A priori estimates and solvability of a generalized multi-point boundary value problem (Q2784695)

The author considers the generalized multipoint boundary value problem NEWLINE\[NEWLINE x''(t)=f(t,x(t),x'(t)) + e(t), \;0<t<1, \quad \sum_{i=1}^m a_i x(\xi_i) = 0, \quad \sum_{j=1}^n b_j x'(\tau_j) = 0, \tag{1}NEWLINE\]NEWLINE where \(f: [0,1]\times \mathbb{R}^2 \to \mathbb{R}\) is a function satisfying Carathéodory conditions and \(e: [0,1]\to \mathbb{R}\) is a function in \(L^1[0,1]\), \(a_i, b_j \in \mathbb{R}\), \(i=1,\dots,m\), \(j=1,\dots,n\), \(0\leq \xi_1<\xi_2< \cdots < \xi_m \leq 1\), \(0\leq \tau_1<\tau_2< \cdots < \tau_n \leq 1\). NEWLINENEWLINENEWLINEThe existence of solutions to this problem is studied. It is shown that the same problem but with the boundary conditions \(\sum_{i=1}^m a_i x(\xi_i) = 0\) and \(\sum_{j=1}^n b_j x(\tau_j) = 0\) can be reduced to (1). New a priori estimates are obtained.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00016].