TAILIEUCHUNG - Self-adjoint boundary value problems on time scales and symmetric Green’s functions
In this note, higher order self-adjoint differential expressions on time scales, and associated with them self-adjoint boundary conditions, are discussed. The symmetry peoperty of the corresponding Green’s functions is emphasized. | Turk J Math 29 (2005) , 365 – 380. ¨ ITAK ˙ c TUB Self-Adjoint Boundary Value Problems on Time Scales and Symmetric Green’s Functions Gusein Sh. Guseinov Abstract In this note, higher order self-adjoint differential expressions on time scales, and associated with them self-adjoint boundary conditions, are discussed. The symmetry peoperty of the corresponding Green’s functions is emphasized. Key Words: Time scales, self-adjoint differential expressions, self-adjoint boundary conditions. 1. Introduction In [3] self-adjoint boundary value problems (BVPs) for second order differential equations on time scales were introduced and examined by making use of both delta and nabla derivatives. Next some BVPs for higher order equations on time scales involving delta and nabla derivatives at the same time were investigated in [1, 2] where, however, the considered BVPs turned out, in general, to be nonselfadjoint because their Green’s functions were found nonsymmetric. Therefore it remained unclear as to how to place the successive delta and nabla derivatives for higher order to get self-adjoint differential expressions that can yield symmetric Green’s functions. In this paper we offer a solution to this problem indicating two classes of higher order differential equations on time scales. These classes of equations can be formulated as follows. Let T be a time scale, p0 , p1 , ., pn are real-valued right-dense continuous functions 2000 Mathematics Subject Classification: 34B05, 39A10 365 GUSEINOV n defined on T with p0 (t) 6= 0 for all t ∈ T, and a ∈ Tκ , b ∈ Tκn , with a 0 is a fixed real number), and the q−numbers (Kq = q Z ∪ {0} = {q k : k ∈ Z} ∪ {0}, where q > 1 is a fixed real number) are examples of time scales, as are [0, 1] ∪ [2, 3], [0, 1] ∪ N, and the Cantor set, where [0, 1] and [2, 3] are real number intervals. In [4, 8] Aulbach and Hilger introduced also dynamic equations (∆-differential equations) on time scales in order to unify and extend the theory
đang nạp các trang xem trước
