CC BY
7
Серия «Математика»
Том 1 (2007), № 1, С. 254-266
Онлайн-доступ к журналу: http://isu.ru/izvestia
ш
УДК 517.968
Difference-differential equations with fredholm operator in the main part
N. A. Sidorov (sidorov@math.isu.runnet.ru)
Irkutsk University, IDSCT SB RAS, Irkutsk
O. A. Romanova (olga@baikal.ru)
Irkutsk University, Irkutsk
Abstract. We consider the reduction of the degenerate difference-differential equations with Fredholm operator in the main expression to the regular problems. It is shown how the question of the choice of boundary conditions is connected with the Jordan structure of operator coefficients of the equations. The problem of the choice of boundary conditions is solved. The theorems of existence and uniqueness of boundary value problems are proved. The abstract theorems are used for statement and investigation of boundary value problems for partial differential equations and difference equations with degeneration.
Keywords: degenerate equations, Jordan sets, Fredholm operator, boundary value problems.
The correct statement and investigation of a boundary value problem for degenerated systems is the major part in the modern theory of partial differential equations [2]. In case of Fredholm operator at the main differential expression the following methods were used: the analytical Nekrasov-Nazarov method [1], [14], the Lyapunov-Schmidt method [23], the Jordan set technique [23], a technique of psedoinverse operators [5] and their modifications [7], [11]-[18], a group symmetry methods [20], [22], a method of differential inclusions [3], a topological method [14], methods of regularization of ill-posed problems [12]. It should be noted that, unlike the classical bifurcation theory [23], for these problems the analog of the Lyapynov-Schmidt bifurcation equation is differential or integral [12]. Finally, the theory of analytical groups and semigroups of generating operators was developed in [19] in order to solve singular problems. The theory of generalized solutions
Introduction
of such problems in the class of Schwartz distributions was constructed in [8], [24]. The results of these investigations have many applications (see references in [12], [19], [24]).
However, in all papers mentioned above the problems were considered, which could be interpreted as ordinary differential equations in Banach spaces. On the other hand, there are a lot of systems of partial difference differential equations with a noninvertible operator in the main part, which can not be interpreted in this way. So, in papers [6],[9], [10], significantly more complex degenerated partial differential equations in Banach spaces are considered. In this paper we develop the similar results for difference-differential equations.
Let us consider the following equation
Au = Lo Bu + Li Am +-----+ Lq Aq u = f (x), (0.1)
where B and Ai,i = 1, q are closed linear operators with the dense domains from E1 to E2; E1,E2 are Banach spaces and D(B) C D(Ai),i = 1,q, x € Q C Rr, B is Fredholm operator with dimN(B) = dimN(B*) = n,R(B) = R(B), f (x) : Q C Rr ^ E2 is a sufficiently smooth function; linear operators Li act on the sets of abstract functions, defined on x € Q C Rr with the values in Banach spaces E2(E1) and satisfy the following conditions:
1)D(Lo) C D(Li) C D(Li),i = 2,...,q;
2)L0Bu(x) = BL0u(x),LiAiu(x) = AiLiu(x) on u(x) € D(B) n D(L0).
Conditions 1) and 2) are carried out, for example, for difference- differential operators of the following form
d
Li(A)= £ ak(x)Dk ^ bk(x)Afc,
\k\<qt \k\<qt
where
Dk
dk
dxk1... dxkr
Ak u = ... ^ (-1)\k\"\i\Ck1... Ct u(xi + iihi, ...,xr + ir hr), il=0 ir=0
q0 > qi > q2 > ... > qq, ak(x),bik(x) : Q C Rr —> R1. In what follows Lj, will denote this concrete form of operators. It will be more convenient not to specify concretely D(L0).
The investigation of singular equation (0.1) is reduced to regular problems, i.e. to equations solvable with respect to the main part L0. A special decompositions of the Banach spaces E1 and E2 in accordance with the generalized Jordan structure of the operator coefficients B,Ai,i = 1,q are
used. This reduction makes it possible to pose boundary value problems for (0.1).
