Preface
This section presents some results about existence and uniqueness of initial value problems for systems of differential equations as well as for single higher order differential equations because the latter can be transfered to the former.
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Introduction to Linear Algebra with Mathematica
Glossary
Existence and Uniqueness
Our main interest in this section is the initial value problem for the system of differential equations:
Theorem: : Suppose f is a vector-valued function of n+1 variables (t, y_{1}, ... , y_{n}) defined for (t,y) on a set Ω of the form
Theorem: : Suppose f is a continuous vector-valued function of n+1 variables (t, y_{1}, ... , y_{n}) defined for (t,y) on a set Ω of the form
If, in addition, f satisfies a Lipschitz condition and bounded \( \| {\bf f}(t, {\bf y}) \| \le M \) on Ω, then the initial avlue problem has a unique solution on the interval \( |t - t_0 | \le \min \left\{ a, b/M \right\} . \) ▣
The idea to proof this theorem is to apply a fixed point theorem. The main obstakle in its application is the derivative operator, which is unbounded one. To get rid of it, we apply the inverse operator \( \texttt{D}^{-1} f(t) = f(t_0 ) + \int_{t_0}^t f(s)\,{\text d}s \) to both sides of the differential equation. Since the inverse operator \( \texttt{D}^{-1} \) is a bounded operator expressed through a definite integral, we can reduce the given initial value problem to the integral equation. Therefore, the first step in establishing this statement is to rewrite the given initial value problem in equivalent form as Volterra integral equation:
The equivalence follows from the Fundamental Theorem of Calculus. It suffices to find a continuous function \( {\bf x}(t) \) that satisfies the integral equation within the interval \( t_0 -h < t < t_0 +h , \) for some small value \( h \) since the right hand-side (integral) will be continuously differentiable in \( t . \)
The proof of existence and uniqueness theorem for the above initial value problem is based on technique known as Picard iteration to construct the required solution:
Theorem: : Suppose \( a_0 (x), a_1 (x) , \ldots , a_{n-1} (x) \) and f(x) are continuous real-valued functions on an interval \( (a,b) \ni x_0 . \) Then for any choice of (initial conditions) \( y_0 (x), y_1 (x) , \ldots , y_{n-1} (x) , \) there exists a unique solution on the whole interval (a,b) to the initial value problem
- Agarwal, Ravi P. and Lakshmikantham, V. Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Scientific. 1993, ISBN 981-02-1357-3; https://doi.org/10.1142/9789814354486_0003
- The Existence and Uniqueness Theorem for Linear Systems MIT.
- Bernfeld, S., Driver, R.D., and Lakshmikantham, V., Uniqueness for ordinary differential equations, Theory of Computing Systems, 1975, Vol. 9, Issue 4, pp. 359--367.
- Bobisud, L.E. and O'Regan, D.O., Existence of solutions to some singular initial value problems, Journal of Mathematical Analysis and Applications, 1988, Vol. 133, Issue 1, pp. 214--230.
- Bownds, J.M. and Metcalf, F.T., An extension of the Nagumo uniqueness theorem, Proceedings of the American Mathematical Society, 1971, Vol. 27, No 2, pp. 313--316.
- Dhar, A., Nonuniqueness in the solutions of Newton's equation of motion, American Journal of Physics, 1993, Vol. 61, No. 1, pp. 58--61; https://doi.org/10.1119/1.17411
- Ramankutty, P., Kamke's uniqueness theorem, Journal of the London Mathematical Society, 1980, Vol. 22, No. 2, pp. 110--116.
- Samimi, M. and Lakshmikantham, V., General uniqueness criteria for ordinary differential equations, Applied Mathematics and Computations, 1983, Vol. 12, Issue 1, pp. 77--88; https://doi.org/10.1016/0096-3003(83)90043-7
- Zubelevich, O.E., Ordinary differential equations with a non-Lipschitz right-hand side, Doklady Mathematics, 2012, Vol. 86, No. 1, pp. 545--548; doi: 10.1134/S1064562412040382
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