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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 508570, 23 pages
Nonpivot and Implicit Projected Dynamical Systems on Hilbert Spaces
1Department of Mathematics & Statistics, University of Guelph, Guelph, ON, N1G 2W1, Canada
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.
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.
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 ; 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 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 ). 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 , 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 , 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 , it is necessary to decide whether or not we identify the topological dual space with . Commonly this identification is made, one of the reasons for this being that the vectors of the polar of a set of are in . 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  for additional information.
First, consider a pre-Hilbert space with an inner product , and its topological dual . It is well known that is a Banach space for the classical dual norm (). It is also known that there exists an isometry such that is linear and for all , . This mapping is called a duality mapping of .
Theorem 2.1 (Theorem 1 page 68, ). Let be a Hilbert space with the inner product and the duality mapping above. Then J is a surjective isometry from to . The dual space is a Hilbert space with the inner product:
Theorem 2.2 (Theorem 2 page 69, ). Let V be a pre-Hilbert space. Then there exists a completion of V, that is, an isometry j from V to the Hilbert space such that is dense in .
Definition 2.3. Let be a Hilbert space. We call , where(i) is a Hilbert space,(ii) is an isometry from to , a dual realization of . We then set where is the duality pairing for .
Remark 2.4. The duality pairing is a nondegenerate bilinear form on and . These properties permit us to prove that is isomorphic to .
We deduce from Theorems 2.1 and 2.2 that is a surjective isometry such that We use the following convention here: when a dual realization of a space has been chosen, we set and . We say that the isometry is the duality operator associated to the inner product on and to the duality pairing on by the relation A special but most frequent case is to choose a dual realization of the couple ; in this case the Hilbert space is called a pivot space. To be more precise, we introduce the following definition.
Definition 2.5. A Hilbert space with an inner product is called a pivot space, if we identify with . In that case
Sometimes it does not make sense to identify the space itself with its topological dual, as the following example shows.
Let us consider (dense subspace of ) endowed with the inner product:An element is also an element of . If we identify to an element , this function does not define a linear form on , and the expression has no meaning on . 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 be an open subset of, , a continuous and strictly positive function called “weight” and , a continuous and strictly positive function called “real time density.” The bilinear form defined on (continuous functions with compact support on ) by is an inner product. We remark here that if is a weight, then is also a weight. Let us introduce the following.
Definition 2.6. We call a completion of for the inner product .
We now introduce an -dimensional version of the previous space. If we denote by and , the space is a non-pivot Hilbert space with the inner product: The space is clearly a non-pivot Hilbert space for the following inner product and the following bilinear form defines a duality between and . More precisely we have the following (see  for a proof).
Proposition 2.7. The bilinear form (2.12) defines a duality mapping between , given by
For applications of these spaces, the reader can refer to .
2.2. Variational Analysis in Non-Pivot H-Spaces
Let be a Hilbert space of arbitrary (finite or infinite) dimension and let be a nonempty, closed, convex subset. We assume the reader is familiar with tangent and normal cones to at (, respectively, ), and with the projection operator of onto , given by . Moreover we use here the following characterization of : 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 is defined, for any and any element , as the limit (for a proof see ): Let be the operator given by . Note that is nonlinear and discontinuous on the boundary of the set . In [1, 29] several characterizations of 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 be a real reflexive strictly convex and smooth Banach space, and let be a non-empty, closed and convex cone of . Then for all and for all the following decompositions hold: Here is the metric projection operator on , and is the generalized projection operator on (for a definition of see ).
Remark 2.9. It is known that and coincide whenever the cone belongs to a Hilbert space. This observation implies the following result.
Corollary 2.10. Let be a nonempty closed convex cone of a non-pivot Hilbert space . Then for all and the following decompositions hold:
We highlight that Zarantonello has shown in  a similar decomposition result in reflexive Banach spaces.
Lemma 2.11 ([26, Lemma 4.5]). For any closed convex set , where as 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 . We give below a similar proof in non-pivot spaces.