1. P, Q-commutability of Linear Operators in Case of Fredholm
Operator
Let Ei = Mi ® Ni,E2 = M2 ® N2, P be a projector on Mi along N1, Q be a projector on M2 along N2, A be a linear closed operator from E1 to E2, D(A) = Ei, A €{Ai,...,Aq }.
Definition 1. If u € D(A),Pu € D(A),APu = QAu then A(P,Q) -commutes.
Suppose that 0i\rn is a basis in N (B),^i\rn is a basis in N * (B), < 0k,Yi >= $ki, i,k = 1,n, < zi,^k >= 5ik, i,k = 1,n. Then based on [23] there is a unique bounded operator r = (B + n=1 < > zi)-1. Suppose the following condition is satisfied:
(j) _
1. Fredholm operator B has a complete A1 -Jordan set 0i , i = 1, n, j = _ (j) _ _
1,pi, B* has a complete A*- Jordan set ,i = 1,n,j = 1,pi, and the
systems jj - A\^¡pi+1-j), zj - Ai^(pi+1-j), here i = 1n,j = 1pi, corresponding to them are biorthogonal [7]. The projectors
n Pi
P=EE<.,Yj ># -(<.,Y>*), (1.1)
i=1j=1
n P i
Q=EE < > Z(j) - (<., * > Z), (1.2)
i=1j=1
where k = p1 + • • • + pn -root number, generate the direct decomposition
E1 = E1k ® E1<x-k, E2 = E2k ® E2^-k
Corollary 1. The operator r(Q,P)-commutes, A1r(Q,Q)-commutes, rA1(P, P)-commutes, E1xi-k, E1k - invariant subspaces of the operator rA1, E2xi-k,E2k- invariant subspaces of the operator A1r.
The proof is carried out by the substitution of the operators r, A1r, TA1 in formulas (1.1), (1.2), taking into account that = (r* A\)j-1^\, =
(rAi)j-1zfi+1-j) = Ai0(j),7i(Pi+1-j) = A*i = 1^,j = ip
Suppose the operator A(P, Q)-commutes, where P, Q are defined by formulas (1.1), (1.2). Then there is a matrix A, such that A$ = AZ, A*^ = A'Y [9]. This matrix is called the matrix of (P, Q)-commutability.
Corollary 2. The operators B,Al (P,Q)-commute and the matrices of (P,Q)-commutability are the symmetrical cell-diagonal matrices:
Ab = diag(B\,.. .,Bn), Ai = diag(An,..Ani),
where
Bi =
0...0 0...1
01...0
Aii
,i = 1, n.
0...1' 1...0
In case when k > n let us introduce definition.
Definition 2. The operator G (P,Q)-commutes quasitriangularly, if AG is upper quasitriangular matrix, whose diagonal blocks An of dimension pi x Pi,Pi > 2 are lower right triangular matrices. If in addition pi* = ... = Pi* = 1, then in matrix AG at the left of bloc [Aj]i j=i*i*+s are nulls.
2. The Reduction of Equation (0.1) to Regular Problems
Suppose:
2. The operators A2,... ,Aq (P, Q)- commute. Then there are matrices Ai, i = 2, q, such that
Ai$ = AiZ, A*^ = A^T
Following formulas (1.1), (1.2) we introduce the projection operators P, Q, which generate the direct decompositions
El = Elk ® El^-k, E2 = E2k ® E2^-k. Note that rE2^-k C Ei^-k
We look for the solution of equation (0.1) in the following form
u(x) = rv(x) + (C(x), $), (2.1)
where r = (B + < .,7(l) > 2(l))-1 is a bounded operator, v €
E2^-k, C (x) = (Ci(x),...,Cn(x))',Ci(x) = (Cii(x),...,CiPi (x)), $ = ($i,..., $n)', $i = (01,..., i = 1Tn.
