Research article

Traffic network analysis via multidimensional split variational inequality problem with multiple output sets

  • Received: 13 November 2023 Revised: 19 January 2024 Accepted: 24 January 2024 Published: 06 February 2024
  • In this study, we introduce and investigate a new class of split inverse problems, comprising a multidimensional parameter of evolution, which we call the multidimensional split variational inequality problem with multiple output sets. To demonstrate its applicability, we formulate the equilibrium flow of multidimensional traffic network models for an arbitrary number of locations. We define a multidimensional split Wardrop condition with multiple output sets and establish its equivalence with the formulated equilibrium flow of multidimensional traffic network models. We then establish the existence and uniqueness of equilibria for our proposed model. In addition, we propose a method for solving the introduced problem. We then validate our results using some numerical experiments.

    Citation: Timilehin O. Alakoya, Bidisha Ghosh, Salissou Moutari, Vikram Pakrashi, Ranganatha B. Ramachandra. Traffic network analysis via multidimensional split variational inequality problem with multiple output sets[J]. Networks and Heterogeneous Media, 2024, 19(1): 169-195. doi: 10.3934/nhm.2024008

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  • In this study, we introduce and investigate a new class of split inverse problems, comprising a multidimensional parameter of evolution, which we call the multidimensional split variational inequality problem with multiple output sets. To demonstrate its applicability, we formulate the equilibrium flow of multidimensional traffic network models for an arbitrary number of locations. We define a multidimensional split Wardrop condition with multiple output sets and establish its equivalence with the formulated equilibrium flow of multidimensional traffic network models. We then establish the existence and uniqueness of equilibria for our proposed model. In addition, we propose a method for solving the introduced problem. We then validate our results using some numerical experiments.



    The variational inequality problem was first introduced independently by Fichera [1] and Stampacchia [2] to model optimization problems arising from mechanics. The concept of multi-time has been employed in optimization theory, namely in the framework of multi-time optimal control problem. This problem is a particular case of the multidimensional variational problems. Several problems, in science and engineering, can be modelled in terms of optimization problems, which are governed by m-flow type partial differential equations (multi-time evolution systems) and cost functionals expressed as path-independent integrals or multiple integrals. Apart from optimization theory, the concept of multidimensional parameters of evolution has also been applied in space theory, where the space coordinates are represented by two-dimensional time parameters t=(t1,t2), where t1 and t2 represent the intrinsic time and the observer time, respectively. For more details and recent studies in this direction, interested readers are referred to the studies in [3,4,5] and the references therein.

    The study of variational inequality problems in finite dimensional spaces was initiated independently by Smith [6] and Dafermos [7]. They set up the traffic assignment problem in terms of a finite dimensional variational inequality problem (VIP). On the other hand, Lawphongpanich and Hearn [8], and Panicucci et al. [9] studied traffic assignment problems based on Wardrop user equilibrium principle via a variational inequality model.

    Lions and Stampacchia [10], and Brezis [11] independently introduced the time-dependent (evolutionary) variational inequality problem, and developed an existence and uniqueness theory of the problem. Daniele et al. [12] formulated a dynamic traffic network equilibrium problem in terms of an evolutionary variational inequality problem. Ever since then, several other economics related problems like Nash equilibrium problem, spatial price equilibrium problems, internet problems, dynamic financial equilibrium problems and environmental network and ecology problems have been studied via time-dependent variational inequality problem (see [13,14,15,16]).

    Censor et al. [17] introduced a new split inverse problem called the split variational inequality problem (SVIP). The authors proposed iterative methods for estimating the solution of the problem, they and analysed the convergence of the proposed iterative schemes. The SVIP has several areas of applications, including network problems, image reconstruction, cancer treatment planning and many more.

    Very recently, Singh et al. [18] introduced another split inverse problem, which they called evolutionary split variational inequality problem. The authors demonstrated the applicability of this new problem through the formulation of the equilibrium flow of dynamic traffic network models, which comprised two given cities. Moreover, they established the existence and uniqueness of equilibria for the proposed model.

    However, Singh in [19] noted that in an economic problem, other parameters in addition to time may also affect the values of the constraints and arguments associated with the problem. Similarly, in a traffic network problem the flow of traffic depends on several economic parameters other than the time parameter. For instance, traffic flow data are known to be strongly influenced by both space (location) and time. In addition, parameters related to road capacity, safety measures for averting road accidents and several other economic parameters could affect traffic flow. Based on this observation, Singh [19] introduced a new split inverse problem, called the multidimensional split variational inequality problem (MSVIP). This new problem includes a multidimensional parameter of evolution. As an application, the author formulated the equilibrium flow within two different traffic network models, e.g., traffic networks for two given cities.

    More recently, Alakoya and Mewomo [20] studied a new class of split inverse problems, known as split variational inequality problem with multiple output sets. This class of split inverse problems is designed such that multiple variational inequality problems are solved simultaneously. The authors proposed an iterative method for estimating the solution of this problem, and they further presented some numerical experiments to demonstrate the feasibility of the proposed iterative method.

    We note that the results of Singh et al. [18] and Singh [19] are only capable of dealing with two different traffic network models simultaneously. In other words, their results are not applicable when the goal is to study multiple (more than two) traffic network models simultaneously. Moreover, we also note that in formulating the split inverse problems introduced in [18,19], the authors needed to define explicitly two inverse problems (one in each of the two spaces under consideration) such that the image of the solution of the first inverse problem under a bounded linear operator is the solution of the second inverse problem. This method of formulation made the proofs of the results in [18,19] lengthy and not easily comprehensible. To overcome these shortcomings, in this study we introduce and study a new class of split inverse problems, which we call the multidimensional split variational inequality problem with multiple output sets. This newly introduced problem also includes a multidimensional parameter of evolution. Moreover, in formulating our problem we demonstrate that the inverse problems involved in the formulation need not to be explicitly defined. Instead, by introducing an index set our problem could be formulated succinctly and the proofs of the results presented more concisely. To demonstrate its applicability in the economic world, we formulate the equilibrium flow of multidimensional traffic network models for an arbitrary number of locations, e.g., traffic network models for different cities. Moreover, we define a multidimensional split Wardrop condition with multiple output sets (MSWC-MOS), and establish its equivalence with the formulated equilibrium flow of multidimensional traffic network models. Furthermore, we establish the existence and uniqueness of equilibria for our proposed model. We propose a method for solving the introduced problem, which will be useful in evaluating the equilibrium flow of multidimensional traffic network models for different cities simultaneously. Finally, we validate our results using some numerical experiments. To further illustrate the utilization of our newly introduced problem, we apply our results to study the network model of a city with heterogeneous networks. More precisely, we consider a city, which comprises connected automated vehicles (CAVs) and legacy (human-driven) vehicles, alongside electricity network, e.g. for charging the CAVs, and we formulate the equilibrium flow of this network model in terms of our newly introduced multidimensional split variational inequality problem with multiple output sets. We note that the results in [18,19] cannot be applied to the numerical examples and application considered in our study.

