CC BY
75
Онлайн-доступ к журналу: http: / / mathizv.isu.ru
Серия «Математика»
2019. Т. 27. С. 55-70
УДК 518.517 MSG 35L05, 45D05
DOI https://doi.org/10.26516/1997-7670.2019.27.55
Classic Solutions of Boundary Value Problems for Partial Differential Equations with Operator of Finite Index in the Main Part of Equation
N. A. Sidorov
Irkutsk State University, Irkutsk, Russian Federation
Abstract. This paper is an attempt to give the review of a part of our results in the area of singular partial differential equations. Using the results of the theory of complete generalized Jordan sets we consider the reduction of the PDE with the irreversible linear operator B of finite index in the main differential expression to the regular problems. Earlier we and other authors applied similar methods to the development of Lyapunov alternative method in singular analysis and numerous applications in mechanics and mathematical physics. In this paper, we show how the problem of the choice of boundary conditions is connected with the B-Jordan structure of coefficients of PDE. The estimation of various approaches shows that the most efficient approach for solving this problem is the functional approach combined with the alternative Lyapunov method, Jordan structure coefficients and skeleton decomposition of irreversible linear operator in the main part of the equation. On this base, the problem of the correct choice of boundary conditions for a wide class of singular PDE can be solved. The aggregated theorems of existence and uniqueness of classical solutions can be proved with continuously depending of experimental definite function. The theory is illustrated by considering the solution of some integro - differential equations with partial derivatives.
Keywords: degenerate PDE, Jordan set, Banach space, Noether operator, boundary value problems.
Let. B and Ai, i = 1 ,q be closed linear operators from E\ to E-i with the dense domains in E\. Let E\,E2 be Banach spaces, D(B) C D(Ai),
This paper is dedicated to the 100th anniversary of
Irkutsk State University
1. Introduction
B the operator of finite index with closed range of values, dim N(B) = n, dim N(B*) = m, u = n — m< oo. The operator
is a partial differential operator of order
q0 > qi > q2 > ■ ■ ■ > qq-
Functions aik{x) : 0 e IR" -> IR1 and f(x) : Q € RN ->■ E2 are sufficiently smooth. The differential equation
L°^BU + Ll OAlU + ''' + Lq^AqU = fix) (L1)
is considered below.
Definition 1. Equation (f.f) is a regular equation, if the operator B has bounded inverse. Otherwise we say that (f.f) is a singular equation.
In sections 2, 3 the investigation of singular equation (f.f) with irreversible operator B in the main part is reduced to the regular problems. The special decomposition of the Banach spaces E\ and in accordance with the generalized B-Jordan structure of the operator coefficients A\,...,Aq is used. In section 4 this reduction makes it possible to pose boundary value problems for singular equation (f.f) with natural conditions on special projections of the solution.
The methods of this investigation interconnect with the functional approach from [8-11;27] (see also references in [f;2;6;24;25] and the extensive bibliographical review of Loginov school papers [3]).
2. Preliminaries: Pk, Qk ~ commutability of linear operators in case of Noetherian operator В
Let Ei = Mi ф Ni, E2 = M2 Ф N2, P be a projector on Mi along NUQ a projector on M2 along N2, A be a linear closed operator from E\ to E2, D(A) = Ei,Ae{Ai,...,Aq}.
Definition 2. If и e D(A), APu = QAu, then A (P,Q) -commutes.
Let {ip^,..., tpn } be a basis in N(B), {ф^,..., фт } a basis in N(B*). Suppose the following condition is satisfied:
f. The Noether operator В has a complete A\ -Jordan set i = 1, n,
_ fj\ _ _
j = I,Pi, and B* has a complete -Jordan set ф[ , i = 1, m, j = l,Pi, and
the systems ^j) = , = AK/>fi+1~j), where i = ~l,j = T~pl,
I = min(m,n) corresponding to them are biortogonal [27]. The projectors
i Pi
pK = ££<-,7?H#d=f(<,7>,$), (2.1)
i=l j=1
I Pi
Qk = EE < > z? = « (2.2)
i=i j=l
where A; = pi H-----Vpi is a root number, generate the direct decomposition
Ei = Eik © ^ioo-fc, E2 = E2k © E2oD-k-
Corollary 1. The bounded pseudoinverse operator B+(Pk,Qk)-commutes, the operator AB+ (Qk,Qk)-commutes, B+A (Pk, Pk)-commutes, Ei^-k, Eitk are the invariant subspaces of operator B+A, E2oo-k,(ind E2k are the invariant subspaces of operator AB+.
