We consider a Sturm-Liouville operator with boundary conditions rationally dependent on the eigenparameter. We study the basis property in $L_p$ of the system of eigenfunctions corresponding to this operator. We determine the explicit form of the biorthogonal system. Using this we establish a theorem on the minimality of the part of the system of eigenfunctions. For the basisness in L₂ we prove that the system of eigenfunctions is quadratically close to trigonometric systems. For the basisness in $L_p$...

Let $T$ be a $\gamma$-contraction on a Banach space $Y$ and let $S$ be an almost $\gamma$-contraction, i.e. sum of an $\left(\epsilon ,\gamma \right)$-contraction with a continuous, bounded function which is less than $\epsilon$ in norm. According to the contraction principle, there is a unique element $u$ in $Y$ for which $u=Tu.$ If moreover there exists $v$ in $Y$ with $v=Sv$, then we will give estimates for $\parallel u-v\parallel .$ Finally, we establish some inequalities related to the Cauchy problem.

We consider a boundary value problem for a differential equation with deviating arguments and p-Laplacian: $-{\left({\varphi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}+d/dtgradF\left(x\right)+g(t,x\left(t\right),x\left(\delta \left(t\right)\right)$, x’(t), x’(τ(t))) = 0, t ∈ [0,1]; $x\left(t\right)=\underline{\phi }\left(t\right),$ t ≤ 0; $x\left(t\right)=\overline{\phi }\left(t\right)$, t ≥ 1. An existence result is obtained with the help of the Leray-Schauder degree theory, with no restriction on the damping forces d/dt grad F(x).

We discuss the spectral properties of the operator ${}_{ℳ}\left(\alpha \right):=-d²/dt²+(1/2t²-\alpha )²$ on the line. We first briefly describe how this operator appears in various problems in the analysis of operators on nilpotent Lie groups, in the spectral properties of a Schrödinger operator with magnetic field and in superconductivity. We then give a new proof that the minimum over α of the groundstate energy is attained at a unique point and also prove that the minimum is non-degenerate. Our study can also be seen as a refinement for a specific...

In this paper, the general ordinary quasi-differential expression $M_p$ of $n$-th order with complex coefficients and its formal adjoint $M_p^{+}$ on any finite number of intervals $I_p=(a_p,b_p)$, $p=1,\cdots ,N$, are considered in the setting of the direct sums of $L_{w_p}^2(a_p,b_p)$-spaces of functions defined on each of the separate intervals, and a number of results concerning the location of the point spectra and the regularity fields of general differential operators generated by such expressions are obtained. Some of these are extensions or generalizations...