Continuity and monotonicity of solutions to a greedy maximization problem

TL;DR

This paper studies the properties of solutions to a greedy maximization problem related to resource sharing networks, focusing on their continuity and monotonicity over time, which impacts fluid network models.

Contribution

It provides new insights into the continuity and monotonicity of solutions to a network-based greedy maximization problem, considering the network topology.

Findings

Solutions are continuous with respect to time under certain conditions.

Monotonicity of solutions depends on network topology and constraints.

Results inform the design of optimal fluid network models.

Abstract

Motivated by an application to resource sharing network modelling, we consider a problem of greedy maximization (i.e., maximization of the consecutive minima) of a vector in $R^n$, with the admissible set indexed by the time parameter. The structure of the constraints depends on the underlying network topology. We investigate continuity and monotonicity of the resulting maximizers with respect to time. Our results have important consequences for fluid models of the corresponding networks which are optimal, in the appropriate sense, with respect to handling real-time transmission requests.

Equations

$$x_i^{\ast }:=\sup \{x\in \mathbb{R}:h_i(x)=0\}\in (-\infty ,0].$$

$$A_{t}=\Big{\{}a=(a_{1},...,a_{I})\in\mathbb{R}^{I}:\;a_{i}\leq t,\;i\in{\bf I},\;\sum_{i\in G_{j}}h_{i}(a_{i})\leq t,\;j\in{\bf J}\Big{\}},$$

$$A_{t}^{t_{0}}=\Big{\{}a=(a_{1},...,a_{I})\in\mathbb{R}^{I}:\;a_{i}\leq t,\;i\in{\bf I},\;\sum_{i\in G_{j}}h_{i}(a_{i})\leq t-t_{0},\;j\in{\bf J}\Big{\}},$$

$$Z(t,s)=Z(0,t+s)+E(t,s)-D(t,s),$$

$$D_i(t,s)=S_i(T_i(t,s),t,s),T_i(t,s)+Y_i(t,s)=t,i\in \mathbf{I},$$

$$\displaystyle\sum_{i\in G_{j}}\big{(}T_{i}(\tilde{t},s-\tilde{t})-T_{i}(t,s-t)\big{)}\leq\tilde{t}-t,\qquad j\in{\bf J},\qquad$$

$$\overline{Z}(t,s)=\overline{Z}(0,t+s)+\alpha (t+(s\wedge 0){)}^{+}-\overline{D}(t,s),$$

$${\overline{D}}_i(t,s)={\overline{T}}_i(t,s)/m_i,{\overline{T}}_i(t,s)+{\overline{Y}}_i(t,s)=t,i\in \mathbf{I},$$

$$\displaystyle\sum_{i\in G_{j}}\big{(}\overline{T}_{i}(\tilde{t},s-\tilde{t})-\overline{T}_{i}(t,s-t)\big{)}\leq\tilde{t}-t,\qquad j\in{\bf J},\qquad\;\;$$

$$\overline{\mathfrak{X}}(t,s)=(\overline{Z}(t,s),\overline{D}(t,s),\overline{T}(t,s),\overline{Y}(t,s)),t\ge 0,s\in \mathbb{R},$$

$$h_i(x)=m_i[{\overline{Z}}_i(0,x)-{\overline{Z}}_i(0,x_i^{\ast }){]}^{+}+{\alpha }_i{m}_i[x^{+}-(x_i^{\ast }{)}^{+}{]}^{+},i\in \mathbf{I},$$

$$\sum _{i\in G_j}{h}_i(x)\le t,j\in \mathbf{J}.$$

$${\mathbf{N}}^{(1)}=\{j\in \mathbf{J}:G_j\subseteq {\mathbf{I}}^{(1)}\}\setminus {\mathbf{J}}^{(1)}.$$

$$\sum _{i\in G_j\cap {\mathbf{I}}^{(1)}}{h}_i(F_i(t))+\sum _{i\in G_j\setminus {\mathbf{I}}^{(1)}}{h}_i(x)\le t,j\in {\mathbf{K}}^{(1)}.$$

$$\Big{(}\bigcup_{l=1}^{k}{\bf J}^{(l)}\Big{)}\cap\Big{(}\bigcup_{l=1}^{k}{\bf N}^{(l)}\Big{)}=\mathchar 31\relax,\qquad\qquad\bigcup_{l=1}^{k}({\bf J}^{(l)}\cup{\bf N}^{(l)})\neq{\bf J},$$

$${\mathbf{N}}^{(l)}=\{j\in \mathbf{J}:G_j\subseteq \bigcup _{p=1}^l{\mathbf{I}}^{(p)}\}\setminus (\bigcup _{p=1}^l{\mathbf{J}}^{(p)}\cup \bigcup _{p=1}^{l-1}{\mathbf{N}}^{(p)}),$$

$$\sum _{i\in G_j}{h}_i(F_i(t))=t,j\in {\mathbf{J}}^{(l)},$$

$$\sum _{i\in G_j\cap {\bigcup }_{p=1}^l{\mathbf{I}}^{(p)}}{h}_i(F_i(t))+\sum _{i\in G_j\setminus {\bigcup }_{p=1}^l{\mathbf{I}}^{(p)}}{h}_i(f^{(l)})<t,j\in {\mathbf{K}}^{(l)},$$

$$F_i(t)=\sup \{x\le t:h_i(x)=h_i(f^{(p)})\},i\in {\mathbf{I}}^{(p)},p=1,...,l,$$

$$\sum _{i\in G_j\cap {\bigcup }_{p=1}^k{\mathbf{I}}^{(p)}}{h}_i(F_i(t))+\sum _{i\in G_j\setminus {\bigcup }_{p=1}^k{\mathbf{I}}^{(p)}}{h}_i(x)\le t,j\in {\mathbf{K}}^{(k)}.$$

$$f^{(1)}(t)={\bar{h}}^{-1}(t)\wedge t,t\ge 0.$$

$$F_i(t)=f^{(k)}(t)\vee x_i^{\ast }.$$

$$\sum _{i\in G_j}{h}_i(F_i(t))=\sum _{l=1}^k\sum _{i\in G_j\cap {\mathbf{I}}^{(l)}}{h}_i(f^{(l)}(t))<t.$$

$${\mathbf{J}}^{(1)}(t)=\{j\in \mathbf{J}:|G_j|=\max \{|G_{j^{\prime }}|:j^{\prime }\in \mathbf{J}\}\},$$

$${\mathbf{K}}^{(1)}=\{j\in \mathbf{J}:|G_j|<\underset{j^{\prime }\in \mathbf{J}}{\max }|G_{j^{\prime }}|,G_j\setminus {\mathbf{I}}^{(1)}\ne \text{mathchar}31\},$$

$$f^{(2)}(t)=\underset{j\in {\mathbf{K}}^{(1)}}{\min }\frac{1-\frac{|G_j\cap {\mathbf{I}}^{(1)}|}{\max \{|G_{j^{\prime }}|:j^{\prime }\in \mathbf{J}\}}}{|G_j\setminus {\mathbf{I}}^{(1)}|}t.$$

$$F_1(t)=t-2,F_2(t)=-1,F_3(t)=0.$$

$$F_1(t)=F_2(t)=\frac{t-3}{2},F_3(t)=\frac{t-1}{10}.$$

$$F_1(t)=F_3(t)=\frac{t}{6}-\frac{1}{3},F_2(t)=\frac{5t}{6}-\frac{8}{3}.$$

$$N_l=\{j\in \mathbf{J}:G_j\subseteq \bigcup _{p=1}^l\bigcup _{j^{\prime }\in J_p}{G}_{j^{\prime }}\}\setminus (\bigcup _{p=1}^l{J}_p\cup \bigcup _{p=1}^{l-1}{N}_p),l=1,...k,$$

$${\mathcal{T}}^{\mathcal{D}}=\{t\in [0,T]:{\mathbf{J}}^{(p)}(t)=J_p,p=1,...,k\},\mathcal{D}=(J_1,...,J_k)\in \mathcal{J}.$$

$$g^{-1}(t)=\inf \{s\ge 0:g(s)>t\},t\ge 0.$$

$$g^{-1}(t)=\sup \{s\ge 0:g(s)\le t\},t\ge 0.$$

Full text

Keywords partial order, greedy maximization, continuity, monotonicity, resource sharing, fluid model

Mathematics subject classification 06A06, 26A15, 26A48, 68M20, 90B10, 90C35.

1 Introduction

Starting from seminal papers of Rybko and Stolyar [14], Dai [5], Bramson [2, 3, 4] and other authors, fluid models have become a standard tool in investigating long-time behaviour of complicated queueing systems. Such models are useful for establishing stability and obtaining hydrodynamic or diffusion limits for multiclass queueing networks and resource sharing networks with various service protocols. Using a similar methodology in the case of real-time Earliest Deadline First (EDF) networks with resource sharing is hindered by the lack of suitable fluid model equations. To overcome this difficulty, Kruk [13] suggested the definition of fluid models for these systems by means of an optimality property, called local edge minimality, which is known to characterize the EDF discipline in stochastic resource sharing networks. It turns out that the success of this approach depends on establishing suitable local properties of a vector-valued mapping $F:[0,\infty)\rightarrow\mathbb{R}^{n}$, resulting from a greedy maximization (i.e., maximization of the consecutive minima) of a vector in $\mathbb{R}^{n}$ over the admissible set $A_{t}$, depending on the underlying network topology and indexed by the time parameter. A convenient way to describe the value $F(t)$ for a given $t\geq 0$ is to define it as the maximal element of $A_{t}$ with respect to a suitable “min-sensitive” partial ordering. Roughtly speaking, the function $F$, when well behaved, determines the so-called frontiers (i.e., the left endpoints of the supports) of the states in the corresponding locally edge minimal fluid model. The idea of using frontiers for the asymptotics of EDF systems dates back to the paper of Doytchinov et al. [7] on a G/G/1 queue, and it has been used several times since then. However, both the application of this idea to resource sharing networks and our approach to determining the frontiers by finding maximal elements of partially ordered sets appear to be new.

In this paper, for any $t\geq 0$, we construct the value of $F(t)$ as a solution of a nested sequence of max-min problems in a $t$-dependent admissible set. While none of these max-min problems is hard to solve, their number and forms vary with $t$ in a complex, discontinuous way, making the analysis of the resulting mapping $F$ on $[0,\infty)$ rather involved. We investigate key properties of $F$, namely, its continuity and monotonicity. Our main results are described in more detail in Section 2.4, to follow, after the introduction of indispensable notation. These results are used in our forthcoming paper to establish fundamental properties of locally edge minimal fluid models, like their existence, uniqueness and stability.

We hope that the theory developed in this paper will be useful not only in the asymptotic analysis of EDF-like disciplines, but also in the case of other “greedy“ scheduling policies for resource sharing networks, for example Longest Queue First [6] or Shortest Remaining Processing Time [15].

1.1 Notation

For sets $A$, $B$, we write $A\subsetneq B$ if $A$ is a proper subset of $B$. For a finite set $A$, let $|A|$ denote the cardinality of $A$. Let $\mathbb{N}$ denote the set of positive integers and let $\mathbb{R}$ denote the set of real numbers. For $a,b\in\mathbb{R}$, we write $a\vee b$ ($a\wedge b$) for the maximum (minimum) of $a$ and $b$, $a^{+}$ for $a\vee 0$. Vector inequalities are to be interpreted componentwise, i.e., for $a,b\in\mathbb{R}^{n}$, $a=(a_{1},..,a_{n})$, $b=(b_{1},...,b_{n})$, $a\leq b$ if and only if $a_{i}\leq b_{i}$ for all $i=1,...,n$. For $a=(a_{1},..,a_{n})\in\mathbb{R}^{n}$, we write $\min a$ for $\min_{i=1,...,n}a_{i}$ and $\rm{Argmin\;}a$ for $\{i\in\{1,...,n\}:a_{i}=\min a\}$. For $a=(a_{1},..,a_{n})\in\mathbb{R}^{n}$ and a set $A\subsetneq\{1,...,n\}$ with $\{1,...,n\}\setminus A=\{i_{1},...,i_{k}\}$, where $k=n-|A|$ and $i_{1}<i_{2}<...<i_{k}$, we identify $(a_{i})_{i\notin A}$ with $(a_{i_{1}},...a_{i_{k}})\in\mathbb{R}^{k}$. By convention, a sum of the form $\sum_{i=n}^{m}$ ($\bigcup_{i=n}^{m}$) with $n>m$ or, more generally, a sum of numbers (resp., sets) over the empty set of indices equals zero (resp., $\mathchar 31\relax$). For a set $A\subseteq\mathbb{R}$, let $\overline{A}$ denote the closure of $A$.

2 The mapping $F$

2.1 “Min-sensitive” partial ordering on $\mathbb{R}^{n}$

We define the relation “$\eqslantless$” inductively on $\mathbb{R}^{n}$ as follows.

Definition 2.1

For $a,b\in\mathbb{R}$, we write $a\eqslantless b$ iff $a\leq b$. If $n\geq 2$, for $a,b\in\mathbb{R}^{n}$, $a=(a_{1},..a_{n})$, $b=(b_{1},...,b_{n})$, we write $a\eqslantless b$ if one of the following four cases holds.

(i) $a=b$,

(ii) $\min a<\min b$,

(iii) $\min a=\min b$ and $\rm{Argmin\;}b\subsetneq\rm{Argmin\;}a$,

(iv) $\min a=\min b$, $\rm{Argmin\;}a=\rm{Argmin\;}b\subsetneq\{1,...,n\}$ and $(a_{i})_{i\notin\rm{Argmin\;}a}\eqslantless(b_{i})_{i\notin\rm{Argmin\;}b}$ in $\mathbb{R}^{|\rm{Argmin\;}a|}$.

Remark 2.2

Clearly, if $a\eqslantless b$, then $\min a\leq\min b$. The converse is, in general, false, unless $n=1$. For example, the relation $(0,1)\eqslantless(1,0)$ does not hold. **

The proof of the following lemmma is elementary and it is left to the reader.

