Singular boundary value problems for first and second order impulsive differential equations
The authors consider the impulsive boundary value problem \[ \begin{aligned} & u''=\lambda f(t,u),\quad t\in J'=J\setminus \{t_1,t_2,\dots,t_k\},\\ & \Delta u(t_i)=\lambda J_i(u(t_i),u'(t_i)),\quad i=1,2,\dots,k,\\ & \Delta u'(t_i)=\lambda I_i(u(t_i),u'(t_i)),\quad i=1,2,\dots,k,\\ & u(0)=u(1)=0,\end{aligned}\tag{1} \] where \(J=[0,1]\), \(0=t_0<t_1<t_2<\dots<t_k<t_{k+1}=1\), \(\Delta u(t_i)=u(t^+_i)-u(t^-_i)\), \(\Delta u'(t_i)=u'(t^+_i)-u'(t^-_i)\), \(u'(t^+_i)\), \(u'(t^-_i)\) are the right limit and left limit of \(u'(t)\) at \(t=t_i\) respectively. Using Krasnoselskij's fixed-point theorem in a cone, the authors find sufficient conditions for the existence of a solution of problem (1).
scientific article; zbMATH DE number 2162493