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

We consider the following nonlinear impulsive delay differential equation with variable coefficients and forcing term: equation(∗) View the MathML source 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 {bk} such that bkx2⩽x(x+Ik(x))⩽x2 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 View the MathML source and bk>−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. View the MathML source, under the assumption bk>−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.