Abstract Generated abstract
This paper proves an existence and uniqueness theorem for a multipoint boundary-value problem for a nonlinear system of ordinary differential equations. The argument compares the nonlinear problem with an auxiliary linear system, using a nonsingular change of variables, a matricant, and the associated Green matrix to formulate integral estimates for deviations from a chosen reference function. Under bounds on the transformed Jacobian, a square-integrability condition with norm less than one, and containment of an error tube in the prescribed domain, the theorem guarantees a unique solution satisfying an explicit error estimate. The result also indicates how the estimate may be used to assess an approximate solution and how the construction can be extended beyond the original boundary interval.
Full Text
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
\[ \{J(t,y)\}_{ik}=f_{ik}(t,y),\qquad i,k=1,2,\ldots,n. \]
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 \(G^*\)
\[ \{a(t)\}_{ik}\leq \{Q(t,y)\}_{ik}\leq \{A(t)\}_{ik}, \]
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:
\[ G_P^*(t,\xi)= \begin{cases} G_P(t,\xi), & t_0\leq t\leq t_\mu,\quad t_0<\xi<t_\mu;\\ 0, & t_0\leq t\leq t_\mu,\quad t_\mu<\xi<T;\\ M_P(t,t_\mu)\cdot G_P(t_\mu,\xi), & t_0<\xi<t_\mu,\quad t_\mu\leq t\leq T;\\ M_P(t,\xi), & t_\mu<\xi<t\leq T;\\ 0, & t_\mu\leq t<\xi<T. \end{cases} \]
Define the function \(K^*(t,\xi)\) by the formula
\[ \|G_P^*(t,\xi)\cdot\{Q(\xi,y)-P(\xi)\}\|\leq K^*(t,\xi). \]
Then, under the condition
\[ \int_{t_0}^{T}\int_{t_0}^{T} K^{*2}(t,\xi)\,d\xi\,dt<1, \]
the theorem guarantees the existence of a solution of problem (1)—(2) for \(t\in[t_0,T]\).
Remark 3. Inequality (12) may be taken as a formula for estimating the error of an approximate solution of problem (1)—(2) on the interval \([t_0,t_\mu]\), if the vector \(Y(t)\) is taken as the approximate solution.
Received
27 VI 1958
CITED LITERATURE
- S. M. Lozinskii, Dokl. Akad. Nauk SSSR, 92, No. 2, 225 (1953).
- V. N. Faddeeva, Computational Methods of Linear Algebra, Moscow, 1950.
- F. R. Gantmakher, Theory of Matrices, Moscow, 1954.