Lemma 2.13. For any , where as over any locally compact cone of increments.
Proof. Clearly, we have in general that Taking we get but using the variational principle (2.14) applied to . By definition of the projection operator we have Therefore we have As (just apply the definition and the variational principle (2.14)), we have but using the Corollary 2.10 we have , and therefore, But by Lemma 2.11, , so we can write Therefore we have,
3. Non-Pivot and Implicit PDS in Hilbert Spaces
3.1. PDS in Pivot H-Spaces
Let be a pivot Hilbert space of arbitrary (finite or infinite) dimension and let be a nonempty, closed, convex subset. The following result has been shown (see ).
Theorem 3.1. Let be a Hilbert space and let be a nonempty, closed, convex subset. Let be a Lipschitz continuous vector field and let . Then the initial value problem associated to the projected differential equation (PrDE) has a unique absolutely continuous solution on the interval .
This result is a generalization of the one in , where , was a convex polyhedron and had linear growth.
Definition 3.2. A projected dynamical system then is given by a mapping which solves the initial value problem: , .
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  we first introduce the following.
Definition 3.3. For , a non-pivot projected differential equation (NpPrDE) is a discontinuous ODE given by
Consequently the associated Cauchy problem is given by Next we define what we mean by a solution for a Cauchy problem of type (3.3).
Definition 3.4. An absolutely continuous function , such that 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 which solves the initial value problem .
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: and we call absolutely continuous a solution to (3.5) if We introduce also the following differential inclusion: where
Obviously, we call absolutely continuous a solution to (3.7) if
Proof. (3.3)⇒(3.9). Let be an absolutely continuous function on such that is a solution to (3.3). Then , for all and ; therefore using Corollary 2.10 we get , . Evidently, . Moreover as is a closed, convex cone, we get that and both contains 0). Therefore such that for a.a , so we have for a.a , and is a solution to (3.9).
(3.9)⇒(3.3). As the trajectory remains in it is clear that . First we show that for almost all we have Let us consider three different cases; first suppose that , we have then and then and (3.10) is automatically satisfied. Suppose now that and in , is smooth. In that case is flat and with not reduced to , if ; then in a neighbourhood the trajectory goes in , so we are in the first case and we can exclude time . Suppose now that and is in a corner point. In that case ; therefore if , (3.10) is satisfied. If , it means that for , with in one of the two previous cases; as we can “exclude” time , we have (3.10). As we can write , we have Using the polarity between and 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 for any neighbourhood of 0 in . 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 .
The methodology of the proofs is completely analogous to that used for pivot Hilbert spaces in . 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 .
The main result can be stated as follows.
Theorem 3.7. Let be a Hilbert space and its topological dual and let be a nonempty, closed and convex subset. Let be a Lipschitz continuous vector field with Lipschitz constant . Let . Then the initial value problem (3.3) has a unique solution on .
Proof Existence of a solution on an interval
For this part of the proof, we need two major results, as follows.
Proposition 3.8. Let be a nonpivot H-space, let be its topological dual, and let be a non-empty, closed and convex subset. Let be a Lipschitz continuous vector field with Lipschitz constant , so that on , with and arbitrarily fixed, we have .
Then the set-valued mapping given by has a closed graph.
Proof . The proof is similar to the one in .
We show first that the mapping given by has a closed graph. It is clear that for each , the set-valued map maps into . Let such that . We want to show that . From , for all , we deduce that there exists such that . Since the set and is weakly compact, then there exists a subsequence and such that for the weak topology , which is equivalent to Suppose now that . This implies that at least one of the following two alternatives should be satisfied.(1)There exists such that .(2). In the first case as for we have . But and as , there exists such that , we have . Thus , for all . But this contradicts the fact that .
In the second case as , we have ([31, Proposition III.12]) which is a contradiction because . The continuity of and the first part of the proof implies that has non-empty, closed and convex values for each and has a closed graph.
