On Volterra Three-Point Problems for the Sturm–Liouville Operator Related to Potential Symmetry (original) (raw)
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On two-point boundary value problems for the Sturm-Liouville operator
arXiv: Spectral Theory, 2015
In this paper, we study spectral problems for the Sturm-Liouville operator with arbitrary complexvalued potential q(x) and two-point boundary conditions. All types of mentioned boundary conditions are considered. We ivestigate in detail the completeness property and the basis property of the root function system.
Characterization of the spectrum of irregular boundary value problem for the
2012
We consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) ∈ L 2 (0, π) and irregular boundary conditions. We establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator. In the present paper, we consider the eigenvalue problem for the Sturm-Liouvulle equation u ′′ − q(x)u + λu = 0 (1) on the interval (0, π) with the boundary conditions u ′ (0) + (−1) θ u ′ (π) + bu(π) = 0, u(0) + (−1) θ+1 u(π) = 0, (2) where b is a complex number, θ = 0, 1, and the function q(x) is an arbitrary complex-valued function of the class L 2 (0, π). Denote by c(x, µ), s(x, µ) (λ = µ 2) the fundamental system of solutions to (1) with the initial conditions c(0, µ) = s ′ (0, µ) = 1, c ′ (0, µ) = s(0, µ) = 0. The following identity is well known c(x, µ)s ′ (x, µ) − c ′ (x, µ)s(x, µ) = 1. (3) Simple calculations show that the characteristic equation of (1), (2) can be reduced to the form ∆(µ) = 0, where
Boundary Value Problems, Integral Equations and Related Problems, 2010
We consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) ∈ L 2 (0, π) and irregular boundary conditions. We establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator. In the present paper, we consider the eigenvalue problem for the Sturm-Liouvulle equation u ′′ − q(x)u + λu = 0 (1) on the interval (0, π) with the boundary conditions u ′ (0) + (−1) θ u ′ (π) + bu(π) = 0, u(0) + (−1) θ+1 u(π) = 0, (2) where b is a complex number, θ = 0, 1, and the function q(x) is an arbitrary complex-valued function of the class L 2 (0, π). Denote by c(x, µ), s(x, µ) (λ = µ 2) the fundamental system of solutions to (1) with the initial conditions c(0, µ) = s ′ (0, µ) = 1, c ′ (0, µ) = s(0, µ) = 0. The following identity is well known c(x, µ)s ′ (x, µ) − c ′ (x, µ)s(x, µ) = 1. (3) Simple calculations show that the characteristic equation of (1), (2) can be reduced to the form ∆(µ) = 0, where
Hacettepe Journal of Mathematics and Statistics, 2019
The spectral problem\[-y''+q(x)y=\lambda y,\ \ \ \ 0<x<1\]\[y(0)=0, \quad y'(0)=\lambda(ay(1)+by'(1)),\]is considered, where lambda\lambdalambda is a spectral parameter, q(x)inL1(0,1)q(x)\in{{L}_{1}}(0,1)q(x)inL1(0,1) is a complex-valued function, aaa and bbb are arbitrary complex numbers which satisfy the condition ∣a∣+∣b∣ne0|a|+|b|\ne 0∣a∣+∣b∣ne0. We study the spectral properties (existence of eigenvalues, asymptotic formulae for eigenvalues and eigenfunctions, minimality and basicity of the system of eigenfunctions in Lp(0,1){{L}_{p}}(0,1)Lp(0,1)) of the above-mentioned Sturm-Liouville problem.
On an eigenvalue problem of Ahmad and Lazer for ordinary differential equations
Proceedings of the American Mathematical Society, 1987
Lazer, we show the existence of a class of nonselfadjoint eigenvalue problems related to the equation y(n) + Xp(x)y = 0 for which the general eigenvalues comparison is not true. We use a comparison principle for the zeros of the corresponding Cauchy problem. This paper provides a contribution to the understanding of a problem raised by S. Ahmad and A. C. Lazer [1] in connection with the comparison of the eigenvalues for some multi-point boundary value problems which are not selfadjoint. One is given the equation (1) Lny + Xp(x)y = 0, where p(x) is a continuous function of constant sign on an interval /, A is a parameter, and Lny is a linear differential disconjugate operator of order n, that is, the only solution of Lny = 0 with n zeros on I (counting multiplicity) is y = 0. Let us consider the eigenvalue problem given by equation (1) and the system of boundary conditions ,, Lzy(a)=0, iG{ii,...,ik}, L]V(b)=0, JGiJu.-.Jn-k}, where o, b G I, 1 < k < n-1, Liy, i = 0,..., n-1, are the quasi-derivatives of y(x) (see [7]), and {t'i,..., ¿fc}, {ji, ■ ■ ■ ,jn-k) are two arbitrary sets of indices from the set {0,... ,n-1}. Problems of this type have been studied extensively (cf. [2, 3, 5]). In particular, Elias [5] has shown that if (-l)n_fcp(x) < 0, then the eigenvalues of problems (1) and (2) are real and nonnegative and form a divergence sequence {Am}m£N-Ahmad and Lazer [1] have considered a particular type of boundary condition (2), that is (3) y(a)=y'(a) =-= yik-1\a) = 0, y(b)=y'(b) =-=y(n-k-i\b)=0, and showed that if we set p = Pi, where p¿, i-1,2, are two continuous functions, considering the corresponding sequence of eigenvalues (A¿,m)m6N, i = 1,2, ordered by magnitude, then the condition (4)_ (-l)n-kp2(x) < (-l)"-fepi(x) < 0
Resolvent operator and spectrum of new type boundary value problems
Filomat, 2015
The aim of this study is to investigate a new type boundary value problems which consist of the equation -y''(x) + (By)(x) = ?y(x) on two disjoint intervals (-1,0) and (0,1) together with transmission conditions at the point of interaction x = 0 and with eigenparameter dependent boundary conditions, where B is an abstract linear operator, unbounded in general, in the direct sum of Lebesgue spaces L2(-1,0)( L2(0,1). By suggesting an own approaches we introduce modified Hilbert space and linear operator in it such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. We establish such properties as isomorphism and coerciveness with respect to spectral parameter, maximal decreasing of the resolvent operator and discreteness of the spectrum. Further we examine asymptotic behaviour of the eigenvalues.
A Uniqueness the Theorem for Singular Sturm-Liouville Problem
2004
In this paper, we show that If q(x) is prescribed on the (] 2, π π then the one spectrum suffices to determine q(x) on the interval (.The potential function q(x) in a Sturm Liouville problem is uniquely determined with one spectra by using the Hochstadt and Lieberman's method [2].
Nonlinear Analysis-theory Methods & Applications, 1998
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On the spectrum of an irregular Sturm-Liouville problem
Doklady Mathematics, 2010
We consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) ∈ L 2 (0, π) and irregular boundary conditions. We establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator.