Topological transversality: Application to an integrodifferential equation
Consider the following integrodifferential equation \[ x'(t) = F \left( t,x(t), \int^t_0 k \bigl[ t,s,x(s) \bigr] ds \right), \quad x(0) = 0, \tag{1} \] where \(k : \mathbb{R}^2_+ \times \mathbb{R} \to \mathbb{R}\), \(F : \mathbb{R}_+ \times \mathbb{R}^2 \to \mathbb{R}\) are continuous functions. The author studies the global existence of the solution of equation (1) under some suitable conditions on the functions \(k\) and \(F\) involved in (1). The main tools employed in the analysis are based on the applications of topological transversality arguments and an integral inequality recently established by the reviewer [J. Math. Anal. Appl. 189, No. 1, 128-144 (1995; Zbl 0824.26010)]. An application to water percolation phenomena is also given.
scientific article; zbMATH DE number 902477