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

S. M. Labovski

Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 136, pp. 382-393

Résumé

Conditions of negativity for the Green's function of a two-point boundary value problem $$ \mathcal{L}_\lambda u := u^{(n)}-\lambda\int_0^l u(s) d_s r(x,s)=f(x), \ \ \ x\in[0,l], \ \ \ B^k(u)=\alpha, $$ where $B^k(u)=(u(0),\ldots,u^{(n-k-1)}(0),u(l),-u'(l),\ldots,(-1)^{(k-1)}u^{(k-1)}(0)),$ $n\ge3,$ $0\!<\!k\!<\!n,$ $k$ 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 $$ u(0)=\cdots=u^{(n-k-2)}(0)=0, \ \ \ u(l)=\cdots=u^{(k-2)}(l)=0, $$ $u^{(n-k-1)}(0)\ge0,$ $u^{(k-1)}(l)\ge0,$ $f(x)\le 0$ 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 $\lambda^k$ be the smallest positive value of $\lambda$ for which the problem $\mathcal{L}_\lambda u=0,$ $B^ku=0$ has a nontrivial solution. Then the pair of conditions $\lambda <\lambda^{k-1}$ and $\lambda <\lambda^{k+1}$ is necessary and sufficient for positivity of the solution of the problem.

Détail

Keywords: Green's function, positivity, functional differential equation.

Bibliographie

[1] S. Labovskiy, “On positivity of Green's functions of a functional-differential equation”, Tambov University Reports. Series: Natural and Technical Sciences, 20:5 (2015), 1246–1249 (In Russian)

[2] M. Krasnosel'skii, E. Lifshits, A. Sobolev, Positive Linear Systems, the Method of Positive Operators, Heldermann–Verlag, Berlin, 1989, 354 pp. | Zbl

[3] M. Kreĭn, M. Rutman, “Linear operators leaving invariant a cone in a Banach space”, Uspekhi Mat. Nauk, 3:1(23) (1948), 3–95 (In Russian) | Zbl

[4] S. M. Labovskiĭ, “Positive solutions of linear functional differential equations”, Differential Equations, 20 (1984), 428–434 | MR | Zbl

[5] S. M. Labovskiĭ, “Positive solutions of a two-point boundary value problem for a linear singular functional-differential equation”, Differential Equations, 24:10 (1988), 1116–1123 | MR | Zbl