Abstract and Applied Analysis

Volume 2013 (2013), Article ID 952915, 12 pages

http://dx.doi.org/10.1155/2013/952915

Research Article

A Quasi-Variational Approach for the Dynamic Oligopolistic Market Equilibrium Problem

Annamaria Barbagallo¹ and Paolo Mauro²

¹Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”, Via Cintia, 80126 Naples, Italy

²Department of Mathematics and Computer Science, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

Received 18 July 2013; Accepted 19 September 2013

Academic Editor: Abdellah Bnouhachem

Copyright © 2013 Annamaria Barbagallo and Paolo Mauro. 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.

The paper is concerned with the dynamic oligopolistic market equilibrium problem in the realistic case in which we allow the presence of capacity constraints and production excesses and, moreover, we assume that the production function depends not only on the time but also on the equilibrium distribution. As a consequence, we introduce the generalized dynamic Cournot-Nash principle in the elastic case and prove the equivalence between this equilibrium definition and a suitable evolutionary quasi-variational inequality. For completeness we make the analysis of existence, regularity, and sensitivity of the solution. In the end, a numerical example is provided.

1. Introduction

The aim of the paper is to improve the results obtained in [1] concerning the dynamic oligopolistic market equilibrium problem in presence of production excesses by introducing the dependence on the equilibrium commodity shipment in the production function (see in (4)) and, as a consequence, studying the so-called elastic model. This is a more realistic situation since it is reasonable to think that the production function is influenced not only by the time, but also by the evaluation of the amount of commodity shipment, namely, the forecasted equilibrium solution. The presence of production excesses may be well justified in periods of economic crisis, so it is possible that some of the amounts of the commodity available are sold out whereas for a part of the products, an excess of production can occur.

In the last decade a lot of problems considering a feasible set depending on equilibrium solutions have been studied (see, e.g., [2–4]). It is well known that the equilibrium models with fixed constraint sets may be expressed in terms of evolutionary variational inequalities, while models with elastic constraint sets are expressed by evolutionary quasi-variational inequalities. Moreover, the dependence on time leads to considering variational and quasi-variational inequalities in an infinite dimensional setting, for example, a Lebesgue space.

Let us remember that a dynamic oligopolistic market equilibrium problem is the problem of finding a trade equilibrium in a supply-demand market between a finite number of spatially separated firms producing homogeneous goods in a fixed time interval. Moreover, the firms act in a noncooperative behavior. This problem has its origin with Cournot [5]. He considered only two firms and for this reason it was called the duopoly problem. Later, Nash [6, 7] extended Cournot's duopoly problem to agents. A more complete and efficient study was done by Nagurney et al. in [8–11], but the problem was still faced in a static case through a finite dimensional variational approach. Finally, in [12] the time dependence was considered in the model. It allows to explore the change of behavior of equilibrium states for the oligopolistic market models over a finite time interval of interest. As Beckmann and Wallace stressed, for the first time, in [13], “the time-dependent formulation of equilibrium problems allows one to explore the dynamics of adjustment processes in which a delay on time response is operating.” Of course a delay on time response always happens because the processes do not have an infinite speed. Usually, such adjustment processes can be represented by means of a memory term which depends on previous equilibrium solutions according to the Volterra operator (see, e.g., [14, 15]).

Furthermore, in [16] the authors describe the behavior of the market by using the Lagrange multipliers of the infinite dimensional duality theory developed in [17–21]. Such results make use of the notion of tangent cone, normal cone, and quasi-relative interior of sets (see [22, 23]), important tools to overcome the difficulty of the emptiness of the topological interior of the ordering cone which defines constraints of several infinite dimensional problems (see [24, 25]). Moreover, a sensitivity result has been obtained which states that, under additional assumptions, small changes of the solution happen in correspondence with small changes of the profit function.

Lately, in [1, 26], the model presented in [12] has been improved with the addition of production excesses and both production and demand excesses, respectively. Another important question is to find some regularity properties for the solution. In [1, 26], the continuity of solution is proved under suitable assumptions, and it results to be very helpful in order to introduce numerical schemes to compute equilibrium solutions (see [27, 28]).

In [29, 30], the authors abandon the study of the problem from a producer's point of view whose purpose is to maximize his own profit and focus their attention on the policy-maker's perspective whose aim is to control the commodity exportations by means of the imposition of taxes or incentives and formulate the resulting optimization problem as an inverse variational inequality.

