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Oscillation of even order nonlinear delay dynamic equations on time scales

Lynn H. Erbe, Raziye Mert, Allan Peterson, Ağacık Zafer (2013)

Czechoslovak Mathematical Journal

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...

Oscillation properties for a scalar linear difference equation of mixed type

Leonid Berezansky, Sandra Pinelas (2016)

Mathematica Bohemica

The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type Δ x ( n ) + k = - p q a k ( n ) x ( n + k ) = 0 , n > n 0 , where Δ x ( n ) = x ( n + 1 ) - x ( n ) is the difference operator and { a k ( n ) } are sequences of real numbers for k = - p , ... , q , and p > 0 , q 0 . We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.

Summation equations with sign changing kernels and applications to discrete fractional boundary value problems

Christopher S. Goodrich (2016)

Commentationes Mathematicae Universitatis Carolinae

We consider the summation equation, for t [ μ - 2 , μ + b ] μ - 2 , y ( t ) = γ 1 ( t ) H 1 i = 1 n a i y ξ i + γ 2 ( t ) H 2 i = 1 m b i y ζ i + λ s = 0 b G ( t , s ) f ( s + μ - 1 , y ( s + μ - 1 ) ) in the case where the map ( t , s ) G ( t , s ) may change sign; here μ ( 1 , 2 ] is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that G 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 H 1 and H 2 . Finally, as an application of the abstract existence result,...

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