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Solvability of multi-point boundary value problem at resonance. III. (English) Zbl 1054.34033

The nonlinear second-order ordinary differential equation

x '' =f(t,x(t),x ' (t))+e(t),t(0,1),

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 f:[0,1]× 2 , 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)].

34B15Nonlinear boundary value problems for ODE
47J05Equations involving nonlinear operators (general)
47N20Applications of operator theory to differential and integral equations