This paper is structured as follows. In Section 2 we present the dynamic oligopolistic market equilibrium problem with the elastic production function and after that we give the definition of equilibrium according to the generalized Cournot-Nash principle. Moreover, we prove the equivalence with a suitable evolutionary quasi-variational inequality. Section 3 is devoted to prove a result of existence of the solution, while in Section 4 Kuratowski's set convergence will be a preliminary property in order to prove the continuity of the equilibrium solution. In Section 5 we establish a sensitivity result that shows how the equilibrium solution can change if the data have been perturbed. In Section 6 a numerical example is provided to make the theoretical model presented in the previous sections more clearer.

2. Quasi-Variational Inequalities in Dynamic Oligopolistic Markets

Let us consider firms , , that produce a homogeneous commodity and demand markets , , that are generally spatially separated. Assume that the homogeneous commodity, produced by the firms and consumed by the markets, is involved during a time interval , .

Let , and , denote the nonnegative commodity shipment between the supply market and the demand market at the time . In particular, let us set the vector , and , as the strategy vector for the firm .

Let us group the commodity shipments into a matrix function and suppose that . Furthermore, we assume that the nonnegative commodity shipment between the producer and the demand market has to satisfy time-dependent constraints, namely, there exist two nonnegative functions such that

and suppose that .

Let us denote

It is easy to verify that is a nonempty, compact, and convex subset of . Let , , denote the nonnegative commodity output produced by firm at the time . Let us group the production output into a vector function and let us suppose that .

Now, let us introduce the production excesses. Let , , be the nonnegative production excess for the commodity of the firm at the time . Let us group the production excess into a vector function and let us assume that .

We consider a formulation of equilibrium problems where the dependence of the production on the unknown solution is in the average sense with respect to the time; namely, the following feasibility condition holds:

Hence, condition (3) states that the average of the quantity produced by each firm , in the time interval , must be equal to the commodity shipments from that firm to all the demand markets plus the production excess, at the time . In fact, the production is supposed to depend on the firms' evaluation of the commodity shipments. So one can expect the producers not to evaluate the market practicability instantly, but by an average with respect to the whole time interval.

The set of feasible vectors is then given by the set-valued map defined as:

Moreover, let us associate with each firm a production cost ,, and assume that the production cost of a firm may depend upon the entire production pattern; namely,

Similarly, let us associate with each demand market a demand price for unity of the commodity , , and assume that the demand price of a demand market may depend, in general, upon the entire consumption pattern; namely,

Let , , denote the storage cost of the commodity produced by the firm and assume that this cost may depend upon the entire production pattern; namely,

Finally, let , and , denote the transaction cost, which includes the transportation cost associated with trading the commodity between firm and demand market . Here we permit the transaction cost to depend upon the entire shipment pattern; namely,

Hence, we have the following mappings:

The profit , , of the firm at the time is, then,

namely, it is equal to the price that the demand markets are disposed to pay minus the production costs, the storage costs, and the transportation costs.

By virtue of (3), we can express the nonnegative production excess at the time in terms of the integral average of the production function and the commodity shipment. As a consequence, we get

Then, the production costs and the storage costs, by virtue of (5) and (7), respectively, become

and, analogously, the profit (10) becomes

As a consequence, the set of feasible vectors becomes the set-valued map , defined as

Let us denote , , and . Let us assume the following assumptions:

(i) is continuously differentiable for each , a.e. in , (ii) is a Carathéodory function, such that

(iii) is pseudoconcave with respect to the variables , , a.e. in .

For the reader's convenience, we recall that a function , continuously differentiable, is called pseudoconcave with respect to , , a.e. in (see [31]), if the following holds a.e. in :

Moreover, we recall that in the Hilbert space , we define the canonical bilinear form on by

where ,, and

Now, let us consider the dynamic oligopolistic market, in which the firms supply the commodity in a noncooperative fashion, each one trying to maximize its own profit at the time . We seek to determine a nonnegative commodity distribution matrix function for which the firms and the demand markets will be in a state of equilibrium according to the dynamic Cournot-Nash principle.

Definition 1. is a dynamic oligopolistic market equilibrium in presence of production excesses if and only if for each and a.e. in

where and , for , a.e. in .

Definition 1 states that each firm maximizes its own profit, at the time , considering the given optimal strategy of the other firms.

