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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 508570, 23 pages
http://dx.doi.org/10.1155/2012/508570
Research Article

Nonpivot and Implicit Projected Dynamical Systems on Hilbert Spaces

1Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1
2Department of Mathematics & Computer Science, University of Catania, 95124 Catania, Italy

Received 9 November 2010; Accepted 17 January 2011

Academic Editor: L.E. Persson

Copyright © 2012 Monica Gabriela Cojocaru and Stephane Pia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents a generalization of the concept and uses of projected dynamical systems to the case of nonpivot Hilbert spaces. These are Hilbert spaces in which the topological dual space is not identified with the base space. The generalization consists of showing the existence of such systems and their relation to variational problems, such as variational inequalities. In the case of usual Hilbert spaces these systems have been extensively studied, and, as in previous works, this new generalization has been motivated by applications, as shown below.

1. Introduction

In this paper we study the existence of solutions for a class of differential equations with discontinuous and nonlinear right-hand side on the class of nonpivot Hilbert spaces. This class of equations (called projected differential equations) was first introduced (in the form we use) in [1]; however have other studies of a similar formulation has been known since [24]. The formulation of the flow of such equations as dynamical systems in 𝑛 is due to [1, 5], and it has been applied to study the dynamics of solutions of finite-dimensional variational inequalities in [5, 6].

Finite-dimensional variational inequalities theory provides solutions to a wide class of equilibrium problems in mathematical economics, optimization, management science, operations research, finance, and so forth (see, e.g., [4, 68] and the references therein). Therefore there has been a steady interest over the years in studying the stability of solutions to finite-dimensional variational inequalities (and consequently the stability of equilibria for various problems). In general, such a study is done by associating a projected dynamical system to a variational inequality problem; however in the past few years the applied problems, as well as the theoretical results, have progressed to a qualitative study of stability of solutions to variational inequality problems on Hilbert spaces and even on Banach spaces. Examples of the kind of variational problems (and their applications) can be found in see [919] and the references therein).

In this paper we present a new step in this study: we show that a projected differential equation has solutions on a non-pivot Hilbert space of any dimension. We prove the existence and uniqueness of integral curves and show they remain in a given constraint set of interest. As in the finite-dimensional case, a dynamics given by solutions to a projected differential equation is interesting because it describes these problems as dynamical systems. Moreover, as shown in this paper, the new results were needed to be developed for the study of the weighted traffic equilibrium problem (see [20]). Our goal in this paper is to present the mathematical techniques involved in proving the existence of solutions to projected differential equations in a non-pivot setting, which is in fact similar to the one in [21], but adapted to a non-pivot space; in addition, there are a number of preliminary results needed prior to obtaining our main result, which are remarkable since they also hold in a larger setting, namely, that of a reflexive Banach space (see the results in [22, 23]). Last but not least, we also present a projected system formulation called implicit. These kinds of systems have been introduced in the literature in [24], but without any existence result being presented in their case. We thus solve this additional problem in this paper as well.

2. Background Material

In this section we present several definitions and results pertinent to the reader and considered essential for the presentation of the later material.

2.1. Dual Realization of a Hilbert Space

Each time we work with a Hilbert space 𝑉, it is necessary to decide whether or not we identify the topological dual space 𝑉=(𝑉,) with 𝑉. Commonly this identification is made, one of the reasons for this being that the vectors of the polar of a set of 𝑉 are in 𝑉. In some cases the identification does not make sense. For clarity of presentation, we remind below of the basic results regarding the dual realization of a Hilbert space. The readers can refer to [25] for additional information.

First, consider a pre-Hilbert space 𝑉 with an inner product ((𝑥,𝑦)), and its topological dual 𝑉=(𝑉,). It is well known that 𝑉 is a Banach space for the classical dual norm (𝑓=sup𝑥𝑉(|𝑓(𝑥)|/𝑥)). It is also known that there exists an isometry 𝐽𝑉𝑉 such that 𝐽 is linear and for all 𝑥𝑉, 𝐽(𝑥)=grad(𝑥2/2). This mapping 𝐽 is called a duality mapping of (𝑉,𝑉).

Theorem 2.1 (Theorem  1 page 68, [25]). Let 𝑉 be a Hilbert space with the inner product ((𝑥,𝑦)) and 𝐽(𝑉,𝑉) the duality mapping above. Then J is a surjective isometry from 𝑉 to 𝑉. The dual space 𝑉 is a Hilbert space with the inner product: ((𝑓,𝑔))=𝐽1𝑓,𝐽1𝑔𝐽=𝑓1𝑔.(2.1)

Theorem 2.2 (Theorem  2 page 69, [25]). Let V be a pre-Hilbert space. Then there exists a completion 𝑉 of V, that is, an isometry j from V to the Hilbert space 𝑉 such that 𝑗(𝑉) is dense in 𝑉.

Definition 2.3. Let 𝑉 be a Hilbert space. We call {𝐹,𝑗}, where(i)𝐹 is a Hilbert space,(ii)𝑗 is an isometry from 𝐹 to (𝑉,), a dual realization of 𝑉. We then set 𝑓,𝑥=𝑗𝑓(𝑥),𝑓𝐹,𝑥𝑉,(2.2) where 𝑓,𝑥 is the duality pairing for 𝐹×𝑉.

Remark 2.4. The duality pairing is a nondegenerate bilinear form on 𝐹×𝑉 and 𝑓𝐹=sup𝑥𝑉(|𝑓,𝑥|/𝑥). These properties permit us to prove that 𝐹 is isomorphic to 𝑉.

We deduce from Theorems 2.1 and 2.2 that 𝑘=𝑗1𝐽(𝑉,𝐹) is a surjective isometry such that(𝑥,𝑦)=𝑘(𝑥),𝑦.(2.3) We use the following convention here: when a dual realization {𝐹,𝑗} of a space has been chosen, we set 𝐹=𝑉 and 𝑗𝑓(𝑥)=𝑓,𝑥. We say that the isometry 𝑘𝑉𝑉 is the duality operator associated to the inner product on 𝑉 and to the duality pairing on 𝑉×𝑉 by the relation(𝑥,𝑦)=𝑘(𝑥),𝑦.(2.4) A special but most frequent case is to choose a dual realization of 𝑉 the couple {𝑉,𝐽}; in this case the Hilbert space 𝑉 is called a pivot space. To be more precise, we introduce the following definition.

Definition 2.5. A Hilbert space 𝐻 with an inner product (𝑥,𝑦) is called a pivot space, if we identify 𝐻 with 𝐻. In that case 𝐻=𝐻,𝑗=𝐽,𝑥,𝑦=(𝑥,𝑦).(2.5)

Sometimes it does not make sense to identify the space itself with its topological dual, as the following example shows.

Let us consider 𝑉=𝐿2(,(1+|𝑥|))𝐿2() (dense subspace of 𝐿2()) endowed with the inner product:(𝑢,𝑣)𝑉=(1+|𝑥|)𝑢(𝑥)𝑣(𝑥)𝑑𝑥.(2.6)An element 𝜑𝐿2() is also an element of 𝑉. If we identify 𝜑 to an element 𝑓𝐿2(), this function does not define a linear form on 𝑉, and the expression 𝜑(𝑣)=𝑓,𝑣𝑉 has no meaning on 𝑉. In this situation it is necessary to work in a non-pivot Hilbert space. We provide now some useful examples of non-pivot H-spaces.

Let Ω𝑛 be an open subset of, 𝑎Ω𝑅+{0}, a continuous and strictly positive function called “weight” and 𝑠Ω𝑅+{0}, a continuous and strictly positive function called “real time density.” The bilinear form defined on 𝒞0(Ω) (continuous functions with compact support on Ω) by(𝑥,𝑦)𝑎,𝑠=Ω𝑥(𝜔)𝑦(𝜔)𝑎(𝜔)𝑠(𝜔)𝑑𝜔(2.7) is an inner product. We remark here that if 𝑎 is a weight, then 𝑎1=1/𝑎 is also a weight. Let us introduce the following.

