Solvability of a Multi-point Boundary Value Problem of Neumann Type

Let f : [0,1]×R2 → R be a function satisfying Carathéodory’s conditions and e(t) ∈ L1[0,1]. Let ξi ∈ (0,1), ai ∈ R, i = 1,2, . . . ,m− 2, 0 < ξ1 < ξ2 < · · · < ξm−2 < 1 be given. This paper is concerned with the problem of existence of a solution for the m-point boundary value problem x′′(t) = f (t,x(t),x′(t))+e(t), 0 < t < 1; x(0) = 0, x′(1) =∑m−2 i=1 aix′(ξi). This paper gives conditions for the existence of a solution for this boundary value problem using some new Poincaré type a priori estimates. This problem was studied earlier by Gupta, Ntouyas, and Tsamatos (1994) when all of the ai ∈ R, i = 1,2, . . . ,m−2, had the same sign. The results of this paper give considerably better existence conditions even in the case when all of the ai ∈ R, i = 1,2, . . . ,m−2, have the same sign. Some examples are given to illustrate this point.