Theorem 2. Suppose that assumptions (i), (ii), and (iii) are satisfied. Then, is a dynamic oligopolistic market equilibrium according to Definition 1 if and only if it satisfies the evolutionary quasi-variational inequality

Proof. First of all, let us prove that the evolutionary quasi-variational inequality (20), that we can write as follows:

is equivalent to the following point-to-point quasi-variational inequality:

where

In fact, let us suppose by absurdum that (22) does not hold, namely, , with such that

Let us choose, now,

Hence, let us consider

that is a contradiction. The vice versa is immediate.

So the equivalence between the evolutionary quasi-variational inequalities (20) and (22) is proved.

Let us prove, now, the equivalence between the dynamic Cournot-Nash principle and the evolutionary quasi-variational inequality (20).

Let us suppose that is an equilibrium point according to Definition 1; namely,

For well known theorems of optimization, we have that the necessary and sufficient condition to get (27) is that for all , for all , a.e. in

By assumption that is a Carathéodory function such that

moreover, and , so we have

Then, we get

from which, by summing up each firm , for , we obtain

Vice versa, let us suppose that is a solution to evolutionary quasi-variational inequality (20), but not an equilibrium solution according to the dynamic Cournot-Nash principle, namely, with , and such that

Since the profit function is pseudoconcave with respect to , we get

If we choose such that

then

so we get the contradiction.

3. An Existence Theorem for Equilibrium Solutions

Now, we prove an existence result for the equilibrium solution to the dynamic elastic oligopolistic market equilibrium problem. To this aim, we recall a general existence result for solutions to quasi-variational inequalities in topological linear locally convex Hausdorff spaces due to Tan [32].

Theorem 3. Let be a topological linear locally convex Hausdorff space and let be a convex compact nonempty subset of . Let be an upper semicontinuous multivalued mapping with , , convex compact nonempty, let be a closed lower semicontinuous multivalued mapping with , , convex compact nonempty, and let be a proper convex lower semicontinuous function. Then, there exists such that:

(i), (ii)there exists for which

Now, we are able to prove our main result.

Theorem 4. Let and be two vector functions such that assumptions (i) and (iii) are satisfied and

(I) is measurable in , for all , continuous in , a.e. in , such that , for all , a.e. in ;(II) is measurable in , for all , continuous in , a.e. in , such that , for all , a.e. in ;(III), a.e. in , such that

Then, evolutionary quasi-variational inequality (20) admits a solution.

Proof. At first, observe that under the hypotheses (I) and (II) and if ,

Moreover, by (I) and (II) it follows that and belong to the class of nemytskii operators. Therefore if is a sequence such that , in , we have

where the functions and are and -continuous, respectively.

Now, in order to show that is a closed multifunction, we prove that the following condition holds. For every two arbitrary sequences and such that and in , with , , then . To this aim, let us consider two arbitrary convergent sequences in , and to and , respectively. Since , , for , , and a.e. in , and the convergence of the sequence in implies that also satisfies the capacity constraints.

Moreover, the following relationship holds:

The left-hand side converges almost everywhere to ; for the right-hand side, meanwhile, , we have

By considering that the convergence of in implies the convergence even in , hence the sequence converges uniformly to in .

Now, let us show the lower semicontinuity of the multifunction . To this aim it suffices to prove that for every such that , in , and for every , there exists a sequence such that , in , with , for all .

Let be an arbitrary sequence such that , in , and let . Let us note that, for and , and if

we obtain, by virtue of the uniform convergence of to in , that

As a consequence, there exists an index such that for one has, for all , for all ,

Then, we consider the sequence such that

(i) for , for all , for all ,

(ii) for , for all , for all ,

where denotes the Hilbertian projection on .

It is easy to verify that if , for (47), . Instead, for , since for (45), , for all , for all , a.e. in ,

Moreover, since , for all , for all , a.e. in , we have

Finally, we get

Hence, we can conclude that belongs to , for all .

Let us prove now the convergence of to in . Let us observe that

As a consequence, we have that the sequence converges to . It is easy to show that is a closed, bounded, and convex subset of and since the space is compact, , for all , is compact too. As a consequence, all the hypotheses of Theorem 3 are satisfied and the existence of at least one solution is guaranteed.

4. Regularity Results for Equilibrium Solutions

In this section, we study the assumptions under which the continuity of solutions to evolutionary quasi-variational inequality, which expresses the equilibrium condition for the dynamic elastic oligopolistic market equilibrium problem in presence of production excesses, is ensured.

