Nonpivot and Implicit Projected Dynamical Systems on Hilbert Spaces

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.

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Monica Gabriela Cojocaru 1 and Stephane Pia 2

Recommended by L.E. Persson

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

Received 9 November 2010; Accepted 17 January 2011

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.

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 [ 2- 4]. 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, 6- 8] 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 [ 9- 19] 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 *} = [Lagrangian (script capital L)] (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 *} = [Lagrangian (script capital L)] (V , ... ) . It is well known that ^{V *} is a Banach space for the classical dual norm ( _{||f || *} = _{sup x ∈V} ( |f (x ) | / ||x || ) ). It is also known that there exists an isometry J :V [arrow right] ^{V *} such that J is linear and for all x ∈V , J (x ) =grad ( ||x ^{|| 2} /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 ∈ [Lagrangian (script capital L)] (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: [figure omitted; refer to PDF]

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 [Lagrangian (script capital L)] (V , ... ) ,

a dual realization of V . We then set [figure omitted; refer to PDF] where Y9;f ,x YA; is the duality pairing for F ×V .

Remark 2.4.

The duality pairing is a nondegenerate bilinear form on F ×V and ||f _{|| F} = _{sup x ∈V} ( | Y9; f ,x YA; | / ||x || ) . These properties permit us to prove that F is isomorphic to ^{V *} .

We deduce from Theorems 2.1and 2.2that k = ^{j -1} [composite function]J ∈ [Lagrangian (script capital L)] (V ,F ) is a surjective isometry such that [figure omitted; refer to PDF] We use the following convention here: when a dual realization {F ,j } of a space has been chosen, we set F = ^{V *} and j [composite function]f (x ) = Y9;f ,x YA; . We say that the isometry k :V [arrow right] ^{V *} is the duality operator associated to the inner product on V and to the duality pairing on ^{V *} ×V by the relation [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF]

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

Let us consider V = ^{L 2} ( ... , (1 + |x | ) ) ⊂ ^{L 2} ( ... ) (dense subspace of ^{L 2} ( ... ) ) endowed with the inner product: [figure omitted; refer to PDF] An element [straight phi] ∈ ^{L 2} ( ... ^{) *} is also an element of ^{V *} . If we identify [straight phi] to an element f ∈ ^{L 2} ( ... ) , this function does not define a linear form on V , and the expression [straight phi] (v ) = Y9;f ,v _{YA; V} 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 Ω ⊂ ^{... n} be an open subset of, a : Ω [arrow right] ^{R +} \ { 0 } , a continuous and strictly positive function called "weight" and s : Ω [arrow right] ^{R +} \ { 0 } , a continuous and strictly positive function called "real time density." The bilinear form defined on _{...9E; 0} ( Ω ) (continuous functions with compact support on Ω ) by [figure omitted; refer to PDF] is an inner product. We remark here that if a is a weight, then ^{a -1} =1 /a is also a weight. Let us introduce the following.

Definition 2.6.

We call ^{L 2} ( Ω ,a ,s ) a completion of _{...9E; 0} ( Ω ) for the inner product Y9;x ,y _{YA; a ,s} .

We now introduce an n -dimensional version of the previous space. If we denote by _{V i} = ^{L 2} ( Ω , ... , _{a i} , _{s i} ) and ^{V i *} = ^{L 2} ( Ω , ... , ^{a i -1} , _{s i} ) , the space [figure omitted; refer to PDF] is a non-pivot Hilbert space with the inner product: [figure omitted; refer to PDF] The space [figure omitted; refer to PDF] is clearly a non-pivot Hilbert space for the following inner product [figure omitted; refer to PDF] and the following bilinear form [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF]