Lemma 2.3

The relation “$\eqslantless$” is a partial ordering on $\mathbb{R}^{n}$.

Remark 2.4

For $a,b\in\mathbb{R}^{n}$, the inequality $a\leq b$ implies $a\eqslantless b$. In dimensions greater than one the converse is in general, false, for example, $(0,2)\eqslantless(1,1)$. **

2.2 The mapping definition

Let $I,J\in\mathbb{N}$ and let ${\bf I}=\{1,...,I\}$, ${\bf J}=\{1,...,J\}$. For $i\in\bf{I}$, let $h_{i}:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous, nonnegative, nondecreasing function with $\lim_{x\rightarrow\infty}h_{i}(x)=\infty$ and

[TABLE]

In other words, each $h_{i}$ is the cumulative distribution function of an atomless, $\sigma$-finite measure in $\mathbb{R}$, with finite, nonpositive infimum of its support. Let $G_{j}$, $j\in{\bf J}$, be a family of distinct, nonempty subsets of ${\bf I}$ (not necessarily pairwise disjoint) such that ${\bf I}=\bigcup_{j\in{\bf J}}G_{j}$.

Definition 2.5

For $t\geq 0$, we denote by $F(t)=(F_{i}(t))_{i\in{\bf I}}$ the maximal element of the set

[TABLE]

with respect to the relation “$\eqslantless$”.

Somewhat informally, $F(t)$ may be thought of as the result of “greedy” maximization of a vector $a\in\mathbb{R}^{I}$, subject to the constraints defining the set $A_{t}$, in the following sense. We first maximize $\min a$ over $a\in A_{t}$, then we maximize the “next minimum” $\min\{a_{i},i\notin\rm{Argmin\;}a\}$ over the set of maximizers of the previous problem, and we continue in this way until all the entries of the maximizer $a^{*}=F(t)$ are determined. In Section 3.1 we formalize and describe in detail this nested max-min procedure which implies, in particular, the existence and uniqueness of the maximizer $F(t)$.

Remark 2.6

A seemingly more general version of Definition 2.5, in which for a fixed (possibly positive) $t_{0}\in\mathbb{R}$ and $t\geq t_{0}$, the set $A_{t}$ is replaced by

[TABLE]

and the half-line $(-\infty,0]$ in (2.1) is replaced by $(-\infty,t_{0}]$, may be easily reduced to the case considered in Definition 2.5 by the change of variables $y=x-t_{0}$, $s=t-t_{0}$ and by using $\tilde{h}_{i}(y)=h_{i}(x)=h_{i}(y+t_{0})$ instead of $h_{i}$. **

2.3 Motivation: fluid models for resource sharing networks

The need for investigating the properties of the mapping $F$ defined above arises in the theory of fluid models for real-time networks with resource sharing. Below, we briefly (and somewhat informally) describe this connection. The reader may consult [13] for more details and references.

Consider a network with a finite number of resources (nodes), labelled by $j=1,...,J$, and a finite set of routes, labelled by $i=1,...,I$. Let ${\bf I}=\{1,...,I\}$, ${\bf J}=\{1,...,J\}$. For $j\in{\bf J}$, let $G_{j}\subseteq{\bf I}$ be the set of routes using the resource $j$. For convenience, we assume that all the resources have a unit service rate. By a flow on route $i$ we mean a continuous transmission of a file through the resources used by this route. We assume that a flow takes simultaneous possession of all the resources on its route during the transmission. Each flow in the network has a deadline for transmission completion. Networks of this type may be used to model, e.g., voice and video transmission, manufacturing systems with order due dates or emergency health care services. In what follows, by the lead time of a flow we mean the difference between its deadline and the current time.

As in [12, 13], the time evolution of such a system may be described by the process $\mathfrak{X}(t,s)=(Z(t,s),D(t,s),T(t,s),Y(t,s))$, $t\geq 0,s\in\mathbb{R},$ where the component processes $Z$, $D$, $T$, $Y$ are defined as follows. For $t\geq 0$ and $s\in\mathbb{R}$, $Z(t,s)=(Z_{i}(t,s))_{i\in{\bf I}}$, where $Z_{i}(t,s)$ is the number of flows on route $i$ with lead times at time $t$ less than or equal to $s$ which are still present in the system at that time. Similarly, the vectors $D(t,s)=(D_{i}(t,s))_{i\in{\bf I}}$, $T(t,s)=(T_{i}(t,s))_{i\in{\bf I}}$ denote the number of departures (i.e., transmission completions) and the cumulative transmission time by time $t$ corresponding to each route $i$ of flows with lead times at time $t$ less than or equal to $s$. Let $Y_{i}(t,s)=t-T_{i}(t,s)$, $i\in{\bf I}$, denote the cumulative idleness by time $t$ with regard to transmission of flows on route $i$ with lead times at time $t$ less than or equal to $s$ and let $Y(t,s)=(Y_{i}(t,s))_{i\in{\bf I}}$. The process $\mathfrak{X}$ satisfies the following network equations, valid for $\tilde{t}\geq t\geq 0$, $s\in\mathbb{R}$:

[TABLE]

where $E(t,s)=(E_{i}(t,s))_{i\in{\bf I}}$ is the corresponding external arrival process and $S_{i}(t^{\prime},t,s)$ denotes the number of transmission completions of flows on route $i\in{\bf I}$ having lead times at time $t$ less than or equal to $s$, by the time the system has spent $t^{\prime}$ units of time transmitting these flows.

Fluid models are deterministic, continuous analogs of resource sharing networks, in which individual flows are replaced by a divisible commodity (fluid), moving along $I$ routes with $J$ resources (nodes). They usually arise from formal functional law of large numbers approximations of the corresponding stochastic flow level models. The analogs of the network equations (2.2)-(2.4) are the following fluid model equations, valid for $\tilde{t}\geq t\geq 0$, $s\in\mathbb{R}$:

[TABLE]

where $m_{i}$ is the mean transmission time of a flow on route $i$ and $\alpha=(\alpha_{i})_{i\in{\bf I}}$ is the vector of flow arrival rates. A system

[TABLE]

with continuous, nonnegative components, satisfying the equations (2.5)-(2.7), together with some natural monotonicity assumptions, is called a fluid model for the resource sharing network under consideration.

To proceed further, we will introduce a class of fluid models which is, in some sense, optimal with respect to handling real-time transmission requests. To this end, we define a partial ordering “$\ll$” on the space of real functions on $\mathbb{R}$, which is extremely sensitive to the behaviour of the functions under comparison for small arguments.

Definition 2.7** ([13], Definition 5)**

Let $f,g:\mathbb{R}\rightarrow\mathbb{R}$ be such that for some $a\in\mathbb{R}$ we have $f\equiv g$ on $(-\infty,a]$ and let $c=\sup\{a\in\mathbb{R}:f(x)=g(x)\;\forall x\leq a\}.$ We write $f\ll g$ if either $c=\infty$ (i.e., $f\equiv g$ on $\mathbb{R}$), or $c<\infty$ and there exists $\epsilon>0$ such that $f\leq g$ on $[c,c+\epsilon]$.

Definition 2.8** (see [13], Definition 11)**

A fluid model $\overline{\mathfrak{X}}$ of the form (2.8) for a resource sharing network with $\sum_{i\in{\bf I}}\overline{Z}_{i}(0,\cdot)\equiv 0$ on $(-\infty,c]$ for some $c\in\mathbb{R}$ is called locally edge minimal at a time $t_{0}\geq 0$ if there exists $h>0$ such that for any fluid model $\overline{\mathfrak{X}}^{\prime}$ with the same $\alpha$, $m_{i}$, $G_{i}$ and the same initial state $\overline{Z}^{\prime}(0,\cdot)=\overline{Z}(0,\cdot)$, satisfying $\overline{\mathfrak{X}}^{\prime}(t_{0},\cdot)=\overline{\mathfrak{X}}(t_{0},\cdot),$ we have $\sum_{i\in{\bf I}}\overline{Y}_{i}(t,\cdot)\ll\sum_{i\in{\bf I}}\overline{Y}_{i}^{\prime}(t,\cdot)$ (equivalently, $\sum_{i\in{\bf I}}\overline{Z}_{i}(t,\cdot)\ll\sum_{i\in{\bf I}}\overline{Z}_{i}^{\prime}(t,\cdot)$) for every $t\in(t_{0},t_{0}+h)$. The fluid model $\overline{\mathfrak{X}}$ is called locally edge minimal, if it is locally edge minimal at every $t_{0}\geq 0$.

The intuition behind these notions is that a locally edge minimal fluid model tries to transmit as much “customer mass” corresponding to the earliest deadlines as possible, and hence its idleness with respect to such mass is as small as possible. Accordingly, such a model may be thought of as a “macroscopic” counterpart of a resource sharing network working under the Earliest-Deadline-First (EDF) protocol. Indeed, the EDF service discipline in such a network may be characterized by an analogous notion of local edge minimality, see [13], Definition 8 and Theorems 5-7.

In a forthcoming paper, we show that existence and local uniqueness of a fluid model for given data $\alpha$, $m_{i}$, $G_{i}$, and initial state $\overline{Z}(0,\cdot)$, which is locally edge minimal at a time $t_{0}\geq 0$, is closely related to local monotonicity of the mapping $F$ introduced in Section 2.2, with

[TABLE]

and suitably defined, not necessarily nonpositive, $x^{*}_{i}$, depending on $t_{0}$ (see Remark 2.6). It turns out that for large $t_{0}$, the points $x^{*}_{i}$ are also large, so under a natural assumption that the supports of $\overline{Z}_{i}(0,\cdot)$ are bounded above, the functions $h_{i}$ in (2.9) are linear on $[x^{*}_{i},\infty)$. Consequently, the linear case, investigated in Section 5.1 of this paper, is of considerable importance, because it determines the long-time behaviour of the corresponding locally edge minimal fluid model, for example, its stability or the form of its invariant manifold. In particular, one of remarkable implications of the formulae developed in Section 5.1 is stability of locally edge minimal fluid models, regardless of the underlying resource sharing network topology. Results along this line may be found in our forthcoming paper

2.4 Overwiev of the main results

In this paper, we investigate key properties of the mapping $F:[0,\infty)\rightarrow\mathbb{R}^{I}$, in particular those which are relevant to the theory of locally edge minimal fluid models. In Section 3, we present a detailed construction of $F(t)$ and we provide some illustrating examples. The main result of Section 4 is Theorem 4.2, stating that if each function $h_{i}$ is strictly increasing in $[x^{*}_{i},\infty)$, then the corresponding map $F$ is continuous. This basic regularity result, together with the method of partions, introduced in Section 3.3, is useful in proving various refinements, like Lipschitz continuity of $F$ for Lipschitz $h_{i}$ (Section 5.2) or an upgrade of the local monotonicity result from linear to $C^{1}$ functions $h_{i}$ (Section 5.3). The main contribution of Section 5 is the explicit evaluation of the mapping $F$ near zero in the linear case, implying, in particular, its local monotonicity in a neighbourhood of zero for piecewise linear $h_{i}$. As we have already mentioned, the latter fact is then generalized to (piecewise) $C^{1}$ functions. Finally, in Section 5.4, we show that, somewhat surprisingly, the mapping $F$ may fail to be globally monotone on $[0,\infty)$, even if the corresponding functions $h_{i}$ are linear in $[x^{*}_{i},\infty)$.

3 The mapping construction algorithm

3.1 Construction

Fix $t\geq 0$. We define the vector $F(t)=(F_{i}(t))_{i\in{\bf I}}$ as follows.

Let $f^{(1)}=f^{(1)}(t)$ be the supremum of $x\leq t$ satisfying the constraints

[TABLE]

If $f^{(1)}=t$, we take $F_{i}(t)=t$ for each $i\in{\bf I}$, ${\bf I}^{(1)}={\bf I}$, ${\bf J}^{(1)}={\bf J}$, ${\bf N}^{(1)}=\mathchar 31\relax$ and $k_{max}=k_{max}(t)=1$. In what follows, we assume that $f^{(1)}<t$. By continuity of $h_{i}$, $x=f^{(1)}$ satisfies (3.1) and the set {\bf J}^{(1)}=\big{\{}j\in{\bf J}:\sum_{i\in G_{j}}h_{i}(f^{(1)})=t\big{\}} of active constraints is nonempty. (Indeed, if ${\bf J}^{(1)}=\mathchar 31\relax$, then $f=f^{(1)}+\epsilon$ also satisfies (3.1) for $\epsilon>0$ small enough, which contradicts the definition of $f^{(1)}$.) Let ${\bf I}^{(1)}=\bigcup_{j\in{\bf J}^{(1)}}G_{j}$ and

[TABLE]

For $i\in{\bf I}^{(1)}$, put $F_{i}(t)=\sup\{x\leq t:h_{i}(x)=h_{i}(f^{(1)})\}.$ If ${\bf I}^{(1)}={\bf I}$, this completely determines the vector $F(t)$. In this case, let $k_{max}=1$. Otherwise, let ${\bf K}^{(1)}={\bf J}\setminus({\bf J}^{(1)}\cup{\bf N}^{(1)})$ and let $f^{(2)}=f^{(2)}(t)$ be the supremum of $x\leq t$ satisfying the constraints

[TABLE]

Note that $G_{j}\setminus{\bf I}^{(1)}\neq\mathchar 31\relax$ for $j\in{\bf K}^{(1)}$, so the second sum in (3.3) is taken over a nonempty set of indices. We also have $f^{(1)}<f^{(2)}$ by definition. If $f^{(2)}=t$, we take ${\bf I}^{(2)}={\bf I}\setminus{\bf I}^{(1)}$, ${\bf J}^{(2)}={\bf K}^{(1)}$, ${\bf N}^{(2)}=\mathchar 31\relax$, $F_{i}(t)=t$ for each $i\in{\bf I}^{(2)}$ and $k_{max}=2$. If $f^{(2)}<t$, then $x=f^{(2)}$ satisfies (3.3) and the set ${\bf J}^{(2)}$ of active constraints in (3.3) (i.e., those $j\in{\bf K}^{(1)}$, for which equality holds in (3.3) with $x=f^{(2)}$) is nonempty. In this case, let ${\bf I}^{(2)}=\bigcup_{j\in{\bf J}^{(2)}}G_{j}\setminus{\bf I}^{(1)}$, ${\bf N}^{(2)}=\{j\in{\bf J}:G_{j}\subseteq{\bf I}^{(1)}\cup{\bf I}^{(2)}\}\setminus({\bf J}^{(1)}\cup{\bf J}^{(2)}\cup{\bf N}^{(1)})$ and put $F_{i}(t)=\sup\{x\leq t:h_{i}(x)=h_{i}(f^{(2)})\}$ for $i\in{\bf I}^{(2)}$. If ${\bf I}^{(1)}\cup{\bf I}^{(2)}={\bf I}$, the definition of the vector $F(t)$ is complete and we take $k_{max}=2$, otherwise we let ${\bf K}^{(2)}={\bf J}\setminus({\bf J}^{(1)}\cup{\bf J}^{(2)}\cup{\bf N}^{(1)}\cup{\bf N}^{(2)})$, and we continue our construction as follows.

