A priori estimates for the existence of a solution for a multi-point boundary value problem. Issue 4 (2000) - Gupta, Chaitan P.; Trofimchuk, Sergei
Abstract : Leta i ∈ R, ζ i ∈ ( 0, 1 ), i = 1, 2, …, m − 2, 0 < ζ 1 < ζ 2 < ⋯ < ζ m − 2 < 1, withα = ∑ i = 1 m − 2 a i ≠ 1 be given. Letx ( t ) ∈ W 2, 1 ( 0, 1 ) be such thatx ′ ( 0 ) = 0, x ( 1 ) = ∑ i = 1 m − 2 a i x ( ζ i ) ( ∗ ) be given. This paper is concerned with the problem of obtaining Poincaré type a priori estimates of the form| | x | | ∞ ≤ C | | x ″ | | 1 . The study of such estimates is motivated by the problem of existence of a solution for the Caratheodory equationx ″ ( t ) = f ( t, x ( t ), x ′ ( t ) ) + e ( t ), 0 < t < 1, satisfying boundary conditions( ∗ ) . This problem was studied earlier by Gupta et al. (Jour. Math. Anal. Appl. 189 (1995), 575–584) when thea i 's, all had the same sign.