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

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 𝕂^∗(𝑥^∗) 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 𝑛 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 𝑚 firms 𝑃_𝑖, 𝑖=1,…,𝑚, that produce a homogeneous commodity and 𝑛 demand markets 𝑄_𝑗, 𝑗=1,…,𝑛, that are generally spatially separated. Assume that the homogeneous commodity, produced by the 𝑚 firms and consumed by the 𝑛 markets, is involved during a time interval [0,𝑇], 𝑇>0.

Let 𝑥_𝑖𝑗(𝑡), 𝑖=1,…,𝑚 and 𝑗=1,…,𝑛, denote the nonnegative commodity shipment between the supply market 𝑃_𝑖 and the demand market 𝑄_𝑗 at the time 𝑡∈[0,𝑇]. In particular, let us set the vector 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),…,𝑥_𝑖𝑛(𝑡)), 𝑖=1,…,𝑚 and 𝑡∈[0,𝑇], as the strategy vector for the firm 𝑃_𝑖.

Let us group the commodity shipments into a matrix function 𝑥∶[0,𝑇]→ℝ₊^𝑚𝑛 and suppose that 𝑥∈𝐿²([0,𝑇],ℝ₊^𝑚𝑛). Furthermore, we assume that the nonnegative commodity shipment 𝑥_𝑖𝑗 between the producer 𝑃_𝑖 and the demand market 𝑄_𝑗 has to satisfy time-dependent constraints, namely, there exist two nonnegative functions 𝑥,𝑥∶[0,𝑇]→ℝ₊^𝑚𝑛 such that 0≤𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡)[],,∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛,a.e.in0,𝑇 (1) and suppose that 𝑥,𝑥∈𝐿²([0,𝑇],ℝ₊^𝑚𝑛).

Let us denote 𝐷=𝑥∈𝐿²[0,𝑇],ℝ^𝑚𝑛∶𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡),[.∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛,a.e.in0,𝑇] (2) It is easy to verify that 𝐷 is a nonempty, compact, and convex subset of 𝐿²([0,𝑇],ℝ^𝑚𝑛). Let 𝑝_𝑖(𝑡,𝑥(𝑡)), 𝑖=1,…,𝑚, denote the nonnegative commodity output produced by firm 𝑃_𝑖 at the time 𝑡∈[0,𝑇]. Let us group the production output into a vector function 𝑝∶[0,𝑇]×𝐷→ℝ^𝑚₊ and let us suppose that 𝑝∈𝐿¹([0,𝑇]×𝐷,ℝ^𝑚₊).

Now, let us introduce the production excesses. Let 𝜖_𝑖(𝑡), 𝑖=1,…,𝑚, be the nonnegative production excess for the commodity of the firm 𝑃_𝑖 at the time 𝑡∈[0,𝑇]. Let us group the production excess into a vector function 𝜖∶[0,𝑇]→ℝ^𝑚₊ and let us assume that 𝜖∈𝐿²([0,𝑇],ℝ^𝑚).

We consider a formulation of equilibrium problems where the dependence of the production on the unknown solution 𝑥^∗ is in the average sense with respect to the time; namely, the following feasibility condition holds: ^𝑛_𝑗=1𝑥_𝑖𝑗(𝑡)+𝜖_𝑖(𝑡)=1𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^∗(𝜏[].)𝑑𝜏,𝑖=1,…,𝑚,a.e.in0,𝑇 (3) Hence, condition (3) states that the average of the quantity produced by each firm 𝑃_𝑖, in the time interval [0,𝑇], must be equal to the commodity shipments from that firm to all the demand markets plus the production excess, at the time 𝑡∈[0,𝑇]. 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 (𝑥,𝜖)∈𝐿²([0,𝑇],ℝ^{𝑚𝑛+𝑚}) is then given by the set-valued map 𝕂∶𝐷→2^{𝐿2([0,𝑇],ℝ+𝑚𝑛+𝑚)} defined as: 𝕂^∗𝑥^∗=(𝑥,𝜖)∈𝐿²[0,𝑇],ℝ^{𝑚𝑛+𝑚}∶𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡)[],,∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛,a.e.in0,𝑇_𝑛𝑗=1𝑥_𝑖𝑗(𝑡)+𝜖_𝑖(𝑡)=1𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^∗(𝜏)[]𝑑𝜏,∀𝑖=1,…,𝑚,a.e.in0,𝑇,𝜖_𝑖(𝑡)[.≥0,∀𝑖=1,…,𝑚,a.e.in0,𝑇] (4)

Moreover, let us associate with each firm 𝑃_𝑖 a production cost 𝑓^∗_𝑖,𝑖=1,…,𝑚, and assume that the production cost of a firm 𝑃_𝑖 may depend upon the entire production pattern; namely, 𝑓^∗_𝑖=𝑓^∗_𝑖(𝑡,𝑥(𝑡),𝜀(𝑡)). (5) Similarly, let us associate with each demand market 𝑄_𝑗 a demand price for unity of the commodity 𝑑_𝑗, 𝑗=1,…,𝑛, and assume that the demand price of a demand market 𝑄_𝑗 may depend, in general, upon the entire consumption pattern; namely, 𝑑_𝑗=𝑑_𝑗(𝑡,𝑥(𝑡)). (6) Let 𝑔^∗_𝑖, 𝑖=1,…,𝑚, denote the storage cost of the commodity produced by the firm 𝑃_𝑖 and assume that this cost may depend upon the entire production pattern; namely, 𝑔^∗_𝑖=𝑔^∗_𝑖(𝑡,𝑥(𝑡),𝜀(𝑡)). (7) Finally, let 𝑐_𝑖𝑗, 𝑖=1,…,𝑚 and 𝑗=1,…,𝑛, denote the transaction cost, which includes the transportation cost associated with trading the commodity between firm 𝑃_𝑖 and demand market 𝑄_𝑗. Here we permit the transaction cost to depend upon the entire shipment pattern; namely, 𝑐_𝑖𝑗=𝑐_𝑖𝑗(𝑡,𝑥(𝑡)). (8) Hence, we have the following mappings: 𝑓^∗∶[0,𝑇]×𝐿²[0,𝑇],ℝ₊^𝑚𝑛×𝐿²[0,𝑇],ℝ^𝑚₊⟶𝐿²[0,𝑇],ℝ^𝑚₊,𝑑∶[0,𝑇]×𝐿²[0,𝑇],ℝ₊^𝑚𝑛⟶𝐿²[0,𝑇],ℝ^𝑛₊,𝑔^∗∶[0,𝑇]×𝐿²[0,𝑇],ℝ₊^𝑚𝑛×𝐿²[0,𝑇],ℝ^𝑚₊⟶𝐿²[0,𝑇],ℝ^𝑚₊,𝑐∶[0,𝑇]×𝐿²[0,𝑇],ℝ₊^𝑚𝑛⟶𝐿²[0,𝑇],ℝ₊^𝑚𝑛. (9) The profit 𝑣^∗_𝑖(𝑡,𝑥(𝑡),𝜀(𝑡)), 𝑖=1,…,𝑚, of the firm 𝑃_𝑖 at the time 𝑡∈[0,𝑇] is, then, 𝑣^∗_𝑖((𝑡)(𝑡=𝑡,𝑥,𝜀))_𝑛𝑗=1𝑑_𝑗(𝑡,𝑥(𝑡))𝑥_𝑖𝑗(𝑡)−𝑓^∗_𝑖(𝑡,𝑥(𝑡),𝜀(𝑡))−𝑔^∗_𝑖((𝑡)(𝑡−𝑡,𝑥,𝜀))_𝑛𝑗=1𝑐_𝑖𝑗(𝑡,𝑥(𝑡))𝑥_𝑖𝑗(𝑡); (10) 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 𝜖_𝑖(𝑡) at the time 𝑡∈[0,𝑇] in terms of the integral average of the production function and the commodity shipment. As a consequence, we get ^𝑛_𝑗=1𝑥_𝑖𝑗(𝑡)≤1𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^∗(𝜏[].)𝑑𝜏,∀𝑖=1,…,𝑚,a.e.in0,𝑇 (11) Then, the production costs and the storage costs, by virtue of (5) and (7), respectively, become 𝑓_𝑖(𝑡,𝑥(𝑡))=𝑓^∗_𝑖((𝑡)(𝑡,𝑔𝑡,𝑥,𝜖))_𝑖(𝑡,𝑥(𝑡))=𝑔^∗_𝑖(𝑡,𝑥(𝑡),𝜖(𝑡)), (12) and, analogously, the profit (10) becomes 𝑣_𝑖(𝑡,𝑥(𝑡))=𝑣^∗_𝑖((𝑡)(𝑡=𝑡,𝑥,𝜖))_𝑛𝑗=1𝑑_𝑗(𝑡,𝑥(𝑡))𝑥_𝑖𝑗(𝑡)−𝑓_𝑖(𝑡,𝑥(𝑡))−𝑔_𝑖(𝑡,𝑥(𝑡))−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡,𝑥(𝑡))𝑥_𝑖𝑗(𝑡). (13)

