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Singular cauchy problems with a large parameter for systems of non-linear ordinary differential equations
U.S.S.R. CompUt.Maths.Math.Phys., Vol.Z7,No.Z,pp,ll8-131,1987 Printed in Great Britain
0041~5553/87 $10.oWo.00 01988 Pergamon Press plc
SINGULAR CAUCHY PROBLEMS WITH A LARGE PARAMETER FOR SYSTEMS OF NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS* N.B. KO~U~OVA
and T.V. PAK
Systems of non-linear ordinary differential equations are considered in the semi-infinite interval Ta6tc-. The coefficients of the equations can have infinite upper limits as t-t-.Theorems of the existence and uniqueness of the solutions of such singular Cauchy problems are given, and the continuous dependence of these solutions on the singularly large parameter occurring in the equations is investigated. For problems with a power "degeneracy" with respect to the parameter p theorems are given on the asymptotic behaviour of the solutions with respect to the parameter, and the asymptotic forms obtained are dual: for fixed t and p+- dtndfor fixed p and t-+-tm. 1. FormuLation of the Cauchy problems at infinity. Theorems of the existence and uniqueness of the solutions. The results of this section for problems without parameters supplement the results obtained in /l-4/ (see:also the description of the history of the problem in /4/). The results obtained in /5, 61 affected the formulation of certain problems in Sects.2 and 3. We have used the notation employed in /4, Sect.1, paragraph 1,'. 1. We consider the Cauchy problem at infinity for a system of n non-linear ordinary differential equations x'=A(t)zff(t, x)-t-g(t),T&&Km,
(1.4)
limx(t) -0.
(1.2)
t-r-
Here Tois a fairly large positive constant, x=R'"'; A, f,g are real or complex,d(t)is a square matrix of order n, continuous in [T,,"), the vector function fk x) is continuous with respect to the set of variables (t,r) in a semi-infinite "tube" @"'(a,To)==e(e)xrro, m), Q*(a)=(x~z?'"'; izl
T&Km.
(1.3)
[To,-), the connection of which Here m,(t) is a non-negative scalar function, ContinUOUS in with the vector function f(f I) will be clear from the later definitions. We will introduce the following notation:
J,(T)=sup j@,(t)@,-'(s) 1(g(s) ids,
ILT,
TaT,.
Definition 2. We will say that fft,f) belongs to the class LipA(&,t) if a non&, Te, negative function m,(t) exists, continuous in [To,m) and for any e>O there are 6.=-O, T.Z=T, such that the following holds: If(t, Z)-!(t, zf jTe,lr(C&, ljc"1<6., where for m,(t) the following condition is satisfied: J,(T.)c+? Definition 2. We will say that j(t,x) belongs to the class Lip,(t) is function m,(t) exists, continuous in ITo,m), such that the following holds:
(1.5) a
If(t, 2)--f(k~) I~mrf~)lz--~l for all t>To,/zl4~,IPJ
118
non-negative
11.6)
119
limJt(T)=O. T-rra
(4.7)
Theorem 1 (existenceand uniqueness of the solution of problem (1.1) and (1.2)). Suppose that either all the non-trivial solutions of system (1.3) are unbounded as t-w and f(t, x)E Lip,(e,1) (case 11, or system (1.3) has no solutions which approach zero as t+m, apart from (case 2). Suppose g(t)& such that the following holds: z)ELip, (t) s(t)=& and f(t, IimJ#(T)=O. T-r_
(1.8)
Then for fairly large Tathere is a unique solution of problem Proof. equation
It is easy to show that problem
x(t)=-
ence
f #A(t)@,-‘(s)
(l.l), (1.2).
(1.11, (1.21 is equivalent to the integral
rf(S,X(S))+g(s) Ids,
DTo.
(1.9)
Using this fact we will apply the principle of compressive mappings to prove the existand uniqueness of the solution of Eq.(1.9). We will put J,=J,(T,), I,=(
J,(T,), if f(t,s)ELiP~(t), efj(To), if f(t,z)ELQh(E,tl.
We wilf fix the number Q, O
J&z(l-q).
(1.10)
In view of (1.81, it is always possible for inequalities (1.10) to hold for 3, due to the choice of T,;we can try to satisfy inequalities (1.10) for 3, by choosing ?',, if fezLip,( or by choosing a,if +Lip,(e, t). We will introduce a Banach space CX[T*,m), continuous and bounded in ITo,-) of the functions x(t)with values in X, X=R'"', and norm I&) Ic=auP Ix(t) I. L>T. We will take in Cx[To,m) a closed sphere &=(z(t)eCJTo, m); Ix(t)I&a}. The non-linear operator
converts& into itself, since IP(x)~c~~,+~,~aq+(1--qfa=a. mreover,
F(X) is
a compression operator on S. with a compression coefficient q, since for any functions s(t)%%, Z(t)=& we have the following:
According to the principle of compressive mappings (see /7, Ch.VIII, Sect.33/), in the sphere S, there is a unique solution z(t)of Eq.(1.9), r(t)~Cx[T~,m). This solution can be found by the method of successive approximations 3.'"*"'(t)=F(1(')(t)), k=O, I,..., where the sequence {s'*)(t)) converges to s(t)beginning from any &')(t)%% at a rate
From (1.9) and (1.10) we also have aup Is(t) I~JG')(~-q)-', T&To,, l>T whence (1.2) also follows from (1.8). The theorem is proved. Co.ro.llary.Suppose that in Eq.(l.l) g(t)=&
while in other respects the propositions
120 of Theorem 1 are satisfied. r(t)==O.
Then
problem
(1.11, (1.2) has no other solutions apart from
Note 1. It is easy to see that Theorem 1 remains true if in Case 1 we put f(&z)=(p(t,z)+ 1 Eq.(l.l)hasnoother 9(k4* where cp(t,z)~Lip~(s,t),and~(t,~)ELiPA(~).Iti~aleoobviousthatinCase solutions lying in the sphere S, apart from the one which satisfies condition (1.2). 2. In /l-4/ singular Cauchy problems were investigated for systems of non-linearordinary differential equations with a power form of "degeneracy" with respect to t as t-too,i.e. the problems of the form t-‘d=A
(t)x+f(t,
s)+g(t),
T,,Gt
(l.lla)
limx(t)=O, *-em
(l.llb)
where --1Sr is a real number, li~~(~)=d~=const, Rea(~*)~O, 1-m
;~g@)=O,
f&0)=0,
and f(t,r) is bounded in norm in the semi-infinite "tube" @")(e, T,,).The assumptions for which the theorems of the existence and uniqueness of solutions of problems of the form (1.11) were proved in /l-4/ are special cases of the assumptions of Theorem 1. 3. In addition to /l-4/ we will consider system (1.1) for fairly simple assumptions for which, however, the power form of the singularity with respect to t as t*- is not fixed for the coefficients of the equations. Such problems arise, in particular, when using the Liouville-Green approximations (or the WKB approximations) for transferring the boundary conditions of a certain class for systems of second-order linear ordinary differential equations from infinity. suppose that in system (1.1) VtaTo. A(t)=diag(hl(t),...,hn(t)) Assw@ions
(1.12)
1:
1) h,(t) in (1.12) are such that there iS a non-negative function p(t) such that for all j=&2,...,n the following inequalities hold: continuous in [T,,m) Rehj(t)+(t)>O Vt>T,, and p(t)satisfiesthe condition
Jp~t)&=ira;
(1.13)
T0 e>O 2) there is a non-negative function na,(t), continuous in [To,m), and for any there are S,,T,,&,>O, T&T,, such that for all t>T,, lr/G&, /.?ZlGS. inequality (1.41 holds, and for m,(t) the condition m&t)=O(p(t)), t-tis satisfied? t-+=J. 3) IdQI=ob(t)), Assumptions 2: 1) h,(t) in (1.12) are such that for all j==l,Z,...,n the following inequalities hold: Rehi(t)>O Vt%To; here it is possible that, at least for one .j?'i
jih,ct)dt<+m: To 2) There is a non-negative functions m,(t) continuous in [Te,m) such that inequality (1.6) holds for all t>T,, Is/G& 131Ga, and for m,(t)the following condition holds:
j
m,(t)dt<+m;
(1.14)
In
3) The function g(t)is
such that j:I&)
Idtc-b.
7”
T0
Suppose assumptions 1 or 2 hold for system (1.1). Then, for fairly large Theorem 2. a unique solution of problem (l*l), (1.2) exists. Proof.
the form
In view of (1.12) the fundamental matrix of the solutions of system (1.3) has
121 Then in assumptions 1 we have the following estimate for the Cauchy matrix of system (1.3):
SW>T,,
O
1 and when assumptions 2 are satisfied we have l~.4(t)@*-'(s)IGL SataT,,
OcK,--con&.
Using
these estimates and the assumptions of Theorem 2, it is then easy to verify that for system (1.1) the assumptions of Theorem 1 hold.
2. Cauchy problems at infinity with a singularly occurring large parameter, 1. We will consider the Cauchy problem at infinity for a system of n non-linear ordinary differential equations which depend on a positive real parameter p: X'==il(f, n)x+f(t,y, x)-+&t,P),
T&cm,
(2.l)
Here the (nXn)-matrix A(t,p) and the n-column matrix g(t, p) are defined and continuous with respect to the set of variables (t,p) in the region [To,m)X[po,m); the vector function f(k P>r) is defined and continuous with respect to the set of variables (t. p, 5) in the region @")(a, To)X[p,, M), where f(t, p, O)=O. The elements of A($ p) and g(t,p) can have infinite upper limits as t+m both for fixed p, and also as p-+m for fixed t; similarly, f(r,p,z) may be unbounded in norm as t+m for fixed &.z, and also as ,p-m for fixed t and X. We will denote by Q)A(&~) the fundamental. matrix of solutions of the system x’=A(t,
11)x,
T&km,
(2.3)
Pa&.
