Nonlinear Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces (Q386811)

The paper establishes the existence of solutions of the nonlinear (dynamic) Sturm-Liouville boundary value problem with mixed derivatives on \([0, \infty)\) with a real-valued nonlinear term. The mixed derivatives used are the \(\Delta-\) and \(\nabla-\) derivatives. After a brief review of time scale calculus, the authors present a result on the measure of noncompactness for bounded subsets \(A\) of the Banach space \(C_1^{\Delta\nabla}(\mathbb{T}, E)\) where \(\mathbb{T}\) is a timescale and \(E\) is a Banach space. The main existence result is obtained by using Mönch's fixed point theorem. As a special case, the results obtained are valid for Sturm-Liouville \(q-\) difference equations.