The next result is constructing the sequence of approximate solutions for the problem (3.7).
Theorem 3.9. Let be a Hilbert space and its topological dual, and let be a non-empty, closed and convex subset. Let be a Lipschitz continuous vector field so that on , with and , we have . Let and . Then there exists a sequence of absolutely continuous functions defined on , with values in , such that for all and for almost all , and (the sequence of its derivatives) have the following property: for every neighbourhood of 0 in there exists such that
Proof. The proof, based on topological properties of the space , can be found in . However, given we are now working in non-pivot H-spaces, then instead of we now construct .
Next we show that the sequence built in Theorem 3.9 is uniformly convergent to some . Again, following closely , by Theorem 3.9 there exists a pair such that where and are vector functions, not necessarily continuous, satisfying and where as and and .
Let be two indexes. Then we evaluate But using the monotonicity of , the isometry property of , and the b-Lipschitz continuity of we get that We now let , so from the previous inequalities we get Using the same technique as in  we get where is the length of . So the Cauchy criteria are satisfied uniformly and we get the conclusion.
From the previous step we know that is uniformly convergent to and as , we now deduce that there exists a such that . Using the arguments in  and the result of , we deduce the existence of a subsequence of weakly*-convergent to .
Finally, we finish this part of the proof by showing that is indeed a solution of the differential inclusion (3.7). From Theorem 3.9, for each and almost every there exists a pair such that and , where when . Let arbitrarily fixed. Then for almost all So for every and for almost all . By Proposition 3.8, we know that is closed, so it follows that for almost all , Since the set is convex and closed, it follows that By Proposition 3.6, is a solution of problem (3.3).
Uniqueness of Solutions on
Step 1 ( is the unique solution). Suppose that we have two solutions and starting at the same initial point. For any fixed we get because the metric projection is a nonexpansive operator in , is a linear isometry, and is b-Lipschitz. By Gronwall’s inequality we obtain , so we have for any .
Existence of Solutions on
From above we can assert the existence of a solution to problem (3.3) on an interval , with fixed and arbitrary. We note that we can choose such that in the following way: if , we let , and if , then we let . In both cases we obtain . Therefore beginning at each initial point , problem (3.3) has a solution on an interval of length at least . Now if we consider problem (3.3) with , applying again all the above, we obtain an extension of the solution on an interval of length at least . By continuing this solution we obtain a solution on .
3.4. Implicit PDS
In this section we consider a generic Hilbert space , 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 be a generic H-space and let be a non-empty, closed subset. Consider a pair such that is convex and , is continuous, injective, and is Lipschitz continuous.
Consider satisfying . Then the pair is called a convexification pair of .
Example 3.11. Here is an example of such a convexification pair in . Let and let be the map of into , namely: We can easily check that is continuous and monotone. Now take to be , where is an arbitrary constant in . Then we have .
We now introduce another type of a projected equation as follows.
Definition 3.12. Let be a generic H-space and let be a non-empty, closed subset. An implicit projected differential equation (ImPrDE) is a (PrDE) given by (3.2) where , that is:
The motivation for the introduction of such an equation comes from the desire to study the dynamics on a set , where 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 of a nonempty, closed , then the Cauchy problem associated to (3.29) and the pair is given by Next we define what we mean by a solution for a Cauchy problem of type (3.30).
Definition 3.13. An absolutely continuous function , such that 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 , problem (3.30) has solutions on , in the sense of Definition 3.4. But since is assumed continuous and strictly monotone, then is invertible and so ; moreover, we see that such a 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 which solves the initial value problem: where is a convexification pair.
Theorem 3.15. Let be a generic Hilbert space, and let be a non-empty closed subset of . Let be non-empty, closed and convex, let be continuous and strictly monotone, and let be Lipschitz continuous such that . Let also and such that . Then the initial value problem (3.30) has a unique solution on the interval , where .