Suppose mj(t,p) is a non-negative scalar function, continuous with respect to (t, p) in the interval [To, m)X[pe, m); its connection with f(t, p, z) will be clear from the later definitions. We will introduce the following notation:
J,(G)l)=s l@A(t.P)@A-l(s,I') Im,(s,fi)ds,
J&&p)=
s
I~.4(t,PL)@A-‘(&I~)IIg(s~Y)I~s,
Definition 3. we will say that f(t, p,s)ELip,(8,6,p) if a non-negative function mf(t, p) exists that is continuous with respect to (t, p) and for any E;~O there are 6,,T,, &>O, T.>T, such that the following holds: IiN CL, Z)--f(& p,aI~emf(h for all t>TT,, papto,
/.z[G&,
jilG&,
where, for
idI=-ZI (2.4) m,(t,p), the following condition is satisfied:
supI,(t,p)c+W P>Lm m.ro sup
(2.5)
Definition 4. We will say that f(f IL, z)ELip,(t,p), if a non-negative function mt(t,P) exists that is continuous in (t,p) so that the following holds: lf(k P, 4--f(4
IL, aI~w(t,
P)l"--21
(2.6)
for all t>To.~>*a, IzlGz,Il]=~a, where for m,(t,P) the following condition is satisfied: IimJ,(t,p)~O unif0rml.ywith respect to l-m
It* P>PO.
(2.7)
Theorem 3. Suppose either all the non-trivial solutions (2.3) for any fixed p are unbounded as t-t- IL f(t, p, x)ELip~(8,t,p) (case 1) , or system (2.3) for any fixed p has no solutions which approach zero as t-m, apart from r(t,p)=O and fit,n, r)~Lip~(t,pi) (case 2). Suppose g(t, p) is such that the following conditions are satisfied:
',it J&t, p)= 0 limJg(t, p)=O
Ir-=Q
uniformly with respect to uniformly with respect to
II, p>pO, t, t>T~o.
(2.8) (2.9)
Then, for fairly large To there is a unique solution r(t, p) of problem (2.1), (2.2). This solution is continuous with respect to the set of variables (t,p) in the region [To,m)X f&b =+I> and condition (2.2) is satified uniformly over p,@p~, and the following holds:
122
lim x(t,p)= 0
uniformly
P-m
with respect
to
(2.10)
t, t> To.
The existence and uniqueness of the solution x(t,p) of problem Proof. (2-l), (2.2) for any fixed p follow from Theorem 1. This solution can be found from the integral equation equivalent to problem by the method of successive approximations. (2-l), (2.2), We will put
We will fix satisfied:
4, O
and choose
a and T,for which
the following
are
(2.11)
vl.l>po
suP~,@,p)~a(l-_q) f>TO
supJ,(t,(1)%, 1>TO
inequalities
(here a, To, and g are independent of p). We will construct mations to find s(t.p) by assuming the following:
the process
of successive
approxi-
z(O)(t, I*)=o,
@A((t, p)@*-‘(s, pL)[f(S, p, x(h’(s,PL))-+g(s,Id I&
Lid*+‘)(t, p) =k=O, 1,.. ., we
obtain,
in particular,
@To,
(2.12)
@po.
that
33’) (t, p)=
-
s
(D,ct, r-l)@A-‘(& PL)B(S, dds.
(2.13)
(t, p) in the The approximation I“'(t, p) is continuous with respect to the set of variables in view of the continuity with respect to (t,p) of the integrands interval [T,, -) X [pO, -) and the uniform convergence with respect to (t,p) of the integral in (2.13) (the latter In view of (2.11) the following holds: follows from (2.8)). sup
l>T0
and from
(2.8) and
Js”)(t, p) IGa(l-q)
(2.9) we have that the following 2
uniformly
~(1) (t,p) = 0
lim&)(t,p)=O
uniformly
VP!%
conditions
with respect with respect
to
are satisfied: c17 IL>Po,
t, t >
to
To.
P-m
We will use the method of mathematical induction: suppose the statements proved for z"'(4 P) hold for d”‘(t,p), k>l; we will prove that they hold for .z"+"(t,p). From (2.12) we have the estimate
Hence, taking (2.81, (2.9), and (2.11) into account as well as the assumption of the induction and uniform boundedness of J,(t,p) in the interval [T",m)X[pa,~). we obtain that the following conditions are satisfied: p)(4a 12(k+l)(t,
whence
@IJo,
uniformly
with respect
to
CL? cL>ro,
lim.zSb+l)(t,p)=O
uniformly
with respect
to
t,
u-0
The sequence with respect
vt>T,,
!i;t("+l) (t,p)= 0
t>Ta.
converges to the solution x(t,P) of problem (t,p) since the following estimate holds:
{z’“‘(t,p)}
to
it follows
(2.1),
that
ti: .zP) (t, p) =x
(t, p)
uniformly
with respect
Then for z(t,p) the conclusion of the theorem 2. Suppose that in system (2.1)
holds.
to
(t,p), t>To,
Theorem
3 proved.
p>p.o
(2.2) uniformly
123
vt>T,, @PO. A(t,~)=diag(hl(t,~L),...,~"(t,~))
(2.14)
Proposition 3: 1) hi(t,p) in (2.241 are such that a non-negatite function p(t,p) exists. continuous with respect to (t,P) in the interval fTo,~)X[pL@,=) such that for all j=1.2...., n the following inequalities hold ReMt, i1)8P(t,P)X 'titlST,, @ILO, the following condition is satisfied: where for p(t,p)
j p(t, &L)&=f-
VP, Pa&
h
2) A non-negative function m,(t,p)exists, continuous with respect to (t,p) in the interval IT,., -)X[~c~,m),and for any e>O there are 6,.T.,8.>0_ T.>To, such that for ,f(t, p,z) and for m,(t,F) the following holds: inequality (2.4) is satisfied for all t>Tc, (s1<6., (EIC6. m,(t, P)=O(P(t,p)), t-+-m,uniformly with respect to p, I*+&; 3) ,&?(&P) satisfies the following requirements:
Ig’(t, Cc)I=4dt, !4),
t-cm,
uniformly with respect to P, CIauLo,
I&, p)I=o(P(t, p)), CL-m, uniformly with respect to t,
Proposition 4: 1) h~ft,p) in (2.143 are such that for all inequalities hold: vt>T,, p&b; Re hj(t, ptf>O here it is possible that at least for one j,lGj
t>To.
j==l,2,...,nthe fallowing
and at least for one Pt @).L, the following
2) A non-negative function m,(t), continuous in the interval (T,,m) and independent of the inequality (2.6) is satisfied for all t>i”,,~~po,~~~da, p exists such that for f(t,p,z) ._ _.. IEIez, where m,(t) satisfies condition (1.14); 3) Continuous non-negative functions q(t)? t=[T,,oo) exist such that and @(IL), FIPolm) for g(t,P) the following inequality holds:
v@Te, papa.
Id4 kt)I%4W(lt)
(2.15)
where cp(t)satisfies tha condition
j v(t) dKfm, while q(p) satisfies the condition
Theorem 4. Suppose that for system (2.1) propositions3or 4 are satisfied. Then, the conclusion of Theorem 3 holds for problem (2-l), (2.2). Tib prove this theorem it is sufficient to check that the propositionsof!rheorem3 are satisfied. 3. Here and in Sect.3 we will study in more detail problems with a power form of degeneracy with respect to P as y-t=, i.e., problems of the form (2.16) ~-rs'--A(t,CL)Z+f(t,~,5f+g(t, I"), T,
v,,
PCca.
(2.17)
Here OCr is a real number, A(t,P) and g(t,p) are continuous with respect to (t,P) in the I&&n)/ are bounded as p-r- and for fixed region IT+m)Xl~io,=)r and the quantities \A(t,g)j, finite t; f(t,p,o)is continuous with respect to (t,p,z) in the region a~-)(%T0)X[Po,-), f(t,P, is bounded in norm as p-+03 and for fixed t,x. O)==O and f(t,p,z) Be1owe.v. stands for eigenvalues. Proposition 5: 1) In (2.16) either A(t,P)=Ao=const,
Reh(A,)>O
(2.18)
(case A), or ,...,A?(t)), ~{t,~)m~(t}= diag(h,@' (t) and V+&~,...,?Z,
Vt%To
R~~~‘(t)~p(t~~O
(2.19)
124 where the function p(t),continuousin the interval [T,,-),satisfies condition (1.13) (case B); 2) A function mt(t,P) exists, continuous with respect to (t,~)and non-negative in the region [T,,m)Xk~~,m),and for any GO there is a 6., T,,&>O,T.ZT,such that inequality (2.4) holds for all t>T,,pap*,/z/G&,, 111G'6,, and T&&p) in case A satisfies the condition m,(t,p)=0(1),t--T uniformly with respect to
p,PbPLor
and in case B satisfies the condition tern,uniformly with respect to p,P>pO; m,(t,P)=W(0)~,
(2.20)
3) g(t,II) is such that in case A the following conditions are satisfied: ;izg(t,p)=O
uniformly with respect to
P., IL>POl
limg(d, p)=O IL--
uniformly with respect to
t, t> To,
while in case B, the following conditions are satisfied: Ig(t,P)I=o(P(t)),t-r-m,uniformly with respect to P, PtaPO, MC p)I=a(p(Q). PL'W> uniformly with respect to t, t>Z',. Propositions 6: 1) In (2.16) either
A(t,p) =Ao=const,
Re h(A,)>O,
where Jordan cells of order not greater than x,x>f values of A, or Otr is an integer, I--L
(case A) correspond to imaginary eigen-
(2.22)
d(t,c)-,i,tt)+CA*(t)/~~, k-t &(t)=diag(ht)' (t),...,hF’ .-I
(t)),
Re (h:*'(b)+ ~~~I(~)/~')~0 k-l j=l,2,..., n
(2.21)
t&To. k=O, 1,. . ., r-i, Vt>T,, PaILor
(2.23) (2.24)
(case B); here it is possible that at least for one j,iGj
cd
5Re(F,"'(t)+CAll'(t)i~~)dl<+m
VP: Pa&
L-I
To
2) A non-negative function mr(t) exists, that is continuous in the interval which satisfies the condition
q-g If(f,r,X)--f(tr IL,f)l
(To,m),
/x-z1
for all t~Z',,p~po, Ir(G, IIlGz, where e=
; i
in case A, in case B,
(2.25)
and m,(t) satisfies the requirement
dt<+m; j mi(t)te-'
(2.26)
7.
3) Non-negative continuous functions q(t),te[T,,m) and q(P),PE[)krn) exist such that for g(t,P) inequality (2.15) holds, and v(t) satisfies the condition m
and *(PI
satisfies the condition limp"'$(p)=O.
