Periodic solutions and stability of differential equations with piecewise constant argument of generalized type

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Tezin Türü: Doktora

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

Tezin Onay Tarihi: 2009

Öğrenci: CEMİL BÜYÜKADALI

Danışman: MARAT AKHMET

Özet:

In this thesis, we study periodic solutions and stability of differential equations with piecewise constant argument of generalized type. These equations can be divided into three main classes: differential equations with retarded, alternately advanced-retarded, and state-dependent piecewise constant argument of generalized type. First, using the method of small parameter due to Poincaré, the existence and stability of periodic solutions of quasilinear differential equations with retarded piecewise constant argument of generalized type in noncritical case, that is, the unperturbed linear ordinary differential equation has not any nontrivial periodic solution, are investigated. The continuous and differential dependence of the solutions on an initial value and a parameter is considered. A new Gronwall-Bellmann type lemma is proved. Next, quasilinear differential equations with alternately advanced-retarded piecewise constant argument of generalized type is addressed. The critical case, when associated linear homogeneous system admits nontrivial periodic solutions, is considered. Using the technique of Poincaré-Malkin, criteria of existence of periodic solutions of such equations are obtained. One of the main auxiliary results is an analogue of Gronwall-Bellmann Lemma for functions with alternately advanced-retarded piecewise constant argument. Dependence of solutions on an initial value and a parameter is investigated. Finally, a new class of differential equations with state-dependent piecewise constant argument is introduced. It is an extension of systems with piecewise constant argument. Fundamental theoretical results for the equations: existence and uniqueness of solutions, the existence of the periodic solutions, the stability of the zero solution are obtained. Appropriate examples are constructed.