Suppose that for some $k\geq 2$, we have defined numbers $f^{(1)}<f^{(2)}<...<f^{(k)}<t$, nonempty, disjoint subsets ${\bf J}^{(1)}$,…,${\bf J}^{(k)}$ of ${\bf J}$, disjoint (not necessarily nonempty) subsets ${\bf N}^{(1)}$,…,${\bf N}^{(k)}$ of ${\bf J}$ with

[TABLE]

and nonempty, disjoint subsets ${\bf I}^{(1)}$, …,${\bf I}^{(k)}$ of ${\bf I}$ with $\bigcup_{l=1}^{k}{\bf I}^{(l)}\neq{\bf I}$ such that for $l=1,..,k$,

[TABLE]

where

[TABLE]

and ${\bf K}^{(l)}={\bf J}\setminus\bigcup_{p=1}^{l}({\bf J}^{(p)}\cup{\bf N}^{(p)})$. Note that ${\bf K}^{(l)}\neq\mathchar 31\relax$ by (3.4) and $G_{j}\setminus\bigcup_{p=1}^{l}{\bf I}^{(p)}\neq\mathchar 31\relax$ for $j\in{\bf K}^{(l)}$, $l=1,...,k$, by (3.5), so that the second sum in (3.6) is taken over a nonempty set of indices. (Such numbers and sets were defined in the last paragraph for $k=2$.) Let $f^{(k+1)}=f^{(k+1)}(t)$ be the supremum of $x\leq t$ satisfying the constraints

[TABLE]

The inequality (3.6) implies that $f^{(k+1)}>f^{(k)}$. If $f^{(k+1)}=t$, we take ${\bf I}^{(k+1)}={\bf I}\setminus\bigcup_{l=1}^{k}{\bf I}^{(l)}$, ${\bf J}^{(k+1)}={\bf K}^{(k)}$, ${\bf N}^{(k+1)}=\mathchar 31\relax$ and $F_{i}(t)=t$ for each $i\in{\bf I}^{(k+1)}$, so the definition of the vector $F(t)$ is complete. In this case, we put $k_{max}=k+1$. If $f^{(k+1)}<t$, then $x=f^{(k+1)}$ satisfies (3.8) and the set ${\bf J}^{(k+1)}$ of active constraints in (3.8) (i.e., these $j\in{\bf K}^{(k)}$, for which equality holds in (3.8) with $x=f^{(k+1)}$) is nonempty. In this case, let ${\bf I}^{(k+1)}=\bigcup_{j\in{\bf J}^{(k+1)}}G_{j}\setminus\bigcup_{l=1}^{k}{\bf I}^{(l)}$, define ${\bf N}^{(k+1)}$ by (3.5) with $l=k+1$ and put $F_{i}(t)=\sup\{x\leq t:h_{i}(x)=h_{i}(f^{(k+1)})\}$ for $i\in{\bf I}^{(k+1)}$. This ends the $k+1$-th step of our construction. If $\bigcup_{l=1}^{k+1}{\bf I}^{(l)}={\bf I}$, the definition of the vector $F(t)$ is complete. In this case, put $k_{max}=k+1$. Otherwise, we make another (i.e., the $k+2$-th) step of our algorithm, taking $k+1$ instead of $k$ and proceeding as above.

When the construction terminates after $k_{max}$ steps, we have defined the vector $F(t)$.

Remark 3.1

The index $k_{max}$ and the sets ${\bf I}^{(k)}$, ${\bf J}^{(k)}$, ${\bf N}^{(k)}$, ${\bf K}^{(k)}$ defined above depend on the time $t$. In what follows, when we want to stress this dependence, we write $k_{max}(t)$, ${\bf I}^{(k)}(t)$, ${\bf J}^{(k)}(t)$, ${\bf N}^{(k)}(t)$, ${\bf K}^{(k)}(t)$, respectively.**

Remark 3.2

Let $\bar{h}(x)=\max_{j\in{\bf J}}\sum_{i\in G_{j}}h_{i}(x)$, $x\in\mathbb{R}$. If $\bar{h}$ is strictly increasing in $[\min_{i\in{\bf I}}x^{*}_{i},\infty)$, then, by definition,

[TABLE]

Remark 3.3

If $i\in{\bf I}^{(k)}(t)$ for some $k\in\{1,...,k_{max}(t)\}$ and if the function $h_{i}$ is strictly increasing in $[x^{*}_{i},\infty)$, then (compare (3.7)),

[TABLE]

Remark 3.4

In general, some of the sets ${\bf N}^{(k)}$, $k=1,...,k_{max}$, may be nonempty, see Example 3.5, to follow. If $j\in{\bf N}^{(k)}$ for some $k$, then

[TABLE]

This strict inequality may be interpreted as an indication of “unavoidable bottleneck idleness” in the corresponding locally edge minimal fluid network - transferring higher priority fluids by other resources does not allow $j$ to use its full capacity on the time interval $[0,t]$. This phenomenon is well known in the theory of resource sharing networks and it was discussed in detail, e.g., by Gurvich and Van Mieghem [8, 9]. A mild sufficient condition for all the sets ${\bf N}^{(k)}$ to be empty is that for each $j\in{\bf J}$, $G_{j}\setminus\bigcup_{j^{\prime}\neq j}G_{j^{\prime}}\neq\mathchar 31\relax$. This corresponds to the so-called local traffic condition for the underlying network topology, under which every resource has at least one route using only that resource, see [10, 11]. The latter requirement is satisfied, for example, by linear networks, for which $I=J+1$ and $G_{j}=\{1,j+1\}$, $j=1,...,J$. There are, however, some important systems that do not satisfy the local traffic assumption, for example ring networks, used as counterexamples for stability of the LQF protocol [1, 6]. **

3.2 Examples

In this subsection, we provide two examples illustrating the construction of the mapping $F:[0,\infty)\rightarrow\mathbb{R}^{I}$ defined in Subsection 3.1. The first one has relatively simple structure, yielding time-independent $k_{max}$, ${\bf J}^{(k)}$, ${\bf I}^{(k)}$, ${\bf N}^{(k)}$ in $(0,\infty)$ and linear function $F$. The second one, in which $k_{max}$, ${\bf J}^{(k)}$, ${\bf I}^{(k)}$ vary in time and $F$ is nonlinear, indicates some of the difficulties encountered in more general situations.

Example 3.5

Let $h_{i}(x)=x^{+}$ for $x\in\mathbb{R}$ and $i\in{\bf I}$. Then $x^{*}_{i}=0$ for each $i\in{\bf I}$. For $t\geq 0$ and $j\in{\bf J}$, (3.1) takes the form $|G_{j}|\;x^{+}\leq t$, so its maximal solution is $f^{(1)}(t)=t/\max\{|G_{j}|:j\in{\bf J}\}$. Hence, for $t=0$ we have $f^{(1)}(0)=0$, ${\bf J}^{(1)}(0)={\bf J}$, ${\bf I}^{(1)}(0)={\bf I}$, ${\bf N}^{(1)}(0)=\mathchar 31\relax$, $k_{\max}(0)=1$ and $F_{i}(0)=0$ for all $i$. In what follows, we assume that $t>0$. Then

[TABLE]

and $F_{i}(t)=\frac{t}{\max\{|G_{j}|:j\in{\bf J}\}}$ for $i\in{\bf I}^{(1)}(t)=\bigcup_{j\in{\bf J}^{(1)}(t)}G_{j}$.

In the remainder of this example, the time argument in ${\bf J}^{(k)}(t)$, ${\bf I}^{(k)}(t)$, ${\bf N}^{(k)}(t)$, ${\bf K}^{(k)}(t)$, will be skipped. If $|G_{j}|=|G_{j^{\prime}}|$ for every $j,j^{\prime}\in{\bf J}$, then ${\bf J}^{(1)}={\bf J}$, ${\bf N}^{(1)}=\mathchar 31\relax$ and the definition of $F$ is complete. Otherwise, let ${\bf N}^{(1)}$ be given by (3.2). For $j$ belonging to the set

[TABLE]

the constraint (3.3) takes the form $\frac{|G_{j}\cap{\bf I}^{(1)}|\;t}{\max\{|G_{j^{\prime}}|:j^{\prime}\in{\bf J}\}}+|G_{j}\setminus{\bf I}^{(1)}|\;x^{+}\leq t,$ yielding the maximal solution

[TABLE]

Moreover, ${\bf J}^{(2)}$ is the set of $j\in{\bf K}^{(1)}$ attaining the minimum in (3.11) and $F_{i}(t)=f^{(2)}(t)$ for $i\in{\bf I}^{(2)}=\bigcup_{j\in{\bf J}^{(2)}}G_{j}\setminus{\bf I}^{(1)}$. If ${\bf I}^{(1)}\cup{\bf I}^{(2)}={\bf I}$ (in particular, if the minimum in (3.11) is attained at every $j\in{\bf K}^{(1)}$), then the construction of $F$ is complete. Otherwise we proceed similarly, until we get $f^{(3)}(t)$,…,$f^{(k_{max})}(t)$, and hence all $F_{i}(t)$, $i\in{\bf I}$, in the form of linear functions of $t$, with slopes depending (in an increasingly complicated way) on the sets $G_{j}$, describing the topology of the corresponding network.

Observe that in this example some sets ${\bf N}^{(k)}$ may, in general, be nonempty. The simplest such case occurs for $I=J=2$, $G_{1}=\{1\}$ and $G_{2}=\{1,2\}={\bf I}$. Then ${\bf J}^{(1)}=\{2\}$, ${\bf I}^{(1)}={\bf I}$, and hence ${\bf N}^{(1)}=\{1\}$ and $k_{max}=1$. **

In general, the main difficulty in analyzing the properties of the mapping $F$, already indicated in Remark 3.1, is the time-dependence of the index $k_{max}$ and the sets ${\bf I}^{(k)}$, ${\bf J}^{(k)}$, $k=1,...,k_{max}$. The following example, corresponding to a simple linear network topology, illustrates this point. Note that, by Remark 3.4, in this case the sets ${\bf N}^{(k)}$ are empty.

Example 3.6

Let $I=3$, $J=2$, $G_{1}=\{1,2\}$ and $G_{2}=\{1,3\}$. Furthermore, for $x\in\mathbb{R}$, let $h_{1}(x)=(x+2)^{+}$, $h_{2}(x)=(x+1)^{+}$ and $h_{3}(x)=5x^{+}$, so that $x^{*}_{1}=-2$, $x^{*}_{2}=-1$, $x^{*}_{3}=0$.

Let $0\leq t\leq 1$. Then the maximal solution of (3.1) is $f^{(1)}(t)=t-2$ and we have ${\bf J}^{(1)}(t)={\bf J}$. Thus, ${\bf I}^{(1)}(t)={\bf I}$, $k_{max}(t)=1$, and (3.10) implies that

[TABLE]

For $1<t<7/2$, the maximal solution of (3.1) is $f^{(1)}(t)=t/2-3/2$, ${\bf J}^{(1)}(t)=\{1\}$, so ${\bf I}^{(1)}(t)=G_{1}=\{1,2\}$. Furthermore, ${\bf K}^{(1)}(t)=\{2\}$ and the maximal solution of (3.3) is $f^{(2)}(t)=(t-1)/10$, so ${\bf J}^{(2)}(t)=\{2\}$, ${\bf I}^{(2)}(t)=\{3\}$, $k_{max}(t)=2$ and (3.10) yields

[TABLE]

For $t=7/2$ we still have $f^{(1)}(t)=t/2-3/2$, but ${\bf J}^{(1)}(t)={\bf J}$, ${\bf I}^{(1)}(t)={\bf I}$, $k_{max}(t)=1$. However, (3.13) still holds. Note that $F_{1}(7/2)=F_{2}(7/2)=F_{3}(7/2)=1/4$.

Finally, for $t>7/2$, the maximal solution of (3.1) is $f^{(1)}(t)=t/6-1/3$, ${\bf J}^{(1)}(t)=\{2\}$ and ${\bf I}^{(1)}(t)=G_{2}=\{1,3\}$. Thus, ${\bf K}^{(1)}(t)=\{1\}$ and the maximal solution of (3.3) is $f^{(2)}(t)=5t/6-8/3$, so ${\bf J}^{(2)}(t)=\{1\}$, ${\bf I}^{(2)}(t)=\{2\}$, $k_{max}(t)=2$ and (3.10) yields

[TABLE]

It is easy to see that the mapping $F$ given by (3.12)-(3.14) is continuous and nondecreasing on $[0,\infty)$. This example also indicates that there is, in general, no hope for obtaining any global results about the auxiliary functions $f^{(p)}$, $p>1$. Here, $f^{(2)}$ is linear and strictly increasing in $(1,7/2)$ and $(7/2,\infty)$, but it does not even exist in $[0,1]\cup\{7/2\}$. **

3.3 Partitions and inverses

Let $\mathcal{J}$ be the set of “ordered partitions” of the set ${\bf J}$, i.e., finite sequences of subsets of ${\bf J}$ in the form $(J_{1},...,J_{k})$, where $J_{i}\neq\mathchar 31\relax$, $i=1,...,k$, $J_{i}\cap J_{l}=\mathchar 31\relax$, $1\leq i<l\leq k$, and $\bigcup_{i=1}^{k}(J_{i}\cup N_{i})={\bf J}$, $(\bigcup_{i=1}^{k}J_{i})\cap(\bigcup_{i=1}^{k}N_{i})=\mathchar 31\relax$, where the sets $N_{1},...,N_{k}$ are defined recursively as

[TABLE]

(compare the first relation in (3.4) and (3.5)). Fix $T>0$ and let

[TABLE]

We label the nonempty sets of the form $\mathcal{T}^{\mathcal{D}}$, $\mathcal{D}\in\mathcal{J}$, as $\mathcal{T}_{1},...\mathcal{T}_{d}$. Clearly, $\bigcup_{i=1}^{d}\mathcal{T}_{i}=[0,T]$.

For a function $g:\mathbb{R}\rightarrow[0,\infty)$ with $\lim_{x\rightarrow\infty}g(x)=\infty$, let $g^{-1}$ denote its (generalized) right-continuous inverse (see, e.g., [16], Section 13.6),

[TABLE]

If the function $g$ is nondecreasing and $g(x_{0})=0$ for some $x_{0}\in\mathbb{R}$, then

[TABLE]

Fix $\mathcal{D}=(J_{1},...,J_{k})\in\mathcal{J}$. For $t\in\mathcal{T}^{\mathcal{D}}$, we can write $f^{(p)}(t)$, $p=1,...,k_{max}(t)$, (and hence $F(t)$) in closed form, using the inverse functions introduced above. To see this, let $t\in\mathcal{T}^{\mathcal{D}}$ and note that $k=k_{max}(t)$. Choose $j_{1},...,j_{k}\in{\bf J}$ so that $j_{p}\in J_{p}={\bf J}^{(p)}(t)$ for each $j=1,...,k$. Then

[TABLE]

and for $p=2,...,k$, we have the recursive formulae

[TABLE]

where

[TABLE]

(Actually, the minimization with $t$ in (3.16)-(3.17) is necessary only in the case of $f^{(k)}(t)$.)

4 Continuity

In general, the function $F$ may have jumps, as the following one-dimensional example indicates.

Example 4.1

Let $I=J=1$ and let $h_{1}(x)=(x^{+}\wedge 1)+(x-2)^{+}$, $x\in\mathbb{R}$. Then $F_{1}(t)=f^{(1)}(t)=t/2$ for $0\leq t<2$ and $F_{1}(t)=f^{(1)}(t)=t/2+1$ for $t\geq 2$. **

Clearly, the discontinuity of $F_{1}=f^{(1)}$ at $t=2$ in this example is caused by a “flat spot”, i.e., the interval $[1,2]$ on which $h_{1}$ takes the constant value $2$, resulting in the jump of $h_{1}^{-1}$ (see (3.16)). In our queueing application, this corresponds to the lack of “customer mass” with deadlines in the interval $[1,2]$ in the system, causing the frontier to jump over this empty interval. The following theorem assures that in the absence of such “flat spots”, the function $F$ is actually continuous.

Theorem 4.2

Assume that for every $i\in{\bf I}$, the function $h_{i}$ is strictly increasing in $[x^{*}_{i},\infty)$. Then the mapping $F$ is continuous in $[0,\infty)$.

In the proof of this result, we will use the following elementary lemma. For the sake of completeness, we provide its justification.

Lemma 4.3

Let $(X,d)$ be a metric space. Let $y\in X$ and let $\{x_{n}\}$ be a sequence of elements of $X$ such that every subsequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ contains a further subsequence $\{x_{n_{k_{l}}}\}$ converging to $y$. Then $\lim_{n\rightarrow\infty}x_{n}=y$.

Proof. Suppose that the sequence $\{x_{n}\}$ does not converge to $y$. This means that there exist $\epsilon>0$ and a subsequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ such that $d(x_{n_{k}},y)\geq\epsilon$ for all $k$. However, we have assumed the existence of a subsequence $\{x_{n_{k_{l}}}\}$ of $\{x_{n_{k}}\}$ such that $\lim_{l\rightarrow\infty}d(x_{n_{k_{l}}},y)=0$, so we have a contradiction. $\hfill\Box$

The proof of Theorem 4.2 is long and somewhat involved, so we only sketch it here, moving most of the technical details to the appendix. It is convenient to introduce additional notation

[TABLE]

Clearly, the sets ${\bf D}^{(p)}={\bf D}^{(p)}(t)$ depend on the time $t$, see Remark 3.1.

Sketch of the proof of Theorem 4.2. Let $t_{n}>0$ be such that $t_{n}\rightarrow t_{0}$ as $n\rightarrow\infty$. We may assume that $t_{n}\leq t_{0}+1$ for all $n$. Our aim is to show that $F(t_{n})\rightarrow F(t_{0})$, i.e., for every $i\in{\bf I}$, we have

[TABLE]

Without loss of generality (passing to a subsequence if necessary) we may assume that for every $m,n\geq 1$, we have $k_{max}(t_{m})=k_{max}(t_{n})$ and ${\bf J}^{(p)}(t_{m})={\bf J}^{(p)}(t_{n})$ (hence ${\bf I}^{(p)}(t_{m})={\bf I}^{(p)}(t_{n})$, ${\bf N}^{(p)}(t_{m})={\bf N}^{(p)}(t_{n})$) for $p=1,...,k_{max}(t_{m})$. Consequently, in what follows, we will simply write $k_{max}$, instead of $k_{max}(t_{n})$, $n\geq 1$, ${\bf J}^{(p)}$ instead of ${\bf J}^{(p)}(t_{n})$, $n\geq 1$, e.t.c..

By definition, for $n\geq 1$ and $p=1,...,k_{max}$, $\min_{i\in{\bf I}}x^{*}_{i}\leq f^{(p)}(t_{n})\leq t_{n}\leq t_{0}+1$. Hence, by Lemma 4.3, without loss of generality (passing to a subsequence if necessary) we may assume that the sequences $\{f^{(p)}(t_{n})\}$, $p=1,...,k_{max}$, converge. Let

[TABLE]

Remark 3.2 implies that $f^{(1)}$ is continuous, hence

[TABLE]

First suppose that $t_{0}=0$. We have

[TABLE]

${\bf J}^{(1)}(0)={\bf J}$, ${\bf I}^{(1)}(0)={\bf I}$, $k_{max}(0)=1$ and $F_{i}(0)=x^{*}_{i}$ for all $i\in{\bf I}$. Let $i\in{\bf I}$. Then $i\in{\bf I}^{(k)}$ for some $k\in\{1,...,k_{max}\}$. By the definition of $f^{(k)}$, $0\leq h_{i}(f^{(k)}(t_{n}))\leq t_{n}$, and hence, as $t_{n}\downarrow 0$, by (2.1),

[TABLE]

so $F_{i}$ is continuous at [math]. Hence, in what follows, we will assume that $t_{0}>0$.

We will first consider the case in which $f^{(1)}(t_{n})=t_{n}$ for all $n\geq 1$. Then $f^{(1)}(t_{0})=t_{0}$ and thus, for each $i\in{\bf I}$, $F_{i}(t_{n})=t_{n}\rightarrow t_{0}=F_{i}(t_{0}).$ Similarly, if $f^{(1)}(t_{0})=t_{0}$ then $f^{(1)}(t_{n})\rightarrow t_{0}$, so for each $i\in{\bf I}$, the inclusion $F_{i}(t_{n})\in[f^{(1)}(t_{n}),t_{n}]$ implies that $F_{i}(t_{n})\rightarrow t_{0}=F_{i}(t_{0})$. Therefore, we may additionally assume

[TABLE]

We prove (4.2) inductively for $i\in{\bf I}^{(l)}(t_{0})$, $l=1,...,k_{max}(t_{0})$. In the $l$-th inductive step, we define

[TABLE]

It is easy to check that

[TABLE]

where $B_{p}^{(l)}={\bf I}^{(p)}\cap{\bf I}^{(l)}(t_{0})$. We show that there exists $\bar{p}_{l}\in\{b_{l},...,p_{l}\}$ such that

[TABLE]

By (3.10) and (4.10), for $i\in B_{p}^{(l)}$, $p=b_{l},...,\bar{p}_{l}$, as $n\rightarrow\infty$, we have

[TABLE]

We also argue that if $\bar{p}_{l}<p_{l}$, then for $i\in B_{p}^{(l)}$, $p=\bar{p}_{l}+1,...,p_{l}$, we have $f^{(l)}(t_{0})<f^{(p)}_{\infty}\leq x^{*}_{i},$ so by (3.10), as $n\rightarrow\infty$,

[TABLE]

Finally, the equations (4.9), (4.11)-(4.12) imply (4.2) for every $i\in{\bf I}^{(l)}(t_{0})$. The details of the above inductive argument may be found in the Appendix.

5 Monotonicity

In this section, we investigate monotonicity of the mapping $F$. It turns out that, in general, $F$ fails to be globally nondecreasing, even if the functions $h_{i}$ are piecewise linear, see Section 5.4. However, under suitable assumptions on $h_{i}$, monotonicity of $F$ in some neighbourhood of [math] may be established. More precisely, our goal is to find $T>0$ such that for every $0\leq t<\tilde{t}<T$ and $i\in{\bf I}$, we have

[TABLE]

This is done in Section 5.1 for piecewise linear functions $h_{i}$, $i\in{\bf I}$.

5.1 Local monotonicity in the linear case

In this subsection we assume that for all $i\in{\bf I}$,

[TABLE]

where $\rho^{i}$, $i\in{\bf I}$, are given positive constants. Without loss of generality we also assume that

[TABLE]

Let $m^{*}\in\{1,...I\}$ and $0=n_{0}<1\leq n_{1}<n_{2}<...<n_{m^{*}}=I$ be such that

[TABLE]

Let $y^{*}_{k}=x^{*}_{n_{k}}$, $k=1,...,m^{*}$, and let $y^{*}_{m^{*}+1}=\infty$. By (4.5) and (5.4), we have $f^{(1)}(0)=x^{*}_{1}=y^{*}_{1}$. It follows from Remark 3.2 that $f^{(1)}(\cdot)$ is continuous and strictly increasing in $[0,\infty)$. Let $t^{*}_{1}=(f^{(1)})^{-1}(y_{2}^{*})$ if $m^{*}\geq 2$ and $t^{*}_{1}=\infty$ otherwise. Let

[TABLE]

We have

[TABLE]

(see Remark 3.2). Using (5.7), we get $t^{*}_{1}=a^{(1)}(y^{*}_{2}-y^{*}_{1})$. Note that ${\bf J}^{(1)}$ (and hence ${\bf I}^{(1)}$, ${\bf N}^{(1)}$) is constant in $(0,t^{*}_{1})$.

If $x^{*}_{1}=0$ and $a^{(1)}\leq 1$, then $m^{*}=1$ and $t^{*}_{1}=\infty$. In this case, (5.7) implies that $f^{(1)}(t)=t$ for $t\geq 0$, so $F_{i}(t)=t$ for all $t\geq 0$, $i\in{\bf I}$. In the remainder of this section we assume that either $x^{*}_{1}<0$, or $a^{(1)}>1$. In particular, we have

[TABLE]

where $\bar{t}_{1}=\infty$ if $a^{(1)}\geq 1$ and $\bar{t}_{1}=a^{(1)}x^{*}_{1}/(a^{(1)}-1)$ otherwise.

In what follows, we consider only $t\in[0,t^{*}_{1}\wedge\bar{t}_{1})$. If ${\bf I}^{(1)}={\bf I}$, then $k_{max}=1$, and hence the numbers $f^{(k)}(t)$, $k=1,..,k_{max}$, have already been defined. Assume that ${\bf I}^{(1)}\neq{\bf I}$. Then we continue our construction by induction as follows.

Assume that for some $k\geq 1$, there exist strictly positive numbers $t^{*}_{l}$, $\bar{t}_{l}$, $l=1,...,k$, such that for all $0<t<\underline{t}_{k}:=\min_{1\leq l\leq k}(t^{*}_{l}\wedge\bar{t}_{l})$ the sets ${\bf I}^{(l)}={\bf I}^{(l)}(t)$, ${\bf J}^{(l)}={\bf J}^{(l)}(t)$, ${\bf N}^{(l)}={\bf N}^{(l)}(t)$, $l=1,...,k$, do not depend on $t$ and $\bigcup_{l=1}^{k}{\bf I}^{(l)}\neq{\bf I}$, i.e., $k_{max}(t)>k$. For $l=1,...,k$, let $i^{(l)}=\min({\bf I}\setminus\bigcup_{p=1}^{l-1}{\bf I}^{(p)})$ and let $m^{(l)}\in\{1,...,m^{*}\}$ be such that $x^{*}_{i^{(l)}}=y^{*}_{m^{(l)}}$. By definition, $1=i^{(1)}<i^{(2)}<...<i^{(k)}\leq I$ and $1=m^{(1)}\leq m^{(2)}\leq...\leq m^{(k)}\leq m^{*}$. Furthermore, we assume that for each $l=1,...,k$, there exists a constant $a^{(l)}>0$ such that

[TABLE]

(For notational convenience, for $l>1$, we have defined $f^{(l)}(0)$ in (5.9) by continuity, i.e., as $f^{(l)}(0+)=x^{*}_{i^{(l)}}$, although ${\bf J}^{(1)}(0)={\bf J}$ and hence $f^{(l)}(0)$ has not been defined by the algorithm from Section 3.1.) Note that all the above assumptions have already been verified for $k=1$.