Substituting expression (2.1) into equation (0.1) and noting that Brv =
v, since Br = I - Yn=i < > z(1), < >= 0, we obtain
q q
Lov + J2 LiAirv + LoB(C, $) + £ LlAl(C, $) = f (x). (2.2)
i=1 i=1
The operator r (Q, P)-commutes, so from condition 2 and corollary 1 it follows that QAJ(I - Q) = 0, (I - Q)AirQ = 0. Hence, QA^v = 0, Vv € E2xi-k. According to corollary 2 B$ = ABZ, where AB = (B1,..., Bm) is a symmetrical cell-diagonal matrix. Consequently,
(I - Q)B$ = 0, (I - Q)Ai$ = 0,i = l7i, (2.3)
because (I - Q)Z = 0. The following equalities hold:
(Ai(C, $), tf) = AiC, (B(C, $), tf) = Abc. (2.4)
Projecting equation (2.2) onto E2xi-k using (2.3) we obtain the regular equation (solved according to operator L0):
Lv = (I - Q)f (x), (2.5)
where
q
L = Lo + £ LiAir. (2.6)
i=1
In order to determine the vector-function C(x) : Rr — Rk, we project
equation (2.2) onto E2k and using (2.4) we obtain
q
LoAbC + J2 LiAC =<f (x), tf >. (2.7)
i=1
So it is proved
Theorem 1. Suppose conditions 1 and 2 are satisfied, f : Q C Rr — E2 - sufficiently smooth function. Then any solution of equation (0.1) can be represented in the form
u = rv + (C, $)
where v satisfies regular equation (2.5), and the vector C(x) is defined from system (2.7).
Since detAB = 0, then system (2.7) is degenerated in the general case. Let us specify the sufficient conditions on which the systems (2.7) is recurrent sequence of regular linear difference- differential equations (solved according to operator L1).
Introduce the following condition:
3 The operators A2,...,Aq commute quasitriangularly (see Def.2), or k = n.
Lemma 1. Suppose conditions 1, 2, 3 are satisfied. Then system (2.7) is a recurrent sequence of linear difference- differential equations of order q1 with the operators of the form
q
Lks = L1 + ^ apk-s+\,sLi-
i=2
In particular, if condition 1 is satisfied and A2 = ... = Aq = 0, system (2.7) takes the form
LiCiPi(x) =<f (x),^(1) >,
LiCipi-s(x) =< f (x),^(s+1) > -LoCiPi-s+i(x),s = l,pi — 1,i = 1,n. Proof follows from lemma 2 [6] or lemma [10].
So if the conditions of theorem 1 and lemma 1 are satisfied then equation (0.1) can be reduced to regular equation (2.5) and to sequence of regular linear difference-differential equations (2.7) with order q1. Thus under condition 3 system (2.7) be pseudosingular system.
Remark 1. Based on lemma 1 the vector components ci in the case pi > 2 are determined from the recurrent sequence of linear equations of order q1. If pi* = ... = pi*+s = 1 then the corresponding elements Ci*(x),..., ci*+s(x) are determined from the regular system of s + 1 equations of order q1. In particular if p1 = • • • = pn = 1, then AB become zero.
3. The Choice of Boundary Conditions
In this section conditions 1,2,3 are satisfied. In the preceding section for the definition of projections (I — P)u,Pu the solution u of equation (0.1)
we constructed equation (2.5) with regular operator
q
L = Lo + £ LiAir,
i=1
and system (2.7), splitting into recurrent sequence of linear equations with
regular difference-differential operators
q
Lks = L1 + a%pk—s+1, sLi. i=2
The main part of operator L has the order q0, the main part of operator Lks - the order q1. Because q1 < q0 then the boundary conditions on
projections Pu, (I — P)u may be different in the general case. If q1 = 0 then boundary conditions may be given only on the projection (I — P)u. Let u = Q1 n Q2 C Q.
We shall find the solution u : u C Rr ^ E1 of equation (0.1), whose projections Pu, (I—P)u are subjected to the following boundary conditions:
7i(I — P)u = a(x),x £ dQi, (3.1)
T2Pu = /3(x), x £ dQ2. (3.2)
Here t1,t2 are linear difference-differential operators in the space of abstract functions, which commute with the projection operator P and with the operators B, r. So the functions a(x),/(x) must satisfy conditions Pa(x) = 0, (I — P)/(x) = 0. Let the conditions of theorem 1 be satisfied. We shall search for the solution u of equation (0.1), which satisfies boundary conditions (3.1), (3.2). Based on theorem 1 the solution has the form
u(x)=rv(x) + (C (x), $), (3.3)
where v satisfies (2.5) and the condition Qv = 0. The components of the vector C can be find from system (2.7).