Definition 2.6. We call 𝐿2(Ω,𝑎,𝑠) a completion of 𝒞0(Ω) for the inner product 𝑥,𝑦𝑎,𝑠.

We now introduce an 𝑛-dimensional version of the previous space. If we denote by 𝑉𝑖=𝐿2(Ω,,𝑎𝑖,𝑠𝑖) and 𝑉𝑖=𝐿2(Ω,,𝑎𝑖1,𝑠𝑖), the space𝑉=𝑚𝑖=1𝑉𝑖(2.8) is a non-pivot Hilbert space with the inner product:(𝐹,𝐺)𝑉=(𝐹,𝐺)𝐚,𝐬=𝑚𝑖=1Ω𝐹𝑖(𝜔)𝐺𝑖(𝜔)𝑎𝑖(𝜔)𝑠𝑖(𝜔)𝑑𝜔.(2.9) The space𝑉=𝑚𝑖=1𝑉𝑖(2.10) is clearly a non-pivot Hilbert space for the following inner product(𝐹,𝐺)𝑉=(𝐹,𝐺)𝐚1,𝐬=𝑚𝑖=1Ω𝐹𝑖(𝜔)𝐺𝑖(𝜔)𝑠𝑖(𝜔)𝑎𝑖(𝜔)𝑑𝜔,(2.11) and the following bilinear form𝑉×𝑉,𝑓,𝑥𝑉×𝑉=𝑓,𝑥𝐬=𝑚𝑖=1Ω𝑓𝑖(𝜔)𝑥𝑖(𝜔)𝑠𝑖(𝜔)𝑑𝜔(2.12) defines a duality between 𝑉 and 𝑉. More precisely we have the following (see [20] for a proof).

Proposition 2.7. The bilinear form (2.12) defines a duality mapping between 𝑉×𝑉, given by 𝐽𝑎(𝐹)=1𝐹1,,𝑎𝑚𝐹𝑚.(2.13)

For applications of these spaces, the reader can refer to [20].

2.2. Variational Analysis in Non-Pivot H-Spaces

Let 𝑋 be a Hilbert space of arbitrary (finite or infinite) dimension and let 𝐾𝑋 be a nonempty, closed, convex subset. We assume the reader is familiar with tangent and normal cones to 𝐾 at 𝑥𝐾 (𝑇𝐾(𝑥), respectively, 𝑁𝐾(𝑥)), and with the projection operator of 𝑋 onto 𝐾, 𝑃𝐾𝑋𝐾 given by 𝑃𝐾(𝑧)𝑧=inf𝑥𝐾𝑥𝑧. Moreover we use here the following characterization of 𝑃𝐾(𝑥):𝑥=𝑃𝐾𝐽(𝑥)𝑥𝑥,𝑦𝑥0,𝑦𝐾.(2.14) The properties of the projection operator on Hilbert and Banach spaces are well known (see e.g., [2628]). The directional derivative of the operator 𝑃𝐾 is defined, for any 𝑥𝐾 and any element 𝑣𝑋, as the limit (for a proof see [26]):𝜋𝐾(𝑥,𝑣)=lim𝛿0+𝑃𝐾(𝑥+𝛿𝑣)𝑥𝛿;moreover𝜋𝐾(𝑥,𝑣)=𝑃𝑇𝐾(𝑥)(𝑣).(2.15) Let 𝜋𝐾𝐾×𝑋𝑋 be the operator given by (𝑥,𝑣)𝜋𝐾(𝑥,𝑣). Note that 𝜋𝐾 is nonlinear and discontinuous on the boundary of the set 𝐾. In [1, 29] several characterizations of 𝜋𝐾 are given.

The following theorem has been proven in the framework of reflexive strictly convex and smooth Banach spaces. We will use it to obtain a decomposition theorem in non-pivot Hilbert spaces (for a proof see [30, Th. 2.4]).

Theorem 2.8. Let 𝑋 be a real reflexive strictly convex and smooth Banach space, and let 𝐶 be a non-empty, closed and convex cone of 𝑋. Then for all 𝑥𝑋 and for all 𝑓𝑋 the following decompositions hold: 𝑥=𝑃𝐶(𝑥)+𝐽1Π𝐶0𝐽(𝑥),Π𝐶0𝐽(𝑥),𝑃𝐶(𝑥)=0,𝑓=𝑃𝐶0(𝑓)+𝐽Π𝐶𝐽1(𝑓),𝑃𝐶0(𝑓),Π𝐶𝐽1(𝑓)=0.(2.16) Here 𝑃𝐶 is the metric projection operator on 𝐾, and Π𝐶0 is the generalized projection operator on 𝐶0 (for a definition of Π𝐶0 see [28]).

Remark 2.9. It is known that 𝑃𝐶 and Π𝐶 coincide whenever the cone 𝐶 belongs to a Hilbert space. This observation implies the following result.

Corollary 2.10. Let 𝐶 be a nonempty closed convex cone of a non-pivot Hilbert space 𝑋. Then for all 𝑥𝑋 and 𝑓𝑋 the following decompositions hold: 𝑥=𝑃𝐶(𝑥)+𝐽1𝑃𝐶0𝐽(𝑥),𝑃𝐶0𝐽(𝑥),𝑃𝐶(𝑥)=0,𝑓=𝑃𝐶0(𝑓)+𝐽𝑃𝐶𝐽1𝑃(𝑓),𝐶0(𝑓),𝑃𝐶𝐽1(𝑓)=0.(2.17)

We highlight that Zarantonello has shown in [27] a similar decomposition result in reflexive Banach spaces.

Lemma 2.11 ([26, Lemma  4.5]). For any closed convex set 𝐾, 𝑃𝐾(𝑥+)=𝑥++(),𝑥𝐾,𝑇𝐾(𝑥),(2.18)where ()/0 as 0 over any locally compact cone of increments.

Remark 2.12. To prove Lemma 2.11 only the properties of the norm in Hilbert spaces are used; therefore the proof is valid in the non-pivot setting.

The following lemma has been proven in the pivot case in [26]. We give below a similar proof in non-pivot spaces.

Lemma 2.13. For any 𝑥𝐾, 𝑃𝐾(𝑥+)=𝑥+𝑃𝑇𝐾(𝑥)()+(),(2.19) where ()/0 as 0 over any locally compact cone of increments.