As a consequence, the set of feasible vectors 𝑥∈𝐿²([0,𝑇],ℝ^𝑚𝑛) becomes the set-valued map 𝕂∶𝐷→2^{𝐿2([0,𝑇],ℝ+𝑚𝑛)}, defined as 𝕂𝑥^∗=𝑥∈𝐿²[0,𝑇],ℝ₊^𝑚𝑛∶𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡)[],,∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛,a.e.in0,𝑇_𝑛𝑗=1𝑥_𝑖𝑗(𝑡)≤1𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^∗(𝜏)[.𝑑𝜏,∀𝑖=1,…,𝑚,a.e.in0,𝑇] (14)

Let us denote 𝑥_𝑖={𝑥_𝑖𝑗}_{𝑗=1,…,𝑛}, 𝑖=1,…,𝑚, and ∇_𝐷𝑣=(𝜕𝑣_𝑖/𝜕𝑥_𝑖𝑗)_{𝑖=1,…,𝑚𝑗=1,…,𝑛}. Let us assume the following assumptions: (i)𝑣_𝑖(𝑡,𝑥(𝑡)) is continuously differentiable for each 𝑖=1,…,𝑚, a.e. in [0,𝑇], (ii)∇_𝐷𝑣=(𝜕𝑣_𝑖/𝜕𝑥_𝑖𝑗)_{𝑖=1,…,𝑚𝑗=1,…,𝑛} is a Carathéodory function, such that ∃𝛾∈𝐿²([0,𝑇])∶∇_𝐷𝑣(𝑡,𝑥)_𝑚𝑛≤𝛾(𝑡)+‖𝑥‖_𝑚𝑛,∀𝑥∈ℝ^𝑚𝑛,a.e.in[0,𝑇], (15)(iii)𝑣_𝑖(𝑡,𝑥(𝑡)) is pseudoconcave with respect to the variables 𝑥_𝑖, 𝑖=1,…,𝑚, a.e. in [0,𝑇].For the reader's convenience, we recall that a function 𝑣, continuously differentiable, is called pseudoconcave with respect to 𝑥_𝑖, 𝑖=1,…,𝑚, a.e. in [0,𝑇] (see [31]), if the following holds a.e. in [0,𝑇]: ∇_𝐷𝑣_𝑖𝑡,𝑥₁,…,𝑥_𝑖,…,𝑥_𝑚,𝑥_𝑖−𝑦_𝑖=^𝑛_𝑗=1𝜕𝑣_𝑖(𝑡,𝑥)𝜕𝑥_𝑖𝑗𝑥_𝑖𝑗−𝑦_𝑖𝑗≥0⟹𝑣_𝑖𝑡,𝑥₁,…,𝑥_𝑖,…,𝑥_𝑚≥𝑣_𝑖𝑡,𝑥₁,…,𝑦_𝑖,…,𝑥_𝑚. (16)

Moreover, we recall that in the Hilbert space 𝐿²([0,𝑇],ℝ^𝑘), we define the canonical bilinear form on 𝐿²([0,𝑇],ℝ^𝑘)^∗×𝐿²([0,𝑇],ℝ^𝑘) by 𝜙,𝑤∶=^𝑇₀𝜙(𝑡),𝑤(𝑡)𝑑𝑡, (17) where 𝜙∈(𝐿²([0,𝑇],ℝ^𝑘))^∗=𝐿²([0,𝑇],ℝ^𝑘),𝑤∈𝐿²([0,𝑇],ℝ^𝑘), and 𝜙(𝑡),𝑤(𝑡)=^𝑘_𝑙=1𝜙_𝑙(𝑡)𝑤_𝑙(𝑡). (18)

Now, let us consider the dynamic oligopolistic market, in which the 𝑚 firms supply the commodity in a noncooperative fashion, each one trying to maximize its own profit at the time 𝑡∈[0,𝑇]. We seek to determine a nonnegative commodity distribution matrix function 𝑥^∗ for which the 𝑚 firms and the 𝑛 demand markets will be in a state of equilibrium according to the dynamic Cournot-Nash principle.

Definition 1. 𝑥^∗∈𝕂(𝑥^∗) is a dynamic oligopolistic market equilibrium in presence of production excesses if and only if for each 𝑖=1,…,𝑚 and a.e. in [0,𝑇]𝑣_𝑖𝑡,𝑥^∗(𝑡)≥𝑣_𝑖𝑡,𝑥_𝑖(𝑡),𝑥^∗_𝑖(𝑡),a.e.in[0,𝑇], (19) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),…,𝑥_𝑖𝑛(𝑡)) and 𝑥^∗_𝑖(𝑡)=(𝑥^∗₁(𝑡),…,𝑥^∗_𝑖−1(𝑡),𝑥^∗_𝑖+1(𝑡),…,𝑥^∗_𝑚(𝑡)), for 𝑖=1,…,𝑚, a.e. in [0,𝑇].

Definition 1 states that each firm 𝑃_𝑖 maximizes its own profit, at the time 𝑡∈[0,𝑇], considering the given optimal strategy 𝑥^∗_𝑖(𝑡) of the other firms.

