Nội dung

FINITE DIFFERENCE SCHEMES WITH MONOTONE OPERATORS N. C. APREUTESEI Received 13 October 2003 and in revised form 10 December 2003 To the memory of my mother, Liliana Several existence theorems are given for some second-order difference equations associated with maximal monotone operators in Hilbert spaces. Boundary conditions of monotone type are attached. The main tool used here is the theory of maximal monotone operators. 1. Introduction In [1, 2], the authors proved the existence of the solution of the boundary value problem p(t)u

(t) + r(t)u
(t) ∈ Au(t) + f (t),

 u (0) ∈ α u(0) − a ,
a.e. on [0, T], T > 0,
 u (T) ∈ −β u(T) − b , (1.1) (1.2) where A : D(A) ⊆ H → H, α : D(α) ⊆ H → H, and β : D(β) ⊆ H → H are maximal monotone operators in the real Hilbert space H (satisfying some specific properties), a, b are given elements in the domain D(A) of A, f ∈ L2 (0,T;H), and p,r : [0,T] → R are continuous functions, p(t) ≥ k > 0 for all t ∈ [0,T]. Particular cases of this problem were considered before in [9, 10, 12, 15, 16]. If p ≡ 1, r ≡ 0, f ≡ 0, T = ∞, and the boundary conditions are u(0) = a and sup{u(t),t ≥ 0} < ∞ instead of (1.2), the solution u(t) of (1.1), (1.2) defines a semigroup of nonlinear contractions {S1/2 (t), t ≥ 0} on the closure D(A) of D(A) (see [9, 10]). This semigroup and its infinitesimal generator A1/2 have some important properties (see [9, 10, 11, 12]). A discretization of (1.1) is pi (ui+1 − 2ui + ui−1 ) + ri (ui+1 − ui ) ∈ ki Aui + gi , i = 1,N, where N is a given natural number, pi ,ri ,ki > 0, gi ∈ H. This leads to the finite difference scheme

 pi + ri ui+1 − 2pi + ri ui + pi ui−1 ∈ ki Aui + gi ,
 u1 − u 0 ∈ α u 0 − a ,
i = 1, N,  uN+1 − uN ∈ −β uN+1 − b , (1.3) (1.4) where a,b ∈ H are given, (pi )i=1,N , (ri )i=1,N , and (ki )i=1,N are sequences of positive numbers, and (gi )i=1,N ∈ H N . Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:1 (2004) 11–22 2000 Mathematics Subject Classification: 39A12, 39A70, 47H05 URL: http://dx.doi.org/10.1155/S1687183904310046 12 Finite difference schemes with monotone operators In this paper, we study the existence and uniqueness of the solution of problem (1.3), (1.4) under various conditions on A, α, and β. The case pi ≡ 1, ri ≡ 0, gi ≡ 0 was discussed in [14] for the boundary conditions u0 = a and uN+1 = b. These boundary conditions can be seen as a particular case of (1.4) with α = β = ∂ j (the subdifferential of j), where j : H → R is the lower-semicontinuous, convex, and proper function:  0, x = 0, j(x) =  +∞, otherwise. (1.5) In [6, 8, 13, 14], one studies the existence, uniqueness, and asymptotic behavior of the solution of the difference equation