    In this section, we formulate our multidimensional split variational inequality problem with multiple output sets. First, we introduce some important notations and mathematical concepts, which are needed for the problem formulation. In what follows, except otherwise stated, the abbreviation "a.e." means "almost everywhere" and Rm+ denotes the set of non-negative vectors in Rm. We assume that our multidimensional traffic network model comprises a multi-parameter of evolution v, which is the multidimensional parameter of evolution, i.e., v=(vα)Ωv0,v1, where α=1,2,,m. Geometrically, Ωv0,v1 is a hyper-parallelepiped in Rm+ with the opposite diagonal points v0=(v10,v20,,vm0) and v1=(v11,v21,,vm1), which by the product order on Rm+ is equivalent to the closed interval v0vv1. Suppose that we have cities denoted by Ci,i=0,1,,M. The traffic network of each city Ci comprises the set of nodes Ni, representing railway stations, airports, crossings, etc., the set of directed links Li between the nodes, the set of origin-destination pairs Wi and the set of routes Vi. Moreover, it is assumed that each route riVi connects exactly one origin-destination pair. We denote by Vi(wi) the set of all riVi, which connects a given wiWi. Let xi(v)R|Vi| be the flow trajectory, and for each riVi, let xri(v) represent the flow trajectory of the route ri over the multidimensional parameter v. We take our functional setting for the flow trajectories to be the reflexive Banach space Lpi(Ωv0,v1,R|Vi|),pi>1, with the dual space Lqi(Ωv0,v1,R|Vi|), where 1pi+1qi=1,i=0,1,,M. We assume that every feasible flow satisfies the following multidimensional capacity constraints for each i=0,1,,M

    λi(v)xi(v)μi(v),a.e. onΩv0,v1,

    and the multidimensional traffic conservation law/demand requirements

    Φixi(v)=ρi(v),a.e. onΩv0,v1,

    where λi(v),μi(v)Lpi(Ωv0,v1,R|Vi|) are given bounds such that λi(v)μi(v) and ρi(v)Lpi(Ωv0,v1,R|Wi|) is the given demand such that ρi(v)0, and Φi=(ϕri,wi) is the pair-route incidence matrix, whose entries are equal to 1 if route ri links the pair wi and 0 otherwise. It is also assumed that

    Φiλi(v)ρi(v)Φiμi(v),a.e. onΩv0,v1.

    This assumption implies the non-emptiness of the set of feasible flows

    Ki={xi(v)Lpi(Ωv0,v1,R|Vi|):λi(v)xi(v)μi(v)andΦixi(v)=ρi(v),a.e. onΩv0,v1,i=0,1,,M}.

    The canonical bilinear form on Lqi(Ωv0,v1,R|Vi|)×Lpi(Ωv0,v1,R|Vi|) is defined as

    fi(v),xi(v)Ci=Ωv0,v1fi(v),xi(v)dv,xi(v)Lpi(Ωv0,v1,R|Vi|)

    and

    fi(v)Lqi(Ωv0,v1,R|Vi|),  i=0,1,,M,

    where , denotes the Euclidean inner product and dv=dv1dv2dvm denotes the volume element of Ωv0,v1.

    Remark 1. It is clear that for each i=0,1,,M, the feasible set Ki is closed, convex and bounded. From this, it follows that each Ki is weakly compact.

    Moreover, for each xi(v)Ki,i=0,1,,M, the cost trajectory is denoted by the mapping Ai:KiLqi(Ωv0,v1,R|Vi|), and we let Ti:Lp0(Ωv0,v1,R|V0|)Lpi(Ωv0,v1,R|Vi|),i=0,1,,M be bounded linear operators, where T0=ILp0(Ωv0,v1,R|V0|) is the identity operator on Lp0(Ωv0,v1,R|V0|).

    Now, we formulate our multidimensional split variational inequality problem with multiple output sets (MSVIP-MOS) as follows:

    find x0(v)K0 such that

    Ωv0,v1A0(x0(v)),y0(v)x0(v)dv0,y0(v)K0, (2.1)

    and such that

    xi(v)=Tix0(v)KisolvesΩv0,v1Ai(xi(v)),yi(v)xi(v)dv0,yi(v)Ki,i=1,2,,M. (2.2)

    Alternatively, the problem can be formulated in a more compact form as follows:

    find x0(v)K0 such that

    Ωv0,v1Ai(Tix0(v)),yi(v)Tix0(v)dv0,yi(v)Ki,Tix0(v)Ki,   i=0,1,,M. (2.3)

    We denote the solution set of the MSVIP-MOS by

    Γ={x0(v)C0such thatTix0(v)Ci,i=1,2,,M}=C0Mi=1T1i(Ci),

    where C0,Ci,i=1,2,,M are the solution sets of VIPs (2.1) and (2.2), respectively.

    We have the following specials cases of our formulated MSVIP-MOS:

    1. if the multidimensional parameter of evolution v=(tα),α=1,2,,m, then the MSVIP-MOS reduces to a multi-time split variational inequality problem with multiple output sets.

    2. if M=1, then our formulated MSVIP-MOS reduces to the multidimensional split variational inequality problem introduced by Singh [19].

    3. if the multidimensional parameter of evolution v=(vα)Ωv0,v1,α=1,,m, is a single or linear dimensional parameter of evolution, that is, m=1, then Ωv0,v1 is simply the closed real interval [v0,v1] in R+ (set of non-negative real numbers). Moreover, for convenience we set v0=0 and v1=T, where T denotes an arbitrary time. Thus, Ωv0,v1=[0,T] (a fixed time interval). In this case, the MSVIP-MOS reduces to an evolutionary split variational inequality problem with multiple output sets. In addition, if M=1, then the MSVIP-MOS reduces to the evolutionary split variational inequality problem studied by Singh et al. [18].

    4. if all the functions are independent of the multidimensional parameter of evolution v, then the MSVIP-MOS reduces to the split variational inequality problem with multiple output sets studied by Alakoya and Mewomo [20]. In addition, if M=1, then the MSVIP-MOS reduces to the split variational inequality problem introduced by Censor et al. [17].

    In line with the definition of an equilibrium flow for a dynamic traffic network problem given by Danielle et al. [12], we put forward the following definition for a multidimensional traffic network model with multiple networks, in terms of the introduced MSVIP-MOS.

    Definition 2.1. x0(v)K0 is an equilibrium flow if and only if x0(v)Γ.

    The equilibrium flow of a traffic network has been investigated by several authors in terms of the Wardrop condition. Danielle et al. [12] modelled the traffic network equilibrium problem as a classical variational inequality problem, thereby establishing an equivalent relationship between the Wardrop condition and the classical variational inequality problem. On the other hand, Raciti [21] examined the vector form of the Wardrop equilibrium condition. Motivated by these results, here we consider the following MSWC-MOS.

    Definition 2.2. For an arbitrary x0(v)K0 and a.e. on Ωv0,v1, the MSWC-MOS is defined as follows:

    Au00(x0(v))<As00(x0(v))xu00(v)=μu00(v)orxs00(v)=λs00(v),w0W0,u0,s0V0(w0)and such thatxi(v)=Tix0(v)Ki,i=1,2,,M,satisfiesAuii(xi(v))<Asii(xi(v))xuii(v)=μuii(v)orxsii(v)=λsii(v),wiWi,ui,siVi(wi).

    Alternatively, we can recast the definition as follows:

    Definition 2.3. For an arbitrary x0(v)K0 and a.e. on Ωv0,v1, the MSWC-MOS can be defined as

    Auii(xi(v))<Asii(xi(v))xuii(v)=μuii(v)orxsii(v)=λsii(v),wiWi,ui,siVi(wi), (2.4)

    where xi(v)=Tix0(v)Ki,i=0,1,,M.

    In this section, we present an equivalent form of the equilibria of our multidimensional traffic network model with multiple networks via the MSWC-MOS. We note that because of the form of the MSWC-MOS, it is more responsive to the user. Hence, we can conclude that it is a user-oriented equilibrium.

    Now, we state and prove the following theorem, which is the main result of this section.

    Theorem 3.1. Let x0(v)K0 be an arbitrary flow. Then x0(v) is an equilibrium flow if and only if it satisfies the conditions of the MSWC-MOS.

