The paper deals with the problem of existence and uniqueness of solutions for the boundary value problems
where f: [0,1] is a function satisfying Caratheodory’s conditions and e: [0,1] is in . There are 4 theorems concerning the existence of the solutions and 2 theorems concerning the uniqueness of the solutions. We give the following theorem as an example:
Theorem 5. Suppose that there exist functions a(x), b(x), c(x) and d(x) in such that .
for all , and a.e. . Suppose further that there exist an -Caratheodory function and such that for all , and a.e. . Then for
1. (1),(2) has exactly one solution if
2. (1), (3) has exactly one solution if
We note that the proof of the existence of the solutions was made by applying the version of Leray-Schauder continuation theorem given by Mawhin. The paper is a continuation and complementation of an earlier paper of C. P. Gupta [Appl. Anal. 26, No.4, 289-304 (1988; Zbl 0611.34015)].