The problem of existence of positive solutions in models of mathematical economics. (Ukrainian) Zbl 1054.91558

Mitropol’skyj, Yu. O.; Bugir, M. K.; Khoma, G. P.

Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky 1999, No. 6, 35-38 (1999).

The authors discuss the possibility of modelling some economical processes by means of 2nd order difference and differential systems. From this point of view the problem of existence of positive solutions for the equations \(y(k+2)=Ay(k)+a(k),\;k=0,1,2,\ldots,\) and \(y''=Hy+f(t)\;(1)\) is studied. Here \(A=\{a_{ij}\}_{i,j=1}^n\) and \(H=\{h_{ij}\}_{i,j=1}^n\) are constant matrices, \(a(\cdot):{\mathbf Z}_+ \mapsto {\mathbb R}_+^n\), \(f:{\mathbb R} \mapsto {\mathbb R}_+^n\). The authors arrives to the following conclusion: in order that solutions of (1) be nonnegative for arbitrary nonnegative initial data and arbitrary nonnegative \(f(t)\) it is necessary that \(h_{ij}\geq 0, i\neq j,\; i,j=1,\ldots,n\), and it is sufficient that \(h_{ij}\geq0,\;i,j=1,\ldots,n\).

Reviewer: I. O. Parasyuk (Kyïv)

MSC:

91B62	Economic growth models
34A30	Linear ordinary differential equations and systems, general
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations

Keywords:

mathematical economics; positive solution; Perron-Frobenius theorem