- Email: [email protected]
Nonlinear evolution operators and wavelets
Nonlinear Analysis 63 (2005) e65 – e75 www.elsevier.com/locate/na
Nonlinear evolution operators and wavelets Nguyen Minh Chuong Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Str., 10307 Hanoi, Vietnam
Abstract Evolution operators and wavelets are very interesting and attractive, not only by their extremely wide range of applications, but also by their theories of great importance. It is very difficult to show the relations between evolution operators, wavelets and other subjects in pure and applied mathematics. However, perhaps taking into account the obvious relations between the microlocal and wavelet analysis and the white noise; the Brownian motion and the index formulae for de Rham complex; even only on archimedean fields, we can partially illustrate such interesting relations. In our brief talk, let us present some recent results on a semilinear non-classical evolution problem and a Galerkin-wavelet method to solve a very complicated linear evolution one. Then some general results on stationary problems, from which we can continue to study the respective non-stationary cases, will be introduced. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Semilinear non-classical evolution problem; Wavelets; Galerkin-wavelet method; Stationary problems; Laplace transform; Fourier transform; Multiresolution approximation; Sobolev spaces
1. A semilinear non-classical evolution problem In [21,22], a semilinear stationary problem of type Egorov–Kondratiev was investigated for elliptic pseudodifferential equations in Sobolev spaces Hl,p , 1 < p < + ∞. Here, a semilinear parabolic evolution problem of the same type in Sobolev spaces Pl,p (, , ) has been solved, too. The last problem is much more generalized, complicated, difficult, and interesting than the preceding ones. Especially, since the method used to solve this problem is the Laplace transform, a very powerful tool in mathematics and in the sciences. E-mail address: [email protected]. 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.02.075
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Here this tool has been made much more effective. It can be applied, from now on, to a very large and important class of Sobolev spaces. The assumptions, as well as the proofs for main theorems are quite new and different from the usual ones, even from ours [21,22]. This part of the talk is taken from the joint paper [18] with Dang Anh Tuan. 1.1. Definitions Denote by Fn+1 [u(x, t)](, ) = (2) Fn U (, q) = (2)
−n/2
= (2)−1/2
−(n+1)/2
Rn+1 x,t
e−ix,−it u(x, t) dx dt,
e−ix, U (x, q) dx, L[u(x, t)](x, q)
Rnx +∞
e−qt u(x, t) dt,
0
where x, =
n
xi i ,
q ∈ C, Re q > 0.
i=1
The following spaces are introduced. Let 0 l, 1 < p < + ∞, 0 < , . • Pl,p (, , R n × (0, ∞)) is the completion of the space P (Rn+1 ) = {u ∈ C0∞ (Rn × (−∞, +∞)) : supp u ⊂ Rn × (0, +∞)} with respect to the norm uPl,p (,,Rn ×(0,∞)) 1/p 1/ lp −t p = (1 + || + |q| ) |Fn+1 [e u(x, t)](, )| d d , Rn+1 ,
where q =+i. The spaces Pl,p (, , Rn+ ×(0, ∞)), Pl,p (, , ×(0, ∞)), Pl,p (, , j × (0, ∞)), are introduced as usually, and is a bounded domain with or without boundary in Rn . • Hl,p,q (Rn ) is the completion of the space C0∞ (Rn ) with respect to the norm ul,p,q,Rn = ( Rn (1 + || + |q|1/ )lp |Fn u(, q)|p d)1/p . As traditionally we define Hl,p,q (Rn+ ), Hl,p,q (). • El,p (, , Rn ) is the completion of the space LP (Rn ) with respect to the norm p U El,p (,,Rn ) =( R U l,p,q 1/ ,Rn d)1/p , where q=+i. The spaces El,p (, , Rn+ ), El,p (, , ) are defined traditionally.
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1.2. The Laplace transform Theorem 1. The Laplace transform is an isometric isomorphism between Pl,p (, ) and El,p (, ). To prove this theorem we need the two following Lemmas. Lemma 1. The Laplace transform is an isometric isomorphism between (P (Rn ), · Pl,p (,,Rn ) ) and (LP (Rn ), · El,p (,,Rn ) ). Lemma 2. Let X, Y be Banach spaces, X1 , Y1 be respective dense subspaces in X, Y . The isometric isomorphism A : X1 → Y1 has an isometric isomorphism extension from X to Y. It is not difficult to prove such problems as the two above-mentioned Lemmas. 1.3. Classical stationary problem Let be a bounded domain in Rn with smooth boundary j. For studying a classical pseudodifferential BVP by means of the Laplace transform, we use the following elliptic problem: A(x, Dx , q 1/ )U (x, q) = A(x, Dx , q)U (x, q) = F (x, q),
x ∈ ,
Bj (x, Dx , q 1/ )U (x, q) = Bj (x, Dx , q)U (x, q) = Gj (x, q), x ∈ j, j = 1, . . . , s,
(1) (2)
where | arg q|/2, and the operators A(x, Dx , q 1/ ), Bj (x, Dx , q 1/ ) are defined as in [22, p. 452]. Let us now study the following evolution problem in × [0, +∞): j u(x, t) = f (x, t), x ∈ , t > 0, (3) A x, Dx , jt j u(x, t) = gj (x, t), x ∈ j, t > 0, j = 1, . . . , s. (4) Bj x, Dx , jt With the initial conditions
l0 jk u = 0, k = 0, 1, . . . , , jt k
(l0 = max{2s, mj + 1}),
(5)
t=0
by applying formally the Laplace transform to (3)–(4) we get A(x, Dx , q 1/ )U (x, q) = A(x, Dx , q)U (x, q) = F (x, q), Bj (x, Dx , q 1/ )U (x, q) = Bj (x, Dx , q)U (x, q) = Gj (x, q), x ∈ j, j = 1, . . . , s.
x ∈ ,
(3 ) (4 )
The problem (3)–(4) is said to be parabolic if is an even natural number and the problem (3 )–(4 ) is elliptic for | arg q|/2.
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By the results from [21,22] it is not difficult to prove: Theorem 2. Assume that (A(x, Dx , q 1/ ), Bj (x, Dx , q 1/ )|j ) is an elliptic operator for | arg q|/2 and l l0 , 1 < p <+∞. Then for =Re q is large enough, the problem (3)–(4) has a unique solution u ∈ Pl,p (, , ×(0, +∞)) for any f ∈ Pl−2s,p (, , ×(0, +∞)), gj ∈ Pl−mj −1+(1/p),p (, , j × (0, +∞)). 1.4. Non-classical linear stationary problem Let be a bounded domain with smooth boundary j in Rn . Assume that the field is tangent to j only on the points of a submanifold 0 of dimension (n − 2) of j but not tangent to 0 . Consider the problem
j A x, Dx , jt
u(x, t) = f (x, t),
x ∈ , t > 0,
j Bj x, Dx , (D u(x, t)) = gj (x, t), jt
(6)
x ∈ j, t > 0, j = 1, . . . , s,
(7)
where A, Bj are the operators in (3)–(4). For 0 , we use the classification made by Egorov and Kondratiev [23]. If 0 belongs to the first class, we add conditions Dnk u(x, t) = u0k (x, t),
x ∈ 0 , t > 0, k = 0, . . . , s − 1.
