Singular Dirichlet problem for ordinary differential equation with impulses Original Research Article

The paper deals with the impulsive Dirichlet problem u″(t)=f(t,u(t),u′(t)),u″(t)=f(t,u(t),u′(t)), Turn MathJax on View the MathML sourceu(0)=A,u(T)=B, Turn MathJax on View the MathML sourceu(tj+)=Ij(u(tj)),u′(tj+)=Mj(u′(tj)),j=1,…,p, Turn MathJax on where f∈Car((0,T)×R2)f∈Car((0,T)×R2), f has time singularities at t=0t=0 and t=Tt=T, IjIj, Mj∈C0(R)Mj∈C0(R), A, B∈RB∈R. We prove the existence of a solution to this problem under the assumption that there exist lower and upper functions associated with the problem. Our proofs are based on the Schauder fixed point theorem and on the method of a priori estimates.