Boundary value problems for higher order linear impulsive differential equations

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2003

Öğrenci: ÖMÜR UĞUR

Danışman: MARAT AKHMET

Özet:

The theory of impulsive di®erential equations has become an important area of research in recent years. Linear equations, meanwhile, are fundamental in most branches of applied mathematics, science, and technology. The theory of higher order linear impulsive equations, however, has not been studied as much as the cor- responding theory of ordinary di®erential equations. In this work, higher order linear impulsive equations at xed moments of impulses together with certain boundary conditions are investigated by making use of a Green's formula, constructed for piecewise di®erentiable functions. Existence and uniqueness of solutions of such boundary value problems are also addressed. Properties of Green's functions for higher order impulsive boundary value prob- lems are introduced, showing a striking di®erence when compared to classical bound- ary value problems of ordinary di®erential equations. Necessarily, instead of an or- dinary Green's function there corresponds a sequence of Green's functions due to impulses. Finally, as a by-product of boundary value problems, eigenvalue problems for higher order linear impulsive di®erential equations are studied. The conditions for the existence of eigenvalues of linear impulsive operators are presented. Basic properties of eigensolutions of self-adjoint operators are also investigated. In particular, a necessary and su±cient condition for the self-adjointness of Sturm-Liouville opera- tors is given. The corresponding integral equations for boundary value and eigenvalue problems are also demonstrated in the present work.