On the nonlocal boundary value problem of geophysical fluid flows (Q2026454)

This paper considers a new mathematical model of ACC with nonlocal boundary conditions of the form \[ \left\{ \begin{array}{l} u''(t)=a(t)F(t,u(t))+b(t),\quad t_1<t<t_2,\\ u(t_1)=\sum_{i=1}^{m-2}\alpha_iu(\xi_i),\\ u(t_2)=\sum_{i=1}^{m-2}\beta_iu(\xi_i), \end{array} \right. \] where \(a(\cdot), b(\cdot):[t_1,t_2]\rightarrow \mathbb{R}\) are continuous, \(F(\cdot,\cdot):[t_1,t_2]\times\mathbb{R}\rightarrow \mathbb{R}\) is continuous, \[ a(t)=\frac{e^t}{(1+e^t)^2},\quad b(t)=\frac{2\omega e^t(1-e^t)}{(1+e^t)^3}, \] and \(\xi_i (i=1,2,\ldots,m-2)\) satisfies \(t_1<\xi_1<\xi_2<\cdots<\xi_{m-2}<t_2\), \(\alpha_i\) and \(\beta_i\) satisfy \(\sum_{i=1}^{m-2}\alpha_i=\sum_{i=1}^{m-2}\beta_i=1\). Using topological degree, zero exponent theory and fixed point technique, the authors established the existence of positive solutions to the above mentioned nonlocal boundary value problems with nonlinear vorticity.