By (3.10), (5.2) and (5.9), for $t\in(0,\underline{t}_{k})$, $j\in{\bf K}^{(k)}$, the equation (3.6) implies

[TABLE]

while the equation (3.8) takes the form

[TABLE]

which, in turn, is equivalent to

[TABLE]

where

[TABLE]

By (5.10), $b^{(k+1)}_{j}>0$ for each $j\in{\bf K}^{(k)}$. Let $i^{(k+1)}=\min({\bf I}\setminus\bigcup_{p=1}^{k}{\bf I}^{(p)})$, let $m^{(k+1)}\in\{1,...,m^{*}\}$ be such that $x^{*}_{i^{(k+1)}}=y^{*}_{m^{(k+1)}}$ and let

[TABLE]

Recall that $f^{(k+1)}(t)$ is the supremum of $x\leq t$ satisfying the constraints (5.12). By an argument similar to the one used for $f^{(1)}$ above, one may check that $f^{(k+1)}(\cdot)$ is continuous and strictly increasing in $(0,\underline{t}_{k})$. Let $t^{*}_{k+1}=\infty$ if either $m^{(k+1)}=m^{*}$, or $f^{(k+1)}(\underline{t}_{k}-)\leq y_{m^{(k+1)}+1}^{*}$, and $t^{*}_{k+1}=(f^{(k+1)})^{-1}(y_{m^{(k+1)}+1}^{*})$ otherwise. By definition, $f^{(k+1)}(0):=f^{(k+1)}(0+)=x^{*}_{i^{(k+1)}}=y^{*}_{m^{(k+1)}}$ (see the notational remark following (5.9)), and, more generally, for $t\leq\underline{t}_{k}\wedge t^{*}_{k+1}$, we have

[TABLE]

Using (5.13), we get $t^{*}_{k+1}=a^{(k+1)}(y^{*}_{m^{(k+1)}+1}-y^{*}_{m^{(k+1)}})$ unless $t^{*}_{k+1}=\infty$. Note that ${\bf J}^{(k+1)}$ (and hence ${\bf I}^{(k+1)}$, ${\bf N}^{(k+1)}$) is constant in $(0,\underline{t}_{k}\wedge t^{*}_{k+1})$.

If $y^{*}_{m^{(k+1)}}=0$ and $a^{(k+1)}<1$, then $m^{(k+1)}=m^{*}$ and $t^{*}_{k+1}=\infty$. In this case, (5.13) implies that $f^{(k+1)}(t)=t$ for $t\in[0,\underline{t}_{k})$, so $k_{max}(t)=k+1$, $F_{i}(t)=t$ for such $t$ and $i\in{\bf I}\setminus\bigcup_{p=1}^{k}{\bf I}^{(p)}$. In what follows, we assume that either $y^{*}_{m^{(k+1)}}<0$, or $a^{(k+1)}\geq 1$. In particular,

[TABLE]

where $\bar{t}_{k+1}=\infty$ if $a^{(k+1)}\geq 1$ and $\bar{t}_{k+1}=a^{(k+1)}x^{*}_{i^{(k+1)}}/(a^{(k+1)}-1)$ otherwise.

This ends the $k+1$-th step of our construction. If $\bigcup_{l=1}^{k+1}{\bf I}^{(l)}={\bf I}$, then for $0<t<\underline{t}_{k+1}=\min_{1\leq l\leq k+1}(t^{*}_{l}\wedge\bar{t}_{l})$, we have $k_{max}(t)=k+1$ and the definition of $f^{(p)}(t)$, $p=1,...,k_{max}(t)$, is complete. Otherwise, we make another (i.e., the $k+2$-th) step of our algorithm, taking $k+1$ instead of $k$ and proceeding as above.

When the construction terminates, in the time interval $(0,\underline{t}_{k_{max}})$, we have the sets ${\bf I}^{(l)}={\bf I}^{(l)}(t)$, $l=1,...,k_{max}$, constant in $t$, and $f^{(l)}(t)$, $l=1,...,k_{max}$, in the form of strictly increasing linear functions. Let $i\in{\bf I}$ and let $k\in\{1,...,k_{max}\}$ be such that $i\in{\bf I}^{(k)}$. Let $0\leq t<\tilde{t}<\underline{t}_{k_{max}}$. Then, by (3.10), $F_{i}(t)=f^{(k)}(t)\vee x^{*}_{i}\leq f^{(k)}(\tilde{t})\vee x^{*}_{i}=F_{i}(\tilde{t}),$ and (5.1) follows.

Remark 5.1

Since the above argument is local in time, it actually requires only that for each $i\in{\bf I}$, (5.2) holds in some neighborhood of $x^{*}_{i}$, i.e., for all $x<X^{*}_{i}$, where $X^{*}_{i}>x^{*}_{i}$ are given constants. (Without loss of generality we may further assume that $X^{*}_{i^{\prime}}=X^{*}_{i^{\prime\prime}}$ if $i^{\prime},i^{\prime\prime}\in{\bf I}$ and $x^{*}_{i^{\prime}}=x^{*}_{i^{\prime\prime}}$.) We only have to restrict $t$ in each step of our construction to the interval $[0,\underline{t}_{k}^{\prime})$ with $\underline{t}_{k}^{\prime}=\min_{1\leq l\leq k}(t^{*}_{l}\wedge\bar{t}_{l}\wedge\bar{\bar{t}}_{l})$, where $\bar{\bar{t}}_{l}=\infty$ if $f^{(l)}(\underline{t}_{l}-)\leq X^{*}_{i^{(l)}}$ and $\bar{\bar{t}}_{l}=(f^{(l)})^{-1}(X^{*}_{i^{(l)}})$ otherwise. In this case, (5.1) holds for all $i\in{\bf I}$ and $0\leq t<\tilde{t}<\underline{t}_{k_{max}}^{\prime}$. **

5.2 Reduction lemmas and Lipschitz continuity

In this subsection we consider the case of general $h_{i}$, assuming only that each $h_{i}$ is strictly increasing in $[x^{*}_{i},\infty)$. Fix $T>0$ and recall the sets $\mathcal{T}_{1},...\mathcal{T}_{d}$ from Subsection 3.3. The following lemma reduces the problem of establishing monotonicity of $F$ to showing its monotonicity under the additional assumption that $k_{max}$ and the sets ${\bf J}^{(p)}$, ${\bf I}^{(p)}$, $p=1,...,k_{max}$, are constant in $t$.

Lemma 5.2

Assume that for $k=1,...,d$, the mapping $F$ is nondecreasing on $\mathcal{T}_{k}$. Then $F$ is nondecreasing on $[0,T]$.

Proof. First note that $F$ is nondecreasing on $\overline{\mathcal{T}_{k}}$ for each $k=1,...,d$. Indeed, let $t$, $\tilde{t}$ be such that $t<\tilde{t}$ and $t,\tilde{t}\in\overline{\mathcal{T}_{k}}$ for some $k$. Then for each $n\in\mathbb{N}$, there exist $t_{n},\tilde{t}_{n}\in\mathcal{T}_{k}$ such that $t_{n}<\tilde{t}_{n}$ and $t_{n}\rightarrow t$, $\tilde{t}_{n}\rightarrow\tilde{t}$ as $n\rightarrow\infty$. By assumption, $F(t_{n})\leq F(\tilde{t}_{n})$ for each $n$. Letting $n\rightarrow\infty$, by Theorem 4.2 we get $F(t)\leq F(\tilde{t})$, so $F$ is indeed monotone on $\overline{\mathcal{T}_{k}}$.

Let $0\leq t<\tilde{t}\leq T$ and let $k_{0}$ be such that $t\in\mathcal{T}_{k_{0}}$. Let $t_{1}=\sup\{s\leq\tilde{t}:s\in\mathcal{T}_{k_{0}}\}$. Then $t_{1}\in\overline{\mathcal{T}_{k_{0}}}$, so $F(t)\leq F(t_{1})$. If $t_{1}=\tilde{t}$, we have (5.1) and the proof is complete. Assume that $t_{1}<\tilde{t}$. In this case, $(t_{1},\tilde{t}]\cap\mathcal{T}_{k_{0}}=\mathchar 31\relax$ by the definition of $t_{1}$. However, there exist $k_{1}\neq k_{0}$ and a sequence $s_{n}\in\mathcal{T}_{k_{1}}$ such that $s_{n}\downarrow t_{1}$. Let $t_{2}=\sup\{s\leq\tilde{t}:s\in\mathcal{T}_{k_{1}}\}$. Then $t_{1},t_{2}\in\overline{\mathcal{T}_{k_{1}}}$, so $F(t_{1})\leq F(t_{2})$, and hence $F(t)\leq F(t_{2})$. If $t_{2}=\tilde{t}$, the proof is complete, otherwise $(t_{2},\tilde{t}]\cap(\mathcal{T}_{k_{0}}\cup\mathcal{T}_{k_{1}})=\mathchar 31\relax$. Let $k_{2}\notin\{k_{0},k_{1}\}$ and a sequence $s_{n}\in\mathcal{T}_{k_{2}}$ be such that $s_{n}\downarrow t_{2}$. Put $t_{3}=\sup\{s\leq\tilde{t}:s\in\mathcal{T}_{k_{2}}\}$. As above, we have $F(t_{2})\leq F(t_{3})$, and hence $F(t)\leq F(t_{3})$. After a finite number $l\leq d$ of such steps we get $F(t)\leq F(t_{l})$ and $t_{l}=\tilde{t}$, so (5.1) holds. $\hfill\Box$

A slight modification of the above argument yields

Lemma 5.3

Assume that for $k=1,...,d$, the mapping $F$ is Lipschitz continuous on $\mathcal{T}_{k}$. Then $F$ is Lipschitz on $[0,T]$.

This lemma, together with (3.16)-(3.18), may be used to provide a simple proof of the following result.

Theorem 5.4

Assume that there exist $0<c<C<\infty$ such that

[TABLE]

Then $F$ is Lipschitz continuous in $[0,\infty)$.

Proof. Fix $T>0$. We will argue that $F$ is Lipschitz in $[0,T]$ (with the Lipschitz constant independent on $T$). According to Lemma 5.3, it suffices to show this in each $\mathcal{T}_{l}$, $l=1,...,d$. Fix $\mathcal{D}=(J_{1},...,J_{k})\in\mathcal{J}$. By (5.15), for $0\leq x\leq y$, we have $(y-x)/(CI)\leq g(y)-g(x)\leq(y-x)/c$, where $g$ is any of the inverses appearing in (3.16)-(3.17). Hence, the function $f^{(1)}$ defined by (3.16) is Lipschitz in $\mathcal{T}^{\mathcal{D}}$. Using this fact, together with (3.17), (5.15), and proceeding by induction, we show Lipschitz continuity of $f^{(p)}$, $p=2,...,k$, in $\mathcal{T}^{\mathcal{D}}$. From this, we get Lipschitz continuity of $F$ in $\mathcal{T}^{\mathcal{D}}$ by (3.10). $\hfill\Box$

Example 4.1 indicates that the lower bound in (5.15) is necessary for Theorem 5.4 to hold.

5.3 Local monotonicity in the $C^{1}$ case

In this subsection, we assume that for each $i\in{\bf I}$, $h_{i}\in C^{1}([x^{*}_{i},\infty))$ and

[TABLE]

where $h_{i}^{\prime}(x^{*}_{i})$ denotes the right derivative of $h_{i}$ at $x^{*}_{i}$. Again, with no loss of generality we may assume (5.3). Define $m^{*}$, $n_{0},...,n_{m^{*}}$, $y^{*}_{1},...,y^{*}_{m^{*}}$ as in Section 5.1. Because our concern here is local monotonicity of $F$, without loss of generality we may assume that $h_{i}^{\prime}(x)>0$ for all $i\in{\bf I}$ and $x\geq x^{*}_{i}$ (compare Remark 5.1).

The main idea of our analysis for this case is to consider it as a small perturbation of the linear problem considered in Section 5.1, with $\rho^{i}$ given by (5.16). In particular, let $\underline{t}_{k_{max}}$ be as in in Section 5.1 and let $k_{max}^{L}$ be the constant $k_{max}(t)$, $0<t<\underline{t}_{k_{max}}$, defined there. Furthermore for $k=1,...,k_{max}^{L}$, let $a^{(k)}$ be as in Section 5.1 and let ${\bf J}^{(k)}_{L}$, ${\bf I}^{(k)}_{L}$, ${\bf N}^{(k)}_{L}$, ${\bf K}^{(k)}_{L}$, $f^{(k)}_{L}$, denote the sets ${\bf J}^{(k)}(t)$, ${\bf I}^{(k)}(t)$, ${\bf N}^{(k)}(t)$, ${\bf K}^{(k)}(t)$, $t\in(0,\underline{t}_{k_{max}})$, and the function $f^{(k)}$ for the above-mentioned linear problem, respectively.

Fix $0<T\leq\underline{t}_{k_{max}}/2$ and the partition $\mathcal{D}=(J_{1},...,J_{k})\in\mathcal{J}$ such that there exists a sequence $t_{n}\downarrow 0$ such that $t_{n}\in\mathcal{T}^{\mathcal{D}}$ for all $n$. Since $f^{(1)}(t)<...<f^{(k-1)}(t)<t$ for $t\in\mathcal{T}^{\mathcal{D}}$, using (3.16)-(3.17) and proceeding by induction, it is easy to see that $f^{(1)}$,…, $f^{(k-1)}$ are the truncations of $C^{1}$ functions to $\mathcal{T}^{\mathcal{D}}$, and $f^{(k)}$ is the truncation to $\mathcal{T}^{\mathcal{D}}$ of a function in the form $g(t)\wedge t$, where $g$ is $C^{1}$. Therefore, in order to prove that the functions $f^{(p)}$, $p=1,...,k=k_{max}$, (and hence, by (3.10), $F_{i}$, $i\in{\bf I}$) are nondecreasing in an intersection of $\mathcal{T}^{\mathcal{D}}$ and a neighborhood of zero, it suffices to check that

[TABLE]

where $f^{(p)}(0):=f^{(p)}(0+)=\lim_{n\rightarrow\infty}f^{(p)}(t_{n})$. Consequently, by Lemma 5.2, if we verify (5.17), then the proof of monotonicity of the mapping $F$ in a neighborhood of zero is complete. In what follows, we consider only $t\in\mathcal{T}^{\mathcal{D}}$.

