Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 1245.34013
Dhage, Bapurao C.
Theoretical approximation methods for hybrid differential equations.
(English)
[J] Dyn. Syst. Appl. 20, No. 4, 455-477 (2011). ISSN 1056-2176

Consider the initial value problem $$\gathered{d\over dt} \Biggl[{x(t)\over f(t,x(t))}\Biggr]= g(t,x(t))\quad\text{for a.e. }t\in (t_0, t_0+ a),\quad a> 0,\\ x(t_0)= x_0\in\bbfR.\endgathered\tag{*}$$ The author defines lower and upper solutions to $(*)$ and shows that under some conditions on $f$ and $g$ the existence of lower and upper solutions implies the existence of a solution to $(*)$. His main interest concerns the construction of monotone sequences converging to extremal solutions of $(*)$. In particular, he considers the case $$g(t,x)= g_1(t,x)+ g_2(t,x),$$ where $g_1$ is nonincreasing in $x$ and $g_2$ is nondecreasing in $x$. Using the notation of mixed lower and upper solutions, he constructs sequences which monotoneously converge to extremal mixed solutions.
[Klaus R. Schneider (Berlin)]
MSC 2000:
*34A45 Theoretical approximation of solutions of ODE
34A12 Initial value problems for ODE

Keywords: mixed lower and upper solutions

Highlights
Master Server