta )$ with for all $n>\tfrac{1}{\delta }$. Then Lemma 2.1(b) guarantees that $\phi ^n:=(\overline{\phi _0(\tfrac{1}{n}),\ldots ,\phi _{\theta -1}(\tfrac{1}{n})})$ is the desired $\theta$-periodic solution to ($\varDelta _n$) whenever $n\ge N_0:=[\tfrac{1}{\delta }]+1$. We establish that the solutions $\phi ^n$ are weakly hyperbolic. For this purpose, let $\varepsilon >0$. First, thanks to (2.8) there exists a $n_1\in \mathbb {N}$ such that

We know from Theorem A.1(c) that $\lim _{n\rightarrow \infty }\sup _{t\in \mathbb {Z}}\left\| \phi _t^n-\phi _t^0\right\| =0$ and since is equicontinuous by assumption (ii), there exists a $n_2\in \mathbb {N}$ such that

Combining the last two inequalities readily yields for all $t\in \mathbb {Z}$ and $n\ge \max \left\{ n_1,n_2\right\}$, which establishes the limit relation

(2.10)

Second, assumption (ii) implies that the period operators $\varXi _\theta ^n$ of ($V_n$), $n\in \mathbb {N}$, contain a compact factor and hence are compact [19, p. 417, Thm. 1.2]. Thus, Theorem 2.1(a) applies to , $n\in \mathbb {N}_0$, and shows that $\phi ^n$ are weakly hyperbolic. Finally, given $N_0$ and ${\hat{\phi }}$ as in (a) one has

$$\begin{aligned} \left\| \phi _t^n-\phi _t^n\right\| {\mathop {\le }\limits ^{(1.2)}} \bigl \Vert {\hat{\phi }}^n-{\hat{\phi }}^0\bigr \Vert \le K_0\varGamma _0(\tfrac{1}{n})\quad \text {for all }n\ge N_0,\,t\in \mathbb {Z}\end{aligned}$$

with , which concludes the proof of (a).

(c) In case the solution $\phi ^0$ is hyperbolic, then due to (2.10) and the compactness of the period operators $\varXi _\theta ^n$ (see above), Theorem 2.1(b) applies to , $n\in \mathbb {N}_0$. It follows that the solutions $\phi ^n$ are hyperbolic as well. $\square$

The dynamics of ($\varDelta _n$) in the vicinity of hyperbolic solutions $\phi ^n$ is determined by a saddle-point structure consisting of local stable and unstable manifolds resp. fiber bundles [24, p. 256ff, Sect. 4.6] (in the periodic case). These sets allow a dynamical characterization and, given some $r_0>0$, we define the local stable fiber bundle

$$\begin{aligned} \mathcal {W}_+^n:= \left\{ (\tau ,u_\tau )\in \mathcal {B}_{r_0}(\phi ^n): \begin{array}{l} \varphi ^n(t;\tau ,u_\tau )\, \text {exists for all}\, t\ge \tau \\ \text {and }\varphi ^n(t;\tau ,u_\tau )-\phi _t^n\xrightarrow [t\rightarrow \infty ]{}0 \end{array}\right\} \end{aligned}$$

and the local unstable fiber bundle

$$\begin{aligned} \mathcal {W}_-^n := \left\{ (\tau ,u_\tau )\in \mathcal {B}_{r_0}(\phi ^n): \begin{array}{l} \text {there exists a solution }(\phi _t)_{t\le \tau } \text { of }(\varDelta _n)\\ \text {with }\phi _\tau =u_\tau \text { and }\phi _t-\phi _t^n\xrightarrow [t\rightarrow -\infty ]{}0 \end{array}\right\} \end{aligned}$$

associate to $\phi ^n$. The following result relates the fiber bundles of the perturbed equations ($\varDelta _n$), $n\in \mathbb {N}$, to that of the initial problem $(\varDelta _0)$. It requires that is equidifferentiable in each $u\in U_t$, that is there exists a such that

holds uniformly in $n\in \mathbb {N}$.

Fig. 1

Persistence of the saddle-point structure near a hyperbolic solution: A $\theta _1$-periodic solution $\phi ^0$ ($\bullet$ black) of a $\theta _0$-periodic equation $(\varDelta _0)$ persists as as $\theta$-periodic hyperbolic solution $\phi ^n$ ($\circ$ grey) to ($\varDelta _n$), $n\ge N_1$ (cf. Theorem 2.2). The corresponding stable bundle $\phi ^0+\mathcal {W}_+^0$ (black fibers) persists as $\phi ^n+\mathcal {W}_+^n$ (grey fibers), both are locally graphs over $R(P_\tau )$ (dashed), while the unstable bundle $\phi ^0+\mathcal {W}_-^0$ (black fibers) persists as $\phi ^n+\mathcal {W}_-^n$ (grey fibers), being locally graphs over $N(P_\tau )$ (dashed, cf. Theorem 2.3)

We can now show that the saddle-point structure near hyperbolic periodic solutions to $(\varDelta _0)$ is preserved under perturbation (see Fig. 1).

Theorem 2.3

(Perturbed stable and unstable fiber bundles) Let $\theta ={{\,\textrm{lcm}\,}}\left\{ \theta _0,\theta _1\right\}$ and $m\in \mathbb {N}$. Suppose that the $\theta _0$-periodic difference equations ($\varDelta _n$), $n\in \mathbb {N}_0$, fulfill:

• (i) are m-times continuously differentiable for all $n\in \mathbb {N}_0$ on a convex, open set $U_t$ and is equidifferentiable for all $0\le t<\theta _0$,

• (ii) , $n\in \mathbb {N}$, are uniformly continuous on bounded sets uniformly in $n\in \mathbb {N}$, the family is equicontinuous for all $0\le t<\theta _0$ and for every $n\in \mathbb {N}$ there exists a $0\le t<\theta _0$ such that has compact values.

If $\phi ^0$ is a hyperbolic $\theta _1$-periodic solution to $(\varDelta _0)$ satisfying (2.7), (2.8) and $(P_t)_{t\in \mathbb {Z}}$ is the invariant projector onto the stable vector bundle $\mathcal {V}_+^0$ of $(V_0)$ (cf. Theorem 2.1), then there exist $\rho _1>0$ and integers $N_1\ge N_0$ so that the following holds for $n\ge N_1$ or $n=0$, and the $\theta$-periodic hyperbolic solutions $\phi ^n$ ensured by Theorem 2.2:

• (a) The local stable fiber bundle $\mathcal {W}_+^n$ of ($\varDelta _n$) allows the representation

  $$\begin{aligned} \mathcal {W}_+^n = \phi ^n+\left\{ (\tau ,v+w_+^n(\tau ,v))\in \mathbb {Z}\times X:\,v\in B_{\rho _1}(0,R(P_\tau ))\right\} \end{aligned}$$

  as graph of a mapping $w_+^n:\mathbb {Z}\times X\rightarrow X$ with

  $$\begin{aligned} w_+^n(\tau +\theta ,u)=w_+^n(\tau ,u)=w_+^n(\tau ,P_\tau u)\in N(P_\tau ) \quad \text {for all }\tau \in \mathbb {Z}\end{aligned}$$

  and $u\in X$. Moreover, $w_+^n(\tau ,0)\equiv 0$ on $\mathbb {Z}$, the Lipschitz mappings $w_+^n(\tau ,\cdot )$ are of class $C^m$ and the stable fiber bundles of ($\varDelta _n$) and $(\varDelta _0)$ are related via

  

  (2.11)

  for all $\tau \in \mathbb {Z}$, $v\in B_{\rho _1}(0,R(P_\tau ))$, where $\phi _t=\varphi ^0(t;\tau ,\phi _\tau ^0+v+w_+^0(\tau ,v))-\phi _t^0$ whenever $\tau \le t$.

• (b) The local unstable fiber bundle $\mathcal {W}_-^n$ of ($\varDelta _n$) allows the representation

  $$\begin{aligned} \mathcal {W}_-^n = \phi ^n+\left\{ (\tau ,v+w_-^n(\tau ,v))\in \mathbb {Z}\times X:\,v\in B_{\rho _1}(0,N(P_\tau ))\right\} \end{aligned}$$

  as graph of a mapping $w_-^n:\mathbb {Z}\times X\rightarrow X$ with

  $$\begin{aligned} w_-^n(\tau +\theta ,u)=w_-^n(\tau ,u)=w_-^n(\tau ,[I_X-P_\tau ]u)\in R(P_\tau ) \quad \text {for all }\tau \in \mathbb {Z}\end{aligned}$$

  and $u\in X$. Moreover, $w_-^n(\tau ,0)\equiv 0$ on $\mathbb {Z}$, the Lipschitz mappings $w_-^n(\tau ,\cdot )$ are of class $C^m$ and the unstable fiber bundles of ($\varDelta _n$) and $(\varDelta _0)$ are related via

  

  (2.12)

  for all $\tau \in \mathbb {Z}$, $v\in B_{\rho _1}(0,N(P_\tau ))$, where $(\phi _t)_{t\le \tau }$ is the (unique) backward solution to $(\varDelta _0)$ starting in $(\tau ,v+w_-^0(\tau ,v))$, and have the same finite dimension.

• (c) $\mathcal {W}_+^n\cap \mathcal {W}_-^n=\phi ^n$,

with the constants $\beta \in (0,1)$, $K\ge 1$ from Theorem 2.1 applied to $(V_0)$.

In order to achieve convergence as $n\rightarrow \infty$ via (2.11) and (2.12) one needs the derivatives to tend to on bounded sets and, thanks to Theorem 2.2, continuity of the derivative , $0\le t<\theta _0$. A concrete illustration follows in Sect. 3.

Remark 2.1

(Alternative representation of $\mathcal {W}_+^n$ and $\mathcal {W}_-^n$) With some ${\tilde{\rho }}_1>0$ the local stable and unstable fiber bundles of $\phi ^n$ allow the alternative characterization

$$\begin{aligned} \mathcal {W}_+^n&= \phi ^n+\left\{ (\tau ,v+{\tilde{w}}_+^n(\tau ,v))\in \mathbb {Z}\times X:\,v\in B_{{\tilde{\rho }}_1}(0,R(P_\tau ^n))\right\} ,\\ \mathcal {W}_-^n&=\phi ^n+\left\{ (\tau ,v+{\tilde{w}}_-^n(\tau ,v))\in \mathbb {Z}\times X:\,v\in B_{{\tilde{\rho }}_1}(0,N(P_\tau ^n))\right\} \end{aligned}$$

as graphs over the vector bundles $\mathcal {V}_+^n$ resp. $\mathcal {V}_-^n$ of the variational equations ($V_n$), rather than over the vector bundles $\mathcal {V}_+^0$ resp. $\mathcal {V}_-^0$ of $(V_0)$ (cf. [24, pp. 256ff, Sect. 4.6]) as in Theorem 2.3. In addition, then the associate mappings ${\tilde{w}}_+^n(\tau ,\cdot ),{\tilde{w}}_-^n(\tau ,\cdot )$ possess values in $N(P_\tau ^n)$ resp. in $R(P_\tau ^n)$ for all $\tau \in \mathbb {Z}$. According to Theorem 2.1(c) the corresponding invariant projectors for ($V_n$) satisfy $\lim _{n\rightarrow \infty }\left\| P_t^n-P_t^0\right\| =0$ for all $t\in \mathbb {Z}$. Therefore, $\mathcal {W}_-^n$ and $\mathcal {W}_-^0$ share their finite dimension.

Proof

Since the existence of $\mathcal {W}_+^0,\mathcal {W}_-^0$ and their properties are well-established in the literature [24, pp. 187ff], we focus on their persistence and the convergence estimates (2.11) and (2.12). Let $\phi ^n=(\phi _t^n)_{t\in \mathbb {Z}}$ denote the $\theta$-periodic solutions of ($\varDelta _n$) guaranteed by Theorem 2.2 for $n\ge N_0$. The associate equations of perturbed motion

are $\theta$-periodic and have the trivial solution. The general solutions $\varphi ^n$ of ($\varDelta _n$) and $\bar{\varphi }^n$ to ($\bar{\varDelta }_{n}$) are related by $\bar{\varphi }^n(t;\tau ,u)=\varphi ^n(t;\tau ,u+\phi _\tau ^n)-\phi _t^n$ for all $\tau \le t$.

• (a) For each fixed $\tau \in \mathbb {Z}$ the sequence space

  $$\begin{aligned} \ell _\tau ^+ := \bigl \{(\phi _t)_{\tau \le t}:\,\phi _t\in X\text { and }\lim _{t\rightarrow \infty }\left\| \phi _t\right\| =0\bigr \}, \end{aligned}$$

  is complete w.r.t. the $\sup$-norm $\left\| \phi \right\| _\infty :=\sup _{\tau \le t}\left\| \phi _t\right\|$. For $\bar{\rho }>0$ so small that $\left\| \phi _t\right\| <\bar{\rho }$ implies $\phi _t+\phi _t^n\in U_t$ for all $t\in \mathbb {Z}$ and $n\ge N_0$ we introduce the operator

  

  for all $\tau \le t$. Then $u_\tau =P_\tau u_\tau +[I_X-P_\tau ]u_\tau \in X$ is contained in the stable bundle of ($\bar{\varDelta }_{n}$) if and only if $\phi :=\bar{\varphi }^n(\cdot ;\tau ,u_\tau )$ satisfies (cf. [6, proof of Thm. 3.1])

  $$\begin{aligned} T_+^n(\phi ) = \begin{pmatrix} P_\tau u_\tau , 0 \end{pmatrix}. \end{aligned}$$

  (2.13)

  Our approach to (2.13) using the Lipschitz inverse function Theorem A.2 is based on the representation $T_+^n=A_++G_+^n$ with

  

  for all $\tau \le t$. Note that the derivatives exist by assumption (i).

  • (I) Claim: $A_+\in GL(\ell _\tau ^+,R(P_\tau )\times \ell _\tau ^+)$ with $\left\| A_+^{-1}\right\| \le \tfrac{2K}{1-\beta }$. First of all, the sequence in L(X) is $\theta$-periodic and therefore $A_+$ is bounded. In order to show that $A_+$ is invertible, given $v_\tau \in R(P_\tau )$ and a sequence $\psi \in \ell _\tau ^+$, we observe that $A_+\phi =(v_\tau ,\psi )$ has the unique solution

    $$\begin{aligned} \phi _t = \varPhi ^0(t,\tau )P_\tau v_\tau +\sum _{s=\tau }^{t-1}\varPhi ^0(t,s+1)P_s\psi _s - \sum _{s=t}^\infty \varPhi ^0(t,s+1)[I_X-P_s]\psi _s \end{aligned}$$

    in $\ell _\tau ^+$ (a proof can be modelled after e.g. [24, pp. 151–152, Thm. 3.5.3(a)]). Using the dichotomy estimates (2.4) it is not hard to show $\left\| \phi _t\right\| \le K\left\| v_\tau \right\| +K\tfrac{1+\beta }{1-\beta }\left\| \psi \right\| _\infty$ for all $\tau \le t$ and therefore $\left\| A_+^{-1}\right\| \le K+K\tfrac{1+\beta }{1-\beta }=\tfrac{2K}{1-\beta }$.

  • (II) Claim: There exist $\rho \in (0,\bar{\rho }]$, $N_1\ge N_0$ such that ${{\,\textrm{lip}\,}}G_+^n|_{B_\rho (0)}\le \tfrac{1-\beta }{4K}$ holds for all $n\ge N_1$. Due to the limit relation (2.10) in the proof of Theorem 2.2 there is an $N_1\ge N_0$ with

    

    We next abbreviate . This function is continuously differentiable The Mean Value Inequality [19, p. 342, Cor. 4.3] and the fact that is equidifferentiable by assumption (i) with continuous derivative thus implies that there exists a $\rho \in (0,\bar{\rho }]$ such that for all $t\in \mathbb {Z}$, $u,{\bar{u}}\in B_\rho (0,X)$ and $n\ge N_1$. In combination, due to the representation

    

    we finally obtain for all $\phi ,\bar{\phi }\in B_\rho (0,\ell _\tau ^+)$ that

    $$\begin{aligned} \left\| G_+^n(\phi )-G_+^n(\bar{\phi })\right\| _\infty \le \tfrac{1-\beta }{4K}\left\| \phi -\bar{\phi }\right\| _\infty \quad \text {for all }n\ge N_1. \end{aligned}$$