Suppose the operator A (Pk,Qk)-commutes, where Pk,Qk are defined by formulas (2.1), (2.2). Then there is a matrix A, such that A§ = AZ, A*^ = Alry. This matrix is called the matrix of (Pk, Qfc)~commutability.
Corollary 2. The operators B and A (Pk, Qk)-commute and the matrices of (Pk, Qk)-commutability are the symmetrical cell-diagonal matrices:
AB = diag(Bi, ■■■ ,Bt) Aa = diag(A\, ■■■ ,At),
where
"0 ... 0
0 1 . . 0
See the detailed proof in the preprint [10] .
Definition 3. The operator G (Pk,Qk)-commutes quasitriangularly, if Ac is upper quasitriangular matrix, whose diagonal blocks An of dimension Pi x pi are lower right triangular matrices.
0 ••• 1 1 ••• 0
3. The reduction of equation (1.1) to the regular PDE
Suppose:
2. The operators A2, ■ ■ ■ ,Aq (Pk, Qfc)~commute. Then there are matrices Ai i = 1, q, such that
Ai$ = AiZ, ApV = A't7 and Ai = (»An, • • • ,An) is a cell-diagonal matrix ,where
0 ••• 1
Ai i
1 ••• 0
г = 1,1.
Let us consider the case m <n.
We introduce the projectors Pk,Qk based on formulas (2.1), (2.2) and the projector
n
Pn-m= E <-,li>4>i
i=m-\-1
generating the direct decompositions Ei = Eik © span{(f)m+1, • • • , (f)n} © Ei
oo—(fc+ri—m) i E2 = E2k © E2oo—k.
Note that B+ : E2oo_k ~> ^ioo-(fc+n-m) c Ei^k, B+ : E2k —> Eik+n-m-We shall seek the solution of equation (f.f) in the following form
n
u(x) = B+v(x) + (C(x), $) + i3-1)
i=m-\-1
where B+ is a bounded pseudoinverse operator for B, v € E2oo_k, C(x) = (Ci(x),--- ,Cm{x))', Ci(x) = (Cti(x),--- ,Ctp(x)), $ = ,$m)',
Substituting the expression (3.1) in equation (f.f) and noting that BB+v = v, because v € E2oo_k c E2oo_k c E2oo_m, we obtain
+ E +ф) + E ф)+
г=1 г=1
д п „
+Е Е =
j= 1 г=т+1
(3.2)
The operator (Qfc, Pfc)-commutes, so from condition 2 and corollary f it follows that - Qfc) = 0, (/ - = 0. Hence,
QkAiB+v = 0, \/v € E2oo—ie. According to corollary 2 БФ = AbZ, where Ab = (-Bi, • • • , Bm) is a symmetrical cell-diagonal matrix. Consequently,
(1-Як)ВФ = Ъ, {1-Як)АгФ = 0, i=~q, (3.3)
because (/ — Qk)Z = 0. The following equalities are fulfilled:
rlef rlef
(MC,*),*) = A'fi, (2?(C, <&),*) = ABC. (3.4)
Projecting equation (3.2) onto E2oo_k by virtue of (3.3) we obtain the regular PDE
q n „
Cv = (I-Qk)f(x)~Y^ Yl Lj(—)Aj(f>i\i(x), (3.5)
j= 1 i=m-\-1
where
i= 1
is a regular differential operator of order qo. In order to determine the vector-function C(x) : RN —> Rk, we project the equation (3.2) onto E2k and by virtue of (3.4) we obtain the PDE-system
F) q F) q n r)
L^)ABC+Y,L%{-)AC=<f{x)-Y, E
i=l j=2i=m-\-l
(3.7)
So we have proved
Theorem 1. Suppose conditions 1 and 2 are satisfied, m < n, / : Q C Rn —> E2 is a sufficiently smooth function. Then any solution of equation (1.1) can be represented in the form
n
u = B+v + (C,$)+ £ Ai0is
i=m-\-1
where v satisfies regular equation (3.5), and the vector C(x) is defined by system (3.7). The functions Xi(x), i = m + 1, n remain arbitrary functions.
