The nonlinear second-order ordinary differential equation
subject multipoint boundary conditions of four types is considered. The purpose of the paper is to study the resonant cases. Under certain growth conditions on the continuous nonlinear function , existence results are established. The main tool is the coincidence degree theory of Mawhin see, e.g., J. Mawhin [Topological degree and boundary value problems for nonlinear differential equations. Berlin: Springer-Verlag. Lect. Notes Math. 1537, 74–142 (1993; Zbl 0798.34025)].
For Parts I and II, see [Indian J. Pure Appl. Math. 33, No. 4, 475–494 (2002; Zbl 1021.34013), Appl. Math. Comput. 136, No. 2-3, 353–377 (2003; Zbl 1053.34016)].