Proof. Clearly, we have in general that 𝑎+𝑏2=𝑎2+𝑏2+2(𝑎,𝑏).(2.20)Taking 𝑎=𝑥+𝑃𝑥+𝑇𝐾(𝑥)(𝑥+),𝑏=𝑃𝑥+𝑇𝐾(𝑥)(𝑥+)𝑃𝐾(𝑥+),(2.21)we get 𝑥+𝑃𝐾(𝑥+)2=𝑥+𝑃𝑥+𝑇𝐾(𝑥)(𝑥+)2+𝑃𝑥+𝑇𝐾(𝑥)(𝑥+)𝑃𝐾(𝑥+)2+2𝑥+𝑃𝑥+𝑇𝐾(𝑥)(𝑥+),𝑃𝑥+𝑇𝐾(𝑥)(𝑥+)𝑃𝐾,(𝑥+)(2.22) but 𝑥+𝑃𝑥+𝑇𝐾(𝑥)(𝑥+),𝑃𝑥+𝑇𝐾(𝑥)(𝑥+)𝑃𝐾=𝐽(𝑥+)𝑥+𝑃𝑥+𝑇𝐾(𝑥)(𝑥+),𝑃𝑥+𝑇𝐾(𝑥)(𝑥+)𝑃𝐾(𝑥+)0(2.23) using the variational principle (2.14) applied to 𝑃𝑥+𝑇𝐾(𝑥)(𝑥+). By definition of the projection operator we have 𝑥+𝑃𝐾(𝑥+)2𝑥+𝑃𝐾𝑃𝑥+𝑇𝐾(𝑥)(𝑥+)2.(2.24)Therefore we have 𝑥+𝑃𝑥+𝑇𝐾(𝑥)(𝑥+)2+𝑃𝑥+𝑇𝐾(𝑥)(𝑥+)𝑃𝐾(𝑥+)2𝑥+𝑃𝐾𝑃𝑥+𝑇𝐾(𝑥)(𝑥+)2.(2.25)As 𝑃𝑥+𝑇𝐾(𝑥)(𝑥+)=𝑥+𝑃𝑇𝐾(𝑥)() (just apply the definition and the variational principle (2.14)), we have 𝑃𝑇𝐾(𝑥)()2+𝑥+𝑃𝑇𝐾(𝑥)()𝑃𝐾(𝑥+)2𝑥+𝑃𝐾𝑥+𝑃𝑇𝐾(𝑥)()2,(2.26)but using the Corollary 2.10 we have =𝑃𝑇𝐶(𝑥)()+𝐽1𝑃𝑁𝐾(𝑥)(𝐽()), and therefore, 𝑃𝐾(𝑥+)𝑥𝑃𝑇𝐾(𝑥)()2𝐽1𝑃𝑁𝐾(𝑥)(𝐽())+𝑥+𝑃𝑇𝐾(𝑥)()𝑃𝐾𝑥+𝑃𝑇𝐾(𝑥)()2𝐽1𝑃𝑁𝐾(𝑥)(𝐽())2𝑥+𝑃𝑇𝐾(𝑥)()𝑃𝐾𝑥+𝑃𝑇𝐾(𝑥)()2𝐽+21𝑃𝑁𝐾(𝑥)(𝐽())𝑥+𝑃𝑇𝐾(𝑥)()𝑃𝐾𝑥+𝑃𝑇𝐾(𝑥).()(2.27)But by Lemma 2.11, 𝑥+𝑃𝑇𝐾(𝑥)()𝑃𝐾(𝑥+𝑃𝑇𝐾(𝑥)())=𝑜(𝑃𝑇𝐾(𝑥)()), so we can write 𝑃𝐾(𝑥+)𝑥𝑃𝑇𝐾(𝑥)()22𝐽1𝑃𝑁𝐾(𝑥)𝑃(𝐽())+𝑜𝑇𝐾(𝑥)𝑜𝑃()𝑇𝐾(𝑥).()(2.28)Therefore we have, 𝑃𝐾(𝑥+)𝑥𝑃𝑇𝐾(𝑥)()2𝑜()2.(2.29)

3. Non-Pivot and Implicit PDS in Hilbert Spaces

3.1. PDS in Pivot H-Spaces

Let 𝑋 be a pivot Hilbert space of arbitrary (finite or infinite) dimension and let 𝐾𝑋 be a nonempty, closed, convex subset. The following result has been shown (see [21]).

Theorem 3.1. Let 𝑋 be a Hilbert space and let 𝐾 be a nonempty, closed, convex subset. Let 𝐹𝐾𝑋 be a Lipschitz continuous vector field and let 𝑥0𝐾. Then the initial value problem associated to the projected differential equation (PrDE) 𝑑𝑥(𝜏)𝑑𝜏=𝜋𝐾(𝑥(𝜏)),𝐹(𝑥(𝜏)),𝑥(0)=𝑥0𝐾(3.1) has a unique absolutely continuous solution on the interval [0,).

This result is a generalization of the one in [6], where 𝑋=𝑛, 𝐾 was a convex polyhedron and 𝐹 had linear growth.

Definition 3.2. A projected dynamical system then is given by a mapping 𝜙+×𝐾𝐾 which solves the initial value problem: ̇𝜙(𝑡,𝑥)=𝜋𝐾(𝜙(𝑡,𝑥),𝐹(𝜙(𝑡,𝑥)))a.a.𝑡, 𝜙(0,𝑥)=𝑥0𝐾.

3.2. PDS in Non-Pivot H-Spaces

In this subsection we show that, with minor modifications, the existence of PDS in non-pivot H-spaces can be obtained. We first introduce non-pivot projected dynamical systems (NpPDSs) and then show their existence. In analogy with [21] we first introduce the following.

Definition 3.3. For 𝐹𝐾𝑋, a non-pivot projected differential equation (NpPrDE) is a discontinuous ODE given by 𝑑𝑥(𝑡)𝑑𝑡=𝜋𝐾𝐽𝑥(𝑡),1𝐹(𝑥(𝑡))=𝑃𝑇𝐾(𝑥(𝑡))𝐽1.𝐹(𝑥(𝑡))(3.2)

Consequently the associated Cauchy problem is given by𝑑𝑥(𝑡)𝑑𝑡=𝜋𝐾𝐽𝑥(𝑡),1𝐹(𝑥(𝑡)),𝑥(0)=𝑥0𝐾.(3.3) Next we define what we mean by a solution for a Cauchy problem of type (3.3).

Definition 3.4. An absolutely continuous function 𝑥𝑋, such that 𝑥(𝑡)𝐾,𝑥(0)=𝑥0𝐾,𝑡,̇𝑥(𝑡)=𝜋𝐾𝐽𝑥(𝑡),1𝐹(𝑥(𝑡)),a.e.on(3.4) is called a solution for the initial value problem (3.3).

Finally, assuming that problem (3.3) has solutions as described above, then we are ready to introduce the following.

Definition 3.5. A non-pivot projected dynamical system (NpPDS) is given by a mapping 𝜙+×𝐾𝐾 which solves the initial value problem ̇𝜙(𝑡,𝑥)=𝜋𝐾(𝜙(𝑡,𝑥),(𝐽1𝐹)(𝜙(𝑡,𝑥))),a.a.𝑡,𝜙(0,𝑥)=𝑥0𝐾.

To end this section we show how problem (3.3) can be equivalently (in the sense of solution set coincidence) formulated as a differential inclusion problem. Finally, in Subsection 3.3 we show that solutions for this new differential inclusion problem exist. We introduce the following differential inclusion:̇𝑥(𝑡)𝐽1𝐹(𝑥)𝑁𝐾(𝑥),𝑥(0)=𝑥0𝐾,(3.5) and we call 𝑥𝑋 absolutely continuous a solution to (3.5) if𝑥(𝑡)𝐾,𝑥(0)=𝑥0𝐾,𝑡,̇𝑥(𝑡)𝐽1𝐹(𝑥)𝑁𝐾(𝑥),a.a.𝑡.(3.6) We introduce also the following differential inclusion:̇𝑥(𝑡)𝐽1𝑁𝐹(𝑥)𝐾(𝑥),𝑥(0)=𝑥0𝐾,(3.7) where𝑁𝐾(𝑥)=𝑛𝑁𝐾.(𝑥)𝑛𝐹(𝑥)(3.8)

Obviously, we call 𝑥𝑋 absolutely continuous a solution to (3.7) if𝑥(𝑡)𝐾,𝑥(0)=𝑥0𝐾,𝑡,̇𝑥(𝑡)𝐽1𝑁𝐹(𝑥)𝐾(𝑥),a.a.𝑡.(3.9)

Proposition 3.6. The solution set of problem (3.3) coincides with the solution set of problem (3.9).

