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.


A priori estimates
i=1 a i x (ξ i ) be given.We are interested in obtaining a priori estimates of the form x ∞ ≤ C x 1 .The following theorem gives such an estimate.We recall that for a ∈ R, a + = max{a, 0}, a − = max{−a, 0} so that a = a + − a − and |a| = a + + a − .
Remark 2.2.We note that if a i ≤ 0 for every i = 1, 2, . . ., m − 2, then τ = 0 and if The following theorem gives a better estimate for the three-point boundary value in the case of the L 2 -norm.

be given. Then the boundary value problem (1.1) has at least one solution in
where τ is as defined in Theorem 2.1.
Proof.Let X denote the Banach space C 1 [0, 1] and Y denote the Banach space L 1 (0, 1) with their usual norms.We define a linear mapping L : D(L) ⊂ X → Y by setting We also define a nonlinear mapping N : X → Y by setting We note that N is a bounded mapping from X into Y .Next, it is easy to see that the linear mapping L : D(L) ⊂ X → Y , is a one-to-one mapping.Next, the linear mapping where A is given by, is such that for y ∈ Y, Ky ∈ D(L), and LKy = y; and for u ∈ D(L), KLu = u.Furthermore, it follows easily using the Arzela-Ascoli theorem that KN maps a bounded subset of X into a relatively compact subset of X. Hence KN : X → X is a compact mapping.We, next, note that x ∈ C 1 [0, 1] is a solution of the boundary value problem (1.2) if and only if x is a solution to the operator equation Now, the operator equation Lx = Nx + e is equivalent to the equation We apply the Leray-Schauder continuation theorem (cf.[6, Corollary IV.7]) to obtain the existence of a solution for x = KNx + Ke or equivalently to the boundary value problem (1.2).
To do this, it suffices to verify that the set of all possible solutions of the family of equations where τ is as defined in Theorem 2.1.
Let x(t) be a solution of (3.10) for some λ ∈ [0, 1], so that x ∈ W 2,1 (0, 1) with x(0) = 0, x (1) = m−2 i=1 a i x (ξ i ).We then get from the equation in (3.10) and Theorem 2.1 that Proof.As in the proof of Theorem 3.2 it suffices to prove that the set of all possible solutions of the family of equations