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 K ⊂X be a nonempty, closed, convex subset. We assume the reader is familiar with tangent and normal cones to K at x ∈K ( _{T K} (x ) , respectively, _{N K} (x ) ), and with the projection operator of X onto K , _{P K} :X [arrow right]K given by || _{P K} (z ) -z || = _{inf x ∈K} ||x -z || . Moreover we use here the following characterization of _{P K} (x ) : [figure omitted; refer to PDF] The properties of the projection operator on Hilbert and Banach spaces are well known (see e.g., [ 26- 28]). The directional derivative of the operator _{P K} is defined, for any x ∈K and any element v ∈X , as the limit (for a proof see [ 26]): [figure omitted; refer to PDF] Let _{π K} :K ×X [arrow right]X be the operator given by (x ,v ) ... _{π K} (x ,v ) . Note that _{π K} is nonlinear and discontinuous on the boundary of the set K . In [ 1, 29] several characterizations of _{π K} 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 x ∈X and for all f ∈ ^{X *} the following decompositions hold: [figure omitted; refer to PDF] Here _{P C} is the metric projection operator on K , and _{Π ^{C 0}} is the generalized projection operator on ^{C 0} (for a definition of _{Π ^{C 0}} see [ 28]).

Remark 2.9.

It is known that _{P C} and _{Π C} 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 x ∈X and f ∈ ^{X *} the following decompositions hold: [figure omitted; refer to PDF]

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 , [figure omitted; refer to PDF] where [composite function] ( ||h || ) / ||h || [arrow right]0 as h [arrow right]0 over any locally compact cone of increments.

Remark 2.12.

To prove Lemma 2.11only 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 x ∈K , [figure omitted; refer to PDF] where [composite function] ( ||h || ) / ||h || [arrow right]0 as h [arrow right]0 over any locally compact cone of increments.

Proof.

Clearly, we have in general that [figure omitted; refer to PDF] Taking [figure omitted; refer to PDF] we get [figure omitted; refer to PDF] but [figure omitted; refer to PDF] using the variational principle ( 2.14) applied to _{P x + T K (x )} (x +h ) . By definition of the projection operator we have [figure omitted; refer to PDF] Therefore we have [figure omitted; refer to PDF] As _{P x + T K (x )} (x +h ) =x + _{P T K (x )} (h ) (just apply the definition and the variational principle ( 2.14)), we have [figure omitted; refer to PDF] but using the Corollary 2.10we have h = _{P T C ( x )} ( h ) + ^{J -1} _{P N K ( x )} ( J ( h ) ) , and therefore, [figure omitted; refer to PDF] But by Lemma 2.11, x + _{P T K ( x )} ( h ) - _{P K} ( x + _{P T K ( x )} ( h ) ) =o ( || _{P T K ( x )} ( h ) || ) , so we can write [figure omitted; refer to PDF] Therefore we have, [figure omitted; refer to PDF]

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 K ⊂X 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 F :K [arrow right]X be a Lipschitz continuous vector field and let _{x 0} ∈K . Then the initial value problem associated to the projected differential equation (PrDE) [figure omitted; refer to PDF] 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 [varphi] : _{... +} ×K [arrow right]K which solves the initial value problem: [varphi] ... ( t ,x ) = _{π K} ( [varphi] ( t ,x ) , -F ( [varphi] ( t ,x ) ) ) a .a . t , [varphi] (0 ,x ) = _{x 0} ∈K .

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 F :K [arrow right] ^{X *} , a non-pivot projected differential equation (NpPrDE) is a discontinuous ODE given by [figure omitted; refer to PDF]

Consequently the associated Cauchy problem is given by [figure omitted; refer to PDF] Next we define what we mean by a solution for a Cauchy problem of type ( 3.3).

Definition 3.4.

An absolutely continuous function x : ... ⊂ ... [arrow right]X , such that [figure omitted; refer to PDF] 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 [varphi] : _{... +} ×K [arrow right]K which solves the initial value problem [varphi] ... ( t ,x ) = _{π K} ( [varphi] ( t ,x ) , - ( ^{J -1} [composite function]F ) ( [varphi] ( t ,x ) ) ) , a .a . t , [varphi] (0 ,x ) = _{x 0} ∈K .

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 3.3we show that solutions for this new differential inclusion problem exist. We introduce the following differential inclusion: [figure omitted; refer to PDF] and we call x : ... ⊂ ... [arrow right]X absolutely continuous a solution to ( 3.5) if [figure omitted; refer to PDF] We introduce also the following differential inclusion: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Obviously, we call x : ... ⊂ ... [arrow right]X absolutely continuous a solution to ( 3.7) if [figure omitted; refer to PDF]

Proposition 3.6.

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

Proof.