#"xl
Everywhere from now on we will mean by case Ar the satisfaction of the conditions of case A of propositions 5, and similarly for cases B5, A6, B6 etc. Theorem 5. Suppose that for system (2.16) propositions5 or 6 are satisfied. Then, for fairly large To a single solution x(&p) of problem (2.16), (2.17) exists. This solution is continuous with respect to the set of variables (t,).~) in the region [T,,m)X[Po,m), and
125 condition (2.17) is satisfied uniformly with respect to (2.10) is satisfied.
and, moreover, condition Cl, 11,>s;cl0
Proof. In assumptions 5 and 6 it is easy to obtain estimates for the Cauchy matrix ~~(t,~.~)=~~(~,p)@; (s, p) of the sy=-tern z’=X(t,
for ;I(t,p)=p'Ajt.n). Thus, for SX&T, AZ it follows from (2.18) that
p.)x,
T&t<-,
and all nap0
) Ui (G s, p) 1d RI eq
(2.27) pay,, the following inequalities hold: in case
[-- pru (8 -
t)l,
(2.28)
O
in case A, from (2.21) and it follows from the limitation on purely imaginary eigenvalues -4, that Ic,(t, s,I")IQK,y""-i's"-', (2.30) O
Iu,(t, 8, ro /4&,
OtK,=const
(2.31)
(4 is independent of &,j=2,3,4). Using these estimates and the propositions of the theorem, it is easy to verify that functions f(l,s,~_~)=p'f(t,z,~f and ~(~,~)=~'g(~,~) satisfy the propositions of Theorem 3. Then the conclusions of Theorem 5 is a corollary of the conclusion of Theorem 3. Note. 2. For estimates of the Cauchy matrices of systems of linear ordinary differential equations in a semi-infinite interval see /4/ for more detail. 3. In case Bawe can introduce the following changes. Suppose that in system (2.16) the matrix A(t,p) has the form (2.22), where (0) Ito(diag(A* (t),...,?$' (t)), t 31To, (2.32) and j=s+l,s+2,...,n. &:'(a)*h:@'(t)vt 3 To, s=i,2,...,n
(2.33)
since, Then, the diagonal form of the matrices AR(~) in (2.22) cannot be fixed for k==l,2,...,r-1 using the well-known approach (see, for example, /6/ and the papers referred to there) we can diagonalize asymptotically with respect to P the principal part of system (2.16) by using the replacement of variables of the form
Here E,is a unit matrix of order R, and 4(t) are square matrices of order n, which are chosen from the condition that in the system for +,, (new) the matrix .4.(&p)(new) has a diagonal form apart from terms of the order of O(I/P~),P+~. More accurately, Lemma 1 given below holds, in the proof of which essentially we discuss an algorithm for an asymptotic diagonalization of the principal part of system (2.16) using replacement (2.34). are square matrices of order n continuous in the interval Lemma. Suppose &(t),...,L,(t) where A,(t) has the form (2.32), and conditions (2.33).are [To,=). Suppose Lo(t)-&(t), satisfied. Suppose that either the following conditions inf 1h.(O) (if - 3,:C’(t) 13 p,>O, s=i,z,. . . , n, j=s+i, s+&...,n, f>T* SUPl~~(O
t>;.
I< +m,
k--1,2,...,p
are satisfied (case A), or the conditions
1A,‘“’(t)- h:n’ (t) 12 p(t)2% poxI, ILcWI=~(p(~)~, t-cm,
1x0,
j=s+i, s+i,.. . . .?, s=1,2,...,n,
k=i,Z,...,p,
where p(t) is a positive function continuous in the interval [Z's=)(case B). Then square matrices &(t),...,&.(t) of order n, continuous and bounded in the interval [To>=) exist such that the following equation holds:
(2.35)
126 Here the matrices A&) have the form (2.23) for conditions are satisfied: suplAr(t)
k=i,
1c: +m,
t>=*
in case A the following
and
k=i,%,...,p,
2,. . . 1p,
while in case B the following conditions are satisfied IAr(r)[=O(p(t)), t+-, k=/Z,...,p; the (nxn) -matrix C(t,P) is such that in case A the following holds: uniformly with respect to
IC(t,P)l--o(UP=+'). p-m,
I, ts=To,
while in case B iC(t,a)/P(t)l=O(ilCl""). Cl+m, uniformly with respect to t, @Tp. k=O,%,...,p, then If G(f) are continuously differentiable functions in the interval [To,m), k=1,2 ,..., p. Bh(t) are also continuously differentiable functions in the interval IT,,-), Proof.
We will put I;r(t) =Ill~'(t)Ili,,-,,a....,". k=l,2,...,P. Sk(t) =llG'(01/1.1=:.2. ..,ll,
We will assume that (2.35) is true for certain Bk(t), Ah(l).Multiplying the brackets and collecting terms of the same powers of P, we obtain n,(t)=Ao(i)Bi(~)-~,(~)Ao(~)+~I(~)r a-l
(2.36a)
[‘L(t)&-a@)- B,-r(t)&(Ol+ W), A.(t)= Ao(t)B,(0S,(t)Aof0+ z
s=2,3,...,p.
(2.36b)
l-l
From these relations we calculate A.(t), B,(t) successively for s=LZ....,pusing the condition that the matrices A.(t)should have the form (2.23). At the first step we put a:"(f)== 4;'(t), j=i,2,...tn. Then, from (2.36a) for the diagonal elements S%(t) we obtain the relations
aj(0)‘(t)bjj(‘1 (t) -
(1)
bjJ ‘(t)hj
(0)
‘(t)= 0.
i=i,z,...,n,
which are satisfied for any functions bj$'~ft), for example, for ~jjll)(f)=O, t>Fo.j=1,2,...,n, and the off-diagonal elements of the matrix B*(t) are uniquely defined by the equations
We will put s-1
c
IL*(t)&-rv)
R&i=
- ~,-r(t)Mt)l,
a-1
R.(t)=li~$’
For ss2
s=2,3 . . ..*P. (t)Il~j-~,....n.
we put successively in Eqs.(2.36b) <*> (I) (0) j=i.2 hj (f)=)j,(t)+T>j (I).
. . . ..n.
s=2,3,.
. . , p.
b;;)(t)$ (f)=O. i= Then, for the diagonal elements S,(t) we obtain the relations $)(t)b;f’(t)t>To,j=1. 1,2,...,% which are satisfied for any functions b,>‘“‘(t), for example, for b,,~n)(t)=O. 2,..., n; the off-diagonal elements of the matrices B,(t) are uniquely defined by the equations ‘I,
hi
(8)
CO=
/rj
I*)
(6)+ rfj (6)
h/“‘(t)- hp’(t) .
i+j,
le;;iszn,
i
s=2,3,.
. , P.
For the "residue" C@,p) we have from (2.35)
The statement of the lemma also follows from the equations obtained. The lemma is proved. 4. Suppose that in system (2.16) for A&p), g&p) propositions 5 or 6 are satisfied. Suppose the vector-function f(t,p,x) has first-order continuous partial derivatives with in the region ~(-~(~,To)~[IIo,-)~ and satisfiescondition (2.20); respect to the variables zti,....s~n in cases Aaand B. the following inequalities hold:
127
B,Jfor any ~-0
in case
8,.T.,&>O,T.STo
there exist
I
c em,@, p)
such that the following
vt*To,
inequality
holds:
P>Po* I~)<&,
m,(t,p),continuous in the interval [To,-)X[PO,-),satisfies where the non-negative function condition (2.20); in cases Aa and Be the following inequalities hold:
mt(t) satisfies condition (2.26). where 0 is the same as in (2.251, while Then, for problem (2.16) and (2.17) the conclusion of Theorem 5 remains Indeed, for any IslGa,Ix"lCa the following inequality holds:
for all
t>TO,@po,
where
~==s+~(f-z),O
The further discussion
true.
is obvious.
3. Asymptotic behaviour of the solutions with respect to the parameter. e(t,p), continuous with respect to the set Definition 5. We will say that the function can be represented for large u by the of variables (t,~) in the region [T,,m)x[~e~), asymptotic series _ 6(&
P(t)W h-P
uniformly with respect to t when t>T,, if for any integer such that the following estimate holds:
Q,Q>p,
a constant
Cu,C,>O
exists
(3.1) for all t>T, and all fairly large p (we mean by the case Q=p’ that there is no sum in inequality (3.1)). The theorems of this section are related to the investigation of the asymptoticbehaviour with respect to the parameter p of the solutions of problems of the form (2.16), (2.17) in which always henceforth I<: is an integer, and A(t,p) satisfies propositions 5 or 6, and x1,....I,, f(C c1,z) is a vector polynomial of power L with respect to the variables
I.
fk
!bx)=
r,
fl(4 PL)“’
(3.2)
(here k(L,..., 1.) is a multi-index, Z+O, iGj), and the coefficients ji(t,p) and g(t,p) can be expanded for large j& in asymptotic series in inverse integer powers of p uniformly with respect to t,tZT,. These expansions have the form
(3.3) if A(t,k)
satisfies
propositions
5, in which case we have W
if A(4 1~) satisfies propositions
6, we have
(3.5) where I3 is the same as in (2.25). Propositions 7: 1) A(t,p) function p(t) is such that
satisfies
propositions
5, and in case B, in (2.19) the
infp(t)>po>O, 1z.h while
kd"'(f) are infinitely
differentiable
function
(3.6) in the interval
[T,,"), which
satisfy
128 the requirements
I~(&y)I=+-&)*
t-+-m,i=l,2
,..., n,
(3.7)
q-1,2,...; 2) In expansions (3.3) and (3.4) the coefficients q(t),g"'(l),tpb(f), fi'"'(t) are infinitely differentiable functions which, in case A, satisfy the conditions (3.8) (3.9a)
If
(cprO)fi@ W) 1 =O(l),
t-t-,
IC[ZIGL,
k=O,f,...,
(3.9b)
[l[+k>2, q=O,i,..., while in case B they satisfy the conditions (3.10a) (3.10~6)
f
tw)P(t))
[=oWt)ht+w,
k=2,2,...,
f3.1fa)
iQj1jGL,
(3.115)
q=o, 1,. . . ,
If
(w(t)P
(WI=Wp(t)),
k-0,1,. . . , lll+k>L?, q-0, I,...
t-rm,
.