(8)
If the initial value conditions (5) are satisfied by the Laplace transform, problem (6)–(7) (or (6)–(8) when 0 is of the first class) becomes, when Re q > 0, A(x, Dx , q 1/ )U (x, t) = A(x, Dx , q)U (x, q) = F (x, q),
x ∈ ,
(6 )
Bj (x, Dx , q 1/ )(D U (x, q)) = Bj (x, Dx , q)(D U (x, q)) = Gj (x, q), x ∈ j, j = 1, . . . , s,
(7 )
Dnk U (x, q) = U0k (x, q),
(8 )
x ∈ 0 , k = 0, . . . , s − 1.
In the sequel, we will use a function h(x) ∈ C ∞ (), h(x) = 1 in a (d/2)-neighborhood d/2 of 0 (d > 0 small enough), h(x) = 0 outside of a d-neighborhood d of 0 and → N = Nd = {x ∈ d |∃y ∈ 0 , − xy is a normal vector of j}. If 0 belongs to the first class, let us denote Pl,p (, , × (0, +∞)) = {u ∈ Pl,p (, , × (0, +∞))|D (hu) ∈ Pl,p (, , × (0, +∞)), (hu)|N×(0,+∞) ∈ Pl,p (, , N × (0, +∞))}
(9)
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with the norm uPl,p (,,×(0,+∞)) = uPl,p (,,×(0,+∞)) + D (hu)Pl,p (,,×(0,+∞)) + (hu)|N×(0,+∞) Pl,p (,,N×(0,+∞)) .
(10)
If 0 belongs to the third class then in (9) and (10) there is not the presence of (hu)|N×(0,+∞) . Similarly, one defines El,p (, , ) by means of El,p (, , ). Denote Pl,p (, , × (0, +∞), j × (0, +∞)) ⎧ ⎨ = (f, gj ) ∈ Pl−2s,p (, , × (0, +∞)) ⎩
× Pl−mj −2+(1/p),p (, , j × (0, +∞)) hg j j =1 ⎫ s ⎬ ∈ Pl−mj −1+(1/p),p (, , j × (0, +∞)) . ⎭ s
j =1
Analogously, we define El,p (, , , j). One can get: Theorem 3. For l l1 (l1 = max{2s, mj + 2}), 1 < p < + ∞, the Laplace transform is an isometric isomorphism between the following pairs of spaces: Pl,p (, , × (0, +∞)) and Pl,p (, , , j × (0, +∞))
El,p (, , × (0, +∞)), and
El,p (, , , j),
Pl,p (, , × (0, +∞), j × (0, +∞)) × ×
s−1 k=0 s−1
Pl−k−1+(1/p),p (, , 0 × (0, +∞)) and
El,p (, , , j)
El−k−1+(1/p),p (, , 0 ).
k=0
It is now not difficult to prove existence and uniqueness theorems off solutions of the problem (6 )–(7 ) (or (6 )–(8 )) as well as of the problem (6)–(7) (or (6)–(8)) (see [21]). 1.5. Non-classical semilinear evolution problem Let us consider the problem j u(x, t) = f (x, t, u (x, t)), A x, Dx , jt
x ∈ , t > 0,
(11)
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j (D u(x, t)) = gj (x, t, u j (x, t)), Bj x, Dx , jt x ∈ j, t > 0, j = 1, . . . , s,
(12)
m where u (x, t) = (u(x, t), . . . , Dx2s−1 u(x, t)), u j (x, t) = (u(x, t), . . . , Dx j u(x, t)), A, Bj are the same operators as in (6)–(7). We will use conditions (8) in the same situation. If the initial value conditions (5) are satisfied, by the Laplace transform from the problem (11), (12) (or (11), (12), (8) when 0 is of the first class) when Re q > 0, we get
A(x, Dx , q 1/ )U (x, q) = A(x, Dx , q)U (x, q) = L[f (x, t, L−1 U (x, t))](x, q), Bj (x, Dx , q 1/ )(D U (x, q)) = Bj (x, Dx , q)(D U (x, q)) = L[gj (x, t, L−1 U j (x, t))](x, q), Dnk U (x, q) = U0k (x, q),
x ∈ ,
x ∈ j, j = 1, . . . , s,
x ∈ 0 , k = 0, . . . , s − 1,
(11 )
(12 ) (8 )
where L−1 U (x, t) = (L−1 U (x, t), . . . , L−1 Dx2s−1 U (x, t)), L−1 U j (x, t) = (L−1 U (x, t), . . . , L−1 Dx j U (x, t)). m
Assumptions. (i) With u = (u1 , . . . , uN ), u j = (u1 , . . . , uN j ), N = 0 | | 2s 1, (the number of the multi-indexes satisfying 0 | |2s, ) Nj = 0 mj 1, the maps (x, t, u ) → f (x, t, u ) (from × R+ × RN to R), (x, t, u j ) → gj (x, t, u j ) (from j × R+ × RNj to R), satisfy the Carathéodory properties, i.e. they are continuous in u , u j , for almost (x, t) and they are measurable in (x, t) for every u , u j . (ii) The map u(x, t) → (f (x, t, u (x, t)), gi (x, t, u j (x, t))) (from Pl,p (, , × (0, +∞)) to Pl,p (, , × (0, +∞, j × (0, +∞)) transforms each bounded set into a relatively compact set. (iii) 1 > lim inf lim sup (A, Bj )(f, gj )M, , where M→+∞ →+∞
(f, gj )M, (f (x, t, u (x, t)), gj (x, t, u(x, t)))Pl,p (,,×(0,+∞),j×(0,+∞)) = sup M uPl,p (,,×(0,+∞) M . To prove Theorem 4, we need:
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Lemma 3. The map U (x, q) → (L[f (x, t, L−1 U (x, t))](x, g), L[gj (x, t, L−1 U j (x, t))](x, q)) is compact from El,p (, , ) to El,p (, , , j). Theorem 4. Assume that the operator (A(x, Dx , q 1/ ), Bj (x, Dx , q 1/ )|j ) is an elliptic operator for | arg q|/2. Suppose that the assumptions (i)–(iii) are satisfied. Let l l1 , 1 < p < + ∞, U0k ∈ El−k−1+(1/p),p (, , 0 ). Then for large enough = Re q, the problem (11 )–(12 ) (or (11 )–(12 )–(8 ) when 0 is of the first class) has a solution U (x, q) ∈ El,p (, , ). Proof (Sketch). We prove Theorem 3 for 0 belonging to the first class. The proof for 0 of the third class is similar. For each W ∈ El,p (, , ) we have (x, t))](x, q), L[gj (x, t, L−1 W j (x, t))](x, q)) (L[f (x, t, L−1 W ∈ El,p (, , , j). So for large enough =Re q, problem (6 )–(8 ) with the right-hand side (Lf, Lgj , U0k ) has a unique solution U ∈ El,p (, , ). By Lemma 3, the map W → U is compact from El,p (, , ) into itself. So with the assumption (iii) there exists R > 0 such that this map is compact from the closed ball BR = {U ∈ El,p (, , )|U El,p (,,) R} into itself. Therefore, by Schauder theorem, there is a fixed point of this map in BR . This is a solution of the problem (11 )–(12 )–(8 ). By Theorem 4 we get: Theorem 5. Under the assumptions of Theorem 5, if 0 belongs to the third class (or the first class), for large enough , problem (11)–(12) (or (11)–(12)–(8) with the additional condition u0k ∈ Pl−k−1+(1/p),p (, , 0 × (0, +∞)) has a solution u ∈ Pl,p (, , × (0, +∞). 2. Galerkin wavelet method The theory of pseudodifferential operators has been intensively developing. It is so not only because it is a very general theory which includes many theories, especially because it is of great interest and difficulties, but also because it is an extremely powerful tool for studying linear and nonlinear problems. Wavelet theory is growing very rapidly as well. Wavelet approximation methods, however, for even only pseudodifferential operators (not an evolution problem) are investigated in a very few works. In this section we study approximately an evolution problem for a class of interesting, complicated pseudodifferential equations as follows: t ju = −Au(x, t) − a(t − )Au(x, ) d + b(x, t), (13) jt 0 u(x, 0) = u0 (x),
