Nonlinear Multivalued Periodic Systems

We consider a first-order periodic system involving a time-dependent maximal monotone map, a subdifferential term, and a multivalued perturbation F(t, x). We prove existence theorems for the “convex” problem (that is, F is convex valued and for the “nonconvex” problem (that is, F is nonconvex valued). Also, we establish the existence of extremal trajectories (that is, solutions when the multivalued perturbation F(t, x) is replaced by ext F(t, x), the extreme points of F(t, x)). Also, we show that every solution of the convex problem can be approximated uniformly by certain extremal trajectories (“strong relaxation” theorem). Finally, we illustrate our result by examining a nonlinear periodic feedback control system.


Introduction
In the present work, we study the following nonlinear multivalued periodic system: −u (t) ∈ A(t, u(t)) + ∂ϕ(u(t)) + F (t, u(t)) for a.a. t ∈ T = [0, b] u(0) = u(b). (1.1) In this problem, A : T × R N −→ 2 R N \ ∅ is a multivalued map which is maximal monotone in x ∈ R N , ϕ ∈ 0 (R N ) (the cone of lower semicontinuous, convex, proper functions; see Section 2) with ∂ϕ being the subdifferential in the sense of convex analysis and F : T × R N −→ 2 R N \ ∅ is a multivalued perturbation. We prove existence theorems for problem (1.1) when F is convex valued ("convex problem") and when F is nonconvex valued ("nonconvex problem"). We also show the existence of extremal trajectories, that is, solutions of Eq. 1.1 when F (t, x) is replaced by ext F (t, x) (the set of extreme points of F (t, x)). Moreover, we show that every solution of the convex problem can be approximated in the C(T ; R N )-norm by certain extremal trajectories ("strong relaxation" theorem). An example of a feedback periodic control system illustrate our results.
Our work here is related to those of Frigon [5] and Qin and Xue [14]. In Frigon [5], ϕ ≡ 0 and A is time-independent with D(A) = R N . The author proves existence theorems for both the convex and nonconvex problems using the notion of L p -solution tube. Qin and Xue [14] assume that that A is time-independent and is a positive definite N × N -matrix. Also, they assume that ϕ : R N −→ R is continuous convex. They deal with the convex and nonconvex problems and also address the question of existence of extremal trajectories. Finally, we mention the work of Bader and Papageorgiou [1], where A ≡ 0, but the inclusion takes place in the context of a general separable Hilbert space.