Propositions 8: 1) A(t,p) satisfies propositions 6 , and in case Aa in (2.211 the matrix A, is non-singular, while in case Bs in (2.23) the functions h,'"'(t) are infinitely differentiable in the interval [To,=) and satisy the conditions (3.12a) (3.12b) q=o, 1,*..; 2) In expansions (3.3) and (3.5) the coefficients 9(t),g'"'(t), q(t), f.‘“‘(t) are infinitely differentiable functions in the interval [TO,=), which satisfy the conditions
j Iw(t)Ite-idt<+m, s Il&(t)lte-ldt<-i-Oo, Te i 1% (li,(t)g’“‘(t)) 1 P-‘dt
k-l,
i=siZi
2,. . .,q-0,
i,.. .,
(3.13a)
(3.13tb)
k=rEI,r@+f,..., q=O, j,.... Theorem 6. Suppose that for system (2.16) propositions 7 (or 8) are satisfied. Then for problem (2.161, (2.17) the conclusion of Theorem 5 holds. The solution r(t,P) of this problem can be expanded, for large P in the asymptotic series (3.14) uniformly with respect to t, t>T0, where the coefficients x")(t) can be found from (2.16) by formal substitution of the expansions (3.2), (3.31, (3.41, and (3.14) (or (3.21, (3.31, (3.5) and (3.14)). The series in (3.14) is asymptotic for fixed t and P-m, and also for fixed p and t-+-m,i.e., it is the double asymptotic form of the solution of problem (2.16)~ (2.17). The coefficients x@'(t)are infinitely differentiable functions in the interval [Tp,m),which for propositions 7 satisfy the conditions
129
I-g+‘“‘(t) i =0(l),
t--,
k-l , 2 a...,
q=O,l,...,
(3.15)
q=O, 1,. . . .
(3.16)
and for propositions 8, satisfy the conditions j,!-$P)(Q f Pdttf~,
k--l,2
,...,
Proof. 1. We will prove the theorem for propositions 7. a. From (3.31, (3.41, in view of definition 5, we obtain that for all t>T,, following inequalities hold:
paw0
the
(3.17a)
(3.17b) Using these inequalities, when also conditions (3.8) in case A, (or conditions (3.10) in caseB,) and Note 4, it is easy to obtain that for system (2.16) propositions AS (or BJ) hold. Then, for problem (2.161, (2.17) the conclusion of Theorem 5 holds. b. The solution s(t,p) of problem (2.161, (2.17) will be sought formally in the form of series (3.14). Substituting expansions (3.31, (3.41, and (3.14) into (2.16) and equating expressions of the same powers of $1~.we obtain recurrence formulas for determining the coefficients of series (3.14) x'"(t)=--A-'~(t)g"'(t),
(3.18a)
s'"'(t)=-A-'(~(t)g(m)(t)+Xm(l, I('),..., d”-“~-~mx(m)‘(t)},
(3.18b)
m-2,3,.,. . Here A==
do
1 Ao(t)
in case A?, in case B?,
L=(
1 f,",' ;;;;
the functions xn(t, L(‘), . ..,x”“-“) are formed from the components xC1', . . ..x(~-‘) and the components of the expansion 13.4) (up to powers no higher than t/fi"-') using additions and multiplications, & @,O,. . ..O)=O. Taking propositions (2.18) and (3.9) into account in case AT (or (2.191, (3.61, (3.71, and (3.11) in case B,),we obtain from the recurrence formulas (3.18) successively that the coefficients of the formally constructed expansion (3.14) sayisfy conditions (3.15). c. We will put
a2& tt)=
r(K)@)/pk, r=1
l
is fixed,
where x@'(t), k=l, 2,...,Q are found from the recurrence formulas f3.18), and we introduce the function rl(t, P) such that (3.20)
s(t*P)=%J(t,P)+rl(t, P)/P,
where x(t,p) is the solution of problem (2.161, (2.17). In view of the fact that (2.17) is satisfied uniformly with respect to ILwhen !.c)i&(on the basis of Theorem S), and (3.15) holds when p=O, it follows from (3.19).,(3.201‘that limIj(t,p)=O uniformly with respect to *-bea
FL, @!Jo.
(3.21a)
we will show that liml(t,p)=O
uniformly with respect to
Ir+=
t, t>,TV,
(3.219
and that, moreover, the following holds: Ill(G r)l=O(G),
I1-f@‘ uniformly with respect to t, t>TT,.
(3.22)
We then obtain from (3.21) that series (3.141, constructed from the recurrence formulas (3.18), is a doubly asymptotic form of the solution of problem (2.16), (2.17): for fixed p and t+m and for fixed t and p-+m. Also, we obtain (3.22) that the solution s(t,b) of problem (2.161, (2.17) can be represented for large p by the asymptotic series (3.14) uniformly with respect to t when t>T (in the sense of definition 5). d. We obtain from (2.16) for n(t,p) an equation of the form p-'s'=A(t, P)$"f(G P,ri)f&GIL)>
T,<‘t
(3.23)
where J(GPL,rl)=[f(t,~,s,+gllLQ)--l(l,~,~P)I~P, USSR27:2-1
(3.24)
130
Lfr--V
1
for for
Qfl-r f(t,P,zQ) is formed
+ f
6, ,b xQ)pQ,
(3.25)
QGr, Oar;
from
by eliminating terms of the expansion in powers of l/p up to l/Pa inclusive. It is further easy to verify that for system (3.23) the propositions of Theorem once again satisfied. Indeed, we have from (3.24)
5 are
Taking (3.3), (3.14), and (3.15) into account for p=O and also conditions (3.8) in case Ai (or conditions (3.10) in case B,), from (3.26) and Note 4 we obtain that f(t,p,q) satisfies proposition A, (or Bs). Similarly, from Eq.(3.25), taking definition 5 into account, the properties of the functions $(t),f(t,P,zy) and condition (3.15) for q=O,l, we obtain that d(t,P) satisfies propositions 5. Then, on the basis of Theorem 5, problem (3.23), (3.21a) has a unique solution and (3.21b) also holds for this. Moreover, the solution of this problem satisfies the integral equation (3.27) where
@A (t,P) is the fundamental We will put
matrix
P@,p)=supsup
G=l T
of the solutions
$S,P,i(S,P))
of system
(2.27).
t
OpO. Then, for fairly large T, in case A, the following inequalities hold,
KF(t, P)/u~%,
(3.28)
KG(t,P)/(sGC,
0,T, p>po, where &and a are constants in the estimate (2.28) for the Cauchy matrix of system (2.271, while in case B? the following inequalities hold: K&t, O
P)/p(t)G'/z, the constant
(3.29)
KG(t, P)Ip(t)GC, in estimate
(2.29).
From
(3.24) and inequality
(2.28) and (3.28) into account Then, from the integral Eq.(3.27), taking estimates A, and estimates (2.29) and (3.29) in case B,, we obtain in the usual way M(t,P)42Cllr,
BT,
in case
PPd,
whence the correctness of (3.22) follows. 2. For propositions 8, the scheme of the proof remains as before with only small changes. We will consider these changes; We will first formally construct series (3.14). In case A, we obtain the recurrence formulas ~‘“‘(t)=--A,-‘~(t)g’“‘(t),
m=l,
2 r....r,
(3.30a)
r’“+“(t)=--A,-‘[~(t)g(m+r,(t)--z(”J’(t)+~m~m+r(t,
cd’), x(Z), . .
Here
L=(
0
for
mdxr--r,
1
for
m>xr-r;
, @+-“)I,
m=l,
2,.
.
.
(3.3Ob)
131 ....X(m+r-')) are formed from the components x1",. . . . z(“‘+‘-‘) and the the functions Xm+,(t,x(", using summations components of the expansion (3.5) (up to powers no higher than ~/IL""'-') and-multiplications, x~+~(~,O,...,O)=O. In caseB, we obtain the recurrence formulas .z"'(t)=-a,-'(t)lp(t)g"'(t), rnlXl(rn,r-l,
(3.31a)
~"(t)=-~-'(l)(~(l)g'"'(t)+ r, 1-1
(3.31b)
tm[Xm(t, 5(",. . . , P-y
A&)r'"-j)(t)+
-29-‘) ml).
m=2,3,. . . .
Here b"=(;
'f", ;z;;
are formed from the components x('~,...,x("-" and the components the functions X~(t,5"',...,X'*-") of the expansion (3.5) (up to powers no higher than l/p"-') using additions and multiplications. Taking (3.12) and (3.13) into account, from Eqs.(3.30) and (3.31) we obtain successively that the coefficients of the formally constructed series (3.14) satisfy conditions (3.16). In system (2.16) we will make the replacement of variables r(t,P)=%(t,P)+2 "'k'(t)l$, D-L
(3.32)
where, as before, El=
in case Aa, i in case Bs,
x
and the coefficients .@'(t), k=l,2,...,?0 are calculated from Eqs.(3.30) or (3.31) respectively. Then, for z,(t,p) we again obtain a problem oftheform (2.16), (2.17), and for g,(t,p) in an expansion of the form (3.3) the first re terms of the series will not be present. Further, as in Sect.1, a, it is easy to show that the system for x,:satisfiespropositions 6. Then, taking the replacement (3.32) into account, we obtain that for the initial problem (2.16), (2.17) the conclusion of Theorem 5 holds. We will fix Q, Q>l+re and calculate the first Q terms of series (3.14). We introduce the function ~)(t,p)such that
is the solution of problem (2.161, (2.17). The further discussion is similar where s(t,k) to that in Sect.1, b. The appropriate changes are obvious. In particular, for the Cauchy matrices of system (2.27) we here use estimates (2.30) or (2.31) respectively. The theorem proved. iS The authors thank A.A. Abramov for discussing the paper. REFERENCES 1. BIRGER E.S. and LYANIKOVAN.B., A methodof finding solutions for certain systems of ordinary differential equations with specified condition at infinity, I, Zh. vych. Mat. mat. Fiz., 5, 6, 979-990, 1965; II, 6, 3, 446-453, 1966. 2. RUSSELL D.L., Numerical solution of singular initial value problems, SIAM J. Numer Analys. 7, 3, 399-417, 1970. 3. BALLA K., The solution of singular boundary value problems for certain systems of ordinary differential equations, Candidate Dissertation, Computing Centre of the Academy of Sciences of the USSR, Moscow, 1978. 4. KONYUKHOVAN.B.,Singular Cauchy problems for systems of ordinary differential equations, Zh. vychsl. Nat. mat. Fiz., 23, 3, 629-645, 1983. 5. FEDORYUK M.V., Asymptotic methods in the theory of ordinary linear differential equations, Mat. Sbornik, 79, 4, 477-516, 1969. 6. FEDORYUK M.V., Asymptotic Methods for Ordinary Linear Differential Equations, Moscow, Nauka, 1983. 7. TRENOGIN V.A., Functional Analysis, Moscow, Nauka, 1980.
