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On a fixed point theorem for weakly sequentially continuous mapping

Ireneusz Kubiaczyk (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Let E be a metrizable locally convex topological vector space x ∈ E, and let D be a closed convex subset of E such that x ∈ D. In this paper we prove that the weakly sequentially continuous mapping F: D ∪ D which satisfies V̅ = c̅o̅n̅v̅({x} ∪ F(V))⇒ V is relatively weakly compact, has a fixed point. Employing the above results we prove the existence theorem for the Cauchy problem x'(t) = f(t,x(t)), x(0) = x₀. As compared with the previous...

On discreteness of spectrum of a functional differential operator

Sergey Labovskiy, Mário Frengue Getimane (2014)

Mathematica Bohemica

We study conditions of discreteness of spectrum of the functional-differential operator ℒ u = - u ' ' + p ( x ) u ( x ) + ∫ - ∞ ∞ ( u ( x ) - u ( s ) ) d s r ( x , s ) on ( - ∞ , ∞ ) . In the absence of the integral term this operator is a one-dimensional Schrödinger operator. In this paper we consider a symmetric operator with real spectrum. Conditions of discreteness are obtained in terms of the first eigenvalue of a truncated operator. We also obtain one simple condition for discreteness of spectrum.

On L w 2 -quasi-derivatives for solutions of perturbed general quasi-differential equations

Sobhy El-sayed Ibrahim (1999)

Czechoslovak Mathematical Journal

This paper is concerned with square integrable quasi-derivatives for any solution of a general quasi-differential equation of n th order with complex coefficients M [ y ] - λ w y = w f ( t , y [ 0 ] , ... , y [ n - 1 ] ) , t ∈ [ a , b ) provided that all r th quasi-derivatives of solutions of M [ y ] - λ w y = 0 and all solutions of its normal adjoint M + [ z ] - λ ¯ w z = 0 are in L w 2 ( a , b ) and under suitable conditions on the function f .

On Spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping∗

Bao-Zhu Guo, Guo-Dong Zhang (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration...

On Spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping∗

Bao-Zhu Guo, Guo-Dong Zhang (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration...

On the basis property of the root functions of differential operators with matrix coefficients

Oktay Veliev (2011)

Open Mathematics

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these asymptotic formulas, we find necessary and sufficient conditions on the coefficients for which the system of eigenfunctions and associated functions of the operator under consideration forms a Riesz basis.

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