Mathematical Background-Hypotheses
Our approach is based on tools from multivalued analysis (see Hu-Papageorgiou [11]) and from the theory of nonlinear operators of monotone type (see  and Zeidler [15]).
Let ( , ) be a measurable space and X a separable Banach space. We use the following notation: P f (c) (X) = {A ⊆ X : A is nonempty, closed (and convex)} P (w)k(c) (X) = {A ⊆ X : A is nonempty, (w-)compact (and convex)}.
A multifunction (set-valued function), F : −→ 2 X \ ∅ is said to be "graph measurable," if Gr F ∈ ⊗ B(X), where with B(X) being the Borel σ -field of X. If = , the universal σ -field (this is the case if there is a σ -finite measure μ on and is μ-complete), then the Yankov-von Neumann-Aumann selection theorem (see Hu and Papageorgiou [11,p.158] or Gasiński and Papageorgiou [7,p. 906]) says that every graph measurable multifunction F : −→ 2 X \ ∅ admits a measurable selection, that is, there exists a -measurable function f : −→ X such that f (ω) ∈ F (ω) for all ω ∈ . In fact, there is a whole sequence {f n : −→ X} n 1 of measurable selections such that F (ω) ⊆ {f n (ω)} n 1 ∀ω ∈ .
The result is true if the separable Banach space X is replaced by a Souslin space. Recall that a Souslin space is always separable but need not be metrizable. For example, if X * is the dual of a separable Banach space and it is equipped with the w * -topology, then it is a nonmetrizable Souslin space.
A multifunction F : −→ P f (X) is said to be "measurable", if for all x ∈ X, the function ω −→ d(x, F (ω)) = inf u∈F (ω) x − u is -measurable. This is equivalent to saying that for every open set U ⊆ X, the set A measurable multifunction F : −→ P f (X) is graph measurable. The converse is true if there is a σ -finite, complete measure defined on . Now, let ( , , μ) be a σ -finite measure space and X a separable Banach space. Given 1 p +∞ and a graph measurable multifunction F : −→ 2 X \ ∅, we define S p F = {f ∈ L p ( ; X) : f (ω) ∈ F (ω) μ − a.e.}. A straightforward application of the Yankov-von Neumann-Aumann selection theorem, reveals that "S p F = ∅ if and only if ω −→ inf{ u : u ∈ F (ω)} belongs in L p ( )." This set is "decomposable," that is, if (A, f 1 Here, for C ∈ , χ C denotes the characteristic function of C, hence Since χ ω\C = 1 − χ C , we see that the notion of decomposability formally looks very similar to that of convexity, only now the coefficients in the linear combination are functions. In fact, decomposable sets exhibit properties which are similar the those of convex sets (see Hu and Papageorgiou [11,Section 2.3]).
Suppose that Y and Z are Hausdorff topological spaces and let G : Y −→ 2 Z \ ∅ be a multifunction. We say that G is "upper semicontinuous" if for every open set U ⊆ Z, the set G An upper semicontinuous multifunction with closed values has closed graph. The converse is true, if G is locally compact (that is, for every x ∈ X, there exists a neighborhood U of x such that G(U ) ∈ P k (Z)). If Z is a metric space, then G is lower semicontinuous if and only if for For a metric space Z (with metric d Z ) on P f (Z), we can define a generalized metric, known as the "Hausdorff metric", by setting Suppose that V is a Banach space and C ⊆ V is nonempty. We set and wlim sup n→+∞ C n = {u ∈ V : u = w lim k→+∞ u n k , u n k ∈ C n k , n 1 < n 2 < . . .}.
Next, let X be a reflexive Banach space and X * its topological dual. By ·, · , we denote the duality brackets for the pair (X * , X). A multivalued map A : X ⊇ D −→ 2 X * is said to be "monotone", if Here, D = {x ∈ X : A(x) = ∅}, the "domain" of A.
We say that a monotone map is "strictly monotone," if This means that Gr A is maximal with respect to inclusion among the graphs of monotone maps. It is easy to see that, if A : X ⊇ D −→ 2 X * is maximal monotone, then Gr A is sequentially closed in X w × X * and in X × X * w (here, by X w and X * w , we denote the spaces X and X * , respectively, furnished with the weak topology). If A is maximal monotone, then for every x ∈ D, A(x) ∈ P f c (X * ).
For a maximal monotone map A : X −→ 2 X * , we define Since for every x ∈ D, A(x) ∈ P f c (X * ) and X is reflexive, then A 0 (x) = ∅. Moreover, if X * is strictly convex, then A 0 is single-valued. The map A 0 is called the "minimal section" of A.
The "duality map" F : X −→ 2 X * is defined by The Hahn-Banach theorem implies that F has nonempty values. In fact, the duality map is defined for any Banach space. However, its properties strongly depend on the geometry of the Banach space X. In particular, if X and X * are both locally uniformly convex, then F is single-valued and a homeomorphism. By 0 (X), we denote the cone of all functions ϕ : X −→ R = R ∪ {+∞} which are lower semicontinuous, convex, and proper (that is, dom ϕ = {x ∈ X : ϕ(x) < ∞} (the effective domain of ϕ) is nonempty). By ∂ϕ : X −→ 2 X * , we denote the subdifferential of ϕ in the sense of convex analysis, that is If ϕ is continuous at u, then ∂ϕ(u) = ∅. If ϕ is Gâteaux differentiable at u, then ∂ϕ(u) = {ϕ G (u)} (ϕ G (u) denotes the Gâteaux derivative of ϕ at u). The map ∂ϕ : X −→ 2 x * is maximal monotone.
By L 1 w (T ; R N ), we denote the Lebesgue space L 1 (T ; R N ) equipped with the weak norm or equivalently by This norm is equivalent to the Pettis norm (see Egghe [4]). The hypotheses on the map A and on the function ϕ are the following: (iii) For every r > 0, there exists a r ∈ L 2 (T ) such that

