An application of Schauder's fixed point theorem to the existence of solutions of impulsive differential equations (Q1892323)

The authors investigate the two-point boundary value problem for second- order differential equation with one impulsive action (P) \(x'' + h(t,x,x')x = f(t,x,x')\), \(t \in (a,b)\), \(t \neq t_ 1\), \(x^{(i)} (t^ +_ 1) = H_ i (x^{(i)} (t_ 1^ -))\), \(i = 0,1\); \(x(a) = A\), \(x(b) = B\). In particular, here \(f\) is bounded and \(H_ i\), \(i = 0,1\) are nondecreasing continuous functions with \(H_ i (R) = R\). Some sufficient conditions under which the system (P) has at least one solution are established in the paper by means of Schauder's fixed point theorem.