Theorem 2. Suppose that assumptions (i), (ii), and (iii) are satisfied. Then, 𝑥^∗∈𝕂(𝑥^∗) is a dynamic oligopolistic market equilibrium according to Definition 1 if and only if it satisfies the evolutionary quasi-variational inequality ^𝑇₀^𝑚𝑛_𝑖=1𝑗=1−𝜕𝑣_𝑖𝑡,𝑥^∗(𝑡)𝜕𝑥_𝑖𝑗𝑥_𝑖𝑗(𝑡)−𝑥^∗_𝑖𝑗(𝑡𝑥)𝑑𝑡≥0∀𝑥∈𝕂^∗. (20)

Proof. First of all, let us prove that the evolutionary quasi-variational inequality (20), that we can write as follows: ^𝑇₀^𝑚𝑛_𝑖=1𝑗=1−𝜕𝑣_𝑖𝑡,𝑥^∗(𝑡)𝜕𝑥_𝑖𝑗𝑥_𝑖𝑗(𝑡)−𝑥^∗_𝑖𝑗(𝑡=)𝑑𝑡−∇_𝐷𝑣𝑥^∗,𝑥−𝑥^∗=^𝑇₀−∇_𝐷𝑣𝑡,𝑥^∗(𝑡),𝑥(𝑡)−𝑥^∗(𝑡𝑥)𝑑𝑡≥0∀𝑥∈𝕂^∗, (21) is equivalent to the following point-to-point quasi-variational inequality: −∇_𝐷𝑣𝑡,𝑥^∗(𝑡),𝑥(𝑡)−𝑥^∗(𝑡=_{𝑚𝑛𝑖=1𝑗=1})−𝜕𝑣_𝑖𝑡,𝑥^∗(𝑡)𝜕𝑥_𝑖𝑗𝑥_i𝑗(𝑡)−𝑥^∗_𝑖𝑗(𝑡)(𝑡)≥0∀𝑥∈𝕂𝑡,𝑥^∗,a.e.in[0,𝑇], (22) where 𝕂𝑡,𝑥^∗=𝑥(𝑡)∈ℝ₊^𝑚𝑛∶𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡),∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛,_𝑛𝑗=1𝑥_𝑖𝑗(𝑡)≤1𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^∗(𝜏).𝑑𝜏,∀𝑖=1,…,𝑚 (23) In fact, let us suppose by absurdum that (22) does not hold, namely, ∃𝑥(𝑡)∈𝕂(𝑥^∗), ∃𝐼⊆[0,𝑇] with 𝑚(𝐼)>0 such that −∇_𝐷𝑣𝑡,𝑥^∗(𝑡),𝑥(𝑡)−𝑥^∗(𝑡)<0a.e.in𝐼. (24) Let us choose, now, 𝑥(𝑡)=𝑥^∗(𝑡),in[0,𝑇]⧵𝐼,𝑥(𝑡),in𝐼. (25) Hence, let us consider −∇_𝐷𝑣𝑡,𝑥^∗(𝑡),𝑥−𝑥^∗=_{[0,𝑇]⧵𝐼}−∇_𝐷𝑣𝑡,𝑥^∗(𝑡),𝑥(𝑡)−𝑥^∗(𝑡)𝑑𝑡+_𝐼−∇_𝐷𝑣𝑡,𝑥^∗(𝑡),𝑥(𝑡)−𝑥^∗(𝑡)𝑑𝑡<0, (26) 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 𝑥^∗∈𝕂(𝑥^∗) is an equilibrium point according to Definition 1; namely, 𝑣_𝑖𝑡,𝑥^∗(𝑡)≥𝑣_𝑖𝑡,𝑥(𝑡),𝑥^∗(𝑡)∀𝑥(𝑡)∈𝕂𝑡,𝑥^∗,a.e.in[0,𝑇],∀𝑖=1,…,𝑚. (27) For well known theorems of optimization, we have that the necessary and sufficient condition to get (27) is that for all 𝑖=1,…,𝑚, for all 𝑥(𝑡)∈𝕂(𝑡,𝑥^∗), a.e. in [0,𝑇]−∇_𝐷𝑣_𝑖𝑡,𝑥^∗(𝑡),𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡=_𝑛𝑗=1)−𝜕𝑣_𝑖𝑡,𝑥^∗(𝑡)𝜕𝑥_𝑖𝑗𝑥_𝑖𝑗(𝑡)−𝑥^∗_𝑖𝑗(𝑡)≥0. (28) By assumption that ∇_𝐷𝑣_𝑖 is a Carathéodory function such that ∃𝛾∈𝐿²([0,𝑇])∶∇_𝐷𝑣_𝑖(𝑡,𝑥)_𝑚𝑛≤𝛾(𝑡)+‖𝑥‖_𝑚𝑛,∀𝑥∈ℝ^𝑚𝑛,a.e.in[0,𝑇], (29) moreover, 𝑥 and 𝑥^∗∈𝐿²([0,𝑇],ℝ^𝑚𝑛), so we have 𝑡⟶−∇_𝐷𝑣_𝑖𝑡,𝑥^∗(𝑡),𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)∈𝐿²([0,𝑇],ℝ). (30) Then, we get −∇_𝐷𝑣_𝑖𝑡,𝑥^∗(𝑡),𝑥_𝑖−𝑥^∗_𝑖≥0∀𝑥∈𝕂𝑥^∗, (31) from which, by summing up each firm 𝑃_𝑖, for 𝑖=1,…,𝑚, we obtain ^𝑚_𝑖=1−∇_𝐷𝑣_𝑖𝑡,𝑥^∗(𝑡),𝑥−𝑥^∗=−∇_𝐷𝑣𝑡,𝑥^∗(𝑡),𝑥_𝑖−𝑥^∗_𝑖≥0∀𝑥∈𝕂𝑥^∗. (32)
Vice versa, let us suppose that 𝑥^∗(𝑡) is a solution to evolutionary quasi-variational inequality (20), but not an equilibrium solution according to the dynamic Cournot-Nash principle, namely, ∃𝐼⊆[0,𝑇] with 𝑚(𝐼)>0, ∃𝑖∈{1,…,𝑚} and ∃𝑥_𝑖 such that 𝑣_𝑖𝑡,𝑥^∗(𝑡)<𝑣_𝑖𝑡,𝑥_𝑖(𝑡),𝑥^∗(𝑡)in𝐼. (33) Since the profit function 𝑣_𝑖(𝑡,𝑥(𝑡)) is pseudoconcave with respect to 𝑥_𝑖, we get −∇_𝐷𝑣_𝑖𝑡,𝑥^∗(𝑡),𝑥^∗_𝑖(𝑡)−𝑥_𝑖(𝑡)<0in𝐼. (34) If we choose 𝑥∈𝕂(𝑥^∗) such that 𝑥_𝑖(𝑡)=𝑥^∗_𝑖(𝑡)[]𝑥in0,𝑇⧵𝐼,∀𝑖=1,…,𝑚,_∗𝑖(𝑡)in𝐼,if𝑖≠𝑥𝑖,_𝑖in𝐼,if𝑖=𝑖, (35) then ^𝑇₀−∇_𝐷𝑣𝑡,𝑥^∗(𝑡),𝑥(𝑡)−𝑥^∗(𝑡=)𝑑𝑡_𝐼−∇_𝐷𝑣_𝑖𝑡,𝑥^∗(𝑡),𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)𝑑𝑡<0, (36) 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 𝑋 be a topological linear locally convex Hausdorff space and let 𝐷 be a convex compact nonempty subset of 𝑋. Let 𝐶∶𝐷→2^𝑋∗ be an upper semicontinuous multivalued mapping with 𝐶(𝑥), 𝑥∈𝐷, convex compact nonempty, let 𝕂∶𝐷→2^𝐷 be a closed lower semicontinuous multivalued mapping with 𝕂(𝑥), 𝑥∈𝐷, convex compact nonempty, and let 𝜑∶𝐷→ℝ be a proper convex lower semicontinuous function. Then, there exists 𝑥^∗∈𝐷 such that:(i)𝑥^∗∈𝕂(𝑥^∗), (ii)there exists 𝑦^∗∈𝕂(𝑥^∗) for which 𝑥−𝑥^∗,𝑦^∗+𝜑(𝑥)−𝜑𝑥^∗≥0,∀𝑥∈𝕂𝑥^∗. (37)

Now, we are able to prove our main result.

