Existence of positive solutions for second-order ODEs with reverse Carathéodory functions
Given real intervals \([x_0,L] \) and \([u_0,R] \), the nonnegative function \[ f:[x_0,L] \times [u_0,R] \to \mathbb{R} \] is defined as a reverse Carathéodory function if it satisfies: (A) for all \(x\in [x_0,L] \), \(f(x,.)\) is measurable; (B) almost everywhere (a.e.) for \(u\in [u_0,R] \), \(f(.,u) \) is continuous; (C) there exists \(m\in L^1(u_0,R)\) such that, for all \(x\in [x_0,L] \) and a.e. \(u\in [u_0,R]\), \(0<m(u)\leq f(x,u)\). With such an \(f\), the existence of a solution is established for the second-order initial value problem \[ \frac {d}{dx}(l(x)k(x,u(x))u'(x))=f(x,u(x))u'(x),\text{ a.e. for }x\in I:= [0,L], \tag{1} \] \[ u(0)=0, \quad l(0)k(0,u(0))u'(0)=0. \] This result is obtained by defining a convenient operator in a closed convex subset of the space of continuous functions on \(I\), equipped with the maximum norm, and showing that the operator satisfies the hypotheses in Schauder's fixed-point theorem. The fixed-point is then a solution to (1). Here, necessary and sufficient conditions are given that ensure existence and uniqueness of a solution for a simplified version of (1), where both \(k\) and \(f\) only depend on \(u\).
scientific article; zbMATH DE number 1929511