Proof. (3.3)(3.9). Let 𝑥() be an absolutely continuous function on 𝐾 such that 𝑥() is a solution to (3.3). Then 𝑥(𝑡)𝐾, for all 𝑡𝐼 and ̇𝑥(𝑡)=𝜋𝐾(𝑥(𝑡),(𝐽1𝐹)(𝑥(𝑡))),a.e.on; therefore using Corollary 2.10 we get ̇𝑥(𝑡)=𝐽1(𝐹(𝑥))𝐽1𝑃𝑁𝐾(𝑥)(𝐹(𝑥)), a.e.𝐼. Evidently, 𝑃𝑁𝐾(𝑥)(𝐹(𝑥))𝑁𝐾(𝑥). Moreover as 𝑁𝐾(𝑥) is a closed, convex cone, we get that 𝑃𝑁𝐾(𝑥)(𝐹(𝑥))𝑋𝐹(𝑥)𝑋)𝑁0𝐾(𝑥)=𝑇𝐾(𝑥) and both contains 0). Therefore ̃𝑛𝐾𝑁(𝑥)𝐾(𝑥),̃𝑛𝐾(𝑥)=𝑃𝑁𝐾(𝑥)(𝐹(𝑥)) such that ̇𝑥(𝑡)=𝐽1(𝐹(𝑥(𝑡))̃𝑛𝐾(𝑥)) for a.a 𝑡𝐼, so we have ̇𝑥(𝑡)𝐽1𝑁(𝐹(𝑥(𝑡))𝐾(𝑥)) for a.a 𝑡𝐼, and 𝑥() is a solution to (3.9).
(3.9)(3.3). As the trajectory remains in 𝐾 it is clear that ̇𝑥(𝑡)𝑇𝐾(𝑥(𝑡)). First we show that for almost all 𝑡𝐼 we have ̇𝑥(𝑡)𝑁𝐾(𝑥(𝑡)).(3.10) Let us consider three different cases; first suppose that 𝑥(𝑡)int(𝐾), we have then 𝑁𝐾(𝑥(𝑡))={0𝑋} and then 𝑁𝐾(𝑥(𝑡))=𝑋 and (3.10) is automatically satisfied. Suppose now that 𝑥(𝑡)𝜕𝐾 and in 𝑥(𝑡), 𝜕𝐾 is smooth. In that case 𝑇𝐾(𝑥(𝑡)) is flat and 𝑁𝐾(𝑥(𝑡))𝑇𝐾(𝑥(𝑡)) with 𝑁𝐾(𝑥(𝑡)) not reduced to {0𝑋}, if ̇𝑥(𝑡)𝑁𝐾(𝑥(𝑡)); then in a neighbourhood 𝒱(𝑡) the trajectory 𝑥(𝑡),𝑡𝒱(𝑡) goes in int(𝐾), so we are in the first case and we can exclude time 𝑡. Suppose now that 𝑥(𝑡)𝜕𝐾 and 𝑥(𝑡) is in a corner point. In that case 𝑁𝐾(𝑥(𝑡))={0}; therefore if ̇𝑥(𝑡)=0, (3.10) is satisfied. If ̇𝑥(𝑡)0, it means that 𝑥(𝑡)𝑥(𝑡) for 𝑡𝒱(𝑡), with 𝑥(𝑡) in one of the two previous cases; as we can “exclude” time 𝑡, we have (3.10). As we can write ̇𝑥(𝑡)=𝐽1(𝐹(𝑥)̃𝑛𝐾(𝑥)), we have 𝐽(̇𝑥(𝑡))𝐽𝐽1(𝐹(𝑥)),̇𝑥(𝑡)=0.(3.11)Using the polarity between 𝑁𝐾(𝑥(𝑡)) and 𝑇𝐾(𝑥(𝑡)) and the variational principle (2.14) we deduce (3.3).

3.3. Existence of NpPDS

In this section we show that problem (3.3) has solutions and consequently that NpPDSs exist in the sense of Definition 3.5, by showing that problem (3.7) has solutions, in the sense of Definition 3.4. To obtain the main result of this paper, we need some preliminary ones, according to the following steps.(1)We first prove the existence of a sequence of approximate solutions with “good” properties such that 𝑘𝑘0,𝑥𝑘(𝑡),̇𝑥𝑘𝐽(𝑡)graph1𝑁𝐹𝐾+,(3.12) for any neighbourhood of 0 in 𝑋×𝑋. This step constitutes Theorem 3.9.(2)we prove next that the sequence obtained in the first step converges to a solution of problem (3.7) and that it has a weakly convergent subsequence whose derivative converges to ̇𝑥().

The methodology of the proofs is completely analogous to that used for pivot Hilbert spaces in [21]. Therefore we present the results with summary proofs, pointing out where they need to be updated for the case of a non-pivot H-space. The main difference in all proofs is made by the presence of the linear mapping 𝐽.

The main result can be stated as follows.

Theorem 3.7. Let 𝑋 be a Hilbert space and 𝑋 its topological dual and let 𝐾𝑋 be a nonempty, closed and convex subset. Let 𝐹𝐾𝑋 be a Lipschitz continuous vector field with Lipschitz constant 𝑏. Let 𝑥0𝐾. Then the initial value problem (3.3) has a unique solution on +.

Proof Existence of a solution on an interval [0,𝑙],𝑙<
For this part of the proof, we need two major results, as follows.
Proposition 3.8. Let 𝑋 be a nonpivot H-space, let 𝑋 be its topological dual, and let 𝐾𝑋 be a non-empty, closed and convex subset. Let 𝐹𝐾𝑋 be a Lipschitz continuous vector field with Lipschitz constant 𝑏, so that on 𝐾𝐵𝑋(𝑥0,𝐿), with 𝐿>0 and 𝑥0𝐾 arbitrarily fixed, we have 𝐹(𝑥)𝑀=𝐹(𝑥0)+𝑏𝐿.
Then the set-valued mapping 𝒩𝑝𝐾𝐵𝑋(𝑥0,𝐿) given by 𝑁𝑥𝐹𝐾(𝑥),𝑝(3.13) has a closed graph.

Proof . The proof is similar to the one in [21].
We show first that the mapping 𝒩𝑝𝐾𝐵𝑋(𝑥0,𝐿) given by 𝑁𝑥𝐾(𝑥),𝑝 has a closed graph. It is clear that for each 𝑝𝑋, the set-valued map 𝒩𝑝𝐾𝐵𝑋(𝑥0,𝐿) maps 𝐾𝐵𝑋(𝑥0,𝐿) into 2[𝑀𝑝,𝑀𝑝]. Let {(𝑥𝑛,𝑧𝑛)}𝑛graph(𝒩𝑝) such that (𝑥𝑛,𝑧𝑛)(𝑥,𝑧)𝑋×2[𝑀𝑝,𝑀𝑝]. We want to show that (𝑥,𝑦)graph(𝒩𝑝). From 𝑧𝑛graph(𝒩𝑝), for all 𝑛, we deduce that there exists 𝑦𝑛𝑁𝐾(𝑥𝑛) such that 𝑧𝑛=𝑦𝑛,𝑝. Since the set 𝑁𝐾(𝑥)𝐵𝑋(0,𝑀) and 𝐵𝑋(0,𝑀) is weakly compact, then there exists a subsequence 𝑦𝑛𝑘 and 𝑦𝑋 such that 𝑦𝑛𝑘𝑦(3.14) for the weak topology 𝜎(𝑋,𝑋)byreexivity=𝜎(𝑋,𝑋), which is equivalent to 𝑦𝑛𝑘,𝛽𝑦,𝛽,𝛽𝑋.(3.15) Suppose now that 𝑁𝑦𝐾(𝑥). This implies that at least one of the following two alternatives should be satisfied.(1)There exists 𝑤𝐾 such that 𝑦,𝑤𝑥<𝜆<0.(2)𝑦>𝜇>𝐹(𝑥). In the first case as 𝑦𝑛𝑘,𝛽𝑦,𝛽,𝛽𝑋 for 𝑘>𝑘0 we have 𝑦𝑛𝑘,𝑤𝑥<𝜆/2. But 𝑦𝑛𝑘,𝑤𝑥𝑛𝑘=𝑦𝑛𝑘,𝑤𝑥+𝑦𝑛𝑘,𝑥𝑥𝑛𝑘 and as 𝑥𝑛𝑘𝑥, there exists 𝑘1>0 such that 𝑘𝑘1, we have 𝑦𝑛𝑘,𝑥𝑥𝑛𝑘𝑥𝑥𝑛𝑘𝑦𝑛𝑘<(|𝜆|/4𝑀)𝑀=|𝜆|/4. Thus 𝑦𝑛𝑘,𝑤𝑥𝑛𝑘<𝜆/4<0, for all 𝑘>𝑚𝑎𝑥(𝑘0,𝑘1). But this contradicts the fact that 𝑦𝑛𝑘𝑁𝐾(𝑥𝑛𝑘).
In the second case as 𝑦𝑛𝑘,𝛽𝑦,𝛽,𝛽𝑋, we have ([31, Proposition III.12]) 𝐹(𝑥)<𝑦liminf𝑘𝑦𝑛𝑘 which is a contradiction because 𝑦𝑛𝑁𝐾(𝑥𝑛),𝑛. The continuity of 𝐹 and the first part of the proof implies that 𝑁𝑥𝐹𝐾(𝑥),𝑝(3.16) has non-empty, closed and convex values for each 𝑥𝐾 and has a closed graph.

