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In this work, an analogue of Pontryagin’s maximum principle for dynamic equations on time scales is given, combining the continuous and the discrete Pontryagin maximum principles and extending them to other cases ‘in between’. We generalize known results to the case when a certain set of admissible values of the control is not necessarily closed (but convex) and the attainable set is not necessarily convex. At the same time, we impose an additional condition on the graininess of the time scale. For linear systems, sufficient conditions in the form of the maximum principle are obtained. [ABSTRACT FROM PUBLISHER]

In this article, we investigate the asymptotic behaviour of solutions of systems of rational difference equations in arbitrary dimensions. We give conditions for the parameters ensuring that the positive solutions of the considered system are bounded, unbounded, increasing, decreasing, and convergent, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we present conditions ensuring that solutions of linear second-order differential equations oscillate, provided solutions of corresponding difference equations oscillate. We also establish the converse result, namely, when oscillation of solutions of difference equations implies oscillation of solutions of corresponding differential equations. [ABSTRACT FROM AUTHOR]

The objective of this note is to present new Hille and Nehari type asymptotic criteria for a class of third-order delay dynamic equations on a time scale. Assumptions in our theorems are less restrictive, whereas the proofs are significantly simpler compared to those reported in the literature. The results obtained extend and improve some previous results. [ABSTRACT FROM AUTHOR]

The Sturm comparison theorem for second-order Sturm–Liouville difference equations on infinite intervals is established and discussed. [ABSTRACT FROM PUBLISHER]

We consider a nonoscillatory second-order linear dynamic equation on a time scale together with a linear perturbation of this equation and improve recently given conditions on the perturbation that guarantee that the perturbed equation is also nonoscillatory and has solutions that behave asymptotically like a recessive and a dominant solution of the unperturbed equation. The presented results are time scales analogues of a 2002 paper on the differential equations case by Trench. The difference equations case has not been treated yet, but is contained in our study together with other cases of dynamic equations. [ABSTRACT FROM AUTHOR]

We obtain some oscillation criteria for solutions to the second-order half-linear dynamic equation[image omitted] when [image omitted] or [image omitted] . These criteria unify and extend known criteria for corresponding half-linear differential and difference equations. Some of our results are new even in the continuous and the discrete cases. [ABSTRACT FROM AUTHOR]

We study second order scalar delta derivative expressions of Sturm-Liouville type on our newly defined Sturmiantime scales. Sturmian time scales include the discrete and continuous cases studied by Sturm. A form of second order differential expression on a Sturmian time scale considered here satisfies a Green's identity and hence is "formally self-adjoint". A unified variable change method is developed which allows simultaneous change of independent and dependent variable for expressions which include continuous and discrete theories as special cases. This unifies a continuous result of Coppel with a discrete result of Voepel. For the fourth order case, we explore unification of acontinuous result of Ahlbrandt, Hinton and Lewis [4] with a discrete result of Voepel [32]. [ABSTRACT FROM AUTHOR]

Profiles professor Allan Peterson. Research and teaching contributions to mathematics; Educational background; Career history.

In this paper, we will give sufficient conditions for a nonhomogeneous dynamic self-adjoint equation on a time scale to have a zero tending solution. We also give sufficient conditions that guarantees that for each constant C there is a unique bounded solution on [ a ,∞) with y ( a )= C . [ABSTRACT FROM AUTHOR]

We give conditions on the coefficient matrix for certain perturbed linear dynamic equations on time scales ensuring that there exists a bounded solution (which is explicitly given) to which all other solutions converge, and similarly conditions ensuring a bounded solution from which all other solutions diverge. We also consider periodic time scales and corresponding linear dynamic equations with periodic coefficients and prove similar statements about periodic solutions to which all other solutions converge or from which all other solutions diverge. [ABSTRACT FROM AUTHOR]

We briefly present the well-studied exponential function on a time scale and pose the problem of finding an appropriate logarithm function on a time scale. [ABSTRACT FROM AUTHOR]