Existence of solutions to the first-order nonlinear boundary value problems (Q532970)

The authors consider the boundary value problem for impulsive system of differential equations of first order \[ \begin{aligned} &x'(t) = f(t,x(t)),\quad t \in (0,T), \;t \neq t_k,\\ &\Delta x(t_k) = I_k(x(t_k)),\;k = 1,\dots,m,\\ &ax(0) + x(T) = b, \end{aligned} \] where \(0 < t_1 < \dots < t_m < T\), \(f : [0,T]\times {\mathbb R}\to {\mathbb R}\), \(I_k : {\mathbb R} \to {\mathbb R}\) are continuous, \(a\), \(b \in {\mathbb R}\). Existence results for various special cases of this BVP are obtained. As the main tool there is used Schaefer is fixed a point theorem and the nonlinear alternative.