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Approximating crossed symmetric solutions of nonlinear dynamic equations via quasilinearization. (English) Zbl 1046.34036

Here, second-order forward dynamic equations \[ u^{\Delta\Delta}=f(\sigma(t),u^\sigma) \] and their companion backward problems \[ u^{\nabla\nabla}=f(\rho(t),u^\rho) \] under the boundary condition \(u(a)=u(b)=0\) are studied. The equations are defined on compact time scales (i.e., compact subset of the reals) with a certain symmetry property, \(u^\Delta\) resp. \(u^\nabla\) denote \(\Delta\)- resp. \(\nabla\)-derivative of \(u\), and \(\sigma,\rho\) are the jump operators.
The primary purpose of the authors is to study the upper and lower solutions of such nonlinear companion dynamic equations that produce crossed symmetric solutions on time scales. Upper and lower solutions for complementary pairs of forward and backward dynamic boundary value problems are introduced, a quasilinearization procedure for approximating the companion dynamic problems associated with the \(\Delta\)- and \(\nabla\)-derivatives is established and qualitative results are given. Finally, several numerical experiments close their discussion.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
39A10 Additive difference equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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