( 3.3) [implies] ( 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 t ∈I and x ... (t ) = _{π K} (x (t ) , - ( ^{J -1} [composite function]F ) (x (t ) ) ) , a .e .on ... ; therefore using Corollary 2.10we get x ... (t ) = - ^{J -1} (F (x ) ) - ^{J -1} _{P N K (x )} ( -F (x ) ) , a .e . ∈I . Evidently, _{P N K (x )} ( -F (x ) ) ∈ _{N K} (x ) . Moreover as _{N K} (x ) is a closed, convex cone, we get that || _{P N K (x )} ( -F (x ) ) _{|| ^{X *}} ...4; _{|| -F ( x ) || ^{X *}} ) ^{N K 0} (x ) = _{T K} (x ) and both contains 0). Therefore ∃ _{n ... K} (x ) ∈ _{N ... K} (x ) , _{n ... K} (x ) : = _{P N K (x )} ( -F (x ) ) such that x ... (t ) = - ^{J -1} (F (x (t ) ) - _{n ... K} (x ) ) for a.a t ∈I , so we have x ... (t ) ∈ - ^{J -1} (F (x (t ) ) - _{N ... K} (x ) ) for a.a t ∈I , and x ( · ) is a solution to ( 3.9).

( 3.9) [implies] ( 3.3). As the trajectory remains in K it is clear that x ... (t ) ∈ _{T K} (x (t ) ) . First we show that for almost all t ∈I we have [figure omitted; refer to PDF] Let us consider three different cases; first suppose that x (t ) ∈int (K ) , we have then _{N K} (x (t ) ) = { _{0 ^{X *}} } and then ^{N K [perpendicular]} (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 _{T K} (x (t ) ) is flat and ^{N K [perpendicular]} (x (t ) ) [subset of not =] _{T K} (x (t ) ) with ^{N K [perpendicular]} (x (t ) ) not reduced to { _{0 ^{X *}} } , if x ... ( t ) ∉ ^{N K [perpendicular]} ( x ( t ) ) ; then in a neighbourhood ...B1; (t ) the trajectory x (t [variant prime] ) , t [variant prime] ∈ ...B1; (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 ^{N K [perpendicular]} ( x ( t ) ) = { 0 } ; therefore if x ... (t ) =0 , ( 3.10) is satisfied. If x ... (t ) ...0;0 , it means that x (t [variant prime] ) ...0;x (t ) for t [variant prime] ∈ ...B1; (t ) , with x (t [variant prime] ) in one of the two previous cases; as we can "exclude" time t , we have ( 3.10). As we can write x ... ( t ) = ^{J -1} ( -F ( x ) - _{n ... K} ( x ) ) , we have [figure omitted; refer to PDF] Using the polarity between _{N K} (x (t ) ) and _{T K} (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 [figure omitted; refer to PDF] for any neighbourhood [physics M-matrix] 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 K ⊂X be a nonempty, closed and convex subset. Let F :K [arrow right] ^{X *} be a Lipschitz continuous vector field with Lipschitz constant b . Let _{x 0} ∈K . 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 K ⊂X be a non-empty, closed and convex subset. Let F :K [arrow right] ^{X *} be a Lipschitz continuous vector field with Lipschitz constant b , so that on K ∩ _{B X} ( _{x 0} ,L ) , with L >0 and _{x 0} ∈K arbitrarily fixed, we have ||F (x ) || ...4;M : = ||F ( _{x 0} ) || +bL .

Then the set-valued mapping _{...A9; p} :K ∩ _{B X} ( _{x 0} ,L ) [arrow right] ... given by [figure omitted; refer to PDF] has a closed graph.

Proof .

The proof is similar to the one in [ 21].

