1 Statement of the problem and basic notation
Let I⊂ℝ be a non-degenerate interval at a point t0∈ℝ and let
It0=I∖{t0},It0-=]-∞,t0[∩I,It0+=]t0,+∞[∩I.
Consider the linear system of generalized ordinary differential equations
(1.1)dx=dA(t)⋅x+df(t) for t∈It0,
where
A=(aik)i,k=1n∈BVloc(It0;ℝn×n) and f=(fk)k=1n∈BVloc(It0;ℝn)
are matrix- and vector-functions, respectively.
Let H=diag(h1,…,hn):It0→ℝn×n be an arbitrary diagonal matrix-function with continuous diagonal elements
hk:It0→]0,+∞[ (k=1,…,n).
We consider the problem of finding a solution x∈BVloc(It0,ℝn) of system (1.1), satisfying the condition
(1.2)limt→t0-(H-1(t)x(t))=0 and limt→t0+(H-1(t)x(t))=0.
Analogous problems for systems of ordinary differential equations with singularities
(1.3)dxdt=P(t)x+q(t) for t∈I,
where
P∈Lloc(It0;ℝn×n), q∈Lloc(It0;ℝn),
have been investigated in the papers [6, 7, 8].
The singularity of system (1.3) is considered in the sense that the matrix-function P and the vector-function q are not, in general, integrable at the point t0. The solution of problem (1.3), (1.2) is not, in general, continuous at the point t0 and, therefore, it is not a solution in the classical sense. But its restriction on every interval from It0 is a solution of system (1.3). To illustrate this we give the example from [8].
Let α>0 and ε∈]0,α[. Then the problem
dxdt=-αxt+ε|t|ε-1α,limt→0(tαx(t))=0
has the unique solution x(t)=|t|ε-αsgnt. This function is not a solution of the equation on the set I=ℝ, but its restrictions on ]-∞,0[ and ]0,+∞[ are solutions of the one.
The singularity of system (1.1) is considered in the sense that A or f may have non-bounded total variations at the point t0, i.e., on some closed interval [a,b] from I such that t0∈[a,b].
As we know, such a problem for the generalized differential system (1.1) has not been investigated. So, the present investigation is quite topical.
Some singular two-point boundary problems for the generalized differential system (1.1) are investigated in [3, 4, 5].
The interest in the theory of
generalized ordinary differential equations has also been
stimulated by the fact that this theory enables one to investigate
ordinary differential, impulsive differential and difference equations from a
unified point of view (see [2, 1, 3, 4, 5, 10, 11] and the references therein).
In the present paper, we give the sufficient
conditions for the unique solvability of problem (1.1), (1.2).
Analogous results for the Cauchy problem for
systems of ordinary differential equations with singularities belong to Kiguradze [7, 8].
Throughout the paper, use will be made of the following notation and
definitions:
ℝ=]-∞,+∞[, ℝ+=[0,+∞[, and [a,b] and ]a,b[
(a,b∈ℝ) are, respectively, closed and open intervals.
ℝn×m is the space of all real n×m matrices
X=(xik)i,k=1n,m with the norm
∥X∥=maxk=1,…,m∑i=1n|xik|.
On×m (or O) is the zero n×m matrix.
If X=(xik)i,k=1n,m∈ℝn×m, then
|X|=(|xik|)i,k=1n,m,[X]+=|X|+X2,[X]-=|X|-X2.
ℝ+n×m={(xik)i,k=1n,m:xik≥0(i=1,…,n,k=1,…,m)}.
ℝn=ℝn×1 is the space of all real column
n-vectors x=(xi)i=1n, and ℝ+n=ℝ+n×1.
If X∈ℝn×n, then X-1, detX and r(X) are,
respectively, the matrix inverse to X, the determinant of X
and the spectral radius of X, and In is the identity n×n-matrix.
The inequalities between the matrices are understood
componentwise.
A matrix-function is said to be continuous, integrable,
nondecreasing etc., if each of its components is such.
If X:ℝ→ℝn×m is a matrix-function, then Vab(X)
is the sum of total variations on [a,b] of its
components xik (i=1,…,n, k=1,…,m). If a>b, then we assume
Vab(X)=-Vba(X).
X(t-) and X(t+) are, respectively, the left and the right
limits of the matrix-function X:[a,b]→ℝn×m at the
point t (X(a-)=X(a),X(b+)=X(b)),
d1X(t)=X(t)-X(t-),d2X(t)=X(t+)-X(t).
BV([a,b],ℝn×m) is the set of all bounded
variation matrix-functions X:[a,b]→ℝn×m, i.e.,
such that
Vab(X)<∞.
BVloc(J;D), where J⊂ℝ is an interval and
D⊂ℝn×m, is the set of all X:J→D for which the restriction on [a,b] belongs to BV([a,b];D) for every closed interval [a,b] from J.
BVloc(It0;D) is the set of all X:I→D for which the restriction on [a,b] belongs to BV([a,b];D) for every closed interval [a,b] from It0.
Everywhere we assume that a1∈It0- and a2∈It0+ are some fixed points.
If X∈BVloc(It0;ℝn×m), then
V(X)(t)=(v(xik)(t))i,k=1n,m for t∈It0,
where
v(xik)(aj)=0,v(xik)(t)≡Vajt(xik) for (t-t0)(aj-t0)>0(j=1,2).
[X(t)]+v≡V(X)(t)+X(t)2 and [X(t)]-v≡V(X)(t)-X(t)2.
s1,s2,sc and 𝒥:BVloc(It0;ℝ)→BVloc(It0;ℝ) are the operators
defined, respectively, by
s1(x)(aj)=s2(x)(aj)=0,
sc(x)(aj)=x(aj) (j=1,2),
s1(x)(t)=s1(x)(s)+∑s<τ≤td1x(τ),
s2(x)(t)=s2(x)(s)+∑s≤τ<td2x(τ),
sc(x)(t)=sc(x)(s)+x(t)-x(s)-∑j=12(sj(x)(t)-sj(x)(s))
for s<t<t0 and for t0<s<t, and
𝒥(x)(aj)=x(aj) (j=1,2),
𝒥(x)(t)=𝒥(x)(s)+s0(x)(t)-s0(x)(s)+∑t<τ≤sln|1-d1x(τ)|-∑t≤τ<sln|1+d2x(τ)| for t<s<t0,
𝒥(x)(t)=𝒥(x)(s)+s0(x)(t)-s0(x)(s)-∑s<τ≤tln|1-d1x(τ)|+∑s≤τ<tln|1+d2x(τ)| for t0<s<t.