  • (III) In this step we apply the Lipschitz inverse function Theorem A.2 to solve the nonlinear equation (2.13) in the Banach spaces , , points $x_0:=0$, $y_0:=(P_\tau u_\tau ,0)$, the Lipschitz constant $l:=\tfrac{1-\beta }{4K}$ and $\sigma :=\tfrac{1-\beta }{2K}$. Therefore, for every $u_\tau \in B_{\tfrac{1-\beta }{4K^2}\rho }(0,X)$ one has

    $$\begin{aligned} \left\| (P_\tau u_\tau ,0)\right\| {\mathop {=}\limits ^{(1.2)}} \left\| P_\tau u_\tau \right\| <\frac{1-\beta }{4K}\rho =:\rho _1 \end{aligned}$$

    and there exists a unique solution $\phi _+^n(u_\tau )\in B_\rho (0,\ell _\tau ^+)$ to (2.13). Then the function $w_+^n$ parametrizing the stable bundle of ($\bar{\varDelta }_{n}$) is $w_+^n(\tau ,v_\tau ):=[I_X-P_\tau ]\phi _+^n(v_\tau )_\tau$, where $v_\tau =P_\tau u_\tau$. We define $\phi :=\bar{\varphi }^0(\cdot ;\tau ,v_\tau +w_+^0(\tau ,v_\tau ))$, $\bar{\phi }:=\phi _+^n(v_\tau )$ and obtain

    

    Because $\bar{\phi }$ solves ($\bar{\varDelta }_{n}$) and $\phi$ solves $(\bar{\varDelta }_0)$, this simplifies to

    

    and it remains to estimate the right-hand side in this inequality. Since $U_t$ is assumed to be convex, we apply the Mean Value Theorem [19, p. 341, Thm. 4.2] and arrive at

    

    for each $v_\tau \in R(P_\tau )$. Here, let us point out that for all integers $\tau \le t$ one has the relation $\phi _t = \bar{\varphi }^0(t;\tau ,v_\tau +w_+^0(\tau ,v_\tau )) = \varphi ^0(t;\tau ,\phi _\tau ^0+v_\tau +w_+^0(\tau ,v_\tau ))-\phi _t^0.$

• (b) The argument is dual to the proof of (a), but now one works in the sequence space $\ell _\tau ^-:=\bigl \{(\phi _t)_{t\le \tau }:\,\phi _t\in X\text { and }\lim _{t\rightarrow -\infty }\left\| \phi _t\right\| =0\bigr \}$ being complete in the $\sup$-norm. One applies Theorem A.2 with , and $x_0:=0$, $y_0:=(u_\tau -P_\tau u_\tau ,0)$, $l:=\tfrac{1-\beta }{4K}$, $\sigma :=\tfrac{1-\beta }{2K}$ to the nonlinear operator

  

  for all $t\le \tau$. If the unique solution to $T_-^n(\phi )=(P_\tau u_\tau ,0)$ is denoted by $\phi _-^n(v_\tau )\in \ell _\tau ^-$ such that $v_\tau =u_\tau -P_\tau u_\tau$, then $w_-^n(\tau ,v_\tau ):=P_\tau \phi _-^n(v_\tau )_\tau$ has the claimed properties.

• (c) is a consequence of [24, pp. 259–260, Thm. 4.6.4]. $\square$

3 Urysohn integrodifference equations

Let us now illustrate the applicability of our abstract perturbation results from Sect. 2, when the initial problem $(\varDelta _0)$ is an integrodifference equation

whose right-hand side is an Urysohn operator over a compact nonempty $\varOmega \subset \mathbb {R}^\kappa$. For the sake of having well-defined and smooth mappings , $t\in \mathbb {Z}$, in an ambient setting, several assumptions on the kernel functions $f_t$ are due:

Hypothesis 3.1

Let $m\in \mathbb {N}$ and $\alpha \in (0,1]$. Suppose there exists a $\theta _0\in \mathbb {N}$ and open, convex sets $Z_t\subseteq \mathbb {R}^d$ such that the kernel functions

$$\begin{aligned} f_t&=f_{t+\theta _0}:\varOmega ^2\times \overline{Z_t}\rightarrow \mathbb {R}^d,&Z_t&=Z_{t+\theta _0}\quad \text {for all }t\in \mathbb {Z} \end{aligned}$$

(3.1)

fulfill the following assumptions for all $0\le t<\theta _0$ and $0\le k\le m$:

• (i) The derivative $D_3^kf_t:\varOmega ^2\times \overline{Z_t}\rightarrow L_k(\mathbb {R}^d)$ exists as continuous function,

• (ii) for all $r>0$ there exists a continuous function $h_r:\varOmega \rightarrow \mathbb {R}_+$ such that

  $$\begin{aligned} \left| D_3^kf_t(x,y,z)-D_3^kf_t({\bar{x}},y,z)\right| _{L_k(\mathbb {R}^d)}&\le h_r(y)\left| x-{\bar{x}}\right| ^\alpha \end{aligned}$$

  (3.2)

  for all $x,{\bar{x}},y\in \varOmega$, $z\in \overline{Z_t}\cap {\bar{B}}_r(0)$,

• (iii) for all $r>0$ there exists a function $c_r:\mathbb {R}_+\times \varOmega \rightarrow \mathbb {R}_+$ satisfying the limit relation $\lim _{\delta \searrow 0}\sup _{y\in \varOmega }c_r(\delta ,y)=0$, such that $\left| z-{\bar{z}}\right| \le \delta$ implies

  $$\begin{aligned}{} & {} \left| D_3^kf_t(x,y,z)-D_3^kf_t(x,y,{\bar{z}})-\left[ D_3^kf_t({\bar{x}},y,z)-D_3^kf_t({\bar{x}},y,{\bar{z}})\right] \right| _{L_k(\mathbb {R}^d)}\nonumber \\{} & {} \quad \le c_r(\delta ,y)\left| x-{\bar{x}}\right| ^\alpha \quad \text {for all }x,{\bar{x}},y\in \varOmega ,\,{\bar{z}}\in \overline{Z_t}\cap {\bar{B}}_r(0). \end{aligned}$$

  (3.3)

Let $C(\varOmega ,\mathbb {R}^d)$ denote the set of continuous functions $u:\varOmega \rightarrow \mathbb {R}^d$ equipped with the norm $\left\| u\right\| _0:=\sup _{x\in \varOmega }\left| u(x)\right|$. If $\alpha \in (0,1]$, then functions $u:\varOmega \rightarrow \mathbb {R}^d$ having a bounded Hölder constant

$$\begin{aligned}{}[u]_\alpha :=\sup _{\begin{array}{c} x,{\bar{x}}\in \varOmega \\ x\ne {\bar{x}} \end{array}}\frac{\left| u(x)-u({\bar{x}})\right| }{\left| x-{\bar{x}}\right| ^\alpha }<\infty \end{aligned}$$

are called $\alpha$-Hölder (Lipschitz in case $\alpha =1$) and $C^\alpha (\varOmega ,\mathbb {R}^d)\subset C(\varOmega ,\mathbb {R}^d)$ denotes the entity of all such functions. It is a Banach space when equipped with the norm

$$\begin{aligned} \left\| u\right\| _\alpha := {\left\{ \begin{array}{ll} \left\| u\right\| _0,&{}\alpha =0,\\ \max \left\{ \left\| u\right\| _0,[u]_\alpha \right\} ,&{}\alpha \in (0,1]. \end{array}\right. } \end{aligned}$$

(3.4)

Since the compact domain $\varOmega$ is fixed throughout, we conveniently abbreviate

$$\begin{aligned} C_d^\alpha&:=C^\alpha (\varOmega ,\mathbb {R}^d),&C_d^0&:=C(\varOmega ,\mathbb {R}^d) \end{aligned}$$

and obtain the open sets $U_t:=\left\{ u\in C_d^\alpha :\,u(\varOmega )\subset Z_t\right\}$ for all $t\in \mathbb {Z}$.

For our subsequent analysis it is important to note that Hypothesis 3.1 implies the corresponding assumptions made in [28, Sect. 2]. In detail, one has:

Proposition 3.1

(Properties of ($I_{0}$)) Let $t\in \mathbb {Z}$. If Hypothesis 3.1 holds, then the Urysohn operator is well-defined, completely continuous and of class $C^m$ with compact derivative

(3.5)

Combined with the solution identity this shows that entire solutions $\phi$ to ($I_{0}$) inherit the smoothness of the kernel function, i.e. $\phi _t\in C_d^\alpha$, $t\in \mathbb {Z}$. Yet for kernel functions of convolution type a higher smoothness can be expected (cf. [28, Sect. 2.3]).

