chapter  11
Chapter 11
Systems of Impulsive Differential Equations on Half-lines
Pages 12

In this chapter we study the existences, uniqueness, continuous dependence on initial conditions, and boundedness of solutions for a system of impulsive diﬀerential equations using a ﬁxed point approach in vector Banach spaces. In addition, the compactness of the solution space and the u.s.c. of solutions are investigated. More precisely we consider the system of impulsive diﬀerential equations

x′(t) = f(t, x, y), t ∈ J := [0,∞), t 6= tk, k = 1, . . . , y′(t) = g(t, x, y), t ∈ J, t 6= tk, k = 1, . . . ,

x(t+k )− x(t−k ) = Ik(x(tk), y(tk)), k = 1, . . . , y(t+k )− y(t−k ) = Ik(x(tk), y(tk)), k = 1, . . . ,

x(0) = x0, y(0) = y0,

(11.1)

where x0, y0 ∈ R, f , g : J × R × R → R are given functions, and Ik, Ik ∈ C(R × R,R). The notations x(t+k ) = lim

h→0+ x(tk + h) and x(t−k ) = lim

h→0+ x(tk − h) stand for the right and

left hand limits of the function y at t = tk, respectively. For all the results in this chapter see [49].