(14)
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n n n where x ∈ J , > 0, A = (D) is a pseudodifferential operator, defined by = R 2/Z ˜ u ∈ C ∞ (Zn ), with symbol () belonging to S 2m (Zn ), (D)u(x)= ∈Zn e ix ()u(), i.e. ∈ C ∞ (Zn ) satisfying | ()|C (1 + ||)2m−| | for all ∈ Zn and for all multiindexes , is the difference operator, which is defined by := (1 −1, 2 −1, . . . , n −1)T , where j f (x) := f (x + ej ), ej = ( j,l )nl=1 is the jth coordinate vector. Here it is assumed also that ∈ C ∞ (Rn \{0}) and (t) = t 2m (), t > 0, (0) = 1. Moreover ()c(1 + ||2 )m , || R > 0. The functions a(t), b(x, t) are given. Note that when = 0, under some assumptions on a(t) the type of equation (1) may be changed. To establish some convergence estimates for this complicated problem, in order to overcome the difficulties, let us use Galerkin-wavelet method and weak solutions and approximative weak solutions, with very effective tools: the Fourier and Laplace transforms (see [9]).
2.1. Notations and preliminaries Let us recall here some usual notations, definitions and some basic facts. For a function u(x) ∈ L2 (Jn ), the discrete Fourier transform is defined by −2ix e u(x) dx = e−2ix u(x) dx, ∈ Zn . F(u)() = u() ˜ := Jn
[0,1]n
(15)
2ix . The discrete inverse Fourier transform is then u(x) := ∈Zn u()e ˜ The Fourier transform of function f ∈ L2 (Rn ) is fˆ() = Rn e−2ix f (x) dx, ∈ Rn , with the inverse Fourier transform f (x) = Rn e2ix fˆ() d, x ∈ Rn . We consider now the space H s (Jn ), (s ∈ R) of functions u ∈ D (Jn ) (the L. Schwartz space of distributions), such that Ds u ∈ L2 (Jn ) and the norm us = ( ∈Zn 2s 2 )1/2 , (∗) is finite, where |u()| ˜ 1 if = 0 = || if = 0. The norm of u ∈ H s (Jn ) is defined by (∗). Definition 1. Let L2 (H q,s0 ) (q 0, s0 > 0) be the space of all functions u(x, t), x ∈ Jn , t 0, satisfying the following conditions: for each 0 q (i) u(x, t) ∈ H (Jn )∀t ∈ [0, +∞). ∞ ˜ t)|2 dt. The norm of the (ii) The following series converges ∈Zn 2 0 e−4s0 t |u(, ∞ q,s function u ∈ L2 (H 0 ) is defined by uL2 (H q,s0 ) = ( ∈Zn 2q 0 e−4s0 t | u(, ˜ t)|2 dt)1/2 . Next we introduce some notations and definitions on wavelets. Definition 2. A multiresolution approximation (M.R.A.) of L2 (Rn ) is, by definition, an increasing sequence Vj , j ∈ Z, of closed linear subspaces of L2 (Rn ) with the following
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properties: j ∈Z Vj = {0}, j ∈Z Vj = L2 (Rn ); for all f ∈ L2 (Rn ) and all j ∈ Z f (x) ∈ Vj ⇔ f (2x) ∈ Vj +1 ; for all f ∈ L2 (Rn ) and k ∈ Zn f (x) ∈ V0 ⇔ f (x − k) ∈ V0 . There exists a function, which is called scaling function (S.F.) (x) ∈ V0 , such that the sequence {(x − k), k ∈ Zn } is a Riesz basic of V0 . An M.R.A. of L2 (Rn ) is said to be r-regular (r ∈ N) if the function is r-regular, that is for each m ∈ N there exists cm such that for all multiindexes , | | r the following condition holds |D (x)| cm (1 + |x|)−m . Let us denote j k (x) = 2nj /2 (2j x − k), k ∈ Zn . Obviously Vj = span{j k (x), k ∈ Zn }. The the periodization operator u(x), that is [u](x) = notation [u](x) stands for of a function k nj /2 j n u(x + k). Denote (x) := [j k ](x) = 2 n (2 (x + l) − k). Similarly let j k∈Z l∈Z us define an M.R.A. of L2 (Jn ) as follows: [Vj ] := span{kj (x), k ∈ Znj }, j 0, where Znj = Zn /2j Zn . It is easy to check [V0 ] ⊂ [V1 ] ⊂ · · · ⊂ [Vn ] ⊂ · · · , j 0 [Vj ]=L2 (Jn ). Furthermore, j
j
if (j k , j l ) = kl , k, l ∈ Zn , then (k , l ) = kl , k, l ∈ Znj , and dim[Vj ] = 2nj . For each j 0, let Pj : L2 (Jn ) → [Vj ] be the orthogonal project of L2 (Jn ) on [Vj ], which has the well known property: Theorem 1. Let −r − 1 s r, −r q r + 1 and s q then u − Pj us c2j (s−q) uq for all u ∈ H q (Jn ), where c is a constant independent of j and u. 2.2. Convergence estimates of solutions Definition 3. A C 1 -mapping u in t, u : [0, ∞) → H 2m (Jn ) satisfying (ju/jt, v) = t −(Au, v) − 0 a(t − )(Au(), v) d + (b, v), (u(x, 0), v) = (u0 , v), ∀v ∈ L2 (Jn ) is called a weak solution of problem (1)–(2). Definition 4. A C 1 -mapping uh in t, uh : [0, ∞)→Vh satisfying (juh /jt, v)=−(Auh , v) t − 0 a(t − )(Au(), v) d + (b, v), (uh (x, 0), v) = (u0h , v), ∀v ∈ Vh , where u0h := Ru0 is a linear approximation of u0 in Vh , is called a weak Galerkin-wavelet solution of problem (1)–(2). The following theorem asserts the stability of the weak solution of problem (1)–(2). Theorem 2. Let u(x, t) ∈ L2 (H 2m,s0 ) be a weak solution of problem (1)–(2) with b = 0. Assume, moreover that there exists a number s0 > 0 such that + Re a(s ˇ 0 + i ) 0, ∀ ∈ R.(∗∗) Then u2L (H 0,s0 ) 1/4s0 u0 2 . 2
Theorem 3. Let u(x, t) ∈ L2 (H q,s0 ) be a weak solution of (1)–(2) and uh (x, t) be its weak Galerkin-wavelet solution. If (∗∗) in Theorem 2 is satisfied and jj u/jt j ∈ H q (Jn ) (j = 0, 1, q 2m) uniformly in t ∈ [0, ∞) then u − uh 2L (H 0,s0 ) 2
C u0 − u0,h + h 2
2(q−2m)
For proofs of Theorems 2 and 3, see [9].
