Constant sign solutions of two-point fourth order problems

The eigenvalue characterization for the constant sign Green’s functions of ( k , n − k ) $(k,n-k)$ problems

“…Therefore, we can affirm without calculating it explicitly, that Green's function related to the operator u (4) These results have been obtained using the explicit form of Green's function in [8] and [9].…”

Multiplicity results for fourth order problems related to the theory of deformations beams

“…Such result has been extended in [8] (and further in [9]) for any n-th order differential operator, coupled to the so-called (k, n − k) boundary conditions, which are defined, for 1 ≤ k ≤ n − 1, as follows: There is proved that m 4 0 is the least positive eigenvalue of operator u (4) on the space {u ∈ C 4 ([0, 1]), u(0) = u (0) = u(1) = u (1) = 0}…”

“…and −m 4 1 is the biggest negative eigenvalue of operator u (4) in the space {u ∈ C 4 ([0, 1]), u(0) = u (0) = u (0) = u(1) = 0}.…”

“…The study of fourth-order boundary value problems are useful for material mechanics because this kind of problems usually characterize the deformations of an elastic beam. They have been studied by many authors via various methods, such as Leray-Schauder continuation method, topological degree theory, shooting method, fixed point theorems on cones, critical point theory, the lower and upper solutions method or spectral theory, see for example [4,6,3,13,11,16,2] and references therein.…”

“…In [4], it is characterized the sign of the Green's function g M related to the fourth order linear problem L M u(t) = u (4) (t) + M u(t) = σ(t), t ∈ I := [0,1] (1) u(0) = u (0) = u (0) = u(1) = 0.…”