Theorem 1 allows generalizations. Suppose the operators A2(x),--- , Aq(x) with the domains independent from x, are subject to the operators B, and for any x € Q satisfy condition 2. Then theorem 1 remains valid. Let us consider system (3.7) with a unknown vector-function C(x).
Lemma 1. Suppose conditions 1,2 are satisfied, operators Ai, i = 1 ,q, (Pk,Qk)-commute quasitriangulariy. Then system (3.7) is a recurrent sequence of linear differential equations of order qi with the regular differential operators of the form
d q d ^ks = Ll(ydx^ + E a%-s+i,sLi(g^)-
i=2
In particular, if condition 1 is satisfied and A2 = ■ ■ ■ = Aq = 0, system (3.7) takes the form
Ll(^)Cm.s(x) =< f(x),4s+1) > -Ь0фсгРг.3+1(х), s = l,pi — 1, i = 1, m.
Corollary 3. Let equation (1.1) has the form
д
L0(—)Bu + AlU = f(x)
and condition 1 be satisfied. Then the vector C(x) is defined by simple recursion.
Proof. The proof is obvious, because in this case = 1 and A2 =
■ ■ ■ = Aq = 0 in equation (f.f).
Let us consider the second case m> n. In this case we use the direct decompositions:
E = E\k © Eioo-fc, E2 = E2k © span{zn+i, • • • , zm} © E2oo
— (fc+m—n)
and also B+ : E2oD_(k+m_n) Егоо_к,В+ : E2k+m_n Elk. We shall seek the solution of equation (f.f) in the following form
u(x) = B+v(x) + (C(x), Ф), (3.8)
where veE2ook,C(x) = (Ci(x), ■ ■ ■ ,Cn{x))', Сг{х) = {Сц{х), ■ ■ ■ ,Cipi{x),)
Ф = (ФЬ...,Ф „)', Фг = {ф[р,--- i=T~n.
By substituting (3.8) in equation (f.f) we obtain
о Q о о
Lo" Qm-n)v + L^di)AlB+V + )Б(С' Ф)+ (3"9)
г=1
+E ьгфмс,Ф) = т
г=1
with the condition < v,ipi >= 0, i = l,n. Let condition 2 is satisfied. By projecting equation (3.9) onto the subspaces E2k, E2m_n, E2oD-(k+m-n) we obtain
В q 8
L0(q^)AbC + 22 =< №,i) >, (3.10)
i= 1
q д
Y, Lt(—)Qm_nAtB+v = Qm-nf(x), (3.11)
"dx' i= 1
д q д L0(—)(I - Qm-n)v + Y Li(—)Qm_nAiB+v = (/ -Qk- Qm-n)f(x),
i= 1
(3.12)
where v = E2oD_k, ^ = C0i>''' ,ipnY,ipi = • • The element
v can be found from the regular equation
¿v = (I — Qk)f (3.13)
in the subspace E2oo-k^-^2oo-(m-n)• Indeed, if Qm-nv = 0, then by virtue of Qm-nQk = 0 the solution v of the equation 3.13 satisfies equations (3.11), (3.12). □
Thus we obtain the following result
Theorem 2. Let n <m, conditions 1, 2 be satisfied, and / : Q С RN —>• E2 be a sufficiently smooth function. Then any solution of equation (1.1) can be represented in the form (3.8), where v € E2oo_k П E2oD-(m-n) is the solution of equation (3.13), the vector С is defined by the system (3.10).
4. Examples
Let operator B be Fredholm (m = ri). Then basing on theorems 1, 2 the choice problem of correct boundary conditions for equations (3.2), (3.7) and (3.13), (3.10) can be solved for a number of differential operators and Li(^). Example 1 Consider the equation
d2
-^^Bu{x,y) + Au{x,y) = f{x,y) (4.1)
This equation with usual Goursat conditions u\ x=o — 0, u\y=o — 0 and an arbitrary right part evidently has no the classical solution.
Let operators B and A satisfy condition 1, k = p\ + • • • +pn, p% are the lengths of A-Jordan chains of operator B.
