Spectral properties of a beam equation with eigenvalue parameter occurring linearly in the boundary conditions

In this paper, we consider an eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary conditions. The location of eigenvalues on real axis, the structure of root subspaces and the oscillation properties of eigenfunctions of this problem are investigated, and asymptotic formulas for the eigenvalues and eigenfunctions are found. Next, by the use of these properties, we establish sufficient conditions for subsystems of root functions of the considered problem to form a basis in the space $L_p,1 < p < \infty$ .

Eigenvalue bifurcation in doubly nonlinear problems with an application to surface plasmon polaritons

We consider a class of generally non-self-adjoint eigenvalue problems which are nonlinear in the solution as well as in the eigenvalue parameter (“doubly” nonlinear). We prove a bifurcation result from simple isolated eigenvalues of the linear problem using a Lyapunov–Schmidt reduction and provide an expansion of both the nonlinear eigenvalue and the solution. We further prove that if the linear eigenvalue is real and the nonlinear problem $${\mathcal {PT}}$$ PT -symmetric, then the bifurcating nonlinear eigenvalue remains real. These general results are then applied in the context of surface plasmon polaritons (SPPs), i.e. localized solutions for the nonlinear Maxwell’s equations in the presence of one or more interfaces between dielectric and metal layers. We obtain the existence of transverse electric SPPs in certain $${\mathcal {PT}}$$ PT -symmetric configurations.

The problem with fuzzy eigenvalue parameter in one of the boundary conditions

In this work, we study the problem with fuzzy eigenvalue parameter in one of the boundary conditions. We find fuzzy eigenvalues of the problem using the Wronskian functions \underline{W}_{\alpha }\left( \lambda \right) and \overline{W}_{\alpha }\left( \lambda \right). Also, we find eigenfunctions associated with eigenvalues. We draw graphics of eigenfunctions.

Important Notes for a Fuzzy Boundary Value Problem

In this paper is studied a fuzzy Sturm-Liouville problem with the eigenvalue parameter in the boundary condition. Important notes are given for the problem. Integral equations are found of the problem.