The quasilinearization method for boundary value problems on time scales (Q1856985)

The authors consider second-order scalar dynamic equations \[ -y^{\Delta\nabla}(t)+q(t)y(t)=f(t,y(t))+g(t,y(t)),\quad t\in[a,b]\cap{\mathbb{T}},\tag{*} \] on a time scale \({\mathbb{T}}\) under the periodic boundary conditions \(y(\rho(a))=y(b)\), \(y^\Delta(\rho(a))=y^\Delta(b)\), as well as under separated boundary conditions \(y(\rho(a))=A\), \(y(\sigma(b))=B\). Here, \(\sigma,\rho\) denote the forward and backward jump operators, respectively, \(y^\Delta\) stands for the \(\Delta\)-derivative, \(y^\nabla\) for the \(\nabla\)-derivative of a real-valued mapping \(y\) and \(q(t)\geq 0\) is continuous. Under the assumption that so-called coupled lower and upper solutions \(\alpha_0,\beta_0\) to \((\ast)\) exist, a uniqueness result for the solutions to the periodic boundary value problem is given, in case \(g(t,\cdot)\) is strictly increasing and \(f(t,\cdot)\) satisfies a one-sided Lipschitz condition. Furthermore, if \(f\) and \(g\) satisfy a certain convexity and concavity condition, respectively, the authors show in their main result that there exist monotone sequences of coupled lower and upper solutions \((\alpha_n)\), \((\beta_n)\) converging uniformly to the unique solution to \((\ast)\) with periodic boundary conditions. Two corollaries simplify the requirements on the mappings \(f\) and \(g\). A third corollary states that the convergence of the sequences \((\alpha_n)\), \((\beta_n)\) is quadratic, and two examples illustrate the obtained results. Finally, without proof, analogous results are stated in the situation of separated boundary conditions.