If X∈BVloc(It0;ℝn×n),
det(In+(-1)ldlX(t))≠0 for t∈It0 (l=1,2), and Y∈BVloc(It0;ℝn×m), then
𝒜(X,Y)(aj)=On×m,
𝒜(X,Y)(t)-𝒜(X,Y)(s)=Y(t)-Y(s)+∑s<τ≤td1X(τ)⋅(In-d1X(τ))-1d1Y(τ)
-∑s≤τ<td2X(τ)⋅(In+d2X(τ))-1d2Y(τ)
for s<t<t0 if aj<t0, and for t0<s<t<t0 if aj>t0(j=1,2).
If g:[a,b]→ℝ is a nondecreasing function, x:[a,b]→ℝ
and a≤s<t≤b, then
∫stx(τ)𝑑g(τ)=∫]s,t[x(τ)𝑑s0(g)(τ)+∑s<τ≤tx(τ)d1g(τ)+∑s≤τ<tx(τ)d2g(τ),
where ∫]s,t[x(τ)𝑑s0(g)(τ) is the
Lebesgue–Stieltjes integral over the open interval ]s,t[ with
respect to the measure μ0(sc(gc)) corresponding to the
function s0(g). So ∫stx(τ)𝑑g(τ) is the Kurzweil integral (see [9, 10, 11]).
If a=b, then we assume
∫abx(t)𝑑g(t)=0,
and if a>b, then we assume
∫abx(t)𝑑g(t)=-∫bax(t)𝑑g(t).
Moreover, we put
∫st+x(τ)𝑑g(τ)=limδ→0+∫st+δx(τ)𝑑g(τ),∫st-x(τ)𝑑g(τ)=limδ→0+∫st-δx(τ)𝑑g(τ).
If g(t)≡g1(t)-g2(t), where g1 and g2 are
nondecreasing functions, then
∫stx(τ)𝑑g(τ)=∫stx(τ)𝑑g1(τ)-∫stx(τ)𝑑g2(τ) for s,t∈ℝ.
If G=(gik)i,k=1l,n:[a,b]→ℝl×n is a
nondecreasing matrix-function and X=(xkj)k,j=1n,m:[a,b]→ℝn×m, then
∫st𝑑G(τ)⋅X(τ)=(∑k=1n∫stxkj(τ)𝑑gik(τ))i,j=1l,m for a≤s≤t≤b,
Sj(G)(t)≡(sj(gik)(t))i,k=1l,n (j=1,2),Sc(G)(t)≡(sc(gik)(t))i,k=1l,n.
If Gj:[a,b]→ℝl×n (j=1,2) are nondecreasing
matrix-functions, G=G1-G2 and X:[a,b]→ℝn×m,
then
∫st𝑑G(τ)⋅X(τ)=∫st𝑑G1(τ)⋅X(τ)-∫st𝑑G2(τ)⋅X(τ) for s,t∈ℝ,
Sk(G)=Sk(G1)-Sk(G2) (k=1,2),Sc(G)=Sc(G1)-Sc(G2).
Here use will be made of the following formulas:
∫abf(t)d(∫ath(s)𝑑g(s))=∫abf(t)h(t)𝑑g(t) (substitution formula),
∫abf(t)𝑑g(t)+∫abg(t)𝑑f(t)=f(b)g(b)-f(a)g(a)+∑a<t≤bd1f(t)⋅d1g(t)-∑a≤t<bd2f(t)⋅d2g(t)
(integration by parts formula),
∫abh(t)d(f(t)g(t))=∫abh(t)f(t)𝑑g(t)+∫abh(t)g(t)𝑑f(t)-∑a<t≤bh(t)d1f(t)⋅d1g(t)+∑a≤t<bh(t)d2f(t)⋅d2g(t)
(general integration by parts formula)
and
dj(∫atf(s)dg(s))=f(t)djg(t) for t∈[a,b](j=1,2),
where f,g,h∈BV([a,b],ℝ) (see [11, Theorems I.4.25 and I.4.33]). In the sequel, we use these formulas without making this reference.
A vector-function x:It0→ℝn is said to be a solution of
system (1.1) if x∈BV([a,b],ℝn) for every closed interval [a,b] from It0 and
x(t)=x(s)+∫st𝑑A(τ)⋅x(τ)+f(t)-f(s) for a≤s<t≤b.
We assume that
det(In+(-1)jdjA(t))≠0 for t∈It0(j=1,2).
The above inequalities guarantee the unique solvability of the
Cauchy problem for the corresponding nonsingular systems (see [9, 10, 11]), i.e., for the case where A∈BVloc(I,ℝn×n) and f∈BVloc(I,ℝn).
Let A0∈BVloc(It0;ℝn×n). Then a matrix-function C0:It0×It0→ℝn×n is said to be the Cauchy matrix of the generalized homogeneous differential system
(1.4)dx=dA0(t)⋅x
if, for every interval J⊂I and τ∈J, the restriction of the matrix-function C0(⋅,τ):It0→ℝn×n on J is the fundamental matrix of system (1.4), satisfying the condition
C0(τ,τ)=In.
Therefore, C0 is the Cauchy matrix of system (1.4) if and only if the restriction of C0 on J×J for every interval J⊂It0 is the Cauchy matrix of the system in the sense of the definition given in [11].
We assume
It0-(δ)=[t0-δ,t0[∩It0,It0+(δ)=]t0,t0+δ]∩It0,It0(δ)=It0-(δ)∪It0+(δ)
for every δ>0.
Theorem 1.1.
Let there exist a matrix-function A0∈BVloc(It0,Rn×n) and constant matrices B0 and B from R+n×n
such that
(1.5)det(In+(-1)jdjA0(t))≠0 for t∈It0(j=1,2),
(1.6)r(B)<1,
and the estimates
(1.7)|C0(t,τ)|≤H(t)B0H-1(τ) for t∈It0(δ),(t-t0)(τ-t0)>0,|τ-t0|≤|t-t0|
and
(1.8)V(CH)(t,⋅)(t)-V(CH)(t,⋅)(t0±)≤H(t)B for t∈It0+(δ) and t∈It0-(δ), respectively,
hold for some δ>0, where C0 is the Cauchy matrix of system (1.4), and
(1.9)CH(t,τ)=∫τtC0(t,s)𝑑𝒜(A0,A-A0)(s)⋅H(s) for (t-t0)(τ-t0)>0,|τ-t0|≤|t-t0|.