Translated by R.C.G.
0041~5553/87 $10.oWo.00 01988 Pergamon Press plc
SINGULAR CAUCHY PROBLEMS WITH A LARGE PARAMETER FOR SYSTEMS OF NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS* N.B. KO~U~OVA
and T.V. PAK
Systems of non-linear ordinary differential equations are considered in the semi-infinite interval Ta6tc-. The coefficients of the equations can have infinite upper limits as t-t-.Theorems of the existence and uniqueness of the solutions of such singular Cauchy problems are given, and the continuous dependence of these solutions on the singularly large parameter occurring in the equations is investigated. For problems with a power "degeneracy" with respect to the parameter p theorems are given on the asymptotic behaviour of the solutions with respect to the parameter, and the asymptotic forms obtained are dual: for fixed t and p+- dtndfor fixed p and t-+-tm. 1. FormuLation of the Cauchy problems at infinity. Theorems of the existence and uniqueness of the solutions. The results of this section for problems without parameters supplement the results obtained in /l-4/ (see:also the description of the history of the problem in /4/). The results obtained in /5, 61 affected the formulation of certain problems in Sects.2 and 3. We have used the notation employed in /4, Sect.1, paragraph 1,'. 1. We consider the Cauchy problem at infinity for a system of n non-linear ordinary differential equations x'=A(t)zff(t, x)-t-g(t),T&&Km,
(1.4)
limx(t) -0.
(1.2)
t-r-
Here Tois a fairly large positive constant, x=R'"'; A, f,g are real or complex,d(t)is a square matrix of order n, continuous in [T,,"), the vector function fk x) is continuous with respect to the set of variables (t,r) in a semi-infinite "tube" @"'(a,To)==e(e)xrro, m), Q*(a)=(x~z?'"'; izl
T&Km.
(1.3)
[To,-), the connection of which Here m,(t) is a non-negative scalar function, ContinUOUS in with the vector function f(f I) will be clear from the later definitions. We will introduce the following notation:
J,(T)=sup j@,(t)@,-'(s) 1(g(s) ids,
ILT,
TaT,.
Definition 2. We will say that fft,f) belongs to the class LipA(&,t) if a non&, Te, negative function m,(t) exists, continuous in [To,m) and for any e>O there are 6.=-O, T.Z=T, such that the following holds: If(t, Z)-!(t, zf j
(1.5) a
If(t, 2)--f(k~) I~mrf~)lz--~l for all t>To,/zl4~,IPJ
118
non-negative
11.6)
119
limJt(T)=O. T-rra
(4.7)
Theorem 1 (existenceand uniqueness of the solution of problem (1.1) and (1.2)). Suppose that either all the non-trivial solutions of system (1.3) are unbounded as t-w and f(t, x)E Lip,(e,1) (case 11, or system (1.3) has no solutions which approach zero as t+m, apart from (case 2). Suppose g(t)& such that the following holds: z)ELip, (t) s(t)=& and f(t, IimJ#(T)=O. T-r_
(1.8)
Then for fairly large Tathere is a unique solution of problem Proof. equation
It is easy to show that problem
x(t)=-
ence
f #A(t)@,-‘(s)
(l.l), (1.2).
(1.11, (1.21 is equivalent to the integral
rf(S,X(S))+g(s) Ids,
DTo.
(1.9)
Using this fact we will apply the principle of compressive mappings to prove the existand uniqueness of the solution of Eq.(1.9). We will put J,=J,(T,), I,=(
J,(T,), if f(t,s)ELiP~(t), efj(To), if f(t,z)ELQh(E,tl.
We wilf fix the number Q, O
J&z(l-q).
(1.10)
In view of (1.81, it is always possible for inequalities (1.10) to hold for 3, due to the choice of T,;we can try to satisfy inequalities (1.10) for 3, by choosing ?',, if fezLip,( or by choosing a,if +Lip,(e, t). We will introduce a Banach space CX[T*,m), continuous and bounded in ITo,-) of the functions x(t)with values in X, X=R'"', and norm I&) Ic=auP Ix(t) I. L>T. We will take in Cx[To,m) a closed sphere &=(z(t)eCJTo, m); Ix(t)I&a}. The non-linear operator
converts& into itself, since IP(x)~c~~,+~,~aq+(1--qfa=a. mreover,
F(X) is
a compression operator on S. with a compression coefficient q, since for any functions s(t)%%, Z(t)=& we have the following:
According to the principle of compressive mappings (see /7, Ch.VIII, Sect.33/), in the sphere S, there is a unique solution z(t)of Eq.(1.9), r(t)~Cx[T~,m). This solution can be found by the method of successive approximations 3.'"*"'(t)=F(1(')(t)), k=O, I,..., where the sequence {s'*)(t)) converges to s(t)beginning from any &')(t)%% at a rate
From (1.9) and (1.10) we also have aup Is(t) I~JG')(~-q)-', T&To,, l>T whence (1.2) also follows from (1.8). The theorem is proved. Co.ro.llary.Suppose that in Eq.(l.l) g(t)=&
while in other respects the propositions
120 of Theorem 1 are satisfied. r(t)==O.
Then
problem
(1.11, (1.2) has no other solutions apart from
Note 1. It is easy to see that Theorem 1 remains true if in Case 1 we put f(&z)=(p(t,z)+ 1 Eq.(l.l)hasnoother 9(k4* where cp(t,z)~Lip~(s,t),and~(t,~)ELiPA(~).Iti~aleoobviousthatinCase solutions lying in the sphere S, apart from the one which satisfies condition (1.2). 2. In /l-4/ singular Cauchy problems were investigated for systems of non-linearordinary differential equations with a power form of "degeneracy" with respect to t as t-too,i.e. the problems of the form t-‘d=A
(t)x+f(t,
s)+g(t),
T,,Gt
(l.lla)
limx(t)=O, *-em
(l.llb)
where --1Sr is a real number, li~~(~)=d~=const, Rea(~*)~O, 1-m
;~g@)=O,
f&0)=0,
and f(t,r) is bounded in norm in the semi-infinite "tube" @")(e, T,,).The assumptions for which the theorems of the existence and uniqueness of solutions of problems of the form (1.11) were proved in /l-4/ are special cases of the assumptions of Theorem 1. 3. In addition to /l-4/ we will consider system (1.1) for fairly simple assumptions for which, however, the power form of the singularity with respect to t as t*- is not fixed for the coefficients of the equations. Such problems arise, in particular, when using the Liouville-Green approximations (or the WKB approximations) for transferring the boundary conditions of a certain class for systems of second-order linear ordinary differential equations from infinity. suppose that in system (1.1) VtaTo. A(t)=diag(hl(t),...,hn(t)) Assw@ions
(1.12)
1:
1) h,(t) in (1.12) are such that there iS a non-negative function p(t) such that for all j=&2,...,n the following inequalities hold: continuous in [T,,m) Rehj(t)+(t)>O Vt>T,, and p(t)satisfiesthe condition
Jp~t)&=ira;
(1.13)
T0 e>O 2) there is a non-negative function na,(t), continuous in [To,m), and for any there are S,,T,,&,>O, T&T,, such that for all t>T,, lr/G&, /.?ZlGS. inequality (1.41 holds, and for m,(t) the condition m&t)=O(p(t)), t-tis satisfied? t-+=J. 3) IdQI=ob(t)), Assumptions 2: 1) h,(t) in (1.12) are such that for all j==l,Z,...,n the following inequalities hold: Rehi(t)>O Vt%To; here it is possible that, at least for one .j?'i
jih,ct)dt<+m: To 2) There is a non-negative functions m,(t) continuous in [Te,m) such that inequality (1.6) holds for all t>T,, Is/G& 131Ga, and for m,(t)the following condition holds:
j
m,(t)dt<+m;
(1.14)
In
3) The function g(t)is
such that j:I&)
Idtc-b.
7”
T0
Suppose assumptions 1 or 2 hold for system (1.1). Then, for fairly large Theorem 2. a unique solution of problem (l*l), (1.2) exists. Proof.
the form
In view of (1.12) the fundamental matrix of the solutions of system (1.3) has
121 Then in assumptions 1 we have the following estimate for the Cauchy matrix of system (1.3):
SW>T,,
O
1 and when assumptions 2 are satisfied we have l~.4(t)@*-'(s)IGL SataT,,
OcK,--con&.
Using
these estimates and the assumptions of Theorem 2, it is then easy to verify that for system (1.1) the assumptions of Theorem 1 hold.
2. Cauchy problems at infinity with a singularly occurring large parameter, 1. We will consider the Cauchy problem at infinity for a system of n non-linear ordinary differential equations which depend on a positive real parameter p: X'==il(f, n)x+f(t,y, x)-+&t,P),
T&cm,
(2.l)
Here the (nXn)-matrix A(t,p) and the n-column matrix g(t, p) are defined and continuous with respect to the set of variables (t,p) in the region [To,m)X[po,m); the vector function f(k P>r) is defined and continuous with respect to the set of variables (t. p, 5) in the region @")(a, To)X[p,, M), where f(t, p, O)=O. The elements of A($ p) and g(t,p) can have infinite upper limits as t+m both for fixed p, and also as p-+m for fixed t; similarly, f(r,p,z) may be unbounded in norm as t+m for fixed &.z, and also as ,p-m for fixed t and X. We will denote by Q)A(&~) the fundamental. matrix of solutions of the system x’=A(t,
11)x,
T&km,
(2.3)
Pa&.
