an:01408739
Zbl 1054.91558
Mitropol'skyj, Yu. O.; Bugir, M. K.; Khoma, G. P.
The problem of existence of positive solutions in models of mathematical economics
UK
Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky 1999, No. 6, 35-38 (1999).
00063849
1999
j
91B62 34A30 34A12
mathematical economics; positive solution; Perron-Frobenius theorem
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 \(h_{ij}\geq0,\;i,j=1,\ldots,n\).
I. O. Parasyuk (Ky??v)