A NEW CLASS OF GLOBAL FRACTIONAL-ORDER PROJECTIVE DYNAMICAL SYSTEM WITH AN APPLICATION

In this article, some existence and uniqueness of solutions for a new class of global fractional-order projective dynamical system with delay and perturbation are proved by employing the Krasnoselskii fixed point theorem and the Banach fixed point theorem. Moreover, an approximating algorithm is also provided to find a solution of the global fractional-order projective dynamical system. Finally, an application to the idealized traveler information systems for day-to-day adjustments processes and a numerical example are given.


1.
Introduction. It is well known that the delay fractional differential equation is a type of fractional differential equation in which the fractional derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. Introduction of delay in the fractional differential equation (system) enriches its dynamics and allows a precise description of the real life phenomena. Since the (delay) fractional differential equations have been proved to be valuable tools to model many phenomena in various fields of physics, physiology, ecology, viscoelasticity, electrochemistry, control, electromagnetic, engineering and economics, there are many authors to study various theoretical results, numerical algorithms and applications concerned with many kinds of (delay) fractional differential equations in the literature (see, for example, [2-4, 9, 20-23, 41, 44] and the references therein).
On the other hand, it was showed by Dupuis and Nagurney [5] in 1993 that the stationary points of a projective dynamical systems are solutions to some associated variational inequalities. Since the variational inequality is a very powerful tool in investigating various network equilibrium problems arising in economic, management, and engineering, much effort has been made in the projective dynamical systems in recent decades. Some related results in these directions can be found in [11-13, 16, 32, 33, 37-40, 42, 45-47].
Motivated by the work in connection with fractional differential equations and projective dynamical systems, Wu and Zou [34] introduced and studied the following fractional-order projective dynamical systems where M is a real matrix. They proved the existence and uniqueness of the solution for this system and showed the stability for the equilibrium point. They also provided a predictor-corrector algorithm to approximating a solution to fractionalorder projective dynamical system. Very recently, based on a network tatonnement model, Wu et al. [35] introduced a system of fractional-order interval projection neural networks as follows where 0 < α ≤ 1 and , and showed the existence and uniqueness of the equilibrium point for the fractionalorder interval projection neural networks under mild conditions. Moreover, some results concerned with the existence and stability of solutions for the following global fractional-order projective dynamical systems involving set-valued perturbations were presented by Wu et al. [36]. Although many contributions for the fractional differential equations and the (fractional-order) projective dynamical systems, to our best knowledge, there is no researchers to study the fractional-order projective dynamical systems with delay appeared as IVP (4). This fact is the motivation of the present work.
In this paper, we consider an initial value problem (for short, IVP) of fractionalorder projective dynamical system with delay and perturbation of the following form is Caputo's fractional derivative of order 0 < α ≤ 1, P K is the projection operator, ρ is a positive constant, g : R n → R n , M : C([−r, 0], R n ) → R n and N : [t 0 , t 0 + T ] × C([−r, 0], R n ) → R n are given maps, K is a nonempty closed convex subset of R n , x(t) ∈ R n , φ ∈ C([−r, 0], R n ), x t stands for the history of state function up to the time t, i.e., We note that IVP (4) is presented in the most abstract form and also covers many important problems in projective dynamical system, variational inequality and fractional differential equation as special cases [11][12][13]16,[32][33][34][35][36][37]39,40,42,[45][46][47]. If M is a linear map, g (x(t)) = x(t), r = 0 and N = 0, then IVP (4) reduces to the general form of (1). Furthermore, if α = 1, then IVP (4) reduces to the global projective dynamical system which has been studied by many authors during the past decades (see, for example, [12,13,33,39,40,46]). It is worth mentioning that model (4) captures the desired features of both the projective dynamical system and the fractional differential equation with delay and perturbation within the same framework. To the best of our knowledge, no author has studied the fractionalorder projective dynamical system with delay and perturbation. Therefore, the study concerned with the fractional-order projective dynamical system with delay and perturbation is important and interesting in theory and practice. The main purpose of this paper is to give some new results in connection with solutions of IVP (4) under some suitable assumptions.
The rest of this paper is organized as follows. Section 2 introduces some definitions and preliminary facts. Section 3 establishes some sufficient conditions for the existence and uniqueness of the solution of IVP (4). Section 4 provides a numerical algorithm for approximating the solution of IVP (4). Finally, we present an application to the idealized traveler information systems for day-to-day adjustments processes and a numerical example in Section 5.

2.
Preliminaries. In this section, we introduce some basic definitions, notations and preliminary facts. Let I ⊂ R and C(I, R n ) be the Banach space of all continuous functions x(t) from I into R n with the norm where · denotes a suitable complete norm on R n . Definition 2.1. If K is a closed convex subset of R n , then P K : R n → K is defined by Definition 2.2. [15,26] The Riemann-Liouville fractional integral of order α > 0 for a function x(t) is defined by [15,26] The Caputo fractional derivative of order α for a function x(t) ∈ C n ([t 0 , +∞), R) is defined as follows: where t > t 0 and n is a positive integer such that n − 1 < α < n. Remark 1. If α = n, then the Caputo fractional derivative of α for a function x(t) is usual derivative x (n) (t).
if and only if Lemma 2.6. (Krasnoselskii Fixed Point Theorem) [17] Let X be a Banach space and E be a bounded closed convex subset of X. Assume that U and V are two