We show first that the mapping _{...A9; p} :K ∩ _{B X} ( _{x 0} ,L ) [arrow right] ... given by x ... Y9; - _{N ... K} (x ) ,p YA; has a closed graph. It is clear that for each p ∈X , the set-valued map _{...A9; p} :K ∩ _{B X} ( _{x 0} ,L ) [arrow right] ... maps K ∩ _{B X} ( _{x 0} ,L ) into ^{2 [ -M ||p || ,M ||p || ]} . Let { ( _{x n} , _{z n} ) _{} n} ∈graph ( _{...A9; p} ) such that ( _{x n} , _{z n} ) [arrow right] (x ,z ) ∈X × ^{2 [ -M ||p || ,M ||p || ]} . We want to show that (x ,y ) ∈graph ( _{...A9; p} ) . From _{z n} ∈graph ( _{...A9; p} ) , for all n , we deduce that there exists _{y n} ∈ - _{N ... K} ( _{x n} ) such that _{z n} = Y9; _{y n} ,p YA; . Since the set - _{N ... K} (x ) ⊂ _{B ¯ ^{X *}} (0 ,M ) and _{B ¯ ^{X *}} (0 ,M ) is weakly compact, then there exists a subsequence _{y n k} and y ∈ ^{X *} such that [figure omitted; refer to PDF] for the weak topology σ ( ^{X *} , ^{X * *} ) = by reflexivity σ ( ^{X *} ,X ) , which is equivalent to [figure omitted; refer to PDF] Suppose now that y ∉ - _{N ... K} (x ) . This implies that at least one of the following two alternatives should be satisfied.

(1) There exists w ∈K such that Y9; y ,w -x YA; < λ <0 .

(2) ||y || > μ > ||F ( x ) || .

In the first case as Y9; _{y n k} , β YA; [arrow right] Y9;y , β YA; , ∀ β ∈X for k > _{k 0} we have Y9; _{y n k} ,w -x YA; < λ /2 . But Y9; _{y n k} ,w - _{x n k} YA; = Y9; _{y n k} ,w -x YA; + Y9; _{y n k} ,x - _{x n k} YA; and as _{x n k} [arrow right]x , there exists _{k 1} >0 such that ∀k ...5; _{k 1} , we have Y9; _{y n k} ,x - _{x n k} YA; ...4; ||x - _{x n k} || || _{y n k} || < ( | λ | /4M ) M = | λ | /4 . Thus Y9; _{y n k} ,w - _{x n k} YA; < λ /4 <0 , for all k >max ( _{k 0} , _{k 1} ) . But this contradicts the fact that _{y n k} ∈ - _{N ... K} ( _{x n k} ) .

In the second case as Y9; _{y n k} , β YA; [arrow right] Y9;y , β YA; , ∀ β ∈X , we have ([ 31, Proposition III.12]) ||F (x ) || < ||y || ...4; _{liminf k [arrow right] ∞} || _{y n k} || which is a contradiction because _{y n} ∈ - _{N ... K} ( _{x n} ) , ∀n ∈ ... . The continuity of F and the first part of the proof implies that [figure omitted; refer to PDF] has non-empty, closed and convex values for each x ∈K 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 K ⊂X be a non-empty, closed and convex subset. Let F :K [arrow right] ^{X *} be a Lipschitz continuous vector field so that on K ∩ _{B X} ( _{x 0} ,L ) , with L >0 and _{x 0} ∈K , we have ||F (x ) || ...4;M : = ||F ( _{x 0} ) || +bL . Let l : =L /M and ... : = [0 ,l ] . Then there exists a sequence { _{x k} ( · ) } of absolutely continuous functions defined on ... , with values in K , such that for all k ...5;0 , _{x k} (0 ) = _{x 0} and for almost all t ∈ ... , { _{x k} (t ) } and { _{x ... k} (t ) } (the sequence of its derivatives) have the following property: for every neighbourhood [physics M-matrix] of 0 in X ×X there exists _{k 0} = _{k 0} (t , [physics M-matrix] ) such that [figure omitted; refer to PDF]

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 _{z p} : = _{P K} (x - _{h p} F (x ) ) we now construct _{z p} : = _{P K} (x - _{h p} ^{J -1} [composite function]F (x ) ) .

Next we show that the sequence { _{x k} ( · ) } built in Theorem 3.9is uniformly convergent to some x ( · ) . Again, following closely [ 21], by Theorem 3.9there exists a pair ( _{u k} , -F ( _{u k} ) - _{n k} ) ∈ graph ( -F - _{N ... K} ) such that [figure omitted; refer to PDF] where _{... 1 ,k} (t ) and _{... 2 ,k} (t ) are vector functions, not necessarily continuous, satisfying || _{... 1 ,k} (t ) || < _{... k} and || _{... 2 ,k} (t ) || < _{... k} where _{... k} [arrow right]0 as k [arrow right] ∞ and _{n k} ∈ _{N ... K} ( _{u k} ) and _{n m} ∈ _{N ... K} ( _{u m} ) .

