Linear bifurcation analysis with applications to relative socio-spatial dynamics.
One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor's Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality...
The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type where is the difference operator and are sequences of real numbers for , and , . We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.
We consider the summation equation, for , in the case where the map may change sign; here is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that is allowed to change sign, by introducing a new cone we are able to establish the existence of at least one positive solution to this problem by imposing some growth conditions on the functions and . Finally, as an application of the abstract existence result,...