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Zbl 1046.34036
Eloe, P. W.; Sheng, Q.
Approximating crossed symmetric solutions of nonlinear dynamic equations via quasilinearization. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 56, No. 2, A, 253-272 (2004). ISSN 0362-546X

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. \par 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.
[Christian Poetzsche (Minneapolis)]

MSC 2000:: *34B15 Nonlinear boundary value problems of ODE
39A10 Difference equations
65M06 Finite difference methods (IVP of PDE)
34B10 Multipoint boundary value problems

Keywords: quasilinearization; upper and lower solutions; crossed symmetry; dynamic equations on time scales; $\Delta$ and $\nabla$ derivatives

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Zentralblatt MATH Berlin [Germany]