×

Singular boundary value problems for first and second order impulsive differential equations. (English) Zbl 1073.34025

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).

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI