Irena Rachunkova and Milan Tvrdy

Irena Rachunkova and Milan Tvrdy

Periodic Boundary Value Problems for

Nonlinear Second Order Differential Equations with Impulses - Part I

The paper deals with the nonlinear impulsive periodic boundary value problem

(1.1) u''=f(t,u,u'),

(1.2) u(t_i+)=J_i(u(t_i)), u'(t_i+)=M_i(u'(t_i)), i=1,2,..., m,

(1.3) u(0)=u(T), u'(0)=u'(T).

We establish the existence results which rely on the presence of a~well ordered pair (σ₁, σ₂) of lower/upper functions (σ₁ ≤ σ₂ on [0,T]) associated with the problem. In contrast to previous papers investigating such problems, the monotonicity of the impulse functions J_i, M_i is not required here.

Mathematics Subject Classification 2000. 34B37, 34B15, 34C25

Keywords. Second order nonlinear ordinary differential equation with impulses, periodic solutions, lower and upper functions, Leray-Schauder topological degree, a priori estimates.