A periodicity result for a nonlinear function integral equation
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.
scientific article; zbMATH DE number 62361