uo 2q
∞
+
e 0
−4s0 t
u2q
2 ju . + jt dt q
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3. From stationary to evolution problems As for pseudodifferential operators, in recent years we also obtained results on abstract stationary problems (see [1,2,6,10–15,25–27] and references therein). It seems that the abstract evolution problems corresponding to these stationary ones may be also studied. Even such evolution problems as in the papers [3–9,17] may be extended to the L(q) , | < q < ∞-theory and to abstract functional spaces, too. References [1] T.Q. Binh, N.M. Chuong, On a fixed point theorem, Funct. Anal. Appl. (Engl. transl. 0) 30 (3) (1996) 220–221. [2] T.Q. Binh, N.M. Chuong, Approximation of nonlinear operator equations, Numer. Funct. Anal. Optim. 22 (7–8) (2001) 831–844. [3] N.M. Chuong, Parabolic pseudodifferential operators of variable order in Sobolev spaces with weighted norms, Dokl. Akad. Nauk SSSR 262 (4) (1982) 804–807. [4] N.M. Chuong, Degenerate parabolic pseudodifferential operators of variable order, Dokl. Akad. Nauk SSSR 268 (5) (1983) 1055–1058. [5] N.M. Chuong, On the theory of parabolic pseudodifferential operators of variable order, Differentialnye Uravneniya 21 (4) (1985) 686–694. [6] N.M. Chuong, N.V. Co, The multidimensional p-adic Green Function, Proc. AMS 127 (3) (1999) 685–694. [7] N.M. Chuong, B.K. Cuong, Galerkin wavelet approximation for a class of partial integro-differential equation, Fractional Calculus Appl. Anal. (Int. J.) 4 (2) (2001) 143–152. [8] N.M. Chuong, B.K. Cuong, The convergence estimates for Galerkin-wavelet method for initial pseudodifferential periodic problems, Int. J. Math. Sci. (14) (2003) 857–867. [9] N.M. Chuong, B.K. Cuong, Convergence estimates of or Galerkin-wavelet solutions to a Cauchy problem for a class of periodic pseudodifferential problems, Proc. AMS 132 (12) (2004) 3589–3597. [10] N.M. Chuong, N.V. Kinh, Regularization of variational inequalities with perturbed non-monotone and discontinuos operators, Diffenrentialnye Uravneniya 27 (12) (1991) 2171–2172. [11] N.M. Chuong, N.X. Thuan, Random equations for semi-H -monotone operators, Random Oper. Stochastic Equations 10 (4) (2002) 1–8. [12] N.M. Chuong, N.X. Thuan, Random equations for weakly semimonotone operators of type (S) and semi-J -monotone of type (J –S), Random Oper. Stochastic Equations 10 (2) (2002) 123–132. [13] N.M. Chuong, N.X. Thuan, The surjectivity of semiregular maximal monotone random mappings, Random Oper. Stochastic Equations 10 (1) (2002) 47–58. [14] N.M. Chuong, N.X. Thuan, Random nonlinear variational inequalities for mappings of monotone type in Banach spaces, Stochastic Anal. Appl., to appear. [15] N.M. Chuong, N.X. Thuan, Some new random fixed point theorems for nonlinear set-valued mappings, Nonlinear Analysis, TMA, submitted for publication. [17] N.M. Chuong, L.Q. Trung, K.V. Ninh, A boundary value problem for nonlinear parabolic equations of infinite order in Sobolev–Orlicz spaces, Mat. Zametki 48 (1) (1990) 78–85. [18] N.M. Chuong, D.A. Tuan,A non-classical semilinear boundary value problem for parabolic pseudodifferential equations in Sobolev spaces Pl,p (, , ), C. R. Acad. Sci, Paris, Ser. I, submitted for publication. [21] Yu.V. Egorov, N.M. Chuong, On some semilinear boundary value problems for singular integro-differential equations, Uspekhi Mat. Nauk 53 324 (6) (1998) 249–250. [22] Yu.V. Egorov, N.M. Chuong, D.A. Tuan, A semilinear non-classical pseudodifferential boundary value problem in Sobolev spaces, C. R. Acad. Sci., Paris, Ser. I 337 (2003) 451–456. [23] Yu.V. Egorov, V.A. Kondratiev, On an oblique derivative problem, Mat. Sb. 78 120 (1) (1969) 148–176. [25] N.Q. Nga, Set-valued nonlinear variational inequalities for H -monotone mappings in non-reflexive Banach spaces, Nonlinear Anal. TMA 52 (2002) 457–465. [26] N.Q. Nga, N.M. Chuong, On a set-valued nonlinear variational inequality, Differentialnye Uravneniya 37 (2001) 128–129.
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[27] N.Q. Nga, N.M. Chuong, Some fixed point theorems for non-compact and weakly asymptotically regular set-valued mappings, Numer. Funct. Anal. Optim. 24 (7–8) (2003) 895–905.
Further reading [16] N.M. Chuong, T.N. Tri, The integral wavelet transform in Lp (R), 1 p ∞, Fractional Calculus Appl. Anal. (Int. J.) 3 (2) (2000) 133–140. [19] I. Dauberchies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. XLI (1988) 906–909. [20] D.L. Donoho, Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data, Proceedings of the Symposium on Applied Mathematics, vol. 47, AMS, Providence, RI, 1993. [24] Y. Meyer, Ondelettes et Opérators, I, II, Herman, Paris, 1990, MR 93i: 42002, 93i: 42003.