Let k ,m be two indexes. Then we evaluate [figure omitted; refer to PDF] But using the monotonicity of x ... _{N K} (x ) , the isometry property of J , and the b-Lipschitz continuity of F we get that [figure omitted; refer to PDF] We now let [varphi] ( t ) : = || _{x k} ( t ) - _{x m} ( t ) || , so from the previous inequalities we get [figure omitted; refer to PDF] Using the same technique as in [ 21] we get [figure omitted; refer to PDF] 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 { _{x k} ( · ) } is uniformly convergent to x ( · ) and as ( _{x k} (t ) , _{x ... k} (t ) ) ∈graph ( -F - _{N ... K} ) + [physics M-matrix] , we now deduce that there exists a θ such that || _{x ... k} (t ) || ...4; θ . Using the arguments in [ 21] and the result of [ 32], we deduce the existence of a subsequence of { _{x ... k} } 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 k ...5; _{k 0} and almost every t ∈ ... there exists a pair [figure omitted; refer to PDF] such that || _{x k} (t ) - _{u k} (t ) || < _{... k} and || _{x ... k} ( t ) - _{v k} ( t ) || < _{... k} , where _{... k} [arrow right]0 when k [arrow right] ∞ . Let p ∈X arbitrarily fixed. Then for almost all t ∈ ... [figure omitted; refer to PDF] So _{u k} (t ) [arrow right]x (t ) for every t ∈ ... and Y9; _{v k} (t ) ,p YA; [arrow right] Y9; _{x ... k} (t ) ,p YA; for almost all t ∈ ... . By Proposition 3.8, we know that graph ( Y9; -F - _{N ... K} ,p YA; ) is closed, so it follows that for almost all t ∈ ... , [figure omitted; refer to PDF] Since the set F (x (t ) ) - _{N ... K} (x (t ) ) is convex and closed, it follows that [figure omitted; refer to PDF] 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 _{x 1} ( · ) and _{x 2} ( · ) starting at the same initial point. For any fixed t ∈ ... we get [figure omitted; refer to PDF] 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 || _{x 1} (t ) - _{x 2} (t ) ^{|| 2} ...4;0 , so we have _{x 1} (t ) = _{x 2} (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 l ...5;1 / (1 +b ) in the following way: if ||F ( _{x 0} ) || =0 , we let L =1 , and if ||F ( _{x 0} ) || ...0;0 , then we let L ...5; ||F ( _{x 0} ) || . In both cases we obtain l ...5;1 / (1 +b ) . Therefore beginning at each initial point _{x 0} ∈K , 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 _{x 0} =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 ^{K [variant prime]} ⊂X be a non-empty, closed subset. Consider a pair (g ,K ) such that K is convex and g : ^{K [variant prime]} [arrow right]K =r ( ^{K [variant prime]} ) ⊂X , is continuous, injective, and ^{g -1} is Lipschitz continuous.

Consider F :X [arrow right] ^{X *} satisfying (F [composite function]g ) (y ) =F (y ) , ∀y ∈ ^{K [variant prime]} . Then the pair (g ,K ) is called a convexification pair of (F , ^{K [variant prime]} ) .

Example 3.11.

Here is an example of such a convexification pair in ^{... 2} . Let ^{K [variant prime]} = { (x ,y ) ∈ ^{... 2} |" 0 ...4;x ...4;1 , 0 ...4;y ...4;x } and let g be the map of ^{K [variant prime]} into K = [0,1 ] × [0,1 ] , namely: [figure omitted; refer to PDF] 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 F [composite function]g (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 ^{K [variant prime]} ⊂X 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 ) ) , g : ^{K [variant prime]} [arrow right]K ⊂X , that is: [figure omitted; refer to PDF]

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

Considering now ( 3.29) and a convexification pair (g ,K ) of a nonempty, closed ^{K [variant prime]} ⊂X , then the Cauchy problem associated to ( 3.29) and the pair (g ,K ) is given by [figure omitted; refer to PDF] Next we define what we mean by a solution for a Cauchy problem of type ( 3.30).