    Proof. First, we suppose that x0(v)K0 satisfies the conditions of the MSWC-MOS. For a given origin-destination pair wiWi,i=0,1,,M, we define the following sets:

    Ri={uiVi(wi):xuii(v)<μuii},i=0,1,,M,Si={siVi(wi):xsii(v)>λsii},i=0,1,,M.

    By the MSWC-MOS, it follows that

    Auii(Tix0(v))Asii(Tix0(v)),uiRi,siSi,i=0,1,,M,a.e. onΩv0,v1. (3.1)

    It follows from Eq (3.1) that there exist real numbers aiR,i=0,1,,M, such that

    supsiSiAsii(Tix0(v))aiinfuiRiAuii(Tix0(v)),a.e. onΩv0,v1.

    Suppose that yi(v)Ki,i=0,1,M, are arbitrary flows. Then, for a.e. on Ωv0,v1 we have

    riVi(wi),Arii(Tix0(v))<airiRi,i=0,1,,M.

    Note that if riRi, then (Tix0(v))ri=μri(v) and (yrii(v)(Tix0(v))ri)0,i=0,1,,M. Hence, it follows that (Arii(Tix0(v))ai)(yrii(v)(Tix0(v))ri)0,i=0,1,,M, a.e. on Ωv0,v1. In a similar manner, for all riVi(wi) such that Arii(Tix0(v))>ai a.e. on Ωv0,v1, we also have that (Arii(Tix0(v))ai)(yrii(v)(Tix0(v))ri)0,i=0,1,,M, a.e. on Ωv0,v1. Consequently, for each i=0,1,,M, we get

    Ai(Tix0(v)),yi(v)Tix0(v)=wiWiriVi(wi)Arii(Tix0(v))(yrii(v)(Tix0(v))ri)=wiWiriVi(wi)(Arii(Tix0(v))ai)(yrii(v)(Tix0(v))ri)+aiwiWiriVi(wi)(yrii(v)(Tix0(v))ri)0,a.e. onΩv0,v1. (3.2)

    Observe that in Eq (3.2), the value of the term wiWiriVi(wi)(yrii(v)(Tix0(v))ri),i=0,1,,M, is zero by the traffic conservation law/demand requirements, i.e., rV(w)xr(v)=ρw(v) for all x(v)K and wW a.e. on Ωv0,v1. Since each yi(v)Ki,i=0,1,,M is arbitrary, it follows from Eq (3.2) that

    Ωv0,v1Ai(Tix0(v)),yi(v)Tix0(v)0,yi(v)Ki,i=0,1,,M.

    Consequently, x0(v) is an equilibrium flow.

    Next, we prove the converse statement by contradiction, that is, we suppose that x0(v) is an equilibrium flow, but it does not satisfy the conditions of the MSWC-MOS. Then, it follows that there exists origin-destination pairs w0W0,wiWi and routes

    u0,s0V0(w0),ui,siVi(wi),i=1,2,,M,

    together with a set ΨΩv0,v1 having a positive measure such that we have the following cases:

    1.

    Au00(x0(v))<As00(x0(v)),xu00(v)<μu00(v),xs00(v)>λs00(v),a.e. onΨ,

    and such that

    xi(v)=Tix0(v)Ki,i=1,2,,M,

    satisfies

    Auii(xi(v))<Asii(xi(v)),xuii(v)<μuii(v),xsii(v)>λsii(v),a.e. onΨ.

    2.

    Au00(x0(v))<As00(x0(v)),xu00(v)<μu00(v),xs00(v)>λs00(v),a.e. onΨ,

    and such that

    xi(v)=Tix0(v)Ki,i=1,2,,M,

    satisfies

    Auii(xi(v))<Asii(xi(v))xuii(v)=μuii(v)orxsii(v)=λsii(v),a.e. onΨ.

    3.

    Au00(x0(v))<As00(x0(v))xu00(v)=μu00(v)orxs00(v)=λs00(v),a.e. onΨ,

    and such that

    xi(v)=Tix0(v)Ki,i=1,2,,M,

    satisfies

    Auii(xi(v))<Asii(xi(v)),xuii(v)<μuii(v),xsii(v)>λsii(v),a.e. onΨ.

    4.

    Au00(x0(v))<As00(x0(v))xu00(v)=μu00(v)orxs00(v)=λs00(v),a.e. onΨ,

    and such that

    xi(v)=Tix0(v)Ki,i=1,2,,M,

    satisfies

    Auii(xi(v))<Asii(xi(v))xuii(v)=μuii(v)orxsii(v)=λsii(v),a.e. onΨ.

    5. Case 1. with xi(v)=Tix0(v)Ki,i=1,2,,M.

    6. Case 2. with xi(v)=Tix0(v)Ki,i=1,2,,M.

    7. Case 3. with xi(v)=Tix0(v)Ki,i=1,2,,M.

    Starting with the Case 1., let

    δ0(v)=min{μu00(v)xu00(v),xs00(v)λs00(v)}andδi(v)=min{μuii(v)xuii(v),xsii(v)λsii(v)},i=0,1,,M,

    where vΨ. Then, δ0(v)>0 and δi(v)>0,i=0,1,,M, a.e. on Ψ. Next, we construct a flow trajectory y0(v)Lp0(Ωv0,v1,R|V0|) as follows:

    yu00(v)=xu00(v)+δ0(v),ys00(v)=xs00(v)δ0(v),yr00(v)=xr00(v),
    forr0u0,s0,a.e. onΨ,andy0(v)=x0(v)outside ofΨ.

    In the same manner, we can define a flow trajectory yi(v)Lpi(Ωv0,v1,R|Vi|),i=1,2,,M as

    yuii(v)=xuii(v)+δi(v),ysii(v)=xsii(v)δi(v),yrii(v)=xrii(v),
    forriui,si,a.e. onΨ,andyi(v)=xi(v)outside ofΨ.

    Hence, it is obvious that y0(v)K0 such that y0(v)=x0(v) outside of Ψ and yi(v)Ki such that yi(v)=xi(v),i=1,2,,M, outside of Ψ. Moreover, we have

    Ωv0,v1A0(x0(v)),y0(v)x0(v)dv=ΨA0(x0(v)),y0(v)x0(v)dv=Ψδ0(v)(Au00(x0(v))As00(x0(v)))dv<0.

    By a similar argument, xi(v)=Tix0(v)Ki,i=1,2,,M, satisfies

    Ωv0,v1Ai(xi(v)),yi(v)xi(v)dv<0,i=1,2,,M.

    It follows that x0(v) is not an equilibrium flow. Using a similar argument, we can easily show that x0(v) is not an equilibrium flow for Case 2 and Case 3. Furthermore, by the fact that xi(v)=Tix0(v)Ki,i=1,2,,M, in Cases 4, 5, 6 and 7, it is clear that x0(v) is not an equilibrium flow. Consequently, we have a contradiction, and this completes the proof of the theorem.

    Here, we establish the existence and uniqueness of equilibria of our multidimensional traffic network model with multiple networks, which is formulated as a MSVIP-MOS. To prove the existence and uniqueness theorem, we will employ the concept of graph theory of operators. First, we present the following definitions and lemma, which will be needed in establishing our results in this section (see [18,19,22]).

    Definition 4.1. The graph of operator Ti,i=1,2,,M is defined by

    Gr T_i ={(x0(v),Tix0(v))K0×Ki:x0(v)K0}.

