Oscillation CriteriaforFunctionalDynamic Equations onTimeScales

Abstract

Study ofdynamicequationsonatimescalegoesbacktoitsfounderStefan Hilger [40].Itisanewareaofstillfairlytheoreticalexplorationinmathematics. Motivatingthesubjectisanotionthatdynamicequationsontimescalescan build bridgesbetweencontinuousanddiscretemathematics.Further,studying the timescalesleadstoseveralimportantapplications,e.g.,insectpopulation models,neuralnetworks,andheattransfer.Atimescale T is anonemptyclosed subset oftherealnumbers.Whenthetimescaleequalsthesetofrealnumbers,the obtained resultsyieldsresultsofordinarydi erentialequations,whilewhenthe time scaleequalsthesetofintegers,theobtainedresultsyieldresultsofdi erence equations. Thenewtheoryoftheso-called"dynamicequation"isnotonlyunify the theoriesofdi erentialanddi erenceequations,butalsoextendstheclassical cases totheso-calledq-di erenceequations(when T = qN0 := fqt : t 2 N0; q > 1g or T = qZ = qZ [ f0g) whichhaveimportantapplicationsinquantum theory (see[43]). A delaydi erentialequation(DDE)isanequationforafunctionofasingle variable,usuallycalledtime,inwhichthederivativesofthefunctionatacertain time aregivenintermsofthevaluesofthefunctionatearliertimes.Inrecent years,therehasbeenanincreasinginterestinstudyingoscillationandnonoscil- lation ofsolutionsofdynamicequationsontimescaleswhichseekstoharmonize the oscillationofcontinuousanddiscretemathematics,andtoeliminateobscurity from both.Sowechoosethetitleofthethesis"OscillationCriteriaforFunc- tional DynamicEquationsonTimeScales"aimingtousethegeneralizedRiccati transformation andtheinequalitytechniqueinestablishingsomenewoscillation criteria fordelaydynamicequations. This thesisisdevotedto 1. IllustratethenewtheoryofStefanHilgerbygivinganintroductiontothe theory ofdynamicequationsontimescales, iii