Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0935.34061
Diblík, Josef
A criterion for existence of positive solutions of systems of retarded functional differential equations. (English)
[J] Nonlinear Anal., Theory Methods Appl. 38, No.3, A, 327-339 (1999). ISSN 0362-546X

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.
[R.R.Akhmerov (Novosibirsk)]

MSC 2000:: *34K12 Properties of solutions of functional-differential equations
34K05 General theory of functional-differential equations

Keywords: retarded functional-differential equation; positive solution; Wa\D{z}ewski principle

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]