Higher Order Boundary Value Problems

In this chapter we shall establish existence theory on semi-infinite and infinite intervals for the n-th order differential equation (2.1.1) $$ x^{(n)} = f\left( {t,x,x', \ldots x^{(q)} } \right),0 \leqslant q \leqslant n - 1,{\text{ but fixed}}{\text{.}}$$ We shall prove results for the (2 ≤) r-point conjugate (Hermite) type boundary conditions (2.1.2) $$ x^{(i)} (a_j ) = A_{i,j,} 0 \leqslant i \leqslant k_j - 1,1 \leqslant j \leqslant r,\sum\limits_{i = 1}^r {k_i = n,}$$ and the (2 ≤) r-point right focal (Abel-Gontscharoff) type boundary conditions (2.1.3) $$ \begin{gathered}x^{(i)} (a_j ) = A_{i,j,} s_j - 1 \leqslant i \leqslant s_j - 1,s_0 = 0,s_j = \sum\limits_{i = 1}^r {k_i ,} \hfill \\k_i \geqslant 1,1 \leqslant j \leqslant r,s_n = n, \hfill \\\end{gathered}$$ where − ∞