### A generalized Fuc̆ik type eigenvalue problem for p-Laplacian.

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We study a Sturm-Liouville problem containing a spectral parameter in the boundary conditions. We associate to this problem a self-adjoint operator in a Pontryagin space Π₁. Using this operator-theoretic formulation and analytic methods, we study the asymptotic behavior of the eigenvalues under the variation of a large physical parameter in the boundary conditions. The spectral analysis is applied to investigate the well-posedness and stability of the wave equation of a string.

We consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary condition. The oscillation properties of eigenfunctions are studied and asymptotic formulae for eigenvalues and eigenfunctions are deduced. The basis properties in L p(0; l); p ∈ (1;∞); of the system of eigenfunctions are investigated.

The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular Dirichlet problem ${\left(\phi \left({u}^{\text{'}}\right)\right)}^{\text{'}}=\lambda f(t,u,{u}^{\text{'}})$, $u\left(0\right)=u\left(T\right)=A$. Here $\lambda $ is the positive parameter, $A>0$, $f$ is singular at the value $0$ of its first phase variable and may be singular at the value $A$ of its first and at the value $0$ of its second phase variable.

We discuss the discrete $p$-Laplacian eigenvalue problem, $$\left\{\begin{array}{c}\Delta \left({\phi}_{p}\left(\Delta u(k-1)\right)\right)+\lambda a\left(k\right)g\left(u\left(k\right)\right)=0,\phantom{\rule{1.0em}{0ex}}k\in \{1,2,...,T\},\hfill \\ u\left(0\right)=u(T+1)=0,\hfill \end{array}\right.$$ where $T>1$ is a given positive integer and ${\phi}_{p}\left(x\right):={\left|x\right|}^{p-2}x$, $p>1$. First, the existence of an unbounded continuum $\mathcal{C}$ of positive solutions emanating from $(\lambda ,u)=(0,0)$ is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any $\lambda >0$ and all solutions are ordered. Thus the continuum $\mathcal{C}$ is a monotone continuous curve globally defined for all $\lambda >0$.

We consider boundary value problems for nonlinear $2m$th-order eigenvalue problem $$\begin{array}{cc}\hfill {(-1)}^{m}{u}^{\left(2m\right)}\left(t\right)& =\lambda a\left(t\right)f\left(u\right(t\left)\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0<t<1,\hfill \\ \hfill {u}^{\left(2i\right)}\left(0\right)& ={u}^{\left(2i\right)}\left(1\right)=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i=0,1,2,\cdots ,m-1.\hfill \end{array}$$ where $a\in C\left(\right[0,1],[0,\infty \left)\right)$ and $a\left({t}_{0}\right)>0$ for some ${t}_{0}\in [0,1]$, $f\in C\left(\right[0,\infty ),[0,\infty \left)\right)$ and $f\left(s\right)>0$ for $s>0$, and ${f}_{0}=\infty $, where ${f}_{0}={lim}_{s\to {0}^{+}}f\left(s\right)/s$. We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.

In this paper we consider a parametric eigenvalue problem related to a vibrating string which is constructed out of two different materials. Using elementary analysis we show that the corresponding principal eigenvalue is increasing with respect to the parameter. Using a rearrangement technique we recapture a part of our main result, in case the difference between the densities of the two materials is sufficiently small. Finally, a simple numerical algorithm will be presented which will also provide...