Then in accordance with our theory we can impose the following conditions on the projections of the solution:
(I - Pk)u(x,y)\x=0 =0, (I - Pk)u(x,y)\y=o = 0
(4-2)
As a result we can construct the following unique classical solution u(x,y) =
= r fy r {Ar)r 0*1 - (yi - VY (/ _ Qk)f(Xl,y1)dy1dx1 +
J0 -/0 r=0
ra Pi i=l i=l
where T = (B + ^¿li < > ¿i)-1 is the bounded operator (see Scmidt lemma in [27]) The functions Cij(x,y) are defined recursively
CiPi(x,y) = Pn (x,y), d2
CiPi-i(x,y) = f3i2(x,y) -
d2
CiPi-2{x,y) = f3i3(x,y) - -^^CiPi-i{x,y),
where pis(x,y) =< f(x,y),ipl > >, i = l,n, s = l,pi.
Evidently our solution of this special Goursat problem continuously depends on the right part if f(x) € where p = max(p\, ■ ■ ■ ,pn). Example 2 Consider the equation
-3 Jo xs^%Tds = u{t'x) + /(i'x) (4-4)
with the condition
u(0,x)—3 / xsu(0, s)ds = 0. (4.5)
Jo
According to (3.8) we can construct the solution as the sum u(t,x) = v(t,x) + c(t)x, where jJ xv(t, x)dx = 0, c(t) = —3 Jq xf(t,x)dx
dv dt
= v + f(t,x) — 3 / xsf(t,s)ds, Jo
v\t=0 = o.
Therefore, we have the unique solution of example 2
u(t,x) = J et~z x) — 3 J xsf(z,s)ds)^Jdz — 3 J xsf(t,s)ds.
Example 3
Consider the integro-differential equation of order 2 d2u(t, x) f1 d2u(t, s) du(t,x)
The continuous function f(t,x) is defined under a; € [0,1], t > 0. The Cauchy problem with standard conditions u\t=o = 0
du
mlt=o = 0
is unsolvable under an arbitrary function f(t,x).
Projector P = 3 /J xs[-]ds corresponds to the Fredholm operator
B = 1-3 [ xs[-]ds. Jo
We can use theorem f. Therefore introducing special conditions u\t=o = 0,
du
= 0 (4.7)
t=o
we can construct the solution as the sum
u(t, x) = v(t, x) + c(t)x,
where Pv = 0. Functions v(t, x) and c(t) can be found from two the simplest Cauchy problems
JS? = & + §* + /(*.*)
dt I
v\t=0 = o §|t=0 = 0,
)%+Pf = 0
\c( 0) = 0.
Conditions (4.7) were induced by our theorem f. As a result we can easily construct the desired classical solution of the problem (4.6), (4.7)
u(t,x) = / (ei_s — f)/(s, x)ds — 3x / / et~sxf(s,x)dxds. Jo Jo Jo
Example 4
Consider the equation
d2 d
Bu(x, y) + y) = f(x, y) (4.8)
Let B be a Fredholm operator, (<pi,--- ,<fin) be a basis in N(B), and (■01, • • • ,ipn) a basis in N(B*).
64 N. A. SIDOROV Let
{1 i—k n n
' \ ~ . p = 22 < '> <3 = E < '' ^ >
(J, i ^ fc, 1 1
Then according to theorem 1 we can impose the following conditions on projections:
ffoi
(/ _ P)u\x=0 = 0, (/-P) —U=o = 0, Pu|y=o = 0. (4.9) As a result we have the unique classical solution in the form of the sum
u(x,y) =Tv(x,y) + V / < f(x,y),tpi> dy(/>i, (4.10) 1 Jo
where
n l
is the bounded operator. The function v(x,y) is the unique solution of the regular Cauchy problem
If f(x,y) is an analytic function, then
Ф,У) = 22Ci^c>
where
i= 2
C2(v) = ^(i-Q)№y),
W'kv-vV I-
Therefore, we have the following asymptotics of the solution
i 2 n u(x,y) = -Ж2Г/(0, 0) + (у - у) < /(0, 0), фг > фг + 0(|/2 + М3).