    We assume that KiTiK0 for each i=1,2,,M, where TiK0={yi(v)Lpi(Ωv0,v1,R|Vi|):x0(v)K0such thatyi(v)=Tix0(v)}. It can easily be shown that Gr  Ti is a convex set. Since Ti is a bounded linear operator for each i=1,2,,M, it follows that Ti is also continuous. Thus, by the closed graph theorem we have that Gr Ti is closed w.r.t. the product topology. Consequently, Gr Ti is a nonempty, closed and convex subset of K0×Ki,i=1,2,,M. By Remark 1, we have that K0×Ki,i=1,2,,M is a weakly compact set. Thus, Gr Ti,i=1,2,,M is a weakly compact set.

    Definition 4.2. The cost operator A is said to be demi-continuous at the point x(v)K0 if it is strongly-weakly sequentially continuous at this point, that is, if the sequence {A(xn(v))} weakly converges to A(x(v)) for each sequence {xn(v)}K0 such that xn(v)x(v), where the symbol "" denotes strong convergence.

    Definition 4.3. The cost operator A is said to be strictly monotone if

    A(x)A(y),xy>0,x,yK0andxy. (4.1)

    Definition 4.4. The convex hull of a finite subset {(x1(v),Tx1(v)),(x2(v),Tx2(v)),,(xn(v),Txn(v))} of Gr T is defined by

    co{(x1(v),Tx1(v)),(x2(v),Tx2(v)),,(xn(v),Txn(v))}={nj=1δj(xj(v),Txj(v)):nj=1δj=1,for someδj[0,1]}.

    Remark 2. Observe that

    co{(x1(v),Tx1(v)),(x2(v),Tx2(v)),,(xn(v),Txn(v))}(co{x1(v),x2(v),,xn(v)},co{Tx1(v),Tx2(v),,Txn(v)}).

    Definition 4.5. ([19]) A set-valued mapping Q:GrT2K0×K1 is said to be a KKM* mapping if, for any finite subset (x1(v),Tx1(v)),(x2(v),Tx2(v)),,(xn(v),Txn(v)) of Gr  T,

    *Knaster–Kuratowski–Mazurkiewicz lemma

    co{(x1(v),Tx1(v)),(x2(v),Tx2(v)),,(xn(v),Txn(v))}nj=1Q(xj(v),Txj(v)).

    Lemma 4.6. ([19] KKM-Fan theorem) Let Q:GrT2K0×K1 be a KKM mapping with closed set values. If Q(x(v),Tx(v)) is compact for at least one (x(v),Tx(v))GrT, then

    (x(v),Tx(v))GrTQ(x(v),Tx(v)).

    We are now in a position to state and prove the existence theorem.

    Theorem 4.7. Suppose that for i=1,2,,M, the cost operators A0,Ai are demi-continuous, and that there exist B0×BiGrTi nonempty and compact, and D0×DiGrTi compact such that for all (x0(v),Tix0(v))GrTiB0×Bi there exists (y0(v),Tiy0(v))D0×Di with Ωv0,v1A0(x0(v)),y0(v)x0(v)dv<0 and Ωv0,v1Ai(Tix0(v)),Tiy0(v)Tix0(v)dv<0. Then, the MSVIP-MOS has a solution.

    Proof. First, we define the following set-valued mappings:

    ● for all x0(v)K0, we define the mapping P0:K02K0 by

    P0(x0(v))={x0(v)K0:Ωv0,v1A0(x0(v)),x0(v)x0(v)dv<0},

    ● for all yi(v)Ki,i=1,2,,M, we define Pi:Ki2Ki by

    Pi(yi(v))={yi(v)Ki:Ωv0,v1Ai(yi(v)),yi(v)yi(v)dv<0},

    ● for all (x0(v),Tix0(v))GrTi,i=1,2,,M, we define the mappings Qi:GrTi2K0×Ki by

    Qi(x0(v),Tix0(v))={(x0(v),Tix0(v))GrTi:Ωv0,v1A0(x0(v)),x0(v)x0(v)dv0

    and

    Ωv0,v1Ai(Tix0(v)),Tix0(v)Tix0(v)dv0}.

    Clearly, (x0(v),Tix0(v))Qi(x0(v),Tix0(v)),i=1,2,,M. Therefore, Qi(x0(v),Tix0(v)) is nonempty for each i=1,2,,M.

    Next, we prove that for each i=1,2,,M,Qi is a KKM mapping. We proceed by contradiction, i.e., by assuming that Qi is not a KKM mapping for each i=1,2,,M. Then for each i=1,2,,M, there exists a finite subset {(x1(v),Tix1(v)),(x2(v),Tix2(v)),,(xn(v),Tixn(v))} of GrTi such that

    co{(x1(v),Tix1(v)),(x2(v),Tix2(v)),,(xn(v),Tixn(v))}nj=1Qi(xj(v),Tixj(v)),i=1,2,,M. (4.2)

    By the definition of a convex hull, there exists the following, for each i=1,2,,M,

    (ˆy0(v),Tiˆy0(v))co{(x1(v),Tix1(v)),(x2(v),Tix2(v)),,(xn(v),Tixn(v))}

    such that

    (ˆy0(v),Tiˆy0(v))=nj=1βji(xj(v),Tixj(v)),i=1,2,,M,

    where βji[0,1] and nj=1βji=1 for each i=1,2,,M. The expression (4.2) implies that

    (ˆy0(v),Tiˆy0(v))nj=1Qi(xj(v),Tixj(v)),i=1,2,,M.

    Consequently, for any j={1,2,,n}, we have the following cases:

    1. Ωv0,v1A0(ˆy0(v)),xj(v)ˆy0(v)dv<0 and Ωv0,v1Ai(Tiˆy0(v)),Tixj(v)Tiˆy0(v)dv<0, i=1,2,,M.

    2. Ωv0,v1A0(ˆy0(v)),xj(v)ˆy0(v)dv0 and Ωv0,v1Ai(Tiˆy0(v)),Tixj(v)Tiˆy0(v)dv<0, i=1,2,,M.

    3. Ωv0,v1A0(ˆy0(v)),xj(v)ˆy0(v)dv<0 and Ωv0,v1Ai(Tiˆy0(v)),Tixj(v)Tiˆy0(v)dv0, i=1,2,,M.

    Case 1 implies that

    {x1(v),x2(v),,xn(v)}P0(ˆy0(v))and{Tix1(v),Tix2(v),,Tixn(v)}Pi(Tiˆy0(v)),i=1,2,,M.

    Moreover, it is clear that P0(x0) and Pi(Tix0) are convex, for each x0K0 and Tix0Ki, i=1,2,,M. Consequently, we have

    co{x1(v),x2(v),,xn(v)}P0(ˆy0(v))

    and

    co{Tix1(v),Tix2(v),,Tixn(v)}Pi(Tiˆy0(v)),for eachi=1,2,,M.

    By the fact that

    (ˆy0(v),Tiˆy0(v))co{(x1(v),Tix1(v)),(x2(v),Tix2(v)),,(xn(v),Tixn(v))},i=1,2,,M

    and by Remark 2, we have

    (ˆy0(v),Tiˆy0(v))(co{x1(v),x2(v),,xn(v)},co{Tix1(v),Tix2(v),,Tixn(v)}),

    which implies that ˆy0(v)P0(ˆy0(v)) and Tiˆy0(v)Pi(Tiˆy0(v)), i=1,2,,M.

    Thus, we have

    Ωv0,v1A0(ˆy0(v)),ˆy0(v)ˆy0(v)dv<0andΩv0,v1Ai(Tiˆy0(v)),Tixj(v)Tiˆy0(v)dv<0,i=1,2,,M,

    which are contradictions.

    By a similar argument, we can easily show that the other cases also lead to contradictions. Hence, for each i=1,2,,M,Qi is a KKM mapping.