Lemma 2. If v,C satisfy the boundary conditions
Tiv = Ba(x),x £ dQi, (3.4)
T2(C, $) = /(x), x £ dQ2, (3.5)
then solution (3.3) satisfies boundary conditions (3.1), (3.2).
Proof. Since Pu = (C, $), then condition (3.2) holds if and only if condition (3.5) holds. Since Qv = 0, Pr = rQ, then (I — P)u = rv. By applying the operator B to the both sides of this equality, we obtain v = B(I — P)u. From here T1v = t1B(I — P)u = Ba(x), and so B[Ti(I — P)u — a(x)] = 0, i.e., t1(I — P)u — a(x) £ J2 So if v satisfies condition (3.4) then v satisfies (3.1). Besides, it is not difficult to understand if v satisfies (3.1) then v satisfies (3.4).
Lemma 3. If KerL = {0} then any solution v of equation (2.5) with condition (3.4) lies in the subspace E2x-k.
Proof. Since QB = BP, Pa(x) = 0, QA,r = A^Q, then the projection Qv satisfies homogeneous problem LQv = 0. As Ker(L) = {0}, then Qv = 0 and v £ E2^-k. Suppose
4. Operators L, Lks have the bounded inverses.
Then based on lemma 2, lemma 3 and theorems 1 we obtain the following result
Theorem 2. Let conditions 1, 2, 3, 4 be satisfied. Then problem (0.1) with conditions (3.1), (3.2) has the unique solution
u(x) = Tv(x) + (C(x), $),
where v satisfies problem (2.5), (3.4), vector C- equations (2.7) with condition (3.5).
4. Examples
Theorem 2 can be used for the statement and the investigation nonclassical boundary value problems for partial differential and difference equations with degeneration.
def
1) Suppose that the coefficients ai (x, y),bj,(x, y) of elliptic equation (A =
d2 + d2 ). dx2 dy2 ''
d d d d (A + a1 (x,y) — + a2(x,y) — + a3(x,y))Bu + (— + h(x,y) — +
+b2(x,y))Au = f (x,y) (4.1)
are defined in bounded domain Q c R2 and belong to the space C\—2,a(H), l > 2, a € (0,1), f : Q ^ E2 belongs to the space Cl—2 ,a(Q). Fredholm operator B has a complete A -Jordan set, A is a compact operator.
Let the domain w C Q, du = l1 U d1 , where l1 is the part of the straight line x = x0(fig.1).
Fig. 1. Example of a domain
We shall consider the "weakened" Dirichlet problem for equation (4.1) in the domain u, i.e. the problem of finding function u, satisfying equation (4.1) in u, and on the boundary du the conditions
where 0o, 0 £ Ci,a, Pfa =0, (I — P)$o(y) = $1(xo, y), fo, 01 £ D(B).
Note that weakening the classical statement of the Dirichlet problem is that there is no condition on projection Pu along d1 of the boundary du. It is connected with that Pu will be defined from the system of the first order.
Based on theorem 1 the solution has form (2.1).
If condition 4 is satisfied and l > 2max1<i<lpi then based on theorem 2 problem (4.1)-(4.3) has the solution in the class Cl, a(w).
Suppose that the homogeneous problem
has only the trivial solution. Notice that this condition will be satisfied if the diameter of the domain u will be sufficiently small. It is not difficult to show that L is continuously invertible. In fact A is a compact operator, r is a bounded operator, so Ar is a compact operator. So the problem
can be reduced to the equation of the second kind with the compact operator. (In the case when E1 = E2 = Rn it was shown in [2]). So the operator L is Fredholm operator, and also N(L) = {0}. Therefore it is continuously invertible. Based on lemma 1 coefficients of the vector C(x,y) are defined from the split recurrent sequence of linear differential equations of the first order with the boundary condition
u\x=xo = MУ),
(I — P )u\d1 = 01(x,y),
(4.2)
(4.3)
u\du = 0
v\x=xo = B (I — P )0o v\di = B01
(C, $)\x=xo = P0o(y).
So the conditions of theorem 2 for problem (4.1), (4.2), (4.3) are satisfied and it has unique solution in the class Cl,a(w).
Remark 2. If operator B has a complete A1- Jordan set, A^A2 are 1ij iiij
compact operators, aij> vYh 1 v — const > 0,
d r d2 r d Lo( dx ) = Ç aij(x) dXidX" +Ç ai(x) dXi +a(x),
Ll( dx) dx\ + Ç bi(x) dxi '
2
i
x € Q C Rr,
then similar results can be obtained for the equation
d d Lo(—)Bu + Li(—)Aiu + A2U = f (x),x € u,
with the boundary conditions (fig.2):
(I — P)u\du = 0o(x), P$0 =0, Pu\xi=x0 = 0l(x2, . . . ,xr) (I — P)$1 = 0.
f: ( ol V d y
\ ' xo xl
Fig. 2. The domain in the remark
2) Now we consider the initial value problem for difference equation of the first order
B(u(x + 1) - u(x))+ Au = f (x), (4.4)
(I - P)u|x=o = uo,Puo = 0. (4.5)
def
Here operators B, A satisfy condition 1, where Ai = A,x € Q = {0,1,2,...}. We look for the solution of (4.4) in the following form
u(x) =rv(x) + (C(x), $).
According to results of paragraph 3 the problem (4.4)-(4.5)can be reduced to following problems:
v(x + 1) — v(x) + Arv(x) = (I — Q)f (x), (4.6)
v\x=o = Buo (4.7)
Ab(C(x + 1) — C(x)) + A1C(x) =< f (x), ^ > . (4.8)
and it follows from (4.5) that there is no boundary condition on C(x). By initial value (4.7) one easily arrives at the next solution
x-1
v(x) = (I — Ar)xBuo + ]T(I — Ar)x-v-1(I — Q)f (v),
v=o
under x = {1, 2, . . .}.
Based on lemma 1 coefficients of the vector C(x) are defined from the split recurrent sequence
CiP„(x) =<f(x),^1} >,
CiPz-s(x) =< f (x),4s+1) > —(CiPz-s+1(x + 1) — CiPz-s+1(x)),
s = 1,pi — 1,i = 1, n.
So the conditions of theorem 2 for problem (4.4)-(4.5) are satisfied and it has unique solution.
5. Conclusion
In regular case, when the operator in the leading part of degenerate equation is continuously invertible, we can apply for investigation the well-known methods [2]. In irregular case, the problems stated for such equations in the standard way in general have no classical solutions [6]. It was shown in [6] that the solvability depends on the lowest terms. Thus, the question on influence of the lowest terms is important for statement of the boundary value problems in the theory of difference- differential equations with a noninvertible operator in the leading part.
Correct statement of boundary value problems for partial differential equations and difference equations with Fredholm operator in the split leading part and their investigations can be simplified significantly if to find a reasonable projection of the solution onto subspaces in accordance with properties of the Jordan structure of the operator coefficients of the equation [6], [9], [10].
In general, the choice of boundary conditions for equation (0.1) which supply the existence of the unique classical solution for arbitrary f (x) is difficult. So, we need to extend the class of solutions, where we seek for the boundary value problem solutions of equation (0.1). For example, we can suppose that coefficients of projection Pu are the elements of the distributions space. This extended notion of the solution for equation (0.1) when x € R1 was investigated in [8].
The method proposed in this paper of reduction of equation (0.1) to regular problems can be applied to investigation in the case, when operator coefficients of equation (0.1) depend of x.
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Н. А. Сидоров, О. А. Романова
Дифференциально-разностные уравнения с фредгольмовым оператором при главной части
Аннотация. Рассматривается редукция вырожденных дифференциально-разностных уравнений с фредгольмовым оператором при главной части к регулярным задачам. Показано, что выбор начальных условий связан с жордановой структурой операторных коэффицинтов уравнений. Решена задача выбора начальных условий. Доказаны теоремы существования и единственности для задач с заданными начальными условиями. Полученные результаты используются для постановки и исследования граничных задач для дифференциальных уравнений в частных производных и разностных уравнений с вырождением.