Proof

Above all, ($I_{0}$) and (3.1) show that is $\theta _0$-periodic in t. The results from [28] formulated in an abstract measure-theoretical set-up apply to with the $\kappa$-dimensional Lebesgue measure $\mu =\lambda _\kappa$. By [28, Thm. 2.6], is well-defined and due to [28, Cor. 2.7(i)] also completely continuous. In [28, Thm. 2.12] it is shown that is of class $C^m$ and [23, p. 89, Prop. 6.5] implies that , $u\in U_t$, is compact. $\square$

Corollary 3.1

Let $t\in \mathbb {Z}$ and $2\le m$. If for every $r>0$ there exists a continuous function $l_r:\varOmega ^2\rightarrow \mathbb {R}_+$ with

$$\begin{aligned} \left| D_3f_t(x,y,z)-D_3f_t(x,y,{\bar{z}})\right| _{L(\mathbb {R}^d)}\le l_r(x,y)\left| z-{\bar{z}}\right| \quad \text {for all }x,y\in \varOmega \end{aligned}$$

and $z,{\bar{z}}\in Z_t\cap {\bar{B}}_r(0)$, then is Lipschitz on $C_d^0$-bounded sets, that is, for each $r>0$ there exists a $L_r\ge 0$ such that

(3.6)

with the Lipschitz constant $L_r:=\max \left\{ \sup _{\xi \in \varOmega }\int _{\varOmega } l_r(\xi ,y)\,{\mathrm d}y,\int _\varOmega h_r(y)\,{\mathrm d}y\right\}$.

Proof

Let $v\in C_d^\alpha$, $r>0$ and $u,{\bar{u}}\in U_t\cap {\bar{B}}_r(0,C_d^0)$.

• (I) We derive that

  

  Hence, holds after passing to the least upper bound over all $x\in \varOmega$.

• (II) With $Z_t\subseteq \mathbb {R}^d$ also $U_t\subseteq C_d^\alpha$ is convex. Therefore, the Mean Value Theorem [19, p. 341, Thm. 4.2] applies and shows for $x,{\bar{x}}\in \varOmega$ that

  

  Consequently Hypothesis 3.1(ii) leads to

  

  which guarantees that . Referring to (3.4) this implies the local Lipschitz estimate (3.6). $\square$

Along with IDEs ($I_{0}$) we now consider their Nyström discretizations. They are based on quadrature ($\kappa =1$) or cubature rules ($\kappa >1$), i.e. a family of mappings

determined by a grid $\varOmega _n\subset \varOmega$ of finitely many nodes $\eta \in \varOmega _n$ and weights $w_\eta \ge 0$; the dependence of $w_\eta$ on $n\in \mathbb {N}$ is suppressed here. A rule ($Q_n$) is called (cf. [16])

• convergent, if $\lim _{n\rightarrow \infty }Q^nu=\int _\varOmega u(y)\,{\mathrm d}y$ holds for all $u\in C_d^0$,

• stable, provided the weights satisfy

  $$\begin{aligned} W&:=\sup _{n\in \mathbb {N}}W_n<\infty ,&W_n&:=\sum _{\eta \in \varOmega _n}w_\eta . \end{aligned}$$

  (3.7)

Thanks to [16, p. 20, Thm. 1.4.17], convergence implies stability.

In order to evaluate the right-hand side of ($I_{0}$) approximately, we replace the integral by a convergent integration rule ($Q_n$), $n\in \mathbb {N}$. The resulting Nyström method (see [4, 16] for integral equations) yields the family of difference equations

Proposition 3.2

(Properties of ($I_n$)) Let $t\in \mathbb {Z}$. If Hypothesis 3.1 holds, then the discrete Urysohn operator , $n\in \mathbb {N}$, is well-defined, completely continuous and of class $C^m$ with compact derivative

(3.8)

Moreover, if ($Q_n$) is stable, then is equidifferentiable, $D$ are uniformly continuous on bounded sets uniformly in $n\in \mathbb {N}$ and is equicontinuous.

Proof

The grids $\varOmega _n$, $n\in \mathbb {N}$, are a family of compact and discrete subsets of $\varOmega$. If we equip them with the measure $\mu (\varOmega _n):=\sum _{\eta \in \varOmega _n}w_\eta$, then due to [28, Ex. 2.2 and Rem. 2.5] the abstract measure-theoretical integral from [28] becomes

$$\begin{aligned} \int _{\varOmega _n}f_t(x,y,u(y))\,{\mathrm d}\mu (y) = \sum _{\eta \in \varOmega _n}w_\eta f_t(x,\eta ,u(\eta ))\quad \text {for all }x\in \varOmega \end{aligned}$$

and leads to the discrete integral operators in ($I_n$). Given this, well-definedness, complete continuity and smoothness of result from [28] as in the proof of Proposition 3.1. From now on, assume that ($Q_n$) is stable and choose $u\in U_t$.

• (I) Claim: is equidifferentiable. For functions $h\in C_d^\alpha$ the remainder terms [28, (16) resp. (18)] become

  $$\begin{aligned} r_0(h)&=\sup _{\vartheta \in [0,1]}\bigg \Vert \sum _{\eta \in \varOmega _n}w_\eta \left[ D_3f_t(\cdot ,\eta ,(u+\vartheta h)(\eta ))-D_3f_t(\cdot ,\eta ,u(\eta ))\right] \bigg \Vert _0,\\ \rho _0(h)&=\int _0^1\sum _{\eta \in \varOmega _n}w_\eta {\bar{c}}_r^1(\vartheta \left\| h\right\| _0,y)\,{\mathrm d}\vartheta \le \sum _{\eta \in \varOmega _n}w_\eta {\bar{c}}_r^1(\left\| h\right\| _0,\eta ). \end{aligned}$$

  Now it follows from (3.7) that $\lim _{h\rightarrow \infty }r_0(h)=\lim _{h\rightarrow \infty }\rho _0(h)=0$ hold uniformly in $n\in \mathbb {N}$. This yields the claimed equidifferentiability.

• (II) Claim: are uniformly continuous on bounded sets uniformly in $n\in \mathbb {N}$ (and thus is equicontinuous). Let $\varepsilon >0$, $v\in C_d^\alpha$ and given $u,{\bar{u}}\in U_t$ choose $r>0$ so large that $\left\| u\right\| _0,\left\| {\bar{u}}\right\| _0\le r$ holds. Because the (extended) derivative $D_3f_t:\varOmega ^2\times \overline{Z_t}\rightarrow L(\mathbb {R}^d)$ is uniformly continuous on the compact set $\varOmega ^2\times (\overline{Z_t}\cap {\bar{B}}_r(0))$, there exists a $\delta _1>0$ such that

  $$\begin{aligned} \left| z-{\bar{z}}\right|<\delta _1 \Rightarrow \left| D_3f_t(x,y,z)-D_3f_t(x,y,{\bar{z}})\right| <\tfrac{\varepsilon }{2W} \quad \text {for all }z,{\bar{z}}\in Z_t\cap {\bar{B}}_r(0) \end{aligned}$$

  and $x,y\in \varOmega$. If $u,{\bar{u}}\in U_t$ satisfy $\left\| u-{\bar{u}}\right\| _0<\delta _1$, then we obtain $\left| u(y)-{\bar{u}}(y)\right| <\delta _1$ for all $y\in \varOmega$. First, this implies

  

  and passing to the supremum over $x\in \varOmega$ yields Second, from Hypothesis 3.1(iii) there exists a $\delta _2>0$ such that $\sup _{y\in \varOmega }c_r(\delta ,y)<\tfrac{\varepsilon }{2W}$ for every $\delta \in (0,\delta _2]$ and consequently $\left\| u-{\bar{u}}\right\| _0<\delta _2$ guarantees for all $x,{\bar{x}}\in {\varOmega }$ that

  

  and therefore . Referring to (3.4) this results in

  

  for all $n\in \mathbb {N}$. Since $v\in C_d^\alpha$ was arbitrary, this readily implies the claim. $\square$

3.1 Hölder continuous kernel functions

We say an integration rule ($Q_n$) has consistency order $\alpha \in (0,1]$ (cf. [16, p. 21, Def. 1.4.19]), if there exists a $c_0\ge 0$ with

$$\begin{aligned} \left| \int _\varOmega u(y)\,{\mathrm d}y-Q^nu\right| \le \frac{c_0}{n^\alpha }\left\| u\right\| _\alpha \quad \text {for all }u\in C_d^\alpha . \end{aligned}$$