    Next, we claim that for each i=1,2,,M,Qi is a closed set-valued mapping for each (x0(v),Tix0(v))GrTi w.r.t. the weak topology of K0×Ki,i=1,2,,M. Let (x0(v),Tix0(v))GrTi be arbitrary and suppose that {(xn0(v),Tixn0(v))}n=0 is a sequence in Qi(x0(v),Tix0(v)), which converges strongly to (y0(v),Tiy0(v)), i=1,2,,M. Since for each nN,(xn0(v),Tixn0(v))Qi(x0(v),Tix0(v)), i=1,2,,M, we have the following for each nN

    Ωv0,v1A0(xn0(v)),x0(v)xn0(v)dv0andΩv0,v1Ai(Tixn0(v)),Tix0(v)Tixn0(v)dv0,i=1,2,,M. (4.3)

    Since A0,Ai,i=1,2,,M are demi-continuous and T0,Ti,i=1,2,,M are continuous, by taking the limit as n in Eq (4.3), we obtain

    Ωv0,v1A0(y0(v)),x0(v)y0(v)dv0

    and

    Ωv0,v1Ai(Tiy0(v)),Tix0(v)Tiy0(v)dv0,i=1,2,,M,

    which implies that

    (y0(v),Tiy0(v))Qi(x0(v),Tix0(v))

    for each i=1,2,,M. Thus, Qi(x0(v),Tix0(v)) is closed (w.r.t. the strong topology) for each

    (x0(v),Tix0(v))GrTi,i=1,2,,M.

    By the hypothesis in Theorem 4.7, it follows that Qi(x0(v),Tix0(v)),i=1,2,,M is compact (w.r.t. the strong topology) for each

    (x0(v),Tix0(v))D0×DiGrTi,i=1,2,,M.

    Consequently, by the KKM-Fan theorem, we have

    (x0(v),Tix0(v))GrTiQi(x0(v),Tix0(v)),i=1,2,,M.

    This implies that there exists

    (x0(v),Tix0(v))GrTi,i=1,2,,M,

    such that

    (x0(v),Tix0(v))Qi(x0(v),Tix0(v))

    for all

    (x0(v),Tix0(v))GrTi,i=1,2,,M.

    Now, we consider the subsets F0K0,FiKi, i=1,2,,M, such that

    (x0(v),Tix0(v))F0×FiGrTi,i=1,2,,M.

    Then, we can write that there exists

    (x0(v),Tix0(v))F0×Fi

    such that

    (x0(v),Tix0(v))Qi(x0(v),Tix0(v))

    for all

    (x0(v),Tix0(v))F0×Fi,i=1,2,,M.

    Consequently, we have that for all (x0(v),Tix0(v))F0×Fi,

    Ωv0,v1A0(x0(v)),x0(v)x0(v)dv0andΩv0,v1Ai(Tix0(v)),Tix0(v)Tix0(v)dv0,i=1,2,,M. (4.4)

    Let

    yi(v)=Tix0(v),yi(v)=Tix0(v),i=1,2,,M,

    and observe that x0(v) and

    yi(v)=Tix0(v),i=1,2,,M

    are fixed in Eq (4.4). Thus, Eq (4.4) can be rewritten as x0(v)F0, such that

    Ωv0,v1A0(x0(v)),x0(v)x0(v)dv0,x0(v)F0,

    and such that

    yi(v)=Tix0(v)FisolvesΩv0,v1Ai(yi(v)),yi(v)yi(v)dv0,yi(v)Fi,i=1,2,,M.

    Hence, it follows that the MSVIP-MOS has a solution x0(v)F0K0.

    Next, we present the result on the uniqueness of the solution of the MSVIP-MOS in the following corollary.

    Corollary 1. If the cost operators Ai,i=0,1,,M are strictly monotone on Ki,i=0,1,,M, then the MSVIP-MOS has a unique solution.

    Proof. Suppose to the contrary that the MSVIP-MOS does not have a unique solution. Let x0(v)K0 be a solution of the MSVIP-MOS. Then, we have

    Ωv0,v1Ai(Tix0(v)),xi(v)Tix0(v)dv0,xi(v)Ki,Tix0(v)Ki,i=0,1,,M. (4.5)

    Let ˆx0(v)K0 be another solution of the MSVIP-MOS such that x0(v)ˆx0(v). Then, it follows that

    Ωv0,v1Ai(Tiˆx0(v)),ˆxi(v)Tiˆx0(v)dv0,ˆxi(v)Ki,Tiˆx0(v)Ki,i=0,1,,M. (4.6)

    We can rewrite Eq (4.5) as

    Ωv0,v1Ai(Tix0(v)),Tiˆx0(v)Tix0(v)dv0,i=0,1,,M. (4.7)

    By the strict monotonicity of the Ai,i=0,1,,M, together with the fact that x0(v)ˆx0(v), we get

    Ωv0,v1Ai(Tix0(v))Ai(Tiˆx0(v)),Tix0(v)Tiˆx0(v)dv>0,i=0,1,,M. (4.8)

    By adding Eqs (4.7) and (4.8), we obtain

    Ωv0,v1Ai(Tiˆx0(v)),Tix0(v)Tiˆx0(v)dv<0,i=0,1,,M,

    which contradicts Eq (4.6). Therefore, it follows that ˆx0(v) is not a solution of the MSVIP-MOS. Consequently, the MSVIP-MOS has a unique solution.

    In this section, motivated by the work of Cojocaru et al. [23], we study our multidimensional traffic model with multiple networks by employing the theory of a projected dynamical system (PDS). Dupuis and Nagurney [24] were the first to introduce and study the PDS. Furthermore, they established the connections of PDS with the classical variational inequality problem. For more details about the various areas of applications of the PDS, we refer interested readers to [23,25].

    Inspired by the results from the aforementioned works, here, we introduce and formulate a multidimensional split projected dynamical system with multiple output sets (MSPDS-MOS) for pi=2,i=0,1,,M as follows:

    Findx0()K0such thatdx0(,τ)dτ=ΠK0(x0(,τ),A0(x0(,τ))),x0(,0)=x00()K0and such thatxi()=Tix0()Kisatisfiesdxi(,τ)dτ=ΠKi(xi(,τ),Ai(xi(,τ))),xi(,0)=x0i()Ki,i=1,2,,M,

    where Ai:KiL2(Ωv0,v1,R|Vi|),i=0,1,,M, are Lipschitz continuous vector fields and the operators ΠKi:Ki×L2(Ωv0,v1,R|Vi|),i=0,1,,M are defined by

    ΠKi(xi(),yi()):=limδ0+projKi(xi()+δyi())xi()δ,xi()Ki,yi()L2(Ωv0,v1,R|Vi|),

    where projKi() are the nearest point projection of a given vector onto the sets given by Ki.

    Alternatively, the MSPDS-MOS can be formulated as follows:

    Find x0()K0 such that

    dTix0(,τ)dτ=ΠKi(Tix0(,τ),Ai(Tix0(,τ))),xi(,0)=x0i()Ki,i=0,1,,M.

    For clarity, here we have represented the elements of the space L2(Ωv0,v1,R|Vi|) at fixed vΩv0,v1 by x(). Observe that for all vΩv0,v1, a solution of the MSVIP-MOS represents a static state of the underlying system and the static states define one or more equilibrium curves when v varies over Ωv0,v1. On the contrary, the time τ defines the dynamics of the system over the interval [0,) until it attains one of the equilibria on the curves. Clearly, the solutions to the MSPDS-MOS lie in the class of absolutely continuous functions with respect to τ, mapping [0,) to Ki,i=0,1,,M. Before we describe the procedure to solve the MSVIP-MOS, we present the following useful definitions motivated by [26,27].