Remark 2.1 Hypothesis H (A)(i) implies that
Hypothesis H (A)(ii) imposes restrictions on the t-dependence of A and permits the use of the theory of evolution equations involving time-dependent operators (see Pavel [13]).
H (ϕ): ϕ ∈ 0 (R N ) with 0 ∈ D(∂ϕ) and D(∂ϕ) = intdom ϕ or ϕ is bounded above on bounded sets. Remark 2.2 Both conditions in the above hypothesis imply that u −→ ∂ϕ(u) is bounded (that is, maps bounded sets to bounded sets). The following function ϕ satisfies the first condition, namely that D(∂ϕ) = intdom ϕ Next, we prove a result which we will need in the sequel and which is of independent interest. For this reason, it is formulated in a more general setting than the one in which it will be used in this paper. We mention that the result is known for Hilbert spaces (see Brézis [3, p. 25]).
So, as before, let X be a reflexive Banach space and X * its topological dual. By | · | and | · | * , we denote the norm on X and X * respectively and by ·, · the duality brackets for the pair (X * , X). Let A : X −→ 2 X * be a maximal monotone map with 0 ∈ A(0). On account of the Troyanski renorming theorem (see, e.g., Gasiński-Papageorgiou [7, p. 911]), without any loss of generality, we may assume that both X and X * are locally uniformly convex.
Let h ∈ L p (T ; X * ) and consider the multifunction K : T −→ 2 X defined by The map x −→ A(x) + |x| p−2 F is maximal monotone and coercive. Hence, it is surjective (see Gasiński and Papageorgiou [7,p. 336]). Therefore, K(t) = ∅ for all t ∈ T \ N , with N being Lebesgue-null. On this exceptional null set, we set K(t) = {0}. Note that We know that the maximal monotonicity of A implies that Gr A ⊆ X × X * is closed. Moreover, the map ξ : T × X −→ X × X * defined by is a Carathéodory mao, that is, for all x ∈ X, t −→ ξ(t, x) is measurable, while for almost all t ∈ T , x −→ ξ(t, x) is continuous. We know that ξ is jointly measurable (see Hu and Papageorgiou [11, p. 142]). Hence with L T being the Lebesgue σ -field on T and B(X) the Borel σ -field of X. Invoking the Yankov-von Neumann-Aumann selection theorem, we can find a measurable map u : ) for a.a. t ∈ T . We act with u(t) ∈ X and recall that by hypothesis 0 ∈ A(0), we obtain This proves the Claim.
Evidently the map A is monotone. We will show that in fact it is maximal monotone. To this end, suppose that (v, g) ∈ L p (T ; X) × L p (T ; X * ) and assume that On account of the Claim, we can find (u 1 , h 1 ) ∈ Gr A such that We return to Eq. 2.1 and choose (u, h) = (u 1 , h 1 ). Then, using Eq. 2.2, we have

The Convex Problem
In this section, we prove an existence theorem for problem (1.1) when the multivalued perturbation F is convex valued. The precise hypotheses on F are the following: To show the nonemptiness of this set, we argue as follows. Let {s n } n 1 be a sequence of step functions such that s n (t) −→ u(t) for almost all t ∈ T and |s n (t)| |u(t)| for almost all t ∈ T , all n ∈ N. Then, hypothesis H (F ) 1 implies that for every n ∈ N, t −→ F (t, s n (t)) is measurable and so by the Yankov-von Neumann-Aumann selection theorem, we can find h n : T −→ R N measurable such that h n (t) ∈ F (t, s n (t)) for a.a. t ∈ T , all n N, Therefore, by passing to a subsequence if necessary, we may assume that with h ∈ L 2 (T ; R N ). Invoking Proposition 3.9 of Hu and Papageorgiou [11, p. 694 . Hypothesis H (F ) 1 (ii) is a multivalued variant of a condition first used by Hartman [10].
Together with H (F ) 1 , we will need the following extra condition on ∂ϕ: Alternatively, instead of H (F ) 1 , H 0 , we can use the following conditions on F : is a multifunction such that hypotheses H (F ) 1 (i) and (ii) are the same as the corresponding hypotheses H (F ) 1 (i) and (ii) and We know that D(∂ϕ) = dom ϕ (see Hu and Papageorgiou [11, p.346]). Let x 0 ∈ dom ϕ, h ∈ L 2 (T ; R N ) and ε > 0. We consider the following auxiliary Cauchy problem: We have the following existence and uniqueness theorem for this problem.