Example 3.1

(Quadrature rules) Let $\varOmega =[a,b]$ and $n\in \mathbb {N}$. The (left) resp. (right) rectangular rules

$$\begin{aligned} Q_{LR}^nu&:=\tfrac{b-a}{n}\sum _{j=0}^{n-1}u(a+j\tfrac{b-a}{n}),&Q_{RR}^nu&:=\tfrac{b-a}{n}\sum _{j=1}^nu(a+j\tfrac{b-a}{n}) \end{aligned}$$

are convergent and satisfy the quadrature error (cf. [8, p. 52, Theorem])

$$\begin{aligned} \left| \int _a^bu(y)\,{\mathrm d}y-Q_i^nu\right| \le \frac{(b-a)^{\alpha +1}}{n^\alpha }[u]_\alpha \quad \text {for }i\in \left\{ LR,RR\right\} . \end{aligned}$$

Also the midpoint rule $Q_M^nu:=\tfrac{b-a}{n}\sum _{j=0}^{n-1}u(a+(j+\tfrac{1}{2})\tfrac{b-a}{n})$ is convergent and as in [8, p. 52, Theorem] one derives the quadrature error

$$\begin{aligned} \left| \int _a^bu(y)\,{\mathrm d}y-Q_M^nu\right| \le \frac{(b-a)^{\alpha +1}}{2^\alpha n^\alpha }[u]_\alpha . \end{aligned}$$

The trapezoidal rule $Q_T^nu:=\tfrac{1}{2}(Q_{LR}^nu+Q_{RR}^nu)$ is convergent with the same quadrature error as for the rectangular rules. Finally, let $n\in \mathbb {N}$ be even. Representing the Simpson rule as convex combination $Q_S^nu:=\tfrac{2}{3}Q_M^{n/2}u+\tfrac{1}{3}Q_T^{n/2}u$, one obtains

$$\begin{aligned} \left| \int _a^bu(y)\,{\mathrm d}y-Q_S^nu\right| \le \frac{2+2^\alpha }{3} \frac{(b-a)^{\alpha +1}}{n^\alpha }[u]_\alpha . \end{aligned}$$

The next two results provide sufficient conditions on the kernel functions $f_t$ such that the assumptions (2.7) or (2.8) are satisfied for Nyström discretizations ($I_n$).

Proposition 3.3

(Convergence of ) Let $t\in \mathbb {Z}$. Suppose Hypothesis 3.1 holds and that for every $r>0$ there exists a $l_r^0\ge 0$ such that

$$\begin{aligned} \left| f_t(x,y,z)-f_t(x,{\bar{y}},{\bar{z}})\right| \le l_r^0\max \left\{ \left| y-{\bar{y}}\right| ^\alpha ,\left| z-{\bar{z}}\right| \right\} \end{aligned}$$

(3.9)

for all $x,y,{\bar{y}}\in \varOmega$ and $z,{\bar{z}}\in Z_t\cap {\bar{B}}_r(0)$. If ($Q_n$) has consistency order $\alpha$, then for every $r>0$ there exists a $c_r^0\ge 0$ such that

(3.10)

The magnitude of the constant $c_r^0$ is increasing in the Hölder norm of $u\in U_t$.

Proof

Let $t\in \mathbb {Z}$, $r>0$ and $u\in U_t\cap {\bar{B}}_r(0,C_d^\alpha )$. Because ($Q_n$) has consistency order $\alpha$, there exists a $c_0\ge 0$ such that for all $x\in \varOmega$. First, one has $\left\| f_t(x,\cdot ,u(\cdot ))\right\| _0\le \sup _{\xi ,y\in \varOmega }\left| f_t(\xi ,y,u(y))\right| =:b_t$ for every $x\in \varOmega$. Second, due to the assumption (3.9) we conclude

$$\begin{aligned} \left| f_t(x,y,u(y))-f_t(x,{\bar{y}},u({\bar{y}}))\right| \le l_r^0\max \left\{ 1,[u]_\alpha \right\} \left| y-{\bar{y}}\right| ^\alpha \quad \text {for all }y,{\bar{y}}\in \varOmega \end{aligned}$$

and thus $[f_t(x,\cdot ,u(\cdot ))]_\alpha \le l_r^0\max \left\{ 1,[u]_\alpha \right\}$ holds. In conclusion, because of (3.4) we arrive at $\left\| f_t(x,\cdot ,u(\cdot ))\right\| _\alpha \le \max \left\{ b_t,l_r^0\max \left\{ 1,[u]_\alpha \right\} \right\}$ for every $x\in \varOmega$ and consequently choose $c_r^0:=\max _{t=1}^{\theta _0}\left\{ b_t,l_r^0\max \left\{ 1,r\right\} \right\}$. $\square$

Proposition 3.4

(Convergence of ) Let $t\in \mathbb {Z}$. Suppose Hypothesis 3.1 holds and that for every $r>0$ there exist constants

• (iv) $l_r^1\ge 0$ such that for all $x,y,{\bar{y}}\in \varOmega$ and $z\in Z_t\cap {\bar{B}}_r(0)$ one has

  $$\begin{aligned} \left| D_3f_t(x,y,z)-D_3f_t(x,{\bar{y}},{\bar{z}})\right| _{L(\mathbb {R}^d)} \le l_r^1\max \left\{ \left| y-{\bar{y}}\right| ^\alpha ,\left| z-{\bar{z}}\right| \right\} , \end{aligned}$$

• (v) $\gamma _r\ge 0$ such that for all $x,{\bar{x}},y,{\bar{y}}\in \varOmega$ and $u\in U_t\cap {\bar{B}}_r(0,C_d^\alpha )$ one has

  $$\begin{aligned}{} & {} \left| D_3f_t(x,y,u(y))-D_3f_t({\bar{x}},y,u(y)) -\left[ D_3f_t(x,{\bar{y}},u({\bar{y}}))-D_3f_t({\bar{x}},{\bar{y}},u({\bar{y}}))\right] \right| _{L(\mathbb {R}^d)}\nonumber \\{} & {} \quad \le \gamma _r\left| x-{\bar{x}}\right| ^\alpha \left| y-{\bar{y}}\right| ^\alpha . \end{aligned}$$

  (3.11)

If ($Q_n$) has consistency order $\alpha$, then for every $r>0$ there exists a $c_r^1\ge 0$ such that

(3.12)

Sufficient conditions for (3.11) to hold were given in [27, Rem. 1] on convex $\varOmega \subset \mathbb {R}^\kappa$. Furthermore, the explicit form of the constant $c_r^1$ can be obtained from [27, (11)].

Proof

Let $t\in \mathbb {Z}$, $r>0$ and $u\in U_t\cap {\bar{B}}_r(0,C_d^\alpha )$ be fixed. By Proposition 3.1 the derivative of is for all $v\in C_d^\alpha$. Given this, our goal is to apply the convergence result [27, Thm. 2] with the corresponding kernel $k_t(x,y):=D_3f_t(x,y,u(y))$, whose assumptions are verified next:

ad (i): Thanks to $\left| u(y)\right| \le r$ it holds $\left| k_t(x,y)-k_t({\bar{x}},y)\right| \le h_r(y)\left| x-{\bar{x}}\right| ^\alpha$ for all $x,{\bar{x}}\in \varOmega$ due to (3.2) and therefore $[k_t(\cdot ,y)]_\alpha \le \sup _{\eta \in \varOmega }h_r(\eta )$ for all $y\in \varOmega$.

ad (ii): The assumption (iv) and $[u]_\alpha \le r$ yield

$$\begin{aligned} \left| k_t(x,y)-k_t(x,{\bar{y}})\right| \le l_r^1\max \left\{ 1,[u]_\alpha \right\} \left| y-{\bar{y}}\right| ^\alpha \quad \text {for all }y,{\bar{y}}\in \varOmega \end{aligned}$$

and thus $[k_t(x,\cdot )]_\alpha \le l_r^1\max \left\{ 1,r\right\}$ for all $x\in \varOmega$.

ad (iii): As consequence of our assumption (3.11) one obtains for $x,{\bar{x}},y,{\bar{y}}\in \varOmega$ that $\left| k_t(x,y)-k_t({\bar{x}},y)-\left[ k_t(x,{\bar{y}})-k_t({\bar{x}},{\bar{y}})\right] \right| \le \gamma _r\left| x-{\bar{x}}\right| ^\alpha \left| y-{\bar{y}}\right| ^\alpha$.