 pi + ri ui+1 − 2pi + ri ui + pi ui−1 ∈ ki Aui + gi , i ≥ 1, (1.6) (pi ≡ 1, ri ≡ 0 in [13, 14] and the general case in [6, 8]), subject to the boundary conditions u0 = a,   sup ui  < ∞. i≥0 (1.7) Here  ·  is the norm of H. In [7], the author establishes the existence for problem (1.3), (1.4) under the hypothesis that A is also strongly monotone. Other classes of difference or differential inclusions in abstract spaces are presented in [3, 4, 5]. In Section 2, we recall some notions and results that we need to show our main existence theorems. They are stated in Section 3 and represent the discrete version of some results obtained in [1, 2] for the continuous case. 2. Preliminary results In this section, we recall some fundamental elements on nonlinear analysis we need in this paper. If H is a real Hilbert space with the scalar product (·, ·) and the norm  · , then the operator A ⊆ H × H (with the domain D(A) and the range R(A)) is called a monotone operator if (x − x
, y − y
) ≥ 0 for all x,x
∈ D(A), y ∈ Ax, and y
∈ Ax
. The monotone operator A ⊆ H × H is said to be maximal monotone if it is not properly enclosed in a monotone operator. A basic result of Minty (see [11, Theorem 1.2, page 9]) asserts that A is maximal monotone if and only if A is monotone and the range of A + λI is the whole space H for all λ > 0 (or equivalently, for only one λ0 > 0). It is also known that a maximal monotone and coercive operator A is surjective, that is, its range R(A) is H. For all x ∈ D(A), we denote by A0 x the element of least norm in Ax:  0    A x = inf  y , y ∈ Ax . (2.1) If A is maximal monotone and A0 x → ∞ as x → ∞, then A is surjective. The operator A ⊆ H × H (possibly multivalued) is said to be one to one if (Ax1 ) ∩ (Ax2 ) = Φ (with x1 , x2 ∈ D(A)) implies x1 = x2 . N. C. Apreutesei 13 If A and B are maximal monotone in H and their domains satisfy the condition (intD(A)) ∩ D(B) = Φ, then A + B is maximal monotone (see [11, Theorem 1.7, page 46]). If A : D(A) ⊆ H → H is maximal monotone, then A is demiclosed, that is, from [xn , yn ] ∈ A, xn
x and yn → y, then [x, y] ∈ A. Here and everywhere below, we denote by “
” the weak convergence and by “→” the strong convergence in H. For every maximal monotone operator A and the scalar λ > 0, we may consider the single-valued and everywhere-defined operators Jλ and Aλ , namely, Jλ = (I + λA)−1 and Aλ = (I − Jλ )/λ. They are called the resolvent and the Yosida approximation of A, respectively. Obviously, we have Jλ x + λAλ x = x for all x ∈ H and for all λ > 0. Properties of these operators can be found in, for example, [11, Proposition 1.1, page 42] or [11, Proposition 3.2, page 73]. Recall now another result concerning the sum of two maximal monotone operators (see [11, Theorem 3.6, page 82]). Theorem 2.1. If A : D(A) ⊆ H → H and B : D(B) ⊆ H → H are maximal monotone operators in H such that D(A) ∩ D(B) = Φ and (y,Aλ x) ≥ 0 for all [x, y] ∈ B and for all λ > 0, then A + B is maximal monotone. We end this section with some remarks on problem (1.3), (1.4). Denoting θi = pi , pi + ri ki , pi + ri ci = fi = gi , pi + ri i = 1,N, (2.2) problem (1.3), (1.4) becomes
 ui+1 − 1 + θi ui + θi ui−1 ∈ ci Aui + fi ,
 u1 − u0 ∈ α u0 − a , i = 1,N,
 uN+1 − uN ∈ −β uN+1 − b . (2.3) If pi ,ri ,ki > 0, i = 1,N, then θi ∈ (0,1) and ci > 0 for all i = 1,N. Let (ai )i=1,N be the finite sequence given by a0 = 1, ai = 1 , θ1 · · · θi i = 1,N, (2.4) and let ᏸ be the product space H N = H × · · · × H (N factors) endowed with the scalar product
 ui
 i=1,N , vi i=1,N N
 ai ui ,vi . = (2.5) i=1 It is clear that H N and ᏸ coincide as sets and their norms are equivalent. Observe that ai θi = ai−1 , i = 1,N. (2.6)
 
