Functional-analytic existence results for an integral equation in geophysics (Q2313403)

Motivated by the mathematical modelling of arctic gyre flows with continuous oceanic vorticity \(F:\mathbb{R}\to\mathbb{R}\), the author studies the differential equation \[u''(t)= \frac{F(u(t))}{\cosh^2(t)}- \frac{2\omega\sinh(t)}{\cosh^3(t)}\quad (t>t_0),\] subject to the boundary condition $u'(t_0) = 0$ and the asymptotic conditions \[\lim_{t\to\infty} u(t)= \psi_0,\qquad\lim_{t\to\infty} u'(t)\cosh(t)= 0.\] This problem may in turn be transformed into the integral equation \[u(t)= \psi_0+ \int^\infty_t (s-t)\frac{F(u(s))}{\cosh^2(s)}\, ds-\cosh^2(t_0)[1- \tanh(t)]\int^\infty_{t_0} \frac{F(u(s))}{\cosh^2(s)}\,ds\] which the author solves in two cases. First, if \(F\) is Lipschitz continuous, one may prove existence and uniqueness of a continuous solution \(u:[t_0,\infty)\to\mathbb{R}\) satisfying \(u(t)\to\psi_0\) as \(t\to\infty\) by means of Banach's fixed point theorem. Second, if \(F\) is merely continuous, but satisfies a suitable growth condition, one may prove existence of a bounded continuous solution \(u: [t_0,\infty)\to \mathbb{R}\) satisfying the same asymptotic condition, by means of Schauder's fixed point theorem.