On solvability of three spectra problem

Three spectral problems generated by the same Sturm–Liouville equation are considered: Neumann–Dirichlet problem (the Neumann condition at the left end and the Dirichlet condition at the right end) on the whole interval [0, a], Neumann–Dirichlet problem on and Dirichlet–Dirichlet problem on . The three spectra inverse problem, i.e. the problem of recovering the Sturm–Liuville equation using the three spectra of these boundary value problems is completely solved.     

State observability of elastic string vibrations under the boundary conditions of the first kind

We solve state observation problems for string vibrations, i.e., problems in which the initial conditions generating the observed string vibrations should be reconstructed from a given string state at two distinct time instants. The observed vibrations are described by the boundary value problem for the wave equation with homogeneous boundary conditions of the first kind. The observation problem is considered for classical and L ₂-generalized solutions of this boundary value problem.     

Inverse indefinite Sturm-Liouville problems with three spectra

We prove that the potential q(x) of an indefinite Sturm-Liouville problem on the closed interval [a,b] with the indefinite weight function w(x) can be determined uniquely by three spectra, which are generated by the indefinite problem defined on [a,b] and two right-definite problems defined on [a,0] and [0,b], where point 0 lies in (a,b) and is the turning point of the weight function w(x).     

On some two-point boundary value problems for two-dimensional systems of ordinary differential equations

Sufficient conditions for the solvability of two-point boundary value problems for the systemx _i =f_i(t, x₁, x₂) (i=1,2) are given, wheref ₁ andf ₂: [₁, ₁] ×R ₂ R are continuous functions.     

Blow-up sets for linear diffusion equations in one dimension

We consider the heat equation in the half-line with Dirichlet boundary data which blow up in finite time. Though the blow-up set may be any interval [0,a], depending on the Dirichlet data, we prove that the effective blow-up set, that is, the set of points where the solution behaves like u(0,t), consists always only of the origin. As an application of our results we consider a system of two heat equations with a nontrivial nonlinear flux coupling at the boundary. We show that by prescribing the non-linearities the two components may have different blow-up sets. However, the effective blow-up sets do not depend on the coupling and coincide with the origin for both components.     

Blow-up sets for linear diffusion equations in one dimension

We consider the heat equation in the half-line with Dirichlet boundary data which blow up in finite time. Though the blow-up set may be any interval [0,a], a ? [0,￥]a\in[0,\infty] depending on the Dirichlet data, we prove that the effective blow-up set, that is, the set of points x 3 0x\ge0 where the solution behaves like u(0,t), consists always only of the origin. As an application of our results we consider a system of two heat equations with a nontrivial nonlinear flux coupling at the boundary. We show that by prescribing the non-linearities the two components may have different blow-up sets. However, the effective blow-up sets do not depend on the coupling and coincide with the origin for both components.     

On the completeness of root vectors generated by systems of coupled hyperbolic equations

The paper is the second in a set of two papers, which are devoted to a unified approach to the problem of completeness of the generalized eigenvectors (the root vectors) for a specific class of linear non‐selfadjoint unbounded matrix differential operators. The list of the problems for which such operators are the dynamics generators includes the following: (a) initial boundary‐value problem (IBVP) for a non‐homogeneous string with both distributed and boundary damping; (b) IBVP for small vibrations of an ideal filament with a one‐parameter family of dissipative boundary conditions at one end and with a heavy load at the other end; this filament problem is treated for two cases of the boundary parameter: non‐singular and singular; (c) IBVP for a three‐dimensional damped wave equation with spherically symmetric coefficients and both distributed and boundary damping; (d) IBVP for a system of two coupled hyperbolic equations constituting a Timoshenko beam model with variable coefficients and boundary damping; (e) IBVP for a coupled Euler‐Bernoulli and Timoshenko beam model with boundary energy dissipation (the model known in engineering literature as bending‐torsion vibration model); (f) IBVP for two coupled Timoshenko beams model, which is currently accepted as an appropriate model describing vibrational behavior of a longer double‐walled carbon nanotube. Problems have been discussed in the first paper of the aforementioned set. Problems are discussed in the present paper.     

Positive solutions for quasilinear second order differential equation

It is well known that the Krasnoselskii's fixed point theorem is very very important. It was extensively used for studying the boundary value problems. In this article, the Krasnoselskii's fixed point theorem is extended. The new fixed point theorem is obtained. The second order quasilinear differential equation (Φ (y′))′+a(t)f(t,y,y′)=0,, 0<t<1 subject to mixed boundary condition is studied, where f is a nonnegative continuous function, Φ (v)= |v|^p-2 v, p>1. We show the existence of at least one positive solution by using the new fixed point theorem in cone.     

Full range solution of half space neutron transport problem

Based on the fact that the complete set of eigenfunctions of a half-range problem in [0,U] is also part of a larger set that is complete in the full-range [ –U,U], a full-range boundary condition is introduced for solving the half-range problem. Specifically, this condition expresses the solution at the boundary valid for allu [ –U,U] as the sum of a given forward component inu [0,U] and the unknown backward component inu [ –U, 0). Thus the basically ill-posed nature of the half-range problem, viz., that is required to find the response in [ —U,U] from given data in [0,U], is formulated over the entire domain at the boundary as compared to the usual approach that expresses the boundary condition only over [0,U]. This allows us, through a two-step process that considers the full-range properties of the eigenfunctions in [ —U,U] only, to obtainnumerically exact extrapolated end-point and CaseX-function. This means, because of the relationship of these fundamental half-range data with standard half-range expansion coefficientsa ₀₊ andA(v) [2], that the transient integral of the half-range solution has been reduced to mechanical quadratures.     