Finally, combining (i–iii) with the consistency order $\alpha$ of ($Q_n$) shows (3.12). $\square$

Combining the assumptions of Propositions 3.1–3.4 and Corrollary 3.1 yields

Theorem 3.1

(Saddle-point structure of ($I_{0}$), $C^\alpha$-case) Suppose Hypothesis 3.1 holds with $2\le m$ and $\phi ^0$ is a weakly hyperbolic $\theta _1$-periodic solution to ($I_{0}$). If

• (iv) ($Q_n$) has consistency order $\alpha \in (0,1]$,

then there exist constants $K_*,K_+,K_-\ge 0$ and $N_1\in \mathbb {N}$ such that the following holds for all $n\ge N_1$: The associate weakly hyperbolic and $\theta$-periodic solutions $\phi ^n$ to ($I_n$) satisfy

$$\begin{aligned} \sup _{t\in \mathbb {Z}}\left\| \phi _t^n-\phi _t^0\right\| _\alpha \le \frac{K_*c_0}{n^\alpha }. \end{aligned}$$

(3.13)

If $\phi ^0$ is even hyperbolic, then for each $\tau \in \mathbb {Z}$ one has the estimates

• (a) $\left\| w_+^n(\tau ,v)-w_+^0(\tau ,v)\right\| _\alpha \le \tfrac{4K}{1-\beta }\tfrac{K_+}{n^\alpha }\sup _{\tau \le t}\left\| \phi _t\right\| _\alpha$ for all $v\in B_{\rho _1}(0,R(P_\tau ))$,

• (b) $\left\| w_-^n(\tau ,v)-w_-^0(\tau ,v)\right\| _\alpha \le \tfrac{4K}{1-\beta }\tfrac{K_-}{n^\alpha }\sup _{t\le \tau }\left\| \phi _t\right\| _\alpha$ for all $v\in B_{\rho _1}(0,N(P_\tau ))$,

with the forward resp. backward solution $\phi$ to the IDE ($I_{0}$) from Theorem 2.3.

Proof

Let $t\in \mathbb {Z}$ and $r:=\max _{t=1}^{\theta _1}\left\| \phi _t^0\right\| _\alpha$. It results from Propositions 3.1 and 3.2 that ($I_n$), $n\in \mathbb {N}_0$, satisfy the assumptions (i), (ii) of Theorem 2.2. Moreover, Proposition 3.3 implies (2.7) with $\varGamma _0(\varrho ):=c_0c_r^0\varrho ^\alpha$, while Proposition 3.4 guarantees that (2.8) holds. Hence, Theorem 2.2 applies and yields (3.13) with $K_*:=K_0c_r^0$. In particular, for $N_0\in \mathbb {N}$ and $\rho _0>0$ from Theorem 2.2 there is a $N_1\ge N_0$ so that $\sup _{t\in \mathbb {Z}}\left\| \phi _t^n-\phi _t^0\right\| _\alpha <\tfrac{\rho _0}{2}$ for all $n\ge N_1$.

• (a) Let $\rho _1>0$ be so small that the sequence $(\phi _t)_{\tau \le t}$ from Theorem 2.3(a) satisfies $\left\| \phi _t\right\| _\alpha <\tfrac{\rho _0}{2}$ for all $\tau \le t$; such a $\rho _1$ exists since the sequence is contained in the stable fiber bundle of $\phi ^0$. Furthermore, for each $\vartheta \in [0,1]$ we obtain

  $$\begin{aligned} \left| \phi _t^n(y)+\vartheta \phi _t(y)-\phi _t^0(y)\right| \le \left\| \phi _t^n+\vartheta \phi _t-\phi _t^0\right\| _0 < \rho _0\quad \text {for all }y\in \varOmega . \end{aligned}$$

  Now set ${\bar{r}}:=r+\rho _0$. Combining the triangle inequality, Corrollary 3.1 and Proposition 3.4 yields that there exist $L_{{\bar{r}}}\ge 0$ such that

  

  and with $K_+:=c_0c_{{\bar{r}}}^1+L_{{\bar{r}}}K_*c_0$ we obtain

  

  $n\ge N_1$ and $\tau \le t$. Hence, the claimed estimate follows from (2.11).

• (b) As in (a), applying (2.12) rather than (2.11) leads to the assertion. $\square$

3.2 Differentiable kernel functions

Convergence rates improving the consistency order $\alpha \in (0,1]$ obtained in Theorem 3.1 can be expected for integrands in ($I_{0}$) being differentiable in $y\in \varOmega$. Here we follow the convention to consider a function on a not necessarily open set $\varOmega \subset \mathbb {R}^\kappa$ as differentiable, if it allows a differentiable extension to an open superset of $\varOmega$.

Given p-times continuously differentiable functions $u:\varOmega \rightarrow \mathbb {R}^d$ assume that ($Q_n$) allows a quadrature or cubature error of the form (see [8])

$$\begin{aligned} \left| \int _\varOmega u(y)\,{\mathrm d}y-Q^n u\right| \le \frac{c_p}{n^p} \sup _{x\in \varOmega }\left| D^{p}u(x)\right| \quad \text {for all }n\in \mathbb {N} \end{aligned}$$

(3.14)

with constants $c_p\ge 0$.

A smooth framework allows the following improvement of Proposition 3.3:

Proposition 3.5

(Higher order convergence of ) Let $t\in \mathbb {Z}$, $p\in \mathbb {N}$ and $\varOmega$ be convex. Suppose the kernel function $f_t:\varOmega ^2\times Z_t\rightarrow \mathbb {R}^d$ fulfills:

• (iv) The partial derivative $D_1f_t:\varOmega ^2\times Z_t\rightarrow L(\mathbb {R}^\kappa ,\mathbb {R}^d)$ exists,

• (v) both $f_t,D_1f_t$ are of class $C_{(2,3)}^{p}$.

If ($Q_n$) satisfies (3.14), then for every $r>0$ there exists a ${\bar{c}}_r^0\ge 0$ such that

(3.15)

and p-times continuously differentiable functions $u\in U_t$.

Proof

Let $t\in \mathbb {Z}$ and with $u\in U_t$ of class $C^p$ it is convenient to define

$$\begin{aligned} F_t^{(1)}:\varOmega ^2&\rightarrow L(\mathbb {R}^\kappa ,\mathbb {R}^d),&F_t^{(1)}(x,y)&:=D_1f_t(x,y,u(y)). \end{aligned}$$

The estimate (3.15) for the $\left\| \cdot \right\| _0$-norm is an immediate consequence of the error estimate (3.14) and the higher-order chain rule. Let $x,{\bar{x}}\in \varOmega$ and the Mean Value Theorem [19, p. 341, Thm. 4.2] gives

from which Fubini’s theorem [19, p. 162, Thm. 8.4] yields

and passing to the norm implies

Hence, and if we abbreviate ${\bar{c}}_r^0:=\max \left\{ 1,({{\,\textrm{diam}\,}}\varOmega )^{1-\alpha }\right\} \max _{i=0}^1\sup _{x,y\in \varOmega } \left| D_2^{p}F_t^{(i)}(x,y)\right|$, then (3.4) implies the claimed estimate (3.15). $\square$

Smooth functions $f_t$ and reference solutions $\phi ^0$ allow better convergence rates. Indeed under the assumptions of Propositions 3.1, 3.2 and 3.4, 3.5, as well as Corrollary 3.1 results:

Theorem 3.2

(Saddle-point structure of ($I_{0}$), $C^p$-case) Let $\varOmega \subset \mathbb {R}^\kappa$ be convex. Suppose Hypothesis 3.1 holds with $\max \left\{ 2,p\right\} \le m$ and $\phi ^0$ is a weakly hyperbolic $\theta _1$-periodic solution to ($I_{0}$). If

• (iv) ($Q_n$) is stable, has consistency order $\alpha \in (0,1]$ and satisfies (3.14),

• (v) the partial derivatives $D_1^kf_t:\varOmega ^2\times Z_t\rightarrow L_k(\mathbb {R}^\kappa ,\mathbb {R}^d)$ exists for $0\le k\le p$,