We will first consider the case of $x^{*}_{1}=y^{*}_{1}=0$ (hence $m^{*}=1$, $n_{1}=I$) and $a^{(1)}\leq 1$. Then for every $j\in{\bf J}$ and $t>0$ small enough,

[TABLE]

with equality for $j\in{\bf J}^{(1)}_{L}$. If $a^{(1)}<1$, then (5.18) implies that $f^{(1)}(t)=t$, and hence $F_{i}(t)=t$, $i\in{\bf I}$, for $t$ small enough. If $a^{(1)}=1$, then, by (5.18), $f^{(1)}(t)=t+o(t)$. Since $f^{(1)}(t)<f^{(p)}(t)\leq t$ for $p=2,...,k$, this implies $(f^{(1)})^{\prime}(0)=(f^{(2)})^{\prime}(0)=...=(f^{(k)})^{\prime}(0)=1$. Therefore, in the remainder of the proof we may assume that either $x^{*}_{1}<0$, or $a^{(1)}>1$.

By (4.5), we have $f^{(1)}(0)=x^{*}_{1}=y^{*}_{1}$. We claim that

[TABLE]

(compare (5.8)). Suppose that (5.19) is false and let $j\in J_{1}\setminus{\bf J}^{(1)}_{L}$, $j^{\prime}\in{\bf J}^{(1)}_{L}$. Then, by (5.5)-(5.6), for small $t>0$ we have

[TABLE]

We have obtained a contradiction, proving (5.19). Using (5.6) and (5.19), for $j\in J_{1}$, for small $t>0$, we get

[TABLE]

yielding (5.20). If $k=1$, then (5.20) implies (5.17) and our proof is complete, otherwise we proceed by induction as follows.

Assume that for some $1\leq l<k$ there are indices $r\in\{1,..,l\wedge k^{L}_{max}\}$, $0=p_{0}<1\leq p_{1}<p_{2}<...<p_{r}=l$ such that

[TABLE]

where the sets $I_{p}$ are as in (3.18). We also assume that for $p=p_{s-1}+1,...p_{s}$, $s=1,...,r$, we have

[TABLE]

By (5.19)-(5.20), the above assumptions hold for $l=1$ (hence $r=1$, $p_{1}=1$). Note that in this case (5.22) is vacuously true.

For $p=0,...,k-1$, let $K_{p}=\{j^{\prime}\in{\bf J}:G_{j}\setminus\bigcup_{q=1}^{p}I_{q}\neq\mathchar 31\relax\}$. Clearly, $K_{0}={\bf J}$. Moreover, (3.18) implies that $K_{p}={\bf K}^{(p)}(t)$ for $p=1,...k-1$. We claim that

[TABLE]

Indeed, the inclusion in (5.24) is obvious, since $p_{r-1}<l$. By (5.22), $\bigcup_{p=1}^{p_{r-1}}I_{p}=\bigcup_{s=1}^{r-1}{\bf I}^{(s)}_{L}$, hence $\bigcup_{p=1}^{p_{r-1}}{\bf D}^{(p)}(t)=\bigcup_{s=1}^{r-1}{\bf D}^{(s)}_{L}$, yielding the equality in (5.24).

By (3.18) and (5.21) with $s=r$, $\bigcup_{p=p_{r-1}+1}^{l}I_{p}\subseteq\bigcup_{j\in{\bf J}^{(r)}_{L}}G_{j}\subseteq\bigcup_{s=1}^{r}{\bf I}^{(s)}_{L}$. This, together with (5.22) and the fact that the sets $I_{p}$ are disjoint, yields

[TABLE]

We first assume that the inclusion in (5.25) is strict. Then

[TABLE]

Indeed, if (5.26) is false, then ${\bf J}^{(r)}_{L}\subseteq\bigcup_{p=1}^{l}{\bf D}^{(p)}(t)$, and hence ${\bf I}^{(r)}_{L}\subseteq\bigcup_{p=1}^{l}I_{p}$. The latter inclusion, together with (5.22) and the fact that the sets $I_{p}$ are disjoint, yields ${\bf I}^{(r)}_{L}\subseteq\bigcup_{p=p_{r-1}+1}^{l}I_{p}$, contrary to the case assumption, so (5.26) follows. By (2.1), (3.7)-(3.8), (5.16), (5.22)-(5.23) and (5.25), for small $t>0$, $f^{(l+1)}(t)$ is the supremum of $x\leq t$ satisfying the constraints

[TABLE]

[TABLE]

for $j\in K_{l}$. Recall that in the corresponding linear problem, for small $t>0$,

[TABLE]

for $j\in{\bf K}^{(r-1)}_{L}$, with equality for $j\in{\bf J}^{(r)}_{L}$ (compare (5.9), (5.1)). Note that the second equality in (5.28) follows from (5.25). Comparing (5.27) to (5.28) and using (5.24), (5.26), we get the inclusion $J_{l+1}\subseteq{\bf J}^{(r)}_{L}$ and (5.23) for $p=l+1$, $s=r$. This ends the inductive step in the case of strict inclusion in (5.25).

It remains to analyze the case in which

[TABLE]

Then (5.22) holds for $s=1,...,r$, so $\bigcup_{p=1}^{l}I_{p}=\bigcup_{s=1}^{r}{\bf I}^{(p)}_{L}$ and $K_{l}={\bf K}^{(r)}_{L}$ (this is the equality in (5.24), with $r$ in the place of $r-1$). Also, (5.22) for $s\leq r$, together with the inequality $l<k$, implies that $r<k^{L}_{max}$. The counterpart of (5.27) in this case is

[TABLE]

for $j\in K_{l}$. If $r+1=k^{L}_{max}$, $y_{m^{(r+1)}}=y_{m^{*}}=0$ and $a^{(r+1)}<1$, then $f^{(r+1}_{L}(t)=t$ for small $t>0$ and the counterpart of (5.28) is

[TABLE]

for $j\in{\bf K}^{(r)}_{L}=K_{l}$. Comparing (5.30) to (5.31) we see that $f^{(l+1)}(t)=t$ and $l+1=k_{max}(t)=k$, so our inductive proof of (5.17) is complete. If $y_{m^{(r+1)}}<0$ or $a^{(r+1)}\geq 1$, then the counterpart of (5.28) is

[TABLE]

for $j\in{\bf K}^{(r)}_{L}=K_{l}$, with equality for $j\in{\bf J}^{(r)}_{L}$. Comparing (5.30) to (5.32), we get (5.23) for $p=l+1$, $s=r+1$, and $J_{l+1}\subseteq{\bf J}^{(r+1)}_{L}$, yielding (5.21) for $s=r+1$, and the proof of the inductive step is complete.

5.4 Lack of global monotonicity

It is not hard to prove that, without any additional assumptions on the functions $h_{i}$, (5.1) holds for every $0\leq t<\tilde{t}$ and $i\in{\bf I}^{(1)}(t)$. In spite of this, the mapping $F$ is, in general, not monotone on $[0,\infty)$, even if $h_{i}$, $i\in{\bf I}$, are given by (5.2), as the following example shows.

Example 5.5

Let $I=7$, $J=4$, $G_{1}=\{1,3,6\}$, $G_{2}=\{2,4,7\}$, $G_{3}=\{3,4,5,6,7\}$ and $G_{4}=\{6,7\}$. Next, let

[TABLE]

so that $x^{*}_{1}=x^{*}_{2}=x^{*}_{3}=x_{4}^{*}=-11$, $x^{*}_{5}=0$, $x^{*}_{6}=x^{*}_{7}=-10$. One may easily check that for $0\leq t\leq 8$, ${\bf J}^{(1)}(t)=\{1,2\}$, ${\bf J}^{(2)}(t)=\{3\}$, ${\bf I}^{(1)}(t)=\{1,2,3,4,6,7\}$, ${\bf I}^{(2)}(t)=\{5\}$, ${\bf N}^{(1)}(t)=\{4\}$ and ${\bf N}^{(2)}(t)=\mathchar 31\relax$. Moreover,

[TABLE]

hence

[TABLE]

and the mapping $F$ fails to be nondecreasing.

The “network topology” in the above example may be somewhat simplified, at the price of making some of the functions $h_{i}(x)$, $x\geq x^{*}_{i}$, nonlinear. Namely, let $I=5$, $J=3$, $G_{1}=\{1,3\}$, $G_{2}=\{2,4\}$ and $G_{3}=\{3,4,5\}$. Assume (5.33)-(5.34) and let $h_{3}(x)=h_{4}(x)=(x+11)^{+}+2(x+10)^{+},$ so that $x^{*}_{1}=x^{*}_{2}=x^{*}_{3}=x_{4}^{*}=-11$, $x^{*}_{5}=0$. (Somewhat informally, this network structure has been obtained from the previous one by removing the fourth server and merging the routes $3,6$ (resp., $4,7$) into a single route $3$ (resp., $4$). It is easy to verify that in this case for $0\leq t\leq 8$, ${\bf J}^{(1)}(t)=\{1,2\}$, ${\bf J}^{(2)}(t)=\{3\}$, ${\bf I}^{(1)}(t)=\{1,2,3,4\}$, ${\bf I}^{(2)}(t)=\{5\}$ and (5.35)-(5.37) still hold. Note that this network satisfies the local traffic condition and hence ${\bf N}^{(1)}(t)={\bf N}^{(2)}(t)=\mathchar 31\relax$ for all $t$. **

6 Further research directions

The results obtained in this paper may be regarded as introductory in nature and there are several important issues regarding our mapping $F$ that remain to be addressed. First, one would like to relax the assumptions on regularity of $h_{i}$ necessary for local monotonicity of $F$. Example 5.5 shows that “kinks” of $h_{i}$ may create problems in this regard, so it is not immediately clear that the monotonicity result of Section 5.3 may be carried over even to the Lipschitz case. A remedy for this problem may be creating a “differential” version of the algorithm from Section 3.1, determining the derivatives of $f^{(p)}$, rather than their values, for Lipschitz $h_{i}$, in a way similar to our analysis for the linear case. This, however, in the absence of $C^{1}$ regularity of $h_{i}$, yields an ODE system with discontinuous right-hand side, so even establishing existence of solutions to such a system may be challenging.

Another direction that appears to be important for applications is to skip the assumption of strict monotonicity of $h_{i}$. As Example 4.1 indicates, this results in jumps of the corresponding process $F$. However, from the point of view of the queueing application described in Section 2.3, with the functions $h_{i}$ given by (2.9), this is not necessarily a problem, because some $F_{i}$ may “jump over the flat spots”, containing no mass of the corresponding initial distributions $\overline{Z}_{i}(0,\cdot)$, without causing discontinuity of the resulting locally edge minimal fluid model. Similarly, it may be useful to investigate functions $h_{i}$ with upward jumps, corrresponding to distributions with atoms. This would open an avenue to using techniques similar to those developed in our forthcoming paper, but for pre-limit stochastic networks, rather than for the corresponding fluid limits.

Example 5.5 shows that there is no hope for global monotonicity of $F$ in the general case. However, it is plausible that for some simple network topologies (e.g., linear or tree networks), the mapping $F$ is monotone on $[0,\infty)$. This would greatly simplify the analysis of the corresponding fluid limits, and aid the investigation of the pre-limit stochastic networks.

Finally, it may be interesting to replace the relation “$\eqslantless$” and/or the set $A_{t}$ in the definition of $F(t)$ by a different partial ordering and/or admissible set, and to investigate properties and possible applications of the resulting mappings.

7 Appendix: Inductive proof of Theorem 4.2

We continue the argument starting in Section 4. As we have already explained, we may assume that $t_{0}>0$ and that (4.3), (4.6) hold.

7.1 The base case: $i\in{\bf I}^{(1)}(t_{0})$

For $j\in{\bf J}^{(1)}$, we have $\sum_{i\in G_{j}}h_{i}(f^{(1)}(t_{n}))=t_{n}$, so $\sum_{i\in G_{j}}h_{i}(f^{(1)}(t_{0}))=t_{0}$. Consequently, ${\bf J}^{(1)}\subseteq{\bf J}^{(1)}(t_{0})$, and thus ${\bf I}^{(1)}\subseteq{\bf I}^{(1)}(t_{0})$. Let $i\in{\bf I}^{(1)}$. Then $i\in{\bf I}^{(1)}(t_{0})$, so, by (3.10), $F_{i}(t_{n})=f^{(1)}(t_{n})\vee x^{*}_{i}\rightarrow f^{(1)}(t_{0})\vee x^{*}_{i}=F_{i}(t_{0})$ as $n\rightarrow\infty$. We have shown that for $i\in{\bf I}^{(1)}$, we have (4.2).

If ${\bf I}^{(1)}={\bf I}^{(1)}(t_{0})$ (in particular, if ${\bf I}^{(1)}={\bf I}$), we have (4.2) for every $i\in{\bf I}^{(1)}(t_{0})$. Assume that ${\bf I}^{(1)}\neq{\bf I}^{(1)}(t_{0})$. Then

[TABLE]

because $\bigcup_{j\in{\bf D}^{(1)}}G_{j}={\bf I}^{(1)}$ (see (4.1)), while $\bigcup_{j\in{\bf J}^{(1)}(t_{0})}G_{j}={\bf I}^{(1)}(t_{0})$. Let

[TABLE]

By (7.1), we have $p_{1}\geq 2$. For $p=1,...,p_{1}$, let

[TABLE]

Clearly,

[TABLE]

and $\bigcup_{p=1}^{p_{1}}B_{p}^{(1)}\subseteq{\bf I}^{(1)}(t_{0})$. We will check that

[TABLE]

Indeed, let $i\in{\bf I}^{(1)}(t_{0})$. Then $i\in G_{j}$ for some $j\in{\bf J}^{(1)}(t_{0})$, so, by (7.4), $j\in A_{p}^{(1)}\subseteq{\bf D}^{(p)}$ for some $p\in\{1,...,p_{1}\}$. Consequently, for some $q\in\{1,...,p\}$, we have $i\in{\bf I}^{(q)}$ and hence $i\in B_{q}^{(1)}$. Thus, ${\bf I}^{(1)}(t_{0})\subseteq\bigcup_{p=1}^{p_{1}}B_{p}^{(1)}$ and (7.5) follows. Moreover, the above argument justifies the inclusion

[TABLE]

where the equality follows from the inclusion $A_{p}^{(1)}\subseteq{\bf D}^{(p)}$.

