Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients

Abstract

We consider the following nonlinear impulsive delay differential equation with variable coefficients and forcing term: (∗) x′(t)+a(t)x(t)+F(t,x(·))=e(t), t⩾0, t≠t k, x(t k +)−x(t k)=I k(x(t k)), k∈N. Using new methods different from those in Tang et al. (Appl. Math. Comput. 131 (2002) 373), Nieto (J. Comput. Appl. Math. Lett. 15 (2002) 489), Franco and Nieto (J. Comput. Appl. Math. 88 (1998) 144), Nieto (J. Math. Anal. Appl. 205 (1997) 423), He and Yu (J. Math. Anal. Appl. 272 (2002) 67), we establish stability theorems for (∗) when e( t)≡0 under the assumption that there is { b k } such that b k x 2⩽ x( x+ I k ( x))⩽ x 2 for k∈ N and x∈ R (Theorems 2.1–2.4), the existence results for three positive periodic solutions and nonexistence results for periodic solution of (∗) when e( t)≡0 under the assumption exp ∫ 0 T a(u) du ≠∏ 0⩽t k<T (1+b k) and b k >−1 for all k (Theorems 3.1–3.2) and the existence results for periodic solution of (∗) at resonance, which is caused by impulses i.e. exp ( ∫ 0 T a(u) du)=∏ 0⩽t k<T (1+b k) , under the assumption b k >−1 for all k (Theorems 4.2–4.3). Some results obtained improve and generalize the known theorems and some other results are new. Examples are presented to illustrate the main results.