Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

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Methods Of Modern Mathematical Physics

Finite element methods for Maxwell's equations.

Reviews - Eigenfunction Expansions Associated with Second-Order Differential Equations. By E. C. Titchmarsh Pp. 175. 20s. 1946. (Oxford University)

Linear Differential Operators By Prof. Cornelius Lanczos. Pp. xvi + 564. (London: D. Van Nostrand Co., Ltd.; New York: D. Van Nostrand Company, Inc., 1961.) 80s.

Linear Differential Operators

The similarity problem for J-nonnegative Sturm–Liouville operators

We consider a singular Sturm-Liouville expression with the indefinite weight sgnx. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of 1. Moreover, in this situation, the point 1 is a regular critical point. We construct an operator A = (sgnx)(id 2 /dx 2 + q) with non-real spectrum accumulating to a real point. The obtained results are applied to several classes of Sturm-Liouville operators.

Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x

/pdf/indefinite-sturm-liouville-operators-sgn-x-d2-dx2-q-x-with-6jjfy8bobb.pdf

Indefinite Sturm-Liouville operators (sgn x)(-d2/dx2+q(x)) with finite-zone potentials

The relation between the following two properties of linear difference equations with infinite delay is investigated: (i) exponential stability and (ii) -input -state stability (Perron's property) which means that solutions of the non-homogeneous equation with zero initial data belong to when non-homogeneous terms are in . It is assumed that at each moment the prehistory (the sequence of preceding states) belongs to some weighted -space with an exponentially fading weight (the phase space). Our main result states that whenever and a certain boundedness condition on coefficients is fulfilled. This condition is sharp and ensures that, to some extent, exponential and -input -state stabilities do not depend on the choice of a phase space and parameters p and q. -input -state stability corresponds to uniform stability. We provide some evidence that similar criteria should not be expected for non-fading memory spaces.

Bohl–Perron-type stability theorems for linear difference equations with infinite delay

Relation between two properties of linear difference equations with infinite delay is investigated: (i) exponential stability, (ii) $\l^p$-input $\l^q$-state stability (sometimes is called Perron's property). The latter means that solutions of the non-homogeneous equation with zero initial data belong to $\l^q$ when non-homogeneous terms are in $\l^p$. It is assumed that at each moment the prehistory (the sequence of preceding states) belongs to some weighted $\l^r$-space with an exponentially fading weight (the phase space). Our main result states that (i) $\Leftrightarrow$ (ii) whenever $(p,q) \neq (1,\infty)$ and a certain boundedness condition on coefficients is fulfilled. This condition is sharp and ensures that, to some extent, exponential and $\l^p$-input $\l^q$-state stabilities does not depend on the choice of a phase space and parameters $p$ and $q$, respectively. $\l^1$-input $\l^\infty$-state stability corresponds to uniform stability. We provide some evidence that similar criteria should not be expected for non-fading memory spaces.

Illya M. Karabash

Papers

The similarity problem for J-nonnegative Sturm–Liouville operators

Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x

Indefinite Sturm-Liouville operators (sgn x)(-d2/dx2+q(x)) with finite-zone potentials

Bohl–Perron-type stability theorems for linear difference equations with infinite delay

Bohl-Perron type stability theorems for linear difference equations with infinite delay