Definition 3.13.

An absolutely continuous function y : ... ⊂ ... [arrow right]X , such that [figure omitted; refer to PDF] 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 ( · ) = ^{g -1} (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 [varphi] : _{... +} × ^{K [variant prime]} [arrow right]K which solves the initial value problem: [figure omitted; refer to PDF] where (g ,K ) is a convexification pair.

Theorem 3.15.

Let X be a generic Hilbert space, and let ^{K [variant prime]} be a non-empty closed subset of X . Let K be non-empty, closed and convex, let g : ^{K [variant prime]} [arrow right]K be continuous and strictly monotone, and let F : ^{K [variant prime]} [arrow right]X be Lipschitz continuous such that (F [composite function]g _{) | ^{K [variant prime]}} =F . Let also _{x 0} ∈K and L >0 such that || _{x 0} || ...4;L . Then the initial value problem ( 3.30) has a unique solution on the interval [0 ,l ] , where l =L / ( ||F ( _{x 0} ) || +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 ^{x *} ∈K is called a critical point for ( 3.2) if [figure omitted; refer to PDF]

Theorem 4.2.

Let X be a generic Hilbert space and let K ⊂X be a non-empty, closed and convex subset. Let F :X [arrow right] ^{X *} be a vector field. Consider the variational inequality problem: [figure omitted; refer to PDF] 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 _{P K} we describe this relation next.

Definition 4.3.

Let X be a generic H-space and let ^{K [variant prime]} ⊂X be a non-empty, closed subset. Let F :X [arrow right] ^{X *} be a mapping. Then we call g-variational inequality on the set K [variant prime] the problem of [figure omitted; refer to PDF] where (g ,K ) is a convexification pair of (F , ^{K [variant prime]} ) .

We highlight the importance of the relation F [composite function]g (y ) =F (y ) from Definition 3.10in order for ( 4.3) to make sense. Under ( 3.5) we can rewrite ( 4.3) as [figure omitted; refer to PDF]

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 ^{y *} ∈ ^{K [variant prime]} is a solution of ( 4.4); then by definition we have [figure omitted; refer to PDF] So by multiplying by a strictly positive constant λ and using the bilinearity of the inner product, we get [figure omitted; refer to PDF] So we deduce that -F ( ^{y *} ) ∈ _{N K} (g ( ^{y *} ) ) ; using the decomposition Theorem 2.8we get _{P T K ( g ( ^{y *} ) )} ( - ^{J -1} F ( ^{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 [figure omitted; refer to PDF] and by the decomposition theorem we get -F ( ^{y *} ) ∈ _{N K} (g ( ^{y *} ) ) . By the definition of the normal cone to K in g ( ^{y *} ) , the following inequality is satisfied: [figure omitted; refer to PDF] which is exactly ( 4.4).

5. Examples and Applications

5.1. Weighted Traffic Problem

Let us introduce a network ...A9; , that means a set ...B2; of origin-destination pair (origin/destination node) and a set ... of routes. Each route r ∈ ... links exactly one origin-destination pair w ∈ ...B2; . The set of all r ∈ ... which link a given w ∈ ...B2; 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 = { _{a 1} , ... , _{a n} } , and by ^{a -1} = { ^{a 1 -1} , ... , ^{a n -1} } two families of weights such that for each 1 ...4;i ...4;n , _{a i} ∈ ...9E; ( Ω , ^{... +} \ {0 } ) . We introduce also the family of real time traffic densities s = { _{s 1} , ... , _{s n} } such that for each 1 ...4;i ...4;n , _{s i} ∈ ...9E; ( Ω , ^{... +} \ {0 } ) .

Let _{r i} correspond to an element of a and s , newly to _{a i} and _{s i} . If we denote by _{V i} = ^{L 2} ( Ω , ... , _{a i} , _{s i} ) and ^{V i *} = ^{L 2} ( Ω , ... , ^{a i -1} , _{s i} ) , the space [figure omitted; refer to PDF] is a Hilbert space for the inner product [figure omitted; refer to PDF] The space ^{V *} = ^{∏ i =1 n V i *} is a Hilbert space for the following inner product [figure omitted; refer to PDF] and the following bilinear form defines a duality between V and ^{V *} : [figure omitted; refer to PDF] [figure omitted; refer to PDF] 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 ) = ( _{a 1 F 1} , ... , _{a n F n} ) .

