The author considers the second-order boundary value problem
where satisfies the Carathéodory conditions. Two more conditions are assumed:
(A1): For any there exists a function which does not vanish identically on any subinterval of and a constant with on
(A2): There exist functions such that
and for a.e. and all The main result asserts that the problem above has at least one solution provided
The Leray-Schauder continuation theorem is used for establishing the result. However, the method of obtaining the needed a priori estimates is different from that of either the pionneer related works [R. P. Agarwal and D. O’Regan, Tohoku Math. J., II. Ser. 51, No. 3, 391–397 (1999; Zbl 0942.34026)] or [A. Constantin, Ann. Mat. Pura Appl., IV. Ser. 176, 379–394 (1999; Zbl 0969.34024)]. The author gives an interesting example to which the main result of his article applies but not those of Agarwal and O’Regan nor Constantin cited above.