On nonlinear multipoint conjugate value problem for feedback control systems in the cone

We consider the nonlinear conjugate multi-point boundary value problem with feedback control that finds functions $ u and q satisfying$ \begin{align*} & \mathcal Lu(t) = q(t)F[t,u(t),u'(t),..., u^{(m-1)}(t)],\,\, q(t) \in \Phi[t,u(t)],\,\,\, t\in I, \hfill \\ u^{(j)}(t_i) 0, \,\,\,\,\, 1\leq i \leq l,\,\,\,\, 0\leq j\leq k_i-1, \end{align*} $where L is a linear differential operator of order m $, I [0,1] 0 t_1<...<t_l 1 2\leq l\leq \sum_{i 1}^lk_i $. By using fixed point theory for multivalued operators we prove existence one or two nontrivial solutions problem.