Let $p\in\{2,...,p_{1}\}$ be such that $A_{p}^{(1)}\neq\mathchar 31\relax$ and let $j\in A_{p}^{(1)}$. Since $j\in{\bf J}^{(1)}(t_{0})$ and (4.6) holds, we have

[TABLE]

If $f^{(p)}(t_{n})=t_{n}$ for infinitely many $n$, then, by the definition of $f^{(p)}$ (compare (3.7)-(3.8)) and the inequality $f^{(1)}<f^{(p)}$, for these $n$ we have

[TABLE]

Letting $n\rightarrow\infty$ in (7.1), we get

[TABLE]

The relations (7.7) and (7.9), together with the inequalities $f^{(1)}(t_{0})\leq t_{0}$, $t_{0}>0\geq\max_{i\in{\bf I}}x^{*}_{i}$, imply that $f^{(1)}(t_{0})=t_{0}$, which contradicts (4.6). Consequently, since $A_{p_{1}}^{(1)}\neq\mathchar 31\relax$ by (7.2)-(7.3), in the remainder of our proof we may assume that for $p\leq p_{1}$,

[TABLE]

Since $j\in A_{p}^{(1)}\subseteq{\bf D}^{(p)}={\bf J}^{(p)}\cup{\bf N}^{(p)}\subseteq{\bf K}^{(p-1)}$, for every $n$ we have

[TABLE]

(compare (3.6)-(3.7)), and

[TABLE]

with equality for $j\in C_{p}^{(1)}:={\bf J}^{(p)}\cap{\bf J}^{(1)}(t_{0})\subseteq A_{p}^{(1)}$. Note that (7.11) and the above-mentioned equality in (7.12) imply that for $n\geq 1$, $j\in C_{p}^{(1)}$,

[TABLE]

By (4.3) and (7.13),

[TABLE]

Letting $n\rightarrow\infty$ in (7.12) for $j\in A_{p}^{(1)}$ and using (4.3), we get

[TABLE]

This, together with (7.7), (4.4) and monotonicity of $h_{i}$, yields the equality in (7.15) for each $j\in A_{p}^{(1)}$ and the equations

[TABLE]

By (7.6) and (7.16) with $p=p_{1}$, we have

[TABLE]

The relations (4.4), (7.13)-(7.14), (7.16) (with $k=p$) and the properties of the functions $h_{i}$, in turn, imply that either

[TABLE]

or

[TABLE]

provided that $C_{p}^{(1)}\neq\mathchar 31\relax$. In the latter case, $f^{*}_{j,p}$ does not depend on $j\in C_{p}^{(1)}$ and hence it will be denoted by $f^{*}_{p}$.

The above argument implies that if $f^{(1)}(t_{0})\geq\min_{j\in C_{p}^{(1)}}f^{*}_{j,p}$ (or if $f^{*}_{j,p}\neq f^{*}_{j^{\prime},p}$ for some $j,j^{\prime}\in C_{p}^{(1)}$), then (7.19) cannot hold, and hence (7.18) holds. Similarly, if $C_{p}^{(1)}\neq\mathchar 31\relax$ and

[TABLE]

then, by (7.14) and (7.19),

[TABLE]

Let $\bar{p}_{1}=\max\{p\in\{1,...,p_{1}\}:f^{(p)}_{\infty}=f^{(1)}(t_{0})\}.$ We have shown that

[TABLE]

and that if $\bar{p}_{1}<p_{1}$, then for every $p>\bar{p}_{1}$ such that $C_{p}^{(1)}\neq\mathchar 31\relax$, the relations (7.20)-(7.21) hold. It may be the case that $\bar{p}_{1}<p_{1}$ and for some $\tilde{p}\in\{\bar{p}_{1}+1,...,p_{1}\}$, we have $C_{\tilde{p}}^{(1)}=\mathchar 31\relax$. In this case, if $i\in B_{\tilde{p}}^{(1)}$, then the relations (4.4) and (7.17) imply that

[TABLE]

Fix $i\in{\bf I}^{(1)}(t_{0})$. If $i\in B_{p}^{(1)}$ for some $p\leq\bar{p}_{1}$, then, by (3.10) and (7.22),

[TABLE]

so (4.2) holds. If $i\in B_{p}^{(1)}$ for some $p>\bar{p}_{1}$, then there exists $k\in\{p,...,p_{1}\}$ such that $i\in G_{j}\cap{\bf I}^{(p)}$ for some $j\in A_{k}^{(1)}$ (see (7.6)). If $C_{p}^{(1)}\neq\mathchar 31\relax$, then, by (7.21), $f^{(p)}_{\infty}=f^{*}_{p}$ and, by the same argument as in (7.23),

[TABLE]

Thus, since $i\in{\bf I}^{(1)}(t_{0})$, by (2.1), (3.10), (7.21) and (7.24), as $n\rightarrow\infty$,

[TABLE]

and again (4.2) holds. Finally, if $C_{p}^{(1)}=\mathchar 31\relax$, then reasoning as in (7.25), but using (7.23) instead of (7.24), we get $\lim_{n\rightarrow\infty}F_{i}(t_{n})=F_{i}(t_{0})$. Since, by (7.5), every $i\in{\bf I}^{(1)}(t_{0})$ belongs to some $B_{p}^{(1)}$, $p\leq p_{1}$, the above argument shows that (4.2) is true for all $i\in{\bf I}^{(1)}(t_{0})$.

If ${\bf D}^{(1)}(t_{0})={\bf J}$, then ${\bf I}^{(1)}(t_{0})={\bf I}$ and (4.2) holds for each $i\in{\bf I}$. In what follows, we assume that ${\bf K}^{(1)}(t_{0})\neq\mathchar 31\relax$. For future reference note that

[TABLE]

Indeed, let $j\in{\bf J}^{(p)}$ for some $p\leq\bar{p}_{1}$. By (7.10), $\sum_{i\in G_{j}}h_{i}(F_{i}(t_{n}))=\sum_{k=1}^{p}\sum_{i\in G_{j}\cap{\bf I}^{(k)}}h_{i}(f^{(k)}(t_{n}))=t_{n},$ which, together with (7.22), implies that $\sum_{i\in G_{j}}h_{i}(f^{(1)}(t_{0}))=\sum_{k=1}^{p}\sum_{i\in G_{j}\cap{\bf I}^{(k)}}h_{i}(f^{(1)}(t_{0}))=t_{0},$ and hence $j\in{\bf J}^{(1)}(t_{0})$. We also have $\bigcup_{p=1}^{\bar{p}_{1}}{\bf D}^{(p)}\subseteq{\bf D}^{(1)}(t_{0}).$ Indeed, (7.26) implies $\bigcup_{p=1}^{\bar{p}_{1}}{\bf I}^{(p)}=\bigcup_{p=1}^{\bar{p}_{1}}\bigcup_{j\in{\bf J}^{(p)}}G_{j}\subseteq\bigcup_{j\in{\bf J}^{(1)}(t_{0})}G_{j}={\bf I}^{(1)}(t_{0}).$ Thus, for $j\in{\bf D}^{(p)}$, $p\leq\bar{p}_{1}$, we have $G_{j}\subseteq\bigcup_{k=1}^{p}{\bf I}^{(k)}\subseteq{\bf I}^{(1)}(t_{0})$, and hence $j\in{\bf D}^{(1)}(t_{0})$.

7.2 Inductive assumption

Fix $m\in\{1,...,k_{max}(t_{0})-1\}$. For $l=1,...,m$, let $b_{l}$, $p_{l}$ be given by (4.7)-(4.8). By definition, $b_{1}=1$ and $b_{l}\leq p_{l}$ for each $l=1,...,m$. For $l=1,...,m-1$, let $q_{l}=b_{l+1}-1$, so that by (4.7) we have

[TABLE]

For $p=b_{l},...,p_{l}$, $l=1,...,m$, let

[TABLE]

The definitions (4.7)-(4.8) imply that

[TABLE]

(see also (7.27)). Using (7.27) and (7.29), one may easily check (4.9).

Suppose that for $l=1,...,m$, the following assertions hold.

There exist indices $\bar{p}_{l}\in\{b_{l},...,p_{l}\}$ such that (4.10) holds and

[TABLE]

Note that (4.7) and (7.30) imply

[TABLE]

If $\bar{p}_{l}<p_{l}$, then for every $p\in\{\bar{p}_{l}+1,...,p_{l}\}$ such that $C_{p}^{(l)}\neq\mathchar 31\relax$, we have

[TABLE]

where $f^{*}_{p}$ is the common value of $f^{*}_{j,p}$ (defined in (7.13)) for $j\in C_{p}^{(l)}$ and, moreover, $f^{*}_{p}\leq x^{*}_{i}$ for $i\in B_{p}^{(l)}$. If $C_{\tilde{p}}^{(l)}=\mathchar 31\relax$ for some $\tilde{p}\in\{\bar{p}_{l}+1,...,p_{l}\}$, then

[TABLE]

For $j\in A_{p}^{(l)}$, $p=b_{l},...,p_{l}$ and $r=1,..,l$,

[TABLE]

For $r=1,...,m$ and $k=b_{r},...,p_{r}$, $B^{(r)}_{k}\subseteq{\bf I}^{(r)}(t_{0})\subseteq\bigcup_{j\in{\bf J}^{(r)}(t_{0})}G_{j}=\bigcup_{p=b_{r}}^{p_{r}}\bigcup_{j\in A^{(r)}_{p}}G_{j},$ where the equality follows from (7.29). Hence, (7.34) implies

[TABLE]

Because $f^{(1)}(t_{0})<...<f^{(l)}(t_{0})\leq f^{(k)}_{\infty}$ for $k=b_{l},...,p_{l}$, (7.34) implies also that for $p=b_{l},...,p_{l}$,

[TABLE]

Finally, we assume that (4.2) holds for all $i\in\bigcup_{k=1}^{m}{\bf I}^{(k)}(t_{0})$.

Note that all the above assumptions have already been verified in the case of $m=1$.

7.3 Inductive step

Define $b_{m+1}$ by (4.7) with $l=m+1$ and let $q_{m}=b_{m+1}-1$. By definition, we have (7.27) for $l=m$, so ${\bf K}^{(m)}(t_{0})\subseteq{\bf K}^{(q_{m})}$. The inclusion (7.30) implies that

[TABLE]

Also, for $j\in{\bf K}^{(q_{m})}$ (and hence for $j\in{\bf K}^{(m)}(t_{0})$), we have

[TABLE]

If

[TABLE]

for some $n$, then $k_{max}=b_{m+1}$, ${\bf J}^{(b_{m+1})}={\bf K}^{(q_{m})}$, ${\bf N}^{(b_{m+1})}=\mathchar 31\relax$ and $F_{i}(t_{n})=t_{n}$ for all $i\in{\bf I}^{(b_{m+1})}={\bf I}\setminus\bigcup_{k=1}^{q_{m}}{\bf I}^{(k)}$. If (7.39) holds for infinitely many $n$, then letting $n\rightarrow\infty$ in (7.38) along this subsequence, by (7.27) for $l=m$, monotonicity of $h_{i}$, the inequality $F_{i}(t_{0})\leq t_{0}$ for all $i\in{\bf I}$, and the inductive assumption (4.2) for $i\in\bigcup_{k=1}^{m}{\bf I}^{(k)}(t_{0})$, we get

[TABLE]

for all $j\in{\bf K}^{(m)}(t_{0})$. This, in turn, implies that $F_{i}(t_{0})=t_{0}$ for all $i\in{\bf I}^{(m+1)}(t_{0})={\bf I}\setminus\bigcup_{p=1}^{m}{\bf I}^{(p)}(t_{0})\subseteq{\bf I}\setminus\bigcup_{k=1}^{q_{m}}{\bf I}^{(k)}$, where the inclusion follows from the second inclusion in (7.27) for $l=m$. Hence, for all $i\in{\bf I}^{(m+1)}(t_{0})$ (and thus for all $i\in{\bf I}$), we have (4.2) along a subsequence $\{t_{n}\}$ satisfying (7.39). Consequently, in the remainder of the proof we may assume that $f^{(b_{m+1})}(t_{n})<t_{n}$ for each $n\geq 1,$ and thus equality holds in (7.38) for $j\in{\bf J}^{(b_{m+1})}$.