The feasible flows have to satisfy the time-dependent capacity constraints and demand requirements; namely, for all r ∈ ... , w ∈ ...B2; and for almost all t ∈ Ω , [figure omitted; refer to PDF] where 0 ...4; λ ...4; μ are given in ^{L 2} ( [0 ,T ] , ^{... n} ) , ρ ∈ ^{L 2} ( [0 ,T ] , ^{... m} ) where m =card ( ...B2; ) , _{F r} , r ∈ ... , denotes the flow in the route r . If Φ = ( _{Φ w ,r} ) is the pair route incidence matrix, with w ∈ ...B2; and r ∈ ... , that is, [figure omitted; refer to PDF] the demand requirements can be written in matrix-vector notation as [figure omitted; refer to PDF] The set of all feasible flows is given by [figure omitted; refer to PDF] 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 determine [figure omitted; refer to PDF] where K is a closed convex subset of V , and C :K [arrow right] ^{V *} is a mapping.

Definition 5.2.

H ∈V is an equilibrium flow if and only if [figure omitted; refer to PDF]

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.

H ∈K is an equilibrium flow in the sense of ( 5.11) if and only if [figure omitted; refer to PDF] [figure omitted; refer to PDF]

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 ...A6; :D [arrow right] ^{2 X} with ...A6; (x ) convex for all x ∈D and F : ...A6; [arrow right] ^{2 X *} a mapping.

Let us introduce the following variational inequality: [figure omitted; refer to PDF] 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) ...A6; :D [arrow right] ^{2 D} is a closed lower semicontinuous correspondence with closed, convex, and nonempty values,

(ii) C :D [arrow right] ^{2 X [variant prime]} is a monotone, finite continuous, and bounded single-valued map,

(iii): there exist a compact, convex, and nonempty set Z ⊂D and a nonempty subset B ⊂Z such that

(a) ...A6; (B ) ⊂Z ,

(b) ...A6; ( z ) ∩Z ...0; ∅ , ∀z ∈Z ,

(c) for every z ∈Z \B , there exist z ... ∈ ...A6; (z ) ∩Z with Y9;C (z ) , z ... -z YA; <0 .

Then there exists x such that [figure omitted; refer to PDF]

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: [figure omitted; refer to PDF]

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 ,w ∈X , _{P ...A6; (u )} satisfies the condition [figure omitted; refer to PDF] 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 ,v ∈C ; we denote by ...A6; (u ) = _{T C} (u ) and by ...A6; (v ) = _{T C} (v ) the tangent cones of C at u and v .

In fact, w ∈X can only be chosen in one of the following four situations:

(1) w ∈ ...A6; ( u ) ∩ ...A6; ( v ) ,

(2) w ∈ ...A6; ( u ) \ ...A6; ( v ) ,

(3) w ∈ ...A6; ( v ) \ ...A6; ( u ) ,

(4) w ∈X \ ( ...A6; ( u ) ∪ ...A6; ( v ) ) .

Suppose now that we have w ∈ ...A6; (u ) \ ...A6; (v ) ; then by Moreau's decomposition theorem we get [figure omitted; refer to PDF] where _{N C} (v ) is the normal cone of C at v . Consider now X = ^{... 2} , C = [0 , ... ^{] 2} , u = (0,0 ) and v = ( ... , ... ) . It is clear that we have the following: [figure omitted; refer to PDF] So for any w ∈ _{N C} (v ) we get [figure omitted; refer to PDF] Since w is arbitrary, let now w : = μw , for any μ >0 . Then, [figure omitted; refer to PDF] should be true for any μ >0 . However this does not hold.

Consider now the special case of a set-valued mapping ...A6; which is the translation of a closed, convex subset K : [figure omitted; refer to PDF]

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 ...A6; (x ) =K +p (x ) as done by Noor for type B PDS [ 24], we have the following equivalent formulations: [figure omitted; refer to PDF] where g (x ) =x -p (x ) , assuming F (g (x ) ) =F (x -p (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 3.15provides 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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