Let, moreover, respectively,
(1.10)limt→t0±∥∫t0±tH-1(t)C0(t,τ)𝑑𝒜(A0,f)(τ)∥=0.
Then problem (1.1), (1.2) has a unique solution.
Theorem 1.2.
Let there exist a constant matrix B=(bik)i,k=1n∈R+n×n such that conditions (1.6) and
(1.11){[d1aii(t)]+<1for t<t0(i=1,…,n),[d2aii(t)]-<1for t>t0(i=1,…,n)
hold, and the estimates
(1.12){ci(t,τ)≤b0hi(t)hi(τ)for t∈It0(δ),(t-t0)(τ-t0)>0,|τ-t0|≤|t-t0|(i=1,…,n),
and
(1.13){|∫t0±tci(t,τ)hi(τ)d[aii(τ)sgn(τ-t0)]+v|≤biihi(t)for t∈It0+(δ) and t∈It0-(δ), respectively(i=1,…,n),|∫t0±tci(t,τ)hk(τ)𝑑V(𝒜(a0ii,aik))(τ)|≤bikhi(t)for t∈It0+(δ) and t∈It0-(δ), respectively(i≠k,i,k=1,…,n),
hold for some b0>0 and δ>0. Let, moreover, respectively,
(1.14)limt→t0±∫t0±tci(t,τ)hi(t)dV(𝒜(a0ii,fi))(τ)=0 (i=1,…,n),
where a0ii(t)≡-[aii(t)sgn(t-t0)]-vsgn(t-t0) (i=1,…,n), and ci is the Cauchy function of the equation dx=xda0ii(t)
for i∈{1,…,n}. Then problem (1.1), (1.2) has the unique solution.
Remark 1.3.
For t,τ∈It0- and t,τ∈It0+, the Cauchy functions ci(t,τ) (i=1,…,n) in Theorem 1.2 have the form
(1.15)ci(t,s)={exp(sc(a0ii)(t)-sc(a0ii)(s))∏s<τ≤t(1-d1a0ii(τ))-1×∏s≤τ<t(1+d2a0ii(τ))for t>s,exp(sc(a0ii)(t)-sc(a0ii)(s))∏t<τ≤s(1-d1a0ii(τ))×∏t≤τ<s(1+d2a0ii(τ))-1for t<s,1for t=s.
Corollary 1.4.
Let there exist a constant matrix B=(bik)i,k=1n∈R+n×n such that conditions (1.6) and (1.11) hold, and
the estimates
(1.16){𝒥(a0ii)(t)-𝒥(a0ii)(τ)≤μilnt-t0τ-t0 for t,τ∈It0,(t-t0)(τ-t0)>0,|τ-t0|≤|t-t0|(i=1,…,n),
and
(1.17){limτ→t0±|[aii(t)sgn(t-t0)]+v-[aii(τ)sgn(τ-t0)]+v|≤biifor t∈It0+(δ) and t∈It0-(δ),𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦(i=1,…,n),limτ→t0±|V(𝒜(a0ii,aik))(t)-V(𝒜(a0ii,aik))(τ)|≤bikfor t∈It0+(δ) and t∈It0-(δ),𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦(i≠k,i,k=1,…,n)
hold for some μi≥0 (i=1,…,n) and δ>0, where a0ii(t)≡-[aii(t)sgn(t-t0)]-vsgn(t-t0) (i=1,…,n). Let, moreover, respectively,
limt→t0±∫t0±t1|τ-t0|μidV(𝒜(a0ii,fi))(τ)=0 (i=1,…,n).
Then system (1.1) has a unique solution satisfying the initial condition
(1.18)limt→t0±xi(t)|t-t0|μi=0 (i=1,…,n).
Remark 1.5.
In the results given above, the strong inequality (1.6) cannot be replaced by the non-strong one. We give the corresponding example from [8] for the ordinary differential case, i.e., when A(t)≡∫atP(τ)𝑑τ and f(t)≡∫atq(τ)𝑑τ.
On the interval ]-1,0[ consider the problem
dxdt=xt+1|ln|t||,limt→0x(t)t=0.
Every solution of the above equation has the form
x(t)=ct+tln|ln|t|| (c∈ℝ).
So, this problem is not solvable. On the other hand, the Cauchy function c(t,τ)=tτ-1 for t≤τ<0, and the conditions of Theorem 1.1, except of condition (1.6), are fulfilled (on ]-1,0[) for n=1, P(t)≡t-1, q(t)≡0 and h(t)≡t only for the case where B≥1, i.e., when r(B)≥1.
Remark 1.6.
Let, in addition to the conditions of Corollary 1.4, the condition
(1.19)lim supt→t0±ξji(t)<+∞ (j=1,2,i=1,…,n)
hold, where
(1.20)ξji(t)=∑τ∈Itj(∑k=1n|τ-t0|μk|djaik(τ)|+|djfi(τ)|) for t∈It0∩]a1,a2[(j=1,2,i=1,…,n),
It1=]a1,t] and It2=[a1,t[ for a1<t<t0, It1=]t,a2] and It2=[t,a2[ for t0<t<a2. Then the solution of problem (1.1), (1.18) belongs to BVloc(I;ℝn).
Corollary 1.7.
Let there exist a constant matrix B=(bik)i,k=1n∈R+n×n such that conditions (1.6) and (1.11) hold, and
estimates (1.16) for μi=1 (i=1,…,n),
(1.21){|∫t0±t|τ-t0|d[aii(τ)sgn(τ-t0)]+v|≤bii|t-t0|for t∈It0+(δ) and t∈It0-(δ), respectively(i=1,…,n),|∫t0±t|τ-t0|𝑑V(𝒜(a0ii,aik))(τ)|≤bik|t-t0|for t∈It0+(δ) and t∈It0-(δ), respectively(i≠k,i,k=1,…,n)
hold for some δ>0, where a0ii(t)≡-[aii(t)sgn(t-t0)]-vsgn(t-t0) (i=1,…,n). Let, moreover, respectively,
(1.22)limt→t0±1|t-t0||Vt0±t(𝒜(a0ii,fi))(τ)|=0 (i=1,…,n).
Then system (1.1) has a unique solution satisfying the initial condition
(1.23)limt→t0±∥x(t)∥|t-t0|=0.
Remark 1.8.
Let, in addition to the conditions of Corollary 1.7, condition (1.19) hold, where the functions ξji (j=1,2,i=1,…,n) are defined by (1.20), where μi=1 (i=1,…,n), and the intervals Itj (j=1,2) are defined as in Remark 1.6. Then the solution of problem (1.1), (1.23) belongs to BVloc(I;ℝn).