1
Example 5
Consider the equation of 5th order
d3 ( d2 \ id2 \
dt3\d^2+1)u("x,y,t^ + [dH2 +X)u(x>y>^ = f(x>y>t) (4-n)
with boundary conditions
u\x=0 = 0, u\x=7T = 0, (4.12)
u\y=o = 0, u\y=7T = 0. (4.13) Introduce initial conditions
^ _ pj&u(x,y,t)
dP where
= 0, i = 0,1,2, (4.14)
t=o
2 r
P = — sina;sins[-]ds
7T JO
is the projector on Ker B.
The operator B = J^p + 1 with condition (4.12) maps from Cj^j in
C[0)i].
Let A / n2, f(x, y, t) be a continuous function in the domain 0 < x < 1, 0 < y < 1, t > 0.
Then from the proof of theorem 1 it follows that equation (4.11) with the initial and boundary conditions (4.12), (4.13), (4.14) has the unique classical solution. Example 6
Consider the differential-difference integral equation
du(t x) _3 f xsdu^s)ds = u(t,x) + \u(t-A,x) + f(t,x), t> 0, (4.15) ot J o ot
u(t, x)
= 0.
-A<t<0
Following example 2, without loss of generality, we are looking for a solution in the form of a sum
u(t,x) = v(t,x) + c(t)x, (4.16)
where
/ sv(t,s)ds = 0. Jo
Using the steps method we uniquely define the functions v(t,x) and c(t) from the following problems
= v(t, x) + Av(t — A, x) + f(t, x) — 3 /J xsf(t, s)ds,
= 0,
—A<i<0
'c(i) + Xc(t - A) = -3 /J s/(i, s)ds,
c(t)
— A<i<0
If A = 0, then (4.16) satisfies the initial condition (4.5), and we come to the result discussed in Example 2.
5. Conclusion
The very first results concerning nonclassical correct boundary conditions for a degenerate differential equation with the Fredholm operator were established in paper [21].
In papers [8-10] the general approach how to construct the set of correct boundary condition for equation (f.f) was considered. For example, some authors effectively exploited Showalter-Sidorov boundary conditions for the mathematical modeling of complex problems (see [25; 26] ). Such conditions can be obtained as a special case of the approach established above. Of particular interest is finding the solution of a irregular PDE system (f.f), where operator В is assumed to enjoy the skeleton decomposition [15].
In paper [15] the first version of skeleton chains of a linear operator is introduced. In this case the problem of finding the solution of a singular PDE (f.f) also can be reduced to the regular split systems of equations. The corresponding systems also can be solved with the respect of taking into account certain initial and boundary conditions. However the effective use of the concept of skeleton chains for applications will be in the future. The development and applications of our functional approach to other linear and nonlinear integral and integro-differentional systems can be found in [12]- [23] and in mathematical reviews (for example, see MR372I762, MR1959647, MR 0810400, MR334364f, MR2920089, MR279574, MR2675324, MR320f397, etc.)
It should be noted that the interest in this field was stimulated by the important problems first posed in the famous article by L. A. Lusternik " Some issues of nonlinear functional analysis" [4].
References
1. Falaleev M.V., Sidorov N.A. Continuous and generalized solutions of singular differential equations. Lobachevskii Journal of Mathematics, 2005, vol. 20, pp. 31-45
2. Leontyev R.Yu. Nonlinear equations in Banach spaces with a vector parameter in irregular cases. Irkutsk, Irkutsk State University Publ., 2011, 101 p. (in Russian)
3. Loginov B.V. Bibliographic Index of Works (compiled by O. Gorshanina). Ulyanovsk, UlSTU Publ., 2008, 59 p., Series "ULTU Scientists).
4. Lusternik L.A. Some issues of nonlinear functional analysis. Russian math, surveys, 1956, vol. 11, no. 6, pp.145-168. (in Russian)
5. Muftahov I.R., Sidorov D.N., Sidorov N.A. On the Perturbation method. Vestnik YuUrGU. Ser. Mat. Model. Progr., 2015, vol. 8, no. 2, pp. 69-80. https: //doi.org/10.14529/mmp150206
6. Orlov S.S. Generalized solutions of high-order integro-differential equations in Banach spaces. Irkutsk, Irkutsk State University Publ., 2014, 149 p. (in Russian)
7. Rendon L., Sinitsyn A.V., Sidorov N.A. Bifurcation points of nonlinear operators: existence theorems, asymptotics and application to the Vlasov-Maxwell system. Rev. Colombiana Math, 2016, vol. 50, no. 1, pp. 85-107. https://doi.org/10.15446/recolma.v50n1.62200
8. Sidorov N.A., Blagodatskaya E.B. Differential equations with a Fredholm operator in the leading differential expression. Soviet math. Dokl., 1992, vol. 44, no. 1. pp. 302-303.
