Bounds on the non-real eigenvalues of nonlocal indefinite Sturm-Liouville problems with coupled boundary conditions
This paper is concerned with the eigenvalue problem of nonlocal indefinite Sturm-Liouville differential equation \[ -y''+q(x)y+v(x)y(1)=\lambda w(x) y,\quad x\in(0,1),\tag{1} \] associated to nonlocal coupled boundary value conditions \[ \left (\begin{matrix} y(1) \\ y'(1)+\int_0^1v(x)y(x)dx\end{matrix}\right )=e^{i\gamma}C\left (\begin{matrix} y(0) \\ y'(0) \end{matrix}\right),\tag{2} \] where \(q\in L^1([0,1],\mathbb{R})\), \(v\in L^1([0,1],\mathbb{R})\) is called the nonlocal potential, \(w\in L^1([0,1],\mathbb{R})\) changes its sign on \([0, 1]\) and \[ C=\left (\begin{matrix} c_{11} & c_{12}\\ c_{21} & c_{22}\end{matrix}\right ) \] with \(c_{ij}\in\mathbb{R}\), \(\operatorname{det}(C)=1\) and \(\gamma\in[-\pi,\pi)\). Let \(q_{\pm}=\max\{0,\pm q\}\) and \[ \Delta_c=\max \begin{cases} \Delta_{c_1}=\frac{|c_{22}|+|c_{11}|+2}{|c_{12}|}&\text{if }c_{12}\neq 0,\\ \Delta_{c_2}=|c_{11}||c_{21}|, &\text{if }c_{12}=0, \end{cases}\tag{3} \] \[ \Delta_{q,v}= \Delta_c+\| q_-\|_1+2\| v\|_1,~~\Delta=\sqrt{\Delta_{q,v}(1+\Delta_{q,v})}+\Delta_{q,v} \] and let \[ g\in H^1(0,1):=\{g\in L^2(0,1): g\in AC_{\mathrm{loc}}(0,1), g'\in L^2(0,1), g(1)=g(0)=0,\operatorname{sgn}g=\operatorname{sgn}w\}. \tag{4} \] The main result of this article is given by the following theorem: Theorem. Let (3) and (4) hold. Suppose that \(\lambda\) is a non-real eigenvalue of (1) and (2). Then \[ |\operatorname{Im} \lambda|\leq\frac{2}{\alpha}\|g'\|_2(\Delta\sqrt{1+2\Delta}+(1+2\Delta)\|v\|_1), \] \[ |\lambda|\leq\frac{2}{\alpha}\Big[\|g'\|_2(\Delta\sqrt{1+2\Delta}+(1+2\Delta)\|v\|_1)+\|g\|_{\infty}(\Delta^2+(1+2\Delta)\|q\|_1)\Big]. \]
scientific article; zbMATH DE number 7489689
scientific article