On a necessary and sufficient condition for the negativeness of the Green’s function of a two-point boundary value problem for a functional differential equation
Conditions of negativity for the Green’s function of a two-point boundary value problem L_λ u≔u^((n) )-λ∫_0^l▒〖u(s) d_s r(x,s)=f(x), x∈[0,l], B^k (u)=α,〗 where B^k (u)=(u(0),…,u^((n-k-1) ) (0),u(l),-u^'(l) ,…,(-1)^((k-1) ) u^((k-1) ) (0) ), n≥3, 0<k<n, k is odd, are considered. The function r(x,s) is assumed to be non-decreasing in the second argument. A necessary and sufficient condition for the nonnegativity of the solution of this boundary value problem on the set E of functions satisfying the conditions u(0)=⋯=u^((n-k-2) ) (0)=0, u(l)=⋯=u^((k-2) ) (l)=0, u^((n-k-1) ) (0)≥0, u^((k-1) ) (l)≥0, f(x)≤0 is obtained. This condition lies in the subcriticality of boundary value problems with vector functionals B^(k-1) and B^(k+1). Let k be even and λ^k be the smallest positive value of λ for which the problem L_λ u=0, B^k u=0 has a nontrivial solution. Then the pair of conditions λ<λ^(k-1) and λ<λ^(k+1) is necessary and sufficient for positivity of the solution of the problem.