Existence and uniqueness. Let
where δ is positive constant.
In what follows, we introduce the following hypotheses. (H 1 ) g is Lipschitz continuous, i.e., for any (H 6 ) The following inequality holds Now, we are ready to state the existence of solution which is based on the Krasnoselskii's fixed point theorem.
Proof. According to Lemma 2.5, IVP (4) is equivalent to the following equation Obviously, y satisfies the following equation Since K is nonempty, we can choose a point x 0 ∈ K. By the definition of the projection operator, we know that . Let Then it is easy to check that E δ is a closed convex bounded subset of C([−r, T ], R n ). Define two mappings U and V on E δ by . Clearly, the operator equation y = U y + V y (7) has a solution y ∈ E δ if and only if y is a solution of equation (6). Thus, if y ∈ E δ is a solution of operator equation (7), then x(t 0 + t) = y(t) + φ(t 0 + t) is a solution of equation (5) on [0, T ]. Therefore, IVP (4) has a solution is equivalent to that U + V has a fixed point in E δ .
Next, we prove that U + V has a fixed point in E δ . The proof is divided into three steps.
Step 2. We prove that U is a contraction mapping on E δ . For any y 1 , y 2 ∈ E δ , similar to the proof in Step 1, we can show that y 1 s + φ t0+s , y 2 s + φ t0+s ∈ A δ for all s ∈ [0, T ]. By Lemma 2.4 and assumption (H 1 )-(H 2 ), we obtain In view of l1+ρl2 Γ(α+1) T α < L * < 1, we know that U is a contraction mapping on E δ . Step 3. V is a completely continuous mapping. Claim 1. The set {V y : y ∈ E δ } is uniformly bounded.
Let {y m } be a sequences such that y m → y in E δ as m → ∞.
Then for each t ∈ [0, T ], we have which means that V 1 is continuous. On the other hand, by the assumption (H 4 ), for any given number > 0, we can choose δ > 0 such that, for ϕ 1 , Choose m 0 ∈ Z + such that y m − y C([0,T ],R n ) < δ for all m > m 0 . Then we know that and, for any t ∈ [0, T ], Thus, one has which implies that V 2 is continuous on E δ . This shows that V is continuous on E δ and so we know that V is a completely continuous operator. Therefore, we show that mappings U and V satisfy all the conditions of Krasnoselskii fixed point theorem. Thus, Lemma 2.6 implies that U + V has a fixed point on E δ and so the IVP (4) has a solution This completes the proof.
Next, we will show the uniqueness result concerned with the solution of IVP (4) by employing the Banach fixed point theorem. To this end, we need the following hypotheses: The following inequality holds L * * := 2l 1 + ρl 2 + l 3 Γ(α + 1) T α < 1. Proof. According to the argument of Theorem 3.1, it suffices to prove that U + V has a unique fixed point on E δ . Obviously, U + V is a mapping from E δ into itself. Similar to the proof of Theorem 3.1, for any y 1 , y 2 ∈ E δ , one has It follows that and so where L * * < 1. By the Banach fixed point theorem, we know that U + V has a unique fixed point on E δ . This completes the proof.
4. An approximating algorithm. Based on the predictor-corrector scheme, Diethelm et al. [6][7][8] introduced Adams-Bashforth-Moulton algorithms for the numerical solution of differential equations of fractional order. They also provided detailed error analysis for the Adams-Bashforth-Moulton algorithms. Later, Bhalekar et al. [4] and Wang [31] modified this numerical method to solve delayed fractionalorder differential equations, respectively. Now, we take the advantage of the modified Adams-Bashforth-Moulton algorithm for solving IVP (4). To explain the method for constructing the approximating algorithm, we first state a brief introduction.
5. An application and a numerical example. First, we will illustrate how this type of system can be put into practice. In 1994, wardropian user equilibrium tatonnement model was constructed to consider idealized traveler information systems for day-to-day adjustments of flows and costs basis in the presence of information, as explained in detail by Friesz et al. [12], who showed that the network tatonnement model can be formulated as the following projective dynamical systems: where Λ is diagonal matrix, the projection operator P K ensures the feasibility onto the closed set of constraints K, and x(t) = (h(t), u(t)) , in which h(t) denotes the flow on day t and u(t) is the travel cost reported on day t. In some real systems, delay and perturbation can be recognised everywhere included traffic assignment models. It was showed that time delays can significantly influence the dynamics of transportation systems (see, for instance, [10,18,28,43] and the references therein). Thus, it is more suitable to consider the influence of delay in the tatonnement model, which means that M (x(t)) should be replaced by M (x(t − r)) in model (19). Moreover, there always exists a variety of unexpected factors (weather, accidents, special events, etc.) bringing perturbation to transportation systems. Hence, the perturbation item N (t, x(t − r)) is needed to add in model (19). In additional, in the travel information systems, we can obtain the travel costs of upcoming day by the estimation based on traffic flow of the previous days with the help of ATIS. Hence, we are able to believe that these adjustment processes have memory. It is worth mentioning that the fractional order model is more appropriate than integer order one to describe the problem involving a memory (see, for example, [1,19,25,27,29,30]). Thus, it is reasonable to replace the integer order derivative by the fractional order derivative in model (19). Therefore, by taking into account all the facts mentioned above, it is important and interesting to extend model (19) as the following form: where C 0 D α t is the Caputo fractional derivative of order α ∈ (0, 1]. In particular, we take Λ = I, where I is an identity matrix. Then model (20) is nothing but a form the fractional-order projective dynamical systems with delay and perturbation.
Next, we will give a numerical example to illustrate validity of the main results presented in Section 3.
By the assumptions, it is easy to check that