The differential equation (1) is considered under the following types of boundary conditions:
(2) , , , ;
(3) , , , ;
(5) ; , , with , .
Existence theorems for all of the boundary value problems (1-2)–(1-5) are obtained by application of Schauder’s fixed point theorem under continuity and boundedness hypotheses on and its partial derivative . A uniqueness theorem is obtained for the problem (1-4) under the additional assumption of a bound on the partial derivative . Further uniqueness results for the problem (1-2)–(1-5) are reduced to uniqueness questions for solutions of corresponding second order boundary value problems for the integrodifferential equation
obtained by setting , with the Green’s function for the problem
Boundary conditions of the form (2)–(5) have been less extensively studied than the familiar conjugate or focal-type problems. R. A. Usmani [Proc. Am. Math. Soc. 77, 329–335 (1979; Zbl 0424.34019)] has studied a problem of the form (1-2) in which the equation is linear and independent of .