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Dynamic equations on time scales
The model given purely by differential equations works well for continuous behavior such as population growth and other biological phenomena, and the models purely given by difference equations work well for discrete behaviors such as traffic flow with finite number of entrances and exits. But we can find many applications which are mixtures of discrete and continuous processes that are very difficult to model with differential equations or difference equations alone. The strength of time scales theory is that it is not restricted to purely continuous or regularly discrete applications and thus it can model any application that requires simultaneous modeling of continuous and discrete data. For example time scales theory can model insect populations that are continuous while in season, die out in, say winter, while their eggs are still incubating or dormant, and then hatch in a new season, giving rise to a non overlapping population.^ In this work, I will be concerned with second order dynamic equations on time scales. I start with a brief introduction to the time scale calculus and some theory necessary for the new results. I will give sufficient conditions under which certain semipositone boundary value problems on time scales have at least a positive solution by constructing a special cone and applying fixed point theorems. I will also give conditions under which certain semipositone boundary value systems have at least a positive solution. Finally I will give conditions under which a certain boundary value system on time scales has multiple positive solutions. ^
Dahal, Rajendra, "Dynamic equations on time scales" (2009). ETD collection for University of Nebraska - Lincoln. AAI3354834.