Theorem 4. Let 𝑣∶[0,𝑇]→ℝ^𝑚 and 𝑝∶[0,𝑇]→ℝ^𝑚 be two vector functions such that assumptions (i) and (iii) are satisfied and(I)∇_𝐷𝑣(𝑡,𝑥) is measurable in 𝑡, for all 𝑥∈ℝ₊^𝑚𝑛, continuous in 𝑥, a.e. in [0,𝑇], such that ∃𝛾∈𝐿²([0,𝑇])∶‖∇_𝐷𝑣(𝑡,𝑥)‖≤𝛾(𝑡)+‖𝑥‖, for all 𝑥∈ℝ₊^𝑚𝑛, a.e. in [0,𝑇];(II)𝑝(𝑡,𝑥) is measurable in 𝑡, for all 𝑥∈ℝ₊^𝑚𝑛, continuous in 𝑥, a.e. in [0,𝑇], such that ∃𝜙∈𝐿¹([0,𝑇])∶‖𝑝(𝑡,𝑥)‖≤𝜙(𝑡)+‖𝑥‖², for all 𝑥∈ℝ₊^𝑚𝑛, a.e. in [0,𝑇];(III)∃𝜈(𝑡)≥0, a.e. in [0,𝑇], 𝜂∈𝐿^∞([0,𝑇]) such that 𝑝𝑡,𝑥₁−𝑝𝑡,𝑥₂≤𝜂(𝑡)𝑥₁−𝑥₂,∀𝑥₁,𝑥₂∈ℝ^m𝑛₊,𝑎.𝑒.𝑖𝑛[0,𝑇]. (38)Then, evolutionary quasi-variational inequality (20) admits a solution.