4.1. Set Convergence

First of all, we recall the notion of Kuratowski's set convergence that has an important role in order to establish regularity results. The classical notion of convergence for subsets of a given metric space is introduced in the 1950s by Kuratowski (see [33]; see also [34, 35]).

Let be a sequence of subsets of . Recall that

where eventually means that there exists such that for any and frequently means that there exists an infinite subset such that for any (in this last case, according to the notation given above, we also write that there exists a subsequence such that ).

In the following, we recall Kuratowski's set convergence.

Definition 5. We say that converges to some subset in Kuratowski's sense and we briefly write , if . Thus, in order to verify that , it suffices to check that

(i), that is, for any sequence frequently in such that for some , then ;(ii), that is, for any there exists a sequence eventually in such that .

The below lemma establishes that the feasible set of the dynamic elastic oligopolistic market equilibrium problem in the presence of production excesses satisfies the property of Kuratowski's set convergence.

Lemma 6. Let , let be such that

and let be a sequence such that , with , as . Then, the sequence of sets

for all , converges to

as , in Kuratowski's sense.

Proof. Firstly, we prove condition (K1). Let be a sequence such that , with as . Making use of the continuity assumptions on , , and , we get , , and as , respectively. Furthermore,

for and. Since and , then we have for and

and by virtue of the continuity of with respect to the first variable we also obtain

for and . Taking into account a well known generalization of Lebesgue's theorem,

for every .

Let be fixed and let us note that, for and , and if

we have

As a consequence, there exists an index such that for , for all , for all ,

As a consequence, we consider the sequence such that

(i) for , for all , for all ,

(ii) for , for all , for all ,

where denotes the Hilbertian projection on .

Obviously if , for (64) we have . Instead, for , since for (62), , for all , for all , we get

Moreover, since , for all , for all , we have

Since

we have

Then, taking into account (68), we obtain

Hence, , for all , and

Then, the proof of condition (K1) is completed.

Let us prove, now, condition (K2). Let be a sequence such that , with , as . Let be a sequence, such that , for all , and converging to , as . We have to prove that .

Since , for all ,

Passing to the limit in (71) as and taking into account the continuity assumption on the functions , , and , we have

Now, passing to the limit for in the left-hand side of (72), we have

Then, from (74) and (59), we obtain

As a consequence,

and, hence, condition (K2) is achieved.

4.2. Continuity of Solutions to Weighted Quasi-Variational Inequalities

In [2, 36–39] some continuity results for variational and quasi-variational inequalities in infinite dimensional spaces have been obtained. It is worth remarking that similar results have been proved for weighted variational and quasi-variational inequalities in nonpivot Hilbert spaces (see [4, 40]).

Now, we show a continuity result for equilibrium solutions to the dynamic elastic oligopolistic market equilibrium problem in presence of production excesses.

Theorem 7. Let , and let be such that

Moreover, let be a vector function satisfying assumption (iii) and such that

Then the dynamic elastic market equilibrium distribution in presence of production excesses is continuous in .

Proof. The existence of equilibrium solution is ensured by Theorem 4. Moreover, by applying Theorem 8 in [2] and taking into account Lemma 6, we obtain the continuity of in .

5. A Sensitivity Result

In this section a theorem about the sensitivity of solution is presented. The following result establishes that a small change in profit function produces a small change in equilibrium distribution.

Theorem 8. Assume that the profit function changes from to the perturbed function and denote by and the correspondent solutions of the following quasi-variational inequalities:

Let be a strongly monotone function of constant , namely, for all , such that

Moreover, let be a Carathéodory function such that

Then, it follows that

Proof. Choosing in (79) and in (80), by summing up the two new inequalities, we have

By adding and subtracting in (84), we have

Moreover, by using the strong monotonicity, inequality (85), and the Cauchy-Schwartz inequality, we get

from which we get

6. A Numerical Example

This section is devoted to provide a numerical example of the theoretical achievements presented.

Let us consider two firms and two demand markets, as in Figure 1. Let be the capacity constraints such that, a.e. in ,

Let us denote

Let be the production function, such that, a.e. in ,

Figure 1: Network structure of the numerical dynamic spatial oligopoly problem.

As a consequence, the feasible set is the set value function defined by

Let us consider the profit function given by

Then, the operator is given by

The dynamic oligopolistic market equilibrium distribution in presence of the excesses is the solution to the evolutionary quasi-variational inequality:

Let us observe that all the hypotheses of Theorem 4 are satisfied; hence, evolutionary quasi-variational inequality (94) admits solutions.

