On the existence and the stability of the solutions of certain nonlinear equations in topological modules of mappings (Q1337849)

This article presents some solvability and continuous dependence results for the equations \[ K(t,x(t), y(t))=0 \] with \(K(t,x,y): \mathbb{R}\times \mathbb{E}\times \mathbb{F}\to \mathbb{G}\) and \[ \int_ t^{\Phi(t)} f(\tau, x(\tau))d\tau= y(t) \] with \(f: \mathbb{R}_ +\times \mathbb{E}\to \mathbb{F}\) with respect to the unknown function \(x(t)\) and the parameter-function \(y(t)\), \(\mathbb{E}\),\(\mathbb{F}\), \(\mathbb{G}\) are Banach spaces; both equations are considered as ones in suitable topological modules of continuous functions from \(\mathbb{R}^ n\) into \(\mathbb{E}\) over the ring of continuous functions defined on \(\mathbb{R}^ n\).