Proof. At first, observe that under the hypotheses (I) and (II) and if 𝑥∈𝐿²([0,𝑇],ℝ₊^𝑚𝑛), 𝑡⟼∇_𝐷𝑣(𝑡,𝑥(𝑡))∈𝐿²[0,𝑇],ℝ₊^𝑚𝑛,𝑡⟼𝑝(𝑡,𝑥(𝑡))∈𝐿¹[0,𝑇],ℝ^𝑚₊. (39) Moreover, by (I) and (II) it follows that ∇_𝐷𝑣 and 𝑝 belong to the class of nemytskii operators. Therefore if {𝑥^𝑘}_𝑘∈ℕ is a sequence such that 𝑥^𝑘→𝑥, in 𝐿²([0,𝑇],ℝ^𝑚𝑛), we have ∇_𝐷𝑣𝑡,𝑥^𝑘−∇_𝐷𝑣(𝑡,𝑥)_𝐿2𝑝⟶0,𝑡,𝑥^𝑘−𝑝(𝑡,𝑥)_𝐿1⟶0, (40) where the functions ∇_𝐷𝑣 and 𝑝 are 𝐿²- and 𝐿¹-continuous, respectively.
Now, in order to show that 𝕂(𝑥^∗) is a closed multifunction, we prove that the following condition holds. For every two arbitrary sequences {𝑥^𝑘}_𝑘∈ℕ and {𝑦^𝑘}_𝑘∈ℕ such that 𝑥^𝑘→𝑥 and 𝑦^𝑘→𝑦 in 𝐿²([0,𝑇],ℝ^𝑚𝑛), with 𝑦^𝑘∈𝕂(𝑥^𝑘), ∀𝑛∈ℕ, then 𝑦∈𝕂(𝑥). To this aim, let us consider two arbitrary convergent sequences in 𝐿²([0,𝑇],ℝ^𝑚𝑛), {𝑥^𝑘}_𝑘∈ℕ and {𝑦^𝑘}_𝑘∈ℕ to 𝑥 and 𝑦, respectively. Since 𝑦^𝑘∈𝕂(𝑥^𝑘), 𝑥_𝑖𝑗(𝑡)≤𝑦^𝑘_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡), for 𝑖=1,…,𝑚, 𝑗=1,…,𝑛, and a.e. in [0,𝑇], and the convergence of the sequence {𝑦^𝑘}_𝑘∈ℕ in 𝐿²([0,𝑇],ℝ^𝑚𝑛) implies that also 𝑦 satisfies the capacity constraints.
Moreover, the following relationship holds: ^𝑚_𝑗=1𝑦^𝑘_𝑖𝑗(𝑡)≤1𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^𝑘(𝜏[].)𝑑𝜏,𝑖=1,…,𝑚,a.e.in0,𝑇 (41) The left-hand side converges almost everywhere to ∑^𝑚_𝑗=1𝑦_𝑖𝑗(𝑡); for the right-hand side, meanwhile, 𝑖=1,…,𝑚, we have sup_[]0,𝑇^𝑇₀𝑝𝑡,𝑥^𝑘(𝜏)𝑑𝜏−^𝑇₀𝑝(𝑡,𝑥(𝜏))𝑑𝜏≤sup_[]0,𝑇^𝑇₀𝑝𝑡,𝑥^𝑘(𝜏)((𝜏≤𝜂−𝑝𝑡,𝑥))𝑑𝜏_{𝐿∞([0,𝑇])𝑇0}𝑥^𝑘(𝜏)−𝑥(𝜏)𝑑𝜏. (42) By considering that the convergence of {𝑥^𝑘} in 𝐿² implies the convergence even in 𝐿¹, hence the sequence {(1/𝑇)∫^𝑇₀𝑝_𝑖(𝑡,𝑥^𝑘(𝜏))𝑑𝜏}_𝑘∈ℕ converges uniformly to (1/𝑇)∫^𝑇₀𝑝_𝑖(𝑡,𝑥(𝜏))𝑑𝜏 in 𝐿¹([0,𝑇],ℝ^𝑚).
Now, let us show the lower semicontinuity of the multifunction 𝕂. To this aim it suffices to prove that for every {𝑥^𝑘}_𝑘∈ℕ such that 𝑥^𝑘→𝑥, in 𝐿²([0,𝑇],ℝ^𝑚𝑛), and for every 𝑦∈𝕂(𝑥), there exists a sequence {𝑦^𝑘}_𝑘∈ℕ such that 𝑦^𝑘→𝑦, in 𝐿²([0,𝑇],ℝ^𝑚𝑛), with 𝑦^𝑘∈𝕂(𝑥^𝑘), for all 𝑘∈ℕ.
Let {𝑥^𝑘}_𝑘∈ℕ be an arbitrary sequence such that 𝑥^𝑘→𝑥, in 𝐿²([0,𝑇],ℝ^𝑚𝑛), and let 𝑦∈𝕂(𝑥). Let us note that, for 𝑖=1,…,𝑚 and 𝑗=1,…,𝑛, and if 𝑎^𝑘_𝑖𝑗(𝑡)=𝑦_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡)+1×𝑛𝑇_𝑇0𝑝_𝑖(𝑡,𝑥(𝜏))𝑑𝜏−^𝑇₀𝑝_𝑖𝑡,𝑥^𝑘(𝜏)𝑑𝜏, (43) we obtain, by virtue of the uniform convergence of {(1/𝑇)∫^𝑇₀𝑝_𝑖(𝑡,𝑥^𝑘(𝜏))𝑑𝜏}_𝑘∈ℕ to (1/𝑇)∫^𝑇₀𝑝_𝑖(𝑡,𝑥(𝜏))𝑑𝜏 in 𝐿¹([0,𝑇],ℝ^𝑚), that lim_𝑘→+∞𝑎^𝑘_𝑖𝑗(𝑡)=𝑦_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡)≥0,a.e.in[0,𝑇]. (44) As a consequence, there exists an index 𝜈 such that for 𝑘>𝜈 one has, for all 𝑖=1,…,𝑚, for all 𝑗=1,…,𝑛, 𝑎^𝑘_𝑖𝑗(𝑡)≥0,a.e.in[0,𝑇]. (45) Then, we consider the sequence {𝑦^𝑘}_𝑘∈ℕ such that(i) for 𝑘>𝜈, for all 𝑖=1,…,𝑚, for all 𝑗=1,…,𝑛, 𝑦^𝑘_𝑖𝑗(𝑡)=𝑥_𝑖𝑗(𝑡)+min𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡),𝑎^𝑘_𝑖𝑗(𝑡,[],)a.e.in0,𝑇 (46)(ii) for 𝑘≤𝜈, for all 𝑖=1,…,𝑚, for all 𝑗=1,…,𝑛, 𝑦^𝑘_𝑖𝑗(𝑡)=𝑃_{𝕂(𝑥𝑘)}𝑦_𝑖𝑗(𝑡),a.e.in[0,𝑇], (47)where 𝑃_{𝕂(𝑥𝑘)}(⋅) denotes the Hilbertian projection on 𝕂(𝑥^𝑘).
It is easy to verify that if 𝑘≤𝜈, for (47), 𝑦^𝑘∈𝕂(𝑥^𝑘). Instead, for 𝑘>𝜈, since for (45), min{𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡),𝑎^𝑘_𝑖𝑗(𝑡)}≥0, for all 𝑖=1,…,𝑚, for all 𝑗=1,…,𝑛, a.e. in [0,𝑇], 𝑦^𝑘_𝑖𝑗(𝑡)≥𝑥_𝑖𝑗(𝑡),∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛,a.e.in[0,𝑇]. (48) Moreover, since min{𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡),𝑎^𝑘_𝑖𝑗(𝑡)}≤𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡), for all 𝑖=1,…,𝑚, for all 𝑗=1,…,𝑛, a.e. in [0,𝑇], we have 𝑦^𝑘_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡),∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛,a.e.in[0,𝑇]. (49) Finally, we get ^𝑛_𝑗=1𝑦^𝑘_𝑖𝑗(𝑡)≤^𝑛_𝑗=1𝑦_𝑖𝑗(𝑡)+1𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^𝑘(𝜏−1)𝑑𝜏𝑇^𝑇₀𝑝_𝑖((𝜏≤1𝑡,𝑥))𝑑𝜏𝑇^𝑇₀𝑝_𝑖((𝜏1𝑡,𝑥))𝑑𝜏+𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^𝑘(𝜏)−1𝑑𝜏𝑇^𝑇₀𝑝_𝑖((𝜏=1𝑡,𝑥))𝑑𝜏𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^𝑘(𝜏)𝑑𝜏,∀𝑖=1,…,𝑚,a.e.in[0,𝑇]. (50) Hence, we can conclude that 𝑦^𝑘 belongs to 𝕂(𝑥^𝑘), for all 𝑘∈ℕ.
Let us prove now the convergence of {𝑦^𝑘}_𝑘∈ℕ to 𝑦 in 𝐿²([0,𝑇],ℝ^𝑚𝑛). Let us observe that lim_𝑘→+∞min𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡),𝑎^𝑘_𝑖𝑗(𝑡)=𝑦_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡),a.e.in[0,𝑇]. (51) As a consequence, we have that the sequence {𝑦^𝑘}_𝑘∈ℕ converges to 𝑦. It is easy to show that 𝕂(𝑥) is a closed, bounded, and convex subset of 𝐷 and since the space 𝐷 is compact, 𝕂(𝑥), for all 𝑥∈𝐷, 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 (𝑋,𝑑) is introduced in the 1950s by Kuratowski (see [33]; see also [34, 35]).

Let {𝕂_𝑛}_𝑛∈ℕ be a sequence of subsets of 𝑋. Recall that 𝑑−lim_𝑛𝕂_𝑛=𝑥∈𝑋∶∃𝑥_{𝑛𝑛∈ℕ}eventuallyin𝕂_𝑛suchthat𝑥⟶_𝑛^𝑑𝑥,𝑑−lim_𝑛𝕂_𝑛=𝑥∈𝑋∶∃𝑥_{𝑛𝑛∈ℕ}frequentlyin𝕂_𝑛suchthat𝑥⟶_𝑛^𝑑𝑥, (52) where eventually means that there exists 𝛿∈ℕ such that 𝑥_𝑛∈𝕂_𝑛 for any 𝑛≥𝛿 and frequently means that there exists an infinite subset 𝑁⊆ℕ such that 𝑥_𝑛∈𝕂_𝑛 for any 𝑛∈𝑁 (in this last case, according to the notation given above, we also write that there exists a subsequence {𝑥_𝑘𝑛}_𝑛∈ℕ⊆{𝑥_𝑛}_𝑛∈ℕ such that 𝑥_𝑘𝑛∈𝕂_𝑘𝑛).

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

Definition 5. We say that {𝕂_𝑛} converges to some subset 𝕂⊆𝑋 in Kuratowski's sense and we briefly write 𝕂_𝑛→𝕂, if 𝑑−lim_𝑛𝕂_𝑛=𝑑−lim_𝑛𝕂_𝑛=𝕂. Thus, in order to verify that 𝕂_𝑛→𝕂, it suffices to check that(i)𝑑−lim_𝑛𝕂_𝑛⊆𝕂, that is, for any sequence {𝑥_𝑛}_𝑛∈ℕ frequently in 𝕂_𝑛 such that 𝑥→−_𝑛^𝑑𝑥 for some 𝑥∈𝑆, then 𝑥∈𝕂;(ii)𝕂⊂𝑑−lim_𝑛𝕂_𝑛, that is, for any 𝑥∈𝕂 there exists a sequence {𝑥_𝑛}_𝑛∈ℕ eventually in 𝕂_𝑛 such that 𝑥→−_𝑛^𝑑𝑥.