In order to compute a solution to (94) we make use of the direct method (see [41]). We consider the following system:

and we get the following solution, a.e. in ,

Let us observe that the solution satisfies all the constraints; in particular, if we compute

we are able to obtain the related production excesses:

7. Conclusions

In [1] the dynamic oligopolistic market equilibrium problem was studied by introducing production excesses, and the dynamic Cournot-Nash equilibrium was characterized as a solution to a suitable evolutionary variational inequality. In this paper, in order to have a model closer to reality, it was supposed that the production function depends on the equilibrium commodity shipment. Hence, an elastic formulation was introduced that leads to an equivalent formulation by means of a suitable evolutionary quasi-variational inequality. By means of this mathematical formulation, results of existence and regularity of solutions were proved. Furthermore, a sensitivity analysis is provided. At last a numerical example was provided in order to clarify the theoretical results. In future work, it is possible to consider also demand excesses and elastic demand function, in order to have a more complete and realistic model.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work has been partially supported by F.A.R.O. 2012 “Metodi matematici per la modellizzazione di fenomeni naturali.”

References

A. Barbagallo and P. Mauro, “Evolutionary variational formulation for oligopolistic market equilibrium problems with production excesses,” Journal of Optimization Theory and Applications, vol. 155, no. 1, pp. 288–314, 2012.

A. Barbagallo, “Regularity results for evolutionary nonlinear variational and quasi-variational inequalities with applications to dynamic equilibrium problems,” Journal of Global Optimization, vol. 40, no. 1-3, pp. 29–39, 2008.

L. Scrimali, “Quasi-variational inequalities in transportation networks,” Mathematical Models & Methods in Applied Sciences, vol. 14, no. 10, pp. 1541–1560, 2004.

A. Barbagallo and S. Pia, “Weighted quasi-variational inequalities in non-pivot Hilbert spaces and applications,” accepted on Journal of Optimization Theory and Applications

A. Cournot, “Researches into the mathematical principles of the theory of wealth,” Competition Policy International, vol. 4, no. 1, pp. 283–305, 2008.

J. F. Nash, Jr., “Equilibrium points in -person games,” Proceedings of the National Academy of Sciences of the United States of America, vol. 36, pp. 48–49, 1950.

J. Nash, “Non-cooperative games,” Annals of Mathematics. Second Series, vol. 54, pp. 286–295, 1951.

S. Dafermos and A. Nagurney, “Oligopolistic and competitive behavior of spatially separated markets,” Regional Science and Urban Economics, vol. 17, no. 2, pp. 245–254, 1987.

A. Nagurney, “Algorithms for oligopolistic market equilibrium problems,” Regional Science and Urban Economics, vol. 18, no. 3, pp. 425–445, 1988.

A. Nagurney, P. Dupuis, and D. Zhang, “A dynamical systems approach for network oligopolies and variational inequalities,” The Annals of Regional Science, vol. 28, no. 3, pp. 263–283, 1994.

A. Nagurney, Network Economics: A Variational Inequality Approach, vol. 1 of Advances in Computational Economics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.

A. Barbagallo and M.-G. Cojocaru, “Dynamic equilibrium formulation of the oligopolistic market problem,” Mathematical and Computer Modelling, vol. 49, no. 5-6, pp. 966–976, 2009.

M. J. Beckmann and J. P. Wallace, “Continuous lags and the stability of market equilibrium,” Economica, vol. 36, pp. 58–68, 1969.

A. Barbagallo and A. Maugeri, “Memory term for dynamic oligopolistic market equilibrium problem,” Aplimat—Journal of Applied Mathematics, vol. 3, no. 13, 23 pages, 2010.

A. Barbagallo and R. Di Vincenzo, “Lipschitz continuity and duality for dynamic oligopolistic market equilibrium problem with memory term,” Journal of Mathematical Analysis and Applications, vol. 382, no. 1, pp. 231–247, 2011.

A. Barbagallo and A. Maugeri, “Duality theory for the dynamic oligopolistic market equilibrium problem,” Optimization, vol. 60, no. 1-2, pp. 29–52, 2011.

P. Daniele, S. Giuffrè, G. Idone, and A. Maugeri, “Infinite dimensional duality and applications,” Mathematische Annalen, vol. 339, no. 1, pp. 221–239, 2007.

