Comparison theorem for a nonlinear boundary value problem on time scales
The author proves a comparison result for lower and upper solutions \(v(t)\) and \(w(t)\), respectively, to a two-point boundary value problem associated with the second-order dynamic equation \[ -y^{\Delta\Delta}=f(t,y^\sigma,(y^\Delta)^\sigma). \] In particular, sufficient conditions are given so that \(v(t)\leq w(t)\) throughout the considered interval. The proof uses time scales versions of the mean value theorem and the second \(\Delta\)-derivative extremum test. Related work on the mean value theorems appeared in [\textit{S. G. Topal}, J. Difference Equ. Appl. 8, No. 4, 333-343 (2002; Zbl 1006.39016)]. This paper will be useful for researchers interested in lower and upper solutions, and quasilinearization technique for the second-order boundary value problems.
scientific article; zbMATH DE number 1758472