The next result is constructing the sequence of approximate solutions for the problem (3.7).
Theorem 3.9. Let 𝑋 be a Hilbert space and 𝑋 its topological dual, and let 𝐾𝑋 be a non-empty, closed and convex subset. Let 𝐹𝐾𝑋 be a Lipschitz continuous vector field so that on 𝐾𝐵𝑋(𝑥0,𝐿), with 𝐿>0 and 𝑥0𝐾, we have 𝐹(𝑥)𝑀=𝐹(𝑥0)+𝑏𝐿. Let 𝑙=𝐿/𝑀 and =[0,𝑙]. Then there exists a sequence {𝑥𝑘()} of absolutely continuous functions defined on , with values in 𝐾, such that for all 𝑘0,𝑥𝑘(0)=𝑥0 and for almost all 𝑡, {𝑥𝑘(𝑡)} and {̇𝑥𝑘(𝑡)} (the sequence of its derivatives) have the following property: for every neighbourhood of 0 in 𝑋×𝑋 there exists 𝑘0=𝑘0(𝑡,) such that 𝑘𝑘0,𝑥𝑘(𝑡),̇𝑥𝑘𝑁(𝑡)graph𝐹𝐾+.(3.17)
Proof. The proof, based on topological properties of the space 𝑋, can be found in [21]. However, given we are now working in non-pivot H-spaces, then instead of 𝑧𝑝=𝑃𝐾(𝑥𝑝𝐹(𝑥)) we now construct 𝑧𝑝=𝑃𝐾(𝑥𝑝𝐽1𝐹(𝑥)).
Next we show that the sequence {𝑥𝑘()} built in Theorem 3.9 is uniformly convergent to some 𝑥(). Again, following closely [21], by Theorem 3.9 there exists a pair (𝑢𝑘,𝐹(𝑢𝑘)𝑛𝑘𝑁)graph(𝐹𝐾) such that 𝑥𝑘(𝑡)𝑢𝑘(𝑡)=𝜖1,𝑘(𝑡),̇𝑥𝑘(𝑡)+𝐽1𝐹𝑢𝑘(𝑡)+𝑛𝑘=𝜖2,𝑘(𝑡),(3.18)where 𝜖1,𝑘(𝑡) and 𝜖2,𝑘(𝑡) are vector functions, not necessarily continuous, satisfying 𝜖1,𝑘(𝑡)<𝜖𝑘 and 𝜖2,𝑘(𝑡)<𝜖𝑘 where 𝜖𝑘0 as 𝑘 and 𝑛𝑘𝑁𝐾(𝑢𝑘) and 𝑛𝑚𝑁𝐾(𝑢𝑚).
Let 𝑘,𝑚 be two indexes. Then we evaluate 12𝑑𝑥𝑑𝑡𝑘(𝑡)𝑥𝑚(𝑡)2=𝐽̇𝑥𝑘(𝑡)̇𝑥𝑚(𝑡),𝑥𝑘(𝑡)𝑥𝑚=𝑢(𝑡)𝐹𝑘𝑥(𝑡)+𝐹𝑘𝑢(𝑡)+𝐹𝑚𝑥(𝑡)𝐹𝑚(𝑡),𝑥𝑘(𝑡)𝑥𝑚+𝑥(𝑡)𝐹𝑘𝑥(𝑡)+𝐹𝑚(𝑡),𝑥𝑘(𝑡)𝑥𝑚(𝑡)+𝑛𝑘+𝑛𝑚,𝑢𝑘(𝑡)𝑢𝑚(𝑡)+𝑛𝑘+𝑛𝑚,𝑢𝑘(𝑡)+𝑥𝑘(𝑡)+𝑢𝑚(𝑡)𝑥𝑚+𝐽𝜖(𝑡)1,𝑘(𝑡)𝜖2,𝑚(𝑡),𝑥𝑘(𝑡)𝑥𝑚(.𝑡)(3.19) But using the monotonicity of 𝑥𝑁𝐾(𝑥), the isometry property of 𝐽, and the b-Lipschitz continuity of 𝐹 we get that 12𝑑𝑥𝑑𝑡𝑘(𝑡)𝑥𝑚(𝑡)2𝑥𝑏𝑘(𝑡)𝑥𝑚(𝑡)2+𝜖𝑘+𝜖𝑚𝑛𝑘𝑛𝑚𝜖+(1+𝑏)𝑘+𝜖𝑚𝑥𝑘(𝑡)𝑥𝑚.(𝑡)(3.20)We now let 𝜙(𝑡)=𝑥𝑘(𝑡)𝑥𝑚(𝑡), so from the previous inequalities we get ̇𝜙(𝑡)𝜙(𝑡)𝑏𝜙(𝑡)2+𝜖𝑘+𝜖𝑚[](1+𝑏)𝜙(𝑡)+2𝑀.(3.21) Using the same technique as in [21] we get 𝜙(𝑡)2𝑎𝑏𝜖𝑘+𝜖𝑚𝑒2𝑏𝑡𝑎1𝑏𝜖𝑘+𝜖𝑚𝑒2𝑏𝑙1,(3.22) where 𝑙 is the length of . So the Cauchy criteria are satisfied uniformly and we get the conclusion.
From the previous step we know that {𝑥𝑘()} is uniformly convergent to 𝑥() and as (𝑥𝑘(𝑡),̇𝑥𝑘𝑁(𝑡))graph(𝐹𝐾)+, we now deduce that there exists a 𝜃 such that ̇𝑥𝑘(𝑡)𝜃. Using the arguments in [21] and the result of [32], we deduce the existence of a subsequence of {̇𝑥𝑘}weakly*-convergent to ̇𝑥()𝐿(𝐼,𝑋).
Finally, we finish this part of the proof by showing that 𝑥() is indeed a solution of the differential inclusion (3.7). From Theorem 3.9, for each 𝑘𝑘0 and almost every 𝑡 there exists a pair 𝑢𝑘(𝑡),𝑣𝑘𝑁(𝑡)graph𝐹𝐾(3.23) such that 𝑥𝑘(𝑡)𝑢𝑘(𝑡)<𝜖𝑘 and ̇𝑥𝑘(𝑡)𝑣𝑘(𝑡)<𝜖𝑘, where 𝜖𝑘0 when 𝑘. Let 𝑝𝑋 arbitrarily fixed. Then for almost all 𝑡𝑢𝑘(𝑡),𝑣𝑘𝑁(𝑡),𝑝graph𝐹𝐾,,𝑝̇𝑥𝑘(𝑡),𝑝𝑣𝑘(𝑡),𝑝𝑝𝜖𝑘.(3.24) So 𝑢𝑘(𝑡)𝑥(𝑡) for every 𝑡 and 𝑣𝑘(𝑡),𝑝̇𝑥𝑘(𝑡),𝑝 for almost all 𝑡. By Proposition 3.8, we know that 𝑁graph(𝐹𝐾,𝑝) is closed, so it follows that for almost all 𝑡, 𝑥(𝑡),̇𝑥𝑘𝑁(𝑡),𝑝graph𝐹𝐾,𝑝.(3.25) Since the set 𝑁𝐹(𝑥(𝑡))𝐾(𝑥(𝑡)) is convex and closed, it follows that ̇𝑥(𝑡)𝐽1𝑁𝐹𝑥(𝑡)𝐾(𝑥(𝑡)).(3.26) By Proposition 3.6, 𝑥(𝑡) is a solution of problem (3.3).
Uniqueness of Solutions on [0,𝑙]
Step 1 (𝑥() is the unique solution). Suppose that we have two solutions 𝑥1() and 𝑥2() starting at the same initial point. For any fixed 𝑡 we get 12𝑑𝑥𝑑𝑡1(𝑡)𝑥1(𝑡)2=𝐽̇𝑥1(𝑡)̇𝑥2(𝑡),𝑥1(𝑡)𝑥2=𝐽(𝑡)̇𝑥1(𝑡)𝐽̇𝑥2(𝑡),𝑥1(𝑡)𝑥2𝐽(𝑡)1𝑥𝐹1(𝑡)𝑛1𝑥+𝐹2(𝑡)+𝑛2,𝑥1(𝑡)𝑥2𝑥(𝑡)>𝑏1(𝑡)𝑥2(𝑡)2,(3.27) because the metric projection is a nonexpansive operator in 𝑋, 𝐽 is a linear isometry, and 𝐹 is b-Lipschitz. By Gronwall’s inequality we obtain 𝑥1(𝑡)𝑥2(𝑡)20, so we have 𝑥1(𝑡)=𝑥2(𝑡) for any 𝑡.