P. Daniele and S. Giuffrè, “General infinite dimensional duality and applications to evolutionary network equilibrium problems,” Optimization Letters, vol. 1, no. 3, pp. 227–243, 2007.

P. Daniele, S. Giuffré, and A. Maugeri, “Remarks on general infinite dimensional duality with cone and equality constraints,” Communications in Applied Analysis for Theory and Applications, vol. 13, no. 4, pp. 567–577, 2009.

A. Maugeri and F. Raciti, “Remarks on infinite dimensional duality,” Journal of Global Optimization, vol. 46, no. 4, pp. 581–588, 2010.

M. B. Donato, “The infinite dimensional Lagrange multiplier rule for convex optimization problems,” Journal of Functional Analysis, vol. 261, no. 8, pp. 2083–2093, 2011.

J. M. Borwein and A. S. Lewis, “Partially finite convex programming. I. Quasi relative interiors and duality theory,” Mathematical Programming, vol. 57, no. 1, pp. 15–48, 1992.

J. Jahn, Introduction to the Theory of Nonlinear Optimization, Springer, Berlin, Germany, 2nd edition, 1996.

M. B. Donato, A. Maugeri, M. Milasi, and C. Vitanza, “Duality theory for a dynamic Walrasian pure exchange economy,” Pacific Journal of Optimization, vol. 4, no. 3, pp. 537–547, 2008.

M. B. Donato and M. Milasi, “Lagrangean variables in infinite dimensional spaces for a dynamic economic equilibrium problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 15, pp. 5048–5056, 2011.

A. Barbagallo and P. Mauro, “Time-dependent variational inequality for an oligopolistic market equilibrium problem with production and demand excesses,” Abstract and Applied Analysis, vol. 2012, Article ID 651975, 35 pages, 2012.

A. Barbagallo and P. Mauro, “On solving dynamic oligopolistic market equilibrium problems in presence of excesses,” Communications in Applied and Industrial Mathematics, vol. 3, no. 1, pp. 1–20, 2012.

A. Barbagallo, “Advanced results on variational inequality formulation in oligopolistic market equilirbrium problem,” Filomat, vol. 5, pp. 935–947, 2012.

A. Barbagallo and P. Mauro, “Inverse variational inequality approach and applications,” submitted

A. Barbagallo and P. Mauro, “An inverse problem for the dynamic oligopolistic market equilibrium problem in presence of excesses,” accepted on Procedia—Social and Behavioral Sciences

O. L. Mangasarian, “Pseudo-convex functions,” Journal of the Society for Industrial and Applied Mathematics, Series A Control, vol. 3, pp. 281–290, 1965.

N. X. Tan, “Quasi-variational inequality in topological linear locally convex Hausdorff spaces,” Mathematische Nachrichten, vol. 122, pp. 231–245, 1985.

K. Kuratowski, Topology. Vol. I, Academic Press, New York, NY, USA, 1966.

G. Salinetti and R. J.-B. Wets, “On the convergence of sequences of convex sets in finite dimensions,” SIAM Review, vol. 21, no. 1, pp. 18–33, 1979.

G. Salinetti and R. J.-B. Wets, “Addendum “On the convergence of convex sets infinite dimensions”,” SIAM Review, vol. 22, p. 86, 1980.

A. Barbagallo, “Regularity results for time-dependent variational and quasi-variational inequalities and application to the calculation of dynamic traffic network,” Mathematical Models & Methods in Applied Sciences, vol. 17, no. 2, pp. 277–304, 2007.

A. Barbagallo, “Existence and regularity of solutions to nonlinear degenerate evolutionary variational inequalities with applications to dynamic network equilibrium problems,” Applied Mathematics and Computation, vol. 208, no. 1, pp. 1–13, 2009.

A. Barbagallo, “On the regularity of retarded equilibrium in time-dependent traffic equilibrium problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e2406–e2417, 2009.

A. Barbagallo and M.-G. Cojocaru, “Continuity of solutions for parametric variational inequalities in Banach space,” Journal of Mathematical Analysis and Applications, vol. 351, no. 2, pp. 707–720, 2009.

A. Barbagallo and S. Pia, “Weighted variational inequalities in non-pivot Hilbert spaces with applications,” Computational Optimization and Applications, vol. 48, no. 3, pp. 487–514, 2011.

A. Maugeri, “Convex programming, variational inequalities, and applications to the traffic equilibrium problem,” Applied Mathematics and Optimization, vol. 16, no. 2, pp. 169–185, 1987.