• (vi) both $f_t$ and $D_1f_t$ are of class $C_{(2,3)}^{p}$,

then there exist constants $K_*,K_+,K_-\ge 0$ and $N_1\in \mathbb {N}$ such that the following holds for all $n\ge N_1$: The associate weakly hyperbolic and $\theta$-periodic solutions $\phi ^n$ to ($I_n$) satisfy

$$\begin{aligned} \sup _{t\in \mathbb {Z}}\left\| \phi _t^n-\phi _t^0\right\| _\alpha \le \frac{K_*c_p}{n^p}. \end{aligned}$$

(3.16)

If $\phi ^0$ is even hyperbolic, then for each $\tau \in \mathbb {Z}$ one has the estimates

• (a) $\left\| w_+^n(\tau ,v)-w_+^0(\tau ,v)\right\| _\alpha \le \tfrac{4K}{1-\beta }\tfrac{K_+}{n^p}(1+\sup _{\tau \le t}\left\| \phi _t\right\| _\alpha )$ for p-times continuously differentiable $v\in B_{\rho _1}(0,R(P_\tau ))$,

• (b) $\left\| w_-^n(\tau ,v)-w_-^0(\tau ,v)\right\| _\alpha \le \tfrac{4K}{1-\beta }\tfrac{K_-}{n^p}(1+\sup _{t\le \tau }\left\| \phi _t\right\| _\alpha )$ for $v\in B_{\rho _1}(0,N(P_\tau ))$

with the forward resp. backward solution $\phi$ to the IDE ($I_{0}$) from Theorem 2.3.

Proof

Let $t\in \mathbb {Z}$. Above all, as entire solutions to ($I_{0}$) the functions $\phi _t^0$ are of class $C^p$ due to (v) and [19, p. 355, Thm. 8.1]. By means of Proposition 3.5 the estimate (3.16) results as in the above proof of Theorem 3.1, with (3.10) replaced by (3.15).

(a) As in the above proof of Theorem 3.1 one obtains

(3.17)

for all $\vartheta \in [0,1]$ and $n\ge N_1$. Proposition 3.1 yields the explicit derivative

and for the integrand on the right-hand side we observe: Thanks to (v) the periodic solution $\phi ^n$ consists of $C^p$-functions $\phi _t^n:\varOmega \rightarrow \mathbb {R}^d$ and also forward solutions to the IDE ($I_{0}$) are of class $C^p$, i.e. $\phi _t$ is a $C^p$-function for all $t>\tau$. For $t=\tau$ we have $\phi _\tau =v+w_+^0(\tau ,v)$ and because $w_+^0(\tau ,\cdot )$ is of class $C^m$ by Theorem 2.3(a) and $p\le m$, with v also the initial function $\phi _\tau$ is p-times continuously differentiable. Due to (vi) this yields that the integrand $D_3f_t(x,\cdot ,\phi _t^n(\cdot )+\vartheta \phi _t(\cdot ))\phi _t(\cdot ):\varOmega \rightarrow \mathbb {R}^d$ is of class $C^p$ and the estimate (3.14) applies. Hence, as in the proof of Proposition 3.5 one shows that there exists a ${\tilde{C}}\ge 0$ so that and whence (3.17) yields for all $\vartheta \in [0,1]$, $n\ge N_1$ and $\tau \le t$ that

Therefore, the estimate (a) follows from (2.11).

(b) As above in (a), applying (2.12) rather than (2.11) leads to the claimed estimate. Note here that $(\phi _t)_{t\le \tau }$ is a backward solution to ($I_{0}$) and consequently consists of $C^p$-solutions. Whence, also the initial value $\phi _\tau =v+w_-^0(\tau ,v)$ is of class $C^p$ and it is not necessary to assume v to be smooth. $\square$

4 Numerical illustrations

In order to illustrate our theoretical results, we consider an autonomous logistic IDE with separable kernel in 1d. It allows to analyze the behavior of periodic solutions under approximation (largely) explicitly. We demonstrate this by means of Nyström discretizations based on several quadrature rules (taken from [11, pp. 361ff]).

Let $L>0$. On the interval $\varOmega =[-\tfrac{L}{2},\tfrac{L}{2}]$ consider the kernel [18, Sect. 6]

$$\begin{aligned} k(x,y):=\frac{\pi }{4L} {\left\{ \begin{array}{ll} \cos \left( \tfrac{\pi }{2L}(x-y)\right) ,&{}\left| x-y\right| \le L,\\ 0,&{}\text {else}, \end{array}\right. } \end{aligned}$$

which is separable, since it allows the representation

$$\begin{aligned} k(x,y)&=\cos \left( \tfrac{\pi }{2L}x\right) \cos \left( \tfrac{\pi }{2L}y\right) +\sin \left( \tfrac{\pi }{2L}x\right) \sin \left( \tfrac{\pi }{2L}y\right) \\&= \sqrt{\tfrac{\pi +2}{2\pi }L}\cos \left( \tfrac{\pi }{2L}y\right) e_1(x) + \sqrt{\tfrac{\pi -2}{2\pi }L}\sin \left( \tfrac{\pi }{2L}y\right) e_2(x) \end{aligned}$$

with the linearly independent functions $e_1,e_2:[-\tfrac{L}{2},\tfrac{L}{2}]\rightarrow \mathbb {R}$,

$$\begin{aligned} e_1(x)&:=\sqrt{\tfrac{2\pi }{(\pi + 2) L}}\cos \left( \tfrac{\pi x}{2L}\right) ,&e_2(x)&:=\sqrt{\tfrac{2\pi }{(\pi - 2) L}}\sin \left( \tfrac{\pi x}{2L}\right) \end{aligned}$$

being $L^2$-orthonormal, i.e. $\int _{-L/2}^{L/2}e_i(y)e_j(y)\,{\mathrm d}y=\delta _{ij}$ for $1\le i,j\le 2$ holds with the Kronecker symbol $\delta _{ij}\in \left\{ 0,1\right\}$. Given this, the autonomous Hammerstein IDE

$$\begin{aligned} u_{t+1}(x)=a\int _{-L/2}^{L/2}k(x,y)u_t(y)(1-u_t(y))\,{\mathrm d}y \end{aligned}$$

(4.1)

depends on a growth parameter $a>0$. The ansatz $u_t:=v_te_1+w_te_2$ with coefficients $v_t,w_t\in \mathbb {R}$ leads to the autonomous planar difference equation

$$\begin{aligned} {\left\{ \begin{array}{ll} v_{t+1}= a\left( \tfrac{\pi +2}{8}v_t- \tfrac{10}{3}\sqrt{\tfrac{\pi }{{(\pi + 2)} L}}v_t^2- \tfrac{2 {(\pi + 2)}}{3{(\pi - 2)}}\sqrt{\tfrac{\pi }{{(\pi + 2)} L}}w_t^2\right) ,\\ w_{t+1}= a\left( \tfrac{\pi -2}{8} -\tfrac{1}{3}\sqrt{\tfrac{\pi }{{(\pi + 2)} L}}v_t\right) w_t, \end{array}\right. } \end{aligned}$$

(4.2)

which fully describes the dynamics of (4.1). It has the invariant set $\mathbb {R}\times \left\{ 0\right\}$ and when restricted to this v-axis the behavior is determined by the scalar difference equation

$$\begin{aligned} v_{t+1} = av_t\Bigl (\tfrac{\pi +2}{8}-\tfrac{10}{3}\sqrt{\tfrac{\pi }{{(\pi + 2)} L}}v_t\bigr ) =: h_a(v_t). \end{aligned}$$

(4.3)

Fig. 2

Left: The solid and the dashed lines are branches of 5-periodic solutions to the scalar difference equation (4.3), which in turn represent solutions to the logistic IDE (4.1) (traversed in the sequence red, orange, green, blue, black) Right: Floquet spectrum for (4.2) along the 5-periodic solutions indicated as solid resp. dashed lines

If we fix $L=\pi$, then (4.3) possesses a pair of 5-periodic solutions emanating from a supercritical fold bifurcation at $a\approx 5.8164$. These solutions and their hyperbolicity are illustrated in Fig. 2 (the graphics were obtained using a simple continuation scheme with Brent’s method [29, p. 256, Sect. 6.2.3] as corrector). In particular, for $a=6$ we solve $h_a^5(v_0^*)=v_0^*$ in order to arrive at the 5-periodic solution

$$\begin{aligned} (v_t^*,w_t^*) ={\left\{ \begin{array}{ll} (1.1402499962803,0),&{}t\mod 5=0,\\ (1.5300677569209,0),&{}t\mod 5=1,\\ (0.73794766757321,0),&{}t\mod 5=2,\\ (1.6448652484936,0),&{}t\mod 5=3,\\ (0.37693991665699,0),&{}t\mod 5=4 \end{array}\right. } \end{aligned}$$

of (4.2) resulting in the 5-periodic solution $\phi _t^0:=v_t^*e_1$ to the IDE (4.1). This provides us with a precisely known periodic reference solution for the logistic IDE (4.1) in order to illustrate Theorem 2.2 when applied to its Nyström discretizations ($I_n$).

