Periodic solutions of weakly nonlinear integrodifferential equations with impulses
type: Item
instance of: scholarly article
title: Periodic solutions of weakly nonlinear integrodifferential equations with impulses
zbMATH Open document ID: 0609.45007
DOI: 10.1007/BF01058478
author: B. E. Turbaev
published in: Ukrainian Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: The author investigates the equation \[ dx/dt=Ax+f(t,x,\int^{t}_{t- \eta}\phi (t,s,x(s))ds), \] where x,f,\(\phi\) are vectors (f is T-periodic in t, \(\phi\) is T-periodic in t and s) and A is a constant matrix, \(\eta >0\), under the condition that x experiences jumps at \(t=t_ i(x)\), i.e., \(\Delta x|_{t=t_ i(x)}=I_ i(x)\), \(i=0,\pm 1,\pm 2,...\), with \(t_{i+p}(x)=t_ i(x)+T\), \(I_{i+p}(x)=I_ i(x)\) for all i and p a fixed integer. Lipschitz and continuity properties on \(f,\phi,I_ i,t_ i\) are imposed. Under certain additional conditions the existence of T- periodic solutions is proved, first with \(t_ i(x)\equiv t_ i\), second, with \(t_ i(x)\) nonconstant. The proof is by iteration.
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45M05
zbMATH DE Number: 3985894
zbMATH Keywords: nonlinear Volterra equations
zbMATH Keywords: existence
zbMATH Keywords: periodic solutions
reviewed by: Stig-Olof Londen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3985894