    Definition 5.1. A point ˆx0()K0 is called a critical point for the MSPDS-MOS if

    ΠK0(ˆx0(),A0(ˆx0()))=0

    and the point ˆyi()=Tiˆx0()Ki satisfies

    ΠKi(ˆyi(),Ai(ˆyi()))=0,i=1,2,,M.

    Alternatively, the critical point for the MSPDS-MOS can be defined as follows: ˆx0()K0 is called a critical point for the MSPDS-MOS if

    ΠKi(Tiˆx0(),Ai(Tiˆx0()))=0,Tiˆx0()Ki,i=0,1,,M.

    Definition 5.2. The polar set Ko associated with K is defined by

    Ko:={x()L2(Ωv0,v1,R|V|):x(),y()0,y()K}.

    Definition 5.3. The tangent cone to the set K at x()K is defined by

    ˆTK(x())=cl(λ>0Kx()λ),

    where cl denotes the closure operation.

    Definition 5.4. The normal cone of K at x()K is defined by

    NK(x()):={y()L2(Ωv0,v1,R|V|):y(),z()x()0,z()K}.

    Alternatively, we can express this as ˆTK(x())=[NK(x())]o.

    Definition 5.5. The projection of x()L2(Ωv0,v1,R|V|) onto K is defined by

    projK(x()):=argminy()Kx()y().

    Remark 3. The projection map projK() satisfies the following property for each x()L2(Ωv0,v1,R|V|):

    x()projK(x()),y()projK(x())0,y()K.

    We have the following results, which follow from Proposition 2.1 and 2.2 in [26].

    Proposition 1. For all x()K and y()L2(Ωv0,v1,R|V|),ΠK(x(),y()) exists and ΠK(x(),y())=projˆTK(x())(y()).

    Proposition 2. For all x()K, there exists n()NK(x()) such that ΠK(x(),y())=y()n(),y()L2(Ωv0,v1,R|V|).

    Now, we prove the following theorem, which establishes the relationship between solutions of MSVIP-MOS and the critical points of the MSPDS-MOS.

    Theorem 5.6. The point x0()K0 is a solution of the MSVIP-MOS if and only if it is a critical point of the MSPDS-MOS.

    Proof. First, we suppose that x0()K0 is a solution to the MSVIP-MOS, that is,

    Ωv0,v1Ai(Tix0()),yi()Tix0()dv0,yi()Ki,i=0,1,,M,

    which implies that

    Ai(Tix0()),yi()Tix0()0,yi()Ki,i=0,1,,M.

    From the last inequality, it follows that

    Ai(Tix0())NKi(Tix0()),i=0,1,,M.

    By Proposition 2, we have

    ΠKi(Tix0(),Ai(Tix0()))=0, (5.1)

    which implies that x0() is a critical point of the MSPDS-MOS.

    Conversely, suppose that x0() is a critical point of the MSPDS-MOS. Then, Eq (5.1) holds. By Proposition 1, it follows that

    projˆTKi(Tix0())(Ai(Tix0()))=0,i=0,1,,M.

    Applying Remark 3, we obtain

    Ai(Tix0()),zi()0,zi()ˆTKi(Tix0()),i=0,1,,M,

    which gives

    Ai(Tix0())NKi(Tix0()),i=0,1,,M.

    From this, it follows that x0() is a solution of the MSVIP-MOS.

    At this point, we present the method for finding the solution of the MSVIP-MOS. In our numerical experiments, we consider the case in which v=(tα),α=1,2,,m, that is, there are m-dimensional time parameters. We have established the existence and uniqueness of equilibria for the MSVIP-MOS in Section 4. Moreover, Theorem 5.6 guarantees that any point on a curve of equilibria in the set Ωv0,v1 is a critical point of the MSPDS-MOS and vice versa. Taking into consideration all of these facts, now we discretize the set Ωv0,v1 as follows: Ωv0,v1:(v10,v20,,vm0)=(t10,t20,,tm0)<(t11,t21,,tm1)<<(t1j,t2j,,tmj)<<(t1n,t2n,,tmn)=(v11,v21,,vm1). Consequently, for each tj=(t1j,t2j,,tmj),j=0,1,,n, we obtain a sequence of the MSPDS-MOS on the distinct, finite-dimensional, closed and convex sets denoted by Ktj. After evaluating all of the critical points of each MSPDS-MOS, we obtain a sequence of critical points and from this, we generate the curves of equilibria by interpolation.

    To demonstrate the implementation of this procedure, we consider the transportation network patterns of three cities C0,C1 and C2 as shown in Figure 1 below.

    Figure 1.  The transportation network patterns of the three cities C0,C1 and C2.

    We suppose that a bus company has stations at nodes P10 and P20 in City C0, at nodes P11 and P41 in City C1 and at nodes P12 and P122 in City C2. In City C0, the buses from stations P10 and P20 have to deserve the locations P30 and P50, respectively. In City C1, the buses from stations P11 and P41 have to deserve the locations P21 and P31, respectively. While in City C2, the buses from stations P12 and P122 have to deserve the locations P62 and P82, respectively.

    Hence, the network of City C0 comprises six nodes and eight links, and we assume that the origin-destination pairs are w10=(P10,P30) and w20=(P20,P50), which are respectively connected by the following routes:

    w10:{r10=(P10,P20)(P20,P30)r20=(P10,P60)(P60,P50)(P50,P20)(P20,P30),
    w20:{r30=(P20,P30)(P30,P40)(P40,P50)r40=(P20,P30)(P30,P60)(P60,P50).

    Let Ωv0,v1=Ω0,3=[0,3]2. The set of feasible flows, K0, is given by

    K0={x(t)L2(Ω0,3,R4):(0,0,0,0)(x1(t),x2(t),x3(t),x4(t))(t1+t2+1,t1+t2+2,2t1+2t2+2,t1+t2+3)andx1(t)+x2(t)=t1+t2+2,x3(t)+x4(t)=2t1+2t2+3,a.e.inΩ0,3},

    the cost function A0:K0L2(Ω0,3,R4) is defined by

    A0(x(t))=(x1(t),x2(t),x3(t),x4(t))

    and the bounded linear operator

    T0:L2(Ω0,3,R4)L2(Ω0,3,R4)

    is defined by T0x(t)=(x1(t),x2(t),x3(t),x4(t)), where x(t)=(x1(t),x2(t),x3(t),x4(t)).

    Moreover, the network of City C1 is made up of five nodes and seven links, and we assume that the origin-destination pairs are w11=(P11,P21) and w21=(P41,P31), which are respectively connected by the following routes:

    w11:{r11=(P11,P21)r21=(P11,P41)(P41,P21),
    w21:{r31=(P41,P21)(P21,P31)r41=(P41,P51)(P51,P31)r51=(P41,P21)(P21,P51)(P51,P31).

    The set of feasible flows, K1, is given by

    K1={y(t)L2(Ω0,3,R5):(0,0,0,0,0)(y1(t),y2(t),y3(t),y4(t),y5(t))(t1+t2+6,t1+t2+6,2t1+2t2+2,t1+t2+4,4t1+4t2+4)andy1(t)+y2(t)=3t1+3t2+5,y3(t)+y4(t)+y5(t)=2t1+4t2+6,a.e.inΩ0,3},

    the cost function A1:K1L2(Ω0,3,R5) is defined as

    A1(y(t))=(y21(t),y22(t),y23(t),y24(t),y25(t))

    and the bounded linear operator

    T1:L2(Ω0,3,R4)L2(Ω0,3,R5)

    is defined by

    T1y(t)=(y1(t)+y4(t),y2(t)+y3(t),y1(t)+y2(t),2y1(t),2y2(t)+y4(t)y3(t)),

    where

    y(t)=(y1(t),y2(t),y3(t),y4(t)).

