Boundary value problems for first order impulsive delay differential equations with a parameter. (Q1426991)

The authors use the method of lower and upper solutions coupled with the monotone iterative technique to establish the existence of extremal solutions lying between the lower and the upper solution of the problem \[ \begin{aligned} &x'(t)=f(t,x(t),x_t,\lambda), \quad t\in J',\\ &x(t_{k+1})-x(t_k)=I_k(x(t_k),\lambda), \quad k=1,\dots,m,\\ &x(t)=\varphi(t), \quad t\in[-\tau,0],\\ &G(x(T),\lambda)=0,\end{aligned} \] where \(f\in C(J\times\mathbb R\times D\times\mathbb R,\mathbb R),\) \(G\in C(\mathbb R\times \mathbb R,\mathbb R),\) \(D=L^1([-r,0],\mathbb R),\) \(I_k\in C(\mathbb R\times\mathbb R,\mathbb R),\) \(0<t_1<\dots<t_m<T,\) \(\tau>0\), \(J=[0,T],\) \(J'=J\backslash\{t_1,\dots,t_m\}\), and \(x_t\) is defined by \(x_t(s)=x(t+s)\) for \(-\tau\leq s\leq 0\).