Corollary 1.9.
Let aii∈BVloc(I;Rn) (i=1,…,n) and let conditions (1.11) and
(1.24){𝒥(a0ii)(t)-𝒥(a0ii)(s)≤-λilnt-t0s-t0+aii*(t)-aii*(s)for t,s∈It0,(t-t0)(s-t0)>0,|s-t0|≤|t-t0|(i=1,…,n)
hold, where a0ii(t)≡-[aii(t)sgn(t-t0)]-vsgn(t-t0) (i=1,…,n), λi≥0 (i=1,…,n), and aii*(t)sgn(t-t0) (i=1,…,n) are the nondecreasing functions on the interval I. Let, moreover,
(1.25){|∫t0±t|τ-t0|λi-λk𝑑V(𝒜(a0ii,aik))(τ)|<+∞for t∈It0+ and t∈It0-, respectively(i≠k,i,k=1,…,n),
and
(1.26)|∫t0±t|τ-t0|λidV(𝒜(a0ii,fi))(τ)|<+∞ for t∈It0+ and t∈It0-, respectively(i=1,…,n).
Then system (1.1) has a unique solution satisfying the initial condition
(1.27)limt→t0±(|t-t0|λixi(t))=0 (i=1,…,n).
Remark 1.10.
Let the conditions of Corollary 1.9 hold, where λi=0 (i=1,…,n). Let further condition (1.19)
hold, where the functions ξji (j=1,2;i=1,…,n) are defined by (1.20), where μi=0 (i=1,…,n), and the intervals Itj (j=1,2) are defined as in Remark 1.6. Then the solution of problem (1.1), (1.27) belongs to BVloc(I,ℝn).
Remark 1.11.
In Remarks 1.6–1.10, condition (1.19) is essential, i.e., if the condition is violated, then the conclusion of the above remarks is not true. Below we present an example.
Let I=[0,1], n=1, t0=0, τl=1/l (l=1,2,…), and let the function A(t)≡a(t) be defined by
a(0)=0,
a(1)=-2ln2,
a(t)=ln(kl(t-τl)+1l)+(ln2-12)l for τl≤t<τl-1(l=2,3,…),
where kl=(l-2)(2l(l-1)(τl-τl-1))-1 (l=2,3,…).
It is not difficult to verify that the modified Cauchy problem
dx=xda(t),limt→0|x(t)|t=0
has the unique solution x defined by the equalities
x(t)=kl(t-τl)+1l for τl≤t<τl-1(l=2,3,…),x(1)=-2ln2.
Indeed, we have x′(t)=kl and a′(t)x(t)=kl/(kl(t-τl)+l-1)⋅x(t)=kl for τl≤t<τl-1 (l=2,3,…).
Moreover, d2x(t)=d2a(t)≡0, x(τl-1-)=kl(τl-1-τl)+l-1=(2(l-1))-1 (l=2,3,…) and d1x(τl)=(2l)-1 (l=1,2,…). On the other hand, a(τl-1-)=-ln(2(l-1))+(ln2-12)l, d1a(τl)=12 and x(τl)d1a(τl)=(2l)-1 (l=1,2,…). Therefore, d1a(τl)=x(τl)d1a(τl) (l=1,2,…).
As for the limit condition, it is evident.
So, we conclude that x∈BVloc(It0;ℝ), but x∉BVloc(I;ℝ).
Besides, taking into account that the function a(t) is nonincreasing on the intervals τl≤t<τl-1 (l=2,3,…), we conclude that [a(t)]+v=0 on the same intervals. Therefore, due to the equalities d2a(t)≡0 and d1a(τl)=12 (l=1,2,…), all conditions of Remarks 1.6–1.10 are fulfilled except of (1.19).
2 Auxiliary propositions
We use the lemma (see Lemma 2.2 below) on the a priori estimate of solutions of system (1.1). To prove the lemma, we use a new type of Cauchy formula for the representation of solutions differing from the previous earlier ones [11].
Lemma 2.1.
Let A*∈BVloc(I,Rn×n) be such that
(2.1)det(In+(-1)jdjA*(t))≠0 for t∈I(j=1,2),
and f*∈BVloc(I,Rn). Then every solution x∈BVloc(I,Rn) of the system
(2.2)dx=dA*(t)⋅x+df*(t) for t∈I
admits the representation
(2.3)x(t)=C*(t,s)x(s)+∫stC*(t,τ)𝑑𝒜(A*,f*)(τ) for s,t∈I,
where C* is the Cauchy matrix of system (2.2), and A is the operator defined above.
Proof.
By (2.1), the integration by parts formula and the equalities
dj(τ)C*(t,τ)≡-C*(t,τ)djA*(τ)⋅(In+(-1)jdjA*(τ))-1 (j=1,2),
it follows from the variation of constant formula (see [11]) that
x(t)=C*(t,τ)x(s)+f*(t)-f*(s)-∫st𝑑C*(t,τ)⋅(f*(τ)-f*(s))
=C*(t,τ)x(s)+∫stC*(t,τ)⋅𝑑f*(τ)-∑s<τ≤td1C*(t,τ)⋅d1f*(τ)+∑s≤τ<td2C*(t,τ)⋅d2f*(τ)
=C*(t,τ)x(s)+∫stC*(t,τ)⋅𝑑f*(τ)+∑s<τ≤tC*(t,τ)d1A*(τ)⋅(In-d1A*(τ))-1⋅d1f*(τ)
-∑s≤τ<tC*(t,τ)d2A*(τ)⋅(In+d2A*(τ))-1⋅d2f*(τ) for t∈ℝ.
Therefore, due to the definition of the operator 𝒜, equality (2.3) holds.
∎
Lemma 2.2.
Let the matrix-function A0∈BVloc(I;Rn×n) and constant matrices B0 and B from R+n×n
be such that conditions (1.5)–(1.8)
hold for some δ>0, where C0 is the Cauchy matrix of system (1.4). Let, moreover,
(2.4){γ(t)=sup{∥∫t0±sH-1(s)C0(s,τ)𝑑𝒜(A0,f)(τ)∥:(s-t0)(t-t0)>0,|s-t0|≤|t-t0|}<+∞for t∈It0(δ).
Then every solution x∈BVloc(J;Rn) of system (1.1) admits the estimate
(2.5){∥H-1(t)x(t)∥≤ρ(∥B0∥⋅∥H-1(s0)x(s0)∥+γ(t))for t∈J,(s0-t0)(t-t0)≥0,|s0-t0|≤|t-t0|,
where ρ=∥(In-B)-1∥, and J⊂It0(δ) and s0∈J are an arbitrary interval and point, respectively.