Higher order multi‐point boundary value problems

We consider the boundary value problem where n ? 2 and m ? 1 are integers, t_j ∈ [0, 1] for j = 1, …, m, and f and g_i, i = 0, …, n ? 1, are continuous. We obtain sufficient conditions for the existence of a solution of the above problem based on the existence of lower and upper solutions. Explicit conditions are also found for the existence of a solution of the problem. The differential equation has dependence on all lower order derivatives of the unknown function, and the boundary conditions cover many multi‐point boundary conditions studied in the literature. Schauder’s fixed point theorem and appropriate Nagumo conditions are employed in the analysis. Examples are given to illustrate the results. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equation of nonhomogeneous damped string

We consider a class of nonselfadjoint quadratic operator pencils generated by the equation, which governs the vibrations of a string with nonconstant bounded density subject to viscous damping with a nonconstant damping coefficient. These pencils depend on a complex parameterh, which enters the boundary conditions. Depending on the values ofh, the eigenvalues of the above pencils may describe the resonances in the scattering of elastic waves on an infinite string or the eingenmodes of a finite string. We obtain the 7asymptotic representations for these eigenvalues. Assuming that the proper multiplicity of each eigenvalue is equal to one, we prove that the eigenfunctions of these pencils form Riesz bases in the weightedL ²-space, whose weight function is exactly the density of the string. The general case of multiple eigenvalues will be treated in another paper, based on the results of the present work.     

Singular Eigenvalue Problems for the Equation y (n) + λp(x)y = 0

The paper studies singular eigenvalue problems for the equation y ⁽ⁿ⁾ +p(x)y=0 with boundary conditions imposed on the derivatives y ⁽ⁱ⁾ at the points x=a and x=. We look for singular problems which are analogous to regular problems on a finite interval. It is characterized when each eigenfunction has a finite number of zeros and when the spectrum is discrete or continuous, respectively.     

The spectral property of dual matrix strings as a consequence of the existence of spectral matrix functions for canonical systems

The term dual string for scalar strings was introduced in [KK1], where some connections between the spectra of a string and its dual were studied. In [KK2] it was shown that if () is a spectral function of a scalar stringS ₁ with nonnegative spectrum (in the sense of [KK2]), then the function

Property of positive solutions and its applications for Sturm–Liouville singular boundary value problems

This paper is concerned with the property of the positive solutions for Sturm–Liouville singular boundary value problems with the linear conditions. We obtain a relation between the solutions and Green’s function. It implies a necessary condition for the C¹[0,1]$C^1[0,1]$ positive solutions. We apply the result to conclude that the given equation has no C¹[0,1]$C^1[0,1]$ positive solutions.     

Determination of a differential pencil from interior spectral data

In this paper, inverse spectra problems for a differential pencil are studied. By using the approach similar to those in Hochstadt and Lieberman (1978) [14] and Ramm (2000) [26], we prove that (1) if p(x) (or q(x)) is full given on the interval [0,π], then a set of values of eigenfunctions at the mid-point of the interval [0,π] and one spectrum suffice to determine q(x) (or p(x)) on the interval [0,π] and all parameters in the boundary conditions; (2) if p(x) (or q(x)) is full given on the interval [0,π], then some information on eigenfunctions at some internal point and parts of two spectra suffice to determine q(x) (or p(x)) on the interval [0,π] and all parameters in the boundary conditions.     

Carleman estimates for one-dimensional degenerate heat equations

In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation $$ w_t + \left( {a\left( x \right)w_x } \right)_x = f,\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where the function a mainly satisfies $$ a \in \mathcal{C}^0 \left( {\left[ {0,1} \right]} \right) \cap \mathcal{C}^1 \left( {\left( {0,1} \right)} \right),a \gt 0 \hbox{on }\left( {0,1} \right) \hbox{and }\frac{1} {{\sqrt a }} \in L^1 \left( {0,1} \right). $$ We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=x^α (1 − x)^β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation $$ u_t - (a(x)u_x )_x = h\chi _w ,\quad (t,x) \in (0,T) \times (0,1),\quad \omega \subset \subset (0,1). $$ The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=x^α with α ∈[0, 2). Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Inverse spectral problems for Sturm-Liouville operators with singular potentials. Part III: Reconstruction by three spectra

We solve the inverse spectral problem of recovering the singular potential from W⁻¹₂(0,1) of a Sturm-Liouville operator by its spectra on the three intervals [0,1], [0,a], and [a,1] for some a∈(0,1). Necessary and sufficient conditions on the spectral data are derived, and uniqueness of the solution is analyzed.     

Necessary and sufficient conditions for the existence of positive solutions of fourth order multi-point boundary value problems

This paper investigates the existence of positive solutions of singular multi-point boundary value problems of fourth order ordinary differential equation with p-Laplacian. A necessary and sufficient condition for the existence of C²[0,1] positive solution as well as pseudo-C³[0,1] positive solution is given by means of the fixed point theorems on cones.