Disconjugacy and Green’s functions sign for some two-point boundary value problems for fourth order ordinary differential equations

Abstract

Abstract In this paper, for the fourth order linear ordinary differential equation u ( 4 ) ⁢ ( t ) = p ⁢ ( t ) ⁢ u ⁢ ( t ) - μ ⁢ u ⁢ ( t ) , u^{(4)}(t)=p(t)u(t)-\mu u(t), where p : I → R {p:I\to R} is a Lebesgue integrable function, we study the question of disconjugacy on the interval [ a , b ] {[a,b]} , and the problem of the Green’s functions sign for some two-point boundary problems. The optimal sufficient conditions of the disconjugacy and necessary and sufficient conditions of non-negativity (non-positivity) of the Green’s function are found for the mentioned two-point boundary problems.