The below lemma establishes that the feasible set 𝕂 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 𝑥,𝑥∈𝐶⁰([0,𝑇],ℝ₊^𝑚𝑛), let 𝑝∈𝐶⁰([0,𝑇]×ℝ₊^𝑚𝑛,ℝ^𝑚₊) be such that ∃𝜙∈𝐶⁰[0,𝑇],ℝ₊∶𝑝(𝑡,𝑥)≤𝜙(𝑡)+‖𝑥‖²,∀𝑥∈ℝ₊^𝑚𝑛,𝑖𝑛[0,𝑇], (53) and let {𝑡_𝑘}_𝑘∈ℕ be a sequence such that 𝑡_𝑘→𝑡, with 𝑡∈[0,𝑇], as 𝑘→+∞. Then, the sequence of sets 𝕂𝑡_𝑘,𝑥^∗=𝑥𝑡_𝑘∈ℝ^𝑚𝑛∶𝑥_𝑖𝑗𝑡_𝑘≤𝑥_𝑖𝑗𝑡_𝑘≤𝑥_𝑖𝑗𝑡_𝑘,∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛,_𝑛𝑗=1𝑥_𝑖𝑗𝑡_𝑘≤1𝑇^𝑇₀𝑝_𝑖𝑡_𝑘,𝑥^∗(𝜏,)𝑑𝜏,∀𝑖=1,…,𝑚 (54) for all 𝑘∈ℕ, converges to 𝕂𝑡,𝑥^∗=𝑥(𝑡)∈ℝ^𝑚𝑛∶𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡),∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛,_𝑛𝑗=1𝑥_𝑖𝑗(𝑡)≤1𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^∗(𝜏),𝑑𝜏,∀𝑖=1,…,𝑚 (55) as 𝑘→+∞, in Kuratowski's sense.