Suppose mj(t,p) is a non-negative scalar function, continuous with respect to (t, p) in the interval [To, m)X[pe, m); its connection with f(t, p, z) will be clear from the later definitions. We will introduce the following notation:
J,(G)l)=s l@A(t.P)@A-l(s,I') Im,(s,fi)ds,
J&&p)=
s
I~.4(t,PL)@A-‘(&I~)IIg(s~Y)I~s,
Definition 3. we will say that f(t, p,s)ELip,(8,6,p) if a non-negative function mf(t, p) exists that is continuous with respect to (t, p) and for any E;~O there are 6,,T,, &>O, T.>T, such that the following holds: IiN CL, Z)--f(& p,aI~emf(h for all t>TT,, papto,
/.z[G&,
jilG&,
where, for
idI=-ZI (2.4) m,(t,p), the following condition is satisfied:
supI,(t,p)c+W P>Lm m.ro sup
(2.5)
Definition 4. We will say that f(f IL, z)ELip,(t,p), if a non-negative function mt(t,P) exists that is continuous in (t,p) so that the following holds: lf(k P, 4--f(4
IL, aI~w(t,
P)l"--21
(2.6)
for all t>To.~>*a, IzlGz,Il]=~a, where for m,(t,P) the following condition is satisfied: IimJ,(t,p)~O unif0rml.ywith respect to l-m
It* P>PO.
(2.7)
Theorem 3. Suppose either all the non-trivial solutions (2.3) for any fixed p are unbounded as t-t- IL f(t, p, x)ELip~(8,t,p) (case 1) , or system (2.3) for any fixed p has no solutions which approach zero as t-m, apart from r(t,p)=O and fit,n, r)~Lip~(t,pi) (case 2). Suppose g(t, p) is such that the following conditions are satisfied:
',it J&t, p)= 0 limJg(t, p)=O
Ir-=Q
uniformly with respect to uniformly with respect to
II, p>pO, t, t>T~o.
(2.8) (2.9)
Then, for fairly large To there is a unique solution r(t, p) of problem (2.1), (2.2). This solution is continuous with respect to the set of variables (t,p) in the region [To,m)X f&b =+I> and condition (2.2) is satified uniformly over p,@p~, and the following holds:
122
lim x(t,p)= 0
uniformly
P-m
with respect
to
(2.10)
t, t> To.
The existence and uniqueness of the solution x(t,p) of problem Proof. (2-l), (2.2) for any fixed p follow from Theorem 1. This solution can be found from the integral equation equivalent to problem by the method of successive approximations. (2-l), (2.2), We will put
We will fix satisfied:
4, O
and choose
a and T,for which
the following
are
(2.11)
vl.l>po
suP~,@,p)~a(l-_q) f>TO
supJ,(t,(1)%, 1>TO
inequalities
(here a, To, and g are independent of p). We will construct mations to find s(t.p) by assuming the following:
the process
of successive
approxi-
z(O)(t, I*)=o,
@A((t, p)@*-‘(s, pL)[f(S, p, x(h’(s,PL))-+g(s,Id I&
Lid*+‘)(t, p) =k=O, 1,.. ., we
obtain,
in particular,
@To,
(2.12)
@po.
that
33’) (t, p)=
-
s
(D,ct, r-l)@A-‘(& PL)B(S, dds.
(2.13)
(t, p) in the The approximation I“'(t, p) is continuous with respect to the set of variables in view of the continuity with respect to (t,p) of the integrands interval [T,, -) X [pO, -) and the uniform convergence with respect to (t,p) of the integral in (2.13) (the latter In view of (2.11) the following holds: follows from (2.8)). sup
l>T0
and from
(2.8) and
Js”)(t, p) IGa(l-q)
(2.9) we have that the following 2
uniformly
~(1) (t,p) = 0
lim&)(t,p)=O
uniformly
VP!%
conditions
with respect with respect
to
are satisfied: c17 IL>Po,
t, t >
to
To.
P-m
We will use the method of mathematical induction: suppose the statements proved for z"'(4 P) hold for d”‘(t,p), k>l; we will prove that they hold for .z"+"(t,p). From (2.12) we have the estimate
Hence, taking (2.81, (2.9), and (2.11) into account as well as the assumption of the induction and uniform boundedness of J,(t,p) in the interval [T",m)X[pa,~). we obtain that the following conditions are satisfied: p)(4a 12(k+l)(t,
whence
@IJo,
uniformly
with respect
to
CL? cL>ro,
lim.zSb+l)(t,p)=O
uniformly
with respect
to
t,
u-0
The sequence with respect
vt>T,,
!i;t("+l) (t,p)= 0
t>Ta.
converges to the solution x(t,P) of problem (t,p) since the following estimate holds:
{z’“‘(t,p)}
to
it follows
(2.1),
that
ti: .zP) (t, p) =x
(t, p)
uniformly
with respect
Then for z(t,p) the conclusion of the theorem 2. Suppose that in system (2.1)
holds.
to
(t,p), t>To,
Theorem
3 proved.
p>p.o
(2.2) uniformly
123
vt>T,, @PO. A(t,~)=diag(hl(t,~L),...,~"(t,~))
(2.14)
Proposition 3: 1) hi(t,p) in (2.241 are such that a non-negatite function p(t,p) exists. continuous with respect to (t,P) in the interval fTo,~)X[pL@,=) such that for all j=1.2...., n the following inequalities hold ReMt, i1)8P(t,P)X 'titlST,, @ILO, the following condition is satisfied: where for p(t,p)
j p(t, &L)&=f-
VP, Pa&
h
2) A non-negative function m,(t,p)exists, continuous with respect to (t,p) in the interval IT,., -)X[~c~,m),and for any e>O there are 6,.T.,8.>0_ T.>To, such that for ,f(t, p,z) and for m,(t,F) the following holds: inequality (2.4) is satisfied for all t>Tc, (s1<6., (EIC6. m,(t, P)=O(P(t,p)), t-+-m,uniformly with respect to p, I*+&; 3) ,&?(&P) satisfies the following requirements:
Ig’(t, Cc)I=4dt, !4),
t-cm,
uniformly with respect to P, CIauLo,
I&, p)I=o(P(t, p)), CL-m, uniformly with respect to t,
Proposition 4: 1) h~ft,p) in (2.143 are such that for all inequalities hold: vt>T,, p&b; Re hj(t, ptf>O here it is possible that at least for one j,lGj
t>To.
j==l,2,...,nthe fallowing
and at least for one Pt @).L, the following
2) A non-negative function m,(t), continuous in the interval (T,,m) and independent of the inequality (2.6) is satisfied for all t>i”,,~~po,~~~da, p exists such that for f(t,p,z) ._ _.. IEIez, where m,(t) satisfies condition (1.14); 3) Continuous non-negative functions q(t)? t=[T,,oo) exist such that and @(IL), FIPolm) for g(t,P) the following inequality holds:
v@Te, papa.
Id4 kt)I%4W(lt)
(2.15)
where cp(t)satisfies tha condition
j v(t) dKfm, while q(p) satisfies the condition
Theorem 4. Suppose that for system (2.1) propositions3or 4 are satisfied. Then, the conclusion of Theorem 3 holds for problem (2-l), (2.2). Tib prove this theorem it is sufficient to check that the propositionsof!rheorem3 are satisfied. 3. Here and in Sect.3 we will study in more detail problems with a power form of degeneracy with respect to P as y-t=, i.e., problems of the form (2.16) ~-rs'--A(t,CL)Z+f(t,~,5f+g(t, I"), T,
v,,
PCca.
(2.17)
Here OCr is a real number, A(t,P) and g(t,p) are continuous with respect to (t,P) in the I&&n)/ are bounded as p-r- and for fixed region IT+m)Xl~io,=)r and the quantities \A(t,g)j, finite t; f(t,p,o)is continuous with respect to (t,p,z) in the region a~-)(%T0)X[Po,-), f(t,P, is bounded in norm as p-+03 and for fixed t,x. O)==O and f(t,p,z) Be1owe.v. stands for eigenvalues. Proposition 5: 1) In (2.16) either A(t,P)=Ao=const,
Reh(A,)>O
(2.18)
(case A), or ,...,A?(t)), ~{t,~)m~(t}= diag(h,@' (t) and V+&~,...,?Z,
Vt%To
R~~~‘(t)~p(t~~O
(2.19)
124 where the function p(t),continuousin the interval [T,,-),satisfies condition (1.13) (case B); 2) A function mt(t,P) exists, continuous with respect to (t,~)and non-negative in the region [T,,m)Xk~~,m),and for any GO there is a 6., T,,&>O,T.ZT,such that inequality (2.4) holds for all t>T,,pap*,/z/G&,, 111G'6,, and T&&p) in case A satisfies the condition m,(t,p)=0(1),t--T uniformly with respect to
p,PbPLor
and in case B satisfies the condition tern,uniformly with respect to p,P>pO; m,(t,P)=W(0)~,
(2.20)
3) g(t,II) is such that in case A the following conditions are satisfied: ;izg(t,p)=O
uniformly with respect to
P., IL>POl
limg(d, p)=O IL--
uniformly with respect to
t, t> To,
while in case B, the following conditions are satisfied: Ig(t,P)I=o(P(t)),t-r-m,uniformly with respect to P, PtaPO, MC p)I=a(p(Q). PL'W> uniformly with respect to t, t>Z',. Propositions 6: 1) In (2.16) either
A(t,p) =Ao=const,
Re h(A,)>O,
where Jordan cells of order not greater than x,x>f values of A, or Otr is an integer, I--L
(case A) correspond to imaginary eigen-
(2.22)
d(t,c)-,i,tt)+CA*(t)/~~, k-t &(t)=diag(ht)' (t),...,hF’ .-I
(t)),
Re (h:*'(b)+ ~~~I(~)/~')~0 k-l j=l,2,..., n
(2.21)
t&To. k=O, 1,. . ., r-i, Vt>T,, PaILor
(2.23) (2.24)
(case B); here it is possible that at least for one j,iGj
cd
5Re(F,"'(t)+CAll'(t)i~~)dl<+m
VP: Pa&
L-I
To
2) A non-negative function mr(t) exists, that is continuous in the interval which satisfies the condition
q-g If(f,r,X)--f(tr IL,f)l
(To,m),
/x-z1
for all t~Z',,p~po, Ir(G, IIlGz, where e=
; i
in case A, in case B,
(2.25)
and m,(t) satisfies the requirement
dt<+m; j mi(t)te-'
(2.26)
7.
3) Non-negative continuous functions q(t),te[T,,m) and q(P),PE[)krn) exist such that for g(t,P) inequality (2.15) holds, and v(t) satisfies the condition m
and *(PI
satisfies the condition limp"'$(p)=O.