In order to determine the 5-periodic solutions $\phi ^n$ of ($I_n$) we apply a Newton solver to the fixed point problem having ${\hat{\phi }}^0$ as initial value. For various common summed integration methods we display the development of the $C^0$-error

$$\begin{aligned} {{\,\textrm{err}\,}}_n=\sup _{t=1}^\theta \sup _{\eta \in \varOmega _n}\left| \phi _t^n(\eta )-\phi _t^0(\eta )\right| , \end{aligned}$$

as well as the $C^\alpha$-error, $\alpha =1$,

$$\begin{aligned} {{\,\textrm{err}\,}}_n^1:=\max \left\{ {{\,\textrm{err}\,}}_n,\sup _{t=1}^\theta \sup _{\begin{array}{c} \eta ,\bar{\eta }\in \varOmega _n\\ \eta \ne \bar{\eta } \end{array}}\tfrac{\left| \phi _t^n(\eta )-\phi _t^0(\bar{\eta })\right| }{\left| \eta -\bar{\eta }\right| }\right\} \end{aligned}$$

over the numbers of nodes $d_n$ in the error diagrams Figs. 3 and 4.

Fig. 3

$C^0$-errors (left) and $C^\alpha$-errors (with $\alpha =1$, right) for the rectangular (square $\square$), midpoint (asterisk $*$), trapezoidal (diamond $\diamond$) and Simpson ($+$) rule

They illustrate that the solutions $\phi ^n$ approximate $\phi ^0$ preserving the order of the particular quadrature rule. This confirms Theorem 2.2 and specifically the estimate (3.16) of Theorem 3.2. An exception is the fast convergence of the Clenshaw-Curtis method (see Fig. 4) due to the fact that the functions to be approximated are cosine shaped.

Fig. 4

$C^0$-errors (left) and $C^\alpha$-errors (with $\alpha =1$, right) for the Chebyshev (square $\square$), 4th order Gauß (asterisk $*$), 6th order Gauß (diamond $\diamond$) and Clenshaw-Curtis ($+$) rule

5 Summary and perspectives

This paper studied the local dynamics of abstract periodic difference equations of the form $(\varDelta _0)$ near periodic solutions under discretizations. For sufficiently accurate approximations, in Theorem 2.1 it is established that hyperbolicity of a periodic linear equation persists. Applied to the variational equations ($V_n$), Theorem 2.2 guarantees that also hyperbolic periodic solutions persist as hyperbolic solutions to the discretizations ($\varDelta _n$). The same holds true for their associated stable and unstable manifolds (fiber bundles), as shown in Theorem 2.3.

These abstract perturbation results might apply to various spatial discretizations of evolutionary equations, such as projection methods. We nevertheless illustrate them in terms of integrodifference equations ($I_{0}$) and their Nyström approximations ($I_n$), which can be immediately implemented in simulations. Indeed, for these full discretizations the integral is replaced by a convergent quadrature/cubature rule. Its convergence rate regarding the error to the discretized objects (hyperbolic periodic solutions, graphs of their stable and unstable fiber bundles) is preserved throughout. This rate in turn depends on the smoothness of the kernel functions $f_t$. In case they are Hölder continuous, then the convergence rate coincides with the corresponding Hölder exponent (cf. Theorem 3.1), while for higher-order smoothness one obtains polynomial convergence as shown in Theorem 3.2.

Our contribution concentrates on Nyström methods. Nevertheless, rather than using quadrature/cubature formulas, an alternative tool to evaluate the right-hand sides of IDEs is the Fast Fourier Transformation (for short FFT, [22, pp. 106ff, Sect. 8.2]). Although this approach is fairly popular in the literature, it is restricted to Hammerstein IDEs of convolution type $u_{t+1}(x)=\int _\varOmega k_t(x-y)g_t(u_t(y))\,{\mathrm d}y$, while we deal with general Urysohn equations ($I_{0}$). In comparison, as pointed out in [22], in order to arrive at a similar accuracy, Nyström methods require less nodes but tend to be slower than FFT methods. Up to the author’s knowledge, questions concerning the Numerical Dynamics for FFT discretizations were not tackled so far.

We finally point out that global dynamical features of integrodifference equations were addressed in [9] (for polynomial growth functions) and [10] (for general smooth growth functions). Using rigorous numerics they establish the existence of periodic solutions, connecting orbits, and ultimately chaotic dynamics. The methods involve set oriented numerics and topological tools such as the Conley index, which are fundamentally different from ours. Regarding connecting orbits, see also [20].

A Quantitative implicit and Lipschitz inverse function theorem

Let be Banach spaces. We formulate an abstract, but tailor-made implicit function theorem, whose parameter set is merely supposed to be a metric space $(\varLambda ,d)$:

Theorem A.1

(Quantitative implicit function theorem) Let be nonempty open, $x_0\in \varOmega$, $\lambda _0\in \varLambda$, , $q\in [0,1)$, and suppose satisfies

• (i’) $T(x_0,\lambda _0)=y_0$,

• (ii’) the partial derivative exists with ,

• (iii’) there exist functions $\varGamma _0:\mathbb {R}_+\rightarrow \mathbb {R}_+$ and $\varGamma :\mathbb {R}_+^2\rightarrow \mathbb {R}_+$ which satisfy $\lim _{\varrho \searrow 0}\varGamma _0(\varrho )=0$, $\lim _{\varrho _1,\varrho _2\searrow 0}\varGamma (\varrho _1,\varrho _2)=0$, such that for all $x\in \varOmega$, $\lambda \in \varLambda$ it holds

  $$\begin{aligned} \left\| T(x_0,\lambda )-T(x_0,\lambda _0)\right\|&\le \varGamma _0(d(\lambda ,\lambda _0)), \end{aligned}$$

  (A.1)
  $$\begin{aligned} \left\| D_1T(x,\lambda )-D_1T(x_0,\lambda _0)\right\|&\le \varGamma (\left\| x-x_0\right\| ,d(\lambda ,\lambda _0)). \end{aligned}$$

  (A.2)

If $K:=\left\| D_1T(x_0,\lambda _0)^{-1}\right\|$ and $\rho _0,\delta >0$ are chosen so small that

$$\begin{aligned} \varGamma _0(\delta )&\le \tfrac{1-q}{K}\rho _0,&\varGamma (\rho _0,\delta )&\le \tfrac{q}{K}, \end{aligned}$$

(A.3)

then there exists a function satisfying

• (a) $\phi (\lambda _0)=x_0$,

• (b) $T(x,\lambda )=y_0$ in if and only if $x=\phi (\lambda )$,

• (c) $\left\| \phi (\lambda )-x_0\right\| \le \tfrac{K}{1-q}\varGamma _0(d(\lambda ,\lambda _0))$ for all $\lambda \in B_\delta (\lambda _0,\varLambda )$.

Proof

The proof is similar to the one of [25, Thm. A.1]. $\square$

Theorem A.2

(Lipschitz inverse function theorem) Let and $\rho >0$ be given. If a mapping is of the form $T=A+G$ with

• (i) ,

• (ii) is Lipschitz with Lipschitz constant $l<\left\| A^{-1}\right\| ^{-1}$,

then the following holds with $\sigma \in \bigl (l,\left\| A^{-1}\right\| ^{-1}\bigr ]$:

• (a) For all one has

  $$\begin{aligned} (\sigma -l)\left\| x-{\bar{x}}\right\| \le \left\| T(x)-T({\bar{x}})\right\| \le (\left\| A\right\| +l)\left\| x-{\bar{x}}\right\| , \end{aligned}$$

  (A.4)

• (b) for all the equation $T(x)=y$ has a unique solution ,

• (c) with also the function is of class $C^m$, $m\in \mathbb {N}_0$.

Proof

See [17, p. 224, (C.11)], with the smoothness assertion resulting from the Uniform Contraction Principle [7, p. 25, Thm. 2.2]. $\square$