Existence of Solutions on +
From above we can assert the existence of a solution to problem (3.3) on an interval [0;𝑙], with 𝑏>0 fixed and 𝐿>0 arbitrary. We note that we can choose 𝐿 such that 𝑙1/(1+𝑏) in the following way: if 𝐹(𝑥0)=0, we let 𝐿=1, and if 𝐹(𝑥0)0, then we let 𝐿𝐹(𝑥0). In both cases we obtain 𝑙1/(1+𝑏). Therefore beginning at each initial point 𝑥0𝐾, problem (3.3) has a solution on an interval of length at least [0;1/(1+𝑏)]. Now if we consider problem (3.3) with 𝑥0=𝑥(1/(1+𝑏)), applying again all the above, we obtain an extension of the solution on an interval of length at least 1/(1+𝑏). By continuing this solution we obtain a solution on [0,).

3.4. Implicit PDS

In this section we consider a generic Hilbert space 𝑋, where generic is taken to mean that the dimensionality could be either finite or infinite, and the space could be either a pivot or a non-pivot space. Let us introduce the following definition.

Definition 3.10. Let 𝑋 be a generic H-space and let 𝐾𝑋 be a non-empty, closed subset. Consider a pair (𝑔,𝐾) such that 𝐾 is convex and 𝑔𝐾𝐾=𝑟(𝐾)𝑋, is continuous, injective, and 𝑔1 is Lipschitz continuous.
Consider 𝐹𝑋𝑋 satisfying (𝐹𝑔)(𝑦)=𝐹(𝑦),𝑦𝐾. Then the pair (𝑔,𝐾) is called a convexification pair of (𝐹,𝐾).

Example 3.11. Here is an example of such a convexification pair in 2. Let 𝐾={(𝑥,𝑦)20𝑥1,0𝑦𝑥} and let 𝑔 be the map of 𝐾 into 𝐾=[0,1]×[0,1], namely: 2𝑔(𝑥,𝑦)=𝑥,1+𝑥𝑦+1𝑥.1+𝑥(3.28) We can easily check that 𝑔 is continuous and monotone. Now take 𝐹 to be 𝐹(𝑥,𝑦)=(𝑥,𝑎), where 𝑎 is an arbitrary constant in . Then we have 𝐹𝑔(𝑥,𝑦)=(𝑥,𝑎)=𝐹(𝑥,𝑦).

We now introduce another type of a projected equation as follows.

Definition 3.12. Let 𝑋 be a generic H-space and let 𝐾𝑋 be a non-empty, closed subset. An implicit projected differential equation (ImPrDE) is a (PrDE) given by (3.2) where 𝑥(𝑡)=𝑔(𝑦(𝑡)),𝑔𝐾𝐾𝑋, that is: 𝑑𝑔(𝑦(𝑡))𝑑𝑡=𝑃𝑇𝐾(𝑔(𝑦(𝑡)))𝐽1.𝐹𝑔(𝑦(𝑡))(3.29)

The motivation for the introduction of such an equation comes from the desire to study the dynamics on a set 𝐾𝑋, where 𝐾 could be nonconvex, and to study as well some dynamic problems on a so-called translated set (see Section 4 below).

Considering now (3.29) and a convexification pair (𝑔,𝐾) of a nonempty, closed 𝐾𝑋, then the Cauchy problem associated to (3.29) and the pair (𝑔,𝐾) is given by𝑑𝑔(𝑦(𝑡))𝑑𝑡=𝜋𝐾𝐽𝑔(𝑦(𝑡)),1𝐹|𝐾(𝑦(𝑡)),𝑔(𝑦(0))=𝑥0𝐾.(3.30) Next we define what we mean by a solution for a Cauchy problem of type (3.30).

Definition 3.13. An absolutely continuous function 𝑦𝑋, such that 𝑦(𝑡)𝐾,𝑔(𝑦(0))=𝑥0𝐾,𝑡,𝑑𝑔(𝑦(𝑡))𝑑𝑡=𝜋𝐾𝐽𝑔(𝑦(𝑡)),1𝐹|𝐾(𝑦(𝑡)),a.e.on(3.31) is called a solution for the initial value problem (3.30).

We claim that problem (3.30) has solutions by Theorem 3.9. It is obvious that by a change of variable 𝑥()=𝑔(𝑦()), problem (3.30) has solutions on 𝐾, in the sense of Definition 3.4. But since 𝑔 is assumed continuous and strictly monotone, then 𝑔 is invertible and so 𝑦()=𝑔1(𝑥()); moreover, we see that such a 𝑦 is a solution to problem (3.30) in the above sense.

Now we are ready to introduce the following.

Definition 3.14. An implicit projected dynamical system (ImPDS) is given by a mapping 𝜙+×𝐾𝐾 which solves the initial value problem: ̇𝜙(𝑡,𝑔(𝑦(𝑡)))=𝜋𝐾𝐽𝜙(𝑡,𝑔(𝑦(𝑡))),1𝐹(𝜙(𝑡,𝑦(𝑡))),a.a.𝑡,𝜙(0,𝑔(𝑦(0)))=𝑥0𝐾,(3.32) where (𝑔,𝐾) is a convexification pair.