= − ui+1 + 1 + θi ui − θi ui−1 i=1,N ,

 ∈ H N , u1 − u0 ∈ α u0 − a , uN+1 − uN ∈ −β uN+1 − b . (2.7) Consider the operator B in H N × H N :
 B ui D(B) = 
 ui i=1,N i=1,N 14 Finite difference schemes with monotone operators This operator is not necessarily monotone in H N , but we have the following auxiliary result (see [7, Proposition 2.1]). Proposition 2.2. The operator B given above is maximal monotone in ᏸ. Recall here an existence theorem from [7], which we use in the sequel. Theorem 2.3. Assume that A, α, and β are maximal monotone operators in H with 0 ∈ D(A) ∩ D(α) ∩ D(β), A is also strongly monotone and
 Aλ x − Aλ y,z ≥ 0 (2.8) for all z ∈ α(x − y) (with x − y ∈ D(α)) and for all z ∈ β(x − y) (with x − y ∈ D(β)). If θi ∈ (0,1), ci > 0, fi ∈ H, i = 1,N, and a,b ∈ H, then problem (2.3) has a unique solution (ui )i=1,N ∈ D(A)N . 3. Existence theorems Let H be a real Hilbert space with the norm  ·  and the scalar product (·, ·). Consider the maximal monotone operators A : D(A) ⊆ H → H, α : D(α) ⊆ H → H, and β : D(β) ⊆ H → H satisfying the properties 0 ∈ D(A) ∩ D(α) ∩ D(β),
 0 ∈ α(0) ∩ β(0), (3.1) Aλ x − Aλ y,z ≥ 0 ∀z ∈ α(x − y) with x − y ∈ D(α), (3.2) Aλ x − Aλ y,z ≤ 0 ∀z ∈ −β(x − y) with x − y ∈ D(β). (3.3)
 Consider the difference inclusion (1.3), (1.4). As we have already discussed, problem (1.3), (1.4) has the equivalent form (2.3). We first study the existence of the solution to problem (1.3), (1.4) in the case a = b = 0, supposing that
 Aλ x,z ≥ 0 ∀z ∈ α(x) with x ∈ D(α), z ∈ β(x) with x ∈ D(β), (3.4) and R(α) is bounded,  0  β (x) −→ ∞ as x −→ ∞, (3.5) R(β) is bounded,  0  α (x) −→ ∞ as x −→ ∞. (3.6) or Theorem 3.1. Let A, α, and β be maximal monotone operators in the real Hilbert space H such that (3.1), (3.4), and (3.5) or (3.6) hold. If pi ,ri ,ki > 0, i = 1,N, and (gi )i=1,N ∈ H N , then the boundary value problem

 pi + ri ui+1 − 2pi + ri ui + pi ui−1 ∈ ki Aui + gi ,
 u1 − u0 ∈ α u0 ,
i = 1,N,  uN+1 − uN ∈ −β uN+1 , (3.7) N. C. Apreutesei 15 has at least one solution (ui )i=1,N ∈ D(A)N . The solution is unique up to an additive constant. If A or α is one to one, then the solution is unique. If A is, in addition, strongly monotone, then again uniqueness is obtained. Proof. We use the form (2.3) of the problem (3.7), where a = b = 0. By Proposition 2.2, we know that the operator
 B ui D(B) = 
 ui

  = − ui+1 + 1 + θi ui − θi ui−1 i=1,N ,

 ∈ H N , u1 − u0 ∈ α u0 , uN+1 − uN ∈ −β uN+1 i=1,N i=1,N (3.8) is maximal monotone in ᏸ. Denote by | · | the norm in ᏸ. We show that 
    B ui i=1,N  −→ ∞ 
  as  ui   i=1,N  −→ ∞. (3.9) Suppose by contradiction that (uni )i=1,N ∈ D(B) such that |(uni )i=1,N | → ∞ as n → ∞ and |B((uni )i=1,N )| ≤ C1 . If (ai )i=1,N is the sequence given in (2.4), this means that N  i=1 N 2 ai uni  −→ ∞, i=1 
2 ai uni+1 − uni − θi uni − uni−1  ≤ C1 . (3.10) Assume that (3.5) holds. By the boundary conditions in (3.7), we obtain that un1 − un0 is bounded, say un1 − un0  ≤ C2 , for all n ∈ N and  n u  n N+1 − uN −→ ∞   as n −→ ∞ if unN+1  −→ ∞. (3.11)  The equality ai (uni+1 − uni ) = un1 − un0 + ik=1 [ak (unk+1 − unk ) − ak−1 (unk − unk−1 )] implies that ai (uni+1 − uni ) ≤ C2 + C3 |B((ui )i=1,N )| and in view of (3.10), we get ai (uni+1 − uni ) ≤ C4 , i = 1,N, n ∈ N. In particular, aN (unN+1 − unN ) ≤ C4 for all n ∈ N and from (3.11), we infer that unN+1  ≤ C5 for all n ∈ N. Using the boundedness of unN+1 and ak (unk+1 − unk ) and the identity uni = unN+1 − N k=i
 unk+1 − unk , i = 1,N, (3.12)  one arrives at uni  ≤ C6 , hence Ni=1 ai uni 2 ≤ C7 for all n ∈ N. But this is in contradiction with (3.10) and therefore (3.9) is true. This shows that B is coercive. Next we show that
 B ui i=1,N
, Aλ ui  i=1,N ≥0
 ∀ ui i=1,N ∈ D(B), λ > 0. (3.13) 16 Finite difference schemes with monotone operators Indeed,
 B ui
i=1,N , Aλ ui N  i=1,N =− i=1 

ai ui+1 − ui ,Aλ ui − ai−1 ui − ui−1 ,Aλ ui−1 N
ai−1 ui − ui−1 ,Aλ ui − Aλ ui−1 +   (3.14) i=1