Proof.
First consider the case where J⊂(t0+δ,supI). Let x=(xi)i=1n be an arbitrary solution of system (1.1) on J. Then x satisfies the system
dx=dA0(t)⋅x+d(A(t)-A0(t))⋅x+df(t) for t∈J,
i.e., the system
dx=dA0(t)⋅x+d(f(t)+f0(t)) for t∈J,
where
f0(t)≡∫s0td(A(τ)-A0(τ))⋅x(τ).
Let the vector-function z(t)=(zi(t))i=1n be defined by
z(t)≡H-1(t)x(t).
According to Lemma 2.1, we have
(2.6)z(t)≡H-1(t)C0(t,s0)x(s0)+∫s0tH-1(t)𝑑C0(t,τ)⋅𝒜(A0,f+f0)(τ).
From (2.6), by the definitions of the operator 𝒜 and the vector-function f0, we obtain
z(t)=H-1(t)C0(t,s0)x(s0)+∫s0tH-1(t)𝑑C0(t,τ)⋅𝒜(A0,f)(τ)
+∫s0tH-1(t)C0(t,τ)d(A(τ)-A0(τ))⋅H(τ)z(τ)
-∑s0<τ≤tH-1(t)d1C0(t,τ)⋅d1(A(τ)-A0(τ))⋅H(τ)z(τ)
+∑s0≤τ<tH-1(t)d2C0(t,τ)⋅d2(A(τ)-A0(τ))⋅H(τ)z(τ)
=H-1(t)C0(t,s0)x(s0)+∫s0tH-1(t)𝑑C0(t,τ)⋅𝒜(A0,f)(τ)
+∫s0tH-1(t)C0(t,τ)𝑑AH(τ)⋅z(τ)-∑s0<τ≤tH-1(t)d1C0(t,τ)⋅d1AH(τ)⋅z(τ)
+∑s0≤τ<tH-1(t)d2C0(t,τ)⋅d2AH(τ)⋅z(τ) for t>s0,
and so, using the general integration by parts formula, we get
z(t)=H-1(t)C0(t,s0)x(s0)+∫s0tH-1(t)𝑑C0(t,τ)⋅𝒜(A0,f)(τ)
+H-1(t)∫s0td(C0(t,τ)AH(τ))⋅z(τ)-H-1(t)∫s0t𝑑C0(t,τ)⋅AH(τ)z(τ) for t≥s0,
where
AH(t)≡∫s0td(A(τ)-A0(τ))⋅H(τ).
Analogously, we show the last inequality for t<s0. Therefore,
z(t)=H-1(t)C0(t,s0)x(s0)+H-1(t)∫s0t𝑑C0(t,τ)⋅𝒜(A0,f)(τ)
(2.7)+H-1(t)∫s0t𝑑C*(t,τ)⋅z(τ) for t∈J,
where
C*(t,τ)≡C0(t,τ)AH(τ)-∫s0τ𝑑C0(t,s)⋅AH(s).
In addition, using the integration by parts formula and taking into account the equality
dj(τ)C0(t,τ)≡-C0(t,τ)djA0(τ)⋅(In+(-1)jdjA0(τ))-1 (j=1,2),
we have
C*(t,τ)≡CH(t,τ), where CH(t,τ) is defined by (1.9).
Therefore, due to (2.7), we have
z(t)=H-1(t)C0(t,s0)x(s0)+H-1(t)∫s0t𝑑CH(t,τ)⋅z(τ)
(2.8)+H-1(t)∫s0t𝑑C0(t,τ)⋅𝒜(A0,f)(τ) for t∈J.
Let the components of the vector-function y(t)=(yi(t))i=1n be defined as follows:
yi(t)=sup{hi-1(s)|xi(s)|:t≤s≤s0} if s0<t0,
yi(t)=sup{hi-1(s)|xi(s)|:s0≤s≤t} if s0>t0.
In view of (2.8), by taking into account (1.8), it is not difficult to verify that
y(t)≤H-1(t)|C0(t,s0)x(s0)|+H-1(t)(V(CH)(t,⋅)(t)-V(CH)(t,⋅)(s0))y(t)+H-1(t)∫s0t𝑑C0(t,τ)⋅𝒜(A0,f)(τ)
≤B0|H-1(s0)x(s0)|+By(t)+g(t) for s0≤t,
where
g(t)≡H-1(t)|∫s0tC0(t,τ)𝑑𝒜(A0,f)(τ)|.
Therefore, thanks to (1.6) and the nonnegativity of the matrix B0, we have
(In-B)y(t)≤B0|H-1(s0)x(s0)|+g(t) for s0≤t,
and
y(t)≤(In-B)-1(B0|H-1(s0)x(s0)|+g(t)) for s0≤t.
Hence estimate (2.5) holds.
Analogously, we can prove the estimate in other cases.
∎
Lemma 2.3.
Let g∈BV([a,b];R). Then
(2.9)∫absgng(t)𝑑g(t)=|g(b)|-|g(a)|+∑a<t≤b(|g(t-)|-g(t-)sgng(t))-∑a≤t<b(|g(t+)|-g(t+)sgng(t))
and
(2.10)∫absgng(t)d|g(t)|=g(b)-g(a)+∑a<t≤b(g(t-)-|g(t-)|sgng(t))-∑a≤t<b(g(t+)-|g(t+)|sgng(t)).
Proof.
Due to the integration by parts formula and [11, Lemma I.4.23], we find
∫absgng(t)𝑑g(t)=g(b)sgng(b)-g(a)sgng(a)-∫abg(t)d(sgng(t))
+∑a<t≤bd1g(t)⋅d1(sgng(t))-∑a≤t<bd2g(t)⋅d2(sgng(t))
=|g(b)|-|g(a)|-∑a<t≤bg(t)⋅d1(sgng(t))-∑a≤t<bg(t)⋅d2(sgng(t))
+∑a<t≤bd1g(t)⋅d1(sgng(t))-∑a≤t<bd2g(t)⋅d2(sgng(t))
=|g(b)|-|g(a)|-∑a<t≤bg(t-)(sgng(t)-sgng(t-))-∑a≤t<bg(t+)(sgng(t+)-sgng(t)).
So equality (2.9) holds.
In the same way we establish equality (2.10).
∎
Lemma 2.4.