We claim that

[TABLE]

Letting $n\rightarrow\infty$ in (7.38), we get

[TABLE]

for $j\in{\bf K}^{(q_{m})}$, with equality for $j\in{\bf J}^{(b_{m}+1)}$ (compare (7.40)). For $k=1,...,m$, by (4.10) and (7.37), we have

[TABLE]

so for $i\in{\bf I}^{(k)}(t_{0})$, $h_{i}(F_{i}(t_{0}))=h_{i}(f^{(k)}(t_{0}))\leq h_{i}(f^{(b_{m+1})}_{\infty})$. This, together with (7.27) for $l=m$ and (7.43) implies

[TABLE]

for $j\in{\bf K}^{(q_{m})}$ (hence for $j\in{\bf K}^{(m)}(t_{0})$), and thus

[TABLE]

In order to prove the opposite inequality, we will first check that

[TABLE]

Suppose, to the converse, that

[TABLE]

Then ${\bf I}^{(b_{m+1})}\subseteq\bigcup_{j\in{\bf J}^{(b_{m+1})}}G_{j}\subseteq\bigcup_{p=1}^{m}\bigcup_{j\in{\bf J}^{(p)}(t_{0})}G_{j}=\bigcup_{p=1}^{m}{\bf I}^{(p)}(t_{0}),$ and thus, by the second inclusion in (7.27) for $l=m$, $\bigcup_{p=1}^{b_{m+1}}{\bf I}^{(p)}\subseteq\bigcup_{p=1}^{m}{\bf I}^{(p)}(t_{0})$. Consequently, for $j\in{\bf N}^{(b_{m+1})}$, $G_{j}\subseteq\bigcup_{p=1}^{m}{\bf I}^{(p)}(t_{0})$, and hence ${\bf N}^{(b_{m+1})}\subseteq\bigcup_{p=1}^{m}{\bf D}^{(p)}(t_{0})$. This, together with (7.48), implies that ${\bf D}^{(b_{m+1})}\subseteq\bigcup_{p=1}^{m}{\bf D}^{(p)}(t_{0})$, which contradicts the definition of $b_{m+1}$. We have proved (7.47).

Fix $j\in{\bf J}^{(b_{m+1})}\setminus\bigcup_{p=1}^{m}{\bf J}^{(p)}(t_{0})$. Then equality holds in (7.43). We will check that equality holds in (7.45) as well. Suppose that $i\in G_{j}\cap\bigcup_{p=1}^{m}{\bf I}^{(p)}(t_{0})\setminus\bigcup_{p=1}^{q_{m}}{\bf I}^{(p)}$. Then $i\in{\bf I}^{(k)}(t_{0})\cap{\bf I}^{(r)}$ for some $k\leq m$ and $r\geq b_{m+1}$. Letting $n\rightarrow\infty$ in the equality $h_{i}(F_{i}(t_{n}))=h_{i}(f^{(r)}(t_{n}))$, and using the inductive assumption (4.2) for $i\in\bigcup_{p=1}^{m}{\bf I}^{(p)}(t_{0})$, we get $h_{i}(f^{(k)}(t_{0}))=h_{i}(F_{i}(t_{0}))=h_{i}(f^{(r)}_{\infty})$, while $f^{(k)}(t_{0})\leq f^{(m)}(t_{0})\leq f^{(b_{m+1})}_{\infty}\leq f^{(r)}_{\infty}$ by (7.44). Hence $h_{i}(F_{i}(t_{0}))=h_{i}(f^{(b_{m+1})}_{\infty})$, so (7.27) for $l=m$ and the equality in (7.43) imply that equality holds in (7.45) as well. Arguing as in (7.13), we get $f^{(m+1)}(t_{0})>\min\{x^{*}_{i}:i\in G_{j}\setminus\bigcup_{p=1}^{m}{\bf I}^{(p)}(t_{0})\},$ so (7.46) and the equality in (7.45) imply that $f^{(b_{m+1})}_{\infty}=f^{(m+1)}(t_{0})$. We have proved (7.41). Replacing $f^{(b_{m+1})}_{\infty}$ by $f^{(m+1)}(t_{0})$ in the equality in (7.45) we get $j\in{\bf J}^{(m+1)}(t_{0})$, so (7.42) holds as well.

Let $i\in{\bf I}^{(b_{m+1})}\setminus\bigcup_{p=1}^{m}{\bf I}^{(p)}(t_{0})$. We will show that (4.2) holds. Since $i\in{\bf I}^{(b_{m+1})}$, we have $i\in G_{j}$ for some $j\in{\bf J}^{(b_{m+1})}$. If $j\in\bigcup_{p=1}^{m}{\bf D}^{(p)}(t_{0})$, then $i\in\bigcup_{p=1}^{m}{\bf I}^{(p)}(t_{0})$, contrary to the choice of $i$, so $j\in{\bf J}^{(b_{m+1})}\setminus\bigcup_{p=1}^{m}{\bf D}^{(p)}(t_{0})\subseteq{\bf J}^{(b_{m+1})}\setminus\bigcup_{p=1}^{m}{\bf J}^{(p)}(t_{0})$ and hence, by (7.42), $j\in{\bf J}^{(m+1)}(t_{0})$. Then $G_{j}\subseteq\bigcup_{p=1}^{m+1}{\bf I}^{(p)}(t_{0})$, so $i\in{\bf I}^{(m+1)}(t_{0})$. Consequently, by (3.10) and (7.41),

[TABLE]

as $n\rightarrow\infty$, so (4.2) holds.

If ${\bf I}^{(m+1)}(t_{0})\subseteq{\bf I}^{(b_{m+1})}$, we have (4.2) for each $i\in{\bf I}^{(m+1)}(t_{0})$. Assume that

[TABLE]

and let $p_{m+1}$ be given by (4.8) with $l=m+1$. By (7.27) for $l=m$ and (7.50), $b_{m+1}<p_{m+1}$. For $p=b_{m+1},...,p_{m+1}$, define the sets $A^{(m+1)}_{p}$, $B^{(m+1)}_{p}$, $C^{(m+1)}_{p}$ by (7.28) with $l=m+1$. By the definitions of $b_{m+1}$, $p_{m+1}$ and (7.27) for $l=m$, we have (7.29) for $l=m+1$. Clearly, $\bigcup_{p=b_{m+1}}^{p_{m+1}}B_{p}^{(m+1)}\subseteq{\bf I}^{(m+1)}(t_{0})$. We will check that (4.9) holds for $l=m+1$. Indeed, let $i\in{\bf I}^{(m+1)}(t_{0})$. Then $i\in G_{j}$ for some $j\in{\bf J}^{(m+1)}(t_{0})$, so, by (7.29) with $l=m+1$, $j\in A_{p}^{(m+1)}\subseteq{\bf D}^{(p)}$ for some $p\in\{b_{m+1},...,p_{m+1}\}$. Consequently, we have $i\in{\bf I}^{(q)}$ for some $q\leq p$. However, $q>q_{m}$ by the second inclusion in (7.27) with $l=m$, and hence $i\in B_{q}^{(m+1)}$ for some $q\in\{b_{m+1},...,p_{m+1}\}$. Thus, ${\bf I}^{(m+1)}(t_{0})\subseteq\bigcup_{p=b_{m+1}}^{p_{m+1}}B_{p}^{(m+1)}$ and (4.9) with $l=m+1$ follows.

We will first consider the case in which $f^{(m+1)}(t_{0})=t_{0}$. Then ${\bf I}^{(m+1)}(t_{0})={\bf I}\setminus\bigcup_{p=1}^{m}{\bf I}^{(p)}(t_{0})$ and $F_{i}(t_{0})=t_{0}$ for each $i\in{\bf I}^{(m+1)}(t_{0})$. Moreover, by (7.41),

[TABLE]

Let $i\in{\bf I}^{(m+1)}(t_{0})$. Then $i\in{\bf I}^{(p)}$ for some $p\geq b_{m+1}$ by the second inclusion in (7.27) with $l=m$, and thus $f^{(b_{m+1})}(t_{n})\vee x^{*}_{i}\;\leq\;F_{i}(t_{n})\;\leq\;t_{n},$ which, by (7.51), implies that $F_{i}(t_{n})\rightarrow t_{0}=F_{i}(t_{0})$, so (4.2) holds for all $i$.

It remains to consider the case of

[TABLE]

Let $p\in\{b_{m+1},...,p_{m+1}\}$ be such that $A_{p}^{(m+1)}\neq\mathchar 31\relax$ and let $j\in A_{p}^{(m+1)}$. Since $j\in{\bf J}^{(m+1)}(t_{0})$ by (7.28) with $l=m+1$, the relation (7.52) implies

[TABLE]

Suppose that $f^{(p)}(t_{n})=t_{n}$ for infinitely many $n$, and thus $p=k_{max}$. Then ${\bf N}^{(p)}=\mathchar 31\relax$, so ${\bf J}^{(p)}={\bf D}^{(p)}$ and $A^{(l)}_{p}=C^{(l)}_{p}$ for each $l\leq m+1$. If $p_{l}=k_{max}$ for some $l\leq m$, then $p_{l}=p\geq b_{m+1}$, and $C^{(l)}_{p_{l}}=A^{(l)}_{p_{l}}\neq\mathchar 31\relax$ by (4.8), (7.28). Hence, by (7.31), (7.37), (7.32), as $n\rightarrow\infty$, we have $0<t_{0}\leftarrow t_{n}=f^{(p_{l})}(t_{n})\rightarrow f^{*}_{p_{l}}\leq 0$, contradiction. Thus, $p>\max\{p_{l},l\leq m\}$, so by (7.28) and (4.9), ${\bf I}^{(p)}\cap\bigcup_{l=1}^{m}{\bf I}^{(l)}(t_{0})=\mathchar 31\relax$. Consequently, for $j\in A^{(m+1)}_{p}\subseteq{\bf J}^{(m+1)}(t_{0})$, $G_{j}\cap{\bf I}^{(p)}\subseteq{\bf I}^{(m+1)}(t_{0})$ and the definition of $f^{(p)}$ (compare (3.8)), together with (4.9) for $l=1,...,m+1$, imply

[TABLE]

Letting $n\rightarrow\infty$ and using (4.10), (7.41), we get

[TABLE]

which, together with (7.53), implies that $f^{(m+1)}(t_{0})=t_{0}$, contrary to (7.52). Consequently, since $A_{p_{m+1}}^{(m+1)}\neq\mathchar 31\relax$ by (4.8) and (7.28) with $l=m+1$, in the remainder of the proof we may assume that (7.10) holds for $p=b_{m+1},...,p_{m+1}$. This, in turn, implies (7.11), (7.13) and the equality in (7.12) for $j\in C_{p}^{(m+1)}$, $p\in\{b_{m+1},...,p_{m+1}\}$, by the same argument as in the case of $j\in C_{p}^{(1)}$, $p\leq p_{1}$.

The relation (7.13) implies

[TABLE]

For $j\in A_{p}^{(m+1)}$, $p=b_{m+1},...,p_{m+1}$, letting $n\rightarrow\infty$ in (7.12), we get (7.15). This, in turn, together with (7.53), (4.4), (4.7)-(4.8), (4.10), (7.41) and monotonicity of $h_{i}$, yields (7.34) for $j\in A_{p}^{(m+1)}$, $p=b_{m+1},...,p_{m+1}$ and $r=1,..,m+1$. Consequently, (7.36) holds with $l=m+1$ and $p=b_{m+1},...,p_{m+1}$.

Using (7.54) and (7.36) (with $l$, $p$ as above) instead of (7.14) and (7.16), respectively, in analogy to the development of (7.18)-(7.23), (7.24), we can derive the following facts. Let

[TABLE]

Then (4.10) holds with $l=m+1$. If $\bar{p}_{m+1}<p_{m+1}$, then for every $p\in\{\bar{p}_{m+1}+1,...,p_{m+1}\}$ such that $C_{p}^{(m+1)}\neq\mathchar 31\relax$, we have (7.32) with $l=m+1$, where $f^{*}_{p}$ is the common value of $f^{*}_{j,p}$ for $j\in C_{p}^{(m+1)}$. Moreover, for such $p$,

[TABLE]

If $C_{\tilde{p}}^{(m+1)}=\mathchar 31\relax$ for some $\tilde{p}\in\{\bar{p}_{m+1}+1,...,p_{m+1}\}$, then (7.33) holds with $l=m+1$.

Let $i\in{\bf I}^{(m+1)}(t_{0})$. We will check that (4.2) holds. By (4.9) with $l=m+1$, $i\in B^{(m+1)}_{p}$ for some $p\in\{b_{m+1},...,p_{m+1}\}$. If $p\leq\bar{p}_{m+1}$, then (4.2) follows from (4.10) with $l=m+1$ by an argument similar to (7.49). If $\bar{p}_{m+1}<p\leq p_{m+1}$, then the argument is analogous to the proof of the corresponding case $\bar{p}_{1}<p\leq p_{1}$ for $i\in{\bf I}^{(1)}(t_{0})$ (see (7.25) and the paragraph surrounding it); we just use (7.32) with $l=m+1$, (7.55), (7.33) with $l=m+1$, instead of (7.21), (7.24), (7.23), respectively.

If $m+1=k_{max}(t_{0})$, then (4.2) holds for all $i\in{\bf I}$ and we are done. Assume that ${\bf K}^{(m+1)}(t_{0})\neq\mathchar 31\relax$. For the sake of the next inductive step, we will show (7.30) for $l=m+1$. Let $j\in{\bf J}^{(p)}\setminus\bigcup_{k=1}^{m}{\bf J}^{(k)}(t_{0})$ for some $b_{m+1}\leq p\leq\bar{p}_{m+1}$. Since (7.10) holds for this $p$, we have $t_{n}=\sum_{k=1}^{p}\sum_{i\in G_{j}\cap{\bf I}^{(k)}}h_{i}(f^{(k)}(t_{n})).$ Letting $n\rightarrow\infty$ and using (7.28), (4.9), (4.7) with $l=m+1$, (7.35) and (4.10) with $l=m+1$, we get

[TABLE]

so $j\in{\bf J}^{(m+1)}(t_{0})$. We have shown that $\bigcup_{p=b_{m+1}}^{\bar{p}_{m+1}}{\bf J}^{(p)}\subseteq\bigcup_{p=1}^{m+1}{\bf J}^{(p)}(t_{0})$, so

[TABLE]

Let $j\in{\bf D}^{(p)}$ for some $b_{m+1}\leq p\leq\bar{p}_{m+1}$. By (7.27) for $l=m$ and (7.56),

[TABLE]

Thus, $j\in\bigcup_{p=1}^{m+1}{\bf D}^{(p)}(t_{0})$ and (7.30) with $l=m+1$ follows.

In summary, we have shown that either (4.2) holds for all $i\in{\bf I}$, or all the assertions of the inductive assumption, including (4.2) for $i\in{\bf I}^{(l)}$, are true for $l=1,...,m+1$. This ends the inductive proof of (4.2) for each $i\in{\bf I}$.