    Also, the network of City C2 is composed of twelve nodes and thirteen links, and we assume that the origin-destination pairs are w12=(P12,P62) and w22=(P122,P82), which are respectively connected by the following routes:

    w12:{r12=(P12,P22)(P22,P32)(P32,P42)(P42,P52)(P52,P62)r22=(P12,P22)(P22,P82)(P82,P72)(P72,P62),
    w22:{r32=(P122,P112)(P112,P102)(P102,P92)(P92,P82)r42=(P122,P12)(P12,P22)(P22,P82).

    The set of feasible flows, K2, is given by

    K2={z(t)L2(Ω0,3,R4):(0,0,0,0)(z1(t),z2(t),z3(t),z4(t))(2t1+2t2+3,t1+t2+7,3t1+3t2+4,2t1+2t2+5)andz1(t)+z2(t)=3t1+3t2+4,z3(t)+z4(t)=2t1+6t2+7,a.e.inΩ0,3},

    the cost function A2:K2L2(Ω0,3,R4) is defined by

    A2(z(t))=(z1(t)+z21(t),z2(t)+z22(t),z3(t)+z23(t),z4(t)+z24(t))

    and the bounded linear operator

    T2:L2(Ω0,3,R4)L2(Ω0,3,R4)

    is defined by

    T2z(t)=(2z3(t)z1(t),2z4(t)z2(t),2z1(t)+z4(t),2z2(t)+z3(t)),

    where

    z(t)=(z1(t),z2(t),z3(t),z4(t)).

    It can easily be verified that all the hypotheses of Theorem 4.7 are satisfied and that the cost operators denoted by Ai,i=0,1,2 are strictly monotone on the sets of feasible flows denoted by Ki,i=0,1,2. Thus, the MSVIP-MOS has a unique solution. We select

    tj{[k6,k6]:k{0,1,2,,18}}.

    Then, we have a sequence of MSPDS-MOS defined on the feasible sets

    K0,tj={x(tj)L2(Ω0,3,R4):(0,0,0,0)(x1(tj),x2(tj),x3(tj),x4(tj))(t1j+t2j+1,t1j+t2j+2,2t1j+2t2j+2,t1j+t2j+3)andx1(tj)+x2(tj)=t1j+t2j+2,x3(tj)+x4(tj)=2t1j+2t2j+3,a.e.inΩ0,3},
    K1,tj={y(tj)L2(Ω0,3,R5):(0,0,0,0,0)(y1(tj),y2(tj),y3(tj),y4(tj),y5(tj))(t1j+t2j+6,t1j+t2j+6,2t1j+2t2j+2,t1j+t2j+4,4t1j+4t2j+4)andy1(tj)+y2(tj)=3t1j+3t2+5,y3(tj)+y4(tj)+y5(tj)=2t1j+4t2j+6,a.e.inΩ0,3},
    K2,tj={z(tj)L2(Ω0,3,R4):(0,0,0,0)(z1(tj),z2(tj),z3(tj),z4(tj))(2t1j+2t2j+3,t1j+t2j+7,3t1j+3t2j+4,2t1j+2t2j+5)andz1(tj)+z2(tj)=3t1j+3t2j+4,z3(tj)+z4(tj)=2t1j+6t2j+7,a.e.inΩ0,3}.

    For evaluating the unique equilibrium, we have the following system at tj:

    findx(tj)K0,tjsuch thatA0(x(tj))NK0,tj(x(tj))andTix(tj)Ki,tjsolvesAi(Tix(tj))NKi,tj(Tix(tj)),i=1,2.

    After some computations, we obtain the equilibrium points which are presented in Tables 13. Then, we interpolate the points in Tables 13 to get the curves of equilibria displayed in Figures 24.

    Table 1.  Numerical results associated with the traffic network pattern of City C0.
    ti={t1i,t2i} x1(ti) x2(ti) x3(ti) x4(ti)
    {0,0} 1.0000 1.0000 1.5000 1.5000
    {16,16} 1.1667 1.1667 1.8333 1.8333
    {13,13} 1.3333 1.3333 2.1667 2.1667
    {12,12} 1.5000 1.5000 2.5000 2.5000
    {23,23} 1.6667 1.6667 2.8333 2.8333
    {56,56} 1.8333 1.8333 3.1667 3.1667
    {1,1} 2.0000 2.0000 3.5000 3.5000
    {76,76} 2.1667 2.1667 3.8333 3.8333
    {43,43} 2.3333 2.3333 4.1667 4.1667
    {32,32} 2.5000 2.5000 4.5000 4.5000
    {53,53} 2.6667 2.6667 4.8333 4.8333
    {116,116} 2.8333 2.8333 5.1667 5.1667
    {2,2} 3.0000 3.0000 5.5000 5.5000
    {136,136} 3.1667 3.1667 5.8333 5.8333
    {73,73} 3.3333 3.3333 6.1667 6.1667
    {52,52} 3.5000 3.5000 6.5000 6.5000
    {83,83} 3.6667 3.6667 6.8333 6.8333
    {176,176} 3.8333 3.8333 7.1667 7.1667
    {3,3} 4.0000 4.0000 7.5000 7.5000

     | Show Table
    DownLoad: CSV
    Table 2.  Numerical results associated with the traffic network pattern of City C1.
    ti={t1i,t2i} y1(ti) y2(ti) y3(ti) y4(ti) y5(ti)
    {0,0} 2.5000 2.5000 2.0000 2.0000 2.0000
    {16,16} 3.0000 3.0000 2.3333 2.3333 2.3333
    {13,13} 3.5000 3.5000 2.6667 2.6667 2.6667
    {12,12} 4.0000 4.0000 3.0000 3.0000 3.0000
    {23,23} 4.5000 4.5000 3.3333 3.3333 3.3333
    {56,56} 5.0000 5.0000 3.6667 3.6667 3.6667
    {1,1} 5.5000 5.5000 4.0000 4.0000 4.0000
    {76,76} 6.0000 6.0000 4.3333 4.3333 4.3333
    {43,43} 6.5000 6.5000 4.6667 4.6667 4.6667
    {32,32} 7.0000 7.0000 5.0000 5.0000 5.0000
    {53,53} 7.5000 7.5000 5.3333 5.3333 5.3333
    {116,116} 8.0000 8.0000 5.6667 5.6667 5.6667
    {2,2} 8.5000 8.5000 6.0000 6.0000 6.0000
    {136,136} 9.0000 9.0000 6.3333 6.3333 6.3333
    {73,73} 9.5000 9.5000 6.6667 6.6667 6.6667
    {52,52} 10.0000 10.0000 7.0000 7.0000 7.0000
    {83,83} 10.5000 10.5000 7.3333 7.3333 7.3333
    {176,176} 11.0000 11.0000 7.6667 7.6667 7.6667
    {3,3} 11.5000 11.5000 8.0000 8.0000 8.0000

     | Show Table
    DownLoad: CSV
    Table 3.  Numerical results associated with the traffic network pattern of City C2.
    ti={t1i,t2i} z1(ti) z2(ti) z3(ti) z4(ti)
    {0,0} 2.0000 2.0000 3.5000 3.5000
    {16,16} 2.5000 2.5000 4.1667 4.1667
    {13,13} 3.0000 3.0000 4.8333 4.8333
    {12,12} 3.5000 3.5000 5.5000 5.5000
    {23,23} 4.0000 4.0000 6.1667 6.1667
    {56,56} 4.5000 4.5000 6.8333 6.8333
    {1,1} 5.0000 5.0000 7.5000 7.5000
    {76,76} 5.5000 5.5000 8.1667 8.1667
    {43,43} 6.0000 6.0000 8.8333 8.8333
    {32,32} 6.5000 6.5000 9.5000 9.5000
    {53,53} 7.0000 7.0000 10.1667 10.1667
    {116,116} 7.5000 7.5000 10.8333 10.8333
    {2,2} 8.0000 8.0000 11.5000 11.5000
    {136,136} 8.5000 8.5000 12.1667 12.1667
    {73,73} 9.0000 9.0000 12.8333 12.8333
    {52,52} 9.5000 9.5000 13.5000 13.5000
    {83,83} 10.0000 10.0000 14.1667 14.1667
    {176,176} 10.5000 10.5000 14.8333 14.8333
    {3,3} 11.0000 11.0000 15.5000 15.5000

     | Show Table
    DownLoad: CSV
    Figure 2.  The traffic network pattern of City C0.
    Figure 3.  The traffic network pattern of City C1.
    Figure 4.  The traffic network pattern of City C2.

