A periodicity result for a nonlinear function integral equation
type: Item
instance of: scholarly article
title: A periodicity result for a nonlinear function integral equation
zbMATH Open document ID: 0754.45007
DOI: 10.1007/BF00160530
author: Abdessamad Mortabit
author: Ovide Arino
published in: Journal of Mathematical Biology
publication date: 1992-09-27T00:00:00Z
review text: This paper is a continuation of the considerations by the first author and \textit{M. Kimmel} [ibid. 27, No. 3, 341-354 (1989; Zbl 0715.92011)]. The authors study the nonlinear functional integral equation \[ n(t,x)=2\lambda\sigma[N(t-1)]\int^{a_ 2}_{a_ 1}f(x,\varphi(x))n(t-\psi(z),z)dz, \] where \(N(t)=\int^{a_ 2}_{a_ 1}\int^ t_{t-\psi(u)}n(v,u)dv du\), for \(t\geq 0\) and \(x\in(a_ 1,a_ 2)\). Here, the functions \(f(\cdot,\cdot),\varphi(\cdot),\psi(\cdot)\) and \(\sigma(\cdot)\) have to satisfy suitable conditions. The existence and uniqueness of a nonnegative solution \(n\) is proved. The asymptotic behavior of the solution is studied. For \(\lambda\) sufficiently large slow oscillations take place. The existence of a nontrivial periodic solution is proved, too.
reviewed by: Walter Petry
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45M05
Mathematics Subject Classification ID: 45M15
Mathematics Subject Classification ID: 45M20
Mathematics Subject Classification ID: 92C99
zbMATH DE Number: 62361
zbMATH Keywords: nonlinear integral equation with delay
zbMATH Keywords: ejective fixed point
zbMATH Keywords: cell proliferation
zbMATH Keywords: nonlinear functional integral equation
zbMATH Keywords: nonnegative solution
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: slow oscillations
zbMATH Keywords: periodic solution
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 62361