MULTIPOINT FOCAL BOUNDARY VALUE PROBLEMS ON INFITE INTERVALS *

For the differential equation y(n)=f(x,y), we state a set of necessary and sufficient conditions for the existence of a solution (i) on a semi-infinite interval for a k-point right focal boundary value problem and (ii) on (−∞,∞) for a (n−1)-point right focal boundary value problem. The conditions are in terms of the existence of a pair of solutions u(x), v(x) satisfying some auxiliary boundary conditions and algebraic inequatilities.

(1.3)These conditions are stated in terms of the existence of a pair of solutions u(x), v(x) of (1.1) satisfying some auxiliary boundary conditions (BCs) and algebraic inequalities.We assume throughout this paper that the differential equation (1.1) satisfies some of the following hypotheses. A.
f is continuous on R2.

UR.
Solutions of n-point right focal BVPs, if they exist, are unique; that is, if y(x), z(x) are solutions of the BVP (1.1), (1.2) with z x <... < xk and k = n then y(x)= z(x) on [a,l-U.
Solutions of initial value problems (IVPs) are unique.

E'.
All solutions of (1.1) exist on (-co, c), where -co < c < co is a constant depending on the solution.Some existence theorems on infinite intervals for conjugate BVPs have been proved for the cases n = 2 and 3 in [5,6] and for arbitrary n in [7].However, existence theorems on infinite intervals for focal BVPs do not seem to be included in the literature so far.
The proofs for (ii) and (iii) are similar.
We also need the following lemma due to Kolmogorov [4] which is stated here for the sake of convenience.Lemma 2.2: Let M > 0, [a,b] C R and y(z) E Cn[a,b] be an arbitrary function with the property that ly(z) <_ M and _< M on ezists a constant K > 0 depending on M and the interval In, b] such that y(')(z) <_ K on In, b] for 1 < r < n-1.

Nec_essity:
This is obvious since we can choose u(z)= v(z)= y(z) where y(z) is the assumed solution of with (r,i) (1,0).
Remark 1: It follows from theorem 3 of [2] and the theorem in [3] that Theorems 2.3 and 3.1 will be true if we replace (1.1) by y(n) : f(a:, y,..., y(n 1)), (I.I)' provided the hypothesis A is replaced by the hypothesis A' and the additional hypothesis C (compactness of solutions of (1.1)') holds where A' and C are as follows: A': f is continuous on R n + t.

C:
If {yk(z)} is a sequence of solutions of (1.1) and [,al   CompaCt subinterval of (a,b) such that {yk(z)} is uniformly bounded on then there exists a subsequence t, t un, o,m,, on lmark 2: In the case n =3, the hypothesis C can be omitted in view of the comments on page 990 of [2]; while in the case n = 2, the hypotheses U and C can be omitted in view of theorem 3.1 of [8].