Abstract
We consider the fourth-order two-point boundary value problem x ���� + kx �� + lx = f (t, x), 0 < t <1 ,x(0) = x(1) = x � (0) = x � (1) = 0, which is not necessarily linearizable. We give conditions on the parameters k, l and f (t, x) that guarantee the existence of positive solutions. The proof of our main result is based upon topological degree theory and global bifurcation techniques. MSC: 34B15
Highlights
The deformations of an elastic beam in an equilibrium state with fixed both endpoints can be described by the fourth-order boundary value problem x + lx = λh(t)f (x), < t
[, ] such that f (t, x) ≥ c(t)x, (t, x) ∈ [, ] × [, ∞). It is the purpose of this paper to study the existence of positive solutions of ( . ) under conditions (A ), (H ), (H ) and (H )
For other results on the existence and multiplicity of positive solutions and nodal solutions for boundary value problems of fourth-order ordinary differential equations based on bifurcation techniques, see [ – ]
Summary
When l = , the existence of positive solutions of problem When l = , Xu and Han [ ] studied the existence of nodal solutions of problem Motivated by [ ], when k, l satisfy (A ), Shen [ ] studied the existence of nodal solutions of a general fourth-order boundary value problem by applying disconjugate operator theory [ , ] and Rabinowitz’s global bifurcation theorem x + kx + lx = f (t, x), < t < ,
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