Proof. The proof consists in the modification of a few easy steps of the proof given in  combined with the results of the present paper.
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 is called a critical point for (3.2) if
Theorem 4.2. Let X be a generic Hilbert space and let be a non-empty, closed and convex subset. Let be a vector field. Consider the variational inequality problem: Then the solution set of (4.2) coincides with the set of critical points of the non-pivot projected dynamical system (3.2).
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 we describe this relation next.
Definition 4.3. Let be a generic H-space and let be a non-empty, closed subset. Let be a mapping. Then we call g-variational inequality on the set the problem of where is a convexification pair of .
Remark 4.4. In , (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.
Proof. Suppose is a solution of (4.4); then by definition we have
So by multiplying by a strictly positive constant and using the bilinearity of the inner product, we get
So we deduce that ; using the decomposition Theorem 2.8 we get , and so is a critical point of (3.30).
Now suppose that is a critical point of (3.30); then by definition we have and by the decomposition theorem we get . By the definition of the normal cone to in , the following inequality is satisfied: which is exactly (4.4).
5. Examples and Applications
5.1. Weighted Traffic Problem
Let us introduce a network , that means a set of origin-destination pair (origin/destination node) and a set of routes. Each route links exactly one origin-destination pair . The set of all which link a given is denoted by . For each time we consider vector flow . Let us denote by an open subset of , by , , and by two families of weights such that for each . We introduce also the family of real time traffic densities such that for each , .
Let correspond to an element of and , newly to and . If we denote by and , the space is a Hilbert space for the inner product The space is a Hilbert space for the following inner product and the following bilinear form defines a duality between and : More exactly we have the following.
Proposition 5.1. The bilinear form (5.5) is defined over and defines a duality between . The duality mapping is given by .
The feasible flows have to satisfy the time-dependent capacity constraints and demand requirements; namely, for all , and for almost all , where are given in , where , denotes the flow in the route . If is the pair route incidence matrix, with and , that is, the demand requirements can be written in matrix-vector notation as The set of all feasible flows is given by 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 is to determine where is a closed convex subset of , and is a mapping.
Definition 5.2. is an equilibrium flow if and only if
Theorem 5.3. is an equilibrium flow in the sense of (5.11) if and only if
Proof. see .
Based on previous results , this solution coincides with the set of critical points of the associated projected dynamical system.
5.2. Quasivariational Inequalities on Translated Sets
Let be a generic H-space, closed, convex, nonempty in . Let with convex for all and a mapping.
Let us introduce the following variational inequality: Note that in this case the set in which we are looking for the solution depends on . For problem (5.14) we can provide the following existence result (see  or ).
Theorem 5.4. Let be a closed convex subset in a locally convex Hausdorff topological vector space . Let us suppose that(i) is a closed lower semicontinuous correspondence with closed, convex, and nonempty values,(ii) is a monotone, finite continuous, and bounded single-valued map,(iii)there exist a compact, convex, and nonempty set and a nonempty subset such that(a),(b),
(c)for every , there exist with .
Then there exists such that
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:
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 , assuming the following fact.
Assumption 5.7. Let be a pivot H-space. For all , satisfies the condition where is a constant.
However, this assumption fails to be true. One counterexample is as follows. We denote by a closed convex set and we take ; we denote by and by the tangent cones of at and .
In fact, can only be chosen in one of the following four situations:(1), (2), (3), (4). Suppose now that we have ; then by Moreau’s decomposition theorem we get where is the normal cone of at . Consider now , , and . It is clear that we have the following: So for any we get Since is arbitrary, let now , for any . Then, should be true for any . However this does not hold.
Consider now the special case of a set-valued mapping which is the translation of a closed, convex subset :
If as done by Noor for type B PDS , we have the following equivalent formulations: where , assuming . We can observe that if , 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.
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  as an open problem.
The work has been supported by the first author’s NSERC Discovery Grant. The support is gratefully acknowledged.
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