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A criterion for existence of positive solutions of systems of retarded functional differential equations. (English) Zbl 0935.34061
The author considers the following functional-differential equation of retarded type $$ y'(t)=f(t,y_t),\quad t\ge t^*,\tag 1$$ with $f: {\bbfR}\times C([-r,0],{\bbfR}^n)\to{\bbfR}^n$, $y_t\in C([-r,0],{\bbfR}^n)$: $y_t(s)=y(t+s)$ ($s\in[-r,0]$), $f(t,0)\equiv 0$. The main result is as follows. Let $p\in\{1,\ldots,n\}$ be a fixed number. If the $i$th coordinate $f_i(t,\cdot)$ of $f$ is decreasing for $i=1,\ldots,p$ and increasing for $i=p+1,\ldots,n$, then equation (1) has a positive solution iff there exist strictly positive functions $\lambda_i\in C[t^*-r,\infty)$ satisfying for a positive vector $k\in{\bbfR}^n$ the inequalities $$ \lambda_i(t)\ge{\mu_i\over k_i} \exp\biggl(-\mu_i\int_{t^*-r}^{t}\lambda_i(s) ds\biggr) f_i\biggl(t,\biggl[k\exp\biggl(\mu\int_{t^*-r}^{t}\lambda_i(s) ds \biggr)\biggr]\biggr), $$ with $\mu_i=-1$ for $i=1,\ldots,p$ and $\mu_i=1$ for $i=p+1,\ldots,n$. As consequences the author obtains some sufficient condition of existence for positive solutions to difference-differential equations.

34K12Growth, boundedness, comparison of solutions of functional-differential equations
34K05General theory of functional-differential equations
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