 ≥ −aN uN+1 − uN ,Aλ uN + u1 − u0 ,Aλ u0 . Hypothesis (3.4) for x = u0 and z = u1 − u0 gives us (u1 − u0 , Aλ u0 ) ≥ 0, while (3.4) for x = uN+1 and z = −uN+1 + uN implies that −(uN+1 − uN ,Aλ uN ) = (uN+1 − uN ,Aλ uN+1 − Aλ uN ) − (uN+1 − uN ,Aλ uN+1 ) ≥ 0. Thus, by (3.14), inequality (3.13) follows. Let Ꮽ : D(A)N → H N be the operator
 Ꮽ ui
 = c1 v1 ,...,cN vN , i=1,N vi ∈ Aui , ui ∈ D(A), i = 1,N. (3.15) Since (0,...,0) ∈ D(Ꮽ) ∩ D(B) and (3.13) takes place, we deduce with the aid of Theorem 2.1 and Proposition 2.2 the maximal monotonicity of B + Ꮽ in ᏸ. Next, we can easily show that B((ui )i=1,N ), Ꮽ((ui )i=1,N ) ≥ 0, so |(B + Ꮽ)(ui )i=1,N | ≥ |B((ui )i=1,N )|, and from (3.9), one obtains the coercivity of B + Ꮽ. This shows that B + Ꮽ is surjective, that is, for all ( fi )i=1,N ∈ H N , there exists (ui )i=1,N ∈ D(Ꮽ) ∩ D(B) such that (B + Ꮽ)((ui )i=1,N ) = (− fi )i=1,N . But this is the abstract form of (3.7). Thus the existence is proved. We show now that the difference of the two solutions (ui )i=1,N and (vi )i=1,N of (3.7) is a constant. Put wi = ui − vi , i = 0,N + 1. Subtracting the corresponding equations of (2.3) for ui and vi , multiplying by ai wi , and summing from i = 1 to i = N, one arrives with the aid of the monotonicity of A at N i=1 

ai wi+1 − wi ,wi − ai θi wi − wi−1 ,wi  ≥0 (3.16) or, in view of (2.6), at N i=1 

ai wi+1 − wi ,wi − ai−1 wi − wi−1 ,wi−1  N ≥ i=1  2 ai−1 wi − wi−1  . (3.17) By the boundary conditions in (2.3), we have N i=1  2

 ai−1 wi − wi−1  ≤ aN wN+1 − wN ,wN − w1 − w0 ,w0 ≤ 0, (3.18) so w0 = w1 = · · · = wN . This implies that ui = vi + C, i = 0,N, where C ∈ H is a constant. If A or α is one to one, then the uniqueness follows easily. If A is maximal monotone and strongly monotone, then we obtain N i=1  2 ai wi  + N i=1  2 ai−1 wi − wi−1  ≤ 0, so the solution is unique and the proof is complete. (3.19)
N. C. Apreutesei 17 Now we replace (3.4) by (3.2) and (3.3) and remove (3.5) and (3.6). Adding the boundedness of the domain of β, we can state the following result. Theorem 3.2. Let A, α, and β be maximal monotone operators in H such that D(β) is bounded and (3.1), (3.2), (3.3) hold. If a,b ∈ H, (gi )i=1,N ∈ H N , and pi ,ri ,ki > 0, i = 1,N, then problem (1.3), (1.4) admits at least one solution (ui )i=1,N ∈ D(A)N and the difference between two solutions is constant. If A or α is one to one, then the solution is unique. If A is also strongly monotone, then again uniqueness is obtained. Proof. We use again the equivalent form (2.3) of problem (1.3), (1.4) and the maximal monotone operator Ꮽ given by (3.15). If Aλ and Ꮽλ are the Yosida approximations of A and Ꮽ, respectively, then Ꮽλ ((ui )i=1,N ) = (c1 Aλ u1 ,...,cN Aλ uN ) for all (ui )i=1,N ∈ H N . By Proposition 2.2, B + Ꮽλ is maximal monotone in ᏸ, therefore, R(B + Ꮽλ + λI) = ᏸ, that is, for all ( fi )i=1,N ∈ H N , for all λ > 0, the problem
 uλi+1 − 1 + θi uλi + θi uλi−1 = ci Aλ uλi + λuλi + fi , uλ1 − uλ0
∈α uλ0 − a  uλN+1 − uλN , ∈ −β i = 1,N,
 uλN+1 − b , (3.20) has a unique solution (uλi )i=1,N ∈ H N . (The uniqueness follows from Theorem 2.3 for the strongly monotone operator Ꮽλ + λI.) We first prove that (uλi )i=1,N is bounded in H with respect to λ. To do this, we multiply (3.20) by ai uλi and sum up from i = 1 to i = N. Without any loss of generality, suppose that 0 ∈ A0. If not, we put A = A + A0 0 and fi = fi − ci A0 0 instead of A and fi , respectively, where A0 x denotes the element of least norm in Ax. Since Aλ is monotone, Aλ 0 = 0, and ai θi = ai−1 , we derive N i=1
 ai uλi+1 − uλi ,uλi − N ≥ i=1 N
i=1 ai−1 uλi − uλi−1 ,uλi−1  2 ai−1 uλi − uλi−1  + λ N   2 ai  u λ  + i i=1 N i=1 (3.21)
 ai fi ,uλi , hence N i=1  2

