A. P. PLEKHOTIN

AN EXISTENCE AND UNIQUENESS THEOREM FOR THE SOLUTION OF A BOUNDARY-VALUE PROBLEM FOR A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS

(Presented by Academician V. I. Smirnov, 10 VII 1958)

Let the following be given:

1) A system of differential equations

[
y'=f(t,y),
\tag{1}
]

where (y(t)) is the unknown (n)-dimensional vector, and (f(t,y)) is a vector-function.

With respect to the functions (f_i,\ i=1,2,\ldots,n,) it is assumed that they take real values, are continuous, and have continuous partial derivatives

[
f_{ik}(t,y)=\frac{\partial f_i(t,y_1,\ldots,y_n)}{\partial y_k},\qquad
i,k=1,\ldots,n,
]

in a domain (G) of the real ((n+1))-dimensional Euclidean space of the variables (t,y_1,y_2,\ldots,y_n).

2) Boundary conditions

[
\sum_{m=0}^{\mu}\alpha_m y(t_m)=b,\qquad
t_0\leq t_1\leq\cdots\leq t_\mu,
\tag{2}
]

where (\alpha_m,\ m=0,1,\ldots,\mu,) are given constant matrices of order (n); (b) is a given constant vector.

3) The vector (Y(t)) has a continuous derivative (Y'(t)) for (t\in[t_0,t_\mu]), and for (t\in[t_0,t_\mu]) the point ((t,Y(t))\in G^), where (G^\subset G) and is convex in all (y_i,\ i=1,2,\ldots,n),

[
\sum_{m=0}^{\mu}\alpha_mY(t_m)=B;
\tag{3}
]

hence it also follows that the interval ([t_0,t_\mu]) is contained in the projection of the domain (G) onto the (t)-axis.

4) The matrix (U(t)) belongs to the class (C') for (t\in[t_0,t_\mu]) and is nonsingular for every (t\in[t_0,t_\mu]) (cf. ((1))).

Put:

1) (J(t,y)) is the Jacobi matrix defined by the formulas

[
{J(t,y)}{ik}=f(t,y),\qquad i,k=1,2,\ldots,n.
]

2) (Q(t,y)) is the matrix defined by the formula

[
Q(t,y)=U^{-1}(t)\cdot J(t,y)\cdot U(t)-U^{-1}(t)\cdot\frac{d}{dt}U(t).
\tag{4}
]

3) (P(t)) is a square matrix of order (n), continuous for (t\in[t_0,t_\mu]), and (M_P(t,t_0)) is its matricant (\left({}^{3}\right))

[
M_P(t,t_0)=E+\int_{t_0}^{t}P(u)\,du+\int_{t_0}^{t}P(u)\int_{t_0}^{u}P(u_1)\,du_1\,du+\cdots,
]

where (E) is the identity matrix of order (n).

4) (D_P) is the matrix defined by the formula

[
D_P=\sum_{m=0}^{\mu}\alpha_m\cdot U(t_m)\cdot M_P(t_m,t_0).
]

5) If (\det D_P\ne 0), then the matrix

[
G_P(t,\xi)=\frac{1}{2}\,M_P(t,t_0)\cdot D_P^{-1}
\left{
\sum_{m=0}^{\mu}
\left[\operatorname{sign}(t-\xi)-\operatorname{sign}(t_m-\xi)\right]\alpha_m\cdot U(t_m)\cdot M_P(t_m,t_0)
\right}
\cdot M_P^{-1}(\xi,t_0),
]

(\xi\ne t) and (\xi\ne t_m,\ m=0,1,\ldots,\mu), will be called the Green matrix corresponding to the boundary-value problem

[
z'=P(t)\cdot z+\varphi(t);
\tag{5}
]

[
\sum_{m=0}^{\mu}\alpha_m\cdot U(t_m)\cdot z(t_m)=0;
\tag{6}
]

the solution of problem (5)—(6) has the form

[
z(t)=\int_{t_0}^{t_\mu}G_P(t,\xi)\cdot\varphi(\xi)\,d\xi.
]

6) (\tau(t)) is the residual vector defined by the formula

[
\tau(t)=Y'(t)-f[t,Y(t)].
]

Then the following theorem holds:

Theorem. Let:

a) (\det D_p\ne 0);

b) in the domain (G^*)

[
\left|G_P(t,\xi){Q(\xi,y)-P(\xi)}\right|\le K(t,\xi),
\tag{7}
]

where (K(t,\xi)) is a real, bounded, nonnegative function, continuous or having discontinuities in the square (t_0\le t,\xi\le t_\mu) on the same straight lines as (G_P(t,\xi)); here and below the norm is understood in the sense of the first norm of a matrix and a vector ((^2));

c)

[
\int_{t_0}^{t_\mu}\int_{t_0}^{t_\mu} K^2(t,\xi)\,d\xi\,dt<1;
\tag{8}
]

d) (u(t)) is the solution of the integral equation

[
u(t)=\int_{t_0}^{t_\mu}K(t,\xi)\cdot u(\xi)\,d\xi+|\varepsilon_0(t)|,
\tag{9}
]

where (\varepsilon_0(t)) is the solution of the boundary-value problem

[
\varepsilon_0'=P(t)\cdot\varepsilon_0-U^{-1}(t)\cdot\tau(t),
]

[
\sum_{m=0}^{\mu}\alpha_m\cdot U(t_m)\cdot\varepsilon_0(t_m)=b-B
\tag{10}
]

or, whence

[
\varepsilon_0(t)=-\int_{t_0}^{t_\mu} G_P(t,\xi)\cdot U^{-1}(\xi)\cdot \tau(\xi)\,d\xi
+M_P(t,t_0)\cdot D_P^{-1}\cdot(b-B);
\tag{11}
]

d) the domain determined by the inequalities

[
t_0 \leq t \leq t_\mu,\qquad |Y(t)-y|\leq |U(t)|\cdot u(t),
]

lies in (G^*).

Then on the interval ([t_0,t_\mu]) there exists a unique solution (y(t)) of system (1) with boundary conditions (2) such that

[
|Y(t)-y(t)|\leq |U(t)|\cdot u(t),\qquad t_0\leq t\leq t_\mu.
\tag{12}
]

Remark 1. The conditions of the theorem include four parameters: the matrices (U(t)), (P(t)), the vector (Y(t)), and the domain (G^). It is advantageous to choose the matrix (U(t)) from the condition that the matrix (Q(t,y)) change its value as little as possible in the section of the domain (G^) by the plane (t=\mathrm{const}), (t_0\leq t\leq t_\mu). It is advantageous to determine the matrix (P(t)) from the condition that the left-hand side of (7) be as small as possible.

If in the domain (G^*)

[
{a(t)}{ik}\leq {Q(t,y)},}\leq {A(t)}_{ik
]

then one usually sets

[
P(t)=\frac12[A(t)+a(t)].
]

The conditions of the theorem are such that, if problem (1)—(2) has a solution in (G), then there can always be found such (U(t)), (P(t)), (Y(t)), and (G^*) that all the conditions will be satisfied.

Remark 2. Let the interval ([t_0,T]), where (t_\mu\leq T), be contained in the projection of the domain (G^*) onto the (t)-axis.

Define the matrix (G_P^*(t,\xi)) as follows:

[
G_P^*(t,\xi)=
\begin{cases}
G_P(t,\xi), & t_0\leq t\leq t_\mu,\quad t_0<\xi