The link between integral equations and higher order ODEs. (Q1426072)

This paper is concerned with the link between integral equations and higher-order ordinary differential equations (ODEs). More precisely, the author shows that any solution of the following fourth-order ordinary differential equation \[ u''''+q_{1}u''+h(u)=0, \] where \(q_{1}\) is a real constant and \(h\) is a real-valued function, which satisfies reasonable growth conditions is a solution of the integral equation \[ u=\int_{-\infty}^{+\infty}v(x-y)g(u(y))\,dy. \] Specific examples from neuroscience are presented.