 ai−1 uλi − uλi−1  ≤ aN uλN+1 − uλN ,uλN − uλ1 − uλ0 ,uλ0 − N i=1
 ai fi ,uλi . (3.22) Since uλ1 − uλ0 ∈ α(uλ0 − a), 0 ∈ α(0), and α is monotone, we infer  

 − uλ1 − uλ0 ,uλ0 ≤ − uλ1 − uλ0 ,a ≤ a · uλ1 − uλ0  (3.23) and, similarly,
  2

 uλN+1 − uλN ,uλN ≤ −uλN+1 − uλN  + uλN+1 − uλN ,uλN+1 − b + uλN+1 − uλN ,b , (3.24) so
   uλN+1 − uλN ,uλN ≤ b · uλN+1 − uλN . (3.25) 18 Finite difference schemes with monotone operators Now (3.22), (3.23), and (3.25) yield N i=1  2    N  2 ai  u λ   ai−1 uλi − uλi−1  ≤ aN b · uλN+1 − uλN  + a · uλ1 − uλ0   N +  2 ai  f i  1/2  i=1 . i i=1 (3.26) 1/2 The hypothesis that D(β) is bounded and the boundary conditions imply the boundedness of uλN+1 with respect to λ. Using this, together with the estimates  λ u  ≤  k  λ  u − u λ  ≤ k k−1  N  2 ai  u λ  i=1 , i i=1 N 1/2  2 ai−1 uλ − uλ  i (3.27) 1/2 i−1 for k = 1,N in (3.26), one deduces N i=1  2 ai−1 uλi − uλi−1  ≤ C1 + C2  N i=1  2 ai−1 uλ − uλ  i 1/2  N  2 ai uλ  + C3 i−1 i i= 1 1/2 , (3.28) with C1 ,C2 ,C3 > 0 independent of λ.  For each i = 1,N, we have uλi = uλ0 + ik=1 (uλk − uλk−1 ), so  λ  λ u  ≤ u  + i  N 0 k=1  1 ak−1 N  2 ak−1 uλ − uλ  k k=1 1/2 , k−1 i = 1,N. (3.29) From the boundary conditions, it follows that  λ 2    u  ≤  a ·  u λ  +  a 0  0 N i=1  2 ai−1 uλ − uλ  i 1/2 , i−1 (3.30) and thus  λ u  ≤ a + a1/2 0  N i=1  2 ai−1 uλ − uλ  i i−1 1/4 . (3.31) Inequalities (3.29) and (3.31) imply that  λ u  ≤ C4 + C5  i N i=1  2 ai−1 uλ − uλ  i i−1 1/2 , i = 1, N, (3.32) which, together with (3.28), leads to the boundedness N i=1  2 ai−1 uλi − uλi−1  ≤ C6 ∀λ > 0. (3.33) N. C. Apreutesei 19 Now (3.31) and (3.32) show that uλi  ≤ C7 , i = 0,N and λ > 0, and therefore, N  i=1 2 ai uλi  ≤ C8 ∀λ > 0. (3.34) All the constants C j > 0 ( j = 1,...,13) here and below are independent of λ. Multiplying (3.20) by ai Aλ uλi and summing from 1 to N, we get via (2.6) N 

ai uλi+1 − uλi ,Aλ uλi − ai−1 uλi − uλi−1 ,Aλ uλi−1 i=1 N = i=1  2 ai ci Aλ uλi  + λ N
i=1  ai uλi ,Aλ uλi + N i=1  N
− i=1 ai−1 uλi − uλi−1 ,Aλ uλi − Aλ uλi−1
  ai fi ,Aλ uλi . (3.35) Let c = inf {ci , i = 1,N }. Then N c i=1  2