Let the restriction on the set It0 of a function g:I→R belong to the set BVloc(It0,R),
(2.11)w(t)-w(τ)≤v(t)-v(τ) for t,τ∈I,t0<τ<t(t<τ<t0),
and let there exist one-sided limits w(t0-) and w(t0+),
where
w(t)=∫atσ(s)sgng(s)dg(s) for t∈It0+(t∈It0-),
a∈It0+ (a∈It0-) is some fixed point,
(2.12)v∈BVloc(I∖It0-;ℝ) (v∈BVloc(I∖It0+;ℝ)),
and σ:It0→{-1,1} is a continuous function. Then
(2.13)g∈BVloc(I∖It0-;ℝ) (g∈BVloc(I∖It0+;ℝ)).
Proof.
First consider the case where t∈It0+ and σ(t)≡1. Let us introduce the function
u(t)=v(t)-w(t).
Then there exists a one-sided limit u(t0+). By (2.12), we have u∈BVloc(I∖It0-,ℝ) because, due to (2.11), the function u is nondecreasing on the set It0+. Therefore, w∈BVloc(I∖It0-,ℝ) as well.
By (2.9), we find
|g(t)|-|g(τ)|=w(t)-w(τ)-∑τ<s≤t(|g(s-)|-g(s-)sgng(s))+∑τ≤s<t(|g(s+)|-g(s+)sgng(s))
=w(t)-w(τ)+∑τ<s≤t(|g(s)|-|g(s-)|)-∑τ<s≤t(|g(s)|-g(s-)sgng(s))
+∑τ≤s<t(|g(s+)|-g(s+)sgng(s))
=w(t)-w(τ)+∑τ<s≤t(|g(s)|-|g(s-)|)-∑τ<s≤t(|g(s)|-g(s-)sgng(s))
+∑τ≤s<t(|g(s+)|-|g(s)|)+∑τ<s≤t(|g(s)|-g(s+)sgng(s))
=w(t)-w(τ)-s1(w)(t)+s1(w)(τ)-s2(w)(t)+s2(w)(τ)
+s1(|g|)(t)-s1(|g|)(τ)+s2(|g|)(t)-s1(|g|)(τ) for t0<τ<t.
Hence
s0(|g|)(t)-s0(|g|)(τ)=s0(w)(t)-s0(w)(τ) for t0<τ<t,
and
(2.14)s0(|g|)∈BV(I∖It0-,ℝ).
Using now the estimates
|dj(|g(t)|)|≤|djg(t)|=|djw(t)| for t>t0(j=1,2)
we conclude that
v(sj(|g|))(t)-v(sj(|g|))(t0)≤v(sj(w))(t)-v(sj(w))(t0)<+∞ for t>t0(j=1,2)
and, therefore,
(2.15)s1(g),s2(g)∈BV(I∖It0-;ℝ)
and
s1(|g|),s2(|g|)∈BV(I∖It0-;ℝ).
By the last inclusions and (2.14), we get
(2.16)|g|∈BV(I∖It0-;ℝ).
Because of (2.16), we have
w¯∈BV(I∖It0-;ℝ),
where
w¯(t)=∫atsgng(s)d|g(s)| for t≥t0.
As above, we conclude that
s0(g)(t)-s0(g)(τ)=s0(w¯)(t)-s0(w¯)(τ) for t0<τ<t,
and
(2.17)s0(g)∈BV(I∖It0-;ℝ).
So inclusion (2.13) follows from (2.15) and (2.17).
∎
Lemma 2.5.
Let X∈BVloc(I;Rn×n),
det(In+(-1)jdjX(t))≠0 for t∈I (j=1,2), and Y∈BVloc(I;Rn×m). Then
Y(t)-Y(s)=𝒜(X,Y)(t)-𝒜(X,Y)(s)-∑s<τ≤td1X(τ)⋅d1𝒜(X,Y)(τ)
(2.18)+∑s≤τ<td2X(τ)⋅d2𝒜(X,Y)(τ) for s,t∈I,s<t,
and
Vab(Y)≤Vab(𝒜(X,Y))+maxa<t≤b{∥d1X(t)∥}∑a<t≤b∥d1𝒜(X,Y)(t)∥
(2.19)+maxa≤t<b{∥d2X(t)∥}∑a≤t<b∥d2𝒜(X,Y)(t)∥ for a,b∈I,a<b.
Proof.
By the definition of the matrix-function 𝒜(X,Y)(t), we have
dj𝒜(X,Y)(τ)=djY(τ)-(-1)jdjX(τ)⋅(In+(-1)jdjX(τ))-1djY(τ) for τ∈I(j=1,2).
So
dj𝒜(X,Y)(τ)=(In+(-1)jdjX(τ))-1djY(τ) for τ∈I(j=1,2).
From this and the definition of the matrix-function 𝒜(X,Y)(t) there immediately follows equality (2.18). On the other hand, (2.18) guarantees equality (2.19).
∎
3 Proofs of the results
Proof of Theorem 1.1.
First consider the case where supIt0>t0
and prove the existence and uniqueness of a solution of problem (1.1), (1.2) on the interval It0+={t∈It0:t>t0}.
First, note that by (1.10) estimate (2.4) holds and
(3.1)limt→t0+γ(t)=0,
where the function γ(t) is defined by (2.4) as in Lemma 2.2.
Moreover, we suppose that
t0+δ∈It0+.
Let tk∈]t0,t0+δ[ (k=1,2,…) be some decreasing sequence such that
(3.2)limk→+∞tk=t0.
According to [11, Theorem III.1.4], for every natural k, system (1.1) has a unique solution xk defined on the interval It0+ and
satisfying the condition
xk(tk)=0.
Moreover, due to Lemma 2.2 we have the estimates
(3.3)∥H-1(t)xk(t)∥≤ργ(t) for tk≤t≤t1(k=1,2,…),
where ρ=∥(In-B)-1∥γ(t1). In particular, from (3.3) it follows that
∥xk(t1)∥≤ρ0 (k=1,2,…),
where ρ0=ρ∥H(t1)∥γ(t1). So without loss of generality we can assume that the sequence xk(t1) (k=1,2,…) converges. Let
limk→+∞xk(t1)=c0.
By the theorems on the well-posedness of the Cauchy problem (see [2]), we conclude
limk→+∞xk(t)=x(t)
uniformly on every closed interval from It0+, where x is a solution of system (1.1) under the condition
x(t1)=c0.
On the other hand, from (3.2) and (3.3) it follows that
∥H-1(t)x(t)∥≤ργ(t) for t0<t≤t1.