9. Sidorov N.A., Romanova O.A., Blagodatskaya E.B. Partial differential equations with an operator of finite index at the principal part. Differ. Equations, 1993, vol. 30, no. 4. pp. 676-678. (in Russian)
10. Sidorov N.A., Blagodatskaya E.B. Differential equations with a Fredholm operator in the lea,ding differential expression. AN SSSR, Irkutsk Computer Thentr Preprint, 1991, no. 1. 36 p. (in Russian)
11. Sidorov N.A., Loginov B.V., Sinithyn A.V., Falaleev M.V.Lyapunov-Schmidt methods in nonlinear analysis and applications. Springer, ser. Mathematica and applications, 2003, vol. 550, 558 p. https://doi.org/ 10.1007/978-94-017-2122-6
12. Sidorov N.A., Sidorov D.N. Existence and construction of generalized solutions of nonlinear Volterra integral equations of the first kind. Differ. Equations, 2006, vol. 42, no. 9, pp.1312-1316. https://doi.org/ 10.1134/S0012266106090096
13. Sidorov N.A., Romanova O.A. Difference-differential equations with fredholm operator in the main part. The Bulletin of Irkutsk State University. Series Mathematics, 2007, vol. 1, no.1, pp. 254-266. (in Russian)
14. Sidorov N.A., Romanova O.A. On the construction of the trajectory of a single dynamic system with initial data on hyperplanes. The Bulletin of Irkutsk State University. Series Mathematics, 2015, no. 12, pp. 93-105. (in Russian)
15. Sidorov D.N., Sidorov N.A. Solution of irregular systems using skeleton decomposition of linear operators. Vestnik YuUrGU. Ser. Mat. Model. Progr., 2017, vol. 10, no. 2, pp. 63-73. https://doi.org/10.14529/mmp170205
16. Sidorov N.A., Sidorov D.N. On the Solvability of a Class of Volterra Operator Equations of the First Kind with Piecewise Continuous Kernels. Math. Notes, 2014, vol. 96, no. 5, pp. 811-826. https://doi.org/10.4213/mzm10220
17. Sidorov N. A., Leontiev R. Yu., Dreglea A.I. On Small Solutions of Nonlinear Equations with Vector Parameter in Sectorial Neighborhoods. Math. Notes, 2012, vol. 91, no. 1, pp. 90-104. https://doi.org/10.4213/mzm8771
18. Sidorov N.A., Sidorov D.N. Small solutions of nonlinear differential equations near branching points. Russian Math. (Iz. VUZ), 2011, vol. 55, no. 5, pp.43-50. https://doi.org/10.3103/S1066369X11050070
19. Sidorov N.A. Parametrization of simple branching solutions of full rank and iterations in nonlinear analysis. Russian Math. (Iz. VUZ), 2001, vol. 45, no. 9, pp.55-61.
20. Sidorov N.A. Explicit and implicit parametrizations in the construction of branching solutions by iterative methods. Sb. Math. J., 1995, vol. 186, no. 2, pp.297-310. https://doi.org/10.1070/SM1995vl86n02ABEH000017
21. Sidorov N.A. A class of degenerate differential equations with convergence. Math. Notes, 1984, vol. 35, no. 4, pp.300-305. https://doi.org/10.1007/BF01139992
22. Sidorov D.N., Sidorov N.A. Convex majorants method in the theory of nonlinear Volterra equations. Banach J. Math. Anal, 2012, vol. 6, no. 1, pp.1-10. https://doi.org/10.15352/bjma/1337014661
23. Sidorov N.A., Sidorov D.N. Solving the Hammerstein integral equation in the irregular case by successive approximations. Siberian Math. J., 2010, vol. 51, no. 2, pp. 325-329. https://doi.org/10.1007/sll202-010-0033-4
24. Sidorov D.N. Integral Dynamical Models: Singularities, Signals and Control. World scientific series, nonlinear science, Series A, vol. 87, ed. by L. O. Chua, 2015, 243 p. https://doi.org/10.1142/9278
25. Sviridyuk G.A., Fedorov V.E. Linear Sobolev type equations and degenerate semigroups of operators. Utrecht, VSP, 2003, 228 p.