#"xl
Everywhere from now on we will mean by case Ar the satisfaction of the conditions of case A of propositions 5, and similarly for cases B5, A6, B6 etc. Theorem 5. Suppose that for system (2.16) propositions5 or 6 are satisfied. Then, for fairly large To a single solution x(&p) of problem (2.16), (2.17) exists. This solution is continuous with respect to the set of variables (t,).~) in the region [T,,m)X[Po,m), and
125 condition (2.17) is satisfied uniformly with respect to (2.10) is satisfied.
and, moreover, condition Cl, 11,>s;cl0
Proof. In assumptions 5 and 6 it is easy to obtain estimates for the Cauchy matrix ~~(t,~.~)=~~(~,p)@; (s, p) of the sy=-tern z’=X(t,
for ;I(t,p)=p'Ajt.n). Thus, for SX&T, AZ it follows from (2.18) that
p.)x,
T&t<-,
and all nap0
) Ui (G s, p) 1d RI eq
(2.27) pay,, the following inequalities hold: in case
[-- pru (8 -
t)l,
(2.28)
O
in case A, from (2.21) and it follows from the limitation on purely imaginary eigenvalues -4, that Ic,(t, s,I")IQK,y""-i's"-', (2.30) O
Iu,(t, 8, ro /4&,
OtK,=const
(2.31)
(4 is independent of &,j=2,3,4). Using these estimates and the propositions of the theorem, it is easy to verify that functions f(l,s,~_~)=p'f(t,z,~f and ~(~,~)=~'g(~,~) satisfy the propositions of Theorem 3. Then the conclusions of Theorem 5 is a corollary of the conclusion of Theorem 3. Note. 2. For estimates of the Cauchy matrices of systems of linear ordinary differential equations in a semi-infinite interval see /4/ for more detail. 3. In case Bawe can introduce the following changes. Suppose that in system (2.16) the matrix A(t,p) has the form (2.22), where (0) Ito(diag(A* (t),...,?$' (t)), t 31To, (2.32) and j=s+l,s+2,...,n. &:'(a)*h:@'(t)vt 3 To, s=i,2,...,n
(2.33)
since, Then, the diagonal form of the matrices AR(~) in (2.22) cannot be fixed for k==l,2,...,r-1 using the well-known approach (see, for example, /6/ and the papers referred to there) we can diagonalize asymptotically with respect to P the principal part of system (2.16) by using the replacement of variables of the form
Here E,is a unit matrix of order R, and 4(t) are square matrices of order n, which are chosen from the condition that in the system for +,, (new) the matrix .4.(&p)(new) has a diagonal form apart from terms of the order of O(I/P~),P+~. More accurately, Lemma 1 given below holds, in the proof of which essentially we discuss an algorithm for an asymptotic diagonalization of the principal part of system (2.16) using replacement (2.34). are square matrices of order n continuous in the interval Lemma. Suppose &(t),...,L,(t) where A,(t) has the form (2.32), and conditions (2.33).are [To,=). Suppose Lo(t)-&(t), satisfied. Suppose that either the following conditions inf 1h.(O) (if - 3,:C’(t) 13 p,>O, s=i,z,. . . , n, j=s+i, s+&...,n, f>T* SUPl~~(O
t>;.
I< +m,
k--1,2,...,p
are satisfied (case A), or the conditions
1A,‘“’(t)- h:n’ (t) 12 p(t)2% poxI, ILcWI=~(p(~)~, t-cm,
1x0,
j=s+i, s+i,.. . . .?, s=1,2,...,n,
k=i,Z,...,p,
where p(t) is a positive function continuous in the interval [Z's=)(case B). Then square matrices &(t),...,&.(t) of order n, continuous and bounded in the interval [To>=) exist such that the following equation holds:
(2.35)
126 Here the matrices A&) have the form (2.23) for conditions are satisfied: suplAr(t)
k=i,
1c: +m,
t>=*
in case A the following
and
k=i,%,...,p,
2,. . . 1p,
while in case B the following conditions are satisfied IAr(r)[=O(p(t)), t+-, k=/Z,...,p; the (nxn) -matrix C(t,P) is such that in case A the following holds: uniformly with respect to
IC(t,P)l--o(UP=+'). p-m,
I, ts=To,
while in case B iC(t,a)/P(t)l=O(ilCl""). Cl+m, uniformly with respect to t, @Tp. k=O,%,...,p, then If G(f) are continuously differentiable functions in the interval [To,m), k=1,2 ,..., p. Bh(t) are also continuously differentiable functions in the interval IT,,-), Proof.
We will put I;r(t) =Ill~'(t)Ili,,-,,a....,". k=l,2,...,P. Sk(t) =llG'(01/1.1=:.2. ..,ll,
We will assume that (2.35) is true for certain Bk(t), Ah(l).Multiplying the brackets and collecting terms of the same powers of P, we obtain n,(t)=Ao(i)Bi(~)-~,(~)Ao(~)+~I(~)r a-l
(2.36a)
[‘L(t)&-a@)- B,-r(t)&(Ol+ W), A.(t)= Ao(t)B,(0S,(t)Aof0+ z
s=2,3,...,p.
(2.36b)
l-l
From these relations we calculate A.(t), B,(t) successively for s=LZ....,pusing the condition that the matrices A.(t)should have the form (2.23). At the first step we put a:"(f)== 4;'(t), j=i,2,...tn. Then, from (2.36a) for the diagonal elements S%(t) we obtain the relations
aj(0)‘(t)bjj(‘1 (t) -
(1)
bjJ ‘(t)hj
(0)
‘(t)= 0.
i=i,z,...,n,
which are satisfied for any functions bj$'~ft), for example, for ~jjll)(f)=O, t>Fo.j=1,2,...,n, and the off-diagonal elements of the matrix B*(t) are uniquely defined by the equations
We will put s-1
c
IL*(t)&-rv)
R&i=
- ~,-r(t)Mt)l,
a-1
R.(t)=li~$’
For ss2
s=2,3 . . ..*P. (t)Il~j-~,....n.
we put successively in Eqs.(2.36b) <*> (I) (0) j=i.2 hj (f)=)j,(t)+T>j (I).
. . . ..n.
s=2,3,.
. . , p.
b;;)(t)$ (f)=O. i= Then, for the diagonal elements S,(t) we obtain the relations $)(t)b;f’(t)t>To,j=1. 1,2,...,% which are satisfied for any functions b,>‘“‘(t), for example, for b,,~n)(t)=O. 2,..., n; the off-diagonal elements of the matrices B,(t) are uniquely defined by the equations ‘I,
hi
(8)
CO=
/rj
I*)
(6)+ rfj (6)
h/“‘(t)- hp’(t) .
i+j,
le;;iszn,
i
s=2,3,.
. , P.
For the "residue" C@,p) we have from (2.35)
The statement of the lemma also follows from the equations obtained. The lemma is proved. 4. Suppose that in system (2.16) for A&p), g&p) propositions 5 or 6 are satisfied. Suppose the vector-function f(t,p,x) has first-order continuous partial derivatives with in the region ~(-~(~,To)~[IIo,-)~ and satisfiescondition (2.20); respect to the variables zti,....s~n in cases Aaand B. the following inequalities hold:
127
B,Jfor any ~-0
in case
8,.T.,&>O,T.STo
there exist
I
c em,@, p)
such that the following
vt*To,
inequality
holds:
P>Po* I~)<&,
m,(t,p),continuous in the interval [To,-)X[PO,-),satisfies where the non-negative function condition (2.20); in cases Aa and Be the following inequalities hold:
mt(t) satisfies condition (2.26). where 0 is the same as in (2.251, while Then, for problem (2.16) and (2.17) the conclusion of Theorem 5 remains Indeed, for any IslGa,Ix"lCa the following inequality holds:
for all
t>TO,@po,
where
~==s+~(f-z),O
The further discussion
true.
is obvious.
3. Asymptotic behaviour of the solutions with respect to the parameter. e(t,p), continuous with respect to the set Definition 5. We will say that the function can be represented for large u by the of variables (t,~) in the region [T,,m)x[~e~), asymptotic series _ 6(&
P(t)W h-P
uniformly with respect to t when t>T,, if for any integer such that the following estimate holds:
Q,Q>p,
a constant
Cu,C,>O
exists
(3.1) for all t>T, and all fairly large p (we mean by the case Q=p’ that there is no sum in inequality (3.1)). The theorems of this section are related to the investigation of the asymptoticbehaviour with respect to the parameter p of the solutions of problems of the form (2.16), (2.17) in which always henceforth I<: is an integer, and A(t,p) satisfies propositions 5 or 6, and x1,....I,, f(C c1,z) is a vector polynomial of power L with respect to the variables
I.
fk
!bx)=
r,
fl(4 PL)“’
(3.2)
(here k(L,..., 1.) is a multi-index, Z+O, iGj
(3.3) if A(t,k)
satisfies
propositions
5, in which case we have W
if A(4 1~) satisfies propositions
6, we have
(3.5) where I3 is the same as in (2.25). Propositions 7: 1) A(t,p) function p(t) is such that
satisfies
propositions
5, and in case B, in (2.19) the
infp(t)>po>O, 1z.h while
kd"'(f) are infinitely
differentiable
function
(3.6) in the interval
[T,,"), which
satisfy
128 the requirements
I~(&y)I=+-&)*
t-+-m,i=l,2
,..., n,
(3.7)
q-1,2,...; 2) In expansions (3.3) and (3.4) the coefficients q(t),g"'(l),tpb(f), fi'"'(t) are infinitely differentiable functions which, in case A, satisfy the conditions (3.8) (3.9a)
If
(cprO)fi@ W) 1 =O(l),
t-t-,
IC[ZIGL,
k=O,f,...,
(3.9b)
[l[+k>2, q=O,i,..., while in case B they satisfy the conditions (3.10a) (3.10~6)
f
tw)P(t))
[=oWt)ht+w,
k=2,2,...,
f3.1fa)
iQj1jGL,
(3.115)
q=o, 1,. . . ,
If
(w(t)P
(WI=Wp(t)),
k-0,1,. . . , lll+k>L?, q-0, I,...
t-rm,
.