Proof. Firstly, we prove condition (K1). Let {𝑡_𝑘}_𝑘∈ℕ be a sequence such that 𝑡_𝑘→𝑡, with 𝑡∈[0,𝑇] as 𝑘→+∞. Making use of the continuity assumptions on 𝑥, 𝑥, and 𝑝, we get 𝑥(𝑡_𝑘)→𝑥(𝑡), 𝑥(𝑡_𝑘)→𝑥(𝑡), and 𝑝(𝑡_𝑘,𝑦)→𝑝(𝑡,𝑦) as 𝑘→+∞, respectively. Furthermore, ∃𝜙∈𝐶⁰[0,𝑇],ℝ₊∶𝑝𝑡,𝑥^∗(𝜏)≤𝜙(𝑡)+𝑥^∗(𝜏)², (56) for 𝑡∈[0,𝑇] and𝜏∈[0,𝑇]. Since 𝜙∈𝐶⁰([0,𝑇]) and 𝑥^∗∈𝐿²([0,𝑇],ℝ₊^𝑚𝑛), then we have for 𝑡∈[0,𝑇] and 𝜏∈[0,𝑇]𝑝𝑡,𝑥^∗(𝜏)≤𝜙(𝑡)+𝑥^∗(𝜏)²∈𝐿¹([0,𝑇]), (57) and by virtue of the continuity of 𝑝 with respect to the first variable we also obtain lim_𝑛→+∞𝑝𝑡_𝑛,𝑥^∗(𝜏)=𝑝𝑡,𝑥^∗(𝜏), (58) for 𝜏∈[0,𝑇] and 𝑥^∗∈𝐿²([0,𝑇],ℝ₊^𝑚𝑛). Taking into account a well known generalization of Lebesgue's theorem, lim_𝑛→+∞^𝑇₀𝑝𝑡_𝑛,𝑥^∗(𝜏)𝑑𝜏=^𝑇₀𝑝𝑡,𝑥^∗(𝜏)𝑑𝜏, (59) for every 𝑥^∗∈𝐿²([0,𝑇],ℝ₊^𝑚𝑛).
Let 𝑥(𝑡)∈𝕂(𝑡,𝑥^∗) be fixed and let us note that, for 𝑖=1,…,𝑚 and 𝑗=1,…,𝑛, and if 𝑎_𝑖𝑗𝑡_𝑘=𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗𝑡_𝑘+1×𝑛𝑇_𝑇0𝑝_𝑖𝑡_𝑘,𝑥^∗(𝜏)𝑑𝜏−^𝑇₀𝑝_𝑖𝑡,𝑥^∗(𝜏)𝑑𝜏, (60) we have lim_𝑘→+∞𝑎_𝑖𝑗𝑡_𝑘=𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡)≥0. (61) As a consequence, there exists an index 𝜈 such that for 𝑘>𝜈, for all 𝑖=1,…,𝑚, for all 𝑗=1,…,𝑛, 𝑎_𝑖𝑗𝑡_𝑘≥0. (62) As a consequence, we consider the sequence {𝑥(𝑡_𝑘)}_𝑘∈ℕ such that(i) for 𝑘>𝜈, for all 𝑖=1,…,𝑚, for all 𝑗=1,…,𝑛, 𝑥_𝑖𝑗𝑡_𝑘=𝑥_𝑖𝑗𝑡_𝑘+min𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡),𝑥_𝑖𝑗𝑡_𝑘−𝑥_𝑖𝑗𝑡_𝑘,𝑎_𝑖𝑗𝑡_𝑘, (63)(ii) for 𝑘≤𝜈, for all 𝑖=1,…,𝑚, for all 𝑗=1,…,𝑛, 𝑥_𝑖𝑗𝑡_𝑘=𝑃_{𝕂(𝑡𝑘,𝑥∗)}𝑥_𝑖𝑗(𝑡), (64)where 𝑃_{𝕂(𝑡𝑘,𝑥∗)}(⋅) denotes the Hilbertian projection on 𝕂(𝑡_𝑘,𝑥^∗).
Obviously if 𝑘≤𝜈, for (64) we have 𝑥(𝑡_𝑘)∈𝕂(𝑡_𝑘,𝑥^∗). Instead, for 𝑘>𝜈, since for (62), min{𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡),𝑥_𝑖𝑗(𝑡_𝑘)−𝑥_𝑖𝑗(𝑡_𝑘),𝑎_𝑖𝑗(𝑡_𝑘)}≥0, for all 𝑖=1,…,𝑚, for all 𝑗=1,…,𝑛, we get 𝑥_𝑖𝑗𝑡_𝑘≤𝑥_𝑖𝑗𝑡_𝑘,∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛. (65) Moreover, since min{𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡),𝑥_𝑖𝑗(𝑡_𝑘)−𝑥_𝑖𝑗(𝑡_𝑘),𝑎_𝑖𝑗(𝑡_𝑘)}≤𝑥_𝑖𝑗(𝑡_𝑘)−𝑥_𝑖𝑗(𝑡_𝑘), for all 𝑖=1,…,𝑚, for all 𝑗=1,…,𝑛, we have 𝑥_𝑖𝑗𝑡_𝑘≤𝑥_𝑖𝑗𝑡_𝑘,∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛. (66) Since min𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡),𝑥_𝑖𝑗𝑡_𝑘−𝑥_𝑖𝑗𝑡_𝑘,𝑎_𝑖𝑗𝑡_𝑘≤𝑎_𝑖𝑗𝑡_𝑘=𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗𝑡_𝑘+1𝑛𝑇^𝑇₀𝑝_𝑖𝑡_𝑘,𝑥^∗(𝜏)𝑑𝜏−^𝑇₀𝑝_𝑖𝑡,𝑥^∗(𝜏)𝑑𝜏∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛, (67) we have 𝑥_𝑖𝑗𝑡_𝑘≤𝑥_𝑖𝑗(𝑡)+1×𝑛𝑇_𝑇0𝑝_𝑖𝑡_𝑘,𝑥^∗(𝜏)𝑑𝜏−^𝑇₀𝑝_𝑖𝑡,𝑥^∗(𝜏)𝑑𝜏,∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛. (68) Then, taking into account (68), we obtain ^𝑛_𝑗=1𝑥_𝑖𝑗𝑡_𝑘≤^𝑛_𝑗=1𝑥_𝑖𝑗(𝑡)+1𝑇^𝑇₀𝑝_𝑖𝑡_𝑘,𝑥^∗(𝜏−1)𝑑𝜏𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^∗(𝜏)≤1𝑑𝜏𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^∗(𝜏)+1𝑑𝜏𝑇^𝑇₀𝑝_𝑖𝑡_𝑘,𝑥^∗(𝜏)−1𝑑𝜏𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^∗(𝜏)=1𝑑𝜏𝑇^𝑇₀𝑝_𝑖𝑡_𝑘,𝑥^∗(𝜏)𝑑𝜏,∀𝑖=1,…,𝑚. (69) Hence, 𝑥(𝑡_𝑘)∈𝕂(𝑡_𝑘,𝑥^∗), for all 𝑘∈ℕ, and lim_𝑘→+∞𝑥_𝑖𝑗𝑡_𝑘=𝑥_𝑖𝑗(𝑡)+min𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡),𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡),𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡)=𝑥_𝑖𝑗(𝑡)+𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗(𝑡)=𝑥_𝑖𝑗(𝑡). (70) Then, the proof of condition (K1) is completed.
Let us prove, now, condition (K2). Let {𝑡_𝑘}_𝑘∈ℕ be a sequence such that 𝑡_𝑘→𝑡, with 𝑡∈[0,𝑇], as 𝑘→+∞. Let {𝑥(𝑡_𝑘)}_𝑘∈ℕ be a sequence, such that 𝑥(𝑡_𝑘)∈𝕂(𝑡_𝑘,𝑥^∗), for all 𝑘∈ℕ, and converging to 𝑥(𝑡), as 𝑘→+∞. We have to prove that 𝑥(𝑡)∈𝕂(𝑡,𝑥^∗).
Since 𝑥(𝑡_𝑘)∈𝕂(𝑡_𝑘,𝑥^∗), for all 𝑘∈ℕ, 𝑥_𝑖𝑗𝑡_𝑘≤𝑥_𝑖𝑗𝑡_𝑘≤𝑥_𝑖𝑗𝑡_𝑘,∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛,∀𝑘∈ℕ,_𝑛𝑗=1𝑥_𝑖𝑗𝑡_𝑘≤1𝑇^𝑇₀𝑝_𝑖𝑡_𝑘,𝑥^∗(𝜏)𝑑𝜏,∀𝑖=1,…,𝑚,∀𝑘∈ℕ. (71)(72) Passing to the limit in (71) as 𝑛→+∞ and taking into account the continuity assumption on the functions 𝑥, 𝑥, and 𝑝, we have 𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡),∀𝑖=1,…,𝑚,∀𝑗=1,…,𝑛. (73) Now, passing to the limit for 𝑛→+∞ in the left-hand side of (72), we have lim^𝑛_{𝑛→+∞𝑗=1}𝑥_𝑖𝑗𝑡_𝑘=^𝑛_𝑗=1𝑥_𝑖𝑗(𝑡),∀𝑖=1,…,𝑚. (74) Then, from (74) and (59), we obtain ^𝑛_𝑗=1𝑥_𝑖𝑗(𝑡)≤1𝑇^𝑇₀𝑝_𝑖𝑡,𝑥^∗(𝜏)𝑑𝜏,∀𝑗=1,…,𝑛. (75)
As a consequence, 𝑥(𝑡)∈𝕂𝑡,𝑥^∗, (76) 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 𝑥,𝑥∈𝐶⁰([0,𝑇],ℝ₊^𝑚𝑛), and let 𝑝∈𝐶⁰([0,𝑇]×ℝ₊^𝑚𝑛,ℝ^𝑚₊) be such that ∃𝜙∈𝐶⁰([0,𝑇])∶𝑝(𝑡,𝑥)≤𝜙(𝑡)+‖𝑥‖²,∀𝑥∈ℝ^𝑚𝑛,𝑖𝑛[0,𝑇],∃𝜈∈𝐶⁰[0,𝑇],ℝ₊∶𝑝𝑡,𝑥₁−𝑝𝑡,𝑥₂≤𝜈𝑥₁−𝑥₂,∀𝑥₁,𝑥₂∈ℝ^𝑚𝑛,𝑖𝑛[0,𝑇]. (77) Moreover, let 𝑣∈𝐶¹([0,𝑇]×ℝ₊^𝑚𝑛,ℝ^𝑚₊) be a vector function satisfying assumption (iii) and such that ∃𝛾∈𝐶⁰([0,𝑇])∶∇_𝐷𝑣(𝑡,𝑥)≤𝛾(𝑡)+‖𝑥‖,∀𝑥∈ℝ^𝑚𝑛,𝑖𝑛[0,𝑇],∃𝜇>0∶−∇_𝐷𝑣(𝑡,𝑥)+∇_𝐷𝑣𝑡,𝑦,𝑥−𝑦≥𝜇𝑥−𝑦²,∀𝑥,𝑦∈ℝ^𝑚𝑛,𝑖𝑛[0,𝑇]. (78) Then the dynamic elastic market equilibrium distribution in presence of production excesses 𝑥^∗∈𝕂(𝑥^∗) is continuous in [0,𝑇].

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 𝑥^∗∈𝕂(𝑥^∗) in [0,𝑇].

