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Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem. (English) Zbl 1200.34023

Consider the fourth-order boundary value problem

u (4) (t)=f(t,u(t),u '' (t)),t(0,1),
u(0)=u(1)=u '' (0)=u '' (1)=0,

where f:[0,1]×[0,+)×(-,0][0,+) is continuous, such that f(t,0,0)=0 and satisfies a technical condition ensuring that, roughly speaking, f is not necessarily linearizable at (0,0) and (+,-)· Moreover, it is assumed that there exist a non-negative function c 1 and a non-negative constant c 2 such that c 1 (t)+c 2 >0 and f(t,u,p)c 1 (t)u-c 2 p for all t,u,p· The authors give a sufficient condition, expressed in terms of the generalized eigenvalues of the associated BVP u (4) =λ(A(t)u-B(t)u '' ), for the existence of at least one positive solution to the given BVP. The proof is performed by applying the global P. H. Rabinowitz bifurcation theorem [Rocky Mountain J. Math. 3, 161–202 (1973; Zbl 0255.47069)].

34B18Positive solutions of nonlinear boundary value problems for ODE
34C23Bifurcation (ODE)