Theorem 3.15. Let 𝑋 be a generic Hilbert space, and let 𝐾 be a non-empty closed subset of 𝑋. Let 𝐾 be non-empty, closed and convex, let 𝑔𝐾𝐾 be continuous and strictly monotone, and let 𝐹𝐾𝑋 be Lipschitz continuous such that (𝐹𝑔)|𝐾=𝐹. Let also 𝑥0𝐾 and 𝐿>0 such that 𝑥0𝐿. Then the initial value problem (3.30) has a unique solution on the interval [0,𝑙], where 𝑙=𝐿/(𝐹(𝑥0)+𝑏𝐿).

Proof. The proof consists in the modification of a few easy steps of the proof given in [21] combined with the results of the present paper.

4. Applications

4.1. NpPDS, ImPDS, and Variational Inequalities

It is worth noting at this point that, as in the pivot case, a NpPDS is also related to a variational inequality (VI) problem. To show this relation, we first define what is meant by a critical point of NpPDS.

Definition 4.1. A point 𝑥𝐾 is called a critical point for (3.2) if 𝜋𝐾𝑥𝐽,1𝑥𝐹=0.(4.1)

Theorem 4.2. Let X be a generic Hilbert space and let 𝐾𝑋 be a non-empty, closed and convex subset. Let 𝐹𝑋𝑋 be a vector field. Consider the variational inequality problem: 𝑥𝐾𝐹(𝑥),𝑣𝑥0,𝑣𝐾.(4.2) Then the solution set of (4.2) coincides with the set of critical points of the non-pivot projected dynamical system (3.2).

Proof. It follows from the decomposition Theorem 2.8 (see also [23]).

The relation between an ImPDS and a VI problem is more interesting, as has been considered before in the literature, but with superfluous conditions on the projection operator 𝑃𝐾 we describe this relation next.

Definition 4.3. Let 𝑋 be a generic H-space and let 𝐾𝑋 be a non-empty, closed subset. Let 𝐹𝑋𝑋 be a mapping. Then we call g-variational inequality on the set 𝐾 the problem of nding𝑦𝐾,𝐹𝑔(𝑦),𝑧𝑔(𝑦)0,𝑧𝐾,(4.3) where (𝑔,𝐾) is a convexification pair of (𝐹,𝐾).

We highlight the importance of the relation 𝐹𝑔(𝑦)=𝐹(𝑦) from Definition 3.10 in order for (4.3) to make sense. Under (3.5) we can rewrite (4.3) asnd𝑦𝐾,𝐹(𝑦),𝑧𝑔(𝑦)0,𝑧𝐾.(4.4)

Remark 4.4. In [24], (4.4) is considered in a pivot H-space and is called a “general variational inequality.” We prefer to use the term “g-variational inequality” in relation to (4.4), in order to avoid confusion with the commonly accepted “generalized variational inequality” which involves multimappings.

Theorem 4.5. If the problems (4.4) and (3.30) admit a solution, then the equilibrium points of (4.4) coincide with the critical points of (3.30).

Proof. Suppose 𝑦𝐾 is a solution of (4.4); then by definition we have 𝐹𝑦𝑦,𝑧𝑔0,𝑧𝐾.(4.5) So by multiplying by a strictly positive constant 𝜆 and using the bilinearity of the inner product, we get 𝑦𝐹,𝑦0,𝑦𝑇𝐾𝑔𝑦.(4.6) So we deduce that 𝐹(𝑦)𝑁𝐾(𝑔(𝑦)); using the decomposition Theorem 2.8 we get 𝑃𝑇𝐾(𝑔(𝑦))(𝐽1𝐹(𝑦))=0, and so 𝑦 is a critical point of (3.30).
Now suppose that 𝑦 is a critical point of (3.30); then by definition we have 𝑃𝑇𝐾(𝑔(𝑦))𝐽1𝐹𝑦=0,(4.7) and by the decomposition theorem we get 𝐹(𝑦)𝑁𝐾(𝑔(𝑦)). By the definition of the normal cone to 𝐾 in 𝑔(𝑦), the following inequality is satisfied: 𝑦𝐹𝑦,𝑧𝑔0,𝑧𝐾,(4.8) which is exactly (4.4).

5. Examples and Applications

5.1. Weighted Traffic Problem

Let us introduce a network 𝒩, that means a set 𝒲 of origin-destination pair (origin/destination node) and a set of routes. Each route 𝑟 links exactly one origin-destination pair 𝑤𝒲. The set of all 𝑟 which link a given 𝑤𝒲 is denoted by (𝑤). For each time 𝑡(0,𝑇) we consider vector flow 𝐹(𝑡)𝑛. Let us denote by Ω an open subset of , by 𝑛=card(), 𝐚={𝑎1,,𝑎𝑛}, and by 𝐚1={𝑎11,,𝑎𝑛1} two families of weights such that for each 1𝑖𝑛,𝑎𝑖𝒞(Ω,+{0}). We introduce also the family of real time traffic densities 𝐬={𝑠1,,𝑠𝑛} such that for each 1𝑖𝑛, 𝑠𝑖𝒞(Ω,+{0}).

Let 𝑟𝑖 correspond to an element of 𝑎 and 𝑠, newly to 𝑎𝑖 and 𝑠𝑖. If we denote by 𝑉𝑖=𝐿2(Ω,,𝑎𝑖,𝑠𝑖) and 𝑉𝑖=𝐿2(Ω,,𝑎𝑖1,𝑠𝑖), the space𝑉=𝑛𝑖=1𝑉𝑖(5.1) is a Hilbert space for the inner product𝐹,𝐺𝐚,𝐬=𝑛𝑖=1Ω𝐹𝑖(𝜔)𝐺𝑖(𝜔)𝑎𝑖(𝜔)𝑠𝑖(𝜔)𝑑𝜔.(5.2) The space 𝑉=𝑛𝑖=1𝑉𝑖 is a Hilbert space for the following inner product𝐹,𝐺𝐚1,𝐬=𝑛𝑖=1Ω𝐹𝑖(𝜔)𝐺𝑖(𝜔)𝑠𝑖(𝜔)𝑎𝑖(𝜔)𝑑𝜔,(5.3) and the following bilinear form defines a duality between 𝑉 and 𝑉:𝑉×𝑉,(5.4)𝑓,𝑥𝐬=𝑛𝑖=1Ω𝑓𝑖(𝜔)𝑥𝑖(𝜔)𝑠𝑖(𝜔)𝑑𝜔.(5.5) More exactly we have the following.

Proposition 5.1. The bilinear form (5.5) is defined over 𝑉×𝑉 and defines a duality between 𝑉×𝑉. The duality mapping is given by 𝐽(𝐹)=(𝑎1𝐹1,,𝑎𝑛𝐹𝑛).

The feasible flows have to satisfy the time-dependent capacity constraints and demand requirements; namely, for all 𝑟, 𝑤𝒲 and for almost all 𝑡Ω,𝜆𝑟(𝑡)𝐹𝑟(𝑡)𝜇𝑟(𝑡),𝑟(𝑤)𝐹𝑟(𝑡)=𝜌𝑤(𝑡),(5.6) where 0𝜆𝜇 are given in 𝐿2([0,𝑇],𝑛), 𝜌𝐿2([0,𝑇],𝑚) where 𝑚=card(𝒲),𝐹𝑟,𝑟, denotes the flow in the route 𝑟. If Φ=(Φ𝑤,𝑟) is the pair route incidence matrix, with 𝑤𝒲 and 𝑟, that is,Φ𝑤,𝑟=𝜒(𝑤)(𝑟),(5.7) the demand requirements can be written in matrix-vector notation asΦ𝐹(𝑡)=𝜌(𝑡).(5.8) The set of all feasible flows is given by𝐾={𝐹𝑉𝜆(𝑡)𝐹(𝑡)𝜇(𝑡),a.e.inΩ;Φ𝐹(𝑡)=𝜌(𝑡),a.einΩ}.(5.9) We provide now the definition of equilibrium for the traffic problem. First we need to define the notion of equilibrium for a variational inequality. A variational inequality (VI) in a Hilbert space 𝑉 is to determine𝑥𝐾𝐶(𝑥),𝑦𝑥𝐬0,𝑦𝐾,(5.10) where 𝐾 is a closed convex subset of 𝑉, and 𝐶𝐾𝑉 is a mapping.