26. Sviridyuk G.A., Zagrebina S.A. The Showalter - Sidorov problem of the Sobolev type equations. The Bulletin of Irkutsk State University. Series Mathematics, 2010, vol. 3, pp. 104-125.
27. Vainberg M.M., Trenogin V.A. The theory of branches of solutions of nonlinear equations. Wolters-Noordhoff, Groningen, 1974, 302 p.
Nikolay Sidorov, Doctor of Sciences (Physics and Mathematics), Professor, Irkutsk State University, f, K. Marx st., Irkutsk, 664003, Russian Federation, tel.: (3952)242210 (e-mail: sidorovisuOgmail. com)
Received 21.01.19
Классические решения граничных задач для дифференциальных уравнений в частных производных с оператором конечного индекса в главной части
Н. А. Сидоров
Иркутский государственный университет, Иркутск, Российская Федерация
Аннотация. В статье дается обзор результатов в области нерегулярных уравнений в банаховых пространствах с частными производными с необратимым оператором в главной части уравнения. Некоторые результаты статьи ранее анонсировались в препринтах. Показано как, используя информацию об обобщенной в смысле Келдыша жордановой структуре операторных коэффициентов уравнения, можно сводить нерегулярные задачи к регулярным. На этой основе демонстрируется решение проблемы подбора корректных граничных условий для широкого класса нерегулярных уравнений в частных производных высокого порядка. Общие теоремы о
редукции, приведенные в статье, дают возможность не только получать достаточные условия существования и единственности классических решений, но и строить решения с входными данными, взятыми из эксперимента. Теория абстрактных уравнений в банаховых пространствах на содержательном уровне иллюстрируется решением граничных задач для ряда конкретных интегро-дифференциальных и разностных уравнений различных порядков.
Ключевые слова: вырожденные дифференциальные уравнения, банаховы пространства, нетеров оператор, граничная задача.
Список литературы
1. Falaleev М. V., Sidorov N. A. Continuous and generalized solutions of singular differential equations // Lobachevskii Journal of Mathematics. 2005. Vol. 20. P. 3145.
2. Леонтьев Р. Ю. Нелинейные уравнения в банаховых пространствах с векторным параметром в нерегулярных случаях. Иркутск : ИГУ, 2011. 101 с.
3. Логинов Б. В. Библиографический указатель трудов / сост. О. Горшанина. Ульяновск : УлГТУ, 2008, 59 с. (Ученые УлТУ).
4. Люстерник Л. А. Некоторые вопросы нелинейного функционального анализа // Успехи мат. наук. 1956. Вып. 11, № 6. С. 145-168.
5. Муфтахов И. Р., Сидоров Д. Н., Сидоров Н. А. О методе возмущений // ЮУрГУ. Сер. Мат. моделирование и программирование. 2015. Т. 8, № 2. С. 69-80.
6. Орлов С. С. Обобщенные решения интегро-дифференциальных уравнений высокого порядка в банаховых пространствах. Иркутск : ИГУ, 2014. 149 с.
7. Rendon L., Sinitsyn А. V., Sidorov N. A. Bifurcation points of nonlinear operators: existence theorems, asymptotics and application to the Vlasov-Maxwell system // Rev. Colombiana Math. 2016. Vol. 50, N 1. P. 85-107. https://doi.org/10.15446/recolma.v50nl.62200
8. Сидоров H. А., Благодатская E. Б. Дифференциальные уравнения с фред-гольмовым оператором при старшем дифференциальном выражении // Докл. Акад. наук СССР. 1991. Т. 319, № 5. С. 302.
9. Сидоров Н. А., Романова О. А., Благодатская Е. Б. Уравнения с частными производными с оператором конечного индекса при главной части // Дифференц. уравнения. 1994. Т. 30, № 4. С. 729.
10. Сидоров Н. А., Благодатская Е. Б. Дифференциальные уравнения с фред-гольмовым оператором при старшем дифференциальном выражении // Препринты ИПМ им. М.В. Келдыша. 1991. № 1. С. 35.