From this, by (3.1) we conclude that the vector-function x is a solution of problem (1.1), (1.2) on the interval It0+.
Now, let us show that problem (1.1), (1.2) has a unique solution on the interval It0+. Let x¯ be an arbitrary solution of the
problem. Then the vector-function x0(t)=x(t)-x¯(t) is a solution of the homogeneous system
dx=dA(t)⋅x
under the condition
(3.4)limt→t0+(H-1(t)x0(t))=0.
In view of Lemma 2.2, we find
∥x0(t0+δ)∥≤ρ0∥H(t0+δ)∥⋅∥H-1(s0)x0(s0)∥ for t0<s0≤t0+δ,
where ρ0=∥(In-B)-1∥⋅∥B0∥. Passing to the limit as s0→t0 in the last estimate and taking into account (3.4), we get
x(t0+δ)=0.
Since A∈BVloc(It0+;ℝn×n) and t0+δ∈It0+, i.e., we have the regular case, by the above-mentioned
theorem from [11] we conclude that x0(t)≡0 and, therefore, x(t)=x¯(t) for t∈It0+.
Analogously, we show that problem (1.1), (1.2) has a unique solution y on the interval It0-={t∈It0:t<t0} as well if
infIt0<t0.
And if
infIt0<t0<supIt0,
it is evident that the vector-function
z(t)={y(t)if t∈It0-,x(t)if t∈It0+,
will be a unique solution of problem (1.1), (1.2) on the interval It0.
∎
Proof of Theorem 1.2.
Let us assume
A0(t)=diag(a011(t),…,a0nn(t)) for t∈It0(δ),
𝒜(A0,A-A0)(t)=(a~ik(t))i,k=1n for t∈It0.
By the definition of the operator 𝒜, we find
a~ik(t)-a~ik(s)=(aii(t)-a0ik(t))-(aii(s)-a0ik(s))
+∑s0<τ≤td1a0ii(τ)⋅(1-d1a0ii(τ))-1d1(aii(τ)-a0ik(τ))
-∑s0≤τ<td2a0ii(τ)⋅(1+d2a0ii(τ))-1d2(aii(τ)-a0ik(τ))
(3.5) for s<t,(s-t0)(t-t0)>0(i,k=1,…,n),
where a0ik(t)=0 if i≠k (i,k=1,…,n).
Consider the case where i=k (i=1,…,n). It is not difficult to verify that
aii(t)-a0ii(t)=-[aii(t)]-v for t<t0(i=1,…,n),
aii(t)-a0ii(t)=[aii(t)]+v for t>t0(i=1,…,n).
Therefore,
(3.6)dja0ii(t)=[djaii(t)]+ and dj(aii(t)-a0ii(t))=-[djaii(t)]- for t<t0(j=1,2,i=1,…,n),
(3.7)dja0ii(t)=-[djaii(t)]- and dj(aii(t)-a0ii(t))=[djaii(t)]+ for t>t0(j=1,2,i=1,…,n).
Hence, due to (3.6) and (3.7),
dja0ii(t)⋅dj(aii(t)-a0ii(t))=0 for t∈It0(δ)(j=1,2,i=1,…,n).
Thus from (3.5) we have
a~ii(t)=-[aii(t)]-v for t<t0(i=1,…,n),
a~ii(t)=[aii(t)]+v for t>t0(i=1,…,n).
On the other hand, it is evident that
a~ik(t)≡𝒜(a0ii,aik)(t) for i≠k(i,k=1,…,n).
The Cauchy matrix of system (1.4) has the form
C(t,τ)≡diag(c1(t,τ),…,cn(t,τ)), and so
CH(t,τ)≡(∫τtci(t,τ)hk(τ)𝑑a~i,k)i,k=1n,
where CH(t,τ) is the matrix-function defined by (1.9).
In addition, note that due to (1.11), (1.15), (3.6) and (3.7), conditions (1.5) and
ci(t,τ)>0 for (t-t0)(τ-t0)>0(i=1,…,n)
hold. By these results and (1.12)– (1.14), we conclude that
conditions (1.7), (1.8) and (1.10) of Theorem 1.1 are valid. Hence, the theorem immediately follows from Theorem 1.1.
∎
Proof of Corollary 1.4.
In view of (1.11), (1.15), (1.16), (3.6) and (3.7), we have
(3.8)0<ci(t,τ)≤|t-t0τ-t0|μi for (t-t0)(τ-t0)>0(i=1,…,n),
since
ci(t,τ)=exp(𝒥(a0ii)(t)-𝒥(a0ii)(τ)) (i=1,…,n).
Thus it is clear that the functions
(3.9)hi(t)=|t-ti|μi (i=1,…,n)
satisfy inequalities (1.12), where b0=1.
On the other hand, with regard to (3.8) and (3.9), from (1.21) and (1.22) there follow conditions (1.13), (1.22) and (1.14). Therefore, according to Theorem 1.2, problem (1.1), (1.23) has the unique
solution x=(xi)i=1n.
∎
Let us show that the statement of Remark 1.6 is valid.
Proof of Remark 1.6.
Let x=(xi)i=1n be the solution of problem (1.1), (1.18). Let y(t)=(yi(t))i=1n for t∈It0, where yi(t)=|t-t0|-μixi(t) (i=1,…,n). Then by condition (1.18) there exists y(t0±)=0. On defining the continuity y at the point t0, i.e., y(t0)=0, we conclude that the vector-function y is bounded on the
set I. Let a number r>0 be such that
(3.10)∥y(t)∥≤r for t∈I.
First consider the case where t>t0.
By condition (1.19), there exist positive numbers δ and r0 such that the functions ξ1 and ξ2 are bounded in the right δ neighborhood of the point t0, i.e.,
|ξji(t)|<r0 for t∈It0+(δ)(j=1,2,i=1,…,n).
Let
wi(t)=∫a2tsgnxi(s)dxi(s) for t>t0(i=1,…,n).
Due to the definition of solutions of system (1.1), using the substitution formula, we have
wi(t)-wi(τ)=∫τt|xi(s)|𝑑aii(s)+∑k=1,k≠in∫τtsgnxi(s)xk(s)𝑑aik(s)+fi(t)-fi(τ)
≤∫τt|yi(s)|d(∫τs|ζ-t0|μid[aii(ζ)]+v)+∑k=1,k≠in|∫τt|yk(s)|d(∫τs|ζ-t0|μi𝑑v(aik)(ζ))|+|fi(t)-fi(τ)|
(3.11)for t0+δ<τ<t(i=1,…,n),
because aii(s)≡[aii(s)]+v-[aii(s)]-v, and the function [aii(s)]-v is nondecreasing.