Definition 5.2. 𝐻𝑉 is an equilibrium flow if and only if 𝐻𝐾,𝐶(𝐻),𝐹𝐻𝐬0,𝐹𝐾.(5.11)

It is possible to prove the equivalence between condition (5.11) and what we will call a weighted Wardrop condition (5.13).

Theorem 5.3. 𝐻𝐾 is an equilibrium flow in the sense of (5.11) if and only if 𝑠𝑤𝒲,𝑞,𝑚(𝑤),a.e.inΩ,(5.12)𝑞(𝑡)𝐶𝑞(𝐻(𝑡))<𝑠𝑚(𝑡)𝐶𝑚(𝐻(𝑡)),𝐻𝑞(𝑡)=𝜇𝑞(𝑡)or𝐻𝑚(𝑡)=𝜆𝑚(𝑡).(5.13)

Proof. see [20].

Based on previous results [20], this solution coincides with the set of critical points of the associated projected dynamical system.

5.2. Quasivariational Inequalities on Translated Sets
5.2.1. QVI

Let 𝑋 be a generic H-space, 𝐷 closed, convex, nonempty in 𝑋. Let 𝒦𝐷2𝑋 with 𝒦(𝑥) convex for all 𝑥𝐷 and 𝐹𝒦2𝑋 a mapping.

Let us introduce the following variational inequality:nd𝑥𝒦(𝑥),𝐹(𝑥),𝑦𝑥0,𝑦𝒦(𝑥).(5.14) Note that in this case the set in which we are looking for the solution depends on 𝑥. For problem (5.14) we can provide the following existence result (see [17] or [33]).

Theorem 5.4. Let 𝐷 be a closed convex subset in a locally convex Hausdorff topological vector space 𝑋. Let us suppose that(i)𝒦𝐷2𝐷 is a closed lower semicontinuous correspondence with closed, convex, and nonempty values,(ii)𝐶𝐷2𝑋 is a monotone, finite continuous, and bounded single-valued map,(iii)there exist a compact, convex, and nonempty set 𝑍𝐷 and a nonempty subset 𝐵𝑍 such that(a)𝒦(𝐵)𝑍,(b)𝒦(𝑧)𝑍,𝑧𝑍, (c)for every 𝑧𝑍𝐵, there exist ̂𝑧𝒦(𝑧)𝑍 with 𝐶(𝑧),̂𝑧𝑧<0.
Then there exists 𝑥 such that 𝑥𝒦(𝑥)𝐶(𝑥),𝑦𝑥0,𝑦𝒦(𝑥).(5.15)

In order to study the disequilibrium behavior of (5.14), we introduce now the following projected differential equation.

Definition 5.5. We call projected dynamical system associated to the quasivariational inequality (5.14) the solution set of the projected differential equation: 𝑑𝑥(𝑡)𝑑𝑡=𝑃𝑇𝒦(𝑥)(𝑥)𝐽1𝐹(𝑥),𝑥(0)=𝑥0𝒦(𝑥).(5.16)

Remark 5.6. In general there are no existence results for problem (5.16). An existence result for a particular case of (5.16) has been given in [24], assuming the following fact.
Assumption 5.7. Let 𝑋 be a pivot H-space. For all 𝑢,𝑣,𝑤𝑋, 𝑃𝒦(𝑢) satisfies the condition𝑃𝒦(𝑢)(𝑤)𝑃𝒦(𝑣)(𝑤)𝜆𝑢𝑣,(5.17) where 𝜆>0 is a constant.
However, this assumption fails to be true. One counterexample is as follows. We denote by 𝐶 a closed convex set and we take 𝑢,𝑣𝐶; we denote by 𝒦(𝑢)=𝑇𝐶(𝑢) and by 𝒦(𝑣)=𝑇𝐶(𝑣) the tangent cones of 𝐶 at 𝑢 and 𝑣.
In fact, 𝑤𝑋 can only be chosen in one of the following four situations:(1)𝑤𝒦(𝑢)𝒦(𝑣), (2)𝑤𝒦(𝑢)𝒦(𝑣), (3)𝑤𝒦(𝑣)𝒦(𝑢), (4)𝑤𝑋(𝒦(𝑢)𝒦(𝑣)). Suppose now that we have 𝑤𝒦(𝑢)𝒦(𝑣); then by Moreau’s decomposition theorem we get 𝑃𝒦(𝑢)(𝑤)𝑃𝒦(𝑣)=(𝑤)𝑤𝑃𝒦(𝑣)=𝑃(𝑤)𝑁𝐶(𝑣)(𝑤)𝜆𝑢𝑣,(5.18) where 𝑁𝐶(𝑣) is the normal cone of 𝐶 at 𝑣. Consider now 𝑋=2, 𝐶=[0,𝜖]2, 𝑢=(0,0) and 𝑣=(𝜖,𝜖). It is clear that we have the following: 𝑇𝐶(𝑢)=2+,𝑇𝐶(𝑣)=2,𝑁𝐶(𝑣)=2+=𝑇𝐶(𝑢).(5.19) So for any 𝑤𝑁𝐶(𝑣) we get 𝑤𝜆𝑢𝑣=2𝜖𝜆.(5.20) Since 𝑤 is arbitrary, let now 𝑤=𝜇𝑤, for any 𝜇>0. Then, 𝜇𝑤𝜆𝑢𝑣=2𝜖𝜆(5.21) should be true for any 𝜇>0. However this does not hold.

Consider now the special case of a set-valued mapping 𝒦 which is the translation of a closed, convex subset 𝐾:𝒦𝑥𝐾+𝑣(𝑥),(5.22)

where 𝑣(𝑥) is a vector linearly dependant on 𝑥; then problems (5.14) and (5.16) can be studied, under certain conditions, respectively, as a g-VI and an implicit PDS as shown below.

If 𝒦(𝑥)=𝐾+𝑝(𝑥) as done by Noor for type B PDS [24], we have the following equivalent formulations:𝑑𝑥(𝑡)𝑑𝑡=𝑃𝑇𝐾+𝑝(𝑥)(𝑥)𝐽1𝐹(𝑥)=𝑃𝑇𝐾(𝑔(𝑥))𝐽1𝐹(𝑥),𝑥(0)=𝑥0𝐾,(5.23) where 𝑔(𝑥)=𝑥𝑝(𝑥), assuming 𝐹(𝑔(𝑥))=𝐹(𝑥𝑝(𝑥))=𝐹(𝑥). We can observe that if (𝑑𝑝(𝑥))/𝑑𝑡=0, then (5.23) is equal to the implicit projected differential equation (3.29), and therefore Theorem 3.15 provides an existence result without assuming any kind of Lipschitz condition of the projection operator.

6. Conclusions

We show in this paper that previous results of existence of projected dynamical systems can be generalized to two new classes, namely, the non-pivot and the implicit PDS. The generalizations came as needed to study a more realistic traffic equilibrium problem, as well as to study the relations between an implicit PDS and a class of variational inequalities as previously introduced in [24] as an open problem.

Acknowledgments

The work has been supported by the first author’s NSERC Discovery Grant. The support is gratefully acknowledged.

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