arXiv subjects
Publications and source records attributed to E. S. Karulina.
For one boundary problem of fourth order with a spectral parameter in the boundary condition we prove the simplicity of the spectrum and the oscillation properties of the system of the eigenfunctions derivatives.
We get the infima and suprema of the first eigenvalue of the problem $y'' + qy + λy = 0$, $y'(0) - k_0^2 y(0) = y'(1) + k_1^2 y(1) = 0$, where $q$ belongs to the set of nonnegative summable functions on [0,1] such that $\int_0^1 q^γdx = 1$, where $γ\in R\setminus \{0\}$.
It is proved that for class $A_γ=\{q\in L_1[0,1]: q\geq 0, \int_0^1 q^γ\,dx=1\}$, where $γ\in (0,1)$, there exists a potential $q_*\in A_γ$ such that minimal eigenvalue $λ_1(q_*)$ of boundary problem $$ -y"+q_*y=λy, y'(0)=y'(1)=0 $$ is equal to $m_γ=\inf_{q\in A_γ}λ_1(q)$. The equality $m_γ=1$ for $γ\leq 1-2π^{-2}$ and the inequality $m_γ<1$ for $γ>1-2π^{-2}$ are also obtained.
On the basis of the theory of Sturm--Liouville problem with distribution coefficients we get the infima and suprema of the first eigenvalue of the problem $-y" + (q-λ) y=0, y'(0) -k_0^2 y(0) = y'(1) + k_1^2 y(1) = 0$, where $q$ belongs to the set of constant-sign summable functions on $[0,1]$ such that $\int_0^1 q dx=\pm 1$.