It is evident that there exist positive numbers r1 and δ such that
(3.12)|djaii(t)|<r1 for t>t0+δ(j=1,2,i=1,…,n).
Therefore, by Lemma 2.5 (see (2.19)) and (3.12), we get
v(aik)(ζ)-v(aik)(ξ)≤(1+r1)(V(𝒜(a0ii,aik)(ζ)-V(𝒜(a0ii,aik))(ξ)))
and
v(fi)(ζ)-v(fi)(ξ)≤(1+r1)(V(𝒜(a0ii,fi)(ζ)-V(𝒜(a0ii,fi)(ξ)))) for t0+δ<ξ<ζ(i,k=1,…,n).
Thanks to the above estimate, from (3.11) we conclude that
(3.13)wi(t)-wi(τ)≤vi(t)-vi(τ) for t0+δ<τ<t(i=1,…,n),
where
vi(t)=bii∫a2t|yi(s)|𝑑s+(1+r1)∑k=1,k≠inbik∫a2t|yk(s)|𝑑s+(1+r1)(V(𝒜(a0ii,fi)(t)-V(𝒜(a0ii,fi))(a2)))
for t>t0+δ(i=1,…,n).
Now consider the case where t∈It0+(δ).
By the definition of a solution of system (1.1), we find
|djxi(t)|≤∑k=1n|xk(t)||djaik(t)|+|djfi(t)|=∑k=1n|t-t0|μi|yk(t)||djaik(t)|+|djfi(t)|
for t∈It0+(δ)(j=1,2,i=1,…,n).
Hence, due to (3.10) and (3.13), we get
σji(t)≤(1+r)ξj(t)<(1+r)r0 for t∈It0+(δ)(j=1,2,i=1,…,n),
where
σji(t)=∑τ∈Itj|djxi(τ)| for t∈It0+(δ)(j=1,2,i=1,…,n).
So, there exist the nonnegative finite limits
(3.14)limt→t0+σji(t)=cji (j=1,2,i=1,…,n).
On the other hand, it is not difficult to verify the equality
wi(t)-wi(τ)=s0(|xi|)(t)-s0(|xi|)(τ)+∑j=12∫τtsgnxi(s)dsj(xi)(s) for τ,t∈It0+(i=1,…,n),
and thus to obtain
wi(t)-wi(τ)≤s0(|xi|)(t)-s0(|xi|)(τ)+∑j=12(σji(τ)-σji(t)) for τ<t,τ,t∈It0+(i=1,…,n),
and
(3.15)|wi(t)-wi(τ)|≤|s0(|xi|)(t)-s0(|xi|)(τ)|+∑j=12|σji(τ)-σji(t)| for τ,t∈It0+(i=1,…,n).
Moreover, by condition (3.14), the equalities
s0(|xi|)(t)-s0(|xi|)(τ)=|xi(t)|-|xi(τ)|-∑j=12(sj(|xi|)(t)-sj(|xi|)(τ))
for τ<t,τ,t∈It0+(i=1,…,n),limt→t0+xi(t)=0(i=1,…,n),
and the inequalities
|dj|xi|(t)|≤|djxi(t)| for t∈It0+(j=1,2,i=1,…,n),
we conclude that there exist the finite limits
(3.16)limt→t0+s0(|xi|)(t)=c0i (i=1,…,n).
Let now
vi(t)=s0(|xi|)(t)-∑j=12σji(t)) for t∈It0+(δ)(i=1,…,n).
Then by (3.14) and (3.16) there exist the finite limits
limt→t0+vi(t)=c0i-cj1-cj2 (i=1,…,n).
Consequently, by (3.15) there exist the finite one-sided limits wi(t0+) (i=1,…,n) as well. So the conditions of Lemma 2.4 are fulfilled on It0+. Analogously, we show the validity of conditions on the set It0 as well. The statement of the remark follows from the lemma.
∎
Proof of Corollary 1.7.
The proof of the corollary is analogous to that of Corollary 1.4. We only note that the estimate
0<ci(t,τ)≤1 for (t-t0)(τ-t0)>0(i=1,…,n)
is fulfilled instead of (3.8), and that conditions (1.21) and (1.22) guarantee the fulfillment of (1.16) and (1.17).
∎
The proof of the statement of Remark 1.8 is analogous to the proof of Remark 1.6 if we assume μi=1 (i=1,…,n).
Proof of Corollary 1.9.
First, consider the case where t>t0.
Since aii (i=1,…,n) are the functions with bounded local variations on I, we conclude that
(3.17)v(αi)(t)-v(αi)(τ)≤v(aii)(t)-v(aii)(τ) for t0≤τ<t(i=1,…,n),
where
αi(t)≡[aii(t)]+v (i=1,…,n).
Moreover, it is evident that
limt→t0+v(aii)(t)=v(aii)(t0+) (i=1,…,n).
From this, by (3.17) and (1.25) we conclude that there exists a small positive number δ such that
(3.18)b0(v(αi)(t)-v(αi)(t0+))≤12n2 for t∈It0+(δ)(i=1,…,n),
and
(3.19)b0|∫t0+t|τ-t0|λi-λkdV(𝒜(a0ii,aik))(τ)|≤12n2 for t∈It0+(δ)(i≠k,i,k=1,…,n),
where
b0=sup{exp(v(aii*)(t)-v(aii*)(t0+)):t∈It0+(δ)} (i=1,…,n).
On the other hand, taking into account (1.11) and (1.24), from (1.15) it follows that
(3.20)0<ci(t,τ)≤b0|t-t0|-λi|τ-t0|λi for t,τ∈It0+(δ),τ<t(i=1,…,n).
Therefore, estimate (1.12) is valid, where
(3.21)hi(t)=|t-t0|-λi (i=1,…,n).
Thanks to (3.20) and (3.21), we see that conditions (1.26), (3.18) and (3.19) guarantee the validity of conditions (1.13) and (1.14), where bik(t)=(2n2)-1 (i,k=1,…,n). Hence the matrix B=(bik)i,k=1n satisfies condition (1.6).
Analogously, we show the validity of the corresponding conditions of Theorem 1.2 for the case t<t0.
So all conditions of Theorem 1.2 hold. The solvability of problem (1.1), (1.2) immediately follows from Theorem 1.2.
∎
Remark 1.10 can be verified in the same way as the above considered remarks.
,
Inga Gabisonia