Propositions 8: 1) A(t,p) satisfies propositions 6 , and in case Aa in (2.211 the matrix A, is non-singular, while in case Bs in (2.23) the functions h,'"'(t) are infinitely differentiable in the interval [To,=) and satisy the conditions (3.12a) (3.12b) q=o, 1,*..; 2) In expansions (3.3) and (3.5) the coefficients 9(t),g'"'(t), q(t), f.‘“‘(t) are infinitely differentiable functions in the interval [TO,=), which satisfy the conditions
j Iw(t)Ite-idt<+m, s Il&(t)lte-ldt<-i-Oo, Te i 1% (li,(t)g’“‘(t)) 1 P-‘dt
k-l,
i=siZi
2,. . .,q-0,
i,.. .,
(3.13a)
(3.13tb)
k=rEI,r@+f,..., q=O, j,.... Theorem 6. Suppose that for system (2.16) propositions 7 (or 8) are satisfied. Then for problem (2.161, (2.17) the conclusion of Theorem 5 holds. The solution r(t,P) of this problem can be expanded, for large P in the asymptotic series (3.14) uniformly with respect to t, t>T0, where the coefficients x")(t) can be found from (2.16) by formal substitution of the expansions (3.2), (3.31, (3.41, and (3.14) (or (3.21, (3.31, (3.5) and (3.14)). The series in (3.14) is asymptotic for fixed t and P-m, and also for fixed p and t-+-m,i.e., it is the double asymptotic form of the solution of problem (2.16)~ (2.17). The coefficients x@'(t)are infinitely differentiable functions in the interval [Tp,m),which for propositions 7 satisfy the conditions
129
I-g+‘“‘(t) i =0(l),
t--,
k-l , 2 a...,
q=O,l,...,
(3.15)
q=O, 1,. . . .
(3.16)
and for propositions 8, satisfy the conditions j,!-$P)(Q f Pdttf~,
k--l,2
,...,
Proof. 1. We will prove the theorem for propositions 7. a. From (3.31, (3.41, in view of definition 5, we obtain that for all t>T,, following inequalities hold:
paw0
the
(3.17a)
(3.17b) Using these inequalities, when also conditions (3.8) in case A, (or conditions (3.10) in caseB,) and Note 4, it is easy to obtain that for system (2.16) propositions AS (or BJ) hold. Then, for problem (2.161, (2.17) the conclusion of Theorem 5 holds. b. The solution s(t,p) of problem (2.161, (2.17) will be sought formally in the form of series (3.14). Substituting expansions (3.31, (3.41, and (3.14) into (2.16) and equating expressions of the same powers of $1~.we obtain recurrence formulas for determining the coefficients of series (3.14) x'"(t)=--A-'~(t)g"'(t),
(3.18a)
s'"'(t)=-A-'(~(t)g(m)(t)+Xm(l, I('),..., d”-“~-~mx(m)‘(t)},
(3.18b)
m-2,3,.,. . Here A==
do
1 Ao(t)
in case A?, in case B?,
L=(
1 f,",' ;;;;
the functions xn(t, L(‘), . ..,x”“-“) are formed from the components xC1', . . ..x(~-‘) and the components of the expansion 13.4) (up to powers no higher than t/fi"-') using additions and multiplications, & @,O,. . ..O)=O. Taking propositions (2.18) and (3.9) into account in case AT (or (2.191, (3.61, (3.71, and (3.11) in case B,),we obtain from the recurrence formulas (3.18) successively that the coefficients of the formally constructed expansion (3.14) sayisfy conditions (3.15). c. We will put
a2& tt)=
r(K)@)/pk, r=1
l
is fixed,
where x@'(t), k=l, 2,...,Q are found from the recurrence formulas f3.18), and we introduce the function rl(t, P) such that (3.20)
s(t*P)=%J(t,P)+rl(t, P)/P,
where x(t,p) is the solution of problem (2.161, (2.17). In view of the fact that (2.17) is satisfied uniformly with respect to ILwhen !.c)i&(on the basis of Theorem S), and (3.15) holds when p=O, it follows from (3.19).,(3.201‘that limIj(t,p)=O uniformly with respect to *-bea
FL, @!Jo.
(3.21a)
we will show that liml(t,p)=O
uniformly with respect to
Ir+=
t, t>,TV,
(3.219
and that, moreover, the following holds: Ill(G r)l=O(G),
I1-f@‘ uniformly with respect to t, t>TT,.
(3.22)
We then obtain from (3.21) that series (3.141, constructed from the recurrence formulas (3.18), is a doubly asymptotic form of the solution of problem (2.16), (2.17): for fixed p and t+m and for fixed t and p-+m. Also, we obtain (3.22) that the solution s(t,b) of problem (2.161, (2.17) can be represented for large p by the asymptotic series (3.14) uniformly with respect to t when t>T (in the sense of definition 5). d. We obtain from (2.16) for n(t,p) an equation of the form p-'s'=A(t, P)$"f(G P,ri)f&GIL)>
T,<‘t
(3.23)
where J(GPL,rl)=[f(t,~,s,+gllLQ)--l(l,~,~P)I~P, USSR27:2-1
(3.24)
130
Lfr--V
1
for for
Qfl-r f(t,P,zQ) is formed
+ f
6, ,b xQ)pQ,
(3.25)
QGr, Oar;
from
by eliminating terms of the expansion in powers of l/p up to l/Pa inclusive. It is further easy to verify that for system (3.23) the propositions of Theorem once again satisfied. Indeed, we have from (3.24)
5 are
Taking (3.3), (3.14), and (3.15) into account for p=O and also conditions (3.8) in case Ai (or conditions (3.10) in case B,), from (3.26) and Note 4 we obtain that f(t,p,q) satisfies proposition A, (or Bs). Similarly, from Eq.(3.25), taking definition 5 into account, the properties of the functions $(t),f(t,P,zy) and condition (3.15) for q=O,l, we obtain that d(t,P) satisfies propositions 5. Then, on the basis of Theorem 5, problem (3.23), (3.21a) has a unique solution and (3.21b) also holds for this. Moreover, the solution of this problem satisfies the integral equation (3.27) where
@A (t,P) is the fundamental We will put
matrix
P@,p)=supsup
G=l T
of the solutions
$S,P,i(S,P))
of system
(2.27).
t
O
KF(t, P)/u~%,
(3.28)
KG(t,P)/(sGC,
0
P)/p(t)G'/z, the constant
(3.29)
KG(t, P)Ip(t)GC, in estimate
(2.29).
From
(3.24) and inequality
(2.28) and (3.28) into account Then, from the integral Eq.(3.27), taking estimates A, and estimates (2.29) and (3.29) in case B,, we obtain in the usual way M(t,P)42Cllr,
BT,
in case
PPd,
whence the correctness of (3.22) follows. 2. For propositions 8, the scheme of the proof remains as before with only small changes. We will consider these changes; We will first formally construct series (3.14). In case A, we obtain the recurrence formulas ~‘“‘(t)=--A,-‘~(t)g’“‘(t),
m=l,
2 r....r,
(3.30a)
r’“+“(t)=--A,-‘[~(t)g(m+r,(t)--z(”J’(t)+~m~m+r(t,
cd’), x(Z), . .
Here
L=(
0
for
mdxr--r,
1
for
m>xr-r;
, @+-“)I,
m=l,
2,.
.
.
(3.3Ob)
131 ....X(m+r-')) are formed from the components x1",. . . . z(“‘+‘-‘) and the the functions Xm+,(t,x(", using summations components of the expansion (3.5) (up to powers no higher than ~/IL""'-') and-multiplications, x~+~(~,O,...,O)=O. In caseB, we obtain the recurrence formulas .z"'(t)=-a,-'(t)lp(t)g"'(t), rnlXl(rn,r-l,
(3.31a)
~"(t)=-~-'(l)(~(l)g'"'(t)+ r, 1-1
(3.31b)
tm[Xm(t, 5(",. . . , P-y
A&)r'"-j)(t)+
-29-‘) ml).
m=2,3,. . . .
Here b"=(;
'f", ;z;;
are formed from the components x('~,...,x("-" and the components the functions X~(t,5"',...,X'*-") of the expansion (3.5) (up to powers no higher than l/p"-') using additions and multiplications. Taking (3.12) and (3.13) into account, from Eqs.(3.30) and (3.31) we obtain successively that the coefficients of the formally constructed series (3.14) satisfy conditions (3.16). In system (2.16) we will make the replacement of variables r(t,P)=%(t,P)+2 "'k'(t)l$, D-L
(3.32)
where, as before, El=
in case Aa, i in case Bs,
x
and the coefficients .@'(t), k=l,2,...,?0 are calculated from Eqs.(3.30) or (3.31) respectively. Then, for z,(t,p) we again obtain a problem oftheform (2.16), (2.17), and for g,(t,p) in an expansion of the form (3.3) the first re terms of the series will not be present. Further, as in Sect.1, a, it is easy to show that the system for x,:satisfiespropositions 6. Then, taking the replacement (3.32) into account, we obtain that for the initial problem (2.16), (2.17) the conclusion of Theorem 5 holds. We will fix Q, Q>l+re and calculate the first Q terms of series (3.14). We introduce the function ~)(t,p)such that
is the solution of problem (2.161, (2.17). The further discussion is similar where s(t,k) to that in Sect.1, b. The appropriate changes are obvious. In particular, for the Cauchy matrices of system (2.27) we here use estimates (2.30) or (2.31) respectively. The theorem proved. iS The authors thank A.A. Abramov for discussing the paper. REFERENCES 1. BIRGER E.S. and LYANIKOVAN.B., A methodof finding solutions for certain systems of ordinary differential equations with specified condition at infinity, I, Zh. vych. Mat. mat. Fiz., 5, 6, 979-990, 1965; II, 6, 3, 446-453, 1966. 2. RUSSELL D.L., Numerical solution of singular initial value problems, SIAM J. Numer Analys. 7, 3, 399-417, 1970. 3. BALLA K., The solution of singular boundary value problems for certain systems of ordinary differential equations, Candidate Dissertation, Computing Centre of the Academy of Sciences of the USSR, Moscow, 1978. 4. KONYUKHOVAN.B.,Singular Cauchy problems for systems of ordinary differential equations, Zh. vychsl. Nat. mat. Fiz., 23, 3, 629-645, 1983. 5. FEDORYUK M.V., Asymptotic methods in the theory of ordinary linear differential equations, Mat. Sbornik, 79, 4, 477-516, 1969. 6. FEDORYUK M.V., Asymptotic Methods for Ordinary Linear Differential Equations, Moscow, Nauka, 1983. 7. TRENOGIN V.A., Functional Analysis, Moscow, Nauka, 1980.
Translated by R.C.G.