N  ai Aλ uλi  ≤ aN uλN+1 − uλN ,Aλ uλN − uλ1 − uλ0 ,Aλ uλ0 − i=1
 ai fi ,Aλ uλi . (3.36) We observe that assumptions (3.2) and (3.3) and the boundary conditions yield
     uλN+1 − uλN ,Aλ uλN ≤ A0 b · uλN+1 − uλN ,   
  − uλ1 − uλ0 ,Aλ uλ0 ≤ A0 a · uλ1 − uλ0 , (3.37) therefore, (3.36) implies N c i=1  2         ai Aλ uλi  ≤ aN A0 b · uλN+1 − uλN  + A0 a · uλ1 − uλ0   N  2 ai  f i  + 1/2  i=1 N  2 ai Aλ uλ  i i=1 (3.38) 1/2 . In view of (3.29), (3.31), and the boundedness of uλN+1 , this means that N i=1  2 ai Aλ uλi  ≤ C9 + C10  N i=1  2 1/2 ai Aλ uλi   N + C11 i=1  2 ai−1 uλi − uλi−1  1/2 . (3.39) According to (3.33), this leads to N i=1  2 ai Aλ uλi  ≤ C12 . (3.40) 20 Finite difference schemes with monotone operators We prove now that uλi − uλi−1 is a Cauchy sequence with respect to λ. Subtracting (3.20) with ν in place of λ from the original equation (3.20), multiplying the result by ai (uλi − uνi ), and summing up from i = 1 to i = N, we find, with the aid of the equality x = Jλ x + λAλ x, N
i=1 N  ai uλi+1 − uνi+1 − uλi + uνi ,uλi − uνi − N  = i=1 i=1 2 ai−1 uλi − uνi − uλi−1 + uνi−1  + N + i=1
ai−1 uλi − uνi − uλi−1 + uνi−1 ,uλi−1 − uνi−1 N
i=1 ai ci Aλ uλi − Aν uνi ,Jλ uλi − Jν uνi
N  ai ci Aλ uλi − Aν uνi ,λAλ uλi − νAν uνi + i=1
  (3.41)  ai λuλi − νuνi ,uλi − uνi , hence N i=1  2 ai−1 uλi − uνi − uλi−1 + uνi−1 

 ≤ aN uλN+1 − uνN+1 − uλN + uνN ,uλN − uνN − uλ1 − uν1 − uλ0 + uν0 ,uλ0 − uν0 N
+ (λ + ν) i=1  ai ci Aλ uλi ,Aν uνi + (λ + ν) N i=1
(3.42)  ai uλi ,uνi . The boundary conditions in (3.20) and the upper bounds (3.34) and (3.40) imply N  i=1 2 ai−1 uλi − uνi − uλi−1 + uνi−1  ≤ C13 (λ + ν), (3.43) and therefore, uλi − uλi−1 is a strongly convergent sequence in H. Let uλi
ui , i = 1,N (on a subsequence denoted again by λ). Then uλi − uλi−1 → ui − ui−1 , so B((uλi )i=1,N ) → B((ui )i=1,N ). In addition, we have Jλ uλi (= uλi − λAλ uλi )
ui as λ → 0, i = 1,N. Since A is demiclosed, this enables us to pass to the limit as λ → 0 in (3.20) written under the form −B
uλi  i=1,N


 − λ uλi i=1,N − fi i=1,N ∈ Ꮽ Jλ uλi i=1,N , (3.44) and one obtains that (ui )i=1,N verifies problem (2.3). The uniqueness follows like in Theorem 3.1. The proof is complete.
We now replace the boundedness of D(β) by the conditions   (y,x) , y ∈ −β(x) −→ ∞ x    (y,x) inf , y ∈ α(x) −→ ∞ x  inf − We get the following result. as x −→ ∞, (3.45) as x −→ ∞. (3.46)