scholarly journals ON A MINIMIZATION PROBLEM FOR A FUNCTIONAL GENERATED BY THE STURM – LIOUVILLE PROBLEM WITH INTEGRAL CONDITION ON THE POTENTIAL

2017 ◽  
Vol 21 (6) ◽  
pp. 57-61
Author(s):  
S.S. Ezhak
Keyword(s):  
Boundary Value Problem ◽  
Minimization Problem ◽  
Nonlinear Boundary ◽  
Dirichlet Boundary ◽  
Liouville Problem ◽  
Integral Condition ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Functional ◽  
Sturm Liouville ◽  
Class Of Functions

In this article we consider the minimization problem of the functional generated by a Sturm — Liouville problem with Dirichlet boundary conditions and with an integral condition on the potential. Estimation of the infimum of functional in some class of functions y and Q(x) isreduced to estimation of a nonlinear functional non depending on the potential Q(x). This leads to related parameterized nonlinear boundary value problem. Upper and lower estimates are obtained for different values of parameter.

Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2265
Author(s):  
Malgorzata Klimek
Keyword(s):  
Boundary Conditions ◽  
Fractional Derivatives ◽  
Dirichlet Boundary ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Dirichlet Boundary Conditions ◽  
Sturm Liouville ◽  
Left And Right ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Liouville Operators

In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operators are rewritten as Hilbert–Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval (1/2,1]. Applying the spectral Hilbert–Schmidt theorem, we prove that the spectrum of integral Sturm–Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm–Liouville operators.

scholarly journals Inverse problems for Sturm-Liouville difference equations

Filomat ◽  
2016 ◽  
Vol 30 (5) ◽  
pp. 1297-1304 ◽  
Author(s):  
Martin Bohner ◽  
Hikmet Koyunbakan
Keyword(s):  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Dirichlet Boundary ◽  
Liouville Problem ◽  
Dirichlet Boundary Conditions ◽  
Sturm Liouville

We consider a discrete Sturm-Liouville problem with Dirichlet boundary conditions. We show that the specification of the eigenvalues and weight numbers uniquely determines the potential. Moreover, we also show that if the potential is symmetric, then it is uniquely determined by the specification of the eigenvalues. These are discrete versions of well-known results for corresponding differential equations.

scholarly journals A counterexample in Sturm–Liouville completeness theory

2004 ◽  
Vol 134 (2) ◽  
pp. 241-248 ◽  
Author(s):  
Paul Binding ◽  
Branko Ćurgus
Keyword(s):  
Boundary Conditions ◽  
Riesz Basis ◽  
Periodic Boundary ◽  
Dirichlet Boundary ◽  
Liouville Problem ◽  
Periodic Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Indefinite Weight ◽  
Sturm Liouville

We give an example of an indefinite weight Sturm-Liouville problem whose eigenfunctions form a Riesz basis under Dirichlet boundary conditions but not under anti-periodic boundary conditions.

scholarly journals Asymptotic Behaviour of Eigenvalues and Eigenfunctions of a Sturm-Liouville Problem with Retarded Argument

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Erdoğan Şen ◽  
Jong Jin Seo ◽  
Serkan Araci
Keyword(s):  
Boundary Value Problem ◽  
Liouville Operator ◽  
Liouville Problem ◽  
Asymptotic Formulas ◽  
Retarded Argument ◽  
Eigenvalues And Eigenfunctions ◽  
Transmission Coefficients ◽  
Discontinuous Boundary ◽  
Sturm Liouville ◽  
Special Case

In the present paper, a discontinuous boundary-value problem with retarded argument at the two points of discontinuities is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. This is the first work containing two discontinuities points in the theory of differential equations with retarded argument. In that special case the transmission coefficients and retarded argument in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.

ESTIMATES ON THE GREEN FUNCTION AND EXISTENCE OF POSITIVE SOLUTIONS FOR SOME NONLINEAR POLYHARMONIC PROBLEMS OUTSIDE THE UNIT BALL

2008 ◽  
Vol 06 (02) ◽  
pp. 121-150 ◽  
Author(s):  
IMED BACHAR ◽  
HABIB MÂAGLI ◽  
NOUREDDINE ZEDDINI
Keyword(s):  
Green Function ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Closed Ball ◽  
Dirichlet Boundary Conditions ◽  
Kato Class ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
The Green Function ◽  
Class Of Functions

Let [Formula: see text] be the Green function of (-Δ)m, m ≥ 1, on the complementary D of the unit closed ball in ℝn, n ≥ 2, with Dirichlet boundary conditions [Formula: see text], 0 ≤ j ≤ m - 1. We establish some estimates on [Formula: see text] including the 3G-Inequality given by (1.3). Next, we introduce a polyharmonic Kato class of functions [Formula: see text] and we exploit the properties of this class to study the existence of positive solutions of some polyharmonic nonlinear elliptic problems.

Asymptotic formulas for Dirichlet boundary value problems

2005 ◽  
Vol 42 (2) ◽  
pp. 153-171 ◽  
Author(s):  
Bülent Yilmaz ◽  
O. A. Veliev
Keyword(s):  
Boundary Conditions ◽  
Main Part ◽  
Dirichlet Boundary ◽  
Arbitrary Order ◽  
Dirichlet Boundary Conditions ◽  
Asymptotic Formulas ◽  
Summable Function ◽  
Special Cases ◽  
Sturm Liouville ◽  
Liouville Operators

In this article we obtain asymptotic formulas of arbitrary order for eigenfunctions and eigenvalues of the nonselfadjoint Sturm-Liouville operators with Dirichlet boundary conditions, when the potential is a summable function. Then using these we compute the main part of the eigenvalues in special cases.

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