Nonlinear evolution operators and wavelets Nguyen Minh Chuong Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Str., 10307 Hanoi, Vietnam
Abstract Evolution operators and wavelets are very interesting and attractive, not only by their extremely wide range of applications, but also by their theories of great importance. It is very difficult to show the relations between evolution operators, wavelets and other subjects in pure and applied mathematics. However, perhaps taking into account the obvious relations between the microlocal and wavelet analysis and the white noise; the Brownian motion and the index formulae for de Rham complex; even only on archimedean fields, we can partially illustrate such interesting relations. In our brief talk, let us present some recent results on a semilinear non-classical evolution problem and a Galerkin-wavelet method to solve a very complicated linear evolution one. Then some general results on stationary problems, from which we can continue to study the respective non-stationary cases, will be introduced. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Semilinear non-classical evolution problem; Wavelets; Galerkin-wavelet method; Stationary problems; Laplace transform; Fourier transform; Multiresolution approximation; Sobolev spaces
1. A semilinear non-classical evolution problem In [21,22], a semilinear stationary problem of type Egorov–Kondratiev was investigated for elliptic pseudodifferential equations in Sobolev spaces Hl,p , 1 < p < + ∞. Here, a semilinear parabolic evolution problem of the same type in Sobolev spaces Pl,p (, , ) has been solved, too. The last problem is much more generalized, complicated, difficult, and interesting than the preceding ones. Especially, since the method used to solve this problem is the Laplace transform, a very powerful tool in mathematics and in the sciences. E-mail address: [email protected]. 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.02.075
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Here this tool has been made much more effective. It can be applied, from now on, to a very large and important class of Sobolev spaces. The assumptions, as well as the proofs for main theorems are quite new and different from the usual ones, even from ours [21,22]. This part of the talk is taken from the joint paper [18] with Dang Anh Tuan. 1.1. Definitions Denote by Fn+1 [u(x, t)](, ) = (2) Fn U (, q) = (2)
−n/2
= (2)−1/2
−(n+1)/2
Rn+1 x,t
e−ix,−it u(x, t) dx dt,
e−ix, U (x, q) dx, L[u(x, t)](x, q)
Rnx +∞
e−qt u(x, t) dt,
0
where x, =
n
xi i ,
q ∈ C, Re q > 0.
i=1
The following spaces are introduced. Let 0 l, 1 < p < + ∞, 0 < , . • Pl,p (, , R n × (0, ∞)) is the completion of the space P (Rn+1 ) = {u ∈ C0∞ (Rn × (−∞, +∞)) : supp u ⊂ Rn × (0, +∞)} with respect to the norm uPl,p (,,Rn ×(0,∞)) 1/p 1/ lp −t p = (1 + || + |q| ) |Fn+1 [e u(x, t)](, )| d d , Rn+1 ,
where q =+i. The spaces Pl,p (, , Rn+ ×(0, ∞)), Pl,p (, , ×(0, ∞)), Pl,p (, , j × (0, ∞)), are introduced as usually, and is a bounded domain with or without boundary in Rn . • Hl,p,q (Rn ) is the completion of the space C0∞ (Rn ) with respect to the norm ul,p,q,Rn = ( Rn (1 + || + |q|1/ )lp |Fn u(, q)|p d)1/p . As traditionally we define Hl,p,q (Rn+ ), Hl,p,q (). • El,p (, , Rn ) is the completion of the space LP (Rn ) with respect to the norm p U El,p (,,Rn ) =( R U l,p,q 1/ ,Rn d)1/p , where q=+i. The spaces El,p (, , Rn+ ), El,p (, , ) are defined traditionally.
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1.2. The Laplace transform Theorem 1. The Laplace transform is an isometric isomorphism between Pl,p (, ) and El,p (, ). To prove this theorem we need the two following Lemmas. Lemma 1. The Laplace transform is an isometric isomorphism between (P (Rn ), · Pl,p (,,Rn ) ) and (LP (Rn ), · El,p (,,Rn ) ). Lemma 2. Let X, Y be Banach spaces, X1 , Y1 be respective dense subspaces in X, Y . The isometric isomorphism A : X1 → Y1 has an isometric isomorphism extension from X to Y. It is not difficult to prove such problems as the two above-mentioned Lemmas. 1.3. Classical stationary problem Let be a bounded domain in Rn with smooth boundary j. For studying a classical pseudodifferential BVP by means of the Laplace transform, we use the following elliptic problem: A(x, Dx , q 1/ )U (x, q) = A(x, Dx , q)U (x, q) = F (x, q),
x ∈ ,
Bj (x, Dx , q 1/ )U (x, q) = Bj (x, Dx , q)U (x, q) = Gj (x, q), x ∈ j, j = 1, . . . , s,
(1) (2)
where | arg q|/2, and the operators A(x, Dx , q 1/ ), Bj (x, Dx , q 1/ ) are defined as in [22, p. 452]. Let us now study the following evolution problem in × [0, +∞): j u(x, t) = f (x, t), x ∈ , t > 0, (3) A x, Dx , jt j u(x, t) = gj (x, t), x ∈ j, t > 0, j = 1, . . . , s. (4) Bj x, Dx , jt With the initial conditions
l0 jk u = 0, k = 0, 1, . . . , , jt k
(l0 = max{2s, mj + 1}),
(5)
t=0
by applying formally the Laplace transform to (3)–(4) we get A(x, Dx , q 1/ )U (x, q) = A(x, Dx , q)U (x, q) = F (x, q), Bj (x, Dx , q 1/ )U (x, q) = Bj (x, Dx , q)U (x, q) = Gj (x, q), x ∈ j, j = 1, . . . , s.
x ∈ ,
(3 ) (4 )
The problem (3)–(4) is said to be parabolic if is an even natural number and the problem (3 )–(4 ) is elliptic for | arg q|/2.
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By the results from [21,22] it is not difficult to prove: Theorem 2. Assume that (A(x, Dx , q 1/ ), Bj (x, Dx , q 1/ )|j ) is an elliptic operator for | arg q|/2 and l l0 , 1 < p <+∞. Then for =Re q is large enough, the problem (3)–(4) has a unique solution u ∈ Pl,p (, , ×(0, +∞)) for any f ∈ Pl−2s,p (, , ×(0, +∞)), gj ∈ Pl−mj −1+(1/p),p (, , j × (0, +∞)). 1.4. Non-classical linear stationary problem Let be a bounded domain with smooth boundary j in Rn . Assume that the field is tangent to j only on the points of a submanifold 0 of dimension (n − 2) of j but not tangent to 0 . Consider the problem
j A x, Dx , jt
u(x, t) = f (x, t),
x ∈ , t > 0,
j Bj x, Dx , (D u(x, t)) = gj (x, t), jt
(6)
x ∈ j, t > 0, j = 1, . . . , s,
(7)
where A, Bj are the operators in (3)–(4). For 0 , we use the classification made by Egorov and Kondratiev [23]. If 0 belongs to the first class, we add conditions Dnk u(x, t) = u0k (x, t),
x ∈ 0 , t > 0, k = 0, . . . , s − 1.
