Existence of solutions and nonnegative solutions for weighted \(p(r)\)-Laplacian impulsive system multi-point boundary value problems (Q1029419)

The coincidence degree theory due to Gaines and Mawhin is applied to show the existence of solutions for the weighted \(p(r)-\)Laplacian system \[ -\Delta_{p(r)}u+f(r,u,(w(r))^{\frac{1}{p(r)-1}}\, u')=0 , \qquad r \in (0,T)\,, \,r\neq r_{i}\,, \] where \(u:[0,T] \rightarrow {\mathbb{R}}^{N}\) \,and \(-\Delta_{p(r)}u:=-(w(r)\,|u'|^{p(r)-2}\,u')'\)\,, with impulsive multipoint boundary value conditions on \( r_{i}\)\,. The function \(\,f\,\) is supposed to be Carathéodory. In particular, a criterion for the existence of nonnegative solutions is given.