11. Lyapunov-Schmidt methods in nonlinear analysis and applications / N. A. Sidorov, В. V. Loginov, А. V. Sinithyn, M. V. Falaleev. Springer, 2003. 558 p. (Mathematica and applications ; vol. 550). https://doi.org/10.1007/978-94-017-2122-6
12. Сидоров H. А., Сидоров Д. H. Существование и построение обобщенных решений нелинейных интегральных уравнений Вольтерра первого рода // Дифференц. уравнения. 2006. Т. 42, № 9. С. 1312-1316.
13. Сидоров Н. А., Романова О. А. Дифференциально-разностные уравнения с фредгольмовым оператором при главной части // Изв. Иркут. гос. ун-та. Сер. Математика. 2007. Т. 1, № 1. С. 254-266.
14. Сидоров Н. А., Романова О. А. О построении траектории одиночной динамической системы с начальными данными на гиперплоскостях // Изв. Иркут. гос. ун-та. Сер. Математика. 2015. Т. 11. С. 93-105.
15. Сидоров Д. H., Сидоров H. А. Решение нерегулярных систем с использованием скелетного разложения линейных операторов // Вестн. ЮУрГУ. Сер. Математика моделирование и программирование. 2017. Т. 10, № 2. С. 63-73.
16. Сидоров Н. А., Сидоров Д. Н. О разрешимости одного класса операторных уравнений Вольтерра первого рода с кусочно-непрерывными ядрами // Мат. заметки. 2014. Т. 96, № 5. С. 773-789.
17. Sidorov N. A., Leontiev R. Yu., Dreglea A. I. On Small Solutions of Nonlinear Equations with Vector Parameter in Sectorial Neighborhoods // Math. Notes. 2012. Vol. 91, N 1, P. 90-104. https://doi.org/10.4213/mzm8771
18. Сидоров H. А., Сидоров Д. H. Малые решения нелинейных дифференциальных уравнений вблизи точек ветвления // Изв. вузов. Математика. 2011. Т. 55, № 5. С. 43-50.
19. Сидоров H.A. Параметризация простых разветвляющихся решений полного ранга и итераций в нелинейном анализе // Матем. (Из. ВУЗ). 2001. Т. 45, № 9. С.55-61.
20. Сидоров Н. А. Явная и неявная параметризация при построении ветвящихся решений итерационными методами // Мат. сб. 1995. Т. 186, № 2. С. 129-141.
21. Сидоров H.A. Об одном классе вырождающихся дифференциальных уравнений со сходимостью // Мат. заметки. 1984. Т. 35, № 4. С. 300-305.
22. Sidorov D. N., Sidorov N. A. Convex majorants method in the theory of nonlinear Volterra equations // Banach J. Math. Anal. 2012. Vol. 6, N 1. P. 1-10. https://doi.org/10.15352/bjma/1337014661
23. Сидоров H. А., Сидоров Д. H. О решении интегрального уравнения Гаммер-штейна в нерегулярном случае методом последовательных приближений // Сиб. мат. журн. 2010. Т. 51, № 2. С. 325-329.
24. Sidorov D. N. Integral Dynamical Models: Singularities, Signals and Control. World scientific series, nonlinear science, Series A, vol. 87, Ed. by L. O. Chua, 2015. 243 p. https://doi.org/10.1142/9278
25. Свиридюк Г. А., Федоров В. E. Линейные уравнения соболевского типа и вырожденные полугруппы операторов. Утрехт : ВСП, 2003. 228 с.
26. Свиридюк Г. А., Загребина С. А. Задача Шовальтера - Сидорова для уравнений соболевского типа // Изв. Иркут. гос. ун-та. Сер. Математика. 2010. Т. 3. С. 104-125.
27. Vainberg M. M., Trenogin V. A. The theory of branches of solutions of nonlinear equations. Wolters-Noordhoff, Groningen, 1974, 302 p.
Николай Александрович Сидоров, доктор физико-математических наук, профессор, Институт математики, экономики и информатики, Иркутский государственный университет, Российская Федерация, 664000, г. Иркутск, ул. К. Маркса, 1, тел.: (3952)242210 (e-mail: [email protected])
Поступила в редакцию 21.01.19
