General Focal Boundary Value Problems

Abstract

Let p ∈ {1, 2,..., n} be fixed and let the sets {r ₁,..., r _p } and {s ₁,..., s _n-p } form a partition of {1, 2,..., n} such that r ₁ <...< r _p and s ₁ <...< s _n-p . In this chapter we shall consider the differential equations

$${y^{\left( n \right)}} + \mu \left( t \right)f\left( y \right) = 0,0 < t < 1$$

(15.1)

and

$${y^{\left( n \right)}} + \mu \left( t \right)Q\left( {t,y,y',...,{y^{\left( {n - 1} \right)}}} \right) = 0,0 < t < 1$$

(15.2)

subject to the general focal boundary conditions

$${y^{\left( {{r_i} - 1} \right)}}\left( 0 \right) = 0,{\mkern 1mu} 1ip$$

(15.3)

$${y^{\left( {{s_i} - 1} \right)}}\left( 1 \right) = 0,{\mkern 1mu} 1in - p$$

(15.4)