Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions
The paper deals with solutions of the nonlinear equation \[ u'(t)= F(t,u(t)), \qquad t\in I= [0,T], \quad T> 0, \tag{1} \] where \(F: [0,T]\times \mathbb{R}\to \mathbb{R}\) and \(g: \mathbb{R}^2\to \mathbb{R}\) are continuous functions. The goal of the authors is to present a new existence result for solutions of (1) satisfying \(g(u(0),u(T))= 0\). To this end, they introduce a new concept of coupled lower and upper solutions that allow them to obtain a pair of quasisolutions. Under suitable assumptions on \(F\) and \(g\), the authors show the existence of such solutions.
scientific article; zbMATH DE number 2082117