Second-order linear constant coefficient dynamic equations with polynomial forcing on time scales
A general solution for a second-order linear constant coefficient dynamic equation with polynomial forcing on time scales is given.
math.CA↗
arXiv subjects
Douglas R. Anderson
Publications and source records attributed to Douglas R. Anderson.
Young's integral inequality with upper and lower bounds
Young's integral inequality is reformulated with upper and lower bounds for the remainder. The new inequalities improve Young's integral inequality on all time scales, such that the case where equality holds becomes particularly transparent in this new presentation. The corresponding results for difference equations are given, and several examples are included. We extend these results to piecewise-monotone functions as well.
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Titchmarsh-Sims-Weyl theory for complex Hamiltonian systems on Sturmian time scales
We study non-self-adjoint Hamiltonian systems on Sturmian time scales, defining Weyl-Sims sets, which replace the classical Weyl circles, and a matrix-valued $M-$function on suitable cone-shaped domains in the complex plane. Furthermore, we characterize realizations of the corresponding dynamic operator and its adjoint, and construct their resolvents. Even-order scalar equations and the Orr-Sommerfeld equation on time scales are given as examples illustrating the theory, which are new even for difference equations. These results unify previous discrete and continuous theories to dynamic equations on Sturmian time scales.
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Spectral Theory for Second-Order Vector Equations on Finite Time-Varying Domains
In this study, we are concerned with spectral problems of second-order vector dynamic equations with two-point boundary value conditions and mixed derivatives, where the matrix-valued coefficient of the leading term may be singular, and the domain is non-uniform but finite. A concept of self-adjointness of the boundary conditions is introduced. The self-adjointness of the corresponding dynamic operator is discussed on a suitable admissible function space, and fundamental spectral results are obtained. The dual orthogonality of eigenfunctions is shown in a special case. Extensions to even-order Sturm-Liouville dynamic equations, linear Hamiltonian and symplectic nabla systems on general time scales are also discussed.
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Unifying discrete and continuous Weyl-Titchmarsh theory via a class of linear Hamiltonian systems on Sturmian time scales
In this study, we are concerned with introducing Weyl-Titchmarsh theory for a class of dynamic linear Hamiltonian nabla systems over a half-line on Sturmian time scales. After developing fundamental properties of solutions and regular spectral problems, we introduce the corresponding maximal and minimal operators for the system. Matrix disks are constructed and proved to be nested and converge to a limiting set. Some precise relationships among the rank of the matrix radius of the limiting set, the number of linearly independent square summable solutions, and the defect indices of the minimal operator are established. Using the above results, a classification of singular dynamic linear Hamiltonian nabla systems is given in terms of the defect indices of the minimal operator, and several equivalent conditions on the cases of limit point and limit circle are obtained, respectively. These results unify and extend certain classic and recent results on the subject in the continuous and discrete cases, respectively, to Sturmian time scales.
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Alternative solutions of inhomogeneous second--order linear dynamic equations on time scales
We exhibit an alternative method for solving inhomogeneous second--order linear ordinary dynamic equations on time scales, based on reduction of order rather than variation of parameters. Our form extends recent (and long-standing) analysis on $\R$ to a new form for difference equations, quantum equations, and arbitrary dynamic equations on time scales.
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A Comparison of Continuously Controlled and Controlled K-theory
We define an unreduced version of the e-controlled lower $K$-theoretic groups of Ranicki and Yamasaki, and Quinn. We show that the reduced versions of our groups coincide (in the inverse limit and its first derived, $\lim^1$) with those of Ranicki and Yamasaki. We also relate the controlled groups to the continuously controlled groups of Anderson and Munkholm, and to the Quinn homology groups of Quinn.
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