(8)
If the initial value conditions (5) are satisfied by the Laplace transform, problem (6)–(7) (or (6)–(8) when 0 is of the first class) becomes, when Re q > 0, A(x, Dx , q 1/ )U (x, t) = A(x, Dx , q)U (x, q) = F (x, q),
x ∈ ,
(6 )
Bj (x, Dx , q 1/ )(D U (x, q)) = Bj (x, Dx , q)(D U (x, q)) = Gj (x, q), x ∈ j, j = 1, . . . , s,
(7 )
Dnk U (x, q) = U0k (x, q),
(8 )
x ∈ 0 , k = 0, . . . , s − 1.
In the sequel, we will use a function h(x) ∈ C ∞ (), h(x) = 1 in a (d/2)-neighborhood d/2 of 0 (d > 0 small enough), h(x) = 0 outside of a d-neighborhood d of 0 and → N = Nd = {x ∈ d |∃y ∈ 0 , − xy is a normal vector of j}. If 0 belongs to the first class, let us denote Pl,p (, , × (0, +∞)) = {u ∈ Pl,p (, , × (0, +∞))|D (hu) ∈ Pl,p (, , × (0, +∞)), (hu)|N×(0,+∞) ∈ Pl,p (, , N × (0, +∞))}
(9)
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with the norm uPl,p (,,×(0,+∞)) = uPl,p (,,×(0,+∞)) + D (hu)Pl,p (,,×(0,+∞)) + (hu)|N×(0,+∞) Pl,p (,,N×(0,+∞)) .
(10)
If 0 belongs to the third class then in (9) and (10) there is not the presence of (hu)|N×(0,+∞) . Similarly, one defines El,p (, , ) by means of El,p (, , ). Denote Pl,p (, , × (0, +∞), j × (0, +∞)) ⎧ ⎨ = (f, gj ) ∈ Pl−2s,p (, , × (0, +∞)) ⎩
× Pl−mj −2+(1/p),p (, , j × (0, +∞)) hg j j =1 ⎫ s ⎬ ∈ Pl−mj −1+(1/p),p (, , j × (0, +∞)) . ⎭ s
j =1
Analogously, we define El,p (, , , j). One can get: Theorem 3. For l l1 (l1 = max{2s, mj + 2}), 1 < p < + ∞, the Laplace transform is an isometric isomorphism between the following pairs of spaces: Pl,p (, , × (0, +∞)) and Pl,p (, , , j × (0, +∞))
El,p (, , × (0, +∞)), and
El,p (, , , j),
Pl,p (, , × (0, +∞), j × (0, +∞)) × ×
s−1 k=0 s−1
Pl−k−1+(1/p),p (, , 0 × (0, +∞)) and
El,p (, , , j)
El−k−1+(1/p),p (, , 0 ).
k=0
It is now not difficult to prove existence and uniqueness theorems off solutions of the problem (6 )–(7 ) (or (6 )–(8 )) as well as of the problem (6)–(7) (or (6)–(8)) (see [21]). 1.5. Non-classical semilinear evolution problem Let us consider the problem j u(x, t) = f (x, t, u (x, t)), A x, Dx , jt
x ∈ , t > 0,
(11)
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j (D u(x, t)) = gj (x, t, u j (x, t)), Bj x, Dx , jt x ∈ j, t > 0, j = 1, . . . , s,
(12)
m where u (x, t) = (u(x, t), . . . , Dx2s−1 u(x, t)), u j (x, t) = (u(x, t), . . . , Dx j u(x, t)), A, Bj are the same operators as in (6)–(7). We will use conditions (8) in the same situation. If the initial value conditions (5) are satisfied, by the Laplace transform from the problem (11), (12) (or (11), (12), (8) when 0 is of the first class) when Re q > 0, we get
A(x, Dx , q 1/ )U (x, q) = A(x, Dx , q)U (x, q) = L[f (x, t, L−1 U (x, t))](x, q), Bj (x, Dx , q 1/ )(D U (x, q)) = Bj (x, Dx , q)(D U (x, q)) = L[gj (x, t, L−1 U j (x, t))](x, q), Dnk U (x, q) = U0k (x, q),
x ∈ ,
x ∈ j, j = 1, . . . , s,
x ∈ 0 , k = 0, . . . , s − 1,
(11 )
(12 ) (8 )
where L−1 U (x, t) = (L−1 U (x, t), . . . , L−1 Dx2s−1 U (x, t)), L−1 U j (x, t) = (L−1 U (x, t), . . . , L−1 Dx j U (x, t)). m
Assumptions. (i) With u = (u1 , . . . , uN ), u j = (u1 , . . . , uN j ), N = 0 | | 2s 1, (the number of the multi-indexes satisfying 0 | |2s, ) Nj = 0 mj 1, the maps (x, t, u ) → f (x, t, u ) (from × R+ × RN to R), (x, t, u j ) → gj (x, t, u j ) (from j × R+ × RNj to R), satisfy the Carathéodory properties, i.e. they are continuous in u , u j , for almost (x, t) and they are measurable in (x, t) for every u , u j . (ii) The map u(x, t) → (f (x, t, u (x, t)), gi (x, t, u j (x, t))) (from Pl,p (, , × (0, +∞)) to Pl,p (, , × (0, +∞, j × (0, +∞)) transforms each bounded set into a relatively compact set. (iii) 1 > lim inf lim sup (A, Bj )(f, gj )M, , where M→+∞ →+∞
(f, gj )M, (f (x, t, u (x, t)), gj (x, t, u(x, t)))Pl,p (,,×(0,+∞),j×(0,+∞)) = sup M uPl,p (,,×(0,+∞) M . To prove Theorem 4, we need:
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Lemma 3. The map U (x, q) → (L[f (x, t, L−1 U (x, t))](x, g), L[gj (x, t, L−1 U j (x, t))](x, q)) is compact from El,p (, , ) to El,p (, , , j). Theorem 4. Assume that the operator (A(x, Dx , q 1/ ), Bj (x, Dx , q 1/ )|j ) is an elliptic operator for | arg q|/2. Suppose that the assumptions (i)–(iii) are satisfied. Let l l1 , 1 < p < + ∞, U0k ∈ El−k−1+(1/p),p (, , 0 ). Then for large enough = Re q, the problem (11 )–(12 ) (or (11 )–(12 )–(8 ) when 0 is of the first class) has a solution U (x, q) ∈ El,p (, , ). Proof (Sketch). We prove Theorem 3 for 0 belonging to the first class. The proof for 0 of the third class is similar. For each W ∈ El,p (, , ) we have (x, t))](x, q), L[gj (x, t, L−1 W j (x, t))](x, q)) (L[f (x, t, L−1 W ∈ El,p (, , , j). So for large enough =Re q, problem (6 )–(8 ) with the right-hand side (Lf, Lgj , U0k ) has a unique solution U ∈ El,p (, , ). By Lemma 3, the map W → U is compact from El,p (, , ) into itself. So with the assumption (iii) there exists R > 0 such that this map is compact from the closed ball BR = {U ∈ El,p (, , )|U El,p (,,) R} into itself. Therefore, by Schauder theorem, there is a fixed point of this map in BR . This is a solution of the problem (11 )–(12 )–(8 ). By Theorem 4 we get: Theorem 5. Under the assumptions of Theorem 5, if 0 belongs to the third class (or the first class), for large enough , problem (11)–(12) (or (11)–(12)–(8) with the additional condition u0k ∈ Pl−k−1+(1/p),p (, , 0 × (0, +∞)) has a solution u ∈ Pl,p (, , × (0, +∞). 2. Galerkin wavelet method The theory of pseudodifferential operators has been intensively developing. It is so not only because it is a very general theory which includes many theories, especially because it is of great interest and difficulties, but also because it is an extremely powerful tool for studying linear and nonlinear problems. Wavelet theory is growing very rapidly as well. Wavelet approximation methods, however, for even only pseudodifferential operators (not an evolution problem) are investigated in a very few works. In this section we study approximately an evolution problem for a class of interesting, complicated pseudodifferential equations as follows: t ju = −Au(x, t) − a(t − )Au(x, ) d + b(x, t), (13) jt 0 u(x, 0) = u0 (x),
(14)
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n n n where x ∈ J , > 0, A = (D) is a pseudodifferential operator, defined by = R 2/Z ˜ u ∈ C ∞ (Zn ), with symbol () belonging to S 2m (Zn ), (D)u(x)= ∈Zn e ix ()u(), i.e. ∈ C ∞ (Zn ) satisfying | ()|C (1 + ||)2m−| | for all ∈ Zn and for all multiindexes , is the difference operator, which is defined by := (1 −1, 2 −1, . . . , n −1)T , where j f (x) := f (x + ej ), ej = ( j,l )nl=1 is the jth coordinate vector. Here it is assumed also that ∈ C ∞ (Rn \{0}) and (t) = t 2m (), t > 0, (0) = 1. Moreover ()c(1 + ||2 )m , || R > 0. The functions a(t), b(x, t) are given. Note that when = 0, under some assumptions on a(t) the type of equation (1) may be changed. To establish some convergence estimates for this complicated problem, in order to overcome the difficulties, let us use Galerkin-wavelet method and weak solutions and approximative weak solutions, with very effective tools: the Fourier and Laplace transforms (see [9]).
2.1. Notations and preliminaries Let us recall here some usual notations, definitions and some basic facts. For a function u(x) ∈ L2 (Jn ), the discrete Fourier transform is defined by −2ix e u(x) dx = e−2ix u(x) dx, ∈ Zn . F(u)() = u() ˜ := Jn
[0,1]n
(15)
2ix . The discrete inverse Fourier transform is then u(x) := ∈Zn u()e ˜ The Fourier transform of function f ∈ L2 (Rn ) is fˆ() = Rn e−2ix f (x) dx, ∈ Rn , with the inverse Fourier transform f (x) = Rn e2ix fˆ() d, x ∈ Rn . We consider now the space H s (Jn ), (s ∈ R) of functions u ∈ D (Jn ) (the L. Schwartz space of distributions), such that Ds u ∈ L2 (Jn ) and the norm us = ( ∈Zn 2s 2 )1/2 , (∗) is finite, where |u()| ˜ 1 if = 0 = || if = 0. The norm of u ∈ H s (Jn ) is defined by (∗). Definition 1. Let L2 (H q,s0 ) (q 0, s0 > 0) be the space of all functions u(x, t), x ∈ Jn , t 0, satisfying the following conditions: for each 0 q (i) u(x, t) ∈ H (Jn )∀t ∈ [0, +∞). ∞ ˜ t)|2 dt. The norm of the (ii) The following series converges ∈Zn 2 0 e−4s0 t |u(, ∞ q,s function u ∈ L2 (H 0 ) is defined by uL2 (H q,s0 ) = ( ∈Zn 2q 0 e−4s0 t | u(, ˜ t)|2 dt)1/2 . Next we introduce some notations and definitions on wavelets. Definition 2. A multiresolution approximation (M.R.A.) of L2 (Rn ) is, by definition, an increasing sequence Vj , j ∈ Z, of closed linear subspaces of L2 (Rn ) with the following
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properties: j ∈Z Vj = {0}, j ∈Z Vj = L2 (Rn ); for all f ∈ L2 (Rn ) and all j ∈ Z f (x) ∈ Vj ⇔ f (2x) ∈ Vj +1 ; for all f ∈ L2 (Rn ) and k ∈ Zn f (x) ∈ V0 ⇔ f (x − k) ∈ V0 . There exists a function, which is called scaling function (S.F.) (x) ∈ V0 , such that the sequence {(x − k), k ∈ Zn } is a Riesz basic of V0 . An M.R.A. of L2 (Rn ) is said to be r-regular (r ∈ N) if the function is r-regular, that is for each m ∈ N there exists cm such that for all multiindexes , | | r the following condition holds |D (x)| cm (1 + |x|)−m . Let us denote j k (x) = 2nj /2 (2j x − k), k ∈ Zn . Obviously Vj = span{j k (x), k ∈ Zn }. The the periodization operator u(x), that is [u](x) = notation [u](x) stands for of a function k nj /2 j n u(x + k). Denote (x) := [j k ](x) = 2 n (2 (x + l) − k). Similarly let j k∈Z l∈Z us define an M.R.A. of L2 (Jn ) as follows: [Vj ] := span{kj (x), k ∈ Znj }, j 0, where Znj = Zn /2j Zn . It is easy to check [V0 ] ⊂ [V1 ] ⊂ · · · ⊂ [Vn ] ⊂ · · · , j 0 [Vj ]=L2 (Jn ). Furthermore, j
j
if (j k , j l ) = kl , k, l ∈ Zn , then (k , l ) = kl , k, l ∈ Znj , and dim[Vj ] = 2nj . For each j 0, let Pj : L2 (Jn ) → [Vj ] be the orthogonal project of L2 (Jn ) on [Vj ], which has the well known property: Theorem 1. Let −r − 1 s r, −r q r + 1 and s q then u − Pj us c2j (s−q) uq for all u ∈ H q (Jn ), where c is a constant independent of j and u. 2.2. Convergence estimates of solutions Definition 3. A C 1 -mapping u in t, u : [0, ∞) → H 2m (Jn ) satisfying (ju/jt, v) = t −(Au, v) − 0 a(t − )(Au(), v) d + (b, v), (u(x, 0), v) = (u0 , v), ∀v ∈ L2 (Jn ) is called a weak solution of problem (1)–(2). Definition 4. A C 1 -mapping uh in t, uh : [0, ∞)→Vh satisfying (juh /jt, v)=−(Auh , v) t − 0 a(t − )(Au(), v) d + (b, v), (uh (x, 0), v) = (u0h , v), ∀v ∈ Vh , where u0h := Ru0 is a linear approximation of u0 in Vh , is called a weak Galerkin-wavelet solution of problem (1)–(2). The following theorem asserts the stability of the weak solution of problem (1)–(2). Theorem 2. Let u(x, t) ∈ L2 (H 2m,s0 ) be a weak solution of problem (1)–(2) with b = 0. Assume, moreover that there exists a number s0 > 0 such that + Re a(s ˇ 0 + i ) 0, ∀ ∈ R.(∗∗) Then u2L (H 0,s0 ) 1/4s0 u0 2 . 2
Theorem 3. Let u(x, t) ∈ L2 (H q,s0 ) be a weak solution of (1)–(2) and uh (x, t) be its weak Galerkin-wavelet solution. If (∗∗) in Theorem 2 is satisfied and jj u/jt j ∈ H q (Jn ) (j = 0, 1, q 2m) uniformly in t ∈ [0, ∞) then u − uh 2L (H 0,s0 ) 2
C u0 − u0,h + h 2
2(q−2m)
For proofs of Theorems 2 and 3, see [9].
uo 2q
∞
+
e 0
−4s0 t
u2q
2 ju . + jt dt q
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N.M. Chuong / Nonlinear Analysis 63 (2005) e65 – e75
3. From stationary to evolution problems As for pseudodifferential operators, in recent years we also obtained results on abstract stationary problems (see [1,2,6,10–15,25–27] and references therein). It seems that the abstract evolution problems corresponding to these stationary ones may be also studied. Even such evolution problems as in the papers [3–9,17] may be extended to the L(q) , | < q < ∞-theory and to abstract functional spaces, too. References [1] T.Q. Binh, N.M. Chuong, On a fixed point theorem, Funct. Anal. Appl. (Engl. transl. 0) 30 (3) (1996) 220–221. [2] T.Q. Binh, N.M. Chuong, Approximation of nonlinear operator equations, Numer. Funct. Anal. Optim. 22 (7–8) (2001) 831–844. [3] N.M. Chuong, Parabolic pseudodifferential operators of variable order in Sobolev spaces with weighted norms, Dokl. Akad. Nauk SSSR 262 (4) (1982) 804–807. [4] N.M. Chuong, Degenerate parabolic pseudodifferential operators of variable order, Dokl. Akad. Nauk SSSR 268 (5) (1983) 1055–1058. [5] N.M. Chuong, On the theory of parabolic pseudodifferential operators of variable order, Differentialnye Uravneniya 21 (4) (1985) 686–694. [6] N.M. Chuong, N.V. Co, The multidimensional p-adic Green Function, Proc. AMS 127 (3) (1999) 685–694. [7] N.M. Chuong, B.K. Cuong, Galerkin wavelet approximation for a class of partial integro-differential equation, Fractional Calculus Appl. Anal. (Int. J.) 4 (2) (2001) 143–152. [8] N.M. Chuong, B.K. Cuong, The convergence estimates for Galerkin-wavelet method for initial pseudodifferential periodic problems, Int. J. Math. Sci. (14) (2003) 857–867. [9] N.M. Chuong, B.K. Cuong, Convergence estimates of or Galerkin-wavelet solutions to a Cauchy problem for a class of periodic pseudodifferential problems, Proc. AMS 132 (12) (2004) 3589–3597. [10] N.M. Chuong, N.V. Kinh, Regularization of variational inequalities with perturbed non-monotone and discontinuos operators, Diffenrentialnye Uravneniya 27 (12) (1991) 2171–2172. [11] N.M. Chuong, N.X. Thuan, Random equations for semi-H -monotone operators, Random Oper. Stochastic Equations 10 (4) (2002) 1–8. [12] N.M. Chuong, N.X. Thuan, Random equations for weakly semimonotone operators of type (S) and semi-J -monotone of type (J –S), Random Oper. Stochastic Equations 10 (2) (2002) 123–132. [13] N.M. Chuong, N.X. Thuan, The surjectivity of semiregular maximal monotone random mappings, Random Oper. Stochastic Equations 10 (1) (2002) 47–58. [14] N.M. Chuong, N.X. Thuan, Random nonlinear variational inequalities for mappings of monotone type in Banach spaces, Stochastic Anal. Appl., to appear. [15] N.M. Chuong, N.X. Thuan, Some new random fixed point theorems for nonlinear set-valued mappings, Nonlinear Analysis, TMA, submitted for publication. [17] N.M. Chuong, L.Q. Trung, K.V. Ninh, A boundary value problem for nonlinear parabolic equations of infinite order in Sobolev–Orlicz spaces, Mat. Zametki 48 (1) (1990) 78–85. [18] N.M. Chuong, D.A. Tuan,A non-classical semilinear boundary value problem for parabolic pseudodifferential equations in Sobolev spaces Pl,p (, , ), C. R. Acad. Sci, Paris, Ser. I, submitted for publication. [21] Yu.V. Egorov, N.M. Chuong, On some semilinear boundary value problems for singular integro-differential equations, Uspekhi Mat. Nauk 53 324 (6) (1998) 249–250. [22] Yu.V. Egorov, N.M. Chuong, D.A. Tuan, A semilinear non-classical pseudodifferential boundary value problem in Sobolev spaces, C. R. Acad. Sci., Paris, Ser. I 337 (2003) 451–456. [23] Yu.V. Egorov, V.A. Kondratiev, On an oblique derivative problem, Mat. Sb. 78 120 (1) (1969) 148–176. [25] N.Q. Nga, Set-valued nonlinear variational inequalities for H -monotone mappings in non-reflexive Banach spaces, Nonlinear Anal. TMA 52 (2002) 457–465. [26] N.Q. Nga, N.M. Chuong, On a set-valued nonlinear variational inequality, Differentialnye Uravneniya 37 (2001) 128–129.
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[27] N.Q. Nga, N.M. Chuong, Some fixed point theorems for non-compact and weakly asymptotically regular set-valued mappings, Numer. Funct. Anal. Optim. 24 (7–8) (2003) 895–905.
Further reading [16] N.M. Chuong, T.N. Tri, The integral wavelet transform in Lp (R), 1 p ∞, Fractional Calculus Appl. Anal. (Int. J.) 3 (2) (2000) 133–140. [19] I. Dauberchies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. XLI (1988) 906–909. [20] D.L. Donoho, Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data, Proceedings of the Symposium on Applied Mathematics, vol. 47, AMS, Providence, RI, 1993. [24] Y. Meyer, Ondelettes et Opérators, I, II, Herman, Paris, 1990, MR 93i: 42002, 93i: 42003.