    Table 1 displays the equilibrium points at each instant for City C0 while the traffic network pattern of City C0 is presented in Figure 2. We observe from Table 1 that at the beginning of the equilibrium flow in City C0, the flow on each of the routes connecting the origin-destination pair w20 is about 1.5 times the flow on each of the routes connecting the origin destination pair w10, and this factor increases gradually over the equilibrium flow time to about 1.9.

    Table 2 shows the equilibrium points at each instant for City C1 and the traffic network pattern of the city is presented in Figure 3. We note from Table 2 that at the beginning of the equilibrium flow in City C1, the flow on each of the routes connecting the origin-destination pair w11 is about 1.3 times the flow on each of the routes connecting the origin destination pair w21, and this factor increases gradually over the equilibrium flow time to about 1.4.

    Table 3 presents the equilibrium points at each instant for City C2 while the traffic network pattern of the city is presented in Figure 4. It is observed from Table 3 that at the beginning of the equilibrium flow in City C2, the flow on each of the routes connecting the origin-destination pair w22 is about 1.8 times the flow on each of the routes connecting the origin destination pair w12. Contrary to the observation in cities C0 and C1, this factor decreases gradually over the equilibrium flow time to about 1.4. We observe from the results that when the system is in equilibrium every route in each of the three cities is in use. Moreover, routes connecting the same origin-destination pair in each city have an equal amount of flow at each instant t within the equilibrium flow time.

    In this section, we illustrate how our results can be applied to study models with heterogeneous networks. For that purpose, we consider a City C, which comprises a traffic network of human-driven vehicles (HDVs), traffic network of connected automated vehicles (CAVs) and an electricity network as shown in Figure 5 below.

    Figure 5.  The network model of the three heterogeneous networks in City C.

    We denote the traffic network of human-driven vehicles by NHDV, while we denote the traffic network of connected automated vehicles by NCAV and the electricity network by EN. Here, it is assumed that the EN is analogous to the traffic network. Suppose that within the network coverage of CAVs, we have commuters such that some of them need to be transported from location P1 to location P3 and others from location P1 to location P4, using CAVs. On the other hand, we assume that within the network coverage of HDVs, we have commuters who need to be transported by HDVs from locations Q1 and Q12 to locations Q6 and Q8, respectively. Also, we suppose that within the EN, electricity needs to be transmitted from point R1 to point R4.

    Therefore, the NCAV consists of four nodes and six links, and we assume that the origin destination pairs are w1p=(P1,P3) and w2p=(P1,P4), which are connected respectively by the following routes:

    w1p:{r1p=(P1,P2)(P2,P3)r2p=(P1,P4)(P4,P3)r3p=(P1,P3),
    w2p:{r4p=(P1,P4)r5p=(P1,P2)(P2,P4).

    The set of feasible flows K0 is given by

    K0={x0(v)Lp0(Ωv0,v1,R5):λ0(v)x0(v)μ0(v)andΦ0x0(v)=ρ0(v),a.e. onΩv0,v1},

    the cost function is given by A0:K0Lq0(Ωv0,v1,R5) and the bounded linear operator T0:Lp0(Ωv0,v1,R5)Lp0(Ωv0,v1,R5).

    Similarly, the NHDV comprises twelve nodes and thirteen links, and we assume that the origin-destination pairs are w1q=(Q1,Q6) and w2q=(Q12,Q8), which are respectively connected by the following routes:

    w1q:{r1q=(Q1,Q2)(Q2,Q3)(Q3,Q4)(Q4,Q5)(Q5,Q6)r2q=(Q1,Q2)(Q2,Q8)(Q8,Q7)(Q7,Q6),
    w2q:{r3q=(Q12,Q11)(Q11,Q10)(Q10,Q9)(Q9,Q8)r4q=(Q12,Q1)(Q1,Q2)(Q2,Q8).

    The set of feasible flows K1 is given by

    K1={x1(v)Lp1(Ωv0,v1,R4):λ1(v)x1(v)μ1(v)andΦ1x1(v)=ρ1(v),a.e. onΩv0,v1},

    the cost function is given by A1:K1Lq1(Ωv0,v1,R4) and the bounded linear operator T1:Lp1(Ωv0,v1,R4)Lp1(Ωv0,v1,R4).

    On the other hand, the EN consists of six nodes and seven links, and we assume that the origin-destination pair is w1r=(R1,R4), which is connected by the following routes:

    w1r:{r1r=(R1,R2)(R2,R3)(R3,R4)r2r=(R1,R6)(R6,R5)(R5,R4)r3r=(R1,R4).

    The set of feasible flows K2 is given by

    K2={x2(v)Lp2(Ωv0,v1,R3):λ2(v)x2(v)μ2(v)andΦ2x2(v)=ρ2(v),a.e. onΩv0,v1},

    the cost function is given by A2:K2Lq2(Ωv0,v1,R3) and the bounded linear operator

    T2:Lp2(Ωv0,v1,R3)Lp2(Ωv0,v1,R3).

    Then, it follows that x0(v)K0 is an equilibrium flow if and only if

    Ωv0,v1A0(x0(v)),y0(v)x0(v)dv0,y0(v)K0,and such thatxi(v)=Tix0(v)KisolvesΩv0,v1Ai(xi(v)),yi(v)xi(v)dv0,yi(v)Ki,i=1,2. (5.2)

    Therefore, by employing the model (5.2), we can determine the equilibrium flows of the NCAV, NHDV and EN simultaneously.

    We introduced and studied a new class of split inverse problem called the MSVIP-MOS. Our proposed model is finite-dimensional and essentially an assignment problem. It comprises a multidimensional parameter of evolution. To demonstrate the applicability of our proposed model in the economic world, we formulated the equilibrium flow of multidimensional traffic network models for an arbitrary number of locations. Moreover, we proposed a method for solving the introduced problem and validated our results with some numerical experiments. Finally, to further demonstrate the usefulness of our newly introduced model, we applied our results to study the network model of a city with heterogeneous networks that comprises CAVs and legacy (human-driven) vehicles, alongside the EN, e.g. for charging the CAVs, and we formulated the equilibrium flow of this network model in terms of the newly introduced MSVIP-MOS. However, we note that the problem investigated in this study belongs to the class of linear (split) inverse problems, and as such our results are not applicable to nonlinear traffic flow models. In our future study, we will be interested in extending our results to this class of models.

    The authors declare that they have not used artificial intelligence tools in the creation of this article.

    This work was supported by the HEA (the Irish Higher Education Authority) North-South Research Programme 2021. The authors would like to thank the reviewers for their constructive comments and recommendations, which have helped to improve the quality of the manuscript. Vikram Pakrashi would like to also acknowledge Science Foundation Ireland NexSys 21/SPP/3756.

    The authors declare that there is no conflict of interest.



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