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 𝑣(⋅) to the perturbed function 𝑣(̃⋅) and denote by 𝑥^∗ and 𝑥 the correspondent solutions of the following quasi-variational inequalities: −∇_𝐷𝑣𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝕂𝑥^∗,−∇_𝐷𝑣̃(𝑥),𝑥−𝑥≥0,∀𝑥∈𝕂𝑥^∗. (79)(80) Let ∇_𝐷𝑣(𝑡,𝑥) be a strongly monotone function of constant 𝛼, namely, for all 𝑥,𝑦∈𝕂(𝑥^∗), ∃𝛼>0 such that −∇_𝐷𝑣(𝑥)+−∇_𝐷𝑣𝑦,𝑥−𝑦≥𝛼𝑥−𝑦²_{𝐿2([]0,𝑇,ℝ𝑚𝑛)}. (81) Moreover, let ∇_𝐷𝑣 be a Carathéodory function such that ∃ℎ∈𝐿²([0,𝑇])∶∇_𝐷𝑣(𝑡,𝑥(𝑡))_𝑚𝑛≤ℎ(𝑡)+‖𝑥(𝑡)‖_𝑚𝑛,𝑎.𝑒.𝑖𝑛[0,𝑇]. (82) Then, it follows that 𝑥^∗−𝑥_{𝐿2([0,𝑇],ℝ𝑚𝑛)}≤1𝛼−∇_𝐷𝑣̃(𝑥)+∇_𝐷𝑣𝑥^∗_{𝐿2([0,𝑇],ℝ𝑚𝑛)}. (83)

Proof. Choosing 𝑥(𝑡)=𝑥(𝑡) in (79) and 𝑥(𝑡)=𝑥^∗(𝑡) in (80), by summing up the two new inequalities, we have −∇_𝐷𝑣̃(𝑥)+∇_𝐷𝑣𝑥^∗,𝑥^∗−𝑥≥0. (84) By adding and subtracting −∇_𝐷𝑣(𝑥) in (84), we have −∇_𝐷𝑣̃(𝑥)+∇_𝐷𝑣(𝑥),𝑥^∗−𝑥≥−∇_𝐷𝑣𝑥^∗+∇_𝐷𝑣(𝑥),𝑥^∗−𝑥. (85) Moreover, by using the strong monotonicity, inequality (85), and the Cauchy-Schwartz inequality, we get 𝛼𝑥^∗−𝑥²_{𝐿2([]0,𝑇,ℝ𝑚𝑛)}≤−∇_𝐷𝑣𝑥^∗+∇_𝐷𝑣(𝑥),𝑥^∗−𝑥≤−∇_𝐷𝑣̃(𝑥)+∇_𝐷𝑣(𝑥),𝑥^∗−𝑥≤−∇_𝐷𝑣̃(𝑥)+∇_𝐷𝑣(𝑥)_{𝐿2([0,𝑇],ℝ𝑚𝑛)}×𝑥^∗−𝑥_{𝐿2([0,𝑇],ℝ𝑚𝑛)}, (86) from which we get 𝑥^∗−𝑥_{𝐿2([0,𝑇],ℝ𝑚𝑛)}≤1𝛼−∇_𝐷𝑣̃(𝑥)+∇_𝐷𝑣𝑥^∗_{𝐿2([0,𝑇],ℝ𝑚𝑛)}. (87)

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 𝑥,𝑥∈𝐿²([0,1],ℝ⁴) be the capacity constraints such that, a.e. in [0,1], 𝑥(𝑡)=025𝑡12,𝑡0𝑥(𝑡)=10𝑡5𝑡12𝑡10𝑡. (88) Let us denote 𝐷=𝑥∈𝐿²[0,1],ℝ⁴∶𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡),∀𝑖=1,2,∀𝑗=1,2,a.e.in[0,1]. (89) Let 𝑝∈𝐿¹([0,1]×𝐷,ℝ²) be the production function, such that, a.e. in [0,1], 𝑝(𝑡)=6𝑡+2𝑥^∗₁₁(𝑡)3𝑡+2𝑥^∗₁₁(𝑡)+𝑥^∗₁₂(𝑡). (90)

Figure 1 

Network structure of the numerical dynamic spatial oligopoly problem.

As a consequence, the feasible set is the set value function 𝕂∶𝐷→2^{𝐿2([0,1],ℝ4)} defined by 𝕂𝑥^∗=𝑥∈𝐿²[0,1],ℝ⁴∶𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡),∀𝑖=1,2,∀𝑗=1,2,a.e.in[0,1_2𝑗=1],𝑥_𝑖𝑗(𝑡)≤¹₀𝑝_𝑖𝑡,𝑥^∗(𝜏)[.𝑑𝜏,∀𝑖=1,2,a.e.in0,1] (91) Let us consider the profit function 𝑣∈𝐶¹([0,1]×𝐷,ℝ²) given by 𝑣₁(𝑡,𝑥(𝑡))=−4𝑥²₁₁(𝑡)−2𝑥²₁₂(𝑡)−𝑥₁₁(𝑡)𝑥₁₂(𝑡)+6𝑡𝑥₁₁(𝑡)+3𝑡𝑥₁₂(𝑡),𝑣₂(𝑡,𝑥(𝑡))=−2𝑥²₂₁(𝑡)−5𝑥²₂₂(𝑡)−2𝑥₂₁(𝑡)𝑥₂₂(𝑡)+6𝑡𝑥₂₁(𝑡)+5𝑡𝑥₂₂(𝑡). (92)

Then, the operator ∇_𝐷𝑣∈𝐿²([0,1]×𝐷,ℝ⁴) is given by −∇_𝐷𝑣((𝑡=𝑡,𝑥))8𝑥₁₁(𝑡)+𝑥₁₂(𝑡)−6𝑡4𝑥₁₂(𝑡)+𝑥₁₁(𝑡)−3𝑡4𝑥₂₁(𝑡)+2𝑥₂₂(𝑡)−6𝑡10𝑥₂₂(𝑡)+2𝑥₂₁(𝑡).−5𝑡 (93) The dynamic oligopolistic market equilibrium distribution in presence of the excesses is the solution to the evolutionary quasi-variational inequality: ¹₀²²_𝑖=1𝑗=1−𝜕𝑣_𝑖𝑡,𝑥^∗(𝑡)𝜕𝑥_𝑖𝑗𝑥_𝑖𝑗(𝑡)−𝑥^∗_𝑖𝑗(𝑡𝑥)𝑑𝑡≥0,∀𝑥∈𝕂^∗. (94) 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: 8𝑥^∗₁₁(𝑡)+𝑥^∗₁₂(𝑡)−6𝑡=0,𝑥^∗₁₁(𝑡)+4𝑥^∗₁₂(𝑡)−3𝑡=0,4𝑥^∗₂₁(𝑡)+2𝑥^∗₂₂(𝑡)−6𝑡=0,2𝑥^∗₂₁(𝑡)+10𝑥^∗₂₂(𝑡)𝑥−5𝑡=0,_𝑖𝑗(𝑡)≤𝑥^∗_𝑖𝑗(𝑡)≤𝑥_𝑖𝑗(𝑡),∀𝑖=1,2,∀𝑗=1,2,a.e.in[0,1_2𝑗=1],𝑥^∗_𝑖𝑗(𝑡)≤¹₀𝑝_𝑖𝑡,𝑥^∗(𝜏)𝑑𝜏,∀𝑖=1,2,a.e.in[0,1], (95) and we get the following solution, a.e. in [0,1], 𝑥^∗(𝑡)=7𝑡3105𝑡25𝑡2189𝑡. (96) Let us observe that the solution satisfies all the constraints; in particular, if we compute ¹₀𝑝₁𝑡,𝑥^∗(𝜏)7𝑑𝜏=6𝑡+,10₁₀𝑝₂𝑡,𝑥^∗(𝜏)𝑑𝜏=3𝑡+1, (97) we are able to obtain the related production excesses: 𝜖(𝑡)=47𝑡+71025𝑡+1818. (98)

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.”