Proposition 3.2 If hypotheses H (A) and H (ϕ) hold, then problem
. Therefore, we can apply Theorem 1.2 of Pavel [13] and have a solution u 0 ∈ C(T ; R N ) of problem (3.1). We have u 0 (t) ∈ D(∂ϕ) for all t ∈ T and recall that on account of hypothesis H (ϕ), ∂ϕ maps bounded sets to bounded sets. Therefore, Next, we show that this solution is unique. So, suppose that We take inner product with u 0 (t) − v 0 (t). The monotonicity of A(t, ·) and ∂ϕ implies that This proves the uniqueness of the solution of problem (3.1).
We consider the Poincaré map P : dom ϕ −→ dom ϕ defined by

Proposition 3.3 If hypotheses H (A) and H (ϕ) hold, then the Poincaré map P is a contraction.
Proof Let x 0 , x ∈ dom ϕ be two distinct initial conditions for problem (3.1) and let u 0 , u ∈ W 1,p ((0, b); R N ) be the corresponding unique solutions of the Cauchy problem. We have As in the proof of Proposition 3.2, we subtract (3.6) from (3.5) and then take inner product with u 0 (t) − u(t) to obtain , we consider the following auxiliary periodic system:
Proof From Proposition 3.3, we know that the Poincaré map P : dom ϕ −→ dom ϕ is a contraction. So, by the Banach fixed point theorem, there is a unique u 0 ∈ dom ϕ such that The corresponding solution u 0 ∈ W 1,2 ((0, b); R N ) of Eq. 3.1 is the unique solution of the periodic system (3.7). Then on account of hypothesis We act with u 0 (t) and recall that 0 ∈ A(t, 0) for all t ∈ T , we obtain 1 2 In Eq. 3.9, we choose t = b. Using the periodic boundary condition, we have (3.10) We return to Eq. 3.9 and use Eq. 3.10. Then, Evidently, F (t, x) satisfies hypotheses H (F ) 1 (i) and (ii) and in addition, we have In the sequel by S ε , we denote the solution set of the following periodic system By S ε , we denote the solution set of In the next proposition, we derive uniform a priori bounds for the elements of these two solution sets. Proof (a) Let u ∈ S ε ⊆ W 1,2 ((0, b); R N ). We have with h ∈ S 2 F (·,u(·)) . Suppose that the result is not true. Then, two situations can occur: Suppose that (I) holds. From Eq. 3.11 for some a ∈ S 2 A(·,u(·)) and some g ∈ S 2 ∂ϕ(u(·)) , we We take inner product with u(t). Then, Since a ∈ S 2 A(·,u(·)) and 0 ∈ A(t, 0) for all t ∈ T , we have (a(t), u(t)) R N 0 for a.a. t ∈ T .
On account of Proposition 3.5, we can always replace F (t, x) by F (t, x) = F (t, p M (x)). Therefore, without any loss of generality, we may assume that with ϑ ∈ L 2 (T ). Proposition 3.4 implies that we can define the solution map γ ε : L 2 (T ; R N ) −→ C(T ; R N ) which to every h ∈ L 2 (T ; R N ) assigns the unique solution γ ε (h) ∈ W 1,2 ((0, b); R N ) ⊆ C(T ; R N ).

Proposition 3.6 If hypotheses H (A) and H (ϕ) hold, then the solution map
Proof Let h n w −→ h in L 2 (T ; R N ) and set u n = γ ε (u n ) ∈ W 1,2 ((0, b); R N ) ⊆ C(T ; R N ) for all n ∈ N and u = γ ε (h) ∈ W 1,2 ((0, b); R N ) ⊆ C(T ; R N ). We have 0 ∈ u n (t) + A(t, u n (t)) + ∂ϕ(u n (t)) + εu n (t) + h n (t) for a.a. t ∈ T u n (0) = u n (b), n ∈ N. (3.14) and As before (see the proof of Proposition 3.2), subtracting (3.15) from Eq. 3.14 and taking inner product with u n (t) − u(t), we obtain In Eq. 3.16, we pass to the limit as n → +∞ and use Eq. 3.20. Then, so u = u. Therefore, for the original sequence, we have u n −→ u in C(T ; R N ); thus, γ ε is completely continuous.

The Nonconvex Problem
In this section, we prove an existence theorem for the "nonconvex problem" (that is, the multivalued perturbation has nonconvex values).
In this case, the hypotheses on the multivalued perturbation F are the following: x) is lower semicontinuous.
(iii) There exist M > 0 and a M ∈ L 2 (T ) such that

|F (t, x)| a M (t) for a.a. t ∈ T , |x| M.
As before (see Section 3), these hypotheses will be combined with H 0 . Alternatively, instead of the pair H (F ) 2 , H 0 , we can use the following hypotheses on F : is a multifunction such that hypotheses H (F ) 2 (i) and (ii) are the same as the corresponding hypotheses H (F ) 2 (i) and (ii) and H (F ) 2 , H 0 , or H (F ) 2 hold, then problem  (1.1) admits a solution u 0 ∈ W 1,2 ((0, b)

Theorem 4.1 If hypotheses H (A), H (ϕ) and
Proof The a priori bounds in Proposition 3.5 remain valid and so without any loss of generality, we may assume that Again, we introduce the sets F (·,u(·)) ∀u ∈ C(T ; R N ). We show that V is a lower semicontinuous multifunction. According to Proposition 2.6 of Hu and Papageorgiou [11, p. 37], to show the lower semicontinuity of V , it suffices to show that, if u n −→ u in C(T ; H ), then So, suppose that u n −→ u in C(T ; H ) and h ∈ V (u). (4.2) For n ∈ N, we consider the multifunction G n : T −→ P k (R N ) defined by Hypothesis is a Carathéodory function. We know that Carathéodory functions are jointly measurable (see Hu and Papageorgiou [11,p. 142]). Therefore, Invoking the Yankov-von Neumann-Aumann selection theorem, we can find a measurable function h n : T −→ R N such that h n (t) ∈ G n (t) for a.a. t ∈ T , all n ∈ N.

|h(t) − h n (t)| d(h(t), F (t, u n (t)))
with h n ∈ V (u n ) for all n ∈ N, thus, V is lower semicontinuous. Clearly, V has decomposable values. Applying the Bressan-Colombo-Fryszkowski selection theorem (see Bressan and Colombo [2], Fryszkowski [6], and Hu and Papageorgiou [11, p. 245, Theorem 8.7]), we can find a continuous map v : We set τ ε = γ ε • v : K ε c −→ K ε c . Evidently, τ ε is continuous (see Proposition 3.6). Since K ε c ∈ P kc (C(T ; R N )), we can apply the Schauder fixed point theorem and produce u ε 0 ∈ K ε c such that u ε 0 = τ ε (u ε 0 ), so u ε 0 ∈ W 1,2 ((0, b); R N )) is a solution of problem (3.21). Now, consider a sequence ε n → 0 + and let u n = u ε n 0 for n ∈ N. From Eqs. 3.8 and 4.1, we see that there exists c ε > 0 such that This implies that Eqs. 3.18 and 3.19 are valid and hence the sequence For all t, s ∈ T , s t, by the Cauchy-Schwarz inequality, we have for some c 4 > 0, so, the sequence {u n } n 1 is equicontinuous. The Arzela-Ascoli theorem implies that the sequence {u ε n 0 } n 1 ⊆ C(T ; R N ) is relatively compact.
So, passing to a subsequence if necessary, we may assume that Exploiting the continuity of V and reasoning as in the last part of the proof of Theorem 3.7, in the limit as n → +∞, we obtain −u 0 (t) ∈ A(t, u 0 (t)) + ∂ϕ(u 0 (t)) + F (t, u 0 (t)) for a.a. t ∈ T u 0 (0) = u 0 (b), is a solution of Eq. 1.1.

Extremal Trajectories
In this section, we deal with the following version of problem (1.1): Here, by ext F (t, x), we denote the set of extreme points of F (t, x). We know that even if F (t, ·) has strong continuity properties, the multifunction x −→ ext F (t, x) need not have any (see Hu and Papageorgiou [11,Section 2.4]). So, the existence of solutions for problem (5.1) cannot be deduced from Theorems 3.7 and 4.1 and a different approach is needed. We need to strengthen the conditions on the multivalued perturbation F (t, x). The new hypotheses are the following: As before, H (F ) 3 will be combined with H 0 . Alternatively, we can replace the pair H (F ) 3 , H 0 with the following hypotheses: 3 : F : T × R N −→ P kc (R N ) \ ∅ is a multifunction such that hypotheses H (F ) 3 (i) and (ii) are the same as the corresponding hypotheses H (F ) 3 (i) and (ii) and H (F ) 3 , H 0 , or H (F ) 3 hold, then problem

Strong Relaxation
In this section, we show that every solution of the convex problem can be obtained as the limit in the C(T ; R N )-norm of certain extremal trajectories. Such a result is known as "strong relaxation." The result is important in many applications. In the context of control systems, it says that we can approximate any state of the system by states which are generated using "bang-bang controls." This way, we can economize in the use of controls. In the context of game theory, the selection of ext F (·, u(·)) are known as "pure strategies" and the strong relaxation theorem implies that any state can be approximated by ones generated using only pure strategies.
To prove such a result, we need to strengthen further the conditions on F . The new hypotheses are the following: (F (t, x), F (t, y)) η r (t)|x − y| for almost all t ∈ T , all x, y ∈ R N with |x|, |y| r. (iii) Tthere exist M > 0 and a M ∈ L 2 (T ) such that These hypotheses go together with H 0 . Alternatively, the pair H (F ) 4 , H 0 can be replaced by the following hypotheses: and (ii) are the same as the corresponding hypotheses H (F ) 4 (i) and (ii) and In what follows by S c , we denote the solution set of the convex problem (that is, in Eq. 1.1, F (·, ·) has values in P kc (R N )). From Proposition 3.8, we know that S c ∈ P k (C(T ; R N )).
Let x 0 ∈ R N and consider the following Cauchy problem: −u (t) ∈ A(t, u(t)) + ∂ϕ(u(t)) + ext F (t, u(t)) for a.a. t ∈ T u(0) = x 0 . Proof From the previous work, we know that without any loss of generality, we may assume that We know that S c ∈ P k (C (T ; R N )). Similarly, S e (u(0)) C(T ;H ) ∈ P k (C(T ; R N )). We set K * = conv (S c ∪ S e (u(0))) ∈ P kc (C(T ; R N )).
Let S e ⊆ W 1,2 ((0, b); R N ) ⊆ C(T ; R N ) be the solution set of problem (5.1). From Theorem 3.7, we know that S e = ∅. If we strengthen the conditions on A(t, ·), we can show that the set S e is dense in S c for the C(T ; R N )-norm topology.
The stronger conditions on A are the following: Proof We follow the proof of Theorem 6.1, using this time instead of problem (6.7), the periodic problem (that is, the boundary condition in Eq. 6.7 will be u(0) = u(b)). So, {u n } n 1 ⊆ S e . Using the periodic boundary condition and hypothesis H (A) (iv), we have (see Eq (see Eqs. 6.20 and 6.8), so u = u (see hypothesis H (A) (iv)). Therefore, u = lim n→+∞ u n in C(T ; R N ) with u n ∈ S e for all n ∈ N (see Eq. 6.8).
Also, R N x −→ ∂|x| denotes the subdifferential in the sense of convex analysis. We know that ∂|x| = x |x| if x = 0, B 1 if x = 0, (7.2) with B 1 = {x ∈ R N : |x| 1} (see Gasiński and Papageorgiou [7]). The function f : T × R N −→ R N is measurable in t ∈ T and locally L 1 (T )-Lipschitz in x ∈ R, that is, for every M > 0, there exists η 1 M ∈ L 1 (T ) such that |f (t, x) − f (t, y)| η 1 M (t)|x − y| for a.a. t ∈ T , all x, y ∈ R N , |x|, |y| M . The function v : T −→ R m is the control function and C : T × R N −→ P kc (R m ) is the control constraint multifunction. The dependence of C on x ∈ R N implies that there is a priori feedback in the system. We assume that • For all x ∈ R N , t −→ C(t, x) is graph measurable.
• For almost all t ∈ T , x −→ C(t, x) is locally L 1 (T ) h-Lipschitz, that is, for every M > 0, there exists η 2 M ∈ L 1 (T ) such that h(C(t, x), C(t, y)) η 2 M (t)|x − y| for a.a. t ∈ T , all |x|, |y| M . Also, L is an N × m-matrix and we assume that there exist M > 0 and a M ∈ L 2 (T ) such that We set F (t, x) = f (t, x) + L(C(t, x)) ∀(t, x) ∈ T × R N .