On Cauchy Problem for First Order Nonlinear Functional Differential Equations of Non-Volterra's Type

Authors: Bravyi E.¹; Hakl R.²; Lomtatidze A.³

Source: Czechoslovak Mathematical Journal, Volume 52, Number 4, December 2002 , pp. 673-690(18)

Publisher: Springer

Abstract:

On the segment I = [a, b] consider the problem <IMG SRC="http://images.ingentaselect.com/absimages/klu/00114642/klu_cmaj_2002_52_4_426270h.1.gif" ALT="uˆ\prime(t)=f(u)(t), \quad u(a)=c," TEXT="a mathematical formula"> where <IMG SRC="http://images.ingentaselect.com/absimages/klu/00114642/klu_cmaj_2002_52_4_426270h.2.gif" ALT="f:\ C(I, {\Bbb R})\rightarrow L(I, {\Bbb R})" TEXT="a mathematical formula"> is a continuous, in general nonlinear operator satisfying Carathéodory condition, and <IMG SRC="http://images.ingentaselect.com/absimages/klu/00114642/klu_cmaj_2002_52_4_426270h.3.gif" ALT="c \in {\Bbb R}" TEXT="a mathematical formula">. The effective sufficient conditions guaranteeing the solvability and unique solvability of the considered problem are established. Examples verifying the optimality of obtained results are given, as well.

Keywords: nonlinear functional differential equation; initial value problem; non–Volterra's type operator

Document Type: Research article

DOI: http://dx.doi.org/10.1023/B:CMAJ.0000027223.33906.f6

Affiliations: 1: Department of Mathematical Analysis, Perm State Technical University, Komsomolsky str. 29a, 614000 Perm, Russia, Email: bravyi@pi.ccl.ru 2: Mathematical Institute, Academy of Sciences of the Czech Republic, ikova 22, 616 62 Brno, Czech Republic, Email: hakl@ipm.cz ikova 22, 616 62 Brno, Czech Republic, Email: hakl@ipm.cz "> 3: Mathematical Institute, Academy of Sciences of the Czech Republic, ikova 22, 616 62 Brno, Czech Republic, and Department of Mathematical Analysis, Masaryk University, Janákovo nám. 2a, 662 95 Brno, Czech Republic, Email: bacho@math.muni.czikova 22, 616 62 Brno, Czech Republic, and Department of Mathematical Analysis, Masaryk University, Janákovo nám. 2a, 662 95